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1806.03575

{'1806.03575-1-0-0': 'Different from traditional hyperspectral super-resolution approaches that focus on improving the spatial resolution, spectral super-resolution aims at producing a high-resolution hyperspectral image from the RGB observation with super-resolution in spectral domain.', '1806.03575-1-0-1': 'However, it is challenging to accurately reconstruct a high-dimensional continuous spectrum from three discrete intensity values at each pixel, since too much information is lost during the procedure where the latent hyperspectral image is downsampled (e.g., with [MATH]10 scaling factor) in spectral domain to produce an RGB observation.', '1806.03575-1-0-2': 'To address this problem, we present a multi-scale deep convolutional neural network (CNN) to explicitly map the input RGB image into a hyperspectral image.', '1806.03575-1-0-3': 'Through symmetrically downsampling and upsampling the intermediate feature maps in a cascading paradigm, the local and non-local image information can be jointly encoded for spectral representation, ultimately improving the spectral reconstruction accuracy.', '1806.03575-1-0-4': 'Extensive experiments on a large hyperspectral dataset demonstrate the effectiveness of the proposed method.', '1806.03575-1-1-0': '# Introduction', '1806.03575-1-2-0': 'Hyperspectral imaging encodes the reflectance of the scene from hundreds or thousands of bands with a narrow wavelength interval (e.g., 10nm) into a hyperspectral image.', '1806.03575-1-2-1': 'Different from conventional images, each pixel in the hyperspectral image contains a continuous spectrum, thus allowing the acquisition of abundant spectral information.', '1806.03575-1-2-2': 'Such information has proven to be quite useful for distinguishing different materials.', '1806.03575-1-2-3': 'Therefore, hyperspectral images have been widely exploited to facilitate various applications in computer vision community, such as visual tracking [CITATION], image segmentation [CITATION], face recognition [CITATION], document analysis [CITATION], scene classification [CITATION], and anomaly detection [CITATION].', '1806.03575-1-3-0': 'The acquisition of spectral information, however, comes at the cost of decreasing the spatial resolution of hyperspectral images.', '1806.03575-1-3-1': 'This is because a fewer number of photons are captured by each detector due to the narrower width of the spectral bands.', '1806.03575-1-3-2': 'In order to maintain a reasonable signal-to-noise ratio (SNR), the instantaneous field of view (IFOV) needs to be increased [CITATION], which renders it difficult to produce hyperspectral images with high spatial resolution.', '1806.03575-1-3-3': 'To address this problem, many efforts have been made for the hyperspectral imagery super-resolution.', '1806.03575-1-4-0': 'Most of the existing methods mainly focus on enhancing the spatial resolution of the observed hyperspectral image.', '1806.03575-1-4-1': 'According to the input images, they can be divided into two categories: [MATH] fusion based methods where a high-resolution conventional image ([MATH], RGB image) and a low-resolution hyperspectral image are fused together to produce a high-resolution hyperspectral image [CITATION] [MATH] single image super-resolution which directly increases the spatial resolution of a hyperspectral image [CITATION].', '1806.03575-1-4-2': 'Although these methods have shown effective performance, the acquisition of the input hyperspectral image often requires specialized hyperspectral sensors as well as extensive imaging cost.', '1806.03575-1-4-3': 'To mitigate this problem, some recent literature [CITATION] turn to investigate a novel hyperspectral imagery super-resolution scheme, termed spectral super-resolution, which aims at improving the spectral resolution of a given RGB image.', '1806.03575-1-4-4': 'Since the input image can be easily captured by conventional RGB sensors, imaging cost can be greatly reduced.', '1806.03575-1-5-0': 'However, it is challenging to accurately reconstruct a hyperspectral image from a single RGB observation, since mapping three discrete intensity values to a continuous spectrum is a highly ill-posed linear inverse problem.', '1806.03575-1-5-1': 'To address this problem, we propose to learn a complicated non-linear mapping function for spectral super-resolution with deep convolutional neural networks (CNN).', '1806.03575-1-5-2': 'It has been shown that the 3-dimensional color vector for a specific pixel can be viewed as the downsampled observation of the corresponding spectrum.', '1806.03575-1-5-3': 'Moreover, for a candidate pixel, there often exist abundant locally and no-locally similar pixels ([MATH] exhibiting similar spectra) in the spatial domain.', '1806.03575-1-5-4': 'As a result, the color vectors corresponding to those similar pixels can be viewed as a group of downsampled observations of the latent spectra for the candidate pixel.', '1806.03575-1-5-5': 'Therefore, accurate spectral reconstruction requires to explicitly consider both the local and non-local information from the input RGB image.', '1806.03575-1-5-6': 'To this end, we develop a novel multi-scale CNN.', '1806.03575-1-5-7': 'Our method jointly encodes the local and non-local image information through symmetrically downsampling and upsampling the intermediate feature maps in a cascading paradigm, thus enhancing the spectral reconstruction accuracy.', '1806.03575-1-5-8': 'We experimentally show that the proposed method can be easily trained in an end-to-end scheme and beat several state-of-the-art methods on a large hyperspectral image dataset with respect to various evaluation metrics.', '1806.03575-1-6-0': 'Our contributions are twofold:', '1806.03575-1-7-0': '# Related Work', '1806.03575-1-8-0': 'This section gives a brief review of the existing spectral super-resolution methods, which can be divided into the following two categories.', '1806.03575-1-9-0': 'Statistic based methods This line of research mainly focus on exploiting the inherent statistical distribution of the latent hyperspectral image as priors to guide the super-resolution.', '1806.03575-1-9-1': 'Most of these methods involve building overcomplete dictionaries and learning sparse coding coefficients to linearly combine the dictionary atoms.', '1806.03575-1-9-2': 'For example, in [CITATION], Arad [MATH] leveraged image priors to build a dictionary using K-SVD [CITATION].', '1806.03575-1-9-3': 'At test time, orthogonal matching pursuit [CITATION] was used to compute a sparse representation of the input RGB image.', '1806.03575-1-9-4': '[CITATION] proposed a new method inspired by A+ [CITATION], where sparse coefficients are computed by explicitly solving a sparse least square problem.', '1806.03575-1-9-5': 'These methods directly exploit the whole image to build image prior, ignoring local and non-local structure information.', '1806.03575-1-9-6': "What's more, since the image prior is often handcrafted or heuristically designed with shallow structure, these methods fail to generalize well in practice.", '1806.03575-1-10-0': 'Learning based methods These methods directly learn a certain mapping function from the RGB image to a corresponding hyperspectral image.', '1806.03575-1-10-1': 'For example, [CITATION] proposed a training based method using a radial basis function network.', '1806.03575-1-10-2': 'The input data was pre-processed with a white balancing function to alleviate the influence of different illumination.', '1806.03575-1-10-3': 'The total reconstruction accuracy is affected by the performance of this pre-processing stage.', '1806.03575-1-10-4': 'Recently, witnessing the great success of deep learning in many other ill-posed inverse problems such as image denoising [CITATION] and single image super-resolution [CITATION], it is natural to consider using deep networks (especially convolutional neural networks) for spectral super-resolution.', '1806.03575-1-10-5': 'In [CITATION], Galliani [MATH] exploited a variant of fully convolutional DenseNets (FC-DenseNets [CITATION]) for spectral super-resolution.', '1806.03575-1-10-6': 'However, this method is sensitive to the hyper-parameters and its performance can still be further improved.', '1806.03575-1-11-0': '# Proposed Method', '1806.03575-1-12-0': 'In this section, we will introduce the proposed multi-scale convolution neural network in details.', '1806.03575-1-12-1': 'Firstly, we introduce some building blocks which will be utilized in our network.', '1806.03575-1-12-2': 'Then, we will illustrate the architecture of the proposed network.', '1806.03575-1-13-0': '## Building Blocks', '1806.03575-1-14-0': 'There are three basic building blocks in our network.', '1806.03575-1-14-1': 'Their structures are shown in Table [REF].', '1806.03575-1-15-0': 'Double convolution (Double Conv) block consists of two [MATH] convolutions.', '1806.03575-1-15-1': 'Each of them is followed by batch normalization, leaky ReLU and dropout.', '1806.03575-1-15-2': 'We exploit batch normalization and dropout to address overfitting.', '1806.03575-1-16-0': 'Downsample block contains a regular max-pooling layer.', '1806.03575-1-16-1': 'It reduces the spatial size of the feature map and enlarges the receptive field of the network.', '1806.03575-1-17-0': 'Upsample block is utilized to upsample the feature map in the spatial domain.', '1806.03575-1-17-1': 'To this end, much previous literature often adopts the transposed convolution.', '1806.03575-1-17-2': 'However, it is prone to generate checkboard artifacts.', '1806.03575-1-17-3': 'To address this problem, we use the pixel shuffle operation [CITATION].', '1806.03575-1-17-4': 'It has been shown that pixel shuffle alleviates the checkboard artifacts.', '1806.03575-1-17-5': 'In addition, due to not introducing any learnable parameters, pixel shuffle also helps improve the robustness against over-fitting.', '1806.03575-1-18-0': '## Network Architecture', '1806.03575-1-19-0': 'The overall architecture of the proposed multi-scale convolution neural network is depicted in Figure [REF].', '1806.03575-1-19-1': 'The network follows the encoder-decoder pattern.', '1806.03575-1-19-2': 'For the encoder part, each downsampling step consists of a "Double Conv" with a downsample block.', '1806.03575-1-19-3': 'The spatial size is progressively reduced, and the number of features is doubled at each step.', '1806.03575-1-19-4': 'The decoder is symmetric to the encoder path.', '1806.03575-1-19-5': 'Every step in the decoder path consists of an upsampling operation followed by a "Double Conv" block.', '1806.03575-1-19-6': 'The spatial size of the features is recovered, while the number of features is halved every step.', '1806.03575-1-19-7': 'Finally, a [MATH] convolution maps the output features to the reconstructed 31-channel hyperspectral image.', '1806.03575-1-19-8': 'In addition to the feedforward path, skip connections are used to concatenate the corresponding feature maps of the encoder and decoder.', '1806.03575-1-20-0': 'Our method naturally fits the task of spectral reconstruction.', '1806.03575-1-20-1': 'The encoder can be interpreted as extracting features from RGB images.', '1806.03575-1-20-2': 'Through downsampling in a cascade way, the receptive field of the network is constantly increased, which allows the network to "see" more pixels in an increasingly larger field of view.', '1806.03575-1-20-3': 'By doing so, both the local and non-local information can be encoded to better represent the latent spectra.', '1806.03575-1-20-4': 'The symmetric decoder procedure is employed to reconstruct the latent hyperspectral images based on these deep and compact features.', '1806.03575-1-20-5': 'The skip connections with concatenations are essential for introducing multi-scale information and yielding better estimation of the spectra.', '1806.03575-1-21-0': '# Experiments', '1806.03575-1-22-0': '## Datasets', '1806.03575-1-23-0': 'In this study, all experiments are performed on the NTIRE2018 dataset [CITATION].', '1806.03575-1-23-1': 'This dataset is extended from the ICVL dataset [CITATION].', '1806.03575-1-23-2': 'The ICVL dataset includes [MATH] images captured using Specim PS Kappa DX4 hyperspectral camera.', '1806.03575-1-23-3': 'Each image is of size [MATH] in spatial resolution and contains [MATH] spectral bands in the range of [MATH].', '1806.03575-1-23-4': 'In experiments, [MATH] successive bands ranging from [MATH] with [MATH] interval are extracted from each image for evaluation.', '1806.03575-1-23-5': 'In the NTIRE2018 challenge, this dataset is further extended by supplementing [MATH] extra images of the same spatial and spectral resolution.', '1806.03575-1-23-6': 'As a result, [MATH] high-resolution hyperspectral images are collected as the training data.', '1806.03575-1-23-7': 'In addition, another [MATH] hyperspectral images are further introduced as the test set.', '1806.03575-1-23-8': 'In the NTIRE2018 dataset, the corresponding RGB rendition is also provided for each image.', '1806.03575-1-23-9': 'In the following, we will employ the RGB-hyperspectral image pairs to evaluate the proposed method.', '1806.03575-1-24-0': '## Comparison Methods Implementation Details', '1806.03575-1-25-0': 'To demonstrate the effectiveness of the proposed method, we compare it with four spectral super-resolution methods, including spline interpolation, the sparse recovery method in [CITATION] (Arad [MATH]), A+ [CITATION], and the deep learning method in [CITATION] (Galliani [MATH]).', '1806.03575-1-25-1': '[CITATION] are implemented by the codes released by the authors.', '1806.03575-1-25-2': 'Since there is no code released for [CITATION], we reimplement it in this study.', '1806.03575-1-25-3': 'In the following, we will give the implementation details of each method.', '1806.03575-1-26-0': 'Spline interpolation The interpolation algorithm serves as the most primitive baseline in this study.', '1806.03575-1-26-1': 'Specifically, for each RGB pixel [MATH], we use spline interpolation to upsample it and obtain a [MATH]-dimensional spectrum ([MATH]).', '1806.03575-1-26-2': 'According to the visible spectrum, the [MATH], [MATH], [MATH] values of an RGB pixel are assigned to [MATH], [MATH], and [MATH], respectively.', '1806.03575-1-27-0': 'Arad [MATH] and A+ The low spectral resolution image is assumed to be a directly downsampled version of the corresponding hyperspectral image using some specific linear projection matrix.', '1806.03575-1-27-1': 'In [CITATION] this matrix is required to be perfectly known.', '1806.03575-1-27-2': 'In our experiments, we fit the projection matrix using training data with conventional linear regression.', '1806.03575-1-28-0': 'Galliani [MATH] and our method We experimentally find the optimal set of hyper-parameters for both methods.', '1806.03575-1-28-1': '[MATH] dropout is applied to Galliani [MATH], while our method utilizes [MATH] dropout rate.', '1806.03575-1-28-2': 'All the leaky ReLU activation functions are applied with a negative slope of 0.2.', '1806.03575-1-28-3': 'We train the networks for 100 epochs using Adam optimizer with [MATH] regularization.', '1806.03575-1-28-4': 'Weight initialization and learning rate vary for different methods.', '1806.03575-1-28-5': 'For Galliani [MATH], the weights are initialized via HeUniform [CITATION], and the learning rate is set to [MATH] for the first 50 epochs, decayed to [MATH] for the next 50 epochs.', '1806.03575-1-28-6': 'As for our method, we use HeNormal initialization [CITATION].', '1806.03575-1-28-7': 'The initial learning rate is [MATH] and is multiplied by 0.93 every 10 epochs.', '1806.03575-1-28-8': 'We perform data augmentation by extracting patches of size [MATH] with a stride of 40 pixels from training data.', '1806.03575-1-28-9': 'The total amount of training samples is over [MATH].', '1806.03575-1-28-10': 'At the test phase, we directly feed the whole image to the network and get the estimated hyperspectral image in one single forward pass.', '1806.03575-1-29-0': '## Evaluation Metrics', '1806.03575-1-30-0': 'To quantitatively evaluate the performance of the proposed method, we adopt the following two categories of evaluation metrics.', '1806.03575-1-31-0': 'Pixel-level reconstruction error We follow [CITATION] to use absolute and relative root-mean-square error (RMSE and rRMSE) as quantitative measurements for reconstruction accuracy.', '1806.03575-1-31-1': 'Let [MATH] and [MATH] denote the [MATH]th element of the real and estimated hyperspectral images, [MATH] is the average of [MATH], and [MATH] is the total number of elements in one hyperspectral image.', '1806.03575-1-31-2': 'There are two formulas for RMSE and rRMSE respectively.', '1806.03575-1-31-3': '[EQUATION]', '1806.03575-1-31-4': 'Spectral similarity Since the key for spectral super-resolution is to reconstruct the spectra, we also use spectral angle mapper ([MATH]) to evaluate the performance of different methods.', '1806.03575-1-31-5': '[MATH] calculates the average spectral angle between the spectra of real and estimated hyperspectral images.', '1806.03575-1-31-6': 'Let [MATH] represents the spectra of the [MATH]th hyperspectral pixel in real and estimated hyperspectral images ([MATH] is the number of bands), and [MATH] is the total number of pixels within an image.', '1806.03575-1-31-7': 'The [MATH] value can be computed as follows.', '1806.03575-1-31-8': '[EQUATION]', '1806.03575-1-32-0': '## Experimental Results', '1806.03575-1-33-0': 'Convergence Analysis We plot the curve of [MATH] loss on the training set and the curves of five evaluation metrics computed on the test set in Figure [REF].', '1806.03575-1-33-1': 'It can be seen that both the training loss and the value of metrics gradually decrease and ultimately converge with the proceeding of the training.', '1806.03575-1-33-2': 'This demonstrates that the proposed multi-scale convolution neural network converges well.', '1806.03575-1-34-0': 'Quantitative Results Table [REF] provides the quantitative results of our method and all baseline methods.', '1806.03575-1-34-1': 'It can be seen that our model outperforms all competitors with regards to [MATH] and [MATH], and produces comparable results to Galliani [MATH] on [MATH] and [MATH].', '1806.03575-1-34-2': 'More importantly, our method surpasses all the others with respect to spectral angle mapper.', '1806.03575-1-34-3': 'This clearly proves that our model reconstructs spectra more accurately than other competitors.', '1806.03575-1-34-4': 'It is worth pointing out that reconstruction error (absolute and relative [MATH]) is not necessarily positively correlated with spectral angle mapper ([MATH]).', '1806.03575-1-34-5': 'For example, when the pixels of an image are shuffled, [MATH] and [MATH] will remain the same, while [MATH] will change completely.', '1806.03575-1-34-6': 'According to the results in Table [REF], we can find that our finely designed network enhances spectral super-resolution from both aspects, [MATH], yielding better results on both average root-mean-square error and spectral angle similarity.', '1806.03575-1-35-0': 'Visual Results To further clarify the superiority in reconstruction accuracy.', '1806.03575-1-35-1': 'We show the absolute reconstruction error of every test image in Figure [REF].', '1806.03575-1-35-2': 'The error is summarized over all bands of the hyperspectral image.', '1806.03575-1-35-3': 'Since A+ outperforms Arad [MATH] in terms of any evaluation metric, we use A+ to represent the sparse coding methods.', '1806.03575-1-35-4': 'It can be seen that our method yields smoother reconstructed images as well as lower reconstruction error than other competitors.', '1806.03575-1-36-0': 'In addition, we randomly choose three test images and plot the real and reconstructed spectra for four pixels in Figure [REF] to further demonstrate the effectiveness of the proposed method in spectrum reconstruction.', '1806.03575-1-36-1': 'It can be seen that only slight difference exists between the reconstructed spectra and the ground truth.', '1806.03575-1-37-0': 'According to these results above, we can conclude that the proposed method is effective in spectral super-resolution and outperforms several state-of-the-art competitors.', '1806.03575-1-38-0': '# Conclusion', '1806.03575-1-39-0': 'In this study, we show that leveraging both the local and non-local information of input images is essential for the accurate spectral reconstruction.', '1806.03575-1-39-1': 'Following this idea, we design a novel multi-scale convolutional neural network, which employs a symmetrically cascaded downsampling-upsampling architecture to jointly encode the local and non-local image information for spectral reconstruction.', '1806.03575-1-39-2': 'With extensive experiments on a large hyperspectral images dataset, the proposed method clearly outperforms several state-of-the-art methods in terms of reconstruction accuracy and spectral similarity.'}

{'1806.03575-2-0-0': 'Different from traditional hyperspectral super-resolution approaches that focus on improving the spatial resolution, spectral super-resolution aims at producing a high-resolution hyperspectral image from the RGB observation with super-resolution in spectral domain.', '1806.03575-2-0-1': 'However, it is challenging to accurately reconstruct a high-dimensional continuous spectrum from three discrete intensity values at each pixel, since too much information is lost during the procedure where the latent hyperspectral image is downsampled (e.g., with [MATH]10 scaling factor) in spectral domain to produce an RGB observation.', '1806.03575-2-0-2': 'To address this problem, we present a multi-scale deep convolutional neural network (CNN) to explicitly map the input RGB image into a hyperspectral image.', '1806.03575-2-0-3': 'Through symmetrically downsampling and upsampling the intermediate feature maps in a cascading paradigm, the local and non-local image information can be jointly encoded for spectral representation, ultimately improving the spectral reconstruction accuracy.', '1806.03575-2-0-4': 'Extensive experiments on a large hyperspectral dataset demonstrate the effectiveness of the proposed method.', '1806.03575-2-1-0': '# Introduction', '1806.03575-2-2-0': 'Hyperspectral imaging encodes the reflectance of the scene from hundreds or thousands of bands with a narrow wavelength interval (e.g., 10nm) into a hyperspectral image.', '1806.03575-2-2-1': 'Different from conventional images, each pixel in the hyperspectral image contains a continuous spectrum, thus allowing the acquisition of abundant spectral information.', '1806.03575-2-2-2': 'Such information has proven to be quite useful for distinguishing different materials.', '1806.03575-2-2-3': 'Therefore, hyperspectral images have been widely exploited to facilitate various applications in computer vision community, such as visual tracking [CITATION], image segmentation [CITATION], face recognition [CITATION], document analysis [CITATION], scene classification [CITATION], and anomaly detection [CITATION].', '1806.03575-2-3-0': 'The acquisition of spectral information, however, comes at the cost of decreasing the spatial resolution of hyperspectral images.', '1806.03575-2-3-1': 'This is because a fewer number of photons are captured by each detector due to the narrower width of the spectral bands.', '1806.03575-2-3-2': 'In order to maintain a reasonable signal-to-noise ratio (SNR), the instantaneous field of view (IFOV) needs to be increased [CITATION], which renders it difficult to produce hyperspectral images with high spatial resolution.', '1806.03575-2-3-3': 'To address this problem, many efforts have been made for the hyperspectral imagery super-resolution.', '1806.03575-2-4-0': 'Most of the existing methods mainly focus on enhancing the spatial resolution of the observed hyperspectral image.', '1806.03575-2-4-1': 'According to the input images, they can be divided into two categories: [MATH] fusion based methods where a high-resolution conventional image ([MATH], RGB image) and a low-resolution hyperspectral image are fused together to produce a high-resolution hyperspectral image [CITATION] [MATH] single image super-resolution which directly increases the spatial resolution of a hyperspectral image [CITATION].', '1806.03575-2-4-2': 'Although these methods have shown effective performance, the acquisition of the input hyperspectral image often requires specialized hyperspectral sensors as well as extensive imaging cost.', '1806.03575-2-4-3': 'To mitigate this problem, some recent literature [CITATION] turn to investigate a novel hyperspectral imagery super-resolution scheme, termed spectral super-resolution, which aims at improving the spectral resolution of a given RGB image.', '1806.03575-2-4-4': 'Since the input image can be easily captured by conventional RGB sensors, imaging cost can be greatly reduced.', '1806.03575-2-5-0': 'However, it is challenging to accurately reconstruct a hyperspectral image from a single RGB observation, since mapping three discrete intensity values to a continuous spectrum is a highly ill-posed linear inverse problem.', '1806.03575-2-5-1': 'To address this problem, we propose to learn a complicated non-linear mapping function for spectral super-resolution with deep convolutional neural networks (CNN).', '1806.03575-2-5-2': 'It has been shown that the 3-dimensional color vector for a specific pixel can be viewed as the downsampled observation of the corresponding spectrum.', '1806.03575-2-5-3': 'Moreover, for a candidate pixel, there often exist abundant locally and no-locally similar pixels ([MATH] exhibiting similar spectra) in the spatial domain.', '1806.03575-2-5-4': 'As a result, the color vectors corresponding to those similar pixels can be viewed as a group of downsampled observations of the latent spectra for the candidate pixel.', '1806.03575-2-5-5': 'Therefore, accurate spectral reconstruction requires to explicitly consider both the local and non-local information from the input RGB image.', '1806.03575-2-5-6': 'To this end, we develop a novel multi-scale CNN.', '1806.03575-2-5-7': 'Our method jointly encodes the local and non-local image information through symmetrically downsampling and upsampling the intermediate feature maps in a cascading paradigm, thus enhancing the spectral reconstruction accuracy.', '1806.03575-2-5-8': 'We experimentally show that the proposed method can be easily trained in an end-to-end scheme and beat several state-of-the-art methods on a large hyperspectral image dataset with respect to various evaluation metrics.', '1806.03575-2-6-0': 'Our contributions are twofold:', '1806.03575-2-7-0': '# Related Work', '1806.03575-2-8-0': 'This section gives a brief review of the existing spectral super-resolution methods, which can be divided into the following two categories.', '1806.03575-2-9-0': 'Statistic based methods This line of research mainly focus on exploiting the inherent statistical distribution of the latent hyperspectral image as priors to guide the super-resolution.', '1806.03575-2-9-1': 'Most of these methods involve building overcomplete dictionaries and learning sparse coding coefficients to linearly combine the dictionary atoms.', '1806.03575-2-9-2': 'For example, in [CITATION], Arad [MATH] leveraged image priors to build a dictionary using K-SVD [CITATION].', '1806.03575-2-9-3': 'At test time, orthogonal matching pursuit [CITATION] was used to compute a sparse representation of the input RGB image.', '1806.03575-2-9-4': '[CITATION] proposed a new method inspired by A+ [CITATION], where sparse coefficients are computed by explicitly solving a sparse least square problem.', '1806.03575-2-9-5': 'These methods directly exploit the whole image to build image prior, ignoring local and non-local structure information.', '1806.03575-2-9-6': "What's more, since the image prior is often handcrafted or heuristically designed with shallow structure, these methods fail to generalize well in practice.", '1806.03575-2-10-0': 'Learning based methods These methods directly learn a certain mapping function from the RGB image to a corresponding hyperspectral image.', '1806.03575-2-10-1': 'For example, [CITATION] proposed a training based method using a radial basis function network.', '1806.03575-2-10-2': 'The input data was pre-processed with a white balancing function to alleviate the influence of different illumination.', '1806.03575-2-10-3': 'The total reconstruction accuracy is affected by the performance of this pre-processing stage.', '1806.03575-2-10-4': 'Recently, witnessing the great success of deep learning in many other ill-posed inverse problems such as image denoising [CITATION] and single image super-resolution [CITATION], it is natural to consider using deep networks (especially convolutional neural networks) for spectral super-resolution.', '1806.03575-2-10-5': 'In [CITATION], Galliani [MATH] exploited a variant of fully convolutional DenseNets (FC-DenseNets [CITATION]) for spectral super-resolution.', '1806.03575-2-10-6': 'However, this method is sensitive to the hyper-parameters and its performance can still be further improved.', '1806.03575-2-11-0': '# Proposed Method', '1806.03575-2-12-0': 'In this section, we will introduce the proposed multi-scale convolution neural network in details.', '1806.03575-2-12-1': 'Firstly, we introduce some building blocks which will be utilized in our network.', '1806.03575-2-12-2': 'Then, we will illustrate the architecture of the proposed network.', '1806.03575-2-13-0': '## Building Blocks', '1806.03575-2-14-0': 'There are three basic building blocks in our network.', '1806.03575-2-14-1': 'Their structures are shown in Table [REF].', '1806.03575-2-15-0': 'Double convolution (Double Conv) block consists of two [MATH] convolutions.', '1806.03575-2-15-1': 'Each of them is followed by batch normalization, leaky ReLU and dropout.', '1806.03575-2-15-2': 'We exploit batch normalization and dropout to address overfitting.', '1806.03575-2-16-0': 'Downsample block contains a regular max-pooling layer.', '1806.03575-2-16-1': 'It reduces the spatial size of the feature map and enlarges the receptive field of the network.', '1806.03575-2-17-0': 'Upsample block is utilized to upsample the feature map in the spatial domain.', '1806.03575-2-17-1': 'To this end, much previous literature often adopts the transposed convolution.', '1806.03575-2-17-2': 'However, it is prone to generate checkboard artifacts.', '1806.03575-2-17-3': 'To address this problem, we use the pixel shuffle operation [CITATION].', '1806.03575-2-17-4': 'It has been shown that pixel shuffle alleviates the checkboard artifacts.', '1806.03575-2-17-5': 'In addition, due to not introducing any learnable parameters, pixel shuffle also helps improve the robustness against over-fitting.', '1806.03575-2-18-0': '## Network Architecture', '1806.03575-2-19-0': 'The overall architecture of the proposed multi-scale convolution neural network is depicted in Figure [REF].', '1806.03575-2-19-1': 'The network follows the encoder-decoder pattern.', '1806.03575-2-19-2': 'For the encoder part, each downsampling step consists of a "Double Conv" with a downsample block.', '1806.03575-2-19-3': 'The spatial size is progressively reduced, and the number of features is doubled at each step.', '1806.03575-2-19-4': 'The decoder is symmetric to the encoder path.', '1806.03575-2-19-5': 'Every step in the decoder path consists of an upsampling operation followed by a "Double Conv" block.', '1806.03575-2-19-6': 'The spatial size of the features is recovered, while the number of features is halved every step.', '1806.03575-2-19-7': 'Finally, a [MATH] convolution maps the output features to the reconstructed 31-channel hyperspectral image.', '1806.03575-2-19-8': 'In addition to the feedforward path, skip connections are used to concatenate the corresponding feature maps of the encoder and decoder.', '1806.03575-2-20-0': 'Our method naturally fits the task of spectral reconstruction.', '1806.03575-2-20-1': 'The encoder can be interpreted as extracting features from RGB images.', '1806.03575-2-20-2': 'Through downsampling in a cascade way, the receptive field of the network is constantly increased, which allows the network to "see" more pixels in an increasingly larger field of view.', '1806.03575-2-20-3': 'By doing so, both the local and non-local information can be encoded to better represent the latent spectra.', '1806.03575-2-20-4': 'The symmetric decoder procedure is employed to reconstruct the latent hyperspectral images based on these deep and compact features.', '1806.03575-2-20-5': 'The skip connections with concatenations are essential for introducing multi-scale information and yielding better estimation of the spectra.', '1806.03575-2-21-0': '# Experiments', '1806.03575-2-22-0': '## Datasets', '1806.03575-2-23-0': 'In this study, all experiments are performed on the NTIRE2018 dataset [CITATION].', '1806.03575-2-23-1': 'This dataset is extended from the ICVL dataset [CITATION].', '1806.03575-2-23-2': 'The ICVL dataset includes [MATH] images captured using Specim PS Kappa DX4 hyperspectral camera.', '1806.03575-2-23-3': 'Each image is of size [MATH] in spatial resolution and contains [MATH] spectral bands in the range of [MATH].', '1806.03575-2-23-4': 'In experiments, [MATH] successive bands ranging from [MATH] with [MATH] interval are extracted from each image for evaluation.', '1806.03575-2-23-5': 'In the NTIRE2018 challenge, this dataset is further extended by supplementing [MATH] extra images of the same spatial and spectral resolution.', '1806.03575-2-23-6': 'As a result, [MATH] high-resolution hyperspectral images are collected as the training data.', '1806.03575-2-23-7': 'In addition, another [MATH] hyperspectral images are further introduced as the test set.', '1806.03575-2-23-8': 'In the NTIRE2018 dataset, the corresponding RGB rendition is also provided for each image.', '1806.03575-2-23-9': 'In the following, we will employ the RGB-hyperspectral image pairs to evaluate the proposed method.', '1806.03575-2-24-0': '## Comparison Methods Implementation Details', '1806.03575-2-25-0': 'To demonstrate the effectiveness of the proposed method, we compare it with four spectral super-resolution methods, including spline interpolation, the sparse recovery method in [CITATION] (Arad [MATH]), A+ [CITATION], and the deep learning method in [CITATION] (Galliani [MATH]).', '1806.03575-2-25-1': '[CITATION] are implemented by the codes released by the authors.', '1806.03575-2-25-2': 'Since there is no code released for [CITATION], we reimplement it in this study.', '1806.03575-2-25-3': 'In the following, we will give the implementation details of each method.', '1806.03575-2-26-0': 'Spline interpolation The interpolation algorithm serves as the most primitive baseline in this study.', '1806.03575-2-26-1': 'Specifically, for each RGB pixel [MATH], we use spline interpolation to upsample it and obtain a [MATH]-dimensional spectrum ([MATH]).', '1806.03575-2-26-2': 'According to the visible spectrum, the [MATH], [MATH], [MATH] values of an RGB pixel are assigned to [MATH], [MATH], and [MATH], respectively.', '1806.03575-2-27-0': 'Arad [MATH] and A+ The low spectral resolution image is assumed to be a directly downsampled version of the corresponding hyperspectral image using some specific linear projection matrix.', '1806.03575-2-27-1': 'In [CITATION] this matrix is required to be perfectly known.', '1806.03575-2-27-2': 'In our experiments, we fit the projection matrix using training data with conventional linear regression.', '1806.03575-2-28-0': 'Galliani [MATH] and our method We experimentally find the optimal set of hyper-parameters for both methods.', '1806.03575-2-28-1': '[MATH] dropout is applied to Galliani [MATH], while our method utilizes [MATH] dropout rate.', '1806.03575-2-28-2': 'All the leaky ReLU activation functions are applied with a negative slope of 0.2.', '1806.03575-2-28-3': 'We train the networks for 100 epochs using Adam optimizer with [MATH] regularization.', '1806.03575-2-28-4': 'Weight initialization and learning rate vary for different methods.', '1806.03575-2-28-5': 'For Galliani [MATH], the weights are initialized via HeUniform [CITATION], and the learning rate is set to [MATH] for the first 50 epochs, decayed to [MATH] for the next 50 epochs.', '1806.03575-2-28-6': 'As for our method, we use HeNormal initialization [CITATION].', '1806.03575-2-28-7': 'The initial learning rate is [MATH] and is multiplied by 0.93 every 10 epochs.', '1806.03575-2-28-8': 'We perform data augmentation by extracting patches of size [MATH] with a stride of 40 pixels from training data.', '1806.03575-2-28-9': 'The total amount of training samples is over [MATH].', '1806.03575-2-28-10': 'At the test phase, we directly feed the whole image to the network and get the estimated hyperspectral image in one single forward pass.', '1806.03575-2-29-0': '## Evaluation Metrics', '1806.03575-2-30-0': 'To quantitatively evaluate the performance of the proposed method, we adopt the following two categories of evaluation metrics.', '1806.03575-2-31-0': 'Pixel-level reconstruction error We follow [CITATION] to use absolute and relative root-mean-square error (RMSE and rRMSE) as quantitative measurements for reconstruction accuracy.', '1806.03575-2-31-1': 'Let [MATH] and [MATH] denote the [MATH]th element of the real and estimated hyperspectral images, [MATH] is the average of [MATH], and [MATH] is the total number of elements in one hyperspectral image.', '1806.03575-2-31-2': 'There are two formulas for RMSE and rRMSE respectively.', '1806.03575-2-31-3': '[EQUATION]', '1806.03575-2-31-4': 'Spectral similarity Since the key for spectral super-resolution is to reconstruct the spectra, we also use spectral angle mapper ([MATH]) to evaluate the performance of different methods.', '1806.03575-2-31-5': '[MATH] calculates the average spectral angle between the spectra of real and estimated hyperspectral images.', '1806.03575-2-31-6': 'Let [MATH] represents the spectra of the [MATH]th hyperspectral pixel in real and estimated hyperspectral images ([MATH] is the number of bands), and [MATH] is the total number of pixels within an image.', '1806.03575-2-31-7': 'The [MATH] value can be computed as follows.', '1806.03575-2-31-8': '[EQUATION]', '1806.03575-2-32-0': '## Experimental Results', '1806.03575-2-33-0': 'Convergence Analysis We plot the curve of [MATH] loss on the training set and the curves of five evaluation metrics computed on the test set in Figure [REF].', '1806.03575-2-33-1': 'It can be seen that both the training loss and the value of metrics gradually decrease and ultimately converge with the proceeding of the training.', '1806.03575-2-33-2': 'This demonstrates that the proposed multi-scale convolution neural network converges well.', '1806.03575-2-34-0': 'Quantitative Results Table [REF] provides the quantitative results of our method and all baseline methods.', '1806.03575-2-34-1': 'It can be seen that our model outperforms all competitors with regards to [MATH] and [MATH], and produces comparable results to Galliani [MATH] on [MATH] and [MATH].', '1806.03575-2-34-2': 'More importantly, our method surpasses all the others with respect to spectral angle mapper.', '1806.03575-2-34-3': 'This clearly proves that our model reconstructs spectra more accurately than other competitors.', '1806.03575-2-34-4': 'It is worth pointing out that reconstruction error (absolute and relative [MATH]) is not necessarily positively correlated with spectral angle mapper ([MATH]).', '1806.03575-2-34-5': 'For example, when the pixels of an image are shuffled, [MATH] and [MATH] will remain the same, while [MATH] will change completely.', '1806.03575-2-34-6': 'According to the results in Table [REF], we can find that our finely designed network enhances spectral super-resolution from both aspects, [MATH], yielding better results on both average root-mean-square error and spectral angle similarity.', '1806.03575-2-35-0': 'Visual Results To further clarify the superiority in reconstruction accuracy.', '1806.03575-2-35-1': 'We show the absolute reconstruction error of every test image in Figure [REF].', '1806.03575-2-35-2': 'The error is summarized over all bands of the hyperspectral image.', '1806.03575-2-35-3': 'Since A+ outperforms Arad [MATH] in terms of any evaluation metric, we use A+ to represent the sparse coding methods.', '1806.03575-2-35-4': 'It can be seen that our method yields smoother reconstructed images as well as lower reconstruction error than other competitors.', '1806.03575-2-36-0': 'In addition, we randomly choose three test images and plot the real and reconstructed spectra for four pixels in Figure [REF] to further demonstrate the effectiveness of the proposed method in spectrum reconstruction.', '1806.03575-2-36-1': 'It can be seen that only slight difference exists between the reconstructed spectra and the ground truth.', '1806.03575-2-37-0': 'According to these results above, we can conclude that the proposed method is effective in spectral super-resolution and outperforms several state-of-the-art competitors.', '1806.03575-2-38-0': '# Conclusion', '1806.03575-2-39-0': 'In this study, we show that leveraging both the local and non-local information of input images is essential for the accurate spectral reconstruction.', '1806.03575-2-39-1': 'Following this idea, we design a novel multi-scale convolutional neural network, which employs a symmetrically cascaded downsampling-upsampling architecture to jointly encode the local and non-local image information for spectral reconstruction.', '1806.03575-2-39-2': 'With extensive experiments on a large hyperspectral images dataset, the proposed method clearly outperforms several state-of-the-art methods in terms of reconstruction accuracy and spectral similarity.'}

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'1806.03575-3-10-2'], ['1806.03575-2-10-3', '1806.03575-3-10-3'], ['1806.03575-2-10-4', '1806.03575-3-10-4'], ['1806.03575-2-10-5', '1806.03575-3-10-5'], ['1806.03575-2-10-6', '1806.03575-3-10-6'], ['1806.03575-2-20-0', '1806.03575-3-20-0'], ['1806.03575-2-20-1', '1806.03575-3-20-1'], ['1806.03575-2-20-2', '1806.03575-3-20-2'], ['1806.03575-2-20-3', '1806.03575-3-20-3'], ['1806.03575-2-20-4', '1806.03575-3-20-4'], ['1806.03575-2-20-5', '1806.03575-3-20-5'], ['1806.03575-2-31-0', '1806.03575-3-31-0'], ['1806.03575-2-31-1', '1806.03575-3-31-1'], ['1806.03575-2-31-2', '1806.03575-3-31-2'], ['1806.03575-2-31-4', '1806.03575-3-31-4'], ['1806.03575-2-31-5', '1806.03575-3-31-5'], ['1806.03575-2-31-6', '1806.03575-3-31-6'], ['1806.03575-2-31-7', '1806.03575-3-31-7'], ['1806.03575-2-35-0', '1806.03575-3-35-0'], ['1806.03575-2-35-2', '1806.03575-3-35-2'], ['1806.03575-2-35-3', '1806.03575-3-35-3'], ['1806.03575-2-35-4', '1806.03575-3-35-4'], ['1806.03575-2-39-0', '1806.03575-3-39-0'], ['1806.03575-2-39-1', '1806.03575-3-39-1'], ['1806.03575-2-39-2', '1806.03575-3-39-2'], ['1806.03575-2-15-0', '1806.03575-3-15-0'], ['1806.03575-2-15-1', '1806.03575-3-15-1'], ['1806.03575-2-12-0', '1806.03575-3-12-0'], ['1806.03575-2-12-1', '1806.03575-3-12-1'], ['1806.03575-2-12-2', '1806.03575-3-12-2'], ['1806.03575-2-4-0', '1806.03575-3-4-0'], ['1806.03575-2-4-1', '1806.03575-3-4-1'], ['1806.03575-2-4-2', '1806.03575-3-4-2'], ['1806.03575-2-4-3', '1806.03575-3-4-3'], ['1806.03575-2-4-4', '1806.03575-3-4-4'], ['1806.03575-2-9-1', '1806.03575-3-9-1'], ['1806.03575-2-9-2', '1806.03575-3-9-2'], ['1806.03575-2-9-3', '1806.03575-3-9-3'], ['1806.03575-2-9-4', '1806.03575-3-9-4'], ['1806.03575-2-9-5', '1806.03575-3-9-5'], ['1806.03575-2-9-6', '1806.03575-3-9-6'], ['1806.03575-2-27-0', '1806.03575-3-27-0'], ['1806.03575-2-27-1', '1806.03575-3-27-1'], ['1806.03575-2-27-2', '1806.03575-3-27-2'], ['1806.03575-2-33-0', '1806.03575-3-33-0'], ['1806.03575-2-33-1', '1806.03575-3-33-1'], ['1806.03575-2-33-2', '1806.03575-3-33-2'], ['1806.03575-2-2-0', '1806.03575-3-2-0'], ['1806.03575-2-2-1', '1806.03575-3-2-1'], ['1806.03575-2-2-2', '1806.03575-3-2-2'], ['1806.03575-2-17-0', '1806.03575-3-17-0'], ['1806.03575-2-17-1', '1806.03575-3-17-1'], ['1806.03575-2-17-2', '1806.03575-3-17-2'], ['1806.03575-2-17-3', '1806.03575-3-17-3'], ['1806.03575-2-17-4', '1806.03575-3-17-4'], ['1806.03575-2-17-5', '1806.03575-3-17-5'], ['1806.03575-2-34-0', '1806.03575-3-34-0'], ['1806.03575-2-34-1', '1806.03575-3-34-1'], ['1806.03575-2-34-2', '1806.03575-3-34-2'], ['1806.03575-2-34-3', '1806.03575-3-34-3'], ['1806.03575-2-34-4', '1806.03575-3-34-4'], ['1806.03575-2-34-5', '1806.03575-3-34-5'], ['1806.03575-2-34-6', '1806.03575-3-34-6'], ['1806.03575-2-3-0', '1806.03575-3-3-0'], ['1806.03575-2-3-1', '1806.03575-3-3-1'], ['1806.03575-2-3-3', '1806.03575-3-3-3'], ['1806.03575-2-19-0', '1806.03575-3-19-1'], ['1806.03575-2-19-1', '1806.03575-3-19-2'], ['1806.03575-2-19-2', '1806.03575-3-19-3'], ['1806.03575-2-19-3', '1806.03575-3-19-4'], ['1806.03575-2-19-4', '1806.03575-3-19-5'], ['1806.03575-2-19-5', '1806.03575-3-19-6'], ['1806.03575-2-19-6', '1806.03575-3-19-7'], ['1806.03575-2-19-7', '1806.03575-3-19-8'], ['1806.03575-2-19-8', '1806.03575-3-19-9'], ['1806.03575-2-16-0', '1806.03575-3-16-0'], ['1806.03575-2-16-1', '1806.03575-3-16-1'], ['1806.03575-2-25-0', '1806.03575-3-25-0'], ['1806.03575-2-25-1', '1806.03575-3-25-1'], ['1806.03575-2-25-2', '1806.03575-3-25-2'], ['1806.03575-2-25-3', '1806.03575-3-25-3'], ['1806.03575-2-23-0', '1806.03575-3-23-0'], ['1806.03575-2-23-1', '1806.03575-3-23-1'], ['1806.03575-2-23-2', '1806.03575-3-23-2'], ['1806.03575-2-23-3', '1806.03575-3-23-3'], ['1806.03575-2-23-4', '1806.03575-3-23-4'], ['1806.03575-2-23-5', '1806.03575-3-23-5'], ['1806.03575-2-23-6', '1806.03575-3-23-6'], ['1806.03575-2-23-7', '1806.03575-3-23-7'], ['1806.03575-2-23-8', '1806.03575-3-23-8'], ['1806.03575-2-23-9', '1806.03575-3-23-9'], ['1806.03575-2-14-0', '1806.03575-3-14-0'], ['1806.03575-2-14-1', '1806.03575-3-14-1'], ['1806.03575-2-36-0', '1806.03575-3-36-0'], ['1806.03575-2-36-1', '1806.03575-3-36-1'], ['1806.03575-2-37-0', '1806.03575-3-37-0'], ['1806.03575-2-30-0', '1806.03575-3-30-0'], ['1806.03575-2-0-0', '1806.03575-3-0-0'], ['1806.03575-2-0-1', '1806.03575-3-0-1'], ['1806.03575-2-0-2', '1806.03575-3-0-2'], ['1806.03575-2-0-3', '1806.03575-3-0-3'], ['1806.03575-2-0-4', '1806.03575-3-0-4']]

[['1806.03575-2-35-1', '1806.03575-3-35-1'], ['1806.03575-2-15-2', '1806.03575-3-15-2'], ['1806.03575-2-9-0', '1806.03575-3-9-0'], ['1806.03575-2-2-3', '1806.03575-3-2-3'], ['1806.03575-2-3-2', '1806.03575-3-3-2']]

[]

[]

[]

['1806.03575-1-6-0', '1806.03575-1-31-3', '1806.03575-1-31-8', '1806.03575-2-6-0', '1806.03575-2-31-3', '1806.03575-2-31-8', '1806.03575-3-6-0', '1806.03575-3-31-3', '1806.03575-3-31-8']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1806.03575

{'1806.03575-3-0-0': 'Different from traditional hyperspectral super-resolution approaches that focus on improving the spatial resolution, spectral super-resolution aims at producing a high-resolution hyperspectral image from the RGB observation with super-resolution in spectral domain.', '1806.03575-3-0-1': 'However, it is challenging to accurately reconstruct a high-dimensional continuous spectrum from three discrete intensity values at each pixel, since too much information is lost during the procedure where the latent hyperspectral image is downsampled (e.g., with [MATH]10 scaling factor) in spectral domain to produce an RGB observation.', '1806.03575-3-0-2': 'To address this problem, we present a multi-scale deep convolutional neural network (CNN) to explicitly map the input RGB image into a hyperspectral image.', '1806.03575-3-0-3': 'Through symmetrically downsampling and upsampling the intermediate feature maps in a cascading paradigm, the local and non-local image information can be jointly encoded for spectral representation, ultimately improving the spectral reconstruction accuracy.', '1806.03575-3-0-4': 'Extensive experiments on a large hyperspectral dataset demonstrate the effectiveness of the proposed method.', '1806.03575-3-1-0': '# Introduction', '1806.03575-3-2-0': 'Hyperspectral imaging encodes the reflectance of the scene from hundreds or thousands of bands with a narrow wavelength interval (e.g., 10nm) into a hyperspectral image.', '1806.03575-3-2-1': 'Different from conventional images, each pixel in the hyperspectral image contains a continuous spectrum, thus allowing the acquisition of abundant spectral information.', '1806.03575-3-2-2': 'Such information has proven to be quite useful for distinguishing different materials.', '1806.03575-3-2-3': 'Therefore, hyperspectral images have been widely exploited to facilitate various applications in computer vision community, such as visual tracking [CITATION], image segmentation [CITATION], face recognition [CITATION], scene classification [CITATION], and anomaly detection [CITATION].', '1806.03575-3-3-0': 'The acquisition of spectral information, however, comes at the cost of decreasing the spatial resolution of hyperspectral images.', '1806.03575-3-3-1': 'This is because a fewer number of photons are captured by each detector due to the narrower width of the spectral bands.', '1806.03575-3-3-2': 'In order to maintain a reasonable signal-to-noise ratio (SNR), the instantaneous field of view (IFOV) needs to be increased, which renders it difficult to produce hyperspectral images with high spatial resolution.', '1806.03575-3-3-3': 'To address this problem, many efforts have been made for the hyperspectral imagery super-resolution.', '1806.03575-3-4-0': 'Most of the existing methods mainly focus on enhancing the spatial resolution of the observed hyperspectral image.', '1806.03575-3-4-1': 'According to the input images, they can be divided into two categories: [MATH] fusion based methods where a high-resolution conventional image ([MATH], RGB image) and a low-resolution hyperspectral image are fused together to produce a high-resolution hyperspectral image [CITATION] [MATH] single image super-resolution which directly increases the spatial resolution of a hyperspectral image [CITATION].', '1806.03575-3-4-2': 'Although these methods have shown effective performance, the acquisition of the input hyperspectral image often requires specialized hyperspectral sensors as well as extensive imaging cost.', '1806.03575-3-4-3': 'To mitigate this problem, some recent literature [CITATION] turn to investigate a novel hyperspectral imagery super-resolution scheme, termed spectral super-resolution, which aims at improving the spectral resolution of a given RGB image.', '1806.03575-3-4-4': 'Since the input image can be easily captured by conventional RGB sensors, imaging cost can be greatly reduced.', '1806.03575-3-5-0': 'However, it is challenging to accurately reconstruct a hyperspectral image from a single RGB observation, since mapping three discrete intensity values to a continuous spectrum is a highly ill-posed linear inverse problem.', '1806.03575-3-5-1': 'To address this problem, we propose to learn a complicated non-linear mapping function for spectral super-resolution with deep convolutional neural networks (CNN).', '1806.03575-3-5-2': 'It has been shown that the 3-dimensional color vector for a specific pixel can be viewed as the downsampled observation of the corresponding spectrum.', '1806.03575-3-5-3': 'Moreover, for a candidate pixel, there often exist abundant locally and no-locally similar pixels ([MATH] exhibiting similar spectra) in the spatial domain.', '1806.03575-3-5-4': 'As a result, the color vectors corresponding to those similar pixels can be viewed as a group of downsampled observations of the latent spectra for the candidate pixel.', '1806.03575-3-5-5': 'Therefore, accurate spectral reconstruction requires to explicitly consider both the local and non-local information from the input RGB image.', '1806.03575-3-5-6': 'To this end, we develop a novel multi-scale CNN.', '1806.03575-3-5-7': 'Our method jointly encodes the local and non-local image information through symmetrically downsampling and upsampling the intermediate feature maps in a cascading paradigm, thus enhancing the spectral reconstruction accuracy.', '1806.03575-3-5-8': 'We experimentally show that the proposed method can be easily trained in an end-to-end scheme and beat several state-of-the-art methods on a large hyperspectral image dataset with respect to various evaluation metrics.', '1806.03575-3-6-0': 'Our contributions are twofold:', '1806.03575-3-7-0': '# Related Work', '1806.03575-3-8-0': 'This section gives a brief review of the existing spectral super-resolution methods, which can be divided into the following two categories.', '1806.03575-3-9-0': 'Statistic based methods This line of research mainly focus on exploiting the inherent statistical distribution of the latent hyperspectral image as priors to guide the super-resolution [CITATION].', '1806.03575-3-9-1': 'Most of these methods involve building overcomplete dictionaries and learning sparse coding coefficients to linearly combine the dictionary atoms.', '1806.03575-3-9-2': 'For example, in [CITATION], Arad [MATH] leveraged image priors to build a dictionary using K-SVD [CITATION].', '1806.03575-3-9-3': 'At test time, orthogonal matching pursuit [CITATION] was used to compute a sparse representation of the input RGB image.', '1806.03575-3-9-4': '[CITATION] proposed a new method inspired by A+ [CITATION], where sparse coefficients are computed by explicitly solving a sparse least square problem.', '1806.03575-3-9-5': 'These methods directly exploit the whole image to build image prior, ignoring local and non-local structure information.', '1806.03575-3-9-6': "What's more, since the image prior is often handcrafted or heuristically designed with shallow structure, these methods fail to generalize well in practice.", '1806.03575-3-10-0': 'Learning based methods These methods directly learn a certain mapping function from the RGB image to a corresponding hyperspectral image.', '1806.03575-3-10-1': 'For example, [CITATION] proposed a training based method using a radial basis function network.', '1806.03575-3-10-2': 'The input data was pre-processed with a white balancing function to alleviate the influence of different illumination.', '1806.03575-3-10-3': 'The total reconstruction accuracy is affected by the performance of this pre-processing stage.', '1806.03575-3-10-4': 'Recently, witnessing the great success of deep learning in many other ill-posed inverse problems such as image denoising [CITATION] and single image super-resolution [CITATION], it is natural to consider using deep networks (especially convolutional neural networks) for spectral super-resolution.', '1806.03575-3-10-5': 'In [CITATION], Galliani [MATH] exploited a variant of fully convolutional DenseNets (FC-DenseNets [CITATION]) for spectral super-resolution.', '1806.03575-3-10-6': 'However, this method is sensitive to the hyper-parameters and its performance can still be further improved.', '1806.03575-3-11-0': '# Proposed Method', '1806.03575-3-12-0': 'In this section, we will introduce the proposed multi-scale convolution neural network in details.', '1806.03575-3-12-1': 'Firstly, we introduce some building blocks which will be utilized in our network.', '1806.03575-3-12-2': 'Then, we will illustrate the architecture of the proposed network.', '1806.03575-3-13-0': '## Building Blocks', '1806.03575-3-14-0': 'There are three basic building blocks in our network.', '1806.03575-3-14-1': 'Their structures are shown in Table [REF].', '1806.03575-3-15-0': 'Double convolution (Double Conv) block consists of two [MATH] convolutions.', '1806.03575-3-15-1': 'Each of them is followed by batch normalization, leaky ReLU and dropout.', '1806.03575-3-15-2': 'We exploit batch normalization and dropout to deal with overfitting.', '1806.03575-3-16-0': 'Downsample block contains a regular max-pooling layer.', '1806.03575-3-16-1': 'It reduces the spatial size of the feature map and enlarges the receptive field of the network.', '1806.03575-3-17-0': 'Upsample block is utilized to upsample the feature map in the spatial domain.', '1806.03575-3-17-1': 'To this end, much previous literature often adopts the transposed convolution.', '1806.03575-3-17-2': 'However, it is prone to generate checkboard artifacts.', '1806.03575-3-17-3': 'To address this problem, we use the pixel shuffle operation [CITATION].', '1806.03575-3-17-4': 'It has been shown that pixel shuffle alleviates the checkboard artifacts.', '1806.03575-3-17-5': 'In addition, due to not introducing any learnable parameters, pixel shuffle also helps improve the robustness against over-fitting.', '1806.03575-3-18-0': '## Network Architecture', '1806.03575-3-19-0': 'Our method is inspired by the well known U-Net architecture for image segmentation [CITATION].', '1806.03575-3-19-1': 'The overall architecture of the proposed multi-scale convolution neural network is depicted in Figure [REF].', '1806.03575-3-19-2': 'The network follows the encoder-decoder pattern.', '1806.03575-3-19-3': 'For the encoder part, each downsampling step consists of a "Double Conv" with a downsample block.', '1806.03575-3-19-4': 'The spatial size is progressively reduced, and the number of features is doubled at each step.', '1806.03575-3-19-5': 'The decoder is symmetric to the encoder path.', '1806.03575-3-19-6': 'Every step in the decoder path consists of an upsampling operation followed by a "Double Conv" block.', '1806.03575-3-19-7': 'The spatial size of the features is recovered, while the number of features is halved every step.', '1806.03575-3-19-8': 'Finally, a [MATH] convolution maps the output features to the reconstructed 31-channel hyperspectral image.', '1806.03575-3-19-9': 'In addition to the feedforward path, skip connections are used to concatenate the corresponding feature maps of the encoder and decoder.', '1806.03575-3-20-0': 'Our method naturally fits the task of spectral reconstruction.', '1806.03575-3-20-1': 'The encoder can be interpreted as extracting features from RGB images.', '1806.03575-3-20-2': 'Through downsampling in a cascade way, the receptive field of the network is constantly increased, which allows the network to "see" more pixels in an increasingly larger field of view.', '1806.03575-3-20-3': 'By doing so, both the local and non-local information can be encoded to better represent the latent spectra.', '1806.03575-3-20-4': 'The symmetric decoder procedure is employed to reconstruct the latent hyperspectral images based on these deep and compact features.', '1806.03575-3-20-5': 'The skip connections with concatenations are essential for introducing multi-scale information and yielding better estimation of the spectra.', '1806.03575-3-21-0': '# Experiments', '1806.03575-3-22-0': '## Datasets', '1806.03575-3-23-0': 'In this study, all experiments are performed on the NTIRE2018 dataset [CITATION].', '1806.03575-3-23-1': 'This dataset is extended from the ICVL dataset [CITATION].', '1806.03575-3-23-2': 'The ICVL dataset includes [MATH] images captured using Specim PS Kappa DX4 hyperspectral camera.', '1806.03575-3-23-3': 'Each image is of size [MATH] in spatial resolution and contains [MATH] spectral bands in the range of [MATH].', '1806.03575-3-23-4': 'In experiments, [MATH] successive bands ranging from [MATH] with [MATH] interval are extracted from each image for evaluation.', '1806.03575-3-23-5': 'In the NTIRE2018 challenge, this dataset is further extended by supplementing [MATH] extra images of the same spatial and spectral resolution.', '1806.03575-3-23-6': 'As a result, [MATH] high-resolution hyperspectral images are collected as the training data.', '1806.03575-3-23-7': 'In addition, another [MATH] hyperspectral images are further introduced as the test set.', '1806.03575-3-23-8': 'In the NTIRE2018 dataset, the corresponding RGB rendition is also provided for each image.', '1806.03575-3-23-9': 'In the following, we will employ the RGB-hyperspectral image pairs to evaluate the proposed method.', '1806.03575-3-24-0': '## Comparison Methods Implementation Details', '1806.03575-3-25-0': 'To demonstrate the effectiveness of the proposed method, we compare it with four spectral super-resolution methods, including spline interpolation, the sparse recovery method in [CITATION] (Arad [MATH]), A+ [CITATION], and the deep learning method in [CITATION] (Galliani [MATH]).', '1806.03575-3-25-1': '[CITATION] are implemented by the codes released by the authors.', '1806.03575-3-25-2': 'Since there is no code released for [CITATION], we reimplement it in this study.', '1806.03575-3-25-3': 'In the following, we will give the implementation details of each method.', '1806.03575-3-26-0': 'Spline interpolation The interpolation algorithm serves as the most primitive baseline in this study.', '1806.03575-3-26-1': 'Specifically, for each RGB pixel [MATH], we use spline interpolation to upsample it and obtain a [MATH]-dimensional spectrum ([MATH]).', '1806.03575-3-26-2': 'According to the visible spectrum, the [MATH], [MATH], [MATH] values of an RGB pixel are assigned to [MATH], [MATH], and [MATH], respectively.', '1806.03575-3-27-0': 'Arad [MATH] and A+ The low spectral resolution image is assumed to be a directly downsampled version of the corresponding hyperspectral image using some specific linear projection matrix.', '1806.03575-3-27-1': 'In [CITATION] this matrix is required to be perfectly known.', '1806.03575-3-27-2': 'In our experiments, we fit the projection matrix using training data with conventional linear regression.', '1806.03575-3-28-0': 'Galliani [MATH] and our method We experimentally find the optimal set of hyper-parameters for both methods.', '1806.03575-3-28-1': '[MATH] dropout is applied to Galliani [MATH], while our method utilizes [MATH] dropout rate.', '1806.03575-3-28-2': 'All the leaky ReLU activation functions are applied with a negative slope of 0.2.', '1806.03575-3-28-3': 'We train the networks for 100 epochs using Adam optimizer with [MATH] regularization.', '1806.03575-3-28-4': 'Weight initialization and learning rate vary for different methods.', '1806.03575-3-28-5': 'For Galliani [MATH], the weights are initialized via HeUniform [CITATION], and the learning rate is set to [MATH] for the first 50 epochs, decayed to [MATH] for the next 50 epochs.', '1806.03575-3-28-6': 'As for our method, we use HeNormal initialization [CITATION].', '1806.03575-3-28-7': 'The initial learning rate is [MATH] and is multiplied by 0.93 every 10 epochs.', '1806.03575-3-28-8': 'We perform data augmentation by extracting patches of size [MATH] with a stride of 40 pixels from training data.', '1806.03575-3-28-9': 'The total amount of training samples is over [MATH].', '1806.03575-3-28-10': 'At the test phase, we directly feed the whole image to the network and get the estimated hyperspectral image in one single forward pass.', '1806.03575-3-29-0': '## Evaluation Metrics', '1806.03575-3-30-0': 'To quantitatively evaluate the performance of the proposed method, we adopt the following two categories of evaluation metrics.', '1806.03575-3-31-0': 'Pixel-level reconstruction error We follow [CITATION] to use absolute and relative root-mean-square error (RMSE and rRMSE) as quantitative measurements for reconstruction accuracy.', '1806.03575-3-31-1': 'Let [MATH] and [MATH] denote the [MATH]th element of the real and estimated hyperspectral images, [MATH] is the average of [MATH], and [MATH] is the total number of elements in one hyperspectral image.', '1806.03575-3-31-2': 'There are two formulas for RMSE and rRMSE respectively.', '1806.03575-3-31-3': '[EQUATION]', '1806.03575-3-31-4': 'Spectral similarity Since the key for spectral super-resolution is to reconstruct the spectra, we also use spectral angle mapper ([MATH]) to evaluate the performance of different methods.', '1806.03575-3-31-5': '[MATH] calculates the average spectral angle between the spectra of real and estimated hyperspectral images.', '1806.03575-3-31-6': 'Let [MATH] represents the spectra of the [MATH]th hyperspectral pixel in real and estimated hyperspectral images ([MATH] is the number of bands), and [MATH] is the total number of pixels within an image.', '1806.03575-3-31-7': 'The [MATH] value can be computed as follows.', '1806.03575-3-31-8': '[EQUATION]', '1806.03575-3-32-0': '## Experimental Results', '1806.03575-3-33-0': 'Convergence Analysis We plot the curve of [MATH] loss on the training set and the curves of five evaluation metrics computed on the test set in Figure [REF].', '1806.03575-3-33-1': 'It can be seen that both the training loss and the value of metrics gradually decrease and ultimately converge with the proceeding of the training.', '1806.03575-3-33-2': 'This demonstrates that the proposed multi-scale convolution neural network converges well.', '1806.03575-3-34-0': 'Quantitative Results Table [REF] provides the quantitative results of our method and all baseline methods.', '1806.03575-3-34-1': 'It can be seen that our model outperforms all competitors with regards to [MATH] and [MATH], and produces comparable results to Galliani [MATH] on [MATH] and [MATH].', '1806.03575-3-34-2': 'More importantly, our method surpasses all the others with respect to spectral angle mapper.', '1806.03575-3-34-3': 'This clearly proves that our model reconstructs spectra more accurately than other competitors.', '1806.03575-3-34-4': 'It is worth pointing out that reconstruction error (absolute and relative [MATH]) is not necessarily positively correlated with spectral angle mapper ([MATH]).', '1806.03575-3-34-5': 'For example, when the pixels of an image are shuffled, [MATH] and [MATH] will remain the same, while [MATH] will change completely.', '1806.03575-3-34-6': 'According to the results in Table [REF], we can find that our finely designed network enhances spectral super-resolution from both aspects, [MATH], yielding better results on both average root-mean-square error and spectral angle similarity.', '1806.03575-3-35-0': 'Visual Results To further clarify the superiority in reconstruction accuracy.', '1806.03575-3-35-1': 'We show the absolute reconstruction error of test images in Figure [REF].', '1806.03575-3-35-2': 'The error is summarized over all bands of the hyperspectral image.', '1806.03575-3-35-3': 'Since A+ outperforms Arad [MATH] in terms of any evaluation metric, we use A+ to represent the sparse coding methods.', '1806.03575-3-35-4': 'It can be seen that our method yields smoother reconstructed images as well as lower reconstruction error than other competitors.', '1806.03575-3-36-0': 'In addition, we randomly choose three test images and plot the real and reconstructed spectra for four pixels in Figure [REF] to further demonstrate the effectiveness of the proposed method in spectrum reconstruction.', '1806.03575-3-36-1': 'It can be seen that only slight difference exists between the reconstructed spectra and the ground truth.', '1806.03575-3-37-0': 'According to these results above, we can conclude that the proposed method is effective in spectral super-resolution and outperforms several state-of-the-art competitors.', '1806.03575-3-38-0': '# Conclusion', '1806.03575-3-39-0': 'In this study, we show that leveraging both the local and non-local information of input images is essential for the accurate spectral reconstruction.', '1806.03575-3-39-1': 'Following this idea, we design a novel multi-scale convolutional neural network, which employs a symmetrically cascaded downsampling-upsampling architecture to jointly encode the local and non-local image information for spectral reconstruction.', '1806.03575-3-39-2': 'With extensive experiments on a large hyperspectral images dataset, the proposed method clearly outperforms several state-of-the-art methods in terms of reconstruction accuracy and spectral similarity.'}

1708.08916

{'1708.08916-1-0-0': 'Phylogenetic species trees typically represent the speciation history as a bifurcating tree.', '1708.08916-1-0-1': 'Speciation events that simultaneously create more than two descendants, thereby creating polytomies, are rare but not impossible.', '1708.08916-1-0-2': 'In this short note, we describe a statistical test that can be used to reject the null hypothesis that a node in an unrooted species tree is a true polytomy.', '1708.08916-1-0-3': 'The test exploits properties of the multi-species coalescent model and is implemented in ASTRARL.', '1708.08916-1-1-0': 'Consider an unrooted tree on a quartet of taxa, [MATH], [MATH], [MATH], and [MATH], where [MATH] and [MATH] are closer to each other than [MATH], and [MATH] (i.e. the topology [MATH]).', '1708.08916-1-1-1': 'Under the Multi Species Coalescent (MSC) model and in the absence of any gene tree error, the frequency of seeing each topology in gene trees follows a multinomial distribution, with probabilities of [MATH] for the main topology and [MATH] for the alternative ones .', '1708.08916-1-1-2': 'In these equations, [MATH] is the internal branch length separating species [MATH], and [MATH] from [MATH] and [MATH].', '1708.08916-1-2-0': 'For a polytomy over a quartet of taxa, the branch length [MATH], and all topologies are expected to be equally likely with [MATH].', '1708.08916-1-2-1': 'To test (reject) if the quartet of taxa formed a polytomy, we observe that empirical distributions of quartet frequencies will follow a multinomial distribution.', '1708.08916-1-2-2': 'Thus, we can use a Pearson?s goodness-of-fit statistical test by noting that [EQUATION] is asymptotically [MATH] distributed with 2 degree of freedom for large enough [MATH] .', '1708.08916-1-2-3': 'Here [MATH], [MATH], and [MATH] are the frequency of observing gene tree quartet topologies [MATH], [MATH], and [MATH], respectively, and [MATH].', '1708.08916-1-2-4': "Note that [MATH] approximation for three equiprobable outcomes is applicable when [MATH] , and for smaller [MATH]'s an exact calculation of the critical value is required .", '1708.08916-1-3-0': 'With more than 4 species, every internal branch of the tree divides the tree into four parts (i.e. clusters), and defines a quadripartition with three possible topologies.', '1708.08916-1-3-1': 'Assuming that (i) the only source of incongruence between the species tree and gene trees is ILS , (ii) there is no gene tree error, and (iii) all four clusters around the internal branch are correct, we can apply results of [CITATION] to quadripartitions around the internal branches .', '1708.08916-1-3-2': 'Consequently, we can apply the [MATH] goodness-of-fit statistical test to a quadripartition.', '1708.08916-1-3-3': 'Therefore, [MATH] values can be simply replaced by the average value computed from all quartets of taxa such that each taxon belongs to one side of the quadripartition.', '1708.08916-1-4-0': 'This statistical test is implemented inside the ASTRAL package since version 4.11.2, available online at https://github.com/smirarab/ASTRAL, and can be applied using the option -t 10 .'}

{'1708.08916-2-0-0': 'Phylogenetic species trees typically represent the speciation history as a bifurcating tree.', '1708.08916-2-0-1': 'Speciation events that simultaneously create more than two descendants, thereby creating polytomies, are rare but not impossible.', '1708.08916-2-0-2': 'In this short note, we describe a statistical test that can be used to reject the null hypothesis that a node in an unrooted species tree is a true polytomy.', '1708.08916-2-0-3': 'The test exploits properties of the multi-species coalescent model and is implemented in ASTRAL.', '1708.08916-2-1-0': 'Consider an unrooted tree on a quartet of taxa, [MATH], [MATH], [MATH], and [MATH], where [MATH] and [MATH] are closer to each other than [MATH], and [MATH] (i.e. the topology [MATH]).', '1708.08916-2-1-1': 'Under the Multi Species Coalescent (MSC) model and in the absence of any gene tree error, the frequency of seeing each topology in gene trees follows a multinomial distribution, with probabilities of [MATH] for the main topology and [MATH] for the alternative ones .', '1708.08916-2-1-2': 'In these equations, [MATH] is the internal branch length separating species [MATH], and [MATH] from [MATH] and [MATH].', '1708.08916-2-2-0': 'For a polytomy over a quartet of taxa, the branch length [MATH], and all topologies are expected to be equally likely with [MATH].', '1708.08916-2-2-1': 'To test (reject) if the quartet of taxa formed a polytomy, we observe that empirical distributions of quartet frequencies will follow a multinomial distribution.', '1708.08916-2-2-2': 'Thus, we can use a Pearson’s goodness-of-fit statistical test by noting that [EQUATION] is asymptotically [MATH] distributed with 2 degree of freedom for large enough [MATH] .', '1708.08916-2-2-3': 'Here [MATH], [MATH], and [MATH] are the frequency of observing gene tree quartet topologies [MATH], [MATH], and [MATH], respectively, and [MATH].', '1708.08916-2-2-4': "Note that [MATH] approximation for three equiprobable outcomes is applicable when [MATH] , and for smaller [MATH]'s an exact calculation of the critical value is required .", '1708.08916-2-3-0': 'With more than 4 species, every internal branch of the tree divides the tree into four parts (i.e. clusters), and defines a quadripartition with three possible topologies.', '1708.08916-2-3-1': 'Assuming that (i) the only source of incongruence between the species tree and gene trees is ILS , (ii) there is no gene tree error, and (iii) all four clusters around the internal branch are correct, we can apply results of [CITATION] to quadripartitions around the internal branches .', '1708.08916-2-3-2': 'Consequently, we can apply the [MATH] goodness-of-fit statistical test to a quadripartition.', '1708.08916-2-3-3': 'Therefore, [MATH] values can be simply replaced by the average value computed from all quartets of taxa such that each taxon belongs to one side of the quadripartition.', '1708.08916-2-4-0': 'This statistical test is implemented inside the ASTRAL package since version 4.11.2 available online at https://github.com/smirarab/ASTRAL, and can be applied using the option -t 10 .'}

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'1708.08916-4-0-2'], ['1708.08916-3-52-0', '1708.08916-4-55-0'], ['1708.08916-3-52-1', '1708.08916-4-55-1'], ['1708.08916-3-52-3', '1708.08916-4-55-3'], ['1708.08916-3-52-4', '1708.08916-4-55-4'], ['1708.08916-3-26-0', '1708.08916-4-22-0'], ['1708.08916-3-26-1', '1708.08916-4-22-1'], ['1708.08916-3-26-2', '1708.08916-4-22-2'], ['1708.08916-3-20-1', '1708.08916-4-16-1'], ['1708.08916-3-35-2', '1708.08916-4-37-3'], ['1708.08916-3-35-3', '1708.08916-4-37-4'], ['1708.08916-3-35-5', '1708.08916-4-37-6'], ['1708.08916-3-35-6', '1708.08916-4-37-7'], ['1708.08916-3-24-0', '1708.08916-4-20-0'], ['1708.08916-3-24-2', '1708.08916-4-20-2'], ['1708.08916-3-24-3', '1708.08916-4-20-3'], ['1708.08916-3-24-4', '1708.08916-4-20-4'], ['1708.08916-3-24-5', '1708.08916-4-20-5'], ['1708.08916-3-24-6', '1708.08916-4-20-6'], ['1708.08916-3-24-8', '1708.08916-4-20-8'], ['1708.08916-3-42-1', '1708.08916-4-44-1'], ['1708.08916-3-56-0', '1708.08916-4-63-0'], ['1708.08916-3-56-1', '1708.08916-4-63-1'], ['1708.08916-3-58-0', '1708.08916-4-65-0'], ['1708.08916-3-58-1', '1708.08916-4-65-1'], ['1708.08916-3-58-2', '1708.08916-4-65-2'], ['1708.08916-3-13-2', '1708.08916-4-10-2'], ['1708.08916-3-13-3', '1708.08916-4-10-3'], ['1708.08916-3-41-2', '1708.08916-4-43-2'], ['1708.08916-3-17-0', '1708.08916-4-13-0'], ['1708.08916-3-57-0', '1708.08916-4-64-0'], ['1708.08916-3-21-0', '1708.08916-4-17-0'], ['1708.08916-3-21-1', '1708.08916-4-17-1'], ['1708.08916-3-21-2', '1708.08916-4-17-2'], ['1708.08916-3-21-3', '1708.08916-4-17-3'], ['1708.08916-3-21-4', '1708.08916-4-17-4'], ['1708.08916-3-21-5', '1708.08916-4-17-5'], ['1708.08916-3-18-0', '1708.08916-4-14-0'], ['1708.08916-3-22-0', '1708.08916-4-18-0'], ['1708.08916-3-22-1', '1708.08916-4-18-1'], ['1708.08916-3-22-2', '1708.08916-4-18-2'], ['1708.08916-3-22-3', '1708.08916-4-18-3'], ['1708.08916-3-22-4', '1708.08916-4-18-4'], ['1708.08916-3-22-5', '1708.08916-4-18-5'], ['1708.08916-3-30-2', '1708.08916-4-28-3'], ['1708.08916-3-30-3', '1708.08916-4-28-4'], ['1708.08916-3-53-0', '1708.08916-4-56-0'], ['1708.08916-3-53-1', '1708.08916-4-56-1'], ['1708.08916-3-53-3', '1708.08916-4-56-3'], ['1708.08916-3-19-0', '1708.08916-4-15-0'], ['1708.08916-3-19-1', '1708.08916-4-15-1'], ['1708.08916-3-59-0', '1708.08916-4-66-0'], ['1708.08916-3-59-1', '1708.08916-4-66-1'], ['1708.08916-3-59-3', '1708.08916-4-66-5'], ['1708.08916-3-59-4', '1708.08916-4-66-6'], ['1708.08916-3-48-1', '1708.08916-4-51-0'], ['1708.08916-3-48-2', '1708.08916-4-51-1'], ['1708.08916-3-48-5', '1708.08916-4-51-4'], ['1708.08916-3-48-6', '1708.08916-4-51-5'], ['1708.08916-3-48-7', '1708.08916-4-51-6'], ['1708.08916-3-48-8', '1708.08916-4-51-7'], ['1708.08916-3-48-9', '1708.08916-4-51-8'], ['1708.08916-3-48-10', '1708.08916-4-51-9'], ['1708.08916-3-36-0', '1708.08916-4-38-0'], ['1708.08916-3-36-1', '1708.08916-4-38-1'], ['1708.08916-3-36-3', '1708.08916-4-38-3'], ['1708.08916-3-49-1', '1708.08916-4-52-2'], ['1708.08916-3-49-2', '1708.08916-4-52-3'], 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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1708.08916

{'1708.08916-3-0-0': 'Phylogenetic species trees typically represent the speciation history as a bifurcating tree.', '1708.08916-3-0-1': 'Speciation events that simultaneously create more than two descendants, thereby creating polytomies in the phylogeny are believed to be possible.', '1708.08916-3-0-2': 'Moreover, the inability to resolve relationships is often shown as a (soft) polytomy.', '1708.08916-3-0-3': 'In this paper, we describe a statistical test to test the null hypothesis that a node in an unrooted species tree is a polytomy.', '1708.08916-3-0-4': 'The test exploits properties of the multi-species coalescent model and is implemented in ASTRAL.', '1708.08916-3-1-0': '# Introduction', '1708.08916-3-2-0': 'Phylogenies are typically modeled as bifurcating trees.', '1708.08916-3-2-1': 'Even when the evolution is fully vertical, which it is not always , the binary model precludes the possibility of several species evolving simultaneously from a progenitor species .', '1708.08916-3-2-2': 'Such events could be modeled in a multifurcating tree where some nodes have more than two children.', '1708.08916-3-2-3': 'Somewhat confusingly, multifurcations, also called polytomies, are also used when the analyst is unsure about some relationships.', '1708.08916-3-2-4': 'The terms hard and soft polytomies are used to distinguish between these two types of multifurcations , with a soft polytomy reserved for the case where relationships are unknown.', '1708.08916-3-3-0': 'The confusion between hard and soft polytomies is not entirely coincidental.', '1708.08916-3-3-1': 'The difficulty in resolving relationships increases as branches become shorter.', '1708.08916-3-3-2': 'Moreover, time is continuous in a mathematical sense but extremely short branches start to lose their meaning in evolutionary time.', '1708.08916-3-3-3': 'It would be, for example, absurd to have two mammal species originating one day apart.', '1708.08916-3-3-4': 'Taking this idea further, one can argue that branches that span many hundred or even several thousand generations are still a blink in evolutionary time and can be treated like polytomies.', '1708.08916-3-3-5': 'And such short branches will be extremely difficult to resolve with limited data, necessitating soft polytomies.', '1708.08916-3-4-0': 'The minimum length where branches become meaningful will depend on organisms.', '1708.08916-3-4-1': 'But a different question can be answered more broadly.', '1708.08916-3-5-0': 'For any branch of a given species tree, we can pose a null hypothesis that the length of that branch is zero, and thus, the branch should be removed to create a hard polytomy.', '1708.08916-3-5-1': 'Given a dataset, we can try to see if this null hypothesis can be rejected.', '1708.08916-3-5-2': 'If our data rejects the null hypothesis, a hard polytomy is refuted and a soft polytomy is unnecessary.', '1708.08916-3-5-3': 'If we fail to reject the null hypothesis, we can replace our branch with a soft polytomy.', '1708.08916-3-5-4': 'That we call the resulting polytomy soft may seem confusing considering that we started with a hypothesis of a hard polytomy.', '1708.08916-3-5-5': 'But note that the inability to reject a null hypothesis is never accepting the alternative hypothesis.', '1708.08916-3-5-6': 'It may simply be the lack of power to reject due to a lack of signal, which, incidentally, is what a soft polytomy represents.', '1708.08916-3-5-7': 'Thus, this framework connects the concepts of soft and hard polytomies.', '1708.08916-3-6-0': 'In this paper, we present a new test of polytomy as the null hypothesis.', '1708.08916-3-6-1': 'The idea of testing a polytomy as the null hypothesis and rejecting it using data is not new .', '1708.08916-3-6-2': 'Likelihood ratio tests against a zero length branch have been developed (e.g., SOWH test ) and are implemented in popular packages such as PAUP* .', '1708.08916-3-6-3': 'Treating polytomies as the null hypothesis has been pioneered by [CITATION] who sought to not only test for polytomies but also to use a power analysis to distinguish soft and hard polytomies.', '1708.08916-3-6-4': 'Appraising their general framework, [CITATION] showed that the power analysis can be sensitive to model complexity (or lack thereof).', '1708.08916-3-6-5': 'Perhaps most relevant to our work, [CITATION] presented an approximate but fast likelihood ratio test for a polytomy null hypothesis; their test, like what we will present looks at each branch and its surrounding branches while ignoring the rest of the tree.', '1708.08916-3-7-0': 'The existing tests of polytomy as the null and also Bayesian methods of modeling polytomies assume that the sequence data follow a single tree.', '1708.08916-3-7-1': 'Therefore, these methods test whether a gene tree includes a polytomy .', '1708.08916-3-7-2': 'However, the species tree can be different from gene trees and the discordance can have several causes .', '1708.08916-3-7-3': 'Arguably, the actual question of interest is whether the species tree includes hard polytomies.', '1708.08916-3-7-4': 'Alternatively, we may be interested to know whether we should treat the relationship between species as unresolved (i.e, a soft polytomy) given with the amount of data at hand.', '1708.08916-3-7-5': 'Neither question can be answered without considering gene tree discordance, an observation made previously by others as well .', '1708.08916-3-8-0': 'A major cause of gene tree discordance is a population-level process called incomplete lineage sorting (ILS), which has been modeled by the multi-species coalescent (MSC) model .', '1708.08916-3-8-1': 'The model tells us that that likelihood of ILS causing discordance increases as branches become shorter; therefore, any test of polytomies should also consider ILS.', '1708.08916-3-8-2': 'The MSC model has been extensively used for reconstructing species trees using many approaches, including summary methods that first reconstruct gene trees individually and then summarize them to build the species tree.', '1708.08916-3-8-3': 'Many approaches that model ILS rely on dividing the dataset into quartets of species.', '1708.08916-3-8-4': 'These quartet-based methods (e.g., ASTRAL , SVDQuartets , and Bucky-Quartet ) rely on the fact that for a quartet of species, where only three unrooted tree topologies are possible, the species tree topology has the highest probability of occurring in unrooted gene trees under the MSC model.', '1708.08916-3-8-5': 'Relying on the known distributions of quartet frequencies under the MSC model, we previously introduced a way of computing the support of a branch using a measure called local posterior probability (localPP) .', '1708.08916-3-9-0': 'In this paper, we further extend the approach used to compute localPP to develop a fast test for the null hypothesis that a branch has length zero.', '1708.08916-3-9-1': 'Under the null hypothesis, we expect that the three unrooted quartet topologies defined around the branch should have roughly equal frequencies .', '1708.08916-3-9-2': 'This can be rigorously tested, resulting in the approach we present.', '1708.08916-3-9-3': 'Similar ideas have been mentioned in passing previously by [CITATION] and by [CITATION] but to our knowledge, these suggestions have never been implemented or tested.', '1708.08916-3-9-4': 'The main previous paper that tests for polytomies while considering discordance is the work of [CITATION]; however, their work, which considers an avian dataset with five genes, does not use quartet frequencies or MSC and instead relies on a random Markovian walk to build the null distribution of gene trees.', '1708.08916-3-9-5': 'Their approach also looks for zero length branches in the gene trees using the SOWH test .', '1708.08916-3-9-6': 'Their reliance is on gene tree polytomies, while perhaps useful in practice, does not match well with the MSC model where all gene trees are expected to be binary even if the species tree includes a polytomy.', '1708.08916-3-10-0': 'The statistical test that we present is implemented inside the ASTRAL package (option -t 10) since version 4.11.2 and is available online at https://github.com/smirarab/ASTRAL.', '1708.08916-3-11-0': '# Materials and Methods', '1708.08916-3-12-0': '## Background', '1708.08916-3-13-0': 'Recall that an unrooted tree defined on a quartet of species [MATH] can have one of three topologies: [MATH] (i.e., [MATH] and [MATH] are closer to each other than [MATH], and [MATH]), [MATH], or [MATH].', '1708.08916-3-13-1': 'Consider an unrooted species tree [MATH] where the internal branch length separating species [MATH] and [MATH] from [MATH] and [MATH] is [MATH] in coalescent units (CU), which is the number of generations divided by the haploid population size .', '1708.08916-3-13-2': 'Under the Multi Species Coalescent (MSC) model, each gene tree matches the species tree with the probability [MATH] and matches each of the two alternative topologies with the probability [MATH] .', '1708.08916-3-13-3': 'Given true (i.e., error-free) gene trees with no recombination within a locus but free recombination across loci, frequencies [MATH] of gene trees matching topologies [MATH] will follow a multinomial distribution with parameters [MATH] and with the number of trials equal [MATH].', '1708.08916-3-13-4': 'Clearly, for a species tree with [MATH] species, the same results are applicable for any of the [MATH] selections of a quartet of the species with [MATH] defined to be the branch length on the species tree restricted to the quartet.', '1708.08916-3-14-0': '## A statistical test of polytomy', '1708.08916-3-15-0': 'A hard polytomy is mathematically identical to a bifurcating node that has at least one pending branch with length zero; the zero length branch can be contracted in the binary tree to introduce the polytomy.', '1708.08916-3-15-1': 'If the true species tree on a quartet has a hard polytomy (i.e., [MATH]), then all gene tree topologies are equally likely with [MATH].', '1708.08916-3-15-2': 'Thus, if we had the true [MATH] values, we would immediately know if the species tree has a polytomy.', '1708.08916-3-15-3': 'However, we can never know true [MATH] parameters; instead, we have observations [MATH] with [MATH].', '1708.08916-3-15-4': 'Luckily, multinomial distributions concentrate around their mean.', '1708.08916-3-15-5': "As the number of genes increases, the probability of quartet frequencies deviating from their mean rapidly drops; for example, according to Hoeffding's inequality, the probability of divergence by [MATH] drops exponentially and is no more than [MATH].", '1708.08916-3-15-6': 'This concentration gives us hope that even though we never know true [MATH] values from limited data, we can design statistical tests for a [MATH] null hypothesis.', '1708.08916-3-16-0': 'For a given internal branch [MATH] in a given bifurcating species tree, consider the following.', '1708.08916-3-17-0': '[Null hypothesis:] The length of the internal branch [MATH] is zero; thus, the species tree has a polytomy.', '1708.08916-3-18-0': 'To test this null hypothesis, we can use quartet gene tree frequencies given three assumptions.', '1708.08916-3-19-0': 'All positive length branches in the given species tree are correct.', '1708.08916-3-19-1': 'Gene trees are a random error-free sample from the distribution defined by the MSC model.', '1708.08916-3-19-2': 'We have [MATH] gene trees.', '1708.08916-3-20-0': 'A1, which we have previously called the locality assumption , can be somewhat relaxed.', '1708.08916-3-20-1': 'For each bipartition (i.e., branch) of the true species tree, either that bipartition or one of its NNI rearrangements should be present in the given species tree.', '1708.08916-3-21-0': 'We now describe expectations under the null hypothesis.', '1708.08916-3-21-1': 'With start with the [MATH] case.', '1708.08916-3-21-2': 'By the A2 assumption, frequencies [MATH] follow a multinomial distribution with parameters [MATH].', '1708.08916-3-21-3': 'Under the null hypothesis [MATH].', '1708.08916-3-21-4': 'Thus,under the null, [EQUATION] is asymptotically a chi-squared random variable with 2 degrees of freedom .', '1708.08916-3-21-5': 'This chi-squared approximation for three equiprobable outcomes is a good approximation when [MATH] , hence our assumption A3.', '1708.08916-3-21-6': "For smaller [MATH]'s an exact calculation of the critical value is required .", '1708.08916-3-21-7': 'Given the chi-squared random variable as the test statistic, we can simply use a Pearson’s goodness-of-fit statistical test.', '1708.08916-3-21-8': 'Thus, the [MATH]-value is the area to the right of the [MATH] test statistic (Eq. [REF]) under the probability density function of the chi-square distribution with two degrees of freedom.', '1708.08916-3-21-9': 'This integral is available in various software packages, including the Java package colt , which we use.', '1708.08916-3-22-0': 'With [MATH], we apply the test described above to each branch of the species tree independently.', '1708.08916-3-22-1': 'For each branch [MATH], we will have multiple quartets around that branch.', '1708.08916-3-22-2': 'We say that a quartet of species [MATH] is around the branch [MATH] when it is chosen as follows: select an arbitrary leaf [MATH] from the subtree under the left child of [MATH], [MATH] from the subtree under the right child of [MATH], [MATH] from the subtree under the sister branch of [MATH], and [MATH] from the subtree under the sister branch of the parent of [MATH] (this can be easily adopted for a branch incident to root).', '1708.08916-3-22-3': 'Note that by assumption A1 (and its relaxed version), the length of the internal branch of the unrooted species tree induced down to a quartet around [MATH] is identical to the length of [MATH].', '1708.08916-3-22-4': 'Thus, under the null hypothesis, for any quartet around [MATH], we expect that the length of the quartet branch should be zero.', '1708.08916-3-22-5': 'Thus, any arbitrary selection of a quartet around the branch would enable us to use the same exact test we described before for [MATH].', '1708.08916-3-23-0': 'Following the approach we previously used for defining localPP , we can go one step further and use all quartets around the branch.', '1708.08916-3-23-1': 'More precisely, let [MATH] for [MATH] be the number of quartets around the branch [MATH] that in gene tree [MATH] have the topology [MATH] and let [MATH].', '1708.08916-3-23-2': 'Then, we define [EQUATION]', '1708.08916-3-23-3': 'Given these [MATH] values, we use the [MATH] test statistic as defined by Equation [REF], just as before.', '1708.08916-3-24-0': 'While we use Equation [REF] mostly for computational expediency, our approach can be justified.', '1708.08916-3-24-1': 'Let [MATH] be an indicator variable that is 1 iff the quartet [MATH] around the branch [MATH] has the topology [MATH] in gene tree [MATH].', '1708.08916-3-24-2': 'Let [MATH] be the number of gene trees where a quartet [MATH] has the topology [MATH].', '1708.08916-3-24-3': 'Note that any quartet [MATH] around [MATH] can be chosen; thus, our hypothesis testing approach would work if we define [MATH] for any [MATH] and use those [MATH] values in Equation [REF].', '1708.08916-3-24-4': 'In particular, the quartet with the median [MATH] is a valid and reasonable choice.', '1708.08916-3-24-5': 'Moreover, note that if all gene trees are complete, Equation [REF] simplifies to [MATH].', '1708.08916-3-24-6': 'We further assume that in the (unknown) distribution of [MATH] values, the mean approximates the median.', '1708.08916-3-24-7': 'Thus, using [MATH] is a justifiable approximation.', '1708.08916-3-24-8': 'We use this approximation because, as it turns out, computing the mean is more computationally efficient than using the median.', '1708.08916-3-25-0': 'It may initially seem that computing the [MATH] values requires computing [MATH] values.', '1708.08916-3-25-1': 'This would lead to a running time that grows as [MATH].', '1708.08916-3-25-2': 'This would be prohibitively costly.', '1708.08916-3-25-3': 'Computing the median quartet score will also require [MATH].', '1708.08916-3-25-4': 'However, when [MATH] values are defined as the mean quartet score, they can be computed efficiently in [MATH] using the same algorithm that we have previously described for the localPP .', '1708.08916-3-25-5': 'We avoid repeating the algorithm here but note that it is based on a postorder traversal of each gene tree and computing the number of quartets shared between the four sides of the branch [MATH] and each tripartition defined by each node of each gene tree.', '1708.08916-3-25-6': 'This traversal is adopted from ASTRAL-II .', '1708.08916-3-26-0': 'When gene trees have missing data, the definition of [MATH] naturally discards missing quartets.', '1708.08916-3-26-1': 'Similarly, if the gene tree [MATH] includes a polytomy for a quartet, it is counted towards neither of the three [MATH] values and so, is discarded.', '1708.08916-3-26-2': 'Then, Equation [REF] effectively assigns a quartet [MATH] missing/unresolved in a gene tree [MATH] to each quartet topology [MATH] proportionally to the number of present and resolved quartets in the gene tree [MATH] with the topology [MATH]; in other words, a missing [MATH] is imputed to [MATH] where [MATH] is the number of quartets present and resolved in the gene tree [MATH].', '1708.08916-3-26-3': 'A final difficulty arises when none of the quartets defined around [MATH] are present and resolved in a gene tree [MATH].', '1708.08916-3-26-4': 'When this happens, we discard such gene trees from the consideration for branch [MATH], reducing the number of genes.', '1708.08916-3-26-5': 'Thus, the effective number of genes can change from one branch to another based on patterns of gene tree taxon occupancy and resolution.', '1708.08916-3-27-0': '## Evaluations', '1708.08916-3-28-0': 'We examine the behavior of our proposed test, implemented in ASTRAL 5.5.9, on both simulated and empirical datasets on conditions that potentially violate our assumptions A1-A3.', '1708.08916-3-29-0': 'The empirical datasets are a transcriptomic insect dataset , a genomic avian dataset with "super" gene trees resulting from statistical binning , two multi-loci Xenoturbella datasets [CITATION] and Cannon [CITATION], and a transcriptomic plant dataset (Table [REF]).', '1708.08916-3-29-1': 'Since on the empirical data, the true branch length or whether a node should be a hard polytomy is not known, we will rely on estimated branch lengths and will also focus on branches that have been difficult to resolve in the literature.', '1708.08916-3-29-2': 'We will also subsample genes and comment on the number of branches required to reject the null hypothesis.', '1708.08916-3-30-0': 'We use a 201-taxon Simphy simulated dataset that we had previously used to test the local posterior probability .', '1708.08916-3-30-1': 'In this dataset, species trees are generated using the Yule process with a maximum tree height of 500K, 2M, or 10M generations (giving three model conditions) and speciation rates are set to [MATH] (50 replicates per model condition) and to [MATH] (another 50 replicates); the population size is fixed to 200000 everywhere.', '1708.08916-3-30-2': 'Thus, we have three conditions, each with 100 replicates and each tree includes 198 branches (59,400 branches in total).', '1708.08916-3-30-3': 'Branch lengths have a wide range (as we will see).', '1708.08916-3-30-4': 'This dataset includes both true gene trees and estimated gene trees with relatively high levels of gene tree error (Table [REF]).', '1708.08916-3-31-0': 'In our simulations, the true species trees, which we will use, are fully binary.', '1708.08916-3-31-1': 'Thus, the null hypothesis is never correct.', '1708.08916-3-31-2': 'Any failure to reject the null hypothesis is, therefore, a type II error due to the lack of power.', '1708.08916-3-31-3': 'The reader is reminded that the inability to reject the null hypothesis should never be taken as accepting the null hypothesis.', '1708.08916-3-31-4': 'It could simply indicate that the available data is insufficient to distinguish a polytomy from a short branch.', '1708.08916-3-31-5': 'When the null is not rejected, thus, we create soft polytomies.', '1708.08916-3-31-6': 'For long branches, the failure to reject the null could indicate that the test is not appropriate.', '1708.08916-3-31-7': 'However, for very small branches, failing to reject the null is the expected behavior.', '1708.08916-3-32-0': 'Strictly speaking, a polytomy is a branch with length zero.', '1708.08916-3-32-1': 'But how about very small branches?', '1708.08916-3-32-2': 'For example, for a haploid population size of [MATH], a branch length of [MATH] CU corresponds to only ten generations, and for all practical purposes could be considered a polytomy.', '1708.08916-3-32-3': 'Thus, for very small branches, it is arguably not even desirable to reject the null hypothesis.', '1708.08916-3-32-4': 'Unfortunately, the dependence of coalescent units on the population size makes it difficult to give general guidelines as the shortest branch where failure to reject the null is desirable.', '1708.08916-3-32-5': 'Nevertheless, a test for polytomies can still be considered useful if its type-II errors are mostly among very short branches.', '1708.08916-3-33-0': '# Results', '1708.08916-3-34-0': '## Simulated datasets', '1708.08916-3-35-0': 'On the simulated datasets, we bin branches according to the log of their CU length into 20 categories and compute the percentage of branches that are rejected according to our test with [MATH] per bing (Fig [REF]).', '1708.08916-3-35-1': 'The ability to reject the null hypothesis mostly depends on three factors: 1) the branch length, 2) the number of genes, and 3) whether true or estimated gene trees are used.', '1708.08916-3-35-2': 'The impact of all three factors is consistent with what one would expect for a reasonable statistical test.', '1708.08916-3-35-3': 'For the longest branches (e.g., [MATH] CU), the null hypothesis is rejected almost always even with as few as 50 genes and with our highly error-prone estimated gene trees.', '1708.08916-3-35-4': 'Using the true gene trees instead of estimated gene trees increases the power universally, as expected.', '1708.08916-3-35-5': 'For example, with 50 true gene trees, branches as short as [MATH] CU are almost always rejected.', '1708.08916-3-35-6': 'Interestingly, the difference between estimated and true gene trees seems to reduce as the number of genes increase.', '1708.08916-3-36-0': 'Reassuringly, as the number of genes increases, the power to reject the null hypothesis also increases.', '1708.08916-3-36-1': 'Thus, with 1000 genes, branches between 0.1-0.2 CU are rejected 99.9% of the times with true gene trees and 90.0% of the times with estimated gene trees.', '1708.08916-3-36-2': 'Branches below [MATH] are considered very short and can produce the anomaly zone .', '1708.08916-3-36-3': 'Branches in the 0.05-0.15 CU range are rejected 90.4% and 67.4% of times with 1000 true and estimated gene trees, respectively.', '1708.08916-3-37-0': '## Biological dataset', '1708.08916-3-38-0': 'On the biological datasets, the ability to reject the null hypothesis depends on the branch length and the effective number of gene trees (Figure [REF]).', '1708.08916-3-38-1': 'Most branches with the estimated length greater than [MATH] CU had [MATH].', '1708.08916-3-38-2': 'Datasets with more than a thousand genes (avian and insects) had higher resolution and have [MATH] for branches with the estimated length as lows as [MATH] CU.', '1708.08916-3-38-3': 'Yet in all datasets except the avian (where all gene trees are complete) there are some ranges of branch length (often above [MATH] CU) where we are able to reject the null hypothesis for some branches but not for the others.', '1708.08916-3-38-4': 'This cannot be just due to random noise because estimated (not the unknown true) branch length is shown and two branches with the same length, have identical [MATH] values.', '1708.08916-3-38-5': 'Instead, the reason is that the effective number of genes changes in from one branch to another due to the missing data.', '1708.08916-3-38-6': 'Note that missing data impact the effective number of genes only when no quartet around that branch is present, a condition that happens most often for branches close to tips.', '1708.08916-3-39-0': 'To further test the impact of the number of genes, we randomly subsampled gene trees ([MATH]) to find out how many genes are needed before we are able to reject the null hypothesis.', '1708.08916-3-39-1': 'We repeat this subsampling procedure 20 times, and show the average [MATH]-values across all 20 runs (Figures [REF] and [REF]).', '1708.08916-3-39-2': 'In these analyses, we focus on important branches of each empirical dataset.', '1708.08916-3-39-3': 'Note that due to missing data, in some downsampled datasets, occasionally branches have an effective number of genes that is smaller than [MATH], violating our assumption A2; we exclude these branches.', '1708.08916-3-40-0': 'For the avian datasets, 6 branches in the species tree could not be rejected as a polytomy at [MATH] even with all super gene trees (Figure [REF] left).', '1708.08916-3-40-1': 'These mostly belong to what has been called the wall-of-death , a hypothesized rapid radiation at the base of Neoaves .', '1708.08916-3-40-2': 'In this dataset, almost all branches are of interest, especially early Neoavian branches.', '1708.08916-3-40-3': 'Interestingly, when we subsample genes, several distinct patterns emerge for various branches (in Fig [REF], seven examples are highlighted: Cursorimorphae, Otidimorphae, Columbimorphae, Extended Waterbirds, Extended Waterbirds+Hoatzin, Columbea+Otidimorphae, and Coraciimorphae).', '1708.08916-3-40-4': 'Most branches are easily rejected as a polytomy even with a small fraction of the data (data not shown).', '1708.08916-3-40-5': 'For some shorter branches (e.g., Otidimorphae and Columbimorphae) rejecting a polytomy requires hundreds of loci but not all available super gene trees.', '1708.08916-3-40-6': 'Yet for others (e.g., Cursorimorphae and perhaps Exended waterbirds+Hoatzin), we cannot reject the polytomy with the full dataset, but the pattern strongly suggests that if we had more loci, we would be able to reject the null hypothesis.', '1708.08916-3-40-7': 'For these branches, one can argue that a soft polytomy is a reasonable choice, indicating that the resolution is uncertain.', '1708.08916-3-40-8': 'Finally, for some branches (e.g., Columbea+Otidimorphae and Extended Waterbirds), increasing the number of genes does not lead to a substantial decrease in the [MATH]-value.', '1708.08916-3-40-9': 'These may be hard polytomies.', '1708.08916-3-40-10': 'For example, the position of the enigmatic Hoatzin has always been extremely difficult to resolve, even with full genomes ; our results suggest that Hoatzin, Cursorimorphae, and a clade including the rest of the waterbirds may form a hard polytomy.', '1708.08916-3-40-11': 'Interestingly, the Columbea+Otidimorphae branch violates with strong results from Jarvis et al. and other analyses of the same data .', '1708.08916-3-40-12': 'Perhaps, the curious pattern where [MATH]-values increase with more genes indicates that this branch is simply wrong, violating our assumption A1.', '1708.08916-3-41-0': 'For the insect dataset we focus on 6 clades, Holometabola, Acercaria+Hymenoptera, Hexapoda, Orthopteroidea, Pterygota, and Psocodea+Holometabola; these all have been classified as having fairly strong support from the literature , indicating that they enjoy robust support in the literature but some analyses reject them.', '1708.08916-3-41-1': 'As we reduce the number of genes, just like the avian dataset, we see three patterns (Figure [REF] left).', '1708.08916-3-41-2': 'For clades Holometabola, Acercaria+Hymenoptera, and Orthopteroidea, we get [MATH] even with fewer than [MATH] genes and for Pterygota with around 250 genes.', '1708.08916-3-41-3': 'We are not able to reject the null hypothesis for Psocodea+Holometabola with all gene trees, but the decreasing [MATH]-values point to a soft polytomy that could be resolved if we had several hundred more loci.', '1708.08916-3-41-4': 'The support for Hexapoda never decreases as we use more genes, suggesting that the relationship between insects and their close relatives (Collembola and Diplura, both considered insects in the past) may be best represented as a hard polytomy.', '1708.08916-3-41-5': 'On the other hand, for this deep (around 450M years old) and undersampled node, [MATH]-values may fail to reduce not because of a true polytomy but because gene trees are estimated with high and perhaps biased error.', '1708.08916-3-42-0': 'In remaining datasets (plants and Xenoturbella), all important branches that we studied saw decreasing [MATH]-values as the number of gene trees increase (Figure [REF] middle).', '1708.08916-3-42-1': 'In the plant dataset, having around 400 genes seems sufficient for most branches of interest, including the monophyly of Bryophytes and the resolution of Amborella as sister to all the remaining flowering plants.', '1708.08916-3-42-2': 'While the branch that puts Zygnematales as sister to land plants is rejected as a polytomy with about 350 genes, the branch that puts Chara as sister to Coleochaetales, then Zygnematales is not rejected even with the full dataset.', '1708.08916-3-42-3': 'However, it seems to have characteristics of a soft polytomy (decreasing [MATH]-value), which would have been resolved had we had more genes.', '1708.08916-3-43-0': 'The Xenoturbella datasets both have three focal branches, surrounding the position of Xenacoelomorpha.', '1708.08916-3-43-1': 'The branch labeled Bilateria, which has Xenacoelomorpha and Nephrozoa as daughters branches in both papers, can be resolved at [MATH] with as few as 50 (Cannon) or 100 (Rouse) gene trees.', '1708.08916-3-43-2': 'However, pinpointing the position of Xenacoelomorpha also depends on the branch labeled Nephrozoa, which puts Xenacoelomorpha as sister to a clade containing Protostomia and Deuterostomia.', '1708.08916-3-43-3': 'This branch is not rejected as polytomy in either dataset, but the decreasing [MATH]-values suggest that if more loci were available, the null hypothesis would be rejected; thus, both datasets are best understood as leaving the relationships between Protostomia, Xenacoelomorpha, and the rest of Deuterostomia as a soft polytomy with strong suggestion that Xenacoelomorpha is at the base of Nephrozoa (Figure [REF] right).', '1708.08916-3-43-4': 'Remarkably, patterns of difficulty in resolving branches are similar across the two independent datasets with different taxon and gene selection.', '1708.08916-3-44-0': '# Discussion', '1708.08916-3-45-0': 'We introduced a new test for detecting polytomies.', '1708.08916-3-45-1': 'In simulations that included a wide range of branch lengths (Fig. [REF]), we showed that the test behaves as expected.', '1708.08916-3-45-2': 'The null hypothesis is rejected for longer branches and is not rejected for the shorter ones.', '1708.08916-3-45-3': 'The power to reject increases with more gene trees and is reduced with gene tree estimation error.', '1708.08916-3-45-4': 'Nevertheless, for any set of gene trees, there are ranges of branch length where the null hypothesis is rejected for some branches in the range but not for others (Fig. [REF]).', '1708.08916-3-45-5': 'This may be due to stochastic nature of the observed [MATH] frequencies.', '1708.08916-3-46-0': '## Power', '1708.08916-3-47-0': 'Overall, even when we have 1000 genes, it is rare that we can reject the null for branches shorter than 0.03 CU.', '1708.08916-3-47-1': 'A branch of length 0.03 corresponds to 6000 generations in our simulations.', '1708.08916-3-47-2': 'One can argue that a branch that corresponds to such short evolutionary times (roughly 60K years with a generation time of 10 years) should be considered a polytomy for all practical purposes.', '1708.08916-3-47-3': 'Given a sufficiently large number of error-free gene trees even these super-short branches can be distinguished from a polytomy but it is not clear that such distinctions will have any biological relevance.', '1708.08916-3-47-4': 'In the presence of gene tree error and given limited data, resolutions over short branches should always be treated with suspicion, an intuition that our test can formalize.', '1708.08916-3-48-0': 'A natural question to ask is how the number of genes impacts the power.', '1708.08916-3-48-1': 'We can easily compute the required number of genes for rejecting the null hypothesis assuming the expected frequencies match observed frequencies (Fig. [REF]a).', '1708.08916-3-48-2': 'For example, while for a branch of length 0.1 CU we only need [MATH]300 genes before we can reject it as a polytomy, for a branch of length 0.02 (i.e., 5 times shorter), we need [MATH]7500 genes (i.e., 25 times more).', '1708.08916-3-48-3': 'For a quartet species tree, [MATH] with arbitrarily high probability if the number of genes grows as [MATH] .', '1708.08916-3-48-4': 'More broadly, the number of genes required for correct species tree estimation using ASTRAL is proven to grow proportionally to [MATH] and to [MATH] .', '1708.08916-3-48-5': 'Similarly, for any given branch length, we can numerically compute the minimum number of required genes to obtain a given [MATH]-value (e.g., [MATH]).', '1708.08916-3-48-6': 'Assuming the observed frequencies match the expectations, we observe that the required number of genes grows linearly with [MATH] (Fig. [REF]b).', '1708.08916-3-48-7': 'In fact, Figure [REF]b gives us a way to estimate the level of "resolution" that a dataset can provide.', '1708.08916-3-48-8': 'For example, 300 genes can reject the null for branches of [MATH]0.1 CU but if we quadruple the number of genes to 1200, our resolution is increased two-folds to branches of [MATH]0.05 CU.', '1708.08916-3-48-9': 'Note that gene tree error would increase these requirements and hence these should be treated as ballpark estimates.', '1708.08916-3-48-10': 'These estimates also assume we have [MATH] species.', '1708.08916-3-49-0': 'While our test appears to be reasonably powerful, other tests, such likelihood ratio, may be more powerful.', '1708.08916-3-49-1': 'Moreover, it can be argued that our test is conservative in how it handles [MATH].', '1708.08916-3-49-2': 'When multiple quartets are available around a branch, we use their fraction supporting the [MATH] topology as the contribution of that gene to [MATH].', '1708.08916-3-49-3': 'Thus, whether we have one quartet or a hundred quartets, we count each gene tree as one observation of our multinomial distribution.', '1708.08916-3-49-4': 'This is the most conservative approach in dealing with the unknown dependencies between quartets.', '1708.08916-3-49-5': 'The most liberal approach would consider quartets to be fully independent, increasing the degrees of the freedom of the chi-square distribution to [MATH] instead of [MATH].', '1708.08916-3-49-6': 'Such a test will be more powerful, but will be based on an invalid independence assumption; thus, it may have too high of a type I error.', '1708.08916-3-49-7': 'The best possible test would need to model the intricate dependence structure of quartets, a task that is very difficult .', '1708.08916-3-50-0': 'Finally, note that our test of polytomy relates to branch lengths in coalescent units.', '1708.08916-3-50-1': 'A branch of length zero in coalescent units will have length zero in the unit of time (or generations) if we keep the population size fixed.', '1708.08916-3-50-2': 'Mathematically, we can let the population size grow infinitely.', '1708.08916-3-50-3': 'For a mathematical model where the population size grows asymptotically faster than the time, one can have branches that converge to zero in length even though the branch length in time goes to infinity.', '1708.08916-3-50-4': 'This is just a mathematical construct with no biological meaning.', '1708.08916-3-50-5': 'Nevertheless, it helps to remind the reader that a very short branch in the coalescent unit (which our test may fail to reject as a polytomy) may be short not because the time was short but because the population size was large.', '1708.08916-3-50-6': 'Branches between 0.1 and 0.2 CU were not rejected as a polytomy by our test [MATH]10% of times even with 1000 genes.', '1708.08916-3-50-7': 'A length of 0.1 CU can correspond to 10M generations if the haploid population size is 100M.', '1708.08916-3-50-8': 'While such large population sizes are rare, they cannot be discounted.', '1708.08916-3-51-0': '## Divergence from the MSC model and connections to LocalPP', '1708.08916-3-52-0': 'The [MATH]-value from our proposed polytomy test has a close connection to the localPP branch support.', '1708.08916-3-52-1': 'Both measures assume the MSC model and both are a function of quartet scores (i.e., [MATH]).', '1708.08916-3-52-2': 'As the quartet score of the species tree topology and the number of genes increase, both localPP and [MATH]-value increase (Fig. [REF]).', '1708.08916-3-52-3': 'When localPP of a branch is close to 1.0, the polytomy null hypothesis is always rejected.', '1708.08916-3-52-4': 'However, the two measures are not identical.', '1708.08916-3-52-5': 'Interestingly, there are some conditions where localPP is higher than 0.95 but the polytomy null hypothesis is not rejected at the 0.05 level (e.g., see Fig. [REF] top left).', '1708.08916-3-52-6': 'In general, when the frequencies follow expectations of the MSC model, [MATH]-value of the polytomy test is smaller than the localPP.', '1708.08916-3-53-0': 'It is important to remember that our test relies on the properties of the MSC model.', '1708.08916-3-53-1': 'If observed quartet frequencies diverge from the expectations of the MSC model systematically (as opposed to by natural variation), the behavior of our proposed test can change.', '1708.08916-3-53-2': 'For example, if [MATH] is substantially larger than [MATH], rejecting the null hypothesis becomes easier (Fig. [REF]; top right).', '1708.08916-3-53-3': 'This should not come as a surprise because this type of deviation from the MSC model makes the quartet frequencies even more diverged from [MATH] than what is expected under the MSC model.', '1708.08916-3-54-0': 'A more interesting case is when gene flow impacts the gene tree distributions.', '1708.08916-3-54-1': '[CITATION] have identified anomaly zone conditions where the species tree topology has lower quartet frequencies compared to the alternative topologies.', '1708.08916-3-54-2': 'Since the localPP measure does not model gene flow, under those conditions, it will be misled, giving low posterior probability to the species tree topology in the presence of gene flow (Fig. [REF]; bottom).', '1708.08916-3-54-3': 'For example, if [MATH] (meaning that 10% of genes are impacted by the horizontal gene flow), for branches of length 0.1 or shorter, localPP will be zero.', '1708.08916-3-54-4': 'The presence of the gene flow also impacts the test of the polytomy.', '1708.08916-3-54-5': 'For the species tree defined by Solís-Lemus et al. (Fig. 1 of ), when internal branches are short enough, there exist conditions where the gene flow and ILS combined result in quartet frequencies being equal to [MATH] for all the three alternatives.', '1708.08916-3-54-6': 'It is clear that our test will not be able to distinguish such a scenario from a real polytomy (Fig. [REF]; bottom).', '1708.08916-3-54-7': 'One is tempted to argue that perhaps high levels of gene flow between sister branches should favor the outcome that the null is not rejected.', '1708.08916-3-54-8': 'However, this argument fails to explain the observation that for any value of [MATH], the null hypothesis is retained only with very specific settings of surrounding internal branch lengths (Fig. [REF]; bottom).', '1708.08916-3-54-9': 'Thus, we simply caution the reader about the interpretation when gene flow and other sources of the systematic bias are suspected.', '1708.08916-3-55-0': '## Interpretation', '1708.08916-3-56-0': 'In the light of the dependence of our test on the MSC properties, we offer an alternative description of the test.', '1708.08916-3-56-1': 'A safe way to interpret the results of the test, regardless of the causes of gene tree discordance, is to formulate the null hypothesis as follows.', '1708.08916-3-57-0': '[Null hypothesis:] The estimated gene tree quartets around the branch [MATH] support all three NNI rearrangements around the branch in equal numbers.', '1708.08916-3-58-0': 'This is the actual null hypothesis that we test.', '1708.08916-3-58-1': 'Under our assumptions, this hypothesis is equivalent to branch [MATH] being a polytomy.', '1708.08916-3-58-2': 'Under more complex models, such as gene flow + ILS, this null hypothesis holds true for polytomies but also for some binary networks.', '1708.08916-3-59-0': 'The judicious application of our test will preselect the branches where a polytomy null hypothesis is tested and examines the [MATH]-value only for those branches.', '1708.08916-3-59-1': 'When many branches are tested, one arguably needs to correct for multiple hypothesis testing, further reducing the power of the test.', '1708.08916-3-59-2': 'Corrections such as Bonferroni or FDR can be employed (but we did not apply them in our large scale tests that did not target specific hypotheses).', '1708.08916-3-59-3': 'The analyst should specify in advance the branches for which a polytomy null hypothesis is reasonable.', '1708.08916-3-59-4': 'This adds subjectivity, but such problems are always encountered with frequentist tests, and ours is no exception.', '1708.08916-3-59-5': 'Our test also suffers from all the various criticisms leveled against the frequentist hypothesis testing and the interpretation has to avoid all the common pitfalls .', '1708.08916-3-60-0': '# Conclusions', '1708.08916-3-61-0': 'We presented a statistical test, implemented in ASTRAL, for the null hypothesis that a branch of a species tree is a polytomy given a set of gene trees.', '1708.08916-3-61-1': 'Our test, which relies on the properties of the multi-species coalescent model, performed well on simulated and real data.', '1708.08916-3-61-2': 'As expected, its power was a function of branch length, the number of genes, and the gene tree estimation error.', '1708.08916-3-62-0': '# supplementary', '1708.08916-3-63-0': 'All data, R script, and results are available at https://github.com/esayyari/polytomytest.'}

{'1708.08916-4-0-0': 'Phylogenetic species trees typically represent the speciation history as a bifurcating tree.', '1708.08916-4-0-1': 'Speciation events that simultaneously create more than two descendants, thereby creating polytomies in the phylogeny, are possible.', '1708.08916-4-0-2': 'Moreover, the inability to resolve relationships is often shown as a (soft) polytomy.', '1708.08916-4-0-3': 'Both types of polytomies have been traditionally studied in the context of gene tree reconstruction from sequence data.', '1708.08916-4-0-4': 'However, polytomies in the species tree cannot be detected or ruled out without considering gene tree discordance.', '1708.08916-4-0-5': 'In this paper, we describe a statistical test based on properties of the multi-species coalescent model to test the null hypothesis that a branch in an estimated species tree should be replaced by a polytomy.', '1708.08916-4-0-6': 'On both simulated and biological datasets, we show that the null hypothesis is rejected for all but the shortest branches, and in most cases, it is retained for true polytomies.', '1708.08916-4-0-7': 'The test, available as part of the ASTRAL package, can help systematists decide whether their datasets are sufficient to resolve specific relationships of interest.', '1708.08916-4-1-0': '# Introduction', '1708.08916-4-2-0': 'Phylogenies are typically modeled as bifurcating trees.', '1708.08916-4-2-1': 'Even when the evolution is fully vertical, which it is not always [CITATION], the binary model precludes the possibility of several species evolving simultaneously from a progenitor species [CITATION].', '1708.08916-4-2-2': 'These events could be modeled in a multifurcating tree where some nodes, called polytomies, have more than two children.', '1708.08916-4-2-3': 'True polytomies have been suggested for several parts of the tree-of-life (e.g., [CITATION]).', '1708.08916-4-2-4': 'Polytomies are also used when the analyst is unsure about some relationships due to a lack of signal in the data to resolve relationships [CITATION].', '1708.08916-4-2-5': 'The terms hard and soft polytomies are used to distinguish between these two cases [CITATION], with a soft polytomy reserved for the case where relationships are unresolved in an estimated tree and a hard polytomy for multifurcations in the true tree (Fig. [REF]).', '1708.08916-4-2-6': 'Distinguishing the two types of polytomies is not easy.', '1708.08916-4-2-7': 'Moreover, the distinction between soft and hard polytomies can be blurred.', '1708.08916-4-2-8': 'The difficulty in resolving relationships increases as branches become shorter.', '1708.08916-4-2-9': 'In the limit, a branch of length zero is equivalent to a hard polytomy, which is not just difficult but impossible to resolve.', '1708.08916-4-2-10': 'Regardless of abstract distinctions, a major difficulty faced by systematists is to detect whether specific resolutions in their inferred trees are sufficiently supported by data to rule out a polytomy (e.g., see [CITATION]).', '1708.08916-4-3-0': 'For any branch of a given species tree, we can pose a null hypothesis that the length of that branch is zero, and thus, the branch should be removed to create a polytomy.', '1708.08916-4-3-1': 'Using observed data, we can try to reject this null hypothesis, and if we fail to reject, we can replace the branch with a polytomy.', '1708.08916-4-3-2': 'The resulting polytomy is best understood as a soft polytomy because the inability to reject a null hypothesis is never accepting the alternative hypothesis.', '1708.08916-4-3-3': 'The inability to reject may be caused by a real (i.e., hard) polytomy, but it may also simply be due to the lack of power (Fig. [REF]).', '1708.08916-4-3-4': 'In this paper, we present a new test with polytomy as the null hypothesis for multi-locus datasets.', '1708.08916-4-4-0': 'The idea of testing a polytomy as the null hypothesis and rejecting it using data has been applied to single-locus data [CITATION].', '1708.08916-4-4-1': 'Likelihood ratio tests against a zero-length branch have been developed (e.g., SOWH test [CITATION]) and are implemented in popular packages such as PAUP* [CITATION].', '1708.08916-4-4-2': 'Treating polytomies as the null hypothesis has been pioneered by Walsh et al. who sought to not only test for polytomies but also to use a power analysis to distinguish soft and hard polytomies [CITATION].', '1708.08916-4-4-3': 'Appraising their general framework, Braun and Kimball [CITATION] showed that the power analysis can be sensitive to model complexity (or lack thereof).', '1708.08916-4-4-4': 'Perhaps most relevant to our work, Anisimova et al. presented an approximate but fast likelihood ratio test for a polytomy null hypothesis [CITATION]; their test, like what we will present looks at each branch and its surrounding branches while ignoring the rest of the tree.', '1708.08916-4-5-0': 'The existing tests of polytomy as the null and also Bayesian methods of modeling polytomies [CITATION] assume that the sequence data follow a single tree.', '1708.08916-4-5-1': 'Therefore, these methods test whether a gene tree includes a polytomy [CITATION].', '1708.08916-4-5-2': 'However, the species tree can be different from gene trees and the discordance can have several causes, including gene duplication and loss, lateral gene transfer, and incomplete lineage sorting [CITATION].', '1708.08916-4-5-3': 'Arguably, the question of interest is whether the species tree includes polytomies.', '1708.08916-4-5-4': 'Moreover, we are often interested to know whether we should treat the relationship between species as unresolved given the amount of data at hand.', '1708.08916-4-5-5': 'These questions cannot be answered without considering gene tree discordance, an observation made previously by others as well [CITATION].', '1708.08916-4-5-6': 'For example, Poe and Chubb [CITATION], in analyzing an avian dataset with five genes, first looked for zero-length branches in the gene trees using the SOWH test [CITATION] and found evidence that some gene trees may include polytomies.', '1708.08916-4-5-7': 'But they also tested if the pairwise similarity between gene trees was greater than a set of random trees and it was not.', '1708.08916-4-5-8': 'Their test of gene tree congruence, however, was not with respect to any particular model of gene tree evolution.', '1708.08916-4-6-0': 'A major cause of gene tree discordance is a population-level process called incomplete lineage sorting (ILS), which has been modeled by the multi-species coalescent (MSC) model [CITATION].', '1708.08916-4-6-1': 'The model tells us that that likelihood of ILS causing discordance increases as branches become shorter; therefore, any test of polytomies should also consider ILS.', '1708.08916-4-6-2': 'The MSC model has been extensively used for reconstructing species trees using many approaches, including Bayesian co-estimation of gene trees and species trees [CITATION] and site-based approaches [CITATION].', '1708.08916-4-6-3': 'A popular approach (due to its scalability) is the summary method, where we first reconstruct gene trees individually and then summarize them to build the species tree.', '1708.08916-4-6-4': 'Many approaches that model ILS rely on dividing the dataset into quartets of species.', '1708.08916-4-6-5': 'These quartet-based methods (e.g., the summary method ASTRAL [CITATION], the site-based method SVDQuartets [CITATION], and a hybrid method called Bucky-Quartet [CITATION]) rely on the fact that for a quartet of species, where only three unrooted tree topologies are possible, the species tree topology has the highest probability of occurring in unrooted gene trees under the MSC model (Fig. [REF]a).', '1708.08916-4-7-0': 'Relying on the known distribution of quartet frequencies under the MSC model, we previously introduced a way of computing the support of a branch using a measure called local posterior probability (localPP) [CITATION].', '1708.08916-4-7-1': 'In this paper, we further extend the approach used to compute localPP to develop a fast test for the null hypothesis that a branch has length zero.', '1708.08916-4-7-2': 'Under the null hypothesis, we expect that the three unrooted quartet topologies defined around the branch should have equal frequencies [CITATION].', '1708.08916-4-7-3': 'This can be rigorously tested, resulting in the approach we present.', '1708.08916-4-7-4': 'Similar ideas have been mentioned in passing previously by Slowinski [CITATION] and by Allman et al. [CITATION] but to our knowledge, these suggestions have never been implemented or tested.', '1708.08916-4-7-5': 'The statistical test that we present is implemented inside the ASTRAL package (option -t 10) since version 4.11.2 and is available online at https://github.com/smirarab/ASTRAL.', '1708.08916-4-8-0': '# Materials and Methods', '1708.08916-4-9-0': '## Background', '1708.08916-4-10-0': 'An unrooted tree defined on a quartet of species [MATH] can have one of three topologies (Fig. [REF]a): [MATH] (i.e., [MATH] and [MATH] are closer to each other than [MATH], and [MATH]), [MATH], or [MATH].', '1708.08916-4-10-1': 'Consider an unrooted species tree [MATH] where the internal branch length separating species [MATH] and [MATH] from [MATH] and [MATH] is [MATH] in coalescent units (CU), which is the number of generations divided by the haploid population size [CITATION].', '1708.08916-4-10-2': 'Under the Multi Species Coalescent (MSC) model, each gene tree matches the species tree with the probability [MATH] and matches each of the two alternative topologies with the probability [MATH] .', '1708.08916-4-10-3': 'Given true (i.e., error-free) gene trees with no recombination within a locus but free recombination across loci, frequencies [MATH] of gene trees matching topologies [MATH] will follow a multinomial distribution with parameters [MATH] and with the number of trials equal [MATH].', '1708.08916-4-10-4': 'Clearly, for a species tree with [MATH] species, the same results are applicable for any of the [MATH] selections of a quartet of the species with [MATH] defined to be the branch length on the species tree restricted to the quartet (see Fig. [REF] for examples).', '1708.08916-4-11-0': '## A statistical test of polytomy', '1708.08916-4-12-0': 'A true polytomy is mathematically identical to a bifurcating node that has at least one adjacent branch with length zero; the zero-length branch can be contracted in the binary tree to introduce the polytomy (e.g., compare branches P4 - P6 in Fig. [REF]a to the multifurcating tree), If the true species tree for a quartet of taxa has a polytomy (i.e., [MATH]), then all gene tree topologies are equally likely with [MATH].', '1708.08916-4-12-1': 'Thus, if we had the true [MATH] values, we would immediately know if the species tree has a polytomy.', '1708.08916-4-12-2': 'However, we can never know true [MATH] parameters; instead, we have observations [MATH] with [MATH].', '1708.08916-4-12-3': 'Luckily, multinomial distributions concentrate around their mean.', '1708.08916-4-12-4': "As the number of genes increases, the probability of quartet frequencies deviating from their mean rapidly drops; for example, according to Hoeffding's inequality, the probability of divergence by [MATH] drops exponentially and is no more than [MATH].", '1708.08916-4-12-5': 'This concentration gives us hope that even though we never know true [MATH] values from limited data, we can design statistical tests for a [MATH] null hypothesis.', '1708.08916-4-12-6': 'For an internal branch [MATH] in a bifurcating species tree, consider the following.', '1708.08916-4-13-0': '[Null hypothesis:] The length of the internal branch [MATH] is zero; thus, the species tree has a polytomy.', '1708.08916-4-14-0': 'To test this null hypothesis, we can use quartet gene tree frequencies given three assumptions.', '1708.08916-4-15-0': 'All positive length branches in the given species tree are correct.', '1708.08916-4-15-1': 'Gene trees are a random error-free sample from the distribution defined by the MSC model.', '1708.08916-4-15-2': 'We have [MATH] gene trees with at least a quartet relevant to [MATH].', '1708.08916-4-16-0': 'A1, which we have previously called the locality assumption [CITATION], can be somewhat relaxed.', '1708.08916-4-16-1': 'For each bipartition (i.e., branch) of the true species tree, either that bipartition or one of its NNI rearrangements should be present in the given species tree.', '1708.08916-4-17-0': 'We now describe expectations under the null hypothesis.', '1708.08916-4-17-1': 'With start with the [MATH] case.', '1708.08916-4-17-2': 'By the A2 assumption, frequencies [MATH] follow a multinomial distribution with parameters [MATH].', '1708.08916-4-17-3': 'Under the null hypothesis [MATH].', '1708.08916-4-17-4': 'Thus,under the null, [EQUATION] is asymptotically a chi-squared random variable with 2 degrees of freedom .', '1708.08916-4-17-5': 'This chi-squared approximation for three equiprobable outcomes is a good approximation when [MATH] , hence our assumption A3.', '1708.08916-4-17-6': "For smaller [MATH]'s an exact calculation of the critical value is required , but we simply avoid applying our test for [MATH].", '1708.08916-4-17-7': "Given the chi-squared random variable as the test statistic, we can simply use a Pearson's goodness-of-fit statistical test.", '1708.08916-4-17-8': 'Thus, the [MATH]-value is the area to the right of the [MATH] test statistic (Eq. [REF]) under the probability density function of the chi-square distribution with two degrees of freedom (Fig. [REF]b).', '1708.08916-4-17-9': 'This integral is available in various software packages, including the Java package colt [CITATION], which we use.', '1708.08916-4-18-0': 'With [MATH], we apply the test described above to each branch of the species tree independently.', '1708.08916-4-18-1': 'For each branch [MATH], we will have multiple quartets around that branch.', '1708.08916-4-18-2': 'We say that a quartet of species [MATH] is around the branch [MATH] when it is chosen as follows: select an arbitrary leaf [MATH] from the subtree under the left child of [MATH], [MATH] from the subtree under the right child of [MATH], [MATH] from the subtree under the sister branch of [MATH], and [MATH] from the subtree under the sister branch of the parent of [MATH] (this can be easily adopted for a branch incident to root).', '1708.08916-4-18-3': 'Note that by assumption A1 (and its relaxed version), the length of the internal branch of the unrooted species tree induced down to a quartet around [MATH] is identical to the length of [MATH].', '1708.08916-4-18-4': 'Thus, under the null hypothesis, for any quartet around [MATH], we expect that the length of the quartet branch should be zero.', '1708.08916-4-18-5': 'Thus, any arbitrary selection of a quartet around the branch would enable us to use the same exact test we described before for [MATH].', '1708.08916-4-19-0': 'Following the approach we previously used for defining localPP [CITATION], we can also use all quartets around the branch.', '1708.08916-4-19-1': 'More precisely, let [MATH] for [MATH] be the number of quartets around the branch [MATH] that in gene tree [MATH] have the topology [MATH] and let [MATH].', '1708.08916-4-19-2': 'Then, we define [EQUATION]', '1708.08916-4-19-3': 'Given these [MATH] values, we use the [MATH] test statistic as defined by Equation [REF], just as before.', '1708.08916-4-20-0': 'While we use Equation [REF] mostly for computational expediency, our approach can be justified.', '1708.08916-4-20-1': 'Let [MATH] be an indicator variable that is 1 if and only if the quartet [MATH] around the branch [MATH] has the topology [MATH] in gene tree [MATH].', '1708.08916-4-20-2': 'Let [MATH] be the number of gene trees where a quartet [MATH] has the topology [MATH].', '1708.08916-4-20-3': 'Note that any quartet [MATH] around [MATH] can be chosen; thus, our hypothesis testing approach would work if we define [MATH] for any [MATH] and use those [MATH] values in Equation [REF].', '1708.08916-4-20-4': 'In particular, the quartet with the median [MATH] is a valid and reasonable choice.', '1708.08916-4-20-5': 'Moreover, note that if all gene trees are complete, Equation [REF] simplifies to [MATH].', '1708.08916-4-20-6': 'We further assume that in the (unknown) distribution of [MATH] values, the mean approximates the median.', '1708.08916-4-20-7': 'Thus, we approximate [MATH].', '1708.08916-4-20-8': 'We use this approximation because, as it turns out, computing the mean is more computationally efficient than using the median.', '1708.08916-4-21-0': 'It may initially seem that computing the [MATH] values requires computing [MATH] values, which would require [MATH] running time.', '1708.08916-4-21-1': 'This would be too slow for large datasets.', '1708.08916-4-21-2': 'Computing the median quartet score also requires [MATH].', '1708.08916-4-21-3': 'However, the mean quartet score can be computed efficiently in [MATH] using the same algorithm that we have previously described for the localPP [CITATION].', '1708.08916-4-21-4': 'We avoid repeating the algorithm here but note that it is based on a postorder traversal of each gene tree and computing the number of quartets shared between the four sides of the branch [MATH] and each tripartition defined by each node of each gene tree.', '1708.08916-4-21-5': 'This traversal is adopted from ASTRAL-II [CITATION].', '1708.08916-4-22-0': 'When gene trees have missing data, the definition of [MATH] naturally discards missing quartets.', '1708.08916-4-22-1': 'Similarly, if the gene tree [MATH] includes a polytomy for a quartet, it is counted towards neither of the three [MATH] values and so, is discarded.', '1708.08916-4-22-2': 'Then, Equation [REF] effectively assigns a quartet [MATH] missing/unresolved in a gene tree [MATH] to each quartet topology [MATH] proportionally to the number of present and resolved quartets in the gene tree [MATH] with the topology [MATH]; in other words, a missing [MATH] is imputed to [MATH] where [MATH] is the number of quartets present and resolved in the gene tree [MATH].', '1708.08916-4-22-3': 'A final difficulty arises when none of the quartets defined around [MATH] are present or if all of the present ones are unresolved in a multifurcating input gene tree.', '1708.08916-4-22-4': 'When this happens, we discard the gene tree for branch [MATH], reducing the number of genes [MATH].', '1708.08916-4-22-5': 'Thus, the effective number of genes (i.e., effective [MATH]) can change from one branch to another based on patterns of gene tree taxon occupancy and resolution.', '1708.08916-4-22-6': 'Note that the A3 assumption is with respect to this effective number of gene trees and not the total number.', '1708.08916-4-23-0': '## Evaluations', '1708.08916-4-24-0': 'We examine the behavior of our proposed test, implemented in ASTRAL 5.5.9, on several simulated and empirical datasets on conditions that potentially violate assumptions A1 and A2.', '1708.08916-4-25-0': 'The empirical datasets are a transcriptomic insect dataset [CITATION], a genomic avian dataset [CITATION] with "super" gene trees resulting from statistical binning [CITATION], two multi-loci Xenoturbella datasets by Rouse et al. [CITATION] and Cannon et al. [CITATION], and a transcriptomic plant dataset [CITATION] (Table [REF]).', '1708.08916-4-25-1': 'Since in the empirical data, the true branch length or whether a node should be a polytomy is not known, we will report the relationship between the estimated branch lengths and [MATH]-values.', '1708.08916-4-25-2': 'We will also randomly subsample gene trees to test how the amount of data impacts the ability to reject the null; for this, we focus on selected branches that have been difficult to resolve in the literature.', '1708.08916-4-26-0': 'S12A and S12B We simulated two datasets starting from two fixed species trees with 12 species (S12A: Fig [REF]a, S12B: Fig [REF]b).', '1708.08916-4-26-1': 'For both species trees, the tree height is 1.6M generations and the population size is [MATH]; thus, the tree height is 8 CU.', '1708.08916-4-26-2': 'The S12A species tree has two polytomies, each with three children, in addition to a short branch (P0) of length 0.2 coalescent units.', '1708.08916-4-26-3': 'The S12B tree has a polytomy with five children.', '1708.08916-4-26-4': 'For both S12A and S12b, Simphy [CITATION] is used to simulate 50 replicates, each with 1000 gene trees.', '1708.08916-4-26-5': 'After generating the true gene trees, we used Indelible [CITATION] and the GTR+[MATH] model of sequence evolution to simulate 250bp sequences down the gene trees.', '1708.08916-4-26-6': 'The GTR+[MATH] parameters are drawn randomly from Dirichlet distributions used in the ASTRAL-II paper (parameters are estimated from a collection of biological datasets [CITATION]).', '1708.08916-4-26-7': 'We then used FastTree2 [CITATION] to estimate gene trees from the sequence data.', '1708.08916-4-26-8': 'Both datasets have around [MATH] gene tree error, measured as the average RF distance between true and estimated gene trees (Table [REF]).', '1708.08916-4-27-0': 'On this datasets, we score an arbitrary resolution of the true multifurcating species trees.', '1708.08916-4-27-1': 'Therefore, we can have both false positive errors (incorrectly rejecting the null for a polytomy) and false negative errors (failing to reject the null for a positive-length branch).', '1708.08916-4-27-2': 'We vary the number of genes between 20 and 1000 by randomly subsampling them and examine the distribution of [MATH]-values across all 50 replicates for each interesting branch using both true and estimated gene trees.', '1708.08916-4-28-0': 'S201 We use a 201-taxon simulated dataset previously generated .', '1708.08916-4-28-1': 'Species trees are generated using the Yule process with a maximum tree height of 500K, 2M, or 10M generations and speciation rates of [MATH] (50 replicates per model condition) and [MATH] (another 50 replicates).', '1708.08916-4-28-2': 'The population size is fixed to [MATH] in all datasets.', '1708.08916-4-28-3': 'Thus, we have three conditions, each with 100 replicates and each tree includes 198 branches (59,400 branches in total).', '1708.08916-4-28-4': 'Branch lengths have a wide range (as we will see).', '1708.08916-4-28-5': 'The estimated gene trees on this dataset have relatively high levels of gene tree error (Table [REF]).', '1708.08916-4-28-6': 'Each replicates has 1000 gene trees, which we also randomly subsample to 50 and 200.', '1708.08916-4-29-0': 'In this dataset, the true species trees are fully binary and therefore, the null hypothesis is never correct.', '1708.08916-4-29-1': 'Any failure to reject the null hypothesis is a false negative error.', '1708.08916-4-29-2': 'The inability to reject the null hypothesis should never be taken as accepting the null hypothesis because it can.', '1708.08916-4-29-3': 'simply indicate that the available data is insufficient to distinguish a polytomy from a short branch.', '1708.08916-4-29-4': 'An ideal test should be able to reject the null for long branches.', '1708.08916-4-29-5': 'However, for very short branches, failing to reject the null would be the expected behavior.', '1708.08916-4-29-6': 'It is worth contemplating the meaning of super short branches.', '1708.08916-4-29-7': 'For a haploid population size of [MATH], a branch length of [MATH] CU corresponds to only ten generations.', '1708.08916-4-29-8': 'One can argue that such short branches, for most practical purposes, can be considered a polytomy.', '1708.08916-4-29-9': 'Thus, false negative errors among super short branches could perhaps be tolerated.', '1708.08916-4-30-0': '# Results', '1708.08916-4-31-0': '## Simulated datasets', '1708.08916-4-32-0': 'We focus our discussions on [MATH], but we show full distributions of [MATH]-values in many places.', '1708.08916-4-33-0': 'S12A and S12B', '1708.08916-4-34-0': 'On the S12A tree, P1 and P2 are zero-length branches and therefore, the test should ideally fail to reject the null hypothesis for them.', '1708.08916-4-34-1': 'As desired, when true gene trees are used, [MATH]-values are uniformly distributed (Fig. [REF]c; note the linear empirical cumulative distribution functions for P1 and P2 with true gene trees).', '1708.08916-4-34-2': 'For example, the null hypothesis is rejected for 4% of replicates with 1000 gene trees.', '1708.08916-4-34-3': 'As expected, since the null is correct, the false positive rate does not increase as we increase the number of gene trees.', '1708.08916-4-34-4': 'Switching to estimated gene trees universally increases false positive errors (Fig [REF]d).', '1708.08916-4-34-5': 'For example for P1, we reject the null hypothesis in 12 of replicates using 1000 gene trees.', '1708.08916-4-34-6': 'The most severe case of false positive error rates occurs for branch P2, where 24 of replicates are rejected with 1000 gene trees.', '1708.08916-4-34-7': 'Thus, gene tree errors can, in fact, increase the false positive error rates, but the extent of the increase depends on the length of branches surrounding the tested branch.', '1708.08916-4-35-0': 'On the S12A tree, we also examine two binary positive-length branches: P0, which is short (0.2 CU length) and the parent of a polytomy, and P3 (1 CU), which is longer and the child of a polytomy.', '1708.08916-4-35-1': 'On these, we desire that the null hypothesis should get rejected.', '1708.08916-4-35-2': 'The P3 branch is easily rejected in all replicates using true gene trees.', '1708.08916-4-35-3': 'With estimated gene trees, given 50 genes or more, the null is rejected in almost all cases, and is rejected in 66% of replicates with 20 genes.', '1708.08916-4-35-4': 'Thus, the power to reject this moderate length branch (corresponding to [MATH] generations) is very high.', '1708.08916-4-35-5': 'For P0, which is rather short, the ability to reject the null hypothesis depends on the number of genes and similar to other branches, the power is higher for true gene trees.', '1708.08916-4-35-6': 'The false negative rates decrease as the number of genes increases; using 1000 gene trees, the null is rejected in all replicates with true gene trees and in 86% of replicates with estimated gene trees.', '1708.08916-4-35-7': 'Overall, the false negative rate is a function of the number of genes, the length of the branch, and gene tree error, as expected.', '1708.08916-4-36-0': 'The S12B tree shows broadly similar results as S12A (Fig [REF]ef) but some differences are noteworthy.', '1708.08916-4-36-1': 'On the zero-length branches (P4, P5, and P6), as desired, the test fails to reject the null.', '1708.08916-4-36-2': 'However, false positives rates are a bit lower than expected by chance when true gene trees are used.', '1708.08916-4-36-3': 'For example, at [MATH], we barely ever reject the null hypothesis for either of these three branches.', '1708.08916-4-36-4': 'These lower than expected false positive rates may be due to the fact that each branch is considered independently in our test, but P4, P5, and P6 are very much dependent (they all resolve one high degree polytomy).', '1708.08916-4-36-5': 'Even using estimated gene trees, the false positive rate remains low.', '1708.08916-4-36-6': 'With estimated gene trees, for P4, we reject the null in 4 of replicates when we use 1000 gene trees and we never reject the null hypothesis otherwise (Fig [REF]f).', '1708.08916-4-36-7': 'For P5 and P6, the false positive rates is at most 8 and 4 with 1000 genes.', '1708.08916-4-36-8': 'While the false positive rates remain low with estimated gene trees, the rate seems to slightly increase with increased numbers of gene trees.', '1708.08916-4-36-9': 'Alongside the zero-length branches, we also study the branch P7 (length: 2 CU), which is adjacent to the polytomy.', '1708.08916-4-36-10': 'For this relatively long branch, we always reject the null hypothesis with true gene trees.', '1708.08916-4-36-11': 'With estimated gene trees, the false negative rate is only 16% with 20 gene trees and gradually drops to 0% at 200 genes or more.', '1708.08916-4-37-0': 'S201 On the S201 datasets, we can only have false negative errors.', '1708.08916-4-37-1': 'We bin branches according to the log of their CU length into 20 categories and compute the percentage of branches that are rejected according to our test with [MATH] per bin (Fig [REF]).', '1708.08916-4-37-2': 'The false negative rate mostly depends on three factors: 1) the branch length, 2) the number of genes, and 3) whether true or estimated gene trees are used.', '1708.08916-4-37-3': 'The impact of all three factors is consistent with what one would expect for a reasonable statistical test.', '1708.08916-4-37-4': 'For the longest branches (e.g., [MATH] CU), the null hypothesis is rejected almost always even with as few as 50 genes and with our highly error-prone estimated gene trees.', '1708.08916-4-37-5': 'Using the true gene trees instead of estimated gene trees increases the power universally.', '1708.08916-4-37-6': 'For example, with 50 true gene trees, branches as short as [MATH] CU are almost always rejected.', '1708.08916-4-37-7': 'Interestingly, the difference between estimated and true gene trees seems to reduce as the number of genes increase.', '1708.08916-4-38-0': 'Reassuringly, as the number of genes increases, the power to reject the null hypothesis also increases.', '1708.08916-4-38-1': 'Thus, with 1000 genes, branches between 0.1-0.2 CU are rejected 99.9% of the times with true gene trees and 90.0% of the times with estimated gene trees.', '1708.08916-4-38-2': 'Branches below [MATH] are considered very short and can produce the anomaly zone [CITATION].', '1708.08916-4-38-3': 'Branches in the 0.05-0.15 CU range are rejected 90.4% and 67.4% of times with 1000 true and estimated gene trees, respectively.', '1708.08916-4-39-0': '## Biological dataset', '1708.08916-4-40-0': 'On the biological datasets, the ability to reject the null hypothesis depends on the branch length and the effective number of gene trees (Figure [REF]).', '1708.08916-4-40-1': 'Most branches with the estimated length greater than [MATH] CU had [MATH].', '1708.08916-4-40-2': 'Datasets with more than a thousand genes (Aves and insects) had higher resolution and have [MATH] for branches with the estimated length as lows as [MATH] CU.', '1708.08916-4-40-3': 'Yet in all datasets except the Aves (where all gene trees include all species) there are some ranges of branch length (often above [MATH] CU) where we are able to reject the null hypothesis for some branches but not for the others.', '1708.08916-4-40-4': 'This cannot be just due to random noise because estimated (not the unknown true) branch length is shown and two branches with the same length, have identical [MATH] values.', '1708.08916-4-40-5': 'Instead, the reason is that the effective number of genes changes from one branch to another because some gene trees may not include enough species to define a quartet around some branches.', '1708.08916-4-40-6': 'The effective number of genes can also decline due to a lack of gene tree resolution, but this does not happen in our datasets, which include only binary gene trees (we will revisit this in the discussion section).', '1708.08916-4-41-0': 'To further test the impact of the number of genes, for each dataset, we randomly subsampled gene trees ([MATH] but no less than 20 gene trees) to find out how many genes are needed before we are able to reject the null hypothesis.', '1708.08916-4-41-1': 'We repeat this subsampling procedure 20 times, and show the average [MATH]-values across all 20 runs (Figures [REF] and [REF]).', '1708.08916-4-41-2': 'In these analyses, we focus on selected branches of each empirical dataset.', '1708.08916-4-41-3': 'Note that in some downsampled datasets, occasionally branches have an effective number of genes that is smaller than [MATH], violating our assumption A3; we exclude these branches.', '1708.08916-4-42-0': 'For the avian datasets, 6 branches in the species tree could not be rejected as a polytomy at [MATH] even with all super gene trees (Figure [REF]a).', '1708.08916-4-42-1': 'These mostly belong to what has been called the wall-of-death [CITATION], a hypothesized rapid radiation at the base of Neoaves [CITATION].', '1708.08916-4-42-2': 'In subsampling super gene trees, we highlight seven selected branches (labeled A-G) as shown in Figure [REF]a. Interestingly, when we subsample super gene trees, several distinct patterns emerge for various branches.', '1708.08916-4-42-3': 'Most branches are easily rejected as a polytomy even with a small fraction of the data (e.g., C).', '1708.08916-4-42-4': 'For some shorter branches (e.g., G and B) rejecting a polytomy requires hundreds of super gene trees.', '1708.08916-4-42-5': 'Yet for others (e.g., D and perhaps F), we cannot reject the polytomy with the full dataset, but the pattern suggests that if we had more super gene trees, we may have been able to reject them as a polytomy.', '1708.08916-4-42-6': 'Finally, for some branches (e.g., A and G), increasing the number of genes does not lead to a substantial decrease in the [MATH]-value, suggesting that increasing the number of input trees may not be sufficient to resolve them.', '1708.08916-4-43-0': 'For the insect dataset we focus on 6 clades, Holometabola, Acercaria+Hymenoptera, Hexapoda, Orthopteroidea, Pterygota, and Psocodea+Holometabola; these all have been classified as having fairly strong support from the literature [CITATION], indicating that they enjoy robust support in the literature but some analyses reject them.', '1708.08916-4-43-1': 'As we reduce the number of genes, just like the avian dataset, we see three patterns (Figure [REF]a).', '1708.08916-4-43-2': 'For clades Holometabola, Acercaria+Hymenoptera, and Orthopteroidea, we get [MATH] even with fewer than [MATH] genes and for Pterygota with around 250 genes.', '1708.08916-4-43-3': 'We are not able to reject the null hypothesis for Psocodea+Holometabola with all gene trees, but the decreasing [MATH]-values suggest that this resolution could perhaps be resolved if we had several hundred more loci.', '1708.08916-4-43-4': 'The support for Hexapoda never decreases as we use more genes, suggesting that the relationship between insects and their close relatives (Collembola and Diplura, both considered insects in the past) may remain unresolved if we simply increase the number of genes.', '1708.08916-4-43-5': 'For this deep (around 450M years old) and undersampled node, [MATH]-values may fail to reduce either because of a true polytomy or because gene trees are estimated with high (perhaps biased) error.', '1708.08916-4-44-0': 'In remaining datasets (plants and Xenoturbella), all important branches that we studied saw decreasing [MATH]-values as the number of gene trees increase (Figure [REF]b).', '1708.08916-4-44-1': 'In the plant dataset, having around 400 genes seems sufficient for most branches of interest, including the monophyly of Bryophytes and the resolution of Amborella as sister to all the remaining flowering plants.', '1708.08916-4-44-2': 'The branch that puts Zygnematales as sister to land plants is rejected as a polytomy with about 350 genes.', '1708.08916-4-44-3': 'However, the correct relationship between Chara and Coleochaetales remains hard to resolve.', '1708.08916-4-44-4': 'Even with the full dataset, a polytomy is not rejected, though the decreasing [MATH]-values point to the possibility that this relationship would have been resolved had we had more genes.', '1708.08916-4-45-0': 'The Xenoturbella datasets both have three focal branches, surrounding the position of Xenacoelomorpha.', '1708.08916-4-45-1': 'The branch labeled Bilateria, which has Xenacoelomorpha and Nephrozoa as daughters branches in both papers, can be resolved at [MATH] with as few as 50 (Cannon) or 100 (Rouse) gene trees (Figure [REF]c).', '1708.08916-4-45-2': 'However, pinpointing the position of Xenacoelomorpha also depends on the branch labeled Nephrozoa, which puts Xenacoelomorpha as sister to a clade containing Protostomia and Deuterostomia.', '1708.08916-4-45-3': 'The null hypothesis that this branch may be a polytomy is not rejected in either dataset, but a pattern of decreasing [MATH]-values with more loci can be discerned.', '1708.08916-4-45-4': 'Thus, both datasets are best understood as leaving the relationships between Protostomia, Xenacoelomorpha, and the rest of Deuterostomia as uncertain with some evidence that Xenacoelomorpha is at the base of Nephrozoa.', '1708.08916-4-45-5': 'Remarkably, patterns of difficulty in resolving branches are similar across the two independent datasets with different taxon and gene selection.', '1708.08916-4-46-0': '# Discussion', '1708.08916-4-47-0': 'We introduced a new test for rejecting the null hypothesis that a branch in the species tree should be replaced by a polytomy.', '1708.08916-4-47-1': 'Unlike existing tests, our new test considers gene tree discordance due to ILS, as modeled by the MSC model.', '1708.08916-4-47-2': 'In several simulations, we showed that the test behaves as expected.', '1708.08916-4-47-3': 'The null hypothesis is often retained for true polytomies and is often rejected for binary nodes, unless when the true branch lengths are very short.', '1708.08916-4-47-4': 'The power to reject the null hypothesis for binary relationships increases with longer branches or with more gene trees and is reduced with gene tree estimation error.', '1708.08916-4-47-5': 'Gene tree error can also, in some cases, increase the false positive rate.', '1708.08916-4-48-0': '## Power', '1708.08916-4-49-0': 'Overall, even when we have 1000 genes, it is rare that we can reject the null for branches shorter than 0.03 CU.', '1708.08916-4-49-1': 'A branch of length 0.03 corresponds to 6000 generations in our simulations.', '1708.08916-4-49-2': 'One can argue that failing to resolve a branch that corresponds to such short evolutionary times (roughly 60K years with a generation time of 10 years) can perhaps be tolerated.', '1708.08916-4-49-3': 'Mathematically, given a sufficiently large unbiased sample of gene trees, even infinitesimally short branches can be distinguished from a polytomy.', '1708.08916-4-49-4': 'In practice, however, extremely short branches should be treated with suspicion as our input gene trees invariably are not perfect samples from the MSC distribution.', '1708.08916-4-50-0': 'In our biological analyses, we saw that subsampling genes and tracking trajectories of the [MATH]-value may be helpful in predicting the number of required genes to resolve a branch.', '1708.08916-4-50-1': 'The approach we presented can be used in other biological data as well.', '1708.08916-4-50-2': 'However, we caution that such predictions should be interpreted with the limitations of our proposed test in mind.', '1708.08916-4-50-3': 'Many factors such as gene tree error and other sources of discordance can contribute to deviations from MSC, and such deviations may render the predictions inaccurate.', '1708.08916-4-50-4': 'But if such predictions are to be made, a natural question arises: How does the number of genes impact the power?', '1708.08916-4-51-0': 'We can easily compute the required number of genes for rejecting the null hypothesis assuming the expected frequencies match observed frequencies (Fig. [REF]a).', '1708.08916-4-51-1': 'For example, while for a branch of length 0.1 CU we only need [MATH]300 genes before we can reject it as a polytomy, for a branch of length 0.02 (i.e., 5 times shorter), we need [MATH]7500 genes (i.e., 25 times more).', '1708.08916-4-51-2': 'For a quartet species tree, [MATH] with arbitrarily high probability if the number of genes grows as [MATH] [CITATION].', '1708.08916-4-51-3': 'More broadly, the number of genes required for correct species tree estimation using ASTRAL is proven to grow proportionally to [MATH] and to [MATH] [CITATION].', '1708.08916-4-51-4': 'Similarly, for any given branch length, we can numerically compute the minimum number of required genes to obtain a given [MATH]-value (e.g., [MATH]).', '1708.08916-4-51-5': 'Assuming the observed frequencies match the expectations, we observe that the required number of genes grows linearly with [MATH] (Fig. [REF]b).', '1708.08916-4-51-6': 'In fact, Figure [REF]b gives us a way to estimate the level of "resolution" that a dataset can provide.', '1708.08916-4-51-7': 'For example, 300 genes can reject the null for branches of [MATH]0.1 CU but if we quadruple the number of genes to 1200, our resolution is increased two-folds to branches of [MATH]0.05 CU.', '1708.08916-4-51-8': 'Note that gene tree error would increase these requirements and hence these should be treated as ballpark estimates.', '1708.08916-4-51-9': 'These estimates also assume we have [MATH] species.', '1708.08916-4-52-0': 'The test we presented has no guarantees of maximal power.', '1708.08916-4-52-1': 'Other tests, such as likelihood ratio, may be more powerful.', '1708.08916-4-52-2': 'Moreover, it can be argued that our test is conservative in how it handles [MATH].', '1708.08916-4-52-3': 'When multiple quartets are available around a branch, we use their fraction supporting the [MATH] topology as the contribution of that gene to [MATH].', '1708.08916-4-52-4': 'Thus, whether we have one quartet or a hundred quartets, we count each gene tree as one observation of our multinomial distribution.', '1708.08916-4-52-5': 'This is the most conservative approach to deal with the unknown dependencies between quartets.', '1708.08916-4-52-6': 'The most liberal approach would consider quartets to be fully independent, increasing the degrees of the freedom of the chi-square distribution to [MATH] instead of [MATH].', '1708.08916-4-52-7': 'Such a test would be more powerful but would be based on invalid independence assumptions that may raise false positive rates.', '1708.08916-4-52-8': 'An ideal test would need to model the intricate dependence structure of quartets, a task that is very difficult [CITATION].', '1708.08916-4-53-0': 'Finally, note that our test of polytomy relates to branch lengths in coalescent units.', '1708.08916-4-53-1': 'A branch of length zero in coalescent units will have length zero in the unit of time (or generations) if we keep the population size fixed.', '1708.08916-4-53-2': 'Mathematically, we can let the population size grow infinitely.', '1708.08916-4-53-3': 'For a mathematical model where the population size grows asymptotically faster than the time, one can have branches that converge to zero in length even though the branch length in time goes to infinity.', '1708.08916-4-53-4': 'This is just a mathematical construct with no biological meaning.', '1708.08916-4-53-5': 'Nevertheless, it helps to remind us that a very short branch in the coalescent unit (which our test may fail to reject as a polytomy) may be short not because the time was short but because the population size was large.', '1708.08916-4-53-6': 'Branches between 0.1 and 0.2 CU were not rejected as a polytomy by our test [MATH]10% of times even with 1000 genes.', '1708.08916-4-53-7': 'A length of 0.1 CU can correspond to 10M generations if the haploid population size is 100M.', '1708.08916-4-54-0': '## Divergence from the MSC model and connections to localPP', '1708.08916-4-55-0': 'The [MATH]-value from our proposed polytomy test has a close connection to the localPP branch support.', '1708.08916-4-55-1': 'Both measures assume the MSC model and both are a function of quartet scores (i.e., [MATH]).', '1708.08916-4-55-2': 'As the quartet score of the species tree topology and the number of genes increases, both localPP and [MATH]-value increase (Fig. [REF]a).', '1708.08916-4-55-3': 'When localPP of a branch is close to 1.0, the polytomy null hypothesis is always rejected.', '1708.08916-4-55-4': 'However, the two measures are not identical.', '1708.08916-4-55-5': 'Interestingly, there are some conditions where localPP is higher than 0.95 but the polytomy null hypothesis is not rejected at the 0.05 level (Fig. [REF]a).', '1708.08916-4-55-6': 'When the frequencies follow expectations of the MSC model, [MATH]-value of the polytomy test is smaller than the localPP.', '1708.08916-4-56-0': 'It is important to remember that our test relies on the properties of the MSC model.', '1708.08916-4-56-1': 'If observed quartet frequencies diverge from the expectations of the MSC model systematically (as opposed to by natural variation), the behavior of our proposed test can change.', '1708.08916-4-56-2': 'For example, if [MATH] is substantially larger than [MATH], rejecting the null hypothesis becomes easier (Fig. [REF]b).', '1708.08916-4-56-3': 'This should not come as a surprise because this type of deviation from the MSC model makes the quartet frequencies even more diverged from [MATH] than what is expected under the MSC model.', '1708.08916-4-56-4': 'On real data, several factors can may contribute to deviations from MSC.', '1708.08916-4-56-5': 'For example, incorrect homology detection in real datasets is possible (e.g., see [CITATION] for possible homology issues with the avian dataset) and can lead to deviations.', '1708.08916-4-57-0': 'Another source of deviation is gene flow, which can impact the gene tree distributions.', '1708.08916-4-57-1': 'Solís-Lemus et al. have identified anomaly zone conditions where the species tree topology has lower quartet frequencies compared to the alternative topologies [CITATION].', '1708.08916-4-57-2': 'Since the localPP measure does not model gene flow, under those conditions, it will be misled, giving low posterior probability to the species tree topology in the presence of gene flow (Fig. [REF]c).', '1708.08916-4-57-3': 'For example, if [MATH] (meaning that 10% of genes are impacted by the horizontal gene flow), for branches of length 0.1 or shorter, localPP will be zero.', '1708.08916-4-57-4': 'The presence of the gene flow also impacts the test of the polytomy.', '1708.08916-4-57-5': 'For the species tree defined by Solís-Lemus et al. (Fig. 1 of [CITATION]), when internal branches are short enough, there exist conditions where the gene flow and ILS combined result in quartet frequencies being equal to [MATH] for all the three alternatives.', '1708.08916-4-57-6': 'It is clear that our test will not be able to distinguish such a scenario from a real polytomy (Fig. [REF]cd).', '1708.08916-4-57-7': 'One is tempted to argue that perhaps high levels of gene flow between sister branches should favor the outcome that the null is not rejected.', '1708.08916-4-57-8': 'However, this argument fails to explain the observation that for any value of [MATH], the null hypothesis is retained only with very specific settings of surrounding internal branch lengths (Fig. [REF]cd ).', '1708.08916-4-57-9': 'Thus, we simply caution the reader about the interpretation when gene flow and other sources of bias are suspected.', '1708.08916-4-58-0': '## The effective number of gene trees', '1708.08916-4-59-0': 'It is important to note that the effective number of gene trees (effective-[MATH]) can change across branches of the same species tree.', '1708.08916-4-59-1': 'Missing data can reduce the number of genes that have at least one taxon from a quartet defined around the branch of interest.', '1708.08916-4-59-2': 'In our biological datasets, various branches of the same dataset often have a wide range of effective [MATH] (Fig [REF]a), especially for the two transcriptomic datasets (insects and plants) with lots of missing data.', '1708.08916-4-59-3': 'The only exception is the avian dataset, where our super gene trees always include all the taxa.', '1708.08916-4-60-0': 'A second factor that can reduce the effective [MATH] is multifurcations in input gene trees.', '1708.08916-4-60-1': 'If all the quartets around a branch are unresolved in an input gene tree, that gene tree does not count towards the effective [MATH].', '1708.08916-4-60-2': 'Our biological datasets had binary gene trees.', '1708.08916-4-60-3': 'However, as recently shown [CITATION], removing branches with very low support can help addressing gene tree error.', '1708.08916-4-60-4': 'To demonstrate this, we revisit the avian dataset.', '1708.08916-4-60-5': 'The purpose of using super gene trees instead of normal (unbinned) gene trees was to reduce the gene tree estimation error.', '1708.08916-4-60-6': 'Alternatively, one can simply remove branches with support at or below a certain threshold in gene trees and use the resulting tree as input to ASTRAL [CITATION].', '1708.08916-4-60-7': 'With this procedure and the support threshold set to 10%, we generated a new ASTRAL tree based on all 14,446 unbinned gene trees from the avian dataset [CITATION] (Fig [REF]b).', '1708.08916-4-60-8': 'The resulting tree was largely congruent with the ASTRAL tree on super gene trees and with reference phylogenies form the original publication [CITATION].', '1708.08916-4-61-0': 'We tested how the effective [MATH] and [MATH]-values change as a result of contracting low support branches.', '1708.08916-4-61-1': 'Simply contracting branches with 0 support reduces the median effective [MATH] from 13,791 to 10,523.', '1708.08916-4-61-2': 'Further contracting branches with support up to 3 - 75 gradually reduces the effective [MATH] all the way to a median of 610 (Fig [REF]c).', '1708.08916-4-61-3': 'The [MATH]-values tend to decrease as we increase the threshold for contraction (Fig [REF]b).', '1708.08916-4-61-4': 'Several branches fail to reject the null hypothesis regardless of the threshold chosen.', '1708.08916-4-61-5': 'Others reject the null hypothesis with lower levels of contraction but not with the higher levels, showing that the reduced effective [MATH] can reduce the power.', '1708.08916-4-61-6': 'For one branch, interestingly, the null is not rejected if we contract up to 0% and 3% support or if we contract up to 75%, but is rejected otherwise.', '1708.08916-4-61-7': 'This pattern may have a subtle explanation.', '1708.08916-4-61-8': 'With gene trees that include low support branches (up to 3%), we are unable to reject the null hypothesis perhaps because gene tree error creates a uniform distribution of quartets around this branch.', '1708.08916-4-61-9': 'As we further remove low support branches from the gene trees, we start to see quartet frequencies that favor the ASTRAL resolution perhaps because noise is removed and the actual signal can be discerned.', '1708.08916-4-61-10': 'Finally, with aggressive filtering of gene tree branches, effective [MATH] becomes so low that the test simply does not have the power to reject the null.', '1708.08916-4-61-11': 'These interesting patterns suggest that dealing with gene tree error by contracting low support branches may be possible, but the choice of the best threshold is not obvious.', '1708.08916-4-61-12': 'Future studies should further consider this question.', '1708.08916-4-62-0': '## Interpretation', '1708.08916-4-63-0': 'In the light of the dependence of our test on the MSC properties, we offer an alternative description of the test.', '1708.08916-4-63-1': 'A safe way to interpret the results of the test, regardless of the causes of gene tree discordance, is to formulate the null hypothesis as follows.', '1708.08916-4-64-0': '[Null hypothesis:] The estimated gene tree quartets around the branch [MATH] support all three NNI rearrangements around the branch in equal numbers.', '1708.08916-4-65-0': 'This is the actual null hypothesis that we test.', '1708.08916-4-65-1': 'Under our assumptions, this hypothesis is equivalent to branch [MATH] being a polytomy.', '1708.08916-4-65-2': 'Under more complex models, such as gene flow + ILS, this null hypothesis holds true for polytomies but also for some binary networks.', '1708.08916-4-66-0': 'The judicious application of our test will preselect the branches where a polytomy null hypothesis is tested and examines the [MATH]-value only for those branches.', '1708.08916-4-66-1': 'When many branches are tested, one arguably needs to correct for multiple hypothesis testing, further reducing the power of the test.', '1708.08916-4-66-2': 'Corrections such as Bonferroni or FDR [CITATION] can be employed (but we did not apply them in our large scale tests that did not target specific hypotheses).', '1708.08916-4-66-3': 'However, note that even though we formulate the polytomy as a null hypothesis, in reality, we expect that in most cases the branch has positive branch length.', '1708.08916-4-66-4': 'Thus, we expect to reject the null often, in contrast to usual applications of the frequentist test.', '1708.08916-4-66-5': 'The analyst should specify in advance the branches for which a polytomy null hypothesis is reasonable.', '1708.08916-4-66-6': 'This adds subjectivity, but such problems are always encountered with frequentist tests, and ours is no exception.', '1708.08916-4-66-7': 'Our test also suffers from all the various criticisms leveled against the frequentist hypothesis testing [CITATION] and the interpretation has to avoid all the common pitfalls [CITATION].', '1708.08916-4-67-0': '# Conclusions', '1708.08916-4-68-0': 'We presented a statistical test, implemented in ASTRAL, for the null hypothesis that a branch of a species tree is a polytomy given a set of gene trees.', '1708.08916-4-68-1': 'Our test, which relies on the properties of the multi-species coalescent model, performed well on simulated and real data.', '1708.08916-4-68-2': 'As expected, its power was a function of branch length, the number of genes, and the gene tree estimation error.', '1708.08916-4-69-0': '# supplementary', '1708.08916-4-70-0': 'All data, R script, and results are available at https://github.com/esayyari/polytomytest.'}

1803.00635

{'1803.00635-1-0-0': 'We report a de Hass-van Alphen effect study on the Dirac type-II semimetallic candidates MAl[MATH] (where, M = V, Nb and Ta).', '1803.00635-1-0-1': 'The angular-dependence of their Fermi surface cross-sectional areas reveals a remarkably good agreement with first-principle calculations.', '1803.00635-1-0-2': 'Hence, quantum oscillations are consistent with the existence of tilted Dirac cones with Dirac type-II nodes located at 100, 230 and 250 meV above the Fermi level [MATH] for VAl[MATH], NbAl[MATH] and TaAl[MATH], respectively.', '1803.00635-1-0-3': 'Hence, our measurements support the prediction of broken Lorentz invariance in these compounds.', '1803.00635-1-0-4': 'However, for all three compounds we find that their Fermi surfaces yield trivial Berry phases which we tentatively attribute to the distance of their Dirac nodes relative to the Fermi level [MATH].', '1803.00635-1-0-5': 'NbAl[MATH] was reported to be a superconductor suggesting that it might be possible to displace [MATH] towards the Dirac type-II nodes while preserving superconductivity in these Lorentz non-invariant systems.', '1803.00635-1-1-0': 'Condensed-matter systems provide accessible platforms for the discovery of quasiparticles with properties akin to particles predicted by high-energy physics.', '1803.00635-1-1-1': 'Dirac type-I compounds such as Cd[MATH]As[MATH] [CITATION] or Na[MATH]Bi, [CITATION]) and Weyl type-I systems like (Ta,Nb)(P,As) [CITATION] were recently discovered and are garnering a lot of attention.', '1803.00635-1-1-2': 'According to Ref. [CITATION], solid state systems would even offer the potential of finding fermionic excitations which have no analog in high-energy physics such as three-component fermions.', '1803.00635-1-1-3': '[CITATION] Very recently, so-called type-II Dirac/Weyl semimetals, which violate Lorentz-symmetry, were discovered and are being intensively studied.', '1803.00635-1-1-4': 'In type-II Dirac/Weyl semimetals, the crossings between energy bands remain protected but the spectra of the Dirac/Weyl cones are strongly tilted due to an additional momentum dependent term, thus they break the Lorentz invariance.', '1803.00635-1-1-5': 'These Weyl/Dirac type-II nodes become singular points connecting electron and hole pockets in the spectral function.', '1803.00635-1-1-6': 'The associated quasiparticles can be observed in condensed matter systems but the analog particles are absent in Lorentz invariant high energy physics.', '1803.00635-1-1-7': 'Type-II Dirac/Weyl semimetallic systems were proposed to display unique properties, such as Klein tunneling in momentum space,[CITATION] orientation-dependent chiral anomaly,[CITATION] and a modified anomalous Hall conductivity.[', '1803.00635-1-2-0': 'A number of Weyl type-II semimetals were already experimentally studied, including MoTe[MATH],[CITATION] WTe[MATH],[CITATION] Ta[MATH]S[MATH],[CITATION] LaAlGe,[CITATION] and TaIrTe[MATH].[', '1803.00635-1-2-1': '[CITATION] And although several Dirac type-II compounds were also reported, e.g. VAl[MATH],[CITATION] YPd[MATH]Sn,[CITATION] KMgBi,[CITATION] and (Pt,Pd)Te[MATH],[CITATION] only the last two compounds were studied experimentally via angle-resolved photoemission spectroscopy (ARPES) and quantum oscillatory phenomena.[', '1803.00635-1-2-2': '[CITATION] Therefore, Dirac type-II systems remain to be unambiguously identified and characterized experimentally.', '1803.00635-1-2-3': 'For instance, and in addition to the aforementioned predictions, Dirac type-II semimetals have been predicted to become Weyl type-II systems or topological crystalline insulators when time-reversal or inversion symmetries are broken.', '1803.00635-1-2-4': '[CITATION] However, exposing the unique transport properties of such compounds is a difficult task due to the distance of the Dirac type-II nodes with respect to the Fermi level which, in addition, is crossed by topologically trivial and non-trivial bands.', '1803.00635-1-3-0': 'Here, we report the synthesis of the chemical analogues (V, Nb, Ta)Al[MATH] which were predicted to display Dirac type-II nodes.', '1803.00635-1-3-1': '[CITATION] The topography of their Fermi surfaces, revealed through the de Haas van Alphen (dHvA) effect, are found to display remarkably good agreement with band structure calculations.', '1803.00635-1-3-2': 'Therefore, our experimental study indicates that these compounds would break Lorentz invariance.', '1803.00635-1-3-3': '[CITATION] The Dirac type-II nodes in VAl[MATH] and NbAl[MATH] are found to be relatively close to Fermi level, i.e. at respectively [MATH] 100 meV and [MATH] 230 meV above it, making these compounds promising candidates for tuning the Fermi level (e.g. via chemical substitution) towards the linearly dispersive regions of their bands.', '1803.00635-1-3-4': 'In TaAl[MATH] however, such a rigid band shift would displace topologically trivial and the non-trivial bands associated with the Dirac type-II node, thus making it a less promising candidate for exposing topologically dependent physical properties.', '1803.00635-1-4-0': 'Details concerning single-crystal growth can be found in the Supplemental Information (SI) file.[', '1803.00635-1-4-1': '[CITATION] Supplemental Figs. S1, S2, S3 and S4 provide X-ray diffraction, resistivity, a discussion on the role of Al inclusions,[CITATION] and values for the extracted mobilities, respectively.[', '1803.00635-1-4-2': '[CITATION] Density functional theory calculations were performed with the Wien2K [CITATION] implementation of Density Functional Theory (DFT) the Perdew-Burke-Ernzerhof parametrization of the generalized gradient approximation (GGA-PBE) [CITATION].', '1803.00635-1-4-3': 'The angular dependency of the Fermi-surface cross-sectional area were computed through SKEAF.[', '1803.00635-1-4-4': '[CITATION] The validity of these results were verified through the Quantum Expresso implementation of DFT obtaining very similar results.[', '1803.00635-1-5-0': 'Here, we report a study on the electronic structure at the Fermi level of the MAl[MATH] family through the dHvA-effect superimposed onto torque magnetometry.', '1803.00635-1-5-1': 'The MAl[MATH] compounds crystalize in a body-centered tetragonal Bravais lattice belonging to the space group [MATH]4/[MATH] (No. 139) as shown in Fig. [REF](a).', '1803.00635-1-5-2': 'Lattice constants determined through X-ray diffraction are given in the SI.', '1803.00635-1-5-3': '[CITATION] According to DFT, there are two tilted Dirac cones along the Z[MATH]Z line within the first Brillouin zone (see, Fig. [REF](c)), with the Dirac node located at the touching point between the electron and hole cones.', '1803.00635-1-5-4': 'As seen through Figs. [REF](c) to [REF](e), the crossing of the hole bands with the Fermi level produces two hole-like Fermi surface sheets, i.e. the [MATH] and the [MATH] orbits while the electron bands lead to the [MATH] pockets.', '1803.00635-1-5-5': 'The proximity of the [MATH] and [MATH] orbits to the Dirac type-II nodes would make them prime candidates for carriers displaying non-trivial Berry phases.', '1803.00635-1-5-6': 'In contrast, we would expect the [MATH]-pocket along the [MATH]Z line, which is absent in NbAl[MATH], to yield topologically trivial orbits.', '1803.00635-1-5-7': 'As we show below, the electron pocket displays a "helix" like shape while the hole one yields a "dumbbell" like sheet which supports two orbits: neck (or the [MATH]-orbit) and belly (or the [MATH]-orbit).', '1803.00635-1-6-0': 'The torque signal [MATH] can be expressed in terms of the component of the magnetization perpendicular to the external field [MATH]: [MATH]/[MATH] where [MATH] is the volume of the sample.', '1803.00635-1-6-1': 'From the Onsager relation, the frequencies [MATH] of the oscillatory signal are proportional to the extremal cross-sectional areas [MATH] of a given Fermi surface sheet: [EQUATION] where [MATH] is the reduced Planck constant and [MATH] the electrical charge.', '1803.00635-1-6-2': 'Figures [REF](a), [REF](c), and [REF](e) display [MATH] as a function of [MATH], collected at a temperature [MATH] K, for the V, Nb and Ta compounds respectively, where a superimposed oscillatory signal, or the dHvA effect, can be observed.', '1803.00635-1-6-3': 'These traces were collected at angle [MATH] (for the V compound), [MATH] (Nb) and [MATH] (Ta) respectively, where [MATH] and 90[MATH] correspond to fields along the c- and the a-axes, respectively.', '1803.00635-1-6-4': 'The anomaly observed in TaAl[MATH] near [MATH] T is most likely an indication for the quantum limit associated to the [MATH]-orbit.', '1803.00635-1-6-5': 'Figures [REF](b), [REF](d), and [REF](f), display the temperature dependence of the main peaks/frequencies observed in the FFT spectra of the oscillatory signal extracted for each compound, see insets.', '1803.00635-1-6-6': 'The dHvA signals were obtained after fitting the background of [MATH] to a polynomial and its subsequent subtraction.', '1803.00635-1-6-7': 'By fitting the amplitude of the FFT peaks as a function of the temperature to the thermal damping term in the Lifshitz-Kosevich formalism [CITATION] (see, Eq. (2)) one can extract the carrier effective masses [MATH], see Figs. [REF](b), [REF](d), and [REF](f).', '1803.00635-1-6-8': 'For the neck [MATH] T and belly [MATH] T orbits on the "dumbbell" hole-pocket of VAl[MATH], we extracted [MATH] = 0.44 [MATH] and [MATH] = 1.12 [MATH], respectively.', '1803.00635-1-6-9': 'Here, [MATH] is the free-electron mass.', '1803.00635-1-6-10': 'As we show below, these orbits were identified after a detailed comparison with the DFT calculations.', '1803.00635-1-6-11': 'For NbAl[MATH] we extracted [MATH] T with [MATH] = 0.19 [MATH] and [MATH] T with corresponding [MATH] = 0.51 [MATH].', '1803.00635-1-6-12': 'For TaAl[MATH] we identified [MATH] = 174 T with [MATH] and [MATH] T with [MATH].', '1803.00635-1-6-13': 'We are also able to detect the large electron [MATH] pocket of NbAl[MATH] through magnetoresistivity or Shubnikov-de Haas (SdH) measurements under fields up to [MATH] T yielding [MATH] T with [MATH].', '1803.00635-1-6-14': 'These [MATH] values are not particularly light which indicates that these orbits are not very close to the linearly dispersive region of the bands.', '1803.00635-1-7-0': 'In order to study the geometry of the Fermi surface, we measured the angular dependence of the dHvA oscillations.', '1803.00635-1-7-1': 'A comparison between the DFT calculations and the angular dependence of the dHvA frequencies shown in Figs. 3(a) to 3(f), indicates that the [MATH] and [MATH] orbits of the MAl[MATH] compounds correspond to the minimum and maximum cross-sectional areas of the "dumbbell" hole-pockets (in blue, around the [MATH]points).', '1803.00635-1-7-2': 'In Figures 3(b), 3(d), and 3(f), markers correspond to the experimental points while lines depict the angular dependence of the FS cross-sectional areas according to DFT.', '1803.00635-1-7-3': 'The topologically trivial [MATH] orbit corresponds to ellipsoids (in blue) extending along the [MATH] lines in VAl[MATH] and in TaAl[MATH] which are absent in NbAl[MATH].', '1803.00635-1-7-4': 'Finally, the [MATH]-orbit corresponds to the electron-like "helix" sheet, depicted in red around the [MATH]-point.', '1803.00635-1-7-5': 'As seen through Figs. 3(b), 3(d), and 3(f), the experimentally obtained values for the [MATH] and [MATH] orbits agree very well with the calculated ones but become unmeasurable for [MATH].', '1803.00635-1-7-6': 'This suggests either very anisotropic effective masses or, most likely, anisotropic scattering rates.', '1803.00635-1-7-7': 'The [MATH] orbit follows the theoretical predictions with some deviations which can be partially attributed to sample misalignment.', '1803.00635-1-7-8': 'As for the [MATH]-orbit, it was observable only in NbAl[MATH], hence we chose not to plot the respective theoretical traces for VAl[MATH] and TaAl[MATH] in Figs. 3(b) and 3(f).', '1803.00635-1-7-9': 'Its non-observation is likely the result of large cross-sectional areas (or frequencies) combined with heavier effective masses leading to low mobilities.', '1803.00635-1-7-10': 'The very good agreement between the experimentally determined and the calculated FS cross-sectional areas supports the existence of a Dirac Type-II node located at [MATH], [MATH], and [MATH] meV above the Fermi level in VAl[MATH], NbAl[MATH], and TaAl[MATH], respectively.', '1803.00635-1-7-11': 'Notice that our calculations do not yield the same FS topography as the ones reported in Refs. [CITATION].', '1803.00635-1-7-12': 'For instance, in their calculations the dumbbell like hole-pocket would have a quite different geometry.', '1803.00635-1-7-13': 'The exact position of the Dirac nodes relative to [MATH] also differs between our calculations and those in Refs. [CITATION].', '1803.00635-1-8-0': 'Now, we address the topological character of the observed orbits; the Lifshitz-Kosevich formalism describing the field and the temperature-dependence of the dHvA oscillations is given by:[CITATION] [EQUATION] where [MATH] for a parabolic band and [MATH] for a linear one.', '1803.00635-1-8-1': 'The phase shift [MATH] is determined by the dimensionality of the Fermi surface taking values [MATH] for minima and maxima cross-sectional areas of a three-dimensional Fermi surface, respectively.', '1803.00635-1-8-2': '[MATH]) is the thermal damping factor, where [MATH], [MATH] is the Dingle damping factor, where [MATH] is the effective mass and [MATH] from which one can evaluate the quasiparticle lifetime [MATH].', '1803.00635-1-8-3': '[MATH] is the spin damping factor, where [MATH] is the Lande [MATH]-factor.', '1803.00635-1-8-4': 'In order to extract the correct phase of the dHvA oscillations, we make use of the magnetic susceptibility [MATH]: [EQUATION]', '1803.00635-1-8-5': 'The phase of the oscillations is given by [MATH] where [MATH], with [MATH] being the Berry phase, and [MATH] or [MATH] for two- and three-dimensional (3D) FSs, respectively.', '1803.00635-1-8-6': 'For trivial 3D bands [MATH], hence one expects [MATH] or 5/8, for maximum and minimum cross-sectional areas, respectively.', '1803.00635-1-8-7': 'Notice that the value of [MATH] can be affected by the sign of the spin damping factor [MATH] and this has to be carefully considered when extracting the Berry phase.', '1803.00635-1-8-8': 'To experimentally extract the phase, we assign integer Landau level indices [MATH] to the peaks in [MATH] (maxima in the density of states [CITATION]) and [MATH] to the valleys.', '1803.00635-1-8-9': 'The phase [MATH] can be extracted from the intercept of the extrapolation of [MATH] as function of [MATH]; Landau fan diagrams shown in Figs. Fig. [REF](a), Fig. [REF](b), and Fig. [REF](c), for VAl[MATH], NbAl[MATH], and TaAl[MATH], respectively.', '1803.00635-1-8-10': 'For all the three compounds, we obtain [MATH] for the [MATH] orbit which encircles the [MATH]-point in the FBZ which, according to DFT, corresponds to a minimal cross-sectional area of a trivial parabolic band.', '1803.00635-1-8-11': 'Hence, this anomalous 1/8 value can be understood as [MATH], where the -1/2 term is attributable to a "minus" sign provided by the spin dephasing term [MATH].', '1803.00635-1-8-12': 'The [MATH] orbit is the one encircling the Dirac node.', '1803.00635-1-8-13': 'However, it is difficult to extract [MATH] for the [MATH]-orbit of both the VAl[MATH] and TaAl[MATH] compounds given that their higher frequencies are superimposed onto those of the [MATH] and [MATH] orbits.', '1803.00635-1-8-14': 'Fortunately, we were able to extract [MATH] for the [MATH] orbit of NbAl[MATH] through torque measurements.', '1803.00635-1-8-15': 'As for the [MATH]-orbit associated with the trivial parabolic band of TaAl[MATH], we find [MATH], see Fig. [REF](c).', '1803.00635-1-8-16': 'This value was confirmed by SQUID magnetometometry measurements.', '1803.00635-1-8-17': 'See, supplemental Figs. S5, S6, S7 and S8 for band pass filter and phase analysis of the dHvA signal, Berry-phase analysis based on resistivity, based on SQUID magnetometry, and analysis of the spin de-phasing term for VAl[MATH], NbAl[MATH], and TaAl[MATH], respectively.[', '1803.00635-1-8-18': '[CITATION] In the same Ref. supplemental we provide a calculation of the Berry-phase based upon the model of Ref. [CITATION].', '1803.00635-1-8-19': 'We conclude that none of these orbits display a clear non-trivial topological character due to their position with respect to the Dirac node(s).', '1803.00635-1-9-0': 'In summary, we unveiled the Fermi surfaces of the MAl[MATH] family through measurements of quantum oscillations combined with band structure calculations.', '1803.00635-1-9-1': 'Among all three compounds, VAl[MATH] displays the closest Dirac type-II node with respect to its Fermi level ([MATH] 100 meV).', '1803.00635-1-9-2': 'The extracted Berry phases for all of the measured Fermi surfaces are consistent with time-reversal-symmetric systems displaying discrete and topologically trivial values.', '1803.00635-1-9-3': 'Although the [MATH] orbit of NbAl[MATH] encircles the Dirac node, it also displays a trivial phase owing to its distance [MATH] meV) with respect to the Fermi level.', '1803.00635-1-9-4': 'This is not surprising since, as discussed in Ref. [CITATION], the Berry phase becomes trivial as the Fermi level is displaced away from the Dirac node(s).', '1803.00635-1-9-5': 'These compounds still offer promising developments: for instance, our calculations indicate that the Dirac node is displaced towards the Fermi level when the ionic size of the transition metal decreases, indicating a role for chemical substitution.', '1803.00635-1-9-6': 'This approach might also help stabilize bulk superconductivity [CITATION] in this Lorentz non-invariant systems.', '1803.00635-1-9-7': 'Magnetic elements such as Cr are predicted to split the Dirac nodes into two Weyl type-II nodes while breaking the lattice [MATH] rotational symmetry would induce a topological phase-transition.', '1803.00635-1-9-8': '[CITATION]', '1803.00635-1-10-0': 'The authors thank H. -Z. Lu at South University of Science and Technology of China for informative discussions and M. Khan at Louisiana State University for help with our measurements.', '1803.00635-1-10-1': 'This work was supported by DOE-BES through award DE-SC0002613.', '1803.00635-1-10-2': 'K.W.C. was partially supported by the NHMFL-UCGP program.', '1803.00635-1-10-3': 'The NHMFL is supported by NSF through NSF-DMR-1157490 and the State of Florida.', '1803.00635-1-10-4': 'Correspondence and requests for materials should be addressed to K.W.C or to L.B.'}

{'1803.00635-2-0-0': 'We report a de Haas-van Alphen (dHvA) effect study on the Dirac type-II semimetallic candidates MAl[MATH] (where, M = V, Nb and Ta).', '1803.00635-2-0-1': 'The angular-dependence of their Fermi surface (FS) cross-sectional areas reveals a remarkably good agreement with first-principle calculations.', '1803.00635-2-0-2': 'Therefore, dHvA supports the existence of tilted Dirac cones with Dirac type-II nodes located at 100, 230 and 250 meV above the Fermi level [MATH] for VAl[MATH], NbAl[MATH] and TaAl[MATH] respectively, in agreement with the prediction of broken Lorentz invariance in these compounds.', '1803.00635-2-0-3': 'However, for all three compounds we find that the cyclotron orbits on their FSs, including an orbit nearly enclosing the Dirac type-II node, yield trivial Berry phases.', '1803.00635-2-0-4': 'We explain this via an analysis of the Berry phase where the position of this orbit, relative to the Dirac node, is adjusted within the error implied by the small disagreement between our calculations and the experiments.', '1803.00635-2-0-5': 'We suggest that a very small amount of doping could displace [MATH] to produce topologically non-trivial orbits encircling their Dirac node(s).', '1803.00635-2-1-0': 'Condensed-matter systems provide accessible platforms for the discovery of quasiparticles with properties akin to particles predicted by high-energy physics.', '1803.00635-2-1-1': 'Dirac type-I compounds such as Cd[MATH]As[MATH] [CITATION] or Na[MATH]Bi, [CITATION]) and Weyl type-I systems like (Ta,Nb)(P,As) [CITATION] were recently discovered and are garnering a lot of attention.', '1803.00635-2-1-2': 'According to Ref. [CITATION], solid state systems would even offer the potential of finding fermionic excitations which have no analog in high-energy physics such as three-component fermions.', '1803.00635-2-1-3': '[CITATION] Very recently, so-called type-II Dirac/Weyl semimetals, which violate Lorentz-symmetry, were discovered and are being intensively studied.', '1803.00635-2-1-4': 'In type-II Dirac/Weyl semimetals, the crossings between energy bands remain protected but the spectra of the Dirac/Weyl cones are strongly tilted due to an additional momentum dependent term, thus they break the Lorentz invariance.', '1803.00635-2-1-5': 'These Weyl/Dirac type-II nodes become singular points connecting electron and hole pockets in the spectral function.', '1803.00635-2-1-6': 'The associated quasiparticles can be observed in condensed matter systems but the analog particles are absent in Lorentz invariant high energy physics.', '1803.00635-2-1-7': 'Type-II Dirac/Weyl semimetallic systems were proposed to display unique properties, such as Klein tunneling in momentum space,[CITATION] orientation-dependent chiral anomaly,[CITATION] and a modified anomalous Hall conductivity.[', '1803.00635-2-2-0': 'A number of Weyl type-II semimetals were already experimentally studied, including MoTe[MATH],[CITATION] WTe[MATH],[CITATION] Ta[MATH]S[MATH],[CITATION] LaAlGe,[CITATION] and TaIrTe[MATH].[', '1803.00635-2-2-1': '[CITATION] And although several Dirac type-II compounds were also reported, e.g. VAl[MATH],[CITATION] YPd[MATH]Sn,[CITATION] KMgBi,[CITATION] and (Pt,Pd)Te[MATH],[CITATION] only the last two compounds were studied experimentally via angle-resolved photoemission spectroscopy (ARPES) and quantum oscillatory phenomena.[', '1803.00635-2-2-2': '[CITATION] Therefore, Dirac type-II systems remain to be unambiguously identified and characterized experimentally.', '1803.00635-2-2-3': 'For instance, and in addition to the aforementioned predictions, Dirac type-II semimetals have been predicted to become Weyl type-II systems or topological crystalline insulators when time-reversal or inversion symmetries are broken.', '1803.00635-2-2-4': '[CITATION] However, exposing the unique transport properties of such compounds is a difficult task due to the distance of the Dirac type-II nodes with respect to the Fermi level which, in addition, is crossed by topologically trivial and non-trivial bands.', '1803.00635-2-3-0': 'Here, we report the synthesis of the chemical analogues (V, Nb, Ta)Al[MATH] which were predicted to display Dirac type-II nodes.', '1803.00635-2-3-1': '[CITATION] The topography of their Fermi surfaces, revealed through the de Haas van Alphen (dHvA) effect, are found to display remarkably good agreement with band structure calculations.', '1803.00635-2-3-2': 'Therefore, our experimental study indicates that these compounds would break Lorentz invariance.', '1803.00635-2-3-3': '[CITATION] The Dirac type-II nodes in VAl[MATH] and NbAl[MATH] are found to be relatively close to Fermi level, i.e. at respectively [MATH] 100 meV and [MATH] 230 meV above it, making these compounds promising candidates for tuning the Fermi level (e.g. via chemical substitution) towards the linearly dispersive regions of their bands.', '1803.00635-2-3-4': 'In fact, we find that one of the observed cyclotron orbits nearly encloses the Dirac type-II node(s), and although it yields a topologically trivial Berry-phase, a small amount of doping or displacement of [MATH] could lead to a topologically non-trivial orbit.', '1803.00635-2-4-0': 'Details concerning single-crystal growth can be found in the Supplemental Information (SI) file.[', '1803.00635-2-4-1': '[CITATION] Supplemental Figs. S1, S2, S3 and S4 provide X-ray diffraction, resistivity, a discussion on the role of Al inclusions,[CITATION] and values for the extracted mobilities, respectively.[', '1803.00635-2-4-2': '[CITATION] Density functional theory calculations were performed with the Wien2K [CITATION] implementation of Density Functional Theory (DFT) the Perdew-Burke-Ernzerhof parametrization of the generalized gradient approximation (GGA-PBE) [CITATION].', '1803.00635-2-4-3': 'The angular dependency of the Fermi-surface cross-sectional area were computed through SKEAF.[', '1803.00635-2-4-4': '[CITATION] The validity of these results were verified through the Quantum Expresso implementation of DFT obtaining very similar results.[', '1803.00635-2-5-0': 'Here, we report a study on the electronic structure at the Fermi level of the MAl[MATH] family through the dHvA-effect superimposed onto torque magnetometry.', '1803.00635-2-5-1': 'The MAl[MATH] compounds crystalize in a body-centered tetragonal Bravais lattice belonging to the space group [MATH]4/[MATH] (No. 139) as shown in Fig. [REF](a).', '1803.00635-2-5-2': 'Lattice constants determined through X-ray diffraction are given in the SI.', '1803.00635-2-5-3': '[CITATION] According to DFT, there are two tilted Dirac cones along the Z[MATH]Z line within the first Brillouin zone (see, Fig. [REF](c)), with the Dirac node located at the touching point between the electron and hole cones.', '1803.00635-2-5-4': 'As seen through Figs. [REF](c) to [REF](e), the crossing of the hole bands with the Fermi level produces two hole-like Fermi surface sheets, i.e. the [MATH] and the [MATH] orbits while the electron bands lead to the [MATH] pockets.', '1803.00635-2-5-5': 'The proximity of the [MATH] and [MATH] orbits to the Dirac type-II nodes would make them prime candidates for carriers displaying non-trivial Berry phases.', '1803.00635-2-5-6': 'In contrast, we would expect the [MATH]-pocket along the [MATH]Z line, which is absent in NbAl[MATH], to yield topologically trivial orbits.', '1803.00635-2-5-7': 'As we show below, the electron pocket displays a "helix" like shape while the hole one yields a "dumbbell" like sheet which supports two orbits: neck (or the [MATH]-orbit) and belly (or the [MATH]-orbit).', '1803.00635-2-6-0': 'The torque signal [MATH] can be expressed in terms of the component of the magnetization perpendicular to the external field [MATH]: [MATH]/[MATH] where [MATH] is the volume of the sample.', '1803.00635-2-6-1': 'From the Onsager relation, the frequencies [MATH] of the oscillatory signal are proportional to the extremal cross-sectional areas [MATH] of a given Fermi surface sheet: [EQUATION] where [MATH] is the reduced Planck constant and [MATH] the electrical charge.', '1803.00635-2-6-2': 'Figures [REF](a), [REF](c), and [REF](e) display [MATH] as a function of [MATH], collected at a temperature [MATH] K, for the V, Nb and Ta compounds respectively, where a superimposed oscillatory signal, or the dHvA effect, can be observed.', '1803.00635-2-6-3': 'These traces were collected at angle [MATH] (for the V compound), [MATH] (Nb) and [MATH] (Ta) respectively, where [MATH] and 90[MATH] correspond to fields along the c- and the a-axes, respectively.', '1803.00635-2-6-4': 'The anomaly observed in TaAl[MATH] near [MATH] T is most likely an indication for the quantum limit associated to the [MATH]-orbit.', '1803.00635-2-6-5': 'Figures [REF](b), [REF](d), and [REF](f), display the temperature dependence of the main peaks/frequencies observed in the FFT spectra of the oscillatory signal extracted for each compound, see insets.', '1803.00635-2-6-6': 'The dHvA signals were obtained after fitting the background of [MATH] to a polynomial and its subsequent subtraction.', '1803.00635-2-6-7': 'By fitting the amplitude of the FFT peaks as a function of the temperature to the thermal damping term in the Lifshitz-Kosevich formalism [CITATION] (see, Eq. (2)) one can extract the carrier effective masses [MATH], see Figs. [REF](b), [REF](d), and [REF](f).', '1803.00635-2-6-8': 'For the neck [MATH] T and belly [MATH] T orbits on the "dumbbell" hole-pocket of VAl[MATH], we extracted [MATH] = 0.44 [MATH] and [MATH] = 1.12 [MATH], respectively.', '1803.00635-2-6-9': 'Here, [MATH] is the free-electron mass.', '1803.00635-2-6-10': 'As we show below, these orbits were identified after a detailed comparison with the DFT calculations.', '1803.00635-2-6-11': 'For NbAl[MATH] we extracted [MATH] T with [MATH] = 0.19 [MATH] and [MATH] T with corresponding [MATH] = 0.51 [MATH].', '1803.00635-2-6-12': 'For TaAl[MATH] we identified [MATH] = 174 T with [MATH] and [MATH] T with [MATH].', '1803.00635-2-6-13': 'We are also able to detect the large electron [MATH] pocket of NbAl[MATH] through magnetoresistivity or Shubnikov-de Haas (SdH) measurements under fields up to [MATH] T yielding [MATH] T with [MATH].', '1803.00635-2-6-14': 'These [MATH] values are not particularly light which indicates that these orbits are not very close to the linearly dispersive region of the bands.', '1803.00635-2-7-0': 'In order to study the geometry of the Fermi surface, we measured the angular dependence of the dHvA oscillations.', '1803.00635-2-7-1': 'A comparison between the DFT calculations and the angular dependence of the dHvA frequencies shown in Figs. 3(a) to 3(f), indicates that the [MATH] and [MATH] orbits of the MAl[MATH] compounds correspond to the minimum and maximum cross-sectional areas of the "dumbbell" hole-pockets (in blue, around the [MATH]points).', '1803.00635-2-7-2': 'In Figures 3(b), 3(d), and 3(f), markers correspond to the experimental points while lines depict the angular dependence of the FS cross-sectional areas according to DFT.', '1803.00635-2-7-3': 'The topologically trivial [MATH] orbit corresponds to ellipsoids (in blue) extending along the [MATH] lines in VAl[MATH] and in TaAl[MATH] which are absent in NbAl[MATH].', '1803.00635-2-7-4': 'Finally, the [MATH]-orbit corresponds to the electron-like "helix" sheet, depicted in red around the [MATH]-point.', '1803.00635-2-7-5': 'As seen through Figs. 3(b), 3(d), and 3(f), the experimentally obtained values for the [MATH] and [MATH] orbits agree very well with the calculated ones but become unmeasurable for [MATH].', '1803.00635-2-7-6': 'This suggests either very anisotropic effective masses or, most likely, anisotropic scattering rates.', '1803.00635-2-7-7': 'The [MATH] orbit follows the theoretical predictions with some deviations which can be partially attributed to sample misalignment.', '1803.00635-2-7-8': 'As for the [MATH]-orbit, it was observable only in NbAl[MATH], hence we chose not to plot the respective theoretical traces for VAl[MATH] and TaAl[MATH] in Figs. 3(b) and 3(f).', '1803.00635-2-7-9': 'Its non-observation is likely the result of large cross-sectional areas (or frequencies) combined with heavier effective masses leading to low mobilities.', '1803.00635-2-7-10': 'The very good agreement between the experimentally determined and the calculated FS cross-sectional areas supports the existence of a Dirac Type-II node located at [MATH], [MATH], and [MATH] meV above the Fermi level in VAl[MATH], NbAl[MATH], and TaAl[MATH], respectively.', '1803.00635-2-7-11': 'Notice that our calculations do not yield the same FS topography as the ones reported in Refs. [CITATION].', '1803.00635-2-7-12': 'For instance, in their calculations the dumbbell like hole-pocket would have a quite different geometry.', '1803.00635-2-7-13': 'The exact position of the Dirac nodes relative to [MATH] also differs between our calculations and those in Refs. [CITATION].', '1803.00635-2-8-0': 'Now, we address the topological character of the observed orbits; the Lifshitz-Kosevich formalism describing the field and the temperature-dependence of the dHvA oscillations is given by:[CITATION] [EQUATION] where [MATH] for a parabolic band and [MATH] for a linear one.', '1803.00635-2-8-1': 'The phase shift [MATH] is determined by the dimensionality of the Fermi surface taking values [MATH] for minima and maxima cross-sectional areas of a three-dimensional Fermi surface, respectively.', '1803.00635-2-8-2': '[MATH]) is the thermal damping factor, where [MATH], [MATH] is the Dingle damping factor, where [MATH] is the effective mass and [MATH] from which one can evaluate the quasiparticle lifetime [MATH].', '1803.00635-2-8-3': '[MATH] is the spin damping factor, where [MATH] is the Lande [MATH]-factor.', '1803.00635-2-8-4': 'In order to extract the correct phase of the dHvA oscillations, we make use of the magnetic susceptibility [MATH]: [EQUATION]', '1803.00635-2-8-5': 'The phase of the oscillations is given by [MATH] where [MATH], with [MATH] being the Berry phase, and [MATH] or [MATH] for two- and three-dimensional (3D) FSs, respectively.', '1803.00635-2-8-6': 'For trivial 3D bands [MATH], hence one expects [MATH] or 5/8, for maximum and minimum cross-sectional areas, respectively.', '1803.00635-2-8-7': 'Notice that the value of [MATH] can be affected by the sign of the spin damping factor [MATH] and this has to be carefully considered when extracting the Berry phase.', '1803.00635-2-8-8': 'To experimentally extract the phase, we assign integer Landau level indices [MATH] to the peaks in [MATH] (maxima in the density of states [CITATION]) and [MATH] to the valleys.', '1803.00635-2-8-9': 'The phase [MATH] can be extracted from the intercept of the extrapolation of [MATH] as function of [MATH]; Landau fan diagrams shown in Figs. Fig. [REF](a), Fig. [REF](b), and Fig. [REF](c), for VAl[MATH], NbAl[MATH], and TaAl[MATH], respectively.', '1803.00635-2-8-10': 'For all the three compounds, we obtain [MATH] for the [MATH] orbit which encircles the [MATH]-point in the FBZ which, according to DFT, corresponds to a minimal cross-sectional area of a trivial parabolic band.', '1803.00635-2-8-11': 'Hence, this anomalous 1/8 value can be understood as [MATH], where the -1/2 term is attributable to a "minus" sign provided by the spin dephasing term [MATH].', '1803.00635-2-8-12': 'The [MATH] orbit is the one encircling the Dirac node.', '1803.00635-2-8-13': 'However, it is difficult to extract [MATH] for the [MATH]-orbit of both the VAl[MATH] and TaAl[MATH] compounds given that their higher frequencies are superimposed onto those of the [MATH] and [MATH] orbits.', '1803.00635-2-8-14': 'Fortunately, we were able to extract [MATH] for the [MATH] orbit of NbAl[MATH] through torque measurements.', '1803.00635-2-8-15': 'As for the [MATH]-orbit associated with the trivial parabolic band of TaAl[MATH], we find [MATH], see Fig. [REF](c).', '1803.00635-2-8-16': 'This value was confirmed by SQUID magnetometometry measurements.', '1803.00635-2-8-17': 'See, supplemental Figs. S5, S6, S7 and S8 for band pass filter and phase analysis of the dHvA signal, Berry-phase analysis based on resistivity, based on SQUID magnetometry, and analysis of the spin de-phasing term for VAl[MATH], NbAl[MATH], and TaAl[MATH], respectively.[', '1803.00635-2-8-18': '[CITATION] In the same Ref. supplemental we provide a calculation of the Berry-phase based upon the model of Ref. [CITATION].', '1803.00635-2-8-19': 'We show that within the error implied by very small displacements in [MATH] introduced into the DFT calculations to match the experimental results, the [MATH] orbit could not enclose the Dirac node(s) yielding a Berry phase [MATH].', '1803.00635-2-9-0': 'In summary, we unveiled the Fermi surfaces of the MAl[MATH] family through quantum oscillations measurements combined with band structure calculations.', '1803.00635-2-9-1': 'Among all three compounds, VAl[MATH] displays the closest Dirac type-II node with respect to its Fermi level ([MATH] 100 meV).', '1803.00635-2-9-2': 'The extracted Berry phases for all of the measured Fermi surfaces are consistent with time-reversal-symmetric systems displaying discrete and topologically trivial values.', '1803.00635-2-9-3': 'Although the [MATH] orbit of NbAl[MATH] nearly encloses the Dirac node, it also leads to the observation of a trivial Berry-phase due to its exact position in the [MATH] plane relative to the [MATH] position of the Dirac type-II point.', '1803.00635-2-9-4': 'As discussed in Ref. [CITATION], the Berry phase can quickly become trivial as the Fermi level is displaced away from the Dirac node(s).', '1803.00635-2-9-5': 'However, our calculations indicate that the Dirac node is displaced towards the Fermi level as the ionic size of the transition metal decreases, indicating a role for chemical substitution.', '1803.00635-2-9-6': 'One needs only a very small displacement in [MATH] to stabilize a topologically non-trivial [MATH] orbit enclosing the Dirac node.', '1803.00635-2-9-7': 'Notice that this approach might also contribute to stabilize bulk superconductivity [CITATION] or a Weyl type-II state if one chose magnetic dopants.', '1803.00635-2-9-8': '[CITATION]', '1803.00635-2-10-0': 'The authors thank H. -Z. Lu at South University of Science and Technology of China for informative discussions and M. Khan at Louisiana State University for help with our measurements.', '1803.00635-2-10-1': 'This work was supported by DOE-BES through award DE-SC0002613.', '1803.00635-2-10-2': 'K.W.C. was partially supported by the NHMFL-UCGP program.', '1803.00635-2-10-3': 'The NHMFL is supported by NSF through NSF-DMR-1157490 and the State of Florida.', '1803.00635-2-10-4': 'Correspondence and requests for materials should be addressed to K.W.C or to L.B.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1803.00635

hep-ph-0012138

{'hep-ph-0012138-1-0-0': 'We develop an effective theory for heavy baryons and their excited states.', 'hep-ph-0012138-1-0-1': 'The approach is based on the contracted [MATH] symmetry recently shown to emerge from QCD for these states in the combined large [MATH] and heavy quark limits.', 'hep-ph-0012138-1-0-2': 'The effective theory is based on perturbations about this limit; a power counting scheme is developed in which the small parameter is [MATH] where [MATH] (with [MATH] being a typical strong interaction scale).', 'hep-ph-0012138-1-0-3': 'We derive the effective Hamiltonian for strong interactions at next-to-leading order.', 'hep-ph-0012138-1-0-4': 'The next-to-leading order effective Hamiltonian depends on only two parameters beyond the known masses of the nucleon and heavy meson.', 'hep-ph-0012138-1-0-5': 'We also show that the effective operators for certain electroweak transitions can be obtained with no unknown parameters at next-to-leading order.', 'hep-ph-0012138-1-1-0': '# Introduction', 'hep-ph-0012138-1-2-0': 'Recently, it was pointed out that QCD when restricted to the space of heavy baryons (i.e. states with baryon number one containing a single heavy - b or c - quark) has a contracted [MATH] [CITATION] or, more generally, [MATH] symmetry which emerges in the combined large [MATH] (the number of colors) and heavy quark limits [CITATION].', 'hep-ph-0012138-1-2-1': 'Of course, in the real world neither [MATH] nor [MATH] (the generic heavy quark mass) is infinite.', 'hep-ph-0012138-1-2-2': 'Nevertheless if they are large enough, QCD will possess an approximate symmetry.', 'hep-ph-0012138-1-2-3': 'Approximate symmetries can provide significant phenomenological information about a system and have had a long and distinguished history in strong interaction physics.', 'hep-ph-0012138-1-2-4': 'Effective field theory (EFT) is a powerful tool that can be used to extract this information.', 'hep-ph-0012138-1-2-5': 'In this article we develop an effective theory to describe heavy baryons based on the approximate contracted [MATH] symmetry.', 'hep-ph-0012138-1-2-6': 'This is important since this allows one to make testable predictions about the phenomenology of heavy baryons.', 'hep-ph-0012138-1-2-7': 'This paper serves the role of bridging the formal developments of Ref. [CITATION] with phenomenological descriptions of heavy baryons.', 'hep-ph-0012138-1-2-8': 'Our ultimate goal is to study the spectroscopy and electroweak decays of heavy baryon states which will be done in Ref. [CITATION].', 'hep-ph-0012138-1-3-0': 'Before discussing how to implement the EFT approach in the context of heavy baryons it is useful to recall how the EFT program works.', 'hep-ph-0012138-1-3-1': 'Effective field theory is useful when there is a separation of scales between the low energy phenomena and higher energy physics.', 'hep-ph-0012138-1-3-2': 'The short-distance physics can greatly influence the long-distance physics.', 'hep-ph-0012138-1-3-3': 'However, when working at low momentum one cannot resolve the details of what is happening at short distances.', 'hep-ph-0012138-1-3-4': 'The idea of EFT is to describe the low-momentum phenomena in a manner which is insensitive to the details of short-distance physics.', 'hep-ph-0012138-1-3-5': 'Formally, one would begin with the full theory and integrate out all of the short-distance degrees of freedom in a functional integral.', 'hep-ph-0012138-1-3-6': 'This yields an effective action to be used in a functional integral over the low momentum (long-distance) degrees of freedom.', 'hep-ph-0012138-1-3-7': 'Clearly, the effective action so obtained cannot be expressed as the integral over a local Lagrangian.', 'hep-ph-0012138-1-3-8': 'However, all of the nonlocalities occur at the scale of the short-distance degrees of freedom which have been integrated.', 'hep-ph-0012138-1-3-9': 'Thus, to good approximation the effective action can be written in terms of a local Lagrangian.', 'hep-ph-0012138-1-3-10': 'The effects of the short-distance physics are now contained in the coefficients in the effective Lagrangian.', 'hep-ph-0012138-1-4-0': 'Clearly such an approach is, at best, approximate.', 'hep-ph-0012138-1-4-1': 'Fortunately, if the scale separation between the long-distance scales and the short-distance scales is large the approximation can be very good.', 'hep-ph-0012138-1-4-2': 'Formally, one can develop a systematic power-counting expansion where the parameter is the ratio of the typical long-distance scale to the typical short-distance scale.', 'hep-ph-0012138-1-4-3': 'As a practical matter one must truncate the effective Lagrangian at some order.', 'hep-ph-0012138-1-4-4': 'The power counting, however, ensures that the effects of the neglected terms are formally down from the leading term by some power in the ratio of the scales.', 'hep-ph-0012138-1-4-5': 'This, in turn, ensures that the principal effects of the short-distance physics are contained in a few coefficients in the effective Lagrangian.', 'hep-ph-0012138-1-4-6': 'In this manner an insensitivity to the details of the short-distance physics is achieved.', 'hep-ph-0012138-1-4-7': 'The low momentum phenomenology depends on a few coefficients summarizing the effects of the short-distance physics; the manner by which the short-distance physics produces these coefficients is irrelevant.', 'hep-ph-0012138-1-5-0': 'In principle, all of the coefficients in the effective Lagrangian can be calculated from the underlying physics.', 'hep-ph-0012138-1-5-1': 'Often, however, the underlying physics is either unknown or is calculationally intractable.', 'hep-ph-0012138-1-5-2': 'EFT, nevertheless provides a conceptual framework for making predictions about the system.', 'hep-ph-0012138-1-5-3': 'The key insight is that one can treat the parameters in the effective Lagrangian as phenomenological inputs fit from experiment.', 'hep-ph-0012138-1-5-4': 'Thus, providing there are more pieces of independent data than there are free parameters at a given order, one can make real, albeit approximate predictions.', 'hep-ph-0012138-1-5-5': 'Chiral perturbation theory is a familiar paradigm for the EFT program.', 'hep-ph-0012138-1-6-0': 'Let us now turn to the problem of heavy baryons.', 'hep-ph-0012138-1-6-1': 'For simplicity we restrict our attention here to the problem of isoscalar heavy baryons - the [MATH] and [MATH] (generically [MATH]) and their excited states.', 'hep-ph-0012138-1-6-2': 'The generalization to include non-isoscalars such as the [MATH] or [MATH] raises no new conceptual issues but complicates the analysis.', 'hep-ph-0012138-1-6-3': 'The approach of Ref. [CITATION] begins with a restriction to a heavy baryon subspace of the full QCD Hilbert space, i.e. we consider only the space of states with baryon number one and one heavy quark.', 'hep-ph-0012138-1-6-4': 'The physical picture that emerges from the model-dependent analysis of Ref. [CITATION] is that low-lying states of the heavy baryon in the combined large [MATH] and heavy quark limits can be represented as harmonic collective motion of the "brown muck" (the technical term for the light degrees of freedom) against the heavy quark.', 'hep-ph-0012138-1-6-5': 'An equivalent picture in the combined limit is that of the model of Refs. [CITATION] in which the heavy baryons are treated as harmonic collective motions of an ordinary baryon against a heavy meson.', 'hep-ph-0012138-1-6-6': 'The symmetry emerges quite simply from this harmonic oscillator picture: the contracted [MATH] algebra is the algebra of all harmonic oscillator creation and annihilation operators and all bilinears made from them.', 'hep-ph-0012138-1-7-0': 'The key to the analysis of Ref. [CITATION] which enables this picture to emerge is a consistent power-counting scheme.', 'hep-ph-0012138-1-7-1': 'Formally, since we are considering the combined large [MATH] and heavy quark limits, it is useful to consider a single parameter which characterizes departures from this limit.', 'hep-ph-0012138-1-7-2': 'Thus, we introduce: [EQUATION] where [MATH] is a typical strong interaction scale and [MATH] is the mass of the heavy quark.', 'hep-ph-0012138-1-7-3': 'The effective expansion parameter turns out not to be [MATH], but [MATH] [CITATION].', 'hep-ph-0012138-1-7-4': 'The low-lying (collective) excitation energies of the heavy baryons are of order [MATH].', 'hep-ph-0012138-1-7-5': 'This should be contrasted with excitation energies for internal excitation of the brown muck and the dissociation energy of the heavy baryon (into an ordinary baryon and a heavy meson) which are each of order [MATH] [CITATION].', 'hep-ph-0012138-1-7-6': 'Thus, provided that [MATH] can be considered to be small, a scale separation exists between the low-lying collective states of the heavy baryons from all other excitations.', 'hep-ph-0012138-1-7-7': 'This, in turn, implies that EFT methods can be applied to this problem.', 'hep-ph-0012138-1-8-0': 'We note at the outset, however, that in practice [MATH] is not a particularly small number so that the scale separation is not particularly large.', 'hep-ph-0012138-1-8-1': 'This implies that one must work at relatively high order in order to get relatively imprecise results.', 'hep-ph-0012138-1-8-2': 'Thus for example, even working at next-to-leading order one only expects relative accuracy nominally only of order [MATH].', 'hep-ph-0012138-1-8-3': 'Clearly, the approach will be at best semi-quantitative.', 'hep-ph-0012138-1-8-4': 'Moreover, the expansion will not converge for all observables.', 'hep-ph-0012138-1-8-5': 'On phenomenological grounds it is clear that at best the approach will be useful for describing the ground state of the [MATH] and the doublet of the first excited orbital excitation.', 'hep-ph-0012138-1-8-6': 'The second orbitally excited states (which have not been observed in either the [MATH] or [MATH] sectors) is either unbound or very near the threshold and clearly beyond the harmonic limit implicit in the effective field theory.', 'hep-ph-0012138-1-8-7': 'The situation is reminiscent of the contracted [MATH] symmetries seen for light baryons which emerge in the large [MATH] limit [CITATION].', 'hep-ph-0012138-1-8-8': 'In that case the large [MATH] limit predicts an infinite tower of low-lying baryons.', 'hep-ph-0012138-1-8-9': 'However, only the nucleon and delta for [MATH] (or the octet and decuplet for [MATH]) are well described in the real world.', 'hep-ph-0012138-1-9-0': 'The construction of the form of the effective Hamiltonian or Lagrangian for the system is very straightforward.', 'hep-ph-0012138-1-9-1': 'Following the standard rules of effective field theory, once the low-energy degrees of freedom are identified, one may simply write down the most general Lagrangian built from them which is consistent with the underlying symmetries and includes all terms up to a fixed order in the power counting.', 'hep-ph-0012138-1-9-2': 'Similarly, one could add external sources to the theory and impose power counting and thereby determine the from of currents.', 'hep-ph-0012138-1-9-3': 'Of course, in order to compute observables one needs more than the form of the effective Lagrangian and effective operators - one needs values for the coefficients.', 'hep-ph-0012138-1-9-4': 'These coefficients can either be obtained purely phenomenologically or by matching with the underlying theory.', 'hep-ph-0012138-1-10-0': 'In this paper we derive the form of the effective Hamiltonian at next-to-leading order (NLO) directly from the structure of QCD and the power-counting rules .', 'hep-ph-0012138-1-10-1': 'We do this formally by directly making a change of variables to the collective degrees of freedom.', 'hep-ph-0012138-1-10-2': 'This has the benefit of relating the coefficients in the effective theory directly to ground state expectation values of known QCD operators.', 'hep-ph-0012138-1-10-3': 'In this derivation the structure of the Hilbert space plays an essential role.', 'hep-ph-0012138-1-10-4': 'In particular, it is necessary to show that in the combined limit, the Hilbert space can be written as a product space of collective excitations of order [MATH] which may be interpreted as excitations of the brown muck moving coherently against the heavy quark, and non-collective excitation of order [MATH] which correspond to excitations of the brown muck itself.', 'hep-ph-0012138-1-10-5': 'In addition, as we will show, the excitations of the two spaces are independent up to corrections of higher order in [MATH].', 'hep-ph-0012138-1-11-0': 'We also derive results about the effective theory of more phenomenological interest.', 'hep-ph-0012138-1-11-1': 'One of the central achievements of this paper is to show that at NLO the dynamics depends on only two free parameters in the effective Hamiltonian beyond known masses of the nucleon and heavy meson.', 'hep-ph-0012138-1-11-2': 'This is of phenomenological significance in that the small value of the expansion parameter suggests that even for quite crude calculations one should work at least at NLO.', 'hep-ph-0012138-1-11-3': 'Had there been a large number of free parameters at this order it would have suggested that the scheme would be difficult to implement in practice as data on heavy baryons becomes available.', 'hep-ph-0012138-1-11-4': 'Another important result of this paper is the demonstration that certain operators that are not contained in the Hamiltonian but which play a role in describing electroweak properties of heavy baryons are completely determined without the need for phenomenological coefficients (up to corrections of higher order in [MATH]) from the known structure of QCD and the counting rules.', 'hep-ph-0012138-1-11-5': 'This increases the predictive power of the effective theory by eliminating free parameters.', 'hep-ph-0012138-1-12-0': 'As noted above, an expansion in [MATH] is unlikely to be rapidly convergent.', 'hep-ph-0012138-1-12-1': 'Thus it seems sensible to work at relatively high order in the expansion.', 'hep-ph-0012138-1-12-2': 'Here we have stopped at NLO for two reasons: one practical and one theoretical.', 'hep-ph-0012138-1-12-3': 'The practical reason is simply that beyond NLO the number of parameters grows to the point where it is likely to exceed the number of observables which have any reasonable hope of being measured in the foreseeable future.', 'hep-ph-0012138-1-12-4': 'The theoretical reason is that the change of variables technique used to derive the effective theory in terms of known operators in QCD is straightforward only up to NLO.', 'hep-ph-0012138-1-12-5': 'Beyond this order, ambiguities in the definition of the heavy quark mass enter as do effects from the coupling of the low-lowing states described by the effective theory with higher energy excitations.', 'hep-ph-0012138-1-12-6': 'Thus, the treatment becomes more subtle and complex beyond this order.', 'hep-ph-0012138-1-12-7': 'Of course, this does not mean that there is no valid effective theory beyond NLO.', 'hep-ph-0012138-1-12-8': 'However considerable effort must be undertaken in going beyond this order and this effort is simply not justified in view of the practical difficulties mentioned above.', 'hep-ph-0012138-1-13-0': 'This paper is organized as follows.', 'hep-ph-0012138-1-13-1': 'In Sec. [REF] we follow Ref. [CITATION] and briefly review the derivation of the low energy collective variables for heavy baryons from QCD with an emphasis on the power counting rules.', 'hep-ph-0012138-1-13-2': 'In Sec. [REF], we show that the structure of the Hilbert space as a product space emerges from standard large [MATH] analysis.', 'hep-ph-0012138-1-13-3': 'The form of the effective Hamiltonian (up to NLO) is derived from QCD in Sec. [REF].', 'hep-ph-0012138-1-13-4': 'Effective operators for electroweak matrix elements are derived in Sec. [REF].', 'hep-ph-0012138-1-13-5': 'Finally a brief summary is given in Sec. [REF]', 'hep-ph-0012138-1-14-0': '# Dynamical Variables from Counting Rules and QCD', 'hep-ph-0012138-1-15-0': 'In this section we review the counting rules derived from QCD for heavy baryons in the combined large [MATH] and heavy quark limits.', 'hep-ph-0012138-1-15-1': 'We will not reproduce the derivations of Ref. [CITATION] in detail.', 'hep-ph-0012138-1-15-2': 'Rather, we will outline the basic strategy of the derivation along with the principal physical and mathematical assumptions.', 'hep-ph-0012138-1-15-3': 'We also collect results which will be needed later in this paper.', 'hep-ph-0012138-1-16-0': 'The first physical point of significance is to restrict our attention to a limited part of the QCD Hilbert space - the part with baryon number one and heavy quark number one.', 'hep-ph-0012138-1-16-1': 'Furthermore we will ultimately restrict our attention to the lowest-lying states with these quantum numbers.', 'hep-ph-0012138-1-16-2': 'For simplicity, in this section we will only consider a single species of heavy quark at a time as the strong interaction does not couple different heavy flavors.', 'hep-ph-0012138-1-16-3': 'We will generalize to two heavy flavors when we consider electroweak matrix elements in the following sections.', 'hep-ph-0012138-1-17-0': 'QCD is a field theory and as such has an infinite number of degrees of freedom.', 'hep-ph-0012138-1-17-1': 'We can envision making a canonical transformation from the original quark and gluon degrees of freedom to a new set of variables (at the quantum mechanical level it becomes a unitary transformation).', 'hep-ph-0012138-1-17-2': 'The goal is to find such a transformation which has the property that a certain set of the variables (which we denote the collective variables), generate the low-lying collective excited states when acting (repeatedly) on the ground state (the [MATH]).', 'hep-ph-0012138-1-17-3': 'Working explicitly with the collective variables one can calculate directly the properties of the low-lying states.', 'hep-ph-0012138-1-17-4': 'Because we are working in the context of canonical transformations it is far more natural to work with the Hamiltonian formulation of the underlying quantum field theory (QCD) rather than the Lagrangian formulation.', 'hep-ph-0012138-1-18-0': 'Unfortunately, there is no straightforward way to generate such a canonical transformation to collective variables exactly.', 'hep-ph-0012138-1-18-1': 'However, it is easy to find a set of variables which behaves this way approximately.', 'hep-ph-0012138-1-18-2': 'That is, using the power-counting scheme of Eq. ([REF]) one can find collective variables which when acting on the ground state (repeatedly) generate the low-lying collective states plus a small component of higher excited states - the amount of higher excited state admixed is suppressed by powers of [MATH].', 'hep-ph-0012138-1-19-0': 'One canonically conjugate pair of collective variables is the total momentum of the system [MATH], the generator of translations (which is well defined in QCD) and its conjugate variable [MATH], the generator of momentum boosts.', 'hep-ph-0012138-1-19-1': 'This pair of collective variables, however, does not induce the internal excitations of interest here.', 'hep-ph-0012138-1-20-0': 'One might also wish to introduce the momentum of the heavy quark as a collective variable.', 'hep-ph-0012138-1-20-1': 'The expression for [MATH] appropriate in the heavy quark limit is [EQUATION] where [MATH] is the heavy quark field, and [MATH] is the three-dimensional covariant derivative.', 'hep-ph-0012138-1-20-2': 'In the heavy quark limit it is apparent that the variable conjugate to [MATH] is [MATH] defined by [EQUATION]', 'hep-ph-0012138-1-20-3': 'The definitions for [MATH] and [MATH] in Eqs. ([REF]), ([REF]) are clearly valid in the heavy quark limit for the subspace of states with a heavy quark number of unity; in this limit the heavy quark behaves as in nonrelativistic quantum mechanics.', 'hep-ph-0012138-1-20-4': 'In the limit [MATH] and [MATH] correspond to the generators of translations and momentum boosts for the heavy quark.', 'hep-ph-0012138-1-20-5': 'Away from this limit the concept of boosting or translating the heavy quark separately from the full system is not strictly well defined and the [MATH] and [MATH] defined above need not be canonically conjugate.', 'hep-ph-0012138-1-20-6': 'However, even away from the formal limit there will exist some canonically conjugate pair of operators which smoothly goes to the variables defined in Eqs. ([REF]), ([REF]) as the limit is approached.', 'hep-ph-0012138-1-20-7': 'Moreover, by standard heavy quark counting considerations this canonically conjugate pair of operators will differ from those defined in Eqs. ([REF]), ([REF]) by an amount of order [MATH].', 'hep-ph-0012138-1-20-8': 'Here we are interested in the theory up to next-to-leading order, i.e. up to relative order [MATH].', 'hep-ph-0012138-1-20-9': 'Accordingly, to the order at which we work we can take Eqs. ([REF]), ([REF]) as unambiguous definitions.', 'hep-ph-0012138-1-21-0': 'Unfortunately, the conjugate pairs [MATH] and [MATH] are not independent.', 'hep-ph-0012138-1-21-1': 'Clearly [MATH] since [MATH] is a generator of translations and translating the system necessarily translates the position of the heavy quark.', 'hep-ph-0012138-1-21-2': 'Thus, we can not directly use [MATH] as our collective variables to generate low-lying states.', 'hep-ph-0012138-1-21-3': 'However, from these two pairs we can generate two independent conjugate pairs.', 'hep-ph-0012138-1-21-4': 'In particular we can construct a linear combination of [MATH] and [MATH], which we denote [MATH] and a linear combination of [MATH] and [MATH], which we denote [MATH], such that [MATH] and [MATH] are conjugate to each other and commute with [MATH] and [MATH] up to corrections at next-to-next-to-leading order (NNLO).', 'hep-ph-0012138-1-21-5': 'Intuitively [MATH] can be thought of as the generator of translations of the brown muck relative to the heavy quark and [MATH] as its conjugate.', 'hep-ph-0012138-1-21-6': 'As we will see below, the variables [MATH] do act as approximate collective variables in the sense outlined above.', 'hep-ph-0012138-1-22-0': 'The explicit construction of [MATH] and [MATH] is detailed in Ref. [CITATION].', 'hep-ph-0012138-1-22-1': 'The key inputs to this construction are Poincare invariance, heavy quark effective theory, and a decomposition of the QCD Hamiltonian organized via power-counting.', 'hep-ph-0012138-1-22-2': 'The decomposition of the QCD Hamiltonian is most easily accomplished by thinking of the heavy baryon as a bound state of a heavy meson and a nucleon.', 'hep-ph-0012138-1-22-3': 'Standard large [MATH] QCD analysis allows us to identify the interaction energy as scaling as [MATH], while the nucleon mass [MATH] scales as [MATH] [CITATION].', 'hep-ph-0012138-1-22-4': 'Standard heavy quark analysis implies that the interaction energy is of order [MATH] while the mass of the heavy meson, [MATH], is given by [EQUATION]', 'hep-ph-0012138-1-22-5': 'Thus, when constrained to the heavy baryon part of the Hilbert space, the QCD Hamiltonian may be decomposed as: [EQUATION] where [MATH] is of order [MATH].', 'hep-ph-0012138-1-22-6': 'Poincare invariance implies that [EQUATION] while the standard heavy quark treatment implies that [EQUATION]', 'hep-ph-0012138-1-22-7': 'Using Eqs. ([REF]), ([REF]), ([REF]) along with Eq. ([REF]) and the fact that [MATH], the methods of Ref. [CITATION] allows one to deduce that [EQUATION]', 'hep-ph-0012138-1-22-8': 'The expressions for [MATH] and [MATH] differ from the equivalent expressions in Ref. [CITATION] in that we have exploited Eq. ([REF]) to express quantities in terms of [MATH] which is directly physically accessible rather than [MATH] which is not.', 'hep-ph-0012138-1-22-9': 'Throughout the remainder of this paper we will often eliminate [MATH] in favor of [MATH].', 'hep-ph-0012138-1-23-0': 'Thus, we have constructed two independent pairs of conjugate variables: [MATH] and [MATH].', 'hep-ph-0012138-1-23-1': 'These two pairs naturally arise in the Hamiltonian.', 'hep-ph-0012138-1-23-2': 'However, as we will see in Sec. [REF], in describing electroweak operators these pairs are not so natural.', 'hep-ph-0012138-1-23-3': 'Instead it is natural to use the positions and momenta of the heavy quarks [MATH] and the corresponding independent set [MATH] which correspond intuitively to the position and momenta of the light degrees of freedom.', 'hep-ph-0012138-1-23-4': 'Formally they are defined as follows: [EQUATION]', 'hep-ph-0012138-1-23-5': 'In order to exploit the collective variables [MATH] and [MATH] to learn about the low-lying spectrum, it is useful to evaluate multiple commutators of [MATH] and [MATH] with [MATH].', 'hep-ph-0012138-1-23-6': 'Following Ref. [CITATION] one can again use Eqs. ([REF]), ([REF]), ([REF]), ([REF]) and the known [MATH] scaling of [MATH] and [MATH], i.e. [MATH], to obtain [EQUATION] where the reduced mass, [MATH], is given by [EQUATION]', 'hep-ph-0012138-1-23-7': 'More generally, it can be seen by the same methods that multiple commutators of [MATH] with [MATH] are suppressed by multiple powers of [MATH].', 'hep-ph-0012138-1-23-8': 'In particular, if one takes [MATH] commutators of components of [MATH] with [MATH] with [MATH] even, then the [MATH] commutator scales as [MATH].', 'hep-ph-0012138-1-23-9': 'For example: [EQUATION]', 'hep-ph-0012138-1-23-10': 'In contrast, standard large [MATH] counting implies that a displacement of the light degrees of freedom relative to the heavy degrees of freedom by a distance of order [MATH] shifts the Hamiltonian by an amount of order [MATH]: [EQUATION] where [MATH] is a c-number of order [MATH].', 'hep-ph-0012138-1-23-11': 'Equation ([REF]) in turn implies that multiple commutators of [MATH] with [MATH] are generically of order unity: [EQUATION]', 'hep-ph-0012138-1-23-12': 'The power-counting rules from QCD are essentially summarized in Eqs. ([REF]) and ([REF]).', 'hep-ph-0012138-1-23-13': 'By themselves, however, they are not sufficient to develop a fully systematic power-counting scheme for observables without additional phenomenological input.', 'hep-ph-0012138-1-23-14': 'In particular, there are two possible scenarios in which the power-counting may be realized selfconsistently.', 'hep-ph-0012138-1-23-15': 'Intuitively, the [MATH] or symmetric realization corresponds to the case where the effective potential between the heavy quark and the brown muck has a minimum at [MATH], while the [MATH] or symmetry broken realization corresponds to the situation where the effective potential has a minimum at [MATH].', 'hep-ph-0012138-1-23-16': 'The [MATH] realization is labeled symmetry broken since it breaks rotational symmetry in the [MATH] limit.', 'hep-ph-0012138-1-23-17': 'As shown in Ref. [CITATION] the [MATH] (symmetric) realization has the generic property that [EQUATION] in the sense that typical matrix elements between low-lying states of polynomial operators which contain [MATH] and [MATH] scale as [MATH].', 'hep-ph-0012138-1-23-18': 'In contrast, in the [MATH] realization, [EQUATION]', 'hep-ph-0012138-1-23-19': 'In this paper we will assume that the dynamics are such that the system is in the [MATH] (symmetric) realization.', 'hep-ph-0012138-1-24-0': '# Structure of the Hilbert Space', 'hep-ph-0012138-1-25-0': 'In this section we will show that in the combined large [MATH] and heavy quark limits, the Hilbert space can be written as a product space composed of collective and intrinsic excitations, with the collective excitations having energies of order [MATH] and intrinsic excitations are of order unity.', 'hep-ph-0012138-1-25-1': 'The physical picture of these two types of excitations is quite clear - the collective excitations describe coherent motion of the brown muck against the heavy quark while the intrinsic excitations are excitations of the brown muck itself.', 'hep-ph-0012138-1-26-0': "The basic strategy in demonstrating that the Hilbert space is of this form is to exploit the fact that our expansion parameter is valid in the pure large [MATH] limit where Witten's Hartree picture is well established [CITATION].", 'hep-ph-0012138-1-26-1': 'The key point is that while nature is far closer to the heavy quark limit than the large [MATH] limit, our expansion is formulated in the combined large [MATH] and heavy quark limits for [MATH] arbitrary.', 'hep-ph-0012138-1-26-2': 'Thus it applies whether one takes the heavy quark limit first, the large [MATH] limit first or takes the limits simultaneously.', 'hep-ph-0012138-1-26-3': 'We will deduce the structure of the Hilbert space by taking the large [MATH] limit first and then argue that the result goes over smoothly to the combined limit.', 'hep-ph-0012138-1-26-4': 'Ultimately, the reason that we can legitimately take the limits in any order stems from the fact that the dynamics of the system are determined by the interaction Hamiltonian, [MATH].', 'hep-ph-0012138-1-26-5': 'Note that [MATH] is non-singular in both the large [MATH] and heavy quark limits, and therefore we can smoothly take either limit or the combined limit first.', 'hep-ph-0012138-1-27-0': 'Now consider taking the large [MATH] limit prior to taking the heavy quark limit.', 'hep-ph-0012138-1-27-1': "In this case we can use Witten's Hartree picture of baryons.", 'hep-ph-0012138-1-27-2': 'Each quark propagates in the mean field of the remaining quarks.', 'hep-ph-0012138-1-27-3': 'Low energy excitations are described by the excitations of a single quark in the background of the remaining quarks.', 'hep-ph-0012138-1-27-4': 'In the present case we can divide these excitations into two classes - excitations of the heavy quark and excitations of one of the light quarks.', 'hep-ph-0012138-1-27-5': 'Of course, for the light quark excitation it is necessary to anti-symmetrize the wave function as there are many light quarks.', 'hep-ph-0012138-1-27-6': 'In contrast there is only one heavy quark.', 'hep-ph-0012138-1-27-7': 'In the large [MATH] limit these two classes of excitations are orthogonal to each other so that the wave function may be written as a direct product: [EQUATION] where [MATH] ) represent the heavy (light) quark part of the Hartree wave function.', 'hep-ph-0012138-1-28-0': 'The excitation energies for both types of excitations are of order [MATH].', 'hep-ph-0012138-1-28-1': 'However, the excitation energies of the heavy quark are of order [MATH].', 'hep-ph-0012138-1-28-2': 'This is straightforward to see; the heavy quark sits in a potential well of depth and width independent of [MATH].', 'hep-ph-0012138-1-28-3': 'As one takes the large [MATH] limit (after having taken the large [MATH] limit), the heavy quark wave function becomes localized at the bottom of the potential well and thus acts like an harmonic oscillator.', 'hep-ph-0012138-1-28-4': 'One sees that the characteristic excitation energies of the heavy and light types of motion scale with [MATH] according to [EQUATION]', 'hep-ph-0012138-1-28-5': 'As noted above, in this picture the excitations of the heavy quark and the light quarks are independent.', 'hep-ph-0012138-1-28-6': 'The total excitation energy is simply the sum of the heavy excitation energy and the light excitation energy.', 'hep-ph-0012138-1-29-0': 'If we now consider the combined limit with the large [MATH] and heavy quark limits taken simultaneously the picture changes.', 'hep-ph-0012138-1-29-1': 'The Hartree approximation is no longer valid.', 'hep-ph-0012138-1-29-2': 'However, the general counting rules remain - they simply have to be reinterpreted.', 'hep-ph-0012138-1-29-3': 'Physically what happens in the combined limit is that instead of the heavy quark oscillating in the static field of the brown muck, as in the Hartree picture, they oscillate against each other.', 'hep-ph-0012138-1-29-4': 'However, the structure of the Hilbert space is not altered in its essence.', 'hep-ph-0012138-1-29-5': 'The space can still be broken up into a product space of independent classes of excitations as in Eq. ([REF]), but now the two classes of excitations are not the heavy and light degrees of freedom as in the Hartree approximation.', 'hep-ph-0012138-1-29-6': 'Rather, they are the collective excitations (between the brown muck and the heavy quark) and the intrinsic excitations (of the brown muck): [EQUATION] where C ( I ) represent the collective (intrinsic) part.', 'hep-ph-0012138-1-29-7': 'Similarly, the excitation energies behave as in Eq. ([REF]) but with the more general description of collective and intrinsic classes of excitations replacing the heavy and light classes of the Hartree approximation: [EQUATION]', 'hep-ph-0012138-1-29-8': 'We see that the structure of the Hilbert space is quite simple in the [MATH] limit.', 'hep-ph-0012138-1-29-9': 'In order to understand the structure of the effective theory, it is necessary to understand how corrections to this leading behavior emerge.', 'hep-ph-0012138-1-29-10': 'In particular, we will need to quantify the mixing between the collective and intrinsic excitations due to sub-leading terms in [MATH].', 'hep-ph-0012138-1-29-11': 'Again, our strategy is to pass to the case where the large [MATH] limit is taken first and we have the Hartree approximation as a well-defined tool.', 'hep-ph-0012138-1-29-12': 'Then use the smoothness of the two limits to pull back to the case of [MATH] keeping [MATH] arbitrary.', 'hep-ph-0012138-1-30-0': 'It is useful to consider the way [MATH] corrections emerge on top of the Hartree picture as these corrections will ultimately become corrections in our [MATH] counting.', 'hep-ph-0012138-1-30-1': 'We begin by assuming that at finite, but large [MATH], a Hartree picture is a good first approximation.', 'hep-ph-0012138-1-30-2': 'We also make the standard large [MATH] assumption [CITATION] that corrections to the energy and to energy splittings are generically of relative order [MATH].', 'hep-ph-0012138-1-30-3': 'Moreover, these leading relative corrections are independent of [MATH] as the heavy quark only contributes 1 out of [MATH] quarks.', 'hep-ph-0012138-1-30-4': 'Thus, these relative corrections due to effects beyond the Hartree approximation are necessarily of order [MATH] and do not contribute at NLO (which is relative order [MATH]).', 'hep-ph-0012138-1-30-5': 'It is straightforward to see that [MATH] corrections to the energies imply that Hamiltonian matrix elements between two different states with different intrinsic states are generically of order [MATH].', 'hep-ph-0012138-1-30-6': 'This can be shown in two ways.', 'hep-ph-0012138-1-30-7': 'First, off-diagonal matrix elements between different intrinsic states of order [MATH] yield [MATH] corrections to the energies and we know corrections of this order exist.', 'hep-ph-0012138-1-30-8': 'The second way is to construct an explicit wave function using a transition induced by the gluon exchange.', 'hep-ph-0012138-1-30-9': 'The gluon exchange contains two factors of the strong coupling and scales as [MATH]; there is a combinatoric factor of [MATH] as the gluon can connect to any quark; and finally, there is a normalization factor of [MATH] because there are [MATH] distinct terms in the excited state wave function as any of the quarks may be excited.', 'hep-ph-0012138-1-30-10': 'Now we can pass from the large [MATH] limit back to the combined [MATH] limit (with [MATH] fixed) by exploiting smoothness.', 'hep-ph-0012138-1-30-11': 'We obtain the following important result: [EQUATION] where the state [MATH] is a product state [EQUATION]', 'hep-ph-0012138-1-31-0': '# Construction of the Effective Hamiltonian', 'hep-ph-0012138-1-32-0': 'In Sec. [REF] we reviewed the power-counting rules for the collective variables [MATH] and [MATH] in the heavy baryon sector of the QCD Hilbert space.', 'hep-ph-0012138-1-32-1': 'As discussed in Sec. [REF], the general principle of effective field theory implies that the form of the effective theory is completely fixed by the symmetries of the theory and the power-counting rules.', 'hep-ph-0012138-1-32-2': 'Rather than simply invoking this general strategy, it is instructive to derive the effective field theory at next-to-leading order directly from QCD through the change in variables discussed in Sec. [REF].', 'hep-ph-0012138-1-32-3': 'We do this because it is useful to verify that the proposed scaling rules are, indeed, selfconsistent; the derivation gives significant insights into the underlying structure of the theory; and finally, it is satisfying to verify explicitly the validity of the general EFT philosophy in this case.', 'hep-ph-0012138-1-33-0': 'In the present context, where we have been implementing canonical transformations to collective variables, it is simpler to work with an effective Hamiltonian rather than an effective Lagrangian.', 'hep-ph-0012138-1-33-1': 'From Eq. ([REF]) and the power-counting rules of Eqs. ([REF]), ([REF]), ([REF]) and ([REF]) it is straightforward to deduce the form of the QCD Hamiltonian in the heavy baryon sector written in terms of the collective variables up to order [MATH] (NLO).', 'hep-ph-0012138-1-33-2': 'It is given by [EQUATION] where [MATH], [MATH], and [MATH].', 'hep-ph-0012138-1-33-3': 'From rotational invariance, the [MATH] are completely symmetric Cartesian tensor operators of rank [MATH].', 'hep-ph-0012138-1-33-4': 'Note that while terms up to [MATH] contribute at this order, only terms up to [MATH] contribute.', 'hep-ph-0012138-1-33-5': 'The difference between these two is due to the essentially nonrelativistic nature of the kinematics.', 'hep-ph-0012138-1-33-6': 'Formally, this difference is implicit in the different scaling relations for [MATH] and [MATH] in Eqs. ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-ph-0012138-1-34-0': 'Since the form of Eq. ([REF]) was deduced from commutation relations, one cannot immediately conclude that the [MATH] operators are c-numbers.', 'hep-ph-0012138-1-34-1': 'Rather, at this stage in the derivation we can only deduce that they are operators which commute with all of the collective variables: [EQUATION]', 'hep-ph-0012138-1-34-2': 'Thus, the [MATH] tensors are, in general, operators which depend only on the non-collective variables, i.e. the [MATH] are purely intrinsic operators.', 'hep-ph-0012138-1-34-3': 'Note that to the extent that the coefficients [MATH] in the Hamiltonian of Eq. ([REF]) are operators as opposed to c-numbers, the Hamiltonian is not strictly an effective Hamiltonian.', 'hep-ph-0012138-1-34-4': 'Rather it is the full QCD Hamiltonian written in terms of our collective degrees of freedom.', 'hep-ph-0012138-1-34-5': 'The non-collective degrees of freedom have not been integrated out; rather their effects are contained in the non-collective operators [MATH].', 'hep-ph-0012138-1-35-0': 'Non-collective operators might be expected to influence the dynamics of the collective degrees of freedom even though they all commute with the collective degrees of freedom.', 'hep-ph-0012138-1-35-1': 'For example, consider the ground state matrix element of [MATH].', 'hep-ph-0012138-1-35-2': 'This can be expressed in terms of off-diagonal matrix elements: [EQUATION] where [MATH] is the ground state of the heavy baryon system and the intermediate states are grouped into purely collective excitations [MATH] (with excitation energies of order [MATH] which excite only the collective degrees of freedom leaving the brown muck unexcited, and non-collective states [MATH] (with excitation energies of order [MATH] or greater) which involve an intrinsic excitation of the brown muck.', 'hep-ph-0012138-1-35-3': 'If the operators in the Hamiltonian had been purely collective as in a true effective Hamiltonian (which by construction has all non-collective degrees of freedom integrated out), then the sum would not contain the non-collective states [MATH].', 'hep-ph-0012138-1-35-4': 'However, the non-collective operators [MATH] induce transitions to the non-collective states.', 'hep-ph-0012138-1-35-5': 'By acting twice - once to go into the non-collective space, and once to come back the collective space - the non-collective operators can affect collective space matrix elements of operators contained in the Hamiltonian.', 'hep-ph-0012138-1-36-0': 'However, from Eq. ([REF]) we know that terms which connect different intrinsic states are suppressed by order [MATH].', 'hep-ph-0012138-1-36-1': 'We can use this to deduce that the contribution of non-collective intermediate states to matrix elements of composite operators such as in Eq. ([REF]) will always be suppressed by order [MATH] relative to the contributions of the collective intermediated states and thus only contribute at NNLO.', 'hep-ph-0012138-1-36-2': 'This in turn justifies their omission at NLO.', 'hep-ph-0012138-1-36-3': 'Let us see how this works for the decomposition in Eq. ([REF]).', 'hep-ph-0012138-1-36-4': 'From the scaling rules of Eq. ([REF]) it is straightforward to see that the contribution from the collective intermediate states goes as [MATH].', 'hep-ph-0012138-1-36-5': 'A typical term in the sum over states outside the purely collective space is [EQUATION] where [MATH] and [MATH] represent the ground states of the collective and intrinsic subspaces, respectively, while [MATH] and [MATH] represent excited states, and the notation [MATH] represents a product state as in Eq. ([REF]) .', 'hep-ph-0012138-1-36-6': 'The equality in Eq. ([REF]) follows since [MATH] only acts on the collective space while [MATH] only acts on the intrinsic space and the scaling behavior follows from Eqs. ([REF]) and ([REF]) so that the collective part of the expectation value scales as [MATH] while the intrinsic part scales as [MATH].', 'hep-ph-0012138-1-36-7': 'Thus, we see that the contribution of non-collective states is down by a factor of [MATH] compared to the contribution of the collective intermediate states.', 'hep-ph-0012138-1-37-0': 'The [MATH] suppression of the effects of non-collective intermediate states relative to the collective ones is generic.', 'hep-ph-0012138-1-37-1': 'Although it was explicitly demonstrated for the case of the composite operator in Eq. ([REF]), it should be apparent that such a suppression occurs for all products of operators contained in the Hamiltonian.', 'hep-ph-0012138-1-37-2': 'When calculating Hamiltonian matrix elements, the effects of the operators [MATH] in Eq. ([REF]) inducing mixing with the non-collective states will always contribute at relative order [MATH] (i.e. at NNLO) or less.', 'hep-ph-0012138-1-37-3': 'Thus, working at NLO one can simply replace the [MATH] operators by their expectation values in the ground state (or more generally in the ground state band - the set of collective states built on the ground state of the intrinsic degrees of freedom).', 'hep-ph-0012138-1-37-4': 'Therefore, the operators [MATH] can be treated as c-numbers in the effective theory.', 'hep-ph-0012138-1-38-0': 'When the coefficients [MATH] are treated as operators, rotational invariance does not constrain the allowable forms in the Hamiltonian beyond what is given in Eq. ([REF]).', 'hep-ph-0012138-1-38-1': 'For example, the term [MATH] is allowable despite the fact that [MATH] is an [MATH] operator; [MATH] could also be [MATH] operator and couple to [MATH] leading to an [MATH] operator in the Hamiltonian.', 'hep-ph-0012138-1-38-2': 'Once the [MATH] operators are restricted to being c-numbers in the effective theory, rotational invariance greatly restricts the allowable forms: c-numbers do not transform under spatial rotations.', 'hep-ph-0012138-1-38-3': 'Accordingly, the only surviving [MATH] coefficients are those which multiply rotational scalars.', 'hep-ph-0012138-1-38-4': 'Thus, for example, the [MATH] terms must vanish.', 'hep-ph-0012138-1-38-5': 'Treating the [MATH] coefficients as c-numbers and imposing rotational symmetry yields the following form of the effective Hamiltonian up to [MATH]: [EQUATION] where [MATH] refers to the center of mass motion of the entire system while [MATH] refers to the piece of the Hamiltonian whose leading contribution is of order [MATH].', 'hep-ph-0012138-1-38-6': 'The coefficients [MATH], [MATH] and [MATH] are of order [MATH] and may be expressed as ground-state expectation values of [MATH], [MATH] and [MATH], respectively: [EQUATION]', 'hep-ph-0012138-1-38-7': 'A few comments about this derivation are in order here.', 'hep-ph-0012138-1-38-8': 'In essence, we did not have to explicitly integrate out the non-collective higher-lying degrees of freedom.', 'hep-ph-0012138-1-38-9': 'By making a change of variables to [MATH], [MATH], [MATH] and [MATH], we already have identified variables which were decoupled from the intrinsic degrees of freedom to the order at which we are working.', 'hep-ph-0012138-1-38-10': 'Imagine one explicitly made this change of variables prior to formulating the problem in terms of a functional integral.', 'hep-ph-0012138-1-38-11': 'At low order the action could be written as the sum of a collective action plus an intrinsic action without coupling between them, and the only effect of integrating out the intrinsic degrees of freedom would be to supply an overall constant multiplying the remaining functional integral over the collective degrees of freedom.', 'hep-ph-0012138-1-38-12': 'By making this change of variables we make the effect of integrating out the intrinsic degrees of freedom trivial - at least up to NLO.', 'hep-ph-0012138-1-38-13': 'Only if we go beyond this order do effects of the the coupling to the intrinsic degrees of freedom play a role, and the effects of integrating out the intrinsic degrees of freedom become nontrivial.', 'hep-ph-0012138-1-39-0': 'From a phenomenological prospective, it is sufficient to start with Eq. ([REF]).', 'hep-ph-0012138-1-39-1': 'The coefficients [MATH], [MATH] and [MATH] could then be determined approximately from fits to experimental data.', 'hep-ph-0012138-1-39-2': 'Given the fact that the masses of the nucleon and the heavy meson are known, the effective theory at NLO depends on three parameters - [MATH], [MATH] and [MATH].', 'hep-ph-0012138-1-39-3': 'The coefficient [MATH] is determined from the difference in energy between the nucleon-heavy meson threshold and the lowest energy state.', 'hep-ph-0012138-1-39-4': 'However, [MATH] is simply an overall constant and plays no role in the dynamics; the dynamics of the problem depends on two parameters at NLO.', 'hep-ph-0012138-1-39-5': 'It is worth noting that at leading order the dynamics depends only on a single parameter, [MATH].', 'hep-ph-0012138-1-39-6': 'Thus, going from leading order to next-to-leading costs only an additional parameter, [MATH].', 'hep-ph-0012138-1-40-0': '# Effective Operators', 'hep-ph-0012138-1-41-0': 'To make predictions of electroweak current matrix elements between heavy baryon states, one needs more than just the effective Hamiltonian.', 'hep-ph-0012138-1-41-1': 'Effective operators are also required.', 'hep-ph-0012138-1-41-2': 'For general composite operators expressed in terms of the quark and glue degrees of freedom of QCD, the effective operators can be derived in a functional integral form by integrating out the intrinsic degrees of freedom.', 'hep-ph-0012138-1-41-3': 'The most straightforward way to formulate this is as an integral over all QCD variables with [MATH]-functions put in to pick out particular values of the collective variables.', 'hep-ph-0012138-1-41-4': 'In practice, however, the preceding description is rather formal as it is not generally possible to perform this functional integration.', 'hep-ph-0012138-1-41-5': 'At a phenomenological level, one can simply write a form for the operator consistent with symmetries and parameterized by some constants and then fit the constants from experimental data.', 'hep-ph-0012138-1-41-6': 'However, if the operators of interest can be expressed entirely in terms of the collective degrees of freedom at the QCD level, the functional integral approach becomes unnecessary (for the same reason it was unnecessary in the derivation of the collective Hamiltonian).', 'hep-ph-0012138-1-41-7': 'At NLO, one can immediately express the effective operators in terms of the same collective degrees of freedom.', 'hep-ph-0012138-1-42-0': 'To see how this works, consider some operator which at the QCD level can be expressed entirely in terms of the collective operators.', 'hep-ph-0012138-1-42-1': 'We seek to find an analogous operator for use in the effective theory whose matrix elements in the effective theory reproduces the matrix elements of the QCD operator in the QCD states (up to correction at NNLO).', 'hep-ph-0012138-1-42-2': 'Let us start by considering the matrix elements at the QCD level.', 'hep-ph-0012138-1-42-3': 'From the discussion of the structure of the Hilbert space at the QCD level in Sec. [REF], it is apparent that a low-energy (i.e. with excitation energy of order [MATH]) heavy baryon state in a [MATH] expansion at NLO can be written as: [EQUATION] where [MATH] is defined as a product state as in Eq. ([REF]), and [MATH] are constants of order unity.', 'hep-ph-0012138-1-43-0': 'The sum in the second term of Eq. ([REF]) is restricted to contributions with [MATH] due to the product structure of the state.', 'hep-ph-0012138-1-43-1': 'If the sum were not restricted, then there would be contributions of the form [EQUATION] where, [MATH] is simply a single state in the purely collective space.', 'hep-ph-0012138-1-43-2': 'Thus, the contribution to the sum from Eq. ([REF]) is of the form of a single product state.', 'hep-ph-0012138-1-43-3': 'If there are contributions of the form as in Eq. ([REF]) we can replace [MATH] by [MATH] and thereby eliminate contributions with [MATH] from the sum.', 'hep-ph-0012138-1-43-4': 'Doing this has the advantage of removing any possibility of double counting these contributions.', 'hep-ph-0012138-1-44-0': 'Now let us evaluate the matrix element of a purely collective operator, [MATH], i.e. an operator constructed entirely of the collective operators [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-1-44-1': 'From the structure of the state in Eq. ([REF]), the matrix element between two low-energy heavy baryon states (given at NLO) is given by [EQUATION]', 'hep-ph-0012138-1-44-2': 'Thus, up to NNLO corrections the QCD matrix element is simply the matrix element in the collective part of the Hilbert space.', 'hep-ph-0012138-1-44-3': 'On the other hand, the effective Hamiltonian we use at this order with the operators [MATH] replaced by c-numbers is designed to correctly describe the collective part of the Hilbert space at this order.', 'hep-ph-0012138-1-45-0': 'The preceding argument shows, that in the case of purely collective operators the operators in the effective theory are related to the QCD operators in a trivial way up to corrections at NNLO.', 'hep-ph-0012138-1-45-1': 'A purely collective operator at the QCD level is of the form [EQUATION] where the superscript QCD indicates that all of the operators are at the QCD level.', 'hep-ph-0012138-1-45-2': 'The effective operator up to order [MATH] corrections is then obtained by keeping the same operator structure of [MATH], but replacing [MATH], [MATH], [MATH] and [MATH] with the collective operators [MATH], [MATH], [MATH] and [MATH]: [EQUATION].', 'hep-ph-0012138-1-45-3': 'This is essentially a nonrenormalization theorem to this order.', 'hep-ph-0012138-1-45-4': 'The effects of integrating out collective degrees of freedom only renormalize the effective operators at NNLO.', 'hep-ph-0012138-1-46-0': 'In a similar way, all purely non-collective operators at the QCD level, i.e. all operators which commute with all of the collective operators, simply become c-numbers in the effective theory with a value equal to the ground state expectation value of the analogous QCD operator.', 'hep-ph-0012138-1-46-1': 'These general principles can be used to deduce the form of several operators in the effective theory which will prove useful in the calculation of electroweak observables for heavy baryon states.', 'hep-ph-0012138-1-47-0': 'Let us first consider the operator which generates Lorentz boosts, [MATH].', 'hep-ph-0012138-1-47-1': 'The unitary operator for a Lorentz boost is [MATH].', 'hep-ph-0012138-1-47-2': 'The boost operator is useful, for example, in calculating form factors which involve matrix elements of states with different momenta.', 'hep-ph-0012138-1-47-3': 'One can use the boost operator to generate these states from a standard state in its rest frame.', 'hep-ph-0012138-1-47-4': 'From Poincare invariance, the boost operator is given at the field theoretic level by [EQUATION] where the second equality follows from Eq. ([REF]).', 'hep-ph-0012138-1-47-5': 'Thus, up to NLO at the QCD level [MATH] and from the arguments given above, this can immediately be taken over in the effective theory.', 'hep-ph-0012138-1-48-0': 'Among the observables which can be studied using the combined expansion are matrix elements of the electroweak current [MATH], where subscripts [MATH] and [MATH] refer to the heavy quark flavors and [MATH] is the Dirac structure of the left-hand current, [MATH].', 'hep-ph-0012138-1-48-1': 'We will show here that the leading term in the combined heavy quark and large [MATH] expansion of the current [MATH] can be expressed in terms of the collective variables [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0012138-1-48-2': 'Moreover, we will show that the lowest order corrections come at order [MATH] and not at [MATH], so that they contribute only at NNLO.', 'hep-ph-0012138-1-49-0': 'The combined expansion of the current [MATH] can be done analogously to the pure heavy quark expansion in the Heavy Quark Effective Theory (HQET) [CITATION].', 'hep-ph-0012138-1-49-1': 'The idea is to decompose the total heavy quark field into "large" and "small" components or "light" and "heavy" fields with the latter being suppressed by a factor of [MATH] relative to the "large" component.', 'hep-ph-0012138-1-49-2': 'It is then possible to eliminate the "small" component in terms of the "large" component by means of a power series expansion.', 'hep-ph-0012138-1-49-3': 'Near the combined limit considered here, the pure heavy quark expansion breaks down.', 'hep-ph-0012138-1-49-4': 'It has been shown in Refs. [CITATION] that the next-to-leading order terms in the pure heavy quark expansion of the matrix elements of the current [MATH] between the heavy baryon states contain coefficients proportional to [MATH] where [MATH] (with [MATH] being the total mass of a heavy baryon).', 'hep-ph-0012138-1-49-5': 'In the HQET the constant [MATH] is treated as being much smaller than [MATH], so that the NLO operators are indeed suppressed.', 'hep-ph-0012138-1-49-6': 'The situation is different, however, near the combined limit where [MATH] is of order [MATH], so that [MATH] and [MATH].', 'hep-ph-0012138-1-49-7': 'Hence, these corrections cannot be neglected near the combined limit.', 'hep-ph-0012138-1-50-0': 'The appearance of large corrections near the combined limit is a result of the way the total heavy quark field is separated into its "large" and "small" components.', 'hep-ph-0012138-1-50-1': 'The definition of these components is necessarily different from the analogous decomposition in HQET.', 'hep-ph-0012138-1-50-2': 'Near the combined limit, as in HQET, the heavy quark is not far off-shell since its interaction with the brown muck is of order [MATH] while its mass is of order [MATH].', 'hep-ph-0012138-1-50-3': 'However, the space-time dependence of the heavy quark is much different near the combined limit.', 'hep-ph-0012138-1-50-4': 'Near the pure heavy quark limit the brown muck mass is formally much less than the heavy quark mass.', 'hep-ph-0012138-1-50-5': 'As a result, the space-time dependence of the heavy quark is essentially determined by the free particle Lagrangian.', 'hep-ph-0012138-1-50-6': 'In HQET, therefore, it is useful to redefine the heavy quark field by removing the phase factor of the solution of the free Dirac equation, namely [MATH].', 'hep-ph-0012138-1-50-7': 'Near the combined limit the heavy quark remains essentially nonrelativistic.', 'hep-ph-0012138-1-50-8': 'However, the space-time dependence of the heavy quark is much different from that of a free particle.', 'hep-ph-0012138-1-50-9': 'The phase factor now should contain a contribution due to the brown muck mass, which is formally of the same order as the heavy quark mass near the combined limit.', 'hep-ph-0012138-1-50-10': 'This naturally leads to the following redefinition of the heavy quark field: [EQUATION] where [MATH] is the total 4-velocity of the heavy baryon, the operators [MATH] in the rest frame of the heavy baryon project out the upper and lower components of the heavy quark field, and [MATH] here represents a space-time point.', 'hep-ph-0012138-1-50-11': 'As in HQET, the fields [MATH] and [MATH] satisfy conditions [MATH] and [MATH].', 'hep-ph-0012138-1-50-12': 'Fields [MATH] and [MATH] are defined for each value of a heavy baryon 4-velocity [MATH] as can be seen from Eq. ([REF]).', 'hep-ph-0012138-1-50-13': 'The redefinition in Eq. ([REF]) has the effect of removing both the heavy quark mass, [MATH], and the nucleon mass, [MATH], from the dynamics, leaving a dynamical scale of order [MATH], i.e., the scale of the interaction Hamiltonian near the combined limit which is [MATH].', 'hep-ph-0012138-1-51-0': 'The heavy quark part of the QCD Lagrangian density in terms of the fields [MATH] and [MATH] has the form, [EQUATION] where [MATH] is the "transverse part" of the covariant derivative [MATH].', 'hep-ph-0012138-1-51-1': 'The heavy quark Lagrangian in Eq. ([REF]) written in terms of the fields [MATH] and [MATH] differs from its analog in HQET due to the additional term proportional to [MATH].', 'hep-ph-0012138-1-51-2': 'This [MATH] term would naively suggest that the fields [MATH] and [MATH] are both heavy with masses [MATH] and [MATH], respectively, which apparently prevents integrating out the "heavy" field [MATH].', 'hep-ph-0012138-1-52-0': 'This problem is easily resolved if one considers the full Lagrangian density of the system: [EQUATION] where the sum is over all light quarks, and [MATH] is the Yang-Mills Lagrangian density.', 'hep-ph-0012138-1-52-1': 'The total Lagrangian density in Eq. ([REF]) should be re-expressed so as to build the brown muck contribution into the heavy degrees of freedom.', 'hep-ph-0012138-1-52-2': 'This is completely analogous to the Hamiltonian treatment in Sec. [REF].', 'hep-ph-0012138-1-52-3': 'In that case it is done by adding and subtracting to the Hamiltonian a quantity which is an overall constant [MATH] and regrouping terms according to their [MATH] counting scaling.', 'hep-ph-0012138-1-52-4': 'In the Lagrangian formalism this can be accomplished in the following way using a Lorentz covariant operator [MATH]: [EQUATION] where [EQUATION] and [EQUATION].', 'hep-ph-0012138-1-52-5': 'In the rest frame of a heavy baryon operator, [MATH] is equal to the heavy quark density operator [MATH] and its spatial integral equals to one in the Hilbert space of heavy baryons.', 'hep-ph-0012138-1-52-6': 'Thus, in the Hamiltonian the additional term corresponds to removing from the light degrees of freedom an overall constant [MATH].', 'hep-ph-0012138-1-53-0': 'The additional operator proportional to the nucleon mass [MATH] can easily be expressed in terms of the fields [MATH] and [MATH]: [EQUATION].', 'hep-ph-0012138-1-53-1': 'This operator cancels the last term in Eq. ([REF]) so that the Lagrangian density [MATH] has the form: [EQUATION] where the condition [MATH] is used.', 'hep-ph-0012138-1-53-2': 'Thus, the "large" component [MATH] describes a massless field while the "small" component [MATH] has the mass [MATH].', 'hep-ph-0012138-1-53-3': 'Using equations of motion the "heavy" field [MATH] can be expressed in terms of the "light" field [MATH] as follows: [EQUATION]', 'hep-ph-0012138-1-53-4': 'The relation between fields [MATH] and [MATH] in Eq. ([REF]) is identical in form to that in HQET between the "large" and "small" components of the heavy quark field [MATH].', 'hep-ph-0012138-1-54-0': 'We can now expand the field [MATH], the Lagrangian density in Eq. ([REF]), and the current [MATH] in powers of [MATH].', 'hep-ph-0012138-1-54-1': 'This can be done by eliminating fields [MATH] (using Eq. ([REF])) and expanding the numerator in powers of [MATH].', 'hep-ph-0012138-1-54-2': 'Thus, the combined expansion of the heavy quark field including terms up to order [MATH] is, [EQUATION]', 'hep-ph-0012138-1-54-3': 'Using this expansion the Lagrangian density [MATH] including terms up to order [MATH] has the form: [EQUATION] where [MATH] is a strong coupling constant and [MATH] is the gluon field strength tensor.', 'hep-ph-0012138-1-54-4': 'In a similar way we can expand the current [MATH]; again keeping terms up to [MATH] we get, [EQUATION]', 'hep-ph-0012138-1-54-5': 'The expressions in Eqs. ([REF]), ([REF]), ([REF]) are identical in form to the analogous quantities in HQET.', 'hep-ph-0012138-1-54-6': 'However, there is an important difference: these expressions represent the combined expansion in powers of [MATH] and not the [MATH] expansion of HQET.', 'hep-ph-0012138-1-54-7': 'In other words, by defining fields [MATH] and [MATH] appropriately we were able to resum implicitly all the [MATH] corrections in HQET.', 'hep-ph-0012138-1-55-0': 'The correction terms in the combined expansion of the electroweak current, Eq. ([REF]), come at order [MATH] and not at [MATH].', 'hep-ph-0012138-1-55-1': 'It will be shown in Ref. [CITATION] that these corrections contribute to the matrix elements at order [MATH] as well, [MATH], they do not induce any anomalously large constants which violate the combined power counting of these operators.', 'hep-ph-0012138-1-55-2': 'Thus, at NLO in our expansion only the leading order operator, [MATH], contributes to the matrix elements.', 'hep-ph-0012138-1-55-3': 'In addition, it will be shown in Ref. [CITATION] that in the combined limit at NLO there is only a single form factor which parameterizes the matrix elements of the current [MATH] in the [MATH] sector.', 'hep-ph-0012138-1-55-4': 'Accordingly, we can consider one Lorentz component of the current [MATH], namely the [MATH] component.', 'hep-ph-0012138-1-55-5': 'Thus, to completely determine the matrix elements at NLO in the combined expansion, we need to know the form of only one operator, namely [MATH].', 'hep-ph-0012138-1-56-0': 'Let us first consider the flavor conserving operator [MATH].', 'hep-ph-0012138-1-56-1': 'As we are working near the heavy quark limit it is legitimate to expand the state in a fock-space decomposition for the number of heavy quarks and antiquarks.', 'hep-ph-0012138-1-56-2': 'For a heavy baryon state standard heavy quark analysis implies that the leading fock component has one heavy quark; the first sub-leading fock component with two quarks and one antiquark is suppressed by a factor of [MATH].', 'hep-ph-0012138-1-56-3': 'Thus at NLO in [MATH] we can neglect the sub-leading fock component and treat the state as having a single heavy quark.', 'hep-ph-0012138-1-56-4': 'This implies that acting in our heavy baryon Hilbert space, the operator [MATH] is [MATH].', 'hep-ph-0012138-1-56-5': 'Since this is true at the QCD level at NLO it immediately holds in the effective theory at that order.', 'hep-ph-0012138-1-57-0': 'In order to describe the flavor changing current we need to enlarge the Hilbert space of our effective theory to account for electroweak transitions between heavy baryons with heavy quarks of different flavors.', 'hep-ph-0012138-1-57-1': 'The new Hilbert space is the direct product of the Hilbert spaces for baryons with a given flavor of a heavy quark described in Sec. [REF].', 'hep-ph-0012138-1-57-2': 'Any state of the enlarged Hilbert space can be written as [MATH]-dimensional vector, where [MATH] is a number of heavy flavors.', 'hep-ph-0012138-1-57-3': 'Each component of this vector describes a state with a given flavor.', 'hep-ph-0012138-1-57-4': 'The effective Hamiltonian can be readily generalized to include baryons of various heavy quark flavors.', 'hep-ph-0012138-1-57-5': 'It can be written as [MATH] diagonal matrix.', 'hep-ph-0012138-1-57-6': 'For example, for two heavy flavors ([MATH] charm and bottom) the effective Hamiltonian including terms up to NLO in [MATH] has the form: [EQUATION] where the diagonal entries [MATH] and [MATH] are the effective Hamiltonians as in Eq. ([REF]).', 'hep-ph-0012138-1-57-7': 'Analogously, QCD operators acting in the enlarged Hilbert space become [MATH] matrices.', 'hep-ph-0012138-1-57-8': 'The flavor changing weak transitions can be described by off-diagonal matrix elements of an operator [MATH], where the subscripts [MATH] and [MATH] can take integer values from [MATH] to [MATH].', 'hep-ph-0012138-1-57-9': 'This operator can be written in terms of collective operators only.', 'hep-ph-0012138-1-57-10': 'It follows from arguments analogous to the case of the flavor conserving electroweak operator - [MATH] - that the flavor changing operator acting on the enlarged Hilbert space has the form (for [MATH]): [EQUATION]', 'hep-ph-0012138-1-57-11': 'The diagonal elements of the operator in Eq. ([REF]) correspond to flavor conserving operators, while the off-diagonal elements correspond to flavor changing operators.', 'hep-ph-0012138-1-57-12': 'Their action is to instantaneously change the heavy quark flavor at the position where a heavy baryon interacts with the current.', 'hep-ph-0012138-1-57-13': 'Since the operator in Eq. ([REF]) is written in terms of the collective operators only, the corresponding effective operator has an identical form.', 'hep-ph-0012138-1-58-0': 'Finally, let us consider the operator defined by [EQUATION] so that [MATH] represents the contribution to the baryon dipole moment coming from light quarks.', 'hep-ph-0012138-1-58-1': 'In calculating electromagentic transitions between heavy baryons it is useful to divide the electromagnetic current into pieces coming from the heavy and the light quarks and to further divide the light quark contribution into isovector and isoscalar pieces.', 'hep-ph-0012138-1-58-2': 'The isoscalar piece is simply the light-quark contribution to the baryon current.', 'hep-ph-0012138-1-58-3': 'Thus, matrix elements of [MATH] must be calculated to describe electric dipole transitions of heavy baryons.', 'hep-ph-0012138-1-59-0': 'At first sight, it might appear to be impossible to express the operator [MATH] in terms of collective operators.', 'hep-ph-0012138-1-59-1': 'Although [MATH] - the collective position operator for light quarks - is intuitively something like the position of the brown muck and clearly transforms in the same way as [MATH], at the operator level [MATH] is manifestly not identical to [MATH].', 'hep-ph-0012138-1-59-2': 'Recall that by construction [MATH] acts as a generator of boosts for the light degrees of freedom (up to order [MATH] corrections).', 'hep-ph-0012138-1-59-3': 'On the other hand, the operator [MATH] acts only on the quarks and not on the gluons.', 'hep-ph-0012138-1-59-4': 'Thus [MATH] cannot boost the gluons and therefore does not act as a boost for the light degrees of freedom (which include both quarks and gluons).', 'hep-ph-0012138-1-59-5': 'Thus, the operator [MATH] is clearly distinct from [MATH] and there is apparently no simple way to express [MATH] entirely in terms of collective operators.', 'hep-ph-0012138-1-60-0': 'Remarkably, however, one can show that [EQUATION] for all [MATH], [MATH] in the collective subspace defined in Sec. [REF].', 'hep-ph-0012138-1-60-1': 'Thus, when one restricts consideration to the collective subspace which dominates the low energy physics, [MATH] is equivalent to [MATH].', 'hep-ph-0012138-1-61-0': 'To see how this comes about, we start by considering the commutation relations of [MATH] with [MATH], [MATH] and [MATH].', 'hep-ph-0012138-1-61-1': 'To determine the commutator with [MATH] we exploit the fact that [MATH] is a generator of translations for the light degrees of freedom so that [MATH], where [MATH] is a generic light quark field.', 'hep-ph-0012138-1-61-2': 'We can use this to deduce that [EQUATION]', 'hep-ph-0012138-1-61-3': 'The first equality in Eq. ([REF]) follows from the definition of [MATH], the second equality follows because [MATH] translates the light quark degrees of freedom, the third equality follows from a change of variables, and the final equality follows from the definition of [MATH] and the fact that [EQUATION] counts the net number of light quarks, which is [MATH] in a heavy baryon containing one heavy quark.', 'hep-ph-0012138-1-61-4': 'Equation ([REF]) implies that [EQUATION] while the fact that from its definition [MATH] has no heavy quark operators implies that [EQUATION]', 'hep-ph-0012138-1-61-5': 'Thus, the commutation relations of [MATH] with [MATH], [MATH] and [MATH] are identical to the commutation relations of [MATH] with [MATH], [MATH] and [MATH].', 'hep-ph-0012138-1-61-6': 'In Eqs. ([REF]), ([REF]) indices [MATH] indicate Cartesian components of [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-1-62-0': 'Next let us express [MATH] in terms of the various collective and non-collective operators in the problem.', 'hep-ph-0012138-1-62-1': 'Generically, one can represent an arbitrary operator [MATH] as a power series in our collective operators: [EQUATION] with [MATH] specifying a particular term in the expansion, and [MATH] indicating the three Cartesian directions.', 'hep-ph-0012138-1-62-2': 'The coefficients in the expansion, [MATH], are generally non-collective operators which commute with [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-1-62-3': 'Expanding one component of [MATH] in this form gives [EQUATION] where the fact that the [MATH] commutes with [MATH] or [MATH] impose the restriction that no nonzero powers of [MATH] or [MATH] can contribute to the series, while the commutation relation of Eq. ([REF]) implies that the only contributing term containing any powers of [MATH] is the linear term and that the coefficient of this term is unity.', 'hep-ph-0012138-1-63-0': 'The known parity and time reversal properties of [MATH] and [MATH] fix the parity and time reversal properties of the non-collective coefficient operators, [MATH].', 'hep-ph-0012138-1-63-1': 'Since [MATH] is time reversal even and parity odd while [MATH] is time reversal odd and parity even, it follows that: [EQUATION]', 'hep-ph-0012138-1-63-2': 'The significant point about Eq. ([REF]) is that every non-collective coefficient operator is odd under PT.', 'hep-ph-0012138-1-63-3': 'From the analysis of Sec. [REF], generic states can be written as superpositions of outer product states containing collective and intrinsic parts.', 'hep-ph-0012138-1-63-4': 'The collective subspace of the theory is the subspace where the intrinsic wave function is in its ground state.', 'hep-ph-0012138-1-63-5': 'Thus, an arbitrary state in the collective subspace can be written as [MATH] where [MATH] indicates the ground state of the intrinsic system.', 'hep-ph-0012138-1-63-6': 'Now consider the matrix element of a typical term in the series of Eq. ([REF]) between two collective states: [EQUATION] where we have separated the matrix element into its collective and intrinsic parts.', 'hep-ph-0012138-1-63-7': 'Clearly, the matrix element in Eq. ([REF]) vanishes: the intrinsic part of the of the matrix element is a diagonal matrix element of a PT odd operator in a state of good PT.', 'hep-ph-0012138-1-63-8': 'Thus, within the collective subspace all terms in the series of Eq. ([REF]) vanish except the first term.', 'hep-ph-0012138-1-63-9': 'This implies that within this subspace all matrix elements of [MATH] are identical to matrix elements of [MATH] and Eq. ([REF]) is established.', 'hep-ph-0012138-1-64-0': 'As written, the equivalence expressed in Eq. ([REF]) only holds for states in the collective subspace, i.e., states with the intrinsic subspace in its ground state.', 'hep-ph-0012138-1-64-1': 'As discussed in Sec. [REF], the non-collective operators in the Hamiltonian induce components with excited intrinsic states in the physical low-energy states.', 'hep-ph-0012138-1-64-2': 'This in turn spoils the formal equivalence in Eq. ([REF]) for the physical states.', 'hep-ph-0012138-1-64-3': 'Fortunately, however, it is straightforward to see that in the physical states, the matrix elements of [MATH] differ from those of [MATH] by an amount of order [MATH] (or less) and hence may be neglected at next-to-leading order in [MATH].', 'hep-ph-0012138-1-64-4': 'In the first place using reasoning analogous to that leading to Eq. ([REF]), it is easy to see that the matrix elements of the [MATH] between the intrinsic ground state and its excited states is of order [MATH].', 'hep-ph-0012138-1-64-5': 'On the other hand, the analysis of Sec. [REF] implies that admixtures of excited intrinsic states in the physical states are suppressed by at least order [MATH].', 'hep-ph-0012138-1-64-6': 'Combining these two facts one sees that the matrix elements of [MATH] and [MATH] in the physical states are equivalent at NLO.', 'hep-ph-0012138-1-65-0': 'The equivalence at NLO between the matrix elements of [MATH] and [MATH] may seem paradoxical.', 'hep-ph-0012138-1-65-1': 'After all, as discussed above, the two operators are clearly quite distinct at the QCD level.', 'hep-ph-0012138-1-65-2': 'It would seem that the fact that the operators are different would imply that the matrix elements of the two operators should be different in general.', 'hep-ph-0012138-1-65-3': 'The resolution of the paradox is quite simple: the matrix elements of the two operators are different in general.', 'hep-ph-0012138-1-65-4': 'They only coincide (up to NNLO corrections) for a very special class of states - the collective states with excitation energies of order [MATH].', 'hep-ph-0012138-1-65-5': 'For states with excitation energies of order unity, the matrix elements of the two operators can be quite different.', 'hep-ph-0012138-1-65-6': 'Fortunately, we will be using our effective theory to study these low-lying excitations only.', 'hep-ph-0012138-1-65-7': 'Thus, for the purpose of the effective theory at NLO, it is legitimate to replace [MATH] with [MATH].', 'hep-ph-0012138-1-66-0': '# Summary', 'hep-ph-0012138-1-67-0': 'In this paper we have derived an effective theory which will be used in Ref. [CITATION] to study the spectroscopy and electroweak decays of the low-energy states of heavy baryons.', 'hep-ph-0012138-1-67-1': 'The analysis is based largely on the formulation of the problem in Ref. [CITATION].', 'hep-ph-0012138-1-67-2': 'A power-counting scheme is developed in [MATH], where [MATH].', 'hep-ph-0012138-1-67-3': 'The kinematical variables of the problem [MATH], [MATH], [MATH] and [MATH] are directly related to QCD operators which are well defined up to corrections at relative order [MATH].', 'hep-ph-0012138-1-67-4': 'This formalism is based on standard large [MATH] and heavy quark analysis.', 'hep-ph-0012138-1-68-0': 'At the theoretical level, the analysis used in our derivation of the effective theory goes well beyond the analysis of Ref. [CITATION] which was largely kinematical and group-theoretical in nature.', 'hep-ph-0012138-1-68-1': 'In our derivation, the structure of the Hilbert space for heavy baryons plays a critical role.', 'hep-ph-0012138-1-68-2': 'In particular, as the [MATH] limit is approached, the heavy baryon states can be written as products of collective states (with excitation energies of order [MATH]) and intrinsic states (with excitation energies of order [MATH]).', 'hep-ph-0012138-1-68-3': 'To deduce this form of the Hilbert space we exploited the fact that our formulation as an expansion in [MATH] does not depend formally on whether the large [MATH] limit is taken first, the heavy quark limit is taken first or they are taken simultaneously.', 'hep-ph-0012138-1-68-4': 'By taking the large [MATH] limit first we can use standard large [MATH] reasoning to deduce the structure of the Hilbert space and then map back to the case where the limits are taken concurrently.', 'hep-ph-0012138-1-68-5': 'One essential point in this analysis is that matrix elements of operators which couple between different intrinsic states are characteristically down by a factor of [MATH] compared to matrix elements of operators with the same intrinsic state.', 'hep-ph-0012138-1-68-6': 'Thus, the effect of coupling to an excited intrinsic state and then back to the ground state is typically of order [MATH] and hence can be neglected at NLO in the [MATH] expansion.', 'hep-ph-0012138-1-69-0': 'From the kinematics of the problem, the underlying heavy quark theory, Poincare invariance, and the structure of the Hilbert space, we showed that the effective Hamiltonian up to order [MATH] can be written in the form of Eq. ([REF]).', 'hep-ph-0012138-1-69-1': 'At order [MATH] one sees that the excitation spectrum is fixed entirely by a single parameter [MATH], and at order [MATH] only one additional parameter - [MATH] - enters.', 'hep-ph-0012138-1-69-2': 'This is encouraging from a phenomenological perspective since we do not have a large number of coefficients to work with at NLO.', 'hep-ph-0012138-1-70-0': 'We also derived the certain effective operators at NLO.', 'hep-ph-0012138-1-70-1': 'It turns out that the operators discussed here are precisely those needed to compute interesting electroweak transitions of isoscalar heavy baryons at NLO.', 'hep-ph-0012138-1-70-2': 'The key point in the derivation is that all of these operators can be written in terms of the collective operators (up to possible NNLO corrections).', 'hep-ph-0012138-1-70-3': 'From this it was easy to show that the effect of the admixture of components only comes in at NNLO so that at NLO one can simply replace the QCD level collective operator by an effective operator of the same structure.', 'hep-ph-0012138-1-70-4': 'One operator, [MATH], which is effectively the dipole moment of the light quark contribution to the baryon density, was somewhat more subtle.', 'hep-ph-0012138-1-70-5': 'Although, as an operator it cannot be written directly in terms of the collective variables, we showed that its matrix elements between low-lying states are identical to those of the collective operator [MATH].', 'hep-ph-0012138-1-71-0': 'The effective theory derived here is suitable for the study of isoscalar heavy baryons at NLO in our [MATH] expansion.', 'hep-ph-0012138-1-71-1': 'In Ref. [CITATION] we will use the effective Hamiltonian of Sec. [REF] to study the spectroscopy and electroweak decays of [MATH] and [MATH] baryons and their excited states.', 'hep-ph-0012138-1-71-2': 'In order to calculate the semi-leptonic matrix elements we will use relations between form-factors analogous to those obtained in HQET [CITATION].', 'hep-ph-0012138-1-71-3': 'These matrix elements can then be determined using the effective operators discussed in Sec. [REF].', 'hep-ph-0012138-1-71-4': 'We will also calculate the radiative decay rates of the first excited states of [MATH] baryons.', 'hep-ph-0012138-1-72-0': 'This work is supported by the U.S. Department of Energy grant DE-FG02-93ER-40762.', 'hep-ph-0012138-1-73-0': 'hb0 C.K. Chow and T.D. Cohen, Phys.', 'hep-ph-0012138-1-73-1': 'A to be published.', 'hep-ph-0012138-1-73-2': 'hb3 A. Baccouche, C.K. Chow, T.D. Cohen and B. Gelman, "Excited Heavy Baryons and Their Symmetries III: Phenomenology", DOE/ER/40762-215, UM PP01-029, in preparation.', 'hep-ph-0012138-1-73-3': 'bst1 Z. Guralnik, M. Luke and A.V. Manohar, Nucl.', 'hep-ph-0012138-1-73-4': 'bst2 E. Jenkins, A.V. Manohar and M.B. Wise, Nucl.', 'hep-ph-0012138-1-73-5': 'bst4 C.K. Chow and M.B. Wise, Phys.', 'hep-ph-0012138-1-73-6': 'bst5 M. Rho, D.O. Riska and N.N. Scoccola, Z. Phys.', 'hep-ph-0012138-1-73-7': 'E4 47 (1995); bst7 J. Schechter, A. Subbaraman, S. Vaidya and H. Weigel, Nucl.', 'hep-ph-0012138-1-73-8': 'Glozman and D.O. Riska, Nucl.', 'hep-ph-0012138-1-73-9': 'SF1 J.L. Gervais and B. Sakita, Phys.', 'hep-ph-0012138-1-73-10': 'SF2 J.L. Gervais and B. Sakita, Phys.', 'hep-ph-0012138-1-73-11': 'SF3 R. Dashen, E. Jenkins and A.V. Manohar, Phys.', 'hep-ph-0012138-1-73-12': 'SF4 C. Carone, H. Georgi, S. Osofsky, Phys.', 'hep-ph-0012138-1-73-13': 'SF5 M. Luty and J. March-Russell, Nucl.', 'hep-ph-0012138-1-73-14': "LN1 G. 't Hooft, Nucl.", 'hep-ph-0012138-1-73-15': 'LN2 E. Witten, Nucl.', 'hep-ph-0012138-1-73-16': 'HQ1 N. Isgur and M.B. Wise, Phys.', 'hep-ph-0012138-1-73-17': 'HQ2 N. Isgur and M.B. Wise, Phys.', 'hep-ph-0012138-1-73-18': 'HQ3 N. Isgur and M.B. Wise, Nucl.', 'hep-ph-0012138-1-73-19': 'HQ4 H. Georgi, Nucl.', 'hep-ph-0012138-1-73-20': 'HQ5 T. Mannel, W. Roberts and Z. Ryzak, Nucl.', 'hep-ph-0012138-1-73-21': 'HQ6 F. Hussain, J.G. Korner, M. Kramer and G. Thompson, Z. Phys.', 'hep-ph-0012138-1-73-22': 'mQ1 M.E. Luke, Phys.', 'hep-ph-0012138-1-73-23': 'mQ2 H. Georgi, B. Grinstein and M.B. Wise, Phys.', 'hep-ph-0012138-1-73-24': 'mQ3 A.F. Falk, B. Grinstein and M.E. Luke, Nucl.', 'hep-ph-0012138-1-73-25': 'mQ4 M. Neubert, Phys.'}

{'hep-ph-0012138-2-0-0': 'We develop an effective theory for heavy baryons and their excited states.', 'hep-ph-0012138-2-0-1': 'The approach is based on the contracted [MATH] symmetry recently shown to emerge from QCD for these states in the combined large [MATH] and heavy quark limits.', 'hep-ph-0012138-2-0-2': 'The effective theory is based on perturbations about this limit; a power counting scheme is developed in which the small parameter is [MATH] where [MATH] (with [MATH] being a typical strong interaction scale).', 'hep-ph-0012138-2-0-3': 'We derive the effective Hamiltonian for strong interactions at next-to-leading order.', 'hep-ph-0012138-2-0-4': 'The next-to-leading order effective Hamiltonian depends on only two parameters beyond the known masses of the nucleon and heavy meson.', 'hep-ph-0012138-2-0-5': 'We also show that the effective operators for certain electroweak transitions can be obtained with no unknown parameters at next-to-leading order.', 'hep-ph-0012138-2-1-0': '# Introduction', 'hep-ph-0012138-2-2-0': 'Recently, it was pointed out that QCD when restricted to the space of heavy baryons (i.e. states with baryon number one containing a single heavy - b or c - quark) has a contracted [MATH] [CITATION] or, more generally, [MATH] symmetry which emerges in the combined large [MATH] (the number of colors) and heavy quark limits [CITATION].', 'hep-ph-0012138-2-2-1': 'Of course, in the real world neither [MATH] nor [MATH] (the generic heavy quark mass) is infinite.', 'hep-ph-0012138-2-2-2': 'Nevertheless if they are large enough, QCD will possess an approximate symmetry.', 'hep-ph-0012138-2-2-3': 'Approximate symmetries can provide significant phenomenological information about a system and have had a long and distinguished history in strong interaction physics.', 'hep-ph-0012138-2-2-4': 'Effective field theory (EFT) is a powerful tool that can be used to extract this information.', 'hep-ph-0012138-2-2-5': 'In this article we develop an effective theory to describe heavy baryons based on the approximate contracted [MATH] symmetry.', 'hep-ph-0012138-2-2-6': 'This is important since this allows one to make testable predictions about the phenomenology of heavy baryons.', 'hep-ph-0012138-2-2-7': 'This paper serves the role of bridging the formal developments of Ref. [CITATION] with phenomenological descriptions of heavy baryons.', 'hep-ph-0012138-2-2-8': 'Our ultimate goal is to study the spectroscopy and electroweak decays of heavy baryon states which will be done in Ref. [CITATION].', 'hep-ph-0012138-2-3-0': 'Before discussing how to implement the EFT approach in the context of heavy baryons it is useful to recall how the EFT program works.', 'hep-ph-0012138-2-3-1': 'Effective field theory is useful when there is a separation of scales between the low energy phenomena and higher energy physics.', 'hep-ph-0012138-2-3-2': 'The short-distance physics can greatly influence the long-distance physics.', 'hep-ph-0012138-2-3-3': 'However, when working at low momentum one cannot resolve the details of what is happening at short distances.', 'hep-ph-0012138-2-3-4': 'The idea of EFT is to describe the low-momentum phenomena in a manner which is insensitive to the details of short-distance physics.', 'hep-ph-0012138-2-3-5': 'Formally, one would begin with the full theory and integrate out all of the short-distance degrees of freedom in a functional integral.', 'hep-ph-0012138-2-3-6': 'This yields an effective action to be used in a functional integral over the low momentum (long-distance) degrees of freedom.', 'hep-ph-0012138-2-3-7': 'Clearly, the effective action so obtained cannot be expressed as the integral over a local Lagrangian.', 'hep-ph-0012138-2-3-8': 'However, all of the nonlocalities occur at the scale of the short-distance degrees of freedom which have been integrated.', 'hep-ph-0012138-2-3-9': 'Thus, to good approximation the effective action can be written in terms of a local Lagrangian.', 'hep-ph-0012138-2-3-10': 'The effects of the short-distance physics are now contained in the coefficients in the effective Lagrangian.', 'hep-ph-0012138-2-4-0': 'Clearly such an approach is, at best, approximate.', 'hep-ph-0012138-2-4-1': 'Fortunately, if the scale separation between the long-distance scales and the short-distance scales is large the approximation can be very good.', 'hep-ph-0012138-2-4-2': 'Formally, one can develop a systematic power-counting expansion where the parameter is the ratio of the typical long-distance scale to the typical short-distance scale.', 'hep-ph-0012138-2-4-3': 'As a practical matter one must truncate the effective Lagrangian at some order.', 'hep-ph-0012138-2-4-4': 'The power counting, however, ensures that the effects of the neglected terms are formally down from the leading term by some power in the ratio of the scales.', 'hep-ph-0012138-2-4-5': 'This, in turn, ensures that the principal effects of the short-distance physics are contained in a few coefficients in the effective Lagrangian.', 'hep-ph-0012138-2-4-6': 'In this manner an insensitivity to the details of the short-distance physics is achieved.', 'hep-ph-0012138-2-4-7': 'The low momentum phenomenology depends on a few coefficients summarizing the effects of the short-distance physics; the manner by which the short-distance physics produces these coefficients is irrelevant.', 'hep-ph-0012138-2-5-0': 'In principle, all of the coefficients in the effective Lagrangian can be calculated from the underlying physics.', 'hep-ph-0012138-2-5-1': 'Often, however, the underlying physics is either unknown or is calculationally intractable.', 'hep-ph-0012138-2-5-2': 'EFT, nevertheless provides a conceptual framework for making predictions about the system.', 'hep-ph-0012138-2-5-3': 'The key insight is that one can treat the parameters in the effective Lagrangian as phenomenological inputs fit from experiment.', 'hep-ph-0012138-2-5-4': 'Thus, providing there are more pieces of independent data than there are free parameters at a given order, one can make real, albeit approximate predictions.', 'hep-ph-0012138-2-5-5': 'Chiral perturbation theory is a familiar paradigm for the EFT program.', 'hep-ph-0012138-2-6-0': 'Let us now turn to the problem of heavy baryons.', 'hep-ph-0012138-2-6-1': 'For simplicity we restrict our attention here to the problem of isoscalar heavy baryons - the [MATH] and [MATH] (generically [MATH]) and their excited states.', 'hep-ph-0012138-2-6-2': 'The generalization to include non-isoscalars such as the [MATH] or [MATH] raises no new conceptual issues but complicates the analysis.', 'hep-ph-0012138-2-6-3': 'The approach of Ref. [CITATION] begins with a restriction to a heavy baryon subspace of the full QCD Hilbert space, i.e. we consider only the space of states with baryon number one and one heavy quark.', 'hep-ph-0012138-2-6-4': 'The physical picture that emerges from the model-dependent analysis of Ref. [CITATION] is that low-lying states of the heavy baryon in the combined large [MATH] and heavy quark limits can be represented as harmonic collective motion of the "brown muck" (the technical term for the light degrees of freedom) against the heavy quark.', 'hep-ph-0012138-2-6-5': 'An equivalent picture in the combined limit is that of the model of Refs. [CITATION] in which the heavy baryons are treated as harmonic collective motions of an ordinary baryon against a heavy meson.', 'hep-ph-0012138-2-6-6': 'The symmetry emerges quite simply from this harmonic oscillator picture: the contracted [MATH] algebra is the algebra of all harmonic oscillator creation and annihilation operators and all bilinears made from them.', 'hep-ph-0012138-2-7-0': 'The key to the analysis of Ref. [CITATION] which enables this picture to emerge is a consistent power-counting scheme.', 'hep-ph-0012138-2-7-1': 'Formally, since we are considering the combined large [MATH] and heavy quark limits, it is useful to consider a single parameter which characterizes departures from this limit.', 'hep-ph-0012138-2-7-2': 'Thus, we introduce: [EQUATION] where [MATH] is a typical strong interaction scale and [MATH] is the mass of the heavy quark.', 'hep-ph-0012138-2-7-3': 'The effective expansion parameter turns out not to be [MATH], but [MATH] [CITATION].', 'hep-ph-0012138-2-7-4': 'The low-lying (collective) excitation energies of the heavy baryons are of order [MATH].', 'hep-ph-0012138-2-7-5': 'This should be contrasted with excitation energies for internal excitation of the brown muck and the dissociation energy of the heavy baryon (into an ordinary baryon and a heavy meson) which are each of order [MATH] [CITATION].', 'hep-ph-0012138-2-7-6': 'Thus, provided that [MATH] can be considered to be small, a scale separation exists between the low-lying collective states of the heavy baryons from all other excitations.', 'hep-ph-0012138-2-7-7': 'This, in turn, implies that EFT methods can be applied to this problem.', 'hep-ph-0012138-2-8-0': 'We note at the outset, however, that in practice [MATH] is not a particularly small number so that the scale separation is not particularly large.', 'hep-ph-0012138-2-8-1': 'This implies that one must work at relatively high order in order to get relatively imprecise results.', 'hep-ph-0012138-2-8-2': 'Thus for example, even working at next-to-leading order one only expects relative accuracy nominally only of order [MATH].', 'hep-ph-0012138-2-8-3': 'Clearly, the approach will be at best semi-quantitative.', 'hep-ph-0012138-2-8-4': 'Moreover, the expansion will not converge for all observables.', 'hep-ph-0012138-2-8-5': 'On phenomenological grounds it is clear that at best the approach will be useful for describing the ground state of the [MATH] and the doublet of the first excited orbital excitation.', 'hep-ph-0012138-2-8-6': 'The second orbitally excited states (which have not been observed in either the [MATH] or [MATH] sectors) is either unbound or very near the threshold and clearly beyond the harmonic limit implicit in the effective field theory.', 'hep-ph-0012138-2-8-7': 'The situation is reminiscent of the contracted [MATH] symmetries seen for light baryons which emerge in the large [MATH] limit [CITATION].', 'hep-ph-0012138-2-8-8': 'In that case the large [MATH] limit predicts an infinite tower of low-lying baryons.', 'hep-ph-0012138-2-8-9': 'However, only the nucleon and delta for [MATH] (or the octet and decuplet for [MATH]) are well described in the real world.', 'hep-ph-0012138-2-9-0': 'The construction of the form of the effective Hamiltonian or Lagrangian for the system is very straightforward.', 'hep-ph-0012138-2-9-1': 'Following the standard rules of effective field theory, once the low-energy degrees of freedom are identified, one may simply write down the most general Lagrangian built from them which is consistent with the underlying symmetries and includes all terms up to a fixed order in the power counting.', 'hep-ph-0012138-2-9-2': 'Similarly, one could add external sources to the theory and impose power counting and thereby determine the from of currents.', 'hep-ph-0012138-2-9-3': 'Of course, in order to compute observables one needs more than the form of the effective Lagrangian and effective operators - one needs values for the coefficients.', 'hep-ph-0012138-2-9-4': 'These coefficients can either be obtained purely phenomenologically or by matching with the underlying theory.', 'hep-ph-0012138-2-10-0': 'In this paper we derive the form of the effective Hamiltonian at next-to-leading order (NLO) directly from the structure of QCD and the power-counting rules .', 'hep-ph-0012138-2-10-1': 'We do this formally by directly making a change of variables to the collective degrees of freedom.', 'hep-ph-0012138-2-10-2': 'This has the benefit of relating the coefficients in the effective theory directly to ground state expectation values of known QCD operators.', 'hep-ph-0012138-2-10-3': 'In this derivation the structure of the Hilbert space plays an essential role.', 'hep-ph-0012138-2-10-4': 'In particular, it is necessary to show that in the combined limit, the Hilbert space can be written as a product space of collective excitations of order [MATH] which may be interpreted as excitations of the brown muck moving coherently against the heavy quark, and non-collective excitation of order [MATH] which correspond to excitations of the brown muck itself.', 'hep-ph-0012138-2-10-5': 'In addition, as we will show, the excitations of the two spaces are independent up to corrections of higher order in [MATH].', 'hep-ph-0012138-2-11-0': 'We also derive results about the effective theory of more phenomenological interest.', 'hep-ph-0012138-2-11-1': 'One of the central achievements of this paper is to show that at NLO the dynamics depends on only two free parameters in the effective Hamiltonian beyond known masses of the nucleon and heavy meson.', 'hep-ph-0012138-2-11-2': 'This is of phenomenological significance in that the small value of the expansion parameter suggests that even for quite crude calculations one should work at least at NLO.', 'hep-ph-0012138-2-11-3': 'Had there been a large number of free parameters at this order it would have suggested that the scheme would be difficult to implement in practice as data on heavy baryons becomes available.', 'hep-ph-0012138-2-11-4': 'Another important result of this paper is the demonstration that certain operators that are not contained in the Hamiltonian but which play a role in describing electroweak properties of heavy baryons are completely determined without the need for phenomenological coefficients (up to corrections of higher order in [MATH]) from the known structure of QCD and the counting rules.', 'hep-ph-0012138-2-11-5': 'This increases the predictive power of the effective theory by eliminating free parameters.', 'hep-ph-0012138-2-12-0': 'As noted above, an expansion in [MATH] is unlikely to be rapidly convergent.', 'hep-ph-0012138-2-12-1': 'Thus it seems sensible to work at relatively high order in the expansion.', 'hep-ph-0012138-2-12-2': 'Here we have stopped at NLO for two reasons: one practical and one theoretical.', 'hep-ph-0012138-2-12-3': 'The practical reason is simply that beyond NLO the number of parameters grows to the point where it is likely to exceed the number of observables which have any reasonable hope of being measured in the foreseeable future.', 'hep-ph-0012138-2-12-4': 'The theoretical reason is that the change of variables technique used to derive the effective theory in terms of known operators in QCD is straightforward only up to NLO.', 'hep-ph-0012138-2-12-5': 'Beyond this order, ambiguities in the definition of the heavy quark mass enter as do effects from the coupling of the low-lowing states described by the effective theory with higher energy excitations.', 'hep-ph-0012138-2-12-6': 'Thus, the treatment becomes more subtle and complex beyond this order.', 'hep-ph-0012138-2-12-7': 'Of course, this does not mean that there is no valid effective theory beyond NLO.', 'hep-ph-0012138-2-12-8': 'However considerable effort must be undertaken in going beyond this order and this effort is simply not justified in view of the practical difficulties mentioned above.', 'hep-ph-0012138-2-13-0': 'This paper is organized as follows.', 'hep-ph-0012138-2-13-1': 'In Sec. [REF] we follow Ref. [CITATION] and briefly review the derivation of the low energy collective variables for heavy baryons from QCD with an emphasis on the power counting rules.', 'hep-ph-0012138-2-13-2': 'In Sec. [REF], we show that the structure of the Hilbert space as a product space emerges from standard large [MATH] analysis.', 'hep-ph-0012138-2-13-3': 'The form of the effective Hamiltonian (up to NLO) is derived from QCD in Sec. [REF].', 'hep-ph-0012138-2-13-4': 'Effective operators for electroweak matrix elements are derived in Sec. [REF].', 'hep-ph-0012138-2-13-5': 'Finally a brief summary is given in Sec. [REF]', 'hep-ph-0012138-2-14-0': '# Dynamical Variables from Counting Rules and QCD', 'hep-ph-0012138-2-15-0': 'In this section we review the counting rules derived from QCD for heavy baryons in the combined large [MATH] and heavy quark limits.', 'hep-ph-0012138-2-15-1': 'We will not reproduce the derivations of Ref. [CITATION] in detail.', 'hep-ph-0012138-2-15-2': 'Rather, we will outline the basic strategy of the derivation along with the principal physical and mathematical assumptions.', 'hep-ph-0012138-2-15-3': 'We also collect results which will be needed later in this paper.', 'hep-ph-0012138-2-16-0': 'The first physical point of significance is to restrict our attention to a limited part of the QCD Hilbert space - the part with baryon number one and heavy quark number one.', 'hep-ph-0012138-2-16-1': 'Furthermore we will ultimately restrict our attention to the lowest-lying states with these quantum numbers.', 'hep-ph-0012138-2-16-2': 'For simplicity, in this section we will only consider a single species of heavy quark at a time as the strong interaction does not couple different heavy flavors.', 'hep-ph-0012138-2-16-3': 'We will generalize to two heavy flavors when we consider electroweak matrix elements in the following sections.', 'hep-ph-0012138-2-17-0': 'QCD is a field theory and as such has an infinite number of degrees of freedom.', 'hep-ph-0012138-2-17-1': 'We can envision making a canonical transformation from the original quark and gluon degrees of freedom to a new set of variables (at the quantum mechanical level it becomes a unitary transformation).', 'hep-ph-0012138-2-17-2': 'The goal is to find such a transformation which has the property that a certain set of the variables (which we denote the collective variables), generate the low-lying collective excited states when acting (repeatedly) on the ground state (the [MATH]).', 'hep-ph-0012138-2-17-3': 'Working explicitly with the collective variables one can calculate directly the properties of the low-lying states.', 'hep-ph-0012138-2-17-4': 'Because we are working in the context of canonical transformations it is far more natural to work with the Hamiltonian formulation of the underlying quantum field theory (QCD) rather than the Lagrangian formulation.', 'hep-ph-0012138-2-18-0': 'Unfortunately, there is no straightforward way to generate such a canonical transformation to collective variables exactly.', 'hep-ph-0012138-2-18-1': 'However, it is easy to find a set of variables which behaves this way approximately.', 'hep-ph-0012138-2-18-2': 'That is, using the power-counting scheme of Eq. ([REF]) one can find collective variables which when acting on the ground state (repeatedly) generate the low-lying collective states plus a small component of higher excited states - the amount of higher excited state admixed is suppressed by powers of [MATH].', 'hep-ph-0012138-2-19-0': 'One canonically conjugate pair of collective variables is the total momentum of the system [MATH], the generator of translations (which is well defined in QCD) and its conjugate variable [MATH], the generator of momentum boosts.', 'hep-ph-0012138-2-19-1': 'This pair of collective variables, however, does not induce the internal excitations of interest here.', 'hep-ph-0012138-2-20-0': 'One might also wish to introduce the momentum of the heavy quark as a collective variable.', 'hep-ph-0012138-2-20-1': 'The expression for [MATH] appropriate in the heavy quark limit is [EQUATION] where [MATH] is the heavy quark field, and [MATH] is the three-dimensional covariant derivative.', 'hep-ph-0012138-2-20-2': 'In the heavy quark limit it is apparent that the variable conjugate to [MATH] is [MATH] defined by [EQUATION]', 'hep-ph-0012138-2-20-3': 'The definitions for [MATH] and [MATH] in Eqs. ([REF]), ([REF]) are clearly valid in the heavy quark limit for the subspace of states with a heavy quark number of unity; in this limit the heavy quark behaves as in nonrelativistic quantum mechanics.', 'hep-ph-0012138-2-20-4': 'In the limit [MATH] and [MATH] correspond to the generators of translations and momentum boosts for the heavy quark.', 'hep-ph-0012138-2-20-5': 'Away from this limit the concept of boosting or translating the heavy quark separately from the full system is not strictly well defined and the [MATH] and [MATH] defined above need not be canonically conjugate.', 'hep-ph-0012138-2-20-6': 'However, even away from the formal limit there will exist some canonically conjugate pair of operators which smoothly goes to the variables defined in Eqs. ([REF]), ([REF]) as the limit is approached.', 'hep-ph-0012138-2-20-7': 'Moreover, by standard heavy quark counting considerations this canonically conjugate pair of operators will differ from those defined in Eqs. ([REF]), ([REF]) by an amount of order [MATH].', 'hep-ph-0012138-2-20-8': 'Here we are interested in the theory up to next-to-leading order, i.e. up to relative order [MATH].', 'hep-ph-0012138-2-20-9': 'Accordingly, to the order at which we work we can take Eqs. ([REF]), ([REF]) as unambiguous definitions.', 'hep-ph-0012138-2-21-0': 'Unfortunately, the conjugate pairs [MATH] and [MATH] are not independent.', 'hep-ph-0012138-2-21-1': 'Clearly [MATH] since [MATH] is a generator of translations and translating the system necessarily translates the position of the heavy quark.', 'hep-ph-0012138-2-21-2': 'Thus, we can not directly use [MATH] as our collective variables to generate low-lying states.', 'hep-ph-0012138-2-21-3': 'However, from these two pairs we can generate two independent conjugate pairs.', 'hep-ph-0012138-2-21-4': 'In particular we can construct a linear combination of [MATH] and [MATH], which we denote [MATH] and a linear combination of [MATH] and [MATH], which we denote [MATH], such that [MATH] and [MATH] are conjugate to each other and commute with [MATH] and [MATH] up to corrections at next-to-next-to-leading order (NNLO).', 'hep-ph-0012138-2-21-5': 'Intuitively [MATH] can be thought of as the generator of translations of the brown muck relative to the heavy quark and [MATH] as its conjugate.', 'hep-ph-0012138-2-21-6': 'As we will see below, the variables [MATH] do act as approximate collective variables in the sense outlined above.', 'hep-ph-0012138-2-22-0': 'The explicit construction of [MATH] and [MATH] is detailed in Ref. [CITATION].', 'hep-ph-0012138-2-22-1': 'The key inputs to this construction are Poincare invariance, heavy quark effective theory, and a decomposition of the QCD Hamiltonian organized via power-counting.', 'hep-ph-0012138-2-22-2': 'The decomposition of the QCD Hamiltonian is most easily accomplished by thinking of the heavy baryon as a bound state of a heavy meson and a nucleon.', 'hep-ph-0012138-2-22-3': 'Standard large [MATH] QCD analysis allows us to identify the interaction energy as scaling as [MATH], while the nucleon mass [MATH] scales as [MATH] [CITATION].', 'hep-ph-0012138-2-22-4': 'Standard heavy quark analysis implies that the interaction energy is of order [MATH] while the mass of the heavy meson, [MATH], is given by [EQUATION]', 'hep-ph-0012138-2-22-5': 'Thus, when constrained to the heavy baryon part of the Hilbert space, the QCD Hamiltonian may be decomposed as: [EQUATION] where [MATH] is of order [MATH].', 'hep-ph-0012138-2-22-6': 'Poincare invariance implies that [EQUATION] while the standard heavy quark treatment implies that [EQUATION]', 'hep-ph-0012138-2-22-7': 'Using Eqs. ([REF]), ([REF]), ([REF]) along with Eq. ([REF]) and the fact that [MATH], the methods of Ref. [CITATION] allows one to deduce that [EQUATION]', 'hep-ph-0012138-2-22-8': 'The expressions for [MATH] and [MATH] differ from the equivalent expressions in Ref. [CITATION] in that we have exploited Eq. ([REF]) to express quantities in terms of [MATH] which is directly physically accessible rather than [MATH] which is not.', 'hep-ph-0012138-2-22-9': 'Throughout the remainder of this paper we will often eliminate [MATH] in favor of [MATH].', 'hep-ph-0012138-2-23-0': 'Thus, we have constructed two independent pairs of conjugate variables: [MATH] and [MATH].', 'hep-ph-0012138-2-23-1': 'These two pairs naturally arise in the Hamiltonian.', 'hep-ph-0012138-2-23-2': 'However, as we will see in Sec. [REF], in describing electroweak operators these pairs are not so natural.', 'hep-ph-0012138-2-23-3': 'Instead it is natural to use the positions and momenta of the heavy quarks [MATH] and the corresponding independent set [MATH] which correspond intuitively to the position and momenta of the light degrees of freedom.', 'hep-ph-0012138-2-23-4': 'Formally they are defined as follows: [EQUATION]', 'hep-ph-0012138-2-23-5': 'In order to exploit the collective variables [MATH] and [MATH] to learn about the low-lying spectrum, it is useful to evaluate multiple commutators of [MATH] and [MATH] with [MATH].', 'hep-ph-0012138-2-23-6': 'Following Ref. [CITATION] one can again use Eqs. ([REF]), ([REF]), ([REF]), ([REF]) and the known [MATH] scaling of [MATH] and [MATH], i.e. [MATH], to obtain [EQUATION] where the reduced mass, [MATH], is given by [EQUATION]', 'hep-ph-0012138-2-23-7': 'More generally, it can be seen by the same methods that multiple commutators of [MATH] with [MATH] are suppressed by multiple powers of [MATH].', 'hep-ph-0012138-2-23-8': 'In particular, if one takes [MATH] commutators of components of [MATH] with [MATH] with [MATH] even, then the [MATH] commutator scales as [MATH].', 'hep-ph-0012138-2-23-9': 'For example: [EQUATION]', 'hep-ph-0012138-2-23-10': 'In contrast, standard large [MATH] counting implies that a displacement of the light degrees of freedom relative to the heavy degrees of freedom by a distance of order [MATH] shifts the Hamiltonian by an amount of order [MATH]: [EQUATION] where [MATH] is a c-number of order [MATH].', 'hep-ph-0012138-2-23-11': 'Equation ([REF]) in turn implies that multiple commutators of [MATH] with [MATH] are generically of order unity: [EQUATION]', 'hep-ph-0012138-2-23-12': 'The power-counting rules from QCD are essentially summarized in Eqs. ([REF]) and ([REF]).', 'hep-ph-0012138-2-23-13': 'By themselves, however, they are not sufficient to develop a fully systematic power-counting scheme for observables without additional phenomenological input.', 'hep-ph-0012138-2-23-14': 'In particular, there are two possible scenarios in which the power-counting may be realized selfconsistently.', 'hep-ph-0012138-2-23-15': 'Intuitively, the [MATH] or symmetric realization corresponds to the case where the effective potential between the heavy quark and the brown muck has a minimum at [MATH], while the [MATH] or symmetry broken realization corresponds to the situation where the effective potential has a minimum at [MATH].', 'hep-ph-0012138-2-23-16': 'The [MATH] realization is labeled symmetry broken since it breaks rotational symmetry in the [MATH] limit.', 'hep-ph-0012138-2-23-17': 'As shown in Ref. [CITATION] the [MATH] (symmetric) realization has the generic property that [EQUATION] in the sense that typical matrix elements between low-lying states of polynomial operators which contain [MATH] and [MATH] scale as [MATH].', 'hep-ph-0012138-2-23-18': 'In contrast, in the [MATH] realization, [EQUATION]', 'hep-ph-0012138-2-23-19': 'In this paper we will assume that the dynamics are such that the system is in the [MATH] (symmetric) realization.', 'hep-ph-0012138-2-24-0': '# Structure of the Hilbert Space', 'hep-ph-0012138-2-25-0': 'In this section we will show that in the combined large [MATH] and heavy quark limits, the Hilbert space can be written as a product space composed of collective and intrinsic excitations, with the collective excitations having energies of order [MATH] and intrinsic excitations are of order unity.', 'hep-ph-0012138-2-25-1': 'The physical picture of these two types of excitations is quite clear - the collective excitations describe coherent motion of the brown muck against the heavy quark while the intrinsic excitations are excitations of the brown muck itself.', 'hep-ph-0012138-2-26-0': "The basic strategy in demonstrating that the Hilbert space is of this form is to exploit the fact that our expansion parameter is valid in the pure large [MATH] limit where Witten's Hartree picture is well established [CITATION].", 'hep-ph-0012138-2-26-1': 'The key point is that while nature is far closer to the heavy quark limit than the large [MATH] limit, our expansion is formulated in the combined large [MATH] and heavy quark limits for [MATH] arbitrary.', 'hep-ph-0012138-2-26-2': 'Thus it applies whether one takes the heavy quark limit first, the large [MATH] limit first or takes the limits simultaneously.', 'hep-ph-0012138-2-26-3': 'We will deduce the structure of the Hilbert space by taking the large [MATH] limit first and then argue that the result goes over smoothly to the combined limit.', 'hep-ph-0012138-2-26-4': 'Ultimately, the reason that we can legitimately take the limits in any order stems from the fact that the dynamics of the system are determined by the interaction Hamiltonian, [MATH].', 'hep-ph-0012138-2-26-5': 'Note that [MATH] is non-singular in both the large [MATH] and heavy quark limits, and therefore we can smoothly take either limit or the combined limit first.', 'hep-ph-0012138-2-27-0': 'Now consider taking the large [MATH] limit prior to taking the heavy quark limit.', 'hep-ph-0012138-2-27-1': "In this case we can use Witten's Hartree picture of baryons.", 'hep-ph-0012138-2-27-2': 'Each quark propagates in the mean field of the remaining quarks.', 'hep-ph-0012138-2-27-3': 'Low energy excitations are described by the excitations of a single quark in the background of the remaining quarks.', 'hep-ph-0012138-2-27-4': 'In the present case we can divide these excitations into two classes - excitations of the heavy quark and excitations of one of the light quarks.', 'hep-ph-0012138-2-27-5': 'Of course, for the light quark excitation it is necessary to anti-symmetrize the wave function as there are many light quarks.', 'hep-ph-0012138-2-27-6': 'In contrast there is only one heavy quark.', 'hep-ph-0012138-2-27-7': 'In the large [MATH] limit these two classes of excitations are orthogonal to each other so that the wave function may be written as a direct product: [EQUATION] where [MATH] ) represent the heavy (light) quark part of the Hartree wave function.', 'hep-ph-0012138-2-28-0': 'The excitation energies for both types of excitations are of order [MATH].', 'hep-ph-0012138-2-28-1': 'However, the excitation energies of the heavy quark are of order [MATH].', 'hep-ph-0012138-2-28-2': 'This is straightforward to see; the heavy quark sits in a potential well of depth and width independent of [MATH].', 'hep-ph-0012138-2-28-3': 'As one takes the large [MATH] limit (after having taken the large [MATH] limit), the heavy quark wave function becomes localized at the bottom of the potential well and thus acts like an harmonic oscillator.', 'hep-ph-0012138-2-28-4': 'One sees that the characteristic excitation energies of the heavy and light types of motion scale with [MATH] according to [EQUATION]', 'hep-ph-0012138-2-28-5': 'As noted above, in this picture the excitations of the heavy quark and the light quarks are independent.', 'hep-ph-0012138-2-28-6': 'The total excitation energy is simply the sum of the heavy excitation energy and the light excitation energy.', 'hep-ph-0012138-2-29-0': 'If we now consider the combined limit with the large [MATH] and heavy quark limits taken simultaneously the picture changes.', 'hep-ph-0012138-2-29-1': 'The Hartree approximation is no longer valid.', 'hep-ph-0012138-2-29-2': 'However, the general counting rules remain - they simply have to be reinterpreted.', 'hep-ph-0012138-2-29-3': 'Physically what happens in the combined limit is that instead of the heavy quark oscillating in the static field of the brown muck, as in the Hartree picture, they oscillate against each other.', 'hep-ph-0012138-2-29-4': 'However, the structure of the Hilbert space is not altered in its essence.', 'hep-ph-0012138-2-29-5': 'The space can still be broken up into a product space of independent classes of excitations as in Eq. ([REF]), but now the two classes of excitations are not the heavy and light degrees of freedom as in the Hartree approximation.', 'hep-ph-0012138-2-29-6': 'Rather, they are the collective excitations (between the brown muck and the heavy quark) and the intrinsic excitations (of the brown muck): [EQUATION] where C ( I ) represent the collective (intrinsic) part.', 'hep-ph-0012138-2-29-7': 'Similarly, the excitation energies behave as in Eq. ([REF]) but with the more general description of collective and intrinsic classes of excitations replacing the heavy and light classes of the Hartree approximation: [EQUATION]', 'hep-ph-0012138-2-29-8': 'We see that the structure of the Hilbert space is quite simple in the [MATH] limit.', 'hep-ph-0012138-2-29-9': 'In order to understand the structure of the effective theory, it is necessary to understand how corrections to this leading behavior emerge.', 'hep-ph-0012138-2-29-10': 'In particular, we will need to quantify the mixing between the collective and intrinsic excitations due to sub-leading terms in [MATH].', 'hep-ph-0012138-2-29-11': 'Again, our strategy is to pass to the case where the large [MATH] limit is taken first and we have the Hartree approximation as a well-defined tool.', 'hep-ph-0012138-2-29-12': 'Then use the smoothness of the two limits to pull back to the case of [MATH] keeping [MATH] arbitrary.', 'hep-ph-0012138-2-30-0': 'It is useful to consider the way [MATH] corrections emerge on top of the Hartree picture as these corrections will ultimately become corrections in our [MATH] counting.', 'hep-ph-0012138-2-30-1': 'We begin by assuming that at finite, but large [MATH], a Hartree picture is a good first approximation.', 'hep-ph-0012138-2-30-2': 'We also make the standard large [MATH] assumption [CITATION] that corrections to the energy and to energy splittings are generically of relative order [MATH].', 'hep-ph-0012138-2-30-3': 'Moreover, these leading relative corrections are independent of [MATH] as the heavy quark only contributes 1 out of [MATH] quarks.', 'hep-ph-0012138-2-30-4': 'Thus, these relative corrections due to effects beyond the Hartree approximation are necessarily of order [MATH] and do not contribute at NLO (which is relative order [MATH]).', 'hep-ph-0012138-2-30-5': 'It is straightforward to see that [MATH] corrections to the energies imply that Hamiltonian matrix elements between two different states with different intrinsic states are generically of order [MATH].', 'hep-ph-0012138-2-30-6': 'This can be shown in two ways.', 'hep-ph-0012138-2-30-7': 'First, off-diagonal matrix elements between different intrinsic states of order [MATH] yield [MATH] corrections to the energies and we know corrections of this order exist.', 'hep-ph-0012138-2-30-8': 'The second way is to construct an explicit wave function using a transition induced by the gluon exchange.', 'hep-ph-0012138-2-30-9': 'The gluon exchange contains two factors of the strong coupling and scales as [MATH]; there is a combinatoric factor of [MATH] as the gluon can connect to any quark; and finally, there is a normalization factor of [MATH] because there are [MATH] distinct terms in the excited state wave function as any of the quarks may be excited.', 'hep-ph-0012138-2-30-10': 'Now we can pass from the large [MATH] limit back to the combined [MATH] limit (with [MATH] fixed) by exploiting smoothness.', 'hep-ph-0012138-2-30-11': 'We obtain the following important result: [EQUATION] where the state [MATH] is a product state [EQUATION]', 'hep-ph-0012138-2-31-0': '# Construction of the Effective Hamiltonian', 'hep-ph-0012138-2-32-0': 'In Sec. [REF] we reviewed the power-counting rules for the collective variables [MATH] and [MATH] in the heavy baryon sector of the QCD Hilbert space.', 'hep-ph-0012138-2-32-1': 'As discussed in Sec. [REF], the general principle of effective field theory implies that the form of the effective theory is completely fixed by the symmetries of the theory and the power-counting rules.', 'hep-ph-0012138-2-32-2': 'Rather than simply invoking this general strategy, it is instructive to derive the effective field theory at next-to-leading order directly from QCD through the change in variables discussed in Sec. [REF].', 'hep-ph-0012138-2-32-3': 'We do this because it is useful to verify that the proposed scaling rules are, indeed, selfconsistent; the derivation gives significant insights into the underlying structure of the theory; and finally, it is satisfying to verify explicitly the validity of the general EFT philosophy in this case.', 'hep-ph-0012138-2-33-0': 'In the present context, where we have been implementing canonical transformations to collective variables, it is simpler to work with an effective Hamiltonian rather than an effective Lagrangian.', 'hep-ph-0012138-2-33-1': 'From Eq. ([REF]) and the power-counting rules of Eqs. ([REF]), ([REF]), ([REF]) and ([REF]) it is straightforward to deduce the form of the QCD Hamiltonian in the heavy baryon sector written in terms of the collective variables up to order [MATH] (NLO).', 'hep-ph-0012138-2-33-2': 'It is given by [EQUATION] where [MATH], [MATH], and [MATH].', 'hep-ph-0012138-2-33-3': 'From rotational invariance, the [MATH] are completely symmetric Cartesian tensor operators of rank [MATH].', 'hep-ph-0012138-2-33-4': 'Note that while terms up to [MATH] contribute at this order, only terms up to [MATH] contribute.', 'hep-ph-0012138-2-33-5': 'The difference between these two is due to the essentially nonrelativistic nature of the kinematics.', 'hep-ph-0012138-2-33-6': 'Formally, this difference is implicit in the different scaling relations for [MATH] and [MATH] in Eqs. ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-ph-0012138-2-34-0': 'Since the form of Eq. ([REF]) was deduced from commutation relations, one cannot immediately conclude that the [MATH] operators are c-numbers.', 'hep-ph-0012138-2-34-1': 'Rather, at this stage in the derivation we can only deduce that they are operators which commute with all of the collective variables: [EQUATION]', 'hep-ph-0012138-2-34-2': 'Thus, the [MATH] tensors are, in general, operators which depend only on the non-collective variables, i.e. the [MATH] are purely intrinsic operators.', 'hep-ph-0012138-2-34-3': 'Note that to the extent that the coefficients [MATH] in the Hamiltonian of Eq. ([REF]) are operators as opposed to c-numbers, the Hamiltonian is not strictly an effective Hamiltonian.', 'hep-ph-0012138-2-34-4': 'Rather it is the full QCD Hamiltonian written in terms of our collective degrees of freedom.', 'hep-ph-0012138-2-34-5': 'The non-collective degrees of freedom have not been integrated out; rather their effects are contained in the non-collective operators [MATH].', 'hep-ph-0012138-2-35-0': 'Non-collective operators might be expected to influence the dynamics of the collective degrees of freedom even though they all commute with the collective degrees of freedom.', 'hep-ph-0012138-2-35-1': 'For example, consider the ground state matrix element of [MATH].', 'hep-ph-0012138-2-35-2': 'This can be expressed in terms of off-diagonal matrix elements: [EQUATION] where [MATH] is the ground state of the heavy baryon system and the intermediate states are grouped into purely collective excitations [MATH] (with excitation energies of order [MATH] which excite only the collective degrees of freedom leaving the brown muck unexcited, and non-collective states [MATH] (with excitation energies of order [MATH] or greater) which involve an intrinsic excitation of the brown muck.', 'hep-ph-0012138-2-35-3': 'If the operators in the Hamiltonian had been purely collective as in a true effective Hamiltonian (which by construction has all non-collective degrees of freedom integrated out), then the sum would not contain the non-collective states [MATH].', 'hep-ph-0012138-2-35-4': 'However, the non-collective operators [MATH] induce transitions to the non-collective states.', 'hep-ph-0012138-2-35-5': 'By acting twice - once to go into the non-collective space, and once to come back the collective space - the non-collective operators can affect collective space matrix elements of operators contained in the Hamiltonian.', 'hep-ph-0012138-2-36-0': 'However, from Eq. ([REF]) we know that terms which connect different intrinsic states are suppressed by order [MATH].', 'hep-ph-0012138-2-36-1': 'We can use this to deduce that the contribution of non-collective intermediate states to matrix elements of composite operators such as in Eq. ([REF]) will always be suppressed by order [MATH] relative to the contributions of the collective intermediated states and thus only contribute at NNLO.', 'hep-ph-0012138-2-36-2': 'This in turn justifies their omission at NLO.', 'hep-ph-0012138-2-36-3': 'Let us see how this works for the decomposition in Eq. ([REF]).', 'hep-ph-0012138-2-36-4': 'From the scaling rules of Eq. ([REF]) it is straightforward to see that the contribution from the collective intermediate states goes as [MATH].', 'hep-ph-0012138-2-36-5': 'A typical term in the sum over states outside the purely collective space is [EQUATION] where [MATH] and [MATH] represent the ground states of the collective and intrinsic subspaces, respectively, while [MATH] and [MATH] represent excited states, and the notation [MATH] represents a product state as in Eq. ([REF]) .', 'hep-ph-0012138-2-36-6': 'The equality in Eq. ([REF]) follows since [MATH] only acts on the collective space while [MATH] only acts on the intrinsic space and the scaling behavior follows from Eqs. ([REF]) and ([REF]) so that the collective part of the expectation value scales as [MATH] while the intrinsic part scales as [MATH].', 'hep-ph-0012138-2-36-7': 'Thus, we see that the contribution of non-collective states is down by a factor of [MATH] compared to the contribution of the collective intermediate states.', 'hep-ph-0012138-2-37-0': 'The [MATH] suppression of the effects of non-collective intermediate states relative to the collective ones is generic.', 'hep-ph-0012138-2-37-1': 'Although it was explicitly demonstrated for the case of the composite operator in Eq. ([REF]), it should be apparent that such a suppression occurs for all products of operators contained in the Hamiltonian.', 'hep-ph-0012138-2-37-2': 'When calculating Hamiltonian matrix elements, the effects of the operators [MATH] in Eq. ([REF]) inducing mixing with the non-collective states will always contribute at relative order [MATH] (i.e. at NNLO) or less.', 'hep-ph-0012138-2-37-3': 'Thus, working at NLO one can simply replace the [MATH] operators by their expectation values in the ground state (or more generally in the ground state band - the set of collective states built on the ground state of the intrinsic degrees of freedom).', 'hep-ph-0012138-2-37-4': 'Therefore, the operators [MATH] can be treated as c-numbers in the effective theory.', 'hep-ph-0012138-2-38-0': 'When the coefficients [MATH] are treated as operators, rotational invariance does not constrain the allowable forms in the Hamiltonian beyond what is given in Eq. ([REF]).', 'hep-ph-0012138-2-38-1': 'For example, the term [MATH] is allowable despite the fact that [MATH] is an [MATH] operator; [MATH] could also be [MATH] operator and couple to [MATH] leading to an [MATH] operator in the Hamiltonian.', 'hep-ph-0012138-2-38-2': 'Once the [MATH] operators are restricted to being c-numbers in the effective theory, rotational invariance greatly restricts the allowable forms: c-numbers do not transform under spatial rotations.', 'hep-ph-0012138-2-38-3': 'Accordingly, the only surviving [MATH] coefficients are those which multiply rotational scalars.', 'hep-ph-0012138-2-38-4': 'Thus, for example, the [MATH] terms must vanish.', 'hep-ph-0012138-2-38-5': 'Treating the [MATH] coefficients as c-numbers and imposing rotational symmetry yields the following form of the effective Hamiltonian up to [MATH]: [EQUATION] where [MATH] refers to the center of mass motion of the entire system while [MATH] refers to the piece of the Hamiltonian whose leading contribution is of order [MATH].', 'hep-ph-0012138-2-38-6': 'The coefficients [MATH], [MATH] and [MATH] are of order [MATH] and may be expressed as ground-state expectation values of [MATH], [MATH] and [MATH], respectively: [EQUATION]', 'hep-ph-0012138-2-38-7': 'A few comments about this derivation are in order here.', 'hep-ph-0012138-2-38-8': 'In essence, we did not have to explicitly integrate out the non-collective higher-lying degrees of freedom.', 'hep-ph-0012138-2-38-9': 'By making a change of variables to [MATH], [MATH], [MATH] and [MATH], we already have identified variables which were decoupled from the intrinsic degrees of freedom to the order at which we are working.', 'hep-ph-0012138-2-38-10': 'Imagine one explicitly made this change of variables prior to formulating the problem in terms of a functional integral.', 'hep-ph-0012138-2-38-11': 'At low order the action could be written as the sum of a collective action plus an intrinsic action without coupling between them, and the only effect of integrating out the intrinsic degrees of freedom would be to supply an overall constant multiplying the remaining functional integral over the collective degrees of freedom.', 'hep-ph-0012138-2-38-12': 'By making this change of variables we make the effect of integrating out the intrinsic degrees of freedom trivial - at least up to NLO.', 'hep-ph-0012138-2-38-13': 'Only if we go beyond this order do effects of the the coupling to the intrinsic degrees of freedom play a role, and the effects of integrating out the intrinsic degrees of freedom become nontrivial.', 'hep-ph-0012138-2-39-0': 'From a phenomenological prospective, it is sufficient to start with Eq. ([REF]).', 'hep-ph-0012138-2-39-1': 'The coefficients [MATH], [MATH] and [MATH] could then be determined approximately from fits to experimental data.', 'hep-ph-0012138-2-39-2': 'Given the fact that the masses of the nucleon and the heavy meson are known, the effective theory at NLO depends on three parameters - [MATH], [MATH] and [MATH].', 'hep-ph-0012138-2-39-3': 'The coefficient [MATH] is determined from the difference in energy between the nucleon-heavy meson threshold and the lowest energy state.', 'hep-ph-0012138-2-39-4': 'However, [MATH] is simply an overall constant and plays no role in the dynamics; the dynamics of the problem depends on two parameters at NLO.', 'hep-ph-0012138-2-39-5': 'It is worth noting that at leading order the dynamics depends only on a single parameter, [MATH].', 'hep-ph-0012138-2-39-6': 'Thus, going from leading order to next-to-leading costs only an additional parameter, [MATH].', 'hep-ph-0012138-2-40-0': '# Effective Operators', 'hep-ph-0012138-2-41-0': 'To make predictions of electroweak current matrix elements between heavy baryon states, one needs more than just the effective Hamiltonian.', 'hep-ph-0012138-2-41-1': 'Effective operators are also required.', 'hep-ph-0012138-2-41-2': 'For general composite operators expressed in terms of the quark and glue degrees of freedom of QCD, the effective operators can be derived in a functional integral form by integrating out the intrinsic degrees of freedom.', 'hep-ph-0012138-2-41-3': 'The most straightforward way to formulate this is as an integral over all QCD variables with [MATH]-functions put in to pick out particular values of the collective variables.', 'hep-ph-0012138-2-41-4': 'In practice, however, the preceding description is rather formal as it is not generally possible to perform this functional integration.', 'hep-ph-0012138-2-41-5': 'At a phenomenological level, one can simply write a form for the operator consistent with symmetries and parameterized by some constants and then fit the constants from experimental data.', 'hep-ph-0012138-2-41-6': 'However, if the operators of interest can be expressed entirely in terms of the collective degrees of freedom at the QCD level, the functional integral approach becomes unnecessary (for the same reason it was unnecessary in the derivation of the collective Hamiltonian).', 'hep-ph-0012138-2-41-7': 'At NLO, one can immediately express the effective operators in terms of the same collective degrees of freedom.', 'hep-ph-0012138-2-42-0': 'To see how this works, consider some operator which at the QCD level can be expressed entirely in terms of the collective operators.', 'hep-ph-0012138-2-42-1': 'We seek to find an analogous operator for use in the effective theory whose matrix elements in the effective theory reproduces the matrix elements of the QCD operator in the QCD states (up to correction at NNLO).', 'hep-ph-0012138-2-42-2': 'Let us start by considering the matrix elements at the QCD level.', 'hep-ph-0012138-2-42-3': 'From the discussion of the structure of the Hilbert space at the QCD level in Sec. [REF], it is apparent that a low-energy (i.e. with excitation energy of order [MATH]) heavy baryon state in a [MATH] expansion at NLO can be written as: [EQUATION] where [MATH] is defined as a product state as in Eq. ([REF]), and [MATH] are constants of order unity.', 'hep-ph-0012138-2-43-0': 'The sum in the second term of Eq. ([REF]) is restricted to contributions with [MATH] due to the product structure of the state.', 'hep-ph-0012138-2-43-1': 'If the sum were not restricted, then there would be contributions of the form [EQUATION] where, [MATH] is simply a single state in the purely collective space.', 'hep-ph-0012138-2-43-2': 'Thus, the contribution to the sum from Eq. ([REF]) is of the form of a single product state.', 'hep-ph-0012138-2-43-3': 'If there are contributions of the form as in Eq. ([REF]) we can replace [MATH] by [MATH] and thereby eliminate contributions with [MATH] from the sum.', 'hep-ph-0012138-2-43-4': 'Doing this has the advantage of removing any possibility of double counting these contributions.', 'hep-ph-0012138-2-44-0': 'Now let us evaluate the matrix element of a purely collective operator, [MATH], i.e. an operator constructed entirely of the collective operators [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-2-44-1': 'From the structure of the state in Eq. ([REF]), the matrix element between two low-energy heavy baryon states (given at NLO) is given by [EQUATION]', 'hep-ph-0012138-2-44-2': 'Thus, up to NNLO corrections the QCD matrix element is simply the matrix element in the collective part of the Hilbert space.', 'hep-ph-0012138-2-44-3': 'On the other hand, the effective Hamiltonian we use at this order with the operators [MATH] replaced by c-numbers is designed to correctly describe the collective part of the Hilbert space at this order.', 'hep-ph-0012138-2-45-0': 'The preceding argument shows, that in the case of purely collective operators the operators in the effective theory are related to the QCD operators in a trivial way up to corrections at NNLO.', 'hep-ph-0012138-2-45-1': 'A purely collective operator at the QCD level is of the form [EQUATION] where the superscript QCD indicates that all of the operators are at the QCD level.', 'hep-ph-0012138-2-45-2': 'The effective operator up to order [MATH] corrections is then obtained by keeping the same operator structure of [MATH], but replacing [MATH], [MATH], [MATH] and [MATH] with the collective operators [MATH], [MATH], [MATH] and [MATH]: [EQUATION].', 'hep-ph-0012138-2-45-3': 'This is essentially a nonrenormalization theorem to this order.', 'hep-ph-0012138-2-45-4': 'The effects of integrating out collective degrees of freedom only renormalize the effective operators at NNLO.', 'hep-ph-0012138-2-46-0': 'In a similar way, all purely non-collective operators at the QCD level, i.e. all operators which commute with all of the collective operators, simply become c-numbers in the effective theory with a value equal to the ground state expectation value of the analogous QCD operator.', 'hep-ph-0012138-2-46-1': 'These general principles can be used to deduce the form of several operators in the effective theory which will prove useful in the calculation of electroweak observables for heavy baryon states.', 'hep-ph-0012138-2-47-0': 'Let us first consider the operator which generates Lorentz boosts, [MATH].', 'hep-ph-0012138-2-47-1': 'The unitary operator for a Lorentz boost is [MATH].', 'hep-ph-0012138-2-47-2': 'The boost operator is useful, for example, in calculating form factors which involve matrix elements of states with different momenta.', 'hep-ph-0012138-2-47-3': 'One can use the boost operator to generate these states from a standard state in its rest frame.', 'hep-ph-0012138-2-47-4': 'From Poincare invariance, the boost operator is given at the field theoretic level by [EQUATION] where the second equality follows from Eq. ([REF]).', 'hep-ph-0012138-2-47-5': 'Thus, up to NLO at the QCD level [MATH] and from the arguments given above, this can immediately be taken over in the effective theory.', 'hep-ph-0012138-2-48-0': 'Among the observables which can be studied using the combined expansion are matrix elements of the electroweak current [MATH], where subscripts [MATH] and [MATH] refer to the heavy quark flavors and [MATH] is the Dirac structure of the left-hand current, [MATH].', 'hep-ph-0012138-2-48-1': 'We will show here that the leading term in the combined heavy quark and large [MATH] expansion of the current [MATH] can be expressed in terms of the collective variables [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0012138-2-48-2': 'Moreover, we will show that the lowest order corrections come at order [MATH] and not at [MATH], so that they contribute only at NNLO.', 'hep-ph-0012138-2-49-0': 'The combined expansion of the current [MATH] can be done analogously to the pure heavy quark expansion in the Heavy Quark Effective Theory (HQET) [CITATION].', 'hep-ph-0012138-2-49-1': 'The idea is to decompose the total heavy quark field into "large" and "small" components or "light" and "heavy" fields with the latter being suppressed by a factor of [MATH] relative to the "large" component.', 'hep-ph-0012138-2-49-2': 'It is then possible to eliminate the "small" component in terms of the "large" component by means of a power series expansion.', 'hep-ph-0012138-2-49-3': 'Near the combined limit considered here, the pure heavy quark expansion breaks down.', 'hep-ph-0012138-2-49-4': 'It has been shown in Refs. [CITATION] that the next-to-leading order terms in the pure heavy quark expansion of the matrix elements of the current [MATH] between the heavy baryon states contain coefficients proportional to [MATH] where [MATH] (with [MATH] being the total mass of a heavy baryon).', 'hep-ph-0012138-2-49-5': 'In the HQET the constant [MATH] is treated as being much smaller than [MATH], so that the NLO operators are indeed suppressed.', 'hep-ph-0012138-2-49-6': 'The situation is different, however, near the combined limit where [MATH] is of order [MATH], so that [MATH] and [MATH].', 'hep-ph-0012138-2-49-7': 'Hence, these corrections cannot be neglected near the combined limit.', 'hep-ph-0012138-2-50-0': 'The appearance of large corrections near the combined limit is a result of the way the total heavy quark field is separated into its "large" and "small" components.', 'hep-ph-0012138-2-50-1': 'The definition of these components is necessarily different from the analogous decomposition in HQET.', 'hep-ph-0012138-2-50-2': 'Near the combined limit, as in HQET, the heavy quark is not far off-shell since its interaction with the brown muck is of order [MATH] while its mass is of order [MATH].', 'hep-ph-0012138-2-50-3': 'However, the space-time dependence of the heavy quark is much different near the combined limit.', 'hep-ph-0012138-2-50-4': 'Near the pure heavy quark limit the brown muck mass is formally much less than the heavy quark mass.', 'hep-ph-0012138-2-50-5': 'As a result, the space-time dependence of the heavy quark is essentially determined by the free particle Lagrangian.', 'hep-ph-0012138-2-50-6': 'In HQET, therefore, it is useful to redefine the heavy quark field by removing the phase factor of the solution of the free Dirac equation, namely [MATH].', 'hep-ph-0012138-2-50-7': 'Near the combined limit the heavy quark remains essentially nonrelativistic.', 'hep-ph-0012138-2-50-8': 'However, the space-time dependence of the heavy quark is much different from that of a free particle.', 'hep-ph-0012138-2-50-9': 'The phase factor now should contain a contribution due to the brown muck mass, which is formally of the same order as the heavy quark mass near the combined limit.', 'hep-ph-0012138-2-50-10': 'This naturally leads to the following redefinition of the heavy quark field: [EQUATION] where [MATH] is the total 4-velocity of the heavy baryon, the operators [MATH] in the rest frame of the heavy baryon project out the upper and lower components of the heavy quark field, and [MATH] here represents a space-time point.', 'hep-ph-0012138-2-50-11': 'As in HQET, the fields [MATH] and [MATH] satisfy conditions [MATH] and [MATH].', 'hep-ph-0012138-2-50-12': 'Fields [MATH] and [MATH] are defined for each value of a heavy baryon 4-velocity [MATH] as can be seen from Eq. ([REF]).', 'hep-ph-0012138-2-50-13': 'The redefinition in Eq. ([REF]) has the effect of removing both the heavy quark mass, [MATH], and the nucleon mass, [MATH], from the dynamics, leaving a dynamical scale of order [MATH], i.e., the scale of the interaction Hamiltonian near the combined limit which is [MATH].', 'hep-ph-0012138-2-51-0': 'The heavy quark part of the QCD Lagrangian density in terms of the fields [MATH] and [MATH] has the form, [EQUATION] where [MATH] is the "transverse part" of the covariant derivative [MATH].', 'hep-ph-0012138-2-51-1': 'The heavy quark Lagrangian in Eq. ([REF]) written in terms of the fields [MATH] and [MATH] differs from its analog in HQET due to the additional term proportional to [MATH].', 'hep-ph-0012138-2-51-2': 'This [MATH] term would naively suggest that the fields [MATH] and [MATH] are both heavy with masses [MATH] and [MATH], respectively, which apparently prevents integrating out the "heavy" field [MATH].', 'hep-ph-0012138-2-52-0': 'This problem is easily resolved if one considers the full Lagrangian density of the system: [EQUATION] where the sum is over all light quarks, and [MATH] is the Yang-Mills Lagrangian density.', 'hep-ph-0012138-2-52-1': 'The total Lagrangian density in Eq. ([REF]) should be re-expressed so as to build the brown muck contribution into the heavy degrees of freedom.', 'hep-ph-0012138-2-52-2': 'This is completely analogous to the Hamiltonian treatment in Sec. [REF].', 'hep-ph-0012138-2-52-3': 'In that case it is done by adding and subtracting to the Hamiltonian a quantity which is an overall constant [MATH] and regrouping terms according to their [MATH] counting scaling.', 'hep-ph-0012138-2-52-4': 'In the Lagrangian formalism this can be accomplished in the following way using a Lorentz covariant operator [MATH]: [EQUATION] where [EQUATION] and [EQUATION].', 'hep-ph-0012138-2-52-5': 'In the rest frame of a heavy baryon operator, [MATH] is equal to the heavy quark density operator [MATH] and its spatial integral equals to one in the Hilbert space of heavy baryons.', 'hep-ph-0012138-2-52-6': 'Thus, in the Hamiltonian the additional term corresponds to removing from the light degrees of freedom an overall constant [MATH].', 'hep-ph-0012138-2-53-0': 'The additional operator proportional to the nucleon mass [MATH] can easily be expressed in terms of the fields [MATH] and [MATH]: [EQUATION].', 'hep-ph-0012138-2-53-1': 'This operator cancels the last term in Eq. ([REF]) so that the Lagrangian density [MATH] has the form: [EQUATION] where the condition [MATH] is used.', 'hep-ph-0012138-2-53-2': 'Thus, the "large" component [MATH] describes a massless field while the "small" component [MATH] has the mass [MATH].', 'hep-ph-0012138-2-53-3': 'Using equations of motion the "heavy" field [MATH] can be expressed in terms of the "light" field [MATH] as follows: [EQUATION]', 'hep-ph-0012138-2-53-4': 'The relation between fields [MATH] and [MATH] in Eq. ([REF]) is identical in form to that in HQET between the "large" and "small" components of the heavy quark field [MATH].', 'hep-ph-0012138-2-54-0': 'We can now expand the field [MATH], the Lagrangian density in Eq. ([REF]), and the current [MATH] in powers of [MATH].', 'hep-ph-0012138-2-54-1': 'This can be done by eliminating fields [MATH] (using Eq. ([REF])) and expanding the numerator in powers of [MATH].', 'hep-ph-0012138-2-54-2': 'Thus, the combined expansion of the heavy quark field including terms up to order [MATH] is, [EQUATION]', 'hep-ph-0012138-2-54-3': 'Using this expansion the Lagrangian density [MATH] including terms up to order [MATH] has the form: [EQUATION] where [MATH] is a strong coupling constant and [MATH] is the gluon field strength tensor.', 'hep-ph-0012138-2-54-4': 'In a similar way we can expand the current [MATH]; again keeping terms up to [MATH] we get, [EQUATION]', 'hep-ph-0012138-2-54-5': 'The expressions in Eqs. ([REF]), ([REF]), ([REF]) are identical in form to the analogous quantities in HQET.', 'hep-ph-0012138-2-54-6': 'However, there is an important difference: these expressions represent the combined expansion in powers of [MATH] and not the [MATH] expansion of HQET.', 'hep-ph-0012138-2-54-7': 'In other words, by defining fields [MATH] and [MATH] appropriately we were able to resum implicitly all the [MATH] corrections in HQET.', 'hep-ph-0012138-2-55-0': 'The correction terms in the combined expansion of the electroweak current, Eq. ([REF]), come at order [MATH] and not at [MATH].', 'hep-ph-0012138-2-55-1': 'It will be shown in Ref. [CITATION] that these corrections contribute to the matrix elements at order [MATH] as well, [MATH], they do not induce any anomalously large constants which violate the combined power counting of these operators.', 'hep-ph-0012138-2-55-2': 'Thus, at NLO in our expansion only the leading order operator, [MATH], contributes to the matrix elements.', 'hep-ph-0012138-2-55-3': 'In addition, it will be shown in Ref. [CITATION] that in the combined limit at NLO there is only a single form factor which parameterizes the matrix elements of the current [MATH] in the [MATH] sector.', 'hep-ph-0012138-2-55-4': 'Accordingly, we can consider one Lorentz component of the current [MATH], namely the [MATH] component.', 'hep-ph-0012138-2-55-5': 'Thus, to completely determine the matrix elements at NLO in the combined expansion, we need to know the form of only one operator, namely [MATH].', 'hep-ph-0012138-2-56-0': 'Let us first consider the flavor conserving operator [MATH].', 'hep-ph-0012138-2-56-1': 'As we are working near the heavy quark limit it is legitimate to expand the state in a fock-space decomposition for the number of heavy quarks and antiquarks.', 'hep-ph-0012138-2-56-2': 'For a heavy baryon state standard heavy quark analysis implies that the leading fock component has one heavy quark; the first sub-leading fock component with two quarks and one antiquark is suppressed by a factor of [MATH].', 'hep-ph-0012138-2-56-3': 'Thus at NLO in [MATH] we can neglect the sub-leading fock component and treat the state as having a single heavy quark.', 'hep-ph-0012138-2-56-4': 'This implies that acting in our heavy baryon Hilbert space, the operator [MATH] is [MATH].', 'hep-ph-0012138-2-56-5': 'Since this is true at the QCD level at NLO it immediately holds in the effective theory at that order.', 'hep-ph-0012138-2-57-0': 'In order to describe the flavor changing current we need to enlarge the Hilbert space of our effective theory to account for electroweak transitions between heavy baryons with heavy quarks of different flavors.', 'hep-ph-0012138-2-57-1': 'The new Hilbert space is the direct product of the Hilbert spaces for baryons with a given flavor of a heavy quark described in Sec. [REF].', 'hep-ph-0012138-2-57-2': 'Any state of the enlarged Hilbert space can be written as [MATH]-dimensional vector, where [MATH] is a number of heavy flavors.', 'hep-ph-0012138-2-57-3': 'Each component of this vector describes a state with a given flavor.', 'hep-ph-0012138-2-57-4': 'The effective Hamiltonian can be readily generalized to include baryons of various heavy quark flavors.', 'hep-ph-0012138-2-57-5': 'It can be written as [MATH] diagonal matrix.', 'hep-ph-0012138-2-57-6': 'For example, for two heavy flavors ([MATH] charm and bottom) the effective Hamiltonian including terms up to NLO in [MATH] has the form: [EQUATION] where the diagonal entries [MATH] and [MATH] are the effective Hamiltonians as in Eq. ([REF]).', 'hep-ph-0012138-2-57-7': 'Analogously, QCD operators acting in the enlarged Hilbert space become [MATH] matrices.', 'hep-ph-0012138-2-57-8': 'The flavor changing weak transitions can be described by off-diagonal matrix elements of an operator [MATH], where the subscripts [MATH] and [MATH] can take integer values from [MATH] to [MATH].', 'hep-ph-0012138-2-57-9': 'This operator can be written in terms of collective operators only.', 'hep-ph-0012138-2-57-10': 'It follows from arguments analogous to the case of the flavor conserving electroweak operator - [MATH] - that the flavor changing operator acting on the enlarged Hilbert space has the form (for [MATH]): [EQUATION]', 'hep-ph-0012138-2-57-11': 'The diagonal elements of the operator in Eq. ([REF]) correspond to flavor conserving operators, while the off-diagonal elements correspond to flavor changing operators.', 'hep-ph-0012138-2-57-12': 'Their action is to instantaneously change the heavy quark flavor at the position where a heavy baryon interacts with the current.', 'hep-ph-0012138-2-57-13': 'Since the operator in Eq. ([REF]) is written in terms of the collective operators only, the corresponding effective operator has an identical form.', 'hep-ph-0012138-2-58-0': 'Finally, let us consider the operator defined by [EQUATION] so that [MATH] represents the contribution to the baryon dipole moment coming from light quarks.', 'hep-ph-0012138-2-58-1': 'In calculating electromagentic transitions between heavy baryons it is useful to divide the electromagnetic current into pieces coming from the heavy and the light quarks and to further divide the light quark contribution into isovector and isoscalar pieces.', 'hep-ph-0012138-2-58-2': 'The isoscalar piece is simply the light-quark contribution to the baryon current.', 'hep-ph-0012138-2-58-3': 'Thus, matrix elements of [MATH] must be calculated to describe electric dipole transitions of heavy baryons.', 'hep-ph-0012138-2-59-0': 'At first sight, it might appear to be impossible to express the operator [MATH] in terms of collective operators.', 'hep-ph-0012138-2-59-1': 'Although [MATH] - the collective position operator for light quarks - is intuitively something like the position of the brown muck and clearly transforms in the same way as [MATH], at the operator level [MATH] is manifestly not identical to [MATH].', 'hep-ph-0012138-2-59-2': 'Recall that by construction [MATH] acts as a generator of boosts for the light degrees of freedom (up to order [MATH] corrections).', 'hep-ph-0012138-2-59-3': 'On the other hand, the operator [MATH] acts only on the quarks and not on the gluons.', 'hep-ph-0012138-2-59-4': 'Thus [MATH] cannot boost the gluons and therefore does not act as a boost for the light degrees of freedom (which include both quarks and gluons).', 'hep-ph-0012138-2-59-5': 'Thus, the operator [MATH] is clearly distinct from [MATH] and there is apparently no simple way to express [MATH] entirely in terms of collective operators.', 'hep-ph-0012138-2-60-0': 'Remarkably, however, one can show that [EQUATION] for all [MATH], [MATH] in the collective subspace defined in Sec. [REF].', 'hep-ph-0012138-2-60-1': 'Thus, when one restricts consideration to the collective subspace which dominates the low energy physics, [MATH] is equivalent to [MATH].', 'hep-ph-0012138-2-61-0': 'To see how this comes about, we start by considering the commutation relations of [MATH] with [MATH], [MATH] and [MATH].', 'hep-ph-0012138-2-61-1': 'To determine the commutator with [MATH] we exploit the fact that [MATH] is a generator of translations for the light degrees of freedom so that [MATH], where [MATH] is a generic light quark field.', 'hep-ph-0012138-2-61-2': 'We can use this to deduce that [EQUATION]', 'hep-ph-0012138-2-61-3': 'The first equality in Eq. ([REF]) follows from the definition of [MATH], the second equality follows because [MATH] translates the light quark degrees of freedom, the third equality follows from a change of variables, and the final equality follows from the definition of [MATH] and the fact that [EQUATION] counts the net number of light quarks, which is [MATH] in a heavy baryon containing one heavy quark.', 'hep-ph-0012138-2-61-4': 'Equation ([REF]) implies that [EQUATION] while the fact that from its definition [MATH] has no heavy quark operators implies that [EQUATION]', 'hep-ph-0012138-2-61-5': 'Thus, the commutation relations of [MATH] with [MATH], [MATH] and [MATH] are identical to the commutation relations of [MATH] with [MATH], [MATH] and [MATH].', 'hep-ph-0012138-2-61-6': 'In Eqs. ([REF]), ([REF]) indices [MATH] indicate Cartesian components of [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-2-62-0': 'Next let us express [MATH] in terms of the various collective and non-collective operators in the problem.', 'hep-ph-0012138-2-62-1': 'Generically, one can represent an arbitrary operator [MATH] as a power series in our collective operators: [EQUATION] with [MATH] specifying a particular term in the expansion, and [MATH] indicating the three Cartesian directions.', 'hep-ph-0012138-2-62-2': 'The coefficients in the expansion, [MATH], are generally non-collective operators which commute with [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-2-62-3': 'Expanding one component of [MATH] in this form gives [EQUATION] where the fact that the [MATH] commutes with [MATH] or [MATH] impose the restriction that no nonzero powers of [MATH] or [MATH] can contribute to the series, while the commutation relation of Eq. ([REF]) implies that the only contributing term containing any powers of [MATH] is the linear term and that the coefficient of this term is unity.', 'hep-ph-0012138-2-63-0': 'The known parity and time reversal properties of [MATH] and [MATH] fix the parity and time reversal properties of the non-collective coefficient operators, [MATH].', 'hep-ph-0012138-2-63-1': 'Since [MATH] is time reversal even and parity odd while [MATH] is time reversal odd and parity even, it follows that: [EQUATION]', 'hep-ph-0012138-2-63-2': 'The significant point about Eq. ([REF]) is that every non-collective coefficient operator is odd under PT.', 'hep-ph-0012138-2-63-3': 'From the analysis of Sec. [REF], generic states can be written as superpositions of outer product states containing collective and intrinsic parts.', 'hep-ph-0012138-2-63-4': 'The collective subspace of the theory is the subspace where the intrinsic wave function is in its ground state.', 'hep-ph-0012138-2-63-5': 'Thus, an arbitrary state in the collective subspace can be written as [MATH] where [MATH] indicates the ground state of the intrinsic system.', 'hep-ph-0012138-2-63-6': 'Now consider the matrix element of a typical term in the series of Eq. ([REF]) between two collective states: [EQUATION] where we have separated the matrix element into its collective and intrinsic parts.', 'hep-ph-0012138-2-63-7': 'Clearly, the matrix element in Eq. ([REF]) vanishes: the intrinsic part of the of the matrix element is a diagonal matrix element of a PT odd operator in a state of good PT.', 'hep-ph-0012138-2-63-8': 'Thus, within the collective subspace all terms in the series of Eq. ([REF]) vanish except the first term.', 'hep-ph-0012138-2-63-9': 'This implies that within this subspace all matrix elements of [MATH] are identical to matrix elements of [MATH] and Eq. ([REF]) is established.', 'hep-ph-0012138-2-64-0': 'As written, the equivalence expressed in Eq. ([REF]) only holds for states in the collective subspace, i.e., states with the intrinsic subspace in its ground state.', 'hep-ph-0012138-2-64-1': 'As discussed in Sec. [REF], the non-collective operators in the Hamiltonian induce components with excited intrinsic states in the physical low-energy states.', 'hep-ph-0012138-2-64-2': 'This in turn spoils the formal equivalence in Eq. ([REF]) for the physical states.', 'hep-ph-0012138-2-64-3': 'Fortunately, however, it is straightforward to see that in the physical states, the matrix elements of [MATH] differ from those of [MATH] by an amount of order [MATH] (or less) and hence may be neglected at next-to-leading order in [MATH].', 'hep-ph-0012138-2-64-4': 'In the first place using reasoning analogous to that leading to Eq. ([REF]), it is easy to see that the matrix elements of the [MATH] between the intrinsic ground state and its excited states is of order [MATH].', 'hep-ph-0012138-2-64-5': 'On the other hand, the analysis of Sec. [REF] implies that admixtures of excited intrinsic states in the physical states are suppressed by at least order [MATH].', 'hep-ph-0012138-2-64-6': 'Combining these two facts one sees that the matrix elements of [MATH] and [MATH] in the physical states are equivalent at NLO.', 'hep-ph-0012138-2-65-0': 'The equivalence at NLO between the matrix elements of [MATH] and [MATH] may seem paradoxical.', 'hep-ph-0012138-2-65-1': 'After all, as discussed above, the two operators are clearly quite distinct at the QCD level.', 'hep-ph-0012138-2-65-2': 'It would seem that the fact that the operators are different would imply that the matrix elements of the two operators should be different in general.', 'hep-ph-0012138-2-65-3': 'The resolution of the paradox is quite simple: the matrix elements of the two operators are different in general.', 'hep-ph-0012138-2-65-4': 'They only coincide (up to NNLO corrections) for a very special class of states - the collective states with excitation energies of order [MATH].', 'hep-ph-0012138-2-65-5': 'For states with excitation energies of order unity, the matrix elements of the two operators can be quite different.', 'hep-ph-0012138-2-65-6': 'Fortunately, we will be using our effective theory to study these low-lying excitations only.', 'hep-ph-0012138-2-65-7': 'Thus, for the purpose of the effective theory at NLO, it is legitimate to replace [MATH] with [MATH].', 'hep-ph-0012138-2-66-0': '# Summary', 'hep-ph-0012138-2-67-0': 'In this paper we have derived an effective theory which will be used in Ref. [CITATION] to study the spectroscopy and electroweak decays of the low-energy states of heavy baryons.', 'hep-ph-0012138-2-67-1': 'The analysis is based largely on the formulation of the problem in Ref. [CITATION].', 'hep-ph-0012138-2-67-2': 'A power-counting scheme is developed in [MATH], where [MATH].', 'hep-ph-0012138-2-67-3': 'The kinematical variables of the problem [MATH], [MATH], [MATH] and [MATH] are directly related to QCD operators which are well defined up to corrections at relative order [MATH].', 'hep-ph-0012138-2-67-4': 'This formalism is based on standard large [MATH] and heavy quark analysis.', 'hep-ph-0012138-2-68-0': 'At the theoretical level, the analysis used in our derivation of the effective theory goes well beyond the analysis of Ref. [CITATION] which was largely kinematical and group-theoretical in nature.', 'hep-ph-0012138-2-68-1': 'In our derivation, the structure of the Hilbert space for heavy baryons plays a critical role.', 'hep-ph-0012138-2-68-2': 'In particular, as the [MATH] limit is approached, the heavy baryon states can be written as products of collective states (with excitation energies of order [MATH]) and intrinsic states (with excitation energies of order [MATH]).', 'hep-ph-0012138-2-68-3': 'To deduce this form of the Hilbert space we exploited the fact that our formulation as an expansion in [MATH] does not depend formally on whether the large [MATH] limit is taken first, the heavy quark limit is taken first or they are taken simultaneously.', 'hep-ph-0012138-2-68-4': 'By taking the large [MATH] limit first we can use standard large [MATH] reasoning to deduce the structure of the Hilbert space and then map back to the case where the limits are taken concurrently.', 'hep-ph-0012138-2-68-5': 'One essential point in this analysis is that matrix elements of operators which couple between different intrinsic states are characteristically down by a factor of [MATH] compared to matrix elements of operators with the same intrinsic state.', 'hep-ph-0012138-2-68-6': 'Thus, the effect of coupling to an excited intrinsic state and then back to the ground state is typically of order [MATH] and hence can be neglected at NLO in the [MATH] expansion.', 'hep-ph-0012138-2-69-0': 'From the kinematics of the problem, the underlying heavy quark theory, Poincare invariance, and the structure of the Hilbert space, we showed that the effective Hamiltonian up to order [MATH] can be written in the form of Eq. ([REF]).', 'hep-ph-0012138-2-69-1': 'At order [MATH] one sees that the excitation spectrum is fixed entirely by a single parameter [MATH], and at order [MATH] only one additional parameter - [MATH] - enters.', 'hep-ph-0012138-2-69-2': 'This is encouraging from a phenomenological perspective since we do not have a large number of coefficients to work with at NLO.', 'hep-ph-0012138-2-70-0': 'We also derived the certain effective operators at NLO.', 'hep-ph-0012138-2-70-1': 'It turns out that the operators discussed here are precisely those needed to compute interesting electroweak transitions of isoscalar heavy baryons at NLO.', 'hep-ph-0012138-2-70-2': 'The key point in the derivation is that all of these operators can be written in terms of the collective operators (up to possible NNLO corrections).', 'hep-ph-0012138-2-70-3': 'From this it was easy to show that the effect of the admixture of components only comes in at NNLO so that at NLO one can simply replace the QCD level collective operator by an effective operator of the same structure.', 'hep-ph-0012138-2-70-4': 'One operator, [MATH], which is effectively the dipole moment of the light quark contribution to the baryon density, was somewhat more subtle.', 'hep-ph-0012138-2-70-5': 'Although, as an operator it cannot be written directly in terms of the collective variables, we showed that its matrix elements between low-lying states are identical to those of the collective operator [MATH].', 'hep-ph-0012138-2-71-0': 'The effective theory derived here is suitable for the study of isoscalar heavy baryons at NLO in our [MATH] expansion.', 'hep-ph-0012138-2-71-1': 'In Ref. [CITATION] we will use the effective Hamiltonian of Sec. [REF] to study the spectroscopy and electroweak decays of [MATH] and [MATH] baryons and their excited states.', 'hep-ph-0012138-2-71-2': 'In order to calculate the semi-leptonic matrix elements we will use relations between form-factors analogous to those obtained in HQET [CITATION].', 'hep-ph-0012138-2-71-3': 'These matrix elements can then be determined using the effective operators discussed in Sec. [REF].', 'hep-ph-0012138-2-71-4': 'We will also calculate the radiative decay rates of the first excited states of [MATH] baryons.', 'hep-ph-0012138-2-72-0': 'This work is supported by the U.S. Department of Energy grant DE-FG02-93ER-40762.', 'hep-ph-0012138-2-73-0': 'hb0 C.K. Chow and T.D. Cohen, Phys.', 'hep-ph-0012138-2-73-1': 'A to be published.', 'hep-ph-0012138-2-73-2': 'hb3 A. Baccouche, C.K. Chow, T.D. Cohen and B.A. Gelman, "Excited Heavy Baryons and Their Symmetries III: Phenomenology", DOE/ER/40762-215, UM PP01-029, in preparation.', 'hep-ph-0012138-2-73-3': 'bst1 Z. Guralnik, M. Luke and A.V. Manohar, Nucl.', 'hep-ph-0012138-2-73-4': 'bst2 E. Jenkins, A.V. Manohar and M.B. Wise, Nucl.', 'hep-ph-0012138-2-73-5': 'bst4 C.K. Chow and M.B. Wise, Phys.', 'hep-ph-0012138-2-73-6': 'bst5 M. Rho, D.O. Riska and N.N. Scoccola, Z. Phys.', 'hep-ph-0012138-2-73-7': 'E4 47 (1995); bst7 J. Schechter, A. Subbaraman, S. Vaidya and H. Weigel, Nucl.', 'hep-ph-0012138-2-73-8': 'Glozman and D.O. Riska, Nucl.', 'hep-ph-0012138-2-73-9': 'SF1 J.L. Gervais and B. Sakita, Phys.', 'hep-ph-0012138-2-73-10': 'SF2 J.L. Gervais and B. Sakita, Phys.', 'hep-ph-0012138-2-73-11': 'SF3 R. Dashen, E. Jenkins and A.V. Manohar, Phys.', 'hep-ph-0012138-2-73-12': 'SF4 C. Carone, H. Georgi, S. Osofsky, Phys.', 'hep-ph-0012138-2-73-13': 'SF5 M. Luty and J. March-Russell, Nucl.', 'hep-ph-0012138-2-73-14': "LN1 G. 't Hooft, Nucl.", 'hep-ph-0012138-2-73-15': 'LN2 E. Witten, Nucl.', 'hep-ph-0012138-2-73-16': 'HQ1 N. Isgur and M.B. Wise, Phys.', 'hep-ph-0012138-2-73-17': 'HQ2 N. Isgur and M.B. Wise, Phys.', 'hep-ph-0012138-2-73-18': 'HQ3 N. Isgur and M.B. Wise, Nucl.', 'hep-ph-0012138-2-73-19': 'HQ4 H. Georgi, Nucl.', 'hep-ph-0012138-2-73-20': 'HQ5 T. Mannel, W. Roberts and Z. Ryzak, Nucl.', 'hep-ph-0012138-2-73-21': 'HQ6 F. Hussain, J.G. Korner, M. Kramer and G. Thompson, Z. Phys.', 'hep-ph-0012138-2-73-22': 'mQ1 M.E. Luke, Phys.', 'hep-ph-0012138-2-73-23': 'mQ2 H. Georgi, B. Grinstein and M.B. Wise, Phys.', 'hep-ph-0012138-2-73-24': 'mQ3 A.F. Falk, B. Grinstein and M.E. Luke, Nucl.', 'hep-ph-0012138-2-73-25': 'mQ4 M. Neubert, Phys.'}

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'hep-ph-0012138-4-44-3'], ['hep-ph-0012138-3-21-0', 'hep-ph-0012138-4-21-0'], ['hep-ph-0012138-3-21-1', 'hep-ph-0012138-4-21-1'], ['hep-ph-0012138-3-21-2', 'hep-ph-0012138-4-21-2'], ['hep-ph-0012138-3-21-3', 'hep-ph-0012138-4-21-3'], ['hep-ph-0012138-3-21-4', 'hep-ph-0012138-4-21-4'], ['hep-ph-0012138-3-21-5', 'hep-ph-0012138-4-21-5'], ['hep-ph-0012138-3-21-6', 'hep-ph-0012138-4-21-6'], ['hep-ph-0012138-3-65-0', 'hep-ph-0012138-4-65-0'], ['hep-ph-0012138-3-65-1', 'hep-ph-0012138-4-65-1'], ['hep-ph-0012138-3-65-2', 'hep-ph-0012138-4-65-2'], ['hep-ph-0012138-3-65-3', 'hep-ph-0012138-4-65-3'], ['hep-ph-0012138-3-65-4', 'hep-ph-0012138-4-65-4'], ['hep-ph-0012138-3-65-5', 'hep-ph-0012138-4-65-5'], ['hep-ph-0012138-3-65-6', 'hep-ph-0012138-4-65-6'], ['hep-ph-0012138-3-65-7', 'hep-ph-0012138-4-65-7'], ['hep-ph-0012138-3-45-0', 'hep-ph-0012138-4-45-0'], ['hep-ph-0012138-3-45-1', 'hep-ph-0012138-4-45-1'], ['hep-ph-0012138-3-45-2', 'hep-ph-0012138-4-45-2'], ['hep-ph-0012138-3-45-3', 'hep-ph-0012138-4-45-3'], ['hep-ph-0012138-3-45-4', 'hep-ph-0012138-4-45-4'], ['hep-ph-0012138-3-25-0', 'hep-ph-0012138-4-25-0'], ['hep-ph-0012138-3-25-1', 'hep-ph-0012138-4-25-1'], ['hep-ph-0012138-3-5-0', 'hep-ph-0012138-4-5-0'], ['hep-ph-0012138-3-5-1', 'hep-ph-0012138-4-5-1'], ['hep-ph-0012138-3-5-2', 'hep-ph-0012138-4-5-2'], ['hep-ph-0012138-3-5-3', 'hep-ph-0012138-4-5-3'], ['hep-ph-0012138-3-5-4', 'hep-ph-0012138-4-5-4'], ['hep-ph-0012138-3-5-5', 'hep-ph-0012138-4-5-5'], ['hep-ph-0012138-3-19-0', 'hep-ph-0012138-4-19-0'], ['hep-ph-0012138-3-19-1', 'hep-ph-0012138-4-19-1']]

[['hep-ph-0012138-1-73-2', 'hep-ph-0012138-2-73-2']]

[]

[]

[]

['hep-ph-0012138-1-23-9', 'hep-ph-0012138-2-23-9', 'hep-ph-0012138-3-23-9', 'hep-ph-0012138-4-23-9']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/', '4': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0012138

{'hep-ph-0012138-3-0-0': 'We develop an effective theory for heavy baryons and their excited states.', 'hep-ph-0012138-3-0-1': 'The approach is based on the contracted [MATH] symmetry recently shown to emerge from QCD for these states in the combined large [MATH] and heavy quark limits.', 'hep-ph-0012138-3-0-2': 'The effective theory is based on perturbations about this limit; a power counting scheme is developed in which the small parameter is [MATH] where [MATH] (with [MATH] being a typical strong interaction scale).', 'hep-ph-0012138-3-0-3': 'We derive the effective Hamiltonian for strong interactions at next-to-leading order.', 'hep-ph-0012138-3-0-4': 'The next-to-leading order effective Hamiltonian depends on only two parameters beyond the known masses of the nucleon and heavy meson.', 'hep-ph-0012138-3-0-5': 'We also show that the effective operators for certain electroweak transitions can be obtained with no unknown parameters at next-to-leading order.', 'hep-ph-0012138-3-1-0': '# Introduction', 'hep-ph-0012138-3-2-0': 'Recently, it was pointed out that QCD when restricted to the space of heavy baryons (i.e. states with baryon number one containing a single heavy - b or c - quark) has a contracted [MATH] [CITATION] or, more generally, [MATH] symmetry which emerges in the combined large [MATH] (the number of colors) and heavy quark limits [CITATION].', 'hep-ph-0012138-3-2-1': 'Of course, in the real world neither [MATH] nor [MATH] (the generic heavy quark mass) is infinite.', 'hep-ph-0012138-3-2-2': 'Nevertheless if they are large enough, QCD will possess an approximate symmetry.', 'hep-ph-0012138-3-2-3': 'Approximate symmetries can provide significant phenomenological information about a system and have had a long and distinguished history in strong interaction physics.', 'hep-ph-0012138-3-2-4': 'Effective field theory (EFT) is a powerful tool that can be used to extract this information.', 'hep-ph-0012138-3-2-5': 'In this article we develop an effective theory to describe heavy baryons based on the approximate contracted [MATH] symmetry.', 'hep-ph-0012138-3-2-6': 'This is important since this allows one to make testable predictions about the phenomenology of heavy baryons.', 'hep-ph-0012138-3-2-7': 'This paper serves the role of bridging the formal developments of Ref. [CITATION] with phenomenological descriptions of heavy baryons.', 'hep-ph-0012138-3-2-8': 'Our ultimate goal is to study the spectroscopy and electroweak decays of heavy baryon states which will be done in Ref. [CITATION].', 'hep-ph-0012138-3-3-0': 'Before discussing how to implement the EFT approach in the context of heavy baryons it is useful to recall how the EFT program works.', 'hep-ph-0012138-3-3-1': 'Effective field theory is useful when there is a separation of scales between the low energy phenomena and higher energy physics.', 'hep-ph-0012138-3-3-2': 'The short-distance physics can greatly influence the long-distance physics.', 'hep-ph-0012138-3-3-3': 'However, when working at low momentum one cannot resolve the details of what is happening at short distances.', 'hep-ph-0012138-3-3-4': 'The idea of EFT is to describe the low-momentum phenomena in a manner which is insensitive to the details of short-distance physics.', 'hep-ph-0012138-3-3-5': 'Formally, one would begin with the full theory and integrate out all of the short-distance degrees of freedom in a functional integral.', 'hep-ph-0012138-3-3-6': 'This yields an effective action to be used in a functional integral over the low momentum (long-distance) degrees of freedom.', 'hep-ph-0012138-3-3-7': 'Clearly, the effective action so obtained cannot be expressed as the integral over a local Lagrangian.', 'hep-ph-0012138-3-3-8': 'However, all of the nonlocalities occur at the scale of the short-distance degrees of freedom which have been integrated.', 'hep-ph-0012138-3-3-9': 'Thus, to good approximation the effective action can be written in terms of a local Lagrangian.', 'hep-ph-0012138-3-3-10': 'The effects of the short-distance physics are now contained in the coefficients in the effective Lagrangian.', 'hep-ph-0012138-3-4-0': 'Clearly such an approach is, at best, approximate.', 'hep-ph-0012138-3-4-1': 'Fortunately, if the scale separation between the long-distance scales and the short-distance scales is large the approximation can be very good.', 'hep-ph-0012138-3-4-2': 'Formally, one can develop a systematic power-counting expansion where the parameter is the ratio of the typical long-distance scale to the typical short-distance scale.', 'hep-ph-0012138-3-4-3': 'As a practical matter one must truncate the effective Lagrangian at some order.', 'hep-ph-0012138-3-4-4': 'The power counting, however, ensures that the effects of the neglected terms are formally down from the leading term by some power in the ratio of the scales.', 'hep-ph-0012138-3-4-5': 'This, in turn, ensures that the principal effects of the short-distance physics are contained in a few coefficients in the effective Lagrangian.', 'hep-ph-0012138-3-4-6': 'In this manner an insensitivity to the details of the short-distance physics is achieved.', 'hep-ph-0012138-3-4-7': 'The low momentum phenomenology depends on a few coefficients summarizing the effects of the short-distance physics; the manner by which the short-distance physics produces these coefficients is irrelevant.', 'hep-ph-0012138-3-5-0': 'In principle, all of the coefficients in the effective Lagrangian can be calculated from the underlying physics.', 'hep-ph-0012138-3-5-1': 'Often, however, the underlying physics is either unknown or is calculationally intractable.', 'hep-ph-0012138-3-5-2': 'EFT, nevertheless provides a conceptual framework for making predictions about the system.', 'hep-ph-0012138-3-5-3': 'The key insight is that one can treat the parameters in the effective Lagrangian as phenomenological inputs fit from experiment.', 'hep-ph-0012138-3-5-4': 'Thus, providing there are more pieces of independent data than there are free parameters at a given order, one can make real, albeit approximate predictions.', 'hep-ph-0012138-3-5-5': 'Chiral perturbation theory is a familiar paradigm for the EFT program.', 'hep-ph-0012138-3-6-0': 'Let us now turn to the problem of heavy baryons.', 'hep-ph-0012138-3-6-1': 'For simplicity we restrict our attention here to the problem of isoscalar heavy baryons - the [MATH] and [MATH] (generically [MATH]) and their excited states.', 'hep-ph-0012138-3-6-2': 'The generalization to include non-isoscalars such as the [MATH] or [MATH] raises no new conceptual issues but complicates the analysis.', 'hep-ph-0012138-3-6-3': 'The approach of Ref. [CITATION] begins with a restriction to a heavy baryon subspace of the full QCD Hilbert space, i.e. we consider only the space of states with baryon number one and one heavy quark.', 'hep-ph-0012138-3-6-4': 'The physical picture that emerges from the model-dependent analysis of Ref. [CITATION] is that low-lying states of the heavy baryon in the combined large [MATH] and heavy quark limits can be represented as harmonic collective motion of the "brown muck" (the technical term for the light degrees of freedom) against the heavy quark.', 'hep-ph-0012138-3-6-5': 'An equivalent picture in the combined limit is that of the model of Refs. [CITATION] in which the heavy baryons are treated as harmonic collective motions of an ordinary baryon against a heavy meson.', 'hep-ph-0012138-3-6-6': 'The symmetry emerges quite simply from this harmonic oscillator picture: the contracted [MATH] algebra is the algebra of all harmonic oscillator creation and annihilation operators and all bilinears made from them.', 'hep-ph-0012138-3-7-0': 'The key to the analysis of Ref. [CITATION] which enables this picture to emerge is a consistent power-counting scheme.', 'hep-ph-0012138-3-7-1': 'Formally, since we are considering the combined large [MATH] and heavy quark limits, it is useful to consider a single parameter which characterizes departures from this limit.', 'hep-ph-0012138-3-7-2': 'Thus, we introduce: [EQUATION] where [MATH] is a typical strong interaction scale and [MATH] is the mass of the heavy quark.', 'hep-ph-0012138-3-7-3': 'The effective expansion parameter turns out not to be [MATH], but [MATH] [CITATION].', 'hep-ph-0012138-3-7-4': 'The low-lying (collective) excitation energies of the heavy baryons are of order [MATH].', 'hep-ph-0012138-3-7-5': 'This should be contrasted with excitation energies for internal excitation of the brown muck and the dissociation energy of the heavy baryon (into an ordinary baryon and a heavy meson) which are each of order [MATH] [CITATION].', 'hep-ph-0012138-3-7-6': 'Thus, provided that [MATH] can be considered to be small, a scale separation exists between the low-lying collective states of the heavy baryons from all other excitations.', 'hep-ph-0012138-3-7-7': 'This, in turn, implies that EFT methods can be applied to this problem.', 'hep-ph-0012138-3-8-0': 'We note at the outset, however, that in practice [MATH] is not a particularly small number so that the scale separation is not particularly large.', 'hep-ph-0012138-3-8-1': 'This implies that one must work at relatively high order in order to get relatively imprecise results.', 'hep-ph-0012138-3-8-2': 'Thus for example, even working at next-to-leading order one only expects relative accuracy nominally only of order [MATH].', 'hep-ph-0012138-3-8-3': 'Clearly, the approach will be at best semi-quantitative.', 'hep-ph-0012138-3-8-4': 'Moreover, the expansion will not converge for all observables.', 'hep-ph-0012138-3-8-5': 'On phenomenological grounds it is clear that at best the approach will be useful for describing the ground state of the [MATH] and the doublet of the first excited orbital excitation.', 'hep-ph-0012138-3-8-6': 'The second orbitally excited states (which have not been observed in either the [MATH] or [MATH] sectors) is either unbound or very near the threshold and clearly beyond the harmonic limit implicit in the effective field theory.', 'hep-ph-0012138-3-8-7': 'The situation is reminiscent of the contracted [MATH] symmetries seen for light baryons which emerge in the large [MATH] limit [CITATION].', 'hep-ph-0012138-3-8-8': 'In that case the large [MATH] limit predicts an infinite tower of low-lying baryons.', 'hep-ph-0012138-3-8-9': 'However, only the nucleon and delta for [MATH] (or the octet and decuplet for [MATH]) are well described in the real world.', 'hep-ph-0012138-3-9-0': 'The construction of the form of the effective Hamiltonian or Lagrangian for the system is very straightforward.', 'hep-ph-0012138-3-9-1': 'Following the standard rules of effective field theory, once the low-energy degrees of freedom are identified, one may simply write down the most general Lagrangian built from them which is consistent with the underlying symmetries and includes all terms up to a fixed order in the power counting.', 'hep-ph-0012138-3-9-2': 'Similarly, one could add external sources to the theory and impose power counting and thereby determine the from of currents.', 'hep-ph-0012138-3-9-3': 'Of course, in order to compute observables one needs more than the form of the effective Lagrangian and effective operators - one needs values for the coefficients.', 'hep-ph-0012138-3-9-4': 'These coefficients can either be obtained purely phenomenologically or by matching with the underlying theory.', 'hep-ph-0012138-3-10-0': 'In this paper we derive the form of the effective Hamiltonian at next-to-leading order (NLO) directly from the structure of QCD and the power-counting rules .', 'hep-ph-0012138-3-10-1': 'We do this formally by directly making a change of variables to the collective degrees of freedom.', 'hep-ph-0012138-3-10-2': 'This has the benefit of relating the coefficients in the effective theory directly to ground state expectation values of known QCD operators.', 'hep-ph-0012138-3-10-3': 'In this derivation the structure of the Hilbert space plays an essential role.', 'hep-ph-0012138-3-10-4': 'In particular, it is necessary to show that in the combined limit, the Hilbert space can be written as a product space of collective excitations of order [MATH] which may be interpreted as excitations of the brown muck moving coherently against the heavy quark, and non-collective excitation of order [MATH] which correspond to excitations of the brown muck itself.', 'hep-ph-0012138-3-10-5': 'In addition, as we will show, the excitations of the two spaces are independent up to corrections of higher order in [MATH].', 'hep-ph-0012138-3-11-0': 'We also derive results about the effective theory of more phenomenological interest.', 'hep-ph-0012138-3-11-1': 'One of the central achievements of this paper is to show that at NLO the dynamics depends on only two free parameters in the effective Hamiltonian beyond known masses of the nucleon and heavy meson.', 'hep-ph-0012138-3-11-2': 'This is of phenomenological significance in that the small value of the expansion parameter suggests that even for quite crude calculations one should work at least at NLO.', 'hep-ph-0012138-3-11-3': 'Had there been a large number of free parameters at this order it would have suggested that the scheme would be difficult to implement in practice as data on heavy baryons becomes available.', 'hep-ph-0012138-3-11-4': 'Another important result of this paper is the demonstration that certain operators that are not contained in the Hamiltonian but which play a role in describing electroweak properties of heavy baryons are completely determined without the need for phenomenological coefficients (up to corrections of higher order in [MATH]) from the known structure of QCD and the counting rules.', 'hep-ph-0012138-3-11-5': 'This increases the predictive power of the effective theory by eliminating free parameters.', 'hep-ph-0012138-3-12-0': 'As noted above, an expansion in [MATH] is unlikely to be rapidly convergent.', 'hep-ph-0012138-3-12-1': 'Thus it seems sensible to work at relatively high order in the expansion.', 'hep-ph-0012138-3-12-2': 'Here we have stopped at NLO for two reasons: one practical and one theoretical.', 'hep-ph-0012138-3-12-3': 'The practical reason is simply that beyond NLO the number of parameters grows to the point where it is likely to exceed the number of observables which have any reasonable hope of being measured in the foreseeable future.', 'hep-ph-0012138-3-12-4': 'The theoretical reason is that the change of variables technique used to derive the effective theory in terms of known operators in QCD is straightforward only up to NLO.', 'hep-ph-0012138-3-12-5': 'Beyond this order, ambiguities in the definition of the heavy quark mass enter as do effects from the coupling of the low-lowing states described by the effective theory with higher energy excitations.', 'hep-ph-0012138-3-12-6': 'Thus, the treatment becomes more subtle and complex beyond this order.', 'hep-ph-0012138-3-12-7': 'Of course, this does not mean that there is no valid effective theory beyond NLO.', 'hep-ph-0012138-3-12-8': 'However considerable effort must be undertaken in going beyond this order and this effort is simply not justified in view of the practical difficulties mentioned above.', 'hep-ph-0012138-3-13-0': 'This paper is organized as follows.', 'hep-ph-0012138-3-13-1': 'In Sec. [REF] we follow Ref. [CITATION] and briefly review the derivation of the low energy collective variables for heavy baryons from QCD with an emphasis on the power counting rules.', 'hep-ph-0012138-3-13-2': 'In Sec. [REF], we show that the structure of the Hilbert space as a product space emerges from standard large [MATH] analysis.', 'hep-ph-0012138-3-13-3': 'The form of the effective Hamiltonian (up to NLO) is derived from QCD in Sec. [REF].', 'hep-ph-0012138-3-13-4': 'Effective operators for electroweak matrix elements are derived in Sec. [REF].', 'hep-ph-0012138-3-13-5': 'Finally a brief summary is given in Sec. [REF]', 'hep-ph-0012138-3-14-0': '# Dynamical Variables from Counting Rules and QCD', 'hep-ph-0012138-3-15-0': 'In this section we review the counting rules derived from QCD for heavy baryons in the combined large [MATH] and heavy quark limits.', 'hep-ph-0012138-3-15-1': 'We will not reproduce the derivations of Ref. [CITATION] in detail.', 'hep-ph-0012138-3-15-2': 'Rather, we will outline the basic strategy of the derivation along with the principal physical and mathematical assumptions.', 'hep-ph-0012138-3-15-3': 'We also collect results which will be needed later in this paper.', 'hep-ph-0012138-3-16-0': 'The first physical point of significance is to restrict our attention to a limited part of the QCD Hilbert space - the part with baryon number one and heavy quark number one.', 'hep-ph-0012138-3-16-1': 'Furthermore we will ultimately restrict our attention to the lowest-lying states with these quantum numbers.', 'hep-ph-0012138-3-16-2': 'For simplicity, in this section we will only consider a single species of heavy quark at a time as the strong interaction does not couple different heavy flavors.', 'hep-ph-0012138-3-16-3': 'We will generalize to two heavy flavors when we consider electroweak matrix elements in the following sections.', 'hep-ph-0012138-3-17-0': 'QCD is a field theory and as such has an infinite number of degrees of freedom.', 'hep-ph-0012138-3-17-1': 'We can envision making a canonical transformation from the original quark and gluon degrees of freedom to a new set of variables (at the quantum mechanical level it becomes a unitary transformation).', 'hep-ph-0012138-3-17-2': 'The goal is to find such a transformation which has the property that a certain set of the variables (which we denote the collective variables), generate the low-lying collective excited states when acting (repeatedly) on the ground state (the [MATH]).', 'hep-ph-0012138-3-17-3': 'Working explicitly with the collective variables one can calculate directly the properties of the low-lying states.', 'hep-ph-0012138-3-17-4': 'Because we are working in the context of canonical transformations it is far more natural to work with the Hamiltonian formulation of the underlying quantum field theory (QCD) rather than the Lagrangian formulation.', 'hep-ph-0012138-3-18-0': 'Unfortunately, there is no straightforward way to generate such a canonical transformation to collective variables exactly.', 'hep-ph-0012138-3-18-1': 'However, it is easy to find a set of variables which behaves this way approximately.', 'hep-ph-0012138-3-18-2': 'That is, using the power-counting scheme of Eq. ([REF]) one can find collective variables which when acting on the ground state (repeatedly) generate the low-lying collective states plus a small component of higher excited states - the amount of higher excited state admixed is suppressed by powers of [MATH].', 'hep-ph-0012138-3-19-0': 'One canonically conjugate pair of collective variables is the total momentum of the system [MATH], the generator of translations (which is well defined in QCD) and its conjugate variable [MATH], the generator of momentum boosts.', 'hep-ph-0012138-3-19-1': 'This pair of collective variables, however, does not induce the internal excitations of interest here.', 'hep-ph-0012138-3-20-0': 'One might also wish to introduce the momentum of the heavy quark as a collective variable.', 'hep-ph-0012138-3-20-1': 'The expression for [MATH] appropriate in the heavy quark limit is [EQUATION] where [MATH] is the heavy quark field, and [MATH] is the three-dimensional covariant derivative.', 'hep-ph-0012138-3-20-2': 'In the heavy quark limit it is apparent that the variable conjugate to [MATH] is [MATH] defined by [EQUATION]', 'hep-ph-0012138-3-20-3': 'The definitions for [MATH] and [MATH] in Eqs. ([REF]), ([REF]) are clearly valid in the heavy quark limit for the subspace of states with a heavy quark number of unity; in this limit the heavy quark behaves as in nonrelativistic quantum mechanics.', 'hep-ph-0012138-3-20-4': 'In the limit [MATH] and [MATH] correspond to the generators of translations and momentum boosts for the heavy quark.', 'hep-ph-0012138-3-20-5': 'Away from this limit the concept of boosting or translating the heavy quark separately from the full system is not strictly well defined and the [MATH] and [MATH] defined above need not be canonically conjugate.', 'hep-ph-0012138-3-20-6': 'However, even away from the formal limit there will exist some canonically conjugate pair of operators which smoothly goes to the variables defined in Eqs. ([REF]), ([REF]) as the limit is approached.', 'hep-ph-0012138-3-20-7': 'Moreover, by standard heavy quark counting considerations this canonically conjugate pair of operators will differ from those defined in Eqs. ([REF]), ([REF]) by an amount of order [MATH].', 'hep-ph-0012138-3-20-8': 'Here we are interested in the theory up to next-to-leading order, i.e. up to relative order [MATH].', 'hep-ph-0012138-3-20-9': 'Accordingly, to the order at which we work we can take Eqs. ([REF]), ([REF]) as unambiguous definitions.', 'hep-ph-0012138-3-21-0': 'Unfortunately, the conjugate pairs [MATH] and [MATH] are not independent.', 'hep-ph-0012138-3-21-1': 'Clearly [MATH] since [MATH] is a generator of translations and translating the system necessarily translates the position of the heavy quark.', 'hep-ph-0012138-3-21-2': 'Thus, we can not directly use [MATH] as our collective variables to generate low-lying states.', 'hep-ph-0012138-3-21-3': 'However, from these two pairs we can generate two independent conjugate pairs.', 'hep-ph-0012138-3-21-4': 'In particular we can construct a linear combination of [MATH] and [MATH], which we denote [MATH] and a linear combination of [MATH] and [MATH], which we denote [MATH], such that [MATH] and [MATH] are conjugate to each other and commute with [MATH] and [MATH] up to corrections at next-to-next-to-leading order (NNLO).', 'hep-ph-0012138-3-21-5': 'Intuitively [MATH] can be thought of as the generator of translations of the brown muck relative to the heavy quark and [MATH] as its conjugate.', 'hep-ph-0012138-3-21-6': 'As we will see below, the variables [MATH] do act as approximate collective variables in the sense outlined above.', 'hep-ph-0012138-3-22-0': 'The explicit construction of [MATH] and [MATH] is detailed in Ref. [CITATION].', 'hep-ph-0012138-3-22-1': 'The key inputs to this construction are Poincare invariance, heavy quark effective theory, and a decomposition of the QCD Hamiltonian organized via power-counting.', 'hep-ph-0012138-3-22-2': 'The decomposition of the QCD Hamiltonian is most easily accomplished by thinking of the heavy baryon as a bound state of a heavy meson and a nucleon.', 'hep-ph-0012138-3-22-3': 'Standard large [MATH] QCD analysis allows us to identify the interaction energy as scaling as [MATH], while the nucleon mass [MATH] scales as [MATH] [CITATION].', 'hep-ph-0012138-3-22-4': 'Standard heavy quark analysis implies that the interaction energy is of order [MATH] while the mass of the heavy meson, [MATH], is given by [EQUATION]', 'hep-ph-0012138-3-22-5': 'Thus, when constrained to the heavy baryon part of the Hilbert space, the QCD Hamiltonian may be decomposed as: [EQUATION] where [MATH] is of order [MATH].', 'hep-ph-0012138-3-22-6': 'Poincare invariance implies that [EQUATION] while the standard heavy quark treatment implies that [EQUATION]', 'hep-ph-0012138-3-22-7': 'Using Eqs. ([REF]), ([REF]), ([REF]) along with Eq. ([REF]) and the fact that [MATH], the methods of Ref. [CITATION] allows one to deduce that [EQUATION]', 'hep-ph-0012138-3-22-8': 'The expressions for [MATH] and [MATH] differ from the equivalent expressions in Ref. [CITATION] in that we have exploited Eq. ([REF]) to express quantities in terms of [MATH] which is directly physically accessible rather than [MATH] which is not.', 'hep-ph-0012138-3-22-9': 'Throughout the remainder of this paper we will often eliminate [MATH] in favor of [MATH].', 'hep-ph-0012138-3-23-0': 'Thus, we have constructed two independent pairs of conjugate variables: [MATH] and [MATH].', 'hep-ph-0012138-3-23-1': 'These two pairs naturally arise in the Hamiltonian.', 'hep-ph-0012138-3-23-2': 'However, as we will see in Sec. [REF], in describing electroweak operators these pairs are not so natural.', 'hep-ph-0012138-3-23-3': 'Instead it is natural to use the positions and momenta of the heavy quarks [MATH] and the corresponding independent set [MATH] which correspond intuitively to the position and momenta of the light degrees of freedom.', 'hep-ph-0012138-3-23-4': 'Formally they are defined as follows: [EQUATION]', 'hep-ph-0012138-3-23-5': 'In order to exploit the collective variables [MATH] and [MATH] to learn about the low-lying spectrum, it is useful to evaluate multiple commutators of [MATH] and [MATH] with [MATH].', 'hep-ph-0012138-3-23-6': 'Following Ref. [CITATION] one can again use Eqs. ([REF]), ([REF]), ([REF]), ([REF]) and the known [MATH] scaling of [MATH] and [MATH], i.e. [MATH], to obtain [EQUATION] where the reduced mass, [MATH], is given by [EQUATION]', 'hep-ph-0012138-3-23-7': 'More generally, it can be seen by the same methods that multiple commutators of [MATH] with [MATH] are suppressed by multiple powers of [MATH].', 'hep-ph-0012138-3-23-8': 'In particular, if one takes [MATH] commutators of components of [MATH] with [MATH] with [MATH] even, then the [MATH] commutator scales as [MATH].', 'hep-ph-0012138-3-23-9': 'For example: [EQUATION]', 'hep-ph-0012138-3-23-10': 'In contrast, standard large [MATH] counting implies that a displacement of the light degrees of freedom relative to the heavy degrees of freedom by a distance of order [MATH] shifts the Hamiltonian by an amount of order [MATH]: [EQUATION] where [MATH] is a c-number of order [MATH].', 'hep-ph-0012138-3-23-11': 'Equation ([REF]) in turn implies that multiple commutators of [MATH] with [MATH] are generically of order unity: [EQUATION]', 'hep-ph-0012138-3-23-12': 'The power-counting rules from QCD are essentially summarized in Eqs. ([REF]) and ([REF]).', 'hep-ph-0012138-3-23-13': 'By themselves, however, they are not sufficient to develop a fully systematic power-counting scheme for observables without additional phenomenological input.', 'hep-ph-0012138-3-23-14': 'In particular, there are two possible scenarios in which the power-counting may be realized selfconsistently.', 'hep-ph-0012138-3-23-15': 'Intuitively, the [MATH] or symmetric realization corresponds to the case where the effective potential between the heavy quark and the brown muck has a minimum at [MATH], while the [MATH] or symmetry broken realization corresponds to the situation where the effective potential has a minimum at [MATH].', 'hep-ph-0012138-3-23-16': 'The [MATH] realization is labeled symmetry broken since it breaks rotational symmetry in the [MATH] limit.', 'hep-ph-0012138-3-23-17': 'As shown in Ref. [CITATION] the [MATH] (symmetric) realization has the generic property that [EQUATION] in the sense that typical matrix elements between low-lying states of polynomial operators which contain [MATH] and [MATH] scale as [MATH].', 'hep-ph-0012138-3-23-18': 'In contrast, in the [MATH] realization, [EQUATION]', 'hep-ph-0012138-3-23-19': 'In this paper we will assume that the dynamics are such that the system is in the [MATH] (symmetric) realization.', 'hep-ph-0012138-3-24-0': '# Structure of the Hilbert Space', 'hep-ph-0012138-3-25-0': 'In this section we will show that in the combined large [MATH] and heavy quark limits, the Hilbert space can be written as a product space composed of collective and intrinsic excitations, with the collective excitations having energies of order [MATH] and intrinsic excitations are of order unity.', 'hep-ph-0012138-3-25-1': 'The physical picture of these two types of excitations is quite clear - the collective excitations describe coherent motion of the brown muck against the heavy quark while the intrinsic excitations are excitations of the brown muck itself.', 'hep-ph-0012138-3-26-0': "The basic strategy in demonstrating that the Hilbert space is of this form is to exploit the fact that our expansion parameter is valid in the pure large [MATH] limit where Witten's Hartree picture is well established [CITATION].", 'hep-ph-0012138-3-26-1': 'The key point is that while nature is far closer to the heavy quark limit than the large [MATH] limit, our expansion is formulated in the combined large [MATH] and heavy quark limits for [MATH] arbitrary.', 'hep-ph-0012138-3-26-2': 'Thus it applies whether one takes the heavy quark limit first, the large [MATH] limit first or takes the limits simultaneously.', 'hep-ph-0012138-3-26-3': 'We will deduce the structure of the Hilbert space by taking the large [MATH] limit first and then argue that the result goes over smoothly to the combined limit.', 'hep-ph-0012138-3-26-4': 'Ultimately, the reason that we can legitimately take the limits in any order stems from the fact that the dynamics of the system are determined by the interaction Hamiltonian, [MATH].', 'hep-ph-0012138-3-26-5': 'Note that [MATH] is non-singular in both the large [MATH] and heavy quark limits, and therefore we can smoothly take either limit or the combined limit first.', 'hep-ph-0012138-3-27-0': 'Now consider taking the large [MATH] limit prior to taking the heavy quark limit.', 'hep-ph-0012138-3-27-1': "In this case we can use Witten's Hartree picture of baryons.", 'hep-ph-0012138-3-27-2': 'Each quark propagates in the mean field of the remaining quarks.', 'hep-ph-0012138-3-27-3': 'Low energy excitations are described by the excitations of a single quark in the background of the remaining quarks.', 'hep-ph-0012138-3-27-4': 'In the present case we can divide these excitations into two classes - excitations of the heavy quark and excitations of one of the light quarks.', 'hep-ph-0012138-3-27-5': 'Of course, for the light quark excitation it is necessary to anti-symmetrize the wave function as there are many light quarks.', 'hep-ph-0012138-3-27-6': 'In contrast there is only one heavy quark.', 'hep-ph-0012138-3-27-7': 'In the large [MATH] limit these two classes of excitations are orthogonal to each other so that the wave function may be written as a direct product: [EQUATION] where [MATH] ) represent the heavy (light) quark part of the Hartree wave function.', 'hep-ph-0012138-3-28-0': 'The excitation energies for both types of excitations are of order [MATH].', 'hep-ph-0012138-3-28-1': 'However, the excitation energies of the heavy quark are of order [MATH].', 'hep-ph-0012138-3-28-2': 'This is straightforward to see; the heavy quark sits in a potential well of depth and width independent of [MATH].', 'hep-ph-0012138-3-28-3': 'As one takes the large [MATH] limit (after having taken the large [MATH] limit), the heavy quark wave function becomes localized at the bottom of the potential well and thus acts like an harmonic oscillator.', 'hep-ph-0012138-3-28-4': 'One sees that the characteristic excitation energies of the heavy and light types of motion scale with [MATH] according to [EQUATION]', 'hep-ph-0012138-3-28-5': 'As noted above, in this picture the excitations of the heavy quark and the light quarks are independent.', 'hep-ph-0012138-3-28-6': 'The total excitation energy is simply the sum of the heavy excitation energy and the light excitation energy.', 'hep-ph-0012138-3-29-0': 'If we now consider the combined limit with the large [MATH] and heavy quark limits taken simultaneously the picture changes.', 'hep-ph-0012138-3-29-1': 'The Hartree approximation is no longer valid.', 'hep-ph-0012138-3-29-2': 'However, the general counting rules remain - they simply have to be reinterpreted.', 'hep-ph-0012138-3-29-3': 'Physically what happens in the combined limit is that instead of the heavy quark oscillating in the static field of the brown muck, as in the Hartree picture, they oscillate against each other.', 'hep-ph-0012138-3-29-4': 'However, the structure of the Hilbert space is not altered in its essence.', 'hep-ph-0012138-3-29-5': 'The space can still be broken up into a product space of independent classes of excitations as in Eq. ([REF]), but now the two classes of excitations are not the heavy and light degrees of freedom as in the Hartree approximation.', 'hep-ph-0012138-3-29-6': 'Rather, they are the collective excitations (between the brown muck and the heavy quark) and the intrinsic excitations (of the brown muck): [EQUATION] where C ( I ) represent the collective (intrinsic) part.', 'hep-ph-0012138-3-29-7': 'Similarly, the excitation energies behave as in Eq. ([REF]) but with the more general description of collective and intrinsic classes of excitations replacing the heavy and light classes of the Hartree approximation: [EQUATION]', 'hep-ph-0012138-3-29-8': 'We see that the structure of the Hilbert space is quite simple in the [MATH] limit.', 'hep-ph-0012138-3-29-9': 'In order to understand the structure of the effective theory, it is necessary to understand how corrections to this leading behavior emerge.', 'hep-ph-0012138-3-29-10': 'In particular, we will need to quantify the mixing between the collective and intrinsic excitations due to sub-leading terms in [MATH].', 'hep-ph-0012138-3-29-11': 'Again, our strategy is to pass to the case where the large [MATH] limit is taken first and we have the Hartree approximation as a well-defined tool.', 'hep-ph-0012138-3-29-12': 'Then use the smoothness of the two limits to pull back to the case of [MATH] keeping [MATH] arbitrary.', 'hep-ph-0012138-3-30-0': 'It is useful to consider the way [MATH] corrections emerge on top of the Hartree picture as these corrections will ultimately become corrections in our [MATH] counting.', 'hep-ph-0012138-3-30-1': 'We begin by assuming that at finite, but large [MATH], a Hartree picture is a good first approximation.', 'hep-ph-0012138-3-30-2': 'We also make the standard large [MATH] assumption [CITATION] that corrections to the energy and to energy splittings are generically of relative order [MATH].', 'hep-ph-0012138-3-30-3': 'Moreover, these leading relative corrections are independent of [MATH] as the heavy quark only contributes 1 out of [MATH] quarks.', 'hep-ph-0012138-3-30-4': 'Thus, these relative corrections due to effects beyond the Hartree approximation are necessarily of order [MATH] and do not contribute at NLO (which is relative order [MATH]).', 'hep-ph-0012138-3-30-5': 'It is straightforward to see that [MATH] corrections to the energies imply that Hamiltonian matrix elements between two different states with different intrinsic states are generically of order [MATH].', 'hep-ph-0012138-3-30-6': 'This can be shown in two ways.', 'hep-ph-0012138-3-30-7': 'First, off-diagonal matrix elements between different intrinsic states of order [MATH] yield [MATH] corrections to the energies and we know corrections of this order exist.', 'hep-ph-0012138-3-30-8': 'The second way is to construct an explicit wave function using a transition induced by the gluon exchange.', 'hep-ph-0012138-3-30-9': 'The gluon exchange contains two factors of the strong coupling and scales as [MATH]; there is a combinatoric factor of [MATH] as the gluon can connect to any quark; and finally, there is a normalization factor of [MATH] because there are [MATH] distinct terms in the excited state wave function as any of the quarks may be excited.', 'hep-ph-0012138-3-30-10': 'Now we can pass from the large [MATH] limit back to the combined [MATH] limit (with [MATH] fixed) by exploiting smoothness.', 'hep-ph-0012138-3-30-11': 'We obtain the following important result: [EQUATION] where the state [MATH] is a product state [EQUATION]', 'hep-ph-0012138-3-31-0': '# Construction of the Effective Hamiltonian', 'hep-ph-0012138-3-32-0': 'In Sec. [REF] we reviewed the power-counting rules for the collective variables [MATH] and [MATH] in the heavy baryon sector of the QCD Hilbert space.', 'hep-ph-0012138-3-32-1': 'As discussed in Sec. [REF], the general principle of effective field theory implies that the form of the effective theory is completely fixed by the symmetries of the theory and the power-counting rules.', 'hep-ph-0012138-3-32-2': 'Rather than simply invoking this general strategy, it is instructive to derive the effective field theory at next-to-leading order directly from QCD through the change in variables discussed in Sec. [REF].', 'hep-ph-0012138-3-32-3': 'We do this because it is useful to verify that the proposed scaling rules are, indeed, selfconsistent; the derivation gives significant insights into the underlying structure of the theory; and finally, it is satisfying to verify explicitly the validity of the general EFT philosophy in this case.', 'hep-ph-0012138-3-33-0': 'In the present context, where we have been implementing canonical transformations to collective variables, it is simpler to work with an effective Hamiltonian rather than an effective Lagrangian.', 'hep-ph-0012138-3-33-1': 'From Eq. ([REF]) and the power-counting rules of Eqs. ([REF]), ([REF]), ([REF]) and ([REF]) it is straightforward to deduce the form of the QCD Hamiltonian in the heavy baryon sector written in terms of the collective variables up to order [MATH] (NLO).', 'hep-ph-0012138-3-33-2': 'It is given by [EQUATION] where [MATH], [MATH], and [MATH].', 'hep-ph-0012138-3-33-3': 'From rotational invariance, the [MATH] are completely symmetric Cartesian tensor operators of rank [MATH].', 'hep-ph-0012138-3-33-4': 'Note that while terms up to [MATH] contribute at this order, only terms up to [MATH] contribute.', 'hep-ph-0012138-3-33-5': 'The difference between these two is due to the essentially nonrelativistic nature of the kinematics.', 'hep-ph-0012138-3-33-6': 'Formally, this difference is implicit in the different scaling relations for [MATH] and [MATH] in Eqs. ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-ph-0012138-3-34-0': 'Since the form of Eq. ([REF]) was deduced from commutation relations, one cannot immediately conclude that the [MATH] operators are c-numbers.', 'hep-ph-0012138-3-34-1': 'Rather, at this stage in the derivation we can only deduce that they are operators which commute with all of the collective variables: [EQUATION]', 'hep-ph-0012138-3-34-2': 'Thus, the [MATH] tensors are, in general, operators which depend only on the non-collective variables, i.e. the [MATH] are purely intrinsic operators.', 'hep-ph-0012138-3-34-3': 'Note that to the extent that the coefficients [MATH] in the Hamiltonian of Eq. ([REF]) are operators as opposed to c-numbers, the Hamiltonian is not strictly an effective Hamiltonian.', 'hep-ph-0012138-3-34-4': 'Rather it is the full QCD Hamiltonian written in terms of our collective degrees of freedom.', 'hep-ph-0012138-3-34-5': 'The non-collective degrees of freedom have not been integrated out; rather their effects are contained in the non-collective operators [MATH].', 'hep-ph-0012138-3-35-0': 'Non-collective operators might be expected to influence the dynamics of the collective degrees of freedom even though they all commute with the collective degrees of freedom.', 'hep-ph-0012138-3-35-1': 'For example, consider the ground state matrix element of [MATH].', 'hep-ph-0012138-3-35-2': 'This can be expressed in terms of off-diagonal matrix elements: [EQUATION] where [MATH] is the ground state of the heavy baryon system and the intermediate states are grouped into purely collective excitations [MATH] (with excitation energies of order [MATH] which excite only the collective degrees of freedom leaving the brown muck unexcited, and non-collective states [MATH] (with excitation energies of order [MATH] or greater) which involve an intrinsic excitation of the brown muck.', 'hep-ph-0012138-3-35-3': 'If the operators in the Hamiltonian had been purely collective as in a true effective Hamiltonian (which by construction has all non-collective degrees of freedom integrated out), then the sum would not contain the non-collective states [MATH].', 'hep-ph-0012138-3-35-4': 'However, the non-collective operators [MATH] induce transitions to the non-collective states.', 'hep-ph-0012138-3-35-5': 'By acting twice - once to go into the non-collective space, and once to come back the collective space - the non-collective operators can affect collective space matrix elements of operators contained in the Hamiltonian.', 'hep-ph-0012138-3-36-0': 'However, from Eq. ([REF]) we know that terms which connect different intrinsic states are suppressed by order [MATH].', 'hep-ph-0012138-3-36-1': 'We can use this to deduce that the contribution of non-collective intermediate states to matrix elements of composite operators such as in Eq. ([REF]) will always be suppressed by order [MATH] relative to the contributions of the collective intermediated states and thus only contribute at NNLO.', 'hep-ph-0012138-3-36-2': 'This in turn justifies their omission at NLO.', 'hep-ph-0012138-3-36-3': 'Let us see how this works for the decomposition in Eq. ([REF]).', 'hep-ph-0012138-3-36-4': 'From the scaling rules of Eq. ([REF]) it is straightforward to see that the contribution from the collective intermediate states goes as [MATH].', 'hep-ph-0012138-3-36-5': 'A typical term in the sum over states outside the purely collective space is [EQUATION] where [MATH] and [MATH] represent the ground states of the collective and intrinsic subspaces, respectively, while [MATH] and [MATH] represent excited states, and the notation [MATH] represents a product state as in Eq. ([REF]) .', 'hep-ph-0012138-3-36-6': 'The equality in Eq. ([REF]) follows since [MATH] only acts on the collective space while [MATH] only acts on the intrinsic space and the scaling behavior follows from Eqs. ([REF]) and ([REF]) so that the collective part of the expectation value scales as [MATH] while the intrinsic part scales as [MATH].', 'hep-ph-0012138-3-36-7': 'Thus, we see that the contribution of non-collective states is down by a factor of [MATH] compared to the contribution of the collective intermediate states.', 'hep-ph-0012138-3-37-0': 'The [MATH] suppression of the effects of non-collective intermediate states relative to the collective ones is generic.', 'hep-ph-0012138-3-37-1': 'Although it was explicitly demonstrated for the case of the composite operator in Eq. ([REF]), it should be apparent that such a suppression occurs for all products of operators contained in the Hamiltonian.', 'hep-ph-0012138-3-37-2': 'When calculating Hamiltonian matrix elements, the effects of the operators [MATH] in Eq. ([REF]) inducing mixing with the non-collective states will always contribute at relative order [MATH] (i.e. at NNLO) or less.', 'hep-ph-0012138-3-37-3': 'Thus, working at NLO one can simply replace the [MATH] operators by their expectation values in the ground state (or more generally in the ground state band - the set of collective states built on the ground state of the intrinsic degrees of freedom).', 'hep-ph-0012138-3-37-4': 'Therefore, the operators [MATH] can be treated as c-numbers in the effective theory.', 'hep-ph-0012138-3-38-0': 'When the coefficients [MATH] are treated as operators, rotational invariance does not constrain the allowable forms in the Hamiltonian beyond what is given in Eq. ([REF]).', 'hep-ph-0012138-3-38-1': 'For example, the term [MATH] is allowable despite the fact that [MATH] is an [MATH] operator; [MATH] could also be [MATH] operator and couple to [MATH] leading to an [MATH] operator in the Hamiltonian.', 'hep-ph-0012138-3-38-2': 'Once the [MATH] operators are restricted to being c-numbers in the effective theory, rotational invariance greatly restricts the allowable forms: c-numbers do not transform under spatial rotations.', 'hep-ph-0012138-3-38-3': 'Accordingly, the only surviving [MATH] coefficients are those which multiply rotational scalars.', 'hep-ph-0012138-3-38-4': 'Thus, for example, the [MATH] terms must vanish.', 'hep-ph-0012138-3-38-5': 'Treating the [MATH] coefficients as c-numbers and imposing rotational symmetry yields the following form of the effective Hamiltonian up to [MATH]: [EQUATION] where [MATH] refers to the center of mass motion of the entire system while [MATH] refers to the piece of the Hamiltonian whose leading contribution is of order [MATH].', 'hep-ph-0012138-3-38-6': 'The coefficients [MATH], [MATH] and [MATH] are of order [MATH] and may be expressed as ground-state expectation values of [MATH], [MATH] and [MATH], respectively: [EQUATION]', 'hep-ph-0012138-3-38-7': 'A few comments about this derivation are in order here.', 'hep-ph-0012138-3-38-8': 'In essence, we did not have to explicitly integrate out the non-collective higher-lying degrees of freedom.', 'hep-ph-0012138-3-38-9': 'By making a change of variables to [MATH], [MATH], [MATH] and [MATH], we already have identified variables which were decoupled from the intrinsic degrees of freedom to the order at which we are working.', 'hep-ph-0012138-3-38-10': 'Imagine one explicitly made this change of variables prior to formulating the problem in terms of a functional integral.', 'hep-ph-0012138-3-38-11': 'At low order the action could be written as the sum of a collective action plus an intrinsic action without coupling between them, and the only effect of integrating out the intrinsic degrees of freedom would be to supply an overall constant multiplying the remaining functional integral over the collective degrees of freedom.', 'hep-ph-0012138-3-38-12': 'By making this change of variables we make the effect of integrating out the intrinsic degrees of freedom trivial - at least up to NLO.', 'hep-ph-0012138-3-38-13': 'Only if we go beyond this order do effects of the the coupling to the intrinsic degrees of freedom play a role, and the effects of integrating out the intrinsic degrees of freedom become nontrivial.', 'hep-ph-0012138-3-39-0': 'From a phenomenological prospective, it is sufficient to start with Eq. ([REF]).', 'hep-ph-0012138-3-39-1': 'The coefficients [MATH], [MATH] and [MATH] could then be determined approximately from fits to experimental data.', 'hep-ph-0012138-3-39-2': 'Given the fact that the masses of the nucleon and the heavy meson are known, the effective theory at NLO depends on three parameters - [MATH], [MATH] and [MATH].', 'hep-ph-0012138-3-39-3': 'The coefficient [MATH] is determined from the difference in energy between the nucleon-heavy meson threshold and the lowest energy state.', 'hep-ph-0012138-3-39-4': 'However, [MATH] is simply an overall constant and plays no role in the dynamics; the dynamics of the problem depends on two parameters at NLO.', 'hep-ph-0012138-3-39-5': 'It is worth noting that at leading order the dynamics depends only on a single parameter, [MATH].', 'hep-ph-0012138-3-39-6': 'Thus, going from leading order to next-to-leading costs only an additional parameter, [MATH].', 'hep-ph-0012138-3-40-0': '# Effective Operators', 'hep-ph-0012138-3-41-0': 'To make predictions of electroweak current matrix elements between heavy baryon states, one needs more than just the effective Hamiltonian.', 'hep-ph-0012138-3-41-1': 'Effective operators are also required.', 'hep-ph-0012138-3-41-2': 'For general composite operators expressed in terms of the quark and glue degrees of freedom of QCD, the effective operators can be derived in a functional integral form by integrating out the intrinsic degrees of freedom.', 'hep-ph-0012138-3-41-3': 'The most straightforward way to formulate this is as an integral over all QCD variables with [MATH]-functions put in to pick out particular values of the collective variables.', 'hep-ph-0012138-3-41-4': 'In practice, however, the preceding description is rather formal as it is not generally possible to perform this functional integration.', 'hep-ph-0012138-3-41-5': 'At a phenomenological level, one can simply write a form for the operator consistent with symmetries and parameterized by some constants and then fit the constants from experimental data.', 'hep-ph-0012138-3-41-6': 'However, if the operators of interest can be expressed entirely in terms of the collective degrees of freedom at the QCD level, the functional integral approach becomes unnecessary (for the same reason it was unnecessary in the derivation of the collective Hamiltonian).', 'hep-ph-0012138-3-41-7': 'At NLO, one can immediately express the effective operators in terms of the same collective degrees of freedom.', 'hep-ph-0012138-3-42-0': 'To see how this works, consider some operator which at the QCD level can be expressed entirely in terms of the collective operators.', 'hep-ph-0012138-3-42-1': 'We seek to find an analogous operator for use in the effective theory whose matrix elements in the effective theory reproduces the matrix elements of the QCD operator in the QCD states (up to correction at NNLO).', 'hep-ph-0012138-3-42-2': 'Let us start by considering the matrix elements at the QCD level.', 'hep-ph-0012138-3-42-3': 'From the discussion of the structure of the Hilbert space at the QCD level in Sec. [REF], it is apparent that a low-energy (i.e. with excitation energy of order [MATH]) heavy baryon state in a [MATH] expansion at NLO can be written as: [EQUATION] where [MATH] is defined as a product state as in Eq. ([REF]), and [MATH] are constants of order unity.', 'hep-ph-0012138-3-43-0': 'The sum in the second term of Eq. ([REF]) is restricted to contributions with [MATH] due to the product structure of the state.', 'hep-ph-0012138-3-43-1': 'If the sum were not restricted, then there would be contributions of the form [EQUATION] where, [MATH] is simply a single state in the purely collective space.', 'hep-ph-0012138-3-43-2': 'Thus, the contribution to the sum from Eq. ([REF]) is of the form of a single product state.', 'hep-ph-0012138-3-43-3': 'If there are contributions of the form as in Eq. ([REF]) we can replace [MATH] by [MATH] and thereby eliminate contributions with [MATH] from the sum.', 'hep-ph-0012138-3-43-4': 'Doing this has the advantage of removing any possibility of double counting these contributions.', 'hep-ph-0012138-3-44-0': 'Now let us evaluate the matrix element of a purely collective operator, [MATH], i.e. an operator constructed entirely of the collective operators [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-3-44-1': 'From the structure of the state in Eq. ([REF]), the matrix element between two low-energy heavy baryon states (given at NLO) is given by [EQUATION]', 'hep-ph-0012138-3-44-2': 'Thus, up to NNLO corrections the QCD matrix element is simply the matrix element in the collective part of the Hilbert space.', 'hep-ph-0012138-3-44-3': 'On the other hand, the effective Hamiltonian we use at this order with the operators [MATH] replaced by c-numbers is designed to correctly describe the collective part of the Hilbert space at this order.', 'hep-ph-0012138-3-45-0': 'The preceding argument shows, that in the case of purely collective operators the operators in the effective theory are related to the QCD operators in a trivial way up to corrections at NNLO.', 'hep-ph-0012138-3-45-1': 'A purely collective operator at the QCD level is of the form [EQUATION] where the superscript QCD indicates that all of the operators are at the QCD level.', 'hep-ph-0012138-3-45-2': 'The effective operator up to order [MATH] corrections is then obtained by keeping the same operator structure of [MATH], but replacing [MATH], [MATH], [MATH] and [MATH] with the collective operators [MATH], [MATH], [MATH] and [MATH]: [EQUATION].', 'hep-ph-0012138-3-45-3': 'This is essentially a nonrenormalization theorem to this order.', 'hep-ph-0012138-3-45-4': 'The effects of integrating out collective degrees of freedom only renormalize the effective operators at NNLO.', 'hep-ph-0012138-3-46-0': 'In a similar way, all purely non-collective operators at the QCD level, i.e. all operators which commute with all of the collective operators, simply become c-numbers in the effective theory with a value equal to the ground state expectation value of the analogous QCD operator.', 'hep-ph-0012138-3-46-1': 'These general principles can be used to deduce the form of several operators in the effective theory which will prove useful in the calculation of electroweak observables for heavy baryon states.', 'hep-ph-0012138-3-47-0': 'Let us first consider the operator which generates Lorentz boosts, [MATH].', 'hep-ph-0012138-3-47-1': 'The unitary operator for a Lorentz boost is [MATH].', 'hep-ph-0012138-3-47-2': 'The boost operator is useful, for example, in calculating form factors which involve matrix elements of states with different momenta.', 'hep-ph-0012138-3-47-3': 'One can use the boost operator to generate these states from a standard state in its rest frame.', 'hep-ph-0012138-3-47-4': 'From Poincare invariance, the boost operator is given at the field theoretic level by [EQUATION] where the second equality follows from Eq. ([REF]).', 'hep-ph-0012138-3-47-5': 'Thus, up to NLO at the QCD level [MATH] and from the arguments given above, this can immediately be taken over in the effective theory.', 'hep-ph-0012138-3-48-0': 'Among the observables which can be studied using the combined expansion are matrix elements of the electroweak current [MATH], where subscripts [MATH] and [MATH] refer to the heavy quark flavors and [MATH] is the Dirac structure of the left-hand current, [MATH].', 'hep-ph-0012138-3-48-1': 'We will show here that the leading term in the combined heavy quark and large [MATH] expansion of the current [MATH] can be expressed in terms of the collective variables [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0012138-3-48-2': 'Moreover, we will show that the lowest order corrections come at order [MATH] and not at [MATH], so that they contribute only at NNLO.', 'hep-ph-0012138-3-49-0': 'The combined expansion of the current [MATH] can be done analogously to the pure heavy quark expansion in the Heavy Quark Effective Theory (HQET) [CITATION].', 'hep-ph-0012138-3-49-1': 'The idea is to decompose the total heavy quark field into "large" and "small" components or "light" and "heavy" fields with the latter being suppressed by a factor of [MATH] relative to the "large" component.', 'hep-ph-0012138-3-49-2': 'It is then possible to eliminate the "small" component in terms of the "large" component by means of a power series expansion.', 'hep-ph-0012138-3-49-3': 'Near the combined limit considered here, the pure heavy quark expansion breaks down.', 'hep-ph-0012138-3-49-4': 'It has been shown in Refs. [CITATION] that the next-to-leading order terms in the pure heavy quark expansion of the matrix elements of the current [MATH] between the heavy baryon states contain coefficients proportional to [MATH] where [MATH] (with [MATH] being the total mass of a heavy baryon).', 'hep-ph-0012138-3-49-5': 'In the HQET the constant [MATH] is treated as being much smaller than [MATH], so that the NLO operators are indeed suppressed.', 'hep-ph-0012138-3-49-6': 'The situation is different, however, near the combined limit where [MATH] is of order [MATH], so that [MATH] and [MATH].', 'hep-ph-0012138-3-49-7': 'Hence, these corrections cannot be neglected near the combined limit.', 'hep-ph-0012138-3-50-0': 'The appearance of large corrections near the combined limit is a result of the way the total heavy quark field is separated into its "large" and "small" components.', 'hep-ph-0012138-3-50-1': 'The definition of these components is necessarily different from the analogous decomposition in HQET.', 'hep-ph-0012138-3-50-2': 'Near the combined limit, as in HQET, the heavy quark is not far off-shell since its interaction with the brown muck is of order [MATH] while its mass is of order [MATH].', 'hep-ph-0012138-3-50-3': 'However, the space-time dependence of the heavy quark is much different near the combined limit.', 'hep-ph-0012138-3-50-4': 'Near the pure heavy quark limit the brown muck mass is formally much less than the heavy quark mass.', 'hep-ph-0012138-3-50-5': 'As a result, the space-time dependence of the heavy quark is essentially determined by the free particle Lagrangian.', 'hep-ph-0012138-3-50-6': 'In HQET, therefore, it is useful to redefine the heavy quark field by removing the phase factor of the solution of the free Dirac equation, namely [MATH].', 'hep-ph-0012138-3-50-7': 'Near the combined limit the heavy quark remains essentially nonrelativistic.', 'hep-ph-0012138-3-50-8': 'However, the space-time dependence of the heavy quark is much different from that of a free particle.', 'hep-ph-0012138-3-50-9': 'The phase factor now should contain a contribution due to the brown muck mass, which is formally of the same order as the heavy quark mass near the combined limit.', 'hep-ph-0012138-3-50-10': 'This naturally leads to the following redefinition of the heavy quark field: [EQUATION] where [MATH] is the total 4-velocity of the heavy baryon, the operators [MATH] in the rest frame of the heavy baryon project out the upper and lower components of the heavy quark field, and [MATH] here represents a space-time point.', 'hep-ph-0012138-3-50-11': 'As in HQET, the fields [MATH] and [MATH] satisfy conditions [MATH] and [MATH].', 'hep-ph-0012138-3-50-12': 'Fields [MATH] and [MATH] are defined for each value of a heavy baryon 4-velocity [MATH] as can be seen from Eq. ([REF]).', 'hep-ph-0012138-3-50-13': 'The redefinition in Eq. ([REF]) has the effect of removing both the heavy quark mass, [MATH], and the nucleon mass, [MATH], from the dynamics, leaving a dynamical scale of order [MATH], i.e., the scale of the interaction Hamiltonian near the combined limit which is [MATH].', 'hep-ph-0012138-3-51-0': 'The heavy quark part of the QCD Lagrangian density in terms of the fields [MATH] and [MATH] has the form, [EQUATION] where [MATH] is the "transverse part" of the covariant derivative [MATH].', 'hep-ph-0012138-3-51-1': 'The heavy quark Lagrangian in Eq. ([REF]) written in terms of the fields [MATH] and [MATH] differs from its analog in HQET due to the additional term proportional to [MATH].', 'hep-ph-0012138-3-51-2': 'This [MATH] term would naively suggest that the fields [MATH] and [MATH] are both heavy with masses [MATH] and [MATH], respectively, which apparently prevents integrating out the "heavy" field [MATH].', 'hep-ph-0012138-3-52-0': 'This problem is easily resolved if one considers the full Lagrangian density of the system: [EQUATION] where the sum is over all light quarks, and [MATH] is the Yang-Mills Lagrangian density.', 'hep-ph-0012138-3-52-1': 'The total Lagrangian density in Eq. ([REF]) should be re-expressed so as to build the brown muck contribution into the heavy degrees of freedom.', 'hep-ph-0012138-3-52-2': 'This is completely analogous to the Hamiltonian treatment in Sec. [REF].', 'hep-ph-0012138-3-52-3': 'In that case it is done by adding and subtracting to the Hamiltonian a quantity which is an overall constant [MATH] and regrouping terms according to their [MATH] counting scaling.', 'hep-ph-0012138-3-52-4': 'In the Lagrangian formalism this can be accomplished in the following way using a Lorentz covariant operator [MATH]: [EQUATION] where [EQUATION] and [EQUATION].', 'hep-ph-0012138-3-52-5': 'In the rest frame of a heavy baryon operator, [MATH] is equal to the heavy quark density operator [MATH] and its spatial integral equals to one in the Hilbert space of heavy baryons.', 'hep-ph-0012138-3-52-6': 'Thus, in the Hamiltonian the additional term corresponds to removing from the light degrees of freedom an overall constant [MATH].', 'hep-ph-0012138-3-53-0': 'The additional operator proportional to the nucleon mass [MATH] can easily be expressed in terms of the fields [MATH] and [MATH]: [EQUATION].', 'hep-ph-0012138-3-53-1': 'This operator cancels the last term in Eq. ([REF]) so that the Lagrangian density [MATH] has the form: [EQUATION] where the condition [MATH] is used.', 'hep-ph-0012138-3-53-2': 'Thus, the "large" component [MATH] describes a massless field while the "small" component [MATH] has the mass [MATH].', 'hep-ph-0012138-3-53-3': 'Using equations of motion the "heavy" field [MATH] can be expressed in terms of the "light" field [MATH] as follows: [EQUATION]', 'hep-ph-0012138-3-53-4': 'The relation between fields [MATH] and [MATH] in Eq. ([REF]) is identical in form to that in HQET between the "large" and "small" components of the heavy quark field [MATH].', 'hep-ph-0012138-3-54-0': 'We can now expand the field [MATH], the Lagrangian density in Eq. ([REF]), and the current [MATH] in powers of [MATH].', 'hep-ph-0012138-3-54-1': 'This can be done by eliminating fields [MATH] (using Eq. ([REF])) and expanding the numerator in powers of [MATH].', 'hep-ph-0012138-3-54-2': 'Thus, the combined expansion of the heavy quark field including terms up to order [MATH] is, [EQUATION]', 'hep-ph-0012138-3-54-3': 'Using this expansion the Lagrangian density [MATH] including terms up to order [MATH] has the form: [EQUATION] where [MATH] is a strong coupling constant and [MATH] is the gluon field strength tensor.', 'hep-ph-0012138-3-54-4': 'In a similar way we can expand the current [MATH]; again keeping terms up to [MATH] we get, [EQUATION]', 'hep-ph-0012138-3-54-5': 'The expressions in Eqs. ([REF]), ([REF]), ([REF]) are identical in form to the analogous quantities in HQET.', 'hep-ph-0012138-3-54-6': 'However, there is an important difference: these expressions represent the combined expansion in powers of [MATH] and not the [MATH] expansion of HQET.', 'hep-ph-0012138-3-54-7': 'In other words, by defining fields [MATH] and [MATH] appropriately we were able to resum implicitly all the [MATH] corrections in HQET.', 'hep-ph-0012138-3-55-0': 'The correction terms in the combined expansion of the electroweak current, Eq. ([REF]), come at order [MATH] and not at [MATH].', 'hep-ph-0012138-3-55-1': 'It will be shown in Ref. [CITATION] that these corrections contribute to the matrix elements at order [MATH] as well, [MATH], they do not induce any anomalously large constants which violate the combined power counting of these operators.', 'hep-ph-0012138-3-55-2': 'Thus, at NLO in our expansion only the leading order operator, [MATH], contributes to the matrix elements.', 'hep-ph-0012138-3-55-3': 'In addition, it will be shown in Ref. [CITATION] that in the combined limit at NLO there is only a single form factor which parameterizes the matrix elements of the current [MATH] in the [MATH] sector.', 'hep-ph-0012138-3-55-4': 'Accordingly, we can consider one Lorentz component of the current [MATH], namely the [MATH] component.', 'hep-ph-0012138-3-55-5': 'Thus, to completely determine the matrix elements at NLO in the combined expansion, we need to know the form of only one operator, namely [MATH].', 'hep-ph-0012138-3-56-0': 'Let us first consider the flavor conserving operator [MATH].', 'hep-ph-0012138-3-56-1': 'As we are working near the heavy quark limit it is legitimate to expand the state in a fock-space decomposition for the number of heavy quarks and antiquarks.', 'hep-ph-0012138-3-56-2': 'For a heavy baryon state standard heavy quark analysis implies that the leading fock component has one heavy quark; the first sub-leading fock component with two quarks and one antiquark is suppressed by a factor of [MATH].', 'hep-ph-0012138-3-56-3': 'Thus at NLO in [MATH] we can neglect the sub-leading fock component and treat the state as having a single heavy quark.', 'hep-ph-0012138-3-56-4': 'This implies that acting in our heavy baryon Hilbert space, the operator [MATH] is [MATH].', 'hep-ph-0012138-3-56-5': 'Since this is true at the QCD level at NLO it immediately holds in the effective theory at that order.', 'hep-ph-0012138-3-57-0': 'In order to describe the flavor changing current we need to enlarge the Hilbert space of our effective theory to account for electroweak transitions between heavy baryons with heavy quarks of different flavors.', 'hep-ph-0012138-3-57-1': 'The new Hilbert space is the direct product of the Hilbert spaces for baryons with a given flavor of a heavy quark described in Sec. [REF].', 'hep-ph-0012138-3-57-2': 'Any state of the enlarged Hilbert space can be written as [MATH]-dimensional vector, where [MATH] is a number of heavy flavors.', 'hep-ph-0012138-3-57-3': 'Each component of this vector describes a state with a given flavor.', 'hep-ph-0012138-3-57-4': 'The effective Hamiltonian can be readily generalized to include baryons of various heavy quark flavors.', 'hep-ph-0012138-3-57-5': 'It can be written as [MATH] diagonal matrix.', 'hep-ph-0012138-3-57-6': 'For example, for two heavy flavors ([MATH] charm and bottom) the effective Hamiltonian including terms up to NLO in [MATH] has the form: [EQUATION] where the diagonal entries [MATH] and [MATH] are the effective Hamiltonians as in Eq. ([REF]).', 'hep-ph-0012138-3-57-7': 'Analogously, QCD operators acting in the enlarged Hilbert space become [MATH] matrices.', 'hep-ph-0012138-3-57-8': 'The flavor changing weak transitions can be described by off-diagonal matrix elements of an operator [MATH], where the subscripts [MATH] and [MATH] can take integer values from [MATH] to [MATH].', 'hep-ph-0012138-3-57-9': 'This operator can be written in terms of collective operators only.', 'hep-ph-0012138-3-57-10': 'It follows from arguments analogous to the case of the flavor conserving electroweak operator - [MATH] - that the flavor changing operator acting on the enlarged Hilbert space has the form (for [MATH]): [EQUATION]', 'hep-ph-0012138-3-57-11': 'The diagonal elements of the operator in Eq. ([REF]) correspond to flavor conserving operators, while the off-diagonal elements correspond to flavor changing operators.', 'hep-ph-0012138-3-57-12': 'Their action is to instantaneously change the heavy quark flavor at the position where a heavy baryon interacts with the current.', 'hep-ph-0012138-3-57-13': 'Since the operator in Eq. ([REF]) is written in terms of the collective operators only, the corresponding effective operator has an identical form.', 'hep-ph-0012138-3-58-0': 'Finally, let us consider the operator defined by [EQUATION] so that [MATH] represents the contribution to the baryon dipole moment coming from light quarks.', 'hep-ph-0012138-3-58-1': 'In calculating electromagentic transitions between heavy baryons it is useful to divide the electromagnetic current into pieces coming from the heavy and the light quarks and to further divide the light quark contribution into isovector and isoscalar pieces.', 'hep-ph-0012138-3-58-2': 'The isoscalar piece is simply the light-quark contribution to the baryon current.', 'hep-ph-0012138-3-58-3': 'Thus, matrix elements of [MATH] must be calculated to describe electric dipole transitions of heavy baryons.', 'hep-ph-0012138-3-59-0': 'At first sight, it might appear to be impossible to express the operator [MATH] in terms of collective operators.', 'hep-ph-0012138-3-59-1': 'Although [MATH] - the collective position operator for light quarks - is intuitively something like the position of the brown muck and clearly transforms in the same way as [MATH], at the operator level [MATH] is manifestly not identical to [MATH].', 'hep-ph-0012138-3-59-2': 'Recall that by construction [MATH] acts as a generator of boosts for the light degrees of freedom (up to order [MATH] corrections).', 'hep-ph-0012138-3-59-3': 'On the other hand, the operator [MATH] acts only on the quarks and not on the gluons.', 'hep-ph-0012138-3-59-4': 'Thus [MATH] cannot boost the gluons and therefore does not act as a boost for the light degrees of freedom (which include both quarks and gluons).', 'hep-ph-0012138-3-59-5': 'Thus, the operator [MATH] is clearly distinct from [MATH] and there is apparently no simple way to express [MATH] entirely in terms of collective operators.', 'hep-ph-0012138-3-60-0': 'Remarkably, however, one can show that [EQUATION] for all [MATH], [MATH] in the collective subspace defined in Sec. [REF].', 'hep-ph-0012138-3-60-1': 'Thus, when one restricts consideration to the collective subspace which dominates the low energy physics, [MATH] is equivalent to [MATH].', 'hep-ph-0012138-3-61-0': 'To see how this comes about, we start by considering the commutation relations of [MATH] with [MATH], [MATH] and [MATH].', 'hep-ph-0012138-3-61-1': 'To determine the commutator with [MATH] we exploit the fact that [MATH] is a generator of translations for the light degrees of freedom so that [MATH], where [MATH] is a generic light quark field.', 'hep-ph-0012138-3-61-2': 'We can use this to deduce that [EQUATION]', 'hep-ph-0012138-3-61-3': 'The first equality in Eq. ([REF]) follows from the definition of [MATH], the second equality follows because [MATH] translates the light quark degrees of freedom, the third equality follows from a change of variables, and the final equality follows from the definition of [MATH] and the fact that [EQUATION] counts the net number of light quarks, which is [MATH] in a heavy baryon containing one heavy quark.', 'hep-ph-0012138-3-61-4': 'Equation ([REF]) implies that [EQUATION] while the fact that from its definition [MATH] has no heavy quark operators implies that [EQUATION]', 'hep-ph-0012138-3-61-5': 'Thus, the commutation relations of [MATH] with [MATH], [MATH] and [MATH] are identical to the commutation relations of [MATH] with [MATH], [MATH] and [MATH].', 'hep-ph-0012138-3-61-6': 'In Eqs. ([REF]), ([REF]) indices [MATH] indicate Cartesian components of [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-3-62-0': 'Next let us express [MATH] in terms of the various collective and non-collective operators in the problem.', 'hep-ph-0012138-3-62-1': 'Generically, one can represent an arbitrary operator [MATH] as a power series in our collective operators: [EQUATION] with [MATH] specifying a particular term in the expansion, and [MATH] indicating the three Cartesian directions.', 'hep-ph-0012138-3-62-2': 'The coefficients in the expansion, [MATH], are generally non-collective operators which commute with [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-3-62-3': 'Expanding one component of [MATH] in this form gives [EQUATION] where the fact that the [MATH] commutes with [MATH] or [MATH] impose the restriction that no nonzero powers of [MATH] or [MATH] can contribute to the series, while the commutation relation of Eq. ([REF]) implies that the only contributing term containing any powers of [MATH] is the linear term and that the coefficient of this term is unity.', 'hep-ph-0012138-3-63-0': 'The known parity and time reversal properties of [MATH] and [MATH] fix the parity and time reversal properties of the non-collective coefficient operators, [MATH].', 'hep-ph-0012138-3-63-1': 'Since [MATH] is time reversal even and parity odd while [MATH] is time reversal odd and parity even, it follows that: [EQUATION]', 'hep-ph-0012138-3-63-2': 'The significant point about Eq. ([REF]) is that every non-collective coefficient operator is odd under PT.', 'hep-ph-0012138-3-63-3': 'From the analysis of Sec. [REF], generic states can be written as superpositions of outer product states containing collective and intrinsic parts.', 'hep-ph-0012138-3-63-4': 'The collective subspace of the theory is the subspace where the intrinsic wave function is in its ground state.', 'hep-ph-0012138-3-63-5': 'Thus, an arbitrary state in the collective subspace can be written as [MATH] where [MATH] indicates the ground state of the intrinsic system.', 'hep-ph-0012138-3-63-6': 'Now consider the matrix element of a typical term in the series of Eq. ([REF]) between two collective states: [EQUATION] where we have separated the matrix element into its collective and intrinsic parts.', 'hep-ph-0012138-3-63-7': 'Clearly, the matrix element in Eq. ([REF]) vanishes: the intrinsic part of the of the matrix element is a diagonal matrix element of a PT odd operator in a state of good PT.', 'hep-ph-0012138-3-63-8': 'Thus, within the collective subspace all terms in the series of Eq. ([REF]) vanish except the first term.', 'hep-ph-0012138-3-63-9': 'This implies that within this subspace all matrix elements of [MATH] are identical to matrix elements of [MATH] and Eq. ([REF]) is established.', 'hep-ph-0012138-3-64-0': 'As written, the equivalence expressed in Eq. ([REF]) only holds for states in the collective subspace, i.e., states with the intrinsic subspace in its ground state.', 'hep-ph-0012138-3-64-1': 'As discussed in Sec. [REF], the non-collective operators in the Hamiltonian induce components with excited intrinsic states in the physical low-energy states.', 'hep-ph-0012138-3-64-2': 'This in turn spoils the formal equivalence in Eq. ([REF]) for the physical states.', 'hep-ph-0012138-3-64-3': 'Fortunately, however, it is straightforward to see that in the physical states, the matrix elements of [MATH] differ from those of [MATH] by an amount of order [MATH] (or less) and hence may be neglected at next-to-leading order in [MATH].', 'hep-ph-0012138-3-64-4': 'In the first place using reasoning analogous to that leading to Eq. ([REF]), it is easy to see that the matrix elements of the [MATH] between the intrinsic ground state and its excited states is of order [MATH].', 'hep-ph-0012138-3-64-5': 'On the other hand, the analysis of Sec. [REF] implies that admixtures of excited intrinsic states in the physical states are suppressed by at least order [MATH].', 'hep-ph-0012138-3-64-6': 'Combining these two facts one sees that the matrix elements of [MATH] and [MATH] in the physical states are equivalent at NLO.', 'hep-ph-0012138-3-65-0': 'The equivalence at NLO between the matrix elements of [MATH] and [MATH] may seem paradoxical.', 'hep-ph-0012138-3-65-1': 'After all, as discussed above, the two operators are clearly quite distinct at the QCD level.', 'hep-ph-0012138-3-65-2': 'It would seem that the fact that the operators are different would imply that the matrix elements of the two operators should be different in general.', 'hep-ph-0012138-3-65-3': 'The resolution of the paradox is quite simple: the matrix elements of the two operators are different in general.', 'hep-ph-0012138-3-65-4': 'They only coincide (up to NNLO corrections) for a very special class of states - the collective states with excitation energies of order [MATH].', 'hep-ph-0012138-3-65-5': 'For states with excitation energies of order unity, the matrix elements of the two operators can be quite different.', 'hep-ph-0012138-3-65-6': 'Fortunately, we will be using our effective theory to study these low-lying excitations only.', 'hep-ph-0012138-3-65-7': 'Thus, for the purpose of the effective theory at NLO, it is legitimate to replace [MATH] with [MATH].', 'hep-ph-0012138-3-66-0': '# Summary', 'hep-ph-0012138-3-67-0': 'In this paper we have derived an effective theory which will be used in Ref. [CITATION] to study the spectroscopy and electroweak decays of the low-energy states of heavy baryons.', 'hep-ph-0012138-3-67-1': 'The analysis is based largely on the formulation of the problem in Ref. [CITATION].', 'hep-ph-0012138-3-67-2': 'A power-counting scheme is developed in [MATH], where [MATH].', 'hep-ph-0012138-3-67-3': 'The kinematical variables of the problem [MATH], [MATH], [MATH] and [MATH] are directly related to QCD operators which are well defined up to corrections at relative order [MATH].', 'hep-ph-0012138-3-67-4': 'This formalism is based on standard large [MATH] and heavy quark analysis.', 'hep-ph-0012138-3-68-0': 'At the theoretical level, the analysis used in our derivation of the effective theory goes well beyond the analysis of Ref. [CITATION] which was largely kinematical and group-theoretical in nature.', 'hep-ph-0012138-3-68-1': 'In our derivation, the structure of the Hilbert space for heavy baryons plays a critical role.', 'hep-ph-0012138-3-68-2': 'In particular, as the [MATH] limit is approached, the heavy baryon states can be written as products of collective states (with excitation energies of order [MATH]) and intrinsic states (with excitation energies of order [MATH]).', 'hep-ph-0012138-3-68-3': 'To deduce this form of the Hilbert space we exploited the fact that our formulation as an expansion in [MATH] does not depend formally on whether the large [MATH] limit is taken first, the heavy quark limit is taken first or they are taken simultaneously.', 'hep-ph-0012138-3-68-4': 'By taking the large [MATH] limit first we can use standard large [MATH] reasoning to deduce the structure of the Hilbert space and then map back to the case where the limits are taken concurrently.', 'hep-ph-0012138-3-68-5': 'One essential point in this analysis is that matrix elements of operators which couple between different intrinsic states are characteristically down by a factor of [MATH] compared to matrix elements of operators with the same intrinsic state.', 'hep-ph-0012138-3-68-6': 'Thus, the effect of coupling to an excited intrinsic state and then back to the ground state is typically of order [MATH] and hence can be neglected at NLO in the [MATH] expansion.', 'hep-ph-0012138-3-69-0': 'From the kinematics of the problem, the underlying heavy quark theory, Poincare invariance, and the structure of the Hilbert space, we showed that the effective Hamiltonian up to order [MATH] can be written in the form of Eq. ([REF]).', 'hep-ph-0012138-3-69-1': 'At order [MATH] one sees that the excitation spectrum is fixed entirely by a single parameter [MATH], and at order [MATH] only one additional parameter - [MATH] - enters.', 'hep-ph-0012138-3-69-2': 'This is encouraging from a phenomenological perspective since we do not have a large number of coefficients to work with at NLO.', 'hep-ph-0012138-3-70-0': 'We also derived the certain effective operators at NLO.', 'hep-ph-0012138-3-70-1': 'It turns out that the operators discussed here are precisely those needed to compute interesting electroweak transitions of isoscalar heavy baryons at NLO.', 'hep-ph-0012138-3-70-2': 'The key point in the derivation is that all of these operators can be written in terms of the collective operators (up to possible NNLO corrections).', 'hep-ph-0012138-3-70-3': 'From this it was easy to show that the effect of the admixture of components only comes in at NNLO so that at NLO one can simply replace the QCD level collective operator by an effective operator of the same structure.', 'hep-ph-0012138-3-70-4': 'One operator, [MATH], which is effectively the dipole moment of the light quark contribution to the baryon density, was somewhat more subtle.', 'hep-ph-0012138-3-70-5': 'Although, as an operator it cannot be written directly in terms of the collective variables, we showed that its matrix elements between low-lying states are identical to those of the collective operator [MATH].', 'hep-ph-0012138-3-71-0': 'The effective theory derived here is suitable for the study of isoscalar heavy baryons at NLO in our [MATH] expansion.', 'hep-ph-0012138-3-71-1': 'In Ref. [CITATION] we will use the effective Hamiltonian of Sec. [REF] to study the spectroscopy and electroweak decays of [MATH] and [MATH] baryons and their excited states.', 'hep-ph-0012138-3-71-2': 'In order to calculate the semi-leptonic matrix elements we will use relations between form-factors analogous to those obtained in HQET [CITATION].', 'hep-ph-0012138-3-71-3': 'These matrix elements can then be determined using the effective operators discussed in Sec. [REF].', 'hep-ph-0012138-3-71-4': 'We will also calculate the radiative decay rates of the first excited states of [MATH] baryons.', 'hep-ph-0012138-3-72-0': 'hb0 C.K. Chow and T.D. Cohen, Phys.', 'hep-ph-0012138-3-72-1': 'A to be published.', 'hep-ph-0012138-3-72-2': 'hb3 A. Baccouche, C.K. Chow, T.D. Cohen and B.A. Gelman, "Excited Heavy Baryons and Their Symmetries III: Phenomenology", DOE/ER/40762-215, UM PP01-029, in preparation.', 'hep-ph-0012138-3-72-3': 'bst1 Z. Guralnik, M. Luke and A.V. Manohar, Nucl.', 'hep-ph-0012138-3-72-4': 'bst2 E. Jenkins, A.V. Manohar and M.B. Wise, Nucl.', 'hep-ph-0012138-3-72-5': 'bst4 C.K. Chow and M.B. Wise, Phys.', 'hep-ph-0012138-3-72-6': 'bst5 M. Rho, D.O. Riska and N.N. Scoccola, Z. Phys.', 'hep-ph-0012138-3-72-7': 'E4 47 (1995); bst7 J. Schechter, A. Subbaraman, S. Vaidya and H. Weigel, Nucl.', 'hep-ph-0012138-3-72-8': 'Glozman and D.O. Riska, Nucl.', 'hep-ph-0012138-3-72-9': 'SF1 J.L. Gervais and B. Sakita, Phys.', 'hep-ph-0012138-3-72-10': 'SF2 J.L. Gervais and B. Sakita, Phys.', 'hep-ph-0012138-3-72-11': 'SF3 R. Dashen, E. Jenkins and A.V. Manohar, Phys.', 'hep-ph-0012138-3-72-12': 'SF4 C. Carone, H. Georgi, S. Osofsky, Phys.', 'hep-ph-0012138-3-72-13': 'SF5 M. Luty and J. March-Russell, Nucl.', 'hep-ph-0012138-3-72-14': "LN1 G. 't Hooft, Nucl.", 'hep-ph-0012138-3-72-15': 'LN2 E. Witten, Nucl.', 'hep-ph-0012138-3-72-16': 'HQ1 N. Isgur and M.B. Wise, Phys.', 'hep-ph-0012138-3-72-17': 'HQ2 N. Isgur and M.B. Wise, Phys.', 'hep-ph-0012138-3-72-18': 'HQ3 N. Isgur and M.B. Wise, Nucl.', 'hep-ph-0012138-3-72-19': 'HQ4 H. Georgi, Nucl.', 'hep-ph-0012138-3-72-20': 'HQ5 T. Mannel, W. Roberts and Z. Ryzak, Nucl.', 'hep-ph-0012138-3-72-21': 'HQ6 F. Hussain, J.G. Korner, M. Kramer and G. Thompson, Z. Phys.', 'hep-ph-0012138-3-72-22': 'mQ1 M.E. Luke, Phys.', 'hep-ph-0012138-3-72-23': 'mQ2 H. Georgi, B. Grinstein and M.B. Wise, Phys.', 'hep-ph-0012138-3-72-24': 'mQ3 A.F. Falk, B. Grinstein and M.E. Luke, Nucl.', 'hep-ph-0012138-3-72-25': 'mQ4 M. Neubert, Phys.'}

{'hep-ph-0012138-4-0-0': 'We develop an effective theory for heavy baryons and their excited states.', 'hep-ph-0012138-4-0-1': 'The approach is based on the contracted [MATH] symmetry recently shown to emerge from QCD for these states in the combined large [MATH] and heavy quark limits.', 'hep-ph-0012138-4-0-2': 'The effective theory is based on perturbations about this limit; a power counting scheme is developed in which the small parameter is [MATH] where [MATH] (with [MATH] being a typical strong interaction scale).', 'hep-ph-0012138-4-0-3': 'We derive the effective Hamiltonian for strong interactions at next-to-leading order.', 'hep-ph-0012138-4-0-4': 'The next-to-leading order effective Hamiltonian depends on only two parameters beyond the known masses of the nucleon and heavy meson.', 'hep-ph-0012138-4-0-5': 'We also show that the effective operators for certain electroweak transitions can be obtained with no unknown parameters at next-to-leading order.', 'hep-ph-0012138-4-1-0': '# Introduction', 'hep-ph-0012138-4-2-0': 'Recently, it was pointed out that QCD when restricted to the space of heavy baryons (i.e. states with baryon number one containing a single heavy - b or c - quark) has a contracted [MATH] [CITATION] or, more generally, [MATH] symmetry which emerges in the combined large [MATH] (the number of colors) and heavy quark limits [CITATION].', 'hep-ph-0012138-4-2-1': 'Of course, in the real world neither [MATH] nor [MATH] (the generic heavy quark mass) is infinite.', 'hep-ph-0012138-4-2-2': 'Nevertheless if they are large enough, QCD will possess an approximate symmetry.', 'hep-ph-0012138-4-2-3': 'Approximate symmetries can provide significant phenomenological information about a system and have had a long and distinguished history in strong interaction physics.', 'hep-ph-0012138-4-2-4': 'Effective field theory (EFT) is a powerful tool that can be used to extract this information.', 'hep-ph-0012138-4-2-5': 'In this article we develop an effective theory to describe heavy baryons based on the approximate contracted [MATH] symmetry.', 'hep-ph-0012138-4-2-6': 'This is important since this allows one to make testable predictions about the phenomenology of heavy baryons.', 'hep-ph-0012138-4-2-7': 'This paper serves the role of bridging the formal developments of Ref. [CITATION] with phenomenological descriptions of heavy baryons.', 'hep-ph-0012138-4-2-8': 'Our ultimate goal is to study the spectroscopy and electroweak decays of heavy baryon states which will be done in Ref. [CITATION].', 'hep-ph-0012138-4-3-0': 'Before discussing how to implement the EFT approach in the context of heavy baryons it is useful to recall how the EFT program works.', 'hep-ph-0012138-4-3-1': 'Effective field theory is useful when there is a separation of scales between the low energy phenomena and higher energy physics.', 'hep-ph-0012138-4-3-2': 'The short-distance physics can greatly influence the long-distance physics.', 'hep-ph-0012138-4-3-3': 'However, when working at low momentum one cannot resolve the details of what is happening at short distances.', 'hep-ph-0012138-4-3-4': 'The idea of EFT is to describe the low-momentum phenomena in a manner which is insensitive to the details of short-distance physics.', 'hep-ph-0012138-4-3-5': 'Formally, one would begin with the full theory and integrate out all of the short-distance degrees of freedom in a functional integral.', 'hep-ph-0012138-4-3-6': 'This yields an effective action to be used in a functional integral over the low momentum (long-distance) degrees of freedom.', 'hep-ph-0012138-4-3-7': 'Clearly, the effective action so obtained cannot be expressed as the integral over a local Lagrangian.', 'hep-ph-0012138-4-3-8': 'However, all of the nonlocalities occur at the scale of the short-distance degrees of freedom which have been integrated.', 'hep-ph-0012138-4-3-9': 'Thus, to good approximation the effective action can be written in terms of a local Lagrangian.', 'hep-ph-0012138-4-3-10': 'The effects of the short-distance physics are now contained in the coefficients in the effective Lagrangian.', 'hep-ph-0012138-4-4-0': 'Clearly such an approach is, at best, approximate.', 'hep-ph-0012138-4-4-1': 'Fortunately, if the scale separation between the long-distance scales and the short-distance scales is large the approximation can be very good.', 'hep-ph-0012138-4-4-2': 'Formally, one can develop a systematic power-counting expansion where the parameter is the ratio of the typical long-distance scale to the typical short-distance scale.', 'hep-ph-0012138-4-4-3': 'As a practical matter one must truncate the effective Lagrangian at some order.', 'hep-ph-0012138-4-4-4': 'The power counting, however, ensures that the effects of the neglected terms are formally down from the leading term by some power in the ratio of the scales.', 'hep-ph-0012138-4-4-5': 'This, in turn, ensures that the principal effects of the short-distance physics are contained in a few coefficients in the effective Lagrangian.', 'hep-ph-0012138-4-4-6': 'In this manner an insensitivity to the details of the short-distance physics is achieved.', 'hep-ph-0012138-4-4-7': 'The low momentum phenomenology depends on a few coefficients summarizing the effects of the short-distance physics; the manner by which the short-distance physics produces these coefficients is irrelevant.', 'hep-ph-0012138-4-5-0': 'In principle, all of the coefficients in the effective Lagrangian can be calculated from the underlying physics.', 'hep-ph-0012138-4-5-1': 'Often, however, the underlying physics is either unknown or is calculationally intractable.', 'hep-ph-0012138-4-5-2': 'EFT, nevertheless provides a conceptual framework for making predictions about the system.', 'hep-ph-0012138-4-5-3': 'The key insight is that one can treat the parameters in the effective Lagrangian as phenomenological inputs fit from experiment.', 'hep-ph-0012138-4-5-4': 'Thus, providing there are more pieces of independent data than there are free parameters at a given order, one can make real, albeit approximate predictions.', 'hep-ph-0012138-4-5-5': 'Chiral perturbation theory is a familiar paradigm for the EFT program.', 'hep-ph-0012138-4-6-0': 'Let us now turn to the problem of heavy baryons.', 'hep-ph-0012138-4-6-1': 'For simplicity we restrict our attention here to the problem of isoscalar heavy baryons - the [MATH] and [MATH] (generically [MATH]) and their excited states.', 'hep-ph-0012138-4-6-2': 'The generalization to include non-isoscalars such as the [MATH] or [MATH] raises no new conceptual issues but complicates the analysis.', 'hep-ph-0012138-4-6-3': 'The approach of Ref. [CITATION] begins with a restriction to a heavy baryon subspace of the full QCD Hilbert space, i.e. we consider only the space of states with baryon number one and one heavy quark.', 'hep-ph-0012138-4-6-4': 'The physical picture that emerges from the model-dependent analysis of Ref. [CITATION] is that low-lying states of the heavy baryon in the combined large [MATH] and heavy quark limits can be represented as harmonic collective motion of the "brown muck" (the technical term for the light degrees of freedom) against the heavy quark.', 'hep-ph-0012138-4-6-5': 'An equivalent picture in the combined limit is that of the model of Refs. [CITATION] in which the heavy baryons are treated as harmonic collective motions of an ordinary baryon against a heavy meson.', 'hep-ph-0012138-4-6-6': 'The symmetry emerges quite simply from this harmonic oscillator picture: the contracted [MATH] algebra is the algebra of all harmonic oscillator creation and annihilation operators and all bilinears made from them.', 'hep-ph-0012138-4-7-0': 'The key to the analysis of Ref. [CITATION] which enables this picture to emerge is a consistent power-counting scheme.', 'hep-ph-0012138-4-7-1': 'Formally, since we are considering the combined large [MATH] and heavy quark limits, it is useful to consider a single parameter which characterizes departures from this limit.', 'hep-ph-0012138-4-7-2': 'Thus, we introduce: [EQUATION] where [MATH] is a typical strong interaction scale and [MATH] is the mass of the heavy quark.', 'hep-ph-0012138-4-7-3': 'The effective expansion parameter turns out not to be [MATH], but [MATH] [CITATION].', 'hep-ph-0012138-4-7-4': 'The low-lying (collective) excitation energies of the heavy baryons are of order [MATH].', 'hep-ph-0012138-4-7-5': 'This should be contrasted with excitation energies for internal excitation of the brown muck and the dissociation energy of the heavy baryon (into an ordinary baryon and a heavy meson) which are each of order [MATH] [CITATION].', 'hep-ph-0012138-4-7-6': 'Thus, provided that [MATH] can be considered to be small, a scale separation exists between the low-lying collective states of the heavy baryons from all other excitations.', 'hep-ph-0012138-4-7-7': 'This, in turn, implies that EFT methods can be applied to this problem.', 'hep-ph-0012138-4-8-0': 'We note at the outset, however, that in practice [MATH] is not a particularly small number so that the scale separation is not particularly large.', 'hep-ph-0012138-4-8-1': 'This implies that one must work at relatively high order in order to get relatively imprecise results.', 'hep-ph-0012138-4-8-2': 'Thus for example, even working at next-to-leading order one only expects relative accuracy nominally only of order [MATH].', 'hep-ph-0012138-4-8-3': 'Clearly, the approach will be at best semi-quantitative.', 'hep-ph-0012138-4-8-4': 'Moreover, the expansion will not converge for all observables.', 'hep-ph-0012138-4-8-5': 'On phenomenological grounds it is clear that at best the approach will be useful for describing the ground state of the [MATH] and the doublet of the first excited orbital excitation.', 'hep-ph-0012138-4-8-6': 'The second orbitally excited states (which have not been observed in either the [MATH] or [MATH] sectors) is either unbound or very near the threshold and clearly beyond the harmonic limit implicit in the effective field theory.', 'hep-ph-0012138-4-8-7': 'The situation is reminiscent of the contracted [MATH] symmetries seen for light baryons which emerge in the large [MATH] limit [CITATION].', 'hep-ph-0012138-4-8-8': 'In that case the large [MATH] limit predicts an infinite tower of low-lying baryons.', 'hep-ph-0012138-4-8-9': 'However, only the nucleon and delta for [MATH] (or the octet and decuplet for [MATH]) are well described in the real world.', 'hep-ph-0012138-4-9-0': 'The construction of the form of the effective Hamiltonian or Lagrangian for the system is very straightforward.', 'hep-ph-0012138-4-9-1': 'Following the standard rules of effective field theory, once the low-energy degrees of freedom are identified, one may simply write down the most general Lagrangian built from them which is consistent with the underlying symmetries and includes all terms up to a fixed order in the power counting.', 'hep-ph-0012138-4-9-2': 'Similarly, one could add external sources to the theory and impose power counting and thereby determine the from of currents.', 'hep-ph-0012138-4-9-3': 'Of course, in order to compute observables one needs more than the form of the effective Lagrangian and effective operators - one needs values for the coefficients.', 'hep-ph-0012138-4-9-4': 'These coefficients can either be obtained purely phenomenologically or by matching with the underlying theory.', 'hep-ph-0012138-4-10-0': 'In this paper we derive the form of the effective Hamiltonian at next-to-leading order (NLO) directly from the structure of QCD and the power-counting rules .', 'hep-ph-0012138-4-10-1': 'We do this formally by directly making a change of variables to the collective degrees of freedom.', 'hep-ph-0012138-4-10-2': 'This has the benefit of relating the coefficients in the effective theory directly to ground state expectation values of known QCD operators.', 'hep-ph-0012138-4-10-3': 'In this derivation the structure of the Hilbert space plays an essential role.', 'hep-ph-0012138-4-10-4': 'In particular, it is necessary to show that in the combined limit, the Hilbert space can be written as a product space of collective excitations of order [MATH] which may be interpreted as excitations of the brown muck moving coherently against the heavy quark, and non-collective excitation of order [MATH] which correspond to excitations of the brown muck itself.', 'hep-ph-0012138-4-10-5': 'In addition, as we will show, the excitations of the two spaces are independent up to corrections of higher order in [MATH].', 'hep-ph-0012138-4-11-0': 'We also derive results about the effective theory of more phenomenological interest.', 'hep-ph-0012138-4-11-1': 'One of the central achievements of this paper is to show that at NLO the dynamics depends on only two free parameters in the effective Hamiltonian beyond known masses of the nucleon and heavy meson.', 'hep-ph-0012138-4-11-2': 'This is of phenomenological significance in that the small value of the expansion parameter suggests that even for quite crude calculations one should work at least at NLO.', 'hep-ph-0012138-4-11-3': 'Had there been a large number of free parameters at this order it would have suggested that the scheme would be difficult to implement in practice as data on heavy baryons becomes available.', 'hep-ph-0012138-4-11-4': 'Another important result of this paper is the demonstration that certain operators that are not contained in the Hamiltonian but which play a role in describing electroweak properties of heavy baryons are completely determined without the need for phenomenological coefficients (up to corrections of higher order in [MATH]) from the known structure of QCD and the counting rules.', 'hep-ph-0012138-4-11-5': 'This increases the predictive power of the effective theory by eliminating free parameters.', 'hep-ph-0012138-4-12-0': 'As noted above, an expansion in [MATH] is unlikely to be rapidly convergent.', 'hep-ph-0012138-4-12-1': 'Thus it seems sensible to work at relatively high order in the expansion.', 'hep-ph-0012138-4-12-2': 'Here we have stopped at NLO for two reasons: one practical and one theoretical.', 'hep-ph-0012138-4-12-3': 'The practical reason is simply that beyond NLO the number of parameters grows to the point where it is likely to exceed the number of observables which have any reasonable hope of being measured in the foreseeable future.', 'hep-ph-0012138-4-12-4': 'The theoretical reason is that the change of variables technique used to derive the effective theory in terms of known operators in QCD is straightforward only up to NLO.', 'hep-ph-0012138-4-12-5': 'Beyond this order, ambiguities in the definition of the heavy quark mass enter as do effects from the coupling of the low-lowing states described by the effective theory with higher energy excitations.', 'hep-ph-0012138-4-12-6': 'Thus, the treatment becomes more subtle and complex beyond this order.', 'hep-ph-0012138-4-12-7': 'Of course, this does not mean that there is no valid effective theory beyond NLO.', 'hep-ph-0012138-4-12-8': 'However considerable effort must be undertaken in going beyond this order and this effort is simply not justified in view of the practical difficulties mentioned above.', 'hep-ph-0012138-4-13-0': 'This paper is organized as follows.', 'hep-ph-0012138-4-13-1': 'In Sec. [REF] we follow Ref. [CITATION] and briefly review the derivation of the low energy collective variables for heavy baryons from QCD with an emphasis on the power counting rules.', 'hep-ph-0012138-4-13-2': 'In Sec. [REF], we show that the structure of the Hilbert space as a product space emerges from standard large [MATH] analysis.', 'hep-ph-0012138-4-13-3': 'The form of the effective Hamiltonian (up to NLO) is derived from QCD in Sec. [REF].', 'hep-ph-0012138-4-13-4': 'Effective operators for electroweak matrix elements are derived in Sec. [REF].', 'hep-ph-0012138-4-13-5': 'Finally a brief summary is given in Sec. [REF]', 'hep-ph-0012138-4-14-0': '# Dynamical Variables from Counting Rules and QCD', 'hep-ph-0012138-4-15-0': 'In this section we review the counting rules derived from QCD for heavy baryons in the combined large [MATH] and heavy quark limits.', 'hep-ph-0012138-4-15-1': 'We will not reproduce the derivations of Ref. [CITATION] in detail.', 'hep-ph-0012138-4-15-2': 'Rather, we will outline the basic strategy of the derivation along with the principal physical and mathematical assumptions.', 'hep-ph-0012138-4-15-3': 'We also collect results which will be needed later in this paper.', 'hep-ph-0012138-4-16-0': 'The first physical point of significance is to restrict our attention to a limited part of the QCD Hilbert space - the part with baryon number one and heavy quark number one.', 'hep-ph-0012138-4-16-1': 'Furthermore we will ultimately restrict our attention to the lowest-lying states with these quantum numbers.', 'hep-ph-0012138-4-16-2': 'For simplicity, in this section we will only consider a single species of heavy quark at a time as the strong interaction does not couple different heavy flavors.', 'hep-ph-0012138-4-16-3': 'We will generalize to two heavy flavors when we consider electroweak matrix elements in the following sections.', 'hep-ph-0012138-4-17-0': 'QCD is a field theory and as such has an infinite number of degrees of freedom.', 'hep-ph-0012138-4-17-1': 'We can envision making a canonical transformation from the original quark and gluon degrees of freedom to a new set of variables (at the quantum mechanical level it becomes a unitary transformation).', 'hep-ph-0012138-4-17-2': 'The goal is to find such a transformation which has the property that a certain set of the variables (which we denote the collective variables), generate the low-lying collective excited states when acting (repeatedly) on the ground state (the [MATH]).', 'hep-ph-0012138-4-17-3': 'Working explicitly with the collective variables one can calculate directly the properties of the low-lying states.', 'hep-ph-0012138-4-17-4': 'Because we are working in the context of canonical transformations it is far more natural to work with the Hamiltonian formulation of the underlying quantum field theory (QCD) rather than the Lagrangian formulation.', 'hep-ph-0012138-4-18-0': 'Unfortunately, there is no straightforward way to generate such a canonical transformation to collective variables exactly.', 'hep-ph-0012138-4-18-1': 'However, it is easy to find a set of variables which behaves this way approximately.', 'hep-ph-0012138-4-18-2': 'That is, using the power-counting scheme of Eq. ([REF]) one can find collective variables which when acting on the ground state (repeatedly) generate the low-lying collective states plus a small component of higher excited states - the amount of higher excited state admixed is suppressed by powers of [MATH].', 'hep-ph-0012138-4-19-0': 'One canonically conjugate pair of collective variables is the total momentum of the system [MATH], the generator of translations (which is well defined in QCD) and its conjugate variable [MATH], the generator of momentum boosts.', 'hep-ph-0012138-4-19-1': 'This pair of collective variables, however, does not induce the internal excitations of interest here.', 'hep-ph-0012138-4-20-0': 'One might also wish to introduce the momentum of the heavy quark as a collective variable.', 'hep-ph-0012138-4-20-1': 'The expression for [MATH] appropriate in the heavy quark limit is [EQUATION] where [MATH] is the heavy quark field, and [MATH] is the three-dimensional covariant derivative.', 'hep-ph-0012138-4-20-2': 'In the heavy quark limit it is apparent that the variable conjugate to [MATH] is [MATH] defined by [EQUATION]', 'hep-ph-0012138-4-20-3': 'The definitions for [MATH] and [MATH] in Eqs. ([REF]), ([REF]) are clearly valid in the heavy quark limit for the subspace of states with a heavy quark number of unity; in this limit the heavy quark behaves as in nonrelativistic quantum mechanics.', 'hep-ph-0012138-4-20-4': 'In the limit [MATH] and [MATH] correspond to the generators of translations and momentum boosts for the heavy quark.', 'hep-ph-0012138-4-20-5': 'Away from this limit the concept of boosting or translating the heavy quark separately from the full system is not strictly well defined and the [MATH] and [MATH] defined above need not be canonically conjugate.', 'hep-ph-0012138-4-20-6': 'However, even away from the formal limit there will exist some canonically conjugate pair of operators which smoothly goes to the variables defined in Eqs. ([REF]), ([REF]) as the limit is approached.', 'hep-ph-0012138-4-20-7': 'Moreover, by standard heavy quark counting considerations this canonically conjugate pair of operators will differ from those defined in Eqs. ([REF]), ([REF]) by an amount of order [MATH].', 'hep-ph-0012138-4-20-8': 'Here we are interested in the theory up to next-to-leading order, i.e. up to relative order [MATH].', 'hep-ph-0012138-4-20-9': 'Accordingly, to the order at which we work we can take Eqs. ([REF]), ([REF]) as unambiguous definitions.', 'hep-ph-0012138-4-21-0': 'Unfortunately, the conjugate pairs [MATH] and [MATH] are not independent.', 'hep-ph-0012138-4-21-1': 'Clearly [MATH] since [MATH] is a generator of translations and translating the system necessarily translates the position of the heavy quark.', 'hep-ph-0012138-4-21-2': 'Thus, we can not directly use [MATH] as our collective variables to generate low-lying states.', 'hep-ph-0012138-4-21-3': 'However, from these two pairs we can generate two independent conjugate pairs.', 'hep-ph-0012138-4-21-4': 'In particular we can construct a linear combination of [MATH] and [MATH], which we denote [MATH] and a linear combination of [MATH] and [MATH], which we denote [MATH], such that [MATH] and [MATH] are conjugate to each other and commute with [MATH] and [MATH] up to corrections at next-to-next-to-leading order (NNLO).', 'hep-ph-0012138-4-21-5': 'Intuitively [MATH] can be thought of as the generator of translations of the brown muck relative to the heavy quark and [MATH] as its conjugate.', 'hep-ph-0012138-4-21-6': 'As we will see below, the variables [MATH] do act as approximate collective variables in the sense outlined above.', 'hep-ph-0012138-4-22-0': 'The explicit construction of [MATH] and [MATH] is detailed in Ref. [CITATION].', 'hep-ph-0012138-4-22-1': 'The key inputs to this construction are Poincare invariance, heavy quark effective theory, and a decomposition of the QCD Hamiltonian organized via power-counting.', 'hep-ph-0012138-4-22-2': 'The decomposition of the QCD Hamiltonian is most easily accomplished by thinking of the heavy baryon as a bound state of a heavy meson and a nucleon.', 'hep-ph-0012138-4-22-3': 'Standard large [MATH] QCD analysis allows us to identify the interaction energy as scaling as [MATH], while the nucleon mass [MATH] scales as [MATH] [CITATION].', 'hep-ph-0012138-4-22-4': 'Standard heavy quark analysis implies that the interaction energy is of order [MATH] while the mass of the heavy meson, [MATH], is given by [EQUATION]', 'hep-ph-0012138-4-22-5': 'Thus, when constrained to the heavy baryon part of the Hilbert space, the QCD Hamiltonian may be decomposed as: [EQUATION] where [MATH] is of order [MATH].', 'hep-ph-0012138-4-22-6': 'Poincare invariance implies that [EQUATION] while the standard heavy quark treatment implies that [EQUATION]', 'hep-ph-0012138-4-22-7': 'Using Eqs. ([REF]), ([REF]), ([REF]) along with Eq. ([REF]) and the fact that [MATH], the methods of Ref. [CITATION] allows one to deduce that [EQUATION]', 'hep-ph-0012138-4-22-8': 'The expressions for [MATH] and [MATH] differ from the equivalent expressions in Ref. [CITATION] in that we have exploited Eq. ([REF]) to express quantities in terms of [MATH] which is directly physically accessible rather than [MATH] which is not.', 'hep-ph-0012138-4-22-9': 'Throughout the remainder of this paper we will often eliminate [MATH] in favor of [MATH].', 'hep-ph-0012138-4-23-0': 'Thus, we have constructed two independent pairs of conjugate variables: [MATH] and [MATH].', 'hep-ph-0012138-4-23-1': 'These two pairs naturally arise in the Hamiltonian.', 'hep-ph-0012138-4-23-2': 'However, as we will see in Sec. [REF], in describing electroweak operators these pairs are not so natural.', 'hep-ph-0012138-4-23-3': 'Instead it is natural to use the positions and momenta of the heavy quarks [MATH] and the corresponding independent set [MATH] which correspond intuitively to the position and momenta of the light degrees of freedom.', 'hep-ph-0012138-4-23-4': 'Formally they are defined as follows: [EQUATION]', 'hep-ph-0012138-4-23-5': 'In order to exploit the collective variables [MATH] and [MATH] to learn about the low-lying spectrum, it is useful to evaluate multiple commutators of [MATH] and [MATH] with [MATH].', 'hep-ph-0012138-4-23-6': 'Following Ref. [CITATION] one can again use Eqs. ([REF]), ([REF]), ([REF]), ([REF]) and the known [MATH] scaling of [MATH] and [MATH], i.e. [MATH], to obtain [EQUATION] where the reduced mass, [MATH], is given by [EQUATION]', 'hep-ph-0012138-4-23-7': 'More generally, it can be seen by the same methods that multiple commutators of [MATH] with [MATH] are suppressed by multiple powers of [MATH].', 'hep-ph-0012138-4-23-8': 'In particular, if one takes [MATH] commutators of components of [MATH] with [MATH] with [MATH] even, then the [MATH] commutator scales as [MATH].', 'hep-ph-0012138-4-23-9': 'For example: [EQUATION]', 'hep-ph-0012138-4-23-10': 'In contrast, standard large [MATH] counting implies that a displacement of the light degrees of freedom relative to the heavy degrees of freedom by a distance of order [MATH] shifts the Hamiltonian by an amount of order [MATH]: [EQUATION] where [MATH] is a c-number of order [MATH].', 'hep-ph-0012138-4-23-11': 'Equation ([REF]) in turn implies that multiple commutators of [MATH] with [MATH] are generically of order unity: [EQUATION]', 'hep-ph-0012138-4-23-12': 'The power-counting rules from QCD are essentially summarized in Eqs. ([REF]) and ([REF]).', 'hep-ph-0012138-4-23-13': 'By themselves, however, they are not sufficient to develop a fully systematic power-counting scheme for observables without additional phenomenological input.', 'hep-ph-0012138-4-23-14': 'In particular, there are two possible scenarios in which the power-counting may be realized selfconsistently.', 'hep-ph-0012138-4-23-15': 'Intuitively, the [MATH] or symmetric realization corresponds to the case where the effective potential between the heavy quark and the brown muck has a minimum at [MATH], while the [MATH] or symmetry broken realization corresponds to the situation where the effective potential has a minimum at [MATH].', 'hep-ph-0012138-4-23-16': 'The [MATH] realization is labeled symmetry broken since it breaks rotational symmetry in the [MATH] limit.', 'hep-ph-0012138-4-23-17': 'As shown in Ref. [CITATION] the [MATH] (symmetric) realization has the generic property that [EQUATION] in the sense that typical matrix elements between low-lying states of polynomial operators which contain [MATH] and [MATH] scale as [MATH].', 'hep-ph-0012138-4-23-18': 'In contrast, in the [MATH] realization, [EQUATION]', 'hep-ph-0012138-4-23-19': 'In this paper we will assume that the dynamics are such that the system is in the [MATH] (symmetric) realization.', 'hep-ph-0012138-4-24-0': '# Structure of the Hilbert Space', 'hep-ph-0012138-4-25-0': 'In this section we will show that in the combined large [MATH] and heavy quark limits, the Hilbert space can be written as a product space composed of collective and intrinsic excitations, with the collective excitations having energies of order [MATH] and intrinsic excitations are of order unity.', 'hep-ph-0012138-4-25-1': 'The physical picture of these two types of excitations is quite clear - the collective excitations describe coherent motion of the brown muck against the heavy quark while the intrinsic excitations are excitations of the brown muck itself.', 'hep-ph-0012138-4-26-0': "The basic strategy in demonstrating that the Hilbert space is of this form is to exploit the fact that our expansion parameter is valid in the pure large [MATH] limit where Witten's Hartree picture is well established [CITATION].", 'hep-ph-0012138-4-26-1': 'The key point is that while nature is far closer to the heavy quark limit than the large [MATH] limit, our expansion is formulated in the combined large [MATH] and heavy quark limits for [MATH] arbitrary.', 'hep-ph-0012138-4-26-2': 'Thus it applies whether one takes the heavy quark limit first, the large [MATH] limit first or takes the limits simultaneously.', 'hep-ph-0012138-4-26-3': 'We will deduce the structure of the Hilbert space by taking the large [MATH] limit first and then argue that the result goes over smoothly to the combined limit.', 'hep-ph-0012138-4-26-4': 'Ultimately, the reason that we can legitimately take the limits in any order stems from the fact that the dynamics of the system are determined by the interaction Hamiltonian, [MATH].', 'hep-ph-0012138-4-26-5': 'Note that [MATH] is non-singular in both the large [MATH] and heavy quark limits, and therefore we can smoothly take either limit or the combined limit first.', 'hep-ph-0012138-4-27-0': 'Now consider taking the large [MATH] limit prior to taking the heavy quark limit.', 'hep-ph-0012138-4-27-1': "In this case we can use Witten's Hartree picture of baryons.", 'hep-ph-0012138-4-27-2': 'Each quark propagates in the mean field of the remaining quarks.', 'hep-ph-0012138-4-27-3': 'Low energy excitations are described by the excitations of a single quark in the background of the remaining quarks.', 'hep-ph-0012138-4-27-4': 'In the present case we can divide these excitations into two classes - excitations of the heavy quark and excitations of one of the light quarks.', 'hep-ph-0012138-4-27-5': 'Of course, for the light quark excitation it is necessary to anti-symmetrize the wave function as there are many light quarks.', 'hep-ph-0012138-4-27-6': 'In contrast there is only one heavy quark.', 'hep-ph-0012138-4-27-7': 'In the large [MATH] limit these two classes of excitations are orthogonal to each other so that the wave function may be written as a direct product: [EQUATION] where [MATH] ) represent the heavy (light) quark part of the Hartree wave function.', 'hep-ph-0012138-4-28-0': 'The excitation energies for both types of excitations are of order [MATH].', 'hep-ph-0012138-4-28-1': 'However, the excitation energies of the heavy quark are of order [MATH].', 'hep-ph-0012138-4-28-2': 'This is straightforward to see; the heavy quark sits in a potential well of depth and width independent of [MATH].', 'hep-ph-0012138-4-28-3': 'As one takes the large [MATH] limit (after having taken the large [MATH] limit), the heavy quark wave function becomes localized at the bottom of the potential well and thus acts like an harmonic oscillator.', 'hep-ph-0012138-4-28-4': 'One sees that the characteristic excitation energies of the heavy and light types of motion scale with [MATH] according to [EQUATION]', 'hep-ph-0012138-4-28-5': 'As noted above, in this picture the excitations of the heavy quark and the light quarks are independent.', 'hep-ph-0012138-4-28-6': 'The total excitation energy is simply the sum of the heavy excitation energy and the light excitation energy.', 'hep-ph-0012138-4-29-0': 'If we now consider the combined limit with the large [MATH] and heavy quark limits taken simultaneously the picture changes.', 'hep-ph-0012138-4-29-1': 'The Hartree approximation is no longer valid.', 'hep-ph-0012138-4-29-2': 'However, the general counting rules remain - they simply have to be reinterpreted.', 'hep-ph-0012138-4-29-3': 'Physically what happens in the combined limit is that instead of the heavy quark oscillating in the static field of the brown muck, as in the Hartree picture, they oscillate against each other.', 'hep-ph-0012138-4-29-4': 'However, the structure of the Hilbert space is not altered in its essence.', 'hep-ph-0012138-4-29-5': 'The space can still be broken up into a product space of independent classes of excitations as in Eq. ([REF]), but now the two classes of excitations are not the heavy and light degrees of freedom as in the Hartree approximation.', 'hep-ph-0012138-4-29-6': 'Rather, they are the collective excitations (between the brown muck and the heavy quark) and the intrinsic excitations (of the brown muck): [EQUATION] where C ( I ) represent the collective (intrinsic) part.', 'hep-ph-0012138-4-29-7': 'Similarly, the excitation energies behave as in Eq. ([REF]) but with the more general description of collective and intrinsic classes of excitations replacing the heavy and light classes of the Hartree approximation: [EQUATION]', 'hep-ph-0012138-4-29-8': 'We see that the structure of the Hilbert space is quite simple in the [MATH] limit.', 'hep-ph-0012138-4-29-9': 'In order to understand the structure of the effective theory, it is necessary to understand how corrections to this leading behavior emerge.', 'hep-ph-0012138-4-29-10': 'In particular, we will need to quantify the mixing between the collective and intrinsic excitations due to sub-leading terms in [MATH].', 'hep-ph-0012138-4-29-11': 'Again, our strategy is to pass to the case where the large [MATH] limit is taken first and we have the Hartree approximation as a well-defined tool.', 'hep-ph-0012138-4-29-12': 'Then use the smoothness of the two limits to pull back to the case of [MATH] keeping [MATH] arbitrary.', 'hep-ph-0012138-4-30-0': 'It is useful to consider the way [MATH] corrections emerge on top of the Hartree picture as these corrections will ultimately become corrections in our [MATH] counting.', 'hep-ph-0012138-4-30-1': 'We begin by assuming that at finite, but large [MATH], a Hartree picture is a good first approximation.', 'hep-ph-0012138-4-30-2': 'We also make the standard large [MATH] assumption [CITATION] that corrections to the energy and to energy splittings are generically of relative order [MATH].', 'hep-ph-0012138-4-30-3': 'Moreover, these leading relative corrections are independent of [MATH] as the heavy quark only contributes 1 out of [MATH] quarks.', 'hep-ph-0012138-4-30-4': 'Thus, these relative corrections due to effects beyond the Hartree approximation are necessarily of order [MATH] and do not contribute at NLO (which is relative order [MATH]).', 'hep-ph-0012138-4-30-5': 'It is straightforward to see that [MATH] corrections to the energies imply that Hamiltonian matrix elements between two different states with different intrinsic states are generically of order [MATH].', 'hep-ph-0012138-4-30-6': 'This can be shown in two ways.', 'hep-ph-0012138-4-30-7': 'First, off-diagonal matrix elements between different intrinsic states of order [MATH] yield [MATH] corrections to the energies and we know corrections of this order exist.', 'hep-ph-0012138-4-30-8': 'The second way is to construct an explicit wave function using a transition induced by the gluon exchange.', 'hep-ph-0012138-4-30-9': 'The gluon exchange contains two factors of the strong coupling and scales as [MATH]; there is a combinatoric factor of [MATH] as the gluon can connect to any quark; and finally, there is a normalization factor of [MATH] because there are [MATH] distinct terms in the excited state wave function as any of the quarks may be excited.', 'hep-ph-0012138-4-30-10': 'Now we can pass from the large [MATH] limit back to the combined [MATH] limit (with [MATH] fixed) by exploiting smoothness.', 'hep-ph-0012138-4-30-11': 'We obtain the following important result: [EQUATION] where the state [MATH] is a product state [EQUATION]', 'hep-ph-0012138-4-31-0': '# Construction of the Effective Hamiltonian', 'hep-ph-0012138-4-32-0': 'In Sec. [REF] we reviewed the power-counting rules for the collective variables [MATH] and [MATH] in the heavy baryon sector of the QCD Hilbert space.', 'hep-ph-0012138-4-32-1': 'As discussed in Sec. [REF], the general principle of effective field theory implies that the form of the effective theory is completely fixed by the symmetries of the theory and the power-counting rules.', 'hep-ph-0012138-4-32-2': 'Rather than simply invoking this general strategy, it is instructive to derive the effective field theory at next-to-leading order directly from QCD through the change in variables discussed in Sec. [REF].', 'hep-ph-0012138-4-32-3': 'We do this because it is useful to verify that the proposed scaling rules are, indeed, selfconsistent; the derivation gives significant insights into the underlying structure of the theory; and finally, it is satisfying to verify explicitly the validity of the general EFT philosophy in this case.', 'hep-ph-0012138-4-33-0': 'In the present context, where we have been implementing canonical transformations to collective variables, it is simpler to work with an effective Hamiltonian rather than an effective Lagrangian.', 'hep-ph-0012138-4-33-1': 'From Eq. ([REF]) and the power-counting rules of Eqs. ([REF]), ([REF]), ([REF]) and ([REF]) it is straightforward to deduce the form of the QCD Hamiltonian in the heavy baryon sector written in terms of the collective variables up to order [MATH] (NLO).', 'hep-ph-0012138-4-33-2': 'It is given by [EQUATION] where [MATH], [MATH], and [MATH].', 'hep-ph-0012138-4-33-3': 'From rotational invariance, the [MATH] are completely symmetric Cartesian tensor operators of rank [MATH].', 'hep-ph-0012138-4-33-4': 'Note that while terms up to [MATH] contribute at this order, only terms up to [MATH] contribute.', 'hep-ph-0012138-4-33-5': 'The difference between these two is due to the essentially nonrelativistic nature of the kinematics.', 'hep-ph-0012138-4-33-6': 'Formally, this difference is implicit in the different scaling relations for [MATH] and [MATH] in Eqs. ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-ph-0012138-4-34-0': 'Since the form of Eq. ([REF]) was deduced from commutation relations, one cannot immediately conclude that the [MATH] operators are c-numbers.', 'hep-ph-0012138-4-34-1': 'Rather, at this stage in the derivation we can only deduce that they are operators which commute with all of the collective variables: [EQUATION]', 'hep-ph-0012138-4-34-2': 'Thus, the [MATH] tensors are, in general, operators which depend only on the non-collective variables, i.e. the [MATH] are purely intrinsic operators.', 'hep-ph-0012138-4-34-3': 'Note that to the extent that the coefficients [MATH] in the Hamiltonian of Eq. ([REF]) are operators as opposed to c-numbers, the Hamiltonian is not strictly an effective Hamiltonian.', 'hep-ph-0012138-4-34-4': 'Rather it is the full QCD Hamiltonian written in terms of our collective degrees of freedom.', 'hep-ph-0012138-4-34-5': 'The non-collective degrees of freedom have not been integrated out; rather their effects are contained in the non-collective operators [MATH].', 'hep-ph-0012138-4-35-0': 'Non-collective operators might be expected to influence the dynamics of the collective degrees of freedom even though they all commute with the collective degrees of freedom.', 'hep-ph-0012138-4-35-1': 'For example, consider the ground state matrix element of [MATH].', 'hep-ph-0012138-4-35-2': 'This can be expressed in terms of off-diagonal matrix elements: [EQUATION] where [MATH] is the ground state of the heavy baryon system and the intermediate states are grouped into purely collective excitations [MATH] (with excitation energies of order [MATH] which excite only the collective degrees of freedom leaving the brown muck unexcited, and non-collective states [MATH] (with excitation energies of order [MATH] or greater) which involve an intrinsic excitation of the brown muck.', 'hep-ph-0012138-4-35-3': 'If the operators in the Hamiltonian had been purely collective as in a true effective Hamiltonian (which by construction has all non-collective degrees of freedom integrated out), then the sum would not contain the non-collective states [MATH].', 'hep-ph-0012138-4-35-4': 'However, the non-collective operators [MATH] induce transitions to the non-collective states.', 'hep-ph-0012138-4-35-5': 'By acting twice - once to go into the non-collective space, and once to come back the collective space - the non-collective operators can affect collective space matrix elements of operators contained in the Hamiltonian.', 'hep-ph-0012138-4-36-0': 'However, from Eq. ([REF]) we know that terms which connect different intrinsic states are suppressed by order [MATH].', 'hep-ph-0012138-4-36-1': 'We can use this to deduce that the contribution of non-collective intermediate states to matrix elements of composite operators such as in Eq. ([REF]) will always be suppressed by order [MATH] relative to the contributions of the collective intermediated states and thus only contribute at NNLO.', 'hep-ph-0012138-4-36-2': 'This in turn justifies their omission at NLO.', 'hep-ph-0012138-4-36-3': 'Let us see how this works for the decomposition in Eq. ([REF]).', 'hep-ph-0012138-4-36-4': 'From the scaling rules of Eq. ([REF]) it is straightforward to see that the contribution from the collective intermediate states goes as [MATH].', 'hep-ph-0012138-4-36-5': 'A typical term in the sum over states outside the purely collective space is [EQUATION] where [MATH] and [MATH] represent the ground states of the collective and intrinsic subspaces, respectively, while [MATH] and [MATH] represent excited states, and the notation [MATH] represents a product state as in Eq. ([REF]) .', 'hep-ph-0012138-4-36-6': 'The equality in Eq. ([REF]) follows since [MATH] only acts on the collective space while [MATH] only acts on the intrinsic space and the scaling behavior follows from Eqs. ([REF]) and ([REF]) so that the collective part of the expectation value scales as [MATH] while the intrinsic part scales as [MATH].', 'hep-ph-0012138-4-36-7': 'Thus, we see that the contribution of non-collective states is down by a factor of [MATH] compared to the contribution of the collective intermediate states.', 'hep-ph-0012138-4-37-0': 'The [MATH] suppression of the effects of non-collective intermediate states relative to the collective ones is generic.', 'hep-ph-0012138-4-37-1': 'Although it was explicitly demonstrated for the case of the composite operator in Eq. ([REF]), it should be apparent that such a suppression occurs for all products of operators contained in the Hamiltonian.', 'hep-ph-0012138-4-37-2': 'When calculating Hamiltonian matrix elements, the effects of the operators [MATH] in Eq. ([REF]) inducing mixing with the non-collective states will always contribute at relative order [MATH] (i.e. at NNLO) or less.', 'hep-ph-0012138-4-37-3': 'Thus, working at NLO one can simply replace the [MATH] operators by their expectation values in the ground state (or more generally in the ground state band - the set of collective states built on the ground state of the intrinsic degrees of freedom).', 'hep-ph-0012138-4-37-4': 'Therefore, the operators [MATH] can be treated as c-numbers in the effective theory.', 'hep-ph-0012138-4-38-0': 'When the coefficients [MATH] are treated as operators, rotational invariance does not constrain the allowable forms in the Hamiltonian beyond what is given in Eq. ([REF]).', 'hep-ph-0012138-4-38-1': 'For example, the term [MATH] is allowable despite the fact that [MATH] is an [MATH] operator; [MATH] could also be [MATH] operator and couple to [MATH] leading to an [MATH] operator in the Hamiltonian.', 'hep-ph-0012138-4-38-2': 'Once the [MATH] operators are restricted to being c-numbers in the effective theory, rotational invariance greatly restricts the allowable forms: c-numbers do not transform under spatial rotations.', 'hep-ph-0012138-4-38-3': 'Accordingly, the only surviving [MATH] coefficients are those which multiply rotational scalars.', 'hep-ph-0012138-4-38-4': 'Thus, for example, the [MATH] terms must vanish.', 'hep-ph-0012138-4-38-5': 'Treating the [MATH] coefficients as c-numbers and imposing rotational symmetry yields the following form of the effective Hamiltonian up to [MATH]: [EQUATION] where [MATH] refers to the center of mass motion of the entire system while [MATH] refers to the piece of the Hamiltonian whose leading contribution is of order [MATH].', 'hep-ph-0012138-4-38-6': 'The coefficients [MATH], [MATH] and [MATH] are of order [MATH] and may be expressed as ground-state expectation values of [MATH], [MATH] and [MATH], respectively: [EQUATION]', 'hep-ph-0012138-4-38-7': 'A few comments about this derivation are in order here.', 'hep-ph-0012138-4-38-8': 'In essence, we did not have to explicitly integrate out the non-collective higher-lying degrees of freedom.', 'hep-ph-0012138-4-38-9': 'By making a change of variables to [MATH], [MATH], [MATH] and [MATH], we already have identified variables which were decoupled from the intrinsic degrees of freedom to the order at which we are working.', 'hep-ph-0012138-4-38-10': 'Imagine one explicitly made this change of variables prior to formulating the problem in terms of a functional integral.', 'hep-ph-0012138-4-38-11': 'At low order the action could be written as the sum of a collective action plus an intrinsic action without coupling between them, and the only effect of integrating out the intrinsic degrees of freedom would be to supply an overall constant multiplying the remaining functional integral over the collective degrees of freedom.', 'hep-ph-0012138-4-38-12': 'By making this change of variables we make the effect of integrating out the intrinsic degrees of freedom trivial - at least up to NLO.', 'hep-ph-0012138-4-38-13': 'Only if we go beyond this order do effects of the the coupling to the intrinsic degrees of freedom play a role, and the effects of integrating out the intrinsic degrees of freedom become nontrivial.', 'hep-ph-0012138-4-39-0': 'From a phenomenological prospective, it is sufficient to start with Eq. ([REF]).', 'hep-ph-0012138-4-39-1': 'The coefficients [MATH], [MATH] and [MATH] could then be determined approximately from fits to experimental data.', 'hep-ph-0012138-4-39-2': 'Given the fact that the masses of the nucleon and the heavy meson are known, the effective theory at NLO depends on three parameters - [MATH], [MATH] and [MATH].', 'hep-ph-0012138-4-39-3': 'The coefficient [MATH] is determined from the difference in energy between the nucleon-heavy meson threshold and the lowest energy state.', 'hep-ph-0012138-4-39-4': 'However, [MATH] is simply an overall constant and plays no role in the dynamics; the dynamics of the problem depends on two parameters at NLO.', 'hep-ph-0012138-4-39-5': 'It is worth noting that at leading order the dynamics depends only on a single parameter, [MATH].', 'hep-ph-0012138-4-39-6': 'Thus, going from leading order to next-to-leading costs only an additional parameter, [MATH].', 'hep-ph-0012138-4-40-0': '# Effective Operators', 'hep-ph-0012138-4-41-0': 'To make predictions of electroweak current matrix elements between heavy baryon states, one needs more than just the effective Hamiltonian.', 'hep-ph-0012138-4-41-1': 'Effective operators are also required.', 'hep-ph-0012138-4-41-2': 'For general composite operators expressed in terms of the quark and glue degrees of freedom of QCD, the effective operators can be derived in a functional integral form by integrating out the intrinsic degrees of freedom.', 'hep-ph-0012138-4-41-3': 'The most straightforward way to formulate this is as an integral over all QCD variables with [MATH]-functions put in to pick out particular values of the collective variables.', 'hep-ph-0012138-4-41-4': 'In practice, however, the preceding description is rather formal as it is not generally possible to perform this functional integration.', 'hep-ph-0012138-4-41-5': 'At a phenomenological level, one can simply write a form for the operator consistent with symmetries and parameterized by some constants and then fit the constants from experimental data.', 'hep-ph-0012138-4-41-6': 'However, if the operators of interest can be expressed entirely in terms of the collective degrees of freedom at the QCD level, the functional integral approach becomes unnecessary (for the same reason it was unnecessary in the derivation of the collective Hamiltonian).', 'hep-ph-0012138-4-41-7': 'At NLO, one can immediately express the effective operators in terms of the same collective degrees of freedom.', 'hep-ph-0012138-4-42-0': 'To see how this works, consider some operator which at the QCD level can be expressed entirely in terms of the collective operators.', 'hep-ph-0012138-4-42-1': 'We seek to find an analogous operator for use in the effective theory whose matrix elements in the effective theory reproduces the matrix elements of the QCD operator in the QCD states (up to correction at NNLO).', 'hep-ph-0012138-4-42-2': 'Let us start by considering the matrix elements at the QCD level.', 'hep-ph-0012138-4-42-3': 'From the discussion of the structure of the Hilbert space at the QCD level in Sec. [REF], it is apparent that a low-energy (i.e. with excitation energy of order [MATH]) heavy baryon state in a [MATH] expansion at NLO can be written as: [EQUATION] where [MATH] is defined as a product state as in Eq. ([REF]), and [MATH] are constants of order unity.', 'hep-ph-0012138-4-43-0': 'The sum in the second term of Eq. ([REF]) is restricted to contributions with [MATH] due to the product structure of the state.', 'hep-ph-0012138-4-43-1': 'If the sum were not restricted, then there would be contributions of the form [EQUATION] where, [MATH] is simply a single state in the purely collective space.', 'hep-ph-0012138-4-43-2': 'Thus, the contribution to the sum from Eq. ([REF]) is of the form of a single product state.', 'hep-ph-0012138-4-43-3': 'If there are contributions of the form as in Eq. ([REF]) we can replace [MATH] by [MATH] and thereby eliminate contributions with [MATH] from the sum.', 'hep-ph-0012138-4-43-4': 'Doing this has the advantage of removing any possibility of double counting these contributions.', 'hep-ph-0012138-4-44-0': 'Now let us evaluate the matrix element of a purely collective operator, [MATH], i.e. an operator constructed entirely of the collective operators [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-4-44-1': 'From the structure of the state in Eq. ([REF]), the matrix element between two low-energy heavy baryon states (given at NLO) is given by [EQUATION]', 'hep-ph-0012138-4-44-2': 'Thus, up to NNLO corrections the QCD matrix element is simply the matrix element in the collective part of the Hilbert space.', 'hep-ph-0012138-4-44-3': 'On the other hand, the effective Hamiltonian we use at this order with the operators [MATH] replaced by c-numbers is designed to correctly describe the collective part of the Hilbert space at this order.', 'hep-ph-0012138-4-45-0': 'The preceding argument shows, that in the case of purely collective operators the operators in the effective theory are related to the QCD operators in a trivial way up to corrections at NNLO.', 'hep-ph-0012138-4-45-1': 'A purely collective operator at the QCD level is of the form [EQUATION] where the superscript QCD indicates that all of the operators are at the QCD level.', 'hep-ph-0012138-4-45-2': 'The effective operator up to order [MATH] corrections is then obtained by keeping the same operator structure of [MATH], but replacing [MATH], [MATH], [MATH] and [MATH] with the collective operators [MATH], [MATH], [MATH] and [MATH]: [EQUATION].', 'hep-ph-0012138-4-45-3': 'This is essentially a nonrenormalization theorem to this order.', 'hep-ph-0012138-4-45-4': 'The effects of integrating out collective degrees of freedom only renormalize the effective operators at NNLO.', 'hep-ph-0012138-4-46-0': 'In a similar way, all purely non-collective operators at the QCD level, i.e. all operators which commute with all of the collective operators, simply become c-numbers in the effective theory with a value equal to the ground state expectation value of the analogous QCD operator.', 'hep-ph-0012138-4-46-1': 'These general principles can be used to deduce the form of several operators in the effective theory which will prove useful in the calculation of electroweak observables for heavy baryon states.', 'hep-ph-0012138-4-47-0': 'Let us first consider the operator which generates Lorentz boosts, [MATH].', 'hep-ph-0012138-4-47-1': 'The unitary operator for a Lorentz boost is [MATH].', 'hep-ph-0012138-4-47-2': 'The boost operator is useful, for example, in calculating form factors which involve matrix elements of states with different momenta.', 'hep-ph-0012138-4-47-3': 'One can use the boost operator to generate these states from a standard state in its rest frame.', 'hep-ph-0012138-4-47-4': 'From Poincare invariance, the boost operator is given at the field theoretic level by [EQUATION] where the second equality follows from Eq. ([REF]).', 'hep-ph-0012138-4-47-5': 'Thus, up to NLO at the QCD level [MATH] and from the arguments given above, this can immediately be taken over in the effective theory.', 'hep-ph-0012138-4-48-0': 'Among the observables which can be studied using the combined expansion are matrix elements of the electroweak current [MATH], where subscripts [MATH] and [MATH] refer to the heavy quark flavors and [MATH] is the Dirac structure of the left-hand current, [MATH].', 'hep-ph-0012138-4-48-1': 'We will show here that the leading term in the combined heavy quark and large [MATH] expansion of the current [MATH] can be expressed in terms of the collective variables [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0012138-4-48-2': 'Moreover, we will show that the lowest order corrections come at order [MATH] and not at [MATH], so that they contribute only at NNLO.', 'hep-ph-0012138-4-49-0': 'The combined expansion of the current [MATH] can be done analogously to the pure heavy quark expansion in the Heavy Quark Effective Theory (HQET) [CITATION].', 'hep-ph-0012138-4-49-1': 'The idea is to decompose the total heavy quark field into "large" and "small" components or "light" and "heavy" fields with the latter being suppressed by a factor of [MATH] relative to the "large" component.', 'hep-ph-0012138-4-49-2': 'It is then possible to eliminate the "small" component in terms of the "large" component by means of a power series expansion.', 'hep-ph-0012138-4-49-3': 'Near the combined limit considered here, the pure heavy quark expansion breaks down.', 'hep-ph-0012138-4-49-4': 'It has been shown in Refs. [CITATION] that the next-to-leading order terms in the pure heavy quark expansion of the matrix elements of the current [MATH] between the heavy baryon states contain coefficients proportional to [MATH] where [MATH] (with [MATH] being the total mass of a heavy baryon).', 'hep-ph-0012138-4-49-5': 'In the HQET the constant [MATH] is treated as being much smaller than [MATH], so that the NLO operators are indeed suppressed.', 'hep-ph-0012138-4-49-6': 'The situation is different, however, near the combined limit where [MATH] is of order [MATH], so that [MATH] and [MATH].', 'hep-ph-0012138-4-49-7': 'Hence, these corrections cannot be neglected near the combined limit.', 'hep-ph-0012138-4-50-0': 'The appearance of large corrections near the combined limit is a result of the way the total heavy quark field is separated into its "large" and "small" components.', 'hep-ph-0012138-4-50-1': 'The definition of these components is necessarily different from the analogous decomposition in HQET.', 'hep-ph-0012138-4-50-2': 'Near the combined limit, as in HQET, the heavy quark is not far off-shell since its interaction with the brown muck is of order [MATH] while its mass is of order [MATH].', 'hep-ph-0012138-4-50-3': 'However, the space-time dependence of the heavy quark is much different near the combined limit.', 'hep-ph-0012138-4-50-4': 'Near the pure heavy quark limit the brown muck mass is formally much less than the heavy quark mass.', 'hep-ph-0012138-4-50-5': 'As a result, the space-time dependence of the heavy quark is essentially determined by the free particle Lagrangian.', 'hep-ph-0012138-4-50-6': 'In HQET, therefore, it is useful to redefine the heavy quark field by removing the phase factor of the solution of the free Dirac equation, namely [MATH].', 'hep-ph-0012138-4-50-7': 'Near the combined limit the heavy quark remains essentially nonrelativistic.', 'hep-ph-0012138-4-50-8': 'However, the space-time dependence of the heavy quark is much different from that of a free particle.', 'hep-ph-0012138-4-50-9': 'The phase factor now should contain a contribution due to the brown muck mass, which is formally of the same order as the heavy quark mass near the combined limit.', 'hep-ph-0012138-4-50-10': 'This naturally leads to the following redefinition of the heavy quark field: [EQUATION] where [MATH] is the total 4-velocity of the heavy baryon, the operators [MATH] in the rest frame of the heavy baryon project out the upper and lower components of the heavy quark field, and [MATH] here represents a space-time point.', 'hep-ph-0012138-4-50-11': 'As in HQET, the fields [MATH] and [MATH] satisfy conditions [MATH] and [MATH].', 'hep-ph-0012138-4-50-12': 'Fields [MATH] and [MATH] are defined for each value of a heavy baryon 4-velocity [MATH] as can be seen from Eq. ([REF]).', 'hep-ph-0012138-4-50-13': 'The redefinition in Eq. ([REF]) has the effect of removing both the heavy quark mass, [MATH], and the nucleon mass, [MATH], from the dynamics, leaving a dynamical scale of order [MATH], i.e., the scale of the interaction Hamiltonian near the combined limit which is [MATH].', 'hep-ph-0012138-4-51-0': 'The heavy quark part of the QCD Lagrangian density in terms of the fields [MATH] and [MATH] has the form, [EQUATION] where [MATH] is the "transverse part" of the covariant derivative [MATH].', 'hep-ph-0012138-4-51-1': 'The heavy quark Lagrangian in Eq. ([REF]) written in terms of the fields [MATH] and [MATH] differs from its analog in HQET due to the additional term proportional to [MATH].', 'hep-ph-0012138-4-51-2': 'This [MATH] term would naively suggest that the fields [MATH] and [MATH] are both heavy with masses [MATH] and [MATH], respectively, which apparently prevents integrating out the "heavy" field [MATH].', 'hep-ph-0012138-4-52-0': 'This problem is easily resolved if one considers the full Lagrangian density of the system: [EQUATION] where the sum is over all light quarks, and [MATH] is the Yang-Mills Lagrangian density.', 'hep-ph-0012138-4-52-1': 'The total Lagrangian density in Eq. ([REF]) should be re-expressed so as to build the brown muck contribution into the heavy degrees of freedom.', 'hep-ph-0012138-4-52-2': 'This is completely analogous to the Hamiltonian treatment in Sec. [REF].', 'hep-ph-0012138-4-52-3': 'In that case it is done by adding and subtracting to the Hamiltonian a quantity which is an overall constant [MATH] and regrouping terms according to their [MATH] counting scaling.', 'hep-ph-0012138-4-52-4': 'In the Lagrangian formalism this can be accomplished in the following way using a Lorentz covariant operator [MATH]: [EQUATION] where [EQUATION] and [EQUATION].', 'hep-ph-0012138-4-52-5': 'In the rest frame of a heavy baryon operator, [MATH] is equal to the heavy quark density operator [MATH] and its spatial integral equals to one in the Hilbert space of heavy baryons.', 'hep-ph-0012138-4-52-6': 'Thus, in the Hamiltonian the additional term corresponds to removing from the light degrees of freedom an overall constant [MATH].', 'hep-ph-0012138-4-53-0': 'The additional operator proportional to the nucleon mass [MATH] can easily be expressed in terms of the fields [MATH] and [MATH]: [EQUATION].', 'hep-ph-0012138-4-53-1': 'This operator cancels the last term in Eq. ([REF]) so that the Lagrangian density [MATH] has the form: [EQUATION] where the condition [MATH] is used.', 'hep-ph-0012138-4-53-2': 'Thus, the "large" component [MATH] describes a massless field while the "small" component [MATH] has the mass [MATH].', 'hep-ph-0012138-4-53-3': 'Using equations of motion the "heavy" field [MATH] can be expressed in terms of the "light" field [MATH] as follows: [EQUATION]', 'hep-ph-0012138-4-53-4': 'The relation between fields [MATH] and [MATH] in Eq. ([REF]) is identical in form to that in HQET between the "large" and "small" components of the heavy quark field [MATH].', 'hep-ph-0012138-4-54-0': 'We can now expand the field [MATH], the Lagrangian density in Eq. ([REF]), and the current [MATH] in powers of [MATH].', 'hep-ph-0012138-4-54-1': 'This can be done by eliminating fields [MATH] (using Eq. ([REF])) and expanding the numerator in powers of [MATH].', 'hep-ph-0012138-4-54-2': 'Thus, the combined expansion of the heavy quark field including terms up to order [MATH] is, [EQUATION]', 'hep-ph-0012138-4-54-3': 'Using this expansion the Lagrangian density [MATH] including terms up to order [MATH] has the form: [EQUATION] where [MATH] is a strong coupling constant and [MATH] is the gluon field strength tensor.', 'hep-ph-0012138-4-54-4': 'In a similar way we can expand the current [MATH]; again keeping terms up to [MATH] we get, [EQUATION]', 'hep-ph-0012138-4-54-5': 'The expressions in Eqs. ([REF]), ([REF]), ([REF]) are identical in form to the analogous quantities in HQET.', 'hep-ph-0012138-4-54-6': 'However, there is an important difference: these expressions represent the combined expansion in powers of [MATH] and not the [MATH] expansion of HQET.', 'hep-ph-0012138-4-54-7': 'In other words, by defining fields [MATH] and [MATH] appropriately we were able to resum implicitly all the [MATH] corrections in HQET.', 'hep-ph-0012138-4-55-0': 'The correction terms in the combined expansion of the electroweak current, Eq. ([REF]), come at order [MATH] and not at [MATH].', 'hep-ph-0012138-4-55-1': 'It will be shown in Ref. [CITATION] that these corrections contribute to the matrix elements at order [MATH] as well, [MATH], they do not induce any anomalously large constants which violate the combined power counting of these operators.', 'hep-ph-0012138-4-55-2': 'Thus, at NLO in our expansion only the leading order operator, [MATH], contributes to the matrix elements.', 'hep-ph-0012138-4-55-3': 'In addition, it will be shown in Ref. [CITATION] that in the combined limit at NLO there is only a single form factor which parameterizes the matrix elements of the current [MATH] in the [MATH] sector.', 'hep-ph-0012138-4-55-4': 'Accordingly, we can consider one Lorentz component of the current [MATH], namely the [MATH] component.', 'hep-ph-0012138-4-55-5': 'Thus, to completely determine the matrix elements at NLO in the combined expansion, we need to know the form of only one operator, namely [MATH].', 'hep-ph-0012138-4-56-0': 'Let us first consider the flavor conserving operator [MATH].', 'hep-ph-0012138-4-56-1': 'As we are working near the heavy quark limit it is legitimate to expand the state in a fock-space decomposition for the number of heavy quarks and antiquarks.', 'hep-ph-0012138-4-56-2': 'For a heavy baryon state standard heavy quark analysis implies that the leading fock component has one heavy quark; the first sub-leading fock component with two quarks and one antiquark is suppressed by a factor of [MATH].', 'hep-ph-0012138-4-56-3': 'Thus at NLO in [MATH] we can neglect the sub-leading fock component and treat the state as having a single heavy quark.', 'hep-ph-0012138-4-56-4': 'This implies that acting in our heavy baryon Hilbert space, the operator [MATH] is [MATH].', 'hep-ph-0012138-4-56-5': 'Since this is true at the QCD level at NLO it immediately holds in the effective theory at that order.', 'hep-ph-0012138-4-57-0': 'In order to describe the flavor changing current we need to enlarge the Hilbert space of our effective theory to account for electroweak transitions between heavy baryons with heavy quarks of different flavors.', 'hep-ph-0012138-4-57-1': 'The new Hilbert space is the direct product of the Hilbert spaces for baryons with a given flavor of a heavy quark described in Sec. [REF].', 'hep-ph-0012138-4-57-2': 'Any state of the enlarged Hilbert space can be written as [MATH]-dimensional vector, where [MATH] is a number of heavy flavors.', 'hep-ph-0012138-4-57-3': 'Each component of this vector describes a state with a given flavor.', 'hep-ph-0012138-4-57-4': 'The effective Hamiltonian can be readily generalized to include baryons of various heavy quark flavors.', 'hep-ph-0012138-4-57-5': 'It can be written as [MATH] diagonal matrix.', 'hep-ph-0012138-4-57-6': 'For example, for two heavy flavors ([MATH] charm and bottom) the effective Hamiltonian including terms up to NLO in [MATH] has the form: [EQUATION] where the diagonal entries [MATH] and [MATH] are the effective Hamiltonians as in Eq. ([REF]).', 'hep-ph-0012138-4-57-7': 'Analogously, QCD operators acting in the enlarged Hilbert space become [MATH] matrices.', 'hep-ph-0012138-4-57-8': 'The flavor changing weak transitions can be described by off-diagonal matrix elements of an operator [MATH], where the subscripts [MATH] and [MATH] can take integer values from [MATH] to [MATH].', 'hep-ph-0012138-4-57-9': 'This operator can be written in terms of collective operators only.', 'hep-ph-0012138-4-57-10': 'It follows from arguments analogous to the case of the flavor conserving electroweak operator - [MATH] - that the flavor changing operator acting on the enlarged Hilbert space has the form (for [MATH]): [EQUATION]', 'hep-ph-0012138-4-57-11': 'The diagonal elements of the operator in Eq. ([REF]) correspond to flavor conserving operators, while the off-diagonal elements correspond to flavor changing operators.', 'hep-ph-0012138-4-57-12': 'Their action is to instantaneously change the heavy quark flavor at the position where a heavy baryon interacts with the current.', 'hep-ph-0012138-4-57-13': 'Since the operator in Eq. ([REF]) is written in terms of the collective operators only, the corresponding effective operator has an identical form.', 'hep-ph-0012138-4-58-0': 'Finally, let us consider the operator defined by [EQUATION] so that [MATH] represents the contribution to the baryon dipole moment coming from light quarks.', 'hep-ph-0012138-4-58-1': 'In calculating electromagentic transitions between heavy baryons it is useful to divide the electromagnetic current into pieces coming from the heavy and the light quarks and to further divide the light quark contribution into isovector and isoscalar pieces.', 'hep-ph-0012138-4-58-2': 'The isoscalar piece is simply the light-quark contribution to the baryon current.', 'hep-ph-0012138-4-58-3': 'Thus, matrix elements of [MATH] must be calculated to describe electric dipole transitions of heavy baryons.', 'hep-ph-0012138-4-59-0': 'At first sight, it might appear to be impossible to express the operator [MATH] in terms of collective operators.', 'hep-ph-0012138-4-59-1': 'Although [MATH] - the collective position operator for light quarks - is intuitively something like the position of the brown muck and clearly transforms in the same way as [MATH], at the operator level [MATH] is manifestly not identical to [MATH].', 'hep-ph-0012138-4-59-2': 'Recall that by construction [MATH] acts as a generator of boosts for the light degrees of freedom (up to order [MATH] corrections).', 'hep-ph-0012138-4-59-3': 'On the other hand, the operator [MATH] acts only on the quarks and not on the gluons.', 'hep-ph-0012138-4-59-4': 'Thus [MATH] cannot boost the gluons and therefore does not act as a boost for the light degrees of freedom (which include both quarks and gluons).', 'hep-ph-0012138-4-59-5': 'Thus, the operator [MATH] is clearly distinct from [MATH] and there is apparently no simple way to express [MATH] entirely in terms of collective operators.', 'hep-ph-0012138-4-60-0': 'Remarkably, however, one can show that [EQUATION] for all [MATH], [MATH] in the collective subspace defined in Sec. [REF].', 'hep-ph-0012138-4-60-1': 'Thus, when one restricts consideration to the collective subspace which dominates the low energy physics, [MATH] is equivalent to [MATH].', 'hep-ph-0012138-4-61-0': 'To see how this comes about, we start by considering the commutation relations of [MATH] with [MATH], [MATH] and [MATH].', 'hep-ph-0012138-4-61-1': 'To determine the commutator with [MATH] we exploit the fact that [MATH] is a generator of translations for the light degrees of freedom so that [MATH], where [MATH] is a generic light quark field.', 'hep-ph-0012138-4-61-2': 'We can use this to deduce that [EQUATION]', 'hep-ph-0012138-4-61-3': 'The first equality in Eq. ([REF]) follows from the definition of [MATH], the second equality follows because [MATH] translates the light quark degrees of freedom, the third equality follows from a change of variables, and the final equality follows from the definition of [MATH] and the fact that [EQUATION] counts the net number of light quarks, which is [MATH] in a heavy baryon containing one heavy quark.', 'hep-ph-0012138-4-61-4': 'Equation ([REF]) implies that [EQUATION] while the fact that from its definition [MATH] has no heavy quark operators implies that [EQUATION]', 'hep-ph-0012138-4-61-5': 'Thus, the commutation relations of [MATH] with [MATH], [MATH] and [MATH] are identical to the commutation relations of [MATH] with [MATH], [MATH] and [MATH].', 'hep-ph-0012138-4-61-6': 'In Eqs. ([REF]), ([REF]) indices [MATH] indicate Cartesian components of [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-4-62-0': 'Next let us express [MATH] in terms of the various collective and non-collective operators in the problem.', 'hep-ph-0012138-4-62-1': 'Generically, one can represent an arbitrary operator [MATH] as a power series in our collective operators: [EQUATION] with [MATH] specifying a particular term in the expansion, and [MATH] indicating the three Cartesian directions.', 'hep-ph-0012138-4-62-2': 'The coefficients in the expansion, [MATH], are generally non-collective operators which commute with [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012138-4-62-3': 'Expanding one component of [MATH] in this form gives [EQUATION] where the fact that the [MATH] commutes with [MATH] or [MATH] impose the restriction that no nonzero powers of [MATH] or [MATH] can contribute to the series, while the commutation relation of Eq. ([REF]) implies that the only contributing term containing any powers of [MATH] is the linear term and that the coefficient of this term is unity.', 'hep-ph-0012138-4-63-0': 'The known parity and time reversal properties of [MATH] and [MATH] fix the parity and time reversal properties of the non-collective coefficient operators, [MATH].', 'hep-ph-0012138-4-63-1': 'Since [MATH] is time reversal even and parity odd while [MATH] is time reversal odd and parity even, it follows that: [EQUATION]', 'hep-ph-0012138-4-63-2': 'The significant point about Eq. ([REF]) is that every non-collective coefficient operator is odd under PT.', 'hep-ph-0012138-4-63-3': 'From the analysis of Sec. [REF], generic states can be written as superpositions of outer product states containing collective and intrinsic parts.', 'hep-ph-0012138-4-63-4': 'The collective subspace of the theory is the subspace where the intrinsic wave function is in its ground state.', 'hep-ph-0012138-4-63-5': 'Thus, an arbitrary state in the collective subspace can be written as [MATH] where [MATH] indicates the ground state of the intrinsic system.', 'hep-ph-0012138-4-63-6': 'Now consider the matrix element of a typical term in the series of Eq. ([REF]) between two collective states: [EQUATION] where we have separated the matrix element into its collective and intrinsic parts.', 'hep-ph-0012138-4-63-7': 'Clearly, the matrix element in Eq. ([REF]) vanishes: the intrinsic part of the of the matrix element is a diagonal matrix element of a PT odd operator in a state of good PT.', 'hep-ph-0012138-4-63-8': 'Thus, within the collective subspace all terms in the series of Eq. ([REF]) vanish except the first term.', 'hep-ph-0012138-4-63-9': 'This implies that within this subspace all matrix elements of [MATH] are identical to matrix elements of [MATH] and Eq. ([REF]) is established.', 'hep-ph-0012138-4-64-0': 'As written, the equivalence expressed in Eq. ([REF]) only holds for states in the collective subspace, i.e., states with the intrinsic subspace in its ground state.', 'hep-ph-0012138-4-64-1': 'As discussed in Sec. [REF], the non-collective operators in the Hamiltonian induce components with excited intrinsic states in the physical low-energy states.', 'hep-ph-0012138-4-64-2': 'This in turn spoils the formal equivalence in Eq. ([REF]) for the physical states.', 'hep-ph-0012138-4-64-3': 'Fortunately, however, it is straightforward to see that in the physical states, the matrix elements of [MATH] differ from those of [MATH] by an amount of order [MATH] (or less) and hence may be neglected at next-to-leading order in [MATH].', 'hep-ph-0012138-4-64-4': 'In the first place using reasoning analogous to that leading to Eq. ([REF]), it is easy to see that the matrix elements of the [MATH] between the intrinsic ground state and its excited states is of order [MATH].', 'hep-ph-0012138-4-64-5': 'On the other hand, the analysis of Sec. [REF] implies that admixtures of excited intrinsic states in the physical states are suppressed by at least order [MATH].', 'hep-ph-0012138-4-64-6': 'Combining these two facts one sees that the matrix elements of [MATH] and [MATH] in the physical states are equivalent at NLO.', 'hep-ph-0012138-4-65-0': 'The equivalence at NLO between the matrix elements of [MATH] and [MATH] may seem paradoxical.', 'hep-ph-0012138-4-65-1': 'After all, as discussed above, the two operators are clearly quite distinct at the QCD level.', 'hep-ph-0012138-4-65-2': 'It would seem that the fact that the operators are different would imply that the matrix elements of the two operators should be different in general.', 'hep-ph-0012138-4-65-3': 'The resolution of the paradox is quite simple: the matrix elements of the two operators are different in general.', 'hep-ph-0012138-4-65-4': 'They only coincide (up to NNLO corrections) for a very special class of states - the collective states with excitation energies of order [MATH].', 'hep-ph-0012138-4-65-5': 'For states with excitation energies of order unity, the matrix elements of the two operators can be quite different.', 'hep-ph-0012138-4-65-6': 'Fortunately, we will be using our effective theory to study these low-lying excitations only.', 'hep-ph-0012138-4-65-7': 'Thus, for the purpose of the effective theory at NLO, it is legitimate to replace [MATH] with [MATH].', 'hep-ph-0012138-4-66-0': '# Summary', 'hep-ph-0012138-4-67-0': 'In this paper we have derived an effective theory which will be used in Ref. [CITATION] to study the spectroscopy and electroweak decays of the low-energy states of heavy baryons.', 'hep-ph-0012138-4-67-1': 'The analysis is based largely on the formulation of the problem in Ref. [CITATION].', 'hep-ph-0012138-4-67-2': 'A power-counting scheme is developed in [MATH], where [MATH].', 'hep-ph-0012138-4-67-3': 'The kinematical variables of the problem [MATH], [MATH], [MATH] and [MATH] are directly related to QCD operators which are well defined up to corrections at relative order [MATH].', 'hep-ph-0012138-4-67-4': 'This formalism is based on standard large [MATH] and heavy quark analysis.', 'hep-ph-0012138-4-68-0': 'At the theoretical level, the analysis used in our derivation of the effective theory goes well beyond the analysis of Ref. [CITATION] which was largely kinematical and group-theoretical in nature.', 'hep-ph-0012138-4-68-1': 'In our derivation, the structure of the Hilbert space for heavy baryons plays a critical role.', 'hep-ph-0012138-4-68-2': 'In particular, as the [MATH] limit is approached, the heavy baryon states can be written as products of collective states (with excitation energies of order [MATH]) and intrinsic states (with excitation energies of order [MATH]).', 'hep-ph-0012138-4-68-3': 'To deduce this form of the Hilbert space we exploited the fact that our formulation as an expansion in [MATH] does not depend formally on whether the large [MATH] limit is taken first, the heavy quark limit is taken first or they are taken simultaneously.', 'hep-ph-0012138-4-68-4': 'By taking the large [MATH] limit first we can use standard large [MATH] reasoning to deduce the structure of the Hilbert space and then map back to the case where the limits are taken concurrently.', 'hep-ph-0012138-4-68-5': 'One essential point in this analysis is that matrix elements of operators which couple between different intrinsic states are characteristically down by a factor of [MATH] compared to matrix elements of operators with the same intrinsic state.', 'hep-ph-0012138-4-68-6': 'Thus, the effect of coupling to an excited intrinsic state and then back to the ground state is typically of order [MATH] and hence can be neglected at NLO in the [MATH] expansion.', 'hep-ph-0012138-4-69-0': 'From the kinematics of the problem, the underlying heavy quark theory, Poincare invariance, and the structure of the Hilbert space, we showed that the effective Hamiltonian up to order [MATH] can be written in the form of Eq. ([REF]).', 'hep-ph-0012138-4-69-1': 'At order [MATH] one sees that the excitation spectrum is fixed entirely by a single parameter [MATH], and at order [MATH] only one additional parameter - [MATH] - enters.', 'hep-ph-0012138-4-69-2': 'This is encouraging from a phenomenological perspective since we do not have a large number of coefficients to work with at NLO.', 'hep-ph-0012138-4-70-0': 'We also derived the certain effective operators at NLO.', 'hep-ph-0012138-4-70-1': 'It turns out that the operators discussed here are precisely those needed to compute interesting electroweak transitions of isoscalar heavy baryons at NLO.', 'hep-ph-0012138-4-70-2': 'The key point in the derivation is that all of these operators can be written in terms of the collective operators (up to possible NNLO corrections).', 'hep-ph-0012138-4-70-3': 'From this it was easy to show that the effect of the admixture of components only comes in at NNLO so that at NLO one can simply replace the QCD level collective operator by an effective operator of the same structure.', 'hep-ph-0012138-4-70-4': 'One operator, [MATH], which is effectively the dipole moment of the light quark contribution to the baryon density, was somewhat more subtle.', 'hep-ph-0012138-4-70-5': 'Although, as an operator it cannot be written directly in terms of the collective variables, we showed that its matrix elements between low-lying states are identical to those of the collective operator [MATH].', 'hep-ph-0012138-4-71-0': 'The effective theory derived here is suitable for the study of isoscalar heavy baryons at NLO in our [MATH] expansion.', 'hep-ph-0012138-4-71-1': 'In Ref. [CITATION] we will use the effective Hamiltonian of Sec. [REF] to study the spectroscopy and electroweak decays of [MATH] and [MATH] baryons and their excited states.', 'hep-ph-0012138-4-71-2': 'In order to calculate the semi-leptonic matrix elements we will use relations between form-factors analogous to those obtained in HQET [CITATION].', 'hep-ph-0012138-4-71-3': 'These matrix elements can then be determined using the effective operators discussed in Sec. [REF].', 'hep-ph-0012138-4-71-4': 'We will also calculate the radiative decay rates of the first excited states of [MATH] baryons.', 'hep-ph-0012138-4-72-0': 'hb0 C.K. Chow and T.D. Cohen, Phys.', 'hep-ph-0012138-4-72-1': 'A to be published.', 'hep-ph-0012138-4-72-2': 'hb3 A. Baccouche, C.K. Chow, T.D. Cohen and B.A. Gelman, "Excited Heavy Baryons and Their Symmetries III: Phenomenology", DOE/ER/40762-215, UM PP01-029, in preparation.', 'hep-ph-0012138-4-72-3': 'bst1 Z. Guralnik, M. Luke and A.V. Manohar, Nucl.', 'hep-ph-0012138-4-72-4': 'bst2 E. Jenkins, A.V. Manohar and M.B. Wise, Nucl.', 'hep-ph-0012138-4-72-5': 'bst4 C.K. Chow and M.B. Wise, Phys.', 'hep-ph-0012138-4-72-6': 'bst5 M. Rho, D.O. Riska and N.N. Scoccola, Z. Phys.', 'hep-ph-0012138-4-72-7': 'E4 47 (1995); bst7 J. Schechter, A. Subbaraman, S. Vaidya and H. Weigel, Nucl.', 'hep-ph-0012138-4-72-8': 'Glozman and D.O. Riska, Nucl.', 'hep-ph-0012138-4-72-9': 'SF1 J.L. Gervais and B. Sakita, Phys.', 'hep-ph-0012138-4-72-10': 'SF2 J.L. Gervais and B. Sakita, Phys.', 'hep-ph-0012138-4-72-11': 'SF3 R. Dashen, E. Jenkins and A.V. Manohar, Phys.', 'hep-ph-0012138-4-72-12': 'SF4 C. Carone, H. Georgi, S. Osofsky, Phys.', 'hep-ph-0012138-4-72-13': 'SF5 M. Luty and J. March-Russell, Nucl.', 'hep-ph-0012138-4-72-14': "LN1 G. 't Hooft, Nucl.", 'hep-ph-0012138-4-72-15': 'LN2 E. Witten, Nucl.', 'hep-ph-0012138-4-72-16': 'HQ1 N. Isgur and M.B. Wise, Phys.', 'hep-ph-0012138-4-72-17': 'HQ2 N. Isgur and M.B. Wise, Phys.', 'hep-ph-0012138-4-72-18': 'HQ3 N. Isgur and M.B. Wise, Nucl.', 'hep-ph-0012138-4-72-19': 'HQ4 H. Georgi, Nucl.', 'hep-ph-0012138-4-72-20': 'HQ5 T. Mannel, W. Roberts and Z. Ryzak, Nucl.', 'hep-ph-0012138-4-72-21': 'HQ6 F. Hussain, J.G. Korner, M. Kramer and G. Thompson, Z. Phys.', 'hep-ph-0012138-4-72-22': 'mQ1 M.E. Luke, Phys.', 'hep-ph-0012138-4-72-23': 'mQ2 H. Georgi, B. Grinstein and M.B. Wise, Phys.', 'hep-ph-0012138-4-72-24': 'mQ3 A.F. Falk, B. Grinstein and M.E. Luke, Nucl.', 'hep-ph-0012138-4-72-25': 'mQ4 M. Neubert, Phys.'}

1807.03070

{'1807.03070-1-0-0': 'We provide evidence for anomalous behavior in the quantum oscillations of the antiferromagnetic semimetal SmSb.', '1807.03070-1-0-1': 'Magnetotransport measurements in high magnetic fields reveal that SmSb has a non-trivial [MATH] Berry phase inferred from the Landau index analysis of Shubnikov-de Haas (SdH) oscillations.', '1807.03070-1-0-2': 'Furthermore, striking differences are found between the temperature dependence of the amplitudes of de Haas-van Alphen effect oscillations, which are well fitted by the Lifshitz-Kosevich (LK) formula across the measured temperature range, and those from SdH measurements which show a significant disagreement with LK behavior at low temperatures.', '1807.03070-1-0-3': 'Our findings of unusual quantum oscillations in an antiferromagnetic, mixed valence semimetal with a non-trivial Berry phase can provide an opportunity for studying the interplay between topology, electronic correlations and magnetism.', '1807.03070-1-1-0': '[PACS number(s)]', '1807.03070-1-2-0': '# Introduction', '1807.03070-1-3-0': 'The discovery of topological insulators with bulk band gaps but robust topological surface states protected by time reversal symmetry triggered much research into topological phases of matter [CITATION].', '1807.03070-1-3-1': 'Topological insulators are bulk insulators yet have conducting surface states as a consequence of the topologically non-trivial band structure, where a Dirac cone lies within the bulk gap [CITATION].', '1807.03070-1-3-2': 'Subsequently, a number of semimetals with non-trivial topologies have also been discovered.', '1807.03070-1-3-3': 'In some cases such as Dirac and Weyl semimetals, the electronic structure is gapless and contains Dirac or Weyl points at the positions where the bands cross with linear dispersions and the excitations are well described as Dirac or Weyl fermions [CITATION].', '1807.03070-1-3-4': 'Meanwhile in other topological semimetals, the band structure contains similar band inversion features to topological insulators, but the Fermi level does not lie within the band gap [CITATION].', '1807.03070-1-3-5': 'The role of electronic correlations in topological systems has also become of particular interest after the proposal that SmB[MATH] is a topological Kondo insulator, where the surface state lies within the gap opened due to the Kondo interaction [CITATION].', '1807.03070-1-3-6': 'One striking feature is the presence of quantum oscillations in SmB[MATH] from de Haas-van Alphen effect (dHvA) measurements [CITATION], despite the material being a bulk insulator at low temperatures, stimulating debate over whether these arise from the surface [CITATION], or if this is a bulk property [CITATION].', '1807.03070-1-3-7': 'Furthermore, Tan et al. found that the dHvA amplitude shows an anomalous increase at low temperatures below 1 K, and the origin of this highly unusual deviation is currently not resolved [CITATION].', '1807.03070-1-4-0': 'Recently, the [MATH](Sb,Bi)([MATH] = lanthanide) series of semimetals with a cubic rocksalt structure have attracted considerable interest.', '1807.03070-1-4-1': 'These materials are commonly found to have an extremely large magnetoresistance (XMR), generally attributed to the nearly perfect compensation of electron and hole carriers [CITATION].', '1807.03070-1-4-2': 'Meanwhile, non-trivial band topologies have been theoretically predicted [CITATION], and evidence for topological states is found experimentally for some compounds from ARPES and transport measurements [CITATION], while other members appear to be topologically trivial [CITATION].', '1807.03070-1-4-3': 'In LaSb, it was suggested that the low-temperature plateau in the temperature dependence of the resistivity in applied magnetic fields was due to a topologically protected surface state, similar to that observed in SmB[MATH] [CITATION].', '1807.03070-1-4-4': 'However, ARPES measurements show conflicting results over whether the band topology is trivial [CITATION], while the resistivity plateau can be explained by a three band model with nearly perfect electron-hole compensation [CITATION].', '1807.03070-1-4-5': 'Furthermore, by substituting for lanthanide elements with partially filled [MATH] electron shells, the effect of tuning the spin-orbit coupling, magnetism and electronic correlations can be studied.', '1807.03070-1-4-6': 'For example, CeSb shows evidence for Weyl fermions in the field-induced ferromagnetic state from both theoretical calculations and the measurement of a negative longitudinal magnetoresistance [CITATION].', '1807.03070-1-4-7': 'NdSb has also been proposed to be a topological semimetal from the observation of a Dirac-like semi-metal phase using ARPES, as well as from band structure calculations and analysis of the Landau indices [CITATION].', '1807.03070-1-5-0': 'In this work, we report magnetoresistance and quantum oscillation measurements of the antiferromagnet SmSb [CITATION].', '1807.03070-1-5-1': 'We find clear evidence for a non-trivial Berry phase in SmSb, which is derived from an analysis of the Landau indices obtained from the results of high field magnetotransport experiments.', '1807.03070-1-5-2': 'Moreover, striking differences are found between the temperature dependences of the quantum oscillation amplitudes from Shubnikov-de Haas (SdH) and dHvA measurements.', '1807.03070-1-5-3': 'The amplitudes of the SdH oscillations show anomalous behavior at low temperatures, which may arise due to the presence of multiple Fermi surface sheets, or correspond to a conduction channel related to the topological state.', '1807.03070-1-6-0': '# Results', '1807.03070-1-7-0': 'The temperature dependence of the electrical resistivity [[MATH]] of SmSb is displayed in Fig. [REF]a, where the data up to 300 K is in the inset.', '1807.03070-1-7-1': 'The [MATH] data show a small hump at around 65 K, which is explained as resulting from the splitting of the crystalline electric fields [CITATION], as well as a sharp drop of [MATH] at the antiferromagnetic transition at [MATH] K [CITATION].', '1807.03070-1-7-2': 'While [MATH] changes very little in applied magnetic fields [CITATION], an upturn appears in [MATH] at low temperatures, which becomes more pronounced as the field is increased.', '1807.03070-1-7-3': 'As shown in Figs. [REF]a and [REF]b, there is a strong increase of [MATH] in-field below [MATH], before reaching a nearly constant value below around 1 K. Figure [REF]c shows the field dependence of the magnetoresistance [[MATH]] of SmSb up to 15 T at various temperatures down to 0.3 K.', '1807.03070-1-7-4': 'At 0.3 K [MATH], which decreases rapidly as the temperature is increased.', '1807.03070-1-7-5': 'Figure [REF]d displays measurements performed up to higher fields at 1.8 K, where the magnetoresistance increases quadratically to about 5558 at 60 T.', '1807.03070-1-7-6': 'The magnetoresistance of SmSb is one of the largest observed among the [MATH]Sb family of compounds [CITATION], and the quadratic field dependence suggests that the XMR most likely results from electron-hole carrier compensation [CITATION].', '1807.03070-1-7-7': 'In this case, the increase of mobility below [MATH] may lead to both the small low temperature values of [MATH] in zero-field, as well as the very large magnetoresistance and low-temperature plateau of [MATH] in high fields.', '1807.03070-1-7-8': 'While this low temperature plateau has also been observed in other [MATH]Sb materials [CITATION], the appearance of the plateau at a lower temperature in SmSb is likely due to the comparatively low [MATH].', '1807.03070-1-8-0': 'To examine the topology of the electronic structure of SmSb, we also measured the SdH effect in the resistivity, so as to determine the residual Landau index.', '1807.03070-1-8-1': 'In particular, to reliably extrapolate to the zeroth level, high field quantum oscillation experiments are necessary, and therefore the magnetoresistance was measured in pulsed magnetic fields up to 60 T. Figure.', '1807.03070-1-8-2': '[REF]a displays the [MATH] dependence of the quantum oscillations for two samples S2 and S6 in applied fields up to 32 T and 60 T respectively along the [001] direction, after subtracting the background contribution.', '1807.03070-1-8-3': 'Landau indices were assigned to the positions of the valleys, as displayed in Fig. [REF]b.', '1807.03070-1-8-4': 'The data were fitted with a linear temperature dependence and the extrapolated residual Landau indices are [MATH] for sample S2 and [MATH] for sample S6.', '1807.03070-1-8-5': 'This corresponds to a [MATH] Berry phase, which is evidence for a non-trivial band topology [CITATION].', '1807.03070-1-8-6': 'Importantly, since the lowest observed Landau index is 6 in the 60 T measurements, the uncertainty associated with the fitted intercept is small, indicating that the determination of the Berry phase is reliable.', '1807.03070-1-8-7': 'Therefore our results strongly suggest that the electronic structure of SmSb is topologically non-trivial.', '1807.03070-1-8-8': 'The origin of this Berry phase is currently not resolved, and further studies are necessary to determine if it originates from a gapless topological state such as in Dirac or Weyl semimetals [CITATION].', '1807.03070-1-8-9': 'It is also noted that evidence for a [MATH] Berry phase was also found in LaBi [CITATION], where both an odd number of band inversions and surface states with an odd number of massless Dirac cones were observed in ARPES measurements [CITATION].', '1807.03070-1-9-0': 'In addition, we also performed field-dependent ac susceptibility measurements down to low temperatures where the presence of dHvA oscillations are clearly observed.', '1807.03070-1-9-1': 'For comparison, SdH and dHvA oscillations are shown for various temperatures down to 0.3 K in Figs. [REF]a and [REF]b respectively, after subtracting the background contribution.', '1807.03070-1-9-2': 'The respective fast Fourier transformations (FFT) are shown in Figs. [REF]c and [REF]d, where two principal frequencies were observed, which remain unchanged below [MATH], similar to previous reports [CITATION].', '1807.03070-1-9-3': 'The observed values are [MATH]334 T and 328 T, and [MATH] 633 T and [MATH] 590 T for SdH and dHvA respectively, where the differences may be due to a small misalignment.', '1807.03070-1-9-4': 'It can be seen that the amplitudes of the SdH oscillations in [MATH] corresponding to both the [MATH] and [MATH] bands increase significantly with decreasing temperature below 3 K, while in dHvA measurements the change is more gradual.', '1807.03070-1-9-5': 'Furthermore, the amplitude of the [MATH]-band oscillations is maximum at around 0.8 K, and upon further decreasing the temperature, the amplitude decreases.', '1807.03070-1-9-6': 'Meanwhile, the angular dependence of [MATH] is consistent with a two-dimensional Fermi surface (see Supplementary Information), although in general it is difficult to distinguish this scenario from that of three-dimensional ellipsoidal pockets [CITATION].', '1807.03070-1-10-0': 'Figures [REF]a and [REF]b present the temperature dependence of oscillation amplitudes from both SdH and dHvA measurements.', '1807.03070-1-10-1': 'Since there is a rapid change of the resistivity both with changing temperature and field, the SdH oscillation amplitudes are displayed as [MATH]/[MATH] is proportional to the density of states and that the temperature dependence is well described by the LK formula, for multiband systems the situation can be more complicated due to the different contributions to the conductivity from each band [CITATION].', '1807.03070-1-10-2': 'This effect will not influence the dHvA results, which can be well described by the LK formula across the whole temperature range.', '1807.03070-1-10-3': 'We note that below around [MATH], the normalized oscillation amplitude of the [MATH]-band is anomalously low relative to the dHvA results, while for the [MATH]-band there is an increase.', '1807.03070-1-10-4': 'This suggests that with decreasing temperature, there may be changes of the relative contributions to the conductivity from the two bands.', '1807.03070-1-10-5': 'Therefore, for systems with multiple FS sheets and large magnetoresistance, analysis of the cyclotron masses using SdH measurements may be difficult.', '1807.03070-1-10-6': 'As a result, the origin of the dramatic departure from conventional behavior in the SdH amplitudes is an open question, and in particular since this deviation onsets near [MATH], the possible relationship between the antiferromagnetic state and the anomalous behavior also needs to be explored.', '1807.03070-1-11-0': 'To summarize, we find evidence for a [MATH] Berry phase in SmSb from the analysis of the Landau indices at high fields.', '1807.03070-1-11-1': 'Meanwhile the presence of an extremely large magnetoresistance which increases quadratically with increasing applied magnetic field without saturation suggests the compensation of two types of carriers with high mobilities.', '1807.03070-1-11-2': 'Furthermore, our quantum oscillation measurements show that the amplitudes of dHvA oscillations are well described by the canonical LK formula, while those from the SdH effect show anomalous behavior at low temperatures.', '1807.03070-1-11-3': 'This unusual behavior in SdH oscillations may be a consequence of the non-trivial band topology, or could arise due to different contributions to the total conductivity from multiple FS sheets.', '1807.03070-1-11-4': 'As a result further studies such as ARPES are required to understand the origin of the non-trivial Berry phase, as well as any relationship with the anomalous SdH oscillations.', '1807.03070-1-12-0': '# Methods', '1807.03070-1-13-0': 'Single crystals of SmSb were synthesized using an Sn flux-method [CITATION].', '1807.03070-1-13-1': 'The elements were combined in a molar ratio Sm:Sb:Sn of 1:1:20 in an Al[MATH]O[MATH] crucible before being sealed in evacuated quartz ampoules.', '1807.03070-1-13-2': 'These were slowly cooled from 1100[MATH]C down to 800[MATH]C, before centrifuging to remove the Sn flux.', '1807.03070-1-13-3': 'The resulting crystals are cubic with typical dimensions of around 3 mm.', '1807.03070-1-13-4': 'The residual resistivity ratio of [MATH], indicates a high sample quality.', '1807.03070-1-14-0': 'Resistivity and magnetoresistance measurements were performed using a Quantum Design Physical Property Measurement System (PPMS) from 300 K to 2 K with a maximum field of 9 T. Four Pt wires were attached to the sample by spot welding so that the current was applied along the [100] direction.', '1807.03070-1-14-1': 'The low temperature resistivity and ac susceptibility were measured in a [MATH]He system with a base temperature of 0.27 K and a maximum applied magnetic field of 15 T.', '1807.03070-1-14-2': 'The high field magnetoresistance measurements were performed up to 60 T using a pulsed field magnet at the Los Alamos National Laboratory, USA, while measurements up to 32 T were carried out using the Cell 5 Water-Cooling Magnet at the High Magnetic Field Laboratory of the Chinese Academy of Sciences.', '1807.03070-1-15-0': 'acknowledgments', '1807.03070-1-16-0': 'We thank Y. Liu, C. Cao, Q. Si, Y. Zhou, F. Steglich, P. Li and Z. Z. Wu for interesting discussions and helpful suggestions.', '1807.03070-1-16-1': 'This work was supported by the National Key RD Program of China (Grants No. 2017YFA0303100 and No. 2016YFA0300202), the National Natural Science Foundation of China (Grants No. U1632275, No. 11604291 and No. 11474251), the Science Challenge Project of China (Project No. TZ2016004), and the Innovative Program of Development Foundation of Hefei Center for Physical Science and Technology (Project No. 2017FXCX001).', '1807.03070-1-16-2': 'A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1644779* and the State of Florida.', '1807.03070-1-16-3': 'JS acknowledges support from the DOE BES program "Science of 100 T".', '1807.03070-1-17-0': '# Additional information Correspondence and requests for materials should be addressed to H. Q. Yuan (hqyuan@zju.edu.cn)', '1807.03070-1-18-0': '# Author contributions The project was conceived by F.W, C.Y.G. and H.Q.Y..', '1807.03070-1-18-1': 'The crystals were grown by F.W..', '1807.03070-1-18-2': 'F.W., C.Y.G.,J.L.Z, Y.C. and J.S. performed the measurements, which were analyzed by F.W, C.Y.G., M.S, and H.Q.Y.', '1807.03070-1-18-3': 'The manuscript were written by F.W, C.Y.G., M.S. and H.Q.Y. All authors participated in discussions.'}

{'1807.03070-2-0-0': 'Topologically non-trivial electronic structures can give rise to a range of unusual physical phenomena, and the interplay of band topology with other effects such as electronic correlations and magnetism requires further exploration.', '1807.03070-2-0-1': 'The rare earth monopnictides [MATH](Sb,Bi) ([MATH] = lanthanide) are a large family of semimetals where these different effects may be tuned by the substitution of rare-earth elements.', '1807.03070-2-0-2': 'Here we observe anomalous behavior in the quantum oscillations of one member of this family, antiferromagnetic SmSb.', '1807.03070-2-0-3': 'The analysis of Shubnikov-de Haas (SdH) oscillations provides evidence for a non-zero Berry phase, indicating a non-trivial topology of the [MATH]-band.', '1807.03070-2-0-4': 'Furthermore, striking differences are found between the temperature dependence of the amplitudes of de Haas-van Alphen effect oscillations, which are well fitted by the Lifshitz-Kosevich (LK) formula across the measured temperature range, and those from SdH measurements which show a significant disagreement with LK behavior at low temperatures.', '1807.03070-2-0-5': 'Our findings of unusual quantum oscillations in an antiferromagnetic, mixed valence semimetal with a possible non-trivial band topology can provide an opportunity for studying the interplay between topology, electronic correlations and magnetism.', '1807.03070-2-1-0': '# Introduction', '1807.03070-2-2-0': 'The discovery of topological insulators with bulk band gaps but robust topological surface states protected by time reversal symmetry triggered much research into topological phases of matter [CITATION].', '1807.03070-2-2-1': 'Topological insulators are bulk insulators yet have conducting surface states as a consequence of the topologically non-trivial band structure, where a Dirac cone lies within the bulk gap [CITATION].', '1807.03070-2-2-2': 'Subsequently, a number of semimetals with non-trivial topologies have also been discovered.', '1807.03070-2-2-3': 'In some cases such as Dirac and Weyl semimetals, the electronic structure is gapless and contains Dirac or Weyl points at the positions where the bands cross with linear dispersions and the excitations are well described as Dirac or Weyl fermions [CITATION].', '1807.03070-2-2-4': 'Meanwhile in other topological semimetals, the band structure contains similar band inversion features to topological insulators, but the Fermi level does not lie within the band gap [CITATION].', '1807.03070-2-2-5': 'The role of electronic correlations in topological systems has also become of particular interest after the proposal that SmB[MATH] is a topological Kondo insulator, where the surface state lies within the gap opened due to the Kondo interaction [CITATION].', '1807.03070-2-2-6': 'One striking feature is the presence of quantum oscillations in SmB[MATH] from de Haas-van Alphen effect (dHvA) measurements [CITATION], despite the material being a bulk insulator at low temperatures, stimulating debate over whether these arise from the surface [CITATION], or if this is a bulk property [CITATION].', '1807.03070-2-2-7': 'Furthermore, Tan et al. found that the dHvA amplitude shows an anomalous increase at low temperatures below 1 K, and the origin of this highly unusual deviation is currently not resolved [CITATION].', '1807.03070-2-3-0': 'Recently, the [MATH](Sb,Bi)([MATH] = lanthanide) series of semimetals with a cubic rocksalt structure have attracted considerable interest.', '1807.03070-2-3-1': 'These materials are commonly found to have an extremely large magnetoresistance (XMR), generally attributed to the nearly perfect compensation of electron and hole carriers [CITATION].', '1807.03070-2-3-2': 'Meanwhile, non-trivial band topologies have been theoretically predicted [CITATION], and evidence for topological states is found experimentally for some compounds from ARPES and transport measurements [CITATION], while other members appear to be topologically trivial [CITATION].', '1807.03070-2-3-3': 'In LaSb, it was suggested that the low-temperature plateau in the temperature dependence of the resistivity in applied magnetic fields was due to a topologically protected surface state, similar to that observed in SmB[MATH] [CITATION].', '1807.03070-2-3-4': 'However, ARPES measurements show conflicting results over whether the band topology is trivial [CITATION], while the resistivity plateau can be explained by a three band model with nearly perfect electron-hole compensation [CITATION].', '1807.03070-2-3-5': 'Furthermore, by substituting for lanthanide elements with partially filled [MATH] electron shells, the effect of tuning the spin-orbit coupling, magnetism and electronic correlations can be studied.', '1807.03070-2-3-6': 'For example, CeSb shows evidence for Weyl fermions in the field-induced ferromagnetic state from both theoretical calculations and the measurement of a negative longitudinal magnetoresistance [CITATION].', '1807.03070-2-3-7': 'NdSb has also been proposed to be a topological semimetal from the observation of a Dirac-like semi-metal phase using ARPES, as well as from band structure calculations and analysis of the Landau indices [CITATION].', '1807.03070-2-4-0': 'In this work, we report magnetoresistance and quantum oscillation measurements of the antiferromagnet SmSb [CITATION].', '1807.03070-2-4-1': 'Based on the analysis of Shubnikov-de Haas (SdH) oscillations, we found that the phase of all the harmonic oscillations corresponding to the [MATH]-band is close to [MATH], while for the other bands it is almost zero.', '1807.03070-2-4-2': 'These results provide evidence that the band topology of the [MATH]-band is non-trivial.', '1807.03070-2-4-3': 'Moreover, striking differences are found between the temperature dependences of the quantum oscillation amplitudes from SdH and dHvA measurements.', '1807.03070-2-4-4': 'The amplitudes of the SdH oscillations show anomalous behavior at low temperatures, which may arise due to the presence of multiple Fermi surface sheets or a conduction channel related to the topological state.', '1807.03070-2-5-0': '# Results and Discussion', '1807.03070-2-6-0': 'The temperature dependence of the electrical resistivity [[MATH]] of SmSb is displayed in Fig. 1 a, where the data up to 300 K is shown in the inset.', '1807.03070-2-6-1': 'The [MATH] data show a small hump at around 65 K, which is explained as resulting from the splitting of the crystalline electric fields [CITATION], as well as a sharp drop of [MATH] at the antiferromagnetic transition at [MATH] K [CITATION].', '1807.03070-2-6-2': 'While [MATH] changes very little in applied magnetic fields [CITATION], an upturn appears in [MATH] at low temperatures, which becomes more pronounced as the field is increased.', '1807.03070-2-6-3': 'As shown in Fig. 1a and 1b, there is a strong increase of [MATH] in-field below [MATH], before reaching a nearly constant value below around 1 K. Figure 1c shows the field dependence of the magnetoresistance [[MATH]] of SmSb up to 15 T at various temperatures down to 0.3 K.', '1807.03070-2-6-4': 'At 0.3 K [MATH], which decreases rapidly as the temperature is increased.', '1807.03070-2-6-5': 'Figure 1d displays measurements performed up to higher fields at 1.8 K, where the magnetoresistance increases quadratically to about 5558 at 60 T.', '1807.03070-2-6-6': 'The magnetoresistance of SmSb is one of the largest observed among the [MATH]Sb family of compounds [CITATION], and the quadratic field dependence suggests that the XMR most likely results from electron-hole carrier compensation [CITATION].', '1807.03070-2-6-7': 'In this case, the increase of mobility below [MATH] may lead to both the small low temperature values of [MATH] in zero-field, as well as the very large magnetoresistance and low-temperature plateau of [MATH] in high fields.', '1807.03070-2-6-8': 'While this low temperature plateau has also been observed in other [MATH]Sb materials [CITATION], the appearance of the plateau at a lower temperature in SmSb is likely due to the comparatively low [MATH].', '1807.03070-2-7-0': 'To examine the topology of the electronic structure of SmSb, we also analyzed the SdH oscillations in the resistivity for the two samples displayed in Fig. 1d.', '1807.03070-2-7-1': 'From the fast Fourier transformation analysis (see Supplementary Figure 1), the most prominent oscillation frequencies consist of three fundamental frequencies ([MATH], [MATH] and [MATH]), and two harmonic frequencies ([MATH] and [MATH]).', '1807.03070-2-7-2': 'In the following, the data are analyzed with five oscillations, corresponding to these frequencies.', '1807.03070-2-7-3': 'Note that for the data measured in a pulsed magnetic field, the data measured at 15 K were analyzed, so as to avoid complications arising from strong harmonic oscillations.', '1807.03070-2-7-4': 'To obtain the oscillation phase factors ([MATH]) corresponding to each fundamental frequency, we have used Lifshitz-Kosevich (LK) theory to fit the data [CITATION]: [EQUATION]', '1807.03070-2-7-5': 'Here [MATH] is a constant corresponding to the [MATH]th harmonic of the [MATH]th band, [MATH] is the corresponding phase factor, and [MATH], [MATH] and [MATH] are the temperature, Dingle and spin damping factors respectively.', '1807.03070-2-7-6': 'These are given by [MATH], [MATH], and [MATH], where [MATH] denotes the cyclotron mass for each band, [MATH] is the free electron mass and [MATH] is the Dingle temperature.', '1807.03070-2-7-7': 'The data were fitted using Eq. [REF], and it can be seen in Fig. 2 that this can well describe the SdH oscillations both measured in a static field up to 35 T (S2) and a pulsed field up to 60 T (S6).', '1807.03070-2-7-8': 'Note that since the data are analyzed for a fixed temperature and field-angle, [MATH] is taken to be a constant value.', '1807.03070-2-7-9': 'Here [MATH] for the five components were fixed to the values from the FFT analysis, and the [MATH] for the [MATH] and [MATH] bands were obtained from the temperature dependence of the oscillation amplitudes from dHvA measurements (see Fig. 5).', '1807.03070-2-7-10': 'On the other hand it is difficult to obtain [MATH] from analyzing the temperature dependence of oscillations corresponding to the [MATH]band, and therefore this was a fitted parameter in the analysis.', '1807.03070-2-7-11': 'In addition, [MATH] and [MATH] for each band were fitted parameters in the analysis, and the results are displayed in Supplementary Table I.', '1807.03070-2-8-0': 'The phase factor corresponding to the [MATH]-band from both samples is 1.16[MATH].', '1807.03070-2-8-1': 'Note that the phase is given by [MATH], where [MATH] corresponds to the LK correction, and [MATH] is the sum of geometric and dynamical phases, which includes the Berry phase, the phase factor resulting from the orbital magnetic moment and the Zeeman coupling phase factor [CITATION].', '1807.03070-2-8-2': 'For a two-dimensional cylinder-like Fermi surface, [MATH] is zero, while for a three-dimensional Fermi surface pocket, [MATH] is [MATH] or [MATH] for oscillations corresponding to the minimum or maximum Fermi surface cross sections respectively.', '1807.03070-2-8-3': 'Since the oscillation frequency is lowest when the field is applied along the [100] direction (see Supplementary Figure 2), this suggests that [MATH], and hence [MATH].', '1807.03070-2-8-4': 'This is close to the value of [MATH] indicating a [MATH] Berry phase, as expected for a topologically non-trivial electronic band [CITATION], while the small deviation from [MATH] suggests the possible influence of dynamical phases [CITATION].', '1807.03070-2-8-5': 'On the other hand, for fundamental oscillations corresponding to the [MATH]-band, [MATH] of -0.15[MATH] and 0.01[MATH] are obtained for samples S2 and S6 respectively, while the values for the [MATH] band are -0.24[MATH] and -0.17[MATH].', '1807.03070-2-8-6': 'These results suggest that both these bands have near zero Berry phase, where the influence of dynamical phase factors leads to a small deviation from zero.', '1807.03070-2-9-0': 'A Landau fan diagram was also utilized to extrapolate the dominant oscillation phase factor (Fig. 3), where Landau indices were assigned to the positions of the SdH oscillations which correspond to the deepest valleys, for both S2 and S6.', '1807.03070-2-9-1': "The uncertainty of the valley positions is small which doesn't significantly affect the results, and the periodicity of these valleys corresponds to that of the [MATH]-band.", '1807.03070-2-9-2': 'From linearly fitting the Landau index [MATH] as a function of [MATH], the extrapolated residual Landau indices are [MATH] for sample S2 and [MATH] for sample S6.', '1807.03070-2-9-3': 'These agree well with the above conclusions that the phase factor of the [MATH]-band, which is the strongest frequency component observed in the quantum oscillations, is close to [MATH].', '1807.03070-2-10-0': 'Though these results are consistent with those expected for a non-trivial band topology, in Ref. [CITATION] it was suggested that the phase obtained from quantum oscillation experiments cannot directly probe the Berry phase in three-dimensional centrosymmetric materials.', '1807.03070-2-10-1': 'In our measurements, the field is applied along the [100] direction, which corresponds to the [MATH] rotation axis of the cubic crystal structure.', '1807.03070-2-10-2': 'Due to the symmetry constraints in this situation (mirror symmetry) [CITATION], the spin degeneracy of each band is preserved, and the overall phase factor will be quantized to either [MATH] or zero, which is not a linear summation of the phase factors of both spin up and down bands, and thus cannot be used to directly obtain the Berry phase or determine the band topology.', '1807.03070-2-10-3': 'Here we note that the same phase factor is found from measurements above and below [MATH].', '1807.03070-2-10-4': 'Since the magnetic structure of SmSb has not yet been reported, we are not able to determine if the same symmetry constraints are present in the magnetically ordered state.', '1807.03070-2-10-5': 'Electronic structure calculations for the case of zero applied field suggest that SmSb has a trivial band topology but is close to the boundary between trivial and non-trivial [CITATION].', '1807.03070-2-10-6': 'However we note that these calculations assume a Sm valence of 3+, while a mixed valence of about 2.75 was reported from x-ray photoemission spectroscopy [CITATION].', '1807.03070-2-10-7': 'As such it is important to perform further studies to reveal the origin of this nontrivial Berry phase.', '1807.03070-2-10-8': 'Evidence for a [MATH] Berry phase was also found in isostructural LaBi [CITATION], where both an odd number of band inversions and surface states with an odd number of massless Dirac cones were observed in ARPES measurements [CITATION].', '1807.03070-2-11-0': 'In addition, we also performed field-dependent ac susceptibility measurements down to low temperatures where the presence of dHvA oscillations is clearly observed.', '1807.03070-2-11-1': 'For comparison, SdH and dHvA oscillations are shown for various temperatures down to 0.3 K in Figs. 4a and 4b respectively, after subtracting the background contribution.', '1807.03070-2-11-2': 'The respective FFT are shown in Figs. 4c and 4d, where two principal frequencies are observed, which remain unchanged below [MATH], similar to previous reports [CITATION].', '1807.03070-2-11-3': 'The observed values are [MATH]334 T and 328 T, and [MATH] 633 T and [MATH] 590 T for SdH and dHvA respectively, where the differences may be due to a small misalignment.', '1807.03070-2-11-4': 'The SdH oscillation amplitudes as [MATH] corresponding to both the [MATH] and [MATH] bands increase significantly with decreasing temperature below 2.5 K, while in dHvA measurements the change is more gradual.', '1807.03070-2-11-5': 'Furthermore, the amplitude of the [MATH]-band oscillations is maximum at around 0.8 K, and upon further decreasing the temperature, the amplitude decreases.', '1807.03070-2-11-6': 'Meanwhile, the angular dependence of [MATH] can be fitted with the equation for anisotropic three-dimensional ellipsoidal pockets [CITATION](see Supplementary Figure 2), which can describe the anisotropic bullet-like Fermi surface pockets around the X-points, corresponding to the bulk Fermi surface of the [MATH]-band [CITATION].', '1807.03070-2-12-0': 'Figures 5a and 5b present the temperature dependence of the oscillation amplitudes from both SdH and dHvA measurements.', '1807.03070-2-12-1': 'Since there is a rapid change of the resistivity both with changing temperature and field, the SdH oscillation amplitudes are displayed as [MATH]/[MATH] is proportional to the density of states and that the temperature dependence is well described by the LK formula, for multiband systems the situation can be more complicated due to the different contributions to the conductivity from each band [CITATION].', '1807.03070-2-12-2': 'This effect will not influence the dHvA results, which can be well described by the LK formula across the whole temperature range.', '1807.03070-2-12-3': 'We note that below around [MATH], the normalized oscillation amplitude of the [MATH]-band is anomalously low relative to the dHvA results, while for the [MATH]-band there is an increase.', '1807.03070-2-12-4': 'This suggests that with decreasing temperature, there may be changes of the relative contributions to the conductivity from the two bands.', '1807.03070-2-12-5': 'Therefore, for systems with multiple FS sheets and large magnetoresistance, analysis of the cyclotron masses using SdH measurements may be difficult.', '1807.03070-2-13-0': 'Another question posed by these results is why this anomalous temperature dependence of SdH amplitudes is not observed in other [MATH]Sb compounds.', '1807.03070-2-13-1': 'This may be related to the fact that in other [MATH]Sb materials, the rapid increase of the magnetoresistance occurs at considerably higher temperatures.', '1807.03070-2-13-2': 'For example, in LaSb and YSb this occurs at around 80 K and 100 K, respectively [CITATION], while the SdH oscillations are usually measured below 20 K for these materials, where the magnetoresistance shows relatively little change with temperature.', '1807.03070-2-13-3': 'However, as displayed in Fig. 1b, the magnetoresistance of SmSb changes strongly below 2 K, which may influence the oscillation amplitudes at low temperatures.', '1807.03070-2-13-4': 'Therefore the origin of the dramatic departure from conventional behavior in the SdH amplitudes is an open question, and in particular since this deviation onsets near [MATH], the possible relationship between the antiferromagnetic state and the anomalous behavior also needs to be explored.', '1807.03070-2-14-0': 'To summarize, we find evidence for a [MATH] Berry phase in the [MATH]-band of SmSb from analyzing high field measurements of SdH oscillations.', '1807.03070-2-14-1': 'Furthermore, our quantum oscillation measurements show that the amplitudes of dHvA oscillations are well described by the standard LK formula, while those from the SdH effect show anomalous behavior at low temperatures.', '1807.03070-2-14-2': 'The origin of this unusual behavior in SdH oscillations remains unclear, and therefore further studies are required to understand the origin of the Berry phase, as well as any relationship with the anomalous SdH oscillations.', '1807.03070-2-15-0': '# Methods', '1807.03070-2-16-0': 'Single crystals of SmSb were synthesized using an Sn flux-method [CITATION].', '1807.03070-2-16-1': 'The elements were combined in a molar ratio Sm:Sb:Sn of 1:1:20 in an Al[MATH]O[MATH] crucible before being sealed in evacuated quartz ampoules.', '1807.03070-2-16-2': 'These were slowly cooled from 1100[MATH]C down to 800[MATH]C, before centrifuging to remove the Sn flux.', '1807.03070-2-16-3': 'The resulting crystals are cubic with typical dimensions of around 3 mm.', '1807.03070-2-16-4': 'The residual resistivity ratio of [MATH], indicates a high sample quality.', '1807.03070-2-17-0': 'Low temperature resistivity and ac susceptibility measurements were performed between 0.27 K and 9 K using an Oxford Instruments [MATH]He system, in applied fields up to 15 T.', '1807.03070-2-17-1': 'The resistivity data in this system was measured using a Lakeshore 370 resistance bridge, with a current of 0.3 mA across the whole temperature range.', '1807.03070-2-17-2': 'Four Pt wires were attached to the sample by spot welding so that the current was applied along the [100] direction.', '1807.03070-2-17-3': 'dHvA results were measured using a commercial susceptometer for ac susceptibility measurements, which consists of three sets of coils, including a drive coil, a pick-up coil, and a coil for compensation.', '1807.03070-2-17-4': 'Additional resistivity measurements in fields up to 9 T, and dc-susceptibility measurements in fields up to 13 T, were also performed using a Quantum Design Physical Property Measurement System (PPMS) from 300 K to 2 K, where the susceptibility was measured using a vibrating sample magnetometer insert.', '1807.03070-2-17-5': 'The high field magnetoresistance measurements were performed up to 60 T using a pulsed field magnet at the Los Alamos National Laboratory, USA, while measurements up to 32 T were carried out using the Cell 5 Water-Cooling Magnet at the High Magnetic Field Laboratory of the Chinese Academy of Sciences.', '1807.03070-2-18-0': 'Data availability statement The data that support the findings of this study are available from the corresponding author upon reasonable request.', '1807.03070-2-19-0': 'acknowledgments', '1807.03070-2-20-0': 'We thank Y. Liu, C. Cao, Q. Si, Y. Zhou, F. Steglich, P. Li and Z. Z. Wu for interesting discussions and helpful suggestions.', '1807.03070-2-20-1': 'This work was supported by the the Science Challenge Project of China (Project No. TZ2016004), National Key RD Program of China (Grants No. 2017YFA0303100 and No. 2016YFA0300202), the National Natural Science Foundation of China (Grants No. U1632275, No. 11604291 and No. 11474251), and the Innovative Program of Development Foundation of Hefei Center for Physical Science and Technology (Project No. 2017FXCX001).', '1807.03070-2-20-2': 'A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1644779* and the State of Florida.', '1807.03070-2-20-3': 'JS acknowledges support from the DOE BES program "Science of 100 T".', '1807.03070-2-21-0': '# Competing financial interests', '1807.03070-2-22-0': 'The Authors declare no Competing Financial or Non-Financial Interests', '1807.03070-2-23-0': '# Author contributions The crystals were grown by F.W..', '1807.03070-2-23-1': 'F.W., C.Y.G., J.L.Z., Y.C. and J.S. performed the measurements, which were analyzed by F.W., C.Y.G., M.S., J.S. and H.Q.Y.', '1807.03070-2-23-2': 'The manuscript were written by F.W., C.Y.G., M.S. and H.Q.Y. All authors participated in discussions.', '1807.03070-2-24-0': '# REFERENCES'}

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[]

[['1807.03070-1-9-6', '1807.03070-2-11-6'], ['1807.03070-1-11-0', '1807.03070-2-14-0'], ['1807.03070-1-11-3', '1807.03070-2-14-2'], ['1807.03070-1-11-4', '1807.03070-2-14-2'], ['1807.03070-1-0-0', '1807.03070-2-0-2'], ['1807.03070-1-14-0', '1807.03070-2-17-4'], ['1807.03070-1-14-1', '1807.03070-2-17-0'], ['1807.03070-1-8-4', '1807.03070-2-9-0'], ['1807.03070-1-8-4', '1807.03070-2-9-2'], ['1807.03070-1-8-7', '1807.03070-2-10-5']]

[]

['1807.03070-1-1-0', '1807.03070-1-15-0', '1807.03070-1-16-1', '1807.03070-1-16-2', '1807.03070-2-19-0', '1807.03070-2-20-1', '1807.03070-2-20-2', '1807.03070-2-22-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1807.03070

cond-mat-0404397

{'cond-mat-0404397-1-0-0': 'We discuss classical magnetotransport in a two-dimensional system with strong scatterers.', 'cond-mat-0404397-1-0-1': 'Even in the limit of very low field, when [MATH] is the cyclotron frequency, [MATH] is the scattering time) such a system demonstrates strong negative magnetoresistance caused by non-Markovian memory effects.', 'cond-mat-0404397-1-0-2': 'A regular method for the calculation of non-Markovian corrections to the Drude conductivity is presented.', 'cond-mat-0404397-1-0-3': 'A quantitative theory of the recently discovered anomalous low-field magnetoresistance is developed for the system of two-dimensional electrons scattered by hard disks of radius [MATH] randomly distributed with concentration [MATH].', 'cond-mat-0404397-1-0-4': 'For small magnetic fields the magentoresistance is found to be parabolic and inversely proportional to the gas parameter, [MATH].', 'cond-mat-0404397-1-0-5': 'In some interval of magnetic fields the magnetoresistance is shown to be linear [MATH] in a good agreement with the experiment and numerical simulations.', 'cond-mat-0404397-1-0-6': 'Magnetoresistance saturates for [MATH], when the anomalous memory effects are totally destroyed by the magnetic field.', 'cond-mat-0404397-1-0-7': 'We also discuss magnetotransport at very low fields and show that at such fields magnetoresistance is determined by the trajectories having a long Lyapunov region.', 'cond-mat-0404397-1-1-0': '# Intorduction', 'cond-mat-0404397-1-2-0': 'The problem of magnetoresistance in metal and semiconductor structures has been intensively discussed in literature during the past three decades.', 'cond-mat-0404397-1-2-1': 'A large number of both theoretical and experimental papers on this subject was published.', 'cond-mat-0404397-1-2-2': 'Most of these works were devoted to the case of the degenerate two dimensional electron gas where the electrons move in the plane perpendicular to the magnetic field and scatter on a random impurity potential.', 'cond-mat-0404397-1-2-3': 'In this situation only the electrons with energy close to the Fermi energy participate in conductance and the usual approach based on the Boltzmann equation leads to vanishing magnetoresistance.', 'cond-mat-0404397-1-2-4': 'In other words, the longitudinal resistance [MATH] of the sample does not depend on the magnetic field [MATH].', 'cond-mat-0404397-1-2-5': 'This implies that the explanation of the experimentally observed [MATH] dependence of the longitudinal resistance should be sought beyond the Boltzmann theory.', 'cond-mat-0404397-1-3-0': 'The intense exploration of this area began from the work [CITATION], where the experimentally observed in 2D metals and semiconductor structures negative magnetoresistance (MR), i.e. decreasing [MATH] with increasing [MATH], was explained by quantum interference effects.', 'cond-mat-0404397-1-3-1': 'It was shown that the magnetic field destroys the negative weak localization correction to the conductivity, thus resulting in decreasing longitudinal resistance.', 'cond-mat-0404397-1-3-2': 'Since the first publication on the subject [CITATION] a vast amount of work has been devoted to its further exploration (see for review e.g. [CITATION]).', 'cond-mat-0404397-1-4-0': 'Two years prior to work [CITATION] there appeared a publication [CITATION] where a classical mechanism of negative magnetoresistance was discussed .', 'cond-mat-0404397-1-4-1': 'The mechanism was investigated on the example of a gas of non-interacting electrons scattering on hard disks (antidots).', 'cond-mat-0404397-1-4-2': 'It was shown that with increasing magnetic field there is an increasing number of closed electron orbits which avoid scatterers and therefore are not diffusive (see also recent discussions of this mechanism in [CITATION]).', 'cond-mat-0404397-1-4-3': 'Electrons occupying these orbits do not participate in diffusion.', 'cond-mat-0404397-1-4-4': 'As a result, the longitudinal resistance turns out to be proportional to the factor [MATH] where [MATH] is the probability of the existance of the circular closed orbit, which avoids scatterers (here [MATH] is the cyclotron frequency , [MATH] is the scattering time).', 'cond-mat-0404397-1-4-5': 'Another classical mechanism was presented in [CITATION], where the MR due to non-markovian dynamics of electrons trapped in some region of space was discussed.', 'cond-mat-0404397-1-5-0': 'Notwithstanding these developments, the role of classical effects in magnetotransport was underappreciated for a long time.', 'cond-mat-0404397-1-5-1': 'A new boost to the research in this direction was given by work [CITATION], where it was shown that if electrons move in a smooth disorder potential and in a sufficiently strong magnetic fields a phenomenon called "classical localization" occurs.', 'cond-mat-0404397-1-5-2': 'This phenomenon leads to the exponential suppression of the longitudinal resistance: most electrons are trapped in localized equipotential trajectories and do not participate in diffusion.', 'cond-mat-0404397-1-5-3': 'This work was followed by a series of works [CITATION], discussing different aspects of classical magnetotransport in 2D systems.', 'cond-mat-0404397-1-5-4': 'In [CITATION] it was shown that for lower magnetic fields near the onset of the classical localization the magnetoresistance is positive, i.e. the longitudinal resistance grows with increasing magnetic field.', 'cond-mat-0404397-1-5-5': 'In [CITATION] the combination of smooth disorder and strong scatterers (antidots) was considered.', 'cond-mat-0404397-1-5-6': 'It was shown that in this system under certain conditions there are several regimes of the behavior of magnetoresistance depending on the strength of the magnetic field: first the longitudinal resistance decreases with growing field, then it saturates and then begins to grow.', 'cond-mat-0404397-1-6-0': 'In Refs. [CITATION] magnetoresistance was studied in a situation where the magnetic field is classically strong , that is where the parameter [MATH] is large.', 'cond-mat-0404397-1-6-1': 'In [CITATION] the region of classically small magnetic fields [MATH] was investigated numerically for the case of electrons scattering on hard disks.', 'cond-mat-0404397-1-6-2': 'In this work it was shown that memory effects due to double scattering of an electron on the same disk lead to a negative parabolic magnetoresistance (in the work [CITATION] where these processes were not taken into account, exponentially small MR was predicted).', 'cond-mat-0404397-1-7-0': 'In recent numerical simulations [CITATION] a low-field classical anomaly of the MR was discovered.', 'cond-mat-0404397-1-7-1': 'The anomaly was attributed to the memory effects specific for backscattering events.', 'cond-mat-0404397-1-7-2': 'The numerical simulations were performed for the 2D Lorenz gas which is a system of 2D electrons scattering on hard disks randomly distributed in plane with average concentration [MATH].', 'cond-mat-0404397-1-7-3': 'Magnetotransport in this system is characterized by two dimensionless parameters: [MATH] and the gas parameter [MATH].', 'cond-mat-0404397-1-7-4': 'Here [MATH] is the disk radius, [MATH] is the cyclotron frequency, [MATH] is the mean free time and [MATH] is the mean free path.', 'cond-mat-0404397-1-7-5': 'The anomaly was observed in the case [MATH].', 'cond-mat-0404397-1-7-6': 'Both the numerical simulations and the qualitative considerations presented in [CITATION] indicated that at zero temperature the MR can be expressed in terms of a dimensionless function [MATH] via [EQUATION] where [MATH] is the resistivity for [MATH].', 'cond-mat-0404397-1-7-7': 'Numerical results [CITATION] suggest that [MATH] as [MATH] yielding [EQUATION]', 'cond-mat-0404397-1-7-8': 'The latter expression is in a very good agreement with experimental measurements of negative linear MR in a random antidot arrays [CITATION].', 'cond-mat-0404397-1-7-9': 'It is anomalous in two senses.', 'cond-mat-0404397-1-7-10': 'First, it has a non-analytic dependence on the magnetic field.', 'cond-mat-0404397-1-7-11': 'Second, it does not vanish in the limit of vanishing [MATH], which is normally regarded as the expansion parameter for the corrections to the Drude-Boltzmann picture.', 'cond-mat-0404397-1-7-12': 'This intriguing behavior calls for a rigorous analytical theory of the effect, which would establish Eq. [REF] and enable one to derive the analytical expression for function [MATH].', 'cond-mat-0404397-1-8-0': 'In this paper we present a detailed theory of the anomaly and give an expression for [MATH] (the brief description of our results was given in [CITATION] ).', 'cond-mat-0404397-1-8-1': 'We find that at some interval, [MATH] function [MATH] is linear in agreement with numerical simulations and experiment, but at [MATH] crosses over to a quadratic dependence.', 'cond-mat-0404397-1-8-2': 'Thus, for [MATH] Eq. [REF] yields [MATH].', 'cond-mat-0404397-1-8-3': 'The limit [MATH] should be taken with care.', 'cond-mat-0404397-1-8-4': 'While the small [MATH] expansion seems to be singular as a function of [MATH] the region of [MATH] where this expansion is valid shrinks as [MATH].', 'cond-mat-0404397-1-8-5': 'For [MATH] saturates at some constant value.', 'cond-mat-0404397-1-8-6': 'Therefore, the full variation of [MATH] is of the order [MATH].', 'cond-mat-0404397-1-8-7': 'In other words, the anomalous MR is strong but it exists in a small region of magnetic fields.', 'cond-mat-0404397-1-9-0': '# Qualitative discussion of the problem', 'cond-mat-0404397-1-10-0': 'The mechanism proposed in [CITATION] is linked to the memory effects arising in backscattering events.', 'cond-mat-0404397-1-10-1': 'It has a close relation to the well known non-analyticity of the virial expansion of transport coefficients [CITATION], which we briefly recall.', 'cond-mat-0404397-1-10-2': 'For [MATH] the leading nonanalytic correction to resistivity, [MATH], is due to the processes of return to a scatterer after a single collision on another scatterer (see Fig. [REF]a).', 'cond-mat-0404397-1-11-0': 'The relative correction, [MATH] is proportional to the corresponding backscattering probability, given by the product of [MATH] (which is the probability to reach scatterer 2 without collision and scatter in the angle [MATH]) and the probability [MATH] to return without collisions from 2 to 1 (here [MATH] is the mean free pass).', 'cond-mat-0404397-1-11-1': 'Assuming [MATH] and integrating over intervals [MATH] one obtains [CITATION] [EQUATION]', 'cond-mat-0404397-1-11-2': 'In Ref. [CITATION] it was shown that the probability [MATH] is actually larger than [MATH].', 'cond-mat-0404397-1-11-3': 'Indeed, the exponent [MATH] can be written as [MATH] where [MATH].', 'cond-mat-0404397-1-11-4': 'It represents the probability of the existence of an empty corridor (free of the centers of the disk) of width [MATH] around the electron trajectory from [MATH] to [MATH].', 'cond-mat-0404397-1-11-5': 'However, the passage of a particle from [MATH] to [MATH] ensures the absence of the disks centers in the region of width [MATH] around this part of trajectory (from [MATH] to [MATH]) This reduces the scattering probability on the way back.', 'cond-mat-0404397-1-11-6': 'The correct value of [MATH] can be estimated as [EQUATION] where [EQUATION] is the area of the overlap of the two corridors (see Fig. [REF]a).', 'cond-mat-0404397-1-11-7': 'For example, for [MATH] we have [MATH] and [MATH] which reflects the obvious fact that the particle cannot scatter, if it travels back along the same path.', 'cond-mat-0404397-1-11-8': 'Taking into account the effect of "empty corridor", we get [EQUATION] where [MATH] is a constant of the order of unity.', 'cond-mat-0404397-1-11-9': 'Thus, for [MATH] the "empty corridor" effect simply changes the constant in the argument of the logarithm.', 'cond-mat-0404397-1-12-0': 'The key idea suggested in [CITATION] was that for [MATH] the area of the overlap of the two corridors, [MATH] sharply depends on [MATH] resulting in the observed MR. Indeed, it is seen from Fig. [REF]b that for [MATH] resulting in sharp negative MR [EQUATION]', 'cond-mat-0404397-1-12-1': 'The following qualitative explanation of the observed linear MR was presented in Ref. [CITATION].', 'cond-mat-0404397-1-12-2': 'The value [MATH] was estimated for [MATH] (see Fig. [REF]c) to the first order in [MATH] as [MATH] where [MATH] is the cyclotron radius.', 'cond-mat-0404397-1-12-3': 'Assuming that this estimate also works at [MATH] and expanding [MATH] to the first order in [MATH] one gets [MATH].', 'cond-mat-0404397-1-13-0': 'In fact, the physical picture of the phenomenon is more subtle.', 'cond-mat-0404397-1-13-1': 'The contribution of any trajectory with [MATH] is cancelled to the first order in [MATH] by the contribution of the time-reversed trajectory, since the values of [MATH] are opposite for these paths [ Fig1.', 'cond-mat-0404397-1-13-2': '(d),(e)].', 'cond-mat-0404397-1-13-3': 'The cancellation does not occur only at very small [MATH].', 'cond-mat-0404397-1-13-4': 'The integration in Eq. [REF] over [MATH] yields [MATH].', 'cond-mat-0404397-1-13-5': 'Larger values of [MATH] also give a quadratic in [MATH] contribution to the MR. This contribution is positive and comes from the second order term in the expansion of [MATH] in [MATH].', 'cond-mat-0404397-1-13-6': 'It follows from our results [see Eqs. [REF],[REF]] that the contribution of small angles is dominant resulting in a negative parabolic MR. We find that the parabolic MR crosses over to linear at very small [MATH] which explains why the parabolic MR was not seen in numerical simulations [CITATION] and experiment [CITATION].', 'cond-mat-0404397-1-13-7': 'Note that for very small [MATH] contribution of the trajectories with a long Lyaponov region (Lyapunov trajectories) becomes important.', 'cond-mat-0404397-1-13-8': 'In the Section [REF] we focus on the region [MATH] where the contribution of the Lyapunov trajectories is parametrically small.', 'cond-mat-0404397-1-13-9': 'The Lyapunov trajectories will be discussed in Section [REF].', 'cond-mat-0404397-1-14-0': '# Calculations', 'cond-mat-0404397-1-15-0': '## Kinetic equation', 'cond-mat-0404397-1-16-0': 'In this subsection we introduce the kinetic equation which is the starting point for the calculation of the diffusion coefficient and the resistivity of the Lorenz gas.', 'cond-mat-0404397-1-17-0': 'We consider the Lorenz gas at zero temperature, assuming that the electrons participating in the conduction have the Fermi velocity [MATH].', 'cond-mat-0404397-1-17-1': 'The diffusion coefficient [MATH] is given by [EQUATION]', 'cond-mat-0404397-1-17-2': "Here [MATH] is the retarded Green's function of the Liouville equation and [MATH] stands for the averaging over the positions of the disks.", 'cond-mat-0404397-1-17-3': 'The equation for [MATH] reads [EQUATION] where [MATH] is the Liouville operator of the free motion in the magnetic field and [MATH] describes scattering on the disks.', 'cond-mat-0404397-1-17-4': 'From Eq. [REF] it follows that we need the time integral of [MATH] rather than the whole time-dependent function.', 'cond-mat-0404397-1-17-5': 'This integral can be written as [EQUATION] where [MATH] are the angles of velocities [MATH] and [MATH] respectively.', 'cond-mat-0404397-1-17-6': 'Here we used the energy conservation, which implies that in scattering processes the absolute value of the velocity does not change.', 'cond-mat-0404397-1-17-7': 'Using Eq. [REF], the diffusion coefficient Eq. [REF] can be rewritten as [EQUATION]', 'cond-mat-0404397-1-17-8': 'Since [MATH] depends on [MATH] and [MATH] only, it is convenient to average over the position of the initial point and over the initial angle.', 'cond-mat-0404397-1-17-9': 'Here [MATH] is the area of the sample.', 'cond-mat-0404397-1-17-10': 'Integrating Eq. [REF] with respect to time the equation for [MATH] is written as [EQUATION] where [EQUATION] [MATH] is the unit vector in the direction of [MATH] and [MATH].', 'cond-mat-0404397-1-17-11': 'The interaction with disks is written in Eq. [REF] in the form of collision integral [CITATION].', 'cond-mat-0404397-1-17-12': 'The scattering operators [MATH] transform arbitrary function [MATH] as follows, [EQUATION] where [EQUATION]', 'cond-mat-0404397-1-17-13': "Here [MATH] are the positions of the disks' centers and [EQUATION] is the differential cross-section of one disk.", 'cond-mat-0404397-1-17-14': 'The vector [MATH] depends on angles [MATH] in the integrals Eq. [REF] [EQUATION] and is pointing from the center of a disk to the scattering point at the disk surface (see Fig. [REF]).', 'cond-mat-0404397-1-17-15': 'Physically, operator [MATH] describes influx of particles to velocity [MATH] at the point [MATH], while operator [MATH] describes the outflux from velocity [MATH] at the point [MATH].', 'cond-mat-0404397-1-17-16': 'The averaged value of the [MATH] is equal to the disks concentartion [EQUATION]', 'cond-mat-0404397-1-17-17': 'The Bolzmann equation is obtained by averaging the Liouville equation with respect to the position of the scatterers and neglecting correlations.', 'cond-mat-0404397-1-17-18': 'Indeed, using Eq. [REF] one finds that the average collision operators Eq. [REF] are given by [EQUATION]', 'cond-mat-0404397-1-17-19': 'Replacing the collision operators in the Liouville equation Eq. [REF] with their averages Eq. [REF] one obtains the Boltzmann equation [EQUATION]', 'cond-mat-0404397-1-17-20': 'The Bolzmann-Drude diffusion coefficient [MATH] in the absence of the magnetic field is found from Eqs. [REF] and [REF] as follows.', 'cond-mat-0404397-1-17-21': "Integrating [REF] with respect to [MATH] and expanding the Green's function [MATH] in angular harmonics one finds [EQUATION] where [EQUATION]", 'cond-mat-0404397-1-17-22': 'In particular, [EQUATION] is the transport length.', 'cond-mat-0404397-1-17-23': 'Substituting Eqs. [REF] and [REF] in Eq. [REF] one finds [EQUATION]', 'cond-mat-0404397-1-18-0': '## Perturbative expansion', 'cond-mat-0404397-1-19-0': "In this subsection we derive the perturbative expansion for the average Green's function in order to take into account non-Markovian corrections which are absent in the Bolzmann-Drude picture.", 'cond-mat-0404397-1-20-0': 'Introduce the operators [EQUATION] which describe the fluctuation of the collision integral with respect to its average value.', 'cond-mat-0404397-1-20-1': 'Using these operators the formal solution of [REF], [MATH] can be written as the following series [EQUATION] where [MATH] is defined by Eq. [REF].', 'cond-mat-0404397-1-20-2': 'Here we took into account that [MATH].', 'cond-mat-0404397-1-20-3': 'The first term on the right hand side of Eq. [REF] gives the Drude-Boltzmann result described in the previous subsection.', 'cond-mat-0404397-1-20-4': 'The rest of the terms in Eq. [REF] provide a regular way for the calculation of correlations, which are absent in the Boltzmann picture.', 'cond-mat-0404397-1-21-0': 'In the subsequent analysis of the perturbative expansion Eq. [REF] we will extensively use the representation of the Boltzmann propagator [MATH] in terms of the ballistic propagator [EQUATION]', 'cond-mat-0404397-1-21-1': 'This is achieved by expanding the Boltzmann propagator [MATH] as a sum over the number of scattering events represented by the operator [MATH] [EQUATION]', 'cond-mat-0404397-1-21-2': 'The diagrammatic representation of the expansion Eq. [REF] is shown in Fig. ([REF]) .', 'cond-mat-0404397-1-21-3': 'Each cross this figure corresponds to [MATH] and each solid line to a ballistic propagator [MATH].', 'cond-mat-0404397-1-21-4': "For future reference we write the explicit expression for the ballistic Green's function in zero magnetic field and in weak magnetic fields.", 'cond-mat-0404397-1-21-5': 'For [MATH] the ballistic propagator conserves the velocity.', 'cond-mat-0404397-1-21-6': 'In this case its kernel is given by [EQUATION]', 'cond-mat-0404397-1-21-7': 'Here [MATH] is the angle of the vector [MATH].', 'cond-mat-0404397-1-21-8': 'Magnetic field rotates the velocity vector with the cyclotron frequency.', 'cond-mat-0404397-1-21-9': 'For small magnetic fields, [MATH], we have [EQUATION]', 'cond-mat-0404397-1-22-0': '## Ballistic returns in perturbative expansion', 'cond-mat-0404397-1-23-0': 'In this subsection we show how the processes of ballistic returns discussed in Section [REF] arise in the perturbative expansion Eq. [REF].', 'cond-mat-0404397-1-23-1': 'Consider the second term in Eq. [REF].', 'cond-mat-0404397-1-23-2': 'This term describes the memory effect due to diffusive returns.', 'cond-mat-0404397-1-23-3': 'As discussed in Section [REF], the main contribution comes from returns after a single scattering.', 'cond-mat-0404397-1-23-4': 'This process is described by the diagram shown in Fig. [REF]a.', 'cond-mat-0404397-1-23-5': 'The dashed line corresponds to the pairings [MATH]), external wavy lines to the diffusion propagators [MATH].', 'cond-mat-0404397-1-23-6': 'The internal line corresponds to the Boltzmann propagator Eq. [REF] truncated at one scattering [MATH].', 'cond-mat-0404397-1-24-0': 'Four combinations of [MATH] at the ends of external dashed lines in the diagrams shown in Fig. [REF]a represent four different types of correlation at a given point [MATH].', 'cond-mat-0404397-1-24-1': 'To see this consider the pairing [MATH].', 'cond-mat-0404397-1-24-2': 'By virtue of Eqs. [REF] and [REF] this pairing is proportional to the density-density correlation function [MATH] for [MATH].', 'cond-mat-0404397-1-24-3': 'Assuming that disks are randomly distributed over the sample we get for these functions [EQUATION]', 'cond-mat-0404397-1-24-4': 'It is natural to interpret the vector [MATH] as the position of the center of the disk on which a double scattering occurs.', 'cond-mat-0404397-1-24-5': 'However, this interpretation is valid for diagram [MATH] only.', 'cond-mat-0404397-1-24-6': 'As shown in Fig. [REF], this diagram corresponds to the situation where an electron experiences two real scattering processes on the disk placed at point [MATH].', 'cond-mat-0404397-1-24-7': 'The physical interpretation of other diagrams is more subtle.', 'cond-mat-0404397-1-24-8': 'The diagram [MATH] [Fig. [REF]] does not correspond to any real scattering at point [MATH].', 'cond-mat-0404397-1-24-9': 'It just allows one to calculate correctly the probability for an electron to pass twice the region of the size [MATH] around point [MATH] without scattering.', 'cond-mat-0404397-1-24-10': 'To interpret the diagrams [MATH] and [MATH] note that in the Boltzmann picture, which neglects correlations, the following process is allowed.', 'cond-mat-0404397-1-24-11': 'An electron scatters on a disk and later on passes through the region occupied by this disk without a scattering [Fig. [REF]] (analogous consideration is valid for diagram shown in Fig. [REF]).', 'cond-mat-0404397-1-24-12': 'The diagrams [MATH] and [MATH] correct the Boltzmann result by subtracting the contribution of such unphysical processes.', 'cond-mat-0404397-1-25-0': 'As follows from the qualitative discussion Section [REF], the processes of ballistic returns after a single scattering give rise to the leading (non-analytic) correction to the Drude-Boltzmann result in zero magnetic field.', 'cond-mat-0404397-1-25-1': 'However, taking into account diagrams shown in Fig. [REF]a (describing processes shown in Figs. [REF],[REF],[REF],[REF]) is not sufficient for calculation of the low field anomaly of the MR. Actually, diagrams in Fig. [REF]a do not contain the "empty corridor" effect.', 'cond-mat-0404397-1-25-2': 'We will show that the correct description of the memory effects, specific for ballistic returns, requires the renormalization of diagrams [REF]a by diagrams [REF]b, in which the internal dashed lines contains [MATH] pairings only.', 'cond-mat-0404397-1-25-3': "Physically, the [MATH]'th order diagram of the type [REF]b represents the [MATH]'th order term in the Taylor expansion of the [MATH] in the qualitative estimate [REF].", 'cond-mat-0404397-1-25-4': 'Such renormalization play the key role in the quantitative description of the anomalous MR.', 'cond-mat-0404397-1-26-0': 'In the next subsection we derive analytical expression for diagrams [REF]a.', 'cond-mat-0404397-1-26-1': 'Then we generalize the calculations to account for the corridor effect and calculation of anomalous MR.', 'cond-mat-0404397-1-27-0': '## Non-analytical corrections to the zero-field resistance neglecting the "empty corridor" effect', 'cond-mat-0404397-1-28-0': 'In this subsection we use the perturbative expansion Eq. [REF] to derive analytically the leading correction to the diffusion coefficient.', 'cond-mat-0404397-1-28-1': 'This correction is due to the processes described by diagram Fig. [REF]a.', 'cond-mat-0404397-1-28-2': "The operator expression of the correction to the Green's function related to this diagram is given by", 'cond-mat-0404397-1-29-0': '(0,30)(50,-15) (250,-7)(1,0)68 (318,-7)(0,1)5 (250,-7)(0,1)5 (207,0) [MATH]', 'cond-mat-0404397-1-30-0': 'where the underbracket stands for the pairings of operators [MATH] and [MATH].', 'cond-mat-0404397-1-30-1': 'Substituting this correction into Eq. [REF], and using Eqs. [REF],[REF] to integrate over [MATH] we get [EQUATION] where [EQUATION]', 'cond-mat-0404397-1-30-2': 'Here we used Eq. [REF] for [MATH].', 'cond-mat-0404397-1-31-0': 'Graphically, these calculations are presented in Fig. [REF].', 'cond-mat-0404397-1-31-1': 'The pairings [MATH] entering Eq. [REF], are given in Appendix A for four possible combinations of [MATH].', 'cond-mat-0404397-1-31-2': 'The pairings only depend on the difference [MATH].', 'cond-mat-0404397-1-31-3': 'Therefore in Eq. [REF] one can remove the integral over [MATH] and the sample area [MATH] in the denominator and put [MATH] in the integrand.', 'cond-mat-0404397-1-31-4': 'In other words we put the origin of the coordinate system at the backscattering point [MATH].', 'cond-mat-0404397-1-31-5': 'Next we integrate Eq. [REF] over the angles [MATH].', 'cond-mat-0404397-1-31-6': 'This integration can be easily done taking into account angle dependent delta functions entering in [MATH] and [MATH] (see Eq. [REF]).', 'cond-mat-0404397-1-31-7': 'As a result we get [MATH] and [EQUATION]', 'cond-mat-0404397-1-31-8': 'In derivation of Eq. [REF] we took into account that the pairing of collision operators [MATH] vanishes for [MATH] which is clear from its physical meaning and also can be seen from the delta functions in the factor [MATH] (see Appendix A).', 'cond-mat-0404397-1-31-9': 'At the same time both [MATH] and [MATH] are of the order of the mean free path [MATH].', 'cond-mat-0404397-1-31-10': 'Therefore, [MATH] and [MATH].', 'cond-mat-0404397-1-31-11': 'Physically this means that the typical scattering angle in the processes of ballistic return are close to [MATH].', 'cond-mat-0404397-1-31-12': 'Next, we use Eq. [REF] to evaluate the contributions of different pairings [MATH]', 'cond-mat-0404397-1-32-0': '## (+,+) pairing', 'cond-mat-0404397-1-33-0': 'Consider first the process in which [MATH].', 'cond-mat-0404397-1-33-1': 'Substitute equations [REF] and [REF] in Eq. [REF] and integrate over [MATH].', 'cond-mat-0404397-1-33-2': 'It is convenient for our purposes to perform this integration for a fixed value of the vector [MATH].', 'cond-mat-0404397-1-33-3': 'Then the delta functions in Eq. [REF] ensure that [MATH] and [MATH] lie on the circle of the radius [MATH] centered in the point [MATH] [EQUATION]', 'cond-mat-0404397-1-33-4': 'The graphical solution of these equations is shown in Fig. [REF] ( vectors [MATH] in this figure are the unit vectors in the directions of [MATH] and [MATH] respectively).', 'cond-mat-0404397-1-33-5': 'Since [MATH] is small ( [MATH]) one has approximately [EQUATION]', 'cond-mat-0404397-1-33-6': 'This accuracy is sufficient for the calculation of the diagram in Fig. [REF]a (since we calculate it in the lowest order in [MATH] ).', 'cond-mat-0404397-1-33-7': 'Upon integrating out the vectors [MATH] the integrand in Eq. [REF] depends on [MATH].', 'cond-mat-0404397-1-33-8': 'Finally we get [EQUATION]', 'cond-mat-0404397-1-33-9': 'Introducing new variables [MATH] and [MATH] and taking into account that [MATH] and [MATH] the relative correction to the diffusion coefficient is written as [EQUATION] >From Fig. [REF] it is seen that [MATH], [MATH] are the scattering angles.', 'cond-mat-0404397-1-34-0': '## (+,-) pairing', 'cond-mat-0404397-1-35-0': 'Next we present calculations for [MATH].', 'cond-mat-0404397-1-35-1': 'Like the calculation of the diagram [MATH] we substitute Eq. [REF] and Eq. [REF] into Eq. [REF] and integrate over [MATH] with the use of Eq. [REF].', 'cond-mat-0404397-1-35-2': 'The values of [MATH] and [MATH] should be found from the following equations [EQUATION]', 'cond-mat-0404397-1-35-3': 'The graphical solution of these equation is presented in Fig. [REF].', 'cond-mat-0404397-1-35-4': 'After integrating out vectors [MATH] and the angle [MATH] we get [EQUATION]', 'cond-mat-0404397-1-35-5': 'In derivation of this equation we have used Eq. [REF].', 'cond-mat-0404397-1-35-6': 'It is convenient to rewrite this equation using as integration variables [MATH] which are the scattering angles for the process [MATH].', 'cond-mat-0404397-1-35-7': 'They are expressed in terms of [MATH] and [MATH] as [MATH].', 'cond-mat-0404397-1-35-8': 'In these variables Eq. [REF] reads [EQUATION]', 'cond-mat-0404397-1-35-9': 'It is seen from Eq. [REF] (see also Fig. [REF]) that the contribution of the process [MATH] can be parameterized by the angles [MATH] and [MATH] which are the scattering angles for the process [MATH].', 'cond-mat-0404397-1-35-10': 'Analogous calculations can be easily done for other types of correlations.', 'cond-mat-0404397-1-36-0': 'Summing the different contributions [MATH] we get the following expression for the diagram Fig. [REF]a [EQUATION]', 'cond-mat-0404397-1-36-1': 'This equation is the exact expression for the non-analytic correction [CITATION] to the Drude-Boltzmann resistivity which was qualitatively given in Eq. [REF].', 'cond-mat-0404397-1-36-2': 'Four terms in the product [MATH] correspond to four types of correlations discussed above.', 'cond-mat-0404397-1-37-0': '## "Empty corridor" effects on the zero-field resistance In this subsection we use the perturbation theory for the quantitative derivation of the effect of the "empty corridor" discussed in section [REF].', 'cond-mat-0404397-1-38-0': 'Eq. [REF] takes into account one pairing of operators [MATH] and [MATH].', 'cond-mat-0404397-1-38-1': 'The terms containing [MATH] pairings ([MATH] dashed lines) are typically small as [MATH].', 'cond-mat-0404397-1-38-2': 'However, there is a series of diagrams, shown in Fig. [REF]b, whose contribution is of the order [MATH] [[CITATION]].', 'cond-mat-0404397-1-38-3': 'The internal dashed lines in this series only contain pairings [MATH].', 'cond-mat-0404397-1-38-4': 'Below we show that the series Fig. [REF]b accounts for the effect of the "empty corridor".', 'cond-mat-0404397-1-38-5': 'More precisely, we prove that the [MATH]th order term in this series corresponds to [MATH] term in the Taylor expansion of the [MATH] in Eq. [REF].', 'cond-mat-0404397-1-39-0': 'The addition of the diagram Fig. [REF]b to the diagram Fig. [REF]a leads to the following replacement in Eq. [REF] [EQUATION] where the function [MATH] can be found from the Dyson equation (See Fig. [REF]).', 'cond-mat-0404397-1-40-0': 'Since the operator [MATH] does not change the velocity angle, one can search the solution of the Dyson equation in the form [EQUATION] where [MATH] is the difference between the backscattering angle and [MATH] (See Fig. ([REF])).', 'cond-mat-0404397-1-40-1': 'While substituting Eq. [REF] in the Dyson equation (which is shown graphically in Fig. ([REF]), integrating over angles [MATH], and using the identity [EQUATION] (and analogous identity for vectors [MATH] )we get [EQUATION] where [MATH] are defined in the Appendix [REF].', 'cond-mat-0404397-1-40-2': 'Further calculations can be done in two different ways.', 'cond-mat-0404397-1-40-3': 'It is useful from methodological point of view to discuss both of them.', 'cond-mat-0404397-1-40-4': 'One way is to first integrate Eq. [REF] over the angles of the vectors [MATH] and [MATH] [see Fig. [REF]].', 'cond-mat-0404397-1-40-5': 'The delta-functions entering in Eq. [REF] ensure that point [MATH] (see Figs. [REF],[REF]) lies on the line segment connecting point [MATH] and the origin and point [MATH] lies on the line segment connecting point [MATH] and the origin.', 'cond-mat-0404397-1-40-6': 'This allows to reduce the Dyson equation to the closed relation for function [MATH] [EQUATION] where the pairing [EQUATION] is calculated in Appendix [REF].', 'cond-mat-0404397-1-40-7': 'Here [EQUATION] is shown in Fig. [REF].', 'cond-mat-0404397-1-41-0': 'As a result we have [EQUATION]', 'cond-mat-0404397-1-41-1': 'This equation has an evident solution [EQUATION]', 'cond-mat-0404397-1-41-2': 'Eq. [REF] may be derived in an alternative way.', 'cond-mat-0404397-1-41-3': 'This way allows to understand on the formal basis why one should renormalize the diagrams in Fig. [REF]a by the [MATH] pairings only.', 'cond-mat-0404397-1-41-4': 'Let us integrate in Eq. [REF] over [MATH] using Eq. [REF].', 'cond-mat-0404397-1-41-5': 'Doing this, we get [EQUATION]', 'cond-mat-0404397-1-41-6': 'Here we took into account that [MATH] is a slowly changing function of [MATH] (it changes on a scale of the order of [MATH]).', 'cond-mat-0404397-1-41-7': 'Since Eq. [REF] provide [MATH], we put [MATH].', 'cond-mat-0404397-1-41-8': 'Next we make use of the identity [EQUATION] to integrate over [MATH].', 'cond-mat-0404397-1-41-9': 'After this integration we get [EQUATION]', 'cond-mat-0404397-1-41-10': 'Integration over [MATH] leads again to Eq. [REF].', 'cond-mat-0404397-1-42-0': '>From Eq. [REF] it follows that we need to know [MATH] for [MATH].', 'cond-mat-0404397-1-42-1': 'Using Eq. [REF] we find [EQUATION] where [MATH] is the overlap between two corridors, given by Eq. [REF].', 'cond-mat-0404397-1-42-2': 'The [MATH]-th order term in the Taylor expansion of the exponential [REF] corresponds to a diagram in Fig. [REF]b with [MATH] dashed lines.', 'cond-mat-0404397-1-42-3': 'Indeed, the [MATH]-th order term in the Taylor expansion contains [MATH] integration over coordinates of [MATH] scatterings of the type [MATH].', 'cond-mat-0404397-1-42-4': 'What remains to do to get the resistivity correction is to express [MATH] in Eq. [REF] via angles [MATH] ,[MATH].', 'cond-mat-0404397-1-42-5': 'To do this, the precision Eq. ([REF]) is not sufficient.', 'cond-mat-0404397-1-42-6': 'More specifically, we will need to know the angle between vectors [MATH] and [MATH] to the order [MATH].', 'cond-mat-0404397-1-42-7': 'For this purpose it will be sufficient to make replacements [MATH] and [MATH] in the arguments of vectors [MATH] in Eq. [REF].', 'cond-mat-0404397-1-42-8': 'Then the angle [MATH] is calculated as (see Figs. [REF],[REF],[REF],[REF]) [EQUATION]', 'cond-mat-0404397-1-42-9': 'This equation is valid for [MATH], [MATH] (since [MATH] is not periodic with [MATH] one should specify the integration limits).', 'cond-mat-0404397-1-42-10': 'It worth noting that Eq. [REF] ensures that the argument of the [MATH] function in Eq. [REF] is positive and the overlap width and overlap area are given by [EQUATION]', 'cond-mat-0404397-1-42-11': 'Summing the diagrams Fig. [REF]b together with Fig. [REF]a, one gets an exact equation [EQUATION] instead of qualitative estimate Eq. [REF].', 'cond-mat-0404397-1-43-0': 'Here [MATH] is a numerical coefficient.', 'cond-mat-0404397-1-43-1': 'Thus, addition of the series Fig. [REF]b to Fig. [REF]a leads to the following renormalization: [MATH]', 'cond-mat-0404397-1-44-0': '## Estimate for neglected diagrams', 'cond-mat-0404397-1-45-0': 'In this subsection we use the derivation presented in the previous subsection to show how to select relevant diagrams.', 'cond-mat-0404397-1-45-1': 'The derivation was based on identity [REF].', 'cond-mat-0404397-1-45-2': 'The left hand side of this equation is proportional to the impurity cross section [MATH].', 'cond-mat-0404397-1-45-3': 'However, the right hand side is parameterically larger [MATH] provided that [MATH].', 'cond-mat-0404397-1-45-4': 'One can show that the function similar to the one on the right hand side [REF] arises each time when one of the index [MATH] or [MATH] in the pairing [MATH] is equal to [MATH] (for a pairing [MATH] we have two functions of the type [REF], see Eq. [REF]).', 'cond-mat-0404397-1-45-5': 'For the case when [MATH] or [MATH] equals to [MATH] we have a differential cross section [MATH] instead of function [REF].', 'cond-mat-0404397-1-45-6': 'Now we are ready to estimate different types of diagrams.', 'cond-mat-0404397-1-45-7': "Insertion of one additional [MATH] pairing into a diagram in Figs. [REF],[REF],[REF],[REF] gives two additional Green's functions and, consequently, multiplier [MATH] and two functions of the type [REF] giving a multiplier [MATH].", 'cond-mat-0404397-1-45-8': 'One should also multiply on disk concentration [MATH] and integrate over [MATH].', 'cond-mat-0404397-1-45-9': 'Due to [MATH] functions in [REF] the integration area [MATH] is of the order [MATH].', 'cond-mat-0404397-1-45-10': 'Combining all the multipliers together we have [MATH].', 'cond-mat-0404397-1-45-11': 'Therefore, addition of [MATH] pairing does not lead to any smallness.', 'cond-mat-0404397-1-45-12': 'In contrast to this, insertion of [MATH] pairing leads to smallness [MATH].', 'cond-mat-0404397-1-45-13': 'Indeed, the only difference from the case of [MATH] pairing is that one should replace one of the function [REF] by the corresponding cross section.', 'cond-mat-0404397-1-45-14': 'This leads to the change of one of the multiplier of the order [MATH] by multiplier of the order [MATH].', 'cond-mat-0404397-1-45-15': 'Following this line of reasoning one could conclude that insertion of [MATH] pairing leads to the smallness of the order of [MATH].', 'cond-mat-0404397-1-45-16': 'Actually, the smallness arising from insertion of [MATH] pairing is of the order of [MATH].', 'cond-mat-0404397-1-45-17': 'Indeed, while the replacement of each of two function of the type [MATH] by corresponding cross section leads to the relative smallness of the order [MATH] the integration area in this case is not restricted by angle dependent [MATH] functions.', 'cond-mat-0404397-1-45-18': 'As a result, the integration area, [MATH] is larger by a parameter [MATH] compared to the case of [MATH] pairing.', 'cond-mat-0404397-1-45-19': 'Note that estimates presented above do not work for the diagrams with one dashed line Figs. [REF],[REF],[REF],[REF].', 'cond-mat-0404397-1-45-20': 'In this case one should integrate over initial and final scattering angles.', 'cond-mat-0404397-1-45-21': 'One can easily see that integration of Eq. [MATH] over angle [MATH] leads to the multiplier of the order of [MATH].', 'cond-mat-0404397-1-45-22': 'This implies that in this case all four pairings, [MATH],[MATH],[MATH], and[MATH] are of the same order, of [MATH] (see Eq. [REF]).', 'cond-mat-0404397-1-45-23': 'Note that the higher order diagrams which are small in the parameter [MATH] may turn out to be relevant for the MR at very low magnetic field (see Section [REF]).', 'cond-mat-0404397-1-46-0': '## Anomalous magnetoresistance', 'cond-mat-0404397-1-47-0': 'In this subsection we generalize our calculations for the [MATH] case assuming that [MATH] is small ([MATH].', 'cond-mat-0404397-1-47-1': 'The main contribution in this case still comes from diagrams in Figs. [REF]a,b. Consider for example diagram [MATH].', 'cond-mat-0404397-1-47-2': 'Let us compare the process of double scattering described by this diagram for [MATH] (see Fig. [REF]) with the same process for [MATH] (see Fig. [REF]).', 'cond-mat-0404397-1-47-3': 'For fixed points [MATH] and [MATH], one can see the following differences.', 'cond-mat-0404397-1-47-4': 'First, the scattering angles [MATH] and [MATH] acquire small corrections of the order of [MATH].', 'cond-mat-0404397-1-47-5': 'Second, the parts of the electron trajectory corresponding to free ends of the picture become curved.', 'cond-mat-0404397-1-47-6': 'The backscattering angle [MATH] increases by the value [MATH] [EQUATION] where [MATH] is the value of backscattering angle for [MATH] given by Eq. [REF].', 'cond-mat-0404397-1-47-7': 'Finally, the overlap area of the corridors changes because the trajectories become curved (see Fig. [REF]).', 'cond-mat-0404397-1-48-0': 'The corrections to [MATH] and [MATH] lead to small relative corrections to the resistivity of the order of [MATH] and can be neglected.', 'cond-mat-0404397-1-48-1': 'The same reason allows one to neglect the curvature of the incoming and outgoing parts of the trajectory.', 'cond-mat-0404397-1-48-2': 'Therefore,the only relevant difference is the change of the overlap area of the corridors.', 'cond-mat-0404397-1-48-3': 'The solution of the Dyson equation is analogous to the [MATH] case.', 'cond-mat-0404397-1-48-4': 'The points of intermediate integration [MATH] and [MATH] lie now at the segments of cyclotron circles (from [MATH] to [MATH] and from [MATH] to [MATH]).', 'cond-mat-0404397-1-48-5': 'The pairing of two operators [MATH] is still given by Eq. [REF] with the replacement [MATH].', 'cond-mat-0404397-1-48-6': 'For [MATH] the overlap width is calculated as [EQUATION] where [MATH] is the Heaviside step function.', 'cond-mat-0404397-1-48-7': 'Therefore, the only difference from Eq. [REF] is that one should replace [MATH] where the overlap area is given by [EQUATION]', 'cond-mat-0404397-1-48-8': 'The value of [MATH] is obtained from Eq. [REF] by replacing [MATH] to [MATH] [EQUATION]', 'cond-mat-0404397-1-48-9': 'Introducing dimensionless variables [MATH] we get Eq. [REF], where function [MATH] is given by [EQUATION]', 'cond-mat-0404397-1-48-10': 'Here [EQUATION]', 'cond-mat-0404397-1-48-11': 'Function [MATH] has the following asymptotics [EQUATION]', 'cond-mat-0404397-1-48-12': 'In the interval [MATH] can be well approximated by linear function [EQUATION]', 'cond-mat-0404397-1-48-13': 'Next we discuss the parabolic asymptotics more carefully.', 'cond-mat-0404397-1-48-14': 'We will show that in this asymptotic region there are two contributions of different signs to the magnetoresistance: a negative contribution coming from the trajectories with very small [MATH] such that [MATH] and a positive contribution coming from larger angles.', 'cond-mat-0404397-1-48-15': 'This considerations will be used in the next section, discussing trajectories with the long Lyapunov region.', 'cond-mat-0404397-1-49-0': 'First, we write the difference of two exponents [MATH] as follows [MATH] and expand the equation in the bracket in the Taylor expansion up to the second order [EQUATION]', 'cond-mat-0404397-1-49-1': 'We consider the case [MATH]).', 'cond-mat-0404397-1-49-2': 'One can easily see that in this case the expression [MATH] entering in the argument of [MATH] function in the Eq. [REF] is positive at all values of [MATH] (we take into account Eq. [REF] and have in mind that [MATH]).', 'cond-mat-0404397-1-49-3': 'Therefore, the difference [MATH] is expressed as [EQUATION]', 'cond-mat-0404397-1-49-4': 'To sum the contributions of different electron trajectories, we take into account that the time reversed trajectories have the same statistical weight.', 'cond-mat-0404397-1-49-5': 'Indeed, as seen from Figs. ([REF]-[REF]),([REF]), time reversion correspond to the change [MATH].', 'cond-mat-0404397-1-49-6': 'This transformation does not affect the factor [MATH] entering Eq. [REF].', 'cond-mat-0404397-1-49-7': 'At the same time, the angle [MATH] changes to [MATH] under this transformation (see Eq. [REF]).', 'cond-mat-0404397-1-49-8': 'From Eq. [REF] we find the variation of the overlap area averaged over two time reversed trajectories [EQUATION]', 'cond-mat-0404397-1-49-9': 'This expression is of the order of [MATH] for [MATH] and is equal to zero for larger angles.', 'cond-mat-0404397-1-49-10': 'We conclude therefore, that time reversed contributions do not cancel only in the region of small angles [MATH].', 'cond-mat-0404397-1-49-11': 'Since the total variation of [MATH] is much larger, of the order of [MATH] (see Eq. [REF]), we can replace the expression in the right hand side of Eq. [REF] by [MATH]-function, writing [MATH].', 'cond-mat-0404397-1-49-12': 'Keeping the leading terms of the order of [MATH] only, we obtain [EQUATION]', 'cond-mat-0404397-1-49-13': 'The second term in this equation comes from the averaging of the quadratic term in the exponent expansion [MATH] over two time reversed trajectories.', 'cond-mat-0404397-1-49-14': 'This term is positive and partly compensates the negative contribution of small angles.', 'cond-mat-0404397-1-49-15': 'Using Eqs. [REF],[REF],[REF], after some algebra we get the low-field asymptotic of MR as the sum of the negative and positive contributions discussed above [EQUATION] where numerical coefficient [MATH] is given by [EQUATION]', 'cond-mat-0404397-1-50-0': '# Contribution of trajectories with long Lyapunov region to magnetoresistance', 'cond-mat-0404397-1-51-0': 'The equations derived in the previous section give the contribution to the MR related to the processes shown in Figs. ([REF]-[REF]), the parabolic asymptotics [REF] starting to work when [MATH] becomes smaller than [MATH].', 'cond-mat-0404397-1-51-1': 'Such processes are related to the correlations specific for returns to the initial point after one scattering.', 'cond-mat-0404397-1-51-2': 'As we show at this section for very low magnetic fields, [MATH] other correlations come into play.', 'cond-mat-0404397-1-51-3': 'Specifically, we consider the contribution to the MR of the trajectories containing long Lyapunov regions.', 'cond-mat-0404397-1-51-4': 'Such trajectories consist of the direct and the return paths and involve real double scatterings on some number of disks as shown in Fig. [REF].', 'cond-mat-0404397-1-51-5': 'The divergency between the direct and the return paths is characterized by the Lyapunov length.', 'cond-mat-0404397-1-51-6': 'We will call such trajectories "Lyapunov trajectories".', 'cond-mat-0404397-1-51-7': 'In the diagrammatic series they are presented by the sum of the diagrams shown in Fig. [REF].', 'cond-mat-0404397-1-51-8': 'Just as in the case discussed above, four different pairings are allowed at the ends of external dashed lines: [MATH],[MATH], [MATH] and [MATH].', 'cond-mat-0404397-1-51-9': 'However, in contrast to the diagrams shown in Fig. [REF], internal lines of "Lyapunov diagrams" contain pairings of [MATH] type as well as of [MATH] type.', 'cond-mat-0404397-1-51-10': 'Physically, [MATH] pairings corresponds to real double scatterings in the Lyapunov region.', 'cond-mat-0404397-1-51-11': 'We will count such diagrams by the number [MATH] of correlated links in the Lyapunov region.', 'cond-mat-0404397-1-51-12': 'The Lyapunov trajectory shown in Fig. [REF] corresponds to [MATH].', 'cond-mat-0404397-1-51-13': 'The diagrams in Figs. [REF] discussed in Section [REF], present a particular case of diagrams with Lyapunov region, corresponding to [MATH].', 'cond-mat-0404397-1-51-14': 'The correlated links of any Lyapunov trajectory are renormalized by the pairings [MATH] as was discussed already for [MATH].', 'cond-mat-0404397-1-51-15': 'The contribution of any diagram of such type to the resistivity is small as [MATH].', 'cond-mat-0404397-1-51-16': 'However, as will be shown below, these diagrams have a sharp dependence on the magnetic field at very small fields.', 'cond-mat-0404397-1-51-17': 'We will show that for [MATH] the diagram of the N-th order gives a contribution to the parabolic MR of the same order as the contribution of the diagrams with [MATH] already calculated above.', 'cond-mat-0404397-1-52-0': 'Consider the Lyapunov trajectory with [MATH] correlated links.', 'cond-mat-0404397-1-52-1': 'Denote by [MATH] the angles between the segments of the direct and the return paths and by [MATH] the scattering angles between successive correlated links as shown in Fig. [REF].', 'cond-mat-0404397-1-52-2': 'The contribution of such a process has a sharp dependence on the magnetic field due to the magnetic field dependence of the overlap of the corridors surrounding the direct and the return path.', 'cond-mat-0404397-1-52-3': 'This dependence is different for different segments of the trajectory due to the difference of the angles [MATH].', 'cond-mat-0404397-1-52-4': 'Indeed as we have seen in the previous section if [MATH] is the typical angle between the direct and the return paths then the characteristic scale for the magnetic field dependence of the overlap of the corridors is [MATH].', 'cond-mat-0404397-1-52-5': 'As one can see from Eq. [REF], the typical value of the angle [MATH] is of the order [MATH].', 'cond-mat-0404397-1-52-6': 'The smallest angle [MATH] corresponds to the last segment of the trajectory.', 'cond-mat-0404397-1-52-7': 'Therefore it is this segment that should lead to the sharpest dependence of the resistivity on the magnetic field.', 'cond-mat-0404397-1-52-8': 'Consider the contribution of the last segment at small magnetic field [MATH].', 'cond-mat-0404397-1-52-9': 'For typical trajectories with [MATH] there is a cancellation of the contributions of the time reversed paths to the MR (see discussion in the previous section).', 'cond-mat-0404397-1-52-10': 'However, for a small fraction of the trajectories with [MATH] the contribution of the time reversed paths does not cancel and is proportional to the change of the overlap area [MATH].', 'cond-mat-0404397-1-52-11': 'The phase space of such trajectories is proportional to [MATH].', 'cond-mat-0404397-1-52-12': 'Thus, the contribution of the Lyapunov trajectory with [MATH] links to the resistivity in the region of small magnetic field is given by [EQUATION] where the factor [MATH] is due to the [MATH]-th order of the corresponding diagram.', 'cond-mat-0404397-1-52-13': 'As we show below, the coefficient in this equation does not depend on the diagram order.', 'cond-mat-0404397-1-52-14': 'It equals [MATH] thus coinciding with the coefficient in the small angles contribution to parabolic asymptotic for [MATH] (see Eq. [REF]).', 'cond-mat-0404397-1-52-15': 'It worth noting that only in the case [MATH] one should take into account the contribution of large angles, presented by the second term in Eq. [REF].', 'cond-mat-0404397-1-52-16': 'For [MATH] such contribution is parametrically small.', 'cond-mat-0404397-1-53-0': 'Before presenting rigorous derivation of above statements, we consider qualitatively the contribution of [MATH]-th order diagram for larger magnetic fields [MATH].', 'cond-mat-0404397-1-53-1': 'At such fields the first order contribution of the time reversed paths to the corridor effect does not cancel for trajectories with any values of [MATH] up to the maximal value [MATH].', 'cond-mat-0404397-1-53-2': 'Thus, the factor [MATH] which counts the phase space of relevant trajectories saturates at some constant value of the order of unity and one has [EQUATION]', 'cond-mat-0404397-1-53-3': 'Consider now the MR in the interval [MATH].', 'cond-mat-0404397-1-53-4': 'In this interval the Lyapunov trajectories with the number of double scatterings smaller than [MATH] give a parabolic contribution Eq. [REF] to the MR. The trajectories with the number of double scatterings bigger than [MATH] give a linear contribution Eq. [REF], the main contribution [MATH] coming from the trajectories with [MATH] double scattering.', 'cond-mat-0404397-1-53-5': 'As a result one has [EQUATION]', 'cond-mat-0404397-1-53-6': 'Here [MATH] and [MATH] is given by Eq. [REF].', 'cond-mat-0404397-1-53-7': 'For [MATH] the second term in the right hand side of Eq. [REF] can be neglected and we get finally [EQUATION]', 'cond-mat-0404397-1-53-8': 'Equation Eq. [REF] indicates that at very low magnetic fields, which correspond to large [MATH] the results of section [REF] become incorrect and logarithmic renormalization of the parabolic MR occurs [EQUATION]', 'cond-mat-0404397-1-53-9': 'Next we present a rigorous derivation of the contribution of the the [MATH]-th order Lyapunov trajectory for [MATH].', 'cond-mat-0404397-1-53-10': 'As follows from qualitative considerations presented above, [MATH] for any [MATH].', 'cond-mat-0404397-1-53-11': 'Indeed, [MATH].', 'cond-mat-0404397-1-53-12': 'Since the minimal angle, [MATH] is on the order of [MATH] the maximal angle is small compared to unity, [MATH].', 'cond-mat-0404397-1-53-13': 'Therefore, the direct and return paths of the relevant Lyapunov trajectories are very close to each other.', 'cond-mat-0404397-1-53-14': 'This allows us to characterize a scattering between [MATH] and [MATH] links of the trajectory by one scattering angle [MATH] instead of two different scattering angles for direct and return paths.', 'cond-mat-0404397-1-53-15': 'The dependence on the magnetic field comes from the last link of the trajectory.', 'cond-mat-0404397-1-53-16': 'Expanding [MATH] like it was done in the previous section, we only keep the first term [EQUATION]', 'cond-mat-0404397-1-53-17': 'To write down the analytical expression for [MATH]-th order contribution we should take into account for combinations of [MATH] at the ends of diagram in Fig. [REF].', 'cond-mat-0404397-1-53-18': 'This correspond to four types of scattering on the impurity [MATH] in the Fig. [REF] and leads to appearing of the factor [MATH].', 'cond-mat-0404397-1-53-19': 'Here we took into account that direct and return paths coincides in the first approximation and, as a consequence, scattering angles [MATH] for direct and return paths are related to each other as follows [MATH].', 'cond-mat-0404397-1-53-20': 'The analytical expression for MR is given by [EQUATION]', 'cond-mat-0404397-1-53-21': 'Here we integrate over distances between disks instead of integration over disks positions.', 'cond-mat-0404397-1-53-22': "The factors [MATH] comes from double scatterings, the factor [MATH] from denominators of the two Green's functions, describing the propagation along [MATH]th link on the direct and return way.", 'cond-mat-0404397-1-53-23': 'In writing Eq. [REF] we also took into account the renormalization of correlated links by the [MATH] pairings.', 'cond-mat-0404397-1-53-24': 'As a consequence of this renormalization [MATH]th link, which is passed twice (on the direct and return paths) comes with the factor [MATH] instead of [MATH].', 'cond-mat-0404397-1-53-25': 'What remains to do is to express final angle [MATH] via initial angle [MATH].', 'cond-mat-0404397-1-53-26': 'From simple geometric considerations we get [EQUATION]', 'cond-mat-0404397-1-53-27': 'Combining Eqs. [REF] and [REF] and performing the integral one gets [EQUATION]', 'cond-mat-0404397-1-54-0': '# Interpretation of the resistivity correction in terms of small change of the effective scattering cross-section', 'cond-mat-0404397-1-55-0': 'In our calculations of resistivity corrections we used Eq. [REF] as a starting point.', 'cond-mat-0404397-1-55-1': 'Here we briefly discuss an alternative approach based on the accounting of the memory effects in terms of small change of the effective cross-section.', 'cond-mat-0404397-1-55-2': 'Consider first the [MATH] ballistic returns with given values of the angles [MATH] and [MATH].', 'cond-mat-0404397-1-55-3': 'The contribution of such processes to the resistivity (both for [MATH] and for [MATH]) contains the factor [MATH].', 'cond-mat-0404397-1-55-4': 'Let us introduce now the scattering angle [MATH] for the process [MATH] considered as a single scattering on a complex scatterer.', 'cond-mat-0404397-1-55-5': 'For small values of [MATH] and [MATH] this angle is evidently given by [MATH].', 'cond-mat-0404397-1-55-6': 'For arbitrary values of [MATH] the scattering angle reads [EQUATION]', 'cond-mat-0404397-1-55-7': 'Such definition ensures that [MATH].', 'cond-mat-0404397-1-55-8': 'Next we introduce the integration over this angle: [MATH].', 'cond-mat-0404397-1-55-9': 'Analogous expressions can be easily written for the processes [MATH] and [MATH].', 'cond-mat-0404397-1-55-10': 'As a result, one can write the contribution of four possible types of correlations as follows [EQUATION]', 'cond-mat-0404397-1-55-11': 'The [MATH] process is represented in Eq. [REF] by [MATH].', 'cond-mat-0404397-1-55-12': 'Here we neglected a small difference (on the order of [MATH]) between the angle [MATH] and [MATH].', 'cond-mat-0404397-1-55-13': 'By virtue of Eq. [REF] one can easily check that the memory effects related to four types of ballistic returns can be accounted quantitatively in the frame of usual Boltzmann equation.', 'cond-mat-0404397-1-55-14': 'One should just replace the scattering cross-section on one disk [MATH] by the effective cross-section [EQUATION] where [MATH] is the field-dependent correction given by [EQUATION]', 'cond-mat-0404397-1-55-15': 'This correction does not change the total cross-section [EQUATION]', 'cond-mat-0404397-1-55-16': 'In other words, the enhancement of cross-section caused by processes [MATH] and [MATH] is accompanied by the reduction of the scattering due to the [MATH] and [MATH] correlations.', 'cond-mat-0404397-1-55-17': 'The resistivity correction is proportional to the change of the inverse transport length [EQUATION]', 'cond-mat-0404397-1-55-18': 'Using Eqs. [REF],[REF], and [REF] one can easily get the Eq. [REF] for MR.', 'cond-mat-0404397-1-56-0': 'Finally we note that this approach is easily generalized for calculation of the contribution of the trajectories having long Lyapunov region.', 'cond-mat-0404397-1-56-1': 'For such trajectories one should replace the angles [MATH] and [MATH] in Eq. [REF] by [MATH] and [MATH] (see Section [REF]).', 'cond-mat-0404397-1-56-2': 'For very small [MATH] such that [MATH] and [MATH] the magnetic field induced correction to the scattering cross-section is expressed in a simple form [EQUATION]', 'cond-mat-0404397-1-57-0': '# Discussion', 'cond-mat-0404397-1-58-0': 'In the previous sections we derived analytical theory of the low-field anomaly in the magnetoresistance, caused by sharp dependence of the memory effects specific for backscattering events on the magnetic field.', 'cond-mat-0404397-1-58-1': 'Next we compare our calculations with the results of simulations and with experiment.', 'cond-mat-0404397-1-58-2': 'Note first that there is a parametrically small nonanomalous correction to Eq. [REF] due to returns after multiple scatterings.', 'cond-mat-0404397-1-58-3': 'This correction is given by [CITATION] [EQUATION].', 'cond-mat-0404397-1-58-4': 'To compare the results of simulations [CITATION] with the theoretical results in a wider region of parameters [MATH] we substract [MATH] from the numerical curves.', 'cond-mat-0404397-1-58-5': 'Theoretical and numerical [CITATION] results are plotted in Fig. [REF].', 'cond-mat-0404397-1-58-6': 'in the universal units, [MATH] versus [MATH].', 'cond-mat-0404397-1-58-7': 'It is seen, that the theoretical and numerical results are in a very good agreement.', 'cond-mat-0404397-1-59-0': 'Experimental measurement of the MR in the system of antidot arrays [CITATION] agrees qualitatevely with our predictions.', 'cond-mat-0404397-1-59-1': 'The experimentally observed MR was linear in magnetic field in a wide region of fields.', 'cond-mat-0404397-1-59-2': 'This region corresponds to the interval of magnetic fields, where our results can be aproximated by linear dependence (see Eq. [REF] ).', 'cond-mat-0404397-1-59-3': 'The magnetic fields used in experiment were relatively strong, and the parabolic asymptotic was not achieved.', 'cond-mat-0404397-1-59-4': 'The quantitative comparison with the experiment [CITATION] is more difficult due to several reasons.', 'cond-mat-0404397-1-59-5': 'First of all, in the structures, used in experiment, besides antidots there were also short range scatterers.', 'cond-mat-0404397-1-59-6': 'Our preliminary estimates show that accounting for a short range disorder can change the results of the calculations.', 'cond-mat-0404397-1-59-7': 'Secondly, the antidot distribution was not fully chaotic in the experiment.', 'cond-mat-0404397-1-59-8': 'To be more specific, the antidots were randomly moved from the regular square lattice distribution by shifts on the order of 30-40 of the lattice constant.', 'cond-mat-0404397-1-59-9': 'Finally, the antidots did not have equal sizes, the uncertainty of the size being on the order of 50 .', 'cond-mat-0404397-1-59-10': 'In spite of this, a very good quantitative agreement with the experiment can be achieved by appropriate choice of the antidot size [MATH] in the uncertainty interval (see also discussion of the experiment in Ref. dmit1).', 'cond-mat-0404397-1-59-11': 'Note also that generalization of the theory for the case when antidot sizes vary is straightforward.', 'cond-mat-0404397-1-59-12': 'What has to be done is to average pairings [MATH] with the distribution function for the antidots sizes.', 'cond-mat-0404397-1-59-13': 'This leads to the following modification of the obtained formulae: one should change [MATH] in Eq. [REF] and [MATH] in Eq. [REF].', 'cond-mat-0404397-1-59-14': 'Here over line means averaging over antidot sizes.', 'cond-mat-0404397-1-59-15': 'We do not present here corresponding calculations, since the distribution function of antidot sizes is not known for experiment [CITATION]', 'cond-mat-0404397-1-60-0': 'Next we briefly discuss several interesting unsolved problems.', 'cond-mat-0404397-1-60-1': 'Note first that above consideration of Lyapunov trajectories is valid, provided that [MATH].', 'cond-mat-0404397-1-60-2': 'Indeed, while calculating the [MATH]th order diagrams we neglected small contributions of [MATH] and [MATH] pairings on the all correlated links.', 'cond-mat-0404397-1-60-3': 'Accounting of such pairings is rather tricky and is out of scope of this paper.', 'cond-mat-0404397-1-60-4': 'We expect, however, that such correlations may lead to a factor on the order of [MATH]) in Eq. [REF].', 'cond-mat-0404397-1-60-5': 'Therefore, for [MATH] the contribution of Lyapunov diagrams might become [MATH] dependent.', 'cond-mat-0404397-1-60-6': 'This gives a low-field limit for our theory [EQUATION]', 'cond-mat-0404397-1-60-7': 'We conclude that the behavior of the MR in the limit [MATH] remains so far unclear.', 'cond-mat-0404397-1-61-0': 'In our calculations we fully neglected quantum effects.', 'cond-mat-0404397-1-61-1': 'Such effects should decrease the "effect of empty corridor" due to the diffraction on the edges of the disks.', 'cond-mat-0404397-1-61-2': 'In the situations where the magnetic field is not very small, [MATH], the neglecting of the diffraction effects is justified if [MATH] is a Fermi wavelength).', 'cond-mat-0404397-1-61-3': 'This criterion ensures that diffraction effects on the edges of the disks are not relevant at the scales of the order of [MATH].', 'cond-mat-0404397-1-61-4': 'In the opposite case, [MATH] the diffraction should destroy the "corridor effect", thus suppressing the anomalous MR. For small magnetic fields, when [MATH] the criterion for negligibility of the diffraction becomes stronger, [MATH] where [MATH] is the characteristic size of the Lyapunov region.', 'cond-mat-0404397-1-61-5': 'Another quantum effect which can be especially important from the point of view of the possible experimental realizations is the weak localization phenomena.', 'cond-mat-0404397-1-61-6': 'The weak localization correction to the conductivity also has an anomalous dependence on the magnetic field.', 'cond-mat-0404397-1-61-7': 'Moreover, the interpretation of the memory effects in terms of small change of effective cross-section discussed in Section [REF] is very close to the interpretation of weak localization phenomenon developed in Ref. [CITATION].', 'cond-mat-0404397-1-61-8': 'Similar to discussion in Section [REF], the coherent enhancement of backscattering amplitude caused by weak localization, is accompanied by reduction of coherent scattering in other directions, the total cross-section do not change.', 'cond-mat-0404397-1-61-9': 'The competition of the non-Markovian and the weak localization effects might result in new interesting phenomena.', 'cond-mat-0404397-1-61-10': 'The detailed analysis of the quantum effects is a challenging problem which will be addressed elsewhere.', 'cond-mat-0404397-1-62-0': 'We did not investigate the temperature dependence of the phenomenon.', 'cond-mat-0404397-1-62-1': 'This dependence is related to the scattering by phonons (or electron-electron scattering) neglected in our calculations.', 'cond-mat-0404397-1-62-2': 'It worth noting that importance of the electron-phonon scattering is expected to increase with decreasing magnetic field.', 'cond-mat-0404397-1-62-3': 'Indeed, the potential of the electron-phonon interaction depends on time, therefore restricting the maximal length of a trajectory with the Lyapunov region [MATH] where [MATH] and [MATH] is the temperature dependent characteristic time of the electron-phonon scattering.', 'cond-mat-0404397-1-62-4': 'This implies that at small magnetic fields one should replace the logarithmic factor [MATH], entering Eq. [REF] by a temperature dependent coefficient of the order of [MATH]', 'cond-mat-0404397-1-63-0': '# Summary', 'cond-mat-0404397-1-64-0': 'We propose a theory of the negative anomalous MR in a system with strong scatterers.', 'cond-mat-0404397-1-64-1': 'It is shown that the anomaly in the MR arises due to suppression of "empty corridor effect" by magnetic field.', 'cond-mat-0404397-1-64-2': 'A detailed description of different types of non-Markovian correlations related to ballistic returns is presented.', 'cond-mat-0404397-1-64-3': 'A method of diagrammatic expansion of the Liouville equation is developed which allows us to describe analytically the effects of "empty corridor" on ballistic returns.', 'cond-mat-0404397-1-64-4': 'The analytical expressions for anomalous MR in different intervals of magnetic fields are derived [see Eqs. [REF], [REF], [REF], [REF]].', 'cond-mat-0404397-1-64-5': 'The MR at very low magnetic fields was shown to be determined by the contribution of electron trajectories having long Lyapunov region.', 'cond-mat-0404397-1-64-6': 'An interpretation of the memory effects in terms of small change of the effective scattering cross-section is discussed.', 'cond-mat-0404397-1-64-7': 'The analytical results are shown to be in very good agreement with the numerical simulations and experiment.', 'cond-mat-0404397-1-65-0': '# Acknolegments', 'cond-mat-0404397-1-66-0': 'We thank M.I. Dyakonov for insightful discussions and R. Jullien for providing us with the numerical data.', 'cond-mat-0404397-1-66-1': 'We are also grateful to I.V. Gornyi and D.G. Polyakov for useful comments.', 'cond-mat-0404397-1-66-2': 'The work was partially supported by RFBR, grant of Russian Scientific School, and by programmes of the RAS.', 'cond-mat-0404397-1-67-0': '#', 'cond-mat-0404397-1-68-0': 'This Appendix contains the explicit expressions for the pairings [MATH].', 'cond-mat-0404397-1-68-1': 'Using Eqs. [REF], [REF], [REF] kernels of operators [MATH] entering in Eq. [REF] can be written as [EQUATION] where [EQUATION]', 'cond-mat-0404397-1-68-2': 'The functions [MATH] and [MATH] in Eq. [REF]depend on vectors [MATH] and [MATH] correspondingly [see Eqs. [REF],[REF]].', 'cond-mat-0404397-1-68-3': 'The pairings of operators [MATH] and [MATH] can be calculated with the use of Eqs. [REF],[REF],[REF].', 'cond-mat-0404397-1-68-4': 'As a ersult we have [EQUATION]', 'cond-mat-0404397-1-68-5': 'Here function [MATH] emerges as a result of pairings of [MATH] (see Eq. [REF]) and is given by [EQUATION]', 'cond-mat-0404397-1-69-0': '#', 'cond-mat-0404397-1-70-0': 'To calculate [MATH] we first do the integral in the right hand side of Eq. [REF].', 'cond-mat-0404397-1-70-1': 'Integral of the term containing average density [MATH] is trivial and reduces to [MATH].', 'cond-mat-0404397-1-70-2': 'In the integral containing [MATH] it is convenient to introduce orthogonal coordinates [MATH] and [MATH] such that [MATH] axis is collinear with the velocity of the incident particle and write the delta function entering the definition [REF] of [MATH] as [MATH].', 'cond-mat-0404397-1-70-3': 'We proceed by lifting the [MATH] dependent delta function by integration over the angle [MATH].', 'cond-mat-0404397-1-70-4': 'It is clear that the integral is vanishing for [MATH].', 'cond-mat-0404397-1-70-5': 'To calculate the integral for smaller [MATH] note that by virtue of Eq. [REF] [MATH] and therefore the integral of the [MATH] dependent delta function cancels the scattering cross-section in Eq. [REF].', 'cond-mat-0404397-1-70-6': 'As a result we get [EQUATION]', 'cond-mat-0404397-1-70-7': 'Here [MATH] is a unit step function which is equal to unity when [MATH] and vanishes otherwise.', 'cond-mat-0404397-1-70-8': 'Using Eq. [REF] we can write for backscattering angles close to [MATH]) [EQUATION] where [EQUATION] is shown in Fig. [REF].'}

{'cond-mat-0404397-2-0-0': 'We discuss classical magnetotransport in a two-dimensional system with strong scatterers.', 'cond-mat-0404397-2-0-1': 'Even in the limit of very low field, when [MATH] is the cyclotron frequency, [MATH] is the scattering time) such a system demonstrates strong negative magnetoresistance caused by non-Markovian memory effects.', 'cond-mat-0404397-2-0-2': 'A regular method for the calculation of non-Markovian corrections to the Drude conductivity is presented.', 'cond-mat-0404397-2-0-3': 'A quantitative theory of the recently discovered anomalous low-field magnetoresistance is developed for the system of two-dimensional electrons scattered by hard disks of radius [MATH] randomly distributed with concentration [MATH].', 'cond-mat-0404397-2-0-4': 'For small magnetic fields the magentoresistance is found to be parabolic and inversely proportional to the gas parameter, [MATH].', 'cond-mat-0404397-2-0-5': 'In some interval of magnetic fields the magnetoresistance is shown to be linear [MATH] in a good agreement with the experiment and numerical simulations.', 'cond-mat-0404397-2-0-6': 'Magnetoresistance saturates for [MATH], when the anomalous memory effects are totally destroyed by the magnetic field.', 'cond-mat-0404397-2-0-7': 'We also discuss magnetotransport at very low fields and show that at such fields magnetoresistance is determined by the trajectories having a long Lyapunov region.', 'cond-mat-0404397-2-1-0': '# Intorduction', 'cond-mat-0404397-2-2-0': 'The problem of magnetoresistance in metal and semiconductor structures has been intensively discussed in literature during the past three decades.', 'cond-mat-0404397-2-2-1': 'A large number of both theoretical and experimental papers on this subject was published.', 'cond-mat-0404397-2-2-2': 'Most of these works were devoted to the case of the degenerate two dimensional electron gas where the electrons move in the plane perpendicular to the magnetic field and scatter on a random impurity potential.', 'cond-mat-0404397-2-2-3': 'In this situation only the electrons with energy close to the Fermi energy participate in conductance and the usual approach based on the Boltzmann equation leads to vanishing magnetoresistance.', 'cond-mat-0404397-2-2-4': 'In other words, the longitudinal resistance [MATH] of the sample does not depend on the magnetic field [MATH].', 'cond-mat-0404397-2-2-5': 'This implies that the explanation of the experimentally observed [MATH] dependence of the longitudinal resistance should be sought beyond the Boltzmann theory.', 'cond-mat-0404397-2-3-0': 'The intense exploration of this area began from the work of Altshuler et al, [CITATION] where the experimentally observed in 2D metals and semiconductor structures negative magnetoresistance (MR), i.e. decreasing [MATH] with increasing [MATH], was explained by quantum interference effects.', 'cond-mat-0404397-2-3-1': 'It was shown that the magnetic field destroys the negative weak localization correction to the conductivity, thus resulting in decreasing longitudinal resistance.', 'cond-mat-0404397-2-3-2': 'Since the first publication on the subject [CITATION] a vast amount of work has been devoted to its further exploration (see for review Ref. lee).', 'cond-mat-0404397-2-4-0': 'Two years prior to Ref. Alt there appeared a publication [CITATION] where a classical mechanism of negative magnetoresistance was discussed.', 'cond-mat-0404397-2-4-1': 'The mechanism was investigated on the example of a gas of non-interacting electrons scattering on hard disks (antidots).', 'cond-mat-0404397-2-4-2': 'It was shown that with increasing magnetic field there is an increasing number of closed electron orbits which avoid scatterers and therefore are not diffusive (see also recent discussions [CITATION] of this mechanism).', 'cond-mat-0404397-2-4-3': 'Electrons occupying these orbits do not participate in diffusion.', 'cond-mat-0404397-2-4-4': 'As a result, the longitudinal resistance turns out to be proportional to the factor [MATH] where [MATH] is the probability of the existance of the circular closed orbit, which avoids scatterers (here [MATH] is the cyclotron frequency, [MATH] is the scattering time).', 'cond-mat-0404397-2-4-5': 'Another classical mechanism was presented in Ref. [CITATION], where the MR due to non-markovian dynamics of electrons trapped in some region of space was discussed.', 'cond-mat-0404397-2-5-0': 'Notwithstanding these developments, the role of classical effects in magnetotransport was underappreciated for a long time.', 'cond-mat-0404397-2-5-1': 'A new boost to the research in this direction was given by Ref. [CITATION], where it was shown that if electrons move in a smooth disorder potential and in a sufficiently strong magnetic fields a phenomenon called "classical localization" occurs.', 'cond-mat-0404397-2-5-2': 'This phenomenon leads to the exponential suppression of the longitudinal resistance: most electrons are trapped in localized equipotential trajectories and do not participate in diffusion.', 'cond-mat-0404397-2-5-3': 'This work was followed by a series of works, [CITATION] discussing different aspects of classical magnetotransport in 2D systems.', 'cond-mat-0404397-2-5-4': 'It was shown [CITATION] that for lower magnetic fields near the onset of the classical localization the magnetoresistance is positive, i.e. the longitudinal resistance grows with increasing magnetic field.', 'cond-mat-0404397-2-5-5': 'In Refs. [CITATION] the combination of smooth disorder and strong scatterers (antidots) was considered.', 'cond-mat-0404397-2-5-6': 'It was shown that in this system under certain conditions there are several regimes of the behavior of magnetoresistance depending on the strength of the magnetic field: first the longitudinal resistance decreases with growing field, then it saturates and then begins to grow.', 'cond-mat-0404397-2-6-0': 'In Refs. [CITATION] magnetoresistance was studied in a situation where the magnetic field is classically strong, that is where the parameter [MATH] is large.', 'cond-mat-0404397-2-6-1': 'Recently, the region of classically small magnetic fields [MATH] was investigated numerically [CITATION] for the case of electrons scattering on strong scatterers.', 'cond-mat-0404397-2-6-2': 'It was shown [CITATION] that memory effects due to double scattering of an electron on the same disk lead to a negative parabolic magnetoresistance (in the Ref. baskin, where these processes were not taken into account, exponentially small MR was predicted).', 'cond-mat-0404397-2-6-3': 'The numerical simulations [CITATION] discovered a low-field classical anomaly of the MR. The anomaly was attributed to the memory effects specific for backscattering events.', 'cond-mat-0404397-2-6-4': 'The simulations were performed for the 2D Lorenz gas which is a system of 2D electrons scattering on hard disks randomly distributed in plane with average concentration [MATH].', 'cond-mat-0404397-2-6-5': 'Magnetotransport in this system is characterized by two dimensionless parameters: [MATH] and the gas parameter [MATH].', 'cond-mat-0404397-2-6-6': 'Here [MATH] is the disk radius, [MATH] is the cyclotron frequency, [MATH] is the mean free time and [MATH] is the mean free path.', 'cond-mat-0404397-2-6-7': 'The anomaly was observed in the case [MATH].', 'cond-mat-0404397-2-6-8': 'Both the numerical simulations and the qualitative considerations [CITATION] indicated that at zero temperature the MR can be expressed in terms of a dimensionless function [MATH] via [EQUATION] where [MATH] is the resistivity for [MATH].', 'cond-mat-0404397-2-6-9': 'Numerical results [CITATION] suggest that [MATH] as [MATH] yielding [EQUATION]', 'cond-mat-0404397-2-6-10': 'The latter expression is in a very good agreement with experimental measurements of negative linear MR in a random antidot arrays.', 'cond-mat-0404397-2-6-11': '[CITATION] It is anomalous in two senses.', 'cond-mat-0404397-2-6-12': 'First, it has a non-analytic dependence on the magnetic field.', 'cond-mat-0404397-2-6-13': 'Second, it does not vanish in the limit of vanishing [MATH], which is normally regarded as the expansion parameter for the corrections to the Drude-Boltzmann picture.', 'cond-mat-0404397-2-6-14': 'This intriguing behavior calls for a rigorous analytical theory of the effect, which would establish Eq. [REF] and enable one to derive the analytical expression for function [MATH].', 'cond-mat-0404397-2-7-0': 'In this paper we present a detailed theory of the anomaly and give an expression for [MATH] (the brief description of our results was given in Ref. [CITATION]).', 'cond-mat-0404397-2-7-1': 'We find that at some interval, [MATH] function [MATH] is linear in agreement with numerical simulations and experiment, but at [MATH] crosses over to a quadratic dependence.', 'cond-mat-0404397-2-7-2': 'Thus, for [MATH] Eq. [REF] yields [MATH].', 'cond-mat-0404397-2-7-3': 'The limit [MATH] should be taken with care.', 'cond-mat-0404397-2-7-4': 'While the small [MATH] expansion seems to be singular as a function of [MATH] the region of [MATH] where this expansion is valid shrinks as [MATH].', 'cond-mat-0404397-2-7-5': 'For [MATH] saturates at some constant value.', 'cond-mat-0404397-2-7-6': 'Therefore, the full variation of [MATH] is of the order [MATH].', 'cond-mat-0404397-2-7-7': 'In other words, the anomalous MR is strong but it exists in a small region of magnetic fields.', 'cond-mat-0404397-2-8-0': '# Qualitative discussion of the problem', 'cond-mat-0404397-2-9-0': 'The mechanism proposed in Ref. [CITATION] is linked to the memory effects arising in backscattering events.', 'cond-mat-0404397-2-9-1': 'It has a close relation to the well known non-analyticity of the virial expansion of transport coefficients, [CITATION] which we briefly recall.', 'cond-mat-0404397-2-9-2': 'For [MATH] the leading nonanalytic correction to resistivity, [MATH], is due to the processes of return to a scatterer after a single collision on another scatterer (see Fig. [REF]a).', 'cond-mat-0404397-2-10-0': 'The relative correction, [MATH] is proportional to the corresponding backscattering probability, given by the product of [MATH] (which is the probability to reach scatterer 2 without collision and scatter in the angle [MATH]) and the probability [MATH] to return without collisions from 2 to 1 (here [MATH] is the mean free pass).', 'cond-mat-0404397-2-10-1': 'Assuming [MATH] and integrating over intervals [MATH] one obtains [CITATION] [EQUATION]', 'cond-mat-0404397-2-10-2': 'In Ref. [CITATION] it was shown that the probability [MATH] is actually larger than [MATH].', 'cond-mat-0404397-2-10-3': 'Indeed, the exponent [MATH] can be written as [MATH] where [MATH].', 'cond-mat-0404397-2-10-4': 'It represents the probability of the existence of an empty corridor (free of the centers of the disk) of width [MATH] around the electron trajectory from [MATH] to [MATH].', 'cond-mat-0404397-2-10-5': 'However, the passage of a particle from [MATH] to [MATH] ensures the absence of the disks centers in the region of width [MATH] around this part of trajectory (from [MATH] to [MATH]) This reduces the scattering probability on the way back.', 'cond-mat-0404397-2-10-6': 'The correct value of [MATH] can be estimated as [EQUATION] where [EQUATION] is the area of the overlap of the two corridors (see Fig. [REF]a).', 'cond-mat-0404397-2-10-7': 'For example, for [MATH] we have [MATH] and [MATH] which reflects the obvious fact that the particle cannot scatter, if it travels back along the same path.', 'cond-mat-0404397-2-10-8': 'Taking into account the effect of "empty corridor", we get [EQUATION] where [MATH] is a constant of the order of unity.', 'cond-mat-0404397-2-10-9': 'Thus, for [MATH] the "empty corridor" effect simply changes the constant in the argument of the logarithm.', 'cond-mat-0404397-2-11-0': 'The key idea suggested in Ref. dmit1 was that for [MATH] the area of the overlap of the two corridors, [MATH] sharply depends on [MATH] resulting in the observed MR. Indeed, it is seen from Fig. [REF]b that for [MATH] resulting in sharp negative MR [EQUATION]', 'cond-mat-0404397-2-11-1': 'The following qualitative explanation of the observed linear MR was presented in Ref. dmit1.', 'cond-mat-0404397-2-11-2': 'The value [MATH] was estimated for [MATH] (see Fig. [REF]c) to the first order in [MATH] as [MATH] where [MATH] is the cyclotron radius.', 'cond-mat-0404397-2-11-3': 'Assuming that this estimate also works at [MATH] and expanding [MATH] to the first order in [MATH] one gets [MATH].', 'cond-mat-0404397-2-12-0': 'In fact, the physical picture of the phenomenon is more subtle.', 'cond-mat-0404397-2-12-1': 'The contribution of any trajectory with [MATH] is cancelled to the first order in [MATH] by the contribution of the time-reversed trajectory, since the values of [MATH] are opposite for these paths (see Fig. 1 d, e).', 'cond-mat-0404397-2-12-2': 'The cancellation does not occur only at very small [MATH].', 'cond-mat-0404397-2-12-3': 'The integration in Eq. [REF] over [MATH] yields [MATH].', 'cond-mat-0404397-2-12-4': 'Larger values of [MATH] also give a quadratic in [MATH] contribution to the MR. This contribution is positive and comes from the second order term in the expansion of [MATH] in [MATH].', 'cond-mat-0404397-2-12-5': 'It follows from our results (see Eqs. [REF],[REF]) that the contribution of small angles is dominant resulting in a negative parabolic MR. We find that the parabolic MR crosses over to linear at very small [MATH] which explains why the parabolic MR was not seen in numerical simulations [CITATION] and experiment.', 'cond-mat-0404397-2-12-6': '[CITATION] Note that for very small [MATH] contribution of the trajectories with a long Lyaponov region (Lyapunov trajectories) becomes important.', 'cond-mat-0404397-2-12-7': 'In the Section [REF] we focus on the region [MATH] where the contribution of the Lyapunov trajectories is parametrically small.', 'cond-mat-0404397-2-12-8': 'The Lyapunov trajectories will be discussed in Section [REF].', 'cond-mat-0404397-2-13-0': '# Calculations', 'cond-mat-0404397-2-14-0': '## Kinetic equation', 'cond-mat-0404397-2-15-0': 'In this subsection we introduce the kinetic equation which is the starting point for the calculation of the diffusion coefficient and the resistivity of the Lorenz gas.', 'cond-mat-0404397-2-16-0': 'We consider the Lorenz gas at zero temperature, assuming that the electrons participating in the conduction have the Fermi velocity [MATH].', 'cond-mat-0404397-2-16-1': 'The diffusion coefficient [MATH] is given by [EQUATION]', 'cond-mat-0404397-2-16-2': "Here [MATH] is the retarded Green's function of the Liouville equation and [MATH] stands for the averaging over the positions of the disks.", 'cond-mat-0404397-2-16-3': 'The equation for [MATH] reads [EQUATION] where [MATH] is the Liouville operator of the free motion in the magnetic field and [MATH] describes scattering on the disks.', 'cond-mat-0404397-2-16-4': 'From Eq. [REF] it follows that we need the time integral of [MATH] rather than the whole time-dependent function.', 'cond-mat-0404397-2-16-5': 'This integral can be written as [EQUATION] where [MATH] are the angles of velocities [MATH] and [MATH] respectively.', 'cond-mat-0404397-2-16-6': 'Here we used the energy conservation, which implies that in scattering processes the absolute value of the velocity does not change.', 'cond-mat-0404397-2-16-7': 'Using Eq. [REF], the diffusion coefficient Eq. [REF] can be rewritten as [EQUATION]', 'cond-mat-0404397-2-16-8': 'Since [MATH] depends on [MATH] and [MATH] only, it is convenient to average over the position of the initial point and over the initial angle.', 'cond-mat-0404397-2-16-9': 'Here [MATH] is the area of the sample.', 'cond-mat-0404397-2-16-10': 'Integrating Eq. [REF] with respect to time the equation for [MATH] is written as [EQUATION] where [EQUATION] [MATH] is the unit vector in the direction of [MATH] and [MATH].', 'cond-mat-0404397-2-16-11': 'The interaction with disks is written in Eq. [REF] in the form of collision integral.', 'cond-mat-0404397-2-16-12': '[CITATION] The scattering operators [MATH] transform arbitrary function [MATH] as follows, [EQUATION] where [EQUATION]', 'cond-mat-0404397-2-16-13': "Here [MATH] are the positions of the disk's centers and [EQUATION] is the differential cross-section of one disk.", 'cond-mat-0404397-2-16-14': 'The vector [MATH] depends on angles [MATH] in the integrals Eq. [REF] [EQUATION] and is pointing from the center of a disk to the scattering point at the disk surface (see Fig. [REF]).', 'cond-mat-0404397-2-16-15': 'Physically, operator [MATH] describes influx of particles to velocity [MATH] at the point [MATH], while operator [MATH] describes the outflux from velocity [MATH] at the point [MATH].', 'cond-mat-0404397-2-16-16': 'The averaged value of the [MATH] is equal to the disks concentartion [EQUATION]', 'cond-mat-0404397-2-16-17': 'The Bolzmann equation is obtained by averaging the Liouville equation with respect to the position of the scatterers and neglecting correlations.', 'cond-mat-0404397-2-16-18': 'Indeed, using Eq. [REF] one finds that the average collision operators Eq. [REF] are given by [EQUATION]', 'cond-mat-0404397-2-16-19': 'Replacing the collision operators in the Liouville equation Eq. [REF] with their averages Eq. [REF] one obtains the Boltzmann equation [EQUATION]', 'cond-mat-0404397-2-16-20': 'The Bolzmann-Drude diffusion coefficient [MATH] in the absence of the magnetic field is found from Eqs. [REF] and [REF] as follows.', 'cond-mat-0404397-2-16-21': "Integrating [REF] with respect to [MATH] and expanding the Green's function [MATH] in angular harmonics one finds [EQUATION] where [EQUATION]", 'cond-mat-0404397-2-16-22': 'In particular, [EQUATION] is the transport length.', 'cond-mat-0404397-2-16-23': 'Substituting Eqs. [REF] and [REF] in Eq. [REF] one finds [EQUATION]', 'cond-mat-0404397-2-17-0': '## Perturbative expansion', 'cond-mat-0404397-2-18-0': "In this subsection we derive the perturbative expansion for the average Green's function in order to take into account non-Markovian corrections which are absent in the Bolzmann-Drude picture.", 'cond-mat-0404397-2-19-0': 'Introduce the operators [EQUATION] which describe the fluctuation of the collision integral with respect to its average value.', 'cond-mat-0404397-2-19-1': 'Using these operators the formal solution of [REF], [MATH] can be written as the following series [EQUATION] where [MATH] is defined by Eq. [REF].', 'cond-mat-0404397-2-19-2': 'Here we took into account that [MATH].', 'cond-mat-0404397-2-19-3': 'The first term on the right hand side of Eq. [REF] gives the Drude-Boltzmann result described in the previous subsection.', 'cond-mat-0404397-2-19-4': 'The rest of the terms in Eq. [REF] provide a regular way for the calculation of correlations, which are absent in the Boltzmann picture.', 'cond-mat-0404397-2-20-0': 'In the subsequent analysis of the perturbative expansion Eq. [REF] we will extensively use the representation of the Boltzmann propagator [MATH] in terms of the ballistic propagator [EQUATION]', 'cond-mat-0404397-2-20-1': 'This is achieved by expanding the Boltzmann propagator [MATH] as a sum over the number of scattering events represented by the operator [MATH] [EQUATION]', 'cond-mat-0404397-2-20-2': 'The diagrammatic representation of the expansion [REF] is shown in Fig. ([REF]).', 'cond-mat-0404397-2-20-3': 'Each cross in this figure corresponds to [MATH] and each solid line to a ballistic propagator [MATH].', 'cond-mat-0404397-2-20-4': "For future reference we write the explicit expression for the ballistic Green's function in zero magnetic field and in weak magnetic fields.", 'cond-mat-0404397-2-20-5': 'For [MATH] the ballistic propagator conserves the velocity.', 'cond-mat-0404397-2-20-6': 'In this case its kernel is given by [EQUATION]', 'cond-mat-0404397-2-20-7': 'Here [MATH] is the angle of the vector [MATH].', 'cond-mat-0404397-2-20-8': 'Magnetic field rotates the velocity vector with the cyclotron frequency.', 'cond-mat-0404397-2-20-9': 'For small magnetic fields, [MATH], we have [EQUATION]', 'cond-mat-0404397-2-21-0': '## Ballistic returns in perturbative expansion', 'cond-mat-0404397-2-22-0': 'In this subsection we show how the processes of ballistic returns discussed in Section [REF] arise in the perturbative expansion Eq. [REF].', 'cond-mat-0404397-2-22-1': 'Consider the second term in Eq. [REF].', 'cond-mat-0404397-2-22-2': 'This term describes the memory effect due to diffusive returns.', 'cond-mat-0404397-2-22-3': 'As discussed in Section [REF], the main contribution comes from returns after a single scattering.', 'cond-mat-0404397-2-22-4': 'This process is described by the diagram shown in Fig. [REF]a.', 'cond-mat-0404397-2-22-5': 'The dashed line corresponds to the pairings [MATH]), external wavy lines to the diffusion propagators [MATH].', 'cond-mat-0404397-2-22-6': 'The internal line corresponds to the Boltzmann propagator Eq. [REF] truncated at one scattering [MATH].', 'cond-mat-0404397-2-23-0': 'Four combinations of [MATH] at the ends of external dashed lines in the diagrams shown in Fig. [REF]a represent four different types of correlation at a given point [MATH].', 'cond-mat-0404397-2-23-1': 'To see this consider the pairing [MATH].', 'cond-mat-0404397-2-23-2': 'By virtue of Eqs. [REF] and [REF] this pairing is proportional to the density-density correlation function [MATH] for [MATH].', 'cond-mat-0404397-2-23-3': 'Assuming that disks are randomly distributed over the sample we get for these functions [EQUATION]', 'cond-mat-0404397-2-23-4': 'It is natural to interpret the vector [MATH] as the position of the center of the disk on which a double scattering occurs.', 'cond-mat-0404397-2-23-5': 'However, this interpretation is valid for diagram [MATH] only.', 'cond-mat-0404397-2-23-6': 'As shown in Fig. [REF], this diagram corresponds to the situation where an electron experiences two real scattering processes on the disk placed at point [MATH].', 'cond-mat-0404397-2-23-7': 'The physical interpretation of other diagrams is more subtle.', 'cond-mat-0404397-2-23-8': 'The diagram [MATH] (see Fig. [REF]) does not correspond to any real scattering at point [MATH].', 'cond-mat-0404397-2-23-9': 'It just allows one to calculate correctly the probability for an electron to pass twice the region of the size [MATH] around point [MATH] without scattering.', 'cond-mat-0404397-2-23-10': 'To interpret the diagrams [MATH] and [MATH] note that in the Boltzmann picture, which neglects correlations, the following process is allowed.', 'cond-mat-0404397-2-23-11': 'An electron scatters on a disk and later on passes through the region occupied by this disk without a scattering (see Fig. [REF]) (analogous consideration is valid for diagram shown in Fig. [REF]).', 'cond-mat-0404397-2-23-12': 'The diagrams [MATH] and [MATH] correct the Boltzmann result by subtracting the contribution of such unphysical processes.', 'cond-mat-0404397-2-24-0': 'As follows from the qualitative discussion Section [REF], the processes of ballistic returns after a single scattering give rise to the leading (non-analytic) correction to the Drude-Boltzmann result in zero magnetic field.', 'cond-mat-0404397-2-24-1': 'However, taking into account diagrams shown in Fig. [REF]a (describing processes shown in Figs. [REF]-[REF]) is not sufficient for calculation of the low field anomaly of the MR. Actually, diagrams in Fig. [REF]a do not contain the "empty corridor" effect.', 'cond-mat-0404397-2-24-2': 'We will show that the correct description of the memory effects, specific for ballistic returns, requires the renormalization of diagrams [REF]a by diagrams [REF]b, in which the internal dashed lines contains [MATH] pairings only.', 'cond-mat-0404397-2-24-3': "Physically, the [MATH]'th order diagram of the type [REF]b represents the [MATH]'th order term in the Taylor expansion of the [MATH] in the qualitative estimate [REF].", 'cond-mat-0404397-2-24-4': 'Such renormalization play the key role in the quantitative description of the anomalous MR.', 'cond-mat-0404397-2-25-0': 'In the next subsection we derive analytical expression for diagrams [REF]a.', 'cond-mat-0404397-2-25-1': 'Then we generalize the calculations to account for the corridor effect and calculation of anomalous MR.', 'cond-mat-0404397-2-26-0': '## Non-analytical corrections to the zero-field resistance neglecting the "empty corridor" effect', 'cond-mat-0404397-2-27-0': 'In this subsection we use the perturbative expansion Eq. [REF] to derive analytically the leading correction to the diffusion coefficient.', 'cond-mat-0404397-2-27-1': 'This correction is due to the processes described by diagram Fig. [REF]a.', 'cond-mat-0404397-2-27-2': "The operator expression of the correction to the Green's function related to this diagram is given by", 'cond-mat-0404397-2-28-0': '(0,30)(50,-15) (250,-7)(1,0)68 (318,-7)(0,1)5 (250,-7)(0,1)5 (207,0) [MATH]', 'cond-mat-0404397-2-29-0': 'where the underbracket stands for the pairings of operators [MATH] and [MATH].', 'cond-mat-0404397-2-29-1': 'Substituting this correction into Eq. [REF], and using Eqs. [REF],[REF] to integrate over [MATH] we get [EQUATION] where [EQUATION]', 'cond-mat-0404397-2-29-2': 'Here we used Eq. [REF] for [MATH].', 'cond-mat-0404397-2-30-0': 'Graphically, these calculations are presented in Fig. [REF].', 'cond-mat-0404397-2-30-1': 'The pairings [MATH] entering Eq. [REF], are given in Appendix A for four possible combinations of [MATH].', 'cond-mat-0404397-2-30-2': 'The pairings only depend on the difference [MATH].', 'cond-mat-0404397-2-30-3': 'Therefore in Eq. [REF] one can remove the integral over [MATH] and the sample area [MATH] in the denominator and put [MATH] in the integrand.', 'cond-mat-0404397-2-30-4': 'In other words we put the origin of the coordinate system at the backscattering point [MATH].', 'cond-mat-0404397-2-30-5': 'Next we integrate Eq. [REF] over the angles [MATH].', 'cond-mat-0404397-2-30-6': 'This integration can be easily done taking into account angle dependent delta functions entering in [MATH] and [MATH] (see Eq. [REF]).', 'cond-mat-0404397-2-30-7': 'As a result we get [MATH] and [EQUATION]', 'cond-mat-0404397-2-30-8': 'In derivation of Eq. [REF] we took into account that the pairing of collision operators [MATH] vanishes for [MATH] which is clear from its physical meaning and also can be seen from the delta functions in the factor [MATH] (see Appendix A).', 'cond-mat-0404397-2-30-9': 'At the same time both [MATH] and [MATH] are of the order of the mean free path [MATH].', 'cond-mat-0404397-2-30-10': 'Therefore, [MATH] and [MATH].', 'cond-mat-0404397-2-30-11': 'Physically this means that the typical scattering angle in the processes of ballistic return are close to [MATH].', 'cond-mat-0404397-2-30-12': 'Next, we use Eq. [REF] to evaluate the contributions of different pairings [MATH]', 'cond-mat-0404397-2-31-0': '## (+,+) pairing', 'cond-mat-0404397-2-32-0': 'Consider first the process in which [MATH].', 'cond-mat-0404397-2-32-1': 'Substitute equations [REF] and [REF] in Eq. [REF] and integrate over [MATH].', 'cond-mat-0404397-2-32-2': 'It is convenient for our purposes to perform this integration for a fixed value of the vector [MATH].', 'cond-mat-0404397-2-32-3': 'Then the delta functions in Eq. [REF] ensure that [MATH] and [MATH] lie on the circle of the radius [MATH] centered in the point [MATH] [EQUATION]', 'cond-mat-0404397-2-32-4': 'The graphical solution of these equations is shown in Fig. [REF] ( vectors [MATH] in this figure are the unit vectors in the directions of [MATH] and [MATH] respectively).', 'cond-mat-0404397-2-32-5': 'Since [MATH] is small ( [MATH]) one has approximately [EQUATION]', 'cond-mat-0404397-2-32-6': 'This accuracy is sufficient for the calculation of the diagram in Fig. [REF]a (since we calculate it in the lowest order in [MATH] ).', 'cond-mat-0404397-2-32-7': 'Upon integrating out the vectors [MATH] the integrand in Eq. [REF] depends on [MATH].', 'cond-mat-0404397-2-32-8': 'Finally we get [EQUATION]', 'cond-mat-0404397-2-32-9': 'Introducing new variables [MATH] and [MATH] and taking into account that [MATH] and [MATH] the relative correction to the diffusion coefficient is written as [EQUATION] >From Fig. [REF] it is seen that [MATH], [MATH] are the scattering angles.', 'cond-mat-0404397-2-33-0': '## (+,-) pairing', 'cond-mat-0404397-2-34-0': 'Next we present calculations for [MATH].', 'cond-mat-0404397-2-34-1': 'Like the calculation of the diagram [MATH] we substitute Eq. [REF] and Eq. [REF] into Eq. [REF] and integrate over [MATH] with the use of Eq. [REF].', 'cond-mat-0404397-2-34-2': 'The values of [MATH] and [MATH] should be found from the following equations [EQUATION]', 'cond-mat-0404397-2-34-3': 'The graphical solution of these equation is presented in Fig. [REF].', 'cond-mat-0404397-2-34-4': 'After integrating out vectors [MATH] and the angle [MATH] we get [EQUATION]', 'cond-mat-0404397-2-34-5': 'In derivation of this equation we have used Eq. [REF].', 'cond-mat-0404397-2-34-6': 'It is convenient to rewrite this equation using as integration variables [MATH] which are the scattering angles for the process [MATH].', 'cond-mat-0404397-2-34-7': 'They are expressed in terms of [MATH] and [MATH] as [MATH].', 'cond-mat-0404397-2-34-8': 'In these variables Eq. [REF] reads [EQUATION]', 'cond-mat-0404397-2-34-9': 'It is seen from Eq. [REF] (see also Fig. [REF]) that the contribution of the process [MATH] can be parameterized by the angles [MATH] and [MATH] which are the scattering angles for the process [MATH].', 'cond-mat-0404397-2-34-10': 'Analogous calculations can be easily done for other types of correlations.', 'cond-mat-0404397-2-35-0': 'Summing the different contributions [MATH] we get the following expression for the diagram Fig. [REF]a [EQUATION]', 'cond-mat-0404397-2-35-1': 'This equation is the exact expression for the non-analytic correction [CITATION] to the Drude-Boltzmann resistivity which was qualitatively given in Eq. [REF].', 'cond-mat-0404397-2-35-2': 'Four terms in the product [MATH] correspond to four types of correlations discussed above.', 'cond-mat-0404397-2-36-0': '## "Empty corridor" effects on the zero-field resistance In this subsection we use the perturbation theory for the quantitative derivation of the effect of the "empty corridor" discussed in section [REF].', 'cond-mat-0404397-2-37-0': 'Eq. [REF] takes into account one pairing of operators [MATH] and [MATH].', 'cond-mat-0404397-2-37-1': 'The terms containing [MATH] pairings ([MATH] dashed lines) are typically small as [MATH].', 'cond-mat-0404397-2-37-2': 'However, there is a series of diagrams, shown in Fig. [REF]b, whose contribution is of the order [MATH] (see Ref. [CITATION]).', 'cond-mat-0404397-2-37-3': 'The internal dashed lines in this series only contain pairings [MATH].', 'cond-mat-0404397-2-37-4': 'Below we show that the series Fig. [REF]b accounts for the effect of the "empty corridor".', 'cond-mat-0404397-2-37-5': 'More precisely, we prove that the [MATH]th order term in this series corresponds to [MATH] term in the Taylor expansion of the [MATH] in Eq. [REF].', 'cond-mat-0404397-2-38-0': 'The addition of the diagram Fig. [REF]b to the diagram Fig. [REF]a leads to the following replacement in Eq. [REF] [EQUATION] where the function [MATH] can be found from the Dyson equation (See Fig. [REF]).', 'cond-mat-0404397-2-39-0': 'Since the operator [MATH] does not change the velocity angle, one can search the solution of the Dyson equation in the form [EQUATION] where [MATH] is the difference between the backscattering angle and [MATH] (See Fig. ([REF])).', 'cond-mat-0404397-2-39-1': 'While substituting Eq. [REF] in the Dyson equation (which is shown graphically in Fig. ([REF]), integrating over angles [MATH], and using the identity [EQUATION] (and analogous identity for vectors [MATH] )we get [EQUATION] where [MATH] are defined in the Appendix [REF].', 'cond-mat-0404397-2-39-2': 'Further calculations can be done in two different ways.', 'cond-mat-0404397-2-39-3': 'It is useful from methodological point of view to discuss both of them.', 'cond-mat-0404397-2-39-4': 'One way is to first integrate Eq. [REF] over the angles of the vectors [MATH] and [MATH] (see Fig. [REF]).', 'cond-mat-0404397-2-39-5': 'The delta-functions entering in Eq. [REF] ensure that point [MATH] (see Figs. [REF],[REF]) lies on the line segment connecting point [MATH] and the origin and point [MATH] lies on the line segment connecting point [MATH] and the origin.', 'cond-mat-0404397-2-39-6': 'This allows to reduce the Dyson equation to the closed relation for function [MATH] [EQUATION] where the pairing [EQUATION] is calculated in Appendix [REF].', 'cond-mat-0404397-2-39-7': 'Here [EQUATION] is shown in Fig. [REF].', 'cond-mat-0404397-2-40-0': 'As a result we have [EQUATION]', 'cond-mat-0404397-2-40-1': 'This equation has an evident solution [EQUATION]', 'cond-mat-0404397-2-40-2': 'Eq. [REF] may be derived in an alternative way.', 'cond-mat-0404397-2-40-3': 'This way allows to understand on the formal basis why one should renormalize the diagrams in Fig. [REF]a by the [MATH] pairings only.', 'cond-mat-0404397-2-40-4': 'Let us integrate in Eq. [REF] over [MATH] using Eq. [REF].', 'cond-mat-0404397-2-40-5': 'Doing this, we get [EQUATION]', 'cond-mat-0404397-2-40-6': 'Here we took into account that [MATH] is a slowly changing function of [MATH] (it changes on a scale of the order of [MATH]).', 'cond-mat-0404397-2-40-7': 'Since Eq. [REF] provide [MATH], we put [MATH].', 'cond-mat-0404397-2-40-8': 'Next we make use of the identity [EQUATION] to integrate over [MATH].', 'cond-mat-0404397-2-40-9': 'After this integration we get [EQUATION]', 'cond-mat-0404397-2-40-10': 'Integration over [MATH] leads again to Eq. [REF].', 'cond-mat-0404397-2-41-0': '>From Eq. [REF] it follows that we need to know [MATH] for [MATH].', 'cond-mat-0404397-2-41-1': 'Using Eq. [REF] we find [EQUATION] where [MATH] is the overlap between two corridors, given by Eq. [REF].', 'cond-mat-0404397-2-41-2': 'The [MATH]-th order term in the Taylor expansion of the exponential [REF] corresponds to a diagram in Fig. [REF]b with [MATH] dashed lines.', 'cond-mat-0404397-2-41-3': 'Indeed, the [MATH]-th order term in the Taylor expansion contains [MATH] integration over coordinates of [MATH] scatterings of the type [MATH].', 'cond-mat-0404397-2-41-4': 'What remains to do to get the resistivity correction is to express [MATH] in Eq. [REF] via angles [MATH] ,[MATH].', 'cond-mat-0404397-2-41-5': 'To do this, the precision Eq. [REF] is not sufficient.', 'cond-mat-0404397-2-41-6': 'More specifically, we will need to know the angle between vectors [MATH] and [MATH] to the order [MATH].', 'cond-mat-0404397-2-41-7': 'For this purpose it will be sufficient to make replacements [MATH] and [MATH] in the arguments of vectors [MATH] in Eq. [REF].', 'cond-mat-0404397-2-41-8': 'Then the angle [MATH] is calculated as (see Figs. [REF],[REF],[REF],[REF]) [EQUATION]', 'cond-mat-0404397-2-41-9': 'This equation is valid for [MATH], [MATH] (since [MATH] is not periodic with [MATH] one should specify the integration limits).', 'cond-mat-0404397-2-41-10': 'It worth noting that Eq. [REF] ensures that the argument of the [MATH] function in Eq. [REF] is positive and the overlap width and overlap area are given by [EQUATION]', 'cond-mat-0404397-2-41-11': 'Summing the diagrams Fig. [REF]b together with Fig. [REF]a, one gets an exact equation [EQUATION] instead of qualitative estimate Eq. [REF].', 'cond-mat-0404397-2-42-0': 'Here [MATH] is a numerical coefficient.', 'cond-mat-0404397-2-42-1': 'Thus, addition of the series Fig. [REF]b to Fig. [REF]a leads to the following renormalization: [MATH]', 'cond-mat-0404397-2-43-0': '## Estimate for neglected diagrams', 'cond-mat-0404397-2-44-0': 'In this subsection we use the derivation presented in the previous subsection to show how to select relevant diagrams.', 'cond-mat-0404397-2-44-1': 'The derivation was based on identity [REF].', 'cond-mat-0404397-2-44-2': 'The left hand side of this equation is proportional to the impurity cross section [MATH].', 'cond-mat-0404397-2-44-3': 'However, the right hand side is parameterically larger [MATH] provided that [MATH].', 'cond-mat-0404397-2-44-4': 'One can show that the function similar to the one on the right hand side [REF] arises each time when one of the index [MATH] or [MATH] in the pairing [MATH] is equal to [MATH] (for a pairing [MATH] we have two functions of the type [REF], see Eq. [REF]).', 'cond-mat-0404397-2-44-5': 'For the case when [MATH] or [MATH] equals to [MATH] we have a differential cross section [MATH] instead of function [REF].', 'cond-mat-0404397-2-44-6': 'Now we are ready to estimate different types of diagrams.', 'cond-mat-0404397-2-44-7': "Insertion of one additional [MATH] pairing into a diagram in Figs. [REF]-[REF] gives two additional Green's functions and, consequently, multiplier [MATH] and two functions of the type [REF] giving a multiplier [MATH].", 'cond-mat-0404397-2-44-8': 'One should also multiply on disk concentration [MATH] and integrate over [MATH].', 'cond-mat-0404397-2-44-9': 'Due to [MATH] functions in Eq. [REF] the integration area [MATH] is of the order [MATH].', 'cond-mat-0404397-2-44-10': 'Combining all the multipliers together we have [MATH].', 'cond-mat-0404397-2-44-11': 'Therefore, addition of [MATH] pairing does not lead to any smallness.', 'cond-mat-0404397-2-44-12': 'In contrast to this, insertion of [MATH] pairing leads to smallness [MATH].', 'cond-mat-0404397-2-44-13': 'Indeed, the only difference from the case of [MATH] pairing is that one should replace one of the function [REF] by the corresponding cross section.', 'cond-mat-0404397-2-44-14': 'This leads to the change of one of the multiplier of the order [MATH] by multiplier of the order [MATH].', 'cond-mat-0404397-2-44-15': 'Following this line of reasoning one could conclude that insertion of [MATH] pairing leads to the smallness of the order of [MATH].', 'cond-mat-0404397-2-44-16': 'Actually, the smallness arising from insertion of [MATH] pairing is of the order of [MATH].', 'cond-mat-0404397-2-44-17': 'Indeed, while the replacement of each of two function of the type [MATH] by corresponding cross section leads to the relative smallness of the order [MATH] the integration area in this case is not restricted by angle dependent [MATH] functions.', 'cond-mat-0404397-2-44-18': 'As a result, the integration area, [MATH] is larger by a parameter [MATH] compared to the case of [MATH] pairing.', 'cond-mat-0404397-2-44-19': 'Note that estimates presented above do not work for the diagrams with one dashed line Figs. [REF]-[REF].', 'cond-mat-0404397-2-44-20': 'In this case one should integrate over initial and final scattering angles.', 'cond-mat-0404397-2-44-21': 'One can easily see that integration of Eq. [MATH] over angle [MATH] leads to the multiplier of the order of [MATH].', 'cond-mat-0404397-2-44-22': 'This implies that in this case all four pairings, [MATH],[MATH],[MATH], and[MATH] are of the same order, of [MATH] (see Eq. [REF]).', 'cond-mat-0404397-2-44-23': 'Note that the higher order diagrams which are small in the parameter [MATH] may turn out to be relevant for the MR at very low magnetic field (see Section [REF]).', 'cond-mat-0404397-2-45-0': '## Anomalous magnetoresistance', 'cond-mat-0404397-2-46-0': 'In this subsection we generalize our calculations for the [MATH] case assuming that [MATH] is small ([MATH].', 'cond-mat-0404397-2-46-1': 'The main contribution in this case still comes from diagrams in Figs. [REF]a,b. Consider for example diagram [MATH].', 'cond-mat-0404397-2-46-2': 'Let us compare the process of double scattering described by this diagram for [MATH] (see Fig. [REF]) with the same process for [MATH] (see Fig. [REF]).', 'cond-mat-0404397-2-46-3': 'For fixed points [MATH] and [MATH], one can see the following differences.', 'cond-mat-0404397-2-46-4': 'First, the scattering angles [MATH] and [MATH] acquire small corrections of the order of [MATH].', 'cond-mat-0404397-2-46-5': 'Second, the parts of the electron trajectory corresponding to free ends of the picture become curved.', 'cond-mat-0404397-2-46-6': 'The backscattering angle [MATH] increases by the value [MATH] [EQUATION] where [MATH] is the value of backscattering angle for [MATH] given by Eq. [REF].', 'cond-mat-0404397-2-46-7': 'Finally, the overlap area of the corridors changes because the trajectories become curved (see Fig. [REF]).', 'cond-mat-0404397-2-47-0': 'The corrections to [MATH] and [MATH] lead to small relative corrections to the resistivity of the order of [MATH] and can be neglected.', 'cond-mat-0404397-2-47-1': 'The same reason allows one to neglect the curvature of the incoming and outgoing parts of the trajectory.', 'cond-mat-0404397-2-47-2': 'Therefore,the only relevant difference is the change of the overlap area of the corridors.', 'cond-mat-0404397-2-47-3': 'The solution of the Dyson equation is analogous to the [MATH] case.', 'cond-mat-0404397-2-47-4': 'The points of intermediate integration [MATH] and [MATH] lie now at the segments of cyclotron circles (from [MATH] to [MATH] and from [MATH] to [MATH]).', 'cond-mat-0404397-2-47-5': 'The pairing of two operators [MATH] is still given by Eq. [REF] with the replacement [MATH].', 'cond-mat-0404397-2-47-6': 'For [MATH] the overlap width is calculated as [EQUATION] where [MATH] is the Heaviside step function.', 'cond-mat-0404397-2-47-7': 'Therefore, the only difference from Eq. [REF] is that one should replace [MATH] where the overlap area is given by [EQUATION]', 'cond-mat-0404397-2-47-8': 'The value of [MATH] is obtained from Eq. [REF] by replacing [MATH] to [MATH] [EQUATION]', 'cond-mat-0404397-2-47-9': 'Introducing dimensionless variables [MATH] we get Eq. [REF], where function [MATH] is given by [EQUATION]', 'cond-mat-0404397-2-47-10': 'Here [EQUATION]', 'cond-mat-0404397-2-47-11': 'Function [MATH] has the following asymptotics [EQUATION]', 'cond-mat-0404397-2-47-12': 'In the interval [MATH] can be well approximated by linear function [EQUATION]', 'cond-mat-0404397-2-47-13': 'Next we discuss the parabolic asymptotics more carefully.', 'cond-mat-0404397-2-47-14': 'We will show that in this asymptotic region there are two contributions of different signs to the magnetoresistance: a negative contribution coming from the trajectories with very small [MATH] such that [MATH] and a positive contribution coming from larger angles.', 'cond-mat-0404397-2-47-15': 'This considerations will be used in the next section, discussing trajectories with the long Lyapunov region.', 'cond-mat-0404397-2-48-0': 'First, we write the difference of two exponents [MATH] as follows [MATH] and expand the equation in the bracket in the Taylor expansion up to the second order [EQUATION]', 'cond-mat-0404397-2-48-1': 'We consider the case [MATH]).', 'cond-mat-0404397-2-48-2': 'One can easily see that in this case the expression [MATH] entering in the argument of [MATH] function in the Eq. [REF] is positive at all values of [MATH] (we take into account Eq. [REF] and have in mind that [MATH]).', 'cond-mat-0404397-2-48-3': 'Therefore, the difference [MATH] is expressed as [EQUATION]', 'cond-mat-0404397-2-48-4': 'To sum the contributions of different electron trajectories, we take into account that the time reversed trajectories have the same statistical weight.', 'cond-mat-0404397-2-48-5': 'Indeed, as seen from Figs. ([REF]-[REF]),([REF]), time reversion correspond to the change [MATH].', 'cond-mat-0404397-2-48-6': 'This transformation does not affect the factor [MATH] entering Eq. [REF].', 'cond-mat-0404397-2-48-7': 'At the same time, the angle [MATH] changes to [MATH] under this transformation (see Eq. [REF]).', 'cond-mat-0404397-2-48-8': 'From Eq. [REF] we find the variation of the overlap area averaged over two time reversed trajectories [EQUATION]', 'cond-mat-0404397-2-48-9': 'This expression is of the order of [MATH] for [MATH] and is equal to zero for larger angles.', 'cond-mat-0404397-2-48-10': 'We conclude therefore, that time reversed contributions do not cancel only in the region of small angles [MATH].', 'cond-mat-0404397-2-48-11': 'Since the total variation of [MATH] is much larger, of the order of [MATH] (see Eq. [REF]), we can replace the expression in the right hand side of Eq. [REF] by [MATH]-function, writing [MATH].', 'cond-mat-0404397-2-48-12': 'Keeping the leading terms of the order of [MATH] only, we obtain [EQUATION]', 'cond-mat-0404397-2-48-13': 'The second term in this equation comes from the averaging of the quadratic term in the exponent expansion [MATH] over two time reversed trajectories.', 'cond-mat-0404397-2-48-14': 'This term is positive and partly compensates the negative contribution of small angles.', 'cond-mat-0404397-2-48-15': 'Using Eqs. [REF],[REF],[REF], after some algebra we get the low-field asymptotic of MR as the sum of the negative and positive contributions discussed above [EQUATION] where numerical coefficient [MATH] is given by [EQUATION]', 'cond-mat-0404397-2-49-0': '# Contribution of trajectories with long Lyapunov region to magnetoresistance The equations derived in the previous section give the contribution to the MR related to the processes shown in Figs. ([REF]-[REF]), the parabolic asymptotics [REF] starting to work when [MATH] becomes smaller than [MATH].', 'cond-mat-0404397-2-49-1': 'Such processes are related to the correlations specific for returns to the initial point after one scattering.', 'cond-mat-0404397-2-49-2': 'As we show at this section for very low magnetic fields, [MATH] other correlations come into play.', 'cond-mat-0404397-2-49-3': 'Specifically, we consider the contribution to the MR of the trajectories containing long Lyapunov regions.', 'cond-mat-0404397-2-49-4': 'Such trajectories consist of the direct and the return paths and involve real double scatterings on some number of disks as shown in Fig. [REF].', 'cond-mat-0404397-2-49-5': 'The divergency between the direct and the return paths is characterized by the Lyapunov length.', 'cond-mat-0404397-2-49-6': 'We will call such trajectories "Lyapunov trajectories".', 'cond-mat-0404397-2-49-7': 'In the diagrammatic series they are presented by the sum of the diagrams shown in Fig. [REF].', 'cond-mat-0404397-2-49-8': 'Just as in the case discussed above, four different pairings are allowed at the ends of external dashed lines: [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0404397-2-49-9': 'However, in contrast to the diagrams shown in Fig. [REF], internal lines of "Lyapunov diagrams" contain pairings of [MATH] type as well as of [MATH] type.', 'cond-mat-0404397-2-49-10': 'Physically, [MATH] pairings corresponds to real double scatterings in the Lyapunov region.', 'cond-mat-0404397-2-49-11': 'We will count such diagrams by the number [MATH] of correlated links in the Lyapunov region.', 'cond-mat-0404397-2-49-12': 'The Lyapunov trajectory shown in Fig. [REF] corresponds to [MATH].', 'cond-mat-0404397-2-49-13': 'The diagrams in Figs. [REF] discussed in Section [REF], present a particular case of diagrams with Lyapunov region, corresponding to [MATH].', 'cond-mat-0404397-2-49-14': 'The correlated links of any Lyapunov trajectory are renormalized by the pairings [MATH] as was discussed already for [MATH].', 'cond-mat-0404397-2-49-15': 'The contribution of any diagram of such type to the resistivity is small as [MATH].', 'cond-mat-0404397-2-49-16': 'However, as will be shown below, these diagrams have a sharp dependence on the magnetic field at very small fields.', 'cond-mat-0404397-2-49-17': 'We will show that for [MATH] the diagram of the N-th order gives a contribution to the parabolic MR of the same order as the contribution of the diagrams with [MATH] already calculated above.', 'cond-mat-0404397-2-50-0': 'Consider the Lyapunov trajectory with [MATH] correlated links.', 'cond-mat-0404397-2-50-1': 'Denote by [MATH] the angles between the segments of the direct and the return paths and by [MATH] the scattering angles between successive correlated links as shown in Fig. [REF].', 'cond-mat-0404397-2-50-2': 'The contribution of such a process has a sharp dependence on the magnetic field due to the magnetic field dependence of the overlap of the corridors surrounding the direct and the return path.', 'cond-mat-0404397-2-50-3': 'This dependence is different for different segments of the trajectory due to the difference of the angles [MATH].', 'cond-mat-0404397-2-50-4': 'Indeed as we have seen in the previous section if [MATH] is the typical angle between the direct and the return paths then the characteristic scale for the magnetic field dependence of the overlap of the corridors is [MATH].', 'cond-mat-0404397-2-50-5': 'As one can see from Eq. [REF], the typical value of the angle [MATH] is of the order [MATH].', 'cond-mat-0404397-2-50-6': 'The smallest angle [MATH] corresponds to the last segment of the trajectory.', 'cond-mat-0404397-2-50-7': 'Therefore it is this segment that should lead to the sharpest dependence of the resistivity on the magnetic field.', 'cond-mat-0404397-2-50-8': 'Consider the contribution of the last segment at small magnetic field [MATH].', 'cond-mat-0404397-2-50-9': 'For typical trajectories with [MATH] there is a cancellation of the contributions of the time reversed paths to the MR (see discussion in the previous section).', 'cond-mat-0404397-2-50-10': 'However, for a small fraction of the trajectories with [MATH] the contribution of the time reversed paths does not cancel and is proportional to the change of the overlap area [MATH].', 'cond-mat-0404397-2-50-11': 'The phase space of such trajectories is proportional to [MATH].', 'cond-mat-0404397-2-50-12': 'Thus, the contribution of the Lyapunov trajectory with [MATH] links to the resistivity in the region of small magnetic field is given by [EQUATION] where the factor [MATH] is due to the [MATH]-th order of the corresponding diagram.', 'cond-mat-0404397-2-50-13': 'As we show below, the coefficient in this equation does not depend on the diagram order.', 'cond-mat-0404397-2-50-14': 'It equals [MATH] thus coinciding with the coefficient in the small angles contribution to parabolic asymptotic for [MATH] (see Eq. [REF]).', 'cond-mat-0404397-2-50-15': 'It worth noting that only in the case [MATH] one should take into account the contribution of large angles, presented by the second term in Eq. [REF].', 'cond-mat-0404397-2-50-16': 'For [MATH] such contribution is parametrically small.', 'cond-mat-0404397-2-51-0': 'Before presenting rigorous derivation of above statements, we consider qualitatively the contribution of [MATH]-th order diagram for larger magnetic fields [MATH].', 'cond-mat-0404397-2-51-1': 'At such fields the first order contribution of the time reversed paths to the corridor effect does not cancel for trajectories with any values of [MATH] up to the maximal value [MATH].', 'cond-mat-0404397-2-51-2': 'Thus, the factor [MATH] which counts the phase space of relevant trajectories saturates at some constant value of the order of unity and one has [EQUATION]', 'cond-mat-0404397-2-51-3': 'Consider now the MR in the interval [MATH].', 'cond-mat-0404397-2-51-4': 'In this interval the Lyapunov trajectories with the number of double scatterings smaller than [MATH] give a parabolic contribution Eq. [REF] to the MR. The trajectories with the number of double scatterings bigger than [MATH] give a linear contribution Eq. [REF], the main contribution [MATH] coming from the trajectories with [MATH] double scattering.', 'cond-mat-0404397-2-51-5': 'As a result one has [EQUATION]', 'cond-mat-0404397-2-51-6': 'Here [MATH] and [MATH] is given by Eq. [REF].', 'cond-mat-0404397-2-51-7': 'For [MATH] the second term in the right hand side of Eq. [REF] can be neglected and we get finally [EQUATION]', 'cond-mat-0404397-2-51-8': 'Equation Eq. [REF] indicates that at very low magnetic fields, which correspond to large [MATH] the results of section [REF] become incorrect and logarithmic renormalization of the parabolic MR occurs [EQUATION]', 'cond-mat-0404397-2-51-9': 'Next we present a rigorous derivation of the contribution of the the [MATH]-th order Lyapunov trajectory for [MATH].', 'cond-mat-0404397-2-51-10': 'As follows from qualitative considerations presented above, [MATH] for any [MATH].', 'cond-mat-0404397-2-51-11': 'Indeed, [MATH].', 'cond-mat-0404397-2-51-12': 'Since the minimal angle, [MATH] is on the order of [MATH] the maximal angle is small compared to unity, [MATH].', 'cond-mat-0404397-2-51-13': 'Therefore, the direct and return paths of the relevant Lyapunov trajectories are very close to each other.', 'cond-mat-0404397-2-51-14': 'This allows us to characterize a scattering between [MATH] and [MATH] links of the trajectory by one scattering angle [MATH] instead of two different scattering angles for direct and return paths.', 'cond-mat-0404397-2-51-15': 'The dependence on the magnetic field comes from the last link of the trajectory.', 'cond-mat-0404397-2-51-16': 'Expanding [MATH] like it was done in the previous section, we only keep the first term [EQUATION]', 'cond-mat-0404397-2-51-17': 'To write down the analytical expression for [MATH]-th order contribution we should take into account for combinations of [MATH] at the ends of diagram in Fig. [REF].', 'cond-mat-0404397-2-51-18': 'This correspond to four types of scattering on the impurity [MATH] in the Fig. [REF] and leads to appearing of the factor [MATH].', 'cond-mat-0404397-2-51-19': 'Here we took into account that direct and return paths coincides in the first approximation and, as a consequence, scattering angles [MATH] for direct and return paths are related to each other as follows [MATH].', 'cond-mat-0404397-2-51-20': 'The analytical expression for MR is given by [EQUATION]', 'cond-mat-0404397-2-51-21': 'Here we integrate over distances between disks instead of integration over disks positions.', 'cond-mat-0404397-2-51-22': "The factors [MATH] comes from double scatterings, the factor [MATH] from denominators of the two Green's functions, describing the propagation along [MATH]th link on the direct and return way.", 'cond-mat-0404397-2-51-23': 'In writing Eq. [REF] we also took into account the renormalization of correlated links by the [MATH] pairings.', 'cond-mat-0404397-2-51-24': 'As a consequence of this renormalization [MATH]th link, which is passed twice (on the direct and return paths) comes with the factor [MATH] instead of [MATH].', 'cond-mat-0404397-2-51-25': 'What remains to do is to express final angle [MATH] via initial angle [MATH].', 'cond-mat-0404397-2-51-26': 'From simple geometric considerations we get [EQUATION]', 'cond-mat-0404397-2-51-27': 'Combining Eqs. [REF] and [REF] and performing the integral one gets [EQUATION]', 'cond-mat-0404397-2-52-0': '# Interpretation of the resistivity correction in terms of small change of the effective scattering cross-section', 'cond-mat-0404397-2-53-0': 'In our calculations of resistivity corrections we used Eq. [REF] as a starting point.', 'cond-mat-0404397-2-53-1': 'Here we briefly discuss an alternative approach based on the accounting of the memory effects in terms of small change of the effective cross-section.', 'cond-mat-0404397-2-53-2': 'Consider first the [MATH] ballistic returns with given values of the angles [MATH] and [MATH].', 'cond-mat-0404397-2-53-3': 'The contribution of such processes to the resistivity (both for [MATH] and for [MATH]) contains the factor [MATH].', 'cond-mat-0404397-2-53-4': 'Let us introduce now the scattering angle [MATH] for the process [MATH] considered as a single scattering on a complex scatterer.', 'cond-mat-0404397-2-53-5': 'For small values of [MATH] and [MATH] this angle is evidently given by [MATH].', 'cond-mat-0404397-2-53-6': 'For arbitrary values of [MATH] the scattering angle reads [EQUATION]', 'cond-mat-0404397-2-53-7': 'Such definition ensures that [MATH].', 'cond-mat-0404397-2-53-8': 'Next we introduce the integration over this angle: [MATH].', 'cond-mat-0404397-2-53-9': 'Analogous expressions can be easily written for the processes [MATH] and [MATH].', 'cond-mat-0404397-2-53-10': 'As a result, one can write the contribution of four possible types of correlations as follows [EQUATION]', 'cond-mat-0404397-2-53-11': 'The [MATH] process is represented in Eq. [REF] by [MATH].', 'cond-mat-0404397-2-53-12': 'Here we neglected a small difference (on the order of [MATH]) between the angle [MATH] and [MATH].', 'cond-mat-0404397-2-53-13': 'By virtue of Eq. [REF] one can easily check that the memory effects related to four types of ballistic returns can be accounted quantitatively in the frame of usual Boltzmann equation.', 'cond-mat-0404397-2-53-14': 'One should just replace the scattering cross-section on one disk [MATH] by the effective cross-section [EQUATION] where [MATH] is the field-dependent correction given by [EQUATION]', 'cond-mat-0404397-2-53-15': 'This correction does not change the total cross-section [EQUATION]', 'cond-mat-0404397-2-53-16': 'In other words, the enhancement of cross-section caused by processes [MATH] and [MATH] is accompanied by the reduction of the scattering due to the [MATH] and [MATH] correlations.', 'cond-mat-0404397-2-53-17': 'The resistivity correction is proportional to the change of the inverse transport length [EQUATION]', 'cond-mat-0404397-2-53-18': 'Using Eqs. [REF], [REF] and [REF] one can easily get the Eq. [REF] for MR.', 'cond-mat-0404397-2-54-0': 'Finally we note that this approach is easily generalized for calculation of the contribution of the trajectories having long Lyapunov region.', 'cond-mat-0404397-2-54-1': 'For such trajectories one should replace the angles [MATH] and [MATH] in Eq. [REF] by [MATH] and [MATH] (see Section [REF]).', 'cond-mat-0404397-2-54-2': 'For very small [MATH] such that [MATH] and [MATH] the magnetic field induced correction to the scattering cross-section is expressed in a simple form [EQUATION]', 'cond-mat-0404397-2-55-0': '# Discussion', 'cond-mat-0404397-2-56-0': 'In the previous sections we derived analytical theory of the low-field anomaly in the magnetoresistance, caused by sharp dependence of the memory effects specific for backscattering events on the magnetic field.', 'cond-mat-0404397-2-56-1': 'Next we compare our calculations with the results of simulations and with experiment.', 'cond-mat-0404397-2-56-2': 'Note first that there is a parametrically small nonanomalous correction to Eq. [REF] due to returns after multiple scatterings.', 'cond-mat-0404397-2-56-3': 'This correction is given by [CITATION] [EQUATION].', 'cond-mat-0404397-2-56-4': 'To compare the results of simulations [CITATION] with the theoretical results in a wider region of parameters [MATH] we substract [MATH] from the numerical curves.', 'cond-mat-0404397-2-56-5': 'Theoretical and numerical [CITATION] results are plotted in Fig. [REF].', 'cond-mat-0404397-2-56-6': 'in the universal units, [MATH] versus [MATH].', 'cond-mat-0404397-2-56-7': 'It is seen, that the theoretical and numerical results are in a very good agreement.', 'cond-mat-0404397-2-57-0': 'Experimental measurement of the MR in the system of antidot arrays [CITATION] agrees qualitatevely with our predictions.', 'cond-mat-0404397-2-57-1': 'The experimentally observed MR was linear in magnetic field in a wide region of fields.', 'cond-mat-0404397-2-57-2': 'This region corresponds to the interval of magnetic fields, where our results can be aproximated by linear dependence (see Eq. [REF]).', 'cond-mat-0404397-2-57-3': 'The magnetic fields used in experiment were relatively strong, and the parabolic asymptotic was not achieved.', 'cond-mat-0404397-2-57-4': 'The quantitative comparison with the experiment [CITATION] is more difficult due to several reasons.', 'cond-mat-0404397-2-57-5': 'First of all, in the structures, used in experiment, besides antidots there were also short range scatterers.', 'cond-mat-0404397-2-57-6': 'Our preliminary estimates show that accounting for a short range disorder can change the results of the calculations.', 'cond-mat-0404397-2-57-7': 'Secondly, the antidot distribution was not fully chaotic in the experiment.', 'cond-mat-0404397-2-57-8': 'To be more specific, the antidots were randomly moved from the regular square lattice distribution by shifts on the order of 30-40 of the lattice constant.', 'cond-mat-0404397-2-57-9': 'Finally, the antidots did not have equal sizes, the uncertainty of the size being on the order of 50 .', 'cond-mat-0404397-2-57-10': 'In spite of this, a very good quantitative agreement with the experiment can be achieved by appropriate choice of the antidot size [MATH] in the uncertainty interval (see also discussion of the experiment in Ref. dmit1).', 'cond-mat-0404397-2-57-11': 'Note also that generalization of the theory for the case when antidot sizes vary is straightforward.', 'cond-mat-0404397-2-57-12': 'What has to be done is to average pairings [MATH] with the distribution function for the antidots sizes.', 'cond-mat-0404397-2-57-13': 'This leads to the following modification of the obtained formulae: one should change [MATH] in Eq. [REF] and [MATH] in Eq. [REF].', 'cond-mat-0404397-2-57-14': 'Here over line means averaging over antidot sizes.', 'cond-mat-0404397-2-57-15': 'We do not present here corresponding calculations, since the distribution function of antidot sizes is not known for experiment.', 'cond-mat-0404397-2-57-16': '[CITATION]', 'cond-mat-0404397-2-58-0': 'Next we briefly discuss several interesting unsolved problems.', 'cond-mat-0404397-2-58-1': 'Note first that above consideration of Lyapunov trajectories is valid, provided that [MATH].', 'cond-mat-0404397-2-58-2': 'Indeed, while calculating the [MATH]th order diagrams we neglected small contributions of [MATH] and [MATH] pairings on the all correlated links.', 'cond-mat-0404397-2-58-3': 'Accounting of such pairings is rather tricky and is out of scope of this paper.', 'cond-mat-0404397-2-58-4': 'We expect, however, that such correlations may lead to a factor on the order of [MATH]) in Eq. [REF].', 'cond-mat-0404397-2-58-5': 'Therefore, for [MATH] the contribution of Lyapunov diagrams might become [MATH] dependent.', 'cond-mat-0404397-2-58-6': 'This gives a low-field limit for our theory [EQUATION]', 'cond-mat-0404397-2-58-7': 'We conclude that the behavior of the MR in the limit [MATH] remains so far unclear.', 'cond-mat-0404397-2-59-0': 'In our calculations we fully neglected quantum effects.', 'cond-mat-0404397-2-59-1': 'Such effects should decrease the "effect of empty corridor" due to the diffraction on the edges of the disks.', 'cond-mat-0404397-2-59-2': 'In the situations where the magnetic field is not very small, [MATH], the neglecting of the diffraction effects is justified if [MATH] is a Fermi wavelength).', 'cond-mat-0404397-2-59-3': 'This criterion ensures that diffraction effects on the edges of the disks are not relevant at the scales of the order of [MATH].', 'cond-mat-0404397-2-59-4': 'In the opposite case, [MATH] the diffraction should destroy the "corridor effect", thus suppressing the anomalous MR. For small magnetic fields, when [MATH] the criterion for negligibility of the diffraction becomes stronger, [MATH] where [MATH] is the characteristic size of the Lyapunov region.', 'cond-mat-0404397-2-59-5': 'Another quantum effect which can be especially important from the point of view of the possible experimental realizations is the weak localization phenomena.', 'cond-mat-0404397-2-59-6': 'The weak localization correction to the conductivity also has an anomalous dependence on the magnetic field.', 'cond-mat-0404397-2-59-7': 'Moreover, the interpretation of the memory effects in terms of small change of effective cross-section discussed in Section [REF] is very close to the interpretation of weak localization phenomenon developed in Ref. [CITATION].', 'cond-mat-0404397-2-59-8': 'Similar to discussion in Section [REF], the coherent enhancement of backscattering amplitude caused by weak localization, is accompanied by reduction of coherent scattering in other directions, the total cross-section do not change.', 'cond-mat-0404397-2-59-9': 'The competition of the non-Markovian and the weak localization effects might result in new interesting phenomena.', 'cond-mat-0404397-2-59-10': 'The study of such competition in a system with spin-orbit interaction can be especially interesting, because, in contrast to the weak localization correction, the "corridor effect" is not very sensitive to spin-orbit coupling.', 'cond-mat-0404397-2-59-11': 'The detailed analysis of the quantum effects is a challenging problem which will be addressed elsewhere.', 'cond-mat-0404397-2-60-0': 'We did not investigate the temperature dependence of the phenomenon.', 'cond-mat-0404397-2-60-1': 'This dependence is related to the scattering by phonons (or electron-electron scattering) neglected in our calculations.', 'cond-mat-0404397-2-60-2': 'It worth noting that importance of the electron-phonon scattering is expected to increase with decreasing magnetic field.', 'cond-mat-0404397-2-60-3': 'Indeed, the potential of the electron-phonon interaction depends on time, therefore restricting the maximal length of a trajectory with the Lyapunov region [MATH] where [MATH] and [MATH] is the temperature dependent characteristic time of the electron-phonon scattering.', 'cond-mat-0404397-2-60-4': 'This implies that at small magnetic fields one should replace the logarithmic factor [MATH], entering Eq. [REF] by a temperature dependent coefficient of the order of [MATH]', 'cond-mat-0404397-2-61-0': '# Summary', 'cond-mat-0404397-2-62-0': 'We propose a theory of the negative anomalous MR in a system with strong scatterers.', 'cond-mat-0404397-2-62-1': 'It is shown that the anomaly in the MR arises due to suppression of "empty corridor effect" by magnetic field.', 'cond-mat-0404397-2-62-2': 'A detailed description of different types of non-Markovian correlations related to ballistic returns is presented.', 'cond-mat-0404397-2-62-3': 'A method of diagrammatic expansion of the Liouville equation is developed which allows us to describe analytically the effects of "empty corridor" on ballistic returns.', 'cond-mat-0404397-2-62-4': 'The analytical expressions for anomalous MR in different intervals of magnetic fields are derived [see Eqs. [REF], [REF], [REF], [REF]].', 'cond-mat-0404397-2-62-5': 'The MR at very low magnetic fields was shown to be determined by the contribution of electron trajectories having long Lyapunov region.', 'cond-mat-0404397-2-62-6': 'An interpretation of the memory effects in terms of small change of the effective scattering cross-section is discussed.', 'cond-mat-0404397-2-62-7': 'The analytical results are shown to be in very good agreement with the numerical simulations and experiment.', 'cond-mat-0404397-2-63-0': '# Acknolegments', 'cond-mat-0404397-2-64-0': 'We thank M.I. Dyakonov for insightful discussions and R. Jullien for providing us with the numerical data.', 'cond-mat-0404397-2-64-1': 'We are also grateful to I.V. Gornyi and D.G. Polyakov for useful comments.', 'cond-mat-0404397-2-64-2': 'The work was partially supported by RFBR, grant of Russian Scientific School, and by programmes of the RAS.', 'cond-mat-0404397-2-65-0': '#', 'cond-mat-0404397-2-66-0': 'This Appendix contains the explicit expressions for the pairings [MATH].', 'cond-mat-0404397-2-66-1': 'Using Eqs. [REF], [REF] and [REF], kernels of operators [MATH] entering in Eq. [REF] can be written as [EQUATION] where [EQUATION]', 'cond-mat-0404397-2-66-2': 'The functions [MATH] and [MATH] in Eq. [REF] depend on vectors [MATH] and [MATH] correspondingly (see Eqs. [REF], [REF]).', 'cond-mat-0404397-2-66-3': 'The pairings of operators [MATH] and [MATH] can be calculated with the use of Eqs. [REF],[REF],[REF].', 'cond-mat-0404397-2-66-4': 'As a result we have [EQUATION]', 'cond-mat-0404397-2-66-5': 'Here function [MATH] emerges as a result of pairings of [MATH] (see Eq. [REF]) and is given by [EQUATION]', 'cond-mat-0404397-2-67-0': '#', 'cond-mat-0404397-2-68-0': 'To calculate [MATH] we first do the integral in the right hand side of Eq. [REF].', 'cond-mat-0404397-2-68-1': 'Integral of the term containing average density [MATH] is trivial and reduces to [MATH].', 'cond-mat-0404397-2-68-2': 'In the integral containing [MATH] it is convenient to introduce orthogonal coordinates [MATH] and [MATH] such that [MATH] axis is collinear with the velocity of the incident particle and write the delta function entering the definition [REF] of [MATH] as [MATH].', 'cond-mat-0404397-2-68-3': 'We proceed by lifting the [MATH] dependent delta function by integration over the angle [MATH].', 'cond-mat-0404397-2-68-4': 'It is clear that the integral is vanishing for [MATH].', 'cond-mat-0404397-2-68-5': 'To calculate the integral for smaller [MATH] note that by virtue of Eq. [REF] [MATH] and therefore the integral of the [MATH] dependent delta function cancels the scattering cross-section in Eq. [REF].', 'cond-mat-0404397-2-68-6': 'As a result we get [EQUATION]', 'cond-mat-0404397-2-68-7': 'Here [MATH] is a unit step function which is equal to unity when [MATH] and vanishes otherwise.', 'cond-mat-0404397-2-68-8': 'Using Eq. [REF] we can write for backscattering angles close to [MATH]) [EQUATION] where [EQUATION] is shown in Fig. [REF].'}

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[]

[['cond-mat-0404397-1-68-2', 'cond-mat-0404397-2-66-2'], ['cond-mat-0404397-1-68-4', 'cond-mat-0404397-2-66-4'], ['cond-mat-0404397-1-7-0', 'cond-mat-0404397-2-6-3'], ['cond-mat-0404397-1-7-1', 'cond-mat-0404397-2-6-3']]

[]

['cond-mat-0404397-1-13-2', 'cond-mat-0404397-1-29-0', 'cond-mat-0404397-1-48-10', 'cond-mat-0404397-1-53-11', 'cond-mat-0404397-1-67-0', 'cond-mat-0404397-1-69-0', 'cond-mat-0404397-2-28-0', 'cond-mat-0404397-2-47-10', 'cond-mat-0404397-2-51-11', 'cond-mat-0404397-2-57-16', 'cond-mat-0404397-2-65-0', 'cond-mat-0404397-2-67-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0404397

1307.5970

{'1307.5970-1-0-0': 'We find a representation of the integral of a Gauss-Markov process in the interval [MATH] in terms of Brownian motion.', '1307.5970-1-0-1': 'Moreover, some connections with first-passage-time problems are discussed, and some examples are reported.', '1307.5970-1-1-0': '# Introduction', '1307.5970-1-2-0': 'In this short note, we consider a real continuous Gauss-Markov process [MATH] of the form: [EQUATION] where: [MATH] is a standard Brownian motion (BM); [MATH] is continuous for every [MATH] the covariance [MATH] is continuous for every [MATH] with [MATH] is a monotonically increasing function and [MATH] Notice that a special case of Gauss-Markov process is the Ornstein-Uhlenbeck (OU) process, and in fact any Gauss-Markov process can be represented in terms of a OU process (see e.g. [CITATION]).', '1307.5970-1-2-1': 'Our aim is to find a representation of [EQUATION] in terms of Brownian motion.', '1307.5970-1-2-2': 'Notice that the integrated process [MATH] is equal to [MATH] where [MATH] is the time average of [MATH] in the interval [MATH] The study of [MATH] has interesting applications in Biology, for instance in the framework of diffusion models for neural activity; if one identifies [MATH] with the neuron voltage at time [MATH] then, [MATH] represents the time average of the neural voltage in the interval [MATH].', '1307.5970-1-2-3': 'Another application can be found in Queueing Theory, if [MATH] represents the length of a queue at time [MATH] then, [MATH] represents the cumulative waiting time experienced by all the "users" till the time [MATH].', '1307.5970-1-2-4': 'As for an example from Economics, let us suppose that the variable [MATH] represents the quantity of a commodity that producers have available for sale, then [MATH] provides a measure of the total value that consumers receive from consuming the amount [MATH] of the product.', '1307.5970-1-2-5': 'Really, in certain applications, it is also interesting to study first-passage time (FPT) problems for the integrated process [MATH] to this end, it is useful to dispose of an explicit representation of [MATH].', '1307.5970-1-2-6': 'The results of this paper generalize those of [CITATION].', '1307.5970-1-3-0': '# Main Results We begin with stating and proving the following:', '1307.5970-1-4-0': 'Let [MATH] a continuous bounded deterministic function, then [EQUATION] is normally distributed with mean zero and variance [MATH] where [MATH] and [MATH].', '1307.5970-1-4-1': 'Moreover, if [MATH] then there exists a BM [MATH] such that [MATH]', '1307.5970-1-5-0': 'Proof.', '1307.5970-1-5-1': 'We observe that [MATH] is a Gaussian process with zero mean and variance [EQUATION].', '1307.5970-1-5-2': 'Since [MATH] we get: [EQUATION] where [MATH] and [MATH].', '1307.5970-1-5-3': 'Thus, by calculation, we obtain: [EQUATION].', '1307.5970-1-5-4': "As easily seen, [MATH] and [MATH] have the same derivative, so the equality [MATH] follows for any [MATH] since [MATH] Moreover, by using [MATH]'s formula we get: [EQUATION].", '1307.5970-1-5-5': 'Notice that [MATH] is a martingale and [MATH] is its quadratic variation; therefore, if [MATH] by the Dambis, Dubins-Schwarz Theorem (see e.g. [CITATION]) we obtain that [MATH] can be written as [MATH] where [MATH] is BM.', '1307.5970-1-6-0': '[MATH]', '1307.5970-1-7-0': 'As a corollary of the previous lemma, we obtain our main result:', '1307.5970-1-8-0': 'Let [MATH] be a Gauss-Markov process given by (1.1), and suppose that [MATH] are continuous, [MATH] is a differentiable increasing function; then [MATH] is normally distributed with mean [MATH] and variance [MATH] where [MATH] and [MATH].', '1307.5970-1-8-1': 'Moreover, if [MATH] then there exists a BM [MATH] such that [MATH].', '1307.5970-1-8-2': 'Thus, [MATH] is still Gauss-Markov.', '1307.5970-1-9-0': 'Proof.', '1307.5970-1-9-1': 'We have: [EQUATION] where we have used a variable change in the integral.', '1307.5970-1-9-2': 'Then, the proof follows by using Lemma [REF] with [MATH]', '1307.5970-1-10-0': 'If we consider the time average [MATH] by Proposition [REF] we get [MATH] namely [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-1-10-1': 'In particular, if [MATH] is BM, one has [MATH] (see Example 1 below), and so [MATH] (cf. [CITATION]).', '1307.5970-1-11-0': 'Notice that, if [MATH] then the FPT of [MATH] over a continuous boundary [MATH] i.e. [MATH] is nothing but the FPT of [MATH] over [MATH] where [MATH] or equivalently [MATH]', '1307.5970-1-12-0': '# A Few Examples', '1307.5970-1-13-0': '[MATH] Example 1 (Brownian motion with drift) Let be [MATH] then [MATH] and [MATH].', '1307.5970-1-13-1': 'Moreover, [MATH] and [MATH].', '1307.5970-1-13-2': 'Thus, [MATH] (cf. [CITATION]).', '1307.5970-1-14-0': 'Example 2 (Ornstein-Uhlenbeck process)Let [MATH] be the solution of the SDE: [EQUATION] where [MATH] and [MATH].', '1307.5970-1-14-1': 'The explicit solution is (see e.g. [CITATION]): [EQUATION] where [MATH] is Brownian motion and [MATH].', '1307.5970-1-14-2': 'So, [MATH] is a Gauss-Markov process with [MATH] and [MATH].', '1307.5970-1-14-3': 'By calculation, we obtain: [EQUATION].', '1307.5970-1-14-4': 'Then, by Proposition [REF], we get that [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-1-14-5': 'Moreover, since [MATH] there exists a BM [MATH] such that [MATH]', '1307.5970-1-15-0': 'Example 3 (Brownian bridge) For [MATH] and given [MATH] let [MATH] be the solution of the SDE: [EQUATION].', '1307.5970-1-15-1': 'This is a transformed BM with fixed values at each end of the interval [MATH] and [MATH].', '1307.5970-1-15-2': 'The explicit solution is (see e.g. [CITATION]): [EQUATION] where [MATH] is BM.', '1307.5970-1-15-3': 'So, [MATH] is a Gauss-Markov process with: [EQUATION].', '1307.5970-1-15-4': 'By calculation, we obtain: [EQUATION].', '1307.5970-1-15-5': 'Then, by Proposition [REF], we get that [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-1-15-6': 'By a straightforward, but boring calculation, it can be verified that [MATH] so there exists a BM [MATH] such that [MATH]', '1307.5970-1-16-0': 'Example 4 (Generalized Gauss-Markov process) Let us consider the diffusion [MATH] which is the solution of the SDE: [EQUATION] where [MATH] and [MATH] are regular enough deterministic function.', '1307.5970-1-16-1': 'We suppose that [MATH] i.e. the quadratic variation of [MATH] is increasing to [MATH].', '1307.5970-1-16-2': 'By using the Dambis, Dubins-Schwarz Theorem, it follows that [MATH] where [MATH] is not necessarily deterministic, but it can be a random function.', '1307.5970-1-16-3': 'For this reason, we call [MATH] a generalized Gauss-Markov process.', '1307.5970-1-16-4': 'Denote by [MATH] the "inverse" of the random function [MATH] that is, [MATH] since [MATH] admits derivative and [MATH] also [MATH] exists and [MATH] we focus on the case when there exist deterministic continuous functions [MATH] (with [MATH] and [MATH] such that, for every [MATH] [EQUATION].', '1307.5970-1-16-5': 'Since [MATH] is not deterministic, we cannot obtain exactly the distribution of [MATH] however we are able to find bounds to it.', '1307.5970-1-16-6': 'In fact, we have: [EQUATION].', '1307.5970-1-16-7': 'We can use the arguments of Lemma [REF] with [MATH] and [MATH] by assumptions we get [MATH].', '1307.5970-1-16-8': 'Thus, we conclude that [MATH] is normally distributed with mean [MATH] and variance [MATH] which is bounded between [MATH] and [MATH].', '1307.5970-1-16-9': 'The closer [MATH] to [MATH] the better the approximation above; for instance, if [MATH] we have [MATH] and so [MATH].', '1307.5970-1-16-10': 'The smaller is [MATH] the closer [MATH] to [MATH]'}

{'1307.5970-2-0-0': 'We find a representation of the integral of a Gauss-Markov process in the interval [MATH] in terms of Brownian motion.', '1307.5970-2-0-1': 'Moreover, some connections with first-passage-time problems are discussed, and some examples are reported.', '1307.5970-2-1-0': 'Keywords: Diffusion, Gauss-Markov process, first-passage-time', '1307.5970-2-2-0': 'Mathematics Subject Classification: 60J60, 60H05, 60H10.', '1307.5970-2-3-0': '# Introduction', '1307.5970-2-4-0': 'In this short note, we consider a real continuous Gauss-Markov process [MATH] of the form: [EQUATION] where: [MATH] is a standard Brownian motion (BM); [MATH] is continuous for every [MATH] the covariance [MATH] is continuous for every [MATH] with [MATH] is a monotonically increasing function and [MATH] Notice that a special case of Gauss-Markov process is the Ornstein-Uhlenbeck (OU) process, and in fact any Gauss-Markov process can be represented in terms of a OU process (see e.g. [CITATION]).', '1307.5970-2-4-1': 'Our aim is to find a representation of [EQUATION] in terms of Brownian motion.', '1307.5970-2-4-2': 'Notice that the integrated process [MATH] is equal to [MATH] where [MATH] is the time average of [MATH] in the interval [MATH] The study of [MATH] has interesting applications in Biology, for instance in the framework of diffusion models for neural activity; if one identifies [MATH] with the neuron voltage at time [MATH] then, [MATH] represents the time average of the neural voltage in the interval [MATH].', '1307.5970-2-4-3': 'Another application can be found in Queueing Theory, if [MATH] represents the length of a queue at time [MATH] then, [MATH] represents the cumulative waiting time experienced by all the "users" till the time [MATH] As for an example from Economics, let us suppose that the variable [MATH] represents the quantity of a commodity that producers have available for sale, then [MATH] provides a measure of the total value that consumers receive from consuming the amount [MATH] of the product.', '1307.5970-2-4-4': 'Really, in certain applications, it is interesting to study first-passage time (FPT) problems for the integrated process [MATH] to this end, it is useful to dispose of an explicit representation of [MATH].', '1307.5970-2-4-5': 'The results of this paper generalize those of [CITATION].', '1307.5970-2-5-0': '# Main Results', '1307.5970-2-6-0': 'We begin with stating and proving the following:', '1307.5970-2-7-0': 'Let [MATH] a continuous bounded deterministic function, with [MATH] for every [MATH] then [EQUATION] is normally distributed with mean zero and variance [MATH] where [MATH] and [MATH].', '1307.5970-2-7-1': 'Moreover, if [MATH] then there exists a BM [MATH] such that [MATH]', '1307.5970-2-8-0': 'Proof.', '1307.5970-2-8-1': 'We observe that [MATH] is a Gaussian process with zero mean and variance [EQUATION].', '1307.5970-2-8-2': 'Since [MATH] we get: [EQUATION] where [MATH] and [MATH].', '1307.5970-2-8-3': 'Thus, by calculation, we obtain: [EQUATION].', '1307.5970-2-8-4': "As easily seen, [MATH] and [MATH] have the same derivative, so the equality [MATH] follows for any [MATH] since [MATH] Let [MATH] fixed; by using [MATH]'s formula we get: [EQUATION].", '1307.5970-2-8-5': 'For [MATH] let us consider now the continuous martingale [MATH] having differential [MATH] for [MATH] we have: [EQUATION] where [MATH] denotes the quadratic variation of [MATH].', '1307.5970-2-8-6': 'Indeed, the inequality [REF] easily follows from the fact that, since [MATH] the function [MATH] is monotone (increasing or decreasing), so [MATH] implies [MATH] If [MATH] from inequality [REF] we obtain [MATH] then, by the Dambis, Dubins-Schwarz Theorem (see e.g. [CITATION]) there exists a BM [MATH] such that [MATH].', '1307.5970-2-8-7': 'Thus, [MATH] finally, taking [MATH] we obtain [MATH]', '1307.5970-2-9-0': 'As a corollary of the previous lemma, we obtain our main result:', '1307.5970-2-10-0': 'Let [MATH] be a Gauss-Markov process given by (1.1), and suppose that [MATH] are continuous, [MATH] is a differentiable increasing function; then [MATH] is normally distributed with mean [MATH] and variance [MATH] where [MATH] and [MATH].', '1307.5970-2-10-1': 'Moreover, if [MATH] then there exists a BM [MATH] such that [MATH].', '1307.5970-2-10-2': 'Thus, [MATH] is still Gauss-Markov.', '1307.5970-2-11-0': 'Proof.', '1307.5970-2-11-1': 'We have: [EQUATION] where we have used a variable change in the integral.', '1307.5970-2-11-2': 'Then, the proof follows by using Lemma [REF] with [MATH]', '1307.5970-2-12-0': 'If we consider the time average [MATH] by Proposition [REF] we get [MATH] namely [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-2-12-1': 'In particular, if [MATH] is BM, one obtains [MATH] (cf. [CITATION]).', '1307.5970-2-13-0': 'Notice that, if [MATH] than the FPT of [MATH] over a continuous boundary [MATH] i.e. [MATH] is nothing but the FPT of [MATH] over [MATH] or equivalently [MATH]', '1307.5970-2-14-0': '# A Few Examples', '1307.5970-2-15-0': 'Example 1 (Brownian motion with drift) Let be [MATH] then [MATH] and [MATH].', '1307.5970-2-15-1': 'Moreover, [MATH] and [MATH].', '1307.5970-2-15-2': 'Thus, [MATH] (cf. [CITATION]).', '1307.5970-2-16-0': 'Example 2 (Ornstein-Uhlenbeck process)Let [MATH] be the solution of the SDE: [EQUATION] where [MATH] and [MATH].', '1307.5970-2-16-1': 'The explicit solution is (see e.g. [CITATION] ): [EQUATION] where [MATH] is Brownian motion and [MATH].', '1307.5970-2-16-2': 'So, [MATH] is a Gauss-Markov process with [MATH] and [MATH].', '1307.5970-2-16-3': 'By calculation, we obtain: [EQUATION].', '1307.5970-2-16-4': 'Then, by Proposition [REF], we get that [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-2-16-5': 'Moreover, since [MATH] there exists a BM [MATH] such that [MATH]', '1307.5970-2-17-0': 'Example 3 (Brownian bridge) For [MATH] and given [MATH] let [MATH] be the solution of the SDE: [EQUATION].', '1307.5970-2-17-1': 'This is a transformed BM with fixed values at each end of the interval [MATH] and [MATH].', '1307.5970-2-17-2': 'The explicit solution is (see e.g. [CITATION]): [EQUATION] where [MATH] is BM.', '1307.5970-2-17-3': 'So, [MATH] is a Gauss-Markov process with: [EQUATION].', '1307.5970-2-17-4': 'By calculation, we obtain: [EQUATION].', '1307.5970-2-17-5': 'Then, by Proposition [REF], we get that [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-2-17-6': 'By a straightforward, but boring calculation, it can be verified that [MATH] so there exists a BM [MATH] such that [MATH]', '1307.5970-2-18-0': 'Example 4 (Generalized Gauss-Markov process) Let us consider the diffusion [MATH] which is the solution of the SDE: [EQUATION] where [MATH] is BM and [MATH] is a smooth deterministic function.', '1307.5970-2-18-1': 'We suppose that [MATH] i.e. the quadratic variation of [MATH] is increasing to [MATH].', '1307.5970-2-18-2': 'By using the Dambis, Dubins-Schwarz Theorem, it follows that [MATH] where [MATH] is not necessarily deterministic, but it can be a random function.', '1307.5970-2-18-3': 'For this reason, we call [MATH] a generalized Gauss-Markov process.', '1307.5970-2-18-4': 'Denote by [MATH] the "inverse" of the random function [MATH] that is, [MATH] since [MATH] admits derivative and [MATH] also [MATH] exists and [MATH] we focus on the case when there exist deterministic continuous functions [MATH] (with [MATH] and [MATH] such that, for every [MATH] [EQUATION].', '1307.5970-2-18-5': 'Since [MATH] is not deterministic, we cannot obtain exactly the distribution of [MATH] however we are able to find bounds to it.', '1307.5970-2-18-6': 'In fact, we have: [EQUATION]', '1307.5970-2-18-7': 'We can use the arguments of Lemma [REF] with [MATH] and [MATH] by assumptions we get [MATH].', '1307.5970-2-18-8': 'Thus, we conclude that [MATH] is normally distributed with mean [MATH] and variance [MATH] which is bounded between [MATH] and [MATH].', '1307.5970-2-18-9': 'The closer [MATH] to [MATH] the better the approximation above; for instance, if [MATH] we have [MATH] and so [MATH].', '1307.5970-2-18-10': 'The smaller is [MATH] the closer [MATH] to [MATH]'}

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[]

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[]

['1307.5970-1-5-0', '1307.5970-1-6-0', '1307.5970-1-7-0', '1307.5970-1-9-0', '1307.5970-1-13-0', '1307.5970-1-13-1', '1307.5970-1-13-2', '1307.5970-2-1-0', '1307.5970-2-2-0', '1307.5970-2-6-0', '1307.5970-2-8-0', '1307.5970-2-9-0', '1307.5970-2-11-0', '1307.5970-2-15-2', '1307.5970-3-1-0', '1307.5970-3-2-0', '1307.5970-3-6-0', '1307.5970-3-8-0', '1307.5970-3-9-0', '1307.5970-3-11-0', '1307.5970-3-15-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1307.5970

{'1307.5970-3-0-0': 'We find a representation of the integral of a Gauss-Markov process in the interval [MATH] in terms of Brownian motion.', '1307.5970-3-0-1': 'Moreover, some connections with first-passage-time problems are discussed, and some examples are reported.', '1307.5970-3-1-0': 'Keywords: Diffusion, Gauss-Markov process, first-passage-time', '1307.5970-3-2-0': 'Mathematics Subject Classification: 60J60, 60H05, 60H10.', '1307.5970-3-3-0': '# Introduction', '1307.5970-3-4-0': 'In this short note, we consider a real continuous Gauss-Markov process [MATH] of the form: [EQUATION] where: [MATH] is a standard Brownian motion (BM); [MATH] is continuous for every [MATH] the covariance [MATH] is continuous for every [MATH] with [MATH] is a monotonically increasing function and [MATH] Notice that a special case of Gauss-Markov process is the Ornstein-Uhlenbeck (OU) process, and in fact any Gauss-Markov process can be represented in terms of a OU process (see e.g. [CITATION]).', '1307.5970-3-4-1': 'Our aim is to find a representation of [EQUATION] in terms of Brownian motion.', '1307.5970-3-4-2': 'Notice that the integrated process [MATH] is equal to [MATH] where [MATH] is the time average of [MATH] in the interval [MATH] The study of [MATH] has interesting applications in Biology, for instance in the framework of diffusion models for neural activity; if one identifies [MATH] with the neuron voltage at time [MATH] then, [MATH] represents the time average of the neural voltage in the interval [MATH].', '1307.5970-3-4-3': 'Another application can be found in Queueing Theory, if [MATH] represents the length of a queue at time [MATH] then, [MATH] represents the cumulative waiting time experienced by all the "users" till the time [MATH] As for an example from Economics, let us suppose that the variable [MATH] represents the quantity of a commodity that producers have available for sale, then [MATH] provides a measure of the total value that consumers receive from consuming the amount [MATH] of the product.', '1307.5970-3-4-4': 'Really, in certain applications, it is interesting to study first-passage time (FPT) problems for the integrated process [MATH] to this end, it is useful to dispose of an explicit representation of [MATH].', '1307.5970-3-4-5': 'The results of this paper generalize those of [CITATION].', '1307.5970-3-5-0': '# Main Results', '1307.5970-3-6-0': 'We begin with stating and proving the following:', '1307.5970-3-7-0': 'Let [MATH] a continuous bounded deterministic function, with [MATH] for every [MATH] then [EQUATION] is normally distributed with mean zero and variance [MATH] where [MATH] and [MATH].', '1307.5970-3-7-1': 'Moreover, if [MATH] then there exists a BM [MATH] such that [MATH]', '1307.5970-3-8-0': 'Proof.', '1307.5970-3-8-1': 'We observe that [MATH] is a Gaussian process with zero mean and variance [EQUATION].', '1307.5970-3-8-2': 'Since [MATH] we get: [EQUATION] where [MATH] and [MATH].', '1307.5970-3-8-3': 'Thus, by calculation, we obtain: [EQUATION].', '1307.5970-3-8-4': "As easily seen, [MATH] and [MATH] have the same derivative, so the equality [MATH] follows for any [MATH] since [MATH] Let [MATH] fixed; by using [MATH]'s formula we get: [EQUATION].", '1307.5970-3-8-5': 'For [MATH] let us consider now the continuous martingale [MATH] having differential [MATH] for [MATH] we have: [EQUATION] where [MATH] denotes the quadratic variation of [MATH].', '1307.5970-3-8-6': 'Indeed, the inequality [REF] easily follows from the fact that, since [MATH] the function [MATH] is monotone (increasing or decreasing), so [MATH] implies [MATH] If [MATH] from inequality [REF] we obtain [MATH] then, by the Dambis, Dubins-Schwarz Theorem (see e.g. [CITATION]) there exists a BM [MATH] such that [MATH].', '1307.5970-3-8-7': 'Thus, [MATH] finally, taking [MATH] we obtain [MATH]', '1307.5970-3-9-0': 'As a corollary of the previous lemma, we obtain our main result:', '1307.5970-3-10-0': 'Let [MATH] be a Gauss-Markov process given by (1.1), and suppose that [MATH] are continuous, [MATH] is a differentiable increasing function; then [MATH] is normally distributed with mean [MATH] and variance [MATH] where [MATH] and [MATH].', '1307.5970-3-10-1': 'Moreover, if [MATH] then there exists a BM [MATH] such that [MATH].', '1307.5970-3-10-2': 'Thus, [MATH] is still Gauss-Markov.', '1307.5970-3-11-0': 'Proof.', '1307.5970-3-11-1': 'We have: [EQUATION] where we have used a variable change in the integral.', '1307.5970-3-11-2': 'Then, the proof follows by using Lemma [REF] with [MATH]', '1307.5970-3-12-0': 'If we consider the time average [MATH] by Proposition [REF] we get [MATH] namely [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-3-12-1': 'In particular, if [MATH] is BM, one obtains [MATH] (cf. [CITATION]).', '1307.5970-3-13-0': 'Notice that, if [MATH] than the FPT of [MATH] over a continuous boundary [MATH] i.e. [MATH] is nothing but the FPT of [MATH] over [MATH] or equivalently [MATH]', '1307.5970-3-14-0': '# A Few Examples', '1307.5970-3-15-0': 'Example 1 (Brownian motion with drift) Let be [MATH] then [MATH] and [MATH].', '1307.5970-3-15-1': 'Moreover, [MATH] and [MATH].', '1307.5970-3-15-2': 'Thus, [MATH] (cf. [CITATION]).', '1307.5970-3-16-0': 'Example 2 (Ornstein-Uhlenbeck process)Let [MATH] be the solution of the SDE: [EQUATION] where [MATH] and [MATH].', '1307.5970-3-16-1': 'The explicit solution is (see e.g. [CITATION] ): [EQUATION] where [MATH] is Brownian motion and [MATH].', '1307.5970-3-16-2': 'So, [MATH] is a Gauss-Markov process with [MATH] and [MATH].', '1307.5970-3-16-3': 'By calculation, we obtain: [EQUATION].', '1307.5970-3-16-4': 'Then, by Proposition [REF], we get that [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-3-16-5': 'Moreover, since [MATH] there exists a BM [MATH] such that [MATH]', '1307.5970-3-17-0': 'Example 3 (Brownian bridge) For [MATH] and given [MATH] let [MATH] be the solution of the SDE: [EQUATION].', '1307.5970-3-17-1': 'This is a transformed BM with fixed values at each end of the interval [MATH] and [MATH].', '1307.5970-3-17-2': 'The explicit solution is (see e.g. [CITATION]): [EQUATION] where [MATH] is BM.', '1307.5970-3-17-3': 'So, [MATH] is a Gauss-Markov process with: [EQUATION].', '1307.5970-3-17-4': 'By calculation, we obtain: [EQUATION].', '1307.5970-3-17-5': 'Then, by Proposition [REF], we get that [MATH] is normally distributed with mean [MATH] and variance [MATH].', '1307.5970-3-17-6': 'By a straightforward, but boring calculation, it can be verified that [MATH] so there exists a BM [MATH] such that [MATH]', '1307.5970-3-18-0': 'Example 4 (Generalized Gauss-Markov process) Let us consider the diffusion [MATH] which is the solution of the SDE: [EQUATION] where [MATH] is BM and [MATH] is a smooth deterministic function.', '1307.5970-3-18-1': 'We suppose that [MATH] i.e. the quadratic variation of [MATH] is increasing to [MATH].', '1307.5970-3-18-2': 'By using the Dambis, Dubins-Schwarz Theorem, it follows that [MATH] where [MATH] is not necessarily deterministic, but it can be a random function.', '1307.5970-3-18-3': 'For this reason, we call [MATH] a generalized Gauss-Markov process.', '1307.5970-3-18-4': 'Denote by [MATH] the "inverse" of the random function [MATH] that is, [MATH] since [MATH] admits derivative and [MATH] also [MATH] exists and [MATH] we focus on the case when there exist deterministic continuous functions [MATH] (with [MATH] and [MATH] such that, for every [MATH] [EQUATION].', '1307.5970-3-18-5': 'Since [MATH] is not deterministic, we cannot obtain exactly the distribution of [MATH] however we are able to find bounds to it.', '1307.5970-3-18-6': 'In fact, we have: [EQUATION]', '1307.5970-3-18-7': 'We can use the arguments of Lemma [REF] with [MATH] and [MATH] by assumptions we get [MATH].', '1307.5970-3-18-8': 'Thus, we conclude that [MATH] is normally distributed with mean [MATH] and variance [MATH] which is bounded between [MATH] and [MATH].', '1307.5970-3-18-9': 'The closer [MATH] to [MATH] the better the approximation above; for instance, if [MATH] we have [MATH] and so [MATH].', '1307.5970-3-18-10': 'The smaller is [MATH] the closer [MATH] to [MATH]'}

1204.1008

{'1204.1008-1-0-0': 'Ground-state behaviour of the frustrated quantum spin-[MATH] two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail.', '1204.1008-1-0-1': 'The investigated model is transformed to the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation.', '1204.1008-1-0-2': "It is shown that the ground state of the Heisenberg-Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the 'classical' Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal-transverse field.", '1204.1008-1-0-3': 'The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization.', '1204.1008-1-0-4': 'The ground-state phase diagram totally consists of five ordered and one quantum disordered phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions.', '1204.1008-1-0-5': 'The order parameters are exactly calculated for all five ordered phases and the quantum disordered phase is characterized through different short-range spin-spin correlations.', '1204.1008-1-1-0': '# Introduction', '1204.1008-1-2-0': 'Quantum spin models with competing interactions represent quite interesting and challenging topic for the current research [CITATION].', '1204.1008-1-2-1': 'Such models show many unusual features in their ground-state properties and are very sensitive to the approximative schemes applied to them.', '1204.1008-1-2-2': 'An existence of exact solutions is therefore quite important, since they provide a rigorous information about the complex behaviour of frustrated models.', '1204.1008-1-2-3': 'However, exact results for such systems are still limited (see [CITATION] and references cited therein).', '1204.1008-1-2-4': 'The simplest examples of rigorously solved quantum spin models are the models with the known dimerized [CITATION] or even trimerized [CITATION] ground states.', '1204.1008-1-2-5': 'Another example represents frustrated quantum spin antiferromagnets in high magnetic fields with the so-called localized-magnon eigenstates (see e.g. [CITATION] for recent review).', '1204.1008-1-2-6': 'It should be nevertheless noticed that the aforementioned exact results are usually derived only under certain constraint laid on the interaction parameters and those models are not tractable quite generally.', '1204.1008-1-3-0': 'On the other hand, the exact solutions for some spin-[MATH] quantum spin ladders with multispin interaction allows to find, besides the ground state, also thermodynamic properties quite rigorously [CITATION].', '1204.1008-1-3-1': 'Exactly solvable models, which admit the inclusion of frustration, can also be constructed from the Ising models where the decorations of quantum spins are included [CITATION].', '1204.1008-1-3-2': 'The decoration-iteration procedure allows one to calculate exactly all thermodynamic properties of the decorated models when the exact solution for the corresponding Ising model is known.', '1204.1008-1-3-3': 'The distinctive feature of these models is that the quantum decorations are separated from each other, so that the Hamiltonian can be decomposed into the sum of commuting parts.', '1204.1008-1-3-4': 'The eigenstates of the total Hamiltonian are then simply factorized into a product of the eigenstates of its commuting parts.', '1204.1008-1-4-0': 'New findings of the exactly solvable frustrated quantum spin models with non-trivial (not simply factorizable) ground states are thus highly desirable.', '1204.1008-1-4-1': 'The present article deals with the frustrated quantum spin-[MATH] two-leg ladder with the Heisenberg intra-rung interaction and the Ising inter-rung interactions between nearest-neighbouring spins from either the same or different leg.', '1204.1008-1-4-2': 'Such a model can be regarded as an extension of the spin-[MATH] Heisenberg-Ising bond alternating chain, which was proposed and rigorously solved by Lieb, Schultz and Mattis [CITATION].', '1204.1008-1-4-3': 'Alternatively, this model can also be viewed as the generalization of the exactly solved quantum compass ladder [CITATION] when considering the Heisenberg rather than XZ-type intra-rung interaction and accounting for the additional frustrating Ising inter-rung interaction.', '1204.1008-1-4-4': 'Moreover, it is quite plausible to suspect that the exact results presented hereafter for the spin-[MATH] Heisenberg-Ising two-leg ladder may also bring insight into the relevant behaviour of the corresponding Heisenberg two-leg ladder, which represents a quite challenging and complex research problem in its own right [CITATION].', '1204.1008-1-4-5': 'The frustrated spin-[MATH] Heisenberg two-leg ladder have been extensively investigated by employing various independent numerical and analytical methods such as density-matrix renormalization group [CITATION], numerical diagonalization [CITATION], series expansion [CITATION], bosonization technique [CITATION], strong- and weak-coupling analysis [CITATION], valence-bond spin-wave theory [CITATION], variational matrix-product approach [CITATION], and bond mean-field theory [CITATION].', '1204.1008-1-4-6': 'Among the most interesting results obtained for this quantum spin chain, one could mention an existence of the columnar-dimer phase discussed in [CITATION] or a presence of the fractional plateau in the magnetization process examined in [CITATION].', '1204.1008-1-5-0': 'The theoretical investigation of two-leg ladder models is motivated not only from the academic point of view, but also from the experimental viewpoint, because the two-leg ladder motif captures the magnetic structure of a certain class of real quasi-one-dimensional magnetic materials.', '1204.1008-1-5-1': 'The most widespread family of two-leg ladder compounds form cuprates, in which one finds both experimental representatives with the dominating intra-rung interaction like SrCu[MATH]O[MATH] [CITATION], Cu[MATH](C[MATH]H[MATH]N[MATH]Cl[MATH] [CITATION], (C[MATH]H[MATH]N)[MATH]CuBr[MATH] [CITATION], (5IAP)[MATH]CuBr[MATH] [CITATION] as well as, the magnetic compounds with the dominating intra-leg interaction such as KCuCl[MATH] [CITATION], TlCuCl[MATH] [CITATION], NH[MATH]CuCl[MATH], KCuBr[MATH].', '1204.1008-1-5-2': 'Understanding the low-temperature magnetism of two-leg ladder models also turns out to be crucial for an explanation of the mechanism, which is responsible for the high-temperature superconductivity of cuprates [CITATION].', '1204.1008-1-5-3': 'Besides the cuprates, another experimental representatives of the two-leg ladder compounds represent vanadates (VO)[MATH]P[MATH]O[MATH] [CITATION], CaV[MATH]O[MATH] and MgV[MATH]O[MATH] [CITATION], as well as, the polyradical BIP-BNO [CITATION].', '1204.1008-1-6-0': 'The outline of the paper is as follows.', '1204.1008-1-6-1': 'In section [REF] the model is defined and the pseudospin representation is considered.', '1204.1008-1-6-2': 'The ground-state properties of the model with and without external field are studied in section [REF].', '1204.1008-1-6-3': 'The most important findings are summarized in section [REF].', '1204.1008-1-7-0': '# Heisenberg-Ising two-leg ladder', '1204.1008-1-8-0': 'Let us define the spin-[MATH] Heisenberg-Ising ladder through the following Hamiltonian: [EQUATION] where [MATH], [MATH] are three spatial projections of spin-[MATH] operator, the first index denotes the number of leg, the second enumerates the site, [MATH] is the [MATH] Heisenberg intra-rung interaction between nearest-neighbour spins from the same rung, [MATH] is the Ising intra-leg interaction between nearest-neighbour spins from the same leg, [MATH] is the crossing (diagonal) Ising inter-rung interaction between next-nearest-neighbour spins from different rungs, [MATH] is the external magnetic field.', '1204.1008-1-8-1': 'We also imply the periodic boundary conditions [MATH] and [MATH] along legs.', '1204.1008-1-8-2': 'The coupling constants [MATH] and [MATH] can be interchanged by renumbering of the sites, as well as their signs can be simultaneously reverted by spin rotations.', '1204.1008-1-8-3': 'Therefore, the Hamiltonians [MATH], [MATH] and [MATH] in zero field have equal eigenvalues and the corresponding models are thermodynamically equivalent.', '1204.1008-1-9-0': 'It can be checked that [MATH]-projection of total spin on a rung [MATH] commutes with the total Hamiltonian [MATH] and hence, it represents a conserved quantity.', '1204.1008-1-9-1': 'For further convenience, it is therefore advisable to take advantage of the bond representation, which has been originally suggested by Lieb, Schultz and Mattis for the Heisenberg-Ising chain [CITATION].', '1204.1008-1-9-2': 'Let us introduce the bond-state basis consisting of four state vectors: [EQUATION]', '1204.1008-1-9-3': 'These states are the eigenstates of the Heisenberg coupling between two spins located on the [MATH]th rung, i.e. [MATH] is the singlet-bond state, [MATH], [MATH] are triplet states.', '1204.1008-1-9-4': 'Following [CITATION] two subspaces can be singled out in the bond space: i) [MATH], [MATH] and ii) [MATH], [MATH].', '1204.1008-1-9-5': 'The first and second subspaces corresponds to [MATH] and [MATH] respectively, and the Hamiltonian can be diagonalized separately in each subspace.', '1204.1008-1-9-6': 'We can introduce the index of subspace at [MATH]-th site: [MATH] if the given state is in the subspace with [MATH]) spanned by states [MATH], [MATH], [MATH]).', '1204.1008-1-9-7': 'According to [CITATION], let us also call the bond states as purity (impurity) states if [MATH].', '1204.1008-1-9-8': 'If one makes pseudospin notations for the states [MATH], [MATH], the action of the spin-[MATH] operators in the subspace [MATH] can be expressed in terms of new raising and lowering operators: [EQUATION]', '1204.1008-1-9-9': 'The operators [MATH], [MATH] satisfy the Pauli algebra ([MATH], [MATH], [MATH] for [MATH]), and half of their sum can be identified as a new pseudospin operator [MATH].', '1204.1008-1-10-0': 'Analogously, one can consider the pseudospin representation of the subspace [MATH], [MATH]) and find the action of spin operators in it as follows: [EQUATION]', '1204.1008-1-10-1': 'Combining ([REF]), ([REF]) we find the general expressions for the pseudospin representation of these operators, which are valid in both subspaces: [EQUATION]', '1204.1008-1-10-2': 'The effective Hamiltonian can be rewritten in terms of new operators as follows: [EQUATION]', '1204.1008-1-10-3': 'It is noteworthy that equation ([REF]) can be considered as some nonlinear spin transformation from [MATH], [MATH] to new operators [MATH], [MATH], where e.g. [MATH], [MATH].', '1204.1008-1-10-4': 'The transformation can be generated by the following unitary operator: [EQUATION]', '1204.1008-1-10-5': 'The crucial point is the second factor, which introduces the nonlinearity.', '1204.1008-1-10-6': 'Other terms are the rotation in spin space, they are used to adjust the transformed operators to the form of ([REF]).', '1204.1008-1-10-7': 'Finally, the spin operators are transformed as follows: [EQUATION]', '1204.1008-1-10-8': 'The transformed Hamiltonian has the form of the quantum Ising chain with composite spins in an effective longitudinal and transverse magnetic field: [EQUATION]', '1204.1008-1-10-9': 'The straightforward correspondence [MATH], [MATH] leads to the equivalence between the Hamiltonians ([REF]) and ([REF]).', '1204.1008-1-10-10': 'It should be noted that such kind of a representation remains valid also in case of the asymmetric ladder having both diagonal (crossing) Ising interactions different from each other.', '1204.1008-1-11-0': '# Ground state of the Heisenberg-Ising two-leg ladder', '1204.1008-1-12-0': 'The Hamiltonian ([REF]) can also be rewritten in the following more symmetric form: [EQUATION]', '1204.1008-1-12-1': 'This form serves in evidence that the effective model splits at [MATH]th site into two independent chains provided that two neighbouring bonds are being in different subspaces, i.e. [MATH].', '1204.1008-1-12-2': 'The uniform Hamiltonians, when all [MATH] or all [MATH], read: [EQUATION]', '1204.1008-1-12-3': 'If all bonds are in purity states ([MATH]), one obtains the effective Hamiltonian of the Ising chain in the transverse field that is exactly solvable within Jordan-Wigner fermionization [CITATION].', '1204.1008-1-12-4': 'If all bonds are in impurity states ([MATH]), one comes to the Ising chain in the longitudinal field that is solvable by the transfer-matrix method (see e.g. [CITATION]).', '1204.1008-1-12-5': 'Accordingly, the ground-state energy of the model in [MATH] subspace is given by [CITATION]: [EQUATION] where [MATH] and [MATH] is the complete elliptic integral of the second kind.', '1204.1008-1-13-0': 'The ground-state energy of the model in [MATH] subspace is given by the ground-state energy of the effective Ising chain in the longitudinal field: [EQUATION]', '1204.1008-1-13-1': 'The upper (lower) case corresponds to the ferromagnetically (antiferromagnetically) ordered state being the ground state at strong (weak) enough magnetic fields.', '1204.1008-1-14-0': '## Ground-state phase diagram in a zero field', '1204.1008-1-15-0': 'If the external field vanishes ([MATH]), it is sufficient to show that the inequality [MATH] holds for any finite transverse Ising chain in order to prove that the ground state always corresponds to a uniform bond configuration.', '1204.1008-1-15-1': 'Indeed, two independent chains of size [MATH] and [MATH] can be represented by the following Hamiltonian: [EQUATION]', '1204.1008-1-15-2': 'Let [MATH] and [MATH] be the lowest eigenvalue and eigenstate of [MATH], thus, [MATH] and [MATH] are the lowest eigenvalue and eigenstate of [MATH].', '1204.1008-1-15-3': 'Now, it is straightforward to show that [EQUATION]', '1204.1008-1-15-4': 'Here, we have used that [MATH] for any finite chain [CITATION], and that the lowest mean value of the operator [MATH] is achieved in its ground state.', '1204.1008-1-16-0': 'It is easy to show by straightforward calculation that the same property is valid for the Ising chain in zero longitudinal field, i.e. [EQUATION]', '1204.1008-1-16-1': 'Now, let us prove that the ground state of the whole model may correspond only to one of uniform bond configurations.', '1204.1008-1-16-2': 'It can be readily understood from ([REF]) that the effective Hamiltonian does not contain an interaction between spins from two neighbouring bonds if they are in different subspaces [CITATION].', '1204.1008-1-16-3': "It means that the effective model is split into two independent parts at each boundary ('domain wall') between the purity and impurity states.", '1204.1008-1-16-4': 'Thus, the Heisenberg-Ising ladder for any given configuration of bonds can be considered as a set of independent chains of two kinds and of different sizes.', '1204.1008-1-16-5': 'Then, the ground-state energy of any randomly chosen bond configuration will be as follows: [EQUATION]', '1204.1008-1-16-6': 'It is quite evident from equations ([REF]) and ([REF]) that the one uniform configuration, which corresponds to the state with lower energy than the other one, must be according to inequality ([REF]) the lowest-energy state (i.e. ground state).', '1204.1008-1-16-7': 'Therefore, the model may show in the ground state the first-order quantum phase transition when the lowest eigenenergies in both subspaces becomes equal [MATH].', '1204.1008-1-16-8': 'Besides, there also may appear the more striking second-order (continuous) quantum phase transition at [MATH] in the ground state, which is inherent to the pure quantum spin chain [CITATION].', '1204.1008-1-16-9': 'From this perspective, the Heisenberg-Ising ladder can show a variety of quantum phase transitions in its phase diagram.', '1204.1008-1-17-0': 'The ground-state phase diagram of the spin-[MATH] Heisenberg-Ising ladder in a zero magnetic field is shown in figure [REF].', '1204.1008-1-18-0': 'It reflects the symmetry of the model, i.e. it is invariant under the exchange of the Ising interactions [MATH] and [MATH], as well as, under the simultaneous change of the signs of [MATH] and [MATH].', '1204.1008-1-18-1': 'Altogether, five different ground states can be recognised in figure [REF]:', '1204.1008-1-19-0': 'Quantum paramagnetic (QPM) state for [MATH], and [MATH]: the equivalent transverse Ising chain ([REF]) is in the gapped disordered state with no spontaneous magnetization [MATH] and non-zero magnetization [MATH] induced by the effective transverse field.', '1204.1008-1-19-1': 'For the initial Heisenberg-Ising ladder it means that the rung singlet dimers are dominating on Heisenberg bonds in the ground state.', '1204.1008-1-19-2': 'Although each bond may possess the antiferromagnetic order, the interaction along legs demolishes it and leads to the disordered quantum paramagnetic state.', '1204.1008-1-19-3': 'Stripe Leg (SL) state for [MATH], and [MATH]: the equivalent transverse Ising chain exhibits the spontaneous ferromagnetic ordering with [MATH].', '1204.1008-1-19-4': 'Due to the relations ([REF]) and ([REF]), one obtains for the Heisenberg-Ising ladder [MATH].', '1204.1008-1-19-5': 'This result is taken to mean that the Heisenberg-Ising ladder shows a ferromagnetic order along legs and antiferromagnetic order along rungs, i.e. the magnetizations of chains are opposite.', '1204.1008-1-19-6': 'The staggered magnetization as the relevant order parameter in this phase is non-zero and it exhibits evident quantum reduction of the magnetization given by: [MATH].', '1204.1008-1-19-7': 'Here we used the result for the spontaneous magnetization of transverse Ising chain [CITATION].', '1204.1008-1-19-8': 'Neel state for [MATH], and [MATH]: the effective transverse Ising chain shows the spontaneous antiferromagnetic order with [MATH].', '1204.1008-1-19-9': 'For the Heisenberg-Ising ladder one consequently obtains [MATH].', '1204.1008-1-19-10': 'Hence, it follows that the nearest-neighbour spins both along legs as well as rungs exhibit predominantly antiferromagnetic ordering.', '1204.1008-1-19-11': 'The dependence of staggered magnetization as the relevant order parameter is quite analogous to the previous case [MATH].', '1204.1008-1-19-12': 'Stripe Rung (SR) state for [MATH], [MATH], [MATH]: the model shows classical ordering in this phase with the antiferromagnetically ordered nearest-neighbour spins along legs and the ferromagnetically ordered nearest-neighbour spins along rungs.', '1204.1008-1-19-13': 'Ferromagnetic (FM) state for [MATH], [MATH], [MATH]: the ground state corresponds to the ideal fully polarized ferromagnetic spin state.', '1204.1008-1-20-0': 'The results displayed in figure [REF]a demonstrate that the Heisenberg-Ising ladder is in the disordered QPM phase whenever a relative strength of both Ising interactions [MATH] and [MATH] is sufficiently small compared to the Heisenberg intra-rung interaction [MATH].', '1204.1008-1-20-1': 'It is noteworthy, moreover, that the QPM phase reduces to a set of fully non-correlated singlet dimers placed on all rungs (the so-called rung singlet-dimer state) along the special line [MATH] up to [MATH], which is depicted in figure [REF] by dotted lines.', '1204.1008-1-20-2': 'Under this special condition, the intra-rung spin-spin correlation represents the only non-zero pair correlation function and all the other short-ranged spin-spin correlations vanish and/or change their sign across the special line [MATH].', '1204.1008-1-20-3': 'It should be remarked that the completely identical ground state can also be found in the symmetric Heisenberg two-leg ladder with [MATH] (see figure [REF]b).', '1204.1008-1-20-4': 'To compare with, the rung singlet-dimer state is being the exact ground state of the symmetric Heisenberg-Ising ladder with [MATH] for [MATH], while the symmetric Heisenberg ladder displays this simple factorizable ground state for [MATH] [CITATION] (this horizontal line is for clarity not shown in figure [REF]b as it exactly coincides with the ground-state boundary of the Heisenberg-Ising ladder extended over larger parameter space).', '1204.1008-1-20-5': 'It should be stressed, however, that the short-range spin-spin correlations become non-zero in QPM whenever [MATH] even if this phase still preserves its disordered nature with the prevailing character of the rung singlet-dimer state.', '1204.1008-1-20-6': 'To support this statement, the zero-temperature dependencies of the order parameters and the nearest-neighbour spin-spin correlation along legs are plotted in figure [REF].', '1204.1008-1-20-7': 'The relevant order parameters evidently disappear in QPM, whereas the nearest-neighbour correlation function changes its sign when passing through the special line [MATH] of the rung singlet-dimer state.', '1204.1008-1-21-0': 'Note furthermore that the Heisenberg-Ising ladder undergoes the second-order quantum phase transition from the disordered QPM phase to the spontaneously long-range ordered Neel or SL phase, which predominantly appear in the parameter region where one from both Ising couplings [MATH] and [MATH] is being antiferromagnetic and the other one is ferromagnetic.', '1204.1008-1-21-1': 'The significant quantum reduction of staggered order parameters implies an obvious quantum character of both Neel as well as SL phases in which the antiferromagnetic correlation on rungs still dominates (see figure [REF]).', '1204.1008-1-21-2': 'Thus, the main difference between both the quantum long-range ordered phases emerges in the character nearest-neighbour correlation along legs, which is ferromagnetic in the SL phase but antiferromagnetic in the Neel phase.', '1204.1008-1-21-3': 'The continuous vanishing of the order parameters depicted in figure [REF]a provides a direct evidence of the second-order quantum phase transition between the disordered QPM phase and the ordered SL (or Neel) phase, which is also accompanied with the weak-singular behaviour of the nearest-neighbour correlation function visualized in figure [REF]b by black dots.', '1204.1008-1-21-4': 'Here it should be noted that the [MATH] correlation function of the Heisenberg-Ising ladder can be easily derived from the result of the [MATH] correlation function of the transverse Ising chain calculated in [CITATION].', '1204.1008-1-21-5': 'Last but not least, the strong enough Ising interactions [MATH] and [MATH] may break the antiferromagnetic correlation along rungs and lead to a presence of the fully ordered ferromagnetic rung states FM or SR depending on whether both Ising interactions are ferromagnetic or antiferromagnetic, respectively.', '1204.1008-1-21-6': 'It should be noticed that this change is accompanied with a discontinuous (first-order) quantum phase transition on behalf of a classical character of both FM and SR phases, which can be also clearly seen in figure [REF] from an abrupt change of the order parameters as well as the nearest-neighbour correlation function.', '1204.1008-1-22-0': 'Finally, let us also provide a more detailed comparison between the ground-state phase diagrams of the Heisenberg, Heisenberg-Ising and Ising two-leg ladders all depicted in figure [REF]b.', '1204.1008-1-22-1': 'One can notice that the displayed ground-state phase diagrams of the Heisenberg and Heisenberg-Ising two-leg ladders have several similar features.', '1204.1008-1-22-2': 'The phase boundary between the classically ordered SR phase and three quantum phases (QPM, SL, Neel) of the Heisenberg-Ising ladder follows quite closely the first-order phase boundary between the Haldane-type phase and the dimerized phase of the pure Heisenberg ladder obtained using the series expansion and exact diagonalization [CITATION].', '1204.1008-1-22-3': 'Of course, the most fundamental difference lies in the character of SR and Haldane phases, because the former phase exhibits a classical long-range order contrary to a more spectacular topological order of the pure quantum Haldane-type phase with a non-zero string order parameter [CITATION] even though the ferromagnetic intra-rung correlation is common for both phases.', '1204.1008-1-22-4': 'The difference between the ground states of the Heisenberg-Ising and pure Heisenberg two-leg ladder becomes much less pronounced in the parameter region with the strong intra-rung interaction [MATH], which evokes the quantum phases in the ground state of both these models.', '1204.1008-1-22-5': 'In the case of a sufficiently strong frustration [MATH], the ground state of the Heisenberg-Ising ladder forms the disordered QPM phase, whereas the antiferromagnetic or ferromagnetic (Neel or SL) long-range order emerges along the legs if the diagonal coupling [MATH] is much weaker or stronger than the intra-leg interaction [MATH].', '1204.1008-1-22-6': 'Note that the quantum Neel and SL phases have several common features (e.g. predominant antiferromagnetic correlations along rungs) with the disordered QPM phase from which they evolve when crossing continuous quantum phase transitions given by the set of equations: [MATH] (for [MATH]) and [MATH] (for [MATH]).', '1204.1008-1-22-7': 'It cannot be definitely ruled out whether or not these quantum ordered phases may become the ground state of the pure Heisenberg ladder, because they are also predicted by the bond-mean-field approximation [CITATION] but have not been found by the most of the numerical methods.', '1204.1008-1-22-8': 'Further numerical investigations of the Heisenberg ladder are therefore needed to clarify this unresolved issue.', '1204.1008-1-23-0': '## Ground-state phase diagram in a non-zero field', '1204.1008-1-24-0': 'When the external field [MATH] is switched on, the inequality for the ground states of the Ising chain in the longitudinal field ([REF]) is generally valid only for [MATH] or [MATH].', '1204.1008-1-24-1': 'Hence, it is necessary to modify the procedure of finding the ground states inside the region where the relation ([REF]) is broken.', '1204.1008-1-24-2': 'However, one may use with a success the method suggested by Shastry and Sutherland [CITATION] in order to find the ground states inside this parameter region.', '1204.1008-1-24-3': 'Let us represent our effective Hamiltonian ([REF]) in the form: [EQUATION]', '1204.1008-1-24-4': 'Then, one can employ the variational principle implying that [MATH], where [MATH] is the lowest eigenenergy of [MATH].', '1204.1008-1-25-0': 'Looking for the eigenenergies of [MATH] it is enough to find the lowest eigenstate of each bond configuration:', '1204.1008-1-26-0': 'The phase corresponding to the alternating bond configuration [MATH], [MATH] will be hereafter referred to as the staggered bond (SB) phase.', '1204.1008-1-26-1': "It should be mentioned that there does not exist in the SB phase any correlations between spins from different rungs and the overall energy comes from the intra-rung spin-spin interactions and the Zeeman's energy of the fully polarized rungs [CITATION].", '1204.1008-1-26-2': 'It can be easily seen from a comparison of [MATH] and [MATH] that the eigenenergy of the SB phase [MATH] has always lower energy than the lowest eigenenergy of the fully polarized state [MATH] inside the stripe: [EQUATION]', '1204.1008-1-26-3': 'Note that the ferromagnetic ordering is preferred for external fields above this stripe [MATH], while the antiferromagnetic ordering becomes the lowest-energy state below this stripe [MATH] .', '1204.1008-1-27-0': 'If one compares the respective eigenenergies of the staggered bond phase [MATH] and the uniform purity phase [MATH]), one gets another condition implying that [MATH] becomes lower than [MATH] only if: [EQUATION]', '1204.1008-1-27-1': 'This means that the SB phase with the overall energy [MATH] becomes the ground state inside the region confined by conditions ([REF]) and ([REF]).', '1204.1008-1-27-2': 'The lowest field, which makes the SB phase favourable, can be also found as: [EQUATION]', '1204.1008-1-27-3': 'Similarly, one may also find the condition under which two eigenenergies corresponding to the uniform impurity configuration [MATH] become lower than the respective eigenenergy of the uniform purity configuration [MATH].', '1204.1008-1-27-4': 'The ferromagnetic state of the uniform impurity configuration becomes lower if the external field exceeds the boundary value: [EQUATION] whereas the condition for the antiferromagnetic state of the uniform impurity configuration is independent of the external field: [EQUATION]', '1204.1008-1-27-5': 'It is worthy of notice that it is not possible to find the ground state outside the boundaries ([REF]), ([REF]), ([REF]) using the variational principle.', '1204.1008-1-27-6': 'However, it is shown in the appendix that the bond configuration, which corresponds to the ground state, cannot exceed period two.', '1204.1008-1-27-7': 'Therefore, one has to search for the ground state just among the states that correspond to the following bond configurations: all [MATH]; all [MATH]; [MATH], [MATH], [MATH]).', '1204.1008-1-27-8': 'In this respect, two phases are possible inside the stripe given by ([REF]).', '1204.1008-1-27-9': 'The ground state of the Heisenberg-Ising ladder forms the SB phase if [MATH]: [EQUATION]', '1204.1008-1-27-10': 'Several ground-state phase diagrams are plotted in figures [REF]-[REF] for a non-zero magnetic field.', '1204.1008-1-27-11': 'The most interesting feature stemming from these phase diagrams is that the SB phase may become the ground state for the magnetic field [MATH], which is sufficiently strong to break the rung singlet-dimer state.', '1204.1008-1-27-12': 'It can be observed from figure [REF] that the SB phase indeed evolves along the line of the rung singlet-dimer state and replaces the QPM phase in the ground-state phase diagram.', '1204.1008-1-27-13': 'The external magnetic field may thus cause an appearance of another peculiar quantum SB phase with the translationally broken symmetry, i.e. the alternating singlet and fully polarized triplet bonds on the rungs of the two-leg ladder.', '1204.1008-1-27-14': 'Hence, it follows that the SB phase emerges at moderate values of the external magnetic field and it consequently leads to a presence of the intermediate magnetization plateau at a half of the saturation magnetization.', '1204.1008-1-27-15': 'It is quite apparent from figures [REF]a, [REF]a that the Heisenberg-Ising ladder without the frustrating diagonal Ising interaction [MATH] exhibits this striking magnetization plateau just for the particular case of the antiferromagnetic Ising intra-leg interaction [MATH].', '1204.1008-1-27-16': 'On the other hand, the fractional magnetization plateau inherent to a presence of the SB phase is substantially stabilized by the spin frustration triggered by the non-zero diagonal Ising interaction [MATH] and hence, the plateau region may even extend over a relatively narrow region of the ferromagnetic Ising intra-leg interaction [MATH] as well (see figure [REF]b).', '1204.1008-1-28-0': 'It should be noted that the same mechanism for an appearance of the magnetization plateau has also been predicted for the spin-[MATH] Heisenberg two-leg ladder by making use of exact numerical diagonalization and DMRG methods [CITATION].', '1204.1008-1-28-1': 'Let us therefore conclude our study by comparing the respective ground-state phase diagrams of the Heisenberg-Ising and Heisenberg two-leg ladders in a presence of the external magnetic field displayed in figure [REF].', '1204.1008-1-29-0': 'According to this plot, both models give essentially the same magnetization process in the strong-coupling limit of the Heisenberg intra-rung interaction [MATH], where two subsequent field-induced transitions in the following order QPM-SB-FM can be observed and consequently, the magnetization exhibits two abrupt jumps at the relevant transition fields.', '1204.1008-1-29-1': 'On the other hand, the fundamental differences can be detected in the relevant magnetization process of the Heisenberg-Ising and Heisenberg ladders in the weak-coupling limit of the intra-rung interaction [MATH].', '1204.1008-1-29-2': 'At low fields, the quantum Haldane-like phase constitutes the ground state of the pure Heisenberg ladder in contrast to the classical SR phase, which is being the low-field ground state of the Heisenberg-Ising ladder.', '1204.1008-1-29-3': 'In addition, the Heisenberg-Ising ladder still exhibits a magnetization plateau corresponding to a gapped SB phase at intermediate values of the magnetic field in this parameter space, while a continuous increase of the magnetization can be observed in the pure Heisenberg ladder due to a presence of the gapless Luttinger-liquid phase at moderate fields.', '1204.1008-1-29-4': 'Finally, it is also noteworthy that the saturation fields towards FM phase are also quite different for the Heisenberg-Ising and Heisenberg ladders in the weak-coupling limit of the Heisenberg intra-rung interaction.', '1204.1008-1-30-0': '# Conclusions', '1204.1008-1-31-0': 'The frustrated Heisenberg-Ising two-leg ladder was considered within two rigorous approaches based on the pseudospin representation and the unitary transformation.', '1204.1008-1-31-1': 'Using this procedure, we have proved the exact mapping correspondence between the investigated model and some generalized spin-[MATH] quantum Ising chain with the composite spins in an effective longitudinal and transverse field.', '1204.1008-1-31-2': 'It has been shown that the ground state of the model under investigation must correspond to a regular bond configuration not exceeding the period two.', '1204.1008-1-31-3': 'Hence, the true ground state will be the lowest eigenstate of either the transverse Ising chain, the Ising chain in the longitudinal field, or the non-interacting spin-chain model in the staggered longitudinal-transverse field.', '1204.1008-1-32-0': 'The most interesting results to emerge from the present study are closely related to an extraordinary diversity of the constructed ground-state phase diagrams and the quantum phase transitions between different ground-state phases.', '1204.1008-1-32-1': 'In an absence of the external magnetic field, the ground-state phase diagram constitute four different ordered phases and one quantum disordered phase.', '1204.1008-1-32-2': 'The disordered phase was characterized through short-range spin correlations, which indicate a dominating character of the rung singlet-dimer state in this phase.', '1204.1008-1-32-3': 'On the other hand, the order parameters have been exactly calculated for all the four ordered phases, two of them having classical character and two purely quantum character as evidenced by the quantum reduction of the staggered magnetization in the latter two phases.', '1204.1008-1-32-4': 'Last but not least, it has been demonstrated that the external magnetic field of a moderate strength may cause an appearance of the peculiar SB phase with alternating character of singlet and triplet bonds on the rungs of two-leg ladder.', '1204.1008-1-32-5': 'This latter finding is consistent with a presence of the fractional magnetization plateau at a half of the saturation magnetization in the relevant magnetization process.', '1204.1008-1-33-0': 'The authors are grateful to Oleg Derzhko for several useful comments and suggestions.', '1204.1008-1-33-1': 'T.V. was supported by the National Scholarship Programme of the Slovak Republic for the Support of Mobility of Students, PhD Students, University Teachers and Researchers.', '1204.1008-1-33-2': 'J.S. acknowledges the financial support under the grant VEGA 1/0234/12.', '1204.1008-1-34-0': '# Ground-state bond configuration in magnetic field', '1204.1008-1-35-0': 'In general each state of the model can be represented as an array of alternating purity and impurity non-interacting clusters of different length.', '1204.1008-1-35-1': 'One can formally write the lowest energy of some configuration as [EQUATION] where [MATH] is the lowest energy of the complex of two independent spin chains which correspond to the purity and impurity bond states, [MATH].', '1204.1008-1-35-2': 'If [MATH] corresponds to the lowest energy per spin among other clusters, it is evident that the ground state configuration is alternating purity and impurity clusters of length [MATH] and [MATH].', '1204.1008-1-36-0': 'Let us consider at first the case when [MATH] is restricted by condition ([REF]).', '1204.1008-1-36-1': 'If the configuration contains the cluster of more than 2 impurity bonds, its energy can be lowered by adding pure bond in-between.', '1204.1008-1-36-2': 'If condition ([REF]) is valid, such a configuration can achieve a lower energy.', '1204.1008-1-36-3': 'Using this procedure we can reduce the number of sites in impurity clusters to 2, i.e. only [MATH] or [MATH] can correspond to the ground state.', '1204.1008-1-37-0': 'Define the energy per spin for each configuration as: [EQUATION]', '1204.1008-1-37-1': 'If [MATH] and so on ([MATH]) the ground state corresponds to the staggered bond configuration.', '1204.1008-1-37-2': 'Suppose that for some [MATH] [EQUATION]', '1204.1008-1-37-3': 'Now it is clear that [EQUATION] if [MATH], i.e. the ground state energy of finite transverse Ising chain is a convex function of [MATH].', '1204.1008-1-37-4': 'It can be shown by numerical calculations for finite chains (see figure [REF]) using the numerical approach [CITATION].', '1204.1008-1-38-0': 'It means that [MATH] may have only one extremum and it is the maximum.', '1204.1008-1-38-1': 'Therefore, [MATH] may take the minimal value only in two limiting cases [MATH], [MATH].', '1204.1008-1-38-2': 'The same result can be analogously obtained for [MATH].', '1204.1008-1-38-3': 'Now we can compare the energies of [MATH] and [MATH] configurations.', '1204.1008-1-38-4': 'For [MATH] [EQUATION] due to the upper boundary of condition ([REF]).', '1204.1008-1-38-5': 'To summarize, the ground state configuration in the region confined by ([REF]) can correspond to either staggered bond or uniform purity configuration.', '1204.1008-1-39-0': 'Let us consider the ground state for [MATH].', '1204.1008-1-39-1': 'Similarly to the previous case we can prove that [MATH] is a convex function of [MATH] and [MATH] if the ground state energies of the uniform chains have the following properties: [EQUATION]', '1204.1008-1-39-2': 'One can find that [MATH], and [MATH].', '1204.1008-1-39-3': 'That is why the minimal value of the ground state can be achieved in one of three cases: staggered bond, purity and impurity configuration.', '1204.1008-1-39-4': 'The staggered bond configuration should be excluded from this list for [MATH], since it cannot be the ground state due to the variational principle.', '1204.1008-1-40-0': '# References'}

{'1204.1008-2-0-0': 'Ground-state behaviour of the frustrated quantum spin-[MATH] two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail.', '1204.1008-2-0-1': 'The investigated model is transformed to the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation.', '1204.1008-2-0-2': "It is shown that the ground state of the Heisenberg-Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the 'classical' Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal-transverse field.", '1204.1008-2-0-3': 'The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization.', '1204.1008-2-0-4': 'The ground-state phase diagram totally consists of five ordered and one quantum paramagnetic phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions.', '1204.1008-2-0-5': 'The order parameters are exactly calculated for all five ordered phases and the quantum paramagnetic phase is characterized through different short-range spin-spin correlations.', '1204.1008-2-1-0': '# Introduction', '1204.1008-2-2-0': 'Quantum spin models with competing interactions represent quite interesting and challenging topic for the current research [CITATION].', '1204.1008-2-2-1': 'Such models show many unusual features in their ground-state properties and are very sensitive to the approximative schemes applied to them.', '1204.1008-2-2-2': 'An existence of exact solutions is therefore quite important, since they provide a rigorous information about the complex behaviour of frustrated models.', '1204.1008-2-2-3': 'However, exact results for such systems are still limited (see [CITATION] and references cited therein).', '1204.1008-2-2-4': 'The simplest examples of rigorously solved quantum spin models are the models with the known dimerized [CITATION] or even trimerized [CITATION] ground states.', '1204.1008-2-2-5': 'Another example represents frustrated quantum spin antiferromagnets in high magnetic fields with the so-called localized-magnon eigenstates (see e.g. [CITATION] for recent review).', '1204.1008-2-2-6': 'It should be nevertheless noticed that the aforementioned exact results are usually derived only under certain constraint laid on the interaction parameters and those models are not tractable quite generally.', '1204.1008-2-3-0': 'On the other hand, the exact solutions for some spin-[MATH] quantum spin ladders with multispin interaction allows to find, besides the ground state, also thermodynamic properties quite rigorously [CITATION].', '1204.1008-2-3-1': 'Exactly solvable models, which admit the inclusion of frustration, can also be constructed from the Ising models where the decorations of quantum spins are included [CITATION].', '1204.1008-2-3-2': 'The decoration-iteration procedure allows one to calculate exactly all thermodynamic properties of the decorated models when the exact solution for the corresponding Ising model is known.', '1204.1008-2-3-3': 'The distinctive feature of these models is that the quantum decorations are separated from each other, so that the Hamiltonian can be decomposed into the sum of commuting parts.', '1204.1008-2-3-4': 'The eigenstates of the total Hamiltonian are then simply factorized into a product of the eigenstates of its commuting parts.', '1204.1008-2-4-0': 'New findings of the exactly solvable frustrated quantum spin models with non-trivial (not simply factorizable) ground states are thus highly desirable.', '1204.1008-2-4-1': 'The present article deals with the frustrated quantum spin-[MATH] two-leg ladder with the Heisenberg intra-rung interaction and the Ising inter-rung interactions between nearest-neighbouring spins from either the same or different leg.', '1204.1008-2-4-2': 'Such a model can be regarded as an extension of the spin-[MATH] Heisenberg-Ising bond alternating chain, which was proposed and rigorously solved by Lieb, Schultz and Mattis [CITATION].', '1204.1008-2-4-3': 'Alternatively, this model can also be viewed as the generalization of the exactly solved quantum compass ladder [CITATION] when considering the Heisenberg rather than XZ-type intra-rung interaction and accounting for the additional frustrating Ising inter-rung interaction.', '1204.1008-2-4-4': 'Moreover, it is quite plausible to suspect that the exact results presented hereafter for the spin-[MATH] Heisenberg-Ising two-leg ladder may also bring insight into the relevant behaviour of the corresponding Heisenberg two-leg ladder, which represents a quite challenging and complex research problem in its own right [CITATION].', '1204.1008-2-4-5': 'The frustrated spin-[MATH] Heisenberg two-leg ladder have been extensively investigated by employing various independent numerical and analytical methods such as density-matrix renormalization group [CITATION], numerical diagonalization [CITATION], series expansion [CITATION], bosonization technique [CITATION], strong- and weak-coupling analysis [CITATION], valence-bond spin-wave theory [CITATION], variational matrix-product approach [CITATION], and bond mean-field theory [CITATION].', '1204.1008-2-4-6': 'Among the most interesting results obtained for this quantum spin chain, one could mention an existence of the columnar-dimer phase discussed in [CITATION] or a presence of the fractional plateau in the magnetization process examined in [CITATION].', '1204.1008-2-5-0': 'The theoretical investigation of two-leg ladder models is motivated not only from the academic point of view, but also from the experimental viewpoint, because the two-leg ladder motif captures the magnetic structure of a certain class of real quasi-one-dimensional magnetic materials.', '1204.1008-2-5-1': 'The most widespread family of two-leg ladder compounds form cuprates, in which one finds both experimental representatives with the dominating intra-rung interaction like SrCu[MATH]O[MATH] [CITATION], Cu[MATH](C[MATH]H[MATH]N[MATH]Cl[MATH] [CITATION], (C[MATH]H[MATH]N)[MATH]CuBr[MATH] [CITATION], (5IAP)[MATH]CuBr[MATH] [CITATION] as well as, the magnetic compounds with the dominating intra-leg interaction such as KCuCl[MATH] [CITATION], TlCuCl[MATH] [CITATION], NH[MATH]CuCl[MATH], KCuBr[MATH].', '1204.1008-2-5-2': 'Understanding the low-temperature magnetism of two-leg ladder models also turns out to be crucial for an explanation of the mechanism, which is responsible for the high-temperature superconductivity of cuprates [CITATION].', '1204.1008-2-5-3': 'Besides the cuprates, another experimental representatives of the two-leg ladder compounds represent vanadates (VO)[MATH]P[MATH]O[MATH] [CITATION], CaV[MATH]O[MATH] and MgV[MATH]O[MATH] [CITATION], as well as, the polyradical BIP-BNO [CITATION].', '1204.1008-2-6-0': 'The outline of the paper is as follows.', '1204.1008-2-6-1': 'In section [REF] the model is defined and the pseudospin representation is considered.', '1204.1008-2-6-2': 'The ground-state properties of the model with and without external field are studied in section [REF].', '1204.1008-2-6-3': 'The most important findings are summarized in section [REF].', '1204.1008-2-7-0': '# Heisenberg-Ising two-leg ladder', '1204.1008-2-8-0': 'Let us define the spin-[MATH] Heisenberg-Ising ladder through the following Hamiltonian: [EQUATION] where [MATH], [MATH] are three spatial projections of spin-[MATH] operator, the first index denotes the number of leg, the second enumerates the site, [MATH] is the [MATH] Heisenberg intra-rung interaction between nearest-neighbour spins from the same rung, [MATH] is the Ising intra-leg interaction between nearest-neighbour spins from the same leg, [MATH] is the crossing (diagonal) Ising inter-rung interaction between next-nearest-neighbour spins from different rungs, [MATH] is the external magnetic field.', '1204.1008-2-8-1': 'We also imply the periodic boundary conditions [MATH] and [MATH] along legs.', '1204.1008-2-8-2': 'The coupling constants [MATH] and [MATH] can be interchanged by renumbering of the sites, as well as their signs can be simultaneously reverted by spin rotations.', '1204.1008-2-8-3': 'Therefore, the Hamiltonians [MATH], [MATH] and [MATH] in zero field have equal eigenvalues and the corresponding models are thermodynamically equivalent.', '1204.1008-2-9-0': 'It can be checked that [MATH]-projection of total spin on a rung [MATH] commutes with the total Hamiltonian [MATH] and hence, it represents a conserved quantity.', '1204.1008-2-9-1': 'For further convenience, it is therefore advisable to take advantage of the bond representation, which has been originally suggested by Lieb, Schultz and Mattis for the Heisenberg-Ising chain [CITATION].', '1204.1008-2-9-2': 'Let us introduce the bond-state basis consisting of four state vectors: [EQUATION]', '1204.1008-2-9-3': 'These states are the eigenstates of the Heisenberg coupling between two spins located on the [MATH]th rung, i.e. [MATH] is the singlet-bond state, [MATH], [MATH] are triplet states.', '1204.1008-2-9-4': 'Following [CITATION] two subspaces can be singled out in the bond space: i) [MATH], [MATH] and ii) [MATH], [MATH].', '1204.1008-2-9-5': 'The first and second subspaces corresponds to [MATH] and [MATH] respectively, and the Hamiltonian can be diagonalized separately in each subspace.', '1204.1008-2-9-6': 'We can introduce the index of subspace at [MATH]-th site: [MATH] if the given state is in the subspace with [MATH]) spanned by states [MATH], [MATH], [MATH]).', '1204.1008-2-9-7': 'According to [CITATION], let us also call the bond states as purity (impurity) states if [MATH].', '1204.1008-2-9-8': 'If one makes pseudospin notations for the states [MATH], [MATH], the action of the spin-[MATH] operators in the subspace [MATH] can be expressed in terms of new raising and lowering operators: [EQUATION]', '1204.1008-2-9-9': 'The operators [MATH], [MATH] satisfy the Pauli algebra ([MATH], [MATH], [MATH] for [MATH]), and half of their sum can be identified as a new pseudospin operator [MATH].', '1204.1008-2-10-0': 'Analogously, one can consider the pseudospin representation of the subspace [MATH], [MATH]) and find the action of spin operators in it as follows: [EQUATION]', '1204.1008-2-10-1': 'Combining ([REF]), ([REF]) we find the general expressions for the pseudospin representation of these operators, which are valid in both subspaces: [EQUATION]', '1204.1008-2-10-2': 'The effective Hamiltonian can be rewritten in terms of new operators as follows: [EQUATION]', '1204.1008-2-10-3': 'It is noteworthy that equation ([REF]) can be considered as some nonlinear spin transformation from [MATH], [MATH] to new operators [MATH], [MATH], where e.g. [MATH], [MATH].', '1204.1008-2-10-4': 'The transformation can be generated by the following unitary operator: [EQUATION]', '1204.1008-2-10-5': 'The crucial point is the second factor, which introduces the nonlinearity.', '1204.1008-2-10-6': 'Other terms are the rotation in spin space, they are used to adjust the transformed operators to the form of ([REF]).', '1204.1008-2-10-7': 'Finally, the spin operators are transformed as follows: [EQUATION]', '1204.1008-2-10-8': 'The transformed Hamiltonian has the form of the quantum Ising chain with composite spins in an effective longitudinal and transverse magnetic field: [EQUATION]', '1204.1008-2-10-9': 'The straightforward correspondence [MATH], [MATH] leads to the equivalence between the Hamiltonians ([REF]) and ([REF]).', '1204.1008-2-10-10': 'It should be noted that such kind of a representation remains valid also in case of the asymmetric ladder having both diagonal (crossing) Ising interactions different from each other.', '1204.1008-2-11-0': '# Ground state of the Heisenberg-Ising two-leg ladder', '1204.1008-2-12-0': 'The Hamiltonian ([REF]) can also be rewritten in the following more symmetric form: [EQUATION]', '1204.1008-2-12-1': 'This form serves in evidence that the effective model splits at [MATH]th site into two independent chains provided that two neighbouring bonds are being in different subspaces, i.e. [MATH].', '1204.1008-2-12-2': 'The uniform Hamiltonians, when all [MATH] or all [MATH], read: [EQUATION]', '1204.1008-2-12-3': 'If all bonds are in purity states ([MATH]), one obtains the effective Hamiltonian of the Ising chain in the transverse field that is exactly solvable within Jordan-Wigner fermionization [CITATION].', '1204.1008-2-12-4': 'If all bonds are in impurity states ([MATH]), one comes to the Ising chain in the longitudinal field that is solvable by the transfer-matrix method (see e.g. [CITATION]).', '1204.1008-2-12-5': 'Accordingly, the ground-state energy of the model in [MATH] subspace is given by [CITATION]: [EQUATION] where [MATH] and [MATH] is the complete elliptic integral of the second kind.', '1204.1008-2-13-0': 'The ground-state energy of the model in [MATH] subspace is given by the ground-state energy of the effective Ising chain in the longitudinal field: [EQUATION]', '1204.1008-2-13-1': 'The upper (lower) case corresponds to the ferromagnetically (antiferromagnetically) ordered state being the ground state at strong (weak) enough magnetic fields.', '1204.1008-2-14-0': '## Ground-state phase diagram in a zero field', '1204.1008-2-15-0': 'If the external field vanishes ([MATH]), it is sufficient to show that the inequality [MATH] holds for any finite transverse Ising chain with free ends in order to prove that the ground state always corresponds to a uniform bond configuration.', '1204.1008-2-15-1': 'Here [MATH] denotes the ground state energy of the Hamiltonian: [EQUATION]', '1204.1008-2-15-2': 'Indeed, two independent chains of size [MATH] and [MATH] can be represented by the following Hamiltonian: [EQUATION]', '1204.1008-2-15-3': 'If [MATH] and [MATH] are the lowest eigenvalue and eigenstate of [MATH], then [MATH] and [MATH] are the lowest eigenvalue and eigenstate of [MATH].', '1204.1008-2-15-4': 'Now, it is straightforward to show that [EQUATION]', '1204.1008-2-15-5': 'Here, we have used that [MATH] for any finite chain [CITATION], and that the lowest mean value of the operator [MATH] is achieved in its ground state.', '1204.1008-2-16-0': 'It is easy to show by straightforward calculation that the same property is valid for the Ising chain with free ends in zero longitudinal field, i.e. [EQUATION]', '1204.1008-2-16-1': 'Now, let us prove that the ground state of the whole model may correspond only to one of uniform bond configurations.', '1204.1008-2-16-2': 'It can be readily understood from ([REF]) that the effective Hamiltonian does not contain an interaction between spins from two neighbouring bonds if they are in different subspaces [CITATION].', '1204.1008-2-16-3': "It means that the effective model splits into two independent parts at each boundary ('domain wall') between the purity and impurity states.", '1204.1008-2-16-4': 'Thus, the Heisenberg-Ising ladder for any given configuration of bonds can be considered as a set of independent chains of two kinds and of different sizes.', '1204.1008-2-16-5': 'Then, the ground-state energy of any randomly chosen bond configuration will be as follows: [EQUATION]', '1204.1008-2-16-6': 'It is quite evident from equations ([REF]) and ([REF]) that the one uniform configuration, which corresponds to the state with lower energy than the other one, must be according to inequality ([REF]) the lowest-energy state (i.e. ground state).', '1204.1008-2-16-7': 'Therefore, the model may show in the ground state the first-order quantum phase transition when the lowest eigenenergies in both subspaces becomes equal [MATH].', '1204.1008-2-16-8': 'Besides, there also may appear the more striking second-order (continuous) quantum phase transition at [MATH] in the ground state, which is inherent to the pure quantum spin chain [CITATION].', '1204.1008-2-16-9': 'From this perspective, the Heisenberg-Ising ladder can show a variety of quantum phase transitions in its phase diagram.', '1204.1008-2-17-0': 'The ground-state phase diagram of the spin-[MATH] Heisenberg-Ising ladder in a zero magnetic field is shown in figure [REF].', '1204.1008-2-18-0': 'It reflects the symmetry of the model, i.e. it is invariant under the exchange of the Ising interactions [MATH] and [MATH], as well as, under the simultaneous change of the signs of [MATH] and [MATH].', '1204.1008-2-18-1': 'Altogether, five different ground states can be recognised in figure [REF]:', '1204.1008-2-19-0': 'Quantum paramagnetic (QPM) state for [MATH], and [MATH]: the equivalent transverse Ising chain ([REF]) is in the gapped disordered state with no spontaneous magnetization [MATH] and non-zero magnetization [MATH] induced by the effective transverse field.', '1204.1008-2-19-1': 'For the initial Heisenberg-Ising ladder it means that the rung singlet dimers are dominating on Heisenberg bonds in the ground state.', '1204.1008-2-19-2': 'Although each bond may possess the antiferromagnetic order, the interaction along legs demolishes it and leads to the disordered quantum paramagnetic state.', '1204.1008-2-19-3': 'Stripe Leg (SL) state for [MATH], and [MATH]: the equivalent transverse Ising chain exhibits the spontaneous ferromagnetic ordering with [MATH].', '1204.1008-2-19-4': 'Due to relations ([REF]) and ([REF]), one obtains for the Heisenberg-Ising ladder [MATH].', '1204.1008-2-19-5': 'This result is taken to mean that the Heisenberg-Ising ladder shows a ferromagnetic order along legs and antiferromagnetic order along rungs, i.e. the magnetizations of chains are opposite.', '1204.1008-2-19-6': 'The staggered magnetization as the relevant order parameter in this phase is non-zero and it exhibits evident quantum reduction of the magnetization given by: [MATH].', '1204.1008-2-19-7': 'Here we used the result for the spontaneous magnetization of transverse Ising chain [CITATION].', '1204.1008-2-19-8': 'Neel state for [MATH], and [MATH]: the effective transverse Ising chain shows the spontaneous antiferromagnetic order with [MATH].', '1204.1008-2-19-9': 'For the Heisenberg-Ising ladder one consequently obtains [MATH].', '1204.1008-2-19-10': 'Hence, it follows that the nearest-neighbour spins both along legs as well as rungs exhibit predominantly antiferromagnetic ordering.', '1204.1008-2-19-11': 'The dependence of staggered magnetization as the relevant order parameter is quite analogous to the previous case [MATH].', '1204.1008-2-19-12': 'Stripe Rung (SR) state for [MATH], [MATH], [MATH]: the model shows classical ordering in this phase with the antiferromagnetically ordered nearest-neighbour spins along legs and the ferromagnetically ordered nearest-neighbour spins along rungs.', '1204.1008-2-19-13': 'Ferromagnetic (FM) state for [MATH], [MATH], [MATH]: the ground state corresponds to the ideal fully polarized ferromagnetic spin state.', '1204.1008-2-20-0': 'The results displayed in figure [REF](a) demonstrate that the Heisenberg-Ising ladder is in the disordered QPM phase whenever a relative strength of both Ising interactions [MATH] and [MATH] is sufficiently small compared to the Heisenberg intra-rung interaction [MATH].', '1204.1008-2-20-1': 'It is noteworthy, moreover, that the QPM phase reduces to a set of fully non-correlated singlet dimers placed on all rungs (the so-called rung singlet-dimer state) along the special line [MATH] up to [MATH], which is depicted in figure [REF] by dotted lines.', '1204.1008-2-20-2': 'Under this special condition, the intra-rung spin-spin correlation represents the only non-zero pair correlation function and all the other short-ranged spin-spin correlations vanish and/or change their sign across the special line [MATH].', '1204.1008-2-20-3': 'It should be remarked that the completely identical ground state can also be found in the symmetric Heisenberg two-leg ladder with [MATH] (see figure [REF](b)).', '1204.1008-2-20-4': 'To compare with, the rung singlet-dimer state is being the exact ground state of the symmetric Heisenberg-Ising ladder with [MATH] for [MATH], while the symmetric Heisenberg ladder displays this simple factorizable ground state for [MATH] [CITATION] (this horizontal line is for clarity not shown in figure [REF](b) as it exactly coincides with the ground-state boundary of the Heisenberg-Ising ladder extended over larger parameter space).', '1204.1008-2-20-5': 'It should be stressed, however, that the short-range spin-spin correlations become non-zero in QPM whenever [MATH] even if this phase still preserves its disordered nature with the prevailing character of the rung singlet-dimer state.', '1204.1008-2-20-6': 'To support this statement, the zero-temperature dependencies of the order parameters and the nearest-neighbour spin-spin correlation along legs are plotted in figure [REF].', '1204.1008-2-20-7': 'The relevant order parameters evidently disappear in QPM, whereas the nearest-neighbour correlation function changes its sign when passing through the special line [MATH] of the rung singlet-dimer state.', '1204.1008-2-21-0': 'Note furthermore that the Heisenberg-Ising ladder undergoes the second-order quantum phase transition from the disordered QPM phase to the spontaneously long-range ordered Neel or SL phase, which predominantly appear in the parameter region where one from both Ising couplings [MATH] and [MATH] is being antiferromagnetic and the other one is ferromagnetic.', '1204.1008-2-21-1': 'The significant quantum reduction of staggered order parameters implies an obvious quantum character of both Neel as well as SL phases in which the antiferromagnetic correlation on rungs still dominates (see figure [REF]).', '1204.1008-2-21-2': 'Thus, the main difference between both the quantum long-range ordered phases emerges in the character nearest-neighbour correlation along legs, which is ferromagnetic in the SL phase but antiferromagnetic in the Neel phase.', '1204.1008-2-21-3': 'The continuous vanishing of the order parameters depicted in figure [REF](a) provides a direct evidence of the second-order quantum phase transition between the disordered QPM phase and the ordered SL (or Neel) phase, which is also accompanied with the weak-singular behaviour of the nearest-neighbour correlation function visualized in figure [REF](b) by black dots.', '1204.1008-2-21-4': 'Here it should be noted that the [MATH] correlation function of the Heisenberg-Ising ladder can be easily derived from the result of the [MATH] correlation function of the transverse Ising chain calculated in [CITATION].', '1204.1008-2-21-5': 'Last but not least, the strong enough Ising interactions [MATH] and [MATH] may break the antiferromagnetic correlation along rungs and lead to a presence of the fully ordered ferromagnetic rung states FM or SR depending on whether both Ising interactions are ferromagnetic or antiferromagnetic, respectively.', '1204.1008-2-21-6': 'It should be noticed that this change is accompanied with a discontinuous (first-order) quantum phase transition on behalf of a classical character of both FM and SR phases, which can be also clearly seen in figure [REF] from an abrupt change of the order parameters as well as the nearest-neighbour correlation function.', '1204.1008-2-22-0': 'Finally, let us also provide a more detailed comparison between the ground-state phase diagrams of the Heisenberg, Heisenberg-Ising and Ising two-leg ladders all depicted in figure [REF](b).', '1204.1008-2-22-1': 'One can notice that the displayed ground-state phase diagrams of the Heisenberg and Heisenberg-Ising two-leg ladders have several similar features.', '1204.1008-2-22-2': 'The phase boundary between the classically ordered SR phase and three quantum phases (QPM, SL, Neel) of the Heisenberg-Ising ladder follows quite closely the first-order phase boundary between the Haldane-type phase and the dimerized phase of the pure Heisenberg ladder obtained using the series expansion and exact diagonalization [CITATION].', '1204.1008-2-22-3': 'Of course, the most fundamental difference lies in the character of SR and Haldane phases, because the former phase exhibits a classical long-range order contrary to a more spectacular topological order of the pure quantum Haldane-type phase with a non-zero string order parameter [CITATION] even though the ferromagnetic intra-rung correlation is common for both phases.', '1204.1008-2-22-4': 'The difference between the ground states of the Heisenberg-Ising and pure Heisenberg two-leg ladder becomes much less pronounced in the parameter region with the strong intra-rung interaction [MATH], which evokes the quantum phases in the ground state of both these models.', '1204.1008-2-22-5': 'In the case of a sufficiently strong frustration [MATH], the ground state of the Heisenberg-Ising ladder forms the disordered QPM phase, whereas the antiferromagnetic or ferromagnetic (Neel or SL) long-range order emerges along the legs if the diagonal coupling [MATH] is much weaker or stronger than the intra-leg interaction [MATH].', '1204.1008-2-22-6': 'Note that the quantum Neel and SL phases have several common features (e.g. predominant antiferromagnetic correlations along rungs) with the disordered QPM phase from which they evolve when crossing continuous quantum phase transitions given by the set of equations: [MATH] (for [MATH]) and [MATH] (for [MATH]).', '1204.1008-2-22-7': 'It cannot be definitely ruled out whether or not these quantum ordered phases may become the ground state of the pure Heisenberg ladder, because they are also predicted by the bond-mean-field approximation [CITATION] but have not been found by the most of the numerical methods.', '1204.1008-2-22-8': 'Further numerical investigations of the Heisenberg ladder are therefore needed to clarify this unresolved issue.', '1204.1008-2-23-0': '## Ground-state phase diagram in a non-zero field', '1204.1008-2-24-0': 'When the external field [MATH] is switched on, the inequality for the ground states of the Ising chain in the longitudinal field ([REF]) is generally valid only for [MATH] or [MATH].', '1204.1008-2-24-1': 'Hence, it is necessary to modify the procedure of finding the ground states inside the region where the relation ([REF]) is broken.', '1204.1008-2-24-2': 'However, one may use with a success the method suggested by Shastry and Sutherland [CITATION] in order to find the ground states inside this parameter region.', '1204.1008-2-24-3': 'Let us represent our effective Hamiltonian ([REF]) in the form: [EQUATION]', '1204.1008-2-24-4': 'Then, one can employ the variational principle implying that [MATH], where [MATH] is the lowest eigenenergy of [MATH].', '1204.1008-2-25-0': 'Looking for the eigenenergies of [MATH] it is enough to find the lowest eigenstate of each bond configuration:', '1204.1008-2-26-0': 'The phase corresponding to the alternating bond configuration [MATH], [MATH] will be hereafter referred to as the staggered bond (SB) phase.', '1204.1008-2-26-1': "It should be mentioned that there does not exist in the SB phase any correlations between spins from different rungs and the overall energy comes from the intra-rung spin-spin interactions and the Zeeman's energy of the fully polarized rungs [CITATION].", '1204.1008-2-26-2': 'It can be easily seen from a comparison of [MATH] and [MATH] that the eigenenergy of the SB phase [MATH] has always lower energy than the lowest eigenenergy of the fully polarized state [MATH] inside the stripe: [EQUATION]', '1204.1008-2-26-3': 'Note that the ferromagnetic ordering is preferred for external fields above this stripe [MATH], while the antiferromagnetic ordering becomes the lowest-energy state below this stripe [MATH] .', '1204.1008-2-27-0': 'If one compares the respective eigenenergies of the staggered bond phase [MATH] and the uniform purity phase [MATH]), one gets another condition implying that [MATH] becomes lower than [MATH] only if: [EQUATION]', '1204.1008-2-27-1': 'This means that the SB phase with the overall energy [MATH] becomes the ground state inside the region confined by conditions ([REF]) and ([REF]).', '1204.1008-2-27-2': 'The lowest field, which makes the SB phase favourable, can be also found as: [EQUATION]', '1204.1008-2-27-3': 'Similarly, one may also find the condition under which two eigenenergies corresponding to the uniform impurity configuration [MATH] become lower than the respective eigenenergy of the uniform purity configuration [MATH].', '1204.1008-2-27-4': 'The ferromagnetic state of the uniform impurity configuration becomes lower if the external field exceeds the boundary value: [EQUATION] whereas the condition for the antiferromagnetic state of the uniform impurity configuration is independent of the external field: [EQUATION]', '1204.1008-2-27-5': 'It is worthy of notice that it is not possible to find the ground state outside the boundaries ([REF]), ([REF]), ([REF]) using the variational principle.', '1204.1008-2-27-6': 'However, it is shown in the appendix that the bond configuration, which corresponds to the ground state, cannot exceed period two.', '1204.1008-2-27-7': 'Therefore, one has to search for the ground state just among the states that correspond to the following bond configurations: all [MATH]; all [MATH]; [MATH], [MATH], [MATH]).', '1204.1008-2-27-8': 'In this respect, two phases are possible inside the stripe given by ([REF]).', '1204.1008-2-27-9': 'The ground state of the Heisenberg-Ising ladder forms the SB phase if [MATH]: [EQUATION]', '1204.1008-2-27-10': 'Several ground-state phase diagrams are plotted in figures [REF]-[REF] for a non-zero magnetic field.', '1204.1008-2-27-11': 'The most interesting feature stemming from these phase diagrams is that the SB phase may become the ground state for the magnetic field [MATH], which is sufficiently strong to break the rung singlet-dimer state.', '1204.1008-2-27-12': 'It can be observed from figure [REF] that the SB phase indeed evolves along the line of the rung singlet-dimer state and replaces the QPM phase in the ground-state phase diagram.', '1204.1008-2-27-13': 'The external magnetic field may thus cause an appearance of another peculiar quantum SB phase with the translationally broken symmetry, i.e. the alternating singlet and fully polarized triplet bonds on the rungs of the two-leg ladder.', '1204.1008-2-27-14': 'Hence, it follows that the SB phase emerges at moderate values of the external magnetic field and it consequently leads to a presence of the intermediate magnetization plateau at a half of the saturation magnetization.', '1204.1008-2-27-15': 'It is quite apparent from figures [REF](a), [REF](a) that the Heisenberg-Ising ladder without the frustrating diagonal Ising interaction [MATH] exhibits this striking magnetization plateau just for the particular case of the antiferromagnetic Ising intra-leg interaction [MATH].', '1204.1008-2-27-16': 'On the other hand, the fractional magnetization plateau inherent to a presence of the SB phase is substantially stabilized by the spin frustration triggered by the non-zero diagonal Ising interaction [MATH] and hence, the plateau region may even extend over a relatively narrow region of the ferromagnetic Ising intra-leg interaction [MATH] as well (see figure [REF](b)).', '1204.1008-2-28-0': 'It should be noted that the same mechanism for an appearance of the magnetization plateau has also been predicted for the spin-[MATH] Heisenberg two-leg ladder by making use of exact numerical diagonalization and DMRG methods [CITATION].', '1204.1008-2-28-1': 'Let us therefore conclude our study by comparing the respective ground-state phase diagrams of the Heisenberg-Ising and Heisenberg two-leg ladders in a presence of the external magnetic field displayed in figure [REF].', '1204.1008-2-29-0': 'According to this plot, both models give essentially the same magnetization process in the strong-coupling limit of the Heisenberg intra-rung interaction [MATH], where two subsequent field-induced transitions in the following order QPM-SB-FM can be observed and consequently, the magnetization exhibits two abrupt jumps at the relevant transition fields.', '1204.1008-2-29-1': 'On the other hand, the fundamental differences can be detected in the relevant magnetization process of the Heisenberg-Ising and Heisenberg ladders in the weak-coupling limit of the intra-rung interaction [MATH].', '1204.1008-2-29-2': 'At low fields, the quantum Haldane-like phase constitutes the ground state of the pure Heisenberg ladder in contrast to the classical SR phase, which is being the low-field ground state of the Heisenberg-Ising ladder.', '1204.1008-2-29-3': 'In addition, the Heisenberg-Ising ladder still exhibits a magnetization plateau corresponding to a gapped SB phase at intermediate values of the magnetic field in this parameter space, while a continuous increase of the magnetization can be observed in the pure Heisenberg ladder due to a presence of the gapless Luttinger-liquid phase at moderate fields.', '1204.1008-2-29-4': 'Finally, it is also noteworthy that the saturation fields towards FM phase are also quite different for the Heisenberg-Ising and Heisenberg ladders in the weak-coupling limit of the Heisenberg intra-rung interaction.', '1204.1008-2-30-0': '# Conclusions', '1204.1008-2-31-0': 'The frustrated Heisenberg-Ising two-leg ladder was considered within the rigorous approach using that [MATH]-projection of total spin on a rung is a conserved quantity.', '1204.1008-2-31-1': 'By means of the pseudospin representation of bond states, we have proved the exact mapping correspondence between the investigated model and some generalized spin-[MATH] quantum Ising chain with the composite spins in an effective longitudinal and transverse field.', '1204.1008-2-31-2': 'We have also found the unitary transformation which reproduces the analogous mapping between the models.', '1204.1008-2-31-3': 'While the bond-state representation has more transparent physical interpretation, the unitary transformation gives the complete relation between spin operators of both models, and it might be useful for searching quantum spin ladders which admit exact solutions.', '1204.1008-2-32-0': 'It has been shown that the ground state of the model under investigation must correspond to a regular bond configuration not exceeding the period two.', '1204.1008-2-32-1': 'Hence, the true ground state will be the lowest eigenstate of either the transverse Ising chain, the Ising chain in the longitudinal field, or the non-interacting spin-chain model in the staggered longitudinal-transverse field.', '1204.1008-2-33-0': 'The most interesting results to emerge from the present study are closely related to an extraordinary diversity of the constructed ground-state phase diagrams and the quantum phase transitions between different ground-state phases.', '1204.1008-2-33-1': 'In an absence of the external magnetic field, the ground-state phase diagram constitute four different ordered phases and one quantum paramagnetic phase.', '1204.1008-2-33-2': 'The disordered phase was characterized through short-range spin correlations, which indicate a dominating character of the rung singlet-dimer state in this phase.', '1204.1008-2-33-3': 'On the other hand, the order parameters have been exactly calculated for all the four ordered phases, two of them having classical character and two purely quantum character as evidenced by the quantum reduction of the staggered magnetization in the latter two phases.', '1204.1008-2-33-4': 'Last but not least, it has been demonstrated that the external magnetic field of a moderate strength may cause an appearance of the peculiar SB phase with alternating character of singlet and triplet bonds on the rungs of two-leg ladder.', '1204.1008-2-33-5': 'This latter finding is consistent with a presence of the fractional magnetization plateau at a half of the saturation magnetization in the relevant magnetization process.', '1204.1008-2-34-0': 'The authors are grateful to Oleg Derzhko for several useful comments and suggestions.', '1204.1008-2-34-1': 'T.V. was supported by the National Scholarship Programme of the Slovak Republic for the Support of Mobility of Students, PhD Students, University Teachers and Researchers.', '1204.1008-2-34-2': 'J.S. acknowledges the financial support under the grant VEGA 1/0234/12.', '1204.1008-2-35-0': '# Ground-state bond configuration in magnetic field', '1204.1008-2-36-0': 'In general each state of the model can be represented as an array of alternating purity and impurity non-interacting clusters of different length.', '1204.1008-2-36-1': 'One can formally write the lowest energy of some configuration as [EQUATION] where [MATH] is the lowest energy of the complex of two independent spin chains which correspond to the purity and impurity bond states, [MATH].', '1204.1008-2-36-2': 'If [MATH] corresponds to the lowest energy per spin among other clusters, it is evident that the ground state configuration is alternating purity and impurity clusters of length [MATH] and [MATH].', '1204.1008-2-37-0': 'Let us consider at first the case when [MATH] is restricted by condition ([REF]).', '1204.1008-2-37-1': 'If the configuration contains the cluster of more than 2 impurity bonds, its energy can be lowered by adding pure bond in-between.', '1204.1008-2-37-2': 'If condition ([REF]) is valid, such a configuration can achieve a lower energy.', '1204.1008-2-37-3': 'Using this procedure we can reduce the number of sites in impurity clusters to 2, i.e. only [MATH] or [MATH] can correspond to the ground state.', '1204.1008-2-38-0': 'Define the energy per spin for each configuration as: [EQUATION]', '1204.1008-2-38-1': 'If [MATH] and so on ([MATH]) the ground state corresponds to the staggered bond configuration.', '1204.1008-2-38-2': 'Suppose that for some [MATH] [EQUATION]', '1204.1008-2-38-3': 'Consequently, it is clear that [EQUATION] if [MATH], i.e. the ground state energy of finite transverse Ising chain is a convex function of [MATH].', '1204.1008-2-38-4': 'The last condition is ensured by non-increasing [MATH]) that can be shown by numerical calculations for finite chains (see figure [REF]) using the numerical approach [CITATION].', '1204.1008-2-38-5': 'For [MATH] we get the free spin Hamiltonian and [MATH] depends linearly on [MATH].', '1204.1008-2-39-0': 'Conditions ([REF]),([REF]) together mean that [MATH] may have only one extremum and it is the maximum.', '1204.1008-2-39-1': 'Therefore, [MATH] may take the minimal value only in two limiting cases [MATH], [MATH].', '1204.1008-2-39-2': 'The same result can be analogously obtained for [MATH].', '1204.1008-2-39-3': 'Now we can compare the energies of [MATH] and [MATH] configurations.', '1204.1008-2-39-4': 'For [MATH] [EQUATION] due to the upper boundary of condition ([REF]).', '1204.1008-2-39-5': 'To summarize, the ground state configuration in the region confined by ([REF]) can correspond to either staggered bond or uniform purity configuration.', '1204.1008-2-40-0': 'Let us consider the ground state for [MATH].', '1204.1008-2-40-1': 'Similarly to the previous case we can prove that [MATH] is a convex function of [MATH] and [MATH] if the ground state energies of the uniform chains have the following properties: [EQUATION]', '1204.1008-2-40-2': 'One can find that [MATH], and [MATH].', '1204.1008-2-40-3': 'That is why the minimal value of the ground state can be achieved in one of three cases: staggered bond, purity and impurity configuration.', '1204.1008-2-40-4': 'The staggered bond configuration should be excluded from this list for [MATH], since it cannot be the ground state due to the variational principle.', '1204.1008-2-41-0': '# References'}

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'1204.1008-2-6-1'], ['1204.1008-1-6-2', '1204.1008-2-6-2'], ['1204.1008-1-6-3', '1204.1008-2-6-3'], ['1204.1008-1-12-0', '1204.1008-2-12-0'], ['1204.1008-1-12-1', '1204.1008-2-12-1'], ['1204.1008-1-12-2', '1204.1008-2-12-2'], ['1204.1008-1-12-3', '1204.1008-2-12-3'], ['1204.1008-1-12-4', '1204.1008-2-12-4'], ['1204.1008-1-12-5', '1204.1008-2-12-5'], ['1204.1008-1-16-1', '1204.1008-2-16-1'], ['1204.1008-1-16-2', '1204.1008-2-16-2'], ['1204.1008-1-16-4', '1204.1008-2-16-4'], ['1204.1008-1-16-5', '1204.1008-2-16-5'], ['1204.1008-1-16-6', '1204.1008-2-16-6'], ['1204.1008-1-16-7', '1204.1008-2-16-7'], ['1204.1008-1-16-8', '1204.1008-2-16-8'], ['1204.1008-1-16-9', '1204.1008-2-16-9'], ['1204.1008-1-21-0', '1204.1008-2-21-0'], ['1204.1008-1-21-1', '1204.1008-2-21-1'], ['1204.1008-1-21-2', '1204.1008-2-21-2'], ['1204.1008-1-21-4', '1204.1008-2-21-4'], ['1204.1008-1-21-5', '1204.1008-2-21-5'], ['1204.1008-1-21-6', '1204.1008-2-21-6'], ['1204.1008-1-20-1', '1204.1008-2-20-1'], ['1204.1008-1-20-2', '1204.1008-2-20-2'], ['1204.1008-1-20-5', '1204.1008-2-20-5'], ['1204.1008-1-20-6', '1204.1008-2-20-6'], ['1204.1008-1-20-7', '1204.1008-2-20-7'], ['1204.1008-1-29-0', '1204.1008-2-29-0'], ['1204.1008-1-29-1', '1204.1008-2-29-1'], ['1204.1008-1-29-2', '1204.1008-2-29-2'], ['1204.1008-1-29-3', '1204.1008-2-29-3'], ['1204.1008-1-29-4', '1204.1008-2-29-4'], ['1204.1008-1-17-0', '1204.1008-2-17-0'], ['1204.1008-1-28-0', '1204.1008-2-28-0'], ['1204.1008-1-28-1', '1204.1008-2-28-1'], ['1204.1008-1-26-0', '1204.1008-2-26-0'], ['1204.1008-1-26-1', '1204.1008-2-26-1'], ['1204.1008-1-26-2', '1204.1008-2-26-2'], ['1204.1008-1-26-3', '1204.1008-2-26-3'], ['1204.1008-1-3-0', '1204.1008-2-3-0'], ['1204.1008-1-3-1', '1204.1008-2-3-1'], ['1204.1008-1-3-2', '1204.1008-2-3-2'], ['1204.1008-1-3-3', '1204.1008-2-3-3'], ['1204.1008-1-3-4', '1204.1008-2-3-4'], ['1204.1008-1-36-0', '1204.1008-2-37-0'], ['1204.1008-1-36-1', '1204.1008-2-37-1'], ['1204.1008-1-36-2', '1204.1008-2-37-2'], ['1204.1008-1-36-3', '1204.1008-2-37-3'], ['1204.1008-1-13-0', '1204.1008-2-13-0'], ['1204.1008-1-13-1', '1204.1008-2-13-1'], ['1204.1008-1-32-0', '1204.1008-2-33-0'], ['1204.1008-1-32-2', '1204.1008-2-33-2'], ['1204.1008-1-32-3', '1204.1008-2-33-3'], ['1204.1008-1-32-4', '1204.1008-2-33-4'], ['1204.1008-1-32-5', '1204.1008-2-33-5'], ['1204.1008-1-24-0', '1204.1008-2-24-0'], ['1204.1008-1-24-1', '1204.1008-2-24-1'], ['1204.1008-1-24-2', '1204.1008-2-24-2'], ['1204.1008-1-24-3', '1204.1008-2-24-3'], ['1204.1008-1-24-4', '1204.1008-2-24-4'], ['1204.1008-1-39-0', '1204.1008-2-40-0'], ['1204.1008-1-39-1', '1204.1008-2-40-1'], ['1204.1008-1-39-2', '1204.1008-2-40-2'], ['1204.1008-1-39-3', '1204.1008-2-40-3'], ['1204.1008-1-39-4', '1204.1008-2-40-4'], ['1204.1008-1-27-0', '1204.1008-2-27-0'], ['1204.1008-1-27-1', '1204.1008-2-27-1'], ['1204.1008-1-27-2', '1204.1008-2-27-2'], ['1204.1008-1-27-3', '1204.1008-2-27-3'], ['1204.1008-1-27-4', '1204.1008-2-27-4'], ['1204.1008-1-27-5', '1204.1008-2-27-5'], ['1204.1008-1-27-6', '1204.1008-2-27-6'], ['1204.1008-1-27-7', '1204.1008-2-27-7'], ['1204.1008-1-27-8', '1204.1008-2-27-8'], ['1204.1008-1-27-9', '1204.1008-2-27-9'], ['1204.1008-1-27-10', '1204.1008-2-27-10'], ['1204.1008-1-27-11', '1204.1008-2-27-11'], ['1204.1008-1-27-12', '1204.1008-2-27-12'], ['1204.1008-1-27-13', '1204.1008-2-27-13'], ['1204.1008-1-27-14', '1204.1008-2-27-14'], ['1204.1008-1-33-0', '1204.1008-2-34-0'], ['1204.1008-1-33-1', '1204.1008-2-34-1'], ['1204.1008-1-33-2', '1204.1008-2-34-2'], ['1204.1008-1-19-0', '1204.1008-2-19-0'], ['1204.1008-1-19-1', '1204.1008-2-19-1'], ['1204.1008-1-19-2', '1204.1008-2-19-2'], ['1204.1008-1-19-3', '1204.1008-2-19-3'], ['1204.1008-1-19-5', '1204.1008-2-19-5'], ['1204.1008-1-19-6', '1204.1008-2-19-6'], ['1204.1008-1-19-7', '1204.1008-2-19-7'], ['1204.1008-1-19-8', '1204.1008-2-19-8'], ['1204.1008-1-19-9', '1204.1008-2-19-9'], ['1204.1008-1-19-10', '1204.1008-2-19-10'], ['1204.1008-1-19-11', '1204.1008-2-19-11'], ['1204.1008-1-19-12', '1204.1008-2-19-12'], ['1204.1008-1-19-13', '1204.1008-2-19-13'], ['1204.1008-1-37-0', '1204.1008-2-38-0'], ['1204.1008-1-37-1', '1204.1008-2-38-1'], ['1204.1008-1-37-2', '1204.1008-2-38-2'], ['1204.1008-1-22-1', '1204.1008-2-22-1'], ['1204.1008-1-22-2', '1204.1008-2-22-2'], ['1204.1008-1-22-3', '1204.1008-2-22-3'], ['1204.1008-1-22-4', '1204.1008-2-22-4'], ['1204.1008-1-22-5', '1204.1008-2-22-5'], ['1204.1008-1-22-6', '1204.1008-2-22-6'], ['1204.1008-1-22-7', '1204.1008-2-22-7'], ['1204.1008-1-22-8', '1204.1008-2-22-8'], ['1204.1008-1-5-0', '1204.1008-2-5-0'], ['1204.1008-1-5-1', '1204.1008-2-5-1'], ['1204.1008-1-5-2', '1204.1008-2-5-2'], ['1204.1008-1-5-3', '1204.1008-2-5-3'], ['1204.1008-1-35-0', '1204.1008-2-36-0'], ['1204.1008-1-35-1', '1204.1008-2-36-1'], ['1204.1008-1-35-2', '1204.1008-2-36-2'], ['1204.1008-1-38-1', '1204.1008-2-39-1'], ['1204.1008-1-38-2', '1204.1008-2-39-2'], ['1204.1008-1-38-3', '1204.1008-2-39-3'], ['1204.1008-1-38-4', '1204.1008-2-39-4'], ['1204.1008-1-38-5', '1204.1008-2-39-5'], ['1204.1008-1-31-2', '1204.1008-2-32-0'], ['1204.1008-1-31-3', '1204.1008-2-32-1']]

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[]

[['1204.1008-1-31-0', '1204.1008-2-31-0']]

[]

['1204.1008-1-18-1', '1204.1008-1-25-0', '1204.1008-2-18-1', '1204.1008-2-25-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1204.1008

1712.09313

{'1712.09313-1-0-0': "Generalizing Weyl's tube formula and building on Chern's work, Alesker constructed a canonical family of valuations (finitely-additive measures with good analytic properties) on any Riemannian manifold, the so-called Lipschitz-Killing algebra, which is universal with respect to isometric embeddings.", '1712.09313-1-0-1': 'In this note, we uncover a similar structure for contact manifolds.', '1712.09313-1-0-2': 'Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings.', '1712.09313-1-0-3': 'Those valuations assign numerical invariants to even-dimensional submanifolds, which in a certain sense measure the curvature at points of tangency to the contact structure.', '1712.09313-1-0-4': 'Moreover, these valuations generalize to the class of manifolds equipped with the structure of a Heisenberg algebra on their cotangent bundle.', '1712.09313-1-0-5': 'Pursuing the analogy with Euclidean integral geometry, we construct symplectic-invariant distributions on Grassmannians to produce Crofton formulas on the contact sphere.', '1712.09313-1-0-6': 'Using closely related distributions, we obtain Crofton formulas also in linear symplectic space.', '1712.09313-1-1-0': '# Introduction', '1712.09313-1-2-0': '## Background and motivation', '1712.09313-1-3-0': "Intrinsic volumes first appeared in convex geometry through Steiner's formula: given a compact convex body [MATH], [MATH] where [MATH] is unit Euclidean ball in [MATH] and [MATH] is its volume.", '1712.09313-1-3-1': 'The coefficient [MATH], the [MATH]-th intrinsic volume, can be written explicitly for smooth [MATH] as [MATH], where [MATH] are the principal curvatures of [MATH].', '1712.09313-1-3-2': 'Alternatively, [MATH] can be given by integral-geometric Crofton formulas: [MATH] where [MATH] is the rigid motion invariant measure on the affine Grassmannian, and [MATH] is the Euler characteristic.', '1712.09313-1-3-3': 'A third, axiomatic definition, was given by Hadwiger, who described the intrinsic volumes as the unique rigid-motion invariant continuous finitely additive measures on compact convex sets.', '1712.09313-1-4-0': "A closely related famous result is Weyl's tube formula [CITATION].", '1712.09313-1-4-1': 'It asserts that the volume of an [MATH]-tube around a Riemannian manifold [MATH] embedded isometrically in Euclidean space [MATH] is a polynomial in [MATH], whose coefficients are, remarkably, intrinsic invariants of the Riemannian manifold [MATH], independent of the isometric embedding.', '1712.09313-1-4-2': 'These coefficients, now known as the intrinsic volumes of [MATH], are intimately linked with the asymptotic expansion of the heat kernel, see [CITATION].', '1712.09313-1-5-0': 'These results fall naturally in the domain of valuations on manifolds, a fairly young branch of valuation theory introduced by Alesker et al. in a sequence of works [CITATION], see also [CITATION] for a survey.', '1712.09313-1-5-1': "Valuation theory itself is a mixture of convex and integral geometry, originating in the early 20th century in works of Steiner, Blaschke, Chern and Santalo, as well as in Dehn's solution to Hilbert's third problem.", '1712.09313-1-5-2': 'Generally speaking, valuations are finitely additive measures on some family of nice subsets.', '1712.09313-1-5-3': 'In this note, there is typically some analytic restriction on the nature of the valuation, such as smoothness or smoothness with singularities, and the subsets are manifolds with corners or differentiable polyhedra.', '1712.09313-1-6-0': "Building on results of Chern, Alesker noticed a natural extension of Weyl's theorem: restricting to [MATH] the intrinsic volumes of [MATH], (considered as valuations), yields an intrinsically defined family of valuations on [MATH], now known as the Lipschitz-Killing valuations.", '1712.09313-1-6-1': "Weyl's intrinsic volumes of [MATH] are then the integrals of the corresponding valuations.", '1712.09313-1-6-2': 'In a recent work, Fu and Wannerer [CITATION] characterized the Lipschitz-Killing valuations as the unique family of valuations attached canonically (in a sense made precise therein) to arbitrary Riemannian manifolds that are universal to isometric embeddings.', '1712.09313-1-7-0': 'Numerous intriguing connections between convex and symplectic geometries are known to exist.', '1712.09313-1-7-1': "To name a few: Viterbo's conjectured isoperimetric inequality for capacities of convex bodies [CITATION], was later shown by Artstein-Avidan, Karasev and Ostrover to imply Mahler's famous conjecture [CITATION].", '1712.09313-1-7-2': 'Capacities have been successfully studied up to a bounded factor using convex techniques, see [CITATION] and [CITATION].', '1712.09313-1-7-3': "In [CITATION], links are established between systolic geometry, contact geometry, Mahler's conjecture and the geometry of numbers.", '1712.09313-1-7-4': "Schaffer's dual girth conjecture for normed spaces has been proved using symplectic techniques [CITATION], and generalized further in [CITATION] using hamiltonian group actions.", '1712.09313-1-7-5': 'In very recent works of Abbondandolo et al. [CITATION], some links are established between the geometry of the group of symplectomorphisms and systolic geometry of the 2-sphere.', '1712.09313-1-7-6': 'For an exposition of some of those connections, see [CITATION].', '1712.09313-1-8-0': "The main objective for this work is to further explore the convex-symplectic link by studying the valuation theory of contact manifolds, using the Riemannian case and Weyl's principle as guides.", '1712.09313-1-9-0': '## Informal summary', '1712.09313-1-10-0': 'We find that like in the Riemannian setting, contact manifolds possess a canonical family of valuations associated to them.', '1712.09313-1-10-1': 'We describe those valuations in two ways: geometrically through a curvature-type formula, and also dynamically through the invariants of a certain vector field at its singular points.', '1712.09313-1-11-0': "The contact valuations satisfy Weyl's principle of universality under embeddings, similarly to the Alesker-Chern extension of the Weyl principle.", '1712.09313-1-11-1': 'Let us emphasize the role played by valuations in this phenomenon: It so happens that contact valuations only assume non-zero values on even dimensional submanifolds, while they live on odd-dimensional contact manifolds.', '1712.09313-1-11-2': "Thus unlike the Riemannian case, Weyl's principle in the contact setting is only manifested in its extended to valuations form, as the statement in the original form becomes vacuous: all integrals of the contact valuations vanish.", '1712.09313-1-12-0': 'Contact valuations in fact extend to the larger class of manifolds whose cotangent spaces admit a smoothly varying structure of Heisenberg algebras.', '1712.09313-1-12-1': 'The Heisenberg algebra provides a unifying link between contact, symplectic and metric geometries.', '1712.09313-1-13-0': 'This path leads us to consider the valuation theory of the dual Heisenberg algebra, invariant under the group of the automorphisms of the Heisenberg algebra, which is closely related to the symplectic group.', '1712.09313-1-13-1': 'From this perspective, this is another step in the study of the valuation theory of non-compact Lie groups, which up to now has only been considered for the indefinite orthogonal group in [CITATION].', '1712.09313-1-14-0': 'Further similarity to the metric setting is exhibited by the contact sphere, where we prove Crofton formulas and a Hadwiger-type theorem, thus establishing an integral-geometric and an axiomatic description of the contact valuations.', '1712.09313-1-15-0': 'Finally in the last part we explore the valuation theory of linear symplectic spaces.', '1712.09313-1-15-1': 'We show that there are no non-trivial invariant valuations, but nevertheless one can write oriented Crofton formulas for the symplectic volume of submanifolds.', '1712.09313-1-16-0': '## Main results', '1712.09313-1-17-0': 'Let us very briefly recall or indicate the relevant notions.', '1712.09313-1-17-1': 'For precise definitions see section [REF].', '1712.09313-1-18-0': 'A contact manifold [MATH] is given by a maximally non-integrable hyperplane distribution, namely a smooth field of tangent hyperplanes [MATH] s.t. locally one can find [MATH] with [MATH] and [MATH] a non-degenerate 2-form.', '1712.09313-1-19-0': 'A smooth valuation [MATH] on an orientable manifold [MATH], denoted [MATH], is a finitely additive measure on the compact differentiable polyhedra of [MATH], denoted [MATH], which has the form [MATH] for some forms [MATH] and [MATH].', '1712.09313-1-19-1': 'Here [MATH] is the cosphere bundle, and [MATH] is the conormal cycle of [MATH], which is just the conormal bundle when [MATH] is a manifold.', '1712.09313-1-19-2': 'Orientability is not essential, and is only assumed to simplify the exposition.', '1712.09313-1-20-0': 'There is a natural filtration [MATH].', '1712.09313-1-20-1': 'Very roughly speaking, [MATH] consists of valuations which are locally homogeneous of degree at least [MATH], for example [MATH] are the smooth measures on [MATH].', '1712.09313-1-21-0': 'The generalized valuations [MATH] are, roughly speaking, distributional valuations: we allow [MATH] and [MATH] to be currents rather than smooth forms.', '1712.09313-1-21-1': 'Generalized valuations can be naturally evaluated on sufficiently nice subsets [MATH].', '1712.09313-1-21-2': 'The filtration [MATH] on [MATH] extends to a filtration [MATH] on [MATH].', '1712.09313-1-22-0': 'To any contact manifold [MATH] with contact distribution [MATH] there are canonically associated, linearly independent generalized valuations [MATH], [MATH].', '1712.09313-1-22-1': 'They have the following properties:', '1712.09313-1-23-0': '[MATH] is the Euler characteristic, and [MATH].', '1712.09313-1-23-1': '[MATH] can be naturally evaluated on submanifolds in general position relative to the contact structure (see Definition [REF]).', '1712.09313-1-23-2': 'For such a hypersurface [MATH],', '1712.09313-1-24-0': '[EQUATION] where the local contact areas [MATH] only depends on the germ of [MATH] at [MATH].', '1712.09313-1-24-1': 'They are described explicitly below in equations [REF] or [REF].', '1712.09313-1-25-0': 'Universality to restriction under embedding: if [MATH] is a contact embedding, then for [MATH] one has [MATH].', '1712.09313-1-25-1': 'For a [MATH]-dimensional submanifold in general position [MATH], [MATH], with equality if and only if there are no contact tangent points.', '1712.09313-1-25-2': 'The space of generalized valuations on [MATH] invariant under all contactomorphisms of [MATH] is spanned by [MATH].', '1712.09313-1-26-0': 'The universality with respect to embeddings is sometimes referred to as the Weyl principle.', '1712.09313-1-26-1': 'Thus we recover a Weyl principle in the contact setting.', '1712.09313-1-27-0': 'The local contact areas [MATH] can be given explicitly in two different ways, through a geometric or a dynamical approach.', '1712.09313-1-28-0': 'We may regularize [MATH] to a non-negative lower semicontinuos functional on all [MATH]-dimensional submanifolds, denoted [MATH], the contact area of [MATH].', '1712.09313-1-28-1': 'We observe that [MATH] if and only if [MATH] can be made nowhere tangent to the contact distribution by an arbitrarily small perturbation.', '1712.09313-1-28-2': 'In Riemannian or Hermitian manifolds, a fair amount of the valuation theory appears already in the corresponding flat space, which can be thought of as the tangent space to the given manifold.', '1712.09313-1-28-3': 'In the contact setting, it is no longer true: the tangent space of a contact manifold does not itself inherit a contact structure.', '1712.09313-1-29-0': 'The main observation guiding this paper is that every cotangent space of a contact manifold is canonically the Heisenberg Lie algebra.', '1712.09313-1-29-1': 'We are thus led to study the valuation theory of general manifolds with such structure.', '1712.09313-1-30-0': 'A manifold [MATH] equipped with a hyperplane distribution [MATH] (called horizontal) and a smooth field of nowhere-degenerate forms [MATH] will be called a dual Heisenberg (DH) manifold.', '1712.09313-1-31-0': 'The space of valuations naturally associated to such manifolds turns out to resemble somewhat the Lipschitz-Killing space of valuations in Riemannian geometry, in particular, they exhibit universality with respect to embeddings.', '1712.09313-1-32-0': 'Theorem [REF] is then the contact instance of the following general result.', '1712.09313-1-33-0': 'To any DH manifold [MATH] with horizontal distribution [MATH] there are canonically associated, generalized valuations [MATH], [MATH].', '1712.09313-1-33-1': 'They have the following properties:', '1712.09313-1-34-0': '[MATH] is the Euler characteristic, and [MATH].', '1712.09313-1-35-0': '[MATH] can be naturally evaluated on submanifolds in general position with respect to the horizontal distribution.', '1712.09313-1-35-1': 'For such a hypersurface [MATH],', '1712.09313-1-36-0': '[EQUATION] where [MATH] only depends on the germ of [MATH] at [MATH].', '1712.09313-1-36-1': 'If [MATH] is a DH embedding, then for [MATH] one has [MATH].', '1712.09313-1-37-0': 'The local contact areas [MATH] are given by the same curvature-type formula [REF] as in the contact case.', '1712.09313-1-38-0': 'Unlike the contact case, we do not have a uniqueness result, since the group of symmetries of a DH manifold is in general trivial.', '1712.09313-1-38-1': 'We conjecture that uniqueness does arise once universality to embeddings is mandated, similarly to the Riemannian setting considered by Fu-Wannerer [CITATION].', '1712.09313-1-39-0': 'As an intermediate step of independent interest, we obtain a Hadwiger-type theorem for the dual Heisenberg algebra itself.', '1712.09313-1-39-1': 'We denote by [MATH] the dual of the Heisenberg Lie algebra [MATH], and by [MATH] its automorphism group.', '1712.09313-1-40-0': 'It holds that [MATH] consists of even valuations.', '1712.09313-1-40-1': 'For [MATH], [MATH] while [MATH].', '1712.09313-1-40-2': 'The corresponding Klain sections are zero-order distributions (that is, regular Borel measures).', '1712.09313-1-41-0': 'We also consider [MATH], the connected component of the identity.', '1712.09313-1-41-1': 'We prove', '1712.09313-1-42-0': 'For [MATH], [MATH] while [MATH].', '1712.09313-1-42-1': 'In the latter space, the [MATH]-invariant valuations are complemented by a one-dimensional space of odd valuations.', '1712.09313-1-43-0': 'We then consider the standard contact sphere [MATH], which we identify with the oriented projectivization [MATH] of a symplectic space [MATH].', '1712.09313-1-43-1': 'We construct a canonical distribution [MATH].', '1712.09313-1-43-2': 'We obtain the following Crofton formulas, establishing further common grounds with the Riemannian setting.', '1712.09313-1-44-0': 'Define for [MATH] the generalized valuations [MATH] on [MATH] given by the Crofton formula [EQUATION].', '1712.09313-1-44-1': 'Then for certain explicit constans [MATH] one has', '1712.09313-1-45-0': '[EQUATION]', '1712.09313-1-45-1': 'We also establish a Hadwiger-type theorem for the contact sphere.', '1712.09313-1-46-0': 'Both [MATH] and [MATH] are bases of [MATH].', '1712.09313-1-47-0': 'Finally, we find that while symplectic space and manifolds do not possess interesting invariant valuations, one can nevertheless write certain integral geometric formulas for symplectic volumes of manifolds.', '1712.09313-1-47-1': 'We construct a canonical distribution on the affine oriented Grassmannian [MATH] which is translation- and [MATH]-invariant, and odd to orientation reversal.', '1712.09313-1-47-2': 'We prove the following Crofton formula on symplectic linear space.', '1712.09313-1-48-0': 'Let [MATH] be a [MATH] compact, oriented submanifold with boundary.', '1712.09313-1-48-1': 'Then [EQUATION] where [MATH] and [MATH] is the oriented intersection index.', '1712.09313-1-49-0': '## Plan of the paper', '1712.09313-1-50-0': 'In section [REF] we introduce notation and present the basic geometric facts we will use.', '1712.09313-1-50-1': 'In section [REF] we recall the basics of valuation theory and prove some lemmas we will need.', '1712.09313-1-50-2': 'In section [REF] we construct the canonical valuations on general DH manifolds and establish their universality to embeddings, proving Theorem [REF].', '1712.09313-1-50-3': 'We also explore some geometric properties of those valuations.', '1712.09313-1-50-4': 'In section [REF] we classify the translation-invariant valuations of the dual Heisenberg algebra, proving Theorem [REF], and note their relation to gaussian curvature.', '1712.09313-1-50-5': 'Apart from its intrinsic interest, the linear classification is needed for the last (uniqueness) statement in Theorem [REF].', '1712.09313-1-50-6': 'In section [REF] we specialize the DH valuations to contact manifolds and prove Theorem [REF].', '1712.09313-1-50-7': 'In particular, we give the dynamical description of the contact valuations.In section [REF] we construct symplectic-invariant distributions on linear and affine Grassmannians in symplectic space, which are used in the following two sections.', '1712.09313-1-50-8': 'In section [REF] we consider the standard contact sphere.', '1712.09313-1-50-9': 'We produce Crofton formulas for [MATH] that are invariant under [MATH], proving theorems [REF] and [REF], and we compute some examples explicitly.', '1712.09313-1-50-10': 'We also bound from below the contact area of a convex set.', '1712.09313-1-50-11': 'Finally in section [REF], we consider the integral geometry of linear symplectic space and prove Theorem [REF]'}

{'1712.09313-2-0-0': "Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings.", '1712.09313-2-0-1': 'In this note, we uncover a similar structure for contact manifolds.', '1712.09313-2-0-2': 'Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings.', '1712.09313-2-0-3': 'Those valuations assign numerical invariants to even-dimensional submanifolds, which in a certain sense measure the curvature at points of tangency to the contact structure.', '1712.09313-2-0-4': 'Moreover, these valuations generalize to the class of manifolds equipped with the structure of a Heisenberg algebra on their cotangent bundle.', '1712.09313-2-0-5': 'Pursuing the analogy with Euclidean integral geometry, we construct symplectic-invariant distributions on Grassmannians to produce Crofton formulas on the contact sphere.', '1712.09313-2-0-6': 'Using closely related distributions, we obtain Crofton formulas also in the linear symplectic space.', '1712.09313-2-1-0': '# Introduction', '1712.09313-2-2-0': '## Background and motivation', '1712.09313-2-3-0': "Intrinsic volumes first appeared in convex geometry through Steiner's formula: given a compact convex body [MATH], [MATH] where [MATH] is unit Euclidean ball in [MATH] and [MATH] is its volume.", '1712.09313-2-3-1': 'The coefficient [MATH], the [MATH]-th intrinsic volume, can be written explicitly for smooth [MATH] as [MATH], where [MATH] are the principal curvatures of [MATH].', '1712.09313-2-3-2': 'Alternatively, [MATH] can be given by integral-geometric Crofton formulas: [MATH] where [MATH] is the rigid motion invariant measure on the affine Grassmannian, and [MATH] is the Euler characteristic.', '1712.09313-2-3-3': 'A third, axiomatic definition, was given by Hadwiger, who described the intrinsic volumes as the unique rigid-motion invariant continuous finitely additive measures on compact convex sets.', '1712.09313-2-4-0': "A closely related famous result is Weyl's tube formula [CITATION].", '1712.09313-2-4-1': 'It asserts that the volume of an [MATH]-tube around a Riemannian manifold [MATH] embedded isometrically in Euclidean space [MATH] is a polynomial in [MATH], whose coefficients are, remarkably, intrinsic invariants of the Riemannian manifold [MATH], independent of the isometric embedding.', '1712.09313-2-4-2': 'These coefficients, now known as the intrinsic volumes of [MATH], are intimately linked with the asymptotic expansion of the heat kernel, see [CITATION].', '1712.09313-2-5-0': 'These results fall naturally in the domain of valuations on manifolds, a fairly young branch of valuation theory introduced by Alesker et al. in a sequence of works [CITATION], see also [CITATION] for a survey.', '1712.09313-2-5-1': "Valuation theory itself is a mixture of convex and integral geometry, originating in the early 20th century in works of Steiner, Blaschke, Chern and Santalo, as well as in Dehn's solution to Hilbert's third problem.", '1712.09313-2-5-2': 'Generally speaking, valuations are finitely additive measures on some family of nice subsets.', '1712.09313-2-5-3': 'In this note, there is typically some analytic restriction on the nature of the valuation, such as smoothness or smoothness with singularities, and the subsets are manifolds with corners or differentiable polyhedra.', '1712.09313-2-6-0': "Building on results of Chern, Alesker noticed a natural extension of Weyl's theorem for valuations: restricting to [MATH] the intrinsic volumes of [MATH], (considered as valuations), yields an intrinsically defined family of valuations on [MATH], now known as the Lipschitz-Killing valuations.", '1712.09313-2-6-1': "Weyl's intrinsic volumes of [MATH] are then the integrals of the corresponding valuations.", '1712.09313-2-6-2': 'In a recent work, Fu and Wannerer [CITATION] characterized the Lipschitz-Killing valuations as the unique family of valuations attached canonically (in a sense made precise therein) to arbitrary Riemannian manifolds that are universal to isometric embeddings.', '1712.09313-2-7-0': 'Other spaces whose (smooth) valuation theories were considered in recent years include complex space forms [CITATION], the quaternionic plane [CITATION], the octonionic plane [CITATION] and exceptional spheres [CITATION].', '1712.09313-2-8-0': 'Numerous intriguing connections between convex and symplectic geometries are known to exist.', '1712.09313-2-8-1': "To name a few: Viterbo's conjectured isoperimetric inequality for capacities of convex bodies [CITATION], was later shown by Artstein-Avidan, Karasev and Ostrover to imply Mahler's famous conjecture [CITATION].", '1712.09313-2-8-2': 'Capacities have been successfully studied up to a bounded factor using convex techniques, see [CITATION] and [CITATION].', '1712.09313-2-8-3': "In [CITATION], links are established between systolic geometry, contact geometry, Mahler's conjecture and the geometry of numbers.", '1712.09313-2-8-4': "Schaffer's dual girth conjecture for normed spaces has been proved using symplectic techniques [CITATION], and generalized further in [CITATION] using hamiltonian group actions.", '1712.09313-2-8-5': 'In very recent works of Abbondandolo et al. [CITATION], some links are established between the geometry of the group of symplectomorphisms and systolic geometry of the 2-sphere.', '1712.09313-2-8-6': 'For an exposition of some of those connections, see [CITATION].', '1712.09313-2-9-0': "The main objective for this work is to further explore the convex-symplectic link by studying the valuation theory of contact manifolds, using the Riemannian case and Weyl's principle as guides.", '1712.09313-2-10-0': '## Informal summary', '1712.09313-2-11-0': 'We find that like in the Riemannian setting, contact manifolds possess a canonical family of valuations associated to them.', '1712.09313-2-11-1': 'We describe those valuations in two ways: geometrically through a curvature-type formula, and also dynamically through the invariants of a certain vector field at its singular points.', '1712.09313-2-12-0': "The contact valuations satisfy Weyl's principle of universality under embeddings, similarly to the valuation extension of the Weyl principle.", '1712.09313-2-12-1': 'Let us emphasize the role played by valuations in this phenomenon: It so happens that contact valuations only assume non-zero values on even dimensional submanifolds, while they live on odd-dimensional contact manifolds.', '1712.09313-2-12-2': "Thus unlike the Riemannian case, Weyl's principle in the contact setting is only manifested in its extended to valuations form, as the statement in the original form becomes vacuous: all integrals of the contact valuations vanish.", '1712.09313-2-13-0': 'Contact valuations in fact extend to the larger class of manifolds whose cotangent spaces admit a smoothly varying structure of Heisenberg algebras.', '1712.09313-2-13-1': 'The Heisenberg algebra provides a unifying link between contact, symplectic and metric geometries.', '1712.09313-2-14-0': 'This path leads us to consider the valuation theory of the dual Heisenberg algebra, invariant under the group of the automorphisms of the Heisenberg algebra, which is closely related to the symplectic group.', '1712.09313-2-14-1': 'From this perspective, this is another step in the study of the valuation theory of non-compact Lie groups, which up to now has only been considered for the indefinite orthogonal group in [CITATION].', '1712.09313-2-15-0': 'Further similarity to the metric setting is exhibited by the contact sphere, where we prove Crofton formulas and a Hadwiger-type theorem, thus establishing an integral-geometric and an axiomatic description of the contact valuations.', '1712.09313-2-16-0': 'Finally in the last part we explore the valuation theory of linear symplectic spaces.', '1712.09313-2-16-1': 'We show that there are no non-trivial invariant valuations, but nevertheless one can write oriented Crofton formulas for the symplectic volume of submanifolds.', '1712.09313-2-17-0': '## Main results', '1712.09313-2-18-0': 'Let us very briefly recall or indicate the relevant notions.', '1712.09313-2-18-1': 'For precise definitions see sections [REF] and [REF].', '1712.09313-2-19-0': 'A contact manifold [MATH] is given by a maximally non-integrable hyperplane distribution, namely a smooth field of tangent hyperplanes [MATH] s.t. locally one can find [MATH] with [MATH] and [MATH] a non-degenerate 2-form.', '1712.09313-2-20-0': 'A smooth valuation [MATH] on an orientable manifold [MATH], denoted [MATH], is a finitely additive measure on the compact differentiable polyhedra of [MATH], denoted [MATH], which has the form [MATH] for some forms [MATH] and [MATH].', '1712.09313-2-20-1': 'Here [MATH] is the cosphere bundle, and [MATH] is the conormal cycle of [MATH], which is just the conormal bundle when [MATH] is a manifold.', '1712.09313-2-20-2': 'Orientability is not essential, and is only assumed to simplify the exposition.', '1712.09313-2-21-0': 'There is a natural filtration [MATH].', '1712.09313-2-21-1': 'Very roughly speaking, [MATH] consists of valuations which are locally homogeneous of degree at least [MATH], for example [MATH] are the smooth measures on [MATH].', '1712.09313-2-22-0': 'The generalized valuations [MATH] are, roughly speaking, distributional valuations: we allow [MATH] and [MATH] to be currents rather than smooth forms.', '1712.09313-2-22-1': 'Generalized valuations can be naturally evaluated on sufficiently nice subsets [MATH].', '1712.09313-2-22-2': 'The filtration [MATH] on [MATH] extends to a filtration [MATH] on [MATH].', '1712.09313-2-23-0': 'To any contact manifold [MATH] with contact distribution [MATH] there are canonically associated, linearly independent generalized valuations [MATH], [MATH].', '1712.09313-2-23-1': 'They have the following properties:', '1712.09313-2-24-0': '[MATH] is the Euler characteristic, and [MATH].', '1712.09313-2-24-1': '[MATH] can be naturally evaluated on submanifolds in generic position relative to the contact structure (see Definition [REF]).', '1712.09313-2-24-2': 'For a generic closed hypersurface [MATH],', '1712.09313-2-25-0': '[EQUATION] where the local contact areas [MATH] only depends on the germ of [MATH] at [MATH].', '1712.09313-2-25-1': 'They are described explicitly below in equations [REF] or [REF].', '1712.09313-2-26-0': 'Universality to restriction under embedding: if [MATH] is a contact embedding, then for [MATH] one has [MATH].', '1712.09313-2-26-1': 'For a [MATH]-dimensional submanifold in general position [MATH], [MATH], with equality if and only if there are no contact tangent points.', '1712.09313-2-26-2': 'The space of generalized valuations on [MATH] invariant under all contactomorphisms of [MATH] is spanned by [MATH].', '1712.09313-2-27-0': 'The universality with respect to embeddings is sometimes referred to as the Weyl principle.', '1712.09313-2-27-1': 'Thus we recover a Weyl principle in the contact setting.', '1712.09313-2-28-0': 'The local contact areas [MATH] can be given explicitly in two different ways, through a geometric or a dynamical approach.', '1712.09313-2-29-0': 'We may extend [MATH] to a non-negative lower semicontinuos functional on all [MATH]-dimensional submanifolds with boundary, denoted [MATH], the contact area of [MATH].', '1712.09313-2-29-1': 'We observe that [MATH] if and only if [MATH] can be made nowhere tangent to the contact distribution by an arbitrarily small perturbation.', '1712.09313-2-29-2': 'In Riemannian or Hermitian manifolds, a fair amount of the valuation theory appears already in the corresponding flat space, which can be thought of as the tangent space to the given manifold.', '1712.09313-2-29-3': 'In the contact setting, it is no longer true: the tangent space of a contact manifold does not itself inherit a contact structure.', '1712.09313-2-30-0': 'The main observation guiding this paper is that every cotangent space of a contact manifold is canonically the Heisenberg Lie algebra.', '1712.09313-2-30-1': 'We are thus led to study the valuation theory of general manifolds with such structure.', '1712.09313-2-31-0': 'A manifold [MATH] equipped with a hyperplane distribution [MATH] (called horizontal) and a smooth field of nowhere-degenerate forms [MATH] will be called a dual Heisenberg (DH) manifold.', '1712.09313-2-32-0': 'The space of valuations naturally associated to such manifolds turns out to resemble somewhat the Lipschitz-Killing space of valuations in Riemannian geometry, in particular, they exhibit universality with respect to embeddings.', '1712.09313-2-33-0': 'Theorem [REF] is then the contact instance of the following general result.', '1712.09313-2-34-0': 'To any DH manifold [MATH] with horizontal distribution [MATH] there are canonically associated, generalized valuations [MATH], [MATH].', '1712.09313-2-34-1': 'They have the following properties:', '1712.09313-2-35-0': '[MATH] is the Euler characteristic, and [MATH].', '1712.09313-2-36-0': '[MATH] can be naturally evaluated on submanifolds in generic position with respect to the horizontal distribution.', '1712.09313-2-36-1': 'For a generic closed hypersurface [MATH],', '1712.09313-2-37-0': '[EQUATION] where [MATH] only depends on the germ of [MATH] at [MATH].', '1712.09313-2-37-1': 'If [MATH] is a DH embedding, then for [MATH] one has [MATH].', '1712.09313-2-38-0': 'The local contact areas [MATH] are given by the same curvature-type formula [REF] as in the contact case.', '1712.09313-2-39-0': 'Unlike the contact case, we do not have a uniqueness result, since the group of symmetries of a DH manifold is in general trivial.', '1712.09313-2-39-1': 'We conjecture that uniqueness does arise once universality to embeddings is mandated, similarly to the Riemannian setting considered by Fu-Wannerer [CITATION].', '1712.09313-2-40-0': 'As an intermediate step of independent interest, we obtain a Hadwiger-type theorem for the dual Heisenberg algebra itself.', '1712.09313-2-40-1': 'We denote by [MATH] the dual of the Heisenberg Lie algebra [MATH], and by [MATH] its automorphism group.', '1712.09313-2-41-0': 'It holds that [MATH] consists of even valuations.', '1712.09313-2-41-1': 'For [MATH], [MATH] while [MATH].', '1712.09313-2-41-2': 'The corresponding Klain sections are zero-order distributions (that is, regular Borel measures).', '1712.09313-2-42-0': 'We also consider [MATH], the connected component of the identity.', '1712.09313-2-42-1': 'We prove', '1712.09313-2-43-0': 'For [MATH], [MATH] while [MATH].', '1712.09313-2-43-1': 'In the latter space, the [MATH]-invariant valuations are complemented by a one-dimensional space of odd valuations.', '1712.09313-2-44-0': 'We then consider the standard contact sphere [MATH], which we identify with the oriented projectivization [MATH] of a symplectic space [MATH].', '1712.09313-2-44-1': 'We construct a canonical distribution [MATH].', '1712.09313-2-44-2': 'We obtain the following Crofton formulas, establishing further common grounds with the Riemannian setting.', '1712.09313-2-45-0': 'Define for [MATH] the generalized valuations [MATH] on [MATH] given by the Crofton formula [EQUATION].', '1712.09313-2-45-1': 'Then for certain explicit constans [MATH] one has', '1712.09313-2-46-0': '[EQUATION]', '1712.09313-2-46-1': 'We also establish a Hadwiger-type theorem for the contact sphere.', '1712.09313-2-47-0': 'Both [MATH] and [MATH] are bases of [MATH].', '1712.09313-2-48-0': 'Finally, we find that while symplectic space and manifolds do not possess interesting invariant valuations, one can nevertheless write certain integral geometric formulas for symplectic volumes of manifolds.', '1712.09313-2-48-1': 'We construct a canonical distribution on the affine oriented Grassmannian [MATH] which is translation- and [MATH]-invariant, and odd to orientation reversal.', '1712.09313-2-48-2': 'We prove the following Crofton formula on symplectic linear space.', '1712.09313-2-49-0': 'Let [MATH] be a [MATH] compact, oriented submanifold with boundary.', '1712.09313-2-49-1': 'Then [EQUATION] where [MATH] and [MATH] is the oriented intersection index.', '1712.09313-2-50-0': '## Plan of the paper', '1712.09313-2-51-0': 'In section [REF] we introduce notation and present the basic geometric facts we will use.', '1712.09313-2-51-1': 'In section [REF] we recall the basics of valuation theory and prove some lemmas we will need.', '1712.09313-2-51-2': 'In section [REF] we construct the canonical valuations on general DH manifolds and establish their universality to embeddings, proving Theorem [REF].', '1712.09313-2-51-3': 'We also explore some geometric properties of those valuations.', '1712.09313-2-51-4': 'In section [REF] we classify the translation-invariant valuations of the dual Heisenberg algebra, proving Theorems [REF] and [REF], and note their relation to gaussian curvature.', '1712.09313-2-51-5': 'Apart from its intrinsic interest, the linear classification is needed for the uniqueness statement in Theorem [REF], as well as for Theorem [REF].', '1712.09313-2-51-6': 'In section [REF] we specialize the DH valuations to contact manifolds and prove Theorem [REF].', '1712.09313-2-51-7': 'In particular, we give the dynamical description of the contact valuations.', '1712.09313-2-51-8': 'In section [REF] we construct symplectic-invariant distributions on linear and affine Grassmannians in symplectic space, which are used in the subsequent two sections.', '1712.09313-2-51-9': 'In section [REF] we consider the standard contact sphere.', '1712.09313-2-51-10': 'We produce Crofton formulas for [MATH] that are invariant under [MATH], proving theorems [REF] and [REF], and compute some examples explicitly.', '1712.09313-2-51-11': 'We also bound from below the contact area of a convex set.', '1712.09313-2-51-12': 'Finally in section [REF], we consider the integral geometry of linear symplectic space and prove Theorem [REF].'}

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'1712.09313-3-16-0'], ['1712.09313-2-16-1', '1712.09313-3-16-1'], ['1712.09313-2-9-0', '1712.09313-3-9-0'], ['1712.09313-2-15-0', '1712.09313-3-15-0'], ['1712.09313-2-31-0', '1712.09313-3-31-0'], ['1712.09313-2-51-0', '1712.09313-3-50-0'], ['1712.09313-2-51-1', '1712.09313-3-50-1'], ['1712.09313-2-51-2', '1712.09313-3-50-2'], ['1712.09313-2-51-3', '1712.09313-3-50-3'], ['1712.09313-2-51-4', '1712.09313-3-50-4'], ['1712.09313-2-51-5', '1712.09313-3-50-5'], ['1712.09313-2-51-6', '1712.09313-3-50-6'], ['1712.09313-2-51-7', '1712.09313-3-50-7'], ['1712.09313-2-51-8', '1712.09313-3-50-8'], ['1712.09313-2-51-9', '1712.09313-3-50-9'], ['1712.09313-2-51-10', '1712.09313-3-50-10'], ['1712.09313-2-12-0', '1712.09313-3-12-0'], ['1712.09313-2-12-1', '1712.09313-3-12-1'], ['1712.09313-2-12-2', '1712.09313-3-12-2'], ['1712.09313-2-38-0', '1712.09313-3-38-0'], ['1712.09313-2-0-0', '1712.09313-3-0-0'], ['1712.09313-2-0-1', '1712.09313-3-0-1'], ['1712.09313-2-0-2', '1712.09313-3-0-2'], ['1712.09313-2-0-3', '1712.09313-3-0-3'], ['1712.09313-2-0-4', '1712.09313-3-0-4'], ['1712.09313-2-0-5', '1712.09313-3-0-5'], ['1712.09313-2-0-6', '1712.09313-3-0-6'], ['1712.09313-2-22-0', '1712.09313-3-22-0'], ['1712.09313-2-22-1', '1712.09313-3-22-1'], ['1712.09313-2-22-2', '1712.09313-3-22-2'], ['1712.09313-2-19-0', '1712.09313-3-19-0'], ['1712.09313-2-44-0', '1712.09313-3-43-0'], ['1712.09313-2-44-1', '1712.09313-3-43-1'], ['1712.09313-2-44-2', '1712.09313-3-43-2'], ['1712.09313-2-26-0', '1712.09313-3-26-0'], ['1712.09313-2-26-1', '1712.09313-3-26-1'], ['1712.09313-2-26-2', '1712.09313-3-26-2'], ['1712.09313-2-21-0', '1712.09313-3-21-0'], ['1712.09313-2-21-1', '1712.09313-3-21-1'], ['1712.09313-2-5-0', '1712.09313-3-5-0'], ['1712.09313-2-5-1', '1712.09313-3-5-1'], ['1712.09313-2-5-2', '1712.09313-3-5-2'], ['1712.09313-2-5-3', '1712.09313-3-5-3'], ['1712.09313-2-36-0', '1712.09313-3-36-0'], ['1712.09313-2-43-1', '1712.09313-3-42-1'], ['1712.09313-2-40-0', '1712.09313-3-39-0'], ['1712.09313-2-40-1', '1712.09313-3-39-1'], ['1712.09313-2-34-0', '1712.09313-3-34-0'], ['1712.09313-2-33-0', '1712.09313-3-33-0'], ['1712.09313-2-41-0', '1712.09313-3-40-0'], ['1712.09313-2-41-2', '1712.09313-3-40-2'], ['1712.09313-2-14-0', '1712.09313-3-14-0'], ['1712.09313-2-8-0', '1712.09313-3-8-0'], ['1712.09313-2-8-1', '1712.09313-3-8-1'], ['1712.09313-2-8-2', '1712.09313-3-8-2'], ['1712.09313-2-8-3', '1712.09313-3-8-3'], ['1712.09313-2-8-4', '1712.09313-3-8-4'], ['1712.09313-2-8-5', '1712.09313-3-8-5'], ['1712.09313-2-8-6', '1712.09313-3-8-6'], ['1712.09313-2-49-0', '1712.09313-3-48-0'], ['1712.09313-2-49-1', '1712.09313-3-48-1'], ['1712.09313-2-3-0', '1712.09313-3-3-0'], ['1712.09313-2-3-1', '1712.09313-3-3-1'], ['1712.09313-2-3-2', '1712.09313-3-3-2'], ['1712.09313-2-3-3', '1712.09313-3-3-3'], ['1712.09313-2-45-0', '1712.09313-3-44-0'], ['1712.09313-2-45-1', '1712.09313-3-44-1'], ['1712.09313-2-18-0', '1712.09313-3-18-0'], ['1712.09313-2-18-1', '1712.09313-3-18-1'], ['1712.09313-2-11-0', '1712.09313-3-11-0'], ['1712.09313-2-11-1', '1712.09313-3-11-1'], ['1712.09313-2-29-0', '1712.09313-3-29-0'], ['1712.09313-2-29-1', '1712.09313-3-29-1'], ['1712.09313-2-29-2', '1712.09313-3-29-2'], ['1712.09313-2-29-3', '1712.09313-3-29-3'], ['1712.09313-2-20-0', '1712.09313-3-20-0'], ['1712.09313-2-20-1', '1712.09313-3-20-1'], ['1712.09313-2-20-2', '1712.09313-3-20-2'], ['1712.09313-2-13-1', '1712.09313-3-13-1']]

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[]

[['1712.09313-1-50-7', '1712.09313-2-51-7'], ['1712.09313-1-50-7', '1712.09313-2-51-8'], ['1712.09313-2-25-1', '1712.09313-3-25-1']]

[]

['1712.09313-1-22-1', '1712.09313-1-23-2', '1712.09313-1-33-1', '1712.09313-1-34-0', '1712.09313-1-35-1', '1712.09313-1-40-1', '1712.09313-1-41-0', '1712.09313-1-41-1', '1712.09313-1-42-0', '1712.09313-1-45-0', '1712.09313-1-45-1', '1712.09313-1-46-0', '1712.09313-2-23-1', '1712.09313-2-24-2', '1712.09313-2-34-1', '1712.09313-2-35-0', '1712.09313-2-36-1', '1712.09313-2-41-1', '1712.09313-2-42-0', '1712.09313-2-42-1', '1712.09313-2-43-0', '1712.09313-2-46-0', '1712.09313-2-46-1', '1712.09313-2-47-0', '1712.09313-3-23-1', '1712.09313-3-24-2', '1712.09313-3-34-1', '1712.09313-3-35-0', '1712.09313-3-36-1', '1712.09313-3-40-1', '1712.09313-3-41-0', '1712.09313-3-41-1', '1712.09313-3-42-0', '1712.09313-3-45-0', '1712.09313-3-45-1', '1712.09313-3-46-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1712.09313

{'1712.09313-3-0-0': "Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings.", '1712.09313-3-0-1': 'In this note, we uncover a similar structure for contact manifolds.', '1712.09313-3-0-2': 'Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings.', '1712.09313-3-0-3': 'Those valuations assign numerical invariants to even-dimensional submanifolds, which in a certain sense measure the curvature at points of tangency to the contact structure.', '1712.09313-3-0-4': 'Moreover, these valuations generalize to the class of manifolds equipped with the structure of a Heisenberg algebra on their cotangent bundle.', '1712.09313-3-0-5': 'Pursuing the analogy with Euclidean integral geometry, we construct symplectic-invariant distributions on Grassmannians to produce Crofton formulas on the contact sphere.', '1712.09313-3-0-6': 'Using closely related distributions, we obtain Crofton formulas also in the linear symplectic space.', '1712.09313-3-1-0': '# Introduction', '1712.09313-3-2-0': '## Background and motivation', '1712.09313-3-3-0': "Intrinsic volumes first appeared in convex geometry through Steiner's formula: given a compact convex body [MATH], [MATH] where [MATH] is unit Euclidean ball in [MATH] and [MATH] is its volume.", '1712.09313-3-3-1': 'The coefficient [MATH], the [MATH]-th intrinsic volume, can be written explicitly for smooth [MATH] as [MATH], where [MATH] are the principal curvatures of [MATH].', '1712.09313-3-3-2': 'Alternatively, [MATH] can be given by integral-geometric Crofton formulas: [MATH] where [MATH] is the rigid motion invariant measure on the affine Grassmannian, and [MATH] is the Euler characteristic.', '1712.09313-3-3-3': 'A third, axiomatic definition, was given by Hadwiger, who described the intrinsic volumes as the unique rigid-motion invariant continuous finitely additive measures on compact convex sets.', '1712.09313-3-4-0': "A closely related famous result is Weyl's tube formula [CITATION].", '1712.09313-3-4-1': 'It asserts that the volume of an [MATH]-tube around a Riemannian manifold [MATH] embedded isometrically in Euclidean space [MATH] is a polynomial in [MATH], whose coefficients are, remarkably, intrinsic invariants of the Riemannian manifold [MATH], independent of the isometric embedding.', '1712.09313-3-4-2': 'These coefficients, now known as the intrinsic volumes of [MATH], are intimately linked with the asymptotic expansion of the heat kernel, see [CITATION].', '1712.09313-3-5-0': 'These results fall naturally in the domain of valuations on manifolds, a fairly young branch of valuation theory introduced by Alesker et al. in a sequence of works [CITATION], see also [CITATION] for a survey.', '1712.09313-3-5-1': "Valuation theory itself is a mixture of convex and integral geometry, originating in the early 20th century in works of Steiner, Blaschke, Chern and Santalo, as well as in Dehn's solution to Hilbert's third problem.", '1712.09313-3-5-2': 'Generally speaking, valuations are finitely additive measures on some family of nice subsets.', '1712.09313-3-5-3': 'In this note, there is typically some analytic restriction on the nature of the valuation, such as smoothness or smoothness with singularities, and the subsets are manifolds with corners or differentiable polyhedra.', '1712.09313-3-6-0': "Building on results of Chern, Alesker noticed a natural extension of Weyl's theorem for valuations: restricting to [MATH] the intrinsic volumes of [MATH], (considered as valuations), yields an intrinsically defined family of valuations on [MATH], now known as the Lipschitz-Killing valuations.", '1712.09313-3-6-1': "Weyl's intrinsic volumes of [MATH] are then the integrals of the corresponding valuations.", '1712.09313-3-6-2': 'In a recent work, Fu and Wannerer [CITATION] characterized the Lipschitz-Killing valuations as the unique family of valuations attached canonically (in a sense made precise therein) to arbitrary Riemannian manifolds that are universal to isometric embeddings.', '1712.09313-3-7-0': 'Other spaces whose (smooth) valuation theories were considered in recent years include complex space forms [CITATION], the quaternionic plane [CITATION], the octonionic plane [CITATION] and exceptional spheres [CITATION].', '1712.09313-3-8-0': 'Numerous intriguing connections between convex and symplectic geometries are known to exist.', '1712.09313-3-8-1': "To name a few: Viterbo's conjectured isoperimetric inequality for capacities of convex bodies [CITATION], was later shown by Artstein-Avidan, Karasev and Ostrover to imply Mahler's famous conjecture [CITATION].", '1712.09313-3-8-2': 'Capacities have been successfully studied up to a bounded factor using convex techniques, see [CITATION] and [CITATION].', '1712.09313-3-8-3': "In [CITATION], links are established between systolic geometry, contact geometry, Mahler's conjecture and the geometry of numbers.", '1712.09313-3-8-4': "Schaffer's dual girth conjecture for normed spaces has been proved using symplectic techniques [CITATION], and generalized further in [CITATION] using hamiltonian group actions.", '1712.09313-3-8-5': 'In very recent works of Abbondandolo et al. [CITATION], some links are established between the geometry of the group of symplectomorphisms and systolic geometry of the 2-sphere.', '1712.09313-3-8-6': 'For an exposition of some of those connections, see [CITATION].', '1712.09313-3-9-0': "The main objective for this work is to further explore the convex-symplectic link by studying the valuation theory of contact manifolds, using the Riemannian case and Weyl's principle as guides.", '1712.09313-3-10-0': '## Informal summary', '1712.09313-3-11-0': 'We find that like in the Riemannian setting, contact manifolds possess a canonical family of valuations associated to them.', '1712.09313-3-11-1': 'We describe those valuations in two ways: geometrically through a curvature-type formula, and also dynamically through the invariants of a certain vector field at its singular points.', '1712.09313-3-12-0': "The contact valuations satisfy Weyl's principle of universality under embeddings, similarly to the valuation extension of the Weyl principle.", '1712.09313-3-12-1': 'Let us emphasize the role played by valuations in this phenomenon: It so happens that contact valuations only assume non-zero values on even dimensional submanifolds, while they live on odd-dimensional contact manifolds.', '1712.09313-3-12-2': "Thus unlike the Riemannian case, Weyl's principle in the contact setting is only manifested in its extended to valuations form, as the statement in the original form becomes vacuous: all integrals of the contact valuations vanish.", '1712.09313-3-13-0': 'Contact valuations are in fact an instance of a natural collection of valuations associated to the larger class of manifolds whose cotangent spaces admit a smoothly varying structure of Heisenberg algebras.', '1712.09313-3-13-1': 'The Heisenberg algebra provides a unifying link between contact, symplectic and metric geometries.', '1712.09313-3-14-0': 'This path leads us to consider the valuation theory of the dual Heisenberg algebra, invariant under the group of the automorphisms of the Heisenberg algebra, which is closely related to the symplectic group.', '1712.09313-3-14-1': 'From this perspective, this is another step in the study of the valuation theory of non-compact Lie groups, which up to now has only been considered for the indefinite orthogonal group in [CITATION], and in a somewhat different framework for the special linear group, see [CITATION].', '1712.09313-3-15-0': 'Further similarity to the metric setting is exhibited by the contact sphere, where we prove Crofton formulas and a Hadwiger-type theorem, thus establishing an integral-geometric and an axiomatic description of the contact valuations.', '1712.09313-3-16-0': 'Finally in the last part we explore the valuation theory of linear symplectic spaces.', '1712.09313-3-16-1': 'We show that there are no non-trivial invariant valuations, but nevertheless one can write oriented Crofton formulas for the symplectic volume of submanifolds.', '1712.09313-3-17-0': '## Main results', '1712.09313-3-18-0': 'Let us very briefly recall or indicate the relevant notions.', '1712.09313-3-18-1': 'For precise definitions see sections [REF] and [REF].', '1712.09313-3-19-0': 'A contact manifold [MATH] is given by a maximally non-integrable hyperplane distribution, namely a smooth field of tangent hyperplanes [MATH] s.t. locally one can find [MATH] with [MATH] and [MATH] a non-degenerate 2-form.', '1712.09313-3-20-0': 'A smooth valuation [MATH] on an orientable manifold [MATH], denoted [MATH], is a finitely additive measure on the compact differentiable polyhedra of [MATH], denoted [MATH], which has the form [MATH] for some forms [MATH] and [MATH].', '1712.09313-3-20-1': 'Here [MATH] is the cosphere bundle, and [MATH] is the conormal cycle of [MATH], which is just the conormal bundle when [MATH] is a manifold.', '1712.09313-3-20-2': 'Orientability is not essential, and is only assumed to simplify the exposition.', '1712.09313-3-21-0': 'There is a natural filtration [MATH].', '1712.09313-3-21-1': 'Very roughly speaking, [MATH] consists of valuations which are locally homogeneous of degree at least [MATH], for example [MATH] are the smooth measures on [MATH].', '1712.09313-3-22-0': 'The generalized valuations [MATH] are, roughly speaking, distributional valuations: we allow [MATH] and [MATH] to be currents rather than smooth forms.', '1712.09313-3-22-1': 'Generalized valuations can be naturally evaluated on sufficiently nice subsets [MATH].', '1712.09313-3-22-2': 'The filtration [MATH] on [MATH] extends to a filtration [MATH] on [MATH].', '1712.09313-3-23-0': 'To any contact manifold [MATH] with contact distribution [MATH] there are canonically associated, linearly independent generalized valuations [MATH], [MATH].', '1712.09313-3-23-1': 'They have the following properties:', '1712.09313-3-24-0': '[MATH] is the Euler characteristic, and [MATH].', '1712.09313-3-24-1': '[MATH] can be naturally evaluated on submanifolds in generic position relative to the contact structure (see Definition [REF]).', '1712.09313-3-24-2': 'For a generic closed hypersurface [MATH],', '1712.09313-3-25-0': '[EQUATION] where the local contact area [MATH] only depends on the germ of [MATH] at [MATH].', '1712.09313-3-25-1': 'It is described explicitly below in equations [REF] and [REF].', '1712.09313-3-26-0': 'Universality to restriction under embedding: if [MATH] is a contact embedding, then for [MATH] one has [MATH].', '1712.09313-3-26-1': 'For a [MATH]-dimensional submanifold in general position [MATH], [MATH], with equality if and only if there are no contact tangent points.', '1712.09313-3-26-2': 'The space of generalized valuations on [MATH] invariant under all contactomorphisms of [MATH] is spanned by [MATH].', '1712.09313-3-27-0': 'The universality with respect to embeddings is sometimes referred to as the Weyl principle.', '1712.09313-3-27-1': 'Thus we recover a Weyl principle in the contact setting.', '1712.09313-3-28-0': 'The local contact areas [MATH] can be given explicitly in two different ways, through a geometric or a dynamical approach.', '1712.09313-3-29-0': 'We may extend [MATH] to a non-negative lower semicontinuos functional on all [MATH]-dimensional submanifolds with boundary, denoted [MATH], the contact area of [MATH].', '1712.09313-3-29-1': 'We observe that [MATH] if and only if [MATH] can be made nowhere tangent to the contact distribution by an arbitrarily small perturbation.', '1712.09313-3-29-2': 'In Riemannian or Hermitian manifolds, a fair amount of the valuation theory appears already in the corresponding flat space, which can be thought of as the tangent space to the given manifold.', '1712.09313-3-29-3': 'In the contact setting, it is no longer true: the tangent space of a contact manifold does not itself inherit a contact structure.', '1712.09313-3-30-0': 'The main observation guiding this paper is that every cotangent space of a contact manifold is canonically the Heisenberg Lie algebra.', '1712.09313-3-30-1': 'We are thus led to study the valuation theory of general manifolds with such structure.', '1712.09313-3-31-0': 'A manifold [MATH] equipped with a hyperplane distribution [MATH] (called horizontal) and a smooth field of nowhere-degenerate forms [MATH] will be called a dual Heisenberg (DH) manifold.', '1712.09313-3-32-0': 'The space of valuations naturally associated to such manifolds turns out to resemble somewhat the Lipschitz-Killing space of valuations in Riemannian geometry, in particular, they exhibit universality with respect to embeddings.', '1712.09313-3-33-0': 'Theorem [REF] is then the contact instance of the following general result.', '1712.09313-3-34-0': 'To any DH manifold [MATH] with horizontal distribution [MATH] there are canonically associated, generalized valuations [MATH], [MATH].', '1712.09313-3-34-1': 'They have the following properties:', '1712.09313-3-35-0': '[MATH] is the Euler characteristic, and [MATH].', '1712.09313-3-36-0': '[MATH] can be naturally evaluated on submanifolds in generic position with respect to the horizontal distribution.', '1712.09313-3-36-1': 'For a generic closed hypersurface [MATH],', '1712.09313-3-37-0': '[EQUATION] where [MATH] only depends on the germ of [MATH] at [MATH].', '1712.09313-3-37-1': 'If [MATH] is a DH embedding, then for [MATH] one has [MATH].', '1712.09313-3-38-0': 'The local contact areas [MATH] are given by the same curvature-type formula [REF] as in the contact case.', '1712.09313-3-39-0': 'As an intermediate step of independent interest, we obtain a Hadwiger-type theorem for the dual Heisenberg algebra itself.', '1712.09313-3-39-1': 'We denote by [MATH] the dual of the Heisenberg Lie algebra [MATH], and by [MATH] its automorphism group.', '1712.09313-3-40-0': 'It holds that [MATH] consists of even valuations.', '1712.09313-3-40-1': 'For [MATH], [MATH] while [MATH].', '1712.09313-3-40-2': 'The corresponding Klain sections are zero-order distributions (that is, regular Borel measures).', '1712.09313-3-41-0': 'We also consider [MATH], the connected component of the identity.', '1712.09313-3-41-1': 'We prove', '1712.09313-3-42-0': 'For [MATH], [MATH] while [MATH].', '1712.09313-3-42-1': 'In the latter space, the [MATH]-invariant valuations are complemented by a one-dimensional space of odd valuations.', '1712.09313-3-43-0': 'We then consider the standard contact sphere [MATH], which we identify with the oriented projectivization [MATH] of a symplectic space [MATH].', '1712.09313-3-43-1': 'We construct a canonical distribution [MATH].', '1712.09313-3-43-2': 'We obtain the following Crofton formulas, establishing further common grounds with the Riemannian setting.', '1712.09313-3-44-0': 'Define for [MATH] the generalized valuations [MATH] on [MATH] given by the Crofton formula [EQUATION].', '1712.09313-3-44-1': 'Then for certain explicit constans [MATH] one has', '1712.09313-3-45-0': '[EQUATION]', '1712.09313-3-45-1': 'We also establish a Hadwiger-type theorem for the contact sphere.', '1712.09313-3-46-0': 'Both [MATH] and [MATH] are bases of [MATH].', '1712.09313-3-47-0': 'Finally, we find that while symplectic space and manifolds do not possess interesting invariant valuations, one can nevertheless write certain integral geometric formulas for symplectic volumes of manifolds.', '1712.09313-3-47-1': 'We construct a canonical distribution on the affine oriented Grassmannian [MATH] which is translation- and [MATH]-invariant, and odd to orientation reversal.', '1712.09313-3-47-2': 'We prove the following Crofton formula on symplectic linear space.', '1712.09313-3-48-0': 'Let [MATH] be a [MATH] compact, oriented submanifold with boundary.', '1712.09313-3-48-1': 'Then [EQUATION] where [MATH] and [MATH] is the oriented intersection index.', '1712.09313-3-49-0': '## Plan of the paper', '1712.09313-3-50-0': 'In section [REF] we introduce notation and present the basic geometric facts we will use.', '1712.09313-3-50-1': 'In section [REF] we recall the basics of valuation theory and prove some lemmas we will need.', '1712.09313-3-50-2': 'In section [REF] we construct the canonical valuations on general DH manifolds and establish their universality to embeddings, proving Theorem [REF].', '1712.09313-3-50-3': 'We also explore some geometric properties of those valuations.', '1712.09313-3-50-4': 'In section [REF] we classify the translation-invariant valuations of the dual Heisenberg algebra, proving Theorems [REF] and [REF], and note their relation to gaussian curvature.', '1712.09313-3-50-5': 'Apart from its intrinsic interest, the linear classification is needed for the uniqueness statement in Theorem [REF], as well as for Theorem [REF].', '1712.09313-3-50-6': 'In section [REF] we specialize the DH valuations to contact manifolds and prove Theorem [REF].', '1712.09313-3-50-7': 'In particular, we give the dynamical description of the contact valuations.', '1712.09313-3-50-8': 'In section [REF] we construct symplectic-invariant distributions on linear and affine Grassmannians in symplectic space, which are used in the subsequent two sections.', '1712.09313-3-50-9': 'In section [REF] we consider the standard contact sphere.', '1712.09313-3-50-10': 'We produce Crofton formulas for [MATH] that are invariant under [MATH], proving theorems [REF] and [REF], and compute some examples explicitly.', '1712.09313-3-50-11': 'We also bound from below the contact valuations of a convex set.', '1712.09313-3-50-12': 'Finally in section [REF], we study the integral geometry of linear symplectic space, proving Theorem [REF].'}

0911.0206

{'0911.0206-1-0-0': 'We calculate the radiative decay widths, two-photon ([MATH]) and one photon-one vector meson ([MATH]), of the dynamically generated resonances from vector meson-vector meson interaction in a unitary approach based on the hidden-gauge Lagrangians.', '0911.0206-1-0-1': 'In the present paper we consider the following dynamically generated resonances: [MATH], [MATH], [MATH], [MATH], [MATH], two strangeness=0 and isospin=1 states, and two strangeness=1 and isospin=1/2 states.', '0911.0206-1-0-2': 'For the [MATH] and [MATH] we reproduce the previous results for the two-photon decay widths and further calculate their one photon-one vector decay widths.', '0911.0206-1-0-3': 'For the [MATH] and [MATH] the calculated two-photon decay widths are found to be consistent with data.', '0911.0206-1-0-4': 'The [MATH], [MATH] and [MATH] decay widths of the [MATH], [MATH], [MATH], [MATH] are compared with the results predicted by other approaches.', '0911.0206-1-0-5': 'The [MATH] and [MATH] decay rates of the [MATH] are also calculated and compared with the results obtained in the framework of the covariant oscillator quark model.', '0911.0206-1-0-6': 'The results for the two states with strangeness=0, isospin=1 and two states with strangeness=1, isospin=1/2 are predictions that need to be tested by future experiments.', '0911.0206-1-1-0': '# Introduction', '0911.0206-1-2-0': 'One of the central topics in studies of low-energy strong interaction is to understand how quarks and gluons combine into hadronic objects that we observe experimentally, in other words, to understand low-energy meson and baryon spectroscopy.', '0911.0206-1-2-1': 'Unfortunately, the non-perturbative nature of QCD at low-energies has made a complete solution of this problem from first principles almost impossible (admittedly, lattice QCD has made remarkable progress in recent years, and may provide a solution in the future).', '0911.0206-1-2-2': 'Furthermore, most of the observed hadronic states are not asymptotic states, and as such, they appear only in invariant mass distributions, phase shifts, etc.', '0911.0206-1-2-3': 'This latter feature then implies that in many cases one can not ignore final state interaction among their decay products.', '0911.0206-1-3-0': 'A prominent example is the existence and nature of the [MATH].', '0911.0206-1-3-1': 'For a comprehensive discussion and references, see the mini-review "Note on scalar mesons" of Ref. [CITATION].', '0911.0206-1-3-2': 'Although its existence has long been hypothesized, it took quite a long time until different experiments have finally pinned it down unanimously.', '0911.0206-1-3-3': 'Its nature is even more troubling, i.e., whether it is a genuine [MATH] state, [MATH] state, or molecular state.', '0911.0206-1-3-4': 'In this context, the unitarization technique in combination with the chiral Lagrangians, the so-called unitary chiral theories, have provided a self-consistent picture where the [MATH] may be due to the [MATH] final state interactions [CITATION].', '0911.0206-1-3-5': 'The same approach has been used to study various other hadronic systems, e.g., the kaon-nucleon system [CITATION], heavy-light systems [CITATION] and three body systems [CITATION].', '0911.0206-1-4-0': 'The unitary chiral approach, however, can only be employed to study interactions among the Goldstone-bosons themselves and those between them and other hadrons, because chiral symmetry only defines the interactions involving the Goldstone-bosons.', '0911.0206-1-4-1': 'One may think about applying the same unitarization technique to study other systems by employing phenomenological Lagrangians.', '0911.0206-1-4-2': 'In Refs. [CITATION], by combining the phenomenologically successful hidden-gauge Lagrangians with the above-mentioned unitarization technique, the interactions of vectors mesons among themselves and with octet- and decuplet-baryons have been studied.', '0911.0206-1-4-3': 'In the framework of this approach many interesting results have been obtained, which all compare rather favorably with existing data.', '0911.0206-1-4-4': 'The dynamically generated resonances should contain sizable meson-meson or meson-baryon components in their wave-functions, thus qualifying as "molecular states."', '0911.0206-1-5-0': 'Whether such a picture is correct or partially correct has ultimately to be judged either by data or by studies based on first principles (e.g., lattice QCD calculations).', '0911.0206-1-5-1': 'From the first perspective, one should test as extensively as possible whether the proposed picture is consistent with (all) existing data, make predictions, and propose experiments where such predictions can be tested.', '0911.0206-1-5-2': 'These would provide further support to, or reject, the proposed nature of these states as being dynamically generated.', '0911.0206-1-6-0': 'In the case of the vector meson-vector meson molecular states obtained in Refs. [CITATION], several such tests have been passed: In Refs. [CITATION] it has been shown that the branching ratios into pseudoscalar-pseudoscalar and vector-vector final states of the [MATH], [MATH], [MATH], [MATH], and [MATH] are all consistent with data.', '0911.0206-1-6-1': 'In Ref. [CITATION], the two-photon decay widths of the [MATH] and [MATH] have been calculated and found to agree with data.', '0911.0206-1-6-2': 'Furthermore, in Ref. [CITATION], the ratios of the [MATH] decay rates into a vector meson ([MATH], [MATH], or [MATH]) and one of the tensor states [[MATH], [MATH], and [MATH]] have been calculated, and the agreement with data is found to be quite reasonable.', '0911.0206-1-6-3': 'Following the same approach, in Ref. [CITATION] it is shown that the ratio of the [MATH] decay rates into [MATH] and [MATH] also agrees with data.', '0911.0206-1-7-0': 'The radiative decay of a mesonic state has long been argued to be crucial in determinations of the nature of the state [CITATION].', '0911.0206-1-7-1': 'For instance, the non-observation of the [MATH] decaying into two photons has been used to support its dominant glue nature [CITATION].', '0911.0206-1-7-2': 'In Ref. [CITATION], the two-photon decay widths of the [MATH] and [MATH] have been calculated and found to agree with data which therefore provides further support to the proposed [MATH] molecular nature of these states [CITATION].', '0911.0206-1-7-3': 'In the present paper, we extend our previous work to the [MATH], [MATH], [MATH], and four other states dynamically generated from vector meson - vector meson interaction [CITATION].', '0911.0206-1-7-4': 'By taking into account all the SU(3) allowed coupled channels, we also recalculate the two-photon decay widths of the [MATH] and [MATH], which confirms the earlier results of Ref. [CITATION] and provides a natural estimate of inherent theoretical uncertainties.', '0911.0206-1-7-5': 'We will also calculate the one photon-one vector meson decay widths of these resonances.', '0911.0206-1-7-6': 'As we will show below, in contrast to the results obtained in other theoretical models, our results show some distinct patterns, which should allow one to distinguish between different models once data is available.', '0911.0206-1-8-0': 'This paper is organized as follows: In Section 2, we explain in detail how to calculate the two-photon and one photon-one vector meson decay widths of the dynamically generated states.', '0911.0206-1-8-1': 'In Section 3, we compare the results with those obtained in other approaches and available data, followed by a brief summary in Section 4.', '0911.0206-1-9-0': '# Formalism', '0911.0206-1-10-0': '## Dynamically generated resonances from the vector meson-vector meson interaction', '0911.0206-1-11-0': 'In the following, we briefly outline the main ingredients of the unitary approach (details can be found in Refs. [CITATION]).', '0911.0206-1-11-1': 'There are two basic building-blocks in this approach: transition amplitudes provided by the hidden-gauge Lagrangians [CITATION] and a unitarization procedure.', '0911.0206-1-11-2': 'We adopt the Bethe-Salpeter equation method [MATH] to unitarize the transition amplitudes [MATH] for [MATH]-wave interactions, where [MATH] is a diagonal matrix of the vector meson-vector meson one-loop function [EQUATION] with [MATH] and [MATH] the masses of the two vector mesons.', '0911.0206-1-12-0': 'In Refs. [CITATION] three mechanisms, as shown in Fig. [REF], have been taken into account for the transition amplitudes [MATH]: the four-vector contact term, the t(u)-channel vector exchange amplitude, and the direct box amplitude with two intermediate pseudoscalar mesons.', '0911.0206-1-12-1': 'Other possible mechanisms, e.g. s-channel vector exchange, crossed box amplitudes and box amplitudes involving anomalous couplings, have been neglected, since their contribution was found to be quite small in the detailed study of [MATH] scattering in Ref. [CITATION].', '0911.0206-1-13-0': 'Among the three mechanisms considered for [MATH], the four-vector contact term and [MATH]-channel vector exchange one are responsible for the formation of resonances or bound states provided that the interaction generated by them is strong enough.', '0911.0206-1-13-1': 'In this sense, the dynamically generated states can be thought of as "vector meson-vector meson molecules."', '0911.0206-1-13-2': 'On the other hand, the consideration of the imaginary part of the direct box amplitude allows the generated states to decay into two pseudoscalars.', '0911.0206-1-13-3': 'It should be stressed that in the present approach these two mechanisms play quite different roles: the four-vector contact interaction and the [MATH]-channel vector exchange term are responsible for generating the resonances or bound states, whereas the direct box amplitude mainly contributes to their decays.', '0911.0206-1-13-4': 'This particular feature has an important consequence for the calculation of the radiative decay widths of the dynamically generated states as shown below.', '0911.0206-1-14-0': 'The one-loop function, Eq. (2.1), is divergent and has to be regularized.', '0911.0206-1-14-1': 'In Ref. [CITATION], both dimensional regularization method and cutoff method have been used.', '0911.0206-1-14-2': 'The couplings of the dynamically generated states to their coupled channels are given in Tables I, II, and III of Ref. [CITATION], which we need to calculate the radiative decay widths of these resonances as explained below.', '0911.0206-1-15-0': '## Radiative decays, [MATH] and [MATH], of the dynamically generated resonances', '0911.0206-1-16-0': 'A detailed explanation of the two-photon decay mechanism has been given in Ref. [CITATION].', '0911.0206-1-16-1': 'Here, we follow closely Ref. [CITATION] and extend it to the case of one photon-one vector meson decay.', '0911.0206-1-17-0': 'The coupling of a photon to a dynamically generated resonance goes through couplings to its coupled-channel components in all possible ways such that gauge invariance is conserved.', '0911.0206-1-17-1': 'See, e.g, Refs. [CITATION] for a relevant discussion in the kaon-nucleon system.', '0911.0206-1-17-2': 'A peculiar feature of the hidden-gauge Lagrangians is that photons do not couple directly to charged vector mesons but indirectly through their conversion to [MATH], [MATH], and [MATH].', '0911.0206-1-17-3': 'This, together with the fact that the four-vector contact and the t(u)-channel exchange diagrams are responsible for the generation of the resonances or bound states, imply that the coupling of a photon to the resonance (or bound state) can be factorized into a strong part and an electromagnetic part [CITATION]: i.e., the resonance first decays into two vector mesons and then one or both of them convert into a photon.', '0911.0206-1-17-4': 'This is demonstrated schematically in Fig. [REF] for the case of the two-photon decay.', '0911.0206-1-17-5': 'In the case of one photon-one vector meson decay, one simply replaces one of the final photons by one vector meson.', '0911.0206-1-18-0': 'Close to a pole position, the vector-vector scattering amplitude given in Fig. [REF] can be parameterized as [EQUATION] where [MATH] is the spin projection operator, which projects the initial (final) vector meson-vector meson pair [MATH]) into spin [MATH] with [EQUATION] where [MATH] [[MATH]] is the polarization vector of particle 1 [2] and [MATH], [MATH], [MATH] runs from 1 to 3 since in line with the approximation made in Refs. [CITATION] that [MATH] is small and hence [MATH].', '0911.0206-1-18-1': 'The couplings [MATH]) are obtained from the resonance pole position on the complex plane and are tabulated in Ref. [CITATION].', '0911.0206-1-18-2': 'To evaluate the two-photon and one photon-one vector partial decay widths of the dynamically generated particles, one needs its coupling to the vector-vector components, i.e., [MATH].', '0911.0206-1-19-0': 'The amplitude of a neutral non-strange vector meson converting into a photon is given by [EQUATION] with [MATH].', '0911.0206-1-19-1': 'Therefore, the whole two-photon and one photon-one vector decay amplitudes for a resonance [MATH] of spin [MATH] are [EQUATION] where [MATH] is a proper isospin coefficient which projects the vector-vector pair in isospin space to that in physical space and [MATH] denotes the coupling of resonance [MATH] to channel [MATH].', '0911.0206-1-19-2': 'Recall that in Ref. [CITATION] we have used the following phase conventions: [MATH] and [MATH], which implies that [EQUATION]', '0911.0206-1-19-3': 'From Eq. ([REF]), one can easily read off the isospin projector [MATH].', '0911.0206-1-20-0': 'Summing over polarization of the intermediate vector mesons in Eqs. ([REF],[REF]) and taking into account symmetry factors and proper normalization, one has the following amplitudes [EQUATION] where [MATH] and [MATH] are defined in Eqs. ([REF],[REF],[REF]) with [MATH]) denoting the polarization vector of a vector-meson (photon); [MATH] is the isospin factor, and [MATH], [MATH] account for both a symmetry factor and the unitary normalization used in Refs. [CITATION] [EQUATION]', '0911.0206-1-20-1': 'The two-photon and one photon-one vector decay widths of a dynamically generated resonance [MATH] of spin [MATH] are then given by [EQUATION] where [MATH] is the resonance mass and [MATH] is the photon momentum in the rest frame of the resonance [MATH].', '0911.0206-1-20-2': 'For the photon, we work in the Coulomb gauge ([MATH] and [MATH]), where the sum over the final polarizations are given by [EQUATION] with [MATH] the three momentum of the photon.', '0911.0206-1-20-3': 'For vector mesons, one has [MATH] [see discussion below Eq. ([REF])] and [EQUATION]', '0911.0206-1-20-4': 'With Eqs. ([REF],[REF]), one can easily verify [EQUATION]', '0911.0206-1-21-0': '# Results and discussions', '0911.0206-1-22-0': 'In this section, we discuss our main results and compare them with available data and the predictions of other approaches.', '0911.0206-1-22-1': 'In Tables [REF], [REF], [REF], we show the calculated one photon-one vector meson and two-photon decay widths of the resonances dynamically generated in Ref. [CITATION].', '0911.0206-1-22-2': 'We have also listed relevant data for the two-photon decay widths from different experiments.', '0911.0206-1-22-3': 'It should be pointed out in our approach that among the 11 dynamically generated resonances [CITATION], the [MATH] state does not decay into [MATH] and [MATH]; the same is true for the [MATH] state; on the other hand, the [MATH], [MATH] and [MATH] resonances only decay into [MATH] but not [MATH].', '0911.0206-1-23-0': 'For the [MATH] and [MATH], Nagahiro et al. have calculated the two-photon decay widths as 2.6 keV and 1.62 keV [CITATION].', '0911.0206-1-23-1': 'Recall in that work among all the SU(3) allowed channels only the [MATH] channel was considered and also the couplings deduced from amplitudes on the real-axis were used.', '0911.0206-1-23-2': 'Therefore, the differences between the two-photon decay widths obtained in the present work and those obtained in Ref. [CITATION] can be viewed as inherent theoretical uncertainties, which are [MATH].', '0911.0206-1-23-3': 'As also discussed in Ref. [CITATION], it is clear from Table [REF] that our two-photon decay width for the [MATH] agrees well with the data.', '0911.0206-1-23-4': 'The experimental situation for the [MATH] is not yet clear, but as discussed in Ref. [CITATION], current experimental results are consistent with our result for [MATH].', '0911.0206-1-24-0': 'In addition to the [MATH] and [MATH] we have calculated the two-photon decay widths of the [MATH] and [MATH].', '0911.0206-1-24-1': 'From Table [REF], it can be seen that they agree reasonably well with available data.', '0911.0206-1-24-2': 'Our calculated two-photon decay width for the [MATH] is slightly smaller than the experimental value quoted in the PDG review.', '0911.0206-1-24-3': 'This is quite acceptable since 1) as discussed earlier we have an inherent theoretical uncertainty of [MATH] and 2) there might be other relevant coupled channels that have not been taken into account in the model of Ref. [CITATION], which can be inferred from the fact that the total decay width of the [MATH] in that model [MATH] MeV is smaller than the experimental value [MATH] MeV.', '0911.0206-1-25-0': 'Note that the significantly small value of the widths of the [MATH] and [MATH] compared to that of the [MATH], for example, has a natural interpretation in our theoretical framework since the former two resonances are mostly [MATH] molecules and therefore the couplings to [MATH], [MATH], [MATH], [MATH], which lead to the final [MATH] decay, are very small.', '0911.0206-1-25-1': 'The advantages of working with coupled channels become obvious in the case of these radiative decays.', '0911.0206-1-25-2': 'While a pure [MATH] assignment would lead to [MATH]=0 keV, our coupled channel analysis gives the right strength for the couplings to the weakly coupled channels.', '0911.0206-1-26-0': 'In the following we shall have a closer look at the radiative decay widths of the [MATH], [MATH], [MATH], [MATH], [MATH], and compare them with the predictions from other theoretical approaches.', '0911.0206-1-27-0': '## Radiative decay widths of [MATH] and [MATH]', '0911.0206-1-28-0': 'In Table [REF], we compare our results for the radiative decay widths for the [MATH] with those obtained in other approaches, including the covariant oscillator quark model (COQM) [CITATION], the tensor-meson dominance (TMD) model [CITATION], the AdS/QCD calculation in [CITATION], the model assuming both tensor-meson dominance and vector-meson dominance (TMDVMD) [CITATION], and the nonrelativistic quark-model (NRQM) [CITATION].', '0911.0206-1-28-1': 'From this comparison, one can see that the AdS/QCD calculation and our present study provide a two-photon decay width consistent the data.', '0911.0206-1-28-2': 'The TMD model result is also consistent with the data (it can use either [MATH] or [MATH] as an input to fix its single parameter), while the TMDVMD model prediction is off by a factor of 3.', '0911.0206-1-28-3': 'Particularly interesting is the fact that although the TMDVMD model predicts [MATH] similar to our prediction, but in contrast their result for [MATH] is much larger than ours, almost a factor of 30.', '0911.0206-1-28-4': 'Therefore, an experimental measurement of the ratio of [MATH] will be very useful to disentangle these two pictures of the [MATH].', '0911.0206-1-28-5': 'Furthermore, one notices that all theoretical approaches predict [MATH] to be of the order of a few 100 keV.', '0911.0206-1-29-0': 'In Table [REF], we compare the radiative decay widths of the [MATH] predicted in the present work with those obtained in the COQM [CITATION].', '0911.0206-1-29-1': 'We notice that the COQM predicts [MATH] while our model gives an estimate of [MATH], which are quite distinct even taking into account model uncertainties.', '0911.0206-1-29-2': 'Furthermore, [MATH] in the COQM is almost zero while it is comparable to [MATH] in our approach.', '0911.0206-1-29-3': 'An experimental measurement of any two of the three decay widths will be able to confirm either the COQM picture or the dynamical picture.', '0911.0206-1-30-0': 'An interesting quantity in this context is the ratio [MATH] since naturally branching ratios suffer less from systematic uncertainties within a model.', '0911.0206-1-30-1': 'In Table [REF], we compare our result with data and those obtained in other approaches.', '0911.0206-1-30-2': 'It is clear that our result lies within the experimental bounds while those of the effective field approach (EF) [CITATION] and the two-state mixing scheme (TMS) [CITATION] are slightly larger than the experimental upper limit, with the latter being almost at the upper limit.', '0911.0206-1-30-3': 'Given the fact that we have no free parameters in this calculation, such an agreement is reasonable.', '0911.0206-1-31-0': '## Radiative decay widths of [MATH] and [MATH]', '0911.0206-1-32-0': 'Now let us turn our attention to the [MATH] and [MATH] mesons.', '0911.0206-1-32-1': 'In table [REF] we compare our results for the radiative decay widths of the [MATH] and [MATH] obtained by the coupled channel model with the predictions of other theoretical approaches, including the nonrelativistic quark model (NRQM) [CITATION], the light-front quark model (LFQM) [CITATION], the calculation of Nagahiro et al. [CITATION], and the chiral approach [CITATION].', '0911.0206-1-32-2': 'In the NRQM and LFQM calculations three numbers are given for each decay channel depending on whether the glueball mass used in the calculation is smaller than the [MATH] mass (Light), between the [MATH] and [MATH] masses (Medium), or larger than the [MATH] mass (Heavy) [CITATION].', '0911.0206-1-33-0': 'First we note that for the [MATH] our predicted two-photon decay width is more consistent with the LFQM result in the light glueball scenario, while the [MATH] decay width lies closer to the LFQM result in the heavy glueball scenario.', '0911.0206-1-33-1': 'Furthermore, the [MATH] decay width in our model is an order of magnitude smaller than that in the LFQM.', '0911.0206-1-34-0': 'For the [MATH], the LFQM two-photon decay width is larger than the current experimental limit (see Table [REF]).', '0911.0206-1-34-1': 'On the other hand, our [MATH] decay width is more consistent with the LFQM in the light gluon scenario while the [MATH] decay width is more consistent with that of the LFQM in the heavy gluon scenario.', '0911.0206-1-34-2': 'Similar to the [MATH] case, here further experimental data are needed to clarify the situation.', '0911.0206-1-35-0': 'Furthermore, we notice that the NRQM and the LFQM in the light and medium glueball mass scenarios and our present study all predict that [MATH] for the [MATH] while [MATH] for the [MATH].', '0911.0206-1-35-1': 'On the other hand, the NRQM and LFQM in the heavy glueball scenario predict [MATH] for the [MATH].', '0911.0206-1-35-2': 'Therefore, an experimental measurement of the ratio of [MATH] not only will distinguish between the quark-model picture and the dynamical picture, but also will put a constraint on the mass of a possible glueball in this mass region.', '0911.0206-1-36-0': 'The chiral approach in Ref. [CITATION] delivers smaller values for the two-photon decay rates of the [MATH] and [MATH].', '0911.0206-1-36-1': 'However, the ratio [MATH] lies much closer to our prediction [MATH] than the LFQM results which range between 1.7-3.0.', '0911.0206-1-37-0': 'The work of Nagahiro et al. [CITATION] evaluates the contribution from loops of [MATH]) using a phenomenological scalar coupling of the [MATH]) to [MATH]).', '0911.0206-1-37-1': 'From the new perspective on these states we have after the work of Ref. [CITATION], the scalar coupling may not be justified.', '0911.0206-1-37-2': 'One rather has the [MATH] coupling to [MATH] while the coupling of the [MATH] channel only occurs indirectly through the further decay [MATH] and [MATH], with [MATH] going into an internal propagator.', '0911.0206-1-37-3': 'As found in Ref. [CITATION], loops containing these [MATH] propagators only lead to small contributions compared to leading terms including vector mesons (four-vector contact and t(u)-channel vector exchange).', '0911.0206-1-38-0': 'Experimentally, there is a further piece of information on the [MATH] that is relevant to the present study.', '0911.0206-1-38-1': 'From the [MATH] decay branching ratios to [MATH] and [MATH], one can deduce [CITATION]', '0911.0206-1-39-0': '(f_0(1710))(f_0(1710)KK)=Br(J/f_0(1710))Br(J/f_0(1710)KK)=(3.11.0)10^-4(8.5^+1.2_-0.9)10^-4=0.365^+0.156_-0.169.', '0911.0206-1-40-0': 'In the same way as we obtain the two-photon decay widths, we can also calculate the two-vector-meson decay width of the dynamically generated resonances.', '0911.0206-1-40-1': 'For the [MATH], its decay width to [MATH] is found to be [EQUATION]', '0911.0206-1-40-2': 'Using [MATH] MeV, already derived in Ref. [CITATION], and the ratio [MATH] % also given in Ref. [CITATION], one obtains the following branching ratio', '0911.0206-1-41-0': '(f_0(1710))(f_0(1710)KK)&=&0.21 ,', '0911.0206-1-42-0': 'which lies within the experimental bound, although close to the lower limit.', '0911.0206-1-43-0': '## Radiative decay widths of the [MATH]', '0911.0206-1-44-0': 'The radiative decay widths of the [MATH] calculated in the present work are compared with those calculated in the covariant oscillator quark model (COQM) [CITATION] in Table [REF].', '0911.0206-1-44-1': 'We notice that the results from these two approaches differ by a factor of 10.', '0911.0206-1-44-2': 'However, there is one thing in common, i.e., both predict a much larger [MATH] than the [MATH].', '0911.0206-1-44-3': 'More specifically, in the COQM [MATH], while in our model this ratio is [MATH].', '0911.0206-1-45-0': 'At present there is no experimental measurement of these decay modes.', '0911.0206-1-45-1': 'On the other hand, the [MATH] and [MATH] decay rates have been measured.', '0911.0206-1-45-2': 'According to PDG [CITATION], [MATH] keV and [MATH] keV.', '0911.0206-1-45-3': 'Comparing these decay rates with those shown in Table [REF], one immediately notices that the [MATH] in the dynamical model is of similar order as the [MATH] despite reduced phase space in the former decay, which is of course closely related with the fact that the [MATH] is built out of the coupled channel interaction between the [MATH], [MATH], and [MATH] components in the dynamical model.', '0911.0206-1-45-4': 'Furthermore, both the COQM and our dynamical model predict [MATH], which is opposite to the decays into a kaon plus a photon where [MATH].', '0911.0206-1-45-5': 'An experimental measurement of those decays would be very interesting and will certainly help distinguish the two different pictures of the [MATH].', '0911.0206-1-46-0': '# Summary and conclusions', '0911.0206-1-47-0': 'We have calculated the radiative decay widths ([MATH] and [MATH]) of the [MATH], [MATH], [MATH], [MATH], [MATH], and four other states that appear dynamically from vector meson-vector meson interaction in a unitary approach.', '0911.0206-1-47-1': 'Within this approach, due to the peculiarities of the hidden-gauge Lagrangians and the assumption that these resonances are mainly formed by vector meson-vector meson interaction, one can factorize the radiative decay process into a strong part and an electromagnetic part.', '0911.0206-1-47-2': 'This way, the calculation is greatly simplified and does not induce loop calculations.', '0911.0206-1-47-3': 'The obtained results are found to be consistent with existing data within theoretical and experimental uncertainties.', '0911.0206-1-48-0': 'When data are not available, we have compared our predictions with those obtained in other approaches.', '0911.0206-1-48-1': 'In particular, we have identified the relevant pattern of decay rates predicted by different theoretical models and found them quite distinct.', '0911.0206-1-48-2': 'For instance, the [MATH] ratio is quite different in the dynamical model from those in the TMDVMD model and the COQM model.', '0911.0206-1-48-3': 'The [MATH] ratio in the COQM model is also distinctly different from that in the dynamical model.', '0911.0206-1-48-4': 'A measurement of the [MATH]/[MATH] decay rates into [MATH] and [MATH] could be used not only to distinguish between the quark model (NRQM and LFQM) picture and the dynamical picture but also to put a constraint on the mass of a possible glueball (in the [MATH]-g mixing scheme of the NRQM and LFQM).', '0911.0206-1-48-5': 'For the [MATH], as we have discussed, a measurement of its [MATH] decay mode will definitely be able to determine to what extent the dynamical picture is correct.', '0911.0206-1-49-0': 'It is necessary to stress that the QCD dynamics is much richer than that contained in our unitary approach.', '0911.0206-1-49-1': 'It is, therefore, not too surprising to us that sometimes agreement with data is not perfect, but the model delivers at least a qualitative insight into the decay pattern.', '0911.0206-1-49-2': 'However, up to now the dynamical picture of the [MATH], [MATH], [MATH], [MATH], and [MATH] has been tested in a number of scenarios, including in the [MATH] decays [CITATION], [MATH] decays [CITATION], in their strong decay modes [CITATION], and in their two-photon decay modes, as shown in Ref. [CITATION] and in the present work.', '0911.0206-1-49-3': 'It will be interesting to see what comes out in their one photon-one vector meson decay modes.', '0911.0206-1-49-4': 'Given their distinct pattern in different theoretical models, an experimental measurement of some of the decay modes will be very suggestive of the nature of these resonances.'}

{'0911.0206-2-0-0': 'We calculate the radiative decay widths, two-photon ([MATH]) and one photon-one vector meson ([MATH]), of the dynamically generated resonances from vector meson-vector meson interaction in a unitary approach based on the hidden-gauge Lagrangians.', '0911.0206-2-0-1': 'In the present paper we consider the following dynamically generated resonances: [MATH], [MATH], [MATH], [MATH], [MATH], two strangeness=0 and isospin=1 states, and two strangeness=1 and isospin=1/2 states.', '0911.0206-2-0-2': 'For the [MATH] and [MATH] we reproduce the previous results for the two-photon decay widths and further calculate their one photon-one vector decay widths.', '0911.0206-2-0-3': 'For the [MATH] and [MATH] the calculated two-photon decay widths are found to be consistent with data.', '0911.0206-2-0-4': 'The [MATH], [MATH] and [MATH] decay widths of the [MATH], [MATH], [MATH], [MATH] are compared with the results predicted by other approaches.', '0911.0206-2-0-5': 'The [MATH] and [MATH] decay rates of the [MATH] are also calculated and compared with the results obtained in the framework of the covariant oscillator quark model.', '0911.0206-2-0-6': 'The results for the two states with strangeness=0, isospin=1 and two states with strangeness=1, isospin=1/2 are predictions that need to be tested by future experiments.', '0911.0206-2-1-0': '# Introduction', '0911.0206-2-2-0': 'One of the central topics in studies of low-energy strong interaction is to understand how quarks and gluons combine into hadronic objects that we observe experimentally, in other words, to understand low-energy meson and baryon spectroscopy.', '0911.0206-2-2-1': 'Unfortunately, the non-perturbative nature of QCD at low-energies has made a complete solution of this problem from first principles almost impossible (admittedly, lattice QCD has made remarkable progress in recent years, and may provide a solution in the future).', '0911.0206-2-2-2': 'Furthermore, most of the observed hadronic states are not asymptotic states, and as such, they appear only in invariant mass distributions, phase shifts, etc.', '0911.0206-2-2-3': 'This latter feature then implies that in many cases one can not ignore final state interaction among their decay products.', '0911.0206-2-3-0': 'A prominent example is the existence and nature of the [MATH].', '0911.0206-2-3-1': 'For a comprehensive discussion and references, see the mini-review "Note on scalar mesons" of Ref. [CITATION].', '0911.0206-2-3-2': 'Although its existence has long been hypothesized, it took quite a long time until different experiments have finally pinned it down unanimously.', '0911.0206-2-3-3': 'Its nature is even more troubling, i.e., whether it is a genuine [MATH] state, [MATH] state, or molecular state.', '0911.0206-2-3-4': 'In this context, the unitarization technique in combination with the chiral Lagrangians, the so-called unitary chiral theories, have provided a self-consistent picture where the [MATH] may be due to the [MATH] final state interactions [CITATION].', '0911.0206-2-3-5': 'The same approach has been used to study various other hadronic systems, e.g., the kaon-nucleon system [CITATION], heavy-light systems [CITATION] and three body systems [CITATION].', '0911.0206-2-4-0': 'The unitary chiral approach, however, can only be employed to study interactions among the Goldstone-bosons themselves and those between them and other hadrons, because chiral symmetry only defines the interactions involving the Goldstone-bosons.', '0911.0206-2-4-1': 'One may think about applying the same unitarization technique to study other systems by employing phenomenological Lagrangians.', '0911.0206-2-4-2': 'In Refs. [CITATION], by combining the phenomenologically successful hidden-gauge Lagrangians with the above-mentioned unitarization technique, the interactions of vectors mesons among themselves and with octet- and decuplet-baryons have been studied.', '0911.0206-2-4-3': 'In the framework of this approach many interesting results have been obtained, which all compare rather favorably with existing data.', '0911.0206-2-4-4': 'The dynamically generated resonances should contain sizable meson-meson or meson-baryon components in their wave-functions, thus qualifying as "molecular states."', '0911.0206-2-5-0': 'Whether such a picture is correct or partially correct has ultimately to be judged either by data or by studies based on first principles (e.g., lattice QCD calculations).', '0911.0206-2-5-1': 'From the first perspective, one should test as extensively as possible whether the proposed picture is consistent with (all) existing data, make predictions, and propose experiments where such predictions can be tested.', '0911.0206-2-5-2': 'These would provide further support to, or reject, the proposed nature of these states as being dynamically generated.', '0911.0206-2-6-0': 'In the case of the vector meson-vector meson molecular states obtained in Refs. [CITATION], several such tests have been passed: In Refs. [CITATION] it has been shown that the branching ratios into pseudoscalar-pseudoscalar and vector-vector final states of the [MATH], [MATH], [MATH], [MATH], and [MATH] are all consistent with data.', '0911.0206-2-6-1': 'In Ref. [CITATION], the two-photon decay widths of the [MATH] and [MATH] have been calculated and found to agree with data.', '0911.0206-2-6-2': 'Furthermore, in Ref. [CITATION], the ratios of the [MATH] decay rates into a vector meson ([MATH], [MATH], or [MATH]) and one of the tensor states [[MATH], [MATH], and [MATH]] have been calculated, and the agreement with data is found to be quite reasonable.', '0911.0206-2-6-3': 'Following the same approach, in Ref. [CITATION] it is shown that the ratio of the [MATH] decay rates into [MATH] and [MATH] also agrees with data.', '0911.0206-2-7-0': 'The radiative decay of a mesonic state has long been argued to be crucial in determinations of the nature of the state [CITATION].', '0911.0206-2-7-1': 'For instance, the non-observation of the [MATH] decaying into two photons has been used to support its dominant glue nature [CITATION].', '0911.0206-2-7-2': 'In Ref. [CITATION], the two-photon decay widths of the [MATH] and [MATH] have been calculated and found to agree with data which therefore provides further support to the proposed [MATH] molecular nature of these states [CITATION].', '0911.0206-2-7-3': 'In the present paper, we extend our previous work to the [MATH], [MATH], [MATH], and four other states dynamically generated from vector meson - vector meson interaction [CITATION].', '0911.0206-2-7-4': 'By taking into account all the SU(3) allowed coupled channels, we also recalculate the two-photon decay widths of the [MATH] and [MATH], which confirms the earlier results of Ref. [CITATION] and provides a natural estimate of inherent theoretical uncertainties.', '0911.0206-2-7-5': 'We will also calculate the one photon-one vector meson decay widths of these resonances.', '0911.0206-2-7-6': 'As we will show below, in contrast to the results obtained in other theoretical models, our results show some distinct patterns, which should allow one to distinguish between different models once data are available.', '0911.0206-2-8-0': 'This paper is organized as follows: In Section 2, we explain in detail how to calculate the two-photon and one photon-one vector meson decay widths of the dynamically generated states.', '0911.0206-2-8-1': 'In Section 3, we compare the results with those obtained in other approaches and available data, followed by a brief summary in Section 4.', '0911.0206-2-9-0': '# Formalism', '0911.0206-2-10-0': '## Dynamically generated resonances from the vector meson-vector meson interaction', '0911.0206-2-11-0': 'In the following, we briefly outline the main ingredients of the unitary approach (details can be found in Refs. [CITATION]).', '0911.0206-2-11-1': 'There are two basic building-blocks in this approach: transition amplitudes provided by the hidden-gauge Lagrangians [CITATION] and a unitarization procedure.', '0911.0206-2-11-2': 'We adopt the Bethe-Salpeter equation method [MATH] to unitarize the transition amplitudes [MATH] for [MATH]-wave interactions, where [MATH] is a diagonal matrix of the vector meson-vector meson one-loop function [EQUATION] with [MATH] and [MATH] the masses of the two vector mesons.', '0911.0206-2-12-0': 'In Refs. [CITATION] three mechanisms, as shown in Fig. [REF], have been taken into account for the transition amplitudes [MATH]: the four-vector contact term, the t(u)-channel vector exchange amplitude, and the direct box amplitude with two intermediate pseudoscalar mesons.', '0911.0206-2-12-1': 'Other possible mechanisms, e.g. s-channel vector exchange, crossed box amplitudes and box amplitudes involving anomalous couplings, have been neglected, since their contribution was found to be quite small in the detailed study of [MATH] scattering in Ref. [CITATION].', '0911.0206-2-13-0': 'Among the three mechanisms considered for [MATH], the four-vector contact term and [MATH]-channel vector exchange one are responsible for the formation of resonances or bound states provided that the interaction generated by them is strong enough.', '0911.0206-2-13-1': 'In this sense, the dynamically generated states can be thought of as "vector meson-vector meson molecules."', '0911.0206-2-13-2': 'On the other hand, the consideration of the imaginary part of the direct box amplitude allows the generated states to decay into two pseudoscalars.', '0911.0206-2-13-3': 'It should be stressed that in the present approach these two mechanisms play quite different roles: the four-vector contact interaction and the [MATH]-channel vector exchange term are responsible for generating the resonances or bound states, whereas the direct box amplitude mainly contributes to their decays.', '0911.0206-2-13-4': 'This particular feature has an important consequence for the calculation of the radiative decay widths of the dynamically generated states as shown below.', '0911.0206-2-14-0': 'The one-loop function, Eq. (1), is divergent and has to be regularized.', '0911.0206-2-14-1': 'In Ref. [CITATION], both dimensional regularization method and cutoff method have been used.', '0911.0206-2-14-2': 'The couplings of the dynamically generated states to their coupled channels are given in Tables I, II, and III of Ref. [CITATION], which we need to calculate the radiative decay widths of these resonances as explained below.', '0911.0206-2-14-3': 'In Ref. [CITATION], the couplings were obtained on the second Riemann sheet using the dimensional regularization method without including the box diagrams in the model.', '0911.0206-2-14-4': 'If, instead, the loop functions were regularized using the cutoff method, one had to calculate the couplings from the modulus of amplitude squared on the real axis as done in Ref. [CITATION].', '0911.0206-2-14-5': 'These two approaches were found to yield consistent values for the couplings.', '0911.0206-2-14-6': 'One has also some freedom in the values of the subtraction constants due to data uncertainty and the coupled-channel nature of the problem.', '0911.0206-2-14-7': 'An analysis of the resulting uncertainties has been performed in Refs. [CITATION].', '0911.0206-2-14-8': 'They, however, were found to translate into small uncertainties (at the order of a few percent) in the present calculation.', '0911.0206-2-15-0': 'We take advantage here to clarify a question often raised in connection with the dynamically generated states.', '0911.0206-2-15-1': 'Since we all accept that quarks are present in the physical mesons, the obvious question is what happens to the ordinary [MATH] states?', '0911.0206-2-15-2': 'The answer to this can be found in the works of Refs. [CITATION].', '0911.0206-2-15-3': 'In those works, where the study of the scalar mesons is addressed, one starts with a seed of [MATH] states representing scalar states around [MATH] GeV.', '0911.0206-2-15-4': 'Yet, these states unavoidably couple to meson meson components.', '0911.0206-2-15-5': 'This is a necessity imposed by unitarity, since the meson meson decay channels certainly couple to the physical states.', '0911.0206-2-15-6': 'Invoking symmetries, like SU(3), other meson meson channels, even those closed for the decay, will also couple to those [MATH] components.', '0911.0206-2-15-7': 'For instance the [MATH] resonance decays into [MATH], so this must be a necessary coupled channel.', '0911.0206-2-15-8': 'However, the underlying SU(3) symmetry of the strong interactions will impose also the coupling to the [MATH] component.', '0911.0206-2-15-9': 'One rightly guesses that other channels with masses far away from that of the [MATH] will play a minor role and can be neglected (actually they can be accounted for, as we shall discuss below).', '0911.0206-2-15-10': 'Then one has a coupled channel problem with [MATH], [MATH] and [MATH].', '0911.0206-2-15-11': 'According to Refs. [CITATION] the solution of the coupled channel problem leads to the scalar states where the original [MATH] states are represented by a component of the wave function of minor importance, since the meson meson cloud has taken over and represents the bulk of the wave function.', '0911.0206-2-15-12': 'In simple words we can give a picture for this situation.', '0911.0206-2-15-13': 'As is well known, when we give energy to a hadron to break it and eventually see the quark components, we do not see the quarks, we see mesons produced.', '0911.0206-2-15-14': 'This seems to be the case not only when we break the hadron but when we excite it, such that the creation of mesons becomes energetically more favorable that the excitation of the quarks.', '0911.0206-2-15-15': 'One can easily visualize this in the baryon spectrum: either the Roper or the N*(1535) resonances would require 500-600 MeV of quark excitation energy, if they correspond to genuine quark excitations.', '0911.0206-2-15-16': 'It is clear that the introduction of a pion on top of the nucleon is energetically more favorable, so one should investigate the pion nucleon dynamics (together with other SU(3) related coupled channels) to see if this dynamics is able to produce these states.', '0911.0206-2-15-17': 'Indeed, the N*(1535) appears as dynamically generated from the meson baryon interaction in coupled channels [CITATION].', '0911.0206-2-16-0': 'One can then still rightfully ask where the quark states go.', '0911.0206-2-16-1': 'Are there within this picture states that are mostly of [MATH] nature?', '0911.0206-2-16-2': 'The answer is yes in principle, but nothing can guarantee it.', '0911.0206-2-16-3': 'One might think that they should appear at higher energies given the large energy needed to excite quarks.', '0911.0206-2-16-4': 'However, this is not necessarily true as we shall comment at the end of this section.', '0911.0206-2-16-5': 'On the other hand, the meson meson channels of smaller energy will be open.', '0911.0206-2-16-6': 'This detail should not go unnoticed.', '0911.0206-2-16-7': 'Indeed, let us think of a single channel problem with an attractive potential.', '0911.0206-2-16-8': 'One can get many discrete bound states in principle.', '0911.0206-2-16-9': 'Let us add another channel with an attractive potential, which by itself also generates discrete bound states.', '0911.0206-2-16-10': 'When we allow some coupling among these two channels then the earlier initial states give rise to two orthogonal combinations of the two channels.', '0911.0206-2-16-11': 'One might expect the same thing when we put together meson and quark channels.', '0911.0206-2-16-12': 'Yet, the counting of states does not follow here because for higher energies the meson meson channel will be unbound and then we can have a continuum of states.', '0911.0206-2-16-13': 'We can of course find out resonances, but this is not guaranteed nor is there any rule on how many resonances should appear.', '0911.0206-2-16-14': 'It all depends on the dynamics.', '0911.0206-2-16-15': 'The problem is indeed very interesting, but as far as one restricts oneself to low-lying resonances the meson meson nature is prominent and the effective Lagrangians used to take care of their interaction lead naturally to some bound states, which are those we consider.', '0911.0206-2-16-16': 'As to whether there are other states of simpler quark nature, in our approach we cannot say anything since these components are not part of our coupled channels states.', '0911.0206-2-16-17': 'However, apart from the works mentioned earlier [CITATION] there are works in this direction in Refs. [CITATION], which also conclude that the states of lower energy are mostly of mesonic nature.', '0911.0206-2-17-0': 'Continuing with these observations, in connection with the quark components one can say that even the small admixture of these quark components could change the mass and other properties of the resonances.', '0911.0206-2-17-1': 'This might be so to some extent, but the studies with chiral dynamics and only hadron components have an element in the formalism which allows one to take this into account in an effective way.', '0911.0206-2-17-2': 'This is the subtraction constant in the [MATH] function when the dimensional regularization formula is used, or the cut off in the cut off method.', '0911.0206-2-17-3': 'The basic idea of having the hadronic components as main building blocks is that the spectra is obtained using a natural value for the cut off or the subtraction constant [CITATION].', '0911.0206-2-17-4': 'Fine tuning of these subtraction constant or cut off can take into account the contribution from additional channels not explicitly considered in the approach, like the quark states [CITATION].', '0911.0206-2-17-5': 'In fact sometimes one needs a massive change of the cut off to reproduce the mass of a particle, which is a clear manifestation that the state under consideration is not of hadronic, but more of quark nature.', '0911.0206-2-17-6': 'This is the case for the [MATH] meson, which does not come as an object made of [MATH].', '0911.0206-2-17-7': 'In the study of [MATH] scattering using the lowest-order chiral Lagrangians it would require a cut off of the order of several TeV, which is obviously far away from the natural scale of 1 GeV in effective theories of the low energy hadron spectra [CITATION].', '0911.0206-2-18-0': 'Actually the case of the [MATH] is a good example of warning concerning the dynamical generation of resonances.', '0911.0206-2-18-1': 'If in the [MATH] interaction one takes the leading- and next-to-leading-order of the [MATH]-channel [MATH]-exchange amplitude and unitarizes it with the IAM (inverse amplitude method) or the Bethe Salpeter equation, one obtains the full amplitude (see section III of Ref. [CITATION]).', '0911.0206-2-18-2': 'This is further elaborated in Ref. [CITATION], which warns that this can happen in unitarization procedures, inducing one to think that one obtains a dynamically generated resonance, when in fact one is merely regenerating a preexisting resonance, which has been integrated out of the original Lagrangian and is not contained in the effective Lagrangian as a fundamental field.', '0911.0206-2-18-3': 'Although this warning should be kept in mind, one should also note that apart from regenerating a preexisting resonance, one can, and does in practice, generate other non-preexisting ones due to other terms in the potential, different from those directly associated to the [MATH]-channel exchange of the preexisting resonance, like contact terms and [MATH]- and [MATH]-channel exchange of those preexisting resonances.', '0911.0206-2-18-4': 'This is particularly clear in the case of the low-lying scalar mesons, which have different quantum numbers than the [MATH].', '0911.0206-2-18-5': 'The latter, as mentioned above, would be "regenerated" in the unitarization scheme using the leading- and next-to-leading-order terms in the potential.', '0911.0206-2-19-0': 'Nevertheless, in spite of all the arguments given in favor of the dynamically generated vector-vector states, the fact remains that the tensor states [MATH], [MATH], [MATH], [MATH] are well reproduced in the quark model, including many of their decay modes (see, e.g., Ref. [CITATION]).', '0911.0206-2-19-1': 'This success in both models may reflect the fact that the constituent quarks in quark models are objects effectively dressed with meson clouds and the overlap between the molecular picture and the quark model picture could be bigger than expected in some cases [CITATION].', '0911.0206-2-19-2': 'Yet, even in this case, using one picture or the other could be more suited for other observables than those where the two models succeed.', '0911.0206-2-19-3': 'It is thus worth working with both models to make predictions.', '0911.0206-2-19-4': 'As we shall see in Section III, there are some observables where the predictions of the two models are indeed rather different.', '0911.0206-2-20-0': '## Radiative decays, [MATH] and [MATH], of the dynamically generated resonances', '0911.0206-2-21-0': 'A detailed explanation of the two-photon decay mechanism has been given in Ref. [CITATION].', '0911.0206-2-21-1': 'Here, we follow closely Ref. [CITATION] and extend it to the case of one photon-one vector meson decay.', '0911.0206-2-22-0': 'The coupling of a photon to a dynamically generated resonance goes through couplings to its coupled-channel components in all possible ways such that gauge invariance is conserved (see, e.g, Refs. [CITATION] for a relevant discussion within the kaon-nucleon system).', '0911.0206-2-22-1': 'A peculiar feature of the hidden-gauge Lagrangians is that photons do not couple directly to charged vector mesons but indirectly through their conversion to [MATH], [MATH], and [MATH].', '0911.0206-2-22-2': 'This, together with the fact that the four-vector contact and the t(u)-channel exchange diagrams are responsible for the generation of the resonances or bound states, imply that the coupling of a photon to the resonance (or bound state) can be factorized into a strong part and an electromagnetic part [CITATION]: i.e., the resonance first decays into two vector mesons and then one or both of them convert into a photon.', '0911.0206-2-22-3': 'This is demonstrated schematically in Fig. [REF] for the case of the two-photon decay.', '0911.0206-2-22-4': 'In the case of one photon-one vector meson decay, one simply replaces one of the final photons by one vector meson.', '0911.0206-2-23-0': 'Close to a pole position, the vector-vector scattering amplitude given in Fig. [REF] can be parameterized as [EQUATION] where [MATH] is the spin projection operator, which projects the initial (final) vector meson-vector meson pair [MATH]) into spin [MATH] with [EQUATION] where [MATH] [[MATH]] is the polarization vector of particle 1 [2] and [MATH], [MATH], [MATH] runs from 1 to 3 since in line with the approximation made in Refs. [CITATION] that [MATH] is small and hence [MATH].', '0911.0206-2-23-1': 'The couplings [MATH]) are obtained from the resonance pole position on the complex plane and are tabulated in Ref. [CITATION].', '0911.0206-2-23-2': 'To evaluate the two-photon and one photon-one vector partial decay widths of the dynamically generated particles, one needs its coupling to the vector-vector components, i.e., [MATH].', '0911.0206-2-24-0': 'The amplitude of a neutral non-strange vector meson converting into a photon is given by [EQUATION] with [MATH].', '0911.0206-2-24-1': 'Therefore, the whole two-photon and one photon-one vector decay amplitudes for a resonance [MATH] of spin [MATH] are [EQUATION] where [MATH] is a proper isospin coefficient which projects the vector-vector pair in isospin space to that in physical space and [MATH] denotes the coupling of resonance [MATH] to channel [MATH].', '0911.0206-2-24-2': 'Recall that in Ref. [CITATION] we have used the following phase conventions: [MATH] and [MATH], which implies that [EQUATION]', '0911.0206-2-24-3': 'From Eq. ([REF]), one can easily read off the isospin projector [MATH].', '0911.0206-2-25-0': 'Summing over polarization of the intermediate vector mesons in Eqs. ([REF],[REF]) and taking into account symmetry factors and proper normalization, one has the following amplitudes [EQUATION] where [MATH] and [MATH] are defined in Eqs. ([REF],[REF],[REF]) with [MATH]) denoting the polarization vector of a vector-meson (photon); [MATH] is the isospin factor, and [MATH], [MATH] account for both a symmetry factor and the unitary normalization used in Refs. [CITATION] [EQUATION]', '0911.0206-2-25-1': 'The two-photon and one photon-one vector decay widths of a dynamically generated resonance [MATH] of spin [MATH] are then given by [EQUATION] where [MATH] is the resonance mass and [MATH] is the photon momentum in the rest frame of the resonance [MATH].', '0911.0206-2-25-2': 'For the photon, we work in the Coulomb gauge ([MATH] and [MATH]), where the sum over the final polarizations are given by [EQUATION] with [MATH] the three momentum of the photon.', '0911.0206-2-25-3': 'For vector mesons, one has [MATH] [see discussion below Eq. ([REF])] and [EQUATION]', '0911.0206-2-25-4': 'With Eqs. ([REF],[REF]), one can easily verify [EQUATION]', '0911.0206-2-26-0': '# Results and discussions', '0911.0206-2-27-0': 'In this section, we discuss our main results and compare them with available data and the predictions of other approaches.', '0911.0206-2-27-1': 'In Tables [REF], [REF], [REF], we show the calculated one photon-one vector meson and two-photon decay widths of the resonances dynamically generated in Ref. [CITATION].', '0911.0206-2-27-2': 'We have also listed relevant data for the two-photon decay widths from different experiments.', '0911.0206-2-27-3': 'It should be pointed out in our approach that among the 11 dynamically generated resonances [CITATION], the [MATH] state does not decay into [MATH] and [MATH]; the same is true for the [MATH] state; on the other hand, the [MATH], [MATH] and [MATH] resonances only decay into [MATH] but not [MATH].', '0911.0206-2-28-0': 'For the [MATH] and [MATH], Nagahiro et al. have calculated the two-photon decay widths as 2.6 keV and 1.62 keV [CITATION].', '0911.0206-2-28-1': 'Recall in that work among all the SU(3) allowed channels only the [MATH] channel was considered and also the couplings deduced from amplitudes on the real-axis were used.', '0911.0206-2-28-2': 'Therefore, the differences between the two-photon decay widths obtained in the present work and those obtained in Ref. [CITATION] can be viewed as inherent theoretical uncertainties, which are [MATH].', '0911.0206-2-28-3': 'As also discussed in Ref. [CITATION], it is clear from Table [REF] that our two-photon decay width for the [MATH] agrees well with the data.', '0911.0206-2-28-4': 'The experimental situation for the [MATH] is not yet clear, but as discussed in Ref. [CITATION], current experimental results are consistent with our result for [MATH].', '0911.0206-2-29-0': 'In addition to the [MATH] and [MATH] we have calculated the two-photon decay widths of the [MATH] and [MATH].', '0911.0206-2-29-1': 'From Table [REF], it can be seen that they agree reasonably well with available data.', '0911.0206-2-29-2': 'Our calculated two-photon decay width for the [MATH] is slightly smaller than the experimental value quoted in the PDG review.', '0911.0206-2-29-3': 'This is quite acceptable since 1) as discussed earlier we have an inherent theoretical uncertainty of [MATH] and 2) there might be other relevant coupled channels that have not been taken into account in the model of Ref. [CITATION], which can be inferred from the fact that the total decay width of the [MATH] in that model [MATH] MeV is smaller than the experimental value [MATH] MeV.', '0911.0206-2-30-0': 'Note that the significantly small value of the widths of the [MATH] and [MATH] compared to that of the [MATH], for example, has a natural interpretation in our theoretical framework since the former two resonances are mostly [MATH] molecules and therefore the couplings to [MATH], [MATH], [MATH], [MATH], which lead to the final [MATH] decay, are very small.', '0911.0206-2-30-1': 'The advantages of working with coupled channels become obvious in the case of these radiative decays.', '0911.0206-2-30-2': 'While a pure [MATH] assignment would lead to [MATH]=0 keV, our coupled channel analysis gives the right strength for the couplings to the weakly coupled channels.', '0911.0206-2-31-0': 'In the following we shall have a closer look at the radiative decay widths of the [MATH], [MATH], [MATH], [MATH], [MATH], and compare them with the predictions from other theoretical approaches.', '0911.0206-2-32-0': '## Radiative decay widths of [MATH] and [MATH]', '0911.0206-2-33-0': 'In Table [REF], we compare our results for the radiative decay widths for the [MATH] with those obtained in other approaches, including the covariant oscillator quark model (COQM) [CITATION], the tensor-meson dominance (TMD) model [CITATION], the AdS/QCD calculation in [CITATION], the model assuming both tensor-meson dominance and vector-meson dominance (TMDVMD) [CITATION], and the nonrelativistic quark-model (NRQM) [CITATION].', '0911.0206-2-33-1': 'From this comparison, one can see that the AdS/QCD calculation and our present study provide a two-photon decay width consistent with the data.', '0911.0206-2-33-2': 'The TMD model result is also consistent with the data (it can use either [MATH] or [MATH] as an input to fix its single parameter), while the TMDVMD model prediction is off by a factor of 3.', '0911.0206-2-33-3': 'Particularly interesting is the fact that although the TMDVMD model predicts [MATH] similar to our prediction, but in contrast their result for [MATH] is much larger than ours, almost a factor of 30.', '0911.0206-2-33-4': 'Therefore, an experimental measurement of the ratio of [MATH] will be very useful to disentangle these two pictures of the [MATH].', '0911.0206-2-33-5': 'Furthermore, one notices that all theoretical approaches predict [MATH] to be of the order of a few 100 keV.', '0911.0206-2-34-0': 'In Table [REF], we compare the radiative decay widths of the [MATH] predicted in the present work with those obtained in the COQM [CITATION].', '0911.0206-2-34-1': 'We notice that the COQM predicts [MATH] while our model gives an estimate of [MATH], which are quite distinct even taking into account model uncertainties.', '0911.0206-2-34-2': 'Furthermore, [MATH] in the COQM is almost zero while it is comparable to [MATH] in our approach.', '0911.0206-2-34-3': 'An experimental measurement of any two of the three decay widths will be able to confirm either the COQM picture or the dynamical picture.', '0911.0206-2-35-0': 'An interesting quantity in this context is the ratio [MATH] since naturally branching ratios suffer less from systematic uncertainties within a model.', '0911.0206-2-35-1': 'In Table [REF], we compare our result with data and those obtained in other approaches.', '0911.0206-2-35-2': 'It is clear that our result lies within the experimental bounds while those of the effective field approach (EF) [CITATION] and the two-state mixing scheme (TMS) [CITATION] are slightly larger than the experimental upper limit, with the latter being almost at the upper limit.', '0911.0206-2-35-3': 'Given the fact that we have no free parameters in this calculation, such an agreement is reasonable.', '0911.0206-2-36-0': '## Radiative decay widths of [MATH] and [MATH]', '0911.0206-2-37-0': 'Now let us turn our attention to the [MATH] and [MATH] mesons.', '0911.0206-2-37-1': 'In table [REF] we compare our results for the radiative decay widths of the [MATH] and [MATH] obtained by the coupled channel model with the predictions of other theoretical approaches, including the nonrelativistic quark model (NRQM) [CITATION], the light-front quark model (LFQM) [CITATION], the calculation of Nagahiro et al. [CITATION], and the chiral approach [CITATION].', '0911.0206-2-37-2': 'In the NRQM and LFQM calculations three numbers are given for each decay channel depending on whether the glueball mass used in the calculation is smaller than the [MATH] mass (Light), between the [MATH] and [MATH] masses (Medium), or larger than the [MATH] mass (Heavy) [CITATION].', '0911.0206-2-38-0': 'First we note that for the [MATH] our predicted two-photon decay width is more consistent with the LFQM result in the light glueball scenario, while the [MATH] decay width lies closer to the LFQM result in the heavy glueball scenario.', '0911.0206-2-38-1': 'Furthermore, the [MATH] decay width in our model is an order of magnitude smaller than that in the LFQM.', '0911.0206-2-39-0': 'For the [MATH], the LFQM two-photon decay width is larger than the current experimental limit (see Table [REF]).', '0911.0206-2-39-1': 'On the other hand, our [MATH] decay width is more consistent with the LFQM in the light gluon scenario while the [MATH] decay width is more consistent with that of the LFQM in the heavy gluon scenario.', '0911.0206-2-39-2': 'Similar to the [MATH] case, here further experimental data are needed to clarify the situation.', '0911.0206-2-40-0': 'Furthermore, we notice that the NRQM and the LFQM in the light and medium glueball mass scenarios and our present study all predict that [MATH] for the [MATH] while [MATH] for the [MATH].', '0911.0206-2-40-1': 'On the other hand, the NRQM and LFQM in the heavy glueball scenario predict [MATH] for the [MATH].', '0911.0206-2-40-2': 'Therefore, an experimental measurement of the ratio of [MATH] not only will distinguish between the quark-model picture and the dynamical picture, but also will put a constraint on the mass of a possible glueball in this mass region.', '0911.0206-2-41-0': 'The chiral approach in Ref. [CITATION] delivers smaller values for the two-photon decay rates of the [MATH] and [MATH].', '0911.0206-2-41-1': 'However, the ratio [MATH] lies much closer to our prediction [MATH] than the LFQM results which range between 1.7-3.0.', '0911.0206-2-42-0': 'The work of Nagahiro et al. [CITATION] evaluates the contribution from loops of [MATH]) using a phenomenological scalar coupling of the [MATH]) to [MATH]).', '0911.0206-2-42-1': 'From the new perspective on these states we have after the work of Ref. [CITATION], the scalar coupling may not be justified.', '0911.0206-2-42-2': 'One rather has the [MATH] coupling to [MATH] while the coupling of the [MATH] channel only occurs indirectly through the further decay [MATH] and [MATH], with [MATH] going into an internal propagator.', '0911.0206-2-42-3': 'As found in Ref. [CITATION], loops containing these [MATH] propagators only lead to small contributions compared to leading terms including vector mesons (four-vector contact and t(u)-channel vector exchange).', '0911.0206-2-43-0': 'Experimentally, there is a further piece of information on the [MATH] that is relevant to the present study.', '0911.0206-2-43-1': 'From the [MATH] decay branching ratios to [MATH] and [MATH], one can deduce [CITATION]', '0911.0206-2-44-0': '(f_0(1710))(f_0(1710)KK)=Br(J/f_0(1710))Br(J/f_0(1710)KK)=(3.11.0)10^-4(8.5^+1.2_-0.9)10^-4=0.365^+0.156_-0.169.', '0911.0206-2-45-0': 'In the same way as we obtain the two-photon decay widths, we can also calculate the two-vector-meson decay width of the dynamically generated resonances.', '0911.0206-2-45-1': 'For the [MATH], its decay width to [MATH] is found to be [EQUATION]', '0911.0206-2-45-2': 'Using [MATH] MeV, already derived in Ref. [CITATION], and the ratio [MATH] % also given in Ref. [CITATION], one obtains the following branching ratio', '0911.0206-2-46-0': '(f_0(1710))(f_0(1710)KK)&=&0.21 ,', '0911.0206-2-47-0': 'which lies within the experimental bound, although close to the lower limit.', '0911.0206-2-48-0': '## Radiative decay widths of the [MATH]', '0911.0206-2-49-0': 'The radiative decay widths of the [MATH] calculated in the present work are compared with those calculated in the covariant oscillator quark model (COQM) [CITATION] in Table [REF].', '0911.0206-2-49-1': 'We notice that the results from these two approaches differ by a factor of 10.', '0911.0206-2-49-2': 'However, there is one thing in common, i.e., both predict a much larger [MATH] than the [MATH].', '0911.0206-2-49-3': 'More specifically, in the COQM [MATH], while in our model this ratio is [MATH].', '0911.0206-2-50-0': 'At present there is no experimental measurement of these decay modes.', '0911.0206-2-50-1': 'On the other hand, the [MATH] and [MATH] decay rates have been measured.', '0911.0206-2-50-2': 'According to PDG [CITATION], [MATH] keV and [MATH] keV.', '0911.0206-2-50-3': 'Comparing these decay rates with those shown in Table [REF], one immediately notices that the [MATH] in the dynamical model is of similar order as the [MATH] despite reduced phase space in the former decay, which is of course closely related with the fact that the [MATH] is built out of the coupled channel interaction between the [MATH], [MATH], and [MATH] components in the dynamical model.', '0911.0206-2-50-4': 'Furthermore, both the COQM and our dynamical model predict [MATH], which is opposite to the decays into a kaon plus a photon where [MATH].', '0911.0206-2-50-5': 'An experimental measurement of those decays would be very interesting and will certainly help distinguish the two different pictures of the [MATH].', '0911.0206-2-51-0': '# Summary and conclusions', '0911.0206-2-52-0': 'We have calculated the radiative decay widths ([MATH] and [MATH]) of the [MATH], [MATH], [MATH], [MATH], [MATH], and four other states that appear dynamically from vector meson-vector meson interaction in a unitary approach.', '0911.0206-2-52-1': 'Within this approach, due to the peculiarities of the hidden-gauge Lagrangians and the assumption that these resonances are mainly formed by vector meson-vector meson interaction, one can factorize the radiative decay process into a strong part and an electromagnetic part.', '0911.0206-2-52-2': 'This way, the calculation is greatly simplified and does not induce loop calculations.', '0911.0206-2-52-3': 'The obtained results are found to be consistent with existing data within theoretical and experimental uncertainties.', '0911.0206-2-53-0': 'When data are not available, we have compared our predictions with those obtained in other approaches.', '0911.0206-2-53-1': 'In particular, we have identified the relevant pattern of decay rates predicted by different theoretical models and found them quite distinct.', '0911.0206-2-53-2': 'For instance, the [MATH] ratio is quite different in the dynamical model from those in the TMDVMD model and the COQM model.', '0911.0206-2-53-3': 'The [MATH] ratio in the COQM model is also distinctly different from that in the dynamical model.', '0911.0206-2-53-4': 'A measurement of the [MATH]/[MATH] decay rates into [MATH] and [MATH] could be used not only to distinguish between the quark model (NRQM and LFQM) picture and the dynamical picture but also to put a constraint on the mass of a possible glueball (in the [MATH]-g mixing scheme of the NRQM and LFQM).', '0911.0206-2-53-5': 'For the [MATH], as we have discussed, a measurement of its [MATH] decay mode will definitely be able to determine to what extent the dynamical picture is correct.', '0911.0206-2-54-0': 'It is necessary to stress that the QCD dynamics is much richer than that contained in our unitary approach.', '0911.0206-2-54-1': 'It is, therefore, not too surprising to us that sometimes agreement with data is not perfect, but the model delivers at least a qualitative insight into the decay pattern.', '0911.0206-2-54-2': 'However, up to now the dynamical picture of the [MATH], [MATH], [MATH], [MATH], and [MATH] has been tested in a number of scenarios, including in the [MATH] decays [CITATION], [MATH] decays [CITATION], in their strong decay modes [CITATION], and in their two-photon decay modes, as shown in Ref. [CITATION] and in the present work.', '0911.0206-2-54-3': 'It will be interesting to see what comes out in their one photon-one vector meson decay modes.', '0911.0206-2-54-4': 'Given their distinct pattern in different theoretical models, an experimental measurement of some of the decay modes will be very suggestive of the nature of these resonances.', '0911.0206-2-54-5': 'Such measurements in principle could be carried out by PANDA at FAIR or BESIII at BEPCII.'}

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['0911.0206-1-45-1', '0911.0206-2-50-1'], ['0911.0206-1-45-2', '0911.0206-2-50-2'], ['0911.0206-1-45-3', '0911.0206-2-50-3'], ['0911.0206-1-45-4', '0911.0206-2-50-4'], ['0911.0206-1-45-5', '0911.0206-2-50-5'], ['0911.0206-1-40-0', '0911.0206-2-45-0'], ['0911.0206-1-40-1', '0911.0206-2-45-1'], ['0911.0206-1-40-2', '0911.0206-2-45-2'], ['0911.0206-1-20-0', '0911.0206-2-25-0'], ['0911.0206-1-20-1', '0911.0206-2-25-1'], ['0911.0206-1-20-2', '0911.0206-2-25-2'], ['0911.0206-1-20-3', '0911.0206-2-25-3'], ['0911.0206-1-20-4', '0911.0206-2-25-4'], ['0911.0206-1-7-0', '0911.0206-2-7-0'], ['0911.0206-1-7-1', '0911.0206-2-7-1'], ['0911.0206-1-7-2', '0911.0206-2-7-2'], ['0911.0206-1-7-3', '0911.0206-2-7-3'], ['0911.0206-1-7-4', '0911.0206-2-7-4'], ['0911.0206-1-7-5', '0911.0206-2-7-5'], ['0911.0206-1-36-0', '0911.0206-2-41-0'], ['0911.0206-1-36-1', '0911.0206-2-41-1'], ['0911.0206-1-48-0', '0911.0206-2-53-0'], ['0911.0206-1-48-1', '0911.0206-2-53-1'], ['0911.0206-1-48-2', '0911.0206-2-53-2'], ['0911.0206-1-48-3', '0911.0206-2-53-3'], ['0911.0206-1-48-4', '0911.0206-2-53-4'], ['0911.0206-1-48-5', '0911.0206-2-53-5'], ['0911.0206-1-25-0', '0911.0206-2-30-0'], ['0911.0206-1-25-1', '0911.0206-2-30-1'], ['0911.0206-1-25-2', '0911.0206-2-30-2'], ['0911.0206-1-13-0', '0911.0206-2-13-0'], ['0911.0206-1-13-1', '0911.0206-2-13-1'], ['0911.0206-1-13-2', '0911.0206-2-13-2'], ['0911.0206-1-13-3', '0911.0206-2-13-3'], ['0911.0206-1-13-4', '0911.0206-2-13-4'], ['0911.0206-1-37-0', '0911.0206-2-42-0'], ['0911.0206-1-37-1', '0911.0206-2-42-1'], ['0911.0206-1-37-2', '0911.0206-2-42-2'], ['0911.0206-1-37-3', '0911.0206-2-42-3'], ['0911.0206-1-23-0', '0911.0206-2-28-0'], ['0911.0206-1-23-1', '0911.0206-2-28-1'], ['0911.0206-1-23-2', '0911.0206-2-28-2'], ['0911.0206-1-23-3', '0911.0206-2-28-3'], ['0911.0206-1-23-4', '0911.0206-2-28-4'], ['0911.0206-1-11-0', '0911.0206-2-11-0'], ['0911.0206-1-11-1', '0911.0206-2-11-1'], ['0911.0206-1-11-2', '0911.0206-2-11-2'], ['0911.0206-1-47-0', '0911.0206-2-52-0'], ['0911.0206-1-47-1', '0911.0206-2-52-1'], ['0911.0206-1-47-2', '0911.0206-2-52-2'], ['0911.0206-1-47-3', '0911.0206-2-52-3'], ['0911.0206-1-29-0', '0911.0206-2-34-0'], ['0911.0206-1-29-1', '0911.0206-2-34-1'], ['0911.0206-1-29-2', '0911.0206-2-34-2'], ['0911.0206-1-29-3', '0911.0206-2-34-3'], ['0911.0206-1-42-0', '0911.0206-2-47-0'], ['0911.0206-1-3-0', '0911.0206-2-3-0'], ['0911.0206-1-3-1', '0911.0206-2-3-1'], ['0911.0206-1-3-2', '0911.0206-2-3-2'], ['0911.0206-1-3-3', '0911.0206-2-3-3'], ['0911.0206-1-3-4', '0911.0206-2-3-4'], ['0911.0206-1-3-5', '0911.0206-2-3-5'], ['0911.0206-1-19-0', '0911.0206-2-24-0'], ['0911.0206-1-19-1', '0911.0206-2-24-1'], ['0911.0206-1-19-2', '0911.0206-2-24-2'], ['0911.0206-1-19-3', '0911.0206-2-24-3'], ['0911.0206-1-33-0', '0911.0206-2-38-0'], ['0911.0206-1-33-1', '0911.0206-2-38-1'], ['0911.0206-1-24-0', '0911.0206-2-29-0'], ['0911.0206-1-24-1', '0911.0206-2-29-1'], ['0911.0206-1-24-2', '0911.0206-2-29-2'], ['0911.0206-1-24-3', '0911.0206-2-29-3'], ['0911.0206-1-30-0', '0911.0206-2-35-0'], ['0911.0206-1-30-1', '0911.0206-2-35-1'], ['0911.0206-1-30-2', '0911.0206-2-35-2'], ['0911.0206-1-30-3', '0911.0206-2-35-3'], ['0911.0206-1-12-0', '0911.0206-2-12-0'], ['0911.0206-1-12-1', '0911.0206-2-12-1'], ['0911.0206-1-5-0', '0911.0206-2-5-0'], ['0911.0206-1-5-1', '0911.0206-2-5-1'], ['0911.0206-1-5-2', '0911.0206-2-5-2'], ['0911.0206-1-26-0', '0911.0206-2-31-0'], ['0911.0206-1-14-1', '0911.0206-2-14-1'], ['0911.0206-1-14-2', '0911.0206-2-14-2'], ['0911.0206-1-0-0', '0911.0206-2-0-0'], ['0911.0206-1-0-1', '0911.0206-2-0-1'], ['0911.0206-1-0-2', '0911.0206-2-0-2'], ['0911.0206-1-0-3', '0911.0206-2-0-3'], ['0911.0206-1-0-4', '0911.0206-2-0-4'], ['0911.0206-1-0-5', '0911.0206-2-0-5'], ['0911.0206-1-0-6', '0911.0206-2-0-6'], ['0911.0206-1-38-0', '0911.0206-2-43-0'], ['0911.0206-1-38-1', '0911.0206-2-43-1'], ['0911.0206-1-32-0', '0911.0206-2-37-0'], ['0911.0206-1-32-1', '0911.0206-2-37-1'], ['0911.0206-1-32-2', '0911.0206-2-37-2'], ['0911.0206-1-4-0', '0911.0206-2-4-0'], ['0911.0206-1-4-1', '0911.0206-2-4-1'], ['0911.0206-1-4-2', '0911.0206-2-4-2'], ['0911.0206-1-4-3', '0911.0206-2-4-3'], ['0911.0206-1-4-4', '0911.0206-2-4-4'], ['0911.0206-1-17-2', '0911.0206-2-22-1'], ['0911.0206-1-17-3', '0911.0206-2-22-2'], ['0911.0206-1-17-4', '0911.0206-2-22-3'], ['0911.0206-1-17-5', '0911.0206-2-22-4'], ['0911.0206-1-34-0', '0911.0206-2-39-0'], ['0911.0206-1-34-1', '0911.0206-2-39-1'], ['0911.0206-1-34-2', '0911.0206-2-39-2'], ['0911.0206-1-8-0', '0911.0206-2-8-0'], ['0911.0206-1-8-1', '0911.0206-2-8-1'], ['0911.0206-1-18-0', '0911.0206-2-23-0'], ['0911.0206-1-18-1', '0911.0206-2-23-1'], ['0911.0206-1-18-2', '0911.0206-2-23-2'], ['0911.0206-1-49-0', '0911.0206-2-54-0'], ['0911.0206-1-49-1', '0911.0206-2-54-1'], ['0911.0206-1-49-2', '0911.0206-2-54-2'], ['0911.0206-1-49-3', '0911.0206-2-54-3'], ['0911.0206-1-49-4', '0911.0206-2-54-4'], ['0911.0206-1-35-0', '0911.0206-2-40-0'], ['0911.0206-1-35-1', '0911.0206-2-40-1'], ['0911.0206-1-35-2', '0911.0206-2-40-2'], ['0911.0206-1-28-1', '0911.0206-2-33-1'], ['0911.0206-1-7-6', '0911.0206-2-7-6'], ['0911.0206-1-14-0', '0911.0206-2-14-0'], ['0911.0206-1-17-0', '0911.0206-2-22-0']]

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['0911.0206-1-45-1', '0911.0206-2-50-1'], ['0911.0206-1-45-2', '0911.0206-2-50-2'], ['0911.0206-1-45-3', '0911.0206-2-50-3'], ['0911.0206-1-45-4', '0911.0206-2-50-4'], ['0911.0206-1-45-5', '0911.0206-2-50-5'], ['0911.0206-1-40-0', '0911.0206-2-45-0'], ['0911.0206-1-40-1', '0911.0206-2-45-1'], ['0911.0206-1-40-2', '0911.0206-2-45-2'], ['0911.0206-1-20-0', '0911.0206-2-25-0'], ['0911.0206-1-20-1', '0911.0206-2-25-1'], ['0911.0206-1-20-2', '0911.0206-2-25-2'], ['0911.0206-1-20-3', '0911.0206-2-25-3'], ['0911.0206-1-20-4', '0911.0206-2-25-4'], ['0911.0206-1-7-0', '0911.0206-2-7-0'], ['0911.0206-1-7-1', '0911.0206-2-7-1'], ['0911.0206-1-7-2', '0911.0206-2-7-2'], ['0911.0206-1-7-3', '0911.0206-2-7-3'], ['0911.0206-1-7-4', '0911.0206-2-7-4'], ['0911.0206-1-7-5', '0911.0206-2-7-5'], ['0911.0206-1-36-0', '0911.0206-2-41-0'], ['0911.0206-1-36-1', '0911.0206-2-41-1'], ['0911.0206-1-48-0', '0911.0206-2-53-0'], ['0911.0206-1-48-1', '0911.0206-2-53-1'], ['0911.0206-1-48-2', '0911.0206-2-53-2'], ['0911.0206-1-48-3', '0911.0206-2-53-3'], ['0911.0206-1-48-4', '0911.0206-2-53-4'], ['0911.0206-1-48-5', '0911.0206-2-53-5'], ['0911.0206-1-25-0', '0911.0206-2-30-0'], ['0911.0206-1-25-1', '0911.0206-2-30-1'], ['0911.0206-1-25-2', '0911.0206-2-30-2'], ['0911.0206-1-13-0', '0911.0206-2-13-0'], ['0911.0206-1-13-1', '0911.0206-2-13-1'], ['0911.0206-1-13-2', '0911.0206-2-13-2'], ['0911.0206-1-13-3', '0911.0206-2-13-3'], ['0911.0206-1-13-4', '0911.0206-2-13-4'], ['0911.0206-1-37-0', '0911.0206-2-42-0'], ['0911.0206-1-37-1', '0911.0206-2-42-1'], ['0911.0206-1-37-2', '0911.0206-2-42-2'], ['0911.0206-1-37-3', '0911.0206-2-42-3'], ['0911.0206-1-23-0', '0911.0206-2-28-0'], ['0911.0206-1-23-1', '0911.0206-2-28-1'], ['0911.0206-1-23-2', '0911.0206-2-28-2'], ['0911.0206-1-23-3', '0911.0206-2-28-3'], ['0911.0206-1-23-4', '0911.0206-2-28-4'], ['0911.0206-1-11-0', '0911.0206-2-11-0'], ['0911.0206-1-11-1', '0911.0206-2-11-1'], ['0911.0206-1-11-2', '0911.0206-2-11-2'], ['0911.0206-1-47-0', '0911.0206-2-52-0'], ['0911.0206-1-47-1', '0911.0206-2-52-1'], ['0911.0206-1-47-2', '0911.0206-2-52-2'], ['0911.0206-1-47-3', '0911.0206-2-52-3'], ['0911.0206-1-29-0', '0911.0206-2-34-0'], ['0911.0206-1-29-1', '0911.0206-2-34-1'], ['0911.0206-1-29-2', '0911.0206-2-34-2'], ['0911.0206-1-29-3', '0911.0206-2-34-3'], ['0911.0206-1-42-0', '0911.0206-2-47-0'], ['0911.0206-1-3-0', '0911.0206-2-3-0'], ['0911.0206-1-3-1', '0911.0206-2-3-1'], ['0911.0206-1-3-2', '0911.0206-2-3-2'], ['0911.0206-1-3-3', '0911.0206-2-3-3'], ['0911.0206-1-3-4', '0911.0206-2-3-4'], ['0911.0206-1-3-5', '0911.0206-2-3-5'], ['0911.0206-1-19-0', '0911.0206-2-24-0'], ['0911.0206-1-19-1', '0911.0206-2-24-1'], ['0911.0206-1-19-2', '0911.0206-2-24-2'], ['0911.0206-1-19-3', '0911.0206-2-24-3'], ['0911.0206-1-33-0', '0911.0206-2-38-0'], ['0911.0206-1-33-1', '0911.0206-2-38-1'], ['0911.0206-1-24-0', '0911.0206-2-29-0'], ['0911.0206-1-24-1', '0911.0206-2-29-1'], ['0911.0206-1-24-2', '0911.0206-2-29-2'], ['0911.0206-1-24-3', '0911.0206-2-29-3'], ['0911.0206-1-30-0', '0911.0206-2-35-0'], ['0911.0206-1-30-1', '0911.0206-2-35-1'], ['0911.0206-1-30-2', '0911.0206-2-35-2'], ['0911.0206-1-30-3', '0911.0206-2-35-3'], ['0911.0206-1-12-0', '0911.0206-2-12-0'], ['0911.0206-1-12-1', '0911.0206-2-12-1'], ['0911.0206-1-5-0', '0911.0206-2-5-0'], ['0911.0206-1-5-1', '0911.0206-2-5-1'], ['0911.0206-1-5-2', '0911.0206-2-5-2'], ['0911.0206-1-26-0', '0911.0206-2-31-0'], ['0911.0206-1-14-1', '0911.0206-2-14-1'], ['0911.0206-1-14-2', '0911.0206-2-14-2'], ['0911.0206-1-0-0', '0911.0206-2-0-0'], ['0911.0206-1-0-1', '0911.0206-2-0-1'], ['0911.0206-1-0-2', '0911.0206-2-0-2'], ['0911.0206-1-0-3', '0911.0206-2-0-3'], ['0911.0206-1-0-4', '0911.0206-2-0-4'], ['0911.0206-1-0-5', '0911.0206-2-0-5'], ['0911.0206-1-0-6', '0911.0206-2-0-6'], ['0911.0206-1-38-0', '0911.0206-2-43-0'], ['0911.0206-1-38-1', '0911.0206-2-43-1'], ['0911.0206-1-32-0', '0911.0206-2-37-0'], ['0911.0206-1-32-1', '0911.0206-2-37-1'], ['0911.0206-1-32-2', '0911.0206-2-37-2'], ['0911.0206-1-4-0', '0911.0206-2-4-0'], ['0911.0206-1-4-1', '0911.0206-2-4-1'], ['0911.0206-1-4-2', '0911.0206-2-4-2'], ['0911.0206-1-4-3', '0911.0206-2-4-3'], ['0911.0206-1-4-4', '0911.0206-2-4-4'], ['0911.0206-1-17-2', '0911.0206-2-22-1'], ['0911.0206-1-17-3', '0911.0206-2-22-2'], ['0911.0206-1-17-4', '0911.0206-2-22-3'], ['0911.0206-1-17-5', '0911.0206-2-22-4'], ['0911.0206-1-34-0', '0911.0206-2-39-0'], ['0911.0206-1-34-1', '0911.0206-2-39-1'], ['0911.0206-1-34-2', '0911.0206-2-39-2'], ['0911.0206-1-8-0', '0911.0206-2-8-0'], ['0911.0206-1-8-1', '0911.0206-2-8-1'], ['0911.0206-1-18-0', '0911.0206-2-23-0'], ['0911.0206-1-18-1', '0911.0206-2-23-1'], ['0911.0206-1-18-2', '0911.0206-2-23-2'], ['0911.0206-1-49-0', '0911.0206-2-54-0'], ['0911.0206-1-49-1', '0911.0206-2-54-1'], ['0911.0206-1-49-2', '0911.0206-2-54-2'], ['0911.0206-1-49-3', '0911.0206-2-54-3'], ['0911.0206-1-49-4', '0911.0206-2-54-4'], ['0911.0206-1-35-0', '0911.0206-2-40-0'], ['0911.0206-1-35-1', '0911.0206-2-40-1'], ['0911.0206-1-35-2', '0911.0206-2-40-2']]

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[]

[['0911.0206-1-17-0', '0911.0206-2-22-0']]

[]

['0911.0206-1-39-0', '0911.0206-1-41-0', '0911.0206-2-44-0', '0911.0206-2-46-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0911.0206

quant-ph-0110136

{'quant-ph-0110136-1-0-0': 'Computation based on the principles of Quantum Mechanics [CITATION] has been shown to offer better performances over classical computation, ranging from the square-root improvement in an unstructured search [CITATION] to the exponential gain in the factorisation of integers [CITATION].', 'quant-ph-0110136-1-0-1': 'However superior in reducing the complexity of hard computation, these quantum algorithms and all the others discovered so far are only applicable to the classically computable functions.', 'quant-ph-0110136-1-0-2': 'That leaves untouched the class of classically noncomputable functions, such as the halting problem for Turing machines [CITATION].', 'quant-ph-0110136-1-0-3': 'It is in fact widely believed that quantum computation cannot offer anything new about computability [CITATION].', 'quant-ph-0110136-1-0-4': 'Contrary to this, I show that quantum computation can indeed compute the noncomputables, provided certain hamiltonian and its ground state can be physically constructed.', 'quant-ph-0110136-1-0-5': "I present a quantum algorithm for the classically noncomputable Hilbert's tenth problem [CITATION].", 'quant-ph-0110136-1-0-6': 'This algorithm can ultimately solve the halting problem for Turing machines in the computation of partial recursive functions.', 'quant-ph-0110136-1-0-7': 'With this result, the notion of effective computability is extended beyond the Church-Turing thesis of classical computability.', 'quant-ph-0110136-1-1-0': "# The Hilbert's tenth problem", 'quant-ph-0110136-1-2-0': 'At the turn of the last century, David Hilbert listed 23 important problems, among which the problem number ten could be rephrased as:', 'quant-ph-0110136-1-3-0': 'Given any polynomial equation with any number of unknowns and with integer coefficients: To devise a universal process according to which it can be determined by a finite number of operations whether the equation has integer solutions.', 'quant-ph-0110136-1-4-0': 'This decision problem for such polynomial equations, which are also known as diophantine equations, has eventually been shown in 1970 to be undecidable [CITATION] in the Turing sense.', 'quant-ph-0110136-1-4-1': 'It is consequently noncomputable/undecidable in the most general sense if one accepts, as almost everyone does, the Church-Turing thesis of computability.', 'quant-ph-0110136-1-4-2': "Since exponentiation with the unknowns occur in the exponents (as occurs in the example of the Fermat's last theorem) can be shown to be diophantine with supplementary equations, the study of diophantine equations essentially covers the class of partial recursive functions, which is at the foundations of classical algorithms.", 'quant-ph-0110136-1-4-3': "The undecidability result is thus singularly important: The Hilbert's tenth problem could be solved if and only if the Turing halting problem could be.", 'quant-ph-0110136-1-5-0': '# A quantum algorithm', 'quant-ph-0110136-1-6-0': 'It suffices to consider nonnegative solutions, if any, of a diophantine equation.', 'quant-ph-0110136-1-6-1': 'Let us consider the example [EQUATION] with unknowns [MATH], [MATH], and [MATH].', 'quant-ph-0110136-1-6-2': 'To find out whether this equation has any nonnegative integer solution by quantum algorithms, it requires the realisation of a Fock space built out of the "vacuum" [MATH] by repeating applications of the creation operators [MATH], [MATH] and [MATH], similarly to that of the 3-D simple harmonic oscillators.', 'quant-ph-0110136-1-6-3': '[EQUATION]', 'quant-ph-0110136-1-6-4': 'Upon this Hilbert space, we construct the hamiltonian corresponding to ([REF]) [EQUATION] which has a spectrum bounded from below - semidefinite, in fact.', 'quant-ph-0110136-1-7-0': 'Note that the operators [MATH] have only nonnegative integer eigenvalues [MATH], and that [MATH] so these observables are compatible - they are simultaneously measurable.', 'quant-ph-0110136-1-7-1': 'The ground state [MATH] of the hamiltonian so constructed has the properties [EQUATION] for some [MATH].', 'quant-ph-0110136-1-8-0': 'Thus a projective measurement of the energy [MATH] of the ground state [MATH] will yield the answer for the decision problem: The diophantine equation has at least one integer solution if and only if [MATH], and has not otherwise.', 'quant-ph-0110136-1-8-1': "(If [MATH] in our example, we know that [MATH] from the Fermat's last theorem.)", 'quant-ph-0110136-1-9-0': 'If there is one unique solution then the projective measurements of the observables corresponding to the operators [MATH] will reveal the values of various unknowns.', 'quant-ph-0110136-1-9-1': 'If there are many solutions, finitely or infinitely as in the case of [MATH], the ground state [MATH] will be a linear superposition of states of the form [MATH], where [MATH] are the solutions.', 'quant-ph-0110136-1-9-2': 'In such situation, the measurement may not yield all the solutions.', 'quant-ph-0110136-1-9-3': 'However, finding all the solutions is not the aim of a decision procedure for this kind of problem.', 'quant-ph-0110136-1-10-0': 'Notwithstanding this, measurements of [MATH] of the ground state would always yield some values [MATH] and a straightforward sustitution would confirm if the equation has a solution or not.', 'quant-ph-0110136-1-10-1': 'Thus the measurement on the ground state either of the energy, provided the zero point can be calibrated, or of the number operators will be sufficient to give the result for the decision problem.', 'quant-ph-0110136-1-11-0': 'The quantum algorithm is thus clear:', 'quant-ph-0110136-1-12-0': "Given a diophantine equation with [MATH] unknowns [MATH]'s [EQUATION] we need to simulate on some appropriate Fock space the quantum hamiltonian [EQUATION]", 'quant-ph-0110136-1-12-1': 'If the ground state could be obtained with high probability, measurements of appropriate observables would provide the answer for our decision problem.', 'quant-ph-0110136-1-13-0': 'The key ingredients are the availability of a countably infinite number of Fock states, the ability to construct/simulate a suitable hamiltonian and to obtain its ground state.', 'quant-ph-0110136-1-13-1': 'As a counterpart of the semi-infinite tape of a Turing machine, the Fock space is employed here instead of the qubits of the more well-known model of quantum computation.', 'quant-ph-0110136-1-13-2': 'Its advantage over the infinitely many qubits which would otherwise be required is obvious.', 'quant-ph-0110136-1-14-0': 'Why should this work?', 'quant-ph-0110136-1-14-1': 'We do not look for the zeroes of the polynomial, [MATH], which may not exist, but instead search for the absolute minimum of its square, [MATH], which exists.', 'quant-ph-0110136-1-14-2': 'While it is equally hard to find either the zeroes or the absolute minimum in classical computation, we have converted the problem to the realisation of the ground state of a quantum hamiltonian and there is no known quantum principle against such act.', 'quant-ph-0110136-1-14-3': 'In fact, there is no known physical principles against it.', 'quant-ph-0110136-1-14-4': 'Let us consider the three laws of thermodynamics concerning energy conservation, entropy of closed systems and the unattainability of absolute zero temperature.', 'quant-ph-0110136-1-14-5': 'The energy involved in our algorithm is finite, being the ground state energy of some hamiltonian.', 'quant-ph-0110136-1-14-6': 'The entropy increase which ultimately connects to decoherence effects is a technical problem for all quantum computation in general, and we will discuss this further below.', 'quant-ph-0110136-1-14-7': 'As we will never obtain the absolute zero temperature, we only need to satisfy ourselves that the required ground state can be achieved with a more-than-even chance.', 'quant-ph-0110136-1-14-8': 'Then there is a probability boosting technique, see later, to bring that chance to as closed to unity as one pleases.', 'quant-ph-0110136-1-15-0': 'The general algorithm above could be realised by, but in no way restricted to, the following methods to simulate the required hamiltonian and to obtain the ground state adiabatically.', 'quant-ph-0110136-1-16-0': '# Simulating the hamiltonians', 'quant-ph-0110136-1-17-0': 'One way to construct any suitable hamiltonian so desired is through the technique of ref. [CITATION].', 'quant-ph-0110136-1-17-1': 'We consider the hermitean operators, where [MATH] is the index of the unknowns of the diophantine equation, [EQUATION]', 'quant-ph-0110136-1-17-2': 'Together with the availability of the fundamental hamiltonians [EQUATION] one could construct the unitary time evolutions corresponding to hamiltonians of arbitrary hermitean polynomials in [MATH], and hence in [MATH], to an arbitrary degree of accuracy.', 'quant-ph-0110136-1-17-3': 'These fundamental hamiltonians correspond to translations, phase shifts, squeezers, beam splitters and Kerr nonlinearity.', 'quant-ph-0110136-1-18-0': 'With the polynomial hamiltonian constructed, we need to obtain its ground state.', 'quant-ph-0110136-1-18-1': 'Any approach that allows us to access the ground state will suffice.', 'quant-ph-0110136-1-18-2': 'One way is perhaps to use that of quantum annealing or cooling [CITATION].', 'quant-ph-0110136-1-18-3': 'Another way is to employ the quantum computation method of adiabatic evolution [CITATION].', 'quant-ph-0110136-1-19-0': '# Adiabatic evolution', 'quant-ph-0110136-1-20-0': 'In the adiabatic approach, one starts with a hamiltonian [MATH] whose ground state [MATH] is readily achievable.', 'quant-ph-0110136-1-20-1': 'Then one forms the slowly varying hamiltonian [MATH], [MATH], which interpolates between [MATH] and [MATH] in the time interval [MATH] [EQUATION]', 'quant-ph-0110136-1-20-2': 'Note that we can replace this linear interpolation by some non-linear one provided the conditions of the adiabatic theorem are observed.', 'quant-ph-0110136-1-20-3': 'According to this theorem, the initial ground state will evolve into our desirable ground state [MATH] up to a phase: [EQUATION]', 'quant-ph-0110136-1-20-4': 'For the hamiltonian ([REF]), an estimate of the time [MATH] after which the system remains with high probability in the ground state is [EQUATION] with [EQUATION] and [EQUATION] where [MATH] and [MATH] are respectively the instantaneous ground state and the first excited state of ([REF]) with instantaneous eigenvalues [MATH], [MATH].', 'quant-ph-0110136-1-21-0': 'The time-ordering operator on the left hand side of ([REF]) can be approximated as [EQUATION] for [CITATION] [EQUATION]', 'quant-ph-0110136-1-21-1': 'Note that we have employed here the "norm" [MATH] as defined in ([REF]) for the various hamiltonians which are unbounded from above.', 'quant-ph-0110136-1-21-2': 'This norm is the relevant measure for the problem is only concerned with the lowest states of the interpolating hamiltonian ([REF]).', 'quant-ph-0110136-1-21-3': 'In each interval [MATH], the unitary operators [MATH], for [MATH], can be expressed through the subdivision of [MATH] into [MATH] subintervals of sufficiently small size [MATH] satisfying [MATH], [EQUATION].', 'quant-ph-0110136-1-21-4': 'Each of the [MATH] factors on the right hand side of the last expression can be now simulated through the approach of [CITATION], where it was shown that the number of steps [MATH] grows as a small polynomial in the order of the polynomial in the hamiltonian [MATH] to be simulated, the accuracy to be enacted, and the time interval [MATH] over which it is to be applied.', 'quant-ph-0110136-1-22-0': 'In this way, the requirements of the adiabatic conditions on the one hand and of, on the other hand, the simulations of the hamiltonians in the time interval [MATH] can be satisfied.', 'quant-ph-0110136-1-23-0': '# Decoherence and error correction', 'quant-ph-0110136-1-24-0': 'Our approach above is in fact a combination of the quantum computation of continuous variables and of adiabatic evolution.', 'quant-ph-0110136-1-24-1': 'There exists some error correction protocol for continuous variables [CITATION] which could be of help here to protect the wavefunctions from decoherence.', 'quant-ph-0110136-1-24-2': 'However, the adiabatic computation we exploit is quite robust in general [CITATION].', 'quant-ph-0110136-1-24-3': 'An imperfect conventional quantum algorithm might have different sorts of errors than an imperfect adiabatic process, where the system is kept close to the instantaneous ground state over time.', 'quant-ph-0110136-1-24-4': 'Decoherence by the environment inducing transition between levels could be controlled in principle at a temperature that is small compared to the gap ([REF]), given its estimate.', 'quant-ph-0110136-1-24-5': 'Errors introduced by the hamiltonian simulation may result in a hamiltonian different from ([REF]) but in a form [MATH].', 'quant-ph-0110136-1-24-6': 'Recent numerical study [CITATION] of small systems has in fact indicated that the adiabatic computation is interestingly robust even for fairly large [MATH], provided [MATH] varies either sufficiently slowly or sufficiently rapidly (which is more likely to be the case considered here due to the nature of our hamiltonian simulations).', 'quant-ph-0110136-1-25-0': '# Gap estimation', 'quant-ph-0110136-1-26-0': 'The question of computational complexity, i.e. how large [MATH] is and how small [MATH], [MATH] are for a high probability of measurement success, as seen from the various bounds above is dependent on [MATH], i.e. on the specific diophantine equation in question, and on the initial hamiltonian [MATH].', 'quant-ph-0110136-1-26-1': 'Some estimate for the energy gap ([REF]) is crucial for those of the total computational time [MATH] and also of the temperature to control level transitions.', 'quant-ph-0110136-1-26-2': 'The authors of [CITATION] have pointed out that typically [MATH] is not zero unless the hamiltonian has special symmetry properties.', 'quant-ph-0110136-1-26-3': 'As the levels will cross if there is a symmetry, a necessary condition for no level crossing in our case is that the operators [MATH] do not commute with [MATH], thus do not commute with [MATH] for [MATH], even though they commute with [MATH].', 'quant-ph-0110136-1-26-4': 'Furthermore, the freedom in choosing the initial hamiltonians [MATH] and their ground states and in performing different adiabatic interpolations, not just linear as in ([REF]), might be exploited to enable the gap estimation and to speed up the computation.', 'quant-ph-0110136-1-27-0': 'We shall employ the simple harmonic oscillators, i.e. gaussian approximations, to obtain an estimate for the energy gap.', 'quant-ph-0110136-1-27-1': 'Needed firstly is a version of the Wick theorem [CITATION] for the ordinary product involving the operators [MATH], ..., [MATH], [EQUATION] where the normal ordering [MATH] is done with respect to some annihilation operators [MATH] which annihilate some common state [MATH] for various [MATH], [MATH] -when the operators [MATH], [MATH], ... can be expressed in terms of [MATH] and [MATH].', 'quant-ph-0110136-1-27-2': 'The equation ([REF]), in which the hatted operators are omitted from the normal ordering and [EQUATION] is an exact result and can be proved by induction.', 'quant-ph-0110136-1-28-0': 'We also employ the Bogoliubov ansatz, with real [MATH] and [MATH], for the various [MATH], [MATH] appearing in the interpolating hamiltonian [MATH] ([REF]) at the time instant [MATH] [EQUATION]', 'quant-ph-0110136-1-28-1': 'The [MATH]-zero occupation state is denoted by [MATH], where the [MATH]-dependence of the state is implicit.', 'quant-ph-0110136-1-28-2': 'The canonicity [MATH] demands [EQUATION] upon which [EQUATION]', 'quant-ph-0110136-1-28-3': 'We pay particular attention to the following terms [EQUATION]', 'quant-ph-0110136-1-28-4': 'We can now list the various steps of our estimation:', 'quant-ph-0110136-1-29-0': "We apply the Wick theorem, with respect to [MATH]'s, to the hamiltonian [MATH] ([REF]) [EQUATION]", 'quant-ph-0110136-1-29-1': 'Note that in this process the higher-order terms contribute to the coefficients of lower-order ones through products of various expectation values in ([REF]).', 'quant-ph-0110136-1-29-2': 'With help from ([REF]) and ([REF]), the various coefficients [MATH], [MATH], [MATH], ... in the right hand side above can be expressed as polynomials in [MATH].', 'quant-ph-0110136-1-29-3': 'We next fix [MATH], and [MATH] numerically from ([REF]) and from the imposition that the coefficients [MATH] in ([REF]).', 'quant-ph-0110136-1-29-4': 'Then we can evaluate the coefficients [MATH] from their polynomial expressions in [MATH].', 'quant-ph-0110136-1-29-5': 'From ([REF]), on the other hand, we see also that [EQUATION] where [MATH].', 'quant-ph-0110136-1-29-6': 'Mathematically, [MATH] thus provides some indication of the size of the energy gaps of [MATH] around the energy level [MATH].', 'quant-ph-0110136-1-29-7': 'Even though [MATH], obtained from the linear Bogoliubov transformations ([REF]), in general may not be the true ground state of [MATH], we could use [MATH] as some indicator for the gap [MATH] in ([REF]) in order to estimate [MATH] from the adiabaticity condition ([REF]).', 'quant-ph-0110136-1-30-0': 'We have tried out the one-dimensional example of [MATH], the method indeed yields the exact result for the required gap, which is one.', 'quant-ph-0110136-1-31-0': 'Some variations of the above method might yield a better estimate.', 'quant-ph-0110136-1-31-1': 'For instance, one could numerically obtain [MATH], and thus [MATH], by minimising the energy [MATH] subjected to the constraints ([REF]) with or without the constraints [MATH].', 'quant-ph-0110136-1-31-2': 'The aim of this minimisation, if possible, is to select a state [MATH] whose energy expectation value at the time [MATH] is as closed to that of the ground state at that instant as allowed by the linear Bogoliubov transformations.', 'quant-ph-0110136-1-31-3': 'Other variations might be involving, instead of ([REF]), some non-linear relations, for the canonicity only requires that [MATH] and [MATH] are to be unitarily transformed, [MATH].', 'quant-ph-0110136-1-32-0': 'The accuracy of the estimation and its higher order corrections can be evaluated systematically from the higher order terms in the last line of ([REF]).', 'quant-ph-0110136-1-32-1': 'Details will apppear elsewhere.', 'quant-ph-0110136-1-32-2': 'Numerical diagonalisation of ([REF]) with a series of truncated bases in [MATH] of increasing sizes could further provide us more information about the gap thus obtained.', 'quant-ph-0110136-1-33-0': 'It turns out that with the method above we have traded the original diophantine equation for a set of polynomials equations in [MATH].', 'quant-ph-0110136-1-33-1': 'The solutions of these latter equations, however, are real, not integers as in the case of the original diophantine, and could be obtained or well approximated through the calculus of continuous variables.', 'quant-ph-0110136-1-33-2': '(Tarski [CITATION] has found that looking for the existence of real solutions of polynomials over the real numbers are less restrictive than that of integer solutions of diophantine equations; he has shown that arithmetics over the reals is, in fact, decidable.)', 'quant-ph-0110136-1-33-3': 'Note that in some cases we may not even need to solve the equations exactly in order to have some appropriate estimate for [MATH].', 'quant-ph-0110136-1-34-0': 'To verify a result obtained from the quantum algorithm, we substitute the set of finally measured integers [MATH] into the diophantine equation.', 'quant-ph-0110136-1-34-1': 'If it gives zero, then our task is completed.', 'quant-ph-0110136-1-34-2': 'If not, we numerically diagonalise the hamiltonians in a truncated but sufficiently large basis that contains [MATH] to see if this state is indeed the ground state and to obtain the gap and compare it with the [MATH] we used.', 'quant-ph-0110136-1-34-3': 'Any inconsistency would warrant a new run with new parameters just obtained.', 'quant-ph-0110136-1-34-4': 'That is, in the no solution case we need to satisfy ourselves that the state obtained is the ground state.', 'quant-ph-0110136-1-35-0': 'More general, we have traded the hard question of integer solutions for some more palatable estimation for the real-valued time [MATH] in ([REF]).', 'quant-ph-0110136-1-35-1': 'This estimation is less restrictive in the sense that it needs only be of order of magnitude for a good success probability, which could be boosted further.', 'quant-ph-0110136-1-36-0': '# Probability boosting', 'quant-ph-0110136-1-37-0': 'Due to the particular nature of the present scheme, the various approximations result in some probability that the final measurement will not find the system in the desired ground state; but appropriate choices of the various time parameters could increase the success probability of the algorithm to more than even.', 'quant-ph-0110136-1-37-1': 'We spell out explicitly here the probability boosting technique, as mentioned in [CITATION], that one could subsequently apply.', 'quant-ph-0110136-1-38-0': 'In our computation, adiabatic or otherwise, the end result, starting from some initial state [MATH], may be contaminated with some excited states other than the desirable ground state [MATH], [EQUATION]', 'quant-ph-0110136-1-38-1': 'If [MATH], that is, if there is a better-than-even chance to obtain [MATH] by measurement then one can boost the success probability to arbitrarily closed to unity by performing a concatenated computation over [MATH] Hilbert spaces [EQUATION] where [MATH] is the normalising factor.', 'quant-ph-0110136-1-38-2': 'Let us consider the majority amplitudes, when more than half of the [MATH] Hilbert spaces return the correct results, [MATH]; and the minority amplitudes, [MATH].', 'quant-ph-0110136-1-38-3': 'The ratio of the majority over the minority, [MATH], is clearly boosted for sufficiently large [MATH] and for [MATH] and [MATH].', 'quant-ph-0110136-1-38-4': 'The probability distribution for a measurement of the end state in ([REF]), as a consequence, is exponentially dominated by the majority results.', 'quant-ph-0110136-1-38-5': 'In other words, the probability to obtain a majority result which contains the true ground state can be made arbitrarily closed to unity, [MATH], provided [MATH] and [MATH].', 'quant-ph-0110136-1-38-6': 'However, the decoherence consideration for such [MATH] concatenated Hilbert spaces will be more crucial.', 'quant-ph-0110136-1-39-0': '# General quantum computation', 'quant-ph-0110136-1-40-0': 'Our result is in contrast to the claim in [CITATION] that quantum Turing machines compute exactly the same class of functions as do Turing machines, albeit perhaps more efficiently.', 'quant-ph-0110136-1-40-1': 'We could only offer here some speculations about this apparent discrepancy.', 'quant-ph-0110136-1-40-2': 'The quantum Turing machine approach is a direct generalisation of that of the classical Turing machines but with qubits and some universal set of one-qubit and two-qubit unitary gates to build up, step by step, dimensionally larger, but still dimensionally finite unitary operations.', 'quant-ph-0110136-1-40-3': 'This universal set is chosen on its ability to evaluate any desirable classical logic function.', 'quant-ph-0110136-1-40-4': 'Our approach, on the other hand, is from the start based on infinite-dimension hamiltonians acting on some Fock space and also based on the special properties and unique status of their ground states.', 'quant-ph-0110136-1-40-5': 'The unitary operations are then followed as the Shrodinger time evolutions.', 'quant-ph-0110136-1-40-6': 'Even at the hamiltonian level higher orders of the operators [MATH] and [MATH], i.e. not just two-body but many-body interactions in a sense, are already present.', 'quant-ph-0110136-1-40-7': 'This proliferation, which is even more pronounced at the level of the time-evolution operators, together with the infinite dimensionality and the unique energetic status of the vacuum could be the reasons behind the ability to compute, in a finite number of steps, what the dimensionally finite unitary operators of the standard quantum Turing computation cannot do in a finite number of steps.', 'quant-ph-0110136-1-40-8': 'Note that it was the general hamiltonian computation that was discussed by Benioff and Feynman [CITATION] in the conception days of quantum computation.', 'quant-ph-0110136-1-41-0': '# Turing halting problem', 'quant-ph-0110136-1-42-0': 'The halting problem for Turing machines is also a manifestation of undecidability: a Turing computation is equivalent to the computation of a partial recursive function, which is only defined for a subset of the integers; as this domain is classically undecidable, one cannot always tell in advance whether the Turing machine will halt (that is, whether the input is in the domain of the partial recursive function) or not (when the input is not in the domain).', 'quant-ph-0110136-1-43-0': 'A version of the proof of the unsolvability of the halting problem based on the Cantor diagonal argument goes as follows.', 'quant-ph-0110136-1-43-1': 'The proof is by contradiction with the assumption of the existence of a computable halting function [MATH] which has two integer arguments - [MATH] is the Godel encoded integer number for the algorithm and [MATH] is its (encoded) integer input: [EQUATION]', 'quant-ph-0110136-1-43-2': 'One can then construct a program [MATH] having one integer argument [MATH] in such a way that it calls the function [MATH] and [EQUATION]', 'quant-ph-0110136-1-43-3': 'The application of the halting function [MATH] on the program [MATH] and input [MATH] results in [EQUATION]', 'quant-ph-0110136-1-43-4': 'A contradiction is clearly manifest once we put [MATH] in the last equation above.', 'quant-ph-0110136-1-44-0': 'The construction of such program [MATH] is transparently possible, unless the existence of a computable [MATH] is wrongly assumed.', 'quant-ph-0110136-1-44-1': 'Thus the contradiction discounts the assumption that there is a classically algorithmic way to determine whether any arbitrarily given program with arbitrary input will halt or not.', 'quant-ph-0110136-1-45-0': 'This contradiction argument might be side stepped if we distinguish and separate the two classes of quantum and classical algorithms.', 'quant-ph-0110136-1-45-1': 'A quantum function [MATH], similar to eq. ([REF]), can conceivably exist to determine whether any classical program [MATH] will halt on any classical input [MATH] or not.', 'quant-ph-0110136-1-45-2': 'The contradiction in eq. ([REF]) would be avoided if the quantum halting [MATH] cannot take as argument the modified program [MATH], which is now of quantum character because it now has quantum [MATH] as a subroutine.', 'quant-ph-0110136-1-45-3': 'This will be the case if [MATH] can only accept integer while quantum algorithms, with proper definitions, cannot in general be themselves encoded as integers.', 'quant-ph-0110136-1-45-4': 'It is clear even in the case of a single qubit that the state [MATH] cannot be encoded as integers for all [MATH] and [MATH] - simply because of different cardinalities.', 'quant-ph-0110136-1-45-5': 'In fact, the no-cloning theorem [CITATION] of quantum mechanics does restrict the type of operations available to quantum algorithms.', 'quant-ph-0110136-1-46-0': 'In essence, the way we break the self-referential reasoning here by the differentiation between quantum and classical algorithms is similar to the way Bertrand Russell resolved the set theory paradox (to do with "The set of all sets which are not a member of themselves") by the introduction of classes as distinct from sets.', 'quant-ph-0110136-1-47-0': '# Implications', 'quant-ph-0110136-1-48-0': 'We here put emphasis on the issue of computability in principle, not on that of computational complexity.', 'quant-ph-0110136-1-48-1': "This broadening of the concept of effective computability, taken into account the quantum mechanical principles, has been shown to be able in principle to probabilistically decide the classically undecidables, the Hilbert's tenth problem and thus the Turing halting problem in this instance.", 'quant-ph-0110136-1-48-2': "If this is realisable, and we don't have any evidence to the contrary, the Church-Turing thesis should be modified accordingly.", 'quant-ph-0110136-1-49-0': 'That some generalisation of the notion of computation could help solving the previous undecidability/noncomputability has been recognised and was considered by Kleene as quoted in [CITATION].', 'quant-ph-0110136-1-49-1': 'But this has not been realisable until now simply because of the non-recognition of quantum physics as the missing ingredient.', 'quant-ph-0110136-1-49-2': 'Our quantum algorithm could in fact be regarded as an infinite search through the integers in a finite amount of time, the type of search required by Kleene to solve the Turing halting problem.', 'quant-ph-0110136-1-50-0': "Our decidability result here only deals with the property of being diophantine, which does not cover the property of being arithmetic in general, and as such has no direct consequences on the Godel's Incompleteness theorem.", 'quant-ph-0110136-1-50-1': "However, it is conceivable that the Godel's theorem may lose its restrictive power once the concept of proof is suitably generalised with quantum principles."}

{'quant-ph-0110136-2-0-0': "We propose a quantum algorithm for the classically non-computable Hilbert's tenth problem, which ultimately links to the Turing halting problem.", 'quant-ph-0110136-2-0-1': 'Quantum continuous variables and quantum adiabatic evolution are employed for an implementation.', 'quant-ph-0110136-2-0-2': 'Also discussed are a method for the time estimation for the adiabatic evolution, and a comparison with more the well-known quantum computation employing a finite number of qubits.', 'quant-ph-0110136-2-0-3': 'Provided certain hamiltonian and its ground state can be physically constructed according to the algorithm, the notion of effective computability is extended beyond the Church-Turing thesis of classical computability.', 'quant-ph-0110136-2-1-0': '# Introduction', 'quant-ph-0110136-2-2-0': 'Computation based on the principles of Quantum Mechanics [CITATION] has been shown to offer better performances over classical computation, ranging from the square-root improvement in an unstructured search [CITATION] to the exponential gain in the factorisation of integers [CITATION].', 'quant-ph-0110136-2-2-1': 'However superior in reducing the complexity of hard computation, these quantum algorithms and all the others discovered so far are only applicable to the classically computable functions.', 'quant-ph-0110136-2-2-2': 'That leaves untouched the class of classically noncomputable functions, such as the halting problem for Turing machines [CITATION].', 'quant-ph-0110136-2-2-3': 'It is in fact widely believed that quantum computation cannot offer anything new about computability [CITATION].', 'quant-ph-0110136-2-2-4': 'Contrary to this, We propose that quantum computation may be able to compute the noncomputables, provided certain hamiltonian and its ground state can be physically constructed.', 'quant-ph-0110136-2-2-5': "We propose a quantum algorithm for the classically noncomputable Hilbert's tenth problem [CITATION].", 'quant-ph-0110136-2-2-6': 'This algorithm ultimately links to the halting problem for Turing machines in the computation of partial recursive functions.', 'quant-ph-0110136-2-3-0': 'In the below we propose a quantum algorithm for this class of problems as a matter of principle.', 'quant-ph-0110136-2-3-1': 'The practical details of implemetation are not consider in this conceptual study.', 'quant-ph-0110136-2-4-0': "# Hilbert's tenth problem", 'quant-ph-0110136-2-5-0': 'At the turn of the last century, David Hilbert listed 23 important problems, among which the problem number ten could be rephrased as:', 'quant-ph-0110136-2-6-0': 'Given any polynomial equation with any number of unknowns and with integer coefficients: To devise a universal process according to which it can be determined by a finite number of operations whether the equation has integer solutions.', 'quant-ph-0110136-2-7-0': 'This decision problem for such polynomial equations, which are also known as diophantine equations, has eventually been shown in 1970 by Matiyasevich to be undecidable [CITATION] in the Turing sense.', 'quant-ph-0110136-2-7-1': 'It is consequently noncomputable/undecidable in the most general sense if one accepts, as almost everyone does, the Church-Turing thesis of computability.', 'quant-ph-0110136-2-7-2': "Since exponential diophantine, with the unknowns in the exponents as in the example of the Fermat's last theorem, can be shown to be diophantine with supplementary equations, the study of diophantine equations essentially covers the class of partial recursive functions, which is at the foundations of classical algorithms.", 'quant-ph-0110136-2-7-3': "The undecidability result is thus singularly important: The Hilbert's tenth problem could be solved if and only if the Turing halting problem could be.", 'quant-ph-0110136-2-8-0': '# Turing halting problem', 'quant-ph-0110136-2-9-0': 'The halting problem for Turing machines is also a manifestation of undecidability: a Turing computation is equivalent to the computation of a partial recursive function, which is only defined for a subset of the integers; as this domain is classically undecidable, one cannot always tell in advance whether the Turing machine will halt (that is, whether the input is in the domain of the partial recursive function) or not (when the input is not in the domain).', 'quant-ph-0110136-2-10-0': 'A version of the proof of the unsolvability of the halting problem based on the Cantor diagonal argument goes as follows.', 'quant-ph-0110136-2-10-1': 'The proof is by contradiction with the assumption of the existence of a computable halting function [MATH] which has two integer arguments - [MATH] is the Godel encoded integer number for the algorithm and [MATH] is its (encoded) integer input: [EQUATION]', 'quant-ph-0110136-2-10-2': 'One can then construct a program [MATH] having one integer argument [MATH] in such a way that it calls the function [MATH] and [EQUATION]', 'quant-ph-0110136-2-10-3': 'The application of the halting function [MATH] on the program [MATH] and input [MATH] results in [EQUATION]', 'quant-ph-0110136-2-10-4': 'A contradiction is clearly manifest once we put [MATH] in the last equation above.', 'quant-ph-0110136-2-11-0': 'The construction of such program [MATH] is transparently possible, unless the existence of a computable [MATH] is wrongly assumed.', 'quant-ph-0110136-2-11-1': 'Thus the contradiction discounts the assumption that there is a classically algorithmic way to determine whether any arbitrarily given program with arbitrary input will halt or not.', 'quant-ph-0110136-2-12-0': 'This contradiction argument might be side stepped if we distinguish and separate the two classes of quantum and classical algorithms.', 'quant-ph-0110136-2-12-1': 'A quantum function [MATH], similar to eq. ([REF]), can conceivably exist to determine whether any classical program [MATH] will halt on any classical input [MATH] or not.', 'quant-ph-0110136-2-12-2': 'The contradiction in eq. ([REF]) would be avoided if the quantum halting [MATH] cannot take as argument the modified program [MATH], which is now of quantum character because it now has quantum [MATH] as a subroutine.', 'quant-ph-0110136-2-12-3': 'This will be the case if [MATH] can only accept integer while quantum algorithms, with proper definitions, cannot in general be themselves encoded as integers.', 'quant-ph-0110136-2-12-4': 'It is clear even in the case of a single qubit that the state [MATH] cannot be encoded as integers for all [MATH] and [MATH] - simply because of different cardinalities.', 'quant-ph-0110136-2-12-5': 'In fact, the no-cloning theorem [CITATION] of quantum mechanics does restrict the type of operations available to quantum algorithms.', 'quant-ph-0110136-2-13-0': 'In essence, the way we will break the self-referential reasoning here by the differentiation between quantum and classical algorithms is similar to the way Bertrand Russell resolved the set theory paradox (to do with "The set of all sets which are not members of themselves") by the introduction of classes as distinct from sets.', 'quant-ph-0110136-2-13-1': '(For other lines of arguments, see [CITATION] for example.)', 'quant-ph-0110136-2-14-0': '# A quantum algorithm', 'quant-ph-0110136-2-15-0': 'It suffices to consider nonnegative solutions, if any, of a diophantine equation.', 'quant-ph-0110136-2-15-1': 'Let us consider the example [EQUATION] with unknowns [MATH], [MATH], and [MATH].', 'quant-ph-0110136-2-16-0': 'To find out whether this equation has any nonnegative integer solution by quantum algorithms, it requires the realisation of a Fock space built out of the "vacuum" [MATH] by repeating applications of the creation operators [MATH], [MATH] and [MATH], similarly to that of the 3-D simple harmonic oscillators.', 'quant-ph-0110136-2-16-1': '[EQUATION]', 'quant-ph-0110136-2-16-2': 'Upon this Hilbert space, we construct the hamiltonian corresponding to ([REF]) [EQUATION] which has a spectrum bounded from below - semidefinite, in fact.', 'quant-ph-0110136-2-17-0': 'Note that the operators [MATH] have only nonnegative integer eigenvalues [MATH], and that [MATH] so these observables are compatible - they are simultaneously measurable.', 'quant-ph-0110136-2-17-1': 'The ground state [MATH] of the hamiltonian so constructed has the properties [EQUATION] for some [MATH].', 'quant-ph-0110136-2-18-0': 'Thus a projective measurement of the energy [MATH] of the ground state [MATH] will yield the answer for the decision problem: The diophantine equation has at least one integer solution if and only if [MATH], and has not otherwise.', 'quant-ph-0110136-2-18-1': "(If [MATH] in our example, we know that [MATH] from the Fermat's last theorem.)", 'quant-ph-0110136-2-19-0': 'If there is one unique solution then the projective measurements of the observables corresponding to the operators [MATH] will reveal the values of various unknowns.', 'quant-ph-0110136-2-19-1': 'If there are many solutions, finitely or infinitely as in the case of [MATH], the ground state [MATH] will be a linear superposition of states of the form [MATH], where [MATH] are the solutions.', 'quant-ph-0110136-2-19-2': 'In such situation, the measurement may not yield all the solutions.', 'quant-ph-0110136-2-19-3': 'However, finding all the solutions is not the aim of a decision procedure for this kind of problem.', 'quant-ph-0110136-2-20-0': 'Notwithstanding this, measurements of [MATH] of the ground state would always yield some values [MATH] and a straightforward substitution would confirm if the equation has a solution or not.', 'quant-ph-0110136-2-20-1': 'Thus the measurement on the ground state either of the energy, provided the zero point can be calibrated, or of the number operators will be sufficient to give the result for the decision problem.', 'quant-ph-0110136-2-21-0': 'The quantum algorithm with the ground-state oracle is thus clear:', 'quant-ph-0110136-2-22-0': "Given a diophantine equation with [MATH] unknowns [MATH]'s [EQUATION] we need to simulate on some appropriate Fock space the quantum hamiltonian [EQUATION]", 'quant-ph-0110136-2-22-1': 'If the ground state could be obtained with high probability, measurements of appropriate observables would provide the answer for our decision problem.', 'quant-ph-0110136-2-23-0': 'The key ingredients are the availability of a countably infinite number of Fock states, the ability to construct/simulate a suitable hamiltonian and to obtain its ground state.', 'quant-ph-0110136-2-23-1': 'As a counterpart of the semi-infinite tape of a Turing machine, the Fock space is employed here instead of the qubits of the more well-known model of quantum computation.', 'quant-ph-0110136-2-23-2': 'Its advantage over the infinitely many qubits which would otherwise be required is obvious.', 'quant-ph-0110136-2-24-0': '# Why should this work?', 'quant-ph-0110136-2-25-0': 'We do not look for the zeroes of the polynomial, [MATH], which may not exist, but instead search for the absolute minimum of its square which exists, [EQUATION] and is finite because [MATH] diverges.', 'quant-ph-0110136-2-26-0': 'While it is equally hard to find either the zeroes or the absolute minimum in classical computation, we have converted the problem to the realisation of the ground state of a quantum hamiltonian and there is no known quantum principle against such act.', 'quant-ph-0110136-2-26-1': 'In fact, there is no known physical principles against it.', 'quant-ph-0110136-2-26-2': 'Let us consider the three laws of thermodynamics concerning energy conservation, entropy of closed systems and the unattainability of absolute zero temperature.', 'quant-ph-0110136-2-26-3': 'The energy involved in our algorithm is finite, being the ground state energy of some hamiltonian.', 'quant-ph-0110136-2-26-4': 'The entropy increase which ultimately connects to decoherence effects is a technical problem for all quantum computation in general, and we will discuss this further below.', 'quant-ph-0110136-2-26-5': 'As we will never obtain the absolute zero temperature, we only need to satisfy ourselves that the required ground state can be achieved with a more-than-even chance.', 'quant-ph-0110136-2-26-6': 'Then there is a probability boosting technique, see later, to bring that chance to as closed to unity as one pleases.', 'quant-ph-0110136-2-27-0': 'It may appear that even the quantum process can only explore a finite domain in a finite time and is thus no better than a classical machine in terms of computability.', 'quant-ph-0110136-2-27-1': 'But there is a crucial difference.', 'quant-ph-0110136-2-28-0': 'In a classical search even if the global minimum is come across, it cannot generally be proved that it is the global minimum (unless it is a zero of the Diophantine equation).', 'quant-ph-0110136-2-28-1': 'Armed only with mathematical logic, we would still have to compare it with all other numbers from the infinite domain yet to come, but we obviously can never complete this comparison in finite time -thus, mathematical noncomputability.', 'quant-ph-0110136-2-29-0': 'In the quantum case, the global minimum is encoded in the ground state.', 'quant-ph-0110136-2-29-1': 'Then, by energetic tagging, the global minimum can be found in finite time and confirmed, if it is the ground state that is obtained at the end of the computation.', 'quant-ph-0110136-2-29-2': 'And the ground state may be identified and/or verified by physical principles.', 'quant-ph-0110136-2-29-3': 'These principles are over and above the mathematics which govern the logic of a classical machine and help differentiating the quantum from the classical.', 'quant-ph-0110136-2-29-4': 'Quantum mechanics could "explore" an infinite domain, but only in the sense that it can select, among an infinite number of states, one single state (or a subspace in case of degeneracy) to be identified as the ground state of some given hamiltonian (which is bounded from below).', 'quant-ph-0110136-2-29-5': 'This "sorting" can be done because of energetic reason, which is a physical principle and is not available to mathematical computability.', 'quant-ph-0110136-2-30-0': '# General quantum computation', 'quant-ph-0110136-2-31-0': 'Our proposal is in contrast to the claim in [CITATION] that quantum Turing machines compute exactly the same class of functions as do Turing machines, albeit perhaps more efficiently.', 'quant-ph-0110136-2-31-1': 'We could only offer here some speculations about this apparent discrepancy.', 'quant-ph-0110136-2-31-2': 'The quantum Turing machine approach is a direct generalisation of that of the classical Turing machines but with qubits and some universal set of one-qubit and two-qubit unitary gates to build up, step by step, dimensionally larger, but still dimensionally finite unitary operations.', 'quant-ph-0110136-2-31-3': 'This universal set is chosen on its ability to evaluate any desirable classical logic function.', 'quant-ph-0110136-2-31-4': 'Our approach, on the other hand, is from the start based on infinite-dimension hamiltonians acting on some Fock space and also based on the special properties and unique status of their ground states.', 'quant-ph-0110136-2-31-5': 'The unitary operations are then followed as the Shrodinger time evolutions.', 'quant-ph-0110136-2-31-6': 'Even at the hamiltonian level higher orders of the operators [MATH] and [MATH], i.e. not just two-body but many-body interactions in a sense, are already present.', 'quant-ph-0110136-2-31-7': 'This proliferation, which is even more pronounced at the level of the time-evolution operators, together with the infinite dimensionality and the unique energetic status of the vacuum could be the reasons behind the ability to compute, in a finite number of steps, what the dimensionally finite unitary operators of the standard quantum Turing computation cannot do in a finite number of steps.', 'quant-ph-0110136-2-31-8': 'Note that it was the general hamiltonian computation that was discussed by Benioff and Feynman [CITATION] in the conception days of quantum computation.', 'quant-ph-0110136-2-32-0': 'Our general algorithm above could be realised by, but in no way restricted to, the following methods to simulate the required hamiltonian and to obtain the ground state adiabatically.', 'quant-ph-0110136-2-33-0': '# Simulating the hamiltonians', 'quant-ph-0110136-2-34-0': 'One way to construct any suitable hamiltonian so desired is through the technique of ref. [CITATION].', 'quant-ph-0110136-2-34-1': 'We consider the hermitean operators, where [MATH] is the index of the unknowns of the diophantine equation, [EQUATION]', 'quant-ph-0110136-2-34-2': 'Together with the availability of the fundamental hamiltonians [EQUATION] one could construct the unitary time evolutions corresponding to hamiltonians of arbitrary hermitean polynomials in [MATH], and hence in [MATH], to an arbitrary degree of accuracy.', 'quant-ph-0110136-2-34-3': 'These fundamental hamiltonians correspond to translations, phase shifts, squeezers, beam splitters and Kerr nonlinearity.', 'quant-ph-0110136-2-35-0': 'With the polynomial hamiltonian constructed, we need to obtain its ground state.', 'quant-ph-0110136-2-35-1': 'Any approach that allows us to access the ground state will suffice.', 'quant-ph-0110136-2-35-2': 'One way is perhaps to use that of quantum annealing or cooling [CITATION].', 'quant-ph-0110136-2-35-3': 'Another way is to employ the quantum computation method of adiabatic evolution [CITATION].', 'quant-ph-0110136-2-36-0': '# Adiabatic evolution', 'quant-ph-0110136-2-37-0': 'In the adiabatic approach, one starts with a hamiltonian [MATH] whose ground state [MATH] is readily achievable.', 'quant-ph-0110136-2-37-1': 'Then one forms the slowly varying hamiltonian [MATH], [MATH], which interpolates between [MATH] and [MATH] in the time interval [MATH] [EQUATION]', 'quant-ph-0110136-2-37-2': 'Note that we can replace this linear interpolation by some non-linear one provided the conditions of the adiabatic theorem are observed.', 'quant-ph-0110136-2-37-3': 'According to this theorem, the initial ground state will evolve into our desirable ground state [MATH] up to a phase: [EQUATION]', 'quant-ph-0110136-2-37-4': 'For the hamiltonian ([REF]), an estimate of the time [MATH] after which the system remains with high probability in the ground state is [EQUATION] with [EQUATION] and [EQUATION] where [MATH] and [MATH] are respectively the instantaneous ground state and the first excited state of ([REF]) with instantaneous eigenvalues [MATH], [MATH].', 'quant-ph-0110136-2-38-0': 'The time-ordering operator on the left hand side of ([REF]) can be approximated as [EQUATION] for [CITATION] [EQUATION]', 'quant-ph-0110136-2-38-1': 'Note that we have employed here the "norm" [MATH] as defined in ([REF]) for the various hamiltonians which are unbounded from above.', 'quant-ph-0110136-2-38-2': 'This norm is the relevant measure for the problem is only concerned with the lowest states of the interpolating hamiltonian ([REF]).', 'quant-ph-0110136-2-38-3': 'In each interval [MATH], the unitary operators [MATH], for [MATH], can be expressed through the subdivision of [MATH] into [MATH] subintervals of sufficiently small size [MATH] satisfying [MATH], [EQUATION].', 'quant-ph-0110136-2-38-4': 'Each of the [MATH] factors on the right hand side of the last expression can be now simulated through the approach of [CITATION], where it was shown that the number of steps [MATH] grows as a small polynomial in the order of the polynomial in the hamiltonian [MATH] to be simulated, the accuracy to be enacted, and the time interval [MATH] over which it is to be applied.', 'quant-ph-0110136-2-39-0': 'In this way, the requirements of the adiabatic conditions on the one hand and of, on the other hand, the simulations of the hamiltonians in the time interval [MATH] can be satisfied.', 'quant-ph-0110136-2-40-0': '# Gap estimation', 'quant-ph-0110136-2-41-0': 'The question of computational complexity, i.e. how large [MATH] is and how small [MATH], [MATH] are for a high probability of measurement success, as seen from the various bounds above is dependent on [MATH], i.e. on the specific diophantine equation in question, and on the initial hamiltonian [MATH].', 'quant-ph-0110136-2-41-1': 'Some estimate for the energy gap ([REF]) is crucial for those of the total computational time [MATH] and also of the temperature to control level transitions.', 'quant-ph-0110136-2-42-0': 'The authors of [CITATION] have pointed out that typically [MATH] is not zero unless the hamiltonian has special symmetry properties.', 'quant-ph-0110136-2-42-1': 'As the levels will cross if there is a symmetry, a necessary condition for no level crossing in our case is that the operators [MATH] do not commute with [MATH], thus do not commute with [MATH] for [MATH], even though they commute with [MATH].', 'quant-ph-0110136-2-42-2': 'We show elsewhere [CITATION] that in general accidental degeneracy (two-level crossing) can be avoided, except at the end point [MATH] where the adiabatic process ends with some obvious symmetry.', 'quant-ph-0110136-2-42-3': 'Furthermore, the freedom in choosing the initial hamiltonians [MATH] and their ground states and in performing different adiabatic interpolations, not just linear as in ([REF]), might be exploited to enable the gap estimation and to speed up the computation.', 'quant-ph-0110136-2-43-0': 'We shall employ the simple harmonic oscillators, i.e. gaussian approximations, to obtain an estimate for the energy gap.', 'quant-ph-0110136-2-44-0': 'For the various [MATH], [MATH] appearing in the interpolating hamiltonian [MATH] ([REF]) at the time instant [MATH], we use the Bogoliubov ansatz, with real [MATH] and [MATH], [EQUATION]', 'quant-ph-0110136-2-44-1': 'The [MATH]-zero occupation state is denoted by [MATH], where the [MATH]-dependence of the state is implicit.', 'quant-ph-0110136-2-44-2': 'The canonicity [MATH] demands [EQUATION] upon which [EQUATION] where [EQUATION]', 'quant-ph-0110136-2-44-3': 'We pay particular attention to the following terms [EQUATION]', 'quant-ph-0110136-2-44-4': 'Needed next is a version of the Wick theorem [CITATION] for the ordinary product involving the operators [MATH], ..., [MATH], [EQUATION] where the normal ordering [MATH] is done with respect to some annihilation operators [MATH] which annihilate some common state [MATH] for various [MATH], [MATH].', 'quant-ph-0110136-2-44-5': 'The equation ([REF]), in which the hatted operators are omitted from the normal ordering, is an exact result and can be proved by induction.', 'quant-ph-0110136-2-45-0': 'We can now list the various steps of our estimation:', 'quant-ph-0110136-2-46-0': "We apply the Wick theorem, with respect to [MATH]'s, to the hamiltonian [MATH] ([REF]) [EQUATION]", 'quant-ph-0110136-2-46-1': 'Note that in this process the higher-order terms contribute to the coefficients of lower-order ones through products of various expectation values in ([REF]).', 'quant-ph-0110136-2-46-2': 'With help from ([REF]) and ([REF]), the various coefficients [MATH], [MATH], [MATH], ... in the right hand side above can be expressed as polynomials in [MATH].', 'quant-ph-0110136-2-46-3': 'We next fix [MATH], and [MATH] numerically from ([REF]) and from the imposition that the coefficients [MATH] in ([REF]).', 'quant-ph-0110136-2-46-4': 'Then we can evaluate the coefficients [MATH] from their polynomial expressions in [MATH].', 'quant-ph-0110136-2-46-5': 'From ([REF]), on the other hand, we see also that [EQUATION] where [MATH].', 'quant-ph-0110136-2-46-6': 'Mathematically, [MATH] thus provides some indication of the size of the energy gaps of [MATH] around the energy level [MATH].', 'quant-ph-0110136-2-46-7': 'Even though [MATH], obtained from the linear Bogoliubov transformations ([REF]), in general may not be the true ground state of [MATH], we could use [MATH] as some indicator for the gap [MATH] in ([REF]) in order to estimate [MATH] from the adiabaticity condition ([REF]).', 'quant-ph-0110136-2-47-0': 'Some variations of the above method might yield a better estimate.', 'quant-ph-0110136-2-47-1': 'For instance, one could numerically obtain [MATH], and thus [MATH], by minimising the energy [MATH] subjected to the constraints ([REF]) with or without the constraints [MATH].', 'quant-ph-0110136-2-48-0': 'The aim of this minimisation, if possible, is to select a state [MATH] whose energy expectation value at the time [MATH] is as closed to that of the ground state at that instant as allowed by the linear Bogoliubov transformations.', 'quant-ph-0110136-2-48-1': 'Other variations might be involving, instead of ([REF]), some non-linear relations, for the canonicity only requires that [MATH] and [MATH] are to be unitarily transformed, [MATH].', 'quant-ph-0110136-2-49-0': 'The accuracy of the estimation and its higher order corrections can be evaluated systematically from the higher order terms in the last line of ([REF]).', 'quant-ph-0110136-2-49-1': 'Numerical diagonalisation of ([REF]) with a series of truncated bases in [MATH] of increasing sizes could further provide us more information about the gap thus obtained.', 'quant-ph-0110136-2-50-0': 'It turns out that with the method above we have traded the original diophantine equation for a set of polynomials equations in [MATH].', 'quant-ph-0110136-2-50-1': 'The solutions of these latter equations, however, are real, not integers as in the case of the original diophantine, and could be obtained or well approximated through the calculus of continuous variables.', 'quant-ph-0110136-2-50-2': '(Tarski [CITATION] has found that looking for the existence of real solutions of polynomials over the real numbers are less restrictive than that of integer solutions of diophantine equations; he has shown that arithmetics over the reals is, in fact, decidable.)', 'quant-ph-0110136-2-50-3': 'Note that in some cases we may not even need to solve the equations exactly in order to have some appropriate estimate for [MATH].', 'quant-ph-0110136-2-51-0': 'More general, we have traded the hard question of integer solutions for some more palatable estimation for the real-valued time [MATH] in ([REF]).', 'quant-ph-0110136-2-51-1': 'This estimation is less restrictive in the sense that it needs only be of order of magnitude for a good success probability, which could be boosted further.', 'quant-ph-0110136-2-52-0': '# Brute-force approach', 'quant-ph-0110136-2-53-0': 'As the minimum gap is generically non-zero [CITATION], it takes only a finite time [MATH] for the adiabatic process.', 'quant-ph-0110136-2-53-1': 'Because of this finiteness, without a refined estimate of [MATH] as in the above, a brute-force approach for the quantum computation could be as follows:', 'quant-ph-0110136-2-54-0': 'Use the initial states as the handle to identify the measured state.', 'quant-ph-0110136-2-54-1': 'Once some final state [MATH] has been measured projectively at the end of the computation and checked by the steps above, we reverse the adiabatic evolution starting from this state [MATH] backward, for a time at least [MATH], to get back to the original space in which contains the state [MATH] and of which we know well.', 'quant-ph-0110136-2-54-2': 'Since this latter space is better known, this method could help tagging and identifying the final state [MATH].', 'quant-ph-0110136-2-54-3': 'Note that this process is not quite reversible because of the intervening measurement of [MATH].', 'quant-ph-0110136-2-54-4': 'Care must also be taken when there is degeneracy for the ground state of [MATH], for the adiabatic theorem may not be applicable to this reversed run.', 'quant-ph-0110136-2-54-5': 'Any inconsistency with Quantum Mechanics would require a rerun of the process with longer time [MATH].', 'quant-ph-0110136-2-55-0': 'The steps above will terminate.', 'quant-ph-0110136-2-55-1': 'The key idea is to employ the theory of Quantum Mechanics in order to satisfy ourselves that we have indeed obtained the ground state.', 'quant-ph-0110136-2-55-2': 'A further cross checking method (applicable for degeneracy case) is to use few widely different initial hamiltonians ([REF]) with [MATH], for all [MATH], for the adiabatic runs.', 'quant-ph-0110136-2-56-0': '# Probability boosting', 'quant-ph-0110136-2-57-0': 'Due to the particular nature of the present scheme, the various approximations result in some probability that the final measurement will not find the system in the desired ground state; but appropriate choices of the various time parameters could increase the success probability of the algorithm to more than even.', 'quant-ph-0110136-2-57-1': 'We spell out explicitly here the probability boosting technique, as mentioned in [CITATION], that one could subsequently apply.', 'quant-ph-0110136-2-58-0': 'In our computation, adiabatic or otherwise, the end result, starting from some initial state [MATH], may be contaminated with some excited states other than the desirable ground state [MATH], [EQUATION]', 'quant-ph-0110136-2-58-1': 'If [MATH], that is, if there is a better-than-even chance to obtain [MATH] by measurement then one can boost the success probability to arbitrarily closed to unity by performing a concatenated computation over [MATH] Hilbert spaces [EQUATION] where [MATH] is the normalising factor.', 'quant-ph-0110136-2-58-2': 'Let us consider the majority amplitudes, when more than half of the [MATH] Hilbert spaces return the correct results, [MATH]; and the minority amplitudes, [MATH].', 'quant-ph-0110136-2-58-3': 'The ratio of the majority over the minority, [MATH], is clearly boosted for sufficiently large [MATH] and for [MATH] and [MATH].', 'quant-ph-0110136-2-58-4': 'The probability distribution for a measurement of the end state in ([REF]), as a consequence, is exponentially dominated by the majority results.', 'quant-ph-0110136-2-58-5': 'In other words, the probability to obtain a majority result which contains the true ground state can be made arbitrarily closed to unity, [MATH], provided [MATH] and [MATH].', 'quant-ph-0110136-2-58-6': 'However, the decoherence control for such [MATH] concatenated Hilbert spaces will be more crucial.', 'quant-ph-0110136-2-59-0': '# Decoherence and error correction', 'quant-ph-0110136-2-60-0': 'Our approach above is in fact a combination of the quantum computation of continuous variables and of adiabatic evolution.', 'quant-ph-0110136-2-60-1': 'There exists some error correction protocol for continuous variables [CITATION] which could be of help here to protect the wave functions from decoherence.', 'quant-ph-0110136-2-60-2': 'However, the adiabatic computation we exploit is quite robust in general [CITATION].', 'quant-ph-0110136-2-60-3': 'An imperfect conventional quantum algorithm might have different sorts of errors than an imperfect adiabatic process, where the system is kept close to the instantaneous ground state over time.', 'quant-ph-0110136-2-60-4': 'Decoherence by the environment inducing transition between levels could be controlled in principle at a temperature that is small compared to the gap ([REF]), given its estimate.', 'quant-ph-0110136-2-60-5': 'Errors introduced by the hamiltonian simulation may result in a hamiltonian different from ([REF]) but in a form [MATH].', 'quant-ph-0110136-2-60-6': 'Recent numerical study [CITATION] of small systems has in fact indicated that the adiabatic computation is interestingly robust even for fairly large [MATH], provided [MATH] varies either sufficiently slowly or sufficiently rapidly (which is more likely to be the case considered here due to the nature of our hamiltonian simulations).', 'quant-ph-0110136-2-61-0': '# Concluding remarks', 'quant-ph-0110136-2-62-0': 'We here put emphasis on the issue of computability in principle, not on that of computational complexity.', 'quant-ph-0110136-2-62-1': "This attempt of broadening of the concept of effective computability, taken into account the quantum mechanical principles, has been argued to be able in principle to decide the classically undecidables, the Hilbert's tenth problem and thus the Turing halting problem in this instance.", 'quant-ph-0110136-2-62-2': "If this is realisable, and we don't have any evidence of fundamental nature to the contrary, the Church-Turing thesis should be modified accordingly.", 'quant-ph-0110136-2-63-0': 'On the other hand, if for any reasons the algorithm is not implementable in principle then it would be an example of information being limited by physics, rather than by logical arguments alone.', 'quant-ph-0110136-2-63-1': 'That is, there might be some fundamental physical principles, not those of practicality, which prohibit the implementation.', 'quant-ph-0110136-2-63-2': 'For example, a precision of, say, [MATH] would be required but that is not allowed in principles, and not just because of present-day limitation on technology.', 'quant-ph-0110136-2-63-3': 'Or, there might be not enough physical resources (ultimately limited by the total energy and the lifetime of the universe) to satisfy the execution of the algorithm.', 'quant-ph-0110136-2-63-4': '(In this case it is likely that the Turing program in consideration, even if it eventually halts upon some input, would take a running time longer than the lifetime of the universe.)', 'quant-ph-0110136-2-63-5': 'In either cases, the whole exercise is still very interesting as the unsolvability of those problems and the limit of mathematics itself are also dictated by physical principles and/or resources.', 'quant-ph-0110136-2-64-0': 'Our study is an illustration of "Information is physical" (see [CITATION] for where the theory of General Relativity is also exploited for computation).', 'quant-ph-0110136-2-65-0': 'That some generalisation of the notion of computation could help solving the previous undecidability/noncomputability has been recognised in mathematics and was considered by Kleene as quoted in [CITATION].', 'quant-ph-0110136-2-65-1': 'But this has not been realisable until now simply because of the non-recognition of quantum physics as the missing ingredient.', 'quant-ph-0110136-2-65-2': 'Our quantum algorithm could in fact be regarded as an infinite search through the integers in a finite amount of time, the type of search required by Kleene to solve the Turing halting problem.', 'quant-ph-0110136-2-66-0': "Our decidability study here only deals with the property of being diophantine, which does not cover the property of being arithmetic in general, and as such has no direct consequences on the Godel's Incompleteness theorem.", 'quant-ph-0110136-2-66-1': "However, it is conceivable that the Godel's theorem may lose its restrictive power once the concept of proof is suitably generalised with quantum principles."}

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'quant-ph-0110136-2-35-0'], ['quant-ph-0110136-1-18-1', 'quant-ph-0110136-2-35-1'], ['quant-ph-0110136-1-18-2', 'quant-ph-0110136-2-35-2'], ['quant-ph-0110136-1-18-3', 'quant-ph-0110136-2-35-3'], ['quant-ph-0110136-1-29-0', 'quant-ph-0110136-2-46-0'], ['quant-ph-0110136-1-29-1', 'quant-ph-0110136-2-46-1'], ['quant-ph-0110136-1-29-2', 'quant-ph-0110136-2-46-2'], ['quant-ph-0110136-1-29-3', 'quant-ph-0110136-2-46-3'], ['quant-ph-0110136-1-29-4', 'quant-ph-0110136-2-46-4'], ['quant-ph-0110136-1-29-5', 'quant-ph-0110136-2-46-5'], ['quant-ph-0110136-1-29-6', 'quant-ph-0110136-2-46-6'], ['quant-ph-0110136-1-29-7', 'quant-ph-0110136-2-46-7'], ['quant-ph-0110136-1-38-0', 'quant-ph-0110136-2-58-0'], ['quant-ph-0110136-1-38-1', 'quant-ph-0110136-2-58-1'], ['quant-ph-0110136-1-38-2', 'quant-ph-0110136-2-58-2'], ['quant-ph-0110136-1-38-3', 'quant-ph-0110136-2-58-3'], ['quant-ph-0110136-1-38-4', 'quant-ph-0110136-2-58-4'], ['quant-ph-0110136-1-38-5', 'quant-ph-0110136-2-58-5'], 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[]

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/quant-ph/0110136

{'quant-ph-0110136-3-0-0': 'We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science.', 'quant-ph-0110136-3-0-1': "A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed.", 'quant-ph-0110136-3-0-2': 'If this algorithm could be physically implemented, as much as it is valid in principle-that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal-quantum computability would surpass classical computability as delimited by the Church-Turing thesis.', 'quant-ph-0110136-3-0-3': 'It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.', 'quant-ph-0110136-3-1-0': '# Introduction', 'quant-ph-0110136-3-2-0': 'Computation based on the principles of Quantum Mechanics [CITATION] has been shown to offer better performances over classical computation, ranging from the square-root improvement in an unstructured search [CITATION] to the exponential gain in the factorisation of integers [CITATION].', 'quant-ph-0110136-3-2-1': 'However superior in reducing the complexity of hard computation, these quantum algorithms and all the others discovered so far are only applicable to the classically computable functions.', 'quant-ph-0110136-3-2-2': 'That leaves untouched the class of classically noncomputable functions, such as the halting problem for Turing machines [CITATION].', 'quant-ph-0110136-3-2-3': 'It is in fact widely believed that quantum computation cannot offer anything new about computability [CITATION].', 'quant-ph-0110136-3-2-4': 'Contrary to this, we propose that quantum computation may be able to compute the noncomputables, provided certain hamiltonian and its ground state can be physically constructed.', 'quant-ph-0110136-3-2-5': "We propose a quantum algorithm for the classically noncomputable Hilbert's tenth problem [CITATION] which ultimately links to the halting problem for Turing machines in the computation of partial recursive functions.", 'quant-ph-0110136-3-3-0': 'The practical details of implementation the quantum algorithm for this class of problems are not considered in this conceptual study, and will be investigated elsewhere.', 'quant-ph-0110136-3-4-0': "# Hilbert's tenth problem", 'quant-ph-0110136-3-5-0': 'At the turn of the last century, David Hilbert listed 23 important problems, among which the problem number ten could be rephrased as:', 'quant-ph-0110136-3-6-0': 'Given any polynomial equation with any number of unknowns and with integer coefficients: To devise a universal process according to which it can be determined by a finite number of operations whether the equation has integer solutions.', 'quant-ph-0110136-3-7-0': 'This decision problem for such polynomial equations, which are also known as Diophantine equations, has eventually been shown in 1970 by Matiyasevich to be undecidable [CITATION] in the Turing sense.', 'quant-ph-0110136-3-7-1': 'It is consequently noncomputable/undecidable in the most general sense if one accepts, as almost everyone does, the Church-Turing thesis of computability.', 'quant-ph-0110136-3-7-2': "Since exponential Diophantine, with the unknowns in the exponents as in the example of the Fermat's last theorem, can be shown to be Diophantine with supplementary equations, the study of Diophantine equations essentially covers the class of partial recursive functions, which is at the foundations of classical algorithms.", 'quant-ph-0110136-3-7-3': "The undecidability result is thus singularly important: Hilbert's tenth problem could be solved if and only if the Turing halting problem could be.", 'quant-ph-0110136-3-8-0': '# Turing halting problem', 'quant-ph-0110136-3-9-0': 'The halting problem for Turing machines is also a manifestation of undecidability: a Turing computation is equivalent to the computation of a partial recursive function, which is only defined for a subset of the integers; as this domain is classically undecidable, one cannot always tell in advance whether the Turing machine will halt (that is, whether the input is in the domain of the partial recursive function) or not (when the input is not in the domain).', 'quant-ph-0110136-3-10-0': 'A version of the proof of the unsolvability of the halting problem based on the Cantor diagonal argument goes as follows.', 'quant-ph-0110136-3-10-1': 'The proof is by contradiction with the assumption of the existence of a computable halting function [MATH] which has two integer arguments - [MATH] is the Godel encoded integer number for the algorithm and [MATH] is its (encoded) integer input: [EQUATION]', 'quant-ph-0110136-3-10-2': 'One can then construct a program [MATH] having one integer argument [MATH] in such a way that it calls the function [MATH] and [EQUATION]', 'quant-ph-0110136-3-10-3': 'The application of the halting function [MATH] on the program [MATH] and input [MATH] results in [EQUATION]', 'quant-ph-0110136-3-10-4': 'A contradiction is clearly manifest once we put [MATH] in the last equation above.', 'quant-ph-0110136-3-11-0': 'The construction of such program [MATH] is transparently possible, unless the existence of a computable [MATH] is wrongly assumed.', 'quant-ph-0110136-3-11-1': 'Thus the contradiction discounts the assumption that there is a classically algorithmic way to determine whether any arbitrarily given program with arbitrary input will halt or not.', 'quant-ph-0110136-3-12-0': 'This contradiction argument might be side stepped if we distinguish and separate the two classes of quantum and classical algorithms.', 'quant-ph-0110136-3-12-1': 'A quantum function [MATH], similar to eq. ([REF]), can conceivably exist to determine whether any classical program [MATH] will halt on any classical input [MATH] or not.', 'quant-ph-0110136-3-12-2': 'The contradiction in eq. ([REF]) would be avoided if the quantum halting [MATH] cannot take as argument the modified program [MATH], which is now of quantum character because it now has quantum [MATH] as a subroutine.', 'quant-ph-0110136-3-12-3': 'This will be the case if [MATH] can only accept integer while quantum algorithms, with proper definitions, cannot in general be themselves encoded as integers.', 'quant-ph-0110136-3-12-4': 'It is clear even in the case of a single qubit that the state [MATH] cannot be encoded as integers for all [MATH] and [MATH] - simply because of different cardinalities.', 'quant-ph-0110136-3-12-5': 'In fact, the no-cloning theorem [CITATION] of quantum mechanics does restrict the type of operations available to quantum algorithms.', 'quant-ph-0110136-3-13-0': 'In essence, the way we will break the self-referential reasoning here by the differentiation between quantum and classical algorithms is similar to the way John von Neumann and Bertrand Russell resolved the set theory paradox (to do with "The set of all sets which are not members of themselves") by the introduction of classes as distinct from sets.', 'quant-ph-0110136-3-13-1': '(For other lines of arguments, see [CITATION].)', 'quant-ph-0110136-3-14-0': '# An observation', 'quant-ph-0110136-3-15-0': 'It suffices to consider nonnegative solutions, if any, of a Diophantine equation.', 'quant-ph-0110136-3-15-1': 'Let us consider the example [EQUATION] with unknowns [MATH], [MATH], and [MATH].', 'quant-ph-0110136-3-16-0': 'To find out whether this equation has any nonnegative integer solution by quantum algorithms, it requires the realisation of a Fock space built out of the "vacuum" [MATH] by repeating applications of the creation operators [MATH], [MATH] and [MATH], similarly to that of the 3-D simple harmonic oscillators.', 'quant-ph-0110136-3-16-1': '[EQUATION]', 'quant-ph-0110136-3-16-2': 'Upon this Hilbert space, we construct the hamiltonian corresponding to ([REF]) [EQUATION] which has a spectrum bounded from below - semidefinite, in fact.', 'quant-ph-0110136-3-17-0': 'Note that the operators [MATH] have only nonnegative integer eigenvalues [MATH], and that [MATH] so these observables are compatible - they are simultaneously measurable.', 'quant-ph-0110136-3-17-1': 'The ground state [MATH] of the hamiltonian so constructed has the properties [EQUATION] for some [MATH].', 'quant-ph-0110136-3-18-0': 'Thus a projective measurement of the energy [MATH] of the ground state [MATH] will yield the answer for the decision problem: The Diophantine equation has at least one integer solution if and only if [MATH], and has not otherwise.', 'quant-ph-0110136-3-18-1': "(If [MATH] in our example, we know that [MATH] from Fermat's last theorem.)", 'quant-ph-0110136-3-19-0': 'If there is one unique solution then the projective measurements of the observables corresponding to the operators [MATH] will reveal the values of various unknowns.', 'quant-ph-0110136-3-19-1': 'If there are many solutions, finitely or infinitely as in the case of [MATH], the ground state [MATH] will be a linear superposition of states of the form [MATH], where [MATH] are the solutions.', 'quant-ph-0110136-3-19-2': 'In such situation, the measurement may not yield all the solutions.', 'quant-ph-0110136-3-19-3': 'However, finding all the solutions is not the aim of a decision procedure for this kind of problem.', 'quant-ph-0110136-3-20-0': 'Notwithstanding this, measurements of [MATH] of the ground state would always yield some values [MATH] and a straightforward substitution would confirm if the equation has a solution or not.', 'quant-ph-0110136-3-20-1': 'Thus the measurement on the ground state either of the energy, provided the zero point can be calibrated, or of the number operators will be sufficient to give the result for the decision problem.', 'quant-ph-0110136-3-21-0': 'The quantum algorithm with the ground-state oracle is thus clear:', 'quant-ph-0110136-3-22-0': "Given a Diophantine equation with [MATH] unknowns [MATH]'s [EQUATION] we need to simulate on some appropriate Fock space the quantum hamiltonian [EQUATION]", 'quant-ph-0110136-3-22-1': 'If the ground state could be obtained with high probability, measurements of appropriate observables would provide the answer for our decision problem.', 'quant-ph-0110136-3-23-0': 'The key ingredients are the availability of a countably infinite number of Fock states, the ability to construct/simulate a suitable hamiltonian and to obtain/identify its ground state via quantum measurements.', 'quant-ph-0110136-3-23-1': 'As a counterpart of the semi-infinite tape of a Turing machine, the Fock space is employed here instead of the qubits of the more well-known model of quantum computation.', 'quant-ph-0110136-3-23-2': 'Its advantage over the infinitely many qubits which would otherwise be required is obvious.', 'quant-ph-0110136-3-24-0': '# Some preliminary comments', 'quant-ph-0110136-3-25-0': 'We do not look for the zeroes of the polynomial, [MATH], which may not exist, but instead search for the absolute minimum of its square which exists, [EQUATION] and is finite because [MATH] diverges.', 'quant-ph-0110136-3-26-0': 'While it is equally hard to find either the zeroes or the absolute minimum in classical computation, we have converted the problem to the realisation of the ground state of a quantum hamiltonian and there is no known quantum principle against such act.', 'quant-ph-0110136-3-26-1': 'In fact, there is no known physical principles against it.', 'quant-ph-0110136-3-26-2': 'Let us consider the three laws of thermodynamics concerning energy conservation, entropy of closed systems and the unattainability of absolute zero temperature.', 'quant-ph-0110136-3-26-3': 'The energy involved in our algorithm is finite, being the ground state energy of some hamiltonian.', 'quant-ph-0110136-3-26-4': 'The entropy increase which ultimately connects to decoherence effects is a technical problem for all quantum computation in general, and we will discuss this further below.', 'quant-ph-0110136-3-26-5': 'As we will never obtain the absolute zero temperature, we only need to satisfy ourselves that the required ground state can be achieved with a more-than-even chance.', 'quant-ph-0110136-3-26-6': 'Then there is a probability boosting technique, see later, to bring that chance to as closed to unity as one pleases.', 'quant-ph-0110136-3-27-0': 'It may appear that even the quantum process can only explore a finite domain in a finite time and is thus no better than a classical machine in terms of computability.', 'quant-ph-0110136-3-27-1': 'But there is a crucial difference.', 'quant-ph-0110136-3-28-0': 'In a classical search even if the global minimum is come across, it cannot generally be proved that it is the global minimum (unless it is a zero of the Diophantine equation).', 'quant-ph-0110136-3-28-1': 'Armed only with mathematical logic, we would still have to compare it with all other numbers from the infinite domain yet to come, but we obviously can never complete this comparison in finite time -thus, mathematical noncomputability.', 'quant-ph-0110136-3-29-0': 'In the quantum case, the global minimum is encoded in the ground state.', 'quant-ph-0110136-3-29-1': 'Then, by energy tagging, the global minimum can be found in finite time and confirmed, if it is the ground state that is obtained at the end of the computation.', 'quant-ph-0110136-3-29-2': 'And the ground state may be identified and/or verified by physical principles.', 'quant-ph-0110136-3-29-3': 'These principles are over and above the mathematics which govern the logic of a classical machine and help differentiating the quantum from the classical.', 'quant-ph-0110136-3-29-4': 'Quantum mechanics could "explore" an infinite domain, but only in the sense that it can select, among an infinite number of states, one single state (or a subspace in case of degeneracy) to be identified as the ground state of some given hamiltonian (which is bounded from below).', 'quant-ph-0110136-3-29-5': 'This "sorting" can be done because of energetic reason, which is a physical principle and is not available to mathematical computability.', 'quant-ph-0110136-3-30-0': 'On the other hand, our proposal is apparently in contrast to the claim in [CITATION] that quantum Turing machines compute exactly the same class of functions as do Turing machines, albeit perhaps more efficiently.', 'quant-ph-0110136-3-30-1': 'We could only offer here some speculations about this apparent discrepancy.', 'quant-ph-0110136-3-30-2': 'The quantum Turing machine approach is a direct generalisation of that of the classical Turing machines but with qubits and some universal set of one-qubit and two-qubit unitary gates to build up, step by step, dimensionally larger, but still dimensionally finite unitary operations.', 'quant-ph-0110136-3-30-3': 'This universal set is chosen on its ability to evaluate any desirable classical logic function.', 'quant-ph-0110136-3-30-4': 'Our approach, on the other hand, is from the start based on infinite-dimension hamiltonians acting on some Fock space and also based on the special properties and unique status of their ground states.', 'quant-ph-0110136-3-30-5': 'The unitary operations are then followed as the Schrodinger time evolutions.', 'quant-ph-0110136-3-30-6': 'Even at the hamiltonian level higher orders of the operators [MATH] and [MATH], i.e. not just two-body but many-body interactions in a sense, are already present.', 'quant-ph-0110136-3-30-7': 'This proliferation, which is even more pronounced at the level of the time-evolution operators, together with the infinite dimensionality and the unique energetic status of the vacuum could be the reasons behind the ability to compute, in a finite number of steps, what the dimensionally finite unitary operators of the standard quantum Turing computation cannot do in a finite number of steps.', 'quant-ph-0110136-3-30-8': 'Note that it was the general hamiltonian computation that was discussed by Benioff and Feynman [CITATION] in the conception days of quantum computation.', 'quant-ph-0110136-3-31-0': 'Indeed, Nielsen [CITATION] has also found no logical contradiction in applying the most general quantum mechanical principles to the computation of the classical noncomputable, unless certain Hermitean operators cannot somehow be realised as observables or certain unitary processes cannot somehow be admitted as quantum dynamics.', 'quant-ph-0110136-3-31-1': 'And up to now we do not have any evidence nor any principles that prohibit these kinds of observables and dynamics.', 'quant-ph-0110136-3-31-2': '(Ozawa [CITATION] has produced some counter arguments but we think they are not quite applicable here.', 'quant-ph-0110136-3-31-3': 'See [CITATION].)', 'quant-ph-0110136-3-32-0': 'Our general algorithm above could be realised by, but in no way restricted to, the following methods to simulate the required hamiltonian and to obtain the ground state adiabatically.', 'quant-ph-0110136-3-33-0': '# Simulating the hamiltonians', 'quant-ph-0110136-3-34-0': 'One way to construct any suitable hamiltonian so desired is through the technique of ref. [CITATION].', 'quant-ph-0110136-3-34-1': 'We consider the hermitean operators, where [MATH] is the index of the unknowns of the Diophantine equation, [EQUATION]', 'quant-ph-0110136-3-34-2': 'Together with the availability of the fundamental hamiltonians [EQUATION] one could construct the unitary time evolutions corresponding to hamiltonians of arbitrary hermitean polynomials in [MATH], and hence in [MATH], to an arbitrary degree of accuracy.', 'quant-ph-0110136-3-34-3': 'These fundamental hamiltonians correspond to translations, phase shifts, squeezers, beam splitters and Kerr nonlinearity.', 'quant-ph-0110136-3-35-0': 'With the polynomial hamiltonian constructed, we need to obtain its ground state.', 'quant-ph-0110136-3-35-1': 'Any approach that allows us to access the ground state will suffice.', 'quant-ph-0110136-3-35-2': 'One way is perhaps to use that of quantum annealing or cooling [CITATION].', 'quant-ph-0110136-3-35-3': 'Another way is to employ the quantum computation method of adiabatic evolution [CITATION].', 'quant-ph-0110136-3-36-0': '# Adiabatic quantum evolution', 'quant-ph-0110136-3-37-0': 'In the adiabatic approach, one starts with a hamiltonian [MATH] whose ground state [MATH] is readily achievable.', 'quant-ph-0110136-3-37-1': 'Then one forms the slowly varying hamiltonian [MATH], [MATH], which interpolates between [MATH] and [MATH] in the time interval [MATH] [EQUATION]', 'quant-ph-0110136-3-37-2': 'Note that we can replace this linear interpolation by some non-linear one provided the conditions of the adiabatic theorem are observed.', 'quant-ph-0110136-3-37-3': 'According to this theorem, the initial ground state will evolve into our desirable ground state [MATH] up to a phase: [EQUATION]', 'quant-ph-0110136-3-37-4': 'For the hamiltonian ([REF]), an estimate of the time [MATH] after which the system remains with high probability in the ground state is [EQUATION] with [EQUATION] and [EQUATION] where [MATH] and [MATH] are respectively the instantaneous ground state and the first excited state of ([REF]) with instantaneous eigenvalues [MATH], [MATH].', 'quant-ph-0110136-3-38-0': 'The time-ordering operator on the left hand side of ([REF]) can be approximated as [EQUATION] for [CITATION] [EQUATION]', 'quant-ph-0110136-3-38-1': 'Note that we have employed here the "norm" [MATH] as defined in ([REF]) for the various hamiltonians which are unbounded from above.', 'quant-ph-0110136-3-38-2': 'This norm is the relevant measure for the problem is only concerned with the lowest states of the interpolating hamiltonian ([REF]).', 'quant-ph-0110136-3-38-3': 'In each interval [MATH], the unitary operators [MATH], for [MATH], can be expressed through the subdivision of [MATH] into [MATH] subintervals of sufficiently small size [MATH] satisfying [MATH], [EQUATION].', 'quant-ph-0110136-3-38-4': 'Each of the [MATH] factors on the right hand side of the last expression can be now simulated through the approach of [CITATION], where it was shown that the number of steps [MATH] grows as a small polynomial in the order of the polynomial in the hamiltonian [MATH] to be simulated, the accuracy to be enacted, and the time interval [MATH] over which it is to be applied.', 'quant-ph-0110136-3-39-0': 'In this way, the requirements of the adiabatic conditions on the one hand and of, on the other hand, the simulations of the hamiltonians in the time interval [MATH] can be satisfied.', 'quant-ph-0110136-3-40-0': '# An adiabatic algorithm', 'quant-ph-0110136-3-41-0': "In order to solve Hilbert's tenth problem we need on the one hand such time-dependent physical (adiabatic) processes.", 'quant-ph-0110136-3-41-1': 'On the other hand, the theory of Quantum Mechanics can be used to identify the ground state through the usual statistical predictions from the Schrodinger equation with a finitely truncated number of energy states of the time-dependent Hamiltonian [MATH].', 'quant-ph-0110136-3-41-2': 'This way, we can overcome the problem of which states are to be included in the truncated basis for a numerical study of Quantum Mechanics.', 'quant-ph-0110136-3-41-3': 'This also reconciles with the Cantor diagonal arguments which state that the problem could not be solved entirely in the framework of classical computation.', 'quant-ph-0110136-3-42-0': 'Below is an algorithm [CITATION] based on this philosophy of exploiting the interplay between the presumably infinite physical world and the theory of Quantum Mechanics calculated in a finite manner on Turing machines.', 'quant-ph-0110136-3-42-1': 'The algorithm presented may not be the most efficient; there could be many other variations making better use of the same philosophy.', 'quant-ph-0110136-3-43-0': 'It is in general easier to implement some Hamiltonian than to obtain its ground state.', 'quant-ph-0110136-3-43-1': 'We thus should start the computation in yet a different and readily obtainable initial ground state, [MATH], of some initial Hamiltonian, [MATH], then deform this Hamiltonian in a time [MATH] into the Hamiltonian whose ground state is the desired one, through a time-dependent process represented by the interpolating Hamiltonian [MATH].', 'quant-ph-0110136-3-44-0': 'In this approach, inspired by the quantum adiabatic approach, one starts, for example, with a Hamiltonian [MATH], [EQUATION] which admits the readily achievable coherent state [MATH] as the ground state.', 'quant-ph-0110136-3-44-1': 'Then, one forms the time-dependent Hamiltonian [MATH] in ([REF]), which interpolates in the time interval [MATH] between the initial [MATH] and [MATH].', 'quant-ph-0110136-3-45-0': 'Our point is on computability and not on computational complexity, which depends on individual polynomials.', 'quant-ph-0110136-3-45-1': 'Computability is based on the arguments that the adiabatic time [MATH] is finite [CITATION] (for a high probability of achieving the ground state) and that the ground state can be verified by employing the theory of Quantum Mechanics.', 'quant-ph-0110136-3-45-2': 'As long as the energy gap is finite so is the computational time.', 'quant-ph-0110136-3-45-3': "In contrast, the most general classical algorithm for Hilbert's tenth problem (by systematic substituting in integers of larger and larger magnitudes) cannot solve it in principle even allowing for exponentially grown, but finite, amount of time - unless infinite amount of time were available, which it is not.", 'quant-ph-0110136-3-46-0': 'Given a Diophantine equation, the substitution in integers of larger and larger magnitudes is not satisfactory as we do not know when the substitution should be terminated.', 'quant-ph-0110136-3-46-1': 'Likewise, if we want to numerically simulate the quantum algorithm proposed, we would have to use a finitely truncated having [MATH] vectors.', 'quant-ph-0110136-3-46-2': 'But we face the same problem of not knowing which [MATH] to choose in general.', 'quant-ph-0110136-3-46-3': 'That is why the problem is noncomputable on classical computers.', 'quant-ph-0110136-3-47-0': 'Even we can estimate [MATH], as in the appendix, for some starting point of the adiabatic computation as we might not be able in general to exactly know [MATH] a priori.', 'quant-ph-0110136-3-47-1': 'Were the required adiabatic time [MATH] somehow known exactly within classical computation and without the help of quantum computation then the problem might be solved classically.', 'quant-ph-0110136-3-47-2': "But as Hilbert's tenth problem cannot be solved by classical computation, we will have to resort to quantum computation without a priori knowing exactly the time [MATH], except the knowledge that it is finite.", 'quant-ph-0110136-3-47-3': "Constructive logicians [CITATION] allow for this algorithmic situation under the so-called Markov's Principle.", 'quant-ph-0110136-3-48-0': '# Discussion of the algorithm', 'quant-ph-0110136-3-49-0': "The quantum algorithm above can be proved to terminate (even though it could be after a very long time) and give us the decision result for Hilbert's tenth problem.", 'quant-ph-0110136-3-50-0': 'The real spectrum of [MATH] is of integer values (in suitable units), and that is what we also get from measurement.', 'quant-ph-0110136-3-50-1': 'But the spectrum calculated from a finitely truncated basis is not of integer values and will fluctuate with fluctuation size depends on the size of the truncated basis employed.', 'quant-ph-0110136-3-50-2': 'The accuracy size [MATH] of the measured probability distribution is chosen such that the off-set of the ground state energy [MATH] should allow us to conclude whether the ground state energy [MATH] is zero or not.', 'quant-ph-0110136-3-50-3': '([MATH] is in general a function of [MATH] and [MATH].)', 'quant-ph-0110136-3-50-4': '[EQUATION] where [MATH] and [MATH].', 'quant-ph-0110136-3-51-0': 'The termination of our algorithm is obtained if and when the adding of higher [MATH]-number states (those created from the coherent state by the application of the creation operator [MATH]) to the truncated basis does not change the approximated ground state of [MATH] beyond certain range of accuracy, [EQUATION]', 'quant-ph-0110136-3-51-1': 'Even we can prove that the approximated ground state [MATH] and its energy will eventually converge to its true values, the mathematical noncomputability results from the fact that their rates of convergence are unknown.', 'quant-ph-0110136-3-51-2': 'Thus, we might not be able to use mathematical reasoning alone to determine when to stop adding more states to the truncated basis in order to approximate the ground state correctly.', 'quant-ph-0110136-3-51-3': 'Different truncated bases would give some estimates for the ground state but we have no control over these estimates and no idea how good they are.', 'quant-ph-0110136-3-51-4': 'They could be anywhere in relation to the true values.', 'quant-ph-0110136-3-51-5': 'This is nothing but mathematical noncomputability.', 'quant-ph-0110136-3-51-6': "(However, this quantum algorithm has inspired us to reformulate Hilbert's tenth problem with continuous variables [CITATION], and the mathematical computibility of this reformulation, or lack of it, should be investigated further.)", 'quant-ph-0110136-3-52-0': 'To know when the truncated basis is sufficiently large to have the estimated ground state values within any given accuracy, that is, to regain computability, we have to exploit the measurability of physical processes.', 'quant-ph-0110136-3-52-1': 'Because of this measurability we can estimate the true accuracy of our measured values.', 'quant-ph-0110136-3-52-2': 'Then a comparison of results from the Schrodinger equation to these measurable quantities will help determining the accuracy of results from the equation, that is, regaining the lost computability through the physical world, presumably infinite.', 'quant-ph-0110136-3-53-0': '# Decoherence and error correction', 'quant-ph-0110136-3-54-0': 'Our approach above is in fact a combination of the quantum computation of continuous variables and of adiabatic evolution.', 'quant-ph-0110136-3-54-1': 'There exists some error correction protocol for continuous variables [CITATION] which could be of help here to protect the wave functions from decoherence.', 'quant-ph-0110136-3-54-2': 'However, the adiabatic computation we exploit is quite robust in general [CITATION].', 'quant-ph-0110136-3-54-3': 'An imperfect conventional quantum algorithm might have different sorts of errors than an imperfect adiabatic process, where the system is kept close to the instantaneous ground state over time.', 'quant-ph-0110136-3-54-4': 'Decoherence by the environment inducing transition between levels could be controlled in principle at a temperature that is small compared to the gap ([REF]), given its estimate.', 'quant-ph-0110136-3-54-5': 'Errors introduced by the hamiltonian simulation may result in a hamiltonian different from ([REF]) but in a form [MATH].', 'quant-ph-0110136-3-54-6': 'Recent numerical study [CITATION] of small systems has in fact indicated that the adiabatic computation is interestingly robust even for fairly large [MATH], provided [MATH] varies either sufficiently slowly or sufficiently rapidly (which is more likely to be the case considered here due to the nature of our hamiltonian simulations).', 'quant-ph-0110136-3-55-0': '# Probability boosting', 'quant-ph-0110136-3-56-0': 'Due to the particular nature of the present scheme, the various approximations result in some probability that the final measurement will not find the system in the desired ground state; but appropriate choices of the various time parameters could increase the success probability of the algorithm to more than even.', 'quant-ph-0110136-3-56-1': 'We spell out explicitly here the probability boosting technique, as mentioned in [CITATION], that one could subsequently apply.', 'quant-ph-0110136-3-57-0': 'In our computation, adiabatic or otherwise, the end result, starting from some initial state [MATH], may be contaminated with some excited states other than the desirable ground state [MATH], [EQUATION]', 'quant-ph-0110136-3-57-1': 'If [MATH], that is, if there is a better-than-even chance to obtain [MATH] by measurement then one can boost the success probability to arbitrarily closed to unity by performing a concatenated computation over [MATH] Hilbert spaces [EQUATION] where [MATH] is the normalising factor.', 'quant-ph-0110136-3-57-2': 'Let us consider the majority amplitudes, when more than half of the [MATH] Hilbert spaces return the correct results, [MATH]; and the minority amplitudes, [MATH].', 'quant-ph-0110136-3-57-3': 'The ratio of the majority over the minority, [MATH], is clearly boosted for sufficiently large [MATH] and for [MATH] and [MATH].', 'quant-ph-0110136-3-57-4': 'The probability distribution for a measurement of the end state in ([REF]), as a consequence, is exponentially dominated by the majority results.', 'quant-ph-0110136-3-57-5': 'In other words, the probability to obtain a majority result which contains the true ground state can be made arbitrarily closed to unity, [MATH], provided [MATH] and [MATH].', 'quant-ph-0110136-3-57-6': 'However, the decoherence control for such [MATH] concatenated Hilbert spaces will be more crucial.', 'quant-ph-0110136-3-58-0': '# Concluding remarks', 'quant-ph-0110136-3-59-0': 'In this paper, we consider and emphasise on the issue of computability in principle, not that of computational complexity.', 'quant-ph-0110136-3-59-1': "This attempt of broadening of the concept of effective computability, taken into account the quantum mechanical principles, has been argued to be able in principle to decide the classically/Turing undecidables, Hilbert's tenth problem and thus the Turing halting problem in this instance.", 'quant-ph-0110136-3-59-2': "If this is realisable, and we don't have any evidence of fundamental nature to the contrary, the Church-Turing thesis should be modified accordingly.", 'quant-ph-0110136-3-60-0': 'In summary, we have encoded the answer to the question about the existence or lack of non-negative integer solutions for any Diophantine equation into the of ground state of some relevant Hamiltonian.', 'quant-ph-0110136-3-60-1': 'The key factor in the ground state verification is the probability distributions, which not only can be calculated in numerical Quantum Mechanics (with a truncated basis) but also are measurable in practice.', 'quant-ph-0110136-3-60-2': 'After all, probability distributions are also physical observables.', 'quant-ph-0110136-3-60-3': 'However, in using the probability distributions as the identification criteria, we have to assume that Quantum Mechanics is able to describe Nature correctly to the precision required.', 'quant-ph-0110136-3-60-4': 'Note also that we have here a peculiar situation in which the computational complexity, that is, the computation time, might not be known exactly before carrying out the quantum computation - although it can be estimated approximately (see the appendix below).', 'quant-ph-0110136-3-61-0': 'On the other hand, if for any reasons the algorithm is not implementable because of physical principles and/or physical resources then it would be an example of information being limited by physics, rather than by logical arguments alone.', 'quant-ph-0110136-3-62-0': 'Our study is an illustration of "Information is physical" (see [CITATION] for another quantum mechanical approach and [CITATION] for where the theory of General Relativity is also exploited for the computation of Turing noncomputables).', 'quant-ph-0110136-3-63-0': 'That some generalisation of the notion of computation could help solving the previous undecidability/noncomputability has been recognised in mathematics and was considered by Kleene as quoted in [CITATION].', 'quant-ph-0110136-3-63-1': 'But this has not been realisable until now simply because of the non-recognition of quantum physics as the missing ingredient.', 'quant-ph-0110136-3-63-2': 'Our quantum algorithm could in fact be regarded as an infinite search through the integers in a finite amount of time, the type of search required by Kleene to solve the Turing halting problem.', 'quant-ph-0110136-3-64-0': "Our decidability study here only deals with the property of being Diophantine, which does not cover the property of being arithmetic in general, and as such has no direct consequences on the Godel's Incompleteness theorem [CITATION].", 'quant-ph-0110136-3-64-1': "However, it is conceivable that the Godel's theorem may lose its restrictive power once the concept of proof is suitably generalised with quantum principles.", 'quant-ph-0110136-3-65-0': '# Note added in proof', 'quant-ph-0110136-3-66-0': 'The author has recently obtained a criterion based on the final propbability distribution, which can be measured to arbitrary precision, for identifying the ground state of [MATH] at time [MATH].', 'quant-ph-0110136-3-66-1': 'See [CITATION] for more details.'}

1409.2514

{'1409.2514-1-0-0': 'We demonstrate that exciton conductance in organic materials can be enhanced by several orders of magnitude when the molecules are strongly coupled to an electromagnetic mode.', '1409.2514-1-0-1': 'Using a 1D model system, we show how the formation of a collective polaritonic mode allows excitons to bypass the disordered array of molecules and jump directly from one end of the structure to the other.', '1409.2514-1-0-2': 'This finding could have important implications in the fields of exciton transistors, heat transport, photosynthesis, and biological systems in which exciton transport plays a key role.', '1409.2514-1-1-0': 'The transport of excitons (bound electron-hole pairs) is a fundamental process that plays a crucial rule both in natural phenomena such as photosynthesis, where energy has to be transported to a reaction center [CITATION], and in artificial devices such as excitonic transistors [CITATION] or organic solar cells, whose power conversion efficiency can be improved significantly when the exciton diffusion length is increased [CITATION].', '1409.2514-1-1-1': 'Similarly, understanding and manipulating the role of excitons in heat transport has become an active field of research, with possible applications ranging from thermo-electric effects to heat-voltage converters, to nanoscale refrigerators, and even thermal logic gates (cf. [CITATION] and references therein).', '1409.2514-1-1-2': 'The exciton transport efficiency depends on a wide range of factors with such surprising features as the occurrence of noise-assisted transport [CITATION].', '1409.2514-1-1-3': 'Pioneering works have even suggested that coherent transport can play an important role in biological systems [CITATION].', '1409.2514-1-1-4': 'However, most systems composed of organic molecules are disordered and possess relatively large dissipation and dephasing rates, such that exciton transport typically becomes diffusive over long distances [CITATION].', '1409.2514-1-2-0': 'An intriguing possibility to modify exciton properties is by strong coupling to an electromagnetic (EM) mode, forming so-called polaritons (hybrid light-matter states).', '1409.2514-1-2-1': 'This is achieved when the Rabi frequency, i.e., the energy exchange rate between exciton and EM modes, becomes faster than the decay and/or decoherence rates of either constituent.', '1409.2514-1-2-2': 'Polaritons combine the properties of their constituents, in particular, mutual interactions and low effective masses, enabling new applications such as polariton condensation in semiconductors [CITATION] and organic materials [CITATION], the modification of molecular chemistry [CITATION] and work functions [CITATION], or the transfer of excitation between different molecular species [CITATION].', '1409.2514-1-2-3': 'Due to the large dipole moments and high densities, organic materials support large Rabi splittings [CITATION], and can also be strongly coupled to surface plasmon polaritons [CITATION].', '1409.2514-1-2-4': 'The dispersion relation can then be tuned to achieve a further reduction of the effective mass [CITATION].', '1409.2514-1-3-0': 'Very recently, an increase of the electrical conductance of an organic material was shown under strong coupling of the excitons to a cavity mode [CITATION].', '1409.2514-1-3-1': 'Inspired by this result, we demonstrate in this Letter that through strong coupling to an electromagnetic mode, i.e., the creation of polaritonic states, the exciton transport efficiency can be improved by many orders of magnitude.', '1409.2514-1-3-2': 'The strong coupling allows the excitons to bypass the disordered organic system, preventing localization and leading to dramatically improved energy transport properties.', '1409.2514-1-3-3': 'We note that while we focus on organic molecules in the following, the results can readily be generalized to other systems such as quantum dots and Rydberg atoms, or even chains of trapped ions, which offer a high degree of controllability [CITATION].', '1409.2514-1-4-0': 'We focus on a model system that captures the essential physics: A 1D chain of two-level emitters inside a cavity (see [REF]).', '1409.2514-1-4-1': 'The emitter dipole transition is coupled to the single cavity mode, and additionally induces Coulombic dipole-dipole interaction between the emitters.', '1409.2514-1-4-2': 'The effect of internal (e.g., rovibrational or phononic) and external environment modes is taken into account through effective dephasing and nonradiative decay rates modeled using a master equation of Lindblad form.', '1409.2514-1-4-3': 'The system Hamiltonian [MATH] in the rotating wave approximation (setting [MATH] here and in the following) is then [EQUATION] where [MATH] is the bosonic annihilation operator of the cavity mode with energy [MATH] and electric field [MATH].', '1409.2514-1-5-0': 'The molecular excitons of energy [MATH] are created and destroyed by the fermionic operators [MATH] and [MATH].', '1409.2514-1-5-1': 'Molecule [MATH] is characterized by its position [MATH] and dipole moment [MATH], giving the cavity-molecule interaction [MATH].', '1409.2514-1-5-2': 'The dipole-dipole interaction (in the quasistatic limit) is [EQUATION] with [MATH] and [MATH].', '1409.2514-1-6-0': 'The system dynamics is described by a Lindblad master equation for the density matrix [EQUATION] where [MATH] runs over all molecules as well as the cavity mode.', '1409.2514-1-6-1': 'The superoperators [MATH] describe decay and dephasing: [EQUATION] where [MATH] is the standard form for Lindblad superoperators.', '1409.2514-1-6-2': 'The total molecule decay rate [MATH] is given by [MATH], with [MATH] and [MATH] the radiative and nonradiative decay rates, while [MATH] is the dephasing rate.', '1409.2514-1-6-3': 'The decay rate [MATH] of the cavity photons is dominated by leakage through the mirrors.', '1409.2514-1-6-4': 'For later reference, we also define the total molecular decoherence rate [MATH].', '1409.2514-1-7-0': 'As mentioned above, we only include one cavity mode and describe the molecules by a linear 1D chain along the longitudinal cavity direction ([MATH] axis), with positions [MATH], such that the cavity electric field is identical for all molecules.', '1409.2514-1-7-1': 'It is polarized along the (out-of-plane) [MATH] axis, leading to [MATH].', '1409.2514-1-7-2': 'The total coupling between molecules and the cavity mode can be characterized by the collective Rabi frequency [MATH].', '1409.2514-1-7-3': 'For zero detuning [MATH], strong coupling is entered for [MATH] and leads to the formation of upper and lower polaritons at energies [MATH].', '1409.2514-1-7-4': 'The Rabi splitting (energy difference between upper and lower polariton) can approach [MATH]eV in experiments [CITATION], and can be tuned by changing either the molecule density or the mode electric field strength.', '1409.2514-1-8-0': 'In the following, we consider two types of molecular configurations: A perfectly ordered distribution, with molecule positions on a regular grid and dipole moments perfectly aligned to the electric field (i.e., along the [MATH] axis), and a random distribution, where Gaussian noise is added to the regular positions and the dipole moments are oriented randomly.', '1409.2514-1-8-1': 'Note that for the present case of a 1D linear chain, randomness is expected to suppress conductance much more efficiently than in higher dimensions.', '1409.2514-1-9-0': 'We next introduce a prescription for calculating an exciton conductance [MATH], a steady-state quantity to characterize the exciton transport efficiency similar to the electrical conductance for charge transport.', '1409.2514-1-9-1': 'We assume that excitation is continuously pumped into the system on the left side and measure the energy leaving the system through the right side (cf. [REF]).', '1409.2514-1-9-2': 'The pumping is represented by an additional incoherent driving term [MATH].', '1409.2514-1-10-0': 'The energy current is obtained from the rate of change of energy [CITATION] [EQUATION]', '1409.2514-1-10-1': 'This is evaluated in the steady state, [MATH], for which the total rate of change in energy is zero ([MATH]).', '1409.2514-1-10-2': 'However, each of the Lindblad superoperators [MATH] can be associated with a specific physical process.', '1409.2514-1-10-3': 'We thus identify the energy current between the two reservoirs with the loss of energy from the last molecule: [EQUATION]', '1409.2514-1-10-4': 'We then define the exciton conductance as the current per driving power, i.e., [MATH] (which has units of energy).', '1409.2514-1-10-5': 'Note that contrary to [CITATION], the energy entering the system through pumping does not necessarily leave through the sink at the end.', '1409.2514-1-10-6': 'It can also be lost through the radiative and nonradiative decay of the molecules, as well as decay of the cavity mode.', '1409.2514-1-11-0': 'We numerically obtain the steady state of the system using the open-source QuTiP package [CITATION].', '1409.2514-1-11-1': 'To do so, we restrict the total superoperator in [REF] to the zero- and single-excitation subspaces.', '1409.2514-1-11-2': 'This truncation of the Hilbert space is an excellent approximation in the presently relevant linear response regime of weak pumping, i.e., for [MATH] smaller than the system decay rates.', '1409.2514-1-12-0': 'We choose the quantum emitter parameters to approximately correspond to TDBC J-aggregates at room temperature [CITATION]: [MATH]eV, [MATH]ps, [MATH]fs, and [MATH]fs.', '1409.2514-1-12-1': 'The cavity lifetime [MATH]fs is typical for experiments using cavities made of thin metal mirrors [CITATION].', '1409.2514-1-12-2': 'The molecule parameters also determine the dipole moment through [MATH], giving [MATH]Debye.', '1409.2514-1-12-3': 'The average intermolecular spacing is taken as [MATH]nm.', '1409.2514-1-12-4': 'Note that while the coupling to the cavity mode is taken into account explicitly, the radiative decay rate into all other electromagnetic modes can also be modified by the presence of a cavity.', '1409.2514-1-12-5': 'Since [MATH] is much smaller than the other rates, this modification can safely be neglected here.', '1409.2514-1-12-6': 'While we only show results for the parameters given above, we checked that the main conclusions drawn in the following apply for a wide range of parameters and do not depend on the specific values chosen here.', '1409.2514-1-13-0': '[REF] shows the exciton conductance [MATH] at zero cavity-molecule detuning [MATH], as a function of the collective Rabi frequency [MATH].', '1409.2514-1-13-1': 'Here, we keep the number of molecules fixed and change the electric field strength, going through the transition from weak to strong coupling.', '1409.2514-1-13-2': 'This can be achieved in an experiment by, e.g., putting the molecules at different positions inside the cavity [CITATION].', '1409.2514-1-13-3': 'We compare regular and random molecule arrangements for chains of 40 and 60 molecules.', '1409.2514-1-13-4': 'For the regular distribution, the strong dipole-dipole interaction leads to an additional small energy shift [MATH] of the molecular bright state coupling to the cavity; zero detuning thus corresponds to [MATH].', '1409.2514-1-14-0': 'For all of the cases shown in [REF], the conductance is approximately constant in the weak coupling limit [MATH], where the cavity mode does not play a role.', '1409.2514-1-14-1': 'Unsurprisingly, the conductance in this limit strongly depends on the molecular configuration-it is almost completely suppressed for the random case, for which 1D systems always show Anderson localization.', '1409.2514-1-14-2': 'The conductance in the random case is calculated as the logarithmic mean of 100 random configurations as appropriate for localized systems, i.e., [MATH] [CITATION].', '1409.2514-1-14-3': 'Note that even in the regular case, transport is quite inefficient due to the relatively large decay and dephasing rates of the molecular excitons, leading to diffusive transport [CITATION].', '1409.2514-1-14-4': 'Strikingly, when the coupling to the cavity mode is increased, an extraordinary increase of the conductance is observed in all cases.', '1409.2514-1-14-5': 'Once strong coupling is reached ([MATH]), the conductance again becomes almost independent of [MATH], indicating that the fully formed polariton channel dominates exciton conductance.', '1409.2514-1-14-6': 'In this limit, the conductance also becomes almost independent of the configuration and only depends on the number of molecules, i.e., length of the 1D chain.', '1409.2514-1-14-7': 'While randomness can suppress conduction almost completely in the weak-coupling limit, the polariton modes are barely affected by it.', '1409.2514-1-14-8': 'As a consequence of their delocalized nature induced by the collective exciton-cavity coupling, the excitation can efficiently bypass the disordered chain of emitters.', '1409.2514-1-15-0': 'This also provides a possible indication for the mechanism behind the enhanced electrical conduction observed under strong coupling in the experiments by Orgiu et al. [CITATION].', '1409.2514-1-15-1': 'However, the connection between exciton transport through polaritons and electrical conduction is currently unclear, as polaritons are in principle neutral quasiparticles.', '1409.2514-1-16-0': 'We also note that while we focus on incoherent driving for simplicity in this work, we have found that under coherent driving/incoupling at frequency [MATH], the same general behavior is observed.', '1409.2514-1-16-1': 'The main difference is an additional resonant enhancement when [MATH] coincides with the eigenfrequencies of the system (see details in the supplemental material [CITATION]).', '1409.2514-1-16-2': 'An interesting aspect is that even when hopping is completely suppressed, resonant transport occurs not only when driving at the polariton eigenfrequencies, but also at the unmodified molecule frequency-a clear signature that the dark states (which are not coupled to the cavity EM field) are still affected by the existence of strong coupling.', '1409.2514-1-17-0': 'We next focus on the case where the cavity mode is detuned from the molecular excitations by an energy [MATH].', '1409.2514-1-17-1': 'As shown in [REF], the onset of the extraordinary conductance is then shifted to larger coupling strengths for increasing detuning [MATH].', '1409.2514-1-17-2': 'However, the final conductance in the strong-coupling limit is independent of the detuning.', '1409.2514-1-17-3': 'This again indicates that the conduction proceeds through the polariton modes, which are only fully formed when the Rabi frequency [MATH] becomes large enough to not only overcome decoherence processes, but also the detuning.', '1409.2514-1-17-4': 'Once this is fulfilled, their character does not strongly depend on the detuning.', '1409.2514-1-17-5': 'However, even for relatively large and experimentally relevant Rabi frequencies, e.g., [MATH]meV, a small change of the detuning can strongly suppress or enhance conductance.', '1409.2514-1-18-0': 'Next, we develop a simplified model to understand the extraordinary increase in exciton conductance under strong coupling.', '1409.2514-1-18-1': 'The main idea behind it is that there are two almost independent transport channels: (i) Direct excitonic transport through hopping between the molecules, which dominates in the weak-coupling limit, and (ii) polaritonic transport through the collective modes created by strong coupling to the cavity, which increases rapidly (polynomially) as the coupling is increased and saturates in the strong-coupling limit.', '1409.2514-1-18-2': 'To expose the contribution of the second channel unambiguously, we remove the dipole-dipole interaction responsible for hopping from the Hamiltonian [REF].', '1409.2514-1-18-3': 'We furthermore assume that the molecules are all aligned along the cavity field polarization axis [MATH].', '1409.2514-1-18-4': 'The interaction with the cavity mode is then identical for all molecules.', '1409.2514-1-19-0': '[REF] shows that the picture of independent channels is indeed valid: For the conductance [MATH] without hopping, the transmission plateau for weak coupling disappears, while the transmission in strong coupling is essentially unchanged.', '1409.2514-1-19-1': 'The polariton contribution without hopping decreases rapidly with decreasing Rabi frequency.', '1409.2514-1-19-2': 'The independence of the two transport channels is further verified by plotting the sum of the exciton conductance in the weak-coupling limit, [MATH], and the cavity-mediated contribution [MATH] without hopping.', '1409.2514-1-19-3': 'This sum, given by the dash-dotted black line in [REF], agrees excellently with the full result, indicating that the two transport channels are indeed effectively independent.', '1409.2514-1-20-0': 'Since in this model, all molecules but the first (due to the pumping) are indistinguishable, they behave identically.', '1409.2514-1-20-1': 'Independent of the number of molecules [MATH], there are thus just 12 independent components of the steady state density matrix within the zero- and single-excitation subspace.', '1409.2514-1-20-2': 'This density matrix can then be obtained analytically, and inserting the solution into [REF] gives an explicit formula for the conductance.', '1409.2514-1-20-3': 'In the linear response limit of weak driving and for zero detuning [MATH], it is given by [EQUATION] where we have defined the total decoherence rate [MATH].', '1409.2514-1-20-4': 'More details and the full expression for arbitrary detuning are given in the supplemental material [CITATION].', '1409.2514-1-21-0': 'As expected, the analytical solution ([REF]) perfectly matches the numerics (cf. [REF]).', '1409.2514-1-21-1': 'For small Rabi frequency, the conductance through the polariton modes grows with [MATH], while it saturates to a constant value for [MATH].', '1409.2514-1-21-2': 'For large [MATH], this constant value is given by [MATH] if [MATH] and by [MATH] if [MATH].', '1409.2514-1-21-3': 'Importantly, the decay with system size is algebraic [MATH], as opposed to the localized [MATH] behavior expected in the absence of strong coupling.', '1409.2514-1-21-4': 'Note that [MATH] occurs here because only a single molecule is connected to each of the baths; for a quasi-1D wire with a transverse extension, the length of the system would be the relevant variable.', '1409.2514-1-21-5': 'The dependence on the fourth power of [MATH] in the weak-coupling limit is explained by the rate obtained from two quantum jumps, to the cavity mode and back, with coupling [MATH].', '1409.2514-1-21-6': 'Interestingly, [REF] shows that the conductance, at least in the simplified model without disorder or direct hopping, is not directly related to the conventional criterion for the onset of strong coupling where the vacuum Rabi splitting becomes real ([MATH]).', '1409.2514-1-21-7': 'Indeed, the difference [MATH] does not occur in [REF].', '1409.2514-1-21-8': 'Instead, the exciton conductance becomes constant when [MATH] (for large [MATH]).', '1409.2514-1-21-9': 'This is related to large values of the cooperativity [MATH], which can occur even if strong coupling is not fully reached.', '1409.2514-1-22-0': 'To conclude, we have demonstrated that the formation of polariton modes, i.e., strong coupling, can dramatically enhance exciton transport.', '1409.2514-1-22-1': 'When the coupling is strong enough and the polaritons are fully formed, the excitons can almost completely bypass the chain of quantum emitters and "jump" directly from one end to the other, leading to large exciton conductance.', '1409.2514-1-22-2': 'This robust effect persists almost independently of the exact parameters of the system, and most notably occurs efficiently even when the underlying excitonic system is strongly disordered and its transport is completely suppressed due to localization.', '1409.2514-1-22-3': 'Through a simple model, we have furthermore shown that transport through direct hopping and through the polariton modes constitute two effectively independent channels, which helps to explain why the polariton conductance is almost independent of the disorder in the system.', '1409.2514-1-22-4': 'These results demonstrate a possible pathway for improving the efficiency of excitonic devices, where the EM mode could be provided by plasmonic structures to enable fully integrated nanometer-scale devices.', '1409.2514-1-22-5': 'We note that related results have simultaneously been obtained by Schachenmayer et al. [CITATION].', '1409.2514-1-23-0': 'We thank T. Ebbesen, G. Pupillo, C. Genes, and L. Martin Moreno for fruitful discussions, and E. Moreno and A. Delga for a careful reading of the manuscript.', '1409.2514-1-23-1': 'This work has been funded by the European Research Council (ERC-2011-AdG Proposal No. 290981).', '1409.2514-1-24-0': 'Supplemental Material', '1409.2514-1-25-0': '# Coherent driving', '1409.2514-1-26-0': 'In this section, we show some results for coherent driving of the system (as opposed to the incoherent driving used in the main text).', '1409.2514-1-26-1': 'As mentioned in the main text and shown below, the main conclusions are similar to those obtained under incoherent driving.', '1409.2514-1-26-2': 'The main difference is that under coherent driving at frequency [MATH], a resonant enhancement of the conductance is observed when [MATH] corresponds to one of the eigenmodes of the system.', '1409.2514-1-26-3': 'We use the same model as in the main text, but instead of an incoherent pumping term described by [MATH], we add a coherent driving term [MATH] (within the rotating wave approximation) to the Hamiltonian.', '1409.2514-1-26-4': 'Here, [MATH] and [MATH] are the driving amplitude and frequency, respectively.', '1409.2514-1-26-5': 'The steady state can then be calculated by going to the rotating frame, in which [MATH] is time-independent, but the system frequencies are shifted by [MATH].', '1409.2514-1-26-6': 'The exciton conductance is calculated analogously to the incoherent case, using [MATH], where the factor [MATH] is added to make [MATH] unitless.', '1409.2514-1-27-0': 'We focus again on the case without dipole-dipole interaction, which allows to calculate only the polariton contribution to exciton conductivity even when relatively short chains are used ([MATH] in the following).', '1409.2514-1-27-1': 'For simplicity, we also choose zero cavity-molecule detuning, [MATH].', '1409.2514-1-27-2': '[REF] demonstrates that the polariton-mediated exciton conductance under coherent driving displays a polynomial increase with [MATH], which saturates to a constant value once strong coupling is fully entered.', '1409.2514-1-27-3': 'This is identical to the behavior under incoherent pumping.', '1409.2514-1-27-4': 'In addition, the conductance is resonantly enhanced when the driving frequency coincides with an eigenfrequency of the system.', '1409.2514-1-27-5': 'The system eigenfrequencies correspond to the upper and lower polariton states ([MATH]), as well as the dark states at [MATH], which are not directly coupled to the cavity.', '1409.2514-1-27-6': 'Interestingly, and somewhat surprisingly, the largest conductance is found when driving the dark modes ([MATH]), not the polariton modes (cmp.', '1409.2514-1-27-7': 'the violet and dashed orange lines in [REF]b).', '1409.2514-1-27-8': 'This effect persists even when dephasing (which couples between dark and bright states) is turned off, and demonstrates that even the eigenstates that are not themselves strongly coupled to the cavity mode become delocalized when the system enters strong coupling.', '1409.2514-1-28-0': '# Simplified model', '1409.2514-1-29-0': 'In this section, we provide more details on the simplified model and its solution described in the text.', '1409.2514-1-29-1': 'The starting point is the Hamiltonian given in Eq. 1 in the main text: [EQUATION] from which we remove the direct dipole-dipole interaction responsible for hopping by setting [MATH].', '1409.2514-1-29-2': 'We next assume that all molecules are oriented identically, such that the [MATH] are constant and given by [MATH].', '1409.2514-1-29-3': 'As explained in the main text, all molecules apart from the first one, which is pumped, are now indistinguishable.', '1409.2514-1-29-4': 'The time evolution of the system density matrix is given by Eq. 2 in the main text: [EQUATION]', '1409.2514-1-29-5': 'Due to the structure and symmetries of the system, we can write the steady-state density matrix within the zero- and single-excitation subspace as [EQUATION] where [MATH] is the probability to be in the ground state (no photons or excitons), [MATH] is the cavity excitation probability, [MATH] is the excitation probability of molecule [MATH], and [MATH] is the coherence matrix element between constituent [MATH] and [MATH].', '1409.2514-1-29-6': 'Since all but the first molecule behave identically, the density matrix elements involving molecule [MATH] with [MATH] are all identical and given with [MATH].', '1409.2514-1-29-7': 'Additionally, [MATH] denotes the coherence element between any pair of molecules not involving the first.', '1409.2514-1-30-0': 'The steady state density matrix can then be obtained from [MATH], which for the specific form of the density matrix corresponds to a set of linear equations for the 4 real probabilities [MATH] and the 4 complex coherence elements [MATH], giving 12 independent real variables.', '1409.2514-1-30-1': 'Note that there are actually [MATH] linear equations for [MATH] molecules.', '1409.2514-1-30-2': 'However, due to the symmetry, only 11 of them are linearly independent.', '1409.2514-1-30-3': 'The steady-state density matrix is then obtained by solving for the null space of the coefficient matrix of the linear equations, and subsequently normalized to [MATH].', '1409.2514-1-31-0': 'Having obtained the steady state density matrix, the exciton conductivity is calculated using Eq. 7 in the main text, which explicitly gives [EQUATION]', '1409.2514-1-31-1': 'Inserting the solution [MATH] into this expression and taking the linear-response limit [MATH] gives the general solution for [MATH], which we give here for completeness: [EQUATION] where in addition to the definitions in the main text, we use [MATH].', '1409.2514-1-31-2': 'Finally, we note that the same approach works equally well under coherent driving (not shown here).'}

{'1409.2514-2-0-0': 'We demonstrate that exciton conductance in organic materials can be enhanced by several orders of magnitude when the molecules are strongly coupled to an electromagnetic mode.', '1409.2514-2-0-1': 'Using a 1D model system, we show how the formation of a collective polaritonic mode allows excitons to bypass the disordered array of molecules and jump directly from one end of the structure to the other.', '1409.2514-2-0-2': 'This finding could have important implications in the fields of exciton transistors, heat transport, photosynthesis, and biological systems in which exciton transport plays a key role.', '1409.2514-2-1-0': 'The transport of excitons (bound electron-hole pairs) is a fundamental process that plays a crucial rule both in natural phenomena such as photosynthesis, where energy has to be transported to a reaction center [CITATION], and in artificial devices such as excitonic transistors [CITATION] or organic solar cells, whose power conversion efficiency can be improved significantly when the exciton diffusion length is increased [CITATION].', '1409.2514-2-1-1': 'Similarly, understanding and manipulating the role of excitons in heat transport has become an active field of research, with possible applications ranging from thermo-electric effects to heat-voltage converters, to nanoscale refrigerators, and even thermal logic gates (cf. [CITATION] and references therein).', '1409.2514-2-1-2': 'The exciton transport efficiency depends on a wide range of factors with such surprising features as the occurrence of noise-assisted transport [CITATION].', '1409.2514-2-1-3': 'Pioneering works have even suggested that coherent transport can play an important role in biological systems [CITATION].', '1409.2514-2-1-4': 'However, most systems composed of organic molecules are disordered and possess relatively large dissipation and dephasing rates, such that exciton transport typically becomes diffusive over long distances [CITATION].', '1409.2514-2-2-0': 'An intriguing possibility to modify exciton properties is by strong coupling to an electromagnetic (EM) mode, forming so-called polaritons (hybrid light-matter states).', '1409.2514-2-2-1': 'This is achieved when the Rabi frequency, i.e., the energy exchange rate between exciton and EM modes, becomes faster than the decay and/or decoherence rates of either constituent.', '1409.2514-2-2-2': 'Polaritons combine the properties of their constituents, in particular, mutual interactions and low effective masses, enabling new applications such as polariton condensation in semiconductors [CITATION] and organic materials [CITATION], the modification of molecular chemistry [CITATION] and work functions [CITATION], or the transfer of excitation between different molecular species [CITATION].', '1409.2514-2-2-3': 'Due to the large dipole moments and high densities, organic materials support large Rabi splittings [CITATION], and can also be strongly coupled to surface plasmon polaritons [CITATION].', '1409.2514-2-2-4': 'The dispersion relation can then be tuned to achieve a further reduction of the effective mass [CITATION].', '1409.2514-2-3-0': 'Very recently, an increase of the electrical conductance of an organic material was shown under strong coupling of the excitons to a cavity mode [CITATION].', '1409.2514-2-3-1': 'Inspired by this result, we demonstrate in this Letter that through strong coupling to an electromagnetic mode, i.e., the creation of polaritonic states, the exciton transport efficiency can be improved by many orders of magnitude.', '1409.2514-2-3-2': 'The strong coupling allows the excitons to bypass the disordered organic system, preventing localization and leading to dramatically improved energy transport properties.', '1409.2514-2-3-3': 'We note that while we focus on organic molecules in the following, the results can readily be generalized to other systems such as quantum dots and Rydberg atoms, or even chains of trapped ions, which offer a high degree of controllability [CITATION].', '1409.2514-2-4-0': 'We focus on a model system that captures the essential physics: A 1D chain of two-level emitters inside a cavity (see [REF]).', '1409.2514-2-4-1': 'The emitter dipole transition is coupled to the single cavity mode, and additionally induces Coulombic dipole-dipole interaction between the emitters.', '1409.2514-2-4-2': 'The effect of internal (e.g., rovibrational or phononic) and external environment modes is taken into account through effective dephasing and nonradiative decay rates modeled using a master equation of Lindblad form.', '1409.2514-2-4-3': 'The system Hamiltonian [MATH] in the rotating wave approximation (setting [MATH] here and in the following) is then [EQUATION] where [MATH] is the bosonic annihilation operator of the cavity mode with energy [MATH] and electric field [MATH].', '1409.2514-2-5-0': 'The molecular excitons of energy [MATH] are created and destroyed by the fermionic operators [MATH] and [MATH].', '1409.2514-2-5-1': 'Molecule [MATH] is characterized by its position [MATH] and dipole moment [MATH], giving the cavity-molecule interaction [MATH].', '1409.2514-2-5-2': 'The dipole-dipole interaction (in the quasistatic limit) is [EQUATION] with [MATH] and [MATH].', '1409.2514-2-6-0': 'The system dynamics is described by a Lindblad master equation for the density matrix [EQUATION] where [MATH] runs over all molecules as well as the cavity mode.', '1409.2514-2-6-1': 'The superoperators [MATH] describe decay and dephasing: [EQUATION] where [MATH] is the standard form for Lindblad superoperators.', '1409.2514-2-6-2': 'The total molecule decay rate [MATH] is given by [MATH], with [MATH] and [MATH] the radiative and nonradiative decay rates, while [MATH] is the dephasing rate.', '1409.2514-2-6-3': 'The decay rate [MATH] of the cavity photons is dominated by leakage through the mirrors.', '1409.2514-2-6-4': 'For later reference, we also define the total molecular decoherence rate [MATH].', '1409.2514-2-7-0': 'As mentioned above, we only include one cavity mode and describe the molecules by a linear 1D chain along the longitudinal cavity direction ([MATH] axis), with positions [MATH], such that the cavity electric field is identical for all molecules.', '1409.2514-2-7-1': 'It is polarized along the (out-of-plane) [MATH] axis, leading to [MATH].', '1409.2514-2-7-2': 'The total coupling between molecules and the cavity mode can be characterized by the collective Rabi frequency [MATH].', '1409.2514-2-7-3': 'For zero detuning [MATH], strong coupling is entered for [MATH] and leads to the formation of upper and lower polaritons at energies [MATH].', '1409.2514-2-7-4': 'The Rabi splitting (energy difference between upper and lower polariton) can approach [MATH]eV in experiments [CITATION], and can be tuned by changing either the molecule density or the mode electric field strength.', '1409.2514-2-8-0': 'In the following, we consider two types of molecular configurations: A perfectly ordered distribution, with molecule positions on a regular grid and dipole moments perfectly aligned to the electric field (i.e., along the [MATH] axis), and a random distribution, where Gaussian noise is added to the regular positions and the dipole moments are oriented randomly.', '1409.2514-2-8-1': 'Note that for the present case of a 1D linear chain, randomness is expected to suppress conductance much more efficiently than in higher dimensions.', '1409.2514-2-9-0': 'We next introduce a prescription for calculating an exciton conductance [MATH], a steady-state quantity to characterize the exciton transport efficiency similar to the electrical conductance for charge transport.', '1409.2514-2-9-1': 'We assume that excitation is continuously pumped into the system on the left side and measure the energy leaving the system through the right side (cf. [REF]).', '1409.2514-2-9-2': 'The pumping is represented by an additional incoherent driving term [MATH].', '1409.2514-2-10-0': 'The energy current is obtained from the rate of change of energy [CITATION] [EQUATION]', '1409.2514-2-10-1': 'This is evaluated in the steady state, [MATH], for which the total rate of change in energy is zero ([MATH]).', '1409.2514-2-10-2': 'However, each of the Lindblad superoperators [MATH] can be associated with a specific physical process.', '1409.2514-2-10-3': 'We thus identify the energy current between the two reservoirs with the loss of energy from the last molecule: [EQUATION]', '1409.2514-2-10-4': 'We then define the exciton conductance as the current per driving power, i.e., [MATH] (which has units of energy).', '1409.2514-2-10-5': 'Note that contrary to [CITATION], the energy entering the system through pumping does not necessarily leave through the sink at the end.', '1409.2514-2-10-6': 'It can also be lost through the radiative and nonradiative decay of the molecules, as well as decay of the cavity mode.', '1409.2514-2-11-0': 'We numerically obtain the steady state of the system using the open-source QuTiP package [CITATION].', '1409.2514-2-11-1': 'To do so, we restrict the total superoperator in [REF] to the zero- and single-excitation subspaces.', '1409.2514-2-11-2': 'This truncation of the Hilbert space is an excellent approximation in the presently relevant linear response regime of weak pumping, i.e., for [MATH] smaller than the system decay rates.', '1409.2514-2-12-0': 'We choose the quantum emitter parameters to approximately correspond to TDBC J-aggregates at room temperature [CITATION]: [MATH]eV, [MATH]ps, [MATH]fs, and [MATH]fs.', '1409.2514-2-12-1': 'The cavity lifetime [MATH]fs is typical for experiments using cavities made of thin metal mirrors [CITATION].', '1409.2514-2-12-2': 'The molecule parameters also determine the dipole moment through [MATH], giving [MATH]Debye.', '1409.2514-2-12-3': 'The average intermolecular spacing is taken as [MATH]nm.', '1409.2514-2-12-4': 'Note that while the coupling to the cavity mode is taken into account explicitly, the radiative decay rate into all other electromagnetic modes can also be modified by the presence of a cavity.', '1409.2514-2-12-5': 'Since [MATH] is much smaller than the other rates, this modification can safely be neglected here.', '1409.2514-2-12-6': 'While we only show results for the parameters given above, we checked that the main conclusions drawn in the following apply for a wide range of parameters and do not depend on the specific values chosen here.', '1409.2514-2-13-0': '[REF] shows the exciton conductance [MATH] at zero cavity-molecule detuning [MATH], as a function of the collective Rabi frequency [MATH].', '1409.2514-2-13-1': 'Here, we keep the number of molecules fixed and change the electric field strength, going through the transition from weak to strong coupling.', '1409.2514-2-13-2': 'This can be achieved in an experiment by, e.g., putting the molecules at different positions inside the cavity [CITATION].', '1409.2514-2-13-3': 'We compare regular and random molecule arrangements for chains of 40 and 60 molecules.', '1409.2514-2-13-4': 'For the regular distribution, the strong dipole-dipole interaction leads to an additional small energy shift [MATH] of the molecular bright state coupling to the cavity; zero detuning thus corresponds to [MATH].', '1409.2514-2-14-0': 'For all of the cases shown in [REF], the conductance is approximately constant in the weak coupling limit [MATH], where the cavity mode does not play a role.', '1409.2514-2-14-1': 'Unsurprisingly, the conductance in this limit strongly depends on the molecular configuration-it is almost completely suppressed for the random case, for which 1D systems always show Anderson localization.', '1409.2514-2-14-2': 'The conductance in the random case is calculated as the logarithmic mean of 100 random configurations as appropriate for localized systems, i.e., [MATH] [CITATION].', '1409.2514-2-14-3': 'Note that even in the regular case, transport is quite inefficient due to the relatively large decay and dephasing rates of the molecular excitons, leading to diffusive transport [CITATION].', '1409.2514-2-14-4': 'Strikingly, when the coupling to the cavity mode is increased, an extraordinary increase of the conductance is observed in all cases.', '1409.2514-2-14-5': 'Once strong coupling is reached ([MATH]), the conductance again becomes almost independent of [MATH], indicating that the fully formed polariton channel dominates exciton conductance.', '1409.2514-2-14-6': 'In this limit, the conductance also becomes almost independent of the configuration and only depends on the number of molecules, i.e., length of the 1D chain.', '1409.2514-2-14-7': 'While randomness can suppress conduction almost completely in the weak-coupling limit, the polariton modes are barely affected by it.', '1409.2514-2-14-8': 'As a consequence of their delocalized nature induced by the collective exciton-cavity coupling, the excitation can efficiently bypass the disordered chain of emitters.', '1409.2514-2-15-0': 'This also provides a possible indication for the mechanism behind the enhanced electrical conduction observed under strong coupling in the experiments by Orgiu et al. [CITATION].', '1409.2514-2-15-1': 'However, the connection between exciton transport through polaritons and electrical conduction is currently unclear, as polaritons are in principle neutral quasiparticles.', '1409.2514-2-16-0': 'We also note that while we focus on incoherent driving for simplicity in this work, we have found that under coherent driving/incoupling at frequency [MATH], the same general behavior is observed.', '1409.2514-2-16-1': 'The main difference is an additional resonant enhancement when [MATH] coincides with the eigenfrequencies of the system (see details in the supplemental material [CITATION]).', '1409.2514-2-16-2': 'An interesting aspect is that even when hopping is completely suppressed, resonant transport occurs not only when driving at the polariton eigenfrequencies, but also at the unmodified molecule frequency-a clear signature that the dark states (which are not coupled to the cavity EM field) are still affected by the existence of strong coupling.', '1409.2514-2-17-0': 'We next focus on the case where the cavity mode is detuned from the molecular excitations by an energy [MATH].', '1409.2514-2-17-1': 'As shown in [REF], the onset of the extraordinary conductance is then shifted to larger coupling strengths for increasing detuning [MATH].', '1409.2514-2-17-2': 'However, the final conductance in the strong-coupling limit is independent of the detuning.', '1409.2514-2-17-3': 'This again indicates that the conduction proceeds through the polariton modes, which are only fully formed when the Rabi frequency [MATH] becomes large enough to not only overcome decoherence processes, but also the detuning.', '1409.2514-2-17-4': 'Once this is fulfilled, their character does not strongly depend on the detuning.', '1409.2514-2-17-5': 'However, even for relatively large and experimentally relevant Rabi frequencies, e.g., [MATH]meV, a small change of the detuning can strongly suppress or enhance conductance.', '1409.2514-2-18-0': 'Next, we develop a simplified model to understand the extraordinary increase in exciton conductance under strong coupling.', '1409.2514-2-18-1': 'The main idea behind it is that there are two almost independent transport channels: (i) Direct excitonic transport through hopping between the molecules, which dominates in the weak-coupling limit, and (ii) polaritonic transport through the collective modes created by strong coupling to the cavity, which increases rapidly (polynomially) as the coupling is increased and saturates in the strong-coupling limit.', '1409.2514-2-18-2': 'To expose the contribution of the second channel unambiguously, we remove the dipole-dipole interaction responsible for hopping from the Hamiltonian [REF].', '1409.2514-2-18-3': 'We furthermore assume that the molecules are all aligned along the cavity field polarization axis [MATH].', '1409.2514-2-18-4': 'The interaction with the cavity mode is then identical for all molecules.', '1409.2514-2-19-0': '[REF] shows that the picture of independent channels is indeed valid: For the conductance [MATH] without hopping, the transmission plateau for weak coupling disappears, while the transmission in strong coupling is essentially unchanged.', '1409.2514-2-19-1': 'The polariton contribution without hopping decreases rapidly with decreasing Rabi frequency.', '1409.2514-2-19-2': 'The independence of the two transport channels is further verified by plotting the sum of the exciton conductance in the weak-coupling limit, [MATH], and the cavity-mediated contribution [MATH] without hopping.', '1409.2514-2-19-3': 'This sum, given by the dash-dotted black line in [REF], agrees excellently with the full result, indicating that the two transport channels are indeed effectively independent.', '1409.2514-2-20-0': 'Since in this model, all molecules but the first (due to the pumping) are indistinguishable, they behave identically.', '1409.2514-2-20-1': 'Independent of the number of molecules [MATH], there are thus just 12 independent components of the steady state density matrix within the zero- and single-excitation subspace.', '1409.2514-2-20-2': 'This density matrix can then be obtained analytically, and inserting the solution into [REF] gives an explicit formula for the conductance.', '1409.2514-2-20-3': 'In the linear response limit of weak driving and for zero detuning [MATH], it is given by [EQUATION] where we have defined the total decoherence rate [MATH].', '1409.2514-2-20-4': 'More details and the full expression for arbitrary detuning are given in the supplemental material [CITATION].', '1409.2514-2-21-0': 'As expected, the analytical solution ([REF]) perfectly matches the numerics (cf. [REF]).', '1409.2514-2-21-1': 'For small Rabi frequency, the conductance through the polariton modes grows with [MATH], while it saturates to a constant value for [MATH].', '1409.2514-2-21-2': 'For large [MATH], this constant value is given by [MATH] if [MATH] and by [MATH] if [MATH].', '1409.2514-2-21-3': 'Importantly, the decay with system size is algebraic [MATH], as opposed to the localized [MATH] behavior expected in the absence of strong coupling.', '1409.2514-2-21-4': 'Note that [MATH] occurs here because only a single molecule is connected to each of the baths; for a quasi-1D wire with a transverse extension, the length of the system would be the relevant variable.', '1409.2514-2-21-5': 'The dependence on the fourth power of [MATH] in the weak-coupling limit is explained by the rate obtained from two quantum jumps, to the cavity mode and back, with coupling [MATH].', '1409.2514-2-21-6': 'Interestingly, [REF] shows that the conductance, at least in the simplified model without disorder or direct hopping, is not directly related to the conventional criterion for the onset of strong coupling where the vacuum Rabi splitting becomes real ([MATH]).', '1409.2514-2-21-7': 'Indeed, the difference [MATH] does not occur in [REF].', '1409.2514-2-21-8': 'Instead, the exciton conductance becomes constant when [MATH] (for large [MATH]).', '1409.2514-2-21-9': 'This is related to large values of the cooperativity [MATH], which can occur even if strong coupling is not fully reached.', '1409.2514-2-22-0': 'To conclude, we have demonstrated that the formation of polariton modes, i.e., strong coupling, can dramatically enhance exciton transport.', '1409.2514-2-22-1': 'When the coupling is strong enough and the polaritons are fully formed, the excitons can almost completely bypass the chain of quantum emitters and "jump" directly from one end to the other, leading to large exciton conductance.', '1409.2514-2-22-2': 'This robust effect persists almost independently of the exact parameters of the system, and most notably occurs efficiently even when the underlying excitonic system is strongly disordered and its transport is completely suppressed due to localization.', '1409.2514-2-22-3': 'Through a simple model, we have furthermore shown that transport through direct hopping and through the polariton modes constitute two effectively independent channels, which helps to explain why the polariton conductance is almost independent of the disorder in the system.', '1409.2514-2-22-4': 'These results demonstrate a possible pathway for improving the efficiency of excitonic devices, where the EM mode could be provided by plasmonic structures to enable fully integrated nanometer-scale devices.', '1409.2514-2-22-5': 'We note that related results have simultaneously been obtained by Schachenmayer et al. [CITATION].', '1409.2514-2-23-0': 'We thank T. Ebbesen, G. Pupillo, C. Genes, and L. Martin Moreno for fruitful discussions, and E. Moreno and A. Delga for a careful reading of the manuscript.', '1409.2514-2-23-1': 'This work has been funded by the European Research Council (ERC-2011-AdG Proposal No. 290981).', '1409.2514-2-24-0': 'Supplemental Material', '1409.2514-2-25-0': '# Coherent driving', '1409.2514-2-26-0': 'In this section, we show some results for coherent driving of the system (as opposed to the incoherent driving used in the main text).', '1409.2514-2-26-1': 'As mentioned in the main text and shown below, the main conclusions are similar to those obtained under incoherent driving.', '1409.2514-2-26-2': 'The main difference is that under coherent driving at frequency [MATH], a resonant enhancement of the conductance is observed when [MATH] corresponds to one of the eigenmodes of the system.', '1409.2514-2-26-3': 'We use the same model as in the main text, but instead of an incoherent pumping term described by [MATH], we add a coherent driving term [MATH] (within the rotating wave approximation) to the Hamiltonian.', '1409.2514-2-26-4': 'Here, [MATH] and [MATH] are the driving amplitude and frequency, respectively.', '1409.2514-2-26-5': 'The steady state can then be calculated by going to the rotating frame, in which [MATH] is time-independent, but the system frequencies are shifted by [MATH].', '1409.2514-2-26-6': 'The exciton conductance is calculated analogously to the incoherent case, using [MATH], where the factor [MATH] is added to make [MATH] unitless.', '1409.2514-2-27-0': 'We focus again on the case without dipole-dipole interaction, which allows to calculate only the polariton contribution to exciton conductivity even when relatively short chains are used ([MATH] in the following).', '1409.2514-2-27-1': 'For simplicity, we also choose zero cavity-molecule detuning, [MATH].', '1409.2514-2-27-2': '[REF] demonstrates that the polariton-mediated exciton conductance under coherent driving displays a polynomial increase with [MATH], which saturates to a constant value once strong coupling is fully entered.', '1409.2514-2-27-3': 'This is identical to the behavior under incoherent pumping.', '1409.2514-2-27-4': 'In addition, the conductance is resonantly enhanced when the driving frequency coincides with an eigenfrequency of the system.', '1409.2514-2-27-5': 'The system eigenfrequencies correspond to the upper and lower polariton states ([MATH]), as well as the dark states at [MATH], which are not directly coupled to the cavity.', '1409.2514-2-27-6': 'Interestingly, and somewhat surprisingly, the largest conductance is found when driving the dark modes ([MATH]), not the polariton modes (cmp.', '1409.2514-2-27-7': 'the violet and dashed orange lines in [REF]b).', '1409.2514-2-27-8': 'This effect persists even when dephasing (which couples between dark and bright states) is turned off, and demonstrates that even the eigenstates that are not themselves strongly coupled to the cavity mode become delocalized when the system enters strong coupling.', '1409.2514-2-28-0': '# Simplified model', '1409.2514-2-29-0': 'In this section, we provide more details on the simplified model and its solution described in the text.', '1409.2514-2-29-1': 'The starting point is the Hamiltonian given in Eq. 1 in the main text: [EQUATION] from which we remove the direct dipole-dipole interaction responsible for hopping by setting [MATH].', '1409.2514-2-29-2': 'We next assume that all molecules are oriented identically, such that the [MATH] are constant and given by [MATH].', '1409.2514-2-29-3': 'As explained in the main text, all molecules apart from the first one, which is pumped, are now indistinguishable.', '1409.2514-2-29-4': 'The time evolution of the system density matrix is given by Eq. 2 in the main text: [EQUATION]', '1409.2514-2-29-5': 'Due to the structure and symmetries of the system, we can write the steady-state density matrix within the zero- and single-excitation subspace as [EQUATION] where [MATH] is the probability to be in the ground state (no photons or excitons), [MATH] is the cavity excitation probability, [MATH] is the excitation probability of molecule [MATH], and [MATH] is the coherence matrix element between constituent [MATH] and [MATH].', '1409.2514-2-29-6': 'Since all but the first molecule behave identically, the density matrix elements involving molecule [MATH] with [MATH] are all identical and given with [MATH].', '1409.2514-2-29-7': 'Additionally, [MATH] denotes the coherence element between any pair of molecules not involving the first.', '1409.2514-2-30-0': 'The steady state density matrix can then be obtained from [MATH], which for the specific form of the density matrix corresponds to a set of linear equations for the 4 real probabilities [MATH] and the 4 complex coherence elements [MATH], giving 12 independent real variables.', '1409.2514-2-30-1': 'Note that there are actually [MATH] linear equations for [MATH] molecules.', '1409.2514-2-30-2': 'However, due to the symmetry, only 11 of them are linearly independent.', '1409.2514-2-30-3': 'The steady-state density matrix is then obtained by solving for the null space of the coefficient matrix of the linear equations, and subsequently normalized to [MATH].', '1409.2514-2-31-0': 'Having obtained the steady state density matrix, the exciton conductivity is calculated using Eq. 7 in the main text, which explicitly gives [EQUATION]', '1409.2514-2-31-1': 'Inserting the solution [MATH] into this expression and taking the linear-response limit [MATH] gives the general solution for [MATH], which we give here for completeness: [EQUATION] where in addition to the definitions in the main text, we use [MATH].', '1409.2514-2-31-2': 'Finally, we note that the same approach works equally well under coherent driving (not shown here).'}

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'1409.2514-2-10-4'], ['1409.2514-1-10-5', '1409.2514-2-10-5'], ['1409.2514-1-10-6', '1409.2514-2-10-6'], ['1409.2514-1-17-0', '1409.2514-2-17-0'], ['1409.2514-1-17-1', '1409.2514-2-17-1'], ['1409.2514-1-17-2', '1409.2514-2-17-2'], ['1409.2514-1-17-3', '1409.2514-2-17-3'], ['1409.2514-1-17-4', '1409.2514-2-17-4'], ['1409.2514-1-17-5', '1409.2514-2-17-5'], ['1409.2514-1-0-0', '1409.2514-2-0-0'], ['1409.2514-1-0-1', '1409.2514-2-0-1'], ['1409.2514-1-0-2', '1409.2514-2-0-2'], ['1409.2514-1-6-0', '1409.2514-2-6-0'], ['1409.2514-1-6-1', '1409.2514-2-6-1'], ['1409.2514-1-6-2', '1409.2514-2-6-2'], ['1409.2514-1-6-3', '1409.2514-2-6-3'], ['1409.2514-1-6-4', '1409.2514-2-6-4'], ['1409.2514-1-11-0', '1409.2514-2-11-0'], ['1409.2514-1-11-1', '1409.2514-2-11-1'], ['1409.2514-1-11-2', '1409.2514-2-11-2'], ['1409.2514-1-13-0', '1409.2514-2-13-0'], ['1409.2514-1-13-1', '1409.2514-2-13-1'], ['1409.2514-1-13-2', '1409.2514-2-13-2'], ['1409.2514-1-13-3', '1409.2514-2-13-3'], ['1409.2514-1-13-4', '1409.2514-2-13-4'], ['1409.2514-2-30-0', '1409.2514-3-30-0'], ['1409.2514-2-30-1', '1409.2514-3-30-1'], ['1409.2514-2-30-2', '1409.2514-3-30-2'], ['1409.2514-2-30-3', '1409.2514-3-30-3'], ['1409.2514-2-4-3', '1409.2514-3-4-3'], ['1409.2514-2-5-0', '1409.2514-3-5-0'], ['1409.2514-2-5-1', '1409.2514-3-5-1'], ['1409.2514-2-5-2', '1409.2514-3-5-2'], ['1409.2514-2-22-0', '1409.2514-3-22-0'], ['1409.2514-2-22-1', '1409.2514-3-22-1'], ['1409.2514-2-22-2', '1409.2514-3-22-2'], ['1409.2514-2-22-3', '1409.2514-3-22-3'], ['1409.2514-2-22-4', '1409.2514-3-22-4'], ['1409.2514-2-22-5', '1409.2514-3-22-5'], ['1409.2514-2-6-0', '1409.2514-3-6-0'], ['1409.2514-2-6-1', '1409.2514-3-6-1'], ['1409.2514-2-6-2', '1409.2514-3-6-2'], ['1409.2514-2-6-3', '1409.2514-3-6-3'], ['1409.2514-2-6-4', '1409.2514-3-6-4'], ['1409.2514-2-14-0', '1409.2514-3-14-0'], ['1409.2514-2-14-1', '1409.2514-3-14-1'], ['1409.2514-2-14-2', '1409.2514-3-14-2'], ['1409.2514-2-14-3', '1409.2514-3-14-3'], ['1409.2514-2-14-4', '1409.2514-3-14-4'], ['1409.2514-2-14-5', '1409.2514-3-14-5'], ['1409.2514-2-14-6', '1409.2514-3-14-6'], ['1409.2514-2-14-7', '1409.2514-3-14-7'], ['1409.2514-2-14-8', '1409.2514-3-14-8'], ['1409.2514-2-8-0', '1409.2514-3-8-0'], ['1409.2514-2-8-1', '1409.2514-3-8-1'], ['1409.2514-2-10-0', '1409.2514-3-10-0'], ['1409.2514-2-10-1', '1409.2514-3-10-1'], ['1409.2514-2-10-2', '1409.2514-3-10-2'], ['1409.2514-2-10-3', '1409.2514-3-10-3'], ['1409.2514-2-10-4', '1409.2514-3-10-4'], ['1409.2514-2-10-5', '1409.2514-3-10-5'], ['1409.2514-2-10-6', '1409.2514-3-10-6'], ['1409.2514-2-23-0', '1409.2514-3-23-0'], ['1409.2514-2-23-1', '1409.2514-3-23-1'], ['1409.2514-2-0-0', '1409.2514-3-0-0'], ['1409.2514-2-0-1', '1409.2514-3-0-1'], ['1409.2514-2-0-2', '1409.2514-3-0-2'], ['1409.2514-2-9-0', '1409.2514-3-9-0'], ['1409.2514-2-9-1', '1409.2514-3-9-1'], ['1409.2514-2-9-2', '1409.2514-3-9-2'], ['1409.2514-2-4-0', '1409.2514-3-4-0'], ['1409.2514-2-4-2', '1409.2514-3-4-2'], ['1409.2514-2-13-0', '1409.2514-3-13-0'], ['1409.2514-2-13-1', '1409.2514-3-13-1'], ['1409.2514-2-13-2', '1409.2514-3-13-2'], ['1409.2514-2-13-3', '1409.2514-3-13-3'], ['1409.2514-2-13-4', '1409.2514-3-13-4'], ['1409.2514-2-15-0', '1409.2514-3-15-0'], ['1409.2514-2-20-0', '1409.2514-3-20-0'], ['1409.2514-2-20-1', '1409.2514-3-20-1'], ['1409.2514-2-20-2', '1409.2514-3-20-2'], ['1409.2514-2-20-3', '1409.2514-3-20-3'], ['1409.2514-2-2-0', '1409.2514-3-2-0'], ['1409.2514-2-2-1', '1409.2514-3-2-1'], ['1409.2514-2-2-2', '1409.2514-3-2-2'], ['1409.2514-2-2-4', '1409.2514-3-2-4'], ['1409.2514-2-7-0', '1409.2514-3-7-0'], ['1409.2514-2-7-1', '1409.2514-3-7-1'], ['1409.2514-2-7-2', '1409.2514-3-7-2'], ['1409.2514-2-7-3', '1409.2514-3-7-3'], ['1409.2514-2-7-4', '1409.2514-3-7-4'], ['1409.2514-2-19-0', '1409.2514-3-19-0'], ['1409.2514-2-19-1', '1409.2514-3-19-1'], ['1409.2514-2-19-2', '1409.2514-3-19-2'], ['1409.2514-2-19-3', '1409.2514-3-19-3'], ['1409.2514-2-3-0', '1409.2514-3-3-0'], ['1409.2514-2-3-1', 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['1409.2514-2-26-2', '1409.2514-3-26-2'], ['1409.2514-2-26-3', '1409.2514-3-26-3'], ['1409.2514-2-26-4', '1409.2514-3-26-4'], ['1409.2514-2-26-5', '1409.2514-3-26-5'], ['1409.2514-2-26-6', '1409.2514-3-26-6'], ['1409.2514-2-31-0', '1409.2514-3-31-0'], ['1409.2514-2-31-1', '1409.2514-3-31-1'], ['1409.2514-2-31-2', '1409.2514-3-31-2'], ['1409.2514-2-21-0', '1409.2514-3-21-0'], ['1409.2514-2-21-1', '1409.2514-3-21-1'], ['1409.2514-2-21-2', '1409.2514-3-21-2'], ['1409.2514-2-21-3', '1409.2514-3-21-3'], ['1409.2514-2-21-4', '1409.2514-3-21-4'], ['1409.2514-2-21-5', '1409.2514-3-21-5'], ['1409.2514-2-21-6', '1409.2514-3-21-6'], ['1409.2514-2-21-7', '1409.2514-3-21-7'], ['1409.2514-2-21-8', '1409.2514-3-21-8'], ['1409.2514-2-21-9', '1409.2514-3-21-9'], ['1409.2514-2-16-2', '1409.2514-3-16-2'], ['1409.2514-2-1-0', '1409.2514-3-1-0'], ['1409.2514-2-1-2', '1409.2514-3-1-2'], ['1409.2514-2-1-3', '1409.2514-3-1-3'], ['1409.2514-2-1-4', '1409.2514-3-1-4'], ['1409.2514-2-27-0', '1409.2514-3-27-0'], 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[]

[]

[]

['1409.2514-1-24-0', '1409.2514-2-24-0', '1409.2514-3-24-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1409.2514

{'1409.2514-3-0-0': 'We demonstrate that exciton conductance in organic materials can be enhanced by several orders of magnitude when the molecules are strongly coupled to an electromagnetic mode.', '1409.2514-3-0-1': 'Using a 1D model system, we show how the formation of a collective polaritonic mode allows excitons to bypass the disordered array of molecules and jump directly from one end of the structure to the other.', '1409.2514-3-0-2': 'This finding could have important implications in the fields of exciton transistors, heat transport, photosynthesis, and biological systems in which exciton transport plays a key role.', '1409.2514-3-1-0': 'The transport of excitons (bound electron-hole pairs) is a fundamental process that plays a crucial rule both in natural phenomena such as photosynthesis, where energy has to be transported to a reaction center [CITATION], and in artificial devices such as excitonic transistors [CITATION] or organic solar cells, whose power conversion efficiency can be improved significantly when the exciton diffusion length is increased [CITATION].', '1409.2514-3-1-1': 'Similarly, understanding and manipulating the role of excitons in heat transport has become an active field of research, with possible applications ranging from thermoelectric effects to heat-voltage converters, to nanoscale refrigerators, and even thermal logic gates (cf. [CITATION] and references therein).', '1409.2514-3-1-2': 'The exciton transport efficiency depends on a wide range of factors with such surprising features as the occurrence of noise-assisted transport [CITATION].', '1409.2514-3-1-3': 'Pioneering works have even suggested that coherent transport can play an important role in biological systems [CITATION].', '1409.2514-3-1-4': 'However, most systems composed of organic molecules are disordered and possess relatively large dissipation and dephasing rates, such that exciton transport typically becomes diffusive over long distances [CITATION].', '1409.2514-3-2-0': 'An intriguing possibility to modify exciton properties is by strong coupling to an electromagnetic (EM) mode, forming so-called polaritons (hybrid light-matter states).', '1409.2514-3-2-1': 'This is achieved when the Rabi frequency, i.e., the energy exchange rate between exciton and EM modes, becomes faster than the decay and/or decoherence rates of either constituent.', '1409.2514-3-2-2': 'Polaritons combine the properties of their constituents, in particular, mutual interactions and low effective masses, enabling new applications such as polariton condensation in semiconductors [CITATION] and organic materials [CITATION], the modification of molecular chemistry [CITATION] and work functions [CITATION], or the transfer of excitation between different molecular species [CITATION].', '1409.2514-3-2-3': 'Because of the large dipole moments and high densities, organic materials support large Rabi splittings [CITATION], and can also be strongly coupled to surface plasmon polaritons [CITATION].', '1409.2514-3-2-4': 'The dispersion relation can then be tuned to achieve a further reduction of the effective mass [CITATION].', '1409.2514-3-3-0': 'Very recently, an increase of the electrical conductance of an organic material was shown under strong coupling of the excitons to a cavity mode [CITATION].', '1409.2514-3-3-1': 'Inspired by this result, we demonstrate in this Letter that through strong coupling to an electromagnetic mode, i.e., the creation of polaritonic states, the exciton transport efficiency can be improved by many orders of magnitude.', '1409.2514-3-3-2': 'The strong coupling allows the excitons to bypass the disordered organic system, preventing localization and leading to dramatically improved energy transport properties.', '1409.2514-3-3-3': 'We note that while we focus on organic molecules in the following, the results can readily be generalized to other systems such as quantum dots and Rydberg atoms, or even chains of trapped ions, which offer a high degree of controllability [CITATION].', '1409.2514-3-4-0': 'We focus on a model system that captures the essential physics: A 1D chain of two-level emitters inside a cavity (see [REF]).', '1409.2514-3-4-1': 'The emitter dipole transition is coupled to the single cavity mode, and, additionally, induces Coulombic dipole-dipole interaction between the emitters.', '1409.2514-3-4-2': 'The effect of internal (e.g., rovibrational or phononic) and external environment modes is taken into account through effective dephasing and nonradiative decay rates modeled using a master equation of Lindblad form.', '1409.2514-3-4-3': 'The system Hamiltonian [MATH] in the rotating wave approximation (setting [MATH] here and in the following) is then [EQUATION] where [MATH] is the bosonic annihilation operator of the cavity mode with energy [MATH] and electric field [MATH].', '1409.2514-3-5-0': 'The molecular excitons of energy [MATH] are created and destroyed by the fermionic operators [MATH] and [MATH].', '1409.2514-3-5-1': 'Molecule [MATH] is characterized by its position [MATH] and dipole moment [MATH], giving the cavity-molecule interaction [MATH].', '1409.2514-3-5-2': 'The dipole-dipole interaction (in the quasistatic limit) is [EQUATION] with [MATH] and [MATH].', '1409.2514-3-6-0': 'The system dynamics is described by a Lindblad master equation for the density matrix [EQUATION] where [MATH] runs over all molecules as well as the cavity mode.', '1409.2514-3-6-1': 'The superoperators [MATH] describe decay and dephasing: [EQUATION] where [MATH] is the standard form for Lindblad superoperators.', '1409.2514-3-6-2': 'The total molecule decay rate [MATH] is given by [MATH], with [MATH] and [MATH] the radiative and nonradiative decay rates, while [MATH] is the dephasing rate.', '1409.2514-3-6-3': 'The decay rate [MATH] of the cavity photons is dominated by leakage through the mirrors.', '1409.2514-3-6-4': 'For later reference, we also define the total molecular decoherence rate [MATH].', '1409.2514-3-6-5': 'We note that while molecular decay and dephasing as included here models exciton-phonon interactions, it cannot represent exciton self-trapping, which can be approximated by a nonlinear term in the energy functional [CITATION].', '1409.2514-3-6-6': 'Self-trapping can lead to a further reduction of the exciton propagation length in the weakly coupled limit, but we have checked that it does not significantly affect the strongly coupled limit.', '1409.2514-3-6-7': 'We thus neglect it in the following.', '1409.2514-3-7-0': 'As mentioned above, we only include one cavity mode and describe the molecules by a linear 1D chain along the longitudinal cavity direction ([MATH] axis), with positions [MATH], such that the cavity electric field is identical for all molecules.', '1409.2514-3-7-1': 'It is polarized along the (out-of-plane) [MATH] axis, leading to [MATH].', '1409.2514-3-7-2': 'The total coupling between molecules and the cavity mode can be characterized by the collective Rabi frequency [MATH].', '1409.2514-3-7-3': 'For zero detuning [MATH], strong coupling is entered for [MATH] and leads to the formation of upper and lower polaritons at energies [MATH].', '1409.2514-3-7-4': 'The Rabi splitting (energy difference between upper and lower polariton) can approach [MATH]eV in experiments [CITATION], and can be tuned by changing either the molecule density or the mode electric field strength.', '1409.2514-3-8-0': 'In the following, we consider two types of molecular configurations: A perfectly ordered distribution, with molecule positions on a regular grid and dipole moments perfectly aligned to the electric field (i.e., along the [MATH] axis), and a random distribution, where Gaussian noise is added to the regular positions and the dipole moments are oriented randomly.', '1409.2514-3-8-1': 'Note that for the present case of a 1D linear chain, randomness is expected to suppress conductance much more efficiently than in higher dimensions.', '1409.2514-3-9-0': 'We next introduce a prescription for calculating an exciton conductance [MATH], a steady-state quantity to characterize the exciton transport efficiency similar to the electrical conductance for charge transport.', '1409.2514-3-9-1': 'We assume that excitation is continuously pumped into the system on the left side and measure the energy leaving the system through the right side (cf. [REF]).', '1409.2514-3-9-2': 'The pumping is represented by an additional incoherent driving term [MATH].', '1409.2514-3-10-0': 'The energy current is obtained from the rate of change of energy [CITATION] [EQUATION]', '1409.2514-3-10-1': 'This is evaluated in the steady state, [MATH], for which the total rate of change in energy is zero ([MATH]).', '1409.2514-3-10-2': 'However, each of the Lindblad superoperators [MATH] can be associated with a specific physical process.', '1409.2514-3-10-3': 'We thus identify the energy current between the two reservoirs with the loss of energy from the last molecule: [EQUATION]', '1409.2514-3-10-4': 'We then define the exciton conductance as the current per driving power, i.e., [MATH] (which has units of energy).', '1409.2514-3-10-5': 'Note that contrary to [CITATION], the energy entering the system through pumping does not necessarily leave through the sink at the end.', '1409.2514-3-10-6': 'It can also be lost through the radiative and nonradiative decay of the molecules, as well as decay of the cavity mode.', '1409.2514-3-11-0': 'We numerically obtain the steady state of the system using the open-source QuTiP package [CITATION].', '1409.2514-3-11-1': 'To do so, we restrict the total superoperator in [REF] to the zero- and single-excitation subspaces.', '1409.2514-3-11-2': 'This truncation of the Hilbert space is an excellent approximation in the presently relevant linear response regime of weak pumping, i.e., for [MATH] smaller than the system decay rates.', '1409.2514-3-12-0': 'We choose the quantum emitter parameters to approximately correspond to TDBC J aggregates at room temperature [CITATION]: [MATH]eV, [MATH]ps, [MATH]fs, and [MATH]fs.', '1409.2514-3-12-1': 'The cavity lifetime [MATH]fs is typical for experiments using cavities made of thin metal mirrors [CITATION].', '1409.2514-3-12-2': 'The molecule parameters also determine the dipole moment through [MATH], giving [MATH]D.', '1409.2514-3-12-3': 'The average intermolecular spacing is taken as [MATH]nm.', '1409.2514-3-12-4': 'Note that while the coupling to the cavity mode is taken into account explicitly, the radiative decay rate into all other electromagnetic modes can also be modified by the presence of a cavity.', '1409.2514-3-12-5': 'Since [MATH] is much smaller than the other rates, this modification can safely be neglected here.', '1409.2514-3-12-6': 'While we only show results for the parameters given above, we checked that the main conclusions drawn in the following apply for a wide range of parameters and do not depend on the specific values chosen here.', '1409.2514-3-13-0': '[REF] shows the exciton conductance [MATH] at zero cavity-molecule detuning [MATH], as a function of the collective Rabi frequency [MATH].', '1409.2514-3-13-1': 'Here, we keep the number of molecules fixed and change the electric field strength, going through the transition from weak to strong coupling.', '1409.2514-3-13-2': 'This can be achieved in an experiment by, e.g., putting the molecules at different positions inside the cavity [CITATION].', '1409.2514-3-13-3': 'We compare regular and random molecule arrangements for chains of 40 and 60 molecules.', '1409.2514-3-13-4': 'For the regular distribution, the strong dipole-dipole interaction leads to an additional small energy shift [MATH] of the molecular bright state coupling to the cavity; zero detuning thus corresponds to [MATH].', '1409.2514-3-14-0': 'For all of the cases shown in [REF], the conductance is approximately constant in the weak coupling limit [MATH], where the cavity mode does not play a role.', '1409.2514-3-14-1': 'Unsurprisingly, the conductance in this limit strongly depends on the molecular configuration-it is almost completely suppressed for the random case, for which 1D systems always show Anderson localization.', '1409.2514-3-14-2': 'The conductance in the random case is calculated as the logarithmic mean of 100 random configurations as appropriate for localized systems, i.e., [MATH] [CITATION].', '1409.2514-3-14-3': 'Note that even in the regular case, transport is quite inefficient due to the relatively large decay and dephasing rates of the molecular excitons, leading to diffusive transport [CITATION].', '1409.2514-3-14-4': 'Strikingly, when the coupling to the cavity mode is increased, an extraordinary increase of the conductance is observed in all cases.', '1409.2514-3-14-5': 'Once strong coupling is reached ([MATH]), the conductance again becomes almost independent of [MATH], indicating that the fully formed polariton channel dominates exciton conductance.', '1409.2514-3-14-6': 'In this limit, the conductance also becomes almost independent of the configuration and only depends on the number of molecules, i.e., length of the 1D chain.', '1409.2514-3-14-7': 'While randomness can suppress conduction almost completely in the weak-coupling limit, the polariton modes are barely affected by it.', '1409.2514-3-14-8': 'As a consequence of their delocalized nature induced by the collective exciton-cavity coupling, the excitation can efficiently bypass the disordered chain of emitters.', '1409.2514-3-15-0': 'This also provides a possible indication for the mechanism behind the enhanced electrical conduction observed under strong coupling in the experiments by Orgiu et al. [CITATION].', '1409.2514-3-15-1': 'However, the connection between exciton transport through polaritons and electrical conduction is currently unclear, as polaritons are, in principle, neutral quasiparticles.', '1409.2514-3-16-0': 'We also note that while we focus on incoherent driving for simplicity in this work, we have found that under coherent driving or incoupling at frequency [MATH], the same general behavior is observed.', '1409.2514-3-16-1': 'The main difference is an additional resonant enhancement when [MATH] coincides with the eigenfrequencies of the system (see details in the Supplemental Material [CITATION]).', '1409.2514-3-16-2': 'An interesting aspect is that even when hopping is completely suppressed, resonant transport occurs not only when driving at the polariton eigenfrequencies, but also at the unmodified molecule frequency-a clear signature that the dark states (which are not coupled to the cavity EM field) are still affected by the existence of strong coupling.', '1409.2514-3-17-0': 'We next focus on the case where the cavity mode is detuned from the molecular excitations by an energy [MATH].', '1409.2514-3-17-1': 'As shown in [REF], the onset of the extraordinary conductance is then shifted to larger coupling strengths for increasing detuning [MATH].', '1409.2514-3-17-2': 'However, the final conductance in the strong-coupling limit is independent of the detuning.', '1409.2514-3-17-3': 'This again indicates that the conduction proceeds through the polariton modes, which are only fully formed when the Rabi frequency [MATH] becomes large enough to not only overcome decoherence processes, but also the detuning.', '1409.2514-3-17-4': 'Once this is fulfilled, their character does not strongly depend on the detuning.', '1409.2514-3-17-5': 'However, even for relatively large and experimentally relevant Rabi frequencies, e.g., [MATH]meV, a small change of the detuning can strongly suppress or enhance conductance.', '1409.2514-3-17-6': 'If the detuning could be modified dynamically in an experiment (e.g., by displacing a cavity mirror with a piezo), this would enable novel applications based on switching of the exciton conductance.', '1409.2514-3-18-0': 'Next, we develop a simplified model to understand the extraordinary increase in exciton conductance under strong coupling.', '1409.2514-3-18-1': 'The main idea behind it is that there are two almost independent transport channels: (i) Direct excitonic transport through hopping between the molecules, which dominates in the weak-coupling limit, and (ii) polaritonic transport through the collective modes created by strong coupling to the cavity, which increases rapidly (polynomially) as the coupling is increased and saturates in the strong-coupling limit.', '1409.2514-3-18-2': 'To expose the contribution of the second channel unambiguously, we remove the dipole-dipole interaction responsible for hopping from the Hamiltonian [REF].', '1409.2514-3-18-3': 'We furthermore assume that the molecules are all aligned along the cavity field polarization axis [MATH].', '1409.2514-3-18-4': 'The interaction with the cavity mode is then identical for all molecules.', '1409.2514-3-19-0': '[REF] shows that the picture of independent channels is indeed valid: For the conductance [MATH] without hopping, the transmission plateau for weak coupling disappears, while the transmission in strong coupling is essentially unchanged.', '1409.2514-3-19-1': 'The polariton contribution without hopping decreases rapidly with decreasing Rabi frequency.', '1409.2514-3-19-2': 'The independence of the two transport channels is further verified by plotting the sum of the exciton conductance in the weak-coupling limit, [MATH], and the cavity-mediated contribution [MATH] without hopping.', '1409.2514-3-19-3': 'This sum, given by the dash-dotted black line in [REF], agrees excellently with the full result, indicating that the two transport channels are indeed effectively independent.', '1409.2514-3-20-0': 'Since in this model, all molecules but the first (due to the pumping) are indistinguishable, they behave identically.', '1409.2514-3-20-1': 'Independent of the number of molecules [MATH], there are thus just 12 independent components of the steady state density matrix within the zero- and single-excitation subspace.', '1409.2514-3-20-2': 'This density matrix can then be obtained analytically, and inserting the solution into [REF] gives an explicit formula for the conductance.', '1409.2514-3-20-3': 'In the linear response limit of weak driving and for zero detuning [MATH], it is given by [EQUATION] where we have defined the total decoherence rate [MATH].', '1409.2514-3-20-4': 'More details and the full expression for arbitrary detuning are given in the Supplemental Material [CITATION].', '1409.2514-3-21-0': 'As expected, the analytical solution ([REF]) perfectly matches the numerics (cf. [REF]).', '1409.2514-3-21-1': 'For small Rabi frequency, the conductance through the polariton modes grows with [MATH], while it saturates to a constant value for [MATH].', '1409.2514-3-21-2': 'For large [MATH], this constant value is given by [MATH] if [MATH] and by [MATH] if [MATH].', '1409.2514-3-21-3': 'Importantly, the decay with system size is algebraic [MATH], as opposed to the localized [MATH] behavior expected in the absence of strong coupling.', '1409.2514-3-21-4': 'Note that [MATH] occurs here because only a single molecule is connected to each of the baths; for a quasi-1D wire with a transverse extension, the length of the system would be the relevant variable.', '1409.2514-3-21-5': 'The dependence on the fourth power of [MATH] in the weak-coupling limit is explained by the rate obtained from two quantum jumps, to the cavity mode and back, with coupling [MATH].', '1409.2514-3-21-6': 'Interestingly, [REF] shows that the conductance, at least in the simplified model without disorder or direct hopping, is not directly related to the conventional criterion for the onset of strong coupling where the vacuum Rabi splitting becomes real ([MATH]).', '1409.2514-3-21-7': 'Indeed, the difference [MATH] does not occur in [REF].', '1409.2514-3-21-8': 'Instead, the exciton conductance becomes constant when [MATH] (for large [MATH]).', '1409.2514-3-21-9': 'This is related to large values of the cooperativity [MATH], which can occur even if strong coupling is not fully reached.', '1409.2514-3-22-0': 'To conclude, we have demonstrated that the formation of polariton modes, i.e., strong coupling, can dramatically enhance exciton transport.', '1409.2514-3-22-1': 'When the coupling is strong enough and the polaritons are fully formed, the excitons can almost completely bypass the chain of quantum emitters and "jump" directly from one end to the other, leading to large exciton conductance.', '1409.2514-3-22-2': 'This robust effect persists almost independently of the exact parameters of the system, and most notably occurs efficiently even when the underlying excitonic system is strongly disordered and its transport is completely suppressed due to localization.', '1409.2514-3-22-3': 'Through a simple model, we have furthermore shown that transport through direct hopping and through the polariton modes constitute two effectively independent channels, which helps to explain why the polariton conductance is almost independent of the disorder in the system.', '1409.2514-3-22-4': 'These results demonstrate a possible pathway for improving the efficiency of excitonic devices, where the EM mode could be provided by plasmonic structures to enable fully integrated nanometer-scale devices.', '1409.2514-3-22-5': 'We note that related results have simultaneously been obtained by Schachenmayer et al. [CITATION].', '1409.2514-3-23-0': 'We thank T. Ebbesen, G. Pupillo, C. Genes, and L. Martin Moreno for fruitful discussions, and E. Moreno and A. Delga for a careful reading of the manuscript.', '1409.2514-3-23-1': 'This work has been funded by the European Research Council (ERC-2011-AdG Proposal No. 290981).', '1409.2514-3-24-0': 'Supplemental Material', '1409.2514-3-25-0': '# Coherent driving', '1409.2514-3-26-0': 'In this section, we show some results for coherent driving of the system (as opposed to the incoherent driving used in the main text).', '1409.2514-3-26-1': 'As mentioned in the main text and shown below, the main conclusions are similar to those obtained under incoherent driving.', '1409.2514-3-26-2': 'The main difference is that under coherent driving at frequency [MATH], a resonant enhancement of the conductance is observed when [MATH] corresponds to one of the eigenmodes of the system.', '1409.2514-3-26-3': 'We use the same model as in the main text, but instead of an incoherent pumping term described by [MATH], we add a coherent driving term [MATH] (within the rotating wave approximation) to the Hamiltonian.', '1409.2514-3-26-4': 'Here, [MATH] and [MATH] are the driving amplitude and frequency, respectively.', '1409.2514-3-26-5': 'The steady state can then be calculated by going to the rotating frame, in which [MATH] is time-independent, but the system frequencies are shifted by [MATH].', '1409.2514-3-26-6': 'The exciton conductance is calculated analogously to the incoherent case, using [MATH], where the factor [MATH] is added to make [MATH] unitless.', '1409.2514-3-27-0': 'We focus again on the case without dipole-dipole interaction, which allows to calculate only the polariton contribution to exciton conductivity even when relatively short chains are used ([MATH] in the following).', '1409.2514-3-27-1': 'For simplicity, we also choose zero cavity-molecule detuning, [MATH].', '1409.2514-3-27-2': '[REF] demonstrates that the polariton-mediated exciton conductance under coherent driving displays a polynomial increase with [MATH], which saturates to a constant value once strong coupling is fully entered.', '1409.2514-3-27-3': 'This is identical to the behavior under incoherent pumping.', '1409.2514-3-27-4': 'In addition, the conductance is resonantly enhanced when the driving frequency coincides with an eigenfrequency of the system.', '1409.2514-3-27-5': 'The system eigenfrequencies correspond to the upper and lower polariton states ([MATH]), as well as the dark states at [MATH], which are not directly coupled to the cavity.', '1409.2514-3-27-6': 'Interestingly, and somewhat surprisingly, the largest conductance is found when driving the dark modes ([MATH]), not the polariton modes (cmp.', '1409.2514-3-27-7': 'the violet and dashed orange lines in [REF]b).', '1409.2514-3-27-8': 'This effect persists even when dephasing (which couples between dark and bright states) is turned off, and demonstrates that even the eigenstates that are not themselves strongly coupled to the cavity mode become delocalized when the system enters strong coupling.', '1409.2514-3-28-0': '# Simplified model', '1409.2514-3-29-0': 'In this section, we provide more details on the simplified model and its solution described in the text.', '1409.2514-3-29-1': 'The starting point is the Hamiltonian given in Eq. 1 in the main text: [EQUATION] from which we remove the direct dipole-dipole interaction responsible for hopping by setting [MATH].', '1409.2514-3-29-2': 'We next assume that all molecules are oriented identically, such that the [MATH] are constant and given by [MATH].', '1409.2514-3-29-3': 'As explained in the main text, all molecules apart from the first one, which is pumped, are now indistinguishable.', '1409.2514-3-29-4': 'The time evolution of the system density matrix is given by Eq. 2 in the main text: [EQUATION]', '1409.2514-3-29-5': 'Due to the structure and symmetries of the system, we can write the steady-state density matrix within the zero- and single-excitation subspace as [EQUATION] where [MATH] is the probability to be in the ground state (no photons or excitons), [MATH] is the cavity excitation probability, [MATH] is the excitation probability of molecule [MATH], and [MATH] is the coherence matrix element between constituent [MATH] and [MATH].', '1409.2514-3-29-6': 'Since all but the first molecule behave identically, the density matrix elements involving molecule [MATH] with [MATH] are all identical and given with [MATH].', '1409.2514-3-29-7': 'Additionally, [MATH] denotes the coherence element between any pair of molecules not involving the first.', '1409.2514-3-30-0': 'The steady state density matrix can then be obtained from [MATH], which for the specific form of the density matrix corresponds to a set of linear equations for the 4 real probabilities [MATH] and the 4 complex coherence elements [MATH], giving 12 independent real variables.', '1409.2514-3-30-1': 'Note that there are actually [MATH] linear equations for [MATH] molecules.', '1409.2514-3-30-2': 'However, due to the symmetry, only 11 of them are linearly independent.', '1409.2514-3-30-3': 'The steady-state density matrix is then obtained by solving for the null space of the coefficient matrix of the linear equations, and subsequently normalized to [MATH].', '1409.2514-3-31-0': 'Having obtained the steady state density matrix, the exciton conductivity is calculated using Eq. 7 in the main text, which explicitly gives [EQUATION]', '1409.2514-3-31-1': 'Inserting the solution [MATH] into this expression and taking the linear-response limit [MATH] gives the general solution for [MATH], which we give here for completeness: [EQUATION] where in addition to the definitions in the main text, we use [MATH].', '1409.2514-3-31-2': 'Finally, we note that the same approach works equally well under coherent driving (not shown here).'}

1508.04284

{'1508.04284-1-0-0': 'We consider the transport of non-interacting electrons on two- and three-dimensional random Voronoi-Delaunay lattices.', '1508.04284-1-0-1': 'It was recently shown that these topologically disordered lattices feature strong disorder anticorrelations between the coordination numbers that qualitatively change the properties of continuous and first-order phase transitions.', '1508.04284-1-0-2': 'To determine whether or not these unusual features also influence Anderson localization, we study the electronic wave functions by multifractal analysis and finite-size scaling.', '1508.04284-1-0-3': 'We observe only localized states for all energies in the two-dimensional system.', '1508.04284-1-0-4': 'In three dimensions, we find two Anderson transitions between localized and extended states very close to the band edges.', '1508.04284-1-0-5': 'The critical exponent of the localization length is about 1.6.', '1508.04284-1-0-6': 'All these results agree with the usual orthogonal universality class.', '1508.04284-1-0-7': 'Additional generic energetic randomness introduced via random potentials does not lead to qualitative changes but allows us to obtain a phase diagram by varying the strength of these potentials.', '1508.04284-1-1-0': '# Introduction', '1508.04284-1-2-0': 'More than 5 decades ago, Anderson [CITATION] showed that random disorder can localize a quantum particle in space.', '1508.04284-1-2-1': 'This phenomenon, now called Anderson localization, and the corresponding Anderson transitions between localized and metallic phases have since attracted lots of experimental and theoretical attention (see, e.g., Refs. [CITATION] for reviews).', '1508.04284-1-2-2': 'Remarkably, important features of Anderson localization, such as the existence of a metallic phase and the qualitative characteristics of the Anderson transition, depend only on the dimensionality and the symmetries of the system, in close analogy to conventional continuous phase transitions.', '1508.04284-1-3-0': 'This leads to a symmetry classification of Anderson transitions.', '1508.04284-1-3-1': 'Originally, three universality classes where identified, based on the invariance of the Hamiltonian under time reversal and spin rotations.', '1508.04284-1-3-2': 'These classes (orthogonal, unitary, and symplectic) correspond to the Wigner-Dyson classification of random matrices [CITATION].', '1508.04284-1-3-3': 'For example, the orthogonal universality class contains systems that are invariant under both time reversal and spin rotation.', '1508.04284-1-3-4': 'Later, this classification scheme was extended by including additional symmetries (for a review, see, e.g., Evers and Mirlin [CITATION]).', '1508.04284-1-3-5': 'Within each class, the properties of the Anderson transition are expected to be universal.', '1508.04284-1-3-6': 'Specifically, all eigenstates in one-dimensional (1D) and two-dimensional (2D) systems in the orthogonal class are expected to be localized, implying the absence of an Anderson transition.', '1508.04284-1-3-7': 'In contrast, the states in three-dimensional (3D) systems in the orthogonal class can undergo a transition from localized to delocalized as energy or disorder strength are varied; and this transition features universal critical exponents.', '1508.04284-1-4-0': 'These universal properties hold for uncorrelated randomness; spatial disorder correlations can lead to different behavior.', '1508.04284-1-4-1': 'Some short-range correlations have been shown to produce extended states at specific, isolated energies in an otherwise localized system, for instance in the so-called dimer model [CITATION].', '1508.04284-1-4-2': 'In contrast, long-range correlations can lead to the appearance of a true metallic phase (and thus an Anderson transition), even in 1D.', '1508.04284-1-4-3': 'Some of these developments are reviewed by Izrailev et al. [CITATION].', '1508.04284-1-5-0': 'In recent years, topologically disordered systems (lattices with random connectivity) have attracted particular attention because phase transitions in such systems feature surprising violations of the expected universal behavior.', '1508.04284-1-5-1': 'This includes transitions in Ising and Potts magnets as well as the contact process, all defined on random Voronoi-Delaunay (VD) lattices [CITATION].', '1508.04284-1-5-2': 'Barghathi and Vojta [CITATION] solved this puzzle by showing that the 2D random VD lattice belongs to a broad class of random lattices whose disorder fluctuations feature strong anticorrelations and therefore decay faster with increasing length scale than those of generic random systems.', '1508.04284-1-5-3': 'Such lattices are ubiquitous in 2D because the Euler equation for a 2D graph imposes a topological constraint on the coordination numbers; however, examples in higher dimensions exist as well.', '1508.04284-1-5-4': 'The suppressed disorder fluctuations lead to important modifications of the Harris [CITATION] and Imry-Ma [CITATION] criteria that govern the effects of disorder on continuous and first-order transitions, respectively.', '1508.04284-1-6-0': 'These results immediately pose the question of whether or not the unusual features of random VD lattices also modify universal properties of Anderson localization.', '1508.04284-1-6-1': 'Grimm et al. [CITATION] studied the energy level distribution of a tight-binding model defined on a random tessellation [CITATION] similar to a VD lattice.', '1508.04284-1-6-2': 'They found level repulsion indicative of extended states in the metallic regime.', '1508.04284-1-6-3': 'However, to the best of our knowledge, a systematic finite-size scaling (FSS) study that would permit the unambiguous identification of the metallic and localized phases (and the Anderson transition between them) has not yet been performed.', '1508.04284-1-7-0': 'In this paper, we therefore investigate noninteracting electrons on 2D and 3D random VD lattices.', '1508.04284-1-7-1': 'We perform FSS based on a multifractal analysis (MFA) of the electronic wave functions.', '1508.04284-1-7-2': 'Our results can be summarized as follows.', '1508.04284-1-7-3': 'In 2D, we observe localized states for all energies.', '1508.04284-1-7-4': 'In contrast, the 3D system features two Anderson transitions between localized and extended states close to the band edges.', '1508.04284-1-7-5': 'The critical exponent of the correlation length takes the value [MATH], in agreement with the standard orthogonal universality class.', '1508.04284-1-7-6': 'This implies that the unusual coordination number anticorrelations of random VD lattices do not lead to qualitatively different behavior than the well-known Anderson model of localization on regular lattices.', '1508.04284-1-8-0': 'The rest of this paper is organized as follows.', '1508.04284-1-8-1': 'In Sec. [REF], we introduce the random VD lattices and define the Hamiltonian.', '1508.04284-1-8-2': 'We also discuss FSS as well as the MFA of the electronic states.', '1508.04284-1-8-3': 'Section [REF] is devoted to the results of our simulations for both 2D and 3D systems.', '1508.04284-1-8-4': 'We conclude in Sec. [REF].', '1508.04284-1-9-0': '# Model and methods', '1508.04284-1-10-0': '## Random Voronoi-Delaunay lattices', '1508.04284-1-11-0': 'The random VD lattice is a prototypical system with topological (connectivity) disorder.', '1508.04284-1-11-1': 'It can be viewed as a simple model for amorphous solids, foams or biological cell structures.', '1508.04284-1-11-2': 'The random VD lattice is defined as a set of lattice sites at random positions together with the bonds that connect nearest neighbor sites.', '1508.04284-1-11-3': 'These neighbors are determined by the VD construction [CITATION] as follows.', '1508.04284-1-11-4': 'The entire system area (or volume, in the 3D case) is subdivided into disjoint polygons (Voronoi cells) containing exactly one lattice site such that each cell contains all points in space that are closer to that lattice site than to any other site.', '1508.04284-1-11-5': 'The Voronoi diagram is the complete set of these Voronoi cells.', '1508.04284-1-11-6': 'Lattice sites whose Voronoi cells share an edge (or a face, in the 3D case) are considered neighbors, irrespective of their real-space distance.', '1508.04284-1-11-7': 'In 2D, the graph of all bonds connecting pairs of neighbors consists of triangles only.', '1508.04284-1-11-8': 'It is called the Delaunay triangulation.', '1508.04284-1-11-9': 'The corresponding construction in 3D leads to the Delaunay tetrahedralization, a lattice consisting of tetrahedra only.', '1508.04284-1-11-10': 'We emphasize that random VD lattices are not bipartite, i.e., they cannot be divided into disjoint sublattices A and B such that each bond connects an A site to a B site.', '1508.04284-1-11-11': 'A 2D example of a VD construction can be seen in Fig. [REF].', '1508.04284-1-12-0': 'In the following, we consider independent, uniformly distributed random lattice sites of density unity contained in a square or cubic box of linear size [MATH].', '1508.04284-1-12-1': 'We employ periodic boundary conditions.', '1508.04284-1-12-2': 'More details of the algorithm that we use to perform the VD construction are given in the Appendix.', '1508.04284-1-13-0': 'In a random VD lattice, the coordination number (number of neighbors) [MATH] fluctuates from site to site.', '1508.04284-1-13-1': 'This topological disorder is illustrated in Fig. [REF].', '1508.04284-1-13-2': 'The corresponding coordination number distribution can be seen in Fig. [REF].', '1508.04284-1-13-3': 'The average coordination number in 2D is exactly [MATH].', '1508.04284-1-13-4': 'This is a consequence of the Euler equation for a 2D graph consisting of triangles only.', '1508.04284-1-13-5': 'In 3D, the average coordination number is given by [MATH] [CITATION].', '1508.04284-1-13-6': 'In both dimensions, the mean coordination numbers are higher than those of the regular square and cubic lattices often used in numerical localization studies.', '1508.04284-1-13-7': 'The standard deviations of the coordinations numbers are [MATH] in 2D and [MATH] in 3D.', '1508.04284-1-13-8': 'This means that the disorder is moderately strong.', '1508.04284-1-14-0': '## Anderson model', '1508.04284-1-15-0': 'We now consider the motion of noninteracting electrons on a random VD lattice.', '1508.04284-1-15-1': 'We describe it by means of a tight-binding model having one (Wannier) orbital per lattice site.', '1508.04284-1-15-2': 'This corresponding Hamiltonian [EQUATION] is analogous to the Anderson model of localization.', '1508.04284-1-15-3': 'The first term describes the hopping of electrons between nearest (Voronoi) neighbors.', '1508.04284-1-15-4': 'The hopping matrix element is constant and fixed at unity.', '1508.04284-1-15-5': 'This term contains the topological disorder.', '1508.04284-1-15-6': 'The second term represents additional energetic randomness.', '1508.04284-1-15-7': 'The [MATH] are random on-site potentials uniformly distributed in the interval [MATH].', '1508.04284-1-15-8': 'We are mostly interested in the case [MATH] for which the disorder in the system is purely topological.', '1508.04284-1-15-9': 'However, for comparison with the usual Anderson model of localization, we also consider nonzero random potential strength [MATH].', '1508.04284-1-16-0': 'To compute the densities of states (DOS), we directly diagonalize the secular matrices.', '1508.04284-1-16-1': 'The system sizes are [MATH] for 2D and [MATH] for 3D, limited by computer memory.', '1508.04284-1-16-2': 'To find the eigenstates of the Hamiltonian ([REF]) close to a particular energy value for larger system sizes, we use a sparse matrix algorithm based on the Jacobi-Davidson method [CITATION].', '1508.04284-1-16-3': 'Here, we treat system sizes up to [MATH] for 2D and [MATH] for 3D.', '1508.04284-1-17-0': 'Figure [REF] shows an overview over the DOS resulting from these calculations for purely topological disorder ([MATH]).', '1508.04284-1-18-0': 'As the VD lattices are not bipartite, the DOS is not symmetric with respect to the energy [MATH].', '1508.04284-1-18-1': 'While the DOS at the low-energy band edge increases rapidly, there is a pronounced tail on the high-energy side, in particular in 3D.', '1508.04284-1-18-2': 'Interestingly, this tail does not stretch much beyond [MATH].', '1508.04284-1-18-3': 'States in the far tail live on rare large clusters of sites with above average coordination numbers.', '1508.04284-1-18-4': 'The fact that there are almost no states with [MATH] reflects the strong disorder anticorrelation of the random VD lattices that prevent the formation of large rare regions [CITATION].', '1508.04284-1-18-5': 'We also note that the DOS in the 3D band tail fluctuates significantly.', '1508.04284-1-18-6': 'We believe this is a finite-size effect.', '1508.04284-1-19-0': '## Finite-size scaling', '1508.04284-1-20-0': 'The scaling approach is a phenomenological description of the behavior close to the critical point of a continuous transition.', '1508.04284-1-20-1': 'We consider a general dimensionless measure [MATH] characterizing the electronic states as function of energy [MATH] and a characteristic length, e.g., the system size [MATH].', '1508.04284-1-20-2': 'Close to the critical energy [MATH], it fulfills the scaling form [CITATION] [EQUATION]', '1508.04284-1-20-3': 'The description contains the relevant scaling variable [MATH] associated with the relevant exponent [MATH] as well as the leading irrelevant scaling variable [MATH] associated with the irrelevant exponent [MATH].', '1508.04284-1-20-4': 'The dependence of the scaling variables on [MATH] can be described in terms of the expansions [EQUATION] where [MATH] is a dimensionless measure of the distance from the critical energy.', '1508.04284-1-20-5': 'The scaling function [EQUATION] is expanded into a Taylor polynomial of the irrelevant variable.', '1508.04284-1-20-6': 'The coefficients are the regular single-variable scaling functions [EQUATION]', '1508.04284-1-20-7': 'They are expanded into Chebyshev polynomials [MATH] of the first kind and the [MATH]th order.', '1508.04284-1-20-8': 'This general expansion depends on the four expansion orders [MATH], [MATH], [MATH], and [MATH] yielding [MATH] free parameters.', '1508.04284-1-20-9': 'Their values are found by weighted nonlinear fits of the numerical data.', '1508.04284-1-20-10': 'The weights are formed by the reciprocal variances of the data points.', '1508.04284-1-20-11': 'This formalism allows us to simultaneously determine [MATH], [MATH], and [MATH].', '1508.04284-1-21-0': 'In order to get error estimates of these values we use a Monte-Carlo method [CITATION].', '1508.04284-1-21-1': 'It consists in building at least [MATH] synthetic data sets by adding noise to the original values.', '1508.04284-1-21-2': 'This noise is created by Gaussian random numbers with a standard deviation [MATH] equal to the individual error of each data point.', '1508.04284-1-21-3': 'By fitting [MATH] to these synthetic data sets we obtain distributions for the parameter values.', '1508.04284-1-21-4': 'These distributions are usually Gaussian.', '1508.04284-1-21-5': 'Large deviations, e.g., superpositions of multiple Gaussians, are interpreted as instabilities.', '1508.04284-1-21-6': 'The corresponding regressions are not well defined and will be neglected.', '1508.04284-1-21-7': '(This could be avoided by increasing the accuracy of the data, changing the number of parameters, or varying the initial conditions.)', '1508.04284-1-21-8': 'We note that this method constitutes an error-propagation calculation for the random errors only.', '1508.04284-1-21-9': 'Systematic errors are not detected completely, but the influence of different expansion orders can be identified.', '1508.04284-1-22-0': '## Multifractal analysis', '1508.04284-1-23-0': 'Multifractal behavior is a feature of eigenstates at critical points [CITATION].', '1508.04284-1-23-1': 'The MFA is based on a standard box-counting algorithm.', '1508.04284-1-23-2': 'The boxes are squares or cubes for 2D and 3D systems, respectively.', '1508.04284-1-23-3': 'The system of size [MATH] is partitioned into boxes of size [MATH] (see Fig. [REF]).', '1508.04284-1-23-4': 'Using the probability [EQUATION] to find the electron in the [MATH]th box, the measure [EQUATION] is constructed from its [MATH]th moment.', '1508.04284-1-23-5': 'Depending on [MATH], this measure is dominated by boxes with either large or small [MATH].', '1508.04284-1-23-6': 'For multifractal wave functions, [MATH] is expected to behave as a power of the normalized box size [MATH], with the scaling exponent (mass exponent) [EQUATION]', '1508.04284-1-23-7': 'The singularity spectrum [MATH] is the Legendre transform of [MATH], and it comprises the scaling exponents of the fractal dimensions of all moments.', '1508.04284-1-23-8': 'A parametric representation of the singularity spectrum can be obtained by calculating the singularity strength [EQUATION] and the fractal dimension [EQUATION] with [EQUATION]', '1508.04284-1-23-9': 'For statistical purposes, we utilize the ensemble average of these exponents [CITATION].', '1508.04284-1-23-10': 'In particular, we use the ensemble averaged singularity strength [EQUATION] with [EQUATION] to perform the finite-size scaling analysis outlined in the last section.', '1508.04284-1-23-11': '[MATH] denotes the average over different eigenstates.', '1508.04284-1-23-12': 'When determining the error of [MATH], one has to take into account that the numerator and denominator are correlated with each other [CITATION].', '1508.04284-1-24-0': 'Only integer box ratios [MATH] are possible when partitioning the original system without overlap.', '1508.04284-1-24-1': 'A general ratio can be used by employing the periodic boundary conditions to fold back into the real system any protruding box parts that arise for noninteger [MATH] [CITATION].', '1508.04284-1-24-2': 'To obtain a uniform sampling, we are then required to average over all possible box origins.', '1508.04284-1-25-0': 'To extrapolate to the thermodynamic limit [MATH] in Eq. ([REF]), we perform a linear least-squares fit of the numerator versus the denominator at fixed [MATH] (box-size scaling).', '1508.04284-1-25-1': 'The standard deviations [MATH] of [MATH] are utilized to determine the inaccuracy of the regression values.', '1508.04284-1-25-2': 'To approach the limit [MATH], sufficiently small values of [MATH] have to be taken into account.', '1508.04284-1-25-3': 'However, for the smallest [MATH], the data deviate from the power-law behavior because corrections to scaling that stem from the irrelevant scaling variables become important.', '1508.04284-1-25-4': 'Their influence on the estimated exponent values can be reduced by optimizing the fit boundaries.', '1508.04284-1-25-5': 'We use [MATH] for 2D and [MATH] for 3D.', '1508.04284-1-26-0': 'In random VD lattices, the cells have varying sizes; the resolution of the MFA given by the smallest box size [MATH] is therefore arbitrary.', '1508.04284-1-26-1': 'We use [MATH], such that the average number of sites per box is [MATH] (see Fig. [REF]).', '1508.04284-1-26-2': 'Since empty boxes affect the MFA results, only box sizes larger than the maximum distance between neighboring sites are included in the MFA analysis.', '1508.04284-1-26-3': 'This guarantees that there are no empty boxes.', '1508.04284-1-26-4': 'In both dimensions, we use [MATH] in accordance with the distributions shown in the right panel of Fig. [REF].', '1508.04284-1-27-0': '# Results', '1508.04284-1-28-0': '## 2D system', '1508.04284-1-29-0': 'Figure [REF] shows five representative eigenstates of the energy spectrum of a 2D system with [MATH] and [MATH].', '1508.04284-1-29-1': 'The wave function of the lowest eigenvalue (Fig. [REF]a) is exponentially localized.', '1508.04284-1-29-2': 'The maximum amplitude is concentrated at the single Voronoi cell with highest coordination number (here [MATH]).', '1508.04284-1-29-3': 'States on the upper band edge also show localized behavior very clearly (Fig. [REF]e).', '1508.04284-1-29-4': 'However, the highest amplitude is orders of magnitude smaller and, correspondingly, the localization length is larger.', '1508.04284-1-29-5': 'More generally, states of smaller energy are influenced by local fluctuations of the coordination number.', '1508.04284-1-29-6': 'States of higher energy are driven by interferences and the probability amplitudes behave as non-integrable Chladni figures [CITATION].', '1508.04284-1-29-7': 'The number of antinodes increases with decreasing eigenenergy.', '1508.04284-1-29-8': 'In summary, this is a visualization of localization in a classical and quantum mechanical manner for the states close to the left and the right band edge, respectively.', '1508.04284-1-29-9': 'Towards the band center the amplitude fluctuations increase and a classification by visual inspection is not easily possible anymore.', '1508.04284-1-30-0': 'We therefore turn to the MFA to quantitatively characterize the eigenstates as function of energy.', '1508.04284-1-30-1': 'Specifically, we consider the FSS behavior of [MATH] for systems from [MATH] up to [MATH] with [MATH] states per data point.', '1508.04284-1-30-2': 'Figure [REF]', '1508.04284-1-31-0': 'shows the box-size scaling of [MATH] used to estimate [MATH] for several [MATH].', '1508.04284-1-31-1': 'Whereas values at higher [MATH] show the expected power-law dependence on [MATH], results for small [MATH] deviate systematically due to finite-size effects.', '1508.04284-1-31-2': 'Therefore, they are neglected as discussed before.', '1508.04284-1-31-3': 'The multifractal spectra [MATH] resulting from the analysis are also shown in Fig. [REF] for representative energies close to the upper band edge.', '1508.04284-1-31-4': 'For the energy [MATH], we obtain a parabola-like shape as expected.', '1508.04284-1-31-5': 'Towards the band center the parabola shrinks (see [MATH]) and approaches the limiting case for completely extended wave functions, namely the single point [MATH].', '1508.04284-1-31-6': 'In the opposite direction, close to the band edge, the spectrum transforms towards the extremely localized limit, the points [MATH] for negative [MATH] and [MATH] for positive [MATH].', '1508.04284-1-32-0': 'The behavior of [MATH] (the position of the maximum of the parabola) is shown in more detail in Fig. [REF].', '1508.04284-1-33-0': 'For energy values close to the band edge, [MATH] generally increases with [MATH].', '1508.04284-1-33-1': 'This means that these states are localized.', '1508.04284-1-33-2': 'The strength of the localization depends on [MATH].', '1508.04284-1-33-3': 'The pronounced localization at the lower band edge seen in Fig. [REF] is reproduced here.', '1508.04284-1-33-4': 'Strongly fluctuating wave functions near the band center have a value close to [MATH] for all system sizes within their accuracy.', '1508.04284-1-33-5': 'The envelopes of corresponding wave functions have a very large localization length (larger than the system size).', '1508.04284-1-33-6': 'Importantly, the curves for different [MATH] do not show a common crossing point, implying that there is no Anderson transition.', '1508.04284-1-33-7': 'All states are localized.', '1508.04284-1-33-8': '(The seeming crossings at very large [MATH] in Fig. [REF] can be attributed to numerical artifacts.)', '1508.04284-1-34-0': 'We have also studied the effects of additional potential disorder [MATH] and [MATH].', '1508.04284-1-34-1': 'It results in a broadening of the DOS and an enhancement of the localization.', '1508.04284-1-34-2': 'However, qualitative changes compared to [MATH] were not found.', '1508.04284-1-34-3': 'All states are localized.', '1508.04284-1-35-0': '## 3D system', '1508.04284-1-36-0': 'The data analysis for the 3D VD lattice proceeds analogously.', '1508.04284-1-36-1': 'We use system sizes between [MATH] and [MATH].', '1508.04284-1-36-2': 'In contrast to the 2D system, we observe two Anderson transitions induced by purely topological disorder.', '1508.04284-1-36-3': 'These transitions are located close to the two energy-band edges.', '1508.04284-1-36-4': 'The corresponding FSS behavior of [MATH] is visualized in Fig. [REF] for the transition near the lower band edge and in Fig. [REF] for the upper edge.', '1508.04284-1-36-5': 'Thus, localized states exist only near the band edges.', '1508.04284-1-36-6': 'The broad central area of the energy band encompasses extended states.', '1508.04284-1-36-7': 'This corresponds to the fact that the topological disorder is only moderately strong and results in localization behavior similar to that of a weakly disordered regular Anderson model.', '1508.04284-1-37-0': 'As shown in Fig. [REF], the singularity strength [MATH] features a smooth energy dependence for the transition near the lower band edge.', '1508.04284-1-37-1': 'Thus, the FSS approach is applicable.', '1508.04284-1-37-2': 'The results are presented in Tab. 1.', '1508.04284-1-37-3': 'It can be seen that the critical parameters [MATH] and [MATH] are influenced by the irrelevant scaling variable.', '1508.04284-1-37-4': 'We compare different expansion orders to demonstrate the stability of the regression results.', '1508.04284-1-37-5': 'In particular, we neglect the data of small systems [MATH] and use a FSS approach without irregular expansion ([MATH]).', '1508.04284-1-37-6': 'Such regressions show a higher robustness when changing initial conditions.', '1508.04284-1-37-7': 'Taking into account all data, we estimate the critical energy [MATH] and the critical exponent [MATH].', '1508.04284-1-37-8': 'The results of regressions without the irrelevant scaling variable deviate slightly ([MATH] and [MATH]) because of the neglected systematic shift of the intersections described by the irrelevant term.', '1508.04284-1-37-9': 'The quality of all fits is very close to unity.', '1508.04284-1-37-10': 'This indicates that the errors of the original data points were overestimated.', '1508.04284-1-38-0': 'For the transition near the upper band edge, the [MATH] curves shown in Fig. [REF] are very noisy.', '1508.04284-1-38-1': 'This is caused by the low DOS (see right panel of Fig. [REF]) and finite size effects.', '1508.04284-1-38-2': 'The existence of a transition can still be inferred because [MATH] decreases with [MATH] for the smaller energies, while it increases with [MATH] for the largest energies.', '1508.04284-1-38-3': 'This indicates a crossing and thus an Anderson transition.', '1508.04284-1-38-4': 'However, a clear transition point cannot be determined from the available data.', '1508.04284-1-38-5': 'In particular, the values are insufficient to perform a scaling analysis.', '1508.04284-1-38-6': '(Also note that for small systems, gaps in the DOS appear.', '1508.04284-1-38-7': 'Therefore the number of considered eigenstates varies from [MATH] to [MATH] between data points, leading to strong variations in the error bars.)', '1508.04284-1-38-8': 'A rough estimate of the critical energy is [MATH].', '1508.04284-1-39-0': 'We now turn to the effects of additional on-site disorder, i.e., [MATH].', '1508.04284-1-39-1': 'As in 2D, on-site disorder leads to a broadening of the DOS with increasing disorder strength [MATH], this is demonstrated in Fig. [REF].', '1508.04284-1-40-0': 'To find the localization phase diagram, we have performed a number of calculations with either fixed [MATH] or fixed [MATH].', '1508.04284-1-40-1': 'System sizes [MATH], and [MATH] were used with [MATH] data points for each [MATH] to determine the critical points.', '1508.04284-1-40-2': 'Figure [REF] shows the resulting phase diagram.', '1508.04284-1-41-0': 'The region of delocalized states is asymmetric.', '1508.04284-1-41-1': 'For positive energies, it extends towards higher disorder with a maximal disorder strength [MATH] at [MATH].', '1508.04284-1-42-0': 'Figure [REF] shows the localization transition near the upper band edge for fixed [MATH].', '1508.04284-1-42-1': 'The additional potential disorder smoothes the density of states and suppresses the artifacts in the [MATH] data seen for the pure 3D VD lattice.', '1508.04284-1-42-2': 'The FSS approach is thus applicable.', '1508.04284-1-42-3': 'We obtain [MATH] and [MATH].', '1508.04284-1-43-0': 'The transition induced by [MATH] at fixed energy [MATH] is studied in detail in order to obtain a more accurate estimate of [MATH] for systems with both topological and potential disorder.', '1508.04284-1-43-1': 'The data analysis resembles the analysis of the transition tuned by [MATH] for [MATH], as described above.', '1508.04284-1-43-2': 'The details are summarized in Tab. 2.', '1508.04284-1-43-3': 'We observe a critical exponent [MATH] in agreement with the result for purely topological disorder.', '1508.04284-1-44-0': '# Conclusion', '1508.04284-1-45-0': 'To summarize, we have studied the effects of topological disorder on Anderson localization.', '1508.04284-1-45-1': 'To this end, we have investigated the wave functions of noninteracting electrons on random Voronoi-Delaunay lattices by multifractal analysis and finite-size scaling.', '1508.04284-1-45-2': 'In two dimensions, there is no Anderson transition as all states are localized, even in the absence of extra potential disorder.', '1508.04284-1-45-3': 'Adding random potentials further enhances the localization.', '1508.04284-1-45-4': 'In contrast, in three dimensions, the topological disorder of the Voronoi-Delaunay lattice induces two Anderson transitions close to the edges of the energy band, with localized states in the tails and extended states in the bulk of the band.', '1508.04284-1-45-5': 'If extra random potentials are added, the region of extended states first broadens with the broadening density of states, but then it shrinks and vanishes at some critical random potential strength.', '1508.04284-1-46-0': 'All these qualitative features agree with those of the usual Anderson model of localization.', '1508.04284-1-46-1': 'This means that the anticorrelations of the topological disorder [CITATION] do not affect the universal properties of Anderson localization.', '1508.04284-1-46-2': 'This also holds for the critical behavior of the localization transition in three dimensions.', '1508.04284-1-46-3': 'The correlation exponents found in the present paper, viz., [MATH] for purely topological disorder and [MATH] for combined topological and energetic disorder, agree within their errors with high-precision results for the usual Anderson model of localization [CITATION].', '1508.04284-1-47-0': 'Why is Anderson localization not (qualitatively) affected by the disorder anticorrelations of the Voronoi-Delaunay lattice even though other continuous and first-order phase transitions are qualitatively changed and violate the usual Harris and Imry-Ma criteria?', '1508.04284-1-47-1': 'In the systems in which these violations have been found [CITATION], the coordination number directly determines the local distance from the transition point because the effective coupling strength is simply the sum over the effects of all neighbors.', '1508.04284-1-47-2': 'Anticorrelations of the coordination numbers thus generate anticorrelated random-mass (or random-[MATH]) disorder.', '1508.04284-1-47-3': 'Anderson localization is more complex.', '1508.04284-1-47-4': 'In particular, quantum interference effects are crucial, at least away from the band edges, and these effects are not captured by the coordination number alone.', '1508.04284-1-47-5': 'Note, however, that the electronic states in the band tails, where the localization is mostly classical, do seem to be influenced by the disorder anticorrelations (see discussion at the end of Sec. [REF]).', '1508.04284-1-47-6': 'Clearly, more work will be necessary to fully resolve the effects of topological disorder on Anderson localization.', '1508.04284-1-48-0': 'This work work was supported by the NSF under Grant Nos.', '1508.04284-1-48-1': 'DMR-1205803.', '1508.04284-1-48-2': 'We acknowledge useful discussions with H. Barghathi.', '1508.04284-1-49-0': '# Appendix: Algorithm for creating random Voronoi-Delaunay lattices', '1508.04284-1-50-0': 'Computing the Voronoi diagram or the Delaunay triangulation of a given set of points (lattice sites) is a standard problem of computational geometry, and many different algorithms are discussed in text books and the research literature (see, e.g., Ref. [CITATION]).', '1508.04284-1-50-1': 'Our algorithm follows a suggestion by Tanemura et al. [CITATION] and is based on the remarkable "empty circumcircle property" of a 2D Delaunay triangulation.', '1508.04284-1-50-2': 'It states that every triangle formed by the bonds (edges) of the Delaunay triangulation has an empty circumcircle, i.e., a circumcircle that does not contain any other lattice sites.', '1508.04284-1-50-3': 'This is illustrated in Fig. [REF].', '1508.04284-1-51-0': 'Analogously, a 3D Delaunay tessellation features an "empty circumsphere property": Each of the tetrahedra making up the tessellation has a circumsphere that does not contain any other lattice sites.', '1508.04284-1-52-0': 'Our algorithm considers the lattice sites one by one and finds the list of its (Voronoi) neighbors.', '1508.04284-1-52-1': 'This is done in a two-step process: (i) We first identify candidates for the neighbors based on their distance.', '1508.04284-1-52-2': 'All sites within a distance [MATH] from the given site are included in the candidate list.', '1508.04284-1-52-3': 'For optimal performance, the cutoff radius [MATH] should be chosen as small as possible without missing neighbors, reasonable values depend on the structure of the set of lattice sites under consideration (see below).', '1508.04284-1-52-4': '(ii) From these candidates, we then construct all triangles (in 2D) with empty circumcirles for which the given site is one of the vertices.', '1508.04284-1-52-5': 'In 3D, we construct all tetrahedra with empty circumspheres for which the given site is a vertex.', '1508.04284-1-52-6': 'The entire algorithm can be coded efficiently in less than 400 lines of Fortran 90 in 2D and less than 500 lines in 3D.', '1508.04284-1-53-0': 'For the current project, we have applied these algorithms to sets of [MATH] (2D) or [MATH] (3D) uniformly distributed random sites of density unity contained in a square or cubic box of linear size [MATH].', '1508.04284-1-53-1': 'The sizes range up to [MATH] in 2D and [MATH] in 3D, and we use periodic boundary conditions.', '1508.04284-1-53-2': 'We found that the necessary values of the cutoff radius [MATH] are quite small because the bond-length distribution of the random VD lattice drops off rapidly with increasing distance, see Fig. [REF] (in 2D, the tail is approximately Gaussian).', '1508.04284-1-53-3': 'Empirically, we found that [MATH] in 2D and [MATH] in 3D are sufficient to find all neighbors in all the lattices we considered.', '1508.04284-1-54-0': 'For system sizes [MATH], the computational effort of our algorithm scales approximately linearly with the number of sites.', '1508.04284-1-54-1': 'To give an example of the performance, finding the Delaunay triangulation of [MATH] sites in 2D takes about 30 seconds on an Intel core i5-3570 CPU while [MATH] sites in 3D take about 3 minutes.'}

{'1508.04284-2-0-0': 'We consider the transport of non-interacting electrons on two- and three-dimensional random Voronoi-Delaunay lattices.', '1508.04284-2-0-1': 'It was recently shown that these topologically disordered lattices feature strong disorder anticorrelations between the coordination numbers that qualitatively change the properties of continuous and first-order phase transitions.', '1508.04284-2-0-2': 'To determine whether or not these unusual features also influence Anderson localization, we study the electronic wave functions by multifractal analysis and finite-size scaling.', '1508.04284-2-0-3': 'We observe only localized states for all energies in the two-dimensional system.', '1508.04284-2-0-4': 'In three dimensions, we find two Anderson transitions between localized and extended states very close to the band edges.', '1508.04284-2-0-5': 'The critical exponent of the localization length is about 1.6.', '1508.04284-2-0-6': 'All these results agree with the usual orthogonal universality class.', '1508.04284-2-0-7': 'Additional generic energetic randomness introduced via random potentials does not lead to qualitative changes but allows us to obtain a phase diagram by varying the strength of these potentials.', '1508.04284-2-1-0': '# Introduction', '1508.04284-2-2-0': 'More than 5 decades ago, Anderson [CITATION] showed that random disorder can localize a quantum particle in space.', '1508.04284-2-2-1': 'This phenomenon, now called Anderson localization, and the corresponding Anderson transitions between localized and metallic phases have since attracted lots of experimental and theoretical attention (see, e.g., Refs. [CITATION] for reviews).', '1508.04284-2-2-2': 'Remarkably, important features of Anderson localization, such as the existence of a metallic phase and the qualitative characteristics of the Anderson transition, depend on the dimensionality and the symmetries of the system, in close analogy to conventional continuous phase transitions.', '1508.04284-2-3-0': 'This leads to a symmetry classification of Anderson transitions.', '1508.04284-2-3-1': 'Originally, three universality classes were identified, based on the invariance of the Hamiltonian under time reversal and spin rotations.', '1508.04284-2-3-2': 'These classes (orthogonal, unitary, and symplectic) correspond to the Wigner-Dyson classification of random matrices [CITATION].', '1508.04284-2-3-3': 'For example, the orthogonal universality class contains systems that are invariant under both time reversal and spin rotation.', '1508.04284-2-3-4': 'Later, this classification scheme was extended by including additional symmetries (for a review, see, e.g., Evers and Mirlin [CITATION]).', '1508.04284-2-3-5': 'Within each class, the properties of the Anderson transition are expected to be universal.', '1508.04284-2-3-6': 'Specifically, all eigenstates in one-dimensional (1D) and two-dimensional (2D) systems in the orthogonal class are expected to be localized, implying the absence of an Anderson transition.', '1508.04284-2-3-7': 'In contrast, the states in three-dimensional (3D) systems in the orthogonal class can undergo a transition from localized to delocalized as energy or disorder strength are varied; and this transition features universal critical exponents.', '1508.04284-2-4-0': 'These universal properties hold for uncorrelated randomness; spatial disorder correlations can lead to different behavior.', '1508.04284-2-4-1': 'Some short-range correlations have been shown to produce extended states at specific, isolated energies in an otherwise localized system, for instance in the so-called dimer model [CITATION].', '1508.04284-2-4-2': 'In contrast, long-range correlations can lead to the appearance of a true metallic phase (and thus an Anderson transition), even in 1D.', '1508.04284-2-4-3': 'Some of these developments are reviewed by Izrailev et al. [CITATION].', '1508.04284-2-5-0': 'In recent years, topologically disordered systems (lattices with random connectivity) have attracted particular attention because phase transitions in such systems feature surprising violations of the expected universal behavior.', '1508.04284-2-5-1': 'This includes transitions in Ising and Potts magnets as well as the contact process, all defined on random Voronoi-Delaunay (VD) lattices [CITATION].', '1508.04284-2-5-2': 'Barghathi and Vojta [CITATION] solved this puzzle by showing that the 2D random VD lattice belongs to a broad class of random lattices whose disorder fluctuations feature strong anticorrelations and therefore decay faster with increasing length scale than those of generic random systems.', '1508.04284-2-5-3': 'Such lattices are ubiquitous in 2D because the Euler equation for a 2D graph imposes a topological constraint on the coordination numbers; however, examples in higher dimensions exist as well.', '1508.04284-2-5-4': 'The suppressed disorder fluctuations lead to important modifications of the Harris [CITATION] and Imry-Ma [CITATION] criteria that govern the effects of disorder on continuous and first-order transitions, respectively.', '1508.04284-2-6-0': 'These results immediately pose the question of whether or not the unusual features of random VD lattices also modify universal properties of Anderson localization.', '1508.04284-2-6-1': 'Grimm et al. [CITATION] studied the energy level distribution of a tight-binding model defined on a random tessellation [CITATION] similar to a VD lattice.', '1508.04284-2-6-2': 'They found level repulsion indicative of extended states in the metallic regime.', '1508.04284-2-6-3': 'However, to the best of our knowledge, a systematic finite-size scaling (FSS) study that would permit the unambiguous identification of the metallic and localized phases (and the Anderson transition between them) has not yet been performed.', '1508.04284-2-7-0': 'In this paper, we therefore investigate noninteracting electrons on 2D and 3D random VD lattices.', '1508.04284-2-7-1': 'We perform FSS based on a multifractal analysis (MFA) of the electronic wave functions.', '1508.04284-2-7-2': 'Our results can be summarized as follows.', '1508.04284-2-7-3': 'In 2D, we observe localized states for all energies.', '1508.04284-2-7-4': 'In contrast, the 3D system features two Anderson transitions between localized and extended states close to the band edges.', '1508.04284-2-7-5': 'The critical exponent of the correlation length takes the value [MATH], in agreement with the standard orthogonal universality class.', '1508.04284-2-7-6': 'This implies that the unusual coordination number anticorrelations of random VD lattices do not lead to qualitatively different behavior compared to the well-known Anderson model of localization on regular lattices.', '1508.04284-2-8-0': 'The rest of this paper is organized as follows.', '1508.04284-2-8-1': 'In Sec. [REF], we introduce the random VD lattices and define the Hamiltonian.', '1508.04284-2-8-2': 'We also discuss FSS as well as the MFA of the electronic states.', '1508.04284-2-8-3': 'Section [REF] is devoted to the results of our simulations for both 2D and 3D systems.', '1508.04284-2-8-4': 'We conclude in Sec. [REF].', '1508.04284-2-9-0': '# Model and methods', '1508.04284-2-10-0': '## Random Voronoi-Delaunay lattices', '1508.04284-2-11-0': 'The random VD lattice is a prototypical system with topological (connectivity) disorder.', '1508.04284-2-11-1': 'It can be viewed as a simple model for amorphous solids, foams or biological cell structures.', '1508.04284-2-11-2': 'The random VD lattice is defined as a set of lattice sites at random positions together with the bonds that connect nearest neighbor sites.', '1508.04284-2-11-3': 'These neighbors are determined by the VD construction [CITATION] as follows.', '1508.04284-2-11-4': 'The entire system area (or volume, in the 3D case) is subdivided into disjoint polygons (Voronoi cells) containing exactly one lattice site such that each cell contains all points in space that are closer to that lattice site than to any other site.', '1508.04284-2-11-5': 'The Voronoi diagram is the complete set of these Voronoi cells.', '1508.04284-2-11-6': 'Lattice sites whose Voronoi cells share an edge (or a face, in the 3D case) are considered neighbors, irrespective of their real-space distance.', '1508.04284-2-11-7': 'In 2D, the graph of all bonds connecting pairs of neighbors consists of triangles only.', '1508.04284-2-11-8': 'It is called the Delaunay triangulation.', '1508.04284-2-11-9': 'The corresponding construction in 3D leads to the Delaunay tetrahedralization, a lattice consisting of tetrahedra only.', '1508.04284-2-11-10': 'We emphasize that random VD lattices are not bipartite, i.e., they cannot be divided into disjoint sublattices A and B such that each bond connects an A site to a B site.', '1508.04284-2-11-11': 'A 2D example of a VD construction can be seen in Fig. [REF].', '1508.04284-2-12-0': 'In the following, we consider independent, uniformly distributed random lattice sites of density unity contained in a square or cubic box of linear size [MATH].', '1508.04284-2-12-1': 'We employ periodic boundary conditions.', '1508.04284-2-12-2': 'More details of the algorithm that we use to perform the VD construction are given in the Appendix.', '1508.04284-2-13-0': 'In a random VD lattice, the coordination number (number of neighbors) [MATH] fluctuates from site to site.', '1508.04284-2-13-1': 'This topological disorder is illustrated in Fig. [REF].', '1508.04284-2-13-2': 'The corresponding coordination number distribution can be seen in Fig. [REF].', '1508.04284-2-13-3': 'The average coordination number in 2D is exactly [MATH].', '1508.04284-2-13-4': 'This is a consequence of the Euler equation for a 2D graph consisting of triangles only.', '1508.04284-2-13-5': 'In 3D, the average coordination number is given by [MATH] [CITATION].', '1508.04284-2-13-6': 'In both dimensions, the mean coordination numbers are higher than those of the regular square and cubic lattices often used in numerical localization studies.', '1508.04284-2-13-7': 'The standard deviations of the coordinations numbers are [MATH] in 2D and [MATH] in 3D.', '1508.04284-2-13-8': 'This means that the disorder is moderately strong.', '1508.04284-2-14-0': '## Anderson model', '1508.04284-2-15-0': 'We now consider the motion of noninteracting electrons on a random VD lattice.', '1508.04284-2-15-1': 'We describe it by means of a tight-binding model having one (Wannier) orbital per lattice site.', '1508.04284-2-15-2': 'The corresponding Hamiltonian [EQUATION] is analogous to the Anderson model of localization.', '1508.04284-2-15-3': 'The first term describes the hopping of electrons between nearest (Voronoi) neighbors.', '1508.04284-2-15-4': 'The hopping matrix element is constant and fixed at unity.', '1508.04284-2-15-5': 'This term contains the topological disorder.', '1508.04284-2-15-6': 'The second term represents additional energetic randomness.', '1508.04284-2-15-7': 'The [MATH] are random on-site potentials uniformly distributed in the interval [MATH].', '1508.04284-2-15-8': 'We are mostly interested in the case [MATH] for which the disorder in the system is purely topological.', '1508.04284-2-15-9': 'However, for comparison with the usual Anderson model of localization, we also consider nonzero random potential strength [MATH].', '1508.04284-2-16-0': 'To compute the densities of states (DOS), we directly diagonalize the secular matrices.', '1508.04284-2-16-1': 'The system sizes are [MATH] for 2D and [MATH] for 3D, limited by computer memory.', '1508.04284-2-16-2': 'To find the eigenstates of the Hamiltonian ([REF]) close to a particular energy value for larger system sizes, we use a sparse matrix algorithm based on the Jacobi-Davidson method [CITATION].', '1508.04284-2-16-3': 'Here, we treat system sizes up to [MATH] for 2D and [MATH] for 3D.', '1508.04284-2-17-0': 'Figure [REF] shows an overview over the DOS resulting from these calculations for purely topological disorder ([MATH]).', '1508.04284-2-18-0': 'As the VD lattices are not bipartite, the DOS is not symmetric with respect to the energy [MATH].', '1508.04284-2-18-1': 'While the DOS near the low-energy band edge increases rapidly, there is a pronounced tail on the high-energy side, in particular in 3D.', '1508.04284-2-18-2': 'Interestingly, this tail does not stretch much beyond [MATH].', '1508.04284-2-18-3': 'States in the far tail live on rare large clusters of sites with above average coordination numbers.', '1508.04284-2-18-4': 'The fact that there are almost no states with [MATH] reflects the strong disorder anticorrelations of the random VD lattices that prevent the formation of large rare regions [CITATION].', '1508.04284-2-18-5': 'We also note that the DOS in the 3D band tail fluctuates significantly.', '1508.04284-2-18-6': 'We believe this is a finite-size effect.', '1508.04284-2-19-0': '## Finite-size scaling', '1508.04284-2-20-0': 'The scaling approach is a phenomenological description of the behavior close to the critical point of a continuous transition.', '1508.04284-2-20-1': 'We consider a general dimensionless measure [MATH] characterizing the electronic states as function of energy [MATH] and a characteristic length, e.g., the system size [MATH].', '1508.04284-2-20-2': 'Close to the critical energy [MATH], it fulfills the scaling form [CITATION] [EQUATION]', '1508.04284-2-20-3': 'The description contains the relevant scaling variable [MATH] associated with the relevant exponent [MATH] as well as the leading irrelevant scaling variable [MATH] associated with the irrelevant exponent [MATH].', '1508.04284-2-20-4': 'The dependence of the scaling variables on [MATH] can be described in terms of the expansions [EQUATION] where [MATH] is a dimensionless measure of the distance from the critical energy.', '1508.04284-2-20-5': 'The scaling function [EQUATION] is expanded into a Taylor polynomial of the irrelevant variable.', '1508.04284-2-20-6': 'The coefficients are the regular single-variable scaling functions [EQUATION]', '1508.04284-2-20-7': 'They are expanded into Chebyshev polynomials [MATH] of the first kind and the [MATH]th order.', '1508.04284-2-20-8': 'This general expansion depends on the four expansion orders [MATH], [MATH], [MATH], and [MATH] yielding [MATH] free parameters.', '1508.04284-2-20-9': 'Their values are found by weighted nonlinear fits to the numerical data.', '1508.04284-2-20-10': 'The weights are formed by the reciprocal variances of the data points.', '1508.04284-2-20-11': 'This formalism allows us to simultaneously determine [MATH], [MATH], and [MATH].', '1508.04284-2-21-0': 'In order to get error estimates of these values we use a Monte-Carlo method [CITATION].', '1508.04284-2-21-1': 'It consists in building at least [MATH] synthetic data sets by adding noise to the original values.', '1508.04284-2-21-2': 'This noise is created by Gaussian random numbers with a standard deviation [MATH] equal to the individual error of each data point.', '1508.04284-2-21-3': 'By fitting [MATH] to these synthetic data sets we obtain distributions for the parameter values.', '1508.04284-2-21-4': 'These distributions are usually Gaussian.', '1508.04284-2-21-5': 'Large deviations, e.g., superpositions of multiple Gaussians, are interpreted as instabilities.', '1508.04284-2-21-6': 'The corresponding regressions are not well defined and will be neglected.', '1508.04284-2-21-7': '(This could be avoided by increasing the accuracy of the data, changing the number of parameters, or varying the initial conditions.)', '1508.04284-2-21-8': 'We note that this method constitutes an error-propagation calculation for the random errors only.', '1508.04284-2-21-9': 'Systematic errors are not detected completely, but the influence of different expansion orders can be identified.', '1508.04284-2-22-0': '## Multifractal analysis', '1508.04284-2-23-0': 'Multifractal behavior is a feature of eigenstates at critical points [CITATION].', '1508.04284-2-23-1': 'The MFA is based on a standard box-counting algorithm.', '1508.04284-2-23-2': 'The boxes are squares or cubes for 2D and 3D systems, respectively.', '1508.04284-2-23-3': 'The [MATH]-dimensional system of size [MATH] is partitioned into boxes of size [MATH] (see Fig. [REF]).', '1508.04284-2-23-4': 'Using the probability [EQUATION] to find the electron in the [MATH]th box, the measure [EQUATION] is constructed from its [MATH]th moment.', '1508.04284-2-23-5': 'Depending on [MATH], this measure is dominated by boxes with either large or small [MATH].', '1508.04284-2-23-6': 'For multifractal wave functions, [MATH] is expected to behave as a power of the normalized box size [MATH], with the scaling exponent (mass exponent) [EQUATION]', '1508.04284-2-23-7': 'The singularity spectrum [MATH] is the Legendre transform of [MATH], and it comprises the scaling exponents of the fractal dimensions of all moments.', '1508.04284-2-23-8': 'A parametric representation of the singularity spectrum can be obtained by calculating the singularity strength [EQUATION] and the fractal dimension [EQUATION] with [EQUATION]', '1508.04284-2-23-9': 'For statistical purposes, we utilize the ensemble average of these exponents [CITATION].', '1508.04284-2-23-10': 'In particular, we use the ensemble averaged singularity strength [EQUATION] with [EQUATION] to perform the finite-size scaling analysis outlined in the last section.', '1508.04284-2-23-11': '[MATH] denotes the average over different eigenstates.', '1508.04284-2-23-12': 'When determining the error of [MATH], one has to take into account that the numerator and denominator are correlated with each other [CITATION].', '1508.04284-2-24-0': 'Only integer box ratios [MATH] are possible when partitioning the original system without overlap.', '1508.04284-2-24-1': 'A general ratio can be used by employing the periodic boundary conditions to fold back into the real system any protruding box parts that arise for noninteger [MATH] [CITATION].', '1508.04284-2-24-2': 'To obtain a uniform sampling, we are then required to average over all possible box origins.', '1508.04284-2-25-0': 'To extrapolate to the thermodynamic limit [MATH] in Eq. ([REF]), we perform a linear least-squares fit of the numerator versus the denominator at fixed [MATH] (box-size scaling).', '1508.04284-2-25-1': 'The standard deviations [MATH] of [MATH] are utilized to determine the inaccuracy of the regression values.', '1508.04284-2-25-2': 'To approach the limit [MATH], sufficiently small values of [MATH] have to be taken into account.', '1508.04284-2-25-3': 'However, for the smallest [MATH], the data deviate from the power-law behavior because corrections to scaling that stem from the irrelevant scaling variables become important.', '1508.04284-2-25-4': 'Their influence on the estimated exponent values can be reduced by optimizing the fit boundaries.', '1508.04284-2-25-5': 'We use [MATH] for 2D and [MATH] for 3D.', '1508.04284-2-26-0': 'In random VD lattices, the cells have varying sizes; the resolution of the MFA given by the smallest box size [MATH] is therefore arbitrary.', '1508.04284-2-26-1': 'We use [MATH], such that the average number of sites per box is [MATH] (see Fig. [REF]).', '1508.04284-2-26-2': 'Since empty boxes affect the MFA results, only box sizes larger than the maximum distance between neighboring sites are included in the MFA analysis.', '1508.04284-2-26-3': 'This guarantees that there are no empty boxes.', '1508.04284-2-26-4': 'In both dimensions, we use [MATH] in accordance with the distributions shown in the right panel of Fig. [REF].', '1508.04284-2-27-0': '# Results', '1508.04284-2-28-0': '## 2D system', '1508.04284-2-29-0': 'Figure [REF] shows five representative eigenstates of the energy spectrum of a 2D system with [MATH] and [MATH].', '1508.04284-2-29-1': 'The wave function of the lowest eigenvalue (Fig. [REF]a) is exponentially localized.', '1508.04284-2-29-2': 'The maximum amplitude is concentrated at the single Voronoi cell with highest coordination number (here [MATH]).', '1508.04284-2-29-3': 'States on the upper band edge also show localized behavior very clearly (Fig. [REF]e).', '1508.04284-2-29-4': 'However, the highest amplitude is orders of magnitude smaller and, correspondingly, the localization length is larger.', '1508.04284-2-29-5': 'More generally, states of smaller energy are influenced by local fluctuations of the coordination number.', '1508.04284-2-29-6': 'States of higher energy are driven by interferences and the probability amplitudes behave as non-integrable Chladni figures [CITATION].', '1508.04284-2-29-7': 'The number of antinodes increases with decreasing eigenenergy.', '1508.04284-2-29-8': 'In summary, this is a visualization of localization in a classical and quantum mechanical manner for the states close to the lower and the upper band edge, respectively.', '1508.04284-2-29-9': 'Towards the band center the amplitude fluctuations increase and a classification by visual inspection is not easily possible anymore.', '1508.04284-2-30-0': 'We therefore turn to the MFA to quantitatively characterize the eigenstates as function of energy.', '1508.04284-2-30-1': 'Specifically, we consider the FSS behavior of [MATH] for systems from [MATH] up to [MATH] with [MATH] states per data point.', '1508.04284-2-30-2': 'Figure [REF]', '1508.04284-2-31-0': 'shows the box-size scaling of [MATH] used to estimate [MATH] for several [MATH].', '1508.04284-2-31-1': 'Whereas values at higher [MATH] show the expected power-law dependence on [MATH], results for small [MATH] deviate systematically due to finite-size effects.', '1508.04284-2-31-2': 'Therefore, they are neglected as discussed before.', '1508.04284-2-31-3': 'The multifractal spectra [MATH] resulting from the analysis are also shown in Fig. [REF] for representative energies close to the upper band edge.', '1508.04284-2-31-4': 'For the energy [MATH], we obtain a parabola-like shape as expected.', '1508.04284-2-31-5': 'Towards the band center the parabola shrinks (see [MATH]) and approaches the limiting case for completely extended wave functions, namely the single point [MATH].', '1508.04284-2-31-6': 'In the opposite direction, close to the band edge, the spectrum transforms towards the extremely localized limit, the points [MATH] for negative [MATH] and [MATH] for positive [MATH].', '1508.04284-2-32-0': 'The behavior of [MATH] (the position of the maximum of the parabola) is shown in more detail in Fig. [REF].', '1508.04284-2-33-0': 'For energy values close to the band edge, [MATH] generally increases with [MATH].', '1508.04284-2-33-1': 'This means that these states are localized.', '1508.04284-2-33-2': 'The strength of the localization depends on [MATH].', '1508.04284-2-33-3': 'The pronounced localization at the lower band edge seen in Fig. [REF] is reproduced here.', '1508.04284-2-33-4': 'Strongly fluctuating wave functions near the band center have a value close to [MATH] for all system sizes within their accuracy.', '1508.04284-2-33-5': 'The envelopes of corresponding wave functions have a very large localization length (larger than the system size).', '1508.04284-2-33-6': 'Importantly, the curves for different [MATH] do not show a common crossing point, implying that there is no Anderson transition.', '1508.04284-2-33-7': 'All states are localized.', '1508.04284-2-33-8': '(The seeming crossings at very large [MATH] in Fig. [REF] can be attributed to numerical artifacts.)', '1508.04284-2-34-0': 'We have also studied the effects of additional potential disorder [MATH] and [MATH].', '1508.04284-2-34-1': 'It results in a broadening of the DOS and an enhancement of the localization.', '1508.04284-2-34-2': 'However, qualitative changes compared to [MATH] were not found.', '1508.04284-2-34-3': 'All states are localized.', '1508.04284-2-35-0': '## 3D system', '1508.04284-2-36-0': 'The data analysis for the 3D VD lattice proceeds analogously.', '1508.04284-2-36-1': 'We use system sizes between [MATH] and [MATH].', '1508.04284-2-36-2': 'In contrast to the 2D system, we observe two Anderson transitions induced by purely topological disorder.', '1508.04284-2-36-3': 'These transitions are located close to the two energy band edges.', '1508.04284-2-36-4': 'The corresponding FSS behavior of [MATH] is visualized in Fig. [REF] for the transition near the lower band edge and in Fig. [REF] for the upper edge.', '1508.04284-2-36-5': 'Thus, localized states exist only near the band edges.', '1508.04284-2-36-6': 'The broad central area of the energy band encompasses extended states.', '1508.04284-2-36-7': 'This corresponds to the fact that the topological disorder is only moderately strong and results in localization behavior similar to that of a weakly disordered regular Anderson model.', '1508.04284-2-37-0': 'As shown in Fig. [REF], the singularity strength [MATH] features a smooth energy dependence for the transition near the lower band edge.', '1508.04284-2-37-1': 'Thus, the FSS approach is applicable.', '1508.04284-2-37-2': 'The results are presented in Tab. 1.', '1508.04284-2-37-3': 'It can be seen that the critical parameters [MATH] and [MATH] are influenced by the irrelevant scaling variable.', '1508.04284-2-37-4': 'We compare different expansion orders to demonstrate the stability of the regression results.', '1508.04284-2-37-5': 'In particular, we neglect the data of small systems [MATH] and use a FSS approach without irregular expansion ([MATH]).', '1508.04284-2-37-6': 'Such regressions show a higher robustness when changing initial conditions.', '1508.04284-2-37-7': 'Taking into account all data, we estimate the critical energy [MATH] and the critical exponent [MATH].', '1508.04284-2-37-8': 'The results of regressions without the irrelevant scaling variable deviate slightly ([MATH] and [MATH]) because of the neglected systematic shift of the intersections described by the irrelevant term.', '1508.04284-2-37-9': 'The quality of all fits is very close to unity.', '1508.04284-2-37-10': 'This indicates that the errors of the original data points were overestimated.', '1508.04284-2-38-0': 'For the transition near the upper band edge, the [MATH] curves shown in Fig. [REF] are very noisy.', '1508.04284-2-38-1': 'This is caused by the low DOS (see right panel of Fig. [REF]) and finite size effects.', '1508.04284-2-38-2': 'The existence of a transition can still be inferred because [MATH] decreases with [MATH] for the smaller energies, while it increases with [MATH] for the largest energies.', '1508.04284-2-38-3': 'This indicates a crossing and thus an Anderson transition.', '1508.04284-2-38-4': 'However, a clear transition point cannot be determined from the available data.', '1508.04284-2-38-5': 'In particular, the values are insufficient to perform a scaling analysis.', '1508.04284-2-38-6': '(Also note that for small systems, gaps in the DOS appear.', '1508.04284-2-38-7': 'Therefore the number of considered eigenstates varies from [MATH] to [MATH] between data points, leading to strong variations in the error bars.)', '1508.04284-2-38-8': 'A rough estimate of the critical energy is [MATH].', '1508.04284-2-39-0': 'We now turn to the effects of additional on-site disorder, i.e., [MATH].', '1508.04284-2-39-1': 'As in 2D, on-site disorder leads to a broadening of the DOS with increasing disorder strength [MATH], this is demonstrated in Fig. [REF].', '1508.04284-2-40-0': 'To find the localization phase diagram, we have performed a number of calculations with either fixed [MATH] or fixed [MATH].', '1508.04284-2-40-1': 'System sizes [MATH], and [MATH] were used with [MATH] data points for each [MATH] to determine the critical points.', '1508.04284-2-40-2': 'Figure [REF] shows the resulting phase diagram.', '1508.04284-2-41-0': 'The region of delocalized states is asymmetric.', '1508.04284-2-41-1': 'For positive energies, it extends towards higher disorder with a maximal disorder strength [MATH] at [MATH].', '1508.04284-2-42-0': 'Figure [REF] shows the localization transition near the upper band edge for fixed [MATH].', '1508.04284-2-42-1': 'The additional potential disorder smoothes the density of states and suppresses the artifacts in the [MATH] data seen for the pure 3D VD lattice.', '1508.04284-2-42-2': 'The FSS approach is thus applicable.', '1508.04284-2-42-3': 'We obtain [MATH] and [MATH].', '1508.04284-2-43-0': 'The transition induced by [MATH] at fixed energy [MATH] is studied in detail in order to obtain a more accurate estimate of [MATH] for systems with both topological and potential disorder.', '1508.04284-2-43-1': 'The data analysis resembles the analysis of the transition tuned by [MATH] for [MATH], as described above.', '1508.04284-2-43-2': 'The details are summarized in Tab. 2.', '1508.04284-2-43-3': 'We observe a critical exponent [MATH] in agreement with the result for purely topological disorder.', '1508.04284-2-44-0': '# Conclusion', '1508.04284-2-45-0': 'To summarize, we have studied the effects of topological disorder on Anderson localization.', '1508.04284-2-45-1': 'To this end, we have investigated the wave functions of noninteracting electrons on random Voronoi-Delaunay lattices by multifractal analysis and finite-size scaling.', '1508.04284-2-45-2': 'In two dimensions, there is no Anderson transition as all states are localized, even in the absence of extra potential disorder.', '1508.04284-2-45-3': 'Adding random potentials further enhances the localization.', '1508.04284-2-45-4': 'In contrast, in three dimensions, the topological disorder of the Voronoi-Delaunay lattice induces two Anderson transitions close to the edges of the energy band, with localized states in the tails and extended states in the bulk of the band.', '1508.04284-2-45-5': 'If extra random potentials are added, the region of extended states first broadens with the broadening density of states, but then it shrinks and vanishes at some critical random potential strength.', '1508.04284-2-46-0': 'All these qualitative features agree with those of the usual Anderson model of localization.', '1508.04284-2-46-1': 'This means that the anticorrelations of the topological disorder [CITATION] do not affect the universal properties of Anderson localization.', '1508.04284-2-46-2': 'This also holds for the critical behavior of the localization transition in three dimensions.', '1508.04284-2-46-3': 'The correlation exponents found in the present paper, viz., [MATH] for purely topological disorder and [MATH] for combined topological and energetic disorder, agree within their errors with high-precision results for the usual Anderson model of localization [CITATION].', '1508.04284-2-47-0': 'Why is Anderson localization not (qualitatively) affected by the disorder anticorrelations of the Voronoi-Delaunay lattice even though other continuous and first-order phase transitions are qualitatively changed and violate the usual Harris and Imry-Ma criteria?', '1508.04284-2-47-1': 'In the systems in which these violations have been found [CITATION], the coordination number directly determines the local distance from the transition point because the effective coupling strength is simply the sum over the effects of all neighbors.', '1508.04284-2-47-2': 'Anticorrelations of the coordination numbers thus generate anticorrelated random-mass (or random-[MATH]) disorder.', '1508.04284-2-47-3': 'Anderson localization is more complex.', '1508.04284-2-47-4': 'In particular, quantum interference effects are crucial, at least away from the band edges, and these effects are not captured by the coordination number alone.', '1508.04284-2-47-5': 'Note, however, that the electronic states in the band tails, where the localization is mostly classical, do seem to be influenced by the disorder anticorrelations (see discussion at the end of Sec. [REF]).', '1508.04284-2-47-6': 'Clearly, more work will be necessary to fully resolve the effects of topological disorder on Anderson localization.', '1508.04284-2-48-0': 'This work was supported by the NSF under Grant Nos.', '1508.04284-2-48-1': 'DMR-1205803.', '1508.04284-2-48-2': 'We acknowledge useful discussions with H. Barghathi.', '1508.04284-2-49-0': '# Appendix: Algorithm for creating random Voronoi-Delaunay lattices', '1508.04284-2-50-0': 'Computing the Voronoi diagram or the Delaunay triangulation of a given set of points (lattice sites) is a standard problem of computational geometry, and many different algorithms are discussed in text books and the research literature (see, e.g., Ref. [CITATION]).', '1508.04284-2-50-1': 'Our algorithm follows a suggestion by Tanemura et al. [CITATION] and is based on the remarkable "empty circumcircle property" of a 2D Delaunay triangulation.', '1508.04284-2-50-2': 'It states that every triangle formed by the bonds (edges) of the Delaunay triangulation has an empty circumcircle, i.e., a circumcircle that does not contain any other lattice sites.', '1508.04284-2-50-3': 'This is illustrated in Fig. [REF].', '1508.04284-2-51-0': 'Analogously, a 3D Delaunay tessellation features an "empty circumsphere property": Each of the tetrahedra making up the tessellation has a circumsphere that does not contain any other lattice sites.', '1508.04284-2-52-0': 'Our algorithm considers the lattice sites one by one and finds the list of its (Voronoi) neighbors.', '1508.04284-2-52-1': 'This is done in a two-step process: (i) We first identify candidates for the neighbors based on their distance.', '1508.04284-2-52-2': 'All sites within a distance [MATH] from the given site are included in the candidate list.', '1508.04284-2-52-3': 'For optimal performance, the cutoff radius [MATH] should be chosen as small as possible without missing neighbors, reasonable values depend on the structure of the set of lattice sites under consideration (see below).', '1508.04284-2-52-4': '(ii) From these candidates, we then construct all triangles (in 2D) with empty circumcirles for which the given site is one of the vertices.', '1508.04284-2-52-5': 'In 3D, we construct all tetrahedra with empty circumspheres for which the given site is a vertex.', '1508.04284-2-52-6': 'The entire algorithm can be coded efficiently in less than 400 lines of Fortran 90 in 2D and less than 500 lines in 3D.', '1508.04284-2-53-0': 'For the current project, we have applied these algorithms to sets of [MATH] (2D) or [MATH] (3D) uniformly distributed random sites of density unity contained in a square or cubic box of linear size [MATH].', '1508.04284-2-53-1': 'The sizes range up to [MATH] in 2D and [MATH] in 3D, and we use periodic boundary conditions.', '1508.04284-2-53-2': 'We found that the necessary values of the cutoff radius [MATH] are quite small because the bond-length distribution of the random VD lattice drops off rapidly with increasing distance, see Fig. [REF] (in 2D, the tail is approximately Gaussian).', '1508.04284-2-53-3': 'Empirically, we found that [MATH] in 2D and [MATH] in 3D are sufficient to find all neighbors in all the lattices we considered.', '1508.04284-2-54-0': 'For system sizes [MATH], the computational effort of our algorithm scales approximately linearly with the number of sites.', '1508.04284-2-54-1': 'To give an example of the performance, finding the Delaunay triangulation of [MATH] sites in 2D takes about 30 seconds on an Intel core i5-3570 CPU while [MATH] sites in 3D take about 3 minutes.'}

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[['1508.04284-1-2-2', '1508.04284-2-2-2'], ['1508.04284-1-20-9', '1508.04284-2-20-9'], ['1508.04284-1-29-8', '1508.04284-2-29-8'], ['1508.04284-1-36-3', '1508.04284-2-36-3'], ['1508.04284-1-3-1', '1508.04284-2-3-1'], ['1508.04284-1-48-0', '1508.04284-2-48-0'], ['1508.04284-1-23-3', '1508.04284-2-23-3'], ['1508.04284-1-18-1', '1508.04284-2-18-1'], ['1508.04284-1-18-4', '1508.04284-2-18-4'], ['1508.04284-1-15-2', '1508.04284-2-15-2'], ['1508.04284-1-7-6', '1508.04284-2-7-6']]

[]

[]

[]

['1508.04284-1-30-2', '1508.04284-1-48-1', '1508.04284-2-30-2', '1508.04284-2-48-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1508.04284

1810.03443

{'1810.03443-1-0-0': '# Introduction', '1810.03443-1-1-0': 'It is well known since 1960s [CITATION] that energetic gamma rays produce electron-positron pairs on the extragalactic background light (EBL).', '1810.03443-1-1-1': 'Because of this process, gamma-ray flux from distant sources is attenuated severely and the propagation distance is limited.', '1810.03443-1-1-2': 'Very-high-energy (VHE) gamma rays (energy [MATH] GeV) scatter most efficiently on the infrared and visible background photons, and information on the intergalactic absorption may be derived from VHE observations of emitters located at cosmologically large distances.', '1810.03443-1-2-0': 'There exists some long-standing controversy in the determination of the EBL intensity.', '1810.03443-1-2-1': 'Direct observations are very difficult to carry out because of strong foreground contamination by the Zodiacal light and by the Galactic foreground.', '1810.03443-1-2-2': 'Nevertheless, numerous attempts gave coherent, though quite uncertain, results.', '1810.03443-1-2-3': 'At the same time, there exist several theoretical and semi-empirical models for the EBL which, being consistent between each other, point to the intensity a factor of a few lower than that derived from direct observations.', '1810.03443-1-2-4': 'Some of the models are based on the galactic counts and represent lower limits on the EBL intensity, given just by the sum of contributions from guaranteed light sources.', '1810.03443-1-2-5': 'Others include more theoretical input from the star formation rate history but point to a similar range of intensities.', '1810.03443-1-2-6': 'The situation remains unsettled: two independent observations, published in 2017 and using different sophisticated techniques to get rid of the foreground systematics, point consistently to the EBL intensities a factor of two higher than the most recent theoretical model.', '1810.03443-1-2-7': 'For the purposes of the present study, we need a conservative (low-absorption) model and use that of Korochkin and Rubtsov (2018) [CITATION] as the most recent available benchmark for the low-EBL theoretical models.', '1810.03443-1-2-8': 'Variations of the absorption model will be discussed in the context of systematic uncertainties of our results.', '1810.03443-1-3-0': 'Distant blazars were observed in energetic gamma rays from the early days of the VHE astronomy.', '1810.03443-1-3-1': 'It has been pointed out long ago that some of them are seen from distances for which the optical depth with respect to the pair production on EBL is significant.', '1810.03443-1-3-2': 'This apparent tension with expectations was dubbed "the infrared/TeV crisis" [CITATION].', '1810.03443-1-3-3': 'However, subsequent improvements in EBL models demonstrated that the tension is not that strong.', '1810.03443-1-3-4': 'Moreover, apparent hardening of the intrinsic spectrum (corrected for the pair-production attenuation, that is "deabsorbed"), seen in several individual sources, might be related to physical conditions in particular objects, though a successful model for these hardenings is missing [CITATION].', '1810.03443-1-4-0': 'The situation has been changed when the number of sources observed at large optical depths became sufficient for studies of their ensembles.', '1810.03443-1-4-1': 'By comparing deabsorbed spectra of two pairs of similar sources, T. Kneiske (unpublished) has pointed out that spectra of distant sources exhibit unusual hardenings at high energies while those of physically similar nearby objects do not.', '1810.03443-1-4-2': 'The study of a sample of 7 sources observed by imaging atmospheric Cerenkov telescopes (IACTs) at optical depths [MATH] by Horns and Meyer [CITATION] demonstrated that the energy at which such hardenings happen is correlated with the distance to the source in such a way that the spectra become harder only when the correction for the pair-production absorption is significant.', '1810.03443-1-4-3': 'This correlation can hardly be caused by any physical reason and suggests that the optical depth is estimated incorrectly.', '1810.03443-1-4-4': 'The statistical significance of this "pair-production anomaly" was estimated at the [MATH] level.', '1810.03443-1-5-0': 'The most complete sample of sources observed at large optical depths ([MATH], 15 objects observed by IACTs and 5 objects observed by Fermi Large Area Telescope, LAT) has been considered [CITATION] in 2014.', '1810.03443-1-5-1': 'There, we first confirmed the existence of significant hardenings in deabsorbed spectra of a number of objects at the energies where the correction for pair production becomes important.', '1810.03443-1-5-2': 'We then considered the full sample of objects, including those with and without statistically significant hardenings, and fit their deabsorbed spectra with the broken power-law function, assuming the break at the energy [MATH], at which the pair-production optical depth [MATH] (these energies are different for sources at different redshifts).', '1810.03443-1-5-3': 'We found that the break strength, that is the difference between power-law indices below and above [MATH], grows with the source redshift.', '1810.03443-1-5-4': 'Statistical significance of this distance dependence, which again suggests an unphysical origin of the hardenings and hence an incorrect account of the absorption, was found to be [MATH].', '1810.03443-1-6-0': 'All these results are of great importance not only for the gamma-ray astronomy, but also for other fields of physics and astrophysics.', '1810.03443-1-6-1': 'As it will be discussed below, overestimation of the absorption can hardly be understood without invoking new physical or astrophysical phenomena, and the most promising explanation of the data requires the existence of new light particles beyond those described by the Standard Model of particle physics.', '1810.03443-1-7-0': 'Since 2014, several important improvements changed the field.', '1810.03443-1-7-1': 'Firstly, many new observations of distant VHE blazars have been performed by IACTs.', '1810.03443-1-7-2': 'Secondly, the Fermi-LAT team not only accumulated additional years of statistics, very important for faint distant VHE sources, but also issued the new "Pass 8" reconstruction [CITATION] improving the data processing.', '1810.03443-1-7-3': 'Another related novelty is the 3FHL catalog [CITATION] of hard-spectrum sources making pre-selection of the sample easier and more uniform than before.', '1810.03443-1-7-4': 'Thirdly, from the EBL side, new and improved conservative theoretical models have become available [CITATION], but at the same time new sophisticated observational studies strengthened the tension in the determination of the background radiation intensity [CITATION].', '1810.03443-1-7-5': 'Finally, for a number of objects, new information on their redshifts became available.', '1810.03443-1-7-6': 'The main purpose of this work is to revisit the claim of Ref. [CITATION] with the enlarged high-quality sample of sources, taking into account all the listed improvements.', '1810.03443-1-7-7': 'In addition, unlike in the letter-like paper of 2014, here we present all details of the analysis so that the readers could repeat the study themselves in case new data are available.', '1810.03443-1-7-8': 'We also present some tests demonstrating independence of our results from potential biases and their stability with respect to systematic uncertainties.', '1810.03443-1-8-0': 'The result of the present study supports previous claims of distance-dependent hardenings in spectra of VHE gamma-ray blazars, corrected for the pair-production attenuation with conservative EBL models.', '1810.03443-1-8-1': 'Absence of these hardenings is excluded with the statistical significance of [MATH].', '1810.03443-1-9-0': 'The rest of the paper is organized as follows.', '1810.03443-1-9-1': 'In Sec. [REF], we discuss the gamma-ray data used in our study.', '1810.03443-1-9-2': 'Section [REF] summarizes general criteria for the sample construction.', '1810.03443-1-9-3': 'Section [REF] detalizes how they are implemented for the objects observed by IACTs, while Sec. [REF] discusses the construction of the Fermi-LAT subsample.', '1810.03443-1-9-4': 'We list the resulting sample of 28 sources and discuss its general properties in Sec. [REF].', '1810.03443-1-10-0': 'Section [REF] describes the data analysis procedure and presents the main results of the paper.', '1810.03443-1-10-1': 'We describe how the correction to the pair-production opacity is implemented and perform a combined analysis of the full sample of 28 sources.', '1810.03443-1-10-2': 'We fit all the spectra with absorbed broken power laws fixing the break energy to [MATH]; clearly, the value of [MATH] varies with the distance to the source.', '1810.03443-1-10-3': 'Here, we obtain the main result of the paper: a [MATH] evidence for the distance dependence of spectral hardenings.', '1810.03443-1-11-0': 'This result, confirming our results of 2014 but obtained with the enlarged and cleaned up most recent data set and a new absorption model, is discussed in detail in Sec. [REF].', '1810.03443-1-11-1': 'There, we first turn, in Sec. [REF], to potential biases and systematic uncertainties which might affect our result.', '1810.03443-1-11-2': 'In Sec. [REF], we discuss variations in the absorption model: besides the most recent model we use throughout the paper, Korochkin and Rubtsov (2018) [CITATION], we repeat our analysis with three other models for EBL, namely: Gilmore et al. (2012) fixed [CITATION], which was used in our 2014 paper; Franceschini et al. (2017) [CITATION]; and a high-absorption model artificially normalized to the most recent direct observational data [CITATION] on the background-light intensity.', '1810.03443-1-11-3': 'The statistical significance of the effect reported in Sec. [REF] is always above [MATH], remaining, for the low-absorption models, consistent with the result for the baseline model we use.', '1810.03443-1-11-4': 'Next, in Sec. [REF], we discuss the possible bias related to the fact that only brightest sources are seen at large distances, the Malmquist bias.', '1810.03443-1-11-5': 'We extend our sample to include numerous sources not detected in the highest-energy spectral bins, but satisfying at the same time all other selection criteria.', '1810.03443-1-11-6': 'We use flux upper limits for the undetected sources and demonstrate that they are consistent with the distance-dependent spectral hardenings found in Sec. [REF].', '1810.03443-1-11-7': 'In Sec. [REF], we address another potential bias related to the fact that, generally, blazars detected at higher redshifts are mostly flat-spectrum radio quasars (FSRQs) while those at lower redshifts are mostly BL Lac type objects (BLLs).', '1810.03443-1-11-8': 'FSRQs might have intrinsic features in the spectrum which mimic the distance dependence of hardenings just because all observed FSRQs are distant.', '1810.03443-1-11-9': 'We remove FSRQs from our sample and consider BLLs separately; the effect is clearly seen in the subsample and the reduction of the statistical significance is fully explained by the decrease of the number of objects in the sample.', '1810.03443-1-11-10': 'In Sec. [REF], we discuss the effect of uncertainties in the energy determination, which may have important consequences for the estimation of the spectral shape at the highest energies for steeply falling spectra.', '1810.03443-1-11-11': 'We perform various tests and conclude that this effect is unlikely to affect our results.', '1810.03443-1-11-12': 'Finally, in Sec. [REF] we attempt to estimate a potential impact of possible wrong determination of distances to the sources.', '1810.03443-1-11-13': 'Since 2014, new redshift data removed 2 of 20 sources from the original sample of Ref. [CITATION].', '1810.03443-1-11-14': 'We artificially remove 1, 2 or 3 sources from our new sample of 28 objects and demonstrate that this does not change significantly our results, therefore suggesting that (a reasonable number of) erroneous redshifts cannot cause the effect we observe.', '1810.03443-1-11-15': 'Therefore, we conclude that none of the biases or uncertainties we are aware of can change the principal result of the paper, and continue in Sec. [REF] with a brief account of novelties and differences of this result in comparison with previous studies.', '1810.03443-1-12-0': 'While the outline of our study and a summary of results are given in this Introduction, we present a brief account of our conclusions and discuss possible approaches to the interpretation of our results in Sec. [REF].', '1810.03443-1-12-1': 'Appendix [REF] presents observed and best-fit deabsorbed spectra of 28 blazars entering our main sample.', '1810.03443-1-13-0': '# The data', '1810.03443-1-14-0': '## Selection criteria', '1810.03443-1-15-0': 'The purpose of the present study implies the use of high-quality gamma-ray spectra of sources located at large, confidently known distances.', '1810.03443-1-15-1': 'The obvious candidates are blazars with firmly measured redshifts.', '1810.03443-1-15-2': 'For various redshifts [MATH], the absorption due to [MATH] pair production becomes important at different energies of photons: for more distant sources, pair production affects the spectra at lower energies.', '1810.03443-1-15-3': 'The benchmark energy, at which the absorption becomes important, corresponds to the optical depth [MATH] and is denoted as [MATH] hereafter, [EQUATION] which determines [MATH] for a given absorption model, see Fig. [REF].', '1810.03443-1-16-0': 'To study the behaviour of a spectrum in the region of strong absorption, one needs observations at [MATH], which requires the use of different instruments for objects located at different redshifts; for large [MATH], [MATH] is typically dozens of GeV, and the relevant instrument is Fermi LAT, while for less distant objects, [MATH] is of order of several hundred GeV, and the data of IACTs should be used.', '1810.03443-1-16-1': 'Though the data are published in quite different ways, it is essential to treat them on equal footing.', '1810.03443-1-16-2': 'Here we discuss our general requirements governing selection of the objects, which will be implemented in the next two subsections for the Fermi-LAT and IACT data.', '1810.03443-1-17-0': 'Redshift criteria.', '1810.03443-1-17-1': 'Incorrectly determined or uncertain redshifts may hurt any of blazar studies.', '1810.03443-1-17-2': 'This happens quite often, especially for BLLs whose spectra do not possess strong emission lines.', '1810.03443-1-17-3': 'We require a firm spectroscopic redshift for the objects included into our sample.', '1810.03443-1-17-4': 'More specifically, we start from a preselected sample of gamma-ray blazars and check, for every object, the relevant redshift information and references in the NASA/IPAC Extragalactic Database (NED).', '1810.03443-1-17-5': 'The redshifts are accepted if they satisfy (R1) and one of (R2A), (R2B), (R2C) criteria:', '1810.03443-1-18-0': '(R1) the redshift is spectroscopic (not photometric)', '1810.03443-1-19-0': 'AND', '1810.03443-1-20-0': 'EITHER (R2A) the redshift is determined in a dedicated study (if several dedicated studies give different results, the latest one is used),', '1810.03443-1-21-0': 'OR (R2B) the redshift quoted in NED is determined in 2dF Galaxy Redshift Survey [CITATION] or 6dF Galaxy Survey [CITATION],', '1810.03443-1-22-0': 'OR (R2C) the redshift quoted in NED is determined in the Sloan Digital Sky Survey (SDSS), the result is unique and does not change from one release to another.', '1810.03443-1-23-0': 'These criteria are based on the previous experience in the use of redshift data and are of course ad hoc ones.', '1810.03443-1-23-1': 'Their application removes many uncertain redshifts.', '1810.03443-1-24-0': 'Spectral criteria.', '1810.03443-1-24-1': 'In our work, we use binned gamma-ray spectra, because only this information is publicly available for the objects observed by IACTs.', '1810.03443-1-24-2': 'We impose the following criteria for the spectra:', '1810.03443-1-25-0': '(S1) information for at least 5 energy bins is available;', '1810.03443-1-26-0': '(S2) the lowest energy [MATH] of the last spectral bin satisfies [MATH] AND the lowest energy of the third from below spectral bin is below [MATH].', '1810.03443-1-27-0': 'These criteria are aimed at the reconstruction of spectra at [MATH] both below and above [MATH] with reasonable accuracy.', '1810.03443-1-28-0': 'Several comments are in order.', '1810.03443-1-28-1': 'Firstly, (S2) depends on the assumed absorption model, and therefore the samples we use in Sec. [REF], when studying the sensitivity of our results to the choice of this model, are slightly different from one to another.', '1810.03443-1-28-2': 'Secondly, while we require a nonzero measurement in the last bin for our main study, we allow for an upper limit when studying potential effects of the Malmquist bias in Sec. [REF].', '1810.03443-1-29-0': '## IACT subsample', '1810.03443-1-30-0': 'To construct the sample of blazars observed by IACTs, we start with the database consolidating results published by different observatories, the TeVCat online source catalog [CITATION].', '1810.03443-1-30-1': 'The preselected sample includes objects classified as blazars there (classes "HBL", "\'IBL", "LBL", "BL Lac (class uncertain)", "FSRQ" and "blazar"), 68 objects in total as of summer 2017.', '1810.03443-1-30-2': 'Then, for each object in the sample, we checked the redshift selection rules (R1) and ((R2A) or (R2B) or (R2C)), using references from NED, and availability of gamma-ray spectra satisfying (S1), using references from TeVCat.', '1810.03443-1-31-0': 'Some objects have been observed several times, often by multiple instruments.', '1810.03443-1-31-1': 'To avoid double counting of information, which may artificially increase or dilute observed effects, we use only one observation for each source.', '1810.03443-1-31-2': 'It is chosen on the basis of better statistics (including the number of available spectral bins) and of better coverage of the energy ranges both below and above [MATH].', '1810.03443-1-31-3': 'Because of strong variability of many blazars, we never combine any two spectra from observations performed at different epochs to enlarge the energy coverage.', '1810.03443-1-31-4': "We also never combine Fermi-LAT spectra with those obtained from IACT observations because of potential systematic differences between the instruments' energy scales.", '1810.03443-1-31-5': 'These requirements represent a major refinement with respect to some of preceding studies.', '1810.03443-1-32-0': 'The next step requires to fix the absorption model.', '1810.03443-1-32-1': 'We use the most recent EBL model [CITATION] as our baseline low-absorption model; when variations of the model are considered (Sec. [REF]), this step is repeated.', '1810.03443-1-32-2': 'At this step, we calculate the value of [MATH] for every source and check the condition (S2).', '1810.03443-1-32-3': 'We are left with 22 objects which, together with the sources observed by Fermi LAT, are listed in Table [REF].', '1810.03443-1-33-0': '## Fermi-LAT subsample', '1810.03443-1-34-0': 'For blazars observed by Fermi LAT, our starting point is the 3rd Fermi-LAT Catalog of High-Energy Sources, 3FHL [CITATION], presenting a list of sources detected confidently above 10 GeV.', '1810.03443-1-34-1': 'From the catalog, we select 1212 objects classified as blazars (classes "BLL", "bll", "FSRQ", "fsrq", "bcu").', '1810.03443-1-34-2': 'Since the redshifts presented in the catalog may be doubtful or erroneous for distant sources, we check them manually to satisfy (R1) and ((R2A) or (R2B) or (R2C)) criteria, using references from NED, and apply the additional constraint [MATH], because for less distant sources, [MATH] GeV is beyond the limit of reasonable sensitivity of Fermi LAT.', '1810.03443-1-34-3': 'This procedure results in a list of 309 blazars.', '1810.03443-1-34-4': 'For each of them, we use publicly available Fermi-LAT [CITATION] data to construct binned spectra satisfying (S1) automatically.', '1810.03443-1-34-5': 'To this end, we use Fermi Pass 8 [CITATION] Release 2 data (version 302).', '1810.03443-1-34-6': 'We use "SOURCE" class events with version 6 instrumental response functions "P8R2[MATH]SOURCE[MATH]V6", recorded between August 4, 2008, and February 26, 2018 (Mission Elapsed Time interval 239557417 - 541338374).', '1810.03443-1-34-7': 'The data are processed with the standard "gtlike" routine from Fermi Science Tools v10r0p5.', '1810.03443-1-34-8': 'We impose standard Fermi-LAT cuts for point-source analysis including the Earth zenith angle cut ([MATH]).', '1810.03443-1-34-9': 'The background models used are "gll[MATH]iem[MATH]v06.fits" for the Galactic component and "iso[MATH]P8R2[MATH]SOURCE[MATH]V6[MATH]v06.txt" for the isotropic one.', '1810.03443-1-34-10': 'For most of the sources (305 of 309), the coordinates of the source are taken from the 3rd Fermi-LAT Source Catalog, 3FGL [CITATION].', '1810.03443-1-34-11': 'However, not all 3FHL sources have 3FGL counterparts, and for the remaining 4 sources, we used 3FHL coordinates.', '1810.03443-1-35-0': 'The observed spectrum was reconstructed for 8 energy bins, each of 03 dex, for [MATH] GeV.', '1810.03443-1-35-1': 'The lower energy limit of 2 GeV was chosen to cut the inverse-Compton peak energy range in order to improve broken power-law fits used in our subsequent analysis.', '1810.03443-1-36-0': 'At the next step, we fix the absorption model and apply the criterion (S2) to the obtained spectra in the same way as it was done for the IACT subsample.', '1810.03443-1-36-1': 'A measurement of the flux in a spectral bin is included if the corresponding number of source photons was nonzero (99% CL).', '1810.03443-1-36-2': 'Only 6 of 309 blazars satisfy (S2) and enter our final sample.', '1810.03443-1-36-3': 'Other sources, however, are used in the Malmquist-bias tests described in Sec. [REF].', '1810.03443-1-37-0': '## The sample', '1810.03443-1-38-0': 'The list of 28 objects selected in the way described above, Sec. [REF], [REF], is presented in Table [REF].', '1810.03443-1-39-0': 'For IACT observations, references for binned spectra used in this work are given there.', '1810.03443-1-39-1': 'We also list classes of the blazars (BLL or FSRQ) as determined in the catalogs used to compile the samples, TeVCat and 3FHL.', '1810.03443-1-40-0': 'Figure [REF]', '1810.03443-1-41-0': 'represents the distribution of blazars in our main sample in redshift.', '1810.03443-1-41-1': 'One may note that, though BLLs dominate at short and moderate distances and FSRQ are in general farther away, both classes cover large ranges of redshifts.', '1810.03443-1-41-2': 'We will return to this subject in Sec. [REF].', '1810.03443-1-42-0': '# Data analysis and results', '1810.03443-1-43-0': 'The key point of our study is the use of blazar spectra corrected for the pair production in a minimal-absorption model.', '1810.03443-1-43-1': 'Since, for IACT observations, only binned spectra are available, we adopt our analysis procedure for this case even for Fermi LAT, in order to process all observations in a uniform way.', '1810.03443-1-43-2': 'In contrast with many previous studies, including our work [CITATION], we do not use bin-by-bin deabsorption because it can introduce systematic biases in the highest-energy bins for the following reasons.', '1810.03443-1-43-3': 'The optical depth [MATH] is a function of the incoming photon energy [MATH] and the source redshift [MATH].', '1810.03443-1-43-4': 'It is important to note that the absorption is not uniform within the bin: photons of higher energy are absorbed stronger.', '1810.03443-1-43-5': 'For the energies and distances at which the absorption is significant, this effect may affect strongly the result.', '1810.03443-1-43-6': 'It can be taken into account if the shape of the observed spectrum within the bin is known, as we have done in Ref. [CITATION]; however, in practice, statistical uncertainties in the highest energy bins often make its determination unreliable.', '1810.03443-1-44-0': 'For the present study, we start from the assumption about the intrinsic spectrum of a blazar, for which we use a broken power law with the break at [MATH].', '1810.03443-1-44-1': 'Recall that this energy depends on the redshift of the source [MATH], cf. Fig. [REF], and therefore we assume breaks at different energies for different sources.', '1810.03443-1-44-2': 'Then we account for the absorption with the selected EBL model, integrate the obtained spectrum over the energy bins and compare resulted bin-by-bin fluxes with the observed data points.', '1810.03443-1-44-3': 'In this way, we fit the data with the three parameters of the intrinsic spectrum, that is the overall normalization and two spectral indices.', '1810.03443-1-44-4': 'We denote as [MATH] the difference between spectral indices at [MATH] and [MATH] in this best-fit assumed spectrum.', '1810.03443-1-44-5': 'By construction, nonzero [MATH] would indicate the presence of distance-dependent breaks because the assumed break position [MATH] is redshift-dependent.', '1810.03443-1-45-0': 'Figure [REF]', '1810.03443-1-46-0': 'demonstrates the main result of this work: the presence of distance-dependent hardenings in deabsorbed spectra.', '1810.03443-1-46-1': 'The assumption of [MATH] results in [MATH] for 28 degrees of freedom, corresponding to the probability [MATH] to occur as a result of a random fluctuation.', '1810.03443-1-46-2': 'This corresponds to the exclusion of the hypothesis of the absence of distance-dependent breaks at 4.5 standard deviations (recall the footnote at p. sigma-footnote for our conventions).', '1810.03443-1-47-0': 'This statistical study excludes the absence of distance-dependent spectral hardenings.', '1810.03443-1-47-1': 'Next, we check that this result is not mimicked by hardenings at a certain energy [MATH], common for all sources in the sample.', '1810.03443-1-47-2': 'To this end, we perform two additional tests.', '1810.03443-1-47-3': 'First, we make the break energy [MATH] a free parameter of the fit, so that it is adjusted independently for every source.', '1810.03443-1-47-4': 'Figure [REF]', '1810.03443-1-48-0': 'compares the fitted break positions [MATH] with the value of [MATH]; we see that they agree well to each other: averaged over the sample, [MATH].', '1810.03443-1-48-1': 'For a combined fit of all 28 spectra with the hypothesis of breaks at [MATH], we obtain [MATH]d.o.f.[MATH], while it is 0.73 for the case when [MATH] is a free parameter.', '1810.03443-1-48-2': 'The second test consists of fitting all the spectra with breaks at a uniformly fixed energy [MATH] (we consider values 100 GeV[MATH] TeV).', '1810.03443-1-48-3': 'Figure [REF]', '1810.03443-1-49-0': 'presents [MATH]d.o.f. for this analysis as a function of the assumed break energy [MATH], together with the results of the first test.', '1810.03443-1-49-1': 'We see that the fits with distance-independent breaks are worse than the fit with breaks at [MATH] for any fixed assumed break energy.', '1810.03443-1-50-0': '# Discussion', '1810.03443-1-51-0': 'In this section, we perform a study of possible systematic effects affecting our results.', '1810.03443-1-51-1': 'We will see that the effect we observed cannot be explained by any potential bias or systematics we are aware of.', '1810.03443-1-51-2': 'Then we compare our result with those of previous studies.', '1810.03443-1-52-0': '## Systematic uncertainties', '1810.03443-1-53-0': '### Absorption models', '1810.03443-1-54-0': 'The result of the present study, that is the presence of distance-dependent features in deabsorbed spectra precisely at the energies for which the correction for absorption becomes important, may indicate some problems with the absorption model used.', '1810.03443-1-54-1': 'For this study, we used the most recent published model by Korochkin and Rubtsov (2018), Ref. [CITATION].', '1810.03443-1-54-2': 'In this section, we repeat our study with several other representative models to see that the effect we found is present independently of the choice of the model.', '1810.03443-1-54-3': 'Besides the baseline model of Ref. [CITATION], we considered the following three models:', '1810.03443-1-55-0': '(i) Gilmore et al. (2012) fixed [CITATION], used in our previous study [CITATION];', '1810.03443-1-56-0': '(ii) Franceschini et al. (2017) [CITATION];', '1810.03443-1-57-0': '(iii) a toy high-absorption model normalized to the most recent direct observations of EBL [CITATION].', '1810.03443-1-58-0': 'The latter model was obtained by scaling the model (ii) by a multiplication factor fitted to the data of Refs. [CITATION].', '1810.03443-1-58-1': 'These two independent studies used different techniques to distinguish the extragalactic background light from the foreground contribution: Ref. [CITATION] used spectral templates, which are different for EBL and for foregrounds, while Ref. [CITATION] benefited from observations of an absorbing cloud to separate foregrounds experimentally.', '1810.03443-1-58-2': 'Figure [REF]', '1810.03443-1-59-0': 'presents 15 data points of Ref. [CITATION] and a single data point of Ref. [CITATION] together.', '1810.03443-1-59-1': 'All the points, corresponding to different background-light wavelengths, are fitted nicely by the intensity of the model [CITATION] multiplied by a wavelength-independent factor [MATH].', '1810.03443-1-59-2': 'For the toy high-absorption model we therefore take the model of Ref. [CITATION] uniformly upscaled by 3.35.', '1810.03443-1-59-3': 'This may be considered as an upper limit on the opacity.', '1810.03443-1-60-0': 'As it has been discussed above, the energies [MATH], at which the absorption becomes important, depend on the EBL model, and therefore the samples of blazars selected for the study are different for different models.', '1810.03443-1-60-1': 'For the models [CITATION], some blazars from Table [REF] do not enter the sample (marked in Table [REF]).', '1810.03443-1-60-2': 'For the toy high-absorption model, the list of 45 sources in the sample is available from the authors by request.', '1810.03443-1-61-0': 'We present the resulting significance of the observed distance dependence of spectral hardenings for various absorption models in Table [REF].', '1810.03443-1-62-0': 'We see that for the low-absorption models, the significance remains very similar to that obtained for the model of Ref. [CITATION], [MATH].', '1810.03443-1-62-1': 'Clearly, a 3.35 times increase in the opacity, suggested by Refs. [CITATION], had to result in strengthening the effect, and we quantified this consistently within our approach.', '1810.03443-1-63-0': '### The Malmquist bias', '1810.03443-1-64-0': 'Inclusion of sources to our sample is limited by observational capabilities of the instruments through the (S2) criterion of Sec. [REF].', '1810.03443-1-64-1': 'Therefore, sources intrinsically more and more luminous are included at larger distances.', '1810.03443-1-64-2': 'In principle, sources may have spectral features correlated with their luminosity, and the flux selection might mimic the observed effect (a particular model for this is not known).', '1810.03443-1-64-3': 'In this hypothetical scenarios, weaker objects not detected above [MATH] would not have distance-dependent spectral hardenings which we observe for the sources included in our sample.', '1810.03443-1-64-4': 'This suggests a way to test this possible bias by a study of objects not detected above [MATH].', '1810.03443-1-65-0': 'To this end, we consider the same pre-selected 3FHL sample described in Sec. [REF] but relax the condition (S2) of Sec. [REF].', '1810.03443-1-65-1': 'Note that we need a flux-limited sample for this study and therefore perform it for Fermi-LAT sources only, not including the sources observed only by IACTs.', '1810.03443-1-65-2': 'We obtain a sample of 31 blazars with redshifts between 0.354 and 2.534, which are detected significantly in spectral bin containing [MATH], but not above.', '1810.03443-1-65-3': 'For each of the objects, we determine upper limits on the strength [MATH] of assumed breaks at [MATH] for these sources in the way described in Sec. Ref. [REF].', '1810.03443-1-66-0': 'The standard Fermi-LAT routine gtlike does not produce reliable upper limits for weak fluxes.', '1810.03443-1-66-1': 'For our purposes, we first use the gtsrcprob routine to calculate the weight of each photon: the probability to originate from a given source.', '1810.03443-1-66-2': 'The sum of weights is counted and gives the estimated numbers of photons from the source and from backgrounds.', '1810.03443-1-66-3': 'The number of source photons is determined as the sum of weights and is typically very small for undetected sources.', '1810.03443-1-66-4': 'Since the background photons are considered separately, this number may be treated as the number of signal events with zero background.', '1810.03443-1-66-5': 'We round this number to an integer (0 in most cases) and estimate the Poisson-based 95% CL upper limit on the flux in the last bin by division to the exposure calculated with the gtexpcube routine.', '1810.03443-1-67-0': 'The results of the calculation of these upper limits on [MATH] are presented in Fig. [REF].', '1810.03443-1-68-0': 'If distance-dependent hardenings were observed in our sample because of the Malmquist bias, then the upper limits on [MATH] would take randomly distributed positive and negative values.', '1810.03443-1-68-1': 'We see that this is not the case, and therefore our main result cannot be explained by the Malmquist bias.', '1810.03443-1-69-0': '### Source classes', '1810.03443-1-70-0': 'Gamma-ray sources observed at large distances, the blazars, belong to various classes.', '1810.03443-1-70-1': 'In this section, we consider a possibility that distant sources are different from nearby ones, and hence may have systematically different spectral features.', '1810.03443-1-70-2': 'In general, broadband spectral energy distributions (SEDs) of blazars have two wide bumps, often associated with the synchrotron and inverse-Compton emissions.', '1810.03443-1-70-3': 'Relative positions of the bumps are correlated, which may be explained if the same population of relativistic electrons is responsible for both processes.', '1810.03443-1-70-4': 'Therefore, the SEDs are, to the first approximation, characterized by the position of the synchrotron peak so that all blazars form "the blazar sequence" [CITATION].', '1810.03443-1-70-5': 'Flat-spectrum radio quasars (FSRQs) have the synchrotron peak frequency, [MATH], in the radio band, while blazars with [MATH] from infrared to X-ray bands represent various subclasses of BL Lac type objects (BLLs).', '1810.03443-1-70-6': 'On average, FSRQs are more powerful and less numerous, which results in a potential bias in flux-limited blazar samples: bright FSRQs are seen at large distances while more numerous weaker BLLs dominate the sample at relatively low redshifts.', '1810.03443-1-70-7': 'This bias might indeed affect our observations because intrinsic absorption in (distant) FSRQs might introduce spectral features not present in (nearby) BLLs.', '1810.03443-1-71-0': 'To demonstrate that this is not the case, we remove FSRQs from our sample (as it was previously done in several other studies of the gamma-ray absorption, e.g. Ref. [CITATION]).', '1810.03443-1-71-1': 'The classification of FSRQ or BLL given in Table [REF] is based on the information given in the 3FHL catalog for Fermi-LAT sources and in TeVCat for sources observed by IACTs.', '1810.03443-1-71-2': 'Since the classification is uncertain and various authors may use different criteria for claiming a source is a FSRQ or a BLL, we obtained approximate values of [MATH] for objects in the sample, based on SEDs available in NED and also on Refs. [CITATION].', '1810.03443-1-71-3': 'We counted objects with [MATH] Hz as FSRQs and others as BLLs, and found that this was in accordance with the classification given in Table [REF] for all objects.', '1810.03443-1-71-4': 'We note that, while FSRQs are indeed concentrated at larger distances, BLLs cover a wide range of redshifts and the sample of BLLs only may be used for the analysis described in Sec. [REF].', '1810.03443-1-71-5': 'The sample of BLLs contains 19 objects.', '1810.03443-1-71-6': 'The significance of distance-dependent spectral hardenings for BLLs only is [MATH], and this reduction in significance with respect to the main analysis is in accordance with the statistical depletion of the sample.', '1810.03443-1-71-7': 'Fig. [REF]', '1810.03443-1-72-0': 'demonstrates that the same quantity [MATH] does not depend on the synchrotron peak frequency.', '1810.03443-1-72-1': 'The analyses therefore support the conclusion that potential BLL/FSRQ selection effects described in the beginning of this section are not responsible for the effect we observe.', '1810.03443-1-72-2': 'Indirectly, this may suggest that the intrinsic absorption in FSRQs is a subdominant effect with respect to the absorption on EBL.', '1810.03443-1-73-0': '### Tail of the falling spectrum', '1810.03443-1-74-0': 'At the energies under discussion, spectra are determined from event-by-event information about observed individual photons.', '1810.03443-1-74-1': 'Determination of the photon energies is subject to statistical and systematic uncertainties.', '1810.03443-1-74-2': 'Statistical uncertainties are usually taken into account in the spectrum reconstruction procedures; however, they might be underestimated, while systematics may affect the overall energy scale of an instrument.', '1810.03443-1-74-3': 'We are interested in the highest energies, where, because of steeply falling spectra, the flux is often estimated from observations of a few photons.', '1810.03443-1-74-4': 'In the case of a systematic shift of their energies upwards, or if the statistical uncertainty is larger than expected, overestimation of the energies of these few photons may affect significantly the shape of the spectrum.', '1810.03443-1-75-0': 'We first address potential systematic errors.', '1810.03443-1-75-1': 'These are instrument-dependent, and we take advantage of having objects observed by four different instruments in our sample.', '1810.03443-1-75-2': 'Table [REF]', '1810.03443-1-76-0': 'presents significances of the observed distance-dependent spectral hardenings in subsamples of objects with data of one of the instruments dropped.', '1810.03443-1-76-1': 'The observed significance is stable and agrees well with the expectations from the sample size.', '1810.03443-1-76-2': 'Therefore, only a coherent upward systematic error in the energy estimation by the three IACTs and Fermi LAT may result in the effect we found.', '1810.03443-1-76-3': 'We note that this situation is unlikely.', '1810.03443-1-77-0': 'We turn now to the possibility of underestimated statistical errors, which may result in occasional overestimation of energies of particular photons.', '1810.03443-1-77-1': 'In the case of the published spectra, this effect is normally already taken into account by the observing collaboration by making use of the "unfolding" procedure.', '1810.03443-1-77-2': 'In any case, this potential systematics would be strongly suppressed for the objects where the spectral hardening happens at energies for which a sufficient number of photons is observed.', '1810.03443-1-77-3': 'To select those from our sample, we replace the (S2) selection criterion from Sec. [REF] by a much stronger one, requiring two (instead of one) spectral bins above [MATH].', '1810.03443-1-77-4': 'Only 18 of 28 objects in our sample satisfy these criterion.', '1810.03443-1-77-5': 'Performing our analysis for these 18 objects, we find the significance of [MATH] for this purified sample.', '1810.03443-1-77-6': 'This disfavours the relation of our observation to potentially underestimated errors in the energy determination; moreover, the fact that a purified, though depleted, sample exhibits a stronger effect gives a serious support to all our conclusions.', '1810.03443-1-78-0': '### Redshifts', '1810.03443-1-79-0': 'Our study of distance-dependent spectral features is sensitive to correct determination of distances to the sources.', '1810.03443-1-79-1': 'However, redshifts of distant blazars are not always determined unambiguously.', '1810.03443-1-79-2': 'In particular (see Sec. [REF]), redshifts of 2 of 20 objects in the sample we used for our previous study [CITATION] in 2014 were found to be unreliable by 2018, and the two objects left our present sample.', '1810.03443-1-79-3': 'Though for the present sample we performed a careful selection of objects based on the quality of their redshifts, see criteria (R1), (R2x) in Sec. [REF], one might imagine that some of the 28 redshifts would still appear wrong in future analyses.', '1810.03443-1-79-4': 'To understand how this may affect our results, we artificially removed 1, 2 or 3 objects in all possible combinations from the sample and repeated our analysis with the reduced samples.', '1810.03443-1-79-5': 'The lowest significance obtained with 1 object removed is [MATH]; with 2 objects removed it is [MATH] and with three objects removed it is [MATH].', '1810.03443-1-79-6': 'Figure [REF]', '1810.03443-1-80-0': 'presents the distribution of significances among samples with 2 objects removed.', '1810.03443-1-80-1': 'This simple procedure, which mimics potential impact of would-be incorrect redshifts, demonstrates that the effect we found is not saturated by a few high-redshift sources and would not be ruined if their redshifts are found to be incorrect.', '1810.03443-1-81-0': '## Comparison to previous studies', '1810.03443-1-82-0': 'In this section, we give a detailed comparison of our methods and results with those of key previous studies of the gamma-ray opacity of the Universe with ensembles of blazar spectra.', '1810.03443-1-83-0': '### Rubtsov and Troitsky (2014) [CITATION]', '1810.03443-1-84-0': 'The present work uses the same approach as our previous work [CITATION] and confirms its results.', '1810.03443-1-84-1': 'Advantages of the present work, besides the use of new observational data and a more detailed report on systematics, are:', '1810.03443-1-85-0': '(i) pre-selection of Fermi-LAT sources with the 3FHL catalog and the use of Pass 8 data processing;', '1810.03443-1-86-0': '(ii) use of the new conservative EBL models [CITATION];', '1810.03443-1-87-0': '(iii) imposing strict criteria on the redshift quality;', '1810.03443-1-88-0': '(iv) use of a more conservative method which is not based on the bin-by-bin deabsorption.', '1810.03443-1-89-0': 'In particular, these changes resulted in the removal of 4 sources from the sample of Ref. [CITATION]: redshifts of 2 objects (RGB J1448[MATH]361 and 1ES 0347[MATH]121) did not satisfy our new quality criteria; the reported value of the redshift of B3 1307[MATH]433 changed in such a way that the object no longer satisfies the (S2) criterion; for PKS J0730[MATH]1141, new Pass 8 Fermi-LAT reconstruction does not result in a significant detection at [MATH], contrary to the Pass 7REP (V15) used in Ref. [CITATION].', '1810.03443-1-89-1': 'This removed the most distant objects in the sample which, together with a more conservative analysis method, resulted in a considerable reduction of significance of the result (was 12.4[MATH], now 4.5[MATH]).', '1810.03443-1-89-2': 'The main result remains qualitatively the same.', '1810.03443-1-90-0': '### Horns and Meyer (2012) [CITATION]', '1810.03443-1-91-0': 'Ref. [CITATION] was the first study addressing quantitatively spectral hardenings in the ensemble of distant gamma-ray sources.', '1810.03443-1-92-0': '(1) Both Ref. [CITATION] and our study address the distance dependence of upward spectral breaks.', '1810.03443-1-93-0': '(2) Only objects observed by IACTs ([MATH]) were included in the sample of Ref. [CITATION].', '1810.03443-1-93-1': 'They used a stricter requirement of detection at optical depths [MATH], resulting in the sample of 7 objects only, while we require [MATH] and include objects for which the hardening was not significantly detected.', '1810.03443-1-94-0': '(3) In Ref. [CITATION], bin-by-bin deabsorption was used but the correction for the mean energy in the bin for deabsorption was not implemented.', '1810.03443-1-95-0': '### Fermi-LAT collaboration (2012) [CITATION]', '1810.03443-1-96-0': 'The interesting work [CITATION] presented an analysis of the ensemble of BLLs detected by Fermi LAT at high redshifts and, for the first time, presented a detection of the EBL absorption in stacked spectra of these distant sources.', '1810.03443-1-96-1': 'The ensemble of 150 BLLs with redshifts [MATH] was split into three distance bins, and intrinsic spectra were extrapolated from Fermi-LAT observations at low energies.', '1810.03443-1-96-2': 'The EBL absorption correction was subsequently determined and stacked within the three redshift bins.', '1810.03443-1-97-0': 'As we know from our present study, only a few blazars (6 with our criteria) have been confidently observed by Fermi LAT above [MATH].', '1810.03443-1-97-1': 'Therefore, only the start of the absorption feature was seen in the stacked sample, and the constraints on the amount of absorption were not very tight.', '1810.03443-1-97-2': 'For instance, they were expressed as the dependence of [MATH] for [MATH], see Fig. 1 of Ref. [CITATION], where the width of the allowed band was [MATH] at 95% (68%) CL, corresponding to the uncertainty of a factor of a few in [MATH].', '1810.03443-1-97-3': 'While available absorption models are consistent with the upper part of the band, a factor of 2 to 3 lower absorption is allowed for all energies.', '1810.03443-1-97-4': 'This is in a good agreement with our results.', '1810.03443-1-98-0': '### Biteau and Williams (2015) [CITATION]', '1810.03443-1-99-0': 'Ref. [CITATION] considered 106 spectra of 38 sources observed by IACTs.', '1810.03443-1-99-1': 'Every spectrum was deabsorbed bin-by-bin, without applying the correction for the mean energy of the bin, and the fitted by a model spectrum for which a concave shape was assumed.', '1810.03443-1-99-2': 'This study did not reveal any anomaly in the absorption on EBL.', '1810.03443-1-100-0': 'We argue that nonobservation of the absorption anomaly in Ref. [CITATION] does not contradict to the results of the present work.', '1810.03443-1-100-1': 'The difference in conclusions is related both to the sample of source spectra used and to the assumptions on the intrinsic spectra.', '1810.03443-1-100-2': 'We point out the following differences:', '1810.03443-1-101-0': 'In the sample of Ref. [CITATION], a different set of sources is used, dominated by nearby ones, while in our study, the anomaly reveals itself stronger for distant sources.', '1810.03443-1-101-1': 'The dominance of nearby sources in Ref. [CITATION] was enhanced by the use of multiple spectra per source, which gave higher statistical weights to better studied nearby objects, see Figure [REF].', '1810.03443-1-101-2': 'The individual-source analysis of Ref. [CITATION] assumed explicitly a concave shape for the deabsorbed spectrum, thus not allowing for the hardenings we study here by construction, see Figure [REF].', '1810.03443-1-101-3': 'In fact, in most cases (91 of 106 spectra, in particular for all sources with [MATH]), the best-fit model corresponded to a power law, that is to the margin of the allowed concave spectra: convex shapes, which give better fits for many of the spectra in our study, were not allowed.', '1810.03443-1-102-0': 'We conclude that the methods chosen by Biteau and Williams in Ref. [CITATION] are insensitive to the anomalous effect we found in Ref. [CITATION] and confirmed here, therefore Ref. [CITATION] and our work are not in a contradiction.', '1810.03443-1-103-0': '### Galanti et al. (2015) [CITATION]', '1810.03443-1-104-0': 'It is interesting to note that the distance-dependent hardenings can still be noticed with the analysis based on a single power-law fit.', '1810.03443-1-104-1': 'Galanti et al. (2015) [CITATION] considered a sample of 39 blazars observed by IACTs ([MATH]), for all tested opacities.', '1810.03443-1-104-2': 'Deabsorbed spectra were described by single power laws, and the distance dependence of the spectral index was studied.', '1810.03443-1-104-3': 'Though this approach does not constructively include spectral breaks, deabsorbed spectra with upward breaks look harder when fitted by a power-law, and this hardness was found to be redshift-dependent.', '1810.03443-1-104-4': 'Therefore, results of Ref. [CITATION] are in agreement with Ref. [CITATION] and the present work.', '1810.03443-1-105-0': '# Conclusions', '1810.03443-1-106-0': 'In the present work, we readdress the problem of the anomalous transparency of the Universe for high-energy gamma rays, the most significant indication to which [CITATION] was based on an unphysical distance dependence of spectral hardenings in deabsorbed spectra for an ensemble of blazars.', '1810.03443-1-106-1': 'By making use of a robust analysis procedure which avoids bin-by-bin deabsorption, of new gamma-ray and redshift data and of most recent EBL models, we confirm the presence of the anomalous distance dependence with the statistical significance of [MATH].', '1810.03443-1-106-2': 'Compared to Ref. [CITATION], the range of the distances spanned by our sample shrinked considerably because we imposed strict criteria on the redshift quality.', '1810.03443-1-106-3': 'Together with other changes discussed above, this explains the reduction of the significance with respect to Ref. [CITATION].', '1810.03443-1-106-4': 'We discuss potential biases and systematic errors and, by performig various tests, conclude that they cannot be responsible for the effect we observe.', '1810.03443-1-107-0': 'Interpretations of the anomalous transparency of the Universe have been widely discussed in the literature, see e.g. Ref. [CITATION] for a brief review and a list of references.', '1810.03443-1-107-1': 'The simplest possible culprit for unphysical distance-dependent hardenings might be an overestimated EBL intensity.', '1810.03443-1-107-2': 'However, reduction of the intensity below the levels predicted by the models we use might be dangerous because the model intensities are very close to the lower limits from galaxy counting.', '1810.03443-1-107-3': 'With the help of the Markov-chain optimization behind the model [CITATION], we plan to attempt to change the EBL model in such a way that the effect we observe in the present work is reduced and to understand how much in conflict this would be with underlying astrophysical data.', '1810.03443-1-107-4': 'This will be a subject of a forthcoming work.', '1810.03443-1-108-0': 'A number of proposals attempted to explain the apparent anomalous transparency of the Universe by adding secondary emission to that arriving directly from the source.', '1810.03443-1-108-1': 'If these secondary photons are born relatively close to the observer, they do not have time to produce [MATH] pairs and arrive unattenuated.', '1810.03443-1-108-2': 'With respect to the origin of this secondary emission, it is useful to distinguish the cases when it is caused by electromagnetic cascades [CITATION] or from interactions of hadronic cosmic-ray particles [CITATION].', '1810.03443-1-108-3': 'The main characteristic feature of both approaches is that they require extremely low magnetic fields all along the propagation path of the cascade, otherwise secondary photons would not point to the sources because of deflections of their progenitor charged particles in the cascade.', '1810.03443-1-109-0': 'Other proposals require modification of the photon interactions.', '1810.03443-1-109-1': 'We note that the pair-production cross section has been well determined experimentally, and the only possibility to change it in the kinematic regimes relevant for our study is to allow for small deviations from the Lorentz invariance.', '1810.03443-1-109-2': 'However, this change would affect also the development of photon-induced atmospheric showers [CITATION] making the work of IACTs impossible, so that the scenario, in which results of Sec. [REF] are explained by the Lorentz-invariance violation, is in fact excluded by the fact that some photons have been detected by these instruments.', '1810.03443-1-109-3': 'The remaining explanation invokes a hypothetical axion-like particle (ALP).', '1810.03443-1-109-4': 'In external magnetic fields, ALP mixes with photons [CITATION] and, since ALP does not attenuate on EBL, this makes it possible to detect photons from more distant sources.', '1810.03443-1-109-5': 'In one of the scenarios, this mixing takes place in intergalactic magnetic fields along the path from the source to the observer [CITATION], while in the other, a part of photons is converted to ALPs near the source and reconverted back to gamma rays in our neighbourhood (in the magnetic fields of galaxies, clusters and filaments) [CITATION].', '1810.03443-1-109-6': 'Observational consequences of the two scenarios are compared to each other and to data in Ref. [CITATION].', '1810.03443-1-109-7': 'The account of interactions of energetic photons with the axion-like particle allows one to explain consistently all our results without contradicting to any other experimental or observational data.', '1810.03443-1-110-0': '# Observed and best-fit intrinsic spectra of blazars in the main sample.', '1810.03443-1-111-0': 'Figures [REF]-[REF]', '1810.03443-1-112-0': 'present the observed and deabsorbed spectra of 28 blazars from our main sample, see Table [REF], together with values of [MATH] and break strengths [MATH] on which the main result of the paper, Fig. [REF], is based.', '1810.03443-1-113-0': 'This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.', '1810.03443-1-113-1': 'This research has made use of the TeVCat online source catalog [CITATION] and of the data and tools provided by the Fermi Science Support Center.', '1810.03443-1-114-0': 'We are indebted to Alberto Franceschini for interesting correspondence and for sharing detailed numerical tables of the absorption model [CITATION] with us; to Manuel Meyer for numerous interesting discussions and helpful comments; to Timur Dzhatdoev, Oleg Kalashev, Mikhail Kuznetsov, Maxim Libanov, Valery Rubakov and Igor Tkachev for interesting discussions.', '1810.03443-1-114-1': 'The work of AK and ST on constraining the intensity of EBL and its applications to astrophysical manifestations of axion-like particles was supported by the Russian Science Foundation, grant 18-12-00258.'}

{'1810.03443-2-0-0': '# Introduction', '1810.03443-2-1-0': 'It is well known since 1960s [CITATION] that energetic gamma rays produce electron-positron pairs on the extragalactic background light (EBL).', '1810.03443-2-1-1': 'Because of this process, gamma-ray flux from distant sources is attenuated severely and the propagation distance is limited.', '1810.03443-2-1-2': 'Very-high-energy (VHE) gamma rays (energy [MATH] GeV) scatter most efficiently on the infrared and visible background photons, and information on the intergalactic absorption may be deduced black from VHE observations of emitters located at cosmologically large distances.', '1810.03443-2-2-0': 'There exists some long-standing controversy in the determination of the EBL intensity, with direct measurements indicating consistently higher EBL than indirect constraints doblack.', '1810.03443-2-2-1': 'Direct observations are very difficult to carry out because of strong foreground contamination by the Zodiacal light and by the Galactic foreground.', '1810.03443-2-2-2': 'Nevertheless, numerous attempts gave coherent, though quite uncertain, results [CITATION].', '1810.03443-2-2-3': 'At the same time, there are two groups of methods that set lower and upper limits on the EBL.', '1810.03443-2-2-4': 'The first group is based on the deep field observations of the Universe.', '1810.03443-2-2-5': 'This approach uses the method of galactic counts and results in lower limits on the EBL intensity, given just by the sum of contributions from guaranteed light sources [CITATION].', '1810.03443-2-2-6': 'On the other hand, upper constraints come from gamma-ray observations of blazars [CITATION].', '1810.03443-2-2-7': 'As the VHE part of the blazars spectra is highly absorbed by the EBL, the requirement that the intrinsic spectrum is not too hard, or is a smooth extrapolation from lower energies, provides constraints on the attenuation and thus on the EBL.', '1810.03443-2-2-8': 'These gamma-ray upper limits assume standard absorption on the EBL and therefore cannot be used in a work testing possible deviations from this standard picture.', '1810.03443-2-2-9': 'Finally, numerical simulations include more theoretical input from the star formation rate history but point to a similar range of intensities as the galaxy-count methods.', '1810.03443-2-2-10': 'The situation remains unsettled: two independent observations, published in 2017 and using different sophisticated techniques to get rid of the foreground systematics [CITATION], point consistently to the EBL intensities considerably higher than the most recent theoretical models.', '1810.03443-2-2-11': 'For the purposes of the present study, we need a conservative (low-absorption) model and use that of Korochkin and Rubtsov (2018) [CITATION] as the most recent available benchmark for the low-EBL theoretical models.', '1810.03443-2-2-12': 'Variations of the absorption model will be discussed in the context of systematic uncertainties of our results.', '1810.03443-2-3-0': 'Distant blazars are observed in energetic gamma rays from the early days of the VHE astronomy.', '1810.03443-2-3-1': 'It has been pointed out long ago that some of them are seen from distances for which the optical depth with respect to the pair production on EBL is significant.', '1810.03443-2-3-2': 'This apparent tension with expectations was dubbed "the infrared/TeV crisis" [CITATION].', '1810.03443-2-3-3': 'However, subsequent improvements in EBL models demonstrated that the tension is not that strong.', '1810.03443-2-3-4': 'While apparent hardening of the intrinsic spectrum (corrected for the pair-production attenuation, that is "deabsorbed"), seen in several individual sources, might be related to physical conditions in particular objects, a successful model for these hardenings is missing [CITATION].', '1810.03443-2-4-0': 'The situation has been changed when the number of sources observed at large optical depths became sufficient for studies of their ensembles.', '1810.03443-2-4-1': 'The study of a sample of 7 sources observed by imaging atmospheric Cerenkov telescopes (IACTs) at optical depths [MATH] by Horns and Meyer [CITATION] suggested that the energy at which these hardenings happen is correlated with the distance to the source in such a way that the spectra become harder only when the correction for the pair-production absorption is significant.', '1810.03443-2-4-2': 'This correlation can hardly be caused by any physical reason and suggests that the optical depth is estimated incorrectly.', '1810.03443-2-4-3': 'The statistical significance of this "pair-production anomaly" was estimated at the [MATH] level', '1810.03443-2-5-0': 'The largest sample of sources detected at large optical depths ([MATH], 15 objects observed by IACTs and 5 objects observed by Fermi Large Area Telescope, LAT) has been considered [CITATION] in 2014.', '1810.03443-2-5-1': 'There, we first confirmed the existence of significant hardenings in deabsorbed spectra of a number of objects at the energies where the correction for pair production becomes important.', '1810.03443-2-5-2': 'We then considered the full sample of objects, including those with and without statistically significant hardenings, and fit their deabsorbed spectra with the broken power-law function, assuming the break at the energy [MATH], at which the pair-production optical depth [MATH] (these energies are different for sources at different redshifts).', '1810.03443-2-5-3': 'We found that the break strength, that is the difference between power-law indices below and above [MATH], grows with the source redshift [MATH].', '1810.03443-2-5-4': 'Assuming the growth linear in [MATH], the statistical significance of this distance dependence, again suggesting an unphysical origin of the hardenings and hence an incorrect account of the absorption, was found to be [MATH].', '1810.03443-2-6-0': 'All these results are of great importance not only for the gamma-ray astronomy, but also for other fields of physics and astrophysics.', '1810.03443-2-6-1': 'As it will be discussed below, overestimation of the absorption can hardly be understood without invoking new physical or astrophysical phenomena, and its most promising explanation requires the existence of new light particles beyond those described by the Standard Model of particle physics.', '1810.03443-2-7-0': 'Since 2014, several important improvements changed the field.', '1810.03443-2-7-1': 'Firstly, many new observations of distant VHE blazars have been performed by IACTs.', '1810.03443-2-7-2': 'Secondly, the Fermi-LAT team not only accumulated additional years of statistics, very important for faint distant VHE sources, but also issued the new "Pass 8" reconstruction [CITATION] improving the data processing.', '1810.03443-2-7-3': 'Another related novelty is the 3FHL catalog [CITATION] of hard-spectrum sources making pre-selection of the sample easier and more uniform than before.', '1810.03443-2-7-4': 'Thirdly, from the EBL side, new and improved conservative theoretical models have become available [CITATION], but at the same time new sophisticated observational studies strengthened the tension in the determination of the background radiation intensity [CITATION].', '1810.03443-2-7-5': 'Finally, for a number of objects, new information on their redshifts became available.', '1810.03443-2-7-6': 'The main purpose of this work is to revisit the claims of Refs. [CITATION] with an extended high-quality sample of sources, taking into account all the listed improvements and making use of a more conservative analysis method.', '1810.03443-2-7-7': 'We also present some tests demonstrating independence of our results from potential biases and their stability with respect to systematic uncertainties.', '1810.03443-2-7-8': 'The result of the present study favours the presence of black spectral features precisely at the energies for which the correction becomes important, though with a significance of only [MATH].', '1810.03443-2-7-9': 'black However, we do not confirm the distanblackce dependence of the break strengths.', '1810.03443-2-8-0': 'The rest of the paper is organized as follows.', '1810.03443-2-8-1': 'In Sec. [REF], we discuss gamma-ray data used in our study.', '1810.03443-2-8-2': 'Section [REF] summarizes general criteria for the sample construction.', '1810.03443-2-8-3': 'Section [REF] detalizes how they are implemented for the objects observed by IACTs, while Sec. [REF] discusses the construction of the Fermi-LAT subsample.', '1810.03443-2-8-4': 'We list the resulting sample of 31 sources and discuss its general properties in Sec. [REF].', '1810.03443-2-9-0': 'Section [REF] describes the data analysis procedure and presents the main results of the paper.', '1810.03443-2-9-1': 'We describe how the correction to the pair-production opacity is implemented and perform a combined analysis of the full sample of 31 sources.', '1810.03443-2-9-2': 'We fit all the spectra with absorbed broken power laws, assuming that the break energy [MATH] is a free parameter, and demonstrate that the fits suggest [MATH].', '1810.03443-2-9-3': 'Then, we repeat the fits with fixed [MATH] and demonstrate that indeed spectral features at the distance-dependent energies are slightly favoured.', '1810.03443-2-9-4': 'This result is discussed in detail in Sec. [REF].', '1810.03443-2-9-5': 'There, we first turn, in Sec. [REF], to potential biases and systematic uncertainties which might affect our result.', '1810.03443-2-9-6': 'In Sec. [REF], we discuss variations in the absorption model: besides the most recent model we use throughout the paper, Korochkin and Rubtsov (2018) [CITATION], we repeat our analysis with three other EBL models, namely: Gilmore et al. (2012) fixed [CITATION], which was used in our 2014 paper; Franceschini and Rodighiero (2017) [CITATION]; and a toy upper-limit model artificially normalized to the most recent direct observational data [CITATION] on the background-light intensity.', '1810.03443-2-9-7': 'Next, in Sec. [REF], we discuss a possible bias related to the fact that only brightest sources are seen at large distances, the Malmquist bias.', '1810.03443-2-9-8': 'We extend our sample to include numerous sources not detected in the highest-energy spectral bins, but satisfying at the same time all other selection criteria.', '1810.03443-2-9-9': 'We use flux upper limits for the undetected sources and demonstrate that they are consistent with the results of Sec. [REF].', '1810.03443-2-9-10': 'In Sec. [REF], we address another potential bias related to the fact that, generally, blazars detected at higher redshifts are mostly flat-spectrum radio quasars (FSRQs) while those at lower redshifts are mostly BL Lac type objects (BLLs).', '1810.03443-2-9-11': 'FSRQs might have intrinsic features in the spectrum which mimic the distance dependence just because all observed FSRQs are distant.', '1810.03443-2-9-12': 'We demonstrate that the problematic spectral features do not depend on the synchrotron peak frequency often used to distinguish between different classes of blazars.', '1810.03443-2-9-13': 'In Sec. [REF], we discuss the effect of uncertainties in the energy determination, which may have important consequences for the estimation of the spectral shape at the highest energies for steeply falling spectra.', '1810.03443-2-9-14': 'We perform various tests and conclude that this effect is unlikely to affect our results.', '1810.03443-2-9-15': 'Finally, in Sec. [REF] we attempt to estimate a potential impact of possible wrong determination of distances to the sources.', '1810.03443-2-9-16': 'Since 2014, new redshift data removed 4 black of 20 sources from the original sample of Ref. [CITATION].', '1810.03443-2-9-17': 'We randomly remove 1, 2 or 3 sources from our new sample of 31 black objects and demonstrate that, in most cases, this does not change significantly our results, therefore suggesting that (a reasonable number of) erroneous redshifts can hardly cause the effect we observe.', '1810.03443-2-9-18': 'Therefore, we conclude that none of the biases or uncertainties we are aware of may affect black the principal result of the paper, and continue in Sec. [REF] with a brief account of novelties and differences of this result in comparison with previous studies.', '1810.03443-2-10-0': 'While the outline of our study and a summary of results are given in this Introduction, we present a brief account of our conclusions and discuss possible approaches to the interpretation of our results in Sec. [REF].', '1810.03443-2-10-1': 'Appendix [REF] presents observed and best-fit deabsorbed spectra of 31 blazars entering our main sample.', '1810.03443-2-11-0': '# The data', '1810.03443-2-12-0': '## Selection criteria', '1810.03443-2-13-0': 'The purpose of the present study implies the use of high-quality gamma-ray spectra of sources located at large, confidently known distances.', '1810.03443-2-13-1': 'The obvious candidates are blazars with firmly measured redshifts.', '1810.03443-2-13-2': 'For various redshifts [MATH], the absorption due to [MATH] pair production becomes important at different energies of photons: for more distant sources, pair production affects the spectra at lower energies.', '1810.03443-2-13-3': 'The benchmark energy, at which the absorption becomes important, corresponds to the optical depth [MATH] and is denoted as [MATH] hereafter, [EQUATION] which determines [MATH] for a given absorption model, see Fig. [REF].', '1810.03443-2-14-0': 'To study the behaviour of a spectrum in the region of strong absorption, one needs observations at [MATH]; therefore, the use of different instruments is required for objects located at different redshifts : for large [MATH], [MATH] is typically dozens of GeV, and the relevant instrument is Fermi LAT, while for less distant objects, [MATH] is of order of several hundred GeV, and the data of IACTs should be used.', '1810.03443-2-14-1': 'Though the data are published in quite different ways, it is essential to treat them on equal footing.', '1810.03443-2-14-2': 'Here we discuss our general requirements governing selection of the objects, which will be implemented in the next two subsections for the Fermi-LAT and IACT data.', '1810.03443-2-15-0': 'Redshift criteria.', '1810.03443-2-15-1': 'Incorrectly determined or uncertain redshifts may hurt any of blazar studies.', '1810.03443-2-15-2': 'This happens quite often, especially for BLLs whose spectra do not possess strong emission lines.', '1810.03443-2-15-3': 'We require a firm spectroscopic redshift for the objects included into our sample.', '1810.03443-2-15-4': 'More specifically, we start from a preselected sample of gamma-ray blazars and check, for every object, the relevant redshift information and references in the NASA/IPAC Extragalactic Database (NED).', '1810.03443-2-15-5': 'The redshifts are accepted if they satisfy (R1) and one of (R2A), (R2B), (R2C) criteria:', '1810.03443-2-16-0': "(R1) the redshift is spectroscopic (not photometric) and is derived from emission linesSince the absorption-line redshift gives a lower limit to the actual one, inclusion of objects with emission lines in the spectra would be conservative in terms of the search of anomalies; however, we removed 4 such objects from the sample following the Reviewer's comment.", '1810.03443-2-17-0': 'AND', '1810.03443-2-18-0': 'EITHER (R2A) the redshift is determined in a dedicated study (if several dedicated studies give different results, the latest one is used),', '1810.03443-2-19-0': 'OR (R2B) the redshift quoted in NED is determined in 2dF Galaxy Redshift Survey [CITATION] or 6dF Galaxy Survey [CITATION],', '1810.03443-2-20-0': 'OR (R2C) the redshift quoted in NED is determined in the Sloan Digital Sky Survey (SDSS), the result is unique and does not change from one release to another.', '1810.03443-2-21-0': 'These criteria are based on the previous experience in the use of redshift data and are of course ad hoc ones.', '1810.03443-2-21-1': 'Their application removes many uncertain redshifts.', '1810.03443-2-22-0': 'Spectral criteria.', '1810.03443-2-22-1': 'In our work, we use binned gamma-ray spectra, because only this information is publicly available for the objects observed by IACTs.', '1810.03443-2-22-2': 'We impose the following criteria for the spectra:', '1810.03443-2-23-0': '(S1) information for at least 5 energy bins is available;', '1810.03443-2-24-0': '(S2) the lowest energy [MATH] of the last spectral bin satisfies [MATH] AND the lowest energy of the third from below spectral bin is below [MATH].', '1810.03443-2-25-0': 'These criteria are aimed at the reconstruction of spectra at [MATH] both below and above [MATH] with reasonable accuracy.', '1810.03443-2-26-0': 'Several comments are in order.', '1810.03443-2-26-1': 'Firstly, (S2) depends on the assumed absorption model, and therefore the samples we use in Sec. [REF], when studying the sensitivity of our results to the choice of this model, are slightly different one from another.', '1810.03443-2-26-2': 'Secondly, while we require a nonzero measurement in the last bin for our main study, we allow for an upper limit when studying potential effects of the Malmquist bias in Sec. [REF].', '1810.03443-2-27-0': '## IACT subsample', '1810.03443-2-28-0': 'To construct the sample of blazars observed by IACTs, we start with the database consolidating results published by different observatories, the TeVCat online source catalog [CITATION].', '1810.03443-2-28-1': 'The preselected sample includes objects classified as blazars there (classes "HBL", "\'IBL", "LBL", "BL Lac (class uncertain)", "FSRQ" and "blazar"), 72 objects in total as of 2019.', '1810.03443-2-28-2': 'Then, for each object in the sample, we checked the redshift selection rules (R1) and ((R2A) or (R2B) or (R2C)), using references from NED, and availability of gamma-ray spectra satisfying (S1), using references from TeVCat.', '1810.03443-2-29-0': 'Some objects have been observed several times, often by multiple instruments.', '1810.03443-2-29-1': 'To avoid double counting of information, which may artificially increase or dilute observed effects, we use only one observation for each source.', '1810.03443-2-29-2': 'It is chosen on the basis of better statistics (larger effective exposure in cm[MATH]s).', '1810.03443-2-29-3': 'Because of strong variability of many blazars, we never combine any two spectra from observations performed at different epochs to extend the energy coverage.', '1810.03443-2-29-4': "We also never combine Fermi-LAT spectra with those obtained from IACT observations because of potential systematic differences between the instruments' energy scales.", '1810.03443-2-29-5': 'These requirements represent a major refinement with respect to some of preceding studies.', '1810.03443-2-30-0': 'The next step requires to fix the absorption model.', '1810.03443-2-30-1': 'We use the most recent EBL model [CITATION] as our baseline low-absorption model; when variations of the model are considered (Sec. [REF]), this step is repeated.', '1810.03443-2-30-2': 'At this step, we calculate the value of [MATH] for every source and check the condition (S2).', '1810.03443-2-30-3': 'We are left with 26 objects which, together with the sources observed by Fermi LAT, are listed in Table [REF].', '1810.03443-2-31-0': '## Fermi-LAT subsample', '1810.03443-2-32-0': 'For blazars observed by Fermi LAT, our starting point is the 3rd Fermi-LAT Catalog of High-Energy Sources, 3FHL [CITATION], presenting a list of sources detected confidently above 10 GeV.', '1810.03443-2-32-1': 'From the catalog, we select 1212 objects classified as blazars (classes "BLL", "bll", "FSRQ", "fsrq", "bcu").', '1810.03443-2-32-2': 'Since the redshifts presented in the catalog may be doubtful or erroneous for distant sources, we check them manually to satisfy (R1) and ((R2A) or (R2B) or (R2C)) criteria, using references from NED, and apply a preselection constraint [MATH], which is a consequence of (S2) and the Fermi-LAT energy coverage: for less distant sources, [MATH] GeV, that is beyond the limit of reasonable sensitivity of Fermi LAT.', '1810.03443-2-32-3': 'This procedure results in a list of 309 blazars.', '1810.03443-2-32-4': 'For each of them, we use publicly available Fermi-LAT [CITATION] data to construct binned spectra satisfying (S1) automatically with the help of standard routines from Fermi Science Tools 1.0.1.', '1810.03443-2-32-5': 'To this end, we use Fermi Pass 8 [CITATION] Release 3 data (version 303).', '1810.03443-2-32-6': 'We use "SOURCE" class events with version 2 instrumental response functions "P8R3[MATH]SOURCE[MATH]V2", recorded between August 4, 2008, and February 26, 2018 (Mission Elapsed Time interval 239557417 - 541338374).', '1810.03443-2-32-7': 'The background models used are "gll[MATH]iem[MATH]v06.fits" for the Galactic component and "iso[MATH]P8R3[MATH]SOURCE[MATH]V2.txt" for the isotropic one.', '1810.03443-2-32-8': 'black The background model also includes other sources from 3FGL catalog within [MATH] angular distance from the given source.', '1810.03443-2-32-9': 'black For most of the sources (305 of 309), the coordinates of the source are taken from the 3rd Fermi-LAT Source Catalog, 3FGL [CITATION].', '1810.03443-2-32-10': 'However, not all 3FHL sources have 3FGL counterparts, and for the remaining 4 sources, we used 3FHL coordinates.', '1810.03443-2-33-0': 'Firstly, we have performed an independent fit with the standard gtlike routine with the power-law spectrum model "PowerLaw2" in each of 8 energy bins with the width of 0.3 dex, for [MATH] GeV.', '1810.03443-2-33-1': 'The lower energy limit of 2 GeV was chosen to cut the inverse-Compton peak energy range in order to improve broken power-law fits used in our subsequent analysis.', '1810.03443-2-34-0': "Despite the fact that the use of gtlike is the preferred method of constructing sources' spectra, it demonstrates an unstable behavior when the estimated number of counts in the energy bin is lower than one.", '1810.03443-2-34-1': 'In particular, the fit may not converge or converge to zero flux with unphysical value of the flux upper limit.', '1810.03443-2-34-2': 'Moreover, even when the number of photons is less than three, the estimated confidence interval for the flux follows the simple symmetric Gaussian distribution but not the Poisson one, as it should.', '1810.03443-2-34-3': 'Clearly these features may affect further analysis and to avoid the peculiar behavior of the fit optimization procedure we have stepped one level back to use the Fermi Science Tools routines which provide an input for gtlike.', '1810.03443-2-34-4': 'Namely, the standard gtsrcprob routine assigned the weight to each photon which is the probability to originate from a given source within the given source model.', '1810.03443-2-34-5': 'The sum of the corresponding weights provides an estimate for the number of photons from the given source and from each of the background sources.', '1810.03443-2-35-0': 'At the next step, we fix the absorption model and apply the criterion (S2) to the obtained spectra.', '1810.03443-2-35-1': 'For the Fermi-LAT subsample, the criterion is used as follows: the source is accepted if it has at least one photon in the bin above [MATH] and the probability that this photon is associated with the source is more than 99% (according to the gtsrcprob output).', '1810.03443-2-35-2': 'Only 5 of 309 blazars satisfy (S2) and enter our final sample.', '1810.03443-2-36-0': 'Finally, for the bins which contain less than three photons we have replaced the gtlike flux with one obtained from summing weights from gtsrcprob.', '1810.03443-2-36-1': 'Since the background photons are considered separately, this sum may be treated as the number of signal events with zero background.', '1810.03443-2-36-2': 'We round this number to an integer (0 or 1 in most cases) and estimate the Poisson-based 68% CL interval for the flux in these bins by division by the exposure calculated with the gtexpcube routine.', '1810.03443-2-36-3': 'We have checked that the above procedure results in the value which agrees within [MATH] with the output of gtlike is cases when the latter converges to a physical value.', '1810.03443-2-37-0': '## The sample', '1810.03443-2-38-0': 'The list of 31 objects selected in the way described above, Sec. [REF], [REF], is presented in Table [REF].', '1810.03443-2-39-0': 'For IACT observations, references for binned spectra used in this work are given there.', '1810.03443-2-39-1': 'We also list classes of the blazars (BLL or FSRQ) as determined in the catalogs used to compile the samples, TeVCat and 3FHL.', '1810.03443-2-39-2': 'In addition, we give also references to original spectroscopic measurements of redshifts.', '1810.03443-2-40-0': 'Figure [REF]', '1810.03443-2-41-0': 'represents the distribution of blazars in our main sample in redshift.', '1810.03443-2-41-1': 'One may note that, though BLLs dominate at short and moderate distances and FSRQs are in general farther away, both classes cover large ranges of redshifts.', '1810.03443-2-41-2': 'We will return to this subject in Sec. [REF].', '1810.03443-2-42-0': '# Data analysis and results', '1810.03443-2-43-0': 'The key point of our study is the use of blazar spectra corrected for the pair production in a minimal-absorption model.', '1810.03443-2-43-1': 'Since, for IACT observations, only binned spectra are available, we adopt our analysis procedure for this case even for Fermi LAT, in order to process all observations in a uniform way.', '1810.03443-2-43-2': 'In contrast with many previous studies, we do not use bin-by-bin deabsorption because it can introduce systematic biases in the highest-energy bins for the following reasons.', '1810.03443-2-43-3': 'The optical depth [MATH] is a function of the incoming photon energy [MATH] and the source redshift [MATH].', '1810.03443-2-43-4': 'It is important to note that the absorption is not uniform within the bin: photons of higher energy are absorbed stronger.', '1810.03443-2-43-5': 'For the energies and distances at which the absorption is significant, this effect may affect strongly the result.', '1810.03443-2-43-6': 'It can be taken into account if the shape of the observed spectrum within the bin is known, as we have done in Ref. [CITATION]; however, in practice, statistical uncertainties in the highest energy bins often make the determination of the shape uncertain.', '1810.03443-2-43-7': 'The uncertainty of the spectral shape within the bin translates into the error of deabsorbed flux.', '1810.03443-2-43-8': 'The latter effect is enhanced in bins, within which the absorption effect grows dramatically.', '1810.03443-2-43-9': 'It is shown that for wide energy bins of 0.3 dex used for the analysis of Fermi LAT sources, the uncertainty related to deabsorption contributes significantly to the flux error in energy bins above [MATH].', '1810.03443-2-43-10': 'Therefore the absorption of the model spectra instead of the deabsorption of the observed one is performed in our analysis.', '1810.03443-2-44-0': 'Following the logic of previous studies, we start from the assumption about the intrinsic spectrum of a blazar, for which we use a broken power law.', '1810.03443-2-44-1': 'First, we make the break energy [MATH] a free parameter of the fit, so that it is adjusted independently for every source.', '1810.03443-2-44-2': 'We account for the absorption with the selected EBL model, integrate the obtained spectrum over the energy bins and compare resulted bin-by-bin fluxes with the observed data points.', '1810.03443-2-44-3': 'In this way, we fit the data with the four parameters of the intrinsic spectrum, that is the overall normalization, the break position [MATH], the spectral index at [MATH] and [MATH], the difference between spectral indices at [MATH] and [MATH].', '1810.03443-2-44-4': 'Technically, all our fits are performed with the method described in detail in Ref. [CITATION].', '1810.03443-2-44-5': 'Figure [REF]', '1810.03443-2-45-0': 'compares the fitted break positions [MATH] with the value of [MATH]; we see that they agree well to each other: averaged over the sample, [MATH] (for 22 sources with positive breaks, i.e. spectral hardenings, we get [MATH]).', '1810.03443-2-46-0': 'Then, we turn to similar fits with the break energy fixed at [MATH].', '1810.03443-2-46-1': 'Recall that this energy depends on the redshift of the source [MATH], cf. Fig. [REF], and therefore we assume breaks at different energies for different sources.', '1810.03443-2-46-2': 'Values of [MATH] obtained in this way for individual sources are given in Appendix [REF] and graphically presented in Fig. [REF].', '1810.03443-2-47-0': 'One sees that while some spectra have [MATH] consistent with zero, indications for [MATH] exist in many cases.', '1810.03443-2-47-1': 'For the entire sample, the assumption of [MATH] results in [MATH] for 31 black degrees of freedom, corresponding to the probability [MATH] for the observed values of [MATH] at [MATH] to occur as a result of a random fluctuation.', '1810.03443-2-47-2': 'Therefore, our results disagree with the hypothesis of the absence of breaks at [MATH] at 1.9 black standard deviations (recall the footnote at p. sigma-footnote for our conventions).', '1810.03443-2-48-0': 'black We can compare the assumption of breaks at [MATH] with that of breaks at some distance-independent energy, common to all sources.', '1810.03443-2-48-1': 'To this end, we perform fits of all spectral data by absorbed broken power laws, fixing the common break energy [MATH] for all objects in the sample, for various values of [MATH].', '1810.03443-2-48-2': 'For every [MATH], we determine the goodness of the fit, [MATH], from the chi-squared distribution with the corresponding number of degrees of freedom (for 225 data points, 31 spectra fitted with 3 parameters each, the number of degrees of freedom is 132).', '1810.03443-2-48-3': 'The plot of [MATH] as a function of [MATH] is shown in Fig. [REF]', '1810.03443-2-49-0': '(the best possible fit would correspond to [MATH], a very bad fit - to [MATH]), together with the same quantity calculated for fits with breaks at [MATH].', '1810.03443-2-49-1': 'The fit with breaks at distance-dependent energies [MATH] is better than a fit with any fixed [MATH] except for a narrow range [MATH] GeV.', '1810.03443-2-50-0': '# Discussion', '1810.03443-2-51-0': 'In this section, we perform a study of possible systematic effects affecting our results.', '1810.03443-2-51-1': 'We will see that the effect we observed cannot be explained by any potential bias or systematics we are aware of.', '1810.03443-2-51-2': 'Then we compare our result with those of previous studies.', '1810.03443-2-52-0': '## Systematic uncertainties', '1810.03443-2-53-0': '### Absorption models', '1810.03443-2-54-0': 'The results of the present study may indicate some problems with the absorption model used.', '1810.03443-2-54-1': 'For this study, we used the most recent published model by Korochkin and Rubtsov (2018), Ref. [CITATION].', '1810.03443-2-54-2': 'In this section, we repeat our study with several other representative models to see that the effect we found is present independently of the choice of the model.', '1810.03443-2-54-3': 'Besides the baseline model of Ref. [CITATION], we considered the following three models:', '1810.03443-2-55-0': '(i) Gilmore et al. (2012) fixed [CITATION], used in our previous study [CITATION];', '1810.03443-2-56-0': '(ii) Franceschini et al. (2017) [CITATION];', '1810.03443-2-57-0': '(iii) a toy upper-limit model normalized to the most recent direct observations of EBL [CITATION].', '1810.03443-2-58-0': 'The latter model was obtained by scaling the model (ii) by a multiplication factor fitted to the data of Refs. [CITATION].', '1810.03443-2-58-1': 'These two independent studies used different techniques to distinguish the extragalactic background light from the foreground contribution: Ref. [CITATION] used spectral templates, which are different for EBL and for foregrounds, while Ref. [CITATION] benefited from observations of an absorbing cloud to separate foregrounds experimentally.', '1810.03443-2-58-2': 'Figure [REF]', '1810.03443-2-59-0': 'presents 15 data points of Ref. [CITATION] and a single data point of Ref. [CITATION] together.', '1810.03443-2-59-1': 'All the points, corresponding to different background-light wavelengths, are fitted nicely by the intensity of the model [CITATION] multiplied by a wavelength-independent factor [MATH].', '1810.03443-2-59-2': 'For the toy high-absorption model we therefore take the model of Ref. [CITATION] uniformly upscaled by 3.35.', '1810.03443-2-59-3': 'The toy absorption model should be considered as an upper limit on the opacity.', '1810.03443-2-60-0': 'As it has been discussed above, the energies [MATH], at which the absorption becomes important, depend on the EBL model, and therefore samples of blazars selected for the study are different for different models.', '1810.03443-2-60-1': 'For the models [CITATION], some blazars from Table [REF] do not enter the sample while one extra blazar joined the set, see Notes black in Table [REF].', '1810.03443-2-60-2': 'For the toy black upper-limit black model, the list of 29 black sources in the sample is available from the authors by request.', '1810.03443-2-61-0': 'We present the resulting significance of the observation of spectral features at [MATH] for various absorption models in Table [REF].', '1810.03443-2-62-0': 'We see that for the low-absorption models, the results are similar to those obtained for the model of Ref. [CITATION].', '1810.03443-2-62-1': 'Clearly, a 3.35 times increase in the opacity, suggested by Refs. [CITATION], had to result in strengthening the effect, and we quantified this consistently within our approach.', '1810.03443-2-63-0': '### The Malmquist bias', '1810.03443-2-64-0': 'black Given the approach we use (selecting objects detected beyond [MATH]), demonstrating that there is no Malmquist bias is a doable but not-so-easy task.', '1810.03443-2-64-1': 'Indeed, black inclusion of sources to our sample is limited by observational capabilities of the instruments through the (S2) criterion of Sec. [REF].', '1810.03443-2-64-2': 'Therefore, sources intrinsically more and more luminous are included at larger distances.', '1810.03443-2-64-3': 'In principle, sources may have spectral features correlated with their luminosity, and the flux selection might mimic the observed effect (a particular model for this is not known).', '1810.03443-2-64-4': 'In this hypothetical scenario, weaker objects not detected above [MATH] would not have spectral hardenings which we observe for some of the sources included in our sample.', '1810.03443-2-64-5': 'This suggests a way to test this possible bias by a study of objects not detected above [MATH].', '1810.03443-2-65-0': 'To this end, we consider the same pre-selected 3FHL sample described in Sec. [REF] but relax the condition (S2) of Sec. [REF].', '1810.03443-2-65-1': 'Note that we perform the test for Fermi-LAT sources only, not including the sources observed only by IACT.', '1810.03443-2-65-2': 'The main reason for the above limitation is that the flux upper limits are not available for the most of the sources observed by IACTs.', '1810.03443-2-65-3': 'We obtain a sample of 47 blazars with redshifts between 0.233 and 2.534, which are detected significantly in the spectral bin containing [MATH], but not above.', '1810.03443-2-65-4': 'For each of the objects, we determine upper limits on the strength [MATH] of assumed breaks at [MATH] for these sources in the way described in Sec. [REF].', '1810.03443-2-65-5': 'Following the procedure described in Sec. [REF] we estimate the Poisson-based flux in the last bins and then calculate 95% CL upper limits on [MATH].', '1810.03443-2-65-6': 'The results of the calculation of these upper limits are presented in Fig. [REF].', '1810.03443-2-66-0': 'We see that the results obtained with the extended data set do not contradict to those obtained with the main sample, and can conclude that the Malmquist bias does not dominate the results.', '1810.03443-2-67-0': '### Source classes', '1810.03443-2-68-0': 'Gamma-ray sources observed at large distances, the blazars, belong to various classes.', '1810.03443-2-68-1': 'In this section, we consider a possibility that distant sources are different from nearby ones, and hence may have systematically different spectral features.', '1810.03443-2-68-2': 'In general, broadband spectral energy distributions (SEDs) of blazars have two wide bumps, often associated with the synchrotron and inverse-Compton emissions.', '1810.03443-2-68-3': 'Relative positions of the bumps are correlated, which may be explained if the same population of relativistic electrons is responsible for both processes.', '1810.03443-2-68-4': 'Therefore, the SEDs are, to the first approximation, characterized by the position of the synchrotron peak so that all blazars form "the blazar sequence" [CITATION].', '1810.03443-2-68-5': 'Flat-spectrum radio quasars (FSRQs) have the synchrotron peak frequency, [MATH], in the radio band, while blazars with [MATH] from infrared to X-ray bands represent various subclasses of BL Lac type objects (BLLs).', '1810.03443-2-68-6': 'On average, FSRQs are more powerful and less numerous, which results in a potential bias in flux-limited blazar samples: bright FSRQs are seen at large distances while more numerous weaker BLLs dominate the sample at relatively low redshifts.', '1810.03443-2-68-7': 'This bias might indeed affect our observations because intrinsic absorption in (distant) FSRQs might introduce spectral features not present in (nearby) BLLs.', '1810.03443-2-69-0': 'The classification of FSRQ or BLL given in Table [REF] is based on the information given in the 3FHL catalog for Fermi-LAT sources and in TeVCat for sources observed by IACTs.', '1810.03443-2-69-1': 'Since the classification is uncertain and various authors may use different criteria for claiming a source is a FSRQ or a BLL, we obtained approximate values of [MATH] for objects in the sample, based on SEDs available in NED and also on Refs. [CITATION].', '1810.03443-2-69-2': 'We counted objects with [MATH] Hz as FSRQs and others as BLLs, and found that this was in accordance with the classification given in Table [REF] for all objects.', '1810.03443-2-69-3': 'Fig. [REF]', '1810.03443-2-70-0': 'demonstrates that the same quantity [MATH] does not depend on the synchrotron peak frequency.', '1810.03443-2-70-1': 'The analyses therefore support the conclusion that potential BLL/FSRQ selection effects described in the beginning of this section are not important.', '1810.03443-2-70-2': 'Indirectly, this may suggest that the intrinsic absorption in FSRQs is a subdominant effect with respect to the absorption on EBL.', '1810.03443-2-71-0': '### Tail of the falling spectrum', '1810.03443-2-72-0': 'At the energies under discussion, spectra are determined from event-by-event information about observed individual photons.', '1810.03443-2-72-1': 'Determination of the photon energies is subject to statistical and systematic uncertainties.', '1810.03443-2-72-2': 'Statistical uncertainties are usually taken into account in the spectrum reconstruction procedures; however, they might be underestimated, while systematics may affect the overall energy scale of an instrument.', '1810.03443-2-72-3': 'We are interested in the highest energies, where, because of steeply falling spectra, the flux is often estimated from observations of a few photons only.', '1810.03443-2-72-4': 'In the case of a systematic shift of their energies upwards, or if the statistical uncertainty is larger than expected, overestimation of the energies of these few photons may affect significantly the shape of the spectrum.', '1810.03443-2-73-0': 'We first address potential systematic errors.', '1810.03443-2-73-1': 'These are instrument-dependent, and we take advantage of having objects observed by four different instruments in our sample.', '1810.03443-2-73-2': 'Table [REF]', '1810.03443-2-74-0': 'presents significances of the observed distance-dependent spectral hardenings in subsamples of objects with data of one of the instruments dropped.', '1810.03443-2-74-1': 'The observed significance is stable and agrees well with the expectations from the sample size.', '1810.03443-2-74-2': 'Therefore, only a coherent upward systematic error in the energy estimation by the three IACTs and Fermi LAT may result in the effect we found.', '1810.03443-2-74-3': 'We note that this situation is unlikely.', '1810.03443-2-75-0': 'We turn now to the possibility of underestimated statistical errors, which may result in occasional overestimation of energies of particular photons.', '1810.03443-2-75-1': 'In the case of the published spectra, this effect is normally already taken into account by the observing collaboration by making use of the "unfolding" procedure, see e.g. Ref. [CITATION].', '1810.03443-2-75-2': 'In any case, this potential systematics would be strongly suppressed for the objects where the spectral hardening happens at energies for which a sufficient number of photons is observed.', '1810.03443-2-75-3': 'To select those from our sample, we replace the (S2) selection criterion from Sec. [REF] by a much stronger one, requiring two (instead of one) spectral bins above [MATH].', '1810.03443-2-75-4': 'We stress that this does not directly select objects observed at larger black optical depths, nor intrinsically brighter sources; this just reduces statistical errors in the determination of [MATH].', '1810.03443-2-75-5': 'Only 17 of 31 black objects in our sample satisfy this criterion.', '1810.03443-2-75-6': 'Performing our analysis for these 17 black objects, we find the significance of [MATH] for this purified sample.', '1810.03443-2-75-7': 'This disfavours the relation of our observation to potentially underestimated errors in the energy determination; moreover, the fact that a purified, though depleted, sample exhibits a somewhat stronger black effect supports its physical originblack.', '1810.03443-2-76-0': '### Redshifts', '1810.03443-2-77-0': 'Our study of distance-dependent spectral features is sensitive to correct determination of distances to the sources.', '1810.03443-2-77-1': 'However, redshifts of distant blazars are not always determined unambiguously.', '1810.03443-2-77-2': 'In particular (see Sec. [REF]), redshifts of 4 black of 20 objects in the sample we used for our previous study [CITATION] in 2014 were found to be unreliable by 2018, and the objects left our present sample.', '1810.03443-2-77-3': 'Though for the present sample we performed a careful selection of objects based on the quality of their redshifts, see criteria (R1), (R2x) in Sec. [REF], one might imagine that some of the 31 black redshifts would still appear wrong in future analyses.', '1810.03443-2-77-4': 'To understand how this might affect our results, we artificially removed 1, 2 or 3 objects in all possible combinations from the sample and repeated our analysis with the reduced samples.', '1810.03443-2-77-5': 'With 1 object removed, the lowest significance was 1.5[MATH] and the highest one was 2.0 [MATH]; for 2 objects removed the interval was [MATH]; for 3 objects removed it was [MATH].', '1810.03443-2-77-6': 'Figure [REF]', '1810.03443-2-78-0': 'presents the distribution of significances among samples with 2 objects removed.', '1810.03443-2-78-1': 'We see that while it is possible to select sources from the sample whose removal would have a significant impact on the result, in most cases this is not expected.', '1810.03443-2-79-0': '## Comparison to previous studies', '1810.03443-2-80-0': 'In this section, we give a detailed comparison of our methods and results with those of key previous studies of the gamma-ray opacity of the Universe with ensembles of blazar spectra.', '1810.03443-2-81-0': '### Rubtsov and Troitsky (2014) [CITATION]', '1810.03443-2-82-0': 'The present work follows the same approach as our previous work [CITATION].', '1810.03443-2-82-1': 'Advantages of the present work, besides the use of new observational data and a more detailed report on systematics, are:', '1810.03443-2-83-0': '(i) pre-selection of Fermi-LAT sources with the 3FHL catalog and the use of Pass 8 data processing;', '1810.03443-2-84-0': '(ii) use of new conservative EBL models [CITATION];', '1810.03443-2-85-0': '(iii) imposing strict criteria on the redshift quality;', '1810.03443-2-86-0': '(iv) use of a more conservative method which is not based on the bin-by-bin deabsorption;', '1810.03443-2-87-0': '(v) use of a more general statistical test of the presence of spectral breaks.', '1810.03443-2-88-0': 'For the point (v), we stress here that the null hypothesis to be tested is the absence of any spectral features at [MATH].', '1810.03443-2-88-1': 'This hypothesis is disfavoured at 1.9 [MATH] in the present work.', '1810.03443-2-88-2': 'In Ref. [CITATION], we observed the linear growth of [MATH] with [MATH], which does not have a known physical model behind it.', '1810.03443-2-89-0': 'Next, the changes black in criteria black resulted in the removal of 5 black sources from the sample of Ref. [CITATION]: redshifts of 3 black objects (RGB J1448[MATH]361, 1ES 1101[MATH]232 black and 1ES 0347[MATH]121) did not satisfy our new quality criteria; the reported value of the redshift of B3 1307[MATH]433 changed in such a way that the object no longer satisfies the (S2) criterion; for PKS J0730[MATH]1141, new Pass 8 Fermi-LAT reconstruction does not result in a significant detection at [MATH], contrary to the Pass 7REP (V15) used in Ref. [CITATION].', '1810.03443-2-89-1': 'Removal of the most distant objects from the sample, together with a more conservative analysis method, resulted in a considerable reduction of the resulting significance (was 12.4[MATH] for the linear-growth test, now 1.9black[MATH] for the exclusion of the null hypothesis).', '1810.03443-2-90-0': 'It is interesting to note that the present study does not confirm the linear growth of spectral breaks with [MATH] observed in Ref. [CITATION] and never explained theoretically within any approach.', '1810.03443-2-90-1': 'Figure [REF]', '1810.03443-2-91-0': 'demonstrates that the significant distance dependence observed earlier was dominated by the effect of distant sources whose redshifts do not pass the quality cuts applied in the present study.', '1810.03443-2-91-1': 'Better measurements of redshifts of distant blazars observed in VHE gamma rays are clearly welcome.', '1810.03443-2-92-0': '### Horns and Meyer (2012) [CITATION]', '1810.03443-2-93-0': 'Ref. [CITATION] was the first study addressing quantitatively spectral hardenings in the ensemble of distant gamma-ray sources.', '1810.03443-2-94-0': '(1) Both Ref. [CITATION] and our study address upward spectral breaks at distance-dependent energies.', '1810.03443-2-95-0': '(2) Only objects observed by IACTs ([MATH]) were included in the sample of Ref. [CITATION].', '1810.03443-2-95-1': 'There, a stricter requirement of detection at optical depths [MATH] was used, resulting in the sample of 7 objects only, while we require [MATH] and include objects for which the break was not significantly detected.', '1810.03443-2-96-0': '(3) In Ref. [CITATION], bin-by-bin deabsorption was used but the correction for the mean energy in the bin for deabsorption was not implemented.', '1810.03443-2-97-0': '### Fermi-LAT collaboration (2012) [CITATION]', '1810.03443-2-98-0': 'The interesting work [CITATION] presented an analysis of the ensemble of BLLs detected by Fermi LAT at high redshifts and, for the first time, presented a detection of the EBL absorption in stacked spectra of these distant sources.', '1810.03443-2-98-1': 'The ensemble of 150 BLLs with redshifts [MATH] was split into three distance bins, and intrinsic spectra were extrapolated from Fermi-LAT observations at low energies.', '1810.03443-2-98-2': 'The EBL absorption correction was subsequently determined from spectra stacked within the three redshift bins.', '1810.03443-2-99-0': 'As we know from our present study, only a few blazars (5 with our criteria) have been confidently observed by Fermi LAT above [MATH].', '1810.03443-2-99-1': 'Constraints on the amount of absorption obtained in Ref. [CITATION] were not very tight.', '1810.03443-2-99-2': 'While available absorption models are consistent with the upper part of the allowed band in [MATH] for [MATH], cf. Fig. 1 of Ref. [CITATION], considerably lower absorption is allowed for all energies.', '1810.03443-2-99-3': 'This is in agreement with our results.', '1810.03443-2-100-0': '### Biteau and Williams (2015) [CITATION]', '1810.03443-2-101-0': 'Ref. [CITATION] considered 106 spectra of 38 sources observed by IACTs.', '1810.03443-2-101-1': 'Every spectrum was deabsorbed bin-by-bin, without applying the correction for the mean energy of the bin, and then fitted by a model spectrum for which a concave shape was assumed.', '1810.03443-2-101-2': 'This study did not reveal any anomaly in the absorption on EBL.', '1810.03443-2-102-0': 'We argue that nonobservation of the absorption anomaly in Ref. [CITATION] does not contradict to the results of the present work.', '1810.03443-2-102-1': 'The difference in conclusions is related both to the sample of source spectra used and to the assumptions on the intrinsic spectra.', '1810.03443-2-102-2': 'We point out the following differences:', '1810.03443-2-103-0': 'In the sample of Ref. [CITATION], a different set of sources is used, dominated by nearby ones, while the observation of anomalous features at various [MATH] requires a good redshift coverage.', '1810.03443-2-103-1': 'The dominance of nearby sources in Ref. [CITATION] was enhanced by the use of multiple spectra per source, which gave higher statistical weights to better studied nearby objects, see Figure [REF].', '1810.03443-2-103-2': 'Different spectra of the same objects were treated as independent data in the statistical analysis of Ref. [CITATION], which may introduce statistical biases.', '1810.03443-2-103-3': 'The individual-source analysis of Ref. [CITATION] assumed explicitly a concave shape for the deabsorbed spectrum, thus not allowing for the hardenings we study here by construction, see Figure [REF].', '1810.03443-2-103-4': 'In fact, in most cases (91 of 106 spectra, in particular for all sources with [MATH]), the best-fit model corresponded to a power law, that is to the margin of the allowed concave spectra: convex shapes, which give better fits for many of the spectra in our study, were not allowed.', '1810.03443-2-104-0': 'We conclude that the methods chosen by Biteau and Williams in Ref. [CITATION] are insensitive to the anomalous effects found in Refs. [CITATION] and studied here, therefore Ref. [CITATION] and our work are not in a contradiction.', '1810.03443-2-105-0': 'In addition to the main study, which did not allow for convex spectra by definition, Biteau and Williams addressed the result of Horns and Meyer [CITATION] discussed above.', '1810.03443-2-105-1': 'While they confirmed that the application of the method of Horns and Meyer to their sample results in a significant dependence of the break position on the redshift (with even stronger significance than in the original Ref. [CITATION]), Biteau and Williams claimed that the reason for the effect found in Ref. [CITATION] was in incorrect treatment of statistical uncertainties of the flux measurements: the method of Horns and Meyer, based on a nonparametric Kolmogorov-Smirnov test, does not take the uncertainties into account and is sensitive to central values of the measured flux only.', '1810.03443-2-105-2': 'To illustrate this, Biteau and Williams searched for observed excesses in residuals over best-fit concave spectra at energies corresponding to large optical depths.', '1810.03443-2-105-3': 'The results of the analysis, performed for their full set of spectra, indicated no significant deviations.', '1810.03443-2-105-4': 'This part of the study [CITATION], however, also suffers from the use of multiple spectra of the same objects treated as statistically independent data.', '1810.03443-2-105-5': 'In addition, these residuals were calculated with respect to the best-fit spectra deabsorbed with the EBL model determined from the very same spectra, which makes the problem of independent treatment of data points even more complicated.', '1810.03443-2-106-0': '### Galanti et al. (2015) [CITATION]', '1810.03443-2-107-0': 'It is interesting to note that distance-dependent hardenings can still be noticed with the analysis based on a single power-law fit.', '1810.03443-2-107-1': 'Galanti et al. (2015) [CITATION] considered a sample of 39 blazars observed by IACTs ([MATH]), for all tested opacities.', '1810.03443-2-107-2': 'Deabsorbed spectra were described by single power laws, and the distance dependence of the spectral index was studied.', '1810.03443-2-107-3': 'Though this approach does not constructively include spectral breaks, deabsorbed spectra with upward breaks look harder when fitted by a power-law, and this hardness was found to be redshift-dependent.', '1810.03443-2-107-4': 'Therefore, results of Ref. [CITATION] are in agreement with Ref. [CITATION].', '1810.03443-2-108-0': '### Studies of high-energy versus very-high-energy spectral indices', '1810.03443-2-109-0': 'Several previous studies used as their observables the difference between spectral indices in the high-energy (roughly, GeV; definition varies) and very-high-energy (roughly, hundreds of GeV) bands.', '1810.03443-2-109-1': 'These works include Stecker et al. (2006) [CITATION], Stecker and Scully (2010) [CITATION], Sanchez et al. (2013) [CITATION], Essey and Kusenko (2012) [CITATION].', '1810.03443-2-109-2': 'The most elaborated study of this kind, Dominguez and Ajello (2015) [CITATION], was based entirely on the Fermi-LAT data and was therefore insensitive to relative systematic errors of the energy determination by different experiments.', '1810.03443-2-109-3': 'It was also the only one working with spectra corrected for the absorption on EBL.', '1810.03443-2-109-4': 'In all these studies, the energy of spectral breaks was a priory fixed to a certain value, uniform for all sources.', '1810.03443-2-109-5': 'Therefore, the results of the papers mentioned, some of which pointed to anomalies, some did not, cannot be directly compared to ours, because we assumed breaks at energies [MATH], different for different sources.', '1810.03443-2-110-0': '# Conclusions', '1810.03443-2-111-0': 'In the present work, we readdress the problem of the anomalous transparency of the Universe for high-energy gamma rays, the most significant indications to which [CITATION] were based on unphysical spectral features in deabsorbed spectra for an ensemble of blazars which showed up precisely at the energies for which the correction for the absorption on EBL became important.', '1810.03443-2-111-1': 'By making use of a more robust analysis procedure which avoids bin-by-bin deabsorption and tests only the null hypothesis, of new gamma-ray and redshift data and of most recent EBL models, we disfavour the absence of anomalous distance-dependent spectral features with the modest statistical significance of [MATH].', '1810.03443-2-111-2': 'At the same time we do not confirm the linear dependence of the break strength with [MATH].', '1810.03443-2-111-3': 'We discuss potential biases and systematic errors and, by performing various tests, conclude that they cannot seriously affect our results.', '1810.03443-2-112-0': 'Interpretations of the anomalous transparency of the Universe have been widely discussed in the literature, see e.g. Ref. [CITATION] for a brief review and a list of references.', '1810.03443-2-112-1': 'The simplest possible culprit for unphysical spectral features might be an overestimated EBL intensity.', '1810.03443-2-112-2': 'However, reduction of the intensity below the levels predicted by the models we use might be dangerous because the model intensities are very close to the lower limits from galaxy counting.', '1810.03443-2-112-3': 'With the help of the Markov-chain optimization behind the model [CITATION], we plan to attempt to change the EBL model in such a way that the effect, hints to which we observe in the present work, is absent and to understand how much in conflict this would be with underlying astrophysical data.', '1810.03443-2-112-4': 'This will be a subject of a forthcoming work.', '1810.03443-2-113-0': 'A number of proposals attempted to explain the apparent anomalous transparency of the Universe by adding secondary emission to that arriving directly from the source.', '1810.03443-2-113-1': 'If these secondary photons are born relatively close to the observer, they do not have time to produce [MATH] pairs and arrive unattenuated.', '1810.03443-2-113-2': 'With respect to the origin of this secondary emission, it is useful to distinguish the cases when it is caused by electromagnetic cascades [CITATION] or from interactions of hadronic cosmic-ray particles [CITATION].', '1810.03443-2-113-3': 'The main characteristic feature of both approaches is that they require extremely low magnetic fields all along the propagation path of the cascade, otherwise secondary photons would not point to the sources because of deflections of their progenitor charged particles in the cascade.', '1810.03443-2-114-0': 'Other proposals require modification of the photon interactions.', '1810.03443-2-114-1': 'We note that the pair-production cross section has been well determined experimentally, and the only possibility to change it in the kinematic regimes relevant for our study is to allow for small deviations from the Lorentz invariance.', '1810.03443-2-114-2': 'However, this change would affect also the development of photon-induced atmospheric showers [CITATION] making the work of IACTs impossible, so that the scenario, in which results of Sec. [REF] are explained by the Lorentz-invariance violation, is in fact excluded by the fact that some high-energy photons have been detected by these instruments (see also Ref. [CITATION]).', '1810.03443-2-114-3': 'The remaining explanation invokes a hypothetical axion-like particle (ALP).', '1810.03443-2-114-4': 'In external magnetic fields, ALP mixes with photons [CITATION] and, since ALP does not attenuate on EBL, this makes it possible to detect photons from more distant sources.', '1810.03443-2-114-5': 'In one of the scenarios, this mixing takes place in intergalactic magnetic fields along the path from the source to the observer [CITATION], while in the other, a part of photons is converted to ALPs near the source and reconverted back to gamma rays in our neighbourhood (in the magnetic fields of galaxies, clusters and filaments) [CITATION].', '1810.03443-2-114-6': 'Observational consequences of the two scenarios are compared to each other and to data in Ref. [CITATION].', '1810.03443-2-114-7': 'The account of interactions of energetic photons with the axion-like particle allows one to explain consistently all indications to the anomalous transparency of the Universe without contradicting to any other experimental or observational data.', '1810.03443-2-114-8': 'Our present results indicate that further studies are required to prove or exclude the need for new-physics effects in the propagation of energetic gamma rays through the Universe.', '1810.03443-2-115-0': '# Observed and best-fit intrinsic spectra of blazars in the main sample.', '1810.03443-2-116-0': 'Figures [REF]-[REF]', '1810.03443-2-117-0': 'present the observed and deabsorbed spectra of 31 blazars from our main sample, see Table [REF], together with values of [MATH] and break strengths [MATH] on which the main result of the paper is based.', '1810.03443-2-117-1': 'The observed spectra and numerical values of spectral breaks are available online as a supplementary material for this paper.', '1810.03443-2-118-0': 'This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.', '1810.03443-2-118-1': 'This research has made use of the TeVCat online source catalog [CITATION] and of the data and tools provided by the Fermi Science Support Center.', '1810.03443-2-119-0': 'We are indebted to Alberto Franceschini for interesting correspondence and for sharing detailed numerical tables of the absorption model [CITATION] with us; to Manuel Meyer for numerous interesting discussions and helpful comments; to Timur Dzhatdoev, Oleg Kalashev, Mikhail Kuznetsov, Maxim Libanov, Valery Rubakov and Igor Tkachev for interesting discussions.', '1810.03443-2-119-1': "We thank the anonymous reviewer for careful reading of the manuscript and the attention to the study which went beyond the usual referee's duties.", '1810.03443-2-119-2': 'The work of AK and ST on constraining the intensity of EBL and its applications to astrophysical manifestations of axion-like particles was supported by the Russian Science Foundation, grant 18-12-00258.', '1810.03443-2-120-0': 'Note added.', '1810.03443-2-120-1': 'Several months after this study was completed and submitted, an update of Ref. [CITATION] was published by the Fermi LAT Collaboration [CITATION].', '1810.03443-2-120-2': 'We will compare our results to those of Ref. [CITATION] in a forthcoming publication based on Fermi-LAT data only.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1810.03443

1901.10691

{'1901.10691-1-0-0': 'The goal of this paper is to provide a unifying view of a wide range of problems of interest in machine learning by framing them as the minimization of functionals defined on the space of probability measures.', '1901.10691-1-0-1': 'In particular, we show that generative adversarial networks, variational inference, and actor-critic methods in reinforcement learning can all be seen through the lens of our framework.', '1901.10691-1-0-2': 'We then discuss a generic optimization algorithm for our formulation, called probability functional descent (PFD), and show how this algorithm recovers existing methods developed independently in the settings mentioned earlier.', '1901.10691-1-1-0': '# Introduction', '1901.10691-1-2-0': 'Deep learning now plays an important role in many domains, for example, in generative modeling, deep reinforcement learning, and variational inference.', '1901.10691-1-2-1': 'In the process, dozens of new algorithms have been proposed for solving these problems with deep neural networks, specific of course to domain at hand.', '1901.10691-1-3-0': 'In this paper, we introduce a conceptual framework which can be used to understand in a unified way a broad class of machine learning problems.', '1901.10691-1-3-1': 'Central to this framework is an abstract optimization problem in the space of probability measures, a formulation that stems from the observation that in many fields, the object of interest is a probability distribution; moreover, the learning process is guided by a probability functional to be minimized, a loss function that conceptually maps a probability distribution to a real number.', '1901.10691-1-3-2': '[REF] lists these correspondences in the case of generative adversarial networks, variational inference, and reinforcement learning.', '1901.10691-1-4-0': 'Because the optimization now takes place in the infinite-dimensional space of probability measures, standard finite-dimensional algorithms like gradient descent are initially unavailable; even the proper notion for the derivative of these functionals is unclear.', '1901.10691-1-4-1': 'We call upon on a body of literature known as von Mises calculus [CITATION], originally developed in the field of asymptotic statistics, to make these functional derivatives precise.', '1901.10691-1-4-2': 'Remarkably, we find that once the connection is made, the resulting generalized descent algorithm, which we call probability functional descent, is intimately compatible with standard deep learning techniques such as stochastic gradient descent [CITATION], the reparameterization trick [CITATION], and adversarial training [CITATION].', '1901.10691-1-5-0': 'When we apply probability functional descent to the aforementioned domains, we find that we recover a wide range of existing algorithms, and the essential distinction between them is simply the way that the functional derivative, the von Mises influence function in this context, is approximated.', '1901.10691-1-5-1': '[REF] lists these algorithms and their corresponding approximation methods.', '1901.10691-1-5-2': 'Probability functional descent therefore acts as a unifying framework for the analysis of existing algorithms as well as the systematic development of new ones.', '1901.10691-1-6-0': '## Related work', '1901.10691-1-7-0': 'The problem of optimizing functionals of probability measures is not new.', '1901.10691-1-7-1': 'For example, [CITATION] and [CITATION] study these types of problems and even propose Frank-Wolfe and steepest descent algorithms to solve these problems.', '1901.10691-1-7-2': 'However, their results do not attempt to unify the areas that we discuss nor are their algorithms immediately practical for the high-dimensional machine learning settings described here.', '1901.10691-1-7-3': 'Owing to our unifying perspective, though, one may be able to adapt these works to the machine learning applications described here.', '1901.10691-1-8-0': 'One part of our work recasts convex optimization problems as saddle-point problems by means of convex duality as a technique for estimating functional derivatives.', '1901.10691-1-8-1': 'This correspondence between convex optimization problems and saddle point problems is an old and general concept [CITATION], and it underlies classical dual optimization techniques [CITATION].', '1901.10691-1-8-2': 'Nevertheless, the use of these min-max representations remains an active topic of research in machine learning.', '1901.10691-1-8-3': 'Most notably, the literature concerning generative adversarial networks has recognized that certain min-max problems are equivalent to certain convex problems [CITATION].', '1901.10691-1-8-4': 'Outside of GANs, [CITATION] have begun using these min-max representations to inspire learning algorithms.', '1901.10691-1-8-5': 'These min-max representations are an important tool for us that allows for practical implementation of our machinery.', '1901.10691-1-9-0': 'Finally, we mention the work of [CITATION], who also casts many machine learning problems, including Bayesian inference and policy evaluation, as infinite-dimensional optimization problems.', '1901.10691-1-9-1': 'These optimization problems are over functions in Hilbert spaces with an objective in the form of an integral operator.', '1901.10691-1-9-2': 'Because probability distributions may often be represented as functions in an appropriate Hilbert space (via their density, for example, if they exist), this approach bears similarity to ours.', '1901.10691-1-9-3': 'Their use of a Hilbert space as the optimization domain represents a tradeoff that allows access to powerful Hilbert space structure but also requires the use of specialized kernel-based [CITATION] or particle-based methods [CITATION] to practically apply functional gradients and obtain samples.', '1901.10691-1-9-4': 'In our work, we work directly in the space of probability measures and associate a dual space without assuming any Hilbert (or even Banach) structure.', '1901.10691-1-9-5': 'This approach, as we shall illustrate, is compatible with standard deep learning techniques and indeed corresponds exactly to many existing deep learning-based algorithms.', '1901.10691-1-10-0': '# Descent on a Probability Functional', '1901.10691-1-11-0': 'We let [MATH] be the space of Borel probability measures on a topological space [MATH].', '1901.10691-1-11-1': 'Our abstract formulation takes the form of a minimization problem over probability distributions: [EQUATION] where [MATH] is called a probability functional.', '1901.10691-1-11-2': 'In order to avoid technical digressions, we assume that [MATH] is a metric space that is compact, complete, and separable (i.e. a compact Polish space).', '1901.10691-1-11-3': 'We endow [MATH] with the topology of weak convergence, also known as the weak* topology.', '1901.10691-1-12-0': 'We now draw upon elements of von Mises calculus [CITATION] to make precise the notion of derivatives of functionals such as [MATH].', '1901.10691-1-12-1': 'See [CITATION] for an in-depth discussion, or [CITATION] for another perspective.', '1901.10691-1-13-0': '[(Gateaux differential)] Let [MATH] be a function.', '1901.10691-1-13-1': 'The Gateaux differential [MATH] at [MATH] in the direction [MATH] is defined by [EQUATION] where [MATH] for some [MATH].', '1901.10691-1-14-0': 'Intuitively, the Gateaux differential is a generalization of the directional derivative, so that [MATH] describes the change in the value of [MATH] when the probability measure [MATH] is infinitesimally perturbed in the direction of [MATH], towards another measure [MATH].', '1901.10691-1-14-1': 'Though powerful, the Gateaux differential is a function of differences of probability measures, which can make it unwieldy to work with.', '1901.10691-1-14-2': 'In many cases, however, the Gateaux differential has a concise representation as an influence function [MATH].', '1901.10691-1-14-3': '[(Influence function)] We say that [MATH] is an influence function for [MATH] at [MATH] if the Gateaux differential has the integral representation [EQUATION] for all [MATH], where [MATH].', '1901.10691-1-15-0': 'The influence function provides a convenient representation for the Gateaux differential.', '1901.10691-1-15-1': 'We note that if [MATH] is an influence function, then so is [MATH] for a constant [MATH], since [MATH].', '1901.10691-1-15-2': 'We also note that under some regularity conditions, the influence function can be computed as [MATH], where [MATH] is the distribution that gives the point [MATH] with probability [MATH].', '1901.10691-1-16-0': 'The Gateaux derivative and the influence function provide the proper notion of a functional derivative, which allows us to generalize first-order descent algorithms to apply to probability functionals such as [MATH].', '1901.10691-1-16-1': 'In particular, they permit a linear approximation to [MATH] around [MATH], which we denote [MATH]: [EQUATION]', '1901.10691-1-16-2': 'This representation, also known as a von Mises representation, yields additional intuition about the influence function.', '1901.10691-1-16-3': 'Concretely, note that [MATH] decreases locally around [MATH] if [MATH] decreases.', '1901.10691-1-16-4': 'Therefore, [MATH] acts as a potential function defined on [MATH] that dictates where samples [MATH] should descend if the goal is to decrease [MATH].', '1901.10691-1-16-5': 'Of course, [MATH] only carries this interpretation around the current value of [MATH]; once [MATH] is changed, the influence function must be recomputed for the new distribution.', '1901.10691-1-17-0': 'Based on this intuition, we now present probability functional descent, a straightforward analogue of finite-dimensional first-order descent algorithms:', '1901.10691-1-18-0': 'Probability functional descent on [MATH]', '1901.10691-1-19-0': 'Initialize [MATH] to a distribution in [MATH] has not converged Set [MATH] (differentiation step) Update [MATH] to decrease [MATH] (descent step)', '1901.10691-1-20-0': 'As we shall see, probability functional descent serves as a blueprint for many existing algorithms: in generative adversarial networks, the differentiation and descent steps correspond to the discriminator and generator updates respectively; in reinforcement learning, they correspond to policy evaluation and policy improvement.', '1901.10691-1-21-0': 'In its abstract form, probability functional descent requires two design choices in order to convert it into a practical algorithm.', '1901.10691-1-21-1': 'In [REF], we discuss different ways to choose the update in the descent step; [REF] provides one generic way.', '1901.10691-1-21-2': 'In [REF], we discuss different ways to approximate the influence function in the differentiation step; [REF] provides one generic way and a surprising connection to adversarial training.', '1901.10691-1-22-0': '# Applying the Descent Step', '1901.10691-1-23-0': 'One straightforward way to apply the descent step of PFD is to adopt a parametrization [MATH] and descend the stochastic gradient of [MATH].', '1901.10691-1-23-1': 'This gradient step is justified by the following analogue of the chain rule: [(Chain rule)] Let [MATH] be continuously differentiable, in the sense that the influence function [MATH] exists and [MATH] is continuous.', '1901.10691-1-23-2': 'Let the parameterization [MATH] be differentiable, in the sense that [MATH] converges to a weak limit as [MATH].', '1901.10691-1-23-3': 'Then [EQUATION] where [MATH] is treated as a function [MATH] that is not dependent on [MATH].', '1901.10691-1-24-0': 'Without loss of generality, assume [MATH], as the gradient is simply a vector of one-dimensional derivatives.', '1901.10691-1-24-1': 'Let [MATH], and let [MATH].', '1901.10691-1-24-2': 'Then [EQUATION]', '1901.10691-1-24-3': 'Assuming for now that [EQUATION] we have that [EQUATION] where the interchange of limits is by the definition of weak convergence (recall we assumed that [MATH] is compact, so [MATH] is continuous and bounded by virtue of being continuous).', '1901.10691-1-25-0': 'The equality we assumed is the definition of a stronger notion of differentiability called Hadamard differentiability of [MATH].', '1901.10691-1-25-1': 'Our conditions imply Hadamard differentiability via Proposition 2.33 of [CITATION], noting that the map [MATH] is continuous by assumption.', '1901.10691-1-26-0': '[REF] converts the computation of [MATH], where [MATH] may be a complicated nonlinear functional, into the computation of a gradient of an expectation, which is easily handled using standard methods (see e.g. [CITATION]).', '1901.10691-1-26-1': 'For example, the reparameterization trick, also known as the pathwise derivative estimator [CITATION], uses the identity [EQUATION] where [MATH] samples [MATH] using [MATH].', '1901.10691-1-26-2': 'Alternatively, the log derivative trick, also known as the score function gradient estimator, likelihood ratio gradient estimator, or REINFORCE [CITATION], uses the identity [EQUATION] where [MATH] is the probability density function of [MATH].', '1901.10691-1-26-3': 'This gradient-based update rule for the descent step is therefore a natural, practical choice in the context of deep learning.', '1901.10691-1-27-0': '# Approximating the Influence Function', '1901.10691-1-28-0': 'The approximation of the influence function in the differentiation step can in principle be accomplished in many different ways.', '1901.10691-1-28-1': 'Indeed, we shall see that the distinguishing factor between many existing algorithms is exactly which influence function estimator used, as shown in [REF].', '1901.10691-1-28-2': 'In some cases, it is possible that the influence function can be evaluated exactly, bypassing the need for approximation.', '1901.10691-1-28-3': 'Otherwise, the influence function, being a function [MATH], may be modeled as a neural network; the precise way in which this neural network needs to be trained will depend on the exact analytical form of the influence function.', '1901.10691-1-29-0': 'Remarkably, a generic approximation technique is available if the functional [MATH] is convex.', '1901.10691-1-29-1': 'In this case, the influence function [MATH] possesses a variational characterization in terms of the convex conjugate [MATH] of [MATH].', '1901.10691-1-29-2': 'To apply this formalism, we now view [MATH] as a convex subset of the vector space of finite signed Borel measures [MATH], equipped with the topology of weak convergence.', '1901.10691-1-29-3': 'Crucial to the analysis will be its dual space, [MATH], the space of continuous functions [MATH].', '1901.10691-1-29-4': 'Finally, [MATH] denotes the extended real line [MATH].', '1901.10691-1-29-5': 'The convex conjugate is then defined as follows:', '1901.10691-1-30-0': 'Let [MATH] be a function.', '1901.10691-1-30-1': 'Its convex conjugate is a function [MATH] defined by [EQUATION].', '1901.10691-1-31-0': 'Note that [MATH] must now be defined on all of [MATH]; it is always possible to simply define [MATH] if [MATH], although sometimes a different extension may be more convenient.', '1901.10691-1-31-1': 'The convex conjugate forms the core of the following representation for the influence function [MATH]:', '1901.10691-1-32-0': '[(Fenchel-Moreau representation)] Let [MATH] be proper, convex, and lower semicontinuous.', '1901.10691-1-32-1': 'Then the maximizer of [MATH], if it exists, is an influence function for [MATH] at [MATH].', '1901.10691-1-32-2': 'With some abuse of notation, we have that [EQUATION].', '1901.10691-1-33-0': 'We will exploit the Fenchel-Moreau theorem, which applies in the setting of locally convex, Hausdorff topological vector spaces (see e.g. [CITATION]).', '1901.10691-1-33-1': 'The space we consider is [MATH], the space of signed, finite measures equipped with the topology of weak convergence, of which [MATH] is a convex subset.', '1901.10691-1-33-2': '[MATH] is indeed locally convex and Hausdorff, and its dual space is [MATH] (see e.g. [CITATION], section 5.14).', '1901.10691-1-34-0': 'We now show that a maximizer [MATH] is an influence function.', '1901.10691-1-34-1': 'By the Fenchel-Moreau theorem, [EQUATION] and [EQUATION].', '1901.10691-1-34-2': 'Because [MATH] is differentiable, [MATH] is differentiable, so by the envelope theorem [CITATION], [EQUATION] so that [MATH] is an influence function.', '1901.10691-1-35-0': 'The abuse of notation stems from the fact that not all influence functions are maximizers.', '1901.10691-1-35-1': 'This is true, though, if [MATH] if [MATH]: [EQUATION] since the convex function [MATH] lies above its tangent line: [EQUATION].', '1901.10691-1-35-2': 'Since [MATH], we have that [EQUATION] [REF] motivates the following influence function approximation strategy: model [MATH] with a neural network and train it using stochastic gradient ascent on the objective [MATH].', '1901.10691-1-35-3': 'The trained neural network is then an approximation to [MATH] suitable for use in the descent step of PFD.', '1901.10691-1-35-4': 'Under this approximation scheme, PFD can be concisely expressed as the saddle-point problem [EQUATION] where the inner supremum solves for the influence function (the differentiation step of PFD), and the outer infimum descends the linear approximation [MATH] (the descent step of PFD), noting that [MATH] is a constant w.r.t. [MATH].', '1901.10691-1-35-5': 'This procedure is highly reminiscent of adversarial training [CITATION]; for this reason, we call PFD with this approximation scheme based on convex duality adversarial PFD.', '1901.10691-1-35-6': 'PFD therefore explains the prevalence of adversarial training as a deep learning technique and extends its applicability to any convex probability functional.', '1901.10691-1-36-0': 'In the following sections, we demonstrate that PFD provides a broad conceptual framework that describes a wide range of existing machine learning algorithms.', '1901.10691-1-37-0': 'The following lemma will come in handy in our computations.', '1901.10691-1-38-0': 'Suppose [MATH] has a representation [EQUATION] where [MATH] is proper, convex, and lower semicontinuous.', '1901.10691-1-38-1': 'Then [MATH].', '1901.10691-1-39-0': 'By definition of the convex conjugate, [MATH].', '1901.10691-1-39-1': 'Then [MATH], by the Fenchel-Moreau theorem.', '1901.10691-1-40-0': 'We note that when applying this lemma, we will often implicitly define the appropriate extension of [MATH] to [MATH] to be [MATH].', '1901.10691-1-40-1': 'The exact choice of extension can certainly affect the exact form of the convex conjugate; see [CITATION] for one example of this phenomenon.', '1901.10691-1-41-0': '# Generative Adversarial Networks', '1901.10691-1-42-0': 'Generative adversarial networks (GANs) are a technique to train a parameterized probability measure [MATH] to mimic a data distribution [MATH].', '1901.10691-1-42-1': 'There are many variants of the GAN algorithm.', '1901.10691-1-42-2': 'They typically take the form of a saddle-point problem, and it is known that many of them correspond to the minimization of different divergences [MATH].', '1901.10691-1-42-3': 'We complete the picture by showing that many GAN variants could have been derived as instances of PFD applied to different divergences.', '1901.10691-1-43-0': '## Minimax GAN', '1901.10691-1-44-0': '[CITATION] originally proposed the following saddle-point problem [EQUATION].', '1901.10691-1-44-1': 'The interpretation of this minimax GAN problem is that the discriminator [MATH] learns to classify between fake samples from [MATH] and real samples from [MATH] via a binary classification loss, while the generator [MATH] is trained to produce counterfeit samples that fool the classifier.', '1901.10691-1-44-2': 'It was shown that the value of the inner optimization problem equals [MATH], where [EQUATION] is the Jensen-Shannon divergence, and therefore the problem corresponds to training [MATH] to minimize the divergence between [MATH] and [MATH].', '1901.10691-1-44-3': "As a practical algorithm, simultaneous stochastic gradient descent steps are performed on the discriminator's parameters [MATH] and the generator's parameters [MATH] using the two loss functions [EQUATION] where [MATH] and [MATH] are parameterized with neural networks.", '1901.10691-1-45-0': 'Our unifying result is the following:', '1901.10691-1-46-0': 'Adversarial PFD on the Jensen-Shannon divergence objective [EQUATION] yields the minimax GAN algorithm [REF].', '1901.10691-1-47-0': 'That is, the minimax GAN could have been derived mechanically and from first principles as an instance of adversarial PFD.', '1901.10691-1-47-1': 'To build intuition, we note that the discriminator plays the role of the approximate influence function:', '1901.10691-1-48-0': 'Suppose [MATH] has density [MATH] and [MATH] has density [MATH].', '1901.10691-1-48-1': 'Then the influence function for [MATH] is [EQUATION]', '1901.10691-1-48-2': 'Recall that in the minimax GAN, the optimal discriminator [MATH] satisfies [MATH], so the influence function [MATH] is approximated using the learned discriminator.', '1901.10691-1-49-0': 'Now, we rederive the minimax GAN problem [REF] as a form of adversarial PFD.', '1901.10691-1-49-1': 'We compute:', '1901.10691-1-50-0': 'The convex conjugate of [MATH] is [EQUATION]', '1901.10691-1-50-1': "Setting the integrand's derivative w.r.t. [MATH] to [MATH], we find that pointwise, the optimal [MATH] satisfies [EQUATION].", '1901.10691-1-50-2': 'We eliminate [MATH] in the integrand.', '1901.10691-1-50-3': 'Notice that the first two terms in the integrand cancel after plugging in [MATH].', '1901.10691-1-50-4': 'Since [EQUATION] we obtain that [EQUATION] [REF] yields the representation [EQUATION] an ascent step on which is the [MATH]-step in [REF] with the substitution [MATH].', '1901.10691-1-50-5': 'The descent step corresponds to updating [MATH] to decrease the linear approximation [MATH], which corresponds to the [MATH]-step in [REF].', '1901.10691-1-50-6': 'In fact, a similar argument can be applied to the [MATH]-GANs of [CITATION], which generalize the minimax GAN.', '1901.10691-1-50-7': 'The observation that [MATH]-GANs (and hence the minimax GAN) can be derived through convex duality was also noted by [CITATION].', '1901.10691-1-51-0': '## Non-saturating GAN', '1901.10691-1-52-0': '[CITATION] also proposed an alternative to [REF] called the non-saturating GAN, which prescribes descent steps on [EQUATION].', '1901.10691-1-52-1': "In the step on the generator's parameters [MATH], the [MATH] in the minimax GAN has been replaced with [MATH].", '1901.10691-1-52-2': 'This heuristic change prevents gradients to [MATH] from converging to [MATH] when the discriminator is too confident, and it is for this reason that the loss for [MATH] is called the non-saturating loss.', '1901.10691-1-53-0': 'We consider a slightly modified problem, in which the original minimax loss and the non-saturating loss are summed (and scaled by a factor of [MATH]): [EQUATION]', '1901.10691-1-53-1': 'This also prevents gradients to [MATH] from saturating, achieving the same goal as the non-saturating GAN.', '1901.10691-1-53-2': '[CITATION] and [CITATION] recognize that this process minimizes [MATH].', '1901.10691-1-54-0': 'We claim the following:', '1901.10691-1-55-0': 'PFD on the reverse Kullback-Liebler divergence objective [EQUATION] using the binary classification likelihood ratio estimator to approximate the influence function, yields the modified non-saturating GAN optimization problem [REF].', '1901.10691-1-56-0': 'Suppose [MATH] has density [MATH] and [MATH] has density [MATH].', '1901.10691-1-56-1': 'The influence function for [MATH] is [EQUATION]', '1901.10691-1-56-2': 'Now, because the binary classification loss [EQUATION] is minimized by [MATH], one estimator for [MATH] is simply [EQUATION] where [MATH] is updated as in the [MATH]-step in [REF].', '1901.10691-1-56-3': 'With this approximation scheme, the differentiation step and the descent step in PFD correspond exactly to the [MATH]-step and [MATH]-step respectively in [REF].', '1901.10691-1-56-4': 'Once again, the discriminator serves to approximate the influence function.', '1901.10691-1-57-0': '## Wasserstein GAN', '1901.10691-1-58-0': '[CITATION] propose solving the following saddle-point problem [EQUATION] where [MATH] denotes the Lipschitz constant of [MATH].', '1901.10691-1-58-1': 'The corresponding practical algorithm amounts to simultaneous descent steps on [EQUATION] where [MATH] is reprojected back to the space of [MATH]-Lipschitz functions after each [MATH]-step.', '1901.10691-1-58-2': 'Here, [MATH] is again the generator, and [MATH] is the discriminator, sometimes called the critic.', '1901.10691-1-58-3': 'This algorithm is called the Wasserstein GAN algorithm, so named because this algorithm approximately minimizes the [MATH]-Wasserstein distance [MATH]; the motivation for the [MATH]-step in [REF] is so that the discriminator learns the Kantorovich potential that describes the optimal transport from [MATH] to [MATH].', '1901.10691-1-58-4': 'See e.g. [CITATION] for the full optimal transport details.', '1901.10691-1-59-0': 'We claim that the Wasserstein GAN too is an instance of PFD, and once again, the discriminator plays the role of approximate influence function:', '1901.10691-1-60-0': 'Adversarial PFD on the Wasserstein distance objective [EQUATION] yields the Wasserstein GAN algorithm [REF].', '1901.10691-1-61-0': 'The influence function for [MATH] is the Kantorovich potential corresponding to the optimal transport from [MATH] to [MATH].', '1901.10691-1-62-0': 'See [CITATION], Proposition 7.17.', '1901.10691-1-63-0': 'We remark that the gradient computation in Theorem 3 of [CITATION] is a corollary of [REF] and [REF].', '1901.10691-1-63-1': 'Now, we show that the Wasserstein GAN algorithm can be derived mechanically via convex duality.', '1901.10691-1-63-2': 'The connection between the Wasserstein GAN and convex duality was also observed by [CITATION].', '1901.10691-1-64-0': 'The convex conjugate of [MATH] is [EQUATION].', '1901.10691-1-65-0': 'We use the notation [MATH] to denote the convex indicator function, which is [MATH] if [MATH] is true and [MATH] if [MATH] is false.', '1901.10691-1-66-0': 'Using Kantorovich-Rubinstein duality, we have that [EQUATION] where we use the notation [EQUATION].', '1901.10691-1-66-1': 'By [REF], [EQUATION]', '1901.10691-1-67-0': '[REF] yields the representation [EQUATION].', '1901.10691-1-67-1': 'The adversarial PFD differentiation step therefore corresponds exactly to the [MATH]-step in [REF], and the PFD descent step is exactly the [MATH]-step in [REF].', '1901.10691-1-68-0': '# Variational Inference', '1901.10691-1-69-0': 'In Bayesian inference, the central object is the posterior distribution [EQUATION] where [MATH] is an observed datapoint, [MATH] is the likelihood, [MATH] is the prior.', '1901.10691-1-69-1': 'Unfortunately, the posterior is difficult to compute due to the presence of the integral.', '1901.10691-1-69-2': 'Variational inference therefore reframes this computation as an optimization problem in which a variational posterior [MATH] approximates the true posterior by solving [EQUATION]', '1901.10691-1-70-0': '## Black-box variational inference', '1901.10691-1-71-0': 'This objective is not directly optimizable, due to the presence of the intractable [MATH] term.', '1901.10691-1-71-1': 'The tool of choice for variational inference is the evidence lower bound (ELBO), which rewrites [EQUATION].', '1901.10691-1-71-2': 'Because [MATH] is fixed, we may maximize the ELBO to minimize the KL divergence.', '1901.10691-1-71-3': 'The advantage of doing so is that all the terms inside the expectation are now tractable to evaluate, and thus the expectation may be approximated through Monte Carlo sampling.', '1901.10691-1-71-4': 'This leads to the following practical algorithm, namely stochastic gradient descent on the objective [EQUATION]', '1901.10691-1-71-5': 'This is called black-box variational inference [CITATION].', '1901.10691-1-71-6': '[CITATION] later recognized that ignoring the [MATH]-dependence of the term in the expectation yields the same gradients in expectation; it is this variant that we consider.', '1901.10691-1-71-7': 'Our unification result is the following:', '1901.10691-1-72-0': 'PFD on the variational inference objective [EQUATION] using exact influence functions, yields the black-box variational inference algorithm [REF].', '1901.10691-1-73-0': 'In fact, the influence function turns out to be precisely the inside of the negative ELBO bound:', '1901.10691-1-74-0': 'The influence function for [MATH] is [EQUATION]', '1901.10691-1-74-1': 'In this context, the influence function can be evaluated exactly, so the differentiation step of PFD may be performed without approximation.', '1901.10691-1-74-2': 'The descent step of PFD becomes exactly the descent step on [MATH] of [REF], where the [MATH]-dependence of the term in the expectation is ignored.', '1901.10691-1-74-3': 'We remark that the argument of [CITATION] that this dependence is removable can be seen as a corollary of [REF] and [REF].', '1901.10691-1-75-0': '## Adversarial variational Bayes', '1901.10691-1-76-0': 'When the density function of the prior [MATH] or the variational posterior [MATH] is not available, adversarial variational Bayes [CITATION] may be employed.', '1901.10691-1-76-1': 'Here, the quantity [MATH] is approximated by a neural network [MATH] through a binary classification problem, much like [REF].', '1901.10691-1-76-2': 'The resulting algorithm applies simultaneous descent steps on [EQUATION]', '1901.10691-1-76-3': 'This algorithm is another instance of PFD:', '1901.10691-1-77-0': 'PFD on the variational inference objective [MATH], using the binary classification likelihood ratio estimator to approximate the influence function, yields adversarial variational Bayes [REF].', '1901.10691-1-78-0': 'It is easily seen that [EQUATION].', '1901.10691-1-78-1': 'Therefore, the [MATH]-step of [REF] is the differentiation step of PFD, and the [MATH]-step of [REF] is the descent step.', '1901.10691-1-78-2': 'We remark that the gradient computation in Proposition 2 of [CITATION] is a corollary of [REF] and [REF].', '1901.10691-1-79-0': '# Reinforcement Learning', '1901.10691-1-80-0': 'In a Markov decision process, the distribution of states [MATH], actions [MATH], and rewards [MATH] is governed by the distribution [EQUATION] where [MATH] is an initial distribution over states, [MATH] gives the transition probability of arriving at state [MATH] with reward [MATH] from a state [MATH] taking an action [MATH], and [MATH] is a policy that gives the distribution of actions taken when in state [MATH].', '1901.10691-1-80-1': 'In reinforcement learning, we are interested in learning the policy [MATH] that maximizes the expected discounted reward [MATH], where [MATH] is a discount factor, while assuming we only have access to samples from [MATH] and [MATH].', '1901.10691-1-81-0': '## Policy iteration', '1901.10691-1-82-0': 'Policy iteration [CITATION] is one scheme that solves the reinforcement learning problem.', '1901.10691-1-82-1': 'It initializes [MATH] arbitrarily and then cycles between two steps, policy evaluation and policy improvement.', '1901.10691-1-82-2': 'In the policy evaluation step, the state-action value function [MATH] is computed.', '1901.10691-1-82-3': 'In the policy improvement step, the policy is updated to the greedy policy, the policy that at state [MATH] takes the action [MATH] with probability [MATH].', '1901.10691-1-83-0': 'Before we present our unification result, we introduce an arbitrary distribution over states [MATH] and consider the joint distribution [MATH], so that [MATH] is one probability distribution rather than one for every state [MATH].', '1901.10691-1-83-1': 'Now:', '1901.10691-1-84-0': 'PFD on the reinforcement learning objective [EQUATION] using exact influence functions and global minimization of the linear approximation, yields the policy iteration algorithm.', '1901.10691-1-85-0': 'Proofs continue on the following page.', '1901.10691-1-86-0': 'The influence function for [MATH] is [EQUATION] where [MATH] is the state-action value function, [MATH] is the state value function, and [MATH] is the marginal distribution of states after [MATH] steps, all under the policy [MATH].', '1901.10691-1-87-0': 'First, we note that [EQUATION] where we abuse notation to denote [MATH].', '1901.10691-1-88-0': 'We have [EQUATION] or, plugging in the measure, [EQUATION].', '1901.10691-1-88-1': 'The integral is over all free variables; we omit them here and in the following derivation for conciseness.', '1901.10691-1-89-0': 'In computing [MATH], the product rule dictates that a term appear for every [MATH], in which [MATH] is replaced with [MATH].', '1901.10691-1-89-1': 'Hence: [EQUATION] reordering the summations.', '1901.10691-1-89-2': 'Note that for [MATH], the summand vanishes: [EQUATION] since all the variables [MATH] integrate away to [MATH].', '1901.10691-1-89-3': 'This yields: [EQUATION]', '1901.10691-1-89-4': 'Then, substituting the marginal distribution (note [MATH] is not integrated) [EQUATION] we obtain [EQUATION]', '1901.10691-1-89-5': 'Let us rename the integration variables by decreasing their indices by [MATH]: [EQUATION]', '1901.10691-1-89-6': 'Substituting in [EQUATION] we obtain [EQUATION]', '1901.10691-1-89-7': 'Finally, we obtain that [EQUATION].', '1901.10691-1-90-0': 'The descent step of PFD corresponds to taking a step on [MATH] to decrease the linear approximation [EQUATION].', '1901.10691-1-90-1': 'Setting [MATH], this simplifies to either [EQUATION]', '1901.10691-1-90-2': 'The most naive way to decrease [REF] is to globally minimize it.', '1901.10691-1-90-3': 'This corresponds to setting [MATH] to be the greedy policy.', '1901.10691-1-90-4': 'Hence, the evaluation of [MATH] in policy iteration corresponds exactly to computing the influence function in the differentiation step of PFD, and the greedy policy update corresponds to applying the descent step.', '1901.10691-1-91-0': '## Policy gradient and actor-critic', '1901.10691-1-92-0': 'Policy iteration exactly computes the linear approximation and nonparametrically minimizes it.', '1901.10691-1-92-1': 'Now we consider algorithms in which the policy is parameterized and the descent step is taken using a gradient step on [REF] or [REF].', '1901.10691-1-92-2': 'If this approach is taken, there is a lot of flexibility in how the influence function can be approximated, but generally speaking, the result is an actor-critic method [CITATION], which describes a class of algorithms that approximates the value function of the current policy and then takes a gradient step on the parameters of the policy using the estimated value function.', '1901.10691-1-92-3': 'We claim:', '1901.10691-1-93-0': 'Approximate PFD on the reinforcement learning objective [MATH], where the influence function is estimated using, for example, Monte Carlo, least squares, or temporal differences, yields an actor-critic algorithm.', '1901.10691-1-94-0': 'There is a huge number of possible approximations to the influence function; we list several and their corresponding algorithms.', '1901.10691-1-94-1': 'The simplest algorithm is the policy gradient algorithm, also known as REINFORCE [CITATION], which directly uses a Monte Carlo estimate of [MATH] as the influence function estimator.', '1901.10691-1-94-2': 'Stochastic value gradients [CITATION] and the closely related deterministic policy gradient [CITATION] fit a neural network to [MATH] using a temporal difference update and use that as the influence function approximation; their use of a neural network makes them compatible with the reparameterization trick.', '1901.10691-1-94-3': 'Advantage actor-critic [CITATION] estimates [MATH] by estimating [MATH] using Monte Carlo and fitting a neural network to [MATH] using least squares.', '1901.10691-1-94-4': 'All of these algorithms are traditionally justified by the celebrated policy gradient theorem [CITATION]; we remark that this theorem is a corollary of [REF] and [REF].', '1901.10691-1-95-0': '## Dual actor-critic', '1901.10691-1-96-0': 'Because [MATH] is not convex, adversarial PFD does not directly apply.', '1901.10691-1-96-1': 'However, the form of [REF] strongly suggests fixing the arbitrary distribution [MATH] to be the discounted marginal distribution of states [MATH].', '1901.10691-1-96-2': 'Closely related to the linear programming formulation of reinforcement learning [CITATION], this choice turns out to convexify [MATH], thus enabling the use of convex duality to approximate its influence function.', '1901.10691-1-96-3': 'We expect to obtain an adversarial formulation of reinforcement learning; one such formulation is the dual actor-critic algorithm [CITATION]: [EQUATION] where [MATH].', '1901.10691-1-96-4': 'Indeed:', '1901.10691-1-97-0': 'Adversarial PFD on the reinforcement learning objective [MATH] yields the dual actor-critic algorithm [REF].', '1901.10691-1-98-0': 'The convex conjugate of [MATH] is [EQUATION] where [MATH] is the unique solution to [MATH], if it exists.', '1901.10691-1-99-0': 'As mentioned in the text, we set the arbitrary distribution [MATH].', '1901.10691-1-99-1': 'In doing so, [MATH] becomes a state-action occupancy measure that describes the frequency of encounters of the state-action pair [MATH] over trajectories governed by the policy [MATH].', '1901.10691-1-99-2': 'It is known that there is a bijection between occupancy measures [MATH] and policies [MATH] [CITATION].', '1901.10691-1-100-0': 'We can enforce this setting by redefining [EQUATION] where again [MATH] is the convex indicator function.', '1901.10691-1-100-1': 'This equation can be rewritten as [EQUATION] where [MATH].', '1901.10691-1-100-2': 'The constraint is known as the Bellman flow equation.', '1901.10691-1-100-3': 'This formulation is convex, as it is the sum of an affine function and an indicator of a convex set (indeed, an affine subspace).', '1901.10691-1-101-0': 'We recall [MATH], where [MATH].', '1901.10691-1-101-1': 'Now, [MATH] is uniquely defined by [MATH] if a solution to the equation exists.', '1901.10691-1-101-2': 'To see this, note that [MATH] is the fixed point of the Bellman operator [MATH] defined by [EQUATION] which is contractive and therefore has a unique fixed point.', '1901.10691-1-101-3': 'A representation of [MATH] may be obtained via fixed point iteration using [MATH] for an arbitrary action [MATH]: [EQUATION] where the expectation is taken under the deterministic policy [MATH].', '1901.10691-1-102-0': 'We rewrite [MATH] using a Lagrange multiplier [MATH] [EQUATION]', '1901.10691-1-102-1': 'Note that [MATH] is convex in [MATH]; this stems from the fact that [EQUATION].', '1901.10691-1-102-2': 'The result follows from [REF].', '1901.10691-1-103-0': 'Using [REF], adversarial PFD therefore recovers [REF]: [EQUATION]', '1901.10691-1-104-0': '# Conclusion', '1901.10691-1-105-0': 'This paper suggests several new research directions.', '1901.10691-1-105-1': 'First is the transfer of insight and specialized techniques from one domain to another.', '1901.10691-1-105-2': 'As just one example, in the context of GANs, [CITATION] claim that constraining the discriminator to be [MATH]-Lipschitz improves the stability of the training algorithm - could similarly constraining the analogous object in reinforcement learning, namely an approximation to the advantage function, lead to improved stability in deep reinforcement learning?', '1901.10691-1-106-0': 'Moreover, the abstract viewpoint taken in this paper allows for the simultaneous development of new algorithms for GANs, variational inference, and reinforcement learning.', '1901.10691-1-106-1': 'General influence function approximation techniques in the spirit of convex duality could improve all three fields at once; more sophisticated descent techniques beyond gradient descent on parameterized probability distributions, such as Frank-Wolfe or steepest descent on measures, could improve learning or yield valuable convergence guarantees.', '1901.10691-1-107-0': 'Finally, this paper unlocks the possibility of applying probability functional descent to new problems.', '1901.10691-1-107-1': 'In principle, the algorithm can be applied mechanically to any situation where one wants to optimize over probability distributions, possibly leading to new, straightforward ways to solve problems in, for example, mathematical finance, mean field games, or POMDPs.', '1901.10691-1-107-2': 'One could argue that the current excitement over deep learning began once researchers realized that to solve a problem, they could simply write a loss function and then rely on automatic differentiation and gradient descent to minimize it.', '1901.10691-1-107-3': 'We hope that probability functional descent provides a similarly turnkey solution for optimizing loss functions defined on probability distributions and leads to a similar burst of research activity.'}

{'1901.10691-2-0-0': 'The goal of this paper is to provide a unifying view of a wide range of problems of interest in machine learning by framing them as the minimization of functionals defined on the space of probability measures.', '1901.10691-2-0-1': 'In particular, we show that generative adversarial networks, variational inference, and actor-critic methods in reinforcement learning can all be seen through the lens of our framework.', '1901.10691-2-0-2': 'We then discuss a generic optimization algorithm for our formulation, called probability functional descent (PFD), and show how this algorithm recovers existing methods developed independently in the settings mentioned earlier.', '1901.10691-2-1-0': '# Introduction', '1901.10691-2-2-0': 'Deep learning now plays an important role in many domains, for example, in generative modeling, deep reinforcement learning, and variational inference.', '1901.10691-2-2-1': 'In the process, dozens of new algorithms have been proposed for solving these problems with deep neural networks, specific of course to domain at hand.', '1901.10691-2-3-0': 'In this paper, we introduce a conceptual framework which can be used to understand in a unified way a broad class of machine learning problems.', '1901.10691-2-3-1': 'Central to this framework is an abstract optimization problem in the space of probability measures, a formulation that stems from the observation that in many fields, the object of interest is a probability distribution; moreover, the learning process is guided by a probability functional to be minimized, a loss function that conceptually maps a probability distribution to a real number.', '1901.10691-2-3-2': '[REF] lists these correspondences in the case of generative adversarial networks, variational inference, and reinforcement learning.', '1901.10691-2-4-0': 'Because the optimization now takes place in the infinite-dimensional space of probability measures, standard finite-dimensional algorithms like gradient descent are initially unavailable; even the proper notion for the derivative of these functionals is unclear.', '1901.10691-2-4-1': 'We call upon on a body of literature known as von Mises calculus [CITATION], originally developed in the field of asymptotic statistics, to make these functional derivatives precise.', '1901.10691-2-4-2': 'Remarkably, we find that once the connection is made, the resulting generalized descent algorithm, which we call probability functional descent, is intimately compatible with standard deep learning techniques such as stochastic gradient descent [CITATION], the reparameterization trick [CITATION], and adversarial training [CITATION].', '1901.10691-2-5-0': 'When we apply probability functional descent to the aforementioned domains, we find that we recover a wide range of existing algorithms, and the essential distinction between them is simply the way that the functional derivative, the von Mises influence function in this context, is approximated.', '1901.10691-2-5-1': '[REF] lists these algorithms and their corresponding approximation methods.', '1901.10691-2-5-2': 'Probability functional descent therefore acts as a unifying framework for the analysis of existing algorithms as well as the systematic development of new ones.', '1901.10691-2-6-0': '## Related work', '1901.10691-2-7-0': 'The problem of optimizing functionals of probability measures is not new.', '1901.10691-2-7-1': 'For example, [CITATION] and [CITATION] study these types of problems and even propose Frank-Wolfe and steepest descent algorithms to solve these problems.', '1901.10691-2-7-2': 'However, their algorithms are not immediately practical for the high-dimensional machine learning settings described here, and it is not clear how to integrate their methods with modern deep learning techniques.', '1901.10691-2-8-0': 'Several others in the machine learning community also adopt the perspective of descent in the space of probability distributions.', '1901.10691-2-8-1': 'In order to introduce functional gradients, these approaches endow the space of probability distributions with either Hilbert structure [CITATION] or Wasserstein structure [CITATION] and rely on gradient descent or Wasserstein gradient flow respectively to decrease the objective value.', '1901.10691-2-8-2': 'Such approaches typically require kernel-based or particle-based methods to implement in practice.', '1901.10691-2-8-3': 'By contrast, our approach foregoes gradients and instead directly considers descent on linear approximations by leveraging the Gateaux derivative.', '1901.10691-2-8-4': 'As we shall illustrate, this approach is more compatible with standard deep learning techniques and indeed leads exactly to many existing deep learning-based algorithms.', '1901.10691-2-8-5': '[CITATION] provide a technical comparison between these differing approaches for defining derivatives in chapter 5.', '1901.10691-2-9-0': 'Finally, one part of our work recasts convex optimization problems as saddle-point problems by means of convex duality as a technique for estimating functional derivatives.', '1901.10691-2-9-1': 'This correspondence between convex optimization problems and saddle point problems is an old and general concept [CITATION], and it underlies classical dual optimization techniques [CITATION].', '1901.10691-2-9-2': 'Nevertheless, the use of these min-max representations remains an active topic of research in machine learning.', '1901.10691-2-9-3': 'Most notably, the literature concerning generative adversarial networks has recognized that certain min-max problems are equivalent to certain convex problems [CITATION].', '1901.10691-2-9-4': 'Outside of GANs, [CITATION] have begun using these min-max representations to inspire learning algorithms.', '1901.10691-2-9-5': 'These min-max representations are an important tool for us that allows for practical implementation of our theory.', '1901.10691-2-10-0': '# Descent on a Probability Functional', '1901.10691-2-11-0': 'We let [MATH] be the space of Borel probability measures on a topological space [MATH].', '1901.10691-2-11-1': 'Our abstract formulation takes the form of a minimization problem over probability distributions: [EQUATION] where [MATH] is called a probability functional.', '1901.10691-2-11-2': 'In order to avoid technical digressions, we assume that [MATH] is a metric space that is compact, complete, and separable (i.e. a compact Polish space).', '1901.10691-2-11-3': 'We endow [MATH] with the topology of weak convergence, also known as the weak* topology.', '1901.10691-2-12-0': 'We now draw upon elements of von Mises calculus [CITATION] to make precise the notion of derivatives of functionals such as [MATH].', '1901.10691-2-12-1': 'See [CITATION] for an in-depth discussion, or [CITATION] for another perspective.', '1901.10691-2-13-0': '[(Gateaux differential)] Let [MATH] be a function.', '1901.10691-2-13-1': 'The Gateaux differential [MATH] at [MATH] in the direction [MATH] is defined by [EQUATION] where [MATH] for some [MATH].', '1901.10691-2-14-0': 'Intuitively, the Gateaux differential is a generalization of the directional derivative, so that [MATH] describes the change in the value of [MATH] when the probability measure [MATH] is infinitesimally perturbed in the direction of [MATH], towards another measure [MATH].', '1901.10691-2-14-1': 'Though powerful, the Gateaux differential is a function of differences of probability measures, which can make it unwieldy to work with.', '1901.10691-2-14-2': 'In many cases, however, the Gateaux differential [MATH] can be concisely represented as an integral of an influence function [MATH], where the integral is taken with respect to the measure [MATH].', '1901.10691-2-14-3': '[(Influence function)] We say that [MATH] is an influence function for [MATH] at [MATH] if the Gateaux differential [MATH] has the integral representation [EQUATION] for all [MATH], where [MATH].', '1901.10691-2-15-0': 'Let [MATH].', '1901.10691-2-15-1': 'Then [MATH] is an influence function of [MATH] at [MATH] if and only if [EQUATION].', '1901.10691-2-16-0': 'The left-hand side equals [REF], which equals [REF].', '1901.10691-2-17-0': 'The influence function provides a convenient representation for the Gateaux differential.', '1901.10691-2-17-1': 'Because [MATH] is a difference of probability distributions, we can also write [EQUATION] by linearity.', '1901.10691-2-17-2': 'We note that if [MATH] is an influence function, then so is [MATH] for a constant [MATH].', '1901.10691-2-18-0': 'The Gateaux derivative and the influence function provide the proper notion of a functional derivative, which allows us to generalize first-order descent algorithms to apply to probability functionals such as [MATH].', '1901.10691-2-18-1': 'In particular, they permit a linear approximation to [MATH] around [MATH], which we denote [MATH]: [EQUATION]', '1901.10691-2-18-2': 'This expression, also known as a von Mises representation, yields additional intuition about the influence function.', '1901.10691-2-18-3': 'Concretely, note that a small pertubation to [MATH] decreases [MATH] if it decreases [MATH].', '1901.10691-2-18-4': 'Therefore, [MATH] acts as a potential function defined on [MATH] that dictates where samples [MATH] should descend if the goal is to decrease [MATH].', '1901.10691-2-18-5': 'Of course, [MATH] only carries this interpretation around the current value of [MATH].', '1901.10691-2-19-0': 'Based on this intuition, we now present probability functional descent, a straightforward analogue of finite-dimensional first-order descent algorithms to probability functionals.', '1901.10691-2-19-1': 'First, a linear approximation to the functional [MATH] is computed at [MATH] in the form of the influence function [MATH], and then a local step is taken from [MATH] so as to decrease the value of the linear approximation.', '1901.10691-2-19-2': 'Concretely:', '1901.10691-2-20-0': 'Probability functional descent on [MATH]', '1901.10691-2-21-0': 'Initialize [MATH] to a distribution in [MATH] has not converged Set [MATH] (differentiation step) Update [MATH] to decrease [MATH] (descent step)', '1901.10691-2-22-0': 'We shall see that probability functional descent serves as a blueprint for many existing algorithms: in generative adversarial networks, the differentiation and descent steps correspond to the discriminator and generator updates respectively; in reinforcement learning, they correspond to policy evaluation and policy improvement.', '1901.10691-2-23-0': 'In its abstract form, probability functional descent requires two design choices in order to convert it into a practical algorithm.', '1901.10691-2-23-1': 'In [REF], we discuss different ways to choose the update in the descent step; [REF] provides one generic way.', '1901.10691-2-23-2': 'In [REF], we discuss different ways to approximate the influence function in the differentiation step; [REF] provides one generic way and an unexpected connection to adversarial training.', '1901.10691-2-24-0': '# Applying the Descent Step', '1901.10691-2-25-0': 'One straightforward way to apply the descent step of PFD is to adopt a parametrization [MATH] and descend the stochastic gradient of [MATH].', '1901.10691-2-25-1': 'This gradient step is justified by the following analogue of the chain rule: [(Chain rule)] Let [MATH] be continuously differentiable, in the sense that the influence function [MATH] exists and [MATH] is continuous.', '1901.10691-2-25-2': 'Let the parameterization [MATH] be differentiable, in the sense that [MATH] converges to a weak limit as [MATH].', '1901.10691-2-25-3': 'Then [EQUATION] where [MATH] is treated as a function [MATH] that is not dependent on [MATH].', '1901.10691-2-26-0': 'Without loss of generality, assume [MATH], as the gradient is simply a vector of one-dimensional derivatives.', '1901.10691-2-26-1': 'Let [MATH], and let [MATH] (weakly).', '1901.10691-2-26-2': 'Then [EQUATION]', '1901.10691-2-26-3': 'Assuming for now that [EQUATION] we have by [REF] that [EQUATION] where the interchange of limits is by the definition of weak convergence (recall we assumed that [MATH] is compact, so [MATH] is continuous and bounded by virtue of being continuous).', '1901.10691-2-27-0': 'The equality we assumed is the definition of a stronger notion of differentiability called Hadamard differentiability of [MATH].', '1901.10691-2-27-1': 'Our conditions imply Hadamard differentiability via Proposition 2.33 of [CITATION], noting that the map [MATH] is continuous by assumption.', '1901.10691-2-28-0': '[REF] converts the computation of [MATH], where [MATH] may be a complicated nonlinear functional, into the computation of a gradient of an expectation, which is easily handled using standard methods (see e.g. [CITATION]).', '1901.10691-2-28-1': 'For example, the reparameterization trick, also known as the pathwise derivative estimator [CITATION], uses the identity [EQUATION] where [MATH] samples [MATH] using [MATH].', '1901.10691-2-28-2': 'Alternatively, the log derivative trick, also known as the score function gradient estimator, likelihood ratio gradient estimator, or REINFORCE [CITATION], uses the identity [EQUATION] where [MATH] is the probability density function of [MATH].', '1901.10691-2-28-3': 'This gradient-based update rule for the descent step is therefore a natural, practical choice in the context of deep learning.', '1901.10691-2-29-0': '# Approximating the Influence Function', '1901.10691-2-30-0': 'The approximation of the influence function in the differentiation step can in principle be accomplished in many different ways.', '1901.10691-2-30-1': 'Indeed, we shall see that the distinguishing factor between many existing algorithms is exactly which influence function estimator used, as shown in [REF].', '1901.10691-2-30-2': 'In some cases, it is possible that the influence function can be evaluated exactly, bypassing the need for approximation.', '1901.10691-2-30-3': 'Otherwise, the influence function, being a function [MATH], may be modeled as a neural network; the precise way in which this neural network needs to be trained will depend on the exact analytical form of the influence function.', '1901.10691-2-31-0': 'Remarkably, a generic approximation technique is available if the functional [MATH] is convex.', '1901.10691-2-31-1': 'In this case, the influence function [MATH] possesses a variational characterization in terms of the convex conjugate [MATH] of [MATH].', '1901.10691-2-31-2': 'To apply this formalism, we now view [MATH] as a convex subset of the vector space of finite signed Borel measures [MATH], equipped with the topology of weak convergence.', '1901.10691-2-31-3': 'Crucial to the analysis will be its dual space, [MATH], the space of continuous functions [MATH].', '1901.10691-2-31-4': 'Finally, [MATH] denotes the extended real line [MATH].', '1901.10691-2-31-5': 'The convex conjugate is then defined as follows:', '1901.10691-2-32-0': 'Let [MATH] be a function.', '1901.10691-2-32-1': 'Its convex conjugate is a function [MATH] defined by [EQUATION].', '1901.10691-2-33-0': 'Note that [MATH] must now be defined on all of [MATH]; it is always possible to simply define [MATH] if [MATH], although sometimes a different extension may be more convenient.', '1901.10691-2-33-1': 'The convex conjugate forms the core of the following representation for the influence function [MATH]:', '1901.10691-2-34-0': '[(Fenchel-Moreau representation)] Let [MATH] be proper, convex, and lower semicontinuous.', '1901.10691-2-34-1': 'Then the maximizer of [MATH], if it exists, is an influence function for [MATH] at [MATH].', '1901.10691-2-34-2': 'With some abuse of notation, we have that [EQUATION].', '1901.10691-2-35-0': 'We will exploit the Fenchel-Moreau theorem, which applies in the setting of locally convex, Hausdorff topological vector spaces (see e.g. [CITATION]).', '1901.10691-2-35-1': 'The space we consider is [MATH], the space of signed, finite measures equipped with the topology of weak convergence, of which [MATH] is a convex subset.', '1901.10691-2-35-2': '[MATH] is indeed locally convex and Hausdorff, and its dual space is [MATH] (see e.g. [CITATION], section 5.14).', '1901.10691-2-36-0': 'We now show that a maximizer [MATH] is an influence function.', '1901.10691-2-36-1': 'By the Fenchel-Moreau theorem, [EQUATION] and [EQUATION].', '1901.10691-2-36-2': 'Because [MATH] is differentiable, [MATH] is differentiable, so by the envelope theorem [CITATION], [EQUATION] so that [MATH] is an influence function by [REF].', '1901.10691-2-37-0': 'The abuse of notation stems from the fact that not all influence functions are maximizers.', '1901.10691-2-37-1': 'This is true, though, if [MATH] if [MATH]: [EQUATION] since the convex function [MATH] lies above its tangent line: [EQUATION].', '1901.10691-2-37-2': 'Since [MATH], we have that [EQUATION] [REF] motivates the following influence function approximation strategy: model [MATH] with a neural network and train it using stochastic gradient ascent on the objective [MATH].', '1901.10691-2-37-3': 'The trained neural network is then an approximation to [MATH] suitable for use in the descent step of PFD.', '1901.10691-2-37-4': 'Under this approximation scheme, PFD can be concisely expressed as the saddle-point problem [EQUATION] where the inner supremum solves for the influence function (the differentiation step of PFD), and the outer infimum descends the linear approximation [MATH] (the descent step of PFD), noting that [MATH] is a constant w.r.t. [MATH].', '1901.10691-2-37-5': 'This procedure is highly reminiscent of adversarial training [CITATION]; for this reason, we call PFD with this approximation scheme based on convex duality adversarial PFD.', '1901.10691-2-37-6': 'PFD therefore explains the prevalence of adversarial training as a deep learning technique and extends its applicability to any convex probability functional.', '1901.10691-2-38-0': 'In the following sections, we demonstrate that PFD provides a broad conceptual framework for understanding a wide range of existing machine learning algorithms.', '1901.10691-2-39-0': 'The following lemma will come in handy in our computations.', '1901.10691-2-40-0': 'Suppose [MATH] has a representation [EQUATION] where [MATH] is proper, convex, and lower semicontinuous.', '1901.10691-2-40-1': 'Then [MATH].', '1901.10691-2-41-0': 'By definition of the convex conjugate, [MATH].', '1901.10691-2-41-1': 'Then [MATH], by the Fenchel-Moreau theorem.', '1901.10691-2-42-0': 'We note that when applying this lemma, we will often implicitly define the appropriate extension of [MATH] to [MATH] to be [MATH].', '1901.10691-2-42-1': 'The exact choice of extension can certainly affect the exact form of the convex conjugate; see [CITATION] for one example of this phenomenon.', '1901.10691-2-43-0': '# Generative Adversarial Networks', '1901.10691-2-44-0': 'Generative adversarial networks (GANs) are a technique to train a parameterized probability measure [MATH] to mimic a data distribution [MATH].', '1901.10691-2-44-1': 'There are many variants of the GAN algorithm.', '1901.10691-2-44-2': 'They typically take the form of a saddle-point problem, and it is known that many of them correspond to the minimization of different divergences [MATH].', '1901.10691-2-44-3': 'We complete the picture by showing that many GAN variants could have been derived as instances of PFD applied to different divergences.', '1901.10691-2-45-0': '## Minimax GAN', '1901.10691-2-46-0': '[CITATION] originally proposed the following saddle-point problem [EQUATION].', '1901.10691-2-46-1': 'The interpretation of this minimax GAN problem is that the discriminator [MATH] learns to classify between fake samples from [MATH] and real samples from [MATH] via a binary classification loss, while the generator [MATH] is trained to produce counterfeit samples that fool the classifier.', '1901.10691-2-46-2': 'It was shown that the value of the inner optimization problem equals [MATH], where [EQUATION] is the Jensen-Shannon divergence, and therefore the problem corresponds to training [MATH] to minimize the divergence between [MATH] and [MATH].', '1901.10691-2-46-3': "As a practical algorithm, simultaneous stochastic gradient descent steps are performed on the discriminator's parameters [MATH] and the generator's parameters [MATH] using the two loss functions [EQUATION] where [MATH] and [MATH] are parameterized with neural networks.", '1901.10691-2-47-0': 'Our unifying result is the following:', '1901.10691-2-48-0': 'Adversarial PFD on the Jensen-Shannon divergence objective [EQUATION] yields the minimax GAN algorithm [REF].', '1901.10691-2-49-0': 'That is, the minimax GAN could have been derived mechanically and from first principles as an instance of adversarial PFD.', '1901.10691-2-49-1': 'To build intuition, we note that the discriminator plays the role of the approximate influence function:', '1901.10691-2-50-0': 'Suppose [MATH] has density [MATH] and [MATH] has density [MATH].', '1901.10691-2-50-1': 'Then the influence function for [MATH] is [EQUATION].', '1901.10691-2-51-0': 'The result follows from [REF]: [EQUATION]', '1901.10691-2-51-1': 'Recall that in the minimax GAN, the optimal discriminator [MATH] satisfies [MATH], so the influence function [MATH] is approximated using the learned discriminator.', '1901.10691-2-52-0': 'Now, we rederive the minimax GAN problem [REF] as a form of adversarial PFD.', '1901.10691-2-52-1': 'We compute:', '1901.10691-2-53-0': 'The convex conjugate of [MATH] is [EQUATION]', '1901.10691-2-53-1': "Setting the integrand's derivative w.r.t. [MATH] to [MATH], we find that pointwise, the optimal [MATH] satisfies [EQUATION].", '1901.10691-2-53-2': 'We eliminate [MATH] in the integrand.', '1901.10691-2-53-3': 'Notice that the first two terms in the integrand cancel after plugging in [MATH].', '1901.10691-2-53-4': 'Since [EQUATION] we obtain that [EQUATION] [REF] yields the representation [EQUATION] an ascent step on which is the [MATH]-step in [REF] with the substitution [MATH].', '1901.10691-2-53-5': 'The descent step corresponds to updating [MATH] to decrease the linear approximation [MATH], which corresponds to the [MATH]-step in [REF].', '1901.10691-2-53-6': 'In fact, a similar argument can be applied to the [MATH]-GANs of [CITATION], which generalize the minimax GAN.', '1901.10691-2-53-7': 'The observation that [MATH]-GANs (and hence the minimax GAN) can be derived through convex duality was also noted by [CITATION].', '1901.10691-2-54-0': '## Non-saturating GAN', '1901.10691-2-55-0': '[CITATION] also proposed an alternative to [REF] called the non-saturating GAN, which prescribes descent steps on [EQUATION].', '1901.10691-2-55-1': "In the step on the generator's parameters [MATH], the [MATH] in the minimax GAN has been replaced with [MATH].", '1901.10691-2-55-2': 'This heuristic change prevents gradients to [MATH] from converging to [MATH] when the discriminator is too confident, and it is for this reason that the loss for [MATH] is called the non-saturating loss.', '1901.10691-2-56-0': 'We consider a slightly modified problem, in which the original minimax loss and the non-saturating loss are summed (and scaled by a factor of [MATH]): [EQUATION]', '1901.10691-2-56-1': 'This also prevents gradients to [MATH] from saturating, achieving the same goal as the non-saturating GAN.', '1901.10691-2-56-2': '[CITATION] and [CITATION] recognize that this process minimizes [MATH].', '1901.10691-2-57-0': 'We claim the following:', '1901.10691-2-58-0': 'PFD on the reverse Kullback-Liebler divergence objective [EQUATION] using the binary classification likelihood ratio estimator to approximate the influence function, yields the modified non-saturating GAN optimization problem [REF].', '1901.10691-2-59-0': 'Suppose [MATH] has density [MATH] and [MATH] has density [MATH].', '1901.10691-2-59-1': 'The influence function for [MATH] is [EQUATION].', '1901.10691-2-60-0': 'The result follows from [REF]: [EQUATION]', '1901.10691-2-60-1': 'Now, because the binary classification loss [EQUATION] is minimized by [MATH], one estimator for [MATH] is simply [EQUATION] where [MATH] is updated as in the [MATH]-step in [REF].', '1901.10691-2-60-2': 'With this approximation scheme, the differentiation step and the descent step in PFD correspond exactly to the [MATH]-step and [MATH]-step respectively in [REF].', '1901.10691-2-60-3': 'Once again, the discriminator serves to approximate the influence function.', '1901.10691-2-61-0': '## Wasserstein GAN', '1901.10691-2-62-0': '[CITATION] propose solving the following saddle-point problem [EQUATION] where [MATH] denotes the Lipschitz constant of [MATH].', '1901.10691-2-62-1': 'The corresponding practical algorithm amounts to simultaneous descent steps on [EQUATION] where [MATH] is reprojected back to the space of [MATH]-Lipschitz functions after each [MATH]-step.', '1901.10691-2-62-2': 'Here, [MATH] is again the generator, and [MATH] is the discriminator, sometimes called the critic.', '1901.10691-2-62-3': 'This algorithm is called the Wasserstein GAN algorithm, so named because this algorithm approximately minimizes the [MATH]-Wasserstein distance [MATH]; the motivation for the [MATH]-step in [REF] is so that the discriminator learns the Kantorovich potential that describes the optimal transport from [MATH] to [MATH].', '1901.10691-2-62-4': 'See e.g. [CITATION] for the full optimal transport details.', '1901.10691-2-63-0': 'We claim that the Wasserstein GAN too is an instance of PFD, and once again, the discriminator plays the role of approximate influence function:', '1901.10691-2-64-0': 'Adversarial PFD on the Wasserstein distance objective [EQUATION] yields the Wasserstein GAN algorithm [REF].', '1901.10691-2-65-0': 'The influence function for [MATH] is the Kantorovich potential corresponding to the optimal transport from [MATH] to [MATH].', '1901.10691-2-66-0': 'See [CITATION], Proposition 7.17.', '1901.10691-2-67-0': 'We remark that the gradient computation in Theorem 3 of [CITATION] is a corollary of [REF] and [REF].', '1901.10691-2-67-1': 'Now, we show that the Wasserstein GAN algorithm can be derived mechanically via convex duality.', '1901.10691-2-67-2': 'The connection between the Wasserstein GAN and convex duality was also observed by [CITATION].', '1901.10691-2-68-0': 'The convex conjugate of [MATH] is [EQUATION].', '1901.10691-2-69-0': 'We use the notation [MATH] to denote the convex indicator function, which is [MATH] if [MATH] is true and [MATH] if [MATH] is false.', '1901.10691-2-70-0': 'Using Kantorovich-Rubinstein duality, we have that [EQUATION] where we use the notation [EQUATION].', '1901.10691-2-70-1': 'By [REF], [EQUATION]', '1901.10691-2-71-0': '[REF] yields the representation [EQUATION].', '1901.10691-2-71-1': 'The adversarial PFD differentiation step therefore corresponds exactly to the [MATH]-step in [REF], and the PFD descent step is exactly the [MATH]-step in [REF].', '1901.10691-2-72-0': '# Variational Inference', '1901.10691-2-73-0': 'In Bayesian inference, the central object is the posterior distribution [EQUATION] where [MATH] is an observed datapoint, [MATH] is the likelihood, [MATH] is the prior.', '1901.10691-2-73-1': 'Unfortunately, the posterior is difficult to compute due to the presence of the integral.', '1901.10691-2-73-2': 'Variational inference therefore reframes this computation as an optimization problem in which a variational posterior [MATH] approximates the true posterior by solving [EQUATION]', '1901.10691-2-74-0': '## Black-box variational inference', '1901.10691-2-75-0': 'This objective is not directly optimizable, due to the presence of the intractable [MATH] term.', '1901.10691-2-75-1': 'The tool of choice for variational inference is the evidence lower bound (ELBO), which rewrites [EQUATION].', '1901.10691-2-75-2': 'Because [MATH] is fixed, we may maximize the ELBO to minimize the KL divergence.', '1901.10691-2-75-3': 'The advantage of doing so is that all the terms inside the expectation are now tractable to evaluate, and thus the expectation may be approximated through Monte Carlo sampling.', '1901.10691-2-75-4': 'This leads to the following practical algorithm, namely stochastic gradient descent on the objective [EQUATION]', '1901.10691-2-75-5': 'This is called black-box variational inference [CITATION].', '1901.10691-2-75-6': '[CITATION] later recognized that ignoring the [MATH]-dependence of the term in the expectation yields the same gradients in expectation; it is this variant that we consider.', '1901.10691-2-75-7': 'Our unification result is the following:', '1901.10691-2-76-0': 'PFD on the variational inference objective [EQUATION] using exact influence functions, yields the black-box variational inference algorithm [REF].', '1901.10691-2-77-0': 'In fact, the influence function turns out to be precisely the inside of the negative ELBO bound:', '1901.10691-2-78-0': 'The influence function for [MATH] is [EQUATION].', '1901.10691-2-79-0': 'The result follows from [REF]: [EQUATION]', '1901.10691-2-79-1': 'In this context, the influence function can be evaluated exactly, so the differentiation step of PFD may be performed without approximation.', '1901.10691-2-79-2': 'The descent step of PFD becomes exactly the descent step on [MATH] of [REF], where the [MATH]-dependence of the term in the expectation is ignored.', '1901.10691-2-79-3': 'We remark that the argument of [CITATION] that this [MATH]-dependence can be ignored can be seen as a corollary of [REF] and [REF].', '1901.10691-2-80-0': '## Adversarial variational Bayes', '1901.10691-2-81-0': 'When the density function of the prior [MATH] or the variational posterior [MATH] is not available, adversarial variational Bayes [CITATION] may be employed.', '1901.10691-2-81-1': 'Here, the quantity [MATH] is approximated by a neural network [MATH] through a binary classification problem, much like [REF].', '1901.10691-2-81-2': 'The resulting algorithm applies simultaneous descent steps on [EQUATION]', '1901.10691-2-81-3': 'This algorithm is another instance of PFD:', '1901.10691-2-82-0': 'PFD on the variational inference objective [MATH], using the binary classification likelihood ratio estimator to approximate the influence function, yields adversarial variational Bayes [REF].', '1901.10691-2-83-0': 'It is easily seen that [EQUATION].', '1901.10691-2-83-1': 'Therefore, the [MATH]-step of [REF] is the differentiation step of PFD, and the [MATH]-step of [REF] is the descent step.', '1901.10691-2-83-2': 'We remark that the gradient computation in Proposition 2 of [CITATION] is a corollary of [REF] and [REF].', '1901.10691-2-84-0': '# Reinforcement Learning', '1901.10691-2-85-0': 'In a Markov decision process, the distribution of states [MATH], actions [MATH], and rewards [MATH] is governed by the distribution [EQUATION] where [MATH] is an initial distribution over states, [MATH] gives the transition probability of arriving at state [MATH] with reward [MATH] from a state [MATH] taking an action [MATH], and [MATH] is a policy that gives the distribution of actions taken when in state [MATH].', '1901.10691-2-85-1': 'In reinforcement learning, we are interested in learning the policy [MATH] that maximizes the expected discounted reward [MATH], where [MATH] is a discount factor, while assuming we only have access to samples from [MATH] and [MATH].', '1901.10691-2-86-0': '## Policy iteration', '1901.10691-2-87-0': 'Policy iteration [CITATION] is one scheme that solves the reinforcement learning problem.', '1901.10691-2-87-1': 'It initializes [MATH] arbitrarily and then cycles between two steps, policy evaluation and policy improvement.', '1901.10691-2-87-2': 'In the policy evaluation step, the state-action value function [MATH] is computed.', '1901.10691-2-87-3': 'In the policy improvement step, the policy is updated to the greedy policy, the policy that at state [MATH] takes the action [MATH] with probability [MATH].', '1901.10691-2-88-0': 'Before we present our unification result, we introduce an arbitrary distribution over states [MATH] and consider the joint distribution [MATH], so that [MATH] is one probability distribution rather than one for every state [MATH].', '1901.10691-2-88-1': 'Now:', '1901.10691-2-89-0': 'PFD on the reinforcement learning objective [EQUATION] using exact influence functions and global minimization of the linear approximation, yields the policy iteration algorithm.', '1901.10691-2-90-0': 'Proofs continue on the following page.', '1901.10691-2-91-0': 'The influence function for [MATH] is [EQUATION] where [MATH] is the state-action value function, [MATH] is the state value function, and [MATH] is the marginal distribution of states after [MATH] steps, all under the policy [MATH].', '1901.10691-2-92-0': 'First, we note that [EQUATION] where we abuse notation to denote [MATH].', '1901.10691-2-93-0': 'We have [EQUATION] or, plugging in the measure, [EQUATION].', '1901.10691-2-93-1': 'The integral is over all free variables; we omit them here and in the following derivation for conciseness.', '1901.10691-2-94-0': 'In computing [MATH], the product rule dictates that a term appear for every [MATH], in which [MATH] is replaced with [MATH].', '1901.10691-2-94-1': 'Hence: [EQUATION] reordering the summations.', '1901.10691-2-94-2': 'Note that for [MATH], the summand vanishes: [EQUATION] since all the variables [MATH] integrate away to [MATH].', '1901.10691-2-94-3': 'This yields: [EQUATION]', '1901.10691-2-94-4': 'Then, substituting the marginal distribution (note [MATH] is not integrated) [EQUATION] we obtain [EQUATION]', '1901.10691-2-94-5': 'Let us rename the integration variables by decreasing their indices by [MATH]: [EQUATION]', '1901.10691-2-94-6': 'Substituting in [EQUATION] we obtain [EQUATION]', '1901.10691-2-94-7': 'Finally, by [REF], we obtain that [EQUATION].', '1901.10691-2-95-0': 'The descent step of PFD corresponds to taking a step on [MATH] to decrease the linear approximation [EQUATION].', '1901.10691-2-95-1': 'Setting [MATH], this simplifies to either [EQUATION]', '1901.10691-2-95-2': 'The most naive way to decrease [REF] is to globally minimize it.', '1901.10691-2-95-3': 'This corresponds to setting [MATH] to be the greedy policy.', '1901.10691-2-95-4': 'Hence, the evaluation of [MATH] in policy iteration corresponds exactly to computing the influence function in the differentiation step of PFD, and the greedy policy update corresponds to applying the descent step.', '1901.10691-2-96-0': '## Policy gradient and actor-critic', '1901.10691-2-97-0': 'Policy iteration exactly computes the linear approximation and nonparametrically minimizes it.', '1901.10691-2-97-1': 'Now we consider algorithms in which the policy is parameterized and the descent step is taken using a gradient step on [REF] or [REF].', '1901.10691-2-97-2': 'If this approach is taken, there is a lot of flexibility in how the influence function can be approximated, but generally speaking, the result is an actor-critic method [CITATION], which describes a class of algorithms that approximates the value function of the current policy and then takes a gradient step on the parameters of the policy using the estimated value function.', '1901.10691-2-97-3': 'We claim:', '1901.10691-2-98-0': 'Approximate PFD on the reinforcement learning objective [MATH], where the influence function is estimated using, for example, Monte Carlo, least squares, or temporal differences, yields an actor-critic algorithm.', '1901.10691-2-99-0': 'There is a huge number of possible approximations to the influence function; we list several and their corresponding algorithms.', '1901.10691-2-99-1': 'The simplest algorithm is the policy gradient algorithm, also known as REINFORCE [CITATION], which directly uses a Monte Carlo estimate of [MATH] as the influence function estimator.', '1901.10691-2-99-2': 'Stochastic value gradients [CITATION] and the closely related deterministic policy gradient [CITATION] fit a neural network to [MATH] using a temporal difference update and use that as the influence function approximation; their use of a neural network makes them compatible with the reparameterization trick.', '1901.10691-2-99-3': 'Advantage actor-critic [CITATION] estimates [MATH] by estimating [MATH] using Monte Carlo and fitting a neural network to [MATH] using least squares.', '1901.10691-2-99-4': 'All of these algorithms are traditionally justified by the celebrated policy gradient theorem [CITATION]; we remark that this theorem is a corollary of [REF] and [REF].', '1901.10691-2-100-0': '## Dual actor-critic', '1901.10691-2-101-0': 'Because [MATH] is not convex, adversarial PFD does not directly apply.', '1901.10691-2-101-1': 'However, the form of [REF] strongly suggests fixing the arbitrary distribution [MATH] to be the discounted marginal distribution of states [MATH].', '1901.10691-2-101-2': 'Closely related to the linear programming formulation of reinforcement learning [CITATION], this choice turns out to convexify [MATH], thus enabling the use of convex duality to approximate its influence function.', '1901.10691-2-101-3': 'We expect to obtain an adversarial formulation of reinforcement learning; one such formulation is the dual actor-critic algorithm [CITATION]: [EQUATION] where [MATH].', '1901.10691-2-101-4': 'Indeed:', '1901.10691-2-102-0': 'Adversarial PFD on the reinforcement learning objective [MATH] yields the dual actor-critic algorithm [REF].', '1901.10691-2-103-0': 'The convex conjugate of [MATH] is [EQUATION] where [MATH] is the unique solution to [MATH], if it exists.', '1901.10691-2-104-0': 'As mentioned in the text, we set the arbitrary distribution [MATH].', '1901.10691-2-104-1': 'In doing so, [MATH] becomes a state-action occupancy measure that describes the frequency of encounters of the state-action pair [MATH] over trajectories governed by the policy [MATH].', '1901.10691-2-104-2': 'It is known that there is a bijection between occupancy measures [MATH] and policies [MATH] [CITATION].', '1901.10691-2-105-0': 'We can enforce this setting by redefining [EQUATION] where again [MATH] is the convex indicator function.', '1901.10691-2-105-1': 'This equation can be rewritten as [EQUATION] where [MATH].', '1901.10691-2-105-2': 'The constraint is known as the Bellman flow equation.', '1901.10691-2-105-3': 'This formulation is convex, as it is the sum of an affine function and an indicator of a convex set (indeed, an affine subspace).', '1901.10691-2-106-0': 'We recall [MATH], where [MATH].', '1901.10691-2-106-1': 'Now, [MATH] is uniquely defined by [MATH] if a solution to the equation exists.', '1901.10691-2-106-2': 'To see this, note that [MATH] is the fixed point of the Bellman operator [MATH] defined by [EQUATION] which is contractive and therefore has a unique fixed point.', '1901.10691-2-106-3': 'A representation of [MATH] may be obtained via fixed point iteration using [MATH] for an arbitrary action [MATH]: [EQUATION] where the expectation is taken under the deterministic policy [MATH].', '1901.10691-2-107-0': 'We rewrite [MATH] using a Lagrange multiplier [MATH] [EQUATION]', '1901.10691-2-107-1': 'Note that [MATH] is convex in [MATH]; this stems from the fact that [EQUATION].', '1901.10691-2-107-2': 'The result follows from [REF].', '1901.10691-2-108-0': 'Using [REF], adversarial PFD therefore recovers [REF]: [EQUATION]', '1901.10691-2-109-0': '# Conclusion', '1901.10691-2-110-0': 'This paper suggests several new research directions.', '1901.10691-2-110-1': 'First is the transfer of insight and specialized techniques from one domain to another.', '1901.10691-2-110-2': 'As just one example, in the context of GANs, [CITATION] claim that constraining the discriminator to be [MATH]-Lipschitz improves the stability of the training algorithm - could similarly constraining the analogous object in reinforcement learning, namely an approximation to the advantage function, lead to improved stability in deep reinforcement learning?', '1901.10691-2-111-0': 'Moreover, the abstract viewpoint taken in this paper allows for the simultaneous development of new algorithms for GANs, variational inference, and reinforcement learning.', '1901.10691-2-111-1': 'General influence function approximation techniques in the spirit of convex duality could improve all three fields at once.', '1901.10691-2-111-2': 'More sophisticated descent techniques beyond gradient descent on parameterized probability distributions, such as Frank-Wolfe or trust-region methods, could improve learning or yield valuable convergence guarantees.', '1901.10691-2-112-0': 'Finally, this paper unlocks the possibility of applying probability functional descent to new problems.', '1901.10691-2-112-1': 'In principle, the algorithm can be applied mechanically to any situation where one wants to optimize over probability distributions, possibly leading to new, straightforward ways to solve problems in, for example, mathematical finance, mean field games, or POMDPs.', '1901.10691-2-112-2': 'One could argue that the current excitement over deep learning began once researchers realized that to solve a problem, they could simply write a loss function and then rely on automatic differentiation and gradient descent to minimize it.', '1901.10691-2-112-3': 'We hope that probability functional descent provides a similarly turnkey solution for optimizing loss functions defined on probability distributions and leads to a similar burst of research activity.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1901.10691

1801.05041

{'1801.05041-1-0-0': 'This paper introduces grouped latent heterogeneity in panel data quantile regression.', '1801.05041-1-0-1': 'More precisely, we assume that the observed individuals come from a heterogeneous population with an unknown, finite number of types.', '1801.05041-1-0-2': 'The number of types and group membership is not assumed to be known in advance and is estimated by means of a convex optimization problem.', '1801.05041-1-0-3': 'We provide conditions under which group membership is estimated consistently and establish asymptotic normality of the resulting estimators.', '1801.05041-1-1-0': 'myheadings Panel Quantile regression with group fixed effectsGu and Volgushev', '1801.05041-1-2-0': '# Introduction', '1801.05041-1-3-0': 'It is widely accepted in applied econometrics that individual latent effects constitute an important feature of many economic applications.', '1801.05041-1-3-1': 'When panel data are available, a common approach is to incorporate latent structures in completely nonrestrictive way, i.e. the fixed effect approach.', '1801.05041-1-3-2': 'The fixed effect approach is attractive as it imposes minimal assumptions on the structure of the latent effects and on the correlation between the latent effects and the observed covariates.', '1801.05041-1-3-3': 'However, it suffers from the well known incidental parameter problem (nounNeymanScott) that leads to severely biased estimates for model parameters in short panels.', '1801.05041-1-3-4': 'Our contribution, which builds uponasnounKoenker2004, is a linear quantile regression method that accommodates grouped fixed effects.', '1801.05041-1-3-5': 'The advantages over existing proposals are twofold.', '1801.05041-1-3-6': 'First, quantile regression provides a flexible way to model the effect of covariates on observed heterogeneity.', '1801.05041-1-3-7': 'Second, grouped fixed effects maintain the merit of unrestricted correlation between the latent effects and the observables and strike a good balance between the classical fixed effects approach and the other extreme which completely ignores latent heterogeneity.', '1801.05041-1-3-8': 'In contrast to [CITATION], where the fixed effects are treated as nuisance parameters and are regularized to achieve a more efficient estimator for the global parameter, we are interested in estimating the particular group structure of the latent effects together with common parameters in the model.', '1801.05041-1-4-0': 'There is ample evidence from empirical studies that it is often reasonable to consider a number of homogeneous groups within a heterogeneous population.', '1801.05041-1-4-1': 'This discrete approach was taken byasnounHeckmanSinger82 for duration analysis of unemployment spells of a heterogeneous population of workers.asnounBesterHansen argue that in many applications individuals or firms are grouped naturally by some observable covariates such as classes, schools or industry codes.', '1801.05041-1-4-2': 'It is also widely accepted in discrete choice model that individual players are classified as a number of latent types (for instanceasnounKeane among many others).', '1801.05041-1-4-3': 'Unfortunately researchers often do not have a priori information on the group structure, which then needs to be estimated from the data.', '1801.05041-1-5-0': 'We do not assume any prior knowledge of the group structure and combine the quantile regression loss function with the recently proposed convex clustering penalty of [CITATION].', '1801.05041-1-5-1': 'The resulting optimization problem remains convex and can be solved in a fast and reliable fashion.', '1801.05041-1-5-2': 'The convex clustering method introduces a [MATH]-constraint on the pair-wise difference of the individual fixed effects, which tends to push the fixed effects into clusters and the number of clusters is controlled by a penalty parameter.', '1801.05041-1-5-3': 'We show consistency of classification for a suitable range of penalty parameters when [MATH] and [MATH] tend to infinity jointly and propose an information criterion for selecting the number of groups.', '1801.05041-1-5-4': 'We also prove asymptotic normality of the resulting estimators.', '1801.05041-1-6-0': '# Related literature', '1801.05041-1-7-0': 'There is a fast growing literature of quantile regression for panel data.', '1801.05041-1-7-1': 'The pioneering paper,asnounKoenker2004, takes the fixed effect approach and introduces individual latent effects as location shifts.', '1801.05041-1-7-2': 'These individual effects are regularized through an [MATH] penalty which shrinks them towards a common value.asnounLamarche proposes an optimal way to choose the corresponding penalty parameter to optimize the asymptotic efficiency of the common parameters of the conditional quantile function, seeasnounharding2017 for an extension of this approach.', '1801.05041-1-7-3': 'Another line of work that focuses on estimating common parameters while treating individual effects as nuisance includes without using penalization includesasnounKato,asnoungalvao2015 andasnoungalvao2016.', '1801.05041-1-8-0': 'Alternative approaches have also emerged.asnounAD take a random effect view of these individual latent effects.', '1801.05041-1-8-1': 'They consider a correlated random-effect model in the spirit ofasnounChamberlain1982 where the individual effects are modeled through a linear regression of some covariates.', '1801.05041-1-8-2': 'This is further developed inasnounAB16 where the conditional quantile function of the unobserved heterogeneity is modeled as a function of observable covariates.', '1801.05041-1-8-3': 'Related literature on non-separable panel data models includesasnounChernozhukov andasnounEvdokimov and references therein.', '1801.05041-1-9-0': 'Estimating cluster structure has a long history in Statistics and Economics, and has generated a rich and mature literature.', '1801.05041-1-9-1': 'A general overview is given inasnounKaufman.', '1801.05041-1-9-2': 'Among the many available clustering algorithms, the k-means algorithm (nounMacQueen) which uses a partition approach to form clusters is one of the most popular methods.', '1801.05041-1-9-3': 'It has been successfully applied in many economic applications, for instanceasnounLinNg,asnounBM andasnounAndoBai.', '1801.05041-1-9-4': 'Finite mixture models provide an alternative, likelihood based approach.', '1801.05041-1-9-5': 'In the latter, optimal grouping is usually achieved by maximizing the likelihood of the observed data.asnounSun builds a multinomial logistic regression model to infer the group pattern while many identification and estimation methods become available with a recent literature on nonparametric finite mixture model (nounAllman andasnounKasahara among others).', '1801.05041-1-9-6': 'Despite being popular and well studied, both the k-means and the finite mixture methods are computationally challenging.', '1801.05041-1-9-7': 'The corresponding optimization problems are non-convex, which leads to computational issues such as solutions converging to local optima and sensitivity to the starting values of parameters.', '1801.05041-1-10-0': 'A new approach, which has emerged more recently, is based on introducing lasso-type penalties which encourage clustering among observations or estimated parameters.asnounHocking propose a convex relaxation of the hierarchical clustering criterion.', '1801.05041-1-10-1': 'Their approach can be regarded as replacing the classical [MATH] constraint on pair-wise differences by the convex [MATH] constraint and is equivalent to introducing an additive [MATH]-type penalty.', '1801.05041-1-10-2': 'A major computational advantage of this approach is that the resulting objective function is convex.', '1801.05041-1-10-3': 'Further modifications and a theoretical analysis of the resulting clustering procedure were considered in [CITATION], [CITATION] and [CITATION], among others.', '1801.05041-1-10-4': 'All of those authors combine [MATH] penalties with the classical [MATH] loss, and their theoretical results are not applicable to the non-smooth quantile loss function which is the main objective in this paper.', '1801.05041-1-10-5': 'An alternative penalization based approach to clustering was recently introduced by [CITATION].', '1801.05041-1-10-6': 'However, the proposed penalty function does not lead to a convex optimization problem and the computational advantage of convex clustering is lost.', '1801.05041-1-11-0': '# Theoretical analysis', '1801.05041-1-12-0': 'Assume that for individuals [MATH] we observe repeated measures [MATH] where [MATH] denote covariates and [MATH] are responses.', '1801.05041-1-12-1': 'For the theoretical developments that follow, we shall maintain the assumption that data are i.i.d. within individuals and independent across individuals.', '1801.05041-1-12-2': 'The main object of interest in this paper is conditional [MATH]-quantile function of [MATH] given [MATH], which we will denote by [MATH].', '1801.05041-1-12-3': 'We assume that [MATH] is of the form [EQUATION] with individual fixed effects [MATH] taking only a finite number of different values.', '1801.05041-1-12-4': 'More precisely, we will make the following assumption', '1801.05041-1-13-0': 'Assumption (C) implies that the individual fixed effects are grouped into [MATH] distinct groups and that the group centers are separated.', '1801.05041-1-13-1': 'In this paper, we explicitly allow the group membership, and even the number of groups to be unknown.', '1801.05041-1-13-2': 'Our main objective is to jointly estimate the number of groups, unknown group structure and parameters [MATH] from the observations.', '1801.05041-1-13-3': 'To achieve this, we consider penalized estimators of the form [EQUATION] where [EQUATION].', '1801.05041-1-13-4': "Here [MATH] denotes the usual 'check function' and the weights [MATH] are allowed to depend on [MATH] and the data, one particular choice is discussed in Remark [REF].", '1801.05041-1-13-5': 'The form of the penalty is motivated by the work of [CITATION].', '1801.05041-1-13-6': 'Intuitively, large values of [MATH] will push different coefficients closer together and result in clustered structure of the estimators [MATH].', '1801.05041-1-13-7': 'In order to analyze the asymptotic properties of resulting estimators, we need to make some technical assumptions on the data generating process.', '1801.05041-1-13-8': 'Define [MATH] and let [MATH] denote the support of [MATH].', '1801.05041-1-14-0': 'Assumptions (A1)-(A3) are fairly standard and routinely imposed in the quantile regression literature.', '1801.05041-1-14-1': 'Similar assumptions have been made, for instance in [CITATION] [see assumptions (B1)-(B3) in that paper].', '1801.05041-1-15-0': 'To state our first main result define [EQUATION].', '1801.05041-1-16-0': 'Let assumptions (A1)-(A3), (C) hold and assume that [MATH], [MATH] and [EQUATION]', '1801.05041-1-16-1': 'Denote the ordered unique values of [MATH] by [MATH] and define the sets [MATH].', '1801.05041-1-16-2': 'Then [EQUATION].', '1801.05041-1-17-0': 'Next we provide a simple choice of the penalty [MATH] which satisfies condition [REF] of Theorem [REF].', '1801.05041-1-18-0': 'In practice, we propose to use weights of the form [MATH] where [MATH] are the fixed effects quantile regression estimator [EQUATION] of nounKato.', '1801.05041-1-18-1': 'This form of weighting by preliminary estimators is motivated by the work of nounzou2006 on adaptive lasso.', '1801.05041-1-18-2': 'Intuitively, weighting by preliminary estimated distances tends to give smaller penalties to coefficients from different groups thus reducing some of the bias that is typically present in the classical lasso.', '1801.05041-1-18-3': 'Define [EQUATION].', '1801.05041-1-18-4': 'Given the results in [CITATION] it is not difficult to prove that, under conditions (A1)-(A3), [EQUATION]', '1801.05041-1-18-5': 'Given this choice of weights, the conditions [MATH] is satisfied provided that [MATH].', '1801.05041-1-18-6': 'The other conditions in [REF] take the form [EQUATION].', '1801.05041-1-18-7': 'Assuming that [MATH], this provides a range of possible values for [MATH] which will ensure that [REF] holds.', '1801.05041-1-19-0': 'Theorem [REF] shows that it is possible to correctly estimate both number of groups and group membership with probability tending to one by choosing a suitable penalty function.', '1801.05041-1-19-1': 'However, using the estimated coefficients [MATH] directly can be problematic because the penalty term can induce a bias, in particular when [MATH] is small.', '1801.05041-1-19-2': 'While the adaptive weighting by [MATH] can reduce the problem and lead to asymptotically unbiased estimates, bias can still be a problem in finite samples.', '1801.05041-1-19-3': 'A typical approach in the literature to reduce bias which results from lasso-type penalties is to view the lasso problem solution as a candidate model (in our case, a candidate grouping of [MATH]) and re-fit based on this candidate model (see [CITATION] or [CITATION]).', '1801.05041-1-19-4': 'An additional challenge for applying the proposed procedure in practice is the choice of the penalty level [MATH].', '1801.05041-1-19-5': 'The asymptotic results in Theorem [REF] and Remark [REF] provide some guidance on this issue.', '1801.05041-1-19-6': 'However, choosing a precise constant in addition to suitable powers of [MATH] remains an important problem which is particularly relevant if [MATH] is relatively small.', '1801.05041-1-20-0': 'We propose to combine the re-fitting idea with a simple information criterion which will simultaneously reduce the bias problem and provide a simple way to select a final model.', '1801.05041-1-20-1': 'A formal description of our approach is as follows', '1801.05041-1-21-0': 'For a grid of candidate values [MATH], compute [here, [MATH] are defined in [REF]] [EQUATION]', '1801.05041-1-21-1': 'For each [MATH], let [MATH] denote the unique values of [MATH] and let [MATH].', '1801.05041-1-21-2': 'Define the re-fitted estimators [EQUATION]', '1801.05041-1-21-3': 'Define [EQUATION] where [MATH] is a constant to be estimated which is discussed in Remark [REF].', '1801.05041-1-21-4': 'Let [MATH] and denote by [MATH] the corresponding number of groups.', '1801.05041-1-21-5': 'The final estimated parameters are given by [MATH] and the corresponding groups are [MATH].', '1801.05041-1-21-6': 'To simplify notation let [MATH].', '1801.05041-1-22-0': 'Our next result shows that, under fairly general conditions on the penalty parameter [MATH], the above procedure selects the correct number of groups with probability tending to one.', '1801.05041-1-22-1': "Moreover, the estimators [MATH] are shown to enjoy the 'oracle property', i.e. they have the same asymptotic distribution as estimators which are based on the true (but unknown) grouping of individuals.", '1801.05041-1-22-2': 'Before making this statement more formal, we need some additional notation.', '1801.05041-1-22-3': "Let [EQUATION] denote the infeasible 'oracle' which uses the true group membership.", '1801.05041-1-22-4': "The asymptotic variance of the oracle estimator is conveniently expressed in terms of the following two limit which we assume to exist [EQUATION] where [MATH] and [MATH] denotes the [MATH]'th unit vector in [MATH].", '1801.05041-1-23-0': 'Additionally, we need the following condition on the grid [MATH]', '1801.05041-1-24-0': '[(G)] For each [MATH], denote the grid values by [MATH] where [MATH] can depend on [MATH].', '1801.05041-1-24-1': 'There exists a sequence [MATH] such that [MATH].', '1801.05041-1-25-0': 'Assumption (G) is fairly mild.', '1801.05041-1-25-1': 'It only requires that among the candidate values for [MATH] there exists one value so that [MATH] satisfies the assumptions of Theorem [REF].', '1801.05041-1-25-2': 'In practice, we recommend choosing a grid of values that results in sufficiently many different numbers of groups.', '1801.05041-1-26-0': 'Let assumptions (A1)-(A3), (C), (G) hold and assume that [MATH] and [MATH] grows at most polynomially in [MATH] and [MATH].', '1801.05041-1-26-1': 'Assume that there exists [MATH] such that [MATH] with probability tending to one and that [MATH].', '1801.05041-1-26-2': 'Then [MATH] and [EQUATION]', '1801.05041-1-26-3': 'In order to make the proposed estimation procedure fully data-driven, we need to specify a choice for the tuning parameters [MATH].', '1801.05041-1-26-4': 'In our simulations, we found that the following choices lead to good results: [MATH] and [MATH] with [EQUATION] where [MATH] denotes the empirical cdf of the regression residuals from the fixed effects quantile regression estimator given in [REF], [MATH] denotes the corresponding empirical quantile function, and [MATH] is a bandwidth parameter which we set to [MATH] in our simulations.', '1801.05041-1-27-0': 'To motivate this particular choice of constant [MATH], observe the following expansion, which is derived in detail in the proof of Theorem [REF] [EQUATION].', '1801.05041-1-27-1': 'This shows that plugging in the estimated (by fixed effects quantile regression) instead of true residuals underestimates the objective function evaluated at the residuals by roughly the first term on the right-hand side in the above expression.', '1801.05041-1-27-2': 'This term needs to be dominated by the penalty if we want to avoid selecting models that are too large, and so it is natural to scale the penalty by a constant which is proportional to [MATH] in order to ensure reasonable performance across different data generating processes.', '1801.05041-1-27-3': 'Under the simplifying assumption that [MATH] does not depend on [MATH], this term equals [MATH], and under the same assumptions [MATH] provides a consistent estimator for the latter, seeasnounkoenker2005.', '1801.05041-1-28-0': "Theorem [REF] and Theorem [REF] hold 'point-wise' in the parameter space, and we expect that deriving a similar result uniformly in the parameter space (in particular, if cluster centers are allowed to depend on [MATH] and if their separation is lost, seeasnounleeb2008 for such findings in the context of classical lasso penalized regression) is impossible.", '1801.05041-1-28-1': "It is a well established fact in the Statistics and Econometrics literature that inference which is based on such 'point-wise' asymptotic results can be unreliable.", '1801.05041-1-28-2': 'Recently, several approaches to alleviate this problem and achieve uniformly valid post-regularization inference have been proposed (see, among others, [CITATION], [CITATION] and [CITATION]).', '1801.05041-1-28-3': 'Applying similar ideas to the present setting is a very important question which we leave for future research.', '1801.05041-1-29-0': '# Monte Carlo Simulations', '1801.05041-1-30-0': 'To assess the finite sample performance of the convex clustering panel quantile regression estimator, we apply the method to several simulated data.', '1801.05041-1-30-1': 'In particular, we consider data generated from two models and two error distributions for a range of [MATH] and [MATH].', '1801.05041-1-30-2': 'The responses, [MATH], are generated by either a location shift model [EQUATION] or a location scale shift model [EQUATION] where the individual latent effect [MATH], are generated from three groups taking values [MATH] with equal proportions.', '1801.05041-1-30-3': 'The covariates [MATH] are iid random variables following a standard normal distribution and the parameters [MATH] and [MATH].', '1801.05041-1-30-4': 'The error term [MATH] are iid following either a standard normal distribution or a student t distribution with three degrees of freedom.', '1801.05041-1-31-0': 'We begin with some computational details for solving the optimization problem stated in ([REF]).', '1801.05041-1-31-1': 'In practice, we solve a normalized version of ([REF]) as follows [EQUATION] this is equivalent to the objective ([REF]) except that [MATH] is adjusted according to [MATH] and [MATH] so that we can use a generic grid in the simulation.', '1801.05041-1-31-2': 'A closer look reveals that it is a linear programming problem that can be efficiently solved by the interior point (IP hereafter) optimizer or simplex optimizer in any reliable solvers.', '1801.05041-1-31-3': 'We solve the Dual problem of ([REF]) using the Mosek optimization software of [CITATION] through the R interface Rmosek of [CITATION].', '1801.05041-1-31-4': 'The Mosek IP routine provides two solutions by default.', '1801.05041-1-31-5': 'One corresponds to the interior point optimum, and the other is an optimal basic solution which is further obtained through a post-processing step which is often more accurate than the IP solution.', '1801.05041-1-31-6': 'We used the basic solution throughout the simulation exercise.', '1801.05041-1-32-0': 'We first investigate the performance of using the information criteria for estimating the number of groups.', '1801.05041-1-32-1': 'Table [REF] and Table [REF] report the proportion of estimated number of groups under the two models for a different combinations of [MATH] and [MATH]for [MATH] and [MATH], respectively.', '1801.05041-1-32-2': 'Throughout the simulations, we used an equally spaced grid of [MATH] values with width [MATH] and support [MATH].', '1801.05041-1-32-3': 'This grid was chosen to ensure that the number of groups estimated for different [MATH] values covers the integers in range [MATH].', '1801.05041-1-32-4': 'For the IC criteria, the sparsity function [MATH] is estimated with bandwidth [MATH].', '1801.05041-1-33-0': 'The results suggest that the performance for [MATH] is better than for [MATH].', '1801.05041-1-33-1': 'When [MATH], the estimates for the number of groups for both error distributions and for different [MATH] are mostly satisfactory.', '1801.05041-1-33-2': 'The performance for [MATH] error deteriorates compared to those with normal error, especially for higher quantiles.', '1801.05041-1-33-3': 'For [MATH] and quantiles other than the median the proposed method should be used with caution.', '1801.05041-1-34-0': 'Table [REF] and Table [REF] summarize the finite sample properties of [MATH] for [MATH] and [MATH], respectively.', '1801.05041-1-34-1': 'The standard errors we used for constructing confidence intervals of nominal size [MATH] is the sandwich form covariance for non-iid data in the quantreg package.', '1801.05041-1-34-2': 'For different combinations of [MATH] and [MATH], the coverage property and the RMSE performance of [MATH] is very satisfactory.', '1801.05041-1-34-3': 'The results do not change much from normal error to [MATH] error and from median to higher quantiles.', '1801.05041-1-35-0': 'Last, we report in Table [REF] and Table [REF] the proportion of perfect classification of individual effects out of 400 repetitions and the average value of the percentage of correct classification together with their standard error.', '1801.05041-1-35-1': 'Since the comparison of the estimated membership and the true membership only makes sense when [MATH], the estimated membership are based on [MATH] for which [MATH] (see [CITATION] for a similar approach).', '1801.05041-1-35-2': 'Results suggest that for [MATH] and [MATH], the group membership estimation is quite satisfactory.', '1801.05041-1-35-3': 'While the proportion of perfect matches is low even for [MATH], the average proportion of correct classification shows that those effects are typically due to very few misclassified individuals.', '1801.05041-1-35-4': 'For [MATH] perfect classification is almost impossible while average correct classification rates remain reasonable.', '1801.05041-1-35-5': 'These results suggest that for small [MATH], there is just not enough information available for each individual to hope for perfect classification.', '1801.05041-1-36-0': '# Conclusions and future extensions', '1801.05041-1-37-0': 'The present paper suggests a simple and computationally efficient way to incorporate group fixed effects into a panel data quantile regression by means of a convex clustering penalty.', '1801.05041-1-37-1': 'We develop theoretical results on consistent group structure estimation and discuss the asymptotic properties of the resulting joint and group-specific estimators.', '1801.05041-1-38-0': 'There are several directions that we plan to explore in the future.', '1801.05041-1-38-1': 'First, our theory focused on individual fixed effects while assuming common slope coefficients.', '1801.05041-1-38-2': 'It is equally interesting to allow for group structure in some of the slope coefficients while keeping other slope coefficients common across individuals, perhaps even allowing for individual fixed effects.', '1801.05041-1-38-3': 'This can be achieved by straightforward modifications of the penalization approach which we explored so far, but a more detailed theoretical analysis of this approach remains beyond the scope of the present paper.', '1801.05041-1-39-0': 'Second, one can take the standpoint that in many applications there is no exact group structure.', '1801.05041-1-39-1': 'In such settings, an alternative interpretation of the penalty which we investigated is as a way of regularizing problems that have too many parameters.', '1801.05041-1-39-2': 'Such an interpretation is in the spirit of the proposals of [CITATION] and [CITATION], and a detailed investigation of the resulting bias-variance trade-off warrants further research.', '1801.05041-1-40-0': 'Finally, a deeper analysis of issues that are related to uniformity of distributional approximation in the entire parameter space was not addressed here, but remains an important theoretical and practical question.', '1801.05041-1-41-0': '# Proofs', '1801.05041-1-42-0': 'We begin by collecting some useful facts and defining additional notation.', '1801.05041-1-42-1': "We will repeatedly make use of Knight's identity (see [CITATION], p. 121)) which holds for [MATH]: [EQUATION]", '1801.05041-1-42-2': 'Additionally, let [MATH].', '1801.05041-1-42-3': 'The symbols [MATH] will mean that there exists a non-random constant [MATH] which is independent of [MATH] such that [MATH] and [MATH], respectively.', '1801.05041-1-42-4': 'Define [MATH] and let [MATH] denote the conditional cdf of [MATH] given [MATH].', '1801.05041-1-42-5': 'When there is no risk of confusion, we will also write [MATH] instead of [MATH].', '1801.05041-1-42-6': 'Define [MATH].', '1801.05041-1-43-0': '## Proof of Theorem [REF]', '1801.05041-1-44-0': 'We begin by stating some useful technical results which will be proved at the end of this section.', '1801.05041-1-45-0': 'For any fixed [MATH] define [MATH].', '1801.05041-1-45-1': 'Then we have under assumptions (A1)-(A3) [EQUATION] where [EQUATION].', '1801.05041-1-46-0': 'Under assumptions (A1)-(A3) there exist [MATH] such that for all [MATH] [EQUATION]', '1801.05041-1-46-1': 'Under assumption (A1) define for fixed [MATH] [EQUATION]', '1801.05041-1-46-2': 'We have for any fixed [MATH], provided that [MATH] [EQUATION]', '1801.05041-1-46-3': 'Proof of Theorem [REF]', '1801.05041-1-47-0': 'Step 1: first bounds In this step we shall prove that [EQUATION]', '1801.05041-1-47-1': 'Combine the results in Lemma [REF] and Lemma [REF] to find that any minimizer of [MATH] must satisfy [EQUATION]', '1801.05041-1-47-2': 'Let [MATH].', '1801.05041-1-47-3': 'Then for any [MATH] [EQUATION] i.e. by Lemma [REF] [EQUATION] and in particular [MATH] as long as [MATH].', '1801.05041-1-47-4': 'Provided that [MATH] we obtain [EQUATION]', '1801.05041-1-47-5': 'Define [MATH].', '1801.05041-1-47-6': 'Next we will prove that, provided [MATH] also [EQUATION].', '1801.05041-1-47-7': 'To this end, it suffices to prove that [MATH] for any [MATH].', '1801.05041-1-47-8': 'Define [EQUATION].', '1801.05041-1-47-9': 'Define the set [MATH].', '1801.05041-1-47-10': 'Observe that [EQUATION]', '1801.05041-1-47-11': 'Thus [EQUATION]', '1801.05041-1-47-12': 'Now since [MATH] and by definition of [MATH] it follows that under (C) [EQUATION] and since by assumption [MATH] we obtain [EQUATION].', '1801.05041-1-48-0': 'Next we note that for any [MATH] with [MATH] we have [EQUATION] with probability tending to one by [REF] and the definition of [MATH].', '1801.05041-1-48-1': 'For [MATH] with [MATH] note that by Lemma [REF] [EQUATION] where the [MATH] terms are uniform in [MATH].', '1801.05041-1-48-2': 'Thus [EQUATION]', '1801.05041-1-48-3': 'Summarizing we have proved that [EQUATION]', '1801.05041-1-48-4': 'Under the conditions [MATH] the last line is strictly positive with probability tending to one unless [MATH] with probability tending to one.', '1801.05041-1-48-5': 'Thus the proof of [REF] is complete.', '1801.05041-1-49-0': 'Step 2: recovery of clusters with probability to one', '1801.05041-1-50-0': 'To simplify notation, assume that individual [MATH] belongs to cluster 1, individual [MATH] to cluster 2 and so on.', '1801.05041-1-50-1': 'Since all cluster can be handled by similar arguments we only consider the first cluster.', '1801.05041-1-50-2': 'Let [MATH] denote the distinct values of [MATH], ordered in increasing order, and let [MATH].', '1801.05041-1-50-3': 'Again, to simplify notation assume w.o.l.g. that [MATH].', '1801.05041-1-50-4': 'To prove the result, we proceed in an iterative way.', '1801.05041-1-50-5': 'We will prove by contradiction that [MATH], i.e. all estimators of individuals from cluster 1 take the same value.', '1801.05041-1-50-6': 'Assume that [MATH].', '1801.05041-1-51-0': 'We will now prove by contradiction that [MATH].', '1801.05041-1-51-1': 'Assume that [MATH].', '1801.05041-1-51-2': 'Define [MATH] for [MATH] and [MATH] for [MATH].', '1801.05041-1-51-3': 'By [REF] Lemma [REF] we find that [EQUATION]', '1801.05041-1-51-4': 'Next, observe that by construction, under (C) and using the fact that [MATH], [EQUATION]', '1801.05041-1-51-5': 'From this we obtain [EQUATION] where the last inequality holds for sufficiently large [MATH] since by assumption [MATH] and since we assumed [MATH] so that [MATH].', '1801.05041-1-51-6': 'However, this is a contradiction to the fact that [MATH] minimizes [MATH].', '1801.05041-1-52-0': 'In a similar fashion, one can prove that [MATH].', '1801.05041-1-52-1': 'Just define [MATH] and proceed as above.', '1801.05041-1-52-2': 'Since [MATH] and we have already proved that [MATH] this leads to a contradiction with [MATH], and hence [MATH].', '1801.05041-1-52-3': 'All other clusters can be handled in a similar fashion and that completes the proof of the second step.', '1801.05041-1-52-4': '[MATH]', '1801.05041-1-53-0': "Proof of Lemma [REF] Apply Knight's identity [REF] to find that [EQUATION]", '1801.05041-1-53-1': 'Hence it follows that [EQUATION]', '1801.05041-1-53-2': 'Now by a Taylor expansion [EQUATION] so the bound on [MATH] is established.', '1801.05041-1-54-0': 'Next define the classes of functions [EQUATION]', '1801.05041-1-54-1': 'Note that the class of functions [MATH] has envelope function [MATH].', '1801.05041-1-54-2': 'Thus by Lemma 2.6.15 and Theorem 2.6.7 of [CITATION] the class of functions [MATH] satisfies, for any probability measure [MATH], [MATH] for some finite constants [MATH](here, [MATH] denotes the covering number, see Section 2.1 of [CITATION]).', '1801.05041-1-54-3': 'Moreover, [MATH], and elementary computations with covering numbers show that [MATH] for some finite constants [MATH].', '1801.05041-1-54-4': 'Hence we find that by Theorem 2.14.9 of [CITATION]), for any [MATH], [EQUATION] for some constant [MATH] that depends only on [MATH] (here, [MATH] denotes outer probability).', '1801.05041-1-54-5': 'Letting [MATH] and applying the union bound for probabilities we obtain [EQUATION]', '1801.05041-1-54-6': 'Hence [EQUATION]', '1801.05041-1-54-7': 'Thus the bound on [MATH] follows and the proof is complete.', '1801.05041-1-54-8': '[MATH]', '1801.05041-1-55-0': "Proof of Lemma [REF] Observe that by Knight's identity [REF] [EQUATION]", '1801.05041-1-55-1': 'Now under assumption (A2) [MATH] a.s., and thus given (A1) [EQUATION]', '1801.05041-1-55-2': 'This shows the upper bound in [REF].', '1801.05041-1-55-3': 'For the lower bound, note that [MATH] is non-decreasing almost surely.', '1801.05041-1-55-4': 'Moreover, [MATH] a.s. by (A3) and thus by (A2) and (A3) we have almost surely [EQUATION].', '1801.05041-1-55-5': 'Define [MATH].', '1801.05041-1-55-6': 'Noting that [MATH] is non-decreasing almost surely, it follows that a.s. [EQUATION] where the last inequality follows since by definition [MATH] a.s. Finally, under assumption (A1), [MATH].', '1801.05041-1-56-0': 'Summarizing, we find [EQUATION] which proves the lower bound in [REF].', '1801.05041-1-56-1': 'Thus the proof of the Lemma is complete.', '1801.05041-1-56-2': '[MATH]', '1801.05041-1-57-0': 'Proof of Lemma [REF] Consider the class of functions [EQUATION].', '1801.05041-1-57-1': 'Note that by construction [MATH] for all [MATH] and moreover [MATH].', '1801.05041-1-57-2': 'This shows the existence of constants [MATH] such that for all [MATH] for [MATH] where [MATH] depends on [MATH] only and [MATH] denotes the measure corresponding to [MATH].', '1801.05041-1-57-3': 'Thus we have by Theorem 2.14.9 of [CITATION], [EQUATION] where the constant [MATH] depends only on [MATH] and [MATH] denotes outer probability.', '1801.05041-1-57-4': 'Set [MATH] to bound the right-hand side above by [MATH].', '1801.05041-1-57-5': 'Defining the events [EQUATION] we obtain [EQUATION].', '1801.05041-1-57-6': 'Finally, note that under (A1) we have a.s. [EQUATION].', '1801.05041-1-57-7': 'This completes the proof.', '1801.05041-1-57-8': '[MATH]', '1801.05041-1-58-0': '## Proof of Theorem [REF]', '1801.05041-1-59-0': 'We begin by stating a useful technical result that will be proved at the end of this section.', '1801.05041-1-60-0': 'Under assumptions (A1)-(A3) [EQUATION] where, defining [MATH], there exists a constant [MATH] independent of [MATH] such that [EQUATION]', '1801.05041-1-60-1': 'Proof of Theorem [REF] The proof proceeds in several steps.', '1801.05041-1-60-2': "First, we note that the 'oracle' estimation problem [REF] corresponds to a classical, fixed-dimensional quantile regression with true parameter vector [MATH] and [MATH] independent observations [MATH] where [MATH] where [MATH] denotes the k'th unit vector in [MATH].", '1801.05041-1-60-3': 'A straightforward extension of classical proof techniques in parametric quantile regression shows that under assumptions (A1)-(A3) and (C) the oracle estimator is asymptotically normal as claimed.', '1801.05041-1-61-0': 'Second, we observe that by definition of the optimization problem the estimated group structure [MATH] is the same for all values of [MATH] with [MATH] that give rise to the same number of groups.', '1801.05041-1-61-1': 'Since the value of [MATH] depends only on [MATH], it suffices to minimize [MATH] over those values of [MATH] that correspond to different numbers of groups.', '1801.05041-1-61-2': 'Denote the distinct estimated numbers of groups by [MATH], the corresponding estimated groupings by [MATH], and the corresponding values of [MATH] by [MATH].', '1801.05041-1-61-3': 'By assumption (G) and Theorem [REF], the probability of the event [EQUATION]', '1801.05041-1-61-4': 'Hence it suffices to prove that [EQUATION]', '1801.05041-1-61-5': 'Once this result is established, we directly obtain [EQUATION] and thus the asymptotic distribution of [MATH] matches that of the oracle estimator.', '1801.05041-1-62-0': 'We will now prove [REF].', '1801.05041-1-62-1': 'From Theorem 3.2 in [CITATION] we know that under (A1)-(A3) and the additional assumptions that [MATH] but [MATH] grows at most polynomially in [MATH] [EQUATION].', '1801.05041-1-62-2': 'If [MATH] and [MATH] grows at most polynomially in [MATH] it follows that [MATH].', '1801.05041-1-62-3': 'Moreover, standard quantile regression arguments show that [EQUATION] where [MATH].', '1801.05041-1-62-4': 'Next apply Lemma [REF] to find that provided [MATH], [EQUATION]', '1801.05041-1-62-5': 'Next, observe that by asymptotic normality of the oracle estimator [EQUATION] where we defined [MATH].', '1801.05041-1-62-6': 'Again applying Lemma [REF] we obtain [EQUATION]', '1801.05041-1-62-7': 'Combining the results obtained so far we have [EQUATION]', '1801.05041-1-62-8': 'Next, let [MATH] denote the set of all disjoint partitions of [MATH] into [MATH] subsets.', '1801.05041-1-62-9': 'Observe that by [REF] we have under assumption (C) [EQUATION]', '1801.05041-1-62-10': 'Finally, note that by Lemma [REF] [EQUATION]', '1801.05041-1-62-11': 'Summarizing, we find that under (C) [EQUATION]', '1801.05041-1-62-12': 'The final result follows from a combination of [REF], [REF] and [REF].', '1801.05041-1-62-13': 'First, observe that for [MATH] we have by [REF], [REF] and the assumptions on [MATH], with probability tending to one [EQUATION].', '1801.05041-1-62-14': 'It follows that, with probability tending to one, [MATH].', '1801.05041-1-62-15': 'Moreover, for [MATH] we have by [REF] and the assumptions on [MATH], with probability tending to one [EQUATION].', '1801.05041-1-62-16': 'Hence, with probability tending to one, [MATH] and thus [REF] follows.', '1801.05041-1-62-17': '[MATH]', '1801.05041-1-63-0': "Proof of Lemma [REF] By Knight's identity [REF] we have [EQUATION]", '1801.05041-1-63-1': 'Define [EQUATION]', '1801.05041-1-63-2': 'By a Taylor expansion we obtain [EQUATION] and thus the bound on [MATH] is established.', '1801.05041-1-63-3': 'Next we note that [EQUATION] since the conditional expectation given [MATH] equals zero almost surely and moreover [EQUATION]', '1801.05041-1-63-4': 'Note that in particular for [MATH] we have [EQUATION].', '1801.05041-1-63-5': 'Define [MATH] and apply the Bernstein inequality to show that for any [MATH] [EQUATION]', '1801.05041-1-63-6': 'For [MATH] the last line above is bounded by [MATH].', '1801.05041-1-63-7': 'Denote by [MATH] a grid of values [MATH] such that [MATH] for all [MATH] and [EQUATION].', '1801.05041-1-63-8': 'Note that it is possible to find such a [MATH] with [MATH].', '1801.05041-1-63-9': 'It follows that [EQUATION]', '1801.05041-1-63-10': 'Finally, note that for [MATH] [EQUATION]', '1801.05041-1-64-0': 'Since [MATH] we can pick [MATH] such that the last line above is [MATH], and hence [EQUATION].', '1801.05041-1-64-1': 'Finally, observe that, denoting by [MATH] the maximum norm of the entries of the matrix [MATH], [EQUATION] where the last line follows by a straightforward application of the Hoeffding inequality.', '1801.05041-1-64-2': 'Thus the proof of Lemma [REF] is complete.', '1801.05041-1-64-3': '[MATH]'}

{'1801.05041-2-0-0': 'This paper introduces estimation methods for grouped latent heterogeneity in panel data quantile regression.', '1801.05041-2-0-1': 'We assume that the observed individuals come from a heterogeneous population with a finite number of types.', '1801.05041-2-0-2': 'The number of types and group membership is not assumed to be known in advance and is estimated by means of a convex optimization problem.', '1801.05041-2-0-3': 'We provide conditions under which group membership is estimated consistently and establish asymptotic normality of the resulting estimators.', '1801.05041-2-0-4': 'Simulations show that the method works well in finite samples when [MATH] is reasonably large.', '1801.05041-2-0-5': 'To illustrate the proposed methodology we study the effects of the adoption of Right-to-Carry concealed weapon laws on violent crime rates using panel data of 51 U.S. states from 1977 - 2010.', '1801.05041-2-1-0': 'myheadings Panel Quantile regression with group fixed effectsGu and Volgushev', '1801.05041-2-2-0': '# Introduction', '1801.05041-2-3-0': 'It is widely accepted in applied Econometrics that individual latent effects constitute an important feature of many economic applications.', '1801.05041-2-3-1': 'When panel data are available, a common approach is to incorporate latent structures in completely nonrestrictive way, i.e. the fixed effect approach.', '1801.05041-2-3-2': 'The fixed effect approach is attractive as it imposes minimal assumptions on the structure of the latent effects and on the correlation between the latent effects and the observed covariates and hence has become a very common empirical tool (see [CITATION] for a textbook treatment).', '1801.05041-2-4-0': 'A major challenge of the fixed effects approach lies in the fact that it introduces a large number of parameters which grows linearly with the number of individuals.', '1801.05041-2-4-1': 'For a few specific models this can be avoided by differencing out individual effects and learning about the common parameter of interest.', '1801.05041-2-4-2': 'However, for most models, including quantile regression, this simple differencing method no longer exists.', '1801.05041-2-4-3': 'The literature contains various approaches that put additional structure on latent effects in order to reduce the number of parameters and obtain more interpretable models.', '1801.05041-2-4-4': 'One popular approach is to introduce some parametric distributional structure on the latent effects, see for example [CITATION], [CITATION] and the correlated random effects literature.', '1801.05041-2-4-5': 'An alternative is to assume that the fixed effects have a group structure and hence only take a few distinct values which is the approach we take in this paper.', '1801.05041-2-5-0': 'There is ample evidence from empirical studies that it is often reasonable to consider a number of homogeneous groups (clusters) within a heterogeneous population.', '1801.05041-2-5-1': 'This discrete approach was taken byasnounHeckmanSinger82 for duration analysis of unemployment spells of a heterogeneous population of workers.asnounBesterHansen argue that in many applications individuals or firms are grouped naturally by some observable covariates such as classes, schools or industry codes.', '1801.05041-2-5-2': 'It is also widely accepted in the discrete choice model literature that individual agents are classified as a number of latent types (for instanceasnounKeane among many others).', '1801.05041-2-6-0': 'Estimating cluster structure has a long history in Statistics and Economics, and has generated a rich and mature literature.', '1801.05041-2-6-1': 'A general overview is given inasnounKaufman.', '1801.05041-2-6-2': 'Among the many available clustering algorithms, the k-means algorithm (nounMacQueen) is one of the most popular methods.', '1801.05041-2-6-3': 'It has been successfully utilized in many economic applications, for instanceasnounLinNg,asnounBM andasnounAndoBai.', '1801.05041-2-6-4': 'Finite mixture models provide an alternative, likelihood based approach.', '1801.05041-2-6-5': 'In the latter, grouping is usually achieved by maximizing the likelihood of the observed data.asnounSun builds a multinomial logistic regression model to infer the group pattern while nonparametric finite mixture models are considered inasnounAllman andasnounKasahara among many others.', '1801.05041-2-7-0': 'The focus of the present paper is on quantile regression for panel data with grouped individual heterogeneity.', '1801.05041-2-7-1': 'Panel data quantile regression has recently attracted a lot of attention, and there is a rich and growing literature that proposes various approaches to dealing with individual heterogeneity in this setting.', '1801.05041-2-7-2': 'In a pioneering contribution,asnounKoenker2004 takes the fixed effect approach and introduces individual latent effects as location shifts.', '1801.05041-2-7-3': 'These individual effects are regularized through an [MATH] penalty which shrinks them towards a common value.asnounLamarche proposes an optimal way to choose the corresponding penalty parameter in order to optimize the asymptotic efficiency of the common parameters of the conditional quantile function, seeasnounharding2017 for an extension of this approach.', '1801.05041-2-7-4': 'Another line of work that focuses on estimating common parameters while putting no structure on individual effects includesasnounKato,asnoungalvao2015 andasnoungalvao2016.', '1801.05041-2-7-5': 'Alternative approaches have also emerged.asnounAD take a random effect view of these individual latent effects.', '1801.05041-2-7-6': 'They consider a correlated random-effect model in the spirit ofasnounChamberlain1982 where the individual effects are modeled through a linear regression of some covariates.', '1801.05041-2-7-7': 'This is further developed inasnounAB16 andasnounCLP where the conditional quantile function of the unobserved heterogeneity is modelled as a function of observable covariates.', '1801.05041-2-8-0': 'Our contribution, which builds uponasnounKoenker2004, is a linear quantile regression method that accommodates grouped fixed effects.', '1801.05041-2-8-1': 'The advantages of our proposal over existing proposals are twofold.', '1801.05041-2-8-2': 'First, grouped fixed effects maintain the merit of unrestricted correlation between the latent effects and the observables and strike a good balance between the classical fixed effects approach and the other extreme which completely ignores latent heterogeneity.', '1801.05041-2-8-3': 'Second, in contrast to [CITATION], where the fixed effects are treated as nuisance parameters and are regularized to achieve a more efficient estimator for the global parameter, our method allows the researcher to learn the particular group structure of the latent effects together with common parameters of interest in the model.', '1801.05041-2-8-4': 'To the best of our knowledge, panel data quantile regression with grouped fixed effects has not been considered in the literature before.', '1801.05041-2-8-5': 'The only paper that goes in this direction is [CITATION].', '1801.05041-2-8-6': 'While the general framework developed in this paper does include a version of quantile regression with smoothed quantile objective function, the theoretical analysis requires the smoothing parameter to be fixed.', '1801.05041-2-8-7': 'This results in a non-vanishing bias and hence does not correspond to quantile regression in a strict sense.', '1801.05041-2-9-0': 'We do not assume any prior knowledge of the group structure and combine the quantile regression loss function with the recently proposed convex clustering penalty of [CITATION].', '1801.05041-2-9-1': 'The convex clustering method introduces a [MATH]-constraint on the pair-wise difference of the individual fixed effects, which tends to push the fixed effects into clusters.', '1801.05041-2-9-2': 'The number of clusters is controlled by a penalty parameter.', '1801.05041-2-9-3': 'The resulting optimization problem remains convex and can be solved in a fast and reliable fashion.', '1801.05041-2-9-4': 'Further modifications and a theoretical analysis of convex clustering were considered in [CITATION], [CITATION] and [CITATION].', '1801.05041-2-9-5': 'All of those authors combine [MATH] penalties with the classical [MATH] loss, and only consider clustering for cross-sectional data.', '1801.05041-2-9-6': 'Their theoretical results are not directly applicable to panel data or the non-smooth quantile loss function which is the main objective in this paper (all of the available theoretical results explicitly make use of the differentiability of the [MATH] loss function in their proofs).', '1801.05041-2-10-0': 'Our main theoretical contribution is to show consistency of the estimated grouping for a suitable range of penalty parameters when [MATH] and [MATH] tend to infinity jointly.', '1801.05041-2-10-1': 'We also propose a completely data-driven information criterion that facilitates the practical implementation of the method and prove its consistency for group selection as well as asymptotic normality of the resulting parameter estimators.', '1801.05041-2-11-0': 'The remaining part of this paper is organized as follows.', '1801.05041-2-11-1': 'Section 2 contains a detailed description of the proposed methodology and provides details on its practical implementation.', '1801.05041-2-11-2': 'Assumptions and theoretical results are included in Section 3.', '1801.05041-2-11-3': 'Section 4 presents the convex optimization problem and its computational details.', '1801.05041-2-11-4': 'Monte Carlo simulation results are included in Section 5 where investigate the final sample behavior of the proposed methodology.', '1801.05041-2-11-5': 'In Section 6 we apply the method to an empirical application in studying the effect of the adoption of Right-to-Carry concealed weapon law on violent crime rate using a panel data of 51 U.S. states from 1977 - 2010.', '1801.05041-2-11-6': 'All proofs are collected in Section [REF] while additional simulation results and details for the empirical application are relegated to the Appendix.', '1801.05041-2-12-0': '# Methodology', '1801.05041-2-13-0': 'Assume that for individuals [MATH] we observe repeated measures [MATH] where [MATH] denote covariates and [MATH] are responses.', '1801.05041-2-13-1': 'We shall maintain the assumption that data are i.i.d. within individuals and independent across individuals.', '1801.05041-2-13-2': 'The main object of interest in this paper is the conditional [MATH]-quantile function of [MATH] given [MATH], which we will denote by [MATH].', '1801.05041-2-13-3': 'We assume that [MATH] is of the form [EQUATION] with individual fixed effects [MATH] taking only a finite number, say [MATH], of different values, say [MATH].', '1801.05041-2-13-4': 'We explicitly allow the group membership, and even the number of groups to be unknown and to depend on [MATH] but will not stress this dependence in the notation for the sake of simplicity.', '1801.05041-2-13-5': 'Our main objective is to jointly estimate the number of groups, unknown group structure, and parameters [MATH] from the observations.', '1801.05041-2-13-6': 'To achieve this, we consider penalized estimators of the form [EQUATION] where [EQUATION].', '1801.05041-2-13-7': "Here [MATH] denotes the usual 'check function' and the weights [MATH] are allowed to depend on [MATH] and the data; one particular choice is discussed in below.", '1801.05041-2-13-8': 'The form of the penalty is motivated by the work of [CITATION].', '1801.05041-2-13-9': 'Intuitively, large values of [MATH] will push different coefficients closer together and result in clustered structure of the estimators [MATH].', '1801.05041-2-13-10': 'High-level conditions on the weights [MATH] which guarantee consistency of the resulting grouping procedure are provided in Theorem [REF].', '1801.05041-2-14-0': 'There are various possible choices for the penalty parameters [MATH].', '1801.05041-2-14-1': 'We propose to use weights of the form [EQUATION] where [MATH] are the fixed effects quantile regression estimators [EQUATION] of nounKato and [MATH] is a tuning parameter.', '1801.05041-2-14-2': 'This form of weighting by preliminary estimators is motivated by the work of nounzou2006 on adaptive lasso.', '1801.05041-2-14-3': 'Intuitively, weighting by preliminary estimated distances tends to give smaller penalties to coefficients from different groups thus reducing some of the bias that is typically present in the classical lasso.', '1801.05041-2-15-0': 'Given the developments above, it remains to find a value for the tuning parameter [MATH].', '1801.05041-2-15-1': 'The high-level results in Theorem [REF] together with findings in [CITATION] provide a theoretical range for those values (see the discussion following Theorem [REF] for additional details), but this range is not directly useful in practice since only rates and not constants are provided.', '1801.05041-2-15-2': 'Moreover, despite the fact that the weights [MATH] lead to asymptotically unbiased estimates, bias can still be a problem in finite samples.', '1801.05041-2-15-3': 'A typical approach in the literature to reduce bias which results from lasso-type penalties is to view the lasso problem solution as a candidate model (in our case, a candidate grouping of [MATH]) and re-fit based on this candidate model (see [CITATION] or [CITATION] among many others).', '1801.05041-2-16-0': 'To deal with bias issues and the choice of [MATH] in practice, we propose to combine the re-fitting idea with a simple information criterion which will simultaneously reduce the bias problem and provide a simple way to select a final model.', '1801.05041-2-16-1': 'A formal description of our approach is given in Algorithm [REF].', '1801.05041-2-17-0': '[h] InputinputOutputoutput Data [MATH], grid of values [MATH], quantile level of interest [MATH] Estimated number of groups [MATH], estimated group membership [MATH], estimated coefficients [MATH], [MATH] compute [MATH] given in [REF]', '1801.05041-2-18-0': '[MATH] Compute [EQUATION]', '1801.05041-2-18-1': 'Let [MATH] denote the unique values of [MATH], and define [MATH] as the estimated groups.', '1801.05041-2-18-2': 'Compute re-fitted estimators [EQUATION].', '1801.05041-2-18-3': 'Compute the IC criterion [EQUATION] where the choice of [MATH] and [MATH] is given in [REF].', '1801.05041-2-19-0': 'Set [MATH] and denote by [MATH] the corresponding number of groups.', '1801.05041-2-19-1': 'Set [MATH], [MATH].', '1801.05041-2-20-0': 'Grouping via IC criterion', '1801.05041-2-21-0': 'Theorem [REF] provides a formal justification of Algorithm [REF] under high-level conditions on [MATH].', '1801.05041-2-21-1': 'In particular we prove that the group structure is estimated consistently with probability tending to one and derive the asymptotic distribution of the resulting estimators [MATH].', '1801.05041-2-21-2': 'In order to make the proposed estimation procedure fully data-driven, we need to specify a choice for the tuning parameters [MATH] and [MATH].', '1801.05041-2-21-3': 'In our simulations, we found that the following choices lead to good results: [EQUATION] with [EQUATION] where [MATH] denotes the empirical cdf of the regression residuals from the fixed effects quantile regression estimator given in [REF], [MATH] denotes the corresponding empirical quantile function, and [MATH] is a bandwidth parameter (we use the Hall-Sheather rule in our simulations, see [CITATION]).', '1801.05041-2-22-0': 'To motivate this particular choice of constant [MATH], observe the following expansion, which is derived in detail in the proof of Theorem [REF] [EQUATION].', '1801.05041-2-22-1': 'This shows that plugging in the estimated (by fixed effects quantile regression) instead of true errors underestimates the objective function evaluated at the residuals by roughly the first term on the right-hand side in the above expression.', '1801.05041-2-22-2': 'This term needs to be dominated by the penalty if we want to avoid selecting models that are too large, and so it is natural to scale the penalty by a constant which is proportional to [MATH] in order to ensure reasonable performance across different data generating processes.', '1801.05041-2-22-3': 'Under the simplifying assumption that [MATH] does not depend on [MATH], this term equals [MATH], and under the same assumptions [MATH] provides a consistent estimator for the latter, see [CITATION].', '1801.05041-2-22-4': 'Note that the sparsity term introduced here plays a similar role as the noise variance in classical information criteria such as AIC and BIC in least squares regression.', '1801.05041-2-23-0': '# Theoretical analysis', '1801.05041-2-24-0': 'In this section we provide a theoretical analysis of the methodology proposed in Section [REF].', '1801.05041-2-24-1': 'We begin by stating an assumption on the true (but unknown) underlying group structure.', '1801.05041-2-25-0': 'Assumption (C) implies that the individual fixed effects are grouped into [MATH] distinct groups and that the group centers are separated.', '1801.05041-2-25-1': 'Note that the number of groups as well as group membership is allowed to differ across quantiles.', '1801.05041-2-25-2': 'For the sake of a concise notation, the dependence of the number of groups and group centers on [MATH] will from now on be dropped unless there is risk of confusion.', '1801.05041-2-25-3': 'Note also that we require the number of groups to be fixed (i.e. independent of [MATH] and non-random) and exogenous, i.e. independent of the covariates [MATH].', '1801.05041-2-26-0': 'Next we collect some technical assumptions on the data generating process.', '1801.05041-2-26-1': 'Define [MATH] and let [MATH] denote the support of [MATH].', '1801.05041-2-27-0': 'Assumptions (A1)-(A3) are fairly standard and routinely imposed in the quantile regression literature.', '1801.05041-2-27-1': 'Similar assumptions have been made, for instance in [CITATION] [see assumptions (B1)-(B3) in that paper].', '1801.05041-2-28-0': '## Analysis of the estimators in [REF]', '1801.05041-2-29-0': 'To state our first main result define [EQUATION].', '1801.05041-2-29-1': 'In words, [MATH] corresponds to the largest penalty corresponding to the difference between two individual effects from different groups while [MATH] describes the smallest penalty between two effects from the same group.', '1801.05041-2-29-2': 'Our first result provides high-level conditions on [MATH] that guarantee asymptotically correct grouping.', '1801.05041-2-30-0': 'Let assumptions (A1)-(A3), (C) hold and assume that [MATH], [MATH] and [EQUATION]', '1801.05041-2-30-1': 'Denote the ordered unique values of [MATH] by [MATH] (i.e. [MATH] denotes the number of distinct values taken by [MATH] which we interpret as the estimated number of groups) and define the sets [MATH].', '1801.05041-2-30-2': 'Then [EQUATION].', '1801.05041-2-31-0': 'Next we discuss the implications of this general result for the specific choice [MATH] given in [REF].', '1801.05041-2-31-1': 'Define [EQUATION].', '1801.05041-2-31-2': 'From [CITATION] we obtain the bound [EQUATION].', '1801.05041-2-31-3': 'Now if [MATH] then [MATH] and thus [EQUATION].', '1801.05041-2-31-4': 'Moreover, under (C) we have [EQUATION] and thus [EQUATION].', '1801.05041-2-32-0': 'Given this choice of weights, the conditions [MATH] is satisfied provided that [MATH].', '1801.05041-2-32-1': 'The other conditions in [REF] take the form [EQUATION].', '1801.05041-2-32-2': 'Assuming that [MATH], this provides a range of possible values for [MATH] which will ensure that [REF] holds.', '1801.05041-2-33-0': '## Analysis of the information criterion in Algorithm [REF]', '1801.05041-2-34-0': 'In this section we provide theoretical guarantees for the performance of the information criterion based estimators [MATH] Our main result shows that, under fairly general conditions on the penalty parameter [MATH], the IC procedure selects the correct number of groups with probability tending to one.', '1801.05041-2-34-1': "Moreover, the estimators [MATH] are shown to enjoy the 'oracle property', i.e. they have the same asymptotic distribution as estimators which are based on the true (but unknown) grouping of individuals.", '1801.05041-2-34-2': 'Before making this statement more formal, we need some additional notation.', '1801.05041-2-34-3': "Let [EQUATION] denote the infeasible 'oracle' which uses the true group membership.", '1801.05041-2-34-4': "The asymptotic variance of the oracle estimator is conveniently expressed in terms of the following two limits which we assume to exist [EQUATION] where [MATH] and [MATH] denotes the [MATH]'th unit vector in [MATH].", '1801.05041-2-34-5': 'Additionally, we need the following condition on the grid [MATH]', '1801.05041-2-35-0': '[(G)] For each [MATH], denote the grid values by [MATH] where [MATH] can depend on [MATH].', '1801.05041-2-35-1': 'There exists a sequence [MATH] such that [MATH].', '1801.05041-2-36-0': 'Assumption (G) is fairly mild.', '1801.05041-2-36-1': 'It only requires that among the candidate values for [MATH] there exists one value so that [MATH] satisfies the assumptions of Theorem [REF].', '1801.05041-2-36-2': 'In practice, we recommend choosing a grid of values that results in sufficiently many different numbers of groups.', '1801.05041-2-37-0': 'Let assumptions (A1)-(A3), (C), (G) hold and assume that [MATH] and [MATH] grows at most polynomially in [MATH] (i.e. [MATH] for some [MATH]) and [MATH].', '1801.05041-2-37-1': 'Assume that there exists [MATH] such that [MATH] with probability tending to one and that [MATH].', '1801.05041-2-37-2': 'Then [MATH] and [EQUATION]', '1801.05041-2-37-3': 'Theorem [REF] and Theorem [REF] hold point-wise in the parameter space, and we expect that deriving a similar result uniformly in the parameter space (in particular, if cluster centers are allowed to depend on [MATH] and if their separation is lost, see [CITATION] for such findings in the context of classical lasso penalized regression) is impossible.', '1801.05041-2-37-4': "It is a well established fact in the Statistics and Econometrics literature that inference which is based on such 'point-wise' asymptotic results can be unreliable.", '1801.05041-2-37-5': 'Recently, several approaches to alleviate this problem and achieve uniformly valid post-regularization inference have been proposed (see, among others, [CITATION], [CITATION] and [CITATION]).', '1801.05041-2-37-6': 'Applying similar ideas to the present setting is a very important question which we leave for future research.', '1801.05041-2-38-0': '# Details on the optimization problem in Algorithm 1', '1801.05041-2-39-0': 'To implement the proposed quantile panel data regression with group fixed effect, we need to solve the optimization problem stated in ([REF]).', '1801.05041-2-39-1': 'A natural normalization of the objective and the penalty function leads to [EQUATION] this is equivalent to the objective ([REF]) except that [MATH] is adjusted according to [MATH] and [MATH] so that we can use a generic grid for [MATH] rather than letting the grid support change with [MATH].', '1801.05041-2-39-2': 'In practice, the grid support of [MATH] is chosen such that the number of distinct values of the solution [MATH] for [MATH] takes all possible integer values in the set [MATH].', '1801.05041-2-39-3': 'This is always achievable as long as the grid width of [MATH] is small enough.', '1801.05041-2-39-4': 'Since for each fixed [MATH], ([REF]) is a linear programming problem which can be efficiently solved in any reliable solvers, this is not computationally expensive.', '1801.05041-2-40-0': 'To make this section self contained, we provide some details of the primal problem stated in ([REF]) and its corresponding dual problem.', '1801.05041-2-40-1': 'Define [MATH] and observe that we can re-write [MATH].', '1801.05041-2-40-2': 'With this notation [REF] can be equivalently expressed as follows [EQUATION] with [MATH] being a vector of length [MATH] that consists entries [MATH] for [MATH].', '1801.05041-2-40-3': 'We can represent [MATH] as [MATH] where [MATH] is a [MATH] matrix taking the form [EQUATION]', '1801.05041-2-40-4': 'The corresponding dual problem of ([REF]) can be stated as: [EQUATION] with [MATH] being the incidence matrix that identifies the [MATH] individuals.', '1801.05041-2-40-5': 'The solution for [MATH] and [MATH] in the primal problem is then the dual solutions of the dual problem.', '1801.05041-2-40-6': 'We implement the dual problem using the Mosek optimization software of [CITATION] through the R interface Rmosek of [CITATION].', '1801.05041-2-40-7': 'We have also implemented the estimation procedure using the quantreg package in R and the code will be made available for public use.', '1801.05041-2-41-0': '# Monte Carlo Simulations', '1801.05041-2-42-0': '## Finite Sample Performance of the Proposed Estimator', '1801.05041-2-43-0': 'To assess the finite sample performance of the proposed convex clustering panel quantile regression estimator, we apply the method to simulated data sets.', '1801.05041-2-43-1': 'In particular, we consider data generated from two models and two error distributions for a range of [MATH] and [MATH].', '1801.05041-2-43-2': 'The responses, [MATH], are generated by either a location shift model [EQUATION] or a location scale shift model [EQUATION] where the individual latent effects [MATH] are generated from three groups taking values [MATH] with equal proportions.', '1801.05041-2-43-3': 'The covariate [MATH] is generated such that it has a non-zero interclass correlation coefficient.', '1801.05041-2-43-4': 'In particular, [EQUATION] with [MATH] and [MATH] independent and identically distributed over [MATH] and [MATH] respectively.', '1801.05041-2-43-5': 'We conduct simulation experiments with [MATH] to investigate both cases where the fixed effect is independent or correlated with the covariate.', '1801.05041-2-43-6': 'The true parameters are [MATH] and [MATH].', '1801.05041-2-43-7': 'The error terms [MATH] are i.i.d. following either a standard normal distribution or a student t distribution with three degrees of freedom.', '1801.05041-2-43-8': 'Results reported are based on 2000 repetitions.', '1801.05041-2-44-0': 'We first investigate the performance of using the information criterion for estimating the number of groups.', '1801.05041-2-44-1': 'Table [REF] and Table [REF] report the proportion of estimated number of groups under the two models for different combinations of [MATH] and [MATH] for [MATH] and [MATH], respectively.', '1801.05041-2-44-2': 'Throughout the simulations, we used an equally spaced grid of [MATH] values with width [MATH] and support [MATH].', '1801.05041-2-44-3': 'This grid was chosen to ensure that the number of groups estimated for different [MATH] values covers the integers in range [MATH].', '1801.05041-2-44-4': 'For the IC criteria, the sparsity function [MATH] is estimated with bandwidth chosen based on the [CITATION] rule implemented in the quantreg package (see discussion in [CITATION]).', '1801.05041-2-45-0': 'The results suggest that the probability of getting the correct number of groups for [MATH] is slightly better than for [MATH].', '1801.05041-2-45-1': 'When [MATH], the estimates for the number of groups for both error distributions and for different [MATH] are mostly satisfactory.', '1801.05041-2-45-2': 'The performance for [MATH] error deteriorates compared to those with normal error, especially for higher quantiles.', '1801.05041-2-45-3': 'For [MATH] and quantiles other than the median the proposed method should be used with caution.', '1801.05041-2-45-4': 'Including correlation between individual effects and predictors does not lead to dramatic changes in the accuracy for estimating the number of groups and group membership.', '1801.05041-2-46-0': 'Table [REF] and Table [REF] summarize the finite sample properties of [MATH] for [MATH] and [MATH], respectively and compare with the QRFE estimator where no penalization on the individual fixed effect is used (i.e. [MATH]).', '1801.05041-2-47-0': 'The standard errors used for constructing confidence intervals (nominal coverage [MATH]) are based on the nid option with Hall-Sheather bandwidth in the package quantreg.', '1801.05041-2-47-1': 'Results based on the Bofinger bandwidth selection are similar and not reported here.', '1801.05041-2-47-2': 'For the QRFE, the Bofinger bandwidth rule was used since the Hall-Sheather rule resulted in substantial under-coverage with [MATH] for some of the models.', '1801.05041-2-48-0': 'When covariates and fixed effects are independent, the RMSE of the PQR-FEgroup estimator, [MATH], is smaller than that of the QRFE for all settings considered.', '1801.05041-2-48-1': 'This shows that our penalization gains efficiency for estimating [MATH] when there is group structure in the fixed effects.', '1801.05041-2-48-2': 'The results do not change much from normal error to [MATH] error and from median to higher quantiles.', '1801.05041-2-49-0': 'Introducing correlation between predictors and group membership leads to a bias for the grouped effect estimator, while the fixed effects estimator does not suffer from additional bias.', '1801.05041-2-49-1': 'This bias can be quite noticeable for small values of [MATH], especially at the [MATH] quantile.', '1801.05041-2-49-2': 'The bias becomes negligible as [MATH] increases, so there is no contradiction to our asymptotic theory.', '1801.05041-2-49-3': 'An intuitive explanation for this behaviour is that for smaller [MATH] it is difficult to get a perfect grouping, and a wrong grouping leads to bias since there is dependence between predictors and group structure.', '1801.05041-2-50-0': 'Last, we report in Table [REF] and Table [REF] the proportion of perfect classification of individual effects and the average value of the percentage of correct classification together with their standard errors.', '1801.05041-2-50-1': 'Since the comparison of the estimated membership and the true membership only makes sense when [MATH], the estimated membership are based on [MATH] for which [MATH] (see [CITATION] for a similar approach).', '1801.05041-2-50-2': 'Results suggest that for [MATH] and [MATH], the group membership estimation is quite satisfactory.', '1801.05041-2-50-3': 'While the proportion of perfect matches is low even for [MATH], the average proportion of correct classification shows that those effects are typically due to very few misclassified individuals.', '1801.05041-2-50-4': 'For [MATH] perfect classification is almost impossible while average correct classification rates remain reasonable.', '1801.05041-2-50-5': 'Adding correlation between individual effects and covariates leads to a deterioration of the probability for achieving a perfect grouping for location-scale models, especially at higher quantiles, but does not have a strong impact on other results.', '1801.05041-2-50-6': 'Overall the simulations suggest that for small [MATH], there is just not enough information available for each individual to hope for perfect classification.', '1801.05041-2-51-0': '## Further analysis of tuning parameters in the IC Criteria', '1801.05041-2-52-0': 'The discussion at the end of Section [REF] provides a motivation for the tuning parameters [MATH] in the IC criteria.', '1801.05041-2-52-1': 'Here we further investigate the impact of rescaling [MATH] by different factors.', '1801.05041-2-52-2': 'For illustration we consider DGP1 in the previous section where data are generated based on the model ([REF]).', '1801.05041-2-52-3': 'We use a grid of constants [MATH] with width 0.01 and plot the associated performance of the estimated number of groups, the RMSE of [MATH] and the coverage rate for [MATH].', '1801.05041-2-52-4': 'Figure [REF] and Figure [REF] contain corresponding results for the location-scale shift model with [MATH] errors and [MATH], respectively.', '1801.05041-2-52-5': 'For [MATH] as small as 15, the performance is quite sensitive to the chosen constant.', '1801.05041-2-52-6': 'As predicted by the theory this dependence becomes somewhat less prominent as [MATH] increases.', '1801.05041-2-52-7': 'Overall the choice [MATH] shows good performance for settings that we tried in this simulation.', '1801.05041-2-52-8': 'The patterns are similar for those with the normal error and the location-shift models and likewise for DGP2, results are reported in Figure [REF]- Figure [REF] in the Appendix for the sake of completeness.', '1801.05041-2-53-0': '# Empirical Example', '1801.05041-2-54-0': 'To further illustrate our proposed methodology, we revisit the empirical inquiry on the much-debated "More Guns Less Crime" hypothesis.asnounLM97 provide the first empirical analysis which claims that the adoption of Right-to-Carry (RTC) laws, which allows local authorities to issue a concealed weapon permit to all applicants that are eligible, reduces crime.', '1801.05041-2-54-1': 'Ever since its publication, there has been much academic and political debate that challenges the findings.asnounAD03 shows that the negative effect of RTC laws has no statistical significance under a more reasonable model specification and inference using both the state and county level data between 1977 - 1999 for 51 U.S. states.', '1801.05041-2-54-2': 'Their conclusion is echoed by the National ResearchasnounNRC2004 (NRC) report which finds little reliable statistical support for the "More Guns Less Crime" hypothesis.', '1801.05041-2-54-3': 'Recently,asnounADZ14 revisited the hypothesis using the updated panel data from 1977 - 2010.', '1801.05041-2-54-4': 'They correct several mistakes in the dataset used in earlier analysis and discuss the shortcoming of using county level crime data.', '1801.05041-2-54-5': 'We refer the readers to more details provided in [CITATION].', '1801.05041-2-54-6': 'For our empirical analysis, we use their updated state level data kindly provided by the authors.', '1801.05041-2-55-0': 'The main parameter of interest is the effect of the indicator of the RTC laws (denoted as lawind) on crime rates.', '1801.05041-2-55-1': 'In our analysis, we focus on the violent crime rate.', '1801.05041-2-55-2': 'A similar analysis is possible for other categories of offense.', '1801.05041-2-55-3': 'In addition to the law indicator, there is also information on the incarceration rate (sentenced prisoners per 100,000 residents; denoted as prisoner) in the states in the previous year, real per capita personal income (denoted as rpcpi) and other demographic variables on population proportions in different age-gender groups for various ethnicities.', '1801.05041-2-55-4': 'As argued in [CITATION] and [CITATION], to avoid multicollinearity and accounting for the fact that 90% of violent crimes in the U.S. are committed by male offenders, we follow their specification and control for only the proportion of African-American male in the age group 10-19, 20-29 and 30-39.', '1801.05041-2-55-5': 'Inevitably, there are many different model specifications that can be considered for this empirical inquiry and it is impossible to report all the results.', '1801.05041-2-55-6': 'Our goal is to emphasize the heterogeneous law adoption effect for states that have low violent crime rate versus those with high crime rate.', '1801.05041-2-55-7': "We also provide evidence of clustering behavior of the state fixed effects which are incorporated to capture states' unobserved heterogeneity and show that by taking advantage of the dimension reduction of grouping the fixed effects, the parameters of interest on other control variables enjoy better statistical precision.", '1801.05041-2-56-0': 'Our main model specification is [EQUATION] where the additional control variables [MATH] include lagged incarceration rate (prisoners), real per capita personal income (rpcpi) and the three demographic variables (afam1019, afam2029, afam3039).', '1801.05041-2-56-1': 'We note that high incarceration rates in a given state may be a feedback towards rising violent crime, and therefore including those as control variables might lead to endogeneity issues.', '1801.05041-2-56-2': 'As a robustness check, we also report the corresponding results in the appendix for model ([REF]) without the incarcerating rate as a control covariate.', '1801.05041-2-56-3': 'The effects stay mostly unchanged.', '1801.05041-2-57-0': 'Figure [REF] reports the panel data quantile regression estimates with state fixed effects for [MATH] and their associated point-wise confidence intervals.', '1801.05041-2-57-1': 'Similar to the findings in [CITATION], we see that the RTC law-adoption has a positive effects on violent crime rate.', '1801.05041-2-57-2': 'In addition this effect is significant for lower quantiles while there is no statistical significance for such an effect for states at higher quantiles of the violent crime rate.', '1801.05041-2-57-3': 'All other control variables have expected signs, with a negative effect of the lagged incarceration rate indicating that states with stricter laws have lower violent crime rates, although this effect is not very precisely estimated.', '1801.05041-2-57-4': 'Higher real per capita personal income has a negative effect on violent crime at lower quantiles; this effect is diminishing for higher quantiles of violent crime rate.', '1801.05041-2-57-5': 'The proportion of African-American population at age group 30 - 39 has a positive effect on the violent crime and this effect becomes more prominent for higher crime rate states.', '1801.05041-2-57-6': 'Figure [REF] shows the corresponding state fixed effect estimates for different quantile levels.', '1801.05041-2-57-7': 'There is some evidence of clustered behavior of these fixed effect estimates, although these are estimates with statistical errors.', '1801.05041-2-57-8': 'Using our proposed methodology, the estimated number of groups for the five quantile levels are respectively [MATH].', '1801.05041-2-58-0': 'Figure [REF] plots the corresponding panel data quantile regression estimates with the estimated optimal grouped state fixed effects for [MATH] and their associated point-wise confidence intervals.', '1801.05041-2-58-1': 'The pattern of the quantile effects for all the control variables stay roughly the same compared to the fixed effect quantile regression estimates, while the variance of these estimates under the grouped fixed effects are noticeably smaller.', '1801.05041-2-58-2': 'This is similar to what we observe in the simulation section where the common parameters in quantile panel data regression with grouped fixed effects have lower variances than those of the estimates based on "individual heterogeneity".', '1801.05041-2-58-3': 'However, we would also like to point out that the standard errors are based on refitted models with estimated group structure, which does not account for the uncertainty of model selection on the fixed effects and should be interpreted with caution.', '1801.05041-2-58-4': 'A proper inference method based on the proposed methodology accounting for such uncertainty is important and is part of our future research agenda.', '1801.05041-2-59-0': '# Conclusions and future extensions', '1801.05041-2-60-0': 'The present paper suggests a simple and computationally efficient way to incorporate group fixed effects into a panel data quantile regression by means of a convex clustering penalty.', '1801.05041-2-60-1': 'We develop theoretical results on consistent group structure estimation and discuss the asymptotic properties of the resulting joint and group-specific estimators.', '1801.05041-2-61-0': 'There are several directions that we plan to explore in the future.', '1801.05041-2-61-1': 'First, our theory focused on individual fixed effects while assuming common slope coefficients.', '1801.05041-2-61-2': 'It is equally interesting to allow for group structure in some of the slope coefficients while keeping other slope coefficients common across individuals, perhaps even allowing for individual fixed effects.', '1801.05041-2-61-3': 'This can be achieved by straightforward modifications of the penalization approach which we explored so far, but a more detailed theoretical analysis of this approach remains beyond the scope of the present paper.', '1801.05041-2-62-0': 'Second, one can take the standpoint that in many applications there is no exact group structure.', '1801.05041-2-62-1': 'In such settings, an alternative interpretation of the penalty which we investigated is as a way of regularizing problems that have too many parameters.', '1801.05041-2-62-2': 'Such an interpretation is in the spirit of the proposals of [CITATION] and [CITATION], and a detailed investigation of the resulting bias-variance trade-off warrants further research.', '1801.05041-2-63-0': 'Finally, a deeper analysis of issues that are related to uniformity of distributional approximation in the entire parameter space was not addressed here, but remains an important theoretical and practical question which we hope to address in the future.', '1801.05041-2-64-0': '# Proofs', '1801.05041-2-65-0': 'We begin by collecting some useful facts and defining additional notation.', '1801.05041-2-65-1': "We will repeatedly make use of Knight's identity (see [CITATION], p. 121)) which holds for [MATH]: [EQUATION]", '1801.05041-2-65-2': 'Additionally, let [MATH].', '1801.05041-2-65-3': 'The symbols [MATH] will mean that there exists a non-random constant [MATH] which is independent of [MATH] such that [MATH] and [MATH], respectively.', '1801.05041-2-65-4': 'Define [MATH] and let [MATH] denote the conditional cdf of [MATH] given [MATH].', '1801.05041-2-65-5': 'When there is no risk of confusion, we will also write [MATH] instead of [MATH].', '1801.05041-2-65-6': 'Define [MATH].', '1801.05041-2-66-0': '## Proof of Theorem [REF]', '1801.05041-2-67-0': 'We begin by stating some useful technical results which will be proved at the end of this section.', '1801.05041-2-68-0': 'For any fixed [MATH] define [MATH].', '1801.05041-2-68-1': 'Then we have under assumptions (A1)-(A3) [EQUATION] where [EQUATION].', '1801.05041-2-69-0': 'Under assumptions (A1)-(A3) there exist [MATH] such that for all [MATH] [EQUATION]', '1801.05041-2-69-1': 'Under assumption (A1) define for fixed [MATH] [EQUATION]', '1801.05041-2-69-2': 'We have for any fixed [MATH], provided that [MATH] [EQUATION]', '1801.05041-2-69-3': 'Proof of Theorem [REF]', '1801.05041-2-70-0': 'Step 1: first bounds In this step we shall prove that [EQUATION]', '1801.05041-2-70-1': 'Combine the results in Lemma [REF] and Lemma [REF] to find that any minimizer of [MATH] must satisfy [EQUATION]', '1801.05041-2-70-2': 'Let [MATH].', '1801.05041-2-70-3': 'Then for any [MATH] [EQUATION] i.e. by Lemma [REF] [EQUATION] and in particular [MATH] as long as [MATH].', '1801.05041-2-70-4': 'Provided that [MATH] we obtain [EQUATION]', '1801.05041-2-70-5': 'Define [MATH].', '1801.05041-2-70-6': 'Next we will prove that, provided [MATH] also [EQUATION].', '1801.05041-2-70-7': 'To this end, it suffices to prove that [MATH] for any [MATH].', '1801.05041-2-70-8': 'Define [EQUATION].', '1801.05041-2-70-9': 'Define the set [MATH].', '1801.05041-2-70-10': 'Observe that [EQUATION]', '1801.05041-2-70-11': 'Thus [EQUATION]', '1801.05041-2-70-12': 'Now since [MATH] and by definition of [MATH] it follows that under (C) [EQUATION] and since by assumption [MATH] we obtain [EQUATION].', '1801.05041-2-71-0': 'Next we note that for any [MATH] with [MATH] we have [EQUATION] with probability tending to one by [REF] and the definition of [MATH].', '1801.05041-2-71-1': 'For [MATH] with [MATH] note that by Lemma [REF] [EQUATION] where the [MATH] terms are uniform in [MATH].', '1801.05041-2-71-2': 'Thus [EQUATION]', '1801.05041-2-71-3': 'Summarizing we have proved that [EQUATION]', '1801.05041-2-71-4': 'Under the conditions [MATH] the last line is strictly positive with probability tending to one unless [MATH] with probability tending to one.', '1801.05041-2-71-5': 'Thus the proof of [REF] is complete.', '1801.05041-2-72-0': 'Step 2: recovery of clusters with probability to one', '1801.05041-2-73-0': 'To simplify notation, assume that individual [MATH] belongs to cluster 1, individual [MATH] to cluster 2 and so on.', '1801.05041-2-73-1': 'Since all cluster can be handled by similar arguments we only consider the first cluster.', '1801.05041-2-73-2': 'Let [MATH] denote the distinct values of [MATH], ordered in increasing order, and let [MATH].', '1801.05041-2-73-3': 'Again, to simplify notation assume w.o.l.g. that [MATH].', '1801.05041-2-73-4': 'To prove the result, we proceed in an iterative way.', '1801.05041-2-73-5': 'We will prove by contradiction that [MATH], i.e. all estimators of individuals from cluster 1 take the same value.', '1801.05041-2-73-6': 'Assume that [MATH].', '1801.05041-2-74-0': 'We will now prove by contradiction that [MATH].', '1801.05041-2-74-1': 'Assume that [MATH].', '1801.05041-2-74-2': 'Define [MATH] for [MATH] and [MATH] for [MATH].', '1801.05041-2-74-3': 'By [REF] Lemma [REF] we find that [EQUATION]', '1801.05041-2-74-4': 'Next, observe that by construction, under (C) and using the fact that [MATH], [EQUATION]', '1801.05041-2-74-5': 'From this we obtain [EQUATION] where the last inequality holds for sufficiently large [MATH] since by assumption [MATH] and since we assumed [MATH] so that [MATH].', '1801.05041-2-74-6': 'However, this is a contradiction to the fact that [MATH] minimizes [MATH].', '1801.05041-2-75-0': 'In a similar fashion, one can prove that [MATH].', '1801.05041-2-75-1': 'Just define [MATH] and proceed as above.', '1801.05041-2-75-2': 'Since [MATH] and we have already proved that [MATH] this leads to a contradiction with [MATH], and hence [MATH].', '1801.05041-2-75-3': 'All other clusters can be handled in a similar fashion and that completes the proof of the second step.', '1801.05041-2-75-4': '[MATH]', '1801.05041-2-76-0': "Proof of Lemma [REF] Apply Knight's identity [REF] to find that [EQUATION]", '1801.05041-2-76-1': 'Hence it follows that [EQUATION]', '1801.05041-2-76-2': 'Now by a Taylor expansion [EQUATION] so the bound on [MATH] is established.', '1801.05041-2-77-0': 'Next define the classes of functions [EQUATION]', '1801.05041-2-77-1': 'Note that the class of functions [MATH] has envelope function [MATH].', '1801.05041-2-77-2': 'Thus by Lemma 2.6.15 and Theorem 2.6.7 of [CITATION] the class of functions [MATH] satisfies, for any probability measure [MATH], [MATH] for some finite constants [MATH](here, [MATH] denotes the covering number, see Section 2.1 of [CITATION]).', '1801.05041-2-77-3': 'Moreover, [MATH], and elementary computations with covering numbers show that [MATH] for some finite constants [MATH].', '1801.05041-2-77-4': 'Hence we find that by Theorem 2.14.9 of [CITATION]), for any [MATH], [EQUATION] for some constant [MATH] that depends only on [MATH] (here, [MATH] denotes outer probability).', '1801.05041-2-77-5': 'Letting [MATH] and applying the union bound for probabilities we obtain [EQUATION]', '1801.05041-2-77-6': 'Hence [EQUATION]', '1801.05041-2-77-7': 'Thus the bound on [MATH] follows and the proof is complete.', '1801.05041-2-77-8': '[MATH]', '1801.05041-2-78-0': "Proof of Lemma [REF] Observe that by Knight's identity [REF] [EQUATION]", '1801.05041-2-78-1': 'Now under assumption (A2) [MATH] a.s., and thus given (A1) [EQUATION]', '1801.05041-2-78-2': 'This shows the upper bound in [REF].', '1801.05041-2-78-3': 'For the lower bound, note that [MATH] is non-decreasing almost surely.', '1801.05041-2-78-4': 'Moreover, [MATH] a.s. by (A3) and thus by (A2) and (A3) we have almost surely [EQUATION].', '1801.05041-2-78-5': 'Define [MATH].', '1801.05041-2-78-6': 'Noting that [MATH] is non-decreasing almost surely, it follows that a.s. [EQUATION] where the last inequality follows since by definition [MATH] a.s. Finally, under assumption (A1), [MATH].', '1801.05041-2-79-0': 'Summarizing, we find [EQUATION] which proves the lower bound in [REF].', '1801.05041-2-79-1': 'Thus the proof of the Lemma is complete.', '1801.05041-2-79-2': '[MATH]', '1801.05041-2-80-0': 'Proof of Lemma [REF] Consider the class of functions [EQUATION].', '1801.05041-2-80-1': 'Note that by construction [MATH] for all [MATH] and moreover [MATH].', '1801.05041-2-80-2': 'This shows the existence of constants [MATH] such that for all [MATH] for [MATH] where [MATH] depends on [MATH] only and [MATH] denotes the measure corresponding to [MATH].', '1801.05041-2-80-3': 'Thus we have by Theorem 2.14.9 of [CITATION], [EQUATION] where the constant [MATH] depends only on [MATH] and [MATH] denotes outer probability.', '1801.05041-2-80-4': 'Set [MATH] to bound the right-hand side above by [MATH].', '1801.05041-2-80-5': 'Defining the events [EQUATION] we obtain [EQUATION].', '1801.05041-2-80-6': 'Finally, note that under (A1) we have a.s. [EQUATION].', '1801.05041-2-80-7': 'This completes the proof.', '1801.05041-2-80-8': '[MATH]', '1801.05041-2-81-0': '## Proof of Theorem [REF]', '1801.05041-2-82-0': 'We begin by stating a useful technical result that will be proved at the end of this section.', '1801.05041-2-83-0': 'Under assumptions (A1)-(A3) [EQUATION] where, defining [MATH], there exists a constant [MATH] independent of [MATH] such that [EQUATION]', '1801.05041-2-83-1': 'Proof of Theorem [REF] The proof proceeds in several steps.', '1801.05041-2-83-2': "First, we note that the 'oracle' estimation problem [REF] corresponds to a classical, fixed-dimensional quantile regression with true parameter vector [MATH] and [MATH] independent observations [MATH] where [MATH] where [MATH] denotes the k'th unit vector in [MATH].", '1801.05041-2-83-3': 'A straightforward extension of classical proof techniques in parametric quantile regression shows that under assumptions (A1)-(A3) and (C) the oracle estimator is asymptotically normal as claimed.', '1801.05041-2-84-0': 'Second, we observe that by definition of the optimization problem the estimated group structure [MATH] is the same for all values of [MATH] with [MATH] that give rise to the same number of groups.', '1801.05041-2-84-1': 'Since the value of [MATH] depends only on [MATH], it suffices to minimize [MATH] over those values of [MATH] that correspond to different numbers of groups.', '1801.05041-2-84-2': 'Denote the distinct estimated numbers of groups by [MATH], the corresponding estimated groupings by [MATH], and the corresponding values of [MATH] by [MATH].', '1801.05041-2-84-3': 'By assumption (G) and Theorem [REF], the probability of the event [EQUATION]', '1801.05041-2-84-4': 'Hence it suffices to prove that [EQUATION]', '1801.05041-2-84-5': 'Once this result is established, we directly obtain [EQUATION] and thus the asymptotic distribution of [MATH] matches that of the oracle estimator.', '1801.05041-2-85-0': 'We will now prove [REF].', '1801.05041-2-85-1': 'From Theorem 3.2 in [CITATION] we know that under (A1)-(A3) and the additional assumptions that [MATH] but [MATH] grows at most polynomially in [MATH] [EQUATION].', '1801.05041-2-85-2': 'If [MATH] and [MATH] grows at most polynomially in [MATH] it follows that [MATH].', '1801.05041-2-85-3': 'Moreover, standard quantile regression arguments show that [EQUATION] where [MATH].', '1801.05041-2-85-4': 'Next apply Lemma [REF] to find that provided [MATH], [EQUATION]', '1801.05041-2-85-5': 'Next, observe that by asymptotic normality of the oracle estimator [EQUATION] where we defined [MATH].', '1801.05041-2-85-6': 'Again applying Lemma [REF] we obtain [EQUATION]', '1801.05041-2-85-7': 'Combining the results obtained so far we have [EQUATION]', '1801.05041-2-85-8': 'Next, let [MATH] denote the set of all disjoint partitions of [MATH] into [MATH] subsets.', '1801.05041-2-85-9': 'Observe that by [REF] we have under assumption (C) [EQUATION]', '1801.05041-2-85-10': 'Finally, note that by Lemma [REF] [EQUATION]', '1801.05041-2-85-11': 'Summarizing, we find that under (C) [EQUATION]', '1801.05041-2-85-12': 'The final result follows from a combination of [REF], [REF] and [REF].', '1801.05041-2-85-13': 'First, observe that for [MATH] we have by [REF], [REF] and the assumptions on [MATH], with probability tending to one [EQUATION].', '1801.05041-2-85-14': 'It follows that, with probability tending to one, [MATH].', '1801.05041-2-85-15': 'Moreover, for [MATH] we have by [REF] and the assumptions on [MATH], with probability tending to one [EQUATION].', '1801.05041-2-85-16': 'Hence, with probability tending to one, [MATH] and thus [REF] follows.', '1801.05041-2-85-17': '[MATH]', '1801.05041-2-86-0': "Proof of Lemma [REF] By Knight's identity [REF] we have [EQUATION]", '1801.05041-2-86-1': 'Define [EQUATION]', '1801.05041-2-86-2': 'By a Taylor expansion we obtain [EQUATION] and thus the bound on [MATH] is established.', '1801.05041-2-86-3': 'Next we note that [EQUATION] since the conditional expectation given [MATH] equals zero almost surely and moreover [EQUATION]', '1801.05041-2-86-4': 'Note that in particular for [MATH] we have [EQUATION].', '1801.05041-2-86-5': 'Define [MATH] and apply the Bernstein inequality to show that for any [MATH] [EQUATION]', '1801.05041-2-86-6': 'For [MATH] the last line above is bounded by [MATH].', '1801.05041-2-86-7': 'Denote by [MATH] a grid of values [MATH] such that [MATH] for all [MATH] and [EQUATION].', '1801.05041-2-86-8': 'Note that it is possible to find such a [MATH] with [MATH].', '1801.05041-2-86-9': 'It follows that [EQUATION]', '1801.05041-2-86-10': 'Finally, note that for [MATH] [EQUATION]', '1801.05041-2-87-0': 'Since [MATH] we can pick [MATH] such that the last line above is [MATH], and hence [EQUATION].', '1801.05041-2-87-1': 'Finally, observe that, denoting by [MATH] the maximum norm of the entries of the matrix [MATH], [EQUATION] where the last line follows by a straightforward application of the Hoeffding inequality.', '1801.05041-2-87-2': 'Thus the proof of Lemma [REF] is complete.', '1801.05041-2-87-3': '[MATH]', '1801.05041-2-88-0': 'a4paper,left=1in,right=1in,top=1in,bottom=1in', '1801.05041-2-89-0': '# Additional simulation results', '1801.05041-2-90-0': '## More investigations on the tuning parameters in the IC criteria', '1801.05041-2-91-0': 'In the main manuscript, we reported the influence of the turning parameters in the IC criteria on the performance of the group and common parameters estimation for location-scale shift model with [MATH] errors on DGP1 where the predictor [MATH] and the fixed effects [MATH] are independent.', '1801.05041-2-91-1': 'Figure [REF] and Figure [REF] shows the corresponding results for location shift model with [MATH] error on DGP1.', '1801.05041-2-91-2': 'The corresponding plots for DGP2 are collected in Figure [REF] - Figure [REF].', '1801.05041-2-92-0': '# Additional Empirical Application Analysis', '1801.05041-2-93-0': 'As a robustness check, we report here the results of the empirical analysis for model ([REF]) without the incarcerating rate as a control covariate.', '1801.05041-2-93-1': 'Figure [REF] presents the corresponding common parameters estimation using the fixed effect approach while Figure [REF] for the results using our proposed grouped fixed effect estimator.', '1801.05041-2-93-2': 'Figure [REF] reports the raw and the grouped fixed effect estimates.'}

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['1801.05041-1-42-1', '1801.05041-2-65-1'], ['1801.05041-1-42-3', '1801.05041-2-65-3'], ['1801.05041-1-42-4', '1801.05041-2-65-4'], ['1801.05041-1-42-5', '1801.05041-2-65-5'], ['1801.05041-1-47-0', '1801.05041-2-70-0'], ['1801.05041-1-47-1', '1801.05041-2-70-1'], ['1801.05041-1-47-3', '1801.05041-2-70-3'], ['1801.05041-1-47-4', '1801.05041-2-70-4'], ['1801.05041-1-47-6', '1801.05041-2-70-6'], ['1801.05041-1-47-7', '1801.05041-2-70-7'], ['1801.05041-1-47-9', '1801.05041-2-70-9'], ['1801.05041-1-47-12', '1801.05041-2-70-12'], ['1801.05041-1-3-1', '1801.05041-2-3-1'], ['1801.05041-1-3-4', '1801.05041-2-8-0'], ['1801.05041-1-34-3', '1801.05041-2-48-2'], ['1801.05041-1-8-0', '1801.05041-2-7-5'], ['1801.05041-1-8-1', '1801.05041-2-7-6'], ['1801.05041-1-18-5', '1801.05041-2-32-0'], ['1801.05041-1-18-6', '1801.05041-2-32-1'], ['1801.05041-1-18-7', '1801.05041-2-32-2'], ['1801.05041-1-18-1', '1801.05041-2-14-2'], ['1801.05041-1-18-2', '1801.05041-2-14-3'], ['1801.05041-1-28-1', '1801.05041-2-37-4'], ['1801.05041-1-28-2', '1801.05041-2-37-5'], ['1801.05041-1-28-3', '1801.05041-2-37-6'], ['1801.05041-1-26-1', '1801.05041-2-37-1'], ['1801.05041-1-26-2', '1801.05041-2-37-2'], ['1801.05041-1-13-3', '1801.05041-2-13-6'], ['1801.05041-1-13-5', '1801.05041-2-13-8'], ['1801.05041-1-13-6', '1801.05041-2-13-9'], ['1801.05041-1-13-8', '1801.05041-2-26-1'], ['1801.05041-1-13-0', '1801.05041-2-25-0'], ['1801.05041-1-12-0', '1801.05041-2-13-0'], ['1801.05041-1-5-0', '1801.05041-2-9-0'], ['1801.05041-1-5-1', '1801.05041-2-9-3'], ['1801.05041-1-9-3', '1801.05041-2-6-3'], ['1801.05041-1-4-0', '1801.05041-2-5-0'], ['1801.05041-1-4-2', '1801.05041-2-5-2'], ['1801.05041-1-19-3', '1801.05041-2-15-3'], ['1801.05041-1-0-0', '1801.05041-2-0-0'], ['1801.05041-1-0-1', '1801.05041-2-0-1'], ['1801.05041-1-27-1', '1801.05041-2-22-1'], ['1801.05041-1-35-0', '1801.05041-2-50-0'], ['1801.05041-1-35-5', '1801.05041-2-50-6'], ['1801.05041-1-40-0', '1801.05041-2-63-0'], ['1801.05041-1-30-0', '1801.05041-2-43-0'], ['1801.05041-1-30-2', '1801.05041-2-43-2'], ['1801.05041-1-30-4', '1801.05041-2-43-7'], ['1801.05041-1-22-4', '1801.05041-2-34-4'], ['1801.05041-1-20-0', '1801.05041-2-16-0'], ['1801.05041-1-32-0', '1801.05041-2-44-0'], ['1801.05041-1-32-1', '1801.05041-2-44-1'], ['1801.05041-1-3-0', '1801.05041-2-3-0'], ['1801.05041-1-3-5', '1801.05041-2-8-1'], ['1801.05041-1-3-7', '1801.05041-2-8-2'], ['1801.05041-1-3-8', '1801.05041-2-8-3'], ['1801.05041-1-21-4', '1801.05041-2-19-0'], ['1801.05041-1-7-1', '1801.05041-2-7-2'], ['1801.05041-1-7-2', '1801.05041-2-7-3'], ['1801.05041-1-8-2', '1801.05041-2-7-7'], ['1801.05041-1-28-0', '1801.05041-2-37-3'], ['1801.05041-1-26-3', '1801.05041-2-21-2'], ['1801.05041-1-26-4', '1801.05041-2-21-3'], ['1801.05041-1-26-0', '1801.05041-2-37-0'], ['1801.05041-1-13-2', '1801.05041-2-13-5'], ['1801.05041-1-13-4', '1801.05041-2-13-7'], ['1801.05041-1-12-1', '1801.05041-2-13-1'], ['1801.05041-1-12-2', '1801.05041-2-13-2'], ['1801.05041-1-12-3', 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'1801.05041-2-9-4'], ['1801.05041-1-10-4', '1801.05041-2-9-5'], ['1801.05041-1-10-4', '1801.05041-2-9-6']]

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[]

[['1801.05041-1-9-2', '1801.05041-2-6-2'], ['1801.05041-1-9-5', '1801.05041-2-6-5'], ['1801.05041-1-31-1', '1801.05041-2-39-1'], ['1801.05041-1-31-2', '1801.05041-2-39-4'], ['1801.05041-1-19-2', '1801.05041-2-15-2'], ['1801.05041-1-16-1', '1801.05041-2-30-1'], ['1801.05041-1-30-3', '1801.05041-2-43-6'], ['1801.05041-1-22-0', '1801.05041-2-34-0'], ['1801.05041-1-20-1', '1801.05041-2-16-1'], ['1801.05041-1-33-0', '1801.05041-2-45-0'], ['1801.05041-1-32-4', '1801.05041-2-44-4'], ['1801.05041-1-3-2', '1801.05041-2-3-2'], ['1801.05041-1-21-1', '1801.05041-2-18-1'], ['1801.05041-1-21-2', '1801.05041-2-18-2'], ['1801.05041-1-34-0', '1801.05041-2-46-0'], ['1801.05041-1-7-0', '1801.05041-2-7-0'], ['1801.05041-1-7-0', '1801.05041-2-7-1'], ['1801.05041-1-7-3', '1801.05041-2-7-4'], ['1801.05041-1-18-0', '1801.05041-2-14-1'], ['1801.05041-1-13-1', '1801.05041-2-13-4'], ['1801.05041-1-13-1', '1801.05041-2-25-1'], ['1801.05041-1-10-3', '1801.05041-2-9-4'], ['1801.05041-1-10-4', '1801.05041-2-9-5'], ['1801.05041-1-10-4', '1801.05041-2-9-6']]

[]

['1801.05041-1-1-0', '1801.05041-1-15-0', '1801.05041-1-16-2', '1801.05041-1-18-3', '1801.05041-1-23-0', '1801.05041-1-42-2', '1801.05041-1-42-6', '1801.05041-1-47-2', '1801.05041-1-47-5', '1801.05041-1-47-8', '1801.05041-1-47-10', '1801.05041-1-47-11', '1801.05041-1-48-2', '1801.05041-1-49-0', '1801.05041-1-50-6', '1801.05041-1-51-1', '1801.05041-1-52-4', '1801.05041-1-54-6', '1801.05041-1-54-8', '1801.05041-1-55-5', '1801.05041-1-56-2', '1801.05041-1-57-8', '1801.05041-1-62-17', '1801.05041-1-63-1', '1801.05041-1-64-3', '1801.05041-2-1-0', '1801.05041-2-18-0', '1801.05041-2-19-1', '1801.05041-2-20-0', '1801.05041-2-30-2', '1801.05041-2-31-1', '1801.05041-2-65-2', '1801.05041-2-65-6', '1801.05041-2-70-2', '1801.05041-2-70-5', '1801.05041-2-70-8', '1801.05041-2-70-10', '1801.05041-2-70-11', '1801.05041-2-71-2', '1801.05041-2-72-0', '1801.05041-2-73-6', '1801.05041-2-74-1', '1801.05041-2-75-4', '1801.05041-2-77-6', '1801.05041-2-77-8', '1801.05041-2-78-5', '1801.05041-2-79-2', '1801.05041-2-80-8', '1801.05041-2-85-17', '1801.05041-2-86-1', '1801.05041-2-87-3', '1801.05041-2-88-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1801.05041

1602.00858

{'1602.00858-1-0-0': '# Introduction', '1602.00858-1-1-0': 'The PandaX project consists of a series of experiments at the China Jin-Ping Underground Laboratory, utiliziing xenon-based time projection chambers (TPC).', '1602.00858-1-1-1': 'The first phase experiment, PandaX-I [CITATION], with a 120-kg dual-phase xenon dark matter detector, completed its data taking in October 2014.', '1602.00858-1-1-2': 'The TPC of PandaX-I had a cylindrical shape with 60 cm diameter and 15 cm height, confined by the cathode plane located at the bottom in the liquid xenon, and a gate and anode planes located 4 mm below and above the liquid level, respectively.', '1602.00858-1-1-3': 'Particle interactions in the xenon produce prompt scintillation photons (S1).', '1602.00858-1-1-4': 'Part of the ionized electrons produced in the same interaction drift vertically upward between the cathode and gate, and get extracted by a stronger field between the gate and the anode into the gas, producing proportional scintillation photons (S2).', '1602.00858-1-1-5': 'Two arrays of photomultiplier tubes (PMTs) consisting of 143 1-inch R8520-406 tubes and 37 3-inch R11410-MOD tubes from Hamamatsu [CITATION] were located at the top and bottom, respectively, looking into the TPC collecting S1s and S2s.', '1602.00858-1-1-6': 'The time difference between S1 and S2 signals gives the vertical position of the interaction, and the charge pattern on the PMT arrays encodes the horizontal position.', '1602.00858-1-2-0': 'Being an underground experiment for very rare event search, the design philosophy of the electronics and data acquisition (DAQ) system of PandaX-I was to maximize the data collection under the lowest possible threshold, and leave the "data mining" work to the offline analysis.', '1602.00858-1-2-1': 'The system digitizes the waveforms from the PMTs, generates triggers on candidate signals, and save data to disks.', '1602.00858-1-2-2': 'The remainder of this paper is organized as follows.', '1602.00858-1-2-3': 'In Sec. [REF], the requirements of the system are presented, followed by the design details of the electronics in Sec. [REF] and DAQ in Sec. [REF].', '1602.00858-1-2-4': 'We summarize the performance of the system during the experiment in Sec. [REF] before conclusion.', '1602.00858-1-3-0': '# The System Requirements', '1602.00858-1-4-0': 'The typical dark matter recoil signals make low-energy deposition from sub-keV to a few tens of keV.', '1602.00858-1-4-1': 'For PandaX-I, this translates into a typical S1 from 1 to 100 photoelectrons (PE) with a time range of 100 ns, combining all PMTs.', '1602.00858-1-4-2': 'The charge of the corresponding S2 signal is typically one hundred times of S1 (100 to 10000 PE) with a time span of [MATH]s determined by the properties of the gas and the gap between the gate and anode.', '1602.00858-1-4-3': 'There are two general requirements for the PandaX-I electronics.', '1602.00858-1-5-0': 'These two general requirements are shown more specifically in Table [REF].', '1602.00858-1-5-1': 'For PandaX-I, the maximum electron drift length is 15 cm, corresponding to a 75 [MATH]s maximum separation between the S1 and S2 with a drift speed of [MATH]2 mm/[MATH]s. To allow the trigger be generated either by S1 or S2, the readout window should have equal pre- and post-trigger window of at least 75 [MATH]s.', '1602.00858-1-5-2': 'The single photoelectron (SPE) signals, assuming an overall gain of 2[MATH], is about 10 mV in amplitude with a few tens of ns timing span (taking into account the bandwidth limitations from the cables, etc).', '1602.00858-1-5-3': 'Therefore, 100 MS/s sampling rate is sufficient for the digitizers.', '1602.00858-1-5-4': 'To achieve a least bit value significantly less than 1 mV, a 14 bit ADC (with full range [MATH]2 V) is required.', '1602.00858-1-6-0': 'The requirement on bandwidth depends on the types of data runs in PandaX-I.', '1602.00858-1-6-1': 'The first type is the low occupancy LED runs to calibrate the gains of the PMTs, in which the LED driving pulse ([MATH]100 Hz) is used as the forced trigger for the digitizers.', '1602.00858-1-6-2': 'Such type of run can have short readout window ([MATH]s), therefore not demanding on the data size.', '1602.00858-1-6-3': 'The second type is the the calibration runs with gammas and neutrons, which is required to achieve a rate of [MATH]30 Hz in order to collect sufficient calibration data within a few days.', '1602.00858-1-6-4': 'The third type is the the normal dark matter search data, in which the trigger rate is expected to be is [MATH]10 Hz based on the Monte Carlo simulation.', '1602.00858-1-6-5': 'Clearly the latter two types are required to have long readout window of [MATH]s, and the source calibration runs dictate the data bandwidth requirement.', '1602.00858-1-6-6': 'Taking the 14 bit ADC and 100 MS/s sampling rate, the estimated bandwidth is 210 MBytes/s if no data reduction is used on the digitizer data.', '1602.00858-1-6-7': 'In reality, the baseline data can be suppressed (see Sec. [REF]), leading to at least a factor 3 reduction in data size (70 MB/s).', '1602.00858-1-6-8': 'Therefore the bandwidth requirement is loosened to be [MATH]80 MBytes/s.', '1602.00858-1-7-0': '# The System Design', '1602.00858-1-8-0': '## The Electronics System', '1602.00858-1-9-0': 'A schematic diagram of the electronics and DAQ system is shown in Fig. [REF].', '1602.00858-1-9-1': 'The system is very similar to the one used in the XENON100 experiment [CITATION].', '1602.00858-1-10-0': 'The high voltage (HV) mainframe SY1527LC with four A1932AP 48-channel positive HV modules were used as the HV supplies for the PMTs in both arrays.', '1602.00858-1-10-1': 'The raw signals from the PMTs were transmitted out of the detector together with the HV on the same coaxial cable and got separated via a capacitance decoupler outside the detector [CITATION].', '1602.00858-1-10-2': 'The raw signals got amplified linearly by a factor of 10 by Phillips 779 before entering into the CAEN V1724 (8 channels per VME module) digitizers.', '1602.00858-1-10-3': 'V1724 stores data in buffers which get read out upon receiving the trigger signal generated by the trigger system.', '1602.00858-1-10-4': 'The scaler unit (CAEN V830) counts both raw and system trigger signals to monitor the system deadtime.', '1602.00858-1-11-0': '## The Digitizers', '1602.00858-1-12-0': 'The central components of the electronics are 23 VME modules of CAEN V1724 digitizers, located in two VME crates, one as the master and the other as the slave, connected through optical fibers.', '1602.00858-1-12-1': 'Each V1724 supplies eight channels of 14-bit 100 MS/s flash ADC (FADC) with 2.25 Vpp dynamic range, leading to 0.137 mV for the least significant bit.', '1602.00858-1-12-2': 'Each of the 180 PMT signals is connected to a FADC channel.', '1602.00858-1-12-3': 'As mentioned in Sec. [REF], the smallest S1 signal is an SPE ([MATH]10 mV, 100 ns).', '1602.00858-1-12-4': 'For the maximum S1 ([MATH]100 PE, 100 ns) and S2 signals ([MATH]10000 PE, 2 [MATH]s) in the dark matter search region, the maximum amplitude is [MATH]2V even if all photons are collected by a single PMT.', '1602.00858-1-13-0': 'Each V1724 has an independent 100 MHz internal clock.', '1602.00858-1-13-1': 'V1724 allows to synchronize among boards by using the CLK-OUT from a given board as the master, and daisy-chaining the CLK-IN and CLK-OUT of the rest of the boards with programmable phase to each CLK-OUT signals.', '1602.00858-1-13-2': 'All the 23 VME boards were synchronized to less than 2 ns by this means.', '1602.00858-1-14-0': '### The Majority Signal', '1602.00858-1-15-0': 'Each V1724 board houses a 12-bit 100 MHz DAC with 1 V range on a 50 Ohm load and the output is available on the "MON/[MATH]" connector on the front panel [CITATION].', '1602.00858-1-15-1': 'Each FADC channel generates a time-over-threshold signal on a preset threshold (high = 125 mV), and the internal sum of eight channels are output by the DAC.', '1602.00858-1-15-2': 'The area of this so-called MAJ signal encodes the overall PE values in this board for small signals when counting assumption is appropriate.', '1602.00858-1-15-3': 'In PandaX-I, the threshold for the majority signal was set to be [MATH]1 PE for each channel.', '1602.00858-1-15-4': 'The MAJ signals are used in the trigger generation, which will be discussed in detail in Sec. [REF].', '1602.00858-1-16-0': '### Zero Suppression', '1602.00858-1-17-0': 'CAEN V1724 allows to record the waveforms in a Zero Length Encoding (ZLE) mode [CITATION], discarding the data under certain threshold set by the users.', '1602.00858-1-17-1': 'In PandaX-I, after the gain balancing, we chose the amplitude of the 1/3 PE as the threshold for the waveform suppression.', '1602.00858-1-17-2': 'The efficiency for such a threshold was studied by comparing the low intensity LED run with and without the ZLE to be nearly 100%.', '1602.00858-1-17-3': 'To allow proper baseline determination, we set the "look back" and "look forward" samples before and after the threshold crossings both to 40.', '1602.00858-1-17-4': 'In PandaX physical runs, ZLE mode compresses the data by a factor of 7.', '1602.00858-1-17-5': 'Even with the 30 Hz source calibration data, the achieved data rate is 30 MB/s, far less compared to the 80 MByte/s data acquisition rate (see Sec. [REF]).', '1602.00858-1-18-0': '## The Trigger System', '1602.00858-1-19-0': 'The general requirement for the trigger system is to achieve a low energy threshold [MATH]1 keV[MATH].', '1602.00858-1-19-1': 'To accomplish this, the actual design has to take into account two important issues.', '1602.00858-1-19-2': 'First, since signals from many PMTs are involved in generating the trigger, the trigger needs to be robust against coherent noises among channels.', '1602.00858-1-19-3': 'In PandaX-I experiment, we observed coherent electromagnetic pickups short and fast oscillating pulses ([MATH]100 ns) occurring at a frequency of 200 kHz, originated from the CAEN SY1527LC power supply.', '1602.00858-1-19-4': 'Under this circumstance, neither number-of-hit nor summed-signal-amplitude work well as a trigger source.', '1602.00858-1-19-5': 'Second, for the few-PE level S1 signals, it is very difficult to discriminate against noises in the trigger hardware.', '1602.00858-1-19-6': 'In this case, triggers on S2 signals (typically one hundred times larger in charge) are much more effective.', '1602.00858-1-19-7': 'However, for small S2 signals, each PMT receives a few PEs but distributed broadly in a few [MATH]s, a challenge for the trigger system to identify.', '1602.00858-1-20-0': 'In Fig. [REF], an illustration of the PandaX-I trigger generation is shown.', '1602.00858-1-21-0': 'In the PandaX-I detector, both S1 and S2 light collections are dominated by the bottom 3-in PMTs, read out by five V1724 modules, each with its own 100 MHz MAJ output (Sec. [REF]).', '1602.00858-1-21-1': 'As shown in Fig. [REF], a Phillips 740 linear Fan-in/out module sums up the MAJ outputs from the five modules producing an analogue signal S0, which is a global time-over-threshold signal with its amplitude representing the number of fired bottom PMTs.', '1602.00858-1-21-2': 'S0 is then integrated by a ORTEC 575A spectroscopy amplifier with the amplitude of the output proportional to the area of S0.', '1602.00858-1-21-3': 'The integrated signal is further discriminated by a CAEN V814 VME discrimator to generate the raw trigger signal.', '1602.00858-1-21-4': 'Such a trigger scheme is effective to address the two issues above.', '1602.00858-1-21-5': 'The short coherent noises, even those that happened to cross the majority threshold on number of channels, would only produce a narrow S0 signal, whose integral was too small to trigger.', '1602.00858-1-21-6': 'For broad and low amplitude S2 signals, the integral of S0 approximately represent the overall charge of the S2 signals, and is more immune to the analogue noises due to the time-over-threshold nature of the MAJ signals.', '1602.00858-1-22-0': 'V1724 uses a circular buffer to store data in its 1 MBytes memory to mitigate the deadtime during data readout.', '1602.00858-1-22-1': 'BUSY signals are generated when the buffer is "almost full".', '1602.00858-1-22-2': 'The raw trigger signal and BUSY of the twenty three V1724 are input into a CAEN V1495 general logic unit.', '1602.00858-1-22-3': 'To avoid multiple trigger signals within the length of a readout window (200 [MATH]s), a holdoff window of 200 [MATH] is set after a raw trigger.', '1602.00858-1-22-4': 'The system trigger is generated by the FPGA in V1495 that requires a raw trigger not being vetoed by any of the BUSY.', '1602.00858-1-23-0': '# The DAQ System', '1602.00858-1-24-0': '## Architecture', '1602.00858-1-25-0': 'The DAQ system reads out and processes the data from the electronics via the VME controller and optical bridges.', '1602.00858-1-25-1': 'As shown in Fig. [REF], the DAQ server communicates with the VME electronics through optical fibers with a PCI-e interface card (CAEN A3818C), capable to host up to 4 optical fiber links.', '1602.00858-1-25-2': 'The DAQ uses one optical link to control the VME bus through the controller module CAEN V2718.', '1602.00858-1-25-3': 'The DAQ reads out the data from V1724 directly via three optical links, each connecting up to 8 daisy-chained V1724.', '1602.00858-1-26-0': '## DAQ Software', '1602.00858-1-27-0': 'The DAQ software was developed in house in C++, with external libraries from CAEN (Fig. [REF]), as well as the XML and MySQL libraries.', '1602.00858-1-27-1': 'The DAQ software works in a run loop.', '1602.00858-1-27-2': 'At the beginning of a run, the system resets all VME modules, clears buffers and releases all digital status.', '1602.00858-1-27-3': 'All modules are initialized with configuration parameters specified in a user XML file.', '1602.00858-1-27-4': 'After the successful initialization, the system accepts the "run start" logic, and writes corresponding run configuration information into a MySQL database.', '1602.00858-1-27-5': 'MBLT (multi block transfer) mode is used to read out every five events stored in the V1724 circular buffer.', '1602.00858-1-28-0': 'DAQ contains an event builder to construct the event bank, consisting of the header such as DAQ status, configuration, event status, as well as the FADC data blocks.', '1602.00858-1-28-1': 'The FADC data blocks contain auxillary information about the VME crate, board number, channel number, and time stamps, etc., to facilitate the offline analysis.', '1602.00858-1-28-2': 'A long run is broken into number of files, each with a limit size of 1 GB.', '1602.00858-1-28-3': 'At the opening and closing of each data file, the DAQ also inserts corresponding information into the MySQL database.', '1602.00858-1-29-0': 'An issue was identified that the data from different V1724 could occassionally get misaligned, making it nearly impossible to reconstruct the original events.', '1602.00858-1-29-1': 'The issue is due to the loss of synchronization between clocks on different V1724.', '1602.00858-1-29-2': 'To combat this, the DAQ system used V1495 to send out trigger time reset signal at the "RUN START" which mitigated the problem.', '1602.00858-1-29-3': 'In addition, a detection was implemented in the DAQ software to check for difference in the timestamps between the first and last V1724 within an events, and forced the run to stop if a problem was found.', '1602.00858-1-30-0': 'In the stable running period, to achieve a complete automation of the DAQ, a background watchdog program restarted the run if there was a crash.', '1602.00858-1-31-0': '## Realtime and Online Data Processing and Monitoring', '1602.00858-1-32-0': 'The online data processing and quality monitoring scheme is shown in Fig. [REF].', '1602.00858-1-32-1': 'The DAQ server was located underground in the electronics rack with disk arrays attached to it.', '1602.00858-1-32-2': 'To achieve real time monitoring of the DAQ performance, a process on the DAQ server parsed the output file as the data was being written onto the disk, producing low-level data-quality figures such as the trigger rate, PMT hit maps and waveforms, etc., viewable from a web page.', '1602.00858-1-32-3': 'As soon as a file was closed, a file copier program on the DAQ server would copy it to a network attached storage server located outside the laboratory, connected to the underground lab through fiber links.', '1602.00858-1-32-4': 'The storage server was attached to a multi-core processing server.', '1602.00858-1-32-5': 'Once the copying of a new file was completed, an analysis program would be triggered, generating output root files available for online analysis.', '1602.00858-1-32-6': 'Data quality figures of the run up to the most recent file would also be automatically generated, viewable on the web page.', '1602.00858-1-33-0': 'During the smooth running period, the maximum data copying rate was tested to be 60 MB/s, corresponding roughly to a trigger rate of 85 Hz with a 200[MATH]s readout windows on all channels and the baseline suppression enabled.', '1602.00858-1-34-0': '# The System Performance', '1602.00858-1-35-0': 'The electronics and DAQ system functioned stably during the PandaX-I data taking.', '1602.00858-1-35-1': 'Detailed analysis for the final dark matter exposure has been published [CITATION].', '1602.00858-1-35-2': 'Here we restrict the discussions to the performance of the system.', '1602.00858-1-36-0': '## The Electronics performance', '1602.00858-1-37-0': 'The central components of the electronics system are the waveform digitizers.', '1602.00858-1-37-1': 'A typical waveform from a 3-in PMT in a given event is shown in Fig. [REF], in which a clear S1 and S2 signal is observed.', '1602.00858-1-37-2': 'The flat regions on the waveform correspond to the baseline which is suppressed by the ZLE.', '1602.00858-1-37-3': 'For each unsuppressed waveform segment, sufficient pre- and post-threshold-crossing samples were saved, based on which the baseline noise could be computed.', '1602.00858-1-37-4': 'On average, the baseline noise was [MATH]0.7 mV, in comparison to the average amplitude of the SPE which was 10 mV.', '1602.00858-1-38-0': 'The most important performance parameter is the trigger threshold.', '1602.00858-1-38-1': 'As mentioned in Sec. [REF], the trigger was primarily on the S2 signals for the low energy events in the dark matter search region, and the corresponding S1s were identified by offline software.', '1602.00858-1-38-2': 'In Fig. [REF], the charge of S1 vs. the leading edge time of S1 is shown.', '1602.00858-1-38-3': 'The bright vertical band around 100 [MATH]s corresponds to the events triggered by S1 with a charge larger than [MATH]65 PE.', '1602.00858-1-38-4': 'The preceding lower band corresponds to the S2 triggered events primarily with lower values of S1.', '1602.00858-1-38-5': 'To study the threshold for S2, the low energy nuclear recoil events in the [MATH]Cf calibration data were taken with their S2 distribution shown in Fig. [REF].', '1602.00858-1-38-6': 'For these events, the true S2 distribution is expected to approximately be an exponential shape according to the Monte Carlo.', '1602.00858-1-38-7': 'The measured S2 distribution was then fit in which the trigger efficiency function was modeled as a Fermi-Dirac function.', '1602.00858-1-38-8': 'The resulting threshold was 89.0[MATH]1.6 PE for a 50% efficiency.', '1602.00858-1-39-0': '## The DAQ Performance', '1602.00858-1-40-0': 'The limitation of the DAQ data rate was measured with the LED calibration runs without zero suppression by either increasing the rate of the LED forced triggers, or by the length of the readout window.', '1602.00858-1-40-1': 'The results are shown in Fig. [REF].', '1602.00858-1-40-2': 'One sees that in both measurements, the DAQ rate saturated consistently at around 70 MB/s.', '1602.00858-1-41-0': 'Shown in Table [REF] are the typical data rates of the four different run modes in PandaX-I, all significantly less than the 70 MB/s measured upper limit of the bandwidth.', '1602.00858-1-41-1': 'One expects nearly deadtime-less operation under such rates.', '1602.00858-1-42-0': '# Summary and Conclusions', '1602.00858-1-43-0': 'We have described the electronics and data acquisition system for the PandaX-I dark matter direct detection experiment.', '1602.00858-1-43-1': 'This system used commercial electronics modules but custom-developed trigger system and DAQ software.', '1602.00858-1-43-2': 'The system was operated steadily for more than nine months and collected about 8 TB of dark matter data.', '1602.00858-1-43-3': 'The performance met the challenging requirements in the search of very low energy dark matter recoil events.', '1602.00858-1-43-4': 'Although designed for PandaX-I, this system is easily expandable to the future phase of the experiment, and maintains flexibility in acquisition strategy to further optimize the performance.'}

{'1602.00858-2-0-0': '# Introduction', '1602.00858-2-1-0': 'The PandaX project consists of a series of experiments at the China Jin-Ping Underground Laboratory, utilizing xenon-based time projection chambers (TPC).', '1602.00858-2-1-1': 'The first phase experiment, PandaX-I [CITATION], with a 120-kg dual-phase xenon dark matter detector, completed its data taking in October 2014.', '1602.00858-2-1-2': 'The TPC of PandaX-I had a cylindrical shape with 60 cm diameter and 15 cm height, confined by the cathode plane located at the bottom in the liquid xenon, and a gate and anode planes located 4 mm below and above the liquid level, respectively.', '1602.00858-2-1-3': 'Particle interactions in the xenon produce prompt scintillation photons (S1).', '1602.00858-2-1-4': 'Some of the electrons from ionization produced in the same interaction drift vertically upward between the cathode and gate, and get extracted by a stronger field between the gate and the anode into the gas, producing proportional scintillation photons (S2).', '1602.00858-2-1-5': 'The background gammas produce electron recoil events, while the dark matter signals interact with xenon atomic nucleus producing nuclear recoil events.', '1602.00858-2-1-6': 'The ratio of S2 to S1 is less for the latter compared to the former due to different ionization capability.', '1602.00858-2-1-7': 'This ratio is therefore used as an effective background discrimination parameter.', '1602.00858-2-1-8': 'Two arrays of photomultiplier tubes (PMTs) consisting of 143 1-inch R8520-406 tubes and 37 3-inch R11410-MOD tubes from Hamamatsu [CITATION] were located at the top and bottom, respectively, looking into the TPC collecting S1s and S2s.', '1602.00858-2-1-9': 'The time difference between S1 and S2 signals gives the vertical position of the interaction, and the charge pattern of S2 signals on the PMT arrays encodes the horizontal position.', '1602.00858-2-2-0': 'Being an underground experiment for very rare event search, the design philosophy of the electronics and data acquisition (DAQ) system of PandaX-I was to maximize the data collection under the lowest possible threshold, and leave the "data mining" work to the offline analysis.', '1602.00858-2-2-1': 'The system digitizes the waveforms from the PMTs, generates triggers on candidate signals, and saves the data to disks.', '1602.00858-2-2-2': 'The remainder of this paper is organized as follows.', '1602.00858-2-2-3': 'In Sec. [REF], the requirements of the system are presented, followed by the design details of the electronics in Sec. [REF] and DAQ in Sec. [REF].', '1602.00858-2-2-4': 'We summarize the performance of the system during the experiment in Sec. [REF] before the conclusion.', '1602.00858-2-3-0': '# System Requirements', '1602.00858-2-4-0': 'The typical dark matter recoil signals deposit small amounts of energy from sub-keV to a few tens of keV.', '1602.00858-2-4-1': 'For PandaX-I, this translates into a typical S1 from 1 to 100 photoelectrons (PE) with a time range of 100 ns, combining all PMTs.', '1602.00858-2-4-2': 'The intensity of the corresponding S2 signal is typically one hundred times that of S1 (100 to 10000 PE) with a time span of [MATH]s determined by the properties of the gas and the gap between the gate and anode.', '1602.00858-2-4-3': 'There are two general requirements for the PandaX-I electronics.', '1602.00858-2-5-0': 'These two general requirements are shown more specifically in Table [REF].', '1602.00858-2-5-1': 'For PandaX-I, the maximum electron drift length is 15 cm, corresponding to an 88 [MATH]s maximum separation between the S1 and S2 with a drift speed of [MATH]1.7 mm/[MATH]s. To allow the trigger to be generated either by S1 or S2, the readout window should have an equal pre- and post-trigger width of at least 88 [MATH]s. To be able to cleanly monitor random light generation in the detector, the readout window was selected to be [MATH]s before and after the trigger.', '1602.00858-2-5-2': 'The single photoelectron (SPE) signals, assuming an overall gain of 2[MATH], are about 10 mV in amplitude with a full-width-half-maximum of about 20 ns and a full range of about 100 ns (see later Fig. [REF]), resulting in an approximate 10 mV[MATH]20 ns conversion factor from pulse area to a SPE.', '1602.00858-2-5-3': 'According to a simulation including digitization and bandwidth effects, in order to achieve a [MATH] bias to the charge of a single photoelectron, a 100 MS/s sampling rate and a 14 bit ADC (with full range [MATH]2 V) is required.', '1602.00858-2-6-0': 'The requirement on data bandwidth depends on the types of data runs in PandaX-I.', '1602.00858-2-6-1': 'The first type is the low occupancy LED run to calibrate the gains of the PMTs, in which the fast light pulses from external LEDs were injected into the detector with the system triggered by the LED driving pulses ([MATH] 100 Hz).', '1602.00858-2-6-2': 'Such a type of run can have a short readout window ([MATH]s) and therefore has a small event size.', '1602.00858-2-6-3': 'The second type is the calibration run with gammas and neutrons, which is required to achieve a rate of [MATH]30 Hz in order to collect sufficient calibration data within a few days.', '1602.00858-2-6-4': 'The third type is the normal dark matter search run, in which the trigger rate is expected to be [MATH]10 Hz based on the Monte Carlo simulation.', '1602.00858-2-6-5': 'Clearly the latter two types are required to have a long readout window of [MATH]s, and the source calibration runs dictate the data bandwidth requirement.', '1602.00858-2-6-6': 'Taking the 14 bit ADC and 100 MS/s sampling rate, the estimated bandwidth is 210 MB/s if no data reduction is used on the digitizer data.', '1602.00858-2-6-7': 'In reality, the baseline data can be suppressed (see Sec. [REF]), leading to at least a factor 3 reduction in data rate (70 MB/s).', '1602.00858-2-7-0': '# System Design', '1602.00858-2-8-0': '## Electronics System', '1602.00858-2-9-0': 'A schematic diagram of the electronics and DAQ system is shown in Fig. [REF].', '1602.00858-2-9-1': 'The system is very similar to the one used in the XENON100 experiment [CITATION].', '1602.00858-2-10-0': 'The high voltage (HV) mainframe SY1527LC with four A1932AP 48-channel positive HV modules were used as the HV supplies for the PMTs in both arrays.', '1602.00858-2-10-1': 'The raw signals from the PMTs were transmitted out of the detector together with the HV on the same coaxial cable and were separated via a capacitive decoupler outside the detector [CITATION].', '1602.00858-2-10-2': 'Since the average gain of the PMTs was 2[MATH], to achieve a 2[MATH] gain as mentioned in Sec. [REF], the raw PMT signals were amplified linearly by a factor of 10 by the Phillips 779 amplifiers before entering into the CAEN V1724 (8 channels per VME module) digitizers [CITATION].', '1602.00858-2-10-3': 'The V1274 digitizer stored data in buffers which were read out upon receiving the trigger signal generated by the trigger system.', '1602.00858-2-10-4': 'The scaler unit (CAEN V830) counted both raw and system trigger signals to monitor the system deadtime.', '1602.00858-2-11-0': '## Digitizers', '1602.00858-2-12-0': 'The central components of the electronics are 23 VME modules of CAEN V1724 digitizers, located in two VME crates, connected through optical fibers.', '1602.00858-2-12-1': 'Each V1724 supplies eight channels of 14-bit 100 MS/s flash ADC (FADC) with 2.25 V[MATH] dynamic range, leading to 0.137 mV for the least significant bit.', '1602.00858-2-12-2': 'Each of the 180 PMT signals is connected to a FADC channel.', '1602.00858-2-12-3': 'Taking the approximate 10 mV [MATH] 20 ns/SPE conversion factor mentioned in Sec. [REF], the maximum S1 ([MATH]100 PE, 100 ns) and S2 signals ([MATH]10000 PE, 2 [MATH]s) in the dark matter search region will have an amplitude of 400 mV and 2 V, respectively, assuming a triangular shape of the waveform, even if all photons are collected by a single PMT.', '1602.00858-2-13-0': 'Each V1724 has an independent 100 MHz internal clock.', '1602.00858-2-13-1': 'It allows synchronization among boards by using the CLK-OUT from a given board as the master, and daisy-chaining the CLK-IN and CLK-OUT of the rest of the boards with a programmable phase for each CLK-OUT signal.', '1602.00858-2-13-2': 'All 23 VME boards were verified to be synchronized to better than 2 ns by routing the CLK-OUT from each board into the oscilloscope.', '1602.00858-2-13-3': 'This is sufficiently precise for this experiment given that a typical S1 signal is roughly 100 ns wide.', '1602.00858-2-14-0': '### Majority Signal', '1602.00858-2-15-0': 'Each V1724 board houses a 12-bit 100 MHz DAC with 1 V range on a 50 Ohm load and the output is available on the "MON/[MATH]" connector on the front panel.', '1602.00858-2-15-1': 'Each FADC channel generates a time-over-threshold signal on a preset threshold (high = 125 mV), and the internal sum of eight channels is output by the DAC.', '1602.00858-2-15-2': 'The area of this so-called MAJ signal encodes the overall PE values in this board for small signals when PEs are not piling up in the waveform.', '1602.00858-2-15-3': 'In PandaX-I, the threshold for the majority signal was set to be [MATH]1 PE for each channel.', '1602.00858-2-15-4': 'The MAJ signals are used in the trigger generation, which will be discussed in detail in Sec. [REF].', '1602.00858-2-16-0': '### Zero Suppression', '1602.00858-2-17-0': 'The CAEN V1724 has the capability to record the waveforms in a Zero Length Encoding (ZLE) mode, discarding the data under a threshold set by the user.', '1602.00858-2-17-1': 'In PandaX-I, after the gain stage, we chose the amplitude of 1/3 PE as the threshold for the waveform suppression.', '1602.00858-2-17-2': 'The efficiency for such a threshold was studied by comparing the low intensity LED run with and without the ZLE to be nearly 100%.', '1602.00858-2-17-3': 'To allow proper baseline determination, we set the "look back" and "look forward" samples before and after the threshold crossings both to 40.', '1602.00858-2-17-4': 'In PandaX physics runs, ZLE mode compresses the data by a factor of 7.', '1602.00858-2-17-5': 'Even with the 30 Hz source calibration data, the achieved data rate is 30 MB/s, far less compared to the 70 MB/s data acquisition bandwidth (see Sec. [REF]).', '1602.00858-2-18-0': '## Trigger System', '1602.00858-2-19-0': 'The general requirement for the trigger system is to achieve a low energy threshold [MATH]1 keV[MATH].', '1602.00858-2-19-1': 'To accomplish this, the actual design has to take into account two important issues.', '1602.00858-2-19-2': 'First, since signals from many PMTs are involved in generating the trigger, the trigger needs to be robust against coherent noise among channels.', '1602.00858-2-19-3': 'In PandaX-I experiment, we observed coherent short and fast oscillating pulses ([MATH]100 ns) from electromagnetic pickups occurring at a frequency of 200 kHz, that originated from the CAEN SY1527LC power supply .', '1602.00858-2-19-4': 'Under these circumstances, neither number-of-hit nor summed-signal-amplitude work well as a trigger source.', '1602.00858-2-19-5': 'Second, for the few-PE level S1 signals, it is very difficult to discriminate against noise in the trigger hardware.', '1602.00858-2-19-6': 'In this case, triggers on S2 signals (typically one hundred times larger) are much more effective.', '1602.00858-2-19-7': 'However, for small S2 signals, each PMT receives a few PEs but distributed broadly in a few [MATH]s, a challenge for the trigger system to identify.', '1602.00858-2-20-0': 'In Fig. [REF], an illustration of the PandaX-I trigger generation is shown.', '1602.00858-2-21-0': 'In the PandaX-I detector, both S1 and S2 light collections are dominated by the bottom 3-inch PMTs, read out by five V1724 modules, each with its own 100 MHz MAJ output (Sec. [REF]).', '1602.00858-2-21-1': 'As shown in Fig. [REF], a Phillips 740 linear Fan-in/out module sums up the MAJ outputs from the five modules producing an analogue signal S0, which is a global time-over-threshold signal with its amplitude representing the number of fired bottom PMTs.', '1602.00858-2-21-2': 'S0 is then integrated by an ORTEC 575A spectroscopy amplifier (gain 25, shaping time 1[MATH]s) with the amplitude of the output proportional to the area of S0.', '1602.00858-2-21-3': 'The integrated signal is further discriminated by a CAEN V814 VME discriminator to generate the raw trigger signal.', '1602.00858-2-21-4': 'Such a trigger scheme is effective to address the two issues above.', '1602.00858-2-21-5': 'The short coherent noise, even if it happened to cross the majority threshold on a number of channels, would only produce a narrow S0 signal, whose integral would be too small to trigger.', '1602.00858-2-21-6': 'Small and narrow S1 would not generate the trigger either.', '1602.00858-2-21-7': 'On the other hand, for the broad and low amplitude S2 signals, the integral of S0 approximately represents the overall charge.', '1602.00858-2-21-8': 'The discriminator threshold is set to 15 mV, which is sufficiently low to trigger [MATH]1 keV[MATH] events efficiently (see Sec. [REF]).', '1602.00858-2-22-0': 'V1724 uses a circular buffer to store data in its 1 MB memory to mitigate the deadtime during data readout.', '1602.00858-2-22-1': 'BUSY signals are generated when the buffer is "almost full".', '1602.00858-2-22-2': 'The raw trigger signal and BUSY signals of the 23 V1724 are input into a CAEN V1495 general logic unit.', '1602.00858-2-22-3': 'To avoid multiple trigger signals within the length of a readout window (200 [MATH]s), a hold off window of 200 [MATH]s is set after a raw trigger.', '1602.00858-2-22-4': 'The system trigger is generated by the FPGA in V1495 that requires a raw trigger not being vetoed by any of the BUSY signals.', '1602.00858-2-23-0': '# DAQ System', '1602.00858-2-24-0': '## Architecture', '1602.00858-2-25-0': 'The DAQ system reads out and processes the data from the electronics via the VME controller and optical bridges.', '1602.00858-2-25-1': 'As shown in Fig. [REF], the DAQ server communicates with the VME electronics through optical fibers with a PCI-e interface card (CAEN A3818C), capable of hosting up to 4 optical fiber links.', '1602.00858-2-25-2': 'The DAQ uses one optical link to control the VME bus through the controller module CAEN V2718.', '1602.00858-2-25-3': 'The DAQ reads out the data from V1724 directly via three optical links, each connecting up to 8 daisy-chained V1724.', '1602.00858-2-26-0': '## DAQ Software', '1602.00858-2-27-0': 'The DAQ software was developed in house in C++, with external libraries from CAEN (Fig. [REF]), as well as the XML and MySQL libraries.', '1602.00858-2-27-1': 'At the beginning of a run, the system resets all VME modules and clears buffers.', '1602.00858-2-27-2': 'All modules are initialized with configuration parameters specified in a user XML file.', '1602.00858-2-27-3': 'After the successful initialization, the system accepts the "run start" logic, and writes corresponding run configuration information into a MySQL database.', '1602.00858-2-27-4': 'MBLT (multi block transfer) mode is used to read out every five events stored in the V1724 circular buffer.', '1602.00858-2-28-0': 'The DAQ contains an event builder to construct the event, consisting of the header such as DAQ status, configuration, event status, as well as the FADC data blocks.', '1602.00858-2-28-1': 'The FADC data blocks contain auxiliary information about the VME crate, board number, channel number, and time stamps, etc., to facilitate the offline analysis.', '1602.00858-2-28-2': 'A long run is broken into number of files, each with a size limit of 1 GB.', '1602.00858-2-28-3': 'At the opening and closing of each data file, the DAQ also inserts corresponding information into the MySQL database.', '1602.00858-2-29-0': 'An issue was identified in which the data from different V1724s could occasionally get misaligned, making it nearly impossible to reconstruct the original events.', '1602.00858-2-29-1': 'The issue was due to the loss of synchronization between clocks on different V1724s.', '1602.00858-2-29-2': 'To combat this, the DAQ system used the V1495 module to send out a trigger time reset signal at the "RUN START".', '1602.00858-2-29-3': 'In addition, a method was implemented in the DAQ software to check for a difference in the timestamps between the first and last V1724 data block within an event, and forced the run to stop if a problem was found.', '1602.00858-2-29-4': 'These implementations fixed the problem.', '1602.00858-2-30-0': 'In the stable running period, to achieve a complete automation of the DAQ, a background watchdog program restarted the run if there was a crash.', '1602.00858-2-31-0': '## Realtime and Online Data Processing and Monitoring', '1602.00858-2-32-0': 'The online data processing and quality monitoring scheme is shown in Fig. [REF].', '1602.00858-2-32-1': 'The DAQ server was located underground in the electronics rack with disk arrays attached to it.', '1602.00858-2-32-2': 'To achieve real time monitoring of the DAQ performance, a process on the DAQ server parsed the output file as the data were being written onto the disk, producing low-level data-quality plots such as the trigger rate, PMT hit maps and waveforms, etc., viewable from a web page.', '1602.00858-2-32-3': 'As soon as a file was closed, a file copier program on the DAQ server would copy it to a network attached storage server located outside the laboratory, connected to the underground lab through fiber links.', '1602.00858-2-32-4': 'The storage server was attached to a multi-core processing server.', '1602.00858-2-32-5': 'Once the copying of a new file was completed, an analysis program would be triggered, generating output root files available for online analysis.', '1602.00858-2-32-6': 'Data quality figures of the run, up to the most recent file, would also be automatically generated, viewable on the web page.', '1602.00858-2-33-0': 'During the smooth running period, the maximum data copying rate was tested to be 60 MB/s, corresponding roughly to a trigger rate of 85 Hz with a 200[MATH]s readout windows on all channels and the baseline suppression enabled .', '1602.00858-2-34-0': '# System Performance', '1602.00858-2-35-0': 'The electronics and DAQ system functioned stably during the PandaX-I data taking.', '1602.00858-2-35-1': 'Detailed analysis for the final dark matter exposure has been published [CITATION].', '1602.00858-2-35-2': 'Here we restrict the discussions to the performance of the system.', '1602.00858-2-36-0': '## Electronics performance', '1602.00858-2-37-0': 'The central components of the electronics system are the waveform digitizers.', '1602.00858-2-37-1': 'A typical waveform from a 3-inch PMT in a given event is shown in Fig. [REF], in which a clear S1 and S2 signal is observed.', '1602.00858-2-37-2': 'The flat regions on the waveform correspond to the baseline which is suppressed by the ZLE.', '1602.00858-2-37-3': 'For each unsuppressed waveform segment, sufficient pre- and post-threshold-crossing samples were saved, based on which the baseline noise could be computed.', '1602.00858-2-37-4': 'On average, the baseline noise was [MATH] 0.7 mV, in comparison to the average amplitude of the SPE which was 10 mV.', '1602.00858-2-37-5': 'The noise level obtained during the LED calibration run with ZLE disabled was consistent.', '1602.00858-2-38-0': 'The most important performance parameter is the trigger threshold.', '1602.00858-2-38-1': 'As mentioned in Sec. [REF], the trigger was generated from the S2 signals for the low energy events in the dark matter search region, and the corresponding S1s were identified by offline software.', '1602.00858-2-38-2': 'In Fig. [REF], the charge of S1 vs. the leading edge time of S1 is shown.', '1602.00858-2-38-3': 'The bright vertical band around 100 [MATH]s corresponds to the events triggered by S1 with a charge larger than [MATH]65 PE.', '1602.00858-2-38-4': 'The preceding lower band corresponds to the S2 triggered events with lower values of S1.', '1602.00858-2-38-5': 'The events with S1 >100 [MATH]s are from the afterpulsing of PMTs after large S2 signals but identified as S1s.', '1602.00858-2-38-6': 'To study the threshold for S2, the low energy nuclear recoil events in the [MATH]Cf calibration data were taken with their S2 distribution shown in Fig. [REF].', '1602.00858-2-38-7': 'For these events, the true S2 distribution is expected to approximately be an exponential shape according to the Monte Carlo.', '1602.00858-2-38-8': 'The measured S2 distribution was then fitted in which the trigger efficiency function was modeled as a Fermi-Dirac function.', '1602.00858-2-38-9': 'The resulting threshold was 89.0[MATH]1.6 PE for a 50% efficiency.', '1602.00858-2-38-10': 'According to the energy model used in Ref. [CITATION], the mean value of (S1, S2) for a 1 keV[MATH] electron recoil (background) and nuclear recoil (signal) events are (1.6, 867) PE and (3.7, 525) PE, respectively.', '1602.00858-2-38-11': 'Clearly the trigger performance satisfies the threshold requirement in Table [REF].', '1602.00858-2-39-0': '## DAQ Performance', '1602.00858-2-40-0': 'The limitation of the DAQ data rate was measured with the LED calibration runs without zero suppression by increasing either the rate of the LED forced triggers, or the length of the readout window.', '1602.00858-2-40-1': 'The results are shown in Fig. [REF].', '1602.00858-2-40-2': 'One sees that in both measurements, the DAQ rate saturated consistently at around 70 MB/s.', '1602.00858-2-41-0': 'Shown in Table [REF] are the typical data rates of the four different run modes in PandaX-I, all significantly less than the 70 MB/s measured upper limit of the bandwidth.', '1602.00858-2-41-1': 'One expects nearly deadtime-less operation under such rates.', '1602.00858-2-42-0': '# Summary and Conclusions', '1602.00858-2-43-0': 'We have described the electronics and data acquisition system for the PandaX-I dark matter direct detection experiment.', '1602.00858-2-43-1': 'This system used commercial electronics modules but custom-developed trigger system and DAQ software.', '1602.00858-2-43-2': 'The system ran steadily for more than nine months and collected about 8 TB of dark matter data.', '1602.00858-2-43-3': 'The performance met the challenging requirements in the search of very low energy dark matter recoil events.', '1602.00858-2-43-4': 'Although designed for PandaX-I, the same system is used in PandaX-II, the second phase experiment with a 500-kg target currently under operation [CITATION].', '1602.00858-2-43-5': 'The increased detector size in fact suppresses the trigger rate due to the self-shielding effects of liquid xenon, and the longer drift time does not lead to larger event size due to zero suppressions in the waveform data.', '1602.00858-2-43-6': 'This system is also expandable to the future phase of the experiment and maintains flexibility in acquisition strategy to further optimize the performance.'}

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'1602.00858-2-22-2'], ['1602.00858-1-22-3', '1602.00858-2-22-3'], ['1602.00858-1-22-4', '1602.00858-2-22-4'], ['1602.00858-1-29-0', '1602.00858-2-29-0'], ['1602.00858-1-29-1', '1602.00858-2-29-1'], ['1602.00858-1-29-3', '1602.00858-2-29-3'], ['1602.00858-1-32-2', '1602.00858-2-32-2'], ['1602.00858-1-32-6', '1602.00858-2-32-6'], ['1602.00858-1-17-1', '1602.00858-2-17-1'], ['1602.00858-1-17-4', '1602.00858-2-17-4'], ['1602.00858-1-17-5', '1602.00858-2-17-5'], ['1602.00858-1-37-1', '1602.00858-2-37-1'], ['1602.00858-1-37-4', '1602.00858-2-37-4'], ['1602.00858-1-43-2', '1602.00858-2-43-2'], ['1602.00858-1-33-0', '1602.00858-2-33-0'], ['1602.00858-1-2-1', '1602.00858-2-2-1'], ['1602.00858-1-2-4', '1602.00858-2-2-4'], ['1602.00858-1-25-1', '1602.00858-2-25-1'], ['1602.00858-1-1-0', '1602.00858-2-1-0'], ['1602.00858-1-1-4', '1602.00858-2-1-4'], ['1602.00858-1-1-6', '1602.00858-2-1-9'], ['1602.00858-1-28-0', '1602.00858-2-28-0'], ['1602.00858-1-28-1', '1602.00858-2-28-1'], ['1602.00858-1-28-2', '1602.00858-2-28-2'], ['1602.00858-1-12-4', '1602.00858-2-12-3'], ['1602.00858-1-6-1', '1602.00858-2-6-1'], ['1602.00858-1-6-2', '1602.00858-2-6-2'], ['1602.00858-1-10-2', '1602.00858-2-10-2'], ['1602.00858-1-21-6', '1602.00858-2-21-7'], ['1602.00858-1-13-2', '1602.00858-2-13-2'], ['1602.00858-1-27-2', '1602.00858-2-27-1'], ['1602.00858-1-29-2', '1602.00858-2-29-2'], ['1602.00858-1-17-0', '1602.00858-2-17-0'], ['1602.00858-1-43-4', '1602.00858-2-43-4'], ['1602.00858-1-43-4', '1602.00858-2-43-6'], ['1602.00858-1-5-1', '1602.00858-2-5-1'], ['1602.00858-1-5-2', '1602.00858-2-5-2'], ['1602.00858-1-5-4', '1602.00858-2-5-3']]

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1602.00858

1802.07826

{'1802.07826-1-0-0': '# Introduction', '1802.07826-1-1-0': 'The standard theory of particle physics are established finally by a discovery of the Higgs boson[CITATION] in 2012.', '1802.07826-1-1-1': 'A current target of particle physics is searching for a more fundamental theory beyond the standard model (BSM).', '1802.07826-1-1-2': 'A keystone along this direction must be the Higgs boson and top quark.', '1802.07826-1-1-3': 'Since the top quark is the heaviest fermion with a mass at the electroweak symmetry-breaking scale, it is naturally expected to have a special role in the BSM.', '1802.07826-1-2-0': 'The international linear collider[CITATION] (ILC), which is an electron-positron colliding experiment with centre of mass (CM) energies above 250 GeV, is proposed and intensively discussed as a future project of high-energy physics.', '1802.07826-1-2-1': 'One of main goals of ILC experiments is a precise measurement of top quark properties.', '1802.07826-1-2-2': 'Detailed Monte Carlo studies have shown that the ILC would be able to measure most of the standard model parameters to within sub-percent levels[CITATION].', '1802.07826-1-2-3': 'Because of the improvement of the experimental accuracy of the ILC, theoretical predictions are required be given with new level of precision.', '1802.07826-1-2-4': 'In particular, a radiative correction due to the electroweak interaction (including spin polarizations) is mandatory for such requirements.', '1802.07826-1-2-5': 'Before the discovery of the top quark, a full electroweak radiative correction was conducted for an [MATH] process at a lower energy[CITATION], and was then obtained independently for higher energies[CITATION].', '1802.07826-1-2-6': 'The same correction fo a [MATH] process has also been reported[CITATION].', '1802.07826-1-2-7': 'Recently, full electroweak radiative corrections for the process [MATH] using a narrow-width approximation for the top quarks including the spin-polarization effects are reported[CITATION] by authors including those of a present report.', '1802.07826-1-2-8': 'A possible search of the minimum SUSY particles trough loop corrections of the top-quark pair-production at the ILC is reported in [CITATION].', '1802.07826-1-3-0': 'Among several sources of the radiative correction, it is known that the initial state photonic correction gives the largest contribution in general, and thus it is important for a precise estimation of production cross sections.', '1802.07826-1-3-1': 'In this report, the precise estimation of the effect due to the initial-state photon-radiations is discussed in detail', '1802.07826-1-4-0': '# Calculation Method', '1802.07826-1-5-0': '## GRACE system', '1802.07826-1-6-0': 'For precise cross-section calculations of the target process, a GRACE-Loop system is used in this study.', '1802.07826-1-6-1': 'The GRACE system is an automatic system to calculate cross sections of scattering processes at one-loop level for the standard theory[CITATION] and the minimum SUSY model[CITATION].', '1802.07826-1-6-2': 'The GRACE system has treated electroweak processes with two, three or four particles in the final state are calculated[CITATION] at the one-loop order.', '1802.07826-1-6-3': 'The renormalization of the electroweak interaction is carried out using on-shell scheme[CITATION].', '1802.07826-1-6-4': 'Infrared divergences are regulated using fictitious photon-mass[CITATION].', '1802.07826-1-6-5': 'The symbolic manipulation package FORM[CITATION] is used to handle all Dirac and tensor algebra in [MATH]-dimensions.', '1802.07826-1-6-6': 'For loop integrations, all tensor one-loop integrals are reduced to scalar integrals using our own formalism[CITATION], then performed integrations using packages FF[CITATION] or LoopTools[CITATION].', '1802.07826-1-6-7': 'Phase-space integrations are done using an adaptive Monte Carlo integration package BASES[CITATION].', '1802.07826-1-6-8': 'For numerical calculations, we use a quartic precision for floating variables.', '1802.07826-1-7-0': 'While using [MATH]-gauge in linear gauge-fixing terms in the GRACE system" the non-linear gauge fixing Lagrangian[CITATION] is also employing for the sake of the system checking.', '1802.07826-1-7-1': 'Before calculating cross sections, we performed numerical tests to confirm that the amplitudes are independent of all redundant parameters around [MATH] digits at several randomly chosen phase points.', '1802.07826-1-7-2': 'In addition to above checks, soft-photon cut-off independence was examined: cross sections at the one-loop level, results must be independent from a head-photon cut-off parameter [MATH].', '1802.07826-1-7-3': 'We confirmed that, while varying a parameter [MATH] from [MATH] GeV to [MATH] GeV, the results of numerical phase-space integrations are consistent each other within the statistical errors of numerical integrations, that is typically around [MATH] order.', '1802.07826-1-8-0': '## Radiator method', '1802.07826-1-9-0': 'The effect of the initial photon emission can be factorized when a total energy of emitted photons are small enough compared with a beam energy or a small angle (co-linear) emission .', '1802.07826-1-9-1': 'The calculations under a such approximation is referred as to the "soft-colinear photon approximation(SPA)".', '1802.07826-1-9-2': 'Under the SPA, the corrected cross sections with the initial state photon radiation(ISR), [MATH], can be obtained from the tree cross sections [MATH] using a structure function [MATH] as follows: [EQUATION] where [MATH] is the CM energy square and [MATH] is a energy fraction of an emitted photon.', '1802.07826-1-9-3': 'The structure function can be calculated using the perturbative method with the SPA.', '1802.07826-1-9-4': 'Concrete formulae of the structure function are calculated up to two loop order[CITATION].', '1802.07826-1-9-5': 'A further improvement of the cross section estimation is possible using the "exponentiation method".', '1802.07826-1-9-6': 'Initial state photon emissions under the SPA, a probability to emit each photon should be independent each other.', '1802.07826-1-9-7': 'Thus, the probability to emitt any number of photons can be calculated as; [EQUATION] where [MATH] is a probability to emit [MATH] photons.', '1802.07826-1-9-8': 'A factor [MATH] is necessary due to [MATH] identical particles (photons) appearing in the final state.', '1802.07826-1-9-9': 'This is nothing more than the Taylor expansion of the exponential function.', '1802.07826-1-9-10': 'Therefore, the effect of the multiple photon emissions can be estimated by putting the one photon emission probability in an argument of the exponential function.', '1802.07826-1-9-11': 'This technique is referred as to the exponentiation method.', '1802.07826-1-9-12': 'When the exponentiation method is applied to the cross section calculations at loop level, the corrected cross sections can not be expressed simply like the formula ([REF]), because the same loop corrections are included in both of the structure function and loop amplitudes.', '1802.07826-1-9-13': 'To avoid a double counting of the same corrections both in the structure function and loop amplitudes, one have to rearrange terms of corrections.', '1802.07826-1-10-0': 'The total cross section at one-loop (fixed) order without the exponentiation, which is denoted as [MATH], can be expressed as; [EQUATION] where [MATH], [MATH] and [MATH] are the cross sections from the loop diagram, and soft and hard real-emission corrections, respectively.', '1802.07826-1-10-1': 'A photon whose energy is greater (less) than the threshold energy [MATH] is defined as a hard (soft) photon, respectively.', '1802.07826-1-10-2': 'The soft-photon cross section can be expressed as [MATH], where a factorized function [MATH] is obtained from the real-radiation diagrams using the SPA.', '1802.07826-1-10-3': 'The SPA consists of three parts, the initial-state, final-state radiations and their interference terms.', '1802.07826-1-10-4': 'For the initial state radiation, it can be written as; [EQUATION] where [MATH], [MATH] and [MATH] are the fine structure constant, electron mass and fictitious photon mass, respectively.', '1802.07826-1-10-5': 'Here two large log-factors appear as [MATH] and [MATH], where [MATH] is a beam energy and [MATH].', '1802.07826-1-10-6': 'The threshold energy [MATH] is included in both [MATH] and [MATH], and then the total cross section must be independent of the value of [MATH] after summing up all contributions.', '1802.07826-1-10-7': 'At the same time, final results are also independent from the photon mass [MATH] due to the cancellation among [MATH] contributions in [MATH] and [MATH].', '1802.07826-1-11-0': 'The cross section with fixed order correction ([REF]) can be improved using the exponentiation method.', '1802.07826-1-11-1': 'To avoid a double counting of the terms appearing in both loop corrections and the structure function, terms must be re-arranged as follows: [EQUATION] where [MATH] is a correction factor from the initial state photon-loop diagrams, that can be given as; [EQUATION]', '1802.07826-1-11-2': 'This term must be subtracted from [MATH] because the same contribution is also included in the structure function [MATH].', '1802.07826-1-11-3': 'A term [MATH] includes only the final-state radiation and interference terms between the initial and final state radiations.', '1802.07826-1-11-4': 'Instead, [MATH] gives the improved cross section including the initial-state radiation using the radiator method[CITATION].', '1802.07826-1-11-5': 'The total cross section can be calculated using the radiator function as; [EQUATION]', '1802.07826-1-11-6': 'The radiator function [MATH], which is corresponding to square root of the structure function, gives a probability to emit a photon with energy fraction of [MATH] at the CM energy square [MATH].', '1802.07826-1-11-7': 'In this method, electron and positron can emit different energies, and thus finite boost of the CM system can be treated.', '1802.07826-1-11-8': 'The radiator function can be obtained as[CITATION].', '1802.07826-1-11-9': '[EQUATION] where [EQUATION]', '1802.07826-1-11-10': 'In this case, the photon mass [MATH] is cancelled between [MATH] and [MATH], and the threshold energy [MATH] is cancelled between [MATH] and [MATH].', '1802.07826-1-11-11': 'This result is obtained based on perturbative calculations for initial-state photon emission diagrams up to two-loop order [CITATION].', '1802.07826-1-11-12': 'Terms with [MATH] in ([REF]) are obtained from two-loop diagrams.', '1802.07826-1-12-0': '# Results and discussions', '1802.07826-1-13-0': '## Input parameters', '1802.07826-1-14-0': 'The input parameters used in this report are listed in Table [REF].', '1802.07826-1-14-1': 'The masses of the light quarks (i.e., other than the top quark) and [MATH] boson are chosen to be consistent with low-energy experiments[CITATION].', '1802.07826-1-14-2': 'Other particle masses are taken from recent measurements[CITATION].', '1802.07826-1-14-3': 'The weak mixing-angle is given using the on-shell condition [MATH] because of our renormalization scheme.', '1802.07826-1-14-4': 'The fine-structure constant [MATH] is taken from the low-energy limit of Thomson scattering, again because of our renormalization scheme.', '1802.07826-1-15-0': 'Because the initial state photonic corrections are independent of the beam polarization, all cross section calculations in this report are performed with [MATH] left (right) polarization of an electron (positron), respectively.', '1802.07826-1-16-0': '## Electroweak radiative corrections', '1802.07826-1-17-0': '### total cross sections', '1802.07826-1-18-0': 'At first, the fixed order correction without using an exponentiation method is investigated.', '1802.07826-1-18-1': 'Total cross sections obtained at leading (tree) and next-to-leading order (NLO) calculations are shown in Figure [REF].', '1802.07826-1-18-2': 'The NLO calculations near the top-quark production threshold (around the CM energy of [MATH] GeV) shows negative corrections about [MATH].', '1802.07826-1-18-3': 'This effect is called "coulomb correction", and it can be large because produced particles are moving slowly and have enough time-duration to interact with each other.', '1802.07826-1-18-4': 'The corrections become very small around the CM energy of [MATH] GeV and increase at the high energy region as to [MATH] at the CM energy of [MATH] GeV.', '1802.07826-1-18-5': 'Among several types of radiative corrections, e.g., the initial and final state photon radiations, the vertex and box correction, and so on, the initial-state photonic correction give the largest contribution at the high energy region.', '1802.07826-1-18-6': 'As shown in Figure [REF], the cross sections at tree level including the ISR correction (a dotted line in the figure) are almost the same as the full order [MATH] electroweak correction (a dashed line in the figure) at CM energies above [MATH] GeV.', '1802.07826-1-18-7': 'This means that the main contribution of the higher order corrections is caused by the initial state photonic corrections.', '1802.07826-1-18-8': 'On the other hand, other corrections from the loop diagrams also give a large correction near the threshold region.', '1802.07826-1-18-9': 'Around the CM energy [MATH] GeV, these effects are cancelled accidentally and give a small correction on the total cross section.', '1802.07826-1-19-0': 'After subtracting a trivial ISR correction from the total corrections, one can discuss the "pure" weak correction in the full [MATH] electroweak radiative corrections.', '1802.07826-1-19-1': 'While the NLO correction degree is defined as [EQUATION] the weak correction degree is defined as [EQUATION]', '1802.07826-1-19-2': 'In the definition of [MATH], the trivial initial-state photonic corrections are subtracted from the full [MATH] electroweak radiative corrections, and thus, [MATH] shows a fraction of the mainly weak-correction in the full [MATH] electroweak radiative corrections.', '1802.07826-1-19-3': 'The behaviors of [MATH] and [MATH] are shown in Figure [REF] with respect to the CM energies.', '1802.07826-1-19-4': 'One can see that the weak correction becomes smaller and smaller at a high energy region, and arrives at almost zero at the CM energy of [MATH] GeV.', '1802.07826-1-19-5': 'On the other hand, at the CM energy of [MATH] GeV, the pure-weak corrections gives [MATH] correction over the trivial ISR corrections.', '1802.07826-1-20-0': '### angular distribution', '1802.07826-1-21-0': 'As mentioned above, the radiative correction does not change the total cross section accidentally at the CM energy [MATH] GeV.', '1802.07826-1-21-1': 'While the total cross section stays the same after the radiative correction, an angular distribution is not the case.', '1802.07826-1-21-2': 'In general, the real-photon emission affects the angular distribution such that a steep peak being mild.', '1802.07826-1-21-3': 'In reality, the ISR correction (a dotted line with rectangle points) makes a forward peak of a top quark production at a tree level (a solid line with triangle points) smaller as shown in Figure [REF].', '1802.07826-1-21-4': 'In addition to that, the weak correction raises the backward scattering as shown by a dashed line with circle points in Figure [REF].', '1802.07826-1-21-5': 'As the result, the total cross section does not change so much.', '1802.07826-1-22-0': '## Photonic correction at two-loop order', '1802.07826-1-23-0': 'The structure function [MATH] given in ([REF]) are including two-loop effects.', '1802.07826-1-23-1': 'All of above results are obtained using full formula ([REF]).', '1802.07826-1-23-2': 'As mentioned in previous subsection, a main contribution of the radiative corrections comes from the initial state photonic-correction at the high energy region.', '1802.07826-1-23-3': 'Therefore, the ISR correction is important among many terms of the radiative corrections around these energies.', '1802.07826-1-23-4': 'If the two-loop contribution has a significant fraction in the full correction, even higher-loop corrections must be considered for the future experiments.', '1802.07826-1-23-5': 'The fraction of two-loop contribution over the one-loop one is defined as [EQUATION] where [MATH] shows the ISR corrected cross sections using the structure function ([REF]) with omitting [MATH] and [MATH] terms.', '1802.07826-1-23-6': 'Numerical results are shown in Figure [REF].', '1802.07826-1-23-7': 'The two-loop contribution is smaller than [MATH] in the energy region between [MATH] GeV to [MATH] GeV as shown in Figure [REF].', '1802.07826-1-24-0': '## Running coupling', '1802.07826-1-25-0': 'Yet another improvement of the cross section estimation is known as a running coupling method.', '1802.07826-1-25-1': 'This method is also taking a higher order effect of vacuum polarization diagrams into account as an effective coupling constant.', '1802.07826-1-25-2': 'After summing up contributions from fermion-loops on gauge-boson propagators, those form solely a gauge-invariant subset, an electro-weak coupling effectively varies according to the four-momentum square of propagators.', '1802.07826-1-25-3': 'The effective coupling [MATH] at the energy scale [MATH] can be written as; [EQUATION] where [MATH] is the four-momentum square of a propagator of the target process.', '1802.07826-1-25-4': 'At the same time, the weak mixing angle [MATH] is obtained from the Fermi weak-coupling constant [MATH] as[CITATION]; [EQUATION]', '1802.07826-1-25-5': 'The improved cross section [MATH] can be written as; [EQUATION] where [MATH] is the tree cross section at the CM energy-square [MATH] with the coupling constant [MATH].', '1802.07826-1-25-6': 'This method is referred to as the improved Born approximation.', '1802.07826-1-25-7': 'Numerical results are summarized in Figure [REF].', '1802.07826-1-25-8': 'Here the measured value of [MATH][CITATION] is used.', '1802.07826-1-25-9': 'Above the CM energy of [MATH] GeV, the improved Born method gives approximated values better than [MATH] with respect to the NLO cross sections with the ISR.', '1802.07826-1-26-0': '# Summary', '1802.07826-1-27-0': 'We calculated the precise cross sections of an [MATH] process at an energy region from 400 GeV to 800 GeV.', '1802.07826-1-27-1': 'Especially, the initial-state photon emissions are discussed in details.', '1802.07826-1-27-2': 'An exponentiation technique is applied for the initial-state photon emission up to two-loop order.', '1802.07826-1-27-3': 'We found that the total cross section of a top quark pair-production at a center of mass energy of 500 GeV receives the weak corrections of [MATH] over the trivial ISR corrections.', '1802.07826-1-27-4': 'Among the ISR contributions, two-loop diagrams gives less than [MATH] correction with respect to the one-loop ones at the CM energies from [MATH] GeV to [MATH] GeV.', '1802.07826-1-28-0': 'The improved Born approximation gives cross sections better than [MATH] compared with the NLO cross sections with the ISR.'}

{'1802.07826-2-0-0': '# Introduction', '1802.07826-2-1-0': 'The standard theory of particle physics are established finally by a discovery of the Higgs boson[CITATION] in 2012.', '1802.07826-2-1-1': 'A current target of particle physics is searching for a more fundamental theory beyond the standard model (BSM).', '1802.07826-2-1-2': 'A keystone along this direction must be the Higgs boson and top quark.', '1802.07826-2-1-3': 'Since the top quark is the heaviest fermion with a mass at the electroweak symmetry-breaking scale, it is naturally expected to have a special role in the BSM.', '1802.07826-2-2-0': 'The international linear collider[CITATION] (ILC), which is an electron-positron colliding experiment with centre of mass (CM) energies above 250 GeV, is proposed and intensively discussed as a future project of high-energy physics.', '1802.07826-2-2-1': 'One of main goals of ILC experiments is a precise measurement of top quark properties.', '1802.07826-2-2-2': 'Detailed Monte Carlo studies have shown that the ILC would be able to measure most of the standard model parameters to within sub-percent levels[CITATION].', '1802.07826-2-2-3': 'Because of the improvement of the experimental accuracy of the ILC, theoretical predictions are required be given with new level of precision.', '1802.07826-2-2-4': 'In particular, a radiative correction due to the electroweak interaction (including spin polarizations) is mandatory for such requirements.', '1802.07826-2-2-5': 'Before the discovery of the top quark, a full electroweak radiative correction was conducted for an [MATH] process at a lower energy[CITATION], and was then obtained independently for higher energies[CITATION].', '1802.07826-2-2-6': 'The same correction fo a [MATH] process has also been reported[CITATION].', '1802.07826-2-2-7': 'Recently, full electroweak radiative corrections for the process [MATH] using a narrow-width approximation for the top quarks including the spin-polarization effects are reported[CITATION] by authors including those of a present report.', '1802.07826-2-2-8': 'A possible search of the minimum SUSY particles trough loop corrections of the top-quark pair-production at the ILC is reported in [CITATION].', '1802.07826-2-3-0': 'Among several sources of the radiative correction, it is known that the initial state photonic correction gives the largest contribution in general, and thus it is important for a precise estimation of production cross sections.', '1802.07826-2-3-1': 'In this report, the precise estimation of the effect due to the initial-state photon-radiations is discussed in detail', '1802.07826-2-4-0': '# Calculation Method', '1802.07826-2-5-0': '## GRACE system', '1802.07826-2-6-0': 'For precise cross-section calculations of the target process, a GRACE-Loop system is used in this study.', '1802.07826-2-6-1': 'The GRACE system is an automatic system to calculate cross sections of scattering processes at one-loop level for the standard theory[CITATION] and the minimum SUSY model[CITATION].', '1802.07826-2-6-2': 'The GRACE system has treated electroweak processes with two, three or four particles in the final state are calculated[CITATION] at the one-loop order.', '1802.07826-2-6-3': 'The renormalization of the electroweak interaction is carried out using on-shell scheme[CITATION].', '1802.07826-2-6-4': 'Infrared divergences are regulated using fictitious photon-mass[CITATION].', '1802.07826-2-6-5': 'The symbolic manipulation package FORM[CITATION] is used to handle all Dirac and tensor algebra in [MATH]-dimensions.', '1802.07826-2-6-6': 'For loop integrations, all tensor one-loop integrals are reduced to scalar integrals using our own formalism[CITATION], then performed integrations using packages FF[CITATION] or LoopTools[CITATION].', '1802.07826-2-6-7': 'Phase-space integrations are done using an adaptive Monte Carlo integration package BASES[CITATION].', '1802.07826-2-6-8': 'For numerical calculations, we use a quartic precision for floating variables.', '1802.07826-2-7-0': 'While using [MATH]-gauge in linear gauge-fixing terms in the GRACE system" the non-linear gauge fixing Lagrangian[CITATION] is also employing for the sake of the system checking.', '1802.07826-2-7-1': 'Before calculating cross sections, we performed numerical tests to confirm that the amplitudes are independent of all redundant parameters around [MATH] digits at several randomly chosen phase points.', '1802.07826-2-7-2': 'In addition to above checks, soft-photon cut-off independence was examined: cross sections at the one-loop level, results must be independent from a head-photon cut-off parameter [MATH].', '1802.07826-2-7-3': 'We confirmed that, while varying a parameter [MATH] from [MATH] GeV to [MATH] GeV, the results of numerical phase-space integrations are consistent each other within the statistical errors of numerical integrations, that is typically around [MATH] order.', '1802.07826-2-8-0': '## Radiator method', '1802.07826-2-9-0': 'The effect of the initial photon emission can be factorized when a total energy of emitted photons are small enough compared with a beam energy or a small angle (co-linear) emission .', '1802.07826-2-9-1': 'The calculations under a such approximation is referred as to the "soft-colinear photon approximation(SPA)".', '1802.07826-2-9-2': 'Under the SPA, the corrected cross sections with the initial state photon radiation(ISR), [MATH], can be obtained from the tree cross sections [MATH] using a structure function [MATH] as follows: [EQUATION] where [MATH] is the CM energy square and [MATH] is a energy fraction of an emitted photon.', '1802.07826-2-9-3': 'The structure function can be calculated using the perturbative method with the SPA.', '1802.07826-2-9-4': 'Concrete formulae of the structure function are calculated up to two loop order[CITATION].', '1802.07826-2-9-5': 'A further improvement of the cross section estimation is possible using the "exponentiation method".', '1802.07826-2-9-6': 'Initial state photon emissions under the SPA, a probability to emit each photon should be independent each other.', '1802.07826-2-9-7': 'Thus, the probability to emitt any number of photons can be calculated as; [EQUATION] where [MATH] is a probability to emit [MATH] photons.', '1802.07826-2-9-8': 'A factor [MATH] is necessary due to [MATH] identical particles (photons) appearing in the final state.', '1802.07826-2-9-9': 'This is nothing more than the Taylor expansion of the exponential function.', '1802.07826-2-9-10': 'Therefore, the effect of the multiple photon emissions can be estimated by putting the one photon emission probability in an argument of the exponential function.', '1802.07826-2-9-11': 'This technique is referred as to the exponentiation method.', '1802.07826-2-9-12': 'When the exponentiation method is applied to the cross section calculations at loop level, the corrected cross sections can not be expressed simply like the formula ([REF]), because the same loop corrections are included in both of the structure function and loop amplitudes.', '1802.07826-2-9-13': 'To avoid a double counting of the same corrections both in the structure function and loop amplitudes, one have to rearrange terms of corrections.', '1802.07826-2-10-0': 'The total cross section at one-loop (fixed) order without the exponentiation, which is denoted as [MATH], can be expressed as; [EQUATION] where [MATH], [MATH] and [MATH] are the cross sections from the loop diagram, and soft and hard real-emission corrections, respectively.', '1802.07826-2-10-1': 'A photon whose energy is greater (less) than the threshold energy [MATH] is defined as a hard (soft) photon, respectively.', '1802.07826-2-10-2': 'The soft-photon cross section can be expressed as [MATH], where a factorized function [MATH] is obtained from the real-radiation diagrams using the SPA.', '1802.07826-2-10-3': 'The SPA consists of three parts, the initial-state, final-state radiations and their interference terms.', '1802.07826-2-10-4': 'For the initial state radiation, it can be written as; [EQUATION] where [MATH], [MATH] and [MATH] are the fine structure constant, electron mass and fictitious photon mass, respectively.', '1802.07826-2-10-5': 'Here two large log-factors appear as [MATH] and [MATH], where [MATH] is a beam energy and [MATH].', '1802.07826-2-10-6': 'The threshold energy [MATH] is included in both [MATH] and [MATH], and then the total cross section must be independent of the value of [MATH] after summing up all contributions.', '1802.07826-2-10-7': 'At the same time, final results are also independent from the photon mass [MATH] due to the cancellation among [MATH] contributions in [MATH] and [MATH].', '1802.07826-2-11-0': 'The cross section with fixed order correction ([REF]) can be improved using the exponentiation method.', '1802.07826-2-11-1': 'To avoid a double counting of the terms appearing in both loop corrections and the structure function, terms must be re-arranged as follows: [EQUATION] where [MATH] is a correction factor from the initial state photon-loop diagrams, that can be given as; [EQUATION]', '1802.07826-2-11-2': 'This term must be subtracted from [MATH] because the same contribution is also included in the structure function [MATH].', '1802.07826-2-11-3': 'A term [MATH] includes only the final-state radiation and interference terms between the initial and final state radiations.', '1802.07826-2-11-4': 'Instead, [MATH] gives the improved cross section including the initial-state radiation using the radiator method[CITATION].', '1802.07826-2-11-5': 'The total cross section can be calculated using the radiator function as; [EQUATION]', '1802.07826-2-11-6': 'The radiator function [MATH], which is corresponding to square root of the structure function, gives a probability to emit a photon with energy fraction of [MATH] at the CM energy square [MATH].', '1802.07826-2-11-7': 'In this method, electron and positron can emit different energies, and thus finite boost of the CM system can be treated.', '1802.07826-2-11-8': 'The radiator function can be obtained as[CITATION].', '1802.07826-2-11-9': '[EQUATION] where [EQUATION]', '1802.07826-2-11-10': 'In this case, the photon mass [MATH] is cancelled between [MATH] and [MATH], and the threshold energy [MATH] is cancelled between [MATH] and [MATH].', '1802.07826-2-11-11': 'This result is obtained based on perturbative calculations for initial-state photon emission diagrams up to two-loop order [CITATION].', '1802.07826-2-11-12': 'Terms with [MATH] in ([REF]) are obtained from two-loop diagrams.', '1802.07826-2-12-0': '# Results and discussions', '1802.07826-2-13-0': '## Input parameters', '1802.07826-2-14-0': 'The input parameters used in this report are listed in Table [REF].', '1802.07826-2-14-1': 'The masses of the light quarks (i.e., other than the top quark) and [MATH] boson are chosen to be consistent with low-energy experiments[CITATION].', '1802.07826-2-14-2': 'Other particle masses are taken from recent measurements[CITATION].', '1802.07826-2-14-3': 'The weak mixing-angle is given using the on-shell condition [MATH] because of our renormalization scheme.', '1802.07826-2-14-4': 'The fine-structure constant [MATH] is taken from the low-energy limit of Thomson scattering, again because of our renormalization scheme.', '1802.07826-2-15-0': 'Because the initial state photonic corrections are independent of the beam polarization, all cross section calculations in this report are performed with [MATH] left (right) polarization of an electron (positron), respectively.', '1802.07826-2-16-0': '## Electroweak radiative corrections', '1802.07826-2-17-0': '### total cross sections', '1802.07826-2-18-0': 'At first, the fixed order correction without using an exponentiation method is investigated.', '1802.07826-2-18-1': 'Total cross sections obtained at leading (tree) and next-to-leading order (NLO) calculations are shown in Figure [REF].', '1802.07826-2-18-2': 'The NLO calculations near the top-quark production threshold (around the CM energy of [MATH] GeV) shows negative corrections about [MATH].', '1802.07826-2-18-3': 'This effect is called "coulomb correction", and it can be large because produced particles are moving slowly and have enough time-duration to interact with each other.', '1802.07826-2-18-4': 'The corrections become very small around the CM energy of [MATH] GeV and increase at the high energy region as to [MATH] at the CM energy of [MATH] GeV.', '1802.07826-2-18-5': 'Among several types of radiative corrections, e.g., the initial and final state photon radiations, the vertex and box correction, and so on, the initial-state photonic correction give the largest contribution at the high energy region.', '1802.07826-2-18-6': 'As shown in Figure [REF], the cross sections at tree level including the ISR correction (a dotted line in the figure) are almost the same as the full order [MATH] electroweak correction (a dashed line in the figure) at CM energies above [MATH] GeV.', '1802.07826-2-18-7': 'This means that the main contribution of the higher order corrections is caused by the initial state photonic corrections.', '1802.07826-2-18-8': 'On the other hand, other corrections from the loop diagrams also give a large correction near the threshold region.', '1802.07826-2-18-9': 'Around the CM energy [MATH] GeV, these effects are cancelled accidentally and give a small correction on the total cross section.', '1802.07826-2-19-0': 'After subtracting a trivial ISR correction from the total corrections, one can discuss the "pure" weak correction in the full [MATH] electroweak radiative corrections.', '1802.07826-2-19-1': 'While the NLO correction degree is defined as [EQUATION] the weak correction degree is defined as [EQUATION]', '1802.07826-2-19-2': 'In the definition of [MATH], the trivial initial-state photonic corrections are subtracted from the full [MATH] electroweak radiative corrections, and thus, [MATH] shows a fraction of the mainly weak-correction in the full [MATH] electroweak radiative corrections.', '1802.07826-2-19-3': 'The behaviors of [MATH] and [MATH] are shown in Figure [REF] with respect to the CM energies.', '1802.07826-2-19-4': 'One can see that the weak correction becomes smaller and smaller at a high energy region, and arrives at almost zero at the CM energy of [MATH] GeV.', '1802.07826-2-19-5': 'On the other hand, at the CM energy of [MATH] GeV, the pure-weak corrections gives [MATH] correction over the trivial ISR corrections.', '1802.07826-2-20-0': '### angular distribution', '1802.07826-2-21-0': 'As mentioned above, the radiative correction does not change the total cross section accidentally at the CM energy [MATH] GeV.', '1802.07826-2-21-1': 'While the total cross section stays the same after the radiative correction, an angular distribution is not the case.', '1802.07826-2-21-2': 'In general, the real-photon emission affects the angular distribution such that a steep peak being mild.', '1802.07826-2-21-3': 'In reality, the ISR correction (a dotted line with rectangle points) makes a forward peak of a top quark production at a tree level (a solid line with triangle points) smaller as shown in Figure [REF].', '1802.07826-2-21-4': 'In addition to that, the weak correction raises the backward scattering as shown by a dashed line with circle points in Figure [REF].', '1802.07826-2-21-5': 'As the result, the total cross section does not change so much.', '1802.07826-2-22-0': '## Photonic correction at two-loop order', '1802.07826-2-23-0': 'The structure function [MATH] given in ([REF]) are including two-loop effects.', '1802.07826-2-23-1': 'All of above results are obtained using full formula ([REF]).', '1802.07826-2-23-2': 'As mentioned in previous subsection, a main contribution of the radiative corrections comes from the initial state photonic-correction at the high energy region.', '1802.07826-2-23-3': 'Therefore, the ISR correction is important among many terms of the radiative corrections around these energies.', '1802.07826-2-23-4': 'If the two-loop contribution has a significant fraction in the full correction, even higher-loop corrections must be considered for the future experiments.', '1802.07826-2-23-5': 'The fraction of two-loop contribution over the one-loop one is defined as [EQUATION] where [MATH] shows the ISR corrected cross sections using the structure function ([REF]) with omitting [MATH] and [MATH] terms.', '1802.07826-2-23-6': 'Numerical results are shown in Figure [REF].', '1802.07826-2-23-7': 'The two-loop contribution is smaller than [MATH] in the energy region between [MATH] GeV to [MATH] GeV as shown in Figure [REF].', '1802.07826-2-24-0': '## Running coupling', '1802.07826-2-25-0': 'Yet another improvement of the cross section estimation is known as a running coupling method.', '1802.07826-2-25-1': 'This method is also taking a higher order effect of vacuum polarization diagrams into account as an effective coupling constant.', '1802.07826-2-25-2': 'After summing up contributions from fermion-loops on gauge-boson propagators, those form solely a gauge-invariant subset, an electro-weak coupling effectively varies according to the four-momentum square of propagators.', '1802.07826-2-25-3': 'The effective coupling [MATH] at the energy scale [MATH] can be written as; [EQUATION] where [MATH] is the four-momentum square of a propagator of the target process.', '1802.07826-2-25-4': 'At the same time, the weak mixing angle [MATH] is obtained from the Fermi weak-coupling constant [MATH] as[CITATION]; [EQUATION]', '1802.07826-2-25-5': 'The improved cross section [MATH] can be written as; [EQUATION] where [MATH] is the tree cross section at the CM energy-square [MATH] with the coupling constant [MATH].', '1802.07826-2-25-6': 'This method is referred to as the improved Born approximation.', '1802.07826-2-25-7': 'Numerical results are summarized in Figure [REF].', '1802.07826-2-25-8': 'Here the measured value of [MATH][CITATION] is used.', '1802.07826-2-25-9': 'Above the CM energy of [MATH] GeV, the improved Born method gives approximated values better than [MATH] with respect to the NLO cross sections with the ISR.', '1802.07826-2-26-0': '# Summary', '1802.07826-2-27-0': 'We calculated the precise cross sections of an [MATH] process at an energy region from 400 GeV to 800 GeV.', '1802.07826-2-27-1': 'Especially, the initial-state photon emissions are discussed in details.', '1802.07826-2-27-2': 'An exponentiation technique is applied for the initial-state photon emission up to two-loop order.', '1802.07826-2-27-3': 'We found that the total cross section of a top quark pair-production at a center of mass energy of 500 GeV receives the weak corrections of [MATH] over the trivial ISR corrections.', '1802.07826-2-27-4': 'Among the ISR contributions, two-loop diagrams gives less than [MATH] correction with respect to the one-loop ones at the CM energies from [MATH] GeV to [MATH] GeV.', '1802.07826-2-28-0': 'The improved Born approximation gives cross sections better than [MATH] compared with the NLO cross sections with the ISR.'}

[['1802.07826-1-3-0', '1802.07826-2-3-0'], ['1802.07826-1-3-1', '1802.07826-2-3-1'], ['1802.07826-1-15-0', '1802.07826-2-15-0'], ['1802.07826-1-2-0', '1802.07826-2-2-0'], ['1802.07826-1-2-1', '1802.07826-2-2-1'], ['1802.07826-1-2-2', '1802.07826-2-2-2'], ['1802.07826-1-2-3', '1802.07826-2-2-3'], ['1802.07826-1-2-4', '1802.07826-2-2-4'], ['1802.07826-1-2-5', '1802.07826-2-2-5'], ['1802.07826-1-2-6', '1802.07826-2-2-6'], ['1802.07826-1-2-7', '1802.07826-2-2-7'], ['1802.07826-1-2-8', '1802.07826-2-2-8'], ['1802.07826-1-14-0', '1802.07826-2-14-0'], ['1802.07826-1-14-1', '1802.07826-2-14-1'], ['1802.07826-1-14-2', '1802.07826-2-14-2'], ['1802.07826-1-14-3', '1802.07826-2-14-3'], ['1802.07826-1-14-4', '1802.07826-2-14-4'], ['1802.07826-1-27-0', '1802.07826-2-27-0'], ['1802.07826-1-27-1', '1802.07826-2-27-1'], ['1802.07826-1-27-2', '1802.07826-2-27-2'], ['1802.07826-1-27-3', '1802.07826-2-27-3'], ['1802.07826-1-27-4', '1802.07826-2-27-4'], ['1802.07826-1-28-0', '1802.07826-2-28-0'], ['1802.07826-1-10-0', '1802.07826-2-10-0'], ['1802.07826-1-10-1', '1802.07826-2-10-1'], ['1802.07826-1-10-2', '1802.07826-2-10-2'], ['1802.07826-1-10-3', '1802.07826-2-10-3'], ['1802.07826-1-10-4', '1802.07826-2-10-4'], ['1802.07826-1-10-5', '1802.07826-2-10-5'], ['1802.07826-1-10-6', '1802.07826-2-10-6'], ['1802.07826-1-10-7', '1802.07826-2-10-7'], ['1802.07826-1-18-0', '1802.07826-2-18-0'], ['1802.07826-1-18-1', '1802.07826-2-18-1'], ['1802.07826-1-18-2', '1802.07826-2-18-2'], ['1802.07826-1-18-3', '1802.07826-2-18-3'], ['1802.07826-1-18-4', '1802.07826-2-18-4'], ['1802.07826-1-18-5', '1802.07826-2-18-5'], ['1802.07826-1-18-6', '1802.07826-2-18-6'], ['1802.07826-1-18-7', '1802.07826-2-18-7'], ['1802.07826-1-18-8', '1802.07826-2-18-8'], ['1802.07826-1-18-9', '1802.07826-2-18-9'], ['1802.07826-1-19-0', '1802.07826-2-19-0'], ['1802.07826-1-19-1', '1802.07826-2-19-1'], ['1802.07826-1-19-2', '1802.07826-2-19-2'], ['1802.07826-1-19-3', '1802.07826-2-19-3'], ['1802.07826-1-19-4', '1802.07826-2-19-4'], ['1802.07826-1-19-5', '1802.07826-2-19-5'], ['1802.07826-1-1-0', '1802.07826-2-1-0'], ['1802.07826-1-1-1', '1802.07826-2-1-1'], ['1802.07826-1-1-2', '1802.07826-2-1-2'], ['1802.07826-1-1-3', '1802.07826-2-1-3'], ['1802.07826-1-6-0', '1802.07826-2-6-0'], ['1802.07826-1-6-1', '1802.07826-2-6-1'], ['1802.07826-1-6-2', '1802.07826-2-6-2'], ['1802.07826-1-6-3', '1802.07826-2-6-3'], ['1802.07826-1-6-4', '1802.07826-2-6-4'], ['1802.07826-1-6-5', '1802.07826-2-6-5'], ['1802.07826-1-6-6', '1802.07826-2-6-6'], ['1802.07826-1-6-7', '1802.07826-2-6-7'], ['1802.07826-1-6-8', '1802.07826-2-6-8'], ['1802.07826-1-23-0', '1802.07826-2-23-0'], ['1802.07826-1-23-1', '1802.07826-2-23-1'], ['1802.07826-1-23-2', '1802.07826-2-23-2'], ['1802.07826-1-23-3', '1802.07826-2-23-3'], ['1802.07826-1-23-4', '1802.07826-2-23-4'], ['1802.07826-1-23-5', '1802.07826-2-23-5'], ['1802.07826-1-23-6', '1802.07826-2-23-6'], ['1802.07826-1-23-7', '1802.07826-2-23-7'], ['1802.07826-1-25-0', '1802.07826-2-25-0'], ['1802.07826-1-25-1', '1802.07826-2-25-1'], ['1802.07826-1-25-2', '1802.07826-2-25-2'], ['1802.07826-1-25-3', '1802.07826-2-25-3'], ['1802.07826-1-25-4', '1802.07826-2-25-4'], ['1802.07826-1-25-5', '1802.07826-2-25-5'], ['1802.07826-1-25-6', '1802.07826-2-25-6'], ['1802.07826-1-25-7', '1802.07826-2-25-7'], ['1802.07826-1-25-8', '1802.07826-2-25-8'], ['1802.07826-1-25-9', '1802.07826-2-25-9'], ['1802.07826-1-9-0', '1802.07826-2-9-0'], ['1802.07826-1-9-1', '1802.07826-2-9-1'], ['1802.07826-1-9-2', '1802.07826-2-9-2'], ['1802.07826-1-9-3', '1802.07826-2-9-3'], ['1802.07826-1-9-4', '1802.07826-2-9-4'], ['1802.07826-1-9-5', '1802.07826-2-9-5'], ['1802.07826-1-9-6', '1802.07826-2-9-6'], ['1802.07826-1-9-7', '1802.07826-2-9-7'], ['1802.07826-1-9-8', '1802.07826-2-9-8'], ['1802.07826-1-9-9', '1802.07826-2-9-9'], ['1802.07826-1-9-10', '1802.07826-2-9-10'], ['1802.07826-1-9-11', '1802.07826-2-9-11'], ['1802.07826-1-9-12', '1802.07826-2-9-12'], ['1802.07826-1-9-13', '1802.07826-2-9-13'], ['1802.07826-1-11-0', '1802.07826-2-11-0'], ['1802.07826-1-11-1', '1802.07826-2-11-1'], ['1802.07826-1-11-2', '1802.07826-2-11-2'], ['1802.07826-1-11-3', '1802.07826-2-11-3'], ['1802.07826-1-11-4', '1802.07826-2-11-4'], ['1802.07826-1-11-5', '1802.07826-2-11-5'], ['1802.07826-1-11-6', '1802.07826-2-11-6'], ['1802.07826-1-11-7', '1802.07826-2-11-7'], ['1802.07826-1-11-8', '1802.07826-2-11-8'], ['1802.07826-1-11-10', '1802.07826-2-11-10'], ['1802.07826-1-11-11', '1802.07826-2-11-11'], ['1802.07826-1-11-12', '1802.07826-2-11-12'], ['1802.07826-1-21-0', '1802.07826-2-21-0'], ['1802.07826-1-21-1', '1802.07826-2-21-1'], ['1802.07826-1-21-2', '1802.07826-2-21-2'], ['1802.07826-1-21-3', '1802.07826-2-21-3'], ['1802.07826-1-21-4', '1802.07826-2-21-4'], ['1802.07826-1-21-5', '1802.07826-2-21-5'], ['1802.07826-1-7-0', '1802.07826-2-7-0'], ['1802.07826-1-7-1', '1802.07826-2-7-1'], ['1802.07826-1-7-2', '1802.07826-2-7-2'], ['1802.07826-1-7-3', '1802.07826-2-7-3']]

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[]

[]

[]

[]

['1802.07826-1-11-9', '1802.07826-2-11-9']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1802.07826

1206.4924

{'1206.4924-1-0-0': 'The phase diagram of a system constituted of neutrons and [MATH]-hyperons in thermal equilibrium is evaluated in the mean-field approximation.', '1206.4924-1-0-1': 'It is shown that this simple system exhibits a complex phase diagram with first and second order phase transitions.', '1206.4924-1-0-2': 'Due to the generic presence of attractive and repulsive couplings, the existence of phase transitions involving strangeness appears independent of the specific interaction model.', '1206.4924-1-0-3': 'In addition we will show under which conditions a phase transition towards strange matter at high density exists, which is expected to persist even within a complete treatment including all the different strange and non- strange baryon states.', '1206.4924-1-0-4': 'The impact of this transition on the composition of matter in the inner core of neutron stars is discussed.', '1206.4924-1-1-0': '# Introduction', '1206.4924-1-2-0': 'With the purpose of better understanding the dynamics of core-collapse supernova and the observed characteristics of neutron stars, a considerable theoretical effort has been undertaken in recent years concerning the modelization of the equation of state of cold dense matter[CITATION] with some extensions to finite temperature, see e.g. [CITATION].', '1206.4924-1-3-0': 'If it is well admitted that hyperonic and deconfined quark matter could exist in the inner core of neutron stars, a complete understanding of the composition and equation of state of dense matter is far from being achieved.', '1206.4924-1-3-1': 'Concerning hyperons, simple energetic considerations suggest that they should be present at high density [CITATION].', '1206.4924-1-3-2': 'However, in the standard picture the opening of hyperon degrees of freedom leads to a considerable softening of the equation of state[CITATION], which in turns leads to maximum neutron star masses smaller than the highest values obtained in recent observations[CITATION].', '1206.4924-1-3-3': 'This puzzling situation implies that the hyperon-hyperon and hyperon-nucleon couplings must be much more repulsive at high density than presently assumed[CITATION], and/or that something is missing in the present modelization.', '1206.4924-1-4-0': 'Apart from neutron star observations, already for purely nuclear matter, stringent constraints on the equation of state from experimental data as well as from theoretical side from ab-initio calculations only exist up to roughly saturation density, mainly for almost symmetric matter at zero temperature.', '1206.4924-1-4-1': 'What makes the description of hyperonic matter even more difficult is first of all the fact that, contrary to nucleons, hyperonic data from hypernuclei (see e.g. [CITATION]), diffusion and production experiments (see e.g. [CITATION]) and nuclear collision experiments (see e.g. [CITATION]) are scarce.', '1206.4924-1-4-2': 'This lack of experimental information induces large uncertainties within the microscopic approaches [CITATION], which suffer in addition probably from theoretical shortcomings, among others due to the unknown hyperonic three-body forces [CITATION].', '1206.4924-1-4-3': 'Phenomenological extrapolations of the low-density behavior within mean field models are subject to large uncertainties, too.', '1206.4924-1-4-4': 'In any case, there is much uncertainty on the hyperonic interactions at high density, where neutron star observations can give the only hint, though not decisive.', '1206.4924-1-5-0': 'Here, we propose to study the phase diagram of hyperonic matter.', '1206.4924-1-5-1': 'The generic presence of attractive and repulsive couplings suggests the existence, in a model-independent manner, of a phase transition involving strangeness.', '1206.4924-1-5-2': 'In order to have an analytically solvable model, we consider the simplified situation where only neutrons, [MATH], and [MATH]-hyperons are allowed in the matter chemical composition.', '1206.4924-1-5-3': 'We find, as argued above, a first order phase transition involving strangeness.', '1206.4924-1-5-4': 'In addition, we will show under which assumptions on the [MATH] and [MATH] interaction, respecting the available constraints, a phase transition towards strange matter, that is with a discontinuity in the strangeness content of matter, exists at high density.', '1206.4924-1-6-0': 'We leave the inclusion of protons and higher mass strange and non-strange baryons to a future study.', '1206.4924-1-6-1': 'Because of this simplification, we will not be able to exploit the results for a predictive quantitative application to neutron stars and dense supernova matter.', '1206.4924-1-6-2': 'However, this simplification will allow us to have an exactly solvable model with perfectly controlled numerics.', '1206.4924-1-6-3': 'We believe that the relevant degrees of freedom to explore the opening of the strangeness channels are included already at this level, and the qualitative features of the strangeness phase transition will not change in the complete model.', '1206.4924-1-7-0': 'Concerning the phenomenological consequences of our findings let us stress that many published work on hyperonic matter makes use of the mean field approximation (e.g. [CITATION]).', '1206.4924-1-7-1': 'However, if the opening of the strangeness degree of freedom at high baryonic density is associated with a first order phase transition, the mean field equations of state should be modified making use of the Gibbs construction [CITATION].', '1206.4924-1-7-2': 'The possible presence of coexisting phases would in addition modify the matter composition with respect to the uncorrected mean-field predictions [CITATION].', '1206.4924-1-8-0': 'Both, matter composition and equation of state, are important ingredients not only in the calculations of neutrons star models, but in the hydrodynamical codes describing core collapse dynamics, too.', '1206.4924-1-8-1': 'The high density and high temperature behavior not only influences the success of the explosion, but among others also the dependence of the final state (neutron star or black hole) on the progenitor mass (see e.g. [CITATION]).', '1206.4924-1-8-2': 'This means that the possible presence of a phase transition towards strange matter has to be studied at finite temperature, too.', '1206.4924-1-9-0': '# The model', '1206.4924-1-10-0': 'In the following we will illustrate the propositions concerning the phase diagram by choosing a specific interaction model for the [MATH] system.', '1206.4924-1-10-1': 'Of course, the quantitative numerical results are not model-independent, but we will argue in which way our findings on the phase diagram are general.', '1206.4924-1-11-0': 'The energetics of the [MATH] mixture is described through the energy density functional proposed by Balberg and Gal[CITATION] [EQUATION] where the interaction couplings, compatible with the Lattimer-Swesty equation of state in the non-strange sector, are given in table I. Parametrizations (I)-(III) are taken from the work by Balberg and Gal [CITATION] and 220g2.8 corresponds to one of the parametrizations compatible with the observation of an almost two solar mass neutron star [CITATION] from Ref. [CITATION].', '1206.4924-1-11-1': 'In the applications shown below we will use mainly parametrization (I), but we have performed the calculations for the other parameter sets, too.', '1206.4924-1-12-0': 'In the non-relativistic mean-field approximation the kinetic energy density [MATH] has the simple ideal Fermi gas form [EQUATION] where [MATH], [MATH] is the particle spin, [MATH] is the inverse temperature, [MATH] for the chosen interaction parameters, and the Fermi integral depends on an effective chemical potential [MATH], shifted with respect to the thermodynamic chemical potential because of the rest mass and the depth of the self-consistent mean field [MATH].', '1206.4924-1-12-1': 'At zero temperature the Fermi integral can be analytically solved giving [EQUATION]', '1206.4924-1-12-2': 'The energy density [MATH] at zero temperature obtained with the parameter set (I) for different hyperon fractions [MATH] is represented in Fig. [REF].', '1206.4924-1-12-3': 'We can see that pure neutron matter, as well as pure [MATH] matter, are never bound.', '1206.4924-1-12-4': 'For pure neutron matter this is well known.', '1206.4924-1-12-5': 'For pure [MATH]-matter it is less obvious, since it involves the [MATH]-interaction, subject to large uncertainties.', '1206.4924-1-12-6': 'A strong attraction at low densities could alter this result.', '1206.4924-1-12-7': 'We think, however, that this is not the case, since parameter set I by Balberg and Gal [CITATION], shown in Fig. [REF], assumes a much stronger attraction than indicated by more recent analysis [CITATION].', '1206.4924-1-12-8': 'There is almost no doubt that the [MATH] interaction should be attractive at low densities and repulsive at high densities.', '1206.4924-1-12-9': 'Due to the attractive part of the [MATH] coupling, a mixture of the two particle species admits a bound state at finite density.', '1206.4924-1-12-10': 'This rather general feature of the energy density is found within the other parametrizations and other models, e.g. the parametrization of the energy density functional from G-matrix calculations by Vidana et al. [CITATION].', '1206.4924-1-12-11': 'Within parametrization (I), the lowest energy corresponds to a symmetric mixture [MATH] and a bound state is predicted for [MATH].', '1206.4924-1-13-0': 'The existence of a minimum in the energy functional for symmetric matter means that such a state represents the stable matter phase at zero temperature.', '1206.4924-1-13-1': 'On the other side, the vanishing density gas phase is always the stable phase in the limit of infinite temperature.', '1206.4924-1-13-2': 'This implies that a dilute-to-dense phase change implying strangeness has to be expected on very general grounds.', '1206.4924-1-13-3': 'The presence of a minimum is however not sufficient to discriminate between a smooth cross-over and a phase transition.', '1206.4924-1-13-4': 'In the next section we will therefore demonstrate the existence of a phase transition by explicitly calculating the [MATH] phase diagram.', '1206.4924-1-14-0': '# Phase diagram of the two-component [MATH] system', '1206.4924-1-15-0': 'First order transitions are signaled by an instability or concavity anomaly in the mean-field thermodynamic potential, which has to be cured by means of the Gibbs phase equilibrium construction at the thermodynamic limit.', '1206.4924-1-15-1': 'For this reason the convexity analysis of the thermodynamical potential in the extensive variable space has been often employed to spot the presence of phase transitions, e.g. for the neutron-proton system [CITATION].', '1206.4924-1-15-2': 'At zero temperature, one thermodynamic potential is given by the total energy [EQUATION] and the curvature matrix is defined by [EQUATION] with [MATH] and real eigenvalues [MATH].', '1206.4924-1-15-3': 'The spinodal region is then recognized as the locus of negative curvature of the energy surface, [MATH].', '1206.4924-1-15-4': 'The corresponding eigenvector defines a direction in the density space given by [EQUATION]', '1206.4924-1-15-5': 'This instability direction physically represents the chemical composition of density fluctuations which are spontaneously and exponentially amplified in the unstable region in order to achieve phase separation, and gives the order parameter of the associated phase transition.', '1206.4924-1-15-6': 'In all the parametrizations we have analyzed one of the eigenvalues is always positive, meaning that the order parameter of the [MATH] transition is always one-dimensional, similar to the liquid-gas nuclear phase transition at sub-saturation densities.', '1206.4924-1-16-0': 'The zero temperature instability region of the [MATH] mixture is shown in Fig. [REF] with parameter set (I).', '1206.4924-1-16-1': 'We can see that a large portion of the phase diagram is concerned by the instability.', '1206.4924-1-16-2': 'We can qualitatively distinguish three regions characterized by different order parameters.', '1206.4924-1-16-3': "Below nuclear saturation density, we observe an isoscalar [MATH] instability, very close to ordinary nuclear liquid-gas, with [MATH]'s playing the role of protons.", '1206.4924-1-16-4': 'This could have been expected, due to the similarity between the energy density for [MATH] (cf Fig. [REF]) and the well-known energy density for low density [MATH] matter.', '1206.4924-1-16-5': 'Neutron matter close to saturation is stabilized by the inclusion of a non-zero [MATH]-fraction, which is consistent with the observation that the neutron drip-line is shifted towards more neutron-rich systems in hypernuclei [CITATION].', '1206.4924-1-17-0': "Increasing neutron density the phase separation progressively changes towards the strangeness [MATH] direction: the two stable phases connected by the instability have close baryon densities but a very different fraction of [MATH]'s.", '1206.4924-1-17-1': 'For high [MATH] we observe the same behavior with the roles of neutrons and [MATH]-hyperons exchanged.', '1206.4924-1-17-2': 'The part of the phase diagram at high neutron density comprises the region physically explored by supernova and neutron star matter, which is characterized by chemical equilibrium for reactions implying strangeness, [MATH].', '1206.4924-1-17-3': 'The strangeness equilibrium trajectory is represented by a dashed line in Fig. [REF].', '1206.4924-1-17-4': 'This physically corresponds to the sudden opening of strangeness, observed in many modelizations of neutron star matter (see e.g. [CITATION]).', '1206.4924-1-18-0': 'Assuming the order parameter to be given exactly by [MATH], which is a very good approximation at high [MATH], we can understand the existence of this high density strangeness phase transition in terms of the [MATH] and [MATH]-interaction.', '1206.4924-1-18-1': 'Under this assumption, the curvature analysis can be performed one-dimensionally, as a function of [MATH] only.', '1206.4924-1-18-2': 'Thus the instability region is determined mainly by the condition [MATH].', '1206.4924-1-18-3': 'In Fig. [REF] the [MATH] chemical potential, with the constant mass subtracted, that is the first derivative of the energy density with respect to [MATH], is displayed for parametrization (I).', '1206.4924-1-18-4': 'A minimum in [MATH] is related to a zero in the curvature in the strangeness direction, indicating thus the border of an instability region.', '1206.4924-1-18-5': 'This minimum is clearly visible in Fig. [REF].', '1206.4924-1-18-6': 'It is more pronounced with increasing neutron density and shifted to higher values of [MATH].', '1206.4924-1-18-7': 'Apart from the trivial kinetic term [MATH], [MATH] contains the [MATH] single particle potential, [MATH], thus reflects the [MATH] and [MATH] interaction.', '1206.4924-1-18-8': 'Within the model by Balberg and Gal [CITATION], the attractive part of the [MATH] interaction and the specific form of the [MATH] interaction contribute to this minimum.', '1206.4924-1-19-0': 'Let us stress that the central region of the instability domain, below roughly saturation density is determined mainly by the fact that pure neutron (and [MATH])-matter is unbound and there is low-density attraction in the [MATH]-channel.', '1206.4924-1-19-1': 'The finding in this region thus seems qualitatively robust.', '1206.4924-1-19-2': 'The existence of a strangeness phase transition at high density, on the contrary, is not a general model-independent feature, although, as mentioned above, many models show it.', '1206.4924-1-19-3': 'There are others, for instance the G-matrix models of Refs. [CITATION], which do not show an instability in this region.', '1206.4924-1-19-4': 'This can be seen from the absence of a minimum in [MATH] as a function of [MATH] for constant [MATH].', '1206.4924-1-19-5': 'The reason is twofold: first, a [MATH] interaction is completely missing from these models and the [MATH]-interaction has a slightly different form than in Balberg and Gal [CITATION].', '1206.4924-1-19-6': 'Owing to the lack of reliable information on the hyperonic interactions, which would discriminate between different models, we cannot affirm the existence of the strangeness phase transition, related to this instability, but, turning the argument around, the presence of this phase transition in a physical system would allow to learn much about the shape of the interaction.', '1206.4924-1-20-0': 'One remark of caution concerning the relation of the instability domain with a phase transition is in order here.', '1206.4924-1-20-1': 'In principle, the presence of an instability is a pathology of mean-field approaches.', '1206.4924-1-20-2': 'It is an indication that the lowest energy (or free-energy at finite temperature) equilibrium solution is different from the unstable mean-field one.', '1206.4924-1-20-3': 'If the equilibrium solution corresponds to macroscopic dishomogeneities, it can and it should be recovered from the mean-field results making use of the Gibbs construction.', '1206.4924-1-20-4': 'In this case the convexity in the curvature matrix reflects a physical instability towards phase separation, and the phase diagram contains a region of phase coexistence.', '1206.4924-1-20-5': 'However, since the mean-field equations of state are by construction analytic infinitely differentiable functions, it is possible that the instability is due to a multiple evaluation of densities in a given point of the phase diagram defined by the set of associated chemical potentials, too.', '1206.4924-1-20-6': 'This may not be pertinent in our simple model allowing only two different particle species, but might very well occur in a more complete model including all hyperons and resonance states [CITATION].', '1206.4924-1-20-7': 'In this case the unstable solution has to be eliminated from the equilibrium landscape, without necessarily any phase transition.', '1206.4924-1-21-0': 'To discriminate between the two scenarii and correct the mean-field instabilities, one has therefore to study the phase diagram in the space of chemical potentials.', '1206.4924-1-21-1': 'A useful trick to spot for phase transitions with more than one conserved charge, is to perform a Legendre transform to the statistical ensemble where all extensive variables but one are replaced by their conjugated intensive Lagrange parameters [CITATION].', '1206.4924-1-21-2': 'In this ensemble the multidimensional Gibbs equilibrium conditions reduce to a simple Maxwell construction.', '1206.4924-1-22-0': 'In our simple two-dimensional case we can work equivalently within the ensemble where the neutron density is controlled, or, alternatively, within the ensemble where the [MATH]-density is controlled, corresponding to the two constrained energies: [EQUATION]', '1206.4924-1-22-1': 'Let us concentrate on the second representation.', '1206.4924-1-22-2': 'The behavior of the [MATH]-chemical potential, [MATH], is shown for some selected values of the neutron chemical potential in Fig. [REF].', '1206.4924-1-22-3': 'At variance with the well-known non-strange nuclear matter [MATH] system, equilibrium can only be defined within a finite interval of density.', '1206.4924-1-22-4': 'Indeed the ending points of the curves correspond to vanishing neutron density, as it can be seen from the iso-[MATH] contours in Fig. [REF] above.', '1206.4924-1-23-0': 'At relatively high neutron chemical potential the curves present a back-bending at low [MATH] densities, similar to the usual Van der Waals phenomenology for the fluid transition at finite temperature.', '1206.4924-1-23-1': 'In this case the mean-field solution is unique, but a more stable solution is obtained by phase mixing.', '1206.4924-1-23-2': 'At low [MATH] values and, independent of [MATH], in the high density region, multiple evaluations are observed, where different values of [MATH] are compatible with the same controlled value of the hyperon density.', '1206.4924-1-23-3': 'In this situation, only the solution leading to the lowest constrained energy has to be retained.', '1206.4924-1-23-4': 'To explore these different possibilities, a zoom of Fig. [REF] for [MATH] MeV is shown in Fig. [REF], together with the associated thermodynamic potential.', '1206.4924-1-23-5': 'For a better visibility, the rest mass contribution [MATH]has been subtracted, which does not alter the convexity properties of the function.', '1206.4924-1-23-6': 'We can see that at low density, after elimination of the least stable solution, the constrained energy can still be minimized by taking a linear combination of two homogeneous solutions (circles in Fig. [REF]) having the same first-order derivative, that is the same chemical potential.', '1206.4924-1-23-7': 'Since these two points have the same value for all the intensive variables ([MATH]), they respect Gibbs equilibrium rules.', '1206.4924-1-23-8': 'One can therefore see that the ensemble of Maxwell constructions in the [MATH] ensemble is equivalent to the construction of the global convex envelope of the energy, that is to the Gibbs construction of the complete system.', '1206.4924-1-24-0': 'In the high density case (right part of Fig.[REF]), the elimination of the multiple evaluation leaves a monotonous equation of state, which does not allow any energy gain by linear interpolations.', '1206.4924-1-24-1': 'Going back again to the iso-[MATH] curves of Fig.[REF] , we can see that the high [MATH] region corresponding to the bi-evaluation can also be explored in the complementary ([MATH] ensemble where it will be associated to low values of [MATH].', '1206.4924-1-24-2': 'This means that from a practical point of view, once the phase mixing is systematically performed on the low density ([MATH] and [MATH]) region, all the unstable mean-field solutions turn out to be automatically removed.', '1206.4924-1-25-0': 'The final result of the Gibbs construction is given in Fig.[REF] together with the corrected iso-[MATH] paths.', '1206.4924-1-25-1': 'We can see that the whole unstable region of Fig.[REF] can be interpreted as the spinodal region of a first order phase transition.', '1206.4924-1-25-2': 'In particular if we follow the physical path [MATH] associated to equilibrium of strangeness (dashed line in Fig.[REF]), the emergence of strangeness with the opening of the hyperon channel corresponds to the mixed-phase region of a first order phase transition, which is not correctly modeled in the mean-field approximation.', '1206.4924-1-26-0': 'Up to now we have only presented results at zero temperature.', '1206.4924-1-26-1': 'The extension to finite temperature, as it is needed for supernova matter, is relatively simple in the mean-field approximation.', '1206.4924-1-26-2': 'The appropriate thermodynamic potential at non-zero temperature is given by the Helmholtz free energy [EQUATION] where [MATH] is the mean-field entropy density.', '1206.4924-1-26-3': 'Chemical potentials can be obtained by differentiating this expression or, in a simpler numerical way, by inverting the Fermi integral associated to the densities[CITATION] from the two coupled equations: [EQUATION] with [MATH].', '1206.4924-1-26-4': 'Then the Gibbs construction is performed, as in the zero temperature case, from the combined analysis of [MATH] at constant [MATH], and [MATH] at constant [MATH].', '1206.4924-1-26-5': 'The same qualitative behaviors as in Fig. [REF] are observed, with the difference that at finite temperature chemical potentials tend to [MATH] with vanishing density.', '1206.4924-1-26-6': 'As a consequence, the lower energy border of the coexistence zone is always at finite non-zero density, and Gibbs constructions are always equivalent to equal-area constructions in the mono-extensive ensemble.', '1206.4924-1-27-0': 'The phase diagram as a function of the temperature is presented in Fig. [REF].', '1206.4924-1-27-1': 'This phase diagram exhibits different interesting features.', '1206.4924-1-27-2': 'We can see that the three regions that we have tentatively defined at zero temperature appear as distinct phase transitions at finite temperature.', '1206.4924-1-27-3': 'The first phase transition (zone II in Fig.[REF]) separates a low density gas phase from a high density more symmetric liquid phase, very similar to ordinary liquid-gas.', '1206.4924-1-27-4': 'The second one (zone III in Fig.[REF]) reflects the instability of dense strange matter towards the appearance of neutrons and has an almost symmetric counterpart (zone I in Fig.[REF]) in the instability of dense neutron matter towards the formation of [MATH]-hyperons.', '1206.4924-1-27-5': 'Up to a certain temperature, this latter phase transition is explored by the [MATH] trajectory, meaning that it is expected to occur in neutron stars and supernova matter.', '1206.4924-1-27-6': 'At variance with other known phase transitions in nuclear matter, this transition exists at any temperature and is not limited in density; it is always associated (except at [MATH] MeV in the present model) with a critical point, which moves towards high density as the temperature increases.', '1206.4924-1-27-7': 'This means that criticality should be observed in hot supernova matter, at a temperature which is estimated as [MATH] MeV in the present schematic model.', '1206.4924-1-28-0': 'The consequences of these findings for the composition of neutron star matter are drawn in Fig. [REF].', '1206.4924-1-28-1': 'The left panel shows the [MATH] fraction [MATH] as a function of the baryon density under the condition [MATH].', '1206.4924-1-28-2': 'The crossing of the mixed-phase region with increasing neutron density implies that, as soon as the lower density transition border is crossed, the system has to be viewed as a dishomogeneous mixture of macroscopic regions composed essentially of neutrons, with other macroscopic regions with around [MATH] of hyperons.', '1206.4924-1-28-3': 'The extension of the [MATH]-rich zone increases with density until the system exits the coexistence zone, and becomes homogeneous again.', '1206.4924-1-29-0': 'On the right panel of Fig. [REF] the [MATH]-densities for the two coexisting phases are displayed as a function of temperature, again under the condition of [MATH], physically relevant for neutron star and supernova matter.', '1206.4924-1-29-1': 'The extension of the coexistence region decreases with increasing temperature.', '1206.4924-1-29-2': 'The latter finally disappears at [MATH] MeV.', '1206.4924-1-29-3': 'This well illustrates the fact already observed in connection with the phase diagram, cf Fig. [REF], that the critical point of the strangeness phase transition moves to higher density, crossing the physical line [MATH] at a given temperature, [MATH] MeV in the present example.', '1206.4924-1-30-0': '# Conclusions', '1206.4924-1-31-0': 'In this paper we have calculated the phase diagram of an interacting system of neutrons and [MATH]-hyperons in the mean field approximation.', '1206.4924-1-31-1': 'We have shown that this simple system presents a complex phase diagram with first and second order phase transitions.', '1206.4924-1-31-2': 'Some of these phase transitions are probably never explored in physical systems.', '1206.4924-1-31-3': 'However, a possible phase transition at super-saturation baryon densities, from non-strange to strange matter is expected to be observed both in the inner core of neutron star and in the dense regions of core-collapse supernova.', '1206.4924-1-31-4': 'For this latter phenomenology, a critical point is predicted and the associated critical opalescence could have an impact on supernova dynamics [CITATION].', '1206.4924-1-31-5': 'The existence of this particular phase transition can be related to the form of the [MATH] and [MATH] interaction and is, in the present model, essentially due to the presence of an attractive low density [MATH] interaction as well as to the high-density part of the [MATH]-interaction.', '1206.4924-1-31-6': 'As such, it is expected to persist in a realistic model of dense matter including more hyperonic and non-hyperonic baryons.', '1206.4924-1-31-7': 'The immediate consequence of that is that the opening of hyperon channels at high density should not be viewed as a continuous (though abrupt) increase of strangeness in the matter, observed in many models of hyperonic matter, cf e.g. [CITATION], but rather as the coexistence of hyperon-poor and hyperon-rich macroscopic domains.', '1206.4924-1-32-0': 'Different steps have however to be achieved before quantitative predictions on neutron star physics can be drawn from this simple model.', '1206.4924-1-32-1': 'The model should be extended to include all possible hyperons and resonances, which could shift the coexistence borders and induce new phenomena in the direction - not explored in this preliminary study - of the electric charge density.', '1206.4924-1-32-2': 'These improvements will be the object of a future publication.', '1206.4924-1-33-0': 'This paper has been partly supported by ANR under the project NEXEN and the project SN2NS, ANR-10-BLAN-0503, and by IFIN-IN2P3 agreement nr. 07-44.', '1206.4924-1-33-1': 'Support from Compstar, a research networking programme of the European Science foundation, is acknowledged, too.', '1206.4924-1-33-2': 'Ad.', '1206.4924-1-33-3': 'R. R acknowledges partial support from the Romanian National Authority for Scientific Research under grant PN-II-ID-PCE-2011-3-0092 and kind hospitality from LPC-Caen.'}

{'1206.4924-2-0-0': 'The phase diagram of a system constituted of neutrons and [MATH]-hyperons in thermal equilibrium is evaluated in the mean-field approximation.', '1206.4924-2-0-1': 'It is shown that this simple system exhibits a complex phase diagram with first and second order phase transitions.', '1206.4924-2-0-2': 'Due to the generic presence of attractive and repulsive couplings, the existence of phase transitions involving strangeness appears independent of the specific interaction model.', '1206.4924-2-0-3': 'In addition we will show under which conditions a phase transition towards strange matter at high density exists, which is expected to persist even within a complete treatment including all the different strange and non-strange baryon states.', '1206.4924-2-0-4': 'The impact of this transition on the composition of matter in the inner core of neutron stars is discussed.', '1206.4924-2-1-0': '# Introduction', '1206.4924-2-2-0': 'With the purpose of better understanding the dynamics of core-collapse supernova and the observed characteristics of neutron stars, a considerable theoretical effort has been undertaken in recent years concerning the modelization of the equation of state of cold dense matter [CITATION] with some extensions to finite temperature, see e.g. [CITATION].', '1206.4924-2-3-0': 'If it is well admitted that hyperonic and deconfined quark matter could exist in the inner core of neutron stars, a complete understanding of the composition and equation of state of dense matter is far from being achieved.', '1206.4924-2-3-1': 'Concerning hyperons, simple energetic considerations suggest that they should be present at high density [CITATION].', '1206.4924-2-3-2': 'However, in the standard picture the opening of hyperon degrees of freedom leads to a considerable softening of the equation of state [CITATION], which in turns leads to maximum neutron star masses smaller than the highest values obtained in recent observations [CITATION].', '1206.4924-2-3-3': 'This puzzling situation implies that the hyperon-hyperon and hyperon-nucleon couplings must be much more repulsive at high density than presently assumed [CITATION], and/or that something is missing in the present modelization.', '1206.4924-2-4-0': 'Apart from neutron star observations, already for purely nuclear matter, stringent constraints on the equation of state from experimental data as well as from theoretical side from ab-initio calculations only exist up to roughly saturation density, mainly for almost symmetric matter at zero temperature.', '1206.4924-2-4-1': 'What makes the description of hyperonic matter even more difficult is first of all the fact that, contrary to nucleons, hyperonic data from hypernuclei (see e.g. [CITATION]), diffusion and production experiments (see e.g. [CITATION]) and nuclear collision experiments (see e.g. [CITATION]) are scarce.', '1206.4924-2-4-2': 'This lack of experimental information induces large uncertainties within the microscopic approaches [CITATION], which suffer in addition probably from theoretical shortcomings, among others due to the unknown hyperonic three-body forces [CITATION].', '1206.4924-2-4-3': 'Phenomenological extrapolations of the low-density behavior within mean field models are subject to large uncertainties, too.', '1206.4924-2-4-4': 'In any case, there is much uncertainty on the hyperonic interactions at high density, where neutron star observations can give the only hint, though not decisive.', '1206.4924-2-5-0': 'Here, we propose to study the phase diagram of hyperonic matter.', '1206.4924-2-5-1': 'The generic presence of attractive and repulsive couplings suggests the existence, in a model-independent manner, of a phase transition involving strangeness.', '1206.4924-2-5-2': 'In order to have an analytically solvable model, we consider the simplified situation where only neutrons, [MATH], and [MATH]-hyperons are allowed in the matter chemical composition.', '1206.4924-2-5-3': 'We find, as argued above, a first order phase transition involving strangeness.', '1206.4924-2-5-4': 'In addition, we will show under which assumptions on the [MATH] and [MATH] interaction, respecting the available constraints, a phase transition towards strange matter, that is with a discontinuity in the strangeness content of matter, exists at high density.', '1206.4924-2-6-0': 'We leave the inclusion of protons and higher mass strange and non-strange baryons to a future study.', '1206.4924-2-6-1': 'Because of this simplification, we will not be able to exploit the results for a predictive quantitative application to neutron stars and dense supernova matter.', '1206.4924-2-6-2': 'However, this simplification will allow us to have an exactly solvable model with perfectly controlled numerics.', '1206.4924-2-6-3': 'We believe that the relevant degrees of freedom to explore the opening of the strangeness channels are included already at this level, and the qualitative features of the strangeness phase transition will not change in the complete model.', '1206.4924-2-7-0': 'Concerning the phenomenological consequences of our findings let us stress that many published work on hyperonic matter makes use of the mean field approximation (e.g. [CITATION]).', '1206.4924-2-7-1': 'However, if the opening of the strangeness degree of freedom at high baryonic density is associated with a first order phase transition, the mean field equations of state should be modified making use of the Gibbs construction [CITATION].', '1206.4924-2-7-2': 'The possible presence of coexisting phases would in addition modify the matter composition with respect to the uncorrected mean-field predictions [CITATION].', '1206.4924-2-8-0': 'Both, matter composition and equation of state, are important ingredients not only in the calculations of neutrons star models, but in the hydrodynamical codes describing core collapse dynamics, too.', '1206.4924-2-8-1': 'The high density and high temperature behavior not only influences the success of the explosion, but among others also the dependence of the final state (neutron star or black hole) on the progenitor mass (see e.g. [CITATION]).', '1206.4924-2-8-2': 'This means that the possible presence of a phase transition towards strange matter has to be studied at finite temperature, too.', '1206.4924-2-9-0': '# The model', '1206.4924-2-10-0': 'In the following we will illustrate the propositions concerning the phase diagram by choosing a specific interaction model for the [MATH] system.', '1206.4924-2-10-1': 'Of course, the quantitative numerical results are not model-independent, but we will argue in which way our findings on the phase diagram are general.', '1206.4924-2-11-0': 'The energetics of the [MATH] mixture is described through the energy density functional proposed by Balberg and Gal [CITATION] [EQUATION] where the interaction couplings, compatible with the Lattimer-Swesty equation of state in the non-strange sector, are given in table I. Parametrizations BGI, BGII and BGIII are taken from the work by Balberg and Gal [CITATION] and 220g2.8 corresponds to one of the parametrizations compatible with the observation of an almost two solar mass neutron star [CITATION] from Ref. [CITATION].', '1206.4924-2-11-1': 'In the applications shown below we will use mainly parametrization (BGI), but we have performed the calculations for the other parameter sets, too.', '1206.4924-2-12-0': 'In the non-relativistic mean-field approximation the kinetic energy density [MATH] has the simple ideal Fermi gas form [EQUATION] where [MATH], [MATH] is the particle spin, [MATH] is the inverse temperature, [MATH] for the chosen interaction parameters, and the Fermi integral depends on an effective chemical potential [MATH], shifted with respect to the thermodynamic chemical potential because of the rest mass and the depth of the self-consistent mean field [MATH].', '1206.4924-2-12-1': 'At zero temperature the Fermi integral can be analytically solved giving [EQUATION]', '1206.4924-2-12-2': 'The energy density [MATH] at zero temperature obtained with the parameter set (BGI) for different hyperon fractions [MATH] is represented in Fig. [REF].', '1206.4924-2-12-3': 'We can see that pure neutron matter, as well as pure [MATH] matter, are never bound.', '1206.4924-2-12-4': 'For pure neutron matter this is well known.', '1206.4924-2-12-5': 'For pure [MATH]-matter it is less obvious, since it involves the [MATH]-interaction, subject to large uncertainties.', '1206.4924-2-12-6': 'A strong attraction at low densities could alter this result.', '1206.4924-2-12-7': 'We think, however, that this is not the case, since parameter set I by Balberg and Gal [CITATION], shown in Fig. [REF], assumes a much stronger attraction than indicated by more recent analysis [CITATION].', '1206.4924-2-12-8': 'There is almost no doubt that the [MATH] interaction should be attractive at low densities and repulsive at high densities.', '1206.4924-2-12-9': 'Due to the attractive part of the [MATH] coupling, a mixture of the two particle species admits a bound state at finite density.', '1206.4924-2-12-10': 'This rather general feature of the energy density is found within the other parametrizations and other models, e.g. the parametrization of the energy density functional from G-matrix calculations by Vidana et al. [CITATION].', '1206.4924-2-12-11': 'Within parametrization (BGI), the lowest energy corresponds to a symmetric mixture [MATH] and a bound state is predicted for [MATH].', '1206.4924-2-13-0': 'The existence of a minimum in the energy functional for symmetric matter means that such a state represents the stable matter phase at zero temperature.', '1206.4924-2-13-1': 'On the other side, the vanishing density gas phase is always the stable phase in the limit of infinite temperature.', '1206.4924-2-13-2': 'This implies that a dilute-to-dense phase change implying strangeness has to be expected on very general grounds.', '1206.4924-2-13-3': 'The presence of a minimum is however not sufficient to discriminate between a smooth cross-over and a phase transition.', '1206.4924-2-13-4': 'In the next section we will therefore demonstrate the existence of a phase transition by explicitly calculating the [MATH] phase diagram.', '1206.4924-2-14-0': '# Phase diagram of the two-component [MATH] system', '1206.4924-2-15-0': 'First order transitions are signaled by an instability or concavity anomaly in the mean-field thermodynamic potential, which has to be cured by means of the Gibbs phase equilibrium construction at the thermodynamic limit.', '1206.4924-2-15-1': 'For this reason the convexity analysis of the thermodynamical potential in the extensive variable space has been often employed to spot the presence of phase transitions, e.g. for the neutron-proton system [CITATION].', '1206.4924-2-15-2': 'At zero temperature, one thermodynamic potential is given by the total energy [EQUATION] and the curvature matrix is defined by [EQUATION] with [MATH] and real eigenvalues [MATH].', '1206.4924-2-15-3': 'The spinodal region is then recognized as the locus of negative curvature of the energy surface, [MATH].', '1206.4924-2-15-4': 'The corresponding eigenvector defines a direction in the density space given by [EQUATION]', '1206.4924-2-15-5': 'This instability direction physically represents the chemical composition of density fluctuations which are spontaneously and exponentially amplified in the unstable region in order to achieve phase separation, and gives the order parameter of the associated phase transition.', '1206.4924-2-15-6': 'In all the parametrizations we have analyzed one of the eigenvalues is always positive, meaning that the order parameter of the [MATH] transition is always one-dimensional, similar to the liquid-gas nuclear phase transition at sub-saturation densities.', '1206.4924-2-16-0': 'The zero temperature instability region of the [MATH] mixture is shown in Fig. [REF] with parameter set (BGI).', '1206.4924-2-16-1': 'We can see that a large portion of the phase diagram is concerned by the instability.', '1206.4924-2-16-2': 'We can qualitatively distinguish three regions characterized by different order parameters.', '1206.4924-2-16-3': "Below nuclear saturation density, we observe an isoscalar [MATH] instability, very close to ordinary nuclear liquid-gas, with [MATH]'s playing the role of protons.", '1206.4924-2-16-4': 'This could have been expected, due to the similarity between the energy density for [MATH] (cf Fig. [REF]) and the well-known energy density for low density [MATH] matter.', '1206.4924-2-16-5': 'Neutron matter close to saturation is stabilized by the inclusion of a non-zero [MATH]-fraction, which is consistent with the observation that the neutron drip-line is shifted towards more neutron-rich systems in hypernuclei [CITATION].', '1206.4924-2-17-0': "Increasing neutron density the phase separation progressively changes towards the strangeness [MATH] direction: the two stable phases connected by the instability have close baryon densities but a very different fraction of [MATH]'s.", '1206.4924-2-17-1': 'For high [MATH] we observe the same behavior with the roles of neutrons and [MATH]-hyperons exchanged.', '1206.4924-2-17-2': 'The part of the phase diagram at high neutron density comprises the region physically explored by supernova and neutron star matter, which is characterized by chemical equilibrium for reactions implying strangeness, [MATH].', '1206.4924-2-17-3': 'The strangeness equilibrium trajectory is represented by a dashed line in Fig. [REF].', '1206.4924-2-17-4': 'This physically corresponds to the sudden opening of strangeness, observed in many modelizations of neutron star matter (see e.g. [CITATION]).', '1206.4924-2-18-0': 'Assuming the order parameter to be given exactly by [MATH], which is a very good approximation at high [MATH], we can understand the existence of this high density strangeness phase transition in terms of the [MATH] and [MATH]-interaction.', '1206.4924-2-18-1': 'Under this assumption, the curvature analysis can be performed one-dimensionally, as a function of [MATH] only.', '1206.4924-2-18-2': 'Thus the instability region is determined mainly by the condition [MATH].', '1206.4924-2-18-3': 'In Fig. [REF] the [MATH] chemical potential, with the constant mass subtracted, that is the first derivative of the energy density with respect to [MATH], is displayed for parametrization (BGI).', '1206.4924-2-18-4': 'A minimum in [MATH] is related to a zero in the curvature in the strangeness direction, indicating thus the border of an instability region.', '1206.4924-2-18-5': 'This minimum is clearly visible in Fig. [REF].', '1206.4924-2-18-6': 'It is more pronounced with increasing neutron density and shifted to higher values of [MATH].', '1206.4924-2-18-7': 'Apart from the trivial kinetic term [MATH], [MATH] contains the [MATH] single particle potential, [MATH], thus reflects the [MATH] and [MATH] interaction.', '1206.4924-2-18-8': 'Within the model by Balberg and Gal [CITATION], the attractive part of the [MATH] interaction and the specific form of the [MATH] interaction contribute to this minimum.', '1206.4924-2-19-0': 'Let us stress that the central region of the instability domain, below roughly saturation density is determined mainly by the fact that pure neutron (and [MATH])-matter is unbound and there is low-density attraction in the [MATH]-channel.', '1206.4924-2-19-1': 'The finding in this region thus seems qualitatively robust.', '1206.4924-2-19-2': 'The existence of a strangeness phase transition at high density, on the contrary, is not a general model-independent feature, although, as mentioned above, many models show it.', '1206.4924-2-19-3': 'There are others, for instance the G-matrix models of Refs. [CITATION], which do not show an instability in this region.', '1206.4924-2-19-4': 'This can be seen from the absence of a minimum in [MATH] as a function of [MATH] for constant [MATH].', '1206.4924-2-19-5': 'The reason is twofold: first, a [MATH] interaction is completely missing from these models and the [MATH]-interaction has a slightly different form than in Balberg and Gal [CITATION].', '1206.4924-2-19-6': 'Owing to the lack of reliable information on the hyperonic interactions, which would discriminate between different models, we cannot affirm the existence of the strangeness phase transition, related to this instability, but, turning the argument around, the presence of this phase transition in a physical system would allow to learn much about the shape of the interaction.', '1206.4924-2-20-0': 'One remark of caution concerning the relation of the instability domain with a phase transition is in order here.', '1206.4924-2-20-1': 'In principle, the presence of an instability is a pathology of mean-field approaches.', '1206.4924-2-20-2': 'It is an indication that the lowest energy (or free-energy at finite temperature) equilibrium solution is different from the unstable mean-field one.', '1206.4924-2-20-3': 'If the equilibrium solution corresponds to macroscopic dishomogeneities, it can and it should be recovered from the mean-field results making use of the Gibbs construction.', '1206.4924-2-20-4': 'In this case the convexity in the curvature matrix reflects a physical instability towards phase separation, and the phase diagram contains a region of phase coexistence.', '1206.4924-2-20-5': 'However, since the mean-field equations of state are by construction analytic infinitely differentiable functions, it is possible that the instability is due to a multiple evaluation of densities in a given point of the phase diagram defined by the set of associated chemical potentials, too.', '1206.4924-2-20-6': 'This may not be pertinent in our simple model allowing only two different particle species, but might very well occur in a more complete model including all hyperons and resonance states [CITATION].', '1206.4924-2-20-7': 'In this case the unstable solution has to be eliminated from the equilibrium landscape, without necessarily any phase transition.', '1206.4924-2-21-0': 'To discriminate between the two scenarii and correct the mean-field instabilities, one has therefore to study the phase diagram in the space of chemical potentials.', '1206.4924-2-21-1': 'A useful trick to spot for phase transitions with more than one conserved charge, is to perform a Legendre transform to the statistical ensemble where all extensive variables but one are replaced by their conjugated intensive Lagrange parameters [CITATION].', '1206.4924-2-21-2': 'In this ensemble the multidimensional Gibbs equilibrium conditions reduce to a simple Maxwell construction.', '1206.4924-2-22-0': 'In our simple two-dimensional case we can work equivalently within the ensemble where the neutron density is controlled, or, alternatively, within the ensemble where the [MATH]-density is controlled, corresponding to the two constrained energies: [EQUATION]', '1206.4924-2-22-1': 'Let us concentrate on the second representation.', '1206.4924-2-22-2': 'The behavior of the [MATH]-chemical potential, [MATH], is shown for some selected values of the neutron chemical potential in Fig. [REF].', '1206.4924-2-22-3': 'At variance with the well-known non-strange nuclear matter [MATH] system, equilibrium can only be defined within a finite interval of density.', '1206.4924-2-22-4': 'Indeed the ending points of the curves correspond to vanishing neutron density, as it can be seen from the iso-[MATH] contours in Fig. [REF] above.', '1206.4924-2-23-0': 'At relatively high neutron chemical potential the curves present a back-bending at low [MATH] densities, similar to the usual Van der Waals phenomenology for the fluid transition at finite temperature.', '1206.4924-2-23-1': 'In this case the mean-field solution is unique, but a more stable solution is obtained by phase mixing.', '1206.4924-2-23-2': 'At low [MATH] values and, independent of [MATH], in the high density region, multiple evaluations are observed, where different values of [MATH] are compatible with the same controlled value of the hyperon density.', '1206.4924-2-23-3': 'In this situation, only the solution leading to the lowest constrained energy has to be retained.', '1206.4924-2-23-4': 'To explore these different possibilities, a zoom of Fig. [REF] for [MATH] MeV is shown in Fig. [REF], together with the associated thermodynamic potential.', '1206.4924-2-23-5': 'For a better visibility, the rest mass contribution [MATH]has been subtracted, which does not alter the convexity properties of the function.', '1206.4924-2-23-6': 'We can see that at low density, after elimination of the least stable solution, the constrained energy can still be minimized by taking a linear combination of two homogeneous solutions (circles in Fig. [REF]) having the same first-order derivative, that is the same chemical potential.', '1206.4924-2-23-7': 'Since these two points have the same value for all the intensive variables ([MATH]), they respect Gibbs equilibrium rules.', '1206.4924-2-23-8': 'One can therefore see that the ensemble of Maxwell constructions in the [MATH] ensemble is equivalent to the construction of the global convex envelope of the energy, that is to the Gibbs construction of the complete system.', '1206.4924-2-24-0': 'In the high density case (right part of Fig. [REF]), the elimination of the multiple evaluation leaves a monotonous equation of state, which does not allow any energy gain by linear interpolations.', '1206.4924-2-24-1': 'Going back again to the iso-[MATH] curves of Fig. [REF] , we can see that the high [MATH] region corresponding to the bi-evaluation can also be explored in the complementary ([MATH] ensemble where it will be associated to low values of [MATH].', '1206.4924-2-24-2': 'This means that from a practical point of view, once the phase mixing is systematically performed on the low density ([MATH] and [MATH]) region, all the unstable mean-field solutions turn out to be automatically removed.', '1206.4924-2-25-0': 'The final result of the Gibbs construction is given in Fig. [REF] together with the corrected iso-[MATH] paths.', '1206.4924-2-25-1': 'We can see that the whole unstable region of Fig. [REF] can be interpreted as the spinodal region of a first order phase transition.', '1206.4924-2-25-2': 'In particular if we follow the physical path [MATH] associated to equilibrium of strangeness (dashed line in Fig. [REF]), the emergence of strangeness with the opening of the hyperon channel corresponds to the mixed-phase region of a first order phase transition, which is not correctly modeled in the mean-field approximation.', '1206.4924-2-26-0': 'Up to now we have only presented results at zero temperature.', '1206.4924-2-26-1': 'The extension to finite temperature, as it is needed for supernova matter, is relatively simple in the mean-field approximation.', '1206.4924-2-26-2': 'The appropriate thermodynamic potential at non-zero temperature is given by the Helmholtz free energy [EQUATION] where [MATH] is the mean-field entropy density.', '1206.4924-2-26-3': 'Chemical potentials can be obtained by differentiating this expression or, in a simpler numerical way, by inverting the Fermi integral associated to the densities [CITATION] from the two coupled equations: [EQUATION] with [MATH].', '1206.4924-2-26-4': 'Then the Gibbs construction is performed, as in the zero temperature case, from the combined analysis of [MATH] at constant [MATH], and [MATH] at constant [MATH].', '1206.4924-2-26-5': 'The same qualitative behaviors as in Fig. [REF] are observed, with the difference that at finite temperature chemical potentials tend to [MATH] with vanishing density.', '1206.4924-2-26-6': 'As a consequence, the lower energy border of the coexistence zone is always at finite non-zero density, and Gibbs constructions are always equivalent to equal-area constructions in the mono-extensive ensemble.', '1206.4924-2-27-0': 'The phase diagram as a function of the temperature is presented in Fig. [REF].', '1206.4924-2-27-1': 'This phase diagram exhibits different interesting features.', '1206.4924-2-27-2': 'We can see that the three regions that we have tentatively defined at zero temperature appear as distinct phase transitions at finite temperature.', '1206.4924-2-27-3': 'The first phase transition (zone II in Fig. [REF]) separates a low density gas phase from a high density more symmetric liquid phase, very similar to ordinary liquid-gas.', '1206.4924-2-27-4': 'The second one (zone III in Fig. [REF]) reflects the instability of dense strange matter towards the appearance of neutrons and has an almost symmetric counterpart (zone I in Fig. [REF]) in the instability of dense neutron matter towards the formation of [MATH]-hyperons.', '1206.4924-2-27-5': 'Up to a certain temperature, this latter phase transition is explored by the [MATH] trajectory, meaning that it is expected to occur in neutron stars and supernova matter.', '1206.4924-2-27-6': 'At variance with other known phase transitions in nuclear matter, this transition exists at any temperature and is not limited in density; it is always associated (except at [MATH] MeV in the present model) with a critical point, which moves towards high density as the temperature increases.', '1206.4924-2-27-7': 'This means that criticality should be observed in hot supernova matter, at a temperature which is estimated as [MATH] MeV in the present schematic model.', '1206.4924-2-28-0': 'The consequences of these findings for the composition of neutron star matter are drawn in Fig. [REF].', '1206.4924-2-28-1': 'The left panel shows the [MATH] fraction [MATH] as a function of the baryon density under the condition [MATH].', '1206.4924-2-28-2': 'The crossing of the mixed-phase region with increasing neutron density implies that, as soon as the lower density transition border is crossed, the system has to be viewed as a dishomogeneous mixture of macroscopic regions composed essentially of neutrons, with other macroscopic regions with around [MATH] of hyperons.', '1206.4924-2-28-3': 'The extension of the [MATH]-rich zone increases with density until the system exits the coexistence zone, and becomes homogeneous again.', '1206.4924-2-29-0': 'On the right panel of Fig. [REF] the [MATH]-densities for the two coexisting phases are displayed as a function of temperature, again under the condition of [MATH], physically relevant for neutron star and supernova matter.', '1206.4924-2-29-1': 'The extension of the coexistence region decreases with increasing temperature.', '1206.4924-2-29-2': 'The latter finally disappears at [MATH] MeV.', '1206.4924-2-29-3': 'This well illustrates the fact already observed in connection with the phase diagram, cf Fig. [REF], that the critical point of the strangeness phase transition moves to higher density, crossing the physical line [MATH] at a given temperature, [MATH] MeV in the present example.', '1206.4924-2-30-0': '# Conclusions', '1206.4924-2-31-0': 'In this paper we have calculated the phase diagram of an interacting system of neutrons and [MATH]-hyperons in the mean field approximation.', '1206.4924-2-31-1': 'We have shown that this simple system presents a complex phase diagram with first and second order phase transitions.', '1206.4924-2-31-2': 'Some of these phase transitions are probably never explored in physical systems.', '1206.4924-2-31-3': 'However, a possible phase transition at super-saturation baryon densities, from non-strange to strange matter is expected to be observed both in the inner core of neutron star and in the dense regions of core-collapse supernova.', '1206.4924-2-31-4': 'For this latter phenomenology, a critical point is predicted and the associated critical opalescence could have an impact on supernova dynamics [CITATION].', '1206.4924-2-31-5': 'The existence of this particular phase transition can be related to the form of the [MATH] and [MATH] interaction and is, in the present model, essentially due to the presence of an attractive low density [MATH] interaction as well as to the high-density part of the [MATH]-interaction.', '1206.4924-2-31-6': 'As such, it is expected to persist in a realistic model of dense matter including more hyperonic and non-hyperonic baryons.', '1206.4924-2-31-7': 'The immediate consequence of that is that the opening of hyperon channels at high density should not be viewed as a continuous (though abrupt) increase of strangeness in the matter, observed in many models of hyperonic matter, cf e.g. [CITATION], but rather as the coexistence of hyperon-poor and hyperon-rich macroscopic domains.', '1206.4924-2-32-0': 'Different steps have however to be achieved before quantitative predictions on neutron star physics can be drawn from this simple model.', '1206.4924-2-32-1': 'The model should be extended to include all possible hyperons and resonances, which could shift the coexistence borders and induce new phenomena in the direction - not explored in this preliminary study - of the electric charge density.', '1206.4924-2-32-2': 'These improvements will be the object of a future publication.', '1206.4924-2-33-0': 'This paper has been partly supported by ANR under the project NEXEN and the project SN2NS, ANR-10-BLAN-0503, and by IFIN-IN2P3 agreement nr. 07-44.', '1206.4924-2-33-1': 'Support from Compstar, a research networking programme of the European Science foundation, is acknowledged, too.', '1206.4924-2-33-2': 'Ad.', '1206.4924-2-33-3': 'R. R acknowledges partial support from the Romanian National Authority for Scientific Research under grant PN-II-ID-PCE-2011-3-0092 and kind hospitality from LPC-Caen.'}

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[]

[]

[]

['1206.4924-1-33-2', '1206.4924-2-33-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1206.4924

1704.08720

{'1704.08720-1-0-0': 'We compute the [MATH] norm of a general Gaussian gauge-covariant multi-mode channel for any [MATH], where [MATH] is a Schatten space.', '1704.08720-1-0-1': 'As a consequence, we verify the Gaussian optimizer conjecture and the multiplicativity conjecture in these cases.', '1704.08720-1-1-0': '# Introduction', '1704.08720-1-2-0': 'Gaussian quantum channels play a fundamental role in quantum information theory and quantum optics.', '1704.08720-1-2-1': 'They appear, for instance, as a model of attenuation, amplification and noise in electromagnetic communications through metal wires, optical fibers or free space.', '1704.08720-1-2-2': 'Despite their ubiquity several fundamental mathematical questions about their structure remain still unsolved.', '1704.08720-1-2-3': 'Among them are the Gaussian optimizer conjecture and the additivity conjecture.', '1704.08720-1-2-4': 'Our goal here is to contribute a new family of special cases in which we can verify both of these conjectures.', '1704.08720-1-3-0': 'The two conjectures are concerned with the norm of a Gaussian channel acting from a Schatten space [MATH] to a Schatten space [MATH].', '1704.08720-1-3-1': '(We recall the definition of Schatten spaces at the beginning of the following section.)', '1704.08720-1-3-2': 'The Gaussian optimizer conjecture states that, in order to compute this norm, it suffices to test the channel on Gaussian states.', '1704.08720-1-3-3': 'An affirmative answer to this question would be a non-commutative analogue of a theorem by one of us (E.H.L.) which says that in order to compute the norm of an integral operator with a Gaussian integral kernel from [MATH] to [MATH] it suffices to test the integral operator on Gaussian functions [CITATION].', '1704.08720-1-3-4': 'The Gaussian optimizer conjecture is known to be true for gauge-covariant multi-mode Gaussian channels if [MATH] [CITATION] (see also [CITATION] for the proof of the entropy version) and for a subclass of gauge-covariant single-mode channels (namely quantum limited attenuators and amplifiers) for any [MATH] and [MATH] [CITATION] (see also [CITATION] for a proof of the entropy version).', '1704.08720-1-3-5': 'Our main result (Theorem [REF]) is that the Gaussian optimizer conjecture is true for gauge-covariant multi-mode channels if [MATH].', '1704.08720-1-3-6': 'Moreover, we are able to compute the corresponding norm explicitly in terms of the parameters of the channel.', '1704.08720-1-4-0': 'The additivity conjecture asks whether the [MATH] norm of an [MATH]-fold tensor product of a Gaussian channel is equal to the [MATH]-th power of the norm of the channel (so the logarithms of the norms are additive, explaining the name of the conjecture).', '1704.08720-1-4-1': 'For a history of this problem and a review of some important results we refer to [CITATION] and the references therein.', '1704.08720-1-4-2': 'For general quantum channels this additivity is known to be false, but it has been suggested that it might be true for Gaussian channels.', '1704.08720-1-4-3': 'Again the conjecture has been verified for gauge-covariant multi-mode channels if [MATH] [CITATION] and for the same subclass of gauge-covariant single-mode channels as before for any [MATH] and [MATH].', '1704.08720-1-4-4': 'As a consequence of our main result (Corollary [REF]), we are able to conclude that the additivity conjecture holds for [MATH] for general gauge-covariant multi-mode channels.', '1704.08720-1-5-0': 'The main ingredient in our proof is an abstract bound on the [MATH] norm of a positive (not necessarily completely positive and not necessarily trace preserving) map on operators (Theorem [REF]).', '1704.08720-1-5-1': 'This bound is strongly motivated by the works [CITATION] and [CITATION] and is obtained by a simple complex interpolation argument.', '1704.08720-1-5-2': 'What is remarkable is that this bound is optimal for gauge-covariant Gaussian channels.', '1704.08720-1-5-3': 'This is verified in the proof of Theorem [REF] using explicit computations with Gaussian states.', '1704.08720-1-5-4': 'We will show there that the norm is attained asymptotically in the limit of an infinite temperature thermal state.', '1704.08720-1-5-5': 'Note that this is in contrast to the case [MATH] where the norm is attained at the vaccuum (which corresponds to zero temperature).', '1704.08720-1-5-6': 'Also, our explicit expression for the [MATH] norm shows that it is completely determined by the amplification/attenuation matrix [MATH] characteristic of the channel, whereas the explicit expression for the [MATH] norm [CITATION] shows that the latter is determined by the noise matrix [MATH] of the channel.', '1704.08720-1-5-7': 'Therefore, our results are in some sense complementary to those in [CITATION], although the mathematical tools are completely different.'}

{'1704.08720-2-0-0': 'We compute the [MATH] norm of a general Gaussian gauge-covariant multi-mode channel for any [MATH], where [MATH] is a Schatten space.', '1704.08720-2-0-1': 'As a consequence, we verify the Gaussian optimizer conjecture and the multiplicativity conjecture in these cases.', '1704.08720-2-1-0': '# Introduction', '1704.08720-2-2-0': 'Gaussian quantum channels play a fundamental role in quantum information theory and quantum optics.', '1704.08720-2-2-1': 'They appear, for instance, as a model of attenuation, amplification and noise in electromagnetic communications through metal wires, optical fibers or free space.', '1704.08720-2-2-2': 'Despite their ubiquity several fundamental mathematical questions about their structure remain still unsolved.', '1704.08720-2-2-3': 'Among them are the Gaussian optimizer conjecture and the additivity conjecture.', '1704.08720-2-2-4': 'Our goal here is to contribute a new family of special cases in which we can verify both of these conjectures.', '1704.08720-2-3-0': 'The two conjectures are concerned with the norm of a Gaussian channel acting from a Schatten space [MATH] to a Schatten space [MATH].', '1704.08720-2-3-1': '(We recall the definition of Schatten spaces at the beginning of the following section.)', '1704.08720-2-3-2': 'The Gaussian optimizer conjecture states that, in order to compute this norm, it suffices to test the channel on Gaussian states.', '1704.08720-2-3-3': 'An affirmative answer to this question would be a non-commutative analogue of a theorem by one of us (E.H.L.) which says that in order to compute the norm of an integral operator with a Gaussian integral kernel from [MATH] to [MATH] it suffices to test the integral operator on Gaussian functions [CITATION].', '1704.08720-2-3-4': 'The Gaussian optimizer conjecture is known to be true for gauge-covariant multi-mode Gaussian channels if [MATH] [CITATION] (see also [CITATION] for the proof of the entropy version) and for a subclass of gauge-covariant single-mode channels (namely quantum limited attenuators and amplifiers) for any [MATH] and [MATH] [CITATION] (see also [CITATION] for a proof of the entropy version).', '1704.08720-2-3-5': 'Our main result (Theorem [REF]) is that the Gaussian optimizer conjecture is true for gauge-covariant multi-mode channels if [MATH].', '1704.08720-2-3-6': 'Moreover, we are able to compute the corresponding norm explicitly in terms of the parameters of the channel.', '1704.08720-2-4-0': 'The additivity conjecture asks whether the [MATH] norm of an [MATH]-fold tensor product of a Gaussian channel is equal to the [MATH]-th power of the norm of the channel (so the logarithms of the norms are additive, explaining the name of the conjecture).', '1704.08720-2-4-1': 'For a history of this problem and a review of some important results we refer to [CITATION] and the references therein.', '1704.08720-2-4-2': 'For general quantum channels this additivity is known to be false, but it has been suggested that it might be true for Gaussian channels.', '1704.08720-2-4-3': 'Again the conjecture has been verified for gauge-covariant multi-mode channels if [MATH] [CITATION] and for the same subclass of gauge-covariant single-mode channels as before for any [MATH] and [MATH].', '1704.08720-2-4-4': 'As a consequence of our main result (Corollary [REF]), we are able to conclude that the additivity conjecture holds for [MATH] for general gauge-covariant multi-mode channels.', '1704.08720-2-5-0': 'The main ingredient in our proof is an abstract bound on the [MATH] norm of a positive (not necessarily completely positive and not necessarily trace preserving) map on operators (Theorem [REF]).', '1704.08720-2-5-1': 'This bound is strongly motivated by the works [CITATION] and [CITATION] and is obtained by a simple complex interpolation argument.', '1704.08720-2-5-2': 'What is remarkable is that this bound is optimal for gauge-covariant Gaussian channels.', '1704.08720-2-5-3': 'This is verified in the proof of Theorem [REF] using explicit computations with Gaussian states.', '1704.08720-2-5-4': 'We will show there that the norm is attained asymptotically in the limit of an infinite temperature thermal state.', '1704.08720-2-5-5': 'Note that this is in contrast to the case [MATH] where the norm is attained at the vaccuum (which corresponds to zero temperature).', '1704.08720-2-5-6': 'Also, our explicit expression for the [MATH] norm shows that it is completely determined by the amplification/attenuation matrix [MATH] characteristic of the channel, whereas the explicit expression for the [MATH] norm [CITATION] shows that the latter is determined by the noise matrix [MATH] of the channel.', '1704.08720-2-5-7': 'Therefore, our results are in some sense complementary to those in [CITATION], although the mathematical tools are completely different.'}

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[['1704.08720-2-5-5', '1704.08720-3-5-5']]

[]

[['1704.08720-2-4-3', '1704.08720-3-4-3']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1704.08720

{'1704.08720-3-0-0': 'We compute the [MATH] norm of a general Gaussian gauge-covariant multi-mode channel for any [MATH], where [MATH] is a Schatten space.', '1704.08720-3-0-1': 'As a consequence, we verify the Gaussian optimizer conjecture and the multiplicativity conjecture in these cases.', '1704.08720-3-1-0': '# Introduction', '1704.08720-3-2-0': 'Gaussian quantum channels play a fundamental role in quantum information theory and quantum optics.', '1704.08720-3-2-1': 'They appear, for instance, as a model of attenuation, amplification and noise in electromagnetic communications through metal wires, optical fibers or free space.', '1704.08720-3-2-2': 'Despite their ubiquity several fundamental mathematical questions about their structure remain still unsolved.', '1704.08720-3-2-3': 'Among them are the Gaussian optimizer conjecture and the additivity conjecture.', '1704.08720-3-2-4': 'Our goal here is to contribute a new family of special cases in which we can verify both of these conjectures.', '1704.08720-3-3-0': 'The two conjectures are concerned with the norm of a Gaussian channel acting from a Schatten space [MATH] to a Schatten space [MATH].', '1704.08720-3-3-1': '(We recall the definition of Schatten spaces at the beginning of the following section.)', '1704.08720-3-3-2': 'The Gaussian optimizer conjecture states that, in order to compute this norm, it suffices to test the channel on Gaussian states.', '1704.08720-3-3-3': 'An affirmative answer to this question would be a non-commutative analogue of a theorem by one of us (E.H.L.) which says that in order to compute the norm of an integral operator with a Gaussian integral kernel from [MATH] to [MATH] it suffices to test the integral operator on Gaussian functions [CITATION].', '1704.08720-3-3-4': 'The Gaussian optimizer conjecture is known to be true for gauge-covariant multi-mode Gaussian channels if [MATH] [CITATION] (see also [CITATION] for the proof of the entropy version) and for a subclass of gauge-covariant single-mode channels (namely quantum limited attenuators and amplifiers) for any [MATH] and [MATH] [CITATION] (see also [CITATION] for a proof of the entropy version).', '1704.08720-3-3-5': 'Our main result (Theorem [REF]) is that the Gaussian optimizer conjecture is true for gauge-covariant multi-mode channels if [MATH].', '1704.08720-3-3-6': 'Moreover, we are able to compute the corresponding norm explicitly in terms of the parameters of the channel.', '1704.08720-3-4-0': 'The additivity conjecture asks whether the [MATH] norm of an [MATH]-fold tensor product of a Gaussian channel is equal to the [MATH]-th power of the norm of the channel (so the logarithms of the norms are additive, explaining the name of the conjecture).', '1704.08720-3-4-1': 'For a history of this problem and a review of some important results we refer to [CITATION] and the references therein.', '1704.08720-3-4-2': 'For general quantum channels this additivity is known to be false, but it has been suggested that it might be true for Gaussian channels.', '1704.08720-3-4-3': 'Again the conjecture has been verified for gauge-covariant multi-mode channels if [MATH] [CITATION].', '1704.08720-3-4-4': 'As a consequence of our main result (Corollary [REF]), we are able to conclude that the additivity conjecture holds for [MATH] for general gauge-covariant multi-mode channels.', '1704.08720-3-5-0': 'The main ingredient in our proof is an abstract bound on the [MATH] norm of a positive (not necessarily completely positive and not necessarily trace preserving) map on operators (Theorem [REF]).', '1704.08720-3-5-1': 'This bound is strongly motivated by the works [CITATION] and [CITATION] and is obtained by a simple complex interpolation argument.', '1704.08720-3-5-2': 'What is remarkable is that this bound is optimal for gauge-covariant Gaussian channels.', '1704.08720-3-5-3': 'This is verified in the proof of Theorem [REF] using explicit computations with Gaussian states.', '1704.08720-3-5-4': 'We will show there that the norm is attained asymptotically in the limit of an infinite temperature thermal state.', '1704.08720-3-5-5': 'Note that this is in contrast to the case [MATH] where the norm is attained at the vacuum (which corresponds to zero temperature).', '1704.08720-3-5-6': 'Also, our explicit expression for the [MATH] norm shows that it is completely determined by the amplification/attenuation matrix [MATH] characteristic of the channel, whereas the explicit expression for the [MATH] norm [CITATION] shows that the latter is determined by the noise matrix [MATH] of the channel.', '1704.08720-3-5-7': 'Therefore, our results are in some sense complementary to those in [CITATION], although the mathematical tools are completely different.'}

hep-th-0309258

{'hep-th-0309258-1-0-0': 'In the plane-wave matrix model, the background configuration of two membrane fuzzy spheres, one of which rotates around the other one in the [MATH] symmetric space, is allowed as a classical solution.', 'hep-th-0309258-1-0-1': 'We study the one-loop quantum corrections to this background in the path integral formulation.', 'hep-th-0309258-1-0-2': 'Firstly, we show that each fuzzy sphere is stable under the quantum correction.', 'hep-th-0309258-1-0-3': 'Secondly, the effective potential describing the interaction between fuzzy spheres is obtained as a function of [MATH], which is the distance between two fuzzy spheres.', 'hep-th-0309258-1-0-4': 'It is shown that the effective potential is flat and hence the fuzzy spheres do not feel any force.', 'hep-th-0309258-1-1-0': '[5mm] Keywords : pp-wave, Matrix model, Fuzzy sphere', 'hep-th-0309258-1-2-0': 'PACS numbers : 11.25.-w, 11.27.', 'hep-th-0309258-1-2-1': '+d, 12.60.', 'hep-th-0309258-1-2-2': 'Jv', 'hep-th-0309258-1-3-0': '6.6mm', 'hep-th-0309258-1-4-0': '# Introduction', 'hep-th-0309258-1-5-0': 'The plane-wave matrix model [CITATION] is a microscopic description of the discrete light cone quantized (DLCQ) M-theory in the eleven-dimensional [MATH]-wave or plane-wave background.', 'hep-th-0309258-1-5-1': 'The eleven-dimensional plane-wave [CITATION] is maximally supersymmetric and the limiting case of the eleven-dimensional [MATH] type geometries [CITATION].', 'hep-th-0309258-1-5-2': 'Its explicit form is given by [EQUATION] where [MATH].', 'hep-th-0309258-1-5-3': 'Due to the effect of the metric component [MATH] and the presence of the four-form field strength, the plane-wave matrix model has some [MATH] dependent terms, which make the difference between the usual flat space matrix model and the plane-wave one.', 'hep-th-0309258-1-6-0': 'The presence of the [MATH] dependent terms makes the plane-wave matrix model have some peculiar properties.', 'hep-th-0309258-1-6-1': 'One of them is that there are various vacuum structures classified by the [MATH] algebra [CITATION].', 'hep-th-0309258-1-6-2': 'The crucial ingredient for vacua is the membrane fuzzy sphere.', 'hep-th-0309258-1-6-3': 'It preserves the full 16 supersymmetries of the plane-wave matrix model and exists even at finite [MATH] which is the size of matrix.', 'hep-th-0309258-1-7-0': 'After the plane-wave matrix model was proposed and its basic aspects were uncovered, there have been lots of investigations in various directions.', 'hep-th-0309258-1-7-1': 'The structure of vacua has been studied in more detail especially related to the protected multiplet [CITATION].', 'hep-th-0309258-1-7-2': 'The possible BPS objects contained in the plane-wave matrix model have been searched [CITATION].', 'hep-th-0309258-1-7-3': 'The algebraic and structural study of the model itself and the various BPS objects present in it has been performed [CITATION].', 'hep-th-0309258-1-7-4': 'Based on the fact that the low energy description of the M theory is the eleven dimensional supergravity, there also have been supergravity side analysis [CITATION].', 'hep-th-0309258-1-8-0': 'If the M theory is compactified on a circle, then we have ten-dimensional Type IIA string theory.', 'hep-th-0309258-1-8-1': 'Under the circle compactification, the [MATH]-wave geometry ([REF]) becomes the IIA [MATH]-wave background which is not maximally supersymmetric and has 24 supersymmetries [CITATION].', 'hep-th-0309258-1-8-2': 'For the purpose of understanding the plane-wave matrix model as well as the string theory itself, the IIA string theory in the [MATH]-wave background has been also extensively studied, in parallel with the progress in the study of the plane-wave matrix model [CITATION].', 'hep-th-0309258-1-9-0': 'However, despite of quite amount of progress in the study of the plane-wave matrix model, there has been lack of the investigation about the dynamical aspects.', 'hep-th-0309258-1-9-1': 'However, see for example [CITATION].', 'hep-th-0309258-1-9-2': 'In fact, the present status of the plane-wave matrix model enables us to study the dynamics of the model.', 'hep-th-0309258-1-9-3': 'In this paper, we consider the basic objects of the plane-wave matrix model and study their interaction.', 'hep-th-0309258-1-10-0': 'For the interacting objects, we take two membrane fuzzy spheres, both of which are supersymmetric.', 'hep-th-0309258-1-10-1': 'In the [MATH] symmetric subspace which one may see in the [MATH]-wave background ([REF]), two fuzzy spheres are taken to be at the origin.', 'hep-th-0309258-1-10-2': 'In the [MATH] symmetric space, one fuzzy sphere is located at the origin, while the other fuzzy sphere is taken to rotate around the origin with a fixed distance.', 'hep-th-0309258-1-10-3': 'It should be noted that this configuration is allowed as a classical solution of the equations of motion.', 'hep-th-0309258-1-10-4': 'We will evaluate the one-loop corrections to the configuration and obtain the effective potential.', 'hep-th-0309258-1-10-5': 'As we will see, the effective potential is flat.', 'hep-th-0309258-1-10-6': 'This is unconventional from the viewpoint of the flat space matrix model and indicates some intriguing properties of the plane-wave matrix model that should be uncovered.', 'hep-th-0309258-1-11-0': 'The organization of this paper is as follows.', 'hep-th-0309258-1-11-1': 'In the next section, we give the action of the plane-wave matrix model and consider its classical solutions focused on our concern.', 'hep-th-0309258-1-11-2': 'The expansion of the action around a given arbitrary background is given in section [REF].', 'hep-th-0309258-1-11-3': 'In section [REF], we set up the background configuration and consider the fluctuations around it.', 'hep-th-0309258-1-11-4': 'In section [REF], the one-loop stability of each fuzzy sphere is checked for arbitrary size.', 'hep-th-0309258-1-11-5': 'In section [REF], we evaluate the path integration of fluctuations responsible for the interaction between fuzzy spheres.', 'hep-th-0309258-1-11-6': 'It will be shown that the one-loop effective potential is flat.', 'hep-th-0309258-1-11-7': 'Thus, the fuzzy spheres do not feel any force.', 'hep-th-0309258-1-11-8': 'Finally, conclusion and discussion will be given in section [REF].', 'hep-th-0309258-1-12-0': '# Plane-wave matrix model and classical solutions', 'hep-th-0309258-1-13-0': 'The plane-wave matrix model is basically composed of two parts.', 'hep-th-0309258-1-13-1': 'One part is the usual matrix model based on eleven-dimensional flat space-time, that is, the flat space matrix model, and another is a set of terms reflecting the structure of the maximally supersymmetric eleven dimensional plane-wave background, Eq. ([REF]).', 'hep-th-0309258-1-13-2': 'Its action is [EQUATION] where each part of the action on the right hand side is given by [EQUATION]', 'hep-th-0309258-1-13-3': 'Here, [MATH] is the radius of circle compactification along [MATH] and [MATH] is the covariant derivative with the gauge field [MATH], [EQUATION]', 'hep-th-0309258-1-13-4': 'For dealing with the problem in this paper, it is convenient to rescale the gauge field and parameters as [EQUATION]', 'hep-th-0309258-1-13-5': 'With this rescaling, the radius parameter [MATH] disappears and the actions in Eq. ([REF]) become [EQUATION]', 'hep-th-0309258-1-13-6': 'The possible backgrounds allowed by the plane-wave matrix model are the classical solutions of the equations of motion for the matrix fields.', 'hep-th-0309258-1-13-7': 'Since the background that we are concerned about is purely bosonic, we concentrate on solutions of the bosonic fields [MATH].', 'hep-th-0309258-1-13-8': 'We would like to note that we will not consider all possible solutions but only those relevant to our interest for the fuzzy sphere interaction.', 'hep-th-0309258-1-13-9': 'Then, from the rescaled action, ([REF]), the bosonic equations of motion are derived as [EQUATION] where the over dot implies the time derivative [MATH].', 'hep-th-0309258-1-14-0': 'Except for the trivial [MATH] solution, the simplest one is the simple harmonic oscillator solution; [EQUATION] where [MATH] and [MATH] are the amplitudes and phases of oscillations respectively, and [MATH] is the [MATH] unit matrix.', 'hep-th-0309258-1-14-1': 'This oscillatory solution is special to the plane-wave matrix model due to the presence of mass terms for [MATH].', 'hep-th-0309258-1-14-2': 'It should be noted that, because of the mass terms, the configuration corresponding to the time dependent straight line motion, say [MATH] with non-zero constants [MATH] and [MATH], is not possible as a solution of ([REF]), that is, a classical background of plane-wave matrix model, contrary to the case of the flat space matrix model.', 'hep-th-0309258-1-14-3': 'As the generalization of the oscillatory solution, Eq. ([REF]), we get the solution of the form of diagonal matrix with each diagonal element having independent amplitude and phase.', 'hep-th-0309258-1-15-0': 'As for the non-trivial constant matrix solution, Eq. ([REF]) allows the following membrane fuzzy sphere or giant graviton solution: [EQUATION] where [MATH] satisfies the [MATH] algebra, [EQUATION]', 'hep-th-0309258-1-15-1': 'The reason why this solution is possible is basically because of the fact that the matrix field [MATH] feels an extra force due to the Myers interaction which may stabilize the oscillatory force.', 'hep-th-0309258-1-15-2': 'The fuzzy sphere solution [MATH] preserves the full 16 dynamical supersymmetries of the plane-wave and hence is 1/2-BPS object.', 'hep-th-0309258-1-15-3': 'We note that actually there is another fuzzy sphere solution of the form [MATH].', 'hep-th-0309258-1-15-4': 'However, it has been shown that such solution does not have quantum stability and is thus non-BPS object.', 'hep-th-0309258-1-16-0': '# Matrix model expansion around general background', 'hep-th-0309258-1-17-0': 'In this section, the plane-wave matrix model is expanded around the general bosonic background, which is supposed to satisfy the classical equations of motion, Eq. ([REF]).', 'hep-th-0309258-1-18-0': 'We first split the matrix quantities into as follows: [EQUATION] where [MATH] and [MATH] are the classical background fields while [MATH] and [MATH] are the quantum fluctuations around them.', 'hep-th-0309258-1-18-1': 'The fermionic background [MATH] is taken to vanish from now on, since we will only consider the purely bosonic background.', 'hep-th-0309258-1-18-2': 'The quantum fluctuations are the fields subject to the path integration.', 'hep-th-0309258-1-18-3': 'In taking into account the quantum fluctuations, we should recall that the matrix model itself is a gauge theory.', 'hep-th-0309258-1-18-4': 'This implies that the gauge fixing condition should be specified before proceed further.', 'hep-th-0309258-1-18-5': 'In this paper, we take the background field gauge which is usually chosen in the matrix model calculation as [EQUATION]', 'hep-th-0309258-1-18-6': 'Then the corresponding gauge-fixing [MATH] and Faddeev-Popov ghost [MATH] terms are given by [EQUATION]', 'hep-th-0309258-1-18-7': 'Now by inserting the decomposition of the matrix fields ([REF]) into Eqs. ([REF]) and ([REF]), we get the gauge fixed plane-wave action [MATH] expanded around the background.', 'hep-th-0309258-1-18-8': 'The resulting acting is read as [EQUATION] where [MATH] represents the action of order [MATH] with respect to the quantum fluctuations and, for each [MATH], its expression is [EQUATION]', 'hep-th-0309258-1-18-9': 'Some comments are in order for the background gauge choice, Eq. ([REF]).', 'hep-th-0309258-1-18-10': 'One advantage of this gauge choice is that the quadratic part of the action in terms of fluctuations, that is, the quadratic action, is simplified.', 'hep-th-0309258-1-18-11': 'In more detail, there appears the term [MATH] in the expansion of the potential [MATH], which is canceled exactly by the same term with the opposite sign coming from the gauge fixing term of Eq. ([REF]) and hence absent in [MATH] of Eq. ([REF]).', 'hep-th-0309258-1-18-12': 'This cancellation has given some benefits in the actual flat space matrix model calculation.', 'hep-th-0309258-1-18-13': 'This is also the case in the present plane-wave matrix model, except however for the potential of [MATH].', 'hep-th-0309258-1-18-14': 'As we will see later, [MATH] is responsible for completing the [MATH] potential into the nice square form.', 'hep-th-0309258-1-18-15': 'Thus the same term with the opposite sign from the gauge fixing term remains in the quadratic action.', 'hep-th-0309258-1-18-16': 'At later stage, the presence of this term will have an important implication in taking into account of the unphysical gauge degrees of freedom which are eventually eliminated by those of ghosts.', 'hep-th-0309258-1-19-0': '# Fuzzy sphere configuration and fluctuations', 'hep-th-0309258-1-20-0': 'We now set up the background configuration for the membrane fuzzy spheres.', 'hep-th-0309258-1-20-1': 'Since we will study the interaction of two fuzzy spheres, the matrices representing the background have the [MATH] block diagonal form as [EQUATION] where [MATH] with [MATH] are [MATH] matrices.', 'hep-th-0309258-1-20-2': 'If we take [MATH] as [MATH] matrices, then [MATH].', 'hep-th-0309258-1-21-0': 'The two fuzzy spheres are taken to be static in the space where they span, and hence represented by the classical solution, ([REF]); [EQUATION] where, for each [MATH], [MATH] is in the [MATH]-dimensional irreducible representation of [MATH] and satisfies the [MATH] algebra, Eq. ([REF]).', 'hep-th-0309258-1-21-1': 'In the [MATH] symmetric transverse space, the fuzzy spheres are regarded as point objects, of course, in a sense of ignoring the matrix nature.', 'hep-th-0309258-1-21-2': 'We first let the second fuzzy sphere given by the background of [MATH] be at the origin in the transverse space and stay there.', 'hep-th-0309258-1-21-3': 'As for the first fuzzy sphere, it is made to move around the second sphere in the form of circular motion with the radius [MATH].', 'hep-th-0309258-1-21-4': 'Obviously, this configuration is one of the classical solutions of the equations of motion as one can see from Eq. ([REF]).', 'hep-th-0309258-1-21-5': 'Recalling that the transverse space is [MATH] symmetric, all the possible choices of two-dimensional sub-plane where the circular motion takes place are equivalent.', 'hep-th-0309258-1-21-6': 'Thus, without loss of generality, we can take a certain plane for the circular motion.', 'hep-th-0309258-1-21-7': 'In this paper, the [MATH]-[MATH] plane is chosen.', 'hep-th-0309258-1-21-8': 'Then the configuration in the transverse space is given by [EQUATION]', 'hep-th-0309258-1-21-9': 'Eqs. ([REF]) and ([REF]) compose the background configuration about which we are concerned, and all other elements of matrices [MATH] are set to zero.', 'hep-th-0309258-1-21-10': 'A schematic view of the background configuration is presented in Fig. [REF].', 'hep-th-0309258-1-22-0': 'If we evaluate the classical value of the action for this background, it is zero; [EQUATION]', 'hep-th-0309258-1-22-1': 'From now on, we are going to compute the one-loop correction to this action, that is, to the background, ([REF]) and ([REF]), due to the quantum fluctuations via the path integration of the quadratic action [MATH], and obtain the one-loop effective action [MATH] or the effective potential [MATH] as a function of [MATH], the radius of the circular motion.', 'hep-th-0309258-1-23-0': 'For the justification of one-loop computation or the semi-classical analysis, it should be made clear that [MATH] and [MATH] of Eq. ([REF]) can be regarded as perturbations.', 'hep-th-0309258-1-23-1': 'For this purpose, following [], we rescale the fluctuations and parameters as', 'hep-th-0309258-1-24-0': 'A ^-1/2 A , Y^I ^-1/2 Y^I , C ^-1/2 C , C ^-1/2 C ,', 'hep-th-0309258-1-25-0': 'r r , t ^-1 t .', 'hep-th-0309258-1-26-0': 'Under this rescaling, the action [MATH] in the background ([REF]) and ([REF]) becomes [EQUATION] where [MATH], [MATH] and [MATH] do not have [MATH] dependence.', 'hep-th-0309258-1-26-1': 'Now it is obvious that, in the large [MATH] limit, [MATH] and [MATH] can be treated as perturbations and the one-loop computation gives the sensible result.', 'hep-th-0309258-1-27-0': 'Based on the structure of ([REF]), we now write the quantum fluctuations in the [MATH] block matrix form as follows.', 'hep-th-0309258-1-28-0': 'A = Z_(1)^0 & ^0', 'hep-th-0309258-1-29-0': '^0 & Z_(2)^0 , Y^I = Z_(1)^I & ^I', 'hep-th-0309258-1-30-0': '^I & Z_(2)^I , = _(1) &', 'hep-th-0309258-1-31-0': '^ & _(2) ,', 'hep-th-0309258-1-32-0': 'C = C_(1) & C', 'hep-th-0309258-1-33-0': 'C^ & C_(2) , C = C_(1) & C', 'hep-th-0309258-1-34-0': 'C^& C_(2) .', 'hep-th-0309258-1-35-0': 'Although we denote the block off-diagonal matrices for the ghosts by the same symbols with those of the original ghost matrices, there will be no confusion since [MATH] matrices will never appear in what follows.', 'hep-th-0309258-1-35-1': 'The above form of matrices is convenient, since the block diagonal and block off-diagonal parts decouple from each other in the quadratic action and thus can be taken into account separately at one-loop level; [EQUATION] where [MATH]) implies the action for the block (off-) diagonal fluctuations.', 'hep-th-0309258-1-35-2': 'We note that there is no [MATH] parameter in [MATH] and [MATH] due to the above rescaling ([REF]).', 'hep-th-0309258-1-36-0': '# Stability of fuzzy sphere', 'hep-th-0309258-1-37-0': 'In this section, the path integration of [MATH] is performed.', 'hep-th-0309258-1-37-1': 'The resulting effective action will enable us to check the one-loop stability of the fuzzy sphere configuration, Eqs. ([REF]) and ([REF]).', 'hep-th-0309258-1-37-2': 'In fact, the background encoding the circular motion, Eq. ([REF]), does not contribute to [MATH] basically because of the fact that it is proportional to the identity matrix.', 'hep-th-0309258-1-37-3': 'This leads to the situation that the quantum fluctuations around one fuzzy sphere do not interact with those around another fuzzy sphere, and the effective action is just the sum of that for each fuzzy sphere.', 'hep-th-0309258-1-37-4': 'This can then be stated as [EQUATION] where three Lagrangians, [MATH], [MATH], and [MATH], are those for the bosonic, fermionic, and ghost fluctuations respectively around the [MATH]-th fuzzy sphere.', 'hep-th-0309258-1-37-5': 'We note that, at quadratic level, there is no mixing between these three kinds of fluctuations.', 'hep-th-0309258-1-38-0': '## Bosonic fluctuation', 'hep-th-0309258-1-39-0': 'We first evaluate the path integral of bosonic fluctuations.', 'hep-th-0309258-1-39-1': 'The corresponding Lagrangian is given by [EQUATION] where, as alluded to at the end of section [REF], the quadratic term of [MATH] coming from the gauge fixing term appears because the same term with opposite sign has been used for making the complete square form [MATH].', 'hep-th-0309258-1-40-0': 'In order to perform the actual path integration, it is useful to diagonalize the fluctuation matrices and obtain the mass spectrum.', 'hep-th-0309258-1-40-1': 'By the way, since the diagonalization itself has been already given in [CITATION], we will be brief in its presentation and present only the essential points.', 'hep-th-0309258-1-41-0': 'The starting point is the observation that the mass terms are written in terms of the commutators with [MATH] satisfying Eq. ([REF]), the [MATH] algebra.', 'hep-th-0309258-1-41-1': 'This indicates that we can use the representation theory of [MATH] for the diagonalization.', 'hep-th-0309258-1-41-2': 'We regard an [MATH] matrix as an [MATH]-dimensional reducible representation of [MATH], which decomposes into irreducible spin [MATH] representations with the range of [MATH] from [MATH] to [MATH], that is, [MATH].', 'hep-th-0309258-1-41-3': 'Based on this decomposition, an [MATH] matrix may be expanded as [EQUATION] where the [MATH] matrix [MATH] is the matrix spherical harmonics transforming in the irreducible spin [MATH] representation and [MATH] is the corresponding spherical mode.', 'hep-th-0309258-1-41-4': 'We note that [MATH] satisfies the following reality condition since the fluctuation matrices are Hermitian.', 'hep-th-0309258-1-41-5': '[EQUATION]', 'hep-th-0309258-1-41-6': 'The expansion of matrices in the [MATH] language enables us to use the properties of the [MATH] generators, which are given by [EQUATION] where [MATH].', 'hep-th-0309258-1-41-7': 'For the normalization of the matrix spherical harmonics, we choose [EQUATION]', 'hep-th-0309258-1-41-8': 'Having equipped with the necessary machinery, we now proceed the diagonalization.', 'hep-th-0309258-1-42-0': 'By direct application of Eqs. ([REF]), ([REF]), and ([REF]), the gauge field fluctuation [MATH] and the fluctuations in the [MATH] directions [MATH] are immediately diagonalized.', 'hep-th-0309258-1-42-1': 'The corresponding spherical modes and their masses are [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-1-43-0': 'As for the fluctuations in the [MATH] directions, we should first consider the potential [MATH] and diagonalize it by solving the eigenvalue problem, [EQUATION]', 'hep-th-0309258-1-43-1': 'By defining the combinations of matrices [MATH] and using Eqs. ([REF]) and ([REF]), it turns out that the eigenvalue [MATH] takes the values of [MATH], [MATH], or [MATH].', 'hep-th-0309258-1-43-2': 'Let us now denote the spherical eigenmodes of matrix eigenvectors as [MATH], [MATH], and [MATH] for [MATH], [MATH], and [MATH], respectively.', 'hep-th-0309258-1-43-3': 'The eigenmodes satisfy the reality condition in the form of Eq. ([REF]).', 'hep-th-0309258-1-43-4': 'Then the fluctuation matrices may be expressed with respect to these eigenmodes for each eigenvalue.', 'hep-th-0309258-1-44-0': 'For [MATH], we have the following expressions: [EQUATION] where [MATH], [MATH], and the normalization constant are chosen so that the kinetic term for the mode [MATH] is of the form [MATH].', 'hep-th-0309258-1-44-1': 'Here and in what follows, the summations over [MATH] and [MATH] with specified ranges are implicit.', 'hep-th-0309258-1-45-0': 'For [MATH], we have [EQUATION] where [MATH], [MATH], and the normalization constant are chosen in the same way with the case of [MATH].', 'hep-th-0309258-1-46-0': 'Finally, for [MATH], [EQUATION] with [MATH] and [MATH].', 'hep-th-0309258-1-46-1': 'As pointed out in [CITATION], the modes for [MATH] case correspond to the degrees of freedom for the gauge transformation and are thus unphysical.', 'hep-th-0309258-1-46-2': 'Since the authors of [CITATION] took the physical Weyl gauge, [MATH], and worked in the operator formulation, it was not necessary to consider these unphysical modes seriously.', 'hep-th-0309258-1-46-3': 'However, they should be involved properly in the present context because our gauge choice is the covariant background gauge and we work in the path integral formulation.', 'hep-th-0309258-1-47-0': 'We turn to the remaining potential term in the Lagrangian ([REF]) which is the square of [MATH].', 'hep-th-0309258-1-47-1': 'Interestingly enough, the matrices expanded in terms of the eigenmodes [MATH] and [MATH], Eqs. ([REF]) and ([REF]), do not give any contribution, since we obtain [EQUATION] for [MATH] and [MATH].', 'hep-th-0309258-1-47-2': 'Only the modes corresponding to [MATH] contribute to the potential, which is evaluated as [EQUATION] for each [MATH] and [MATH].', 'hep-th-0309258-1-48-0': 'Having diagonalized the fluctuations [MATH], we get the following list of spherical modes in the [MATH] directions with their masses and the ranges of spin [MATH]: [EQUATION] where [MATH].', 'hep-th-0309258-1-49-0': 'With respect to the spherical modes of Eqs. ([REF]) and ([REF]), the Lagrangian ([REF]) is then written in the diagonalized form as [EQUATION] where the sum over [MATH] with the range [MATH] is understood.', 'hep-th-0309258-1-49-1': 'The Lagrangian is just the sum of various harmonic oscillator Lagrangians, which are non-interacting with each other, and therefore the path integration is now straightforward.', 'hep-th-0309258-1-49-2': 'As a result, what we obtain is [EQUATION]', 'hep-th-0309258-1-50-0': '## Fermionic fluctuation', 'hep-th-0309258-1-51-0': 'We turn to the path integral of fermionic fluctuations.', 'hep-th-0309258-1-51-1': 'The Lagrangian is written as [EQUATION]', 'hep-th-0309258-1-51-2': 'It is convenient for our calculation of fermionic part to introduce the [MATH] formulation since the preserved symmetry in the plane-wave matrix model is [MATH] rather than [MATH].', 'hep-th-0309258-1-51-3': 'In this formulation the [MATH] spinor [MATH] is decomposed as [EQUATION] where [MATH] implies a fundamental [MATH] index and [MATH] is a fundamental [MATH] index.', 'hep-th-0309258-1-51-4': 'According to this decomposition, we may take the expression of [MATH] as [EQUATION]', 'hep-th-0309258-1-51-5': "We also rewrite the [MATH] gamma matrices [MATH]'s in terms of [MATH] and [MATH] ones as follows: [EQUATION] where the [MATH]'s are the standard [MATH] Pauli matrices and six of [MATH] are taken to form a basis of [MATH] anti-symmetric matrices.", 'hep-th-0309258-1-51-6': 'The original [MATH] Clifford algebra is satisfied as long as we take normalizations so that the gamma matrices [MATH] with [MATH] indices satisfy the algebra [EQUATION]', 'hep-th-0309258-1-51-7': 'By introducing the above [MATH] formulation, the Lagrangian for the fermionic fluctuations is rewritten as [EQUATION]', 'hep-th-0309258-1-51-8': 'The diagonalization of the Lagrangian proceeds in the same way as in the previous subsection.', 'hep-th-0309258-1-51-9': 'In the present case, it is achieved by solving the following eigenvalue problem: [EQUATION]', 'hep-th-0309258-1-51-10': 'We first expand the [MATH] fermionic matrix [MATH] in terms of the matrix spherical harmonics as [EQUATION]', 'hep-th-0309258-1-51-11': 'If we plug this expansion into the above eigenvalue equation and use the [MATH] algebra Eq. ([REF]), we see that the eigenvalues are [MATH] and [MATH] [CITATION].', 'hep-th-0309258-1-51-12': 'Let us now introduce the fermionic spherical eigenmodes [MATH] and [MATH] corresponding to [MATH] and [MATH] respectively.', 'hep-th-0309258-1-51-13': 'As for the spinorial structure, [MATH]) carries the (anti) fundamental [MATH] index.', 'hep-th-0309258-1-51-14': 'We note that, from now on, we suppress the [MATH] indices.', 'hep-th-0309258-1-52-0': 'Then, as the eigenstate for [MATH], the matrix [MATH] has the expansion in terms of the eigenmode [MATH] as [EQUATION] where [MATH], [MATH], and the subscripts [MATH] denote the [MATH] indices measured by [MATH].', 'hep-th-0309258-1-53-0': 'On the other hand, for [MATH], we have [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-1-54-0': 'By using the mode-expansions, Eqs. ([REF]) and ([REF]), and the [MATH] algebra ([REF]), the fermionic Lagrangian ([REF]) becomes [EQUATION] where it should be understood that there is the summation over [MATH] with the range [MATH].', 'hep-th-0309258-1-54-1': 'Now, the path integration for this Lagrangian may be evaluated immediately, and gives [EQUATION]', 'hep-th-0309258-1-55-0': '## Ghost fluctuation', 'hep-th-0309258-1-56-0': 'As the final part of the diagonal fluctuations, we consider the Lagrangian for the ghost fluctuations, which is given by [EQUATION]', 'hep-th-0309258-1-56-1': 'By using the [MATH] algebra ([REF]) and the following expansions in terms of the matrix spherical harmonics [EQUATION] with [MATH] and [MATH], we may rewrite the above Lagrangian as [EQUATION] where the sum over [MATH] is implicit.', 'hep-th-0309258-1-56-2': 'Then the result of the path integration for this Lagrangian is [EQUATION]', 'hep-th-0309258-1-57-0': '## One-loop stability', 'hep-th-0309258-1-58-0': 'In the previous subsections, we have evaluated the path integrals for the block diagonal fluctuations of the action [MATH], ([REF]).', 'hep-th-0309258-1-58-1': 'Thus, we may now consider the effective action.', 'hep-th-0309258-1-58-2': 'As can be inferred from Eq. ([REF]), it is enough to consider only the effective action of a given fuzzy sphere.', 'hep-th-0309258-1-58-3': 'If we let [MATH] be the effective action for the [MATH]-th fuzzy sphere, then it is given by [EQUATION]', 'hep-th-0309258-1-58-4': 'The right hand side may be viewed just as the product of determinants of non-interacting quantum mechanical simple harmonic oscillators with various frequencies.', 'hep-th-0309258-1-58-5': 'In fact, as we will see in the next section, this is also the case for the effective action describing the interaction between two fuzzy spheres.', 'hep-th-0309258-1-58-6': 'Thus, it is worthwhile to consider a generic situation, which is useful both in the present and the next section.', 'hep-th-0309258-1-59-0': 'Let us then consider a situation, [EQUATION] where [MATH] is included for the later usage.', 'hep-th-0309258-1-59-1': 'The formal expression of the effective action is read as [EQUATION]', 'hep-th-0309258-1-59-2': 'By using the relation [MATH] for a given matrix [MATH], we may present a prototype calculation of single determinant as [EQUATION] where the momentum integration has appeared by inserting the momentum space identity, [MATH], between bra and ket vectors for time, and the final result has been derived by using the formula [EQUATION]', 'hep-th-0309258-1-59-3': 'If we apply the result of this single determinant calculation to Eq. ([REF]), then the effective action is finally obtained as [EQUATION]', 'hep-th-0309258-1-59-4': 'We now return to the present case.', 'hep-th-0309258-1-59-5': 'First of all, we observe that the ghost part ([REF]) eliminates two determinant factors of the bosonic result ([REF]) which are actually the contributions from the unphysical modes.', 'hep-th-0309258-1-59-6': 'Thus, what we get from the bosonic and the ghost parts is only the determinant factors from the physical bosonic modes.', 'hep-th-0309258-1-59-7': 'By using the generic expression, Eq. ([REF]), it turns out that their contributions to the effective action are [EQUATION]', 'hep-th-0309258-1-59-8': 'On the other hand, the contribution from the fermionic part ([REF]) is obtained as [EQUATION]', 'hep-th-0309258-1-59-9': 'Therefore, there is no net contribution to the effective action and we can conclude that the one loop effective action obtained after integrating out each of the block diagonal fluctuations vanishes; [EQUATION]', 'hep-th-0309258-1-59-10': 'This means that each membrane fuzzy sphere has quantum stability at least at one-loop level.', 'hep-th-0309258-1-59-11': 'We note that, if we take [MATH] in the above two contributions, the result of the previous path integral computation [CITATION] is recovered.', 'hep-th-0309258-1-60-0': '# Interaction between fuzzy spheres', 'hep-th-0309258-1-61-0': 'We turn to the action [MATH] in Eq. ([REF]) for the block off-diagonal fluctuations and compute the one-loop effective potential describing the interaction between two fuzzy spheres.', 'hep-th-0309258-1-61-1': 'The action is given by [EQUATION] where [MATH], [MATH] and [MATH] are the Lagrangians for the off-diagonal bosonic, fermionic and ghost fluctuations of Eq. ([REF]) respectively and their explicit expressions will be presented in due course.', 'hep-th-0309258-1-61-2': 'Since, as in the previous section, there is no mixing between different kinds of fluctuations, each Lagrangian can be considered independently.', 'hep-th-0309258-1-62-0': '## Bosonic fluctuation', 'hep-th-0309258-1-63-0': 'Let us first consider the bosonic Lagrangian and evaluate its path integral.', 'hep-th-0309258-1-63-1': 'The Lagrangian is [EQUATION]', 'hep-th-0309258-1-63-2': 'Here, adopting the notation of [*], we have defined [EQUATION] where [MATH] is the [MATH] matrix which is a block at [MATH]-th row and [MATH]-th column in the blocked form of a given matrix [MATH].', 'hep-th-0309258-1-63-3': 'In the present case, [MATH] and [MATH] take values of [MATH] and [MATH].', 'hep-th-0309258-1-63-4': 'For example, if we look at the [MATH] block matrix form of the gauge field fluctuation [MATH] in Eq. ([REF]), then [MATH], [MATH], and [MATH].', 'hep-th-0309258-1-64-0': 'The matrix fields, [MATH], [MATH], and [MATH] are coupled with each other through the circular motion background.', 'hep-th-0309258-1-64-1': 'Since the time dependent trigonometric functions may make the formulation annoying, we consider the newly defined matrix variables as [EQUATION] where [MATH] may be interpreted as the fluctuation tangential to the circular motion at time [MATH] and [MATH] as the normal fluctuation.', 'hep-th-0309258-1-64-2': 'In terms of these fluctuations, the terms in the Lagrangian ([REF]), which are dependent on [MATH] and [MATH], are rewritten as [EQUATION] where we no longer see the explicit time dependent classical functions.', 'hep-th-0309258-1-65-0': 'We are now in a position to consider the diagonalization of the Lagrangian [MATH].', 'hep-th-0309258-1-65-1': 'We note that, compared to the case in the previous section, we are in a somewhat different situation.', 'hep-th-0309258-1-65-2': 'The fluctuation matrices are [MATH] or [MATH] ones, while those in the last section are rectangular [MATH] matrices.', 'hep-th-0309258-1-65-3': 'This means that, when we regard an [MATH] block off-diagonal matrix as an [MATH]-dimensional reducible representation of [MATH], it has the decomposition into irreducible spin [MATH] representations with the range [MATH], that is, [MATH], and may be expanded as [EQUATION] where [MATH] is the [MATH] matrix spherical harmonics transforming in the irreducible spin [MATH] representation and [MATH] is the corresponding spherical mode.', 'hep-th-0309258-1-65-4': 'The basic operation between [MATH] generators [MATH] and [MATH] is given not by the commutator but by the [MATH] operator ([REF]).', 'hep-th-0309258-1-65-5': 'Thus the algebraic properties for the present situation are given by Eq. ([REF]) where the commutator is replaced by the [MATH] operator.', 'hep-th-0309258-1-66-0': 'Having the expansion and algebraic properties, the diagonalization proceeds in exactly the same way as in the previous section.', 'hep-th-0309258-1-66-1': 'Hence, by noting that one may find a detailed procedure in [CITATION], we will present only the results of diagonalization for the Lagrangian.', 'hep-th-0309258-1-66-2': 'We observe that, because of the background for the circular motion in [MATH]-[MATH] plane, the [MATH] symmetry is broken to [MATH], while the [MATH] symmetry remains intact.', 'hep-th-0309258-1-66-3': 'This fact naturally leads us to break the bosonic Lagrangian ([REF]) into three parts as follows: [EQUATION] where [MATH] is the Lagrangian for [MATH], [MATH] is for [MATH] with [MATH], and [MATH] represents the [MATH] rotational part described by [MATH], [MATH], and the gauge fluctuation [MATH].', 'hep-th-0309258-1-67-0': 'We first consider [MATH] and its path integration.', 'hep-th-0309258-1-67-1': 'Its diagonalized form is obtained by [EQUATION] where the sum of [MATH] over the range [MATH] is implicit and the spherical modes [MATH] are the degrees of freedom for the gauge transformation, the block off-diagonal counterpart of [MATH], ([REF]), in the previous section.', 'hep-th-0309258-1-67-2': '[MATH] and [MATH] are the block off-diagonal counterparts of [MATH] and [MATH], respectively.', 'hep-th-0309258-1-67-3': 'The path integral of this Lagrangian is straightforward and results in [EQUATION]', 'hep-th-0309258-1-67-4': 'As for [MATH], we have as its diagonalized form [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-1-67-5': 'Its path integration leads us to have [EQUATION]', 'hep-th-0309258-1-67-6': 'For the rotational part, the diagonalized Lagrangian is obtained as [EQUATION] where [MATH].', 'hep-th-0309258-1-67-7': 'Since all the coupling coefficients between the modes are time-independent constants, the path integral of this Lagrangian is readily evaluated and gives [EQUATION]', 'hep-th-0309258-1-68-0': '## Fermionic fluctuation', 'hep-th-0309258-1-69-0': 'The fermionic Lagrangian of the block off-diagonal action [MATH], ([REF]), is [EQUATION]', 'hep-th-0309258-1-69-1': 'As was done in Eq. ([REF]), we decompose the fermion [MATH] into [MATH] and [MATH] according to [MATH].', 'hep-th-0309258-1-69-2': 'Then the expression of [MATH] may be taken as [EQUATION] where [MATH].', 'hep-th-0309258-1-69-3': 'It should be noted that, contrary to the block diagonal fermionic matrix [MATH] of ([REF]), [MATH] is the two copy of the [MATH] representation [MATH] as one may see from Eq. ([REF]).', 'hep-th-0309258-1-69-4': 'This means that [MATH] and [MATH] should be treated as independent spinors not related in any way.', 'hep-th-0309258-1-69-5': 'By plugging the decomposition ([REF]) into the Lagrangian ([REF]), we have [EQUATION]', 'hep-th-0309258-1-69-6': 'The Lagrangian has explicit time dependence due to the presence of the circular motion background.', 'hep-th-0309258-1-69-7': 'In order to hide it, we take the fermionic field [MATH] as [EQUATION] where we have introduced a new fermionic field [MATH] which is in the [MATH] of [MATH].', 'hep-th-0309258-1-69-8': 'Then, by using the following identities, [EQUATION] which are proved via the Clifford algebra ([REF]), we may show that the Lagrangian ([REF]) becomes [EQUATION]', 'hep-th-0309258-1-69-9': 'The explicit time dependent classical functions disappear and the term containing [MATH] appears, which originates from the kinetic term of [MATH] in ([REF]).', 'hep-th-0309258-1-70-0': 'With the above Lagrangian ([REF]), the diagonalization proceeds in the same manner with that for the block diagonal fermionic Lagrangian ([REF]), except for some differences pointed out in the previous subsection.', 'hep-th-0309258-1-70-1': 'In the expansion of [MATH] and [MATH] indices are suppressed.)', 'hep-th-0309258-1-70-2': 'in terms of the matrix spherical harmonics like ([REF]), let us denote their spherical modes as [MATH] and [MATH] respectively.', 'hep-th-0309258-1-70-3': 'Then the diagonalization results in [MATH] and [MATH].', 'hep-th-0309258-1-70-4': '[MATH] and [MATH] have the same mass of [MATH] with [MATH].', 'hep-th-0309258-1-70-5': 'For the modes [MATH] and [MATH], their mass is [MATH] with [MATH].', 'hep-th-0309258-1-70-6': 'All the modes have the same range of [MATH] as [MATH].', 'hep-th-0309258-1-70-7': 'With respect to these diagonalized spherical modes, the Lagrangian ([REF]) is written as [EQUATION] where [MATH] and the [MATH] indices are suppressed.', 'hep-th-0309258-1-71-0': 'The product [MATH] measures the [MATH] chirality in the [MATH]-[MATH] plane where the circular motion takes place.', 'hep-th-0309258-1-71-1': 'Since [MATH], its eigenvalues are [MATH].', 'hep-th-0309258-1-71-2': 'Each spherical mode may split into modes having definite [MATH] eigenvalues as follows: [EQUATION] where the modes on the right hand sides satisfy [EQUATION]', 'hep-th-0309258-1-71-3': 'One may see that the Lagrangian ([REF]) composed of two independent parts.', 'hep-th-0309258-1-71-4': 'One is for [MATH] and [MATH], and the other one for [MATH] and [MATH].', 'hep-th-0309258-1-71-5': 'If we first consider the part for [MATH] and [MATH], then, according to the above splitting of modes, we have [EQUATION] where again [MATH].', 'hep-th-0309258-1-71-6': 'The path integration of this part is straightforward and gives [EQUATION]', 'hep-th-0309258-1-71-7': 'The other part for [MATH] and [MATH] is obtained as [EQUATION] where the sum over [MATH] for [MATH] is implicit.', 'hep-th-0309258-1-71-8': 'The path integration of this part results in [EQUATION]', 'hep-th-0309258-1-72-0': '## Ghost fluctuation', 'hep-th-0309258-1-73-0': 'Finally, we consider the path integration for the ghost part of the action [MATH] ([REF]).', 'hep-th-0309258-1-73-1': 'The block off-diagonal Lagrangian for the ghosts is [EQUATION]', 'hep-th-0309258-1-73-2': 'The diagonalization may be carried out by using the same logic in the previous subsections.', 'hep-th-0309258-1-73-3': 'We expand the ghost fields [MATH] and [MATH] in terms of the matrix spherical harmonics according to ([REF]), and denote their spherical modes as [MATH] and [MATH] respectively.', 'hep-th-0309258-1-73-4': 'Then the diagonalized Lagrangian is obtained as [EQUATION] where the sum over [MATH] for the range [MATH] is implicit.', 'hep-th-0309258-1-74-0': 'The path integral for the above diagonalized Lagrangian is then immediately evaluated as follows: [EQUATION]', 'hep-th-0309258-1-75-0': '## Effective potential', 'hep-th-0309258-1-76-0': 'Having evaluated the path integral for each part of the block off-diagonal action [MATH] ([REF]), the one-loop effective action, [MATH], which describes the interaction between two membrane fuzzy spheres with the classical configuration ([REF]) and ([REF]), is now given by [EQUATION] where the subscripts on the right hand side denote the bosonic, the fermionic, and the ghost contributions.', 'hep-th-0309258-1-76-1': 'We see that the ghost contribution, Eq. ([REF]), eliminate those of unphysical gauge degrees of freedom present in bosonic contributions, Eqs. ([REF]) and ([REF]).', 'hep-th-0309258-1-76-2': 'Thus only the physical degrees of freedom contribute to the effective action, as it should be.', 'hep-th-0309258-1-77-0': 'The explicit expression of the effective action is obtained by consulting the generic result presented from Eq. ([REF]) to ([REF]).', 'hep-th-0309258-1-77-1': 'The effective potential, [MATH], about which we are concerned in this subsection, is then given by [MATH].', 'hep-th-0309258-1-77-2': 'In expressing the effective potential, it is convenient to write [MATH] as [EQUATION] where [MATH]) is the contribution of the physical bosonic (fermionic) degrees of freedom to the effective potential.', 'hep-th-0309258-1-77-3': 'The expression that we obtain for [MATH] is then [EQUATION] while, for the fermionic contribution [MATH], we obtain [EQUATION]', 'hep-th-0309258-1-77-4': 'The above expressions show that [MATH] and [MATH] have the same structure except for the ranges of [MATH].', 'hep-th-0309258-1-77-5': 'This leads us to expect a great amount of cancellation.', 'hep-th-0309258-1-77-6': 'Indeed, what we have found is that they are exactly the same.', 'hep-th-0309258-1-77-7': 'One way to see the cancellation is to adjust all the ranges of the summation parameter [MATH] to the range [MATH].', 'hep-th-0309258-1-77-8': 'Therefore, the one-loop effective potential [MATH], ([REF]), as a function of the distance [MATH] between two fuzzy spheres is just flat potential; [EQUATION]', 'hep-th-0309258-1-78-0': '# Conclusion and discussion', 'hep-th-0309258-1-79-0': 'We have studied the one-loop quantum corrections to a classical background of the plane-wave matrix model in the framework of the path integration.', 'hep-th-0309258-1-79-1': 'The background is composed of two supersymmetric membrane fuzzy spheres, ([REF]) and ([REF]).', 'hep-th-0309258-1-79-2': 'One fuzzy sphere is located at the origin in the [MATH] symmetric space and the other one rotates around it with the distance [MATH].', 'hep-th-0309258-1-80-0': 'Firstly, the quantum stability of each fuzzy sphere has been shown.', 'hep-th-0309258-1-80-1': 'In fact, the stability check already has been done in the operator [CITATION] as well as in the path integral [CITATION] formulation.', 'hep-th-0309258-1-80-2': 'However, while the size [MATH] of the fuzzy sphere has been taken arbitrary in the operator formulation, it has been restricted to the minimal one, that is [MATH], in the path integral formulation.', 'hep-th-0309258-1-80-3': 'Since the fuzzy sphere in this paper has an arbitrary size, our stability check can be regarded as the full generalization of the previous path integral result.', 'hep-th-0309258-1-81-0': 'Secondly, the one-loop effective potential describing the interaction between two fuzzy spheres has been calculated as a function of the distance [MATH].', 'hep-th-0309258-1-81-1': 'Remarkable result is that the effective potential [MATH] is flat and thus the fuzzy spheres do not feel any force.', 'hep-th-0309258-1-81-2': 'Since the circular motion is taken as the background, [MATH] is constant.', 'hep-th-0309258-1-81-3': 'However, if we slightly deform the circular motion to the elliptic one, [MATH] can be made to have time dependence.', 'hep-th-0309258-1-81-4': 'We can also make the time variation of it, [MATH], arbitrarily small by controlling the degree of deformation.', 'hep-th-0309258-1-81-5': 'In this case, the effective action [MATH] may be written as [EQUATION] where the kinetic term for [MATH] is the value of the classical action [MATH] with the rescaling ([REF]), [MATH] is the would-be one-loop contribution to the effective action, and the term of order [MATH] implies the higher loop corrections.', 'hep-th-0309258-1-81-6': 'One may now argue that the flat effective potential is very natural since [MATH] in our fuzzy sphere configuration.', 'hep-th-0309258-1-81-7': 'However, this makes our result puzzling.', 'hep-th-0309258-1-81-8': 'Let us recall the effective action for graviton-graviton scattering in the flat space matrix model [CITATION].', 'hep-th-0309258-1-81-9': 'If the relative velocity between the gravitons is taken to be zero, the effective potential surely vanishes.', 'hep-th-0309258-1-81-10': 'This indicates that the BPS configuration is recovered.', 'hep-th-0309258-1-81-11': 'In our case, however, even if [MATH], there seems to be no chance to have BPS configuration.', 'hep-th-0309258-1-81-12': 'Although we have not investigated the supersymmetric structure of our background, it is hard to expect it, because the time dependent circular motion contained in the background gives anyway the non-vanishing background energy [MATH].', 'hep-th-0309258-1-81-13': 'We note that the situation in this paper is close to that considered in [CITATION].', 'hep-th-0309258-1-82-0': 'In view of the gauge/gravity duality, it is interesting to study the same situation in the supergravity side.', 'hep-th-0309258-1-82-1': 'Since, at the present status, it is unclear whether our result is just accident or has some reason, we expect that the supergravity side analysis is helpful and provides clearer understanding about the flatness of the effective potential.', 'hep-th-0309258-1-83-0': 'The calculation in this paper has been carried out in the Minkowskian time signature.', 'hep-th-0309258-1-83-1': 'The basic reason why we have not taken the Wick rotation for some convenience in the actual calculation is the periodic nature of the circular motion background ([REF]).', 'hep-th-0309258-1-83-2': 'The naive change of the Minkowskian time to the Euclidean one in the process of calculation breaks the periodicity of the background and the reality of the action.', 'hep-th-0309258-1-83-3': 'If one wants to study the time dependent background with the Euclidean time signature, he or she should begin with the Euclidean action at the first setup.', 'hep-th-0309258-1-83-4': 'In the Euclidean case, the time dependent solutions to the equations of motion are given by hyperbolic functions, which give open paths not periodic ones.', 'hep-th-0309258-1-83-5': 'In the sense that there is no classical background solution corresponding to open path in the Minkowskian time plane-wave matrix model, it would be interesting to consider the model in the Euclidean time for studying such a background.', 'hep-th-0309258-1-84-0': 'In the situation where the one-loop effective potential is flat, it is a natural step to calculate the higher loop corrections to the effective potential for the purpose of knowing the form of interaction between fuzzy spheres.', 'hep-th-0309258-1-84-1': 'We hope to return to this issue in the near future.'}

{'hep-th-0309258-2-0-0': 'In the plane-wave matrix model, the background configuration of two membrane fuzzy spheres, one of which rotates around the other one in the [MATH] symmetric space, is allowed as a classical solution.', 'hep-th-0309258-2-0-1': 'We study the one-loop quantum corrections to this background in the path integral formulation.', 'hep-th-0309258-2-0-2': 'Firstly, we show that each fuzzy sphere is stable under the quantum correction.', 'hep-th-0309258-2-0-3': 'Secondly, the effective potential describing the interaction between fuzzy spheres is obtained as a function of [MATH], which is the distance between two fuzzy spheres.', 'hep-th-0309258-2-0-4': 'It is shown that the effective potential is flat and hence the fuzzy spheres do not feel any force.', 'hep-th-0309258-2-0-5': 'The possibility on the existence of flat directions is discussed.', 'hep-th-0309258-2-1-0': '[5mm] Keywords : pp-wave, Matrix model, Fuzzy sphere', 'hep-th-0309258-2-2-0': 'PACS numbers : 11.25.-w, 11.27.', 'hep-th-0309258-2-2-1': '+d, 12.60.', 'hep-th-0309258-2-2-2': 'Jv', 'hep-th-0309258-2-3-0': '6.6mm', 'hep-th-0309258-2-4-0': '# Introduction', 'hep-th-0309258-2-5-0': 'The plane-wave matrix model [CITATION] is a microscopic description of the discrete light cone quantized (DLCQ) M-theory in the eleven-dimensional [MATH]-wave or plane-wave background.', 'hep-th-0309258-2-5-1': 'The eleven-dimensional plane-wave [CITATION] is maximally supersymmetric and the limiting case of the eleven-dimensional [MATH] type geometries [CITATION].', 'hep-th-0309258-2-5-2': 'Its explicit form is given by [EQUATION] where [MATH].', 'hep-th-0309258-2-5-3': 'Due to the effect of the [MATH] component of the metric and the presence of the four-form field strength, the plane-wave matrix model has some [MATH] dependent terms, which make the difference between the usual flat space matrix model and the plane-wave one.', 'hep-th-0309258-2-6-0': 'The presence of the [MATH] dependent terms makes the plane-wave matrix model have some peculiar properties.', 'hep-th-0309258-2-6-1': 'One of them is that there are various vacuum structures classified by the [MATH] algebra [CITATION].', 'hep-th-0309258-2-6-2': 'The crucial ingredient for vacua is the membrane fuzzy sphere.', 'hep-th-0309258-2-6-3': 'It preserves the full 16 supersymmetries of the plane-wave matrix model and exists even at finite [MATH] which is the size of matrix.', 'hep-th-0309258-2-7-0': 'After the plane-wave matrix model was proposed and its basic aspects were uncovered, there have been lots of investigations in various directions.', 'hep-th-0309258-2-7-1': 'The structure of vacua has been studied in more detail especially related to the protected multiplet [CITATION].', 'hep-th-0309258-2-7-2': 'The possible BPS objects contained in the plane-wave matrix model have been searched [CITATION].', 'hep-th-0309258-2-7-3': 'The algebraic and structural study of the model itself and the various BPS objects present in it has been performed [CITATION].', 'hep-th-0309258-2-7-4': 'Based on the fact that the low energy description of the M theory is the eleven dimensional supergravity, there also have been supergravity side analysis [CITATION].', 'hep-th-0309258-2-8-0': 'If the M theory is compactified on a circle, then we have ten-dimensional Type IIA string theory.', 'hep-th-0309258-2-8-1': 'Under the circle compactification, the [MATH]-wave geometry ([REF]) becomes the IIA [MATH]-wave background which is not maximally supersymmetric and has 24 supersymmetries [CITATION].', 'hep-th-0309258-2-8-2': 'For the purpose of understanding the plane-wave matrix model as well as the string theory itself, the IIA string theory in the [MATH]-wave background has been also extensively studied, in parallel with the progress in the study of the plane-wave matrix model [CITATION].', 'hep-th-0309258-2-9-0': 'However, despite of quite amount of progress in the study of the plane-wave matrix model, there has been lack of the investigation about the dynamical aspects.', 'hep-th-0309258-2-9-1': 'However, see for example [CITATION].', 'hep-th-0309258-2-9-2': 'In fact, the present status of the plane-wave matrix model enables us to study the dynamics of the model.', 'hep-th-0309258-2-9-3': 'In this paper, we consider the basic objects of the plane-wave matrix model and study their interaction.', 'hep-th-0309258-2-10-0': 'For the interacting objects, we take two membrane fuzzy spheres, both of which are supersymmetric.', 'hep-th-0309258-2-10-1': 'In the [MATH] symmetric subspace which one may see in the [MATH]-wave background ([REF]), two fuzzy spheres are taken to be at the origin.', 'hep-th-0309258-2-10-2': 'In the [MATH] symmetric space, one fuzzy sphere is located at the origin, while the other fuzzy sphere is taken to rotate around the origin with a fixed distance.', 'hep-th-0309258-2-10-3': 'It should be noted that this configuration is allowed as a classical solution of the equations of motion and furthermore the rotating fuzzy sphere itself is supersymmetric.', 'hep-th-0309258-2-10-4': 'We will evaluate the one-loop corrections to the configuration and obtain the effective potential.', 'hep-th-0309258-2-10-5': 'As we will see, the effective potential is flat.', 'hep-th-0309258-2-10-6': 'This implies that the whole configuration of fuzzy spheres is also supersymmetric.', 'hep-th-0309258-2-10-7': 'One may argue that the flat potential is natural since each fuzzy sphere configuration is supersymmetric.', 'hep-th-0309258-2-10-8': 'However, the situation is unconventional from the viewpoint of the flat space matrix model [CITATION] and indicates one of intriguing properties of the plane-wave matrix model.', 'hep-th-0309258-2-10-9': 'Moreover, the flat potential shows us the possibility that the plane-wave matrix model has the flat directions which have not been observed in it.', 'hep-th-0309258-2-11-0': 'The organization of this paper is as follows.', 'hep-th-0309258-2-11-1': 'In the next section, we give the action of the plane-wave matrix model and consider its classical solutions focused on our concern.', 'hep-th-0309258-2-11-2': 'The expansion of the action around a given arbitrary background is given in section [REF].', 'hep-th-0309258-2-11-3': 'In section [REF], we set up the background configuration and consider the fluctuations around it.', 'hep-th-0309258-2-11-4': 'In section [REF], the one-loop stability of each fuzzy sphere is checked for arbitrary size.', 'hep-th-0309258-2-11-5': 'In section [REF], we evaluate the path integration of fluctuations responsible for the interaction between fuzzy spheres.', 'hep-th-0309258-2-11-6': 'It will be shown that the one-loop effective potential is flat.', 'hep-th-0309258-2-11-7': 'Thus, the fuzzy spheres do not feel any force.', 'hep-th-0309258-2-11-8': 'Finally, conclusion and discussion will be given in section [REF].', 'hep-th-0309258-2-11-9': 'We discuss the possibility on the existence of flat directions.', 'hep-th-0309258-2-12-0': '# Plane-wave matrix model and classical solutions', 'hep-th-0309258-2-13-0': 'The plane-wave matrix model is basically composed of two parts.', 'hep-th-0309258-2-13-1': 'One part is the usual matrix model based on eleven-dimensional flat space-time, that is, the flat space matrix model, and another is a set of terms reflecting the structure of the maximally supersymmetric eleven dimensional plane-wave background, Eq. ([REF]).', 'hep-th-0309258-2-13-2': 'Its action is [EQUATION] where each part of the action on the right hand side is given by [EQUATION]', 'hep-th-0309258-2-13-3': 'Here, [MATH] is the radius of circle compactification along [MATH] and [MATH] is the covariant derivative with the gauge field [MATH], [EQUATION]', 'hep-th-0309258-2-13-4': 'For dealing with the problem in this paper, it is convenient to rescale the gauge field and parameters as [EQUATION]', 'hep-th-0309258-2-13-5': 'With this rescaling, the radius parameter [MATH] disappears and the actions in Eq. ([REF]) become [EQUATION]', 'hep-th-0309258-2-13-6': 'The possible backgrounds allowed by the plane-wave matrix model are the classical solutions of the equations of motion for the matrix fields.', 'hep-th-0309258-2-13-7': 'Since the background that we are concerned about is purely bosonic, we concentrate on solutions of the bosonic fields [MATH].', 'hep-th-0309258-2-13-8': 'We would like to note that we will not consider all possible solutions but only those relevant to our interest for the fuzzy sphere interaction.', 'hep-th-0309258-2-13-9': 'Then, from the rescaled action, ([REF]), the bosonic equations of motion are derived as [EQUATION] where the over dot implies the time derivative [MATH].', 'hep-th-0309258-2-14-0': 'Except for the trivial [MATH] solution, the simplest one is the simple harmonic oscillator solution; [EQUATION] where [MATH] and [MATH] are the amplitudes and phases of oscillations respectively, and [MATH] is the [MATH] unit matrix.', 'hep-th-0309258-2-14-1': 'This oscillatory solution is special to the plane-wave matrix model due to the presence of mass terms for [MATH].', 'hep-th-0309258-2-14-2': 'It should be noted that, because of the mass terms, the configuration corresponding to the time dependent straight line motion, say [MATH] with non-zero constants [MATH] and [MATH], is not possible as a solution of ([REF]), that is, a classical background of plane-wave matrix model, contrary to the case of the flat space matrix model.', 'hep-th-0309258-2-14-3': 'As the generalization of the oscillatory solution, Eq. ([REF]), we get the solution of the form of diagonal matrix with each diagonal element having independent amplitude and phase.', 'hep-th-0309258-2-15-0': 'As for the non-trivial constant matrix solution, Eq. ([REF]) allows the following membrane fuzzy sphere or giant graviton solution: [EQUATION] where [MATH] satisfies the [MATH] algebra, [EQUATION]', 'hep-th-0309258-2-15-1': 'The reason why this solution is possible is basically because of the fact that the matrix field [MATH] feels an extra force due to the Myers interaction which may stabilize the oscillatory force.', 'hep-th-0309258-2-15-2': 'The fuzzy sphere solution [MATH] preserves the full 16 dynamical supersymmetries of the plane-wave and hence is 1/2-BPS object.', 'hep-th-0309258-2-15-3': 'We note that actually there is another fuzzy sphere solution of the form [MATH].', 'hep-th-0309258-2-15-4': 'However, it has been shown that such solution does not have quantum stability and is thus non-BPS object [CITATION].', 'hep-th-0309258-2-16-0': '# Matrix model expansion around general background', 'hep-th-0309258-2-17-0': 'In this section, the plane-wave matrix model is expanded around the general bosonic background, which is supposed to satisfy the classical equations of motion, Eq. ([REF]).', 'hep-th-0309258-2-18-0': 'We first split the matrix quantities into as follows: [EQUATION] where [MATH] and [MATH] are the classical background fields while [MATH] and [MATH] are the quantum fluctuations around them.', 'hep-th-0309258-2-18-1': 'The fermionic background [MATH] is taken to vanish from now on, since we will only consider the purely bosonic background.', 'hep-th-0309258-2-18-2': 'The quantum fluctuations are the fields subject to the path integration.', 'hep-th-0309258-2-18-3': 'In taking into account the quantum fluctuations, we should recall that the matrix model itself is a gauge theory.', 'hep-th-0309258-2-18-4': 'This implies that the gauge fixing condition should be specified before proceed further.', 'hep-th-0309258-2-18-5': 'In this paper, we take the background field gauge which is usually chosen in the matrix model calculation as [EQUATION]', 'hep-th-0309258-2-18-6': 'Then the corresponding gauge-fixing [MATH] and Faddeev-Popov ghost [MATH] terms are given by [EQUATION]', 'hep-th-0309258-2-18-7': 'Now by inserting the decomposition of the matrix fields ([REF]) into Eqs. ([REF]) and ([REF]), we get the gauge fixed plane-wave action [MATH] expanded around the background.', 'hep-th-0309258-2-18-8': 'The resulting acting is read as [EQUATION] where [MATH] represents the action of order [MATH] with respect to the quantum fluctuations and, for each [MATH], its expression is [EQUATION]', 'hep-th-0309258-2-18-9': 'Some comments are in order for the background gauge choice, Eq. ([REF]).', 'hep-th-0309258-2-18-10': 'One advantage of this gauge choice is that the quadratic part of the action in terms of fluctuations, that is, the quadratic action, is simplified.', 'hep-th-0309258-2-18-11': 'In more detail, there appears the term [MATH] in the expansion of the potential [MATH], which is canceled exactly by the same term with the opposite sign coming from the gauge fixing term of Eq. ([REF]) and hence absent in [MATH] of Eq. ([REF]).', 'hep-th-0309258-2-18-12': 'This cancellation has given some benefits in the actual flat space matrix model calculation.', 'hep-th-0309258-2-18-13': 'This is also the case in the present plane-wave matrix model, except however for the potential of [MATH].', 'hep-th-0309258-2-18-14': 'As we will see later, [MATH] is responsible for completing the [MATH] potential into the nice square form.', 'hep-th-0309258-2-18-15': 'Thus the same term with the opposite sign from the gauge fixing term remains in the quadratic action.', 'hep-th-0309258-2-18-16': 'At later stage, the presence of this term will have an important implication in taking into account of the unphysical gauge degrees of freedom which are eventually eliminated by those of ghosts.', 'hep-th-0309258-2-19-0': '# Fuzzy sphere configuration and fluctuations', 'hep-th-0309258-2-20-0': 'We now set up the background configuration for the membrane fuzzy spheres.', 'hep-th-0309258-2-20-1': 'Since we will study the interaction of two fuzzy spheres, the matrices representing the background have the [MATH] block diagonal form as [EQUATION] where [MATH] with [MATH] are [MATH] matrices.', 'hep-th-0309258-2-20-2': 'If we take [MATH] as [MATH] matrices, then [MATH].', 'hep-th-0309258-2-21-0': 'The two fuzzy spheres are taken to be static in the space where they span, and hence represented by the classical solution, ([REF]); [EQUATION] where, for each [MATH], [MATH] is in the [MATH]-dimensional irreducible representation of [MATH] and satisfies the [MATH] algebra, Eq. ([REF]).', 'hep-th-0309258-2-21-1': 'In the [MATH] symmetric transverse space, the fuzzy spheres are regarded as point objects, of course, in a sense of ignoring the matrix nature.', 'hep-th-0309258-2-21-2': 'We first let the second fuzzy sphere given by the background of [MATH] be at the origin in the transverse space and stay there.', 'hep-th-0309258-2-21-3': 'As for the first fuzzy sphere, it is made to move around the second sphere in the form of circular motion with the radius [MATH].', 'hep-th-0309258-2-21-4': 'Obviously, this configuration is one of the classical solutions of the equations of motion as one can see from Eq. ([REF]).', 'hep-th-0309258-2-21-5': 'Recalling that the transverse space is [MATH] symmetric, all the possible choices of two-dimensional sub-plane where the circular motion takes place are equivalent.', 'hep-th-0309258-2-21-6': 'Thus, without loss of generality, we can take a certain plane for the circular motion.', 'hep-th-0309258-2-21-7': 'In this paper, the [MATH]-[MATH] plane is chosen.', 'hep-th-0309258-2-21-8': 'Then the configuration in the transverse space is given by [EQUATION]', 'hep-th-0309258-2-21-9': 'Eqs. ([REF]) and ([REF]) compose the background configuration about which we are concerned, and all other elements of matrices [MATH] are set to zero.', 'hep-th-0309258-2-21-10': 'We would like to note that not only the fuzzy sphere at the origin given by [MATH] but also the rotating one, [MATH], is supersymmetric [CITATION].', 'hep-th-0309258-2-21-11': 'A schematic view of the background configuration is presented in Fig. [REF].', 'hep-th-0309258-2-22-0': 'If we evaluate the classical value of the action for this background, it is zero; [EQUATION]', 'hep-th-0309258-2-22-1': 'From now on, we are going to compute the one-loop correction to this action, that is, to the background, ([REF]) and ([REF]), due to the quantum fluctuations via the path integration of the quadratic action [MATH], and obtain the one-loop effective action [MATH] or the effective potential [MATH] as a function of [MATH], the radius of the circular motion.', 'hep-th-0309258-2-23-0': 'For the justification of one-loop computation or the semi-classical analysis, it should be made clear that [MATH] and [MATH] of Eq. ([REF]) can be regarded as perturbations.', 'hep-th-0309258-2-23-1': 'For this purpose, following [CITATION], we rescale the fluctuations and parameters as', 'hep-th-0309258-2-24-0': 'A ^-1/2 A , Y^I ^-1/2 Y^I , C ^-1/2 C , C ^-1/2 C ,', 'hep-th-0309258-2-25-0': 'r r , t ^-1 t .', 'hep-th-0309258-2-26-0': 'Under this rescaling, the action [MATH] in the background ([REF]) and ([REF]) becomes [EQUATION] where [MATH], [MATH] and [MATH] do not have [MATH] dependence.', 'hep-th-0309258-2-26-1': 'Now it is obvious that, in the large [MATH] limit, [MATH] and [MATH] can be treated as perturbations and the one-loop computation gives the sensible result.', 'hep-th-0309258-2-27-0': 'Based on the structure of ([REF]), we now write the quantum fluctuations in the [MATH] block matrix form as follows.', 'hep-th-0309258-2-28-0': 'A = Z_(1)^0 & ^0', 'hep-th-0309258-2-29-0': '^0 & Z_(2)^0 , Y^I = Z_(1)^I & ^I', 'hep-th-0309258-2-30-0': '^I & Z_(2)^I , = _(1) &', 'hep-th-0309258-2-31-0': '^ & _(2) ,', 'hep-th-0309258-2-32-0': 'C = C_(1) & C', 'hep-th-0309258-2-33-0': 'C^ & C_(2) , C = C_(1) & C', 'hep-th-0309258-2-34-0': 'C^& C_(2) .', 'hep-th-0309258-2-35-0': 'Although we denote the block off-diagonal matrices for the ghosts by the same symbols with those of the original ghost matrices, there will be no confusion since [MATH] matrices will never appear in what follows.', 'hep-th-0309258-2-35-1': 'The above form of matrices is convenient, since the block diagonal and block off-diagonal parts decouple from each other in the quadratic action and thus can be taken into account separately at one-loop level; [EQUATION] where [MATH]) implies the action for the block (off-) diagonal fluctuations.', 'hep-th-0309258-2-35-2': 'We note that there is no [MATH] parameter in [MATH] and [MATH] due to the above rescaling ([REF]).', 'hep-th-0309258-2-36-0': '# Stability of fuzzy sphere', 'hep-th-0309258-2-37-0': 'In this section, the path integration of [MATH] is performed.', 'hep-th-0309258-2-37-1': 'The resulting effective action will enable us to check the one-loop stability of the fuzzy sphere configuration, Eqs. ([REF]) and ([REF]).', 'hep-th-0309258-2-37-2': 'In fact, the background encoding the circular motion, Eq. ([REF]), does not contribute to [MATH] basically because of the fact that it is proportional to the identity matrix.', 'hep-th-0309258-2-37-3': 'This leads to the situation that the quantum fluctuations around one fuzzy sphere do not interact with those around another fuzzy sphere, and the effective action is just the sum of that for each fuzzy sphere.', 'hep-th-0309258-2-37-4': 'This can then be stated as [EQUATION] where three Lagrangians, [MATH], [MATH], and [MATH], are those for the bosonic, fermionic, and ghost fluctuations respectively around the [MATH]-th fuzzy sphere.', 'hep-th-0309258-2-37-5': 'We note that, at quadratic level, there is no mixing between these three kinds of fluctuations.', 'hep-th-0309258-2-38-0': '## Bosonic fluctuation', 'hep-th-0309258-2-39-0': 'We first evaluate the path integral of bosonic fluctuations.', 'hep-th-0309258-2-39-1': 'The corresponding Lagrangian is given by [EQUATION] where, as alluded to at the end of section [REF], the quadratic term of [MATH] coming from the gauge fixing term appears because the same term with opposite sign has been used for making the complete square form [MATH].', 'hep-th-0309258-2-40-0': 'In order to perform the actual path integration, it is useful to diagonalize the fluctuation matrices and obtain the mass spectrum.', 'hep-th-0309258-2-40-1': 'By the way, since the diagonalization itself has been already given in [CITATION], we will be brief in its presentation and present only the essential points.', 'hep-th-0309258-2-41-0': 'The starting point is the observation that the mass terms are written in terms of the commutators with [MATH] satisfying Eq. ([REF]), the [MATH] algebra.', 'hep-th-0309258-2-41-1': 'This indicates that we can use the representation theory of [MATH] for the diagonalization.', 'hep-th-0309258-2-41-2': 'We regard an [MATH] matrix as an [MATH]-dimensional reducible representation of [MATH], which decomposes into irreducible spin [MATH] representations with the range of [MATH] from [MATH] to [MATH], that is, [MATH].', 'hep-th-0309258-2-41-3': 'Based on this decomposition, an [MATH] matrix may be expanded as [EQUATION] where the [MATH] matrix [MATH] is the matrix spherical harmonics transforming in the irreducible spin [MATH] representation and [MATH] is the corresponding spherical mode.', 'hep-th-0309258-2-41-4': 'We note that [MATH] satisfies the following reality condition since the fluctuation matrices are Hermitian.', 'hep-th-0309258-2-41-5': '[EQUATION]', 'hep-th-0309258-2-41-6': 'The expansion of matrices in the [MATH] language enables us to use the properties of the [MATH] generators, which are given by [EQUATION] where [MATH].', 'hep-th-0309258-2-41-7': 'For the normalization of the matrix spherical harmonics, we choose [EQUATION]', 'hep-th-0309258-2-41-8': 'Having equipped with the necessary machinery, we now proceed the diagonalization.', 'hep-th-0309258-2-42-0': 'By direct application of Eqs. ([REF]), ([REF]), and ([REF]), the gauge field fluctuation [MATH] and the fluctuations in the [MATH] directions [MATH] are immediately diagonalized.', 'hep-th-0309258-2-42-1': 'The corresponding spherical modes and their masses are [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-2-43-0': 'As for the fluctuations in the [MATH] directions, we should first consider the potential [MATH] and diagonalize it by solving the eigenvalue problem, [EQUATION]', 'hep-th-0309258-2-43-1': 'By defining the combinations of matrices [MATH] and using Eqs. ([REF]) and ([REF]), it turns out that the eigenvalue [MATH] takes the values of [MATH], [MATH], or [MATH].', 'hep-th-0309258-2-43-2': 'Let us now denote the spherical eigenmodes of matrix eigenvectors as [MATH], [MATH], and [MATH] for [MATH], [MATH], and [MATH], respectively.', 'hep-th-0309258-2-43-3': 'The eigenmodes satisfy the reality condition in the form of Eq. ([REF]).', 'hep-th-0309258-2-43-4': 'Then the fluctuation matrices may be expressed with respect to these eigenmodes for each eigenvalue.', 'hep-th-0309258-2-44-0': 'For [MATH], we have the following expressions: [EQUATION] where [MATH], [MATH], and the normalization constant are chosen so that the kinetic term for the mode [MATH] is of the form [MATH].', 'hep-th-0309258-2-44-1': 'Here and in what follows, the summations over [MATH] and [MATH] with specified ranges are implicit.', 'hep-th-0309258-2-45-0': 'For [MATH], we have [EQUATION] where [MATH], [MATH], and the normalization constant are chosen in the same way with the case of [MATH].', 'hep-th-0309258-2-46-0': 'Finally, for [MATH], [EQUATION] with [MATH] and [MATH].', 'hep-th-0309258-2-46-1': 'As pointed out in [CITATION], the modes for [MATH] case correspond to the degrees of freedom for the gauge transformation and are thus unphysical.', 'hep-th-0309258-2-46-2': 'Since the authors of [CITATION] took the physical Weyl gauge, [MATH], and worked in the operator formulation, it was not necessary to consider these unphysical modes seriously.', 'hep-th-0309258-2-46-3': 'However, they should be involved properly in the present context because our gauge choice is the covariant background gauge and we work in the path integral formulation.', 'hep-th-0309258-2-47-0': 'We turn to the remaining potential term in the Lagrangian ([REF]) which is the square of [MATH].', 'hep-th-0309258-2-47-1': 'Interestingly enough, the matrices expanded in terms of the eigenmodes [MATH] and [MATH], Eqs. ([REF]) and ([REF]), do not give any contribution, since we obtain [EQUATION] for [MATH] and [MATH].', 'hep-th-0309258-2-47-2': 'Only the modes corresponding to [MATH] contribute to the potential, which is evaluated as [EQUATION] for each [MATH] and [MATH].', 'hep-th-0309258-2-48-0': 'Having diagonalized the fluctuations [MATH], we get the following list of spherical modes in the [MATH] directions with their masses and the ranges of spin [MATH]: [EQUATION] where [MATH].', 'hep-th-0309258-2-49-0': 'With respect to the spherical modes of Eqs. ([REF]) and ([REF]), the Lagrangian ([REF]) is then written in the diagonalized form as [EQUATION] where the sum over [MATH] with the range [MATH] is understood.', 'hep-th-0309258-2-49-1': 'The Lagrangian is just the sum of various harmonic oscillator Lagrangians, which are non-interacting with each other, and therefore the path integration is now straightforward.', 'hep-th-0309258-2-49-2': 'As a result, what we obtain is [EQUATION]', 'hep-th-0309258-2-50-0': '## Fermionic fluctuation', 'hep-th-0309258-2-51-0': 'We turn to the path integral of fermionic fluctuations.', 'hep-th-0309258-2-51-1': 'The Lagrangian is written as [EQUATION]', 'hep-th-0309258-2-51-2': 'It is convenient for our calculation of fermionic part to introduce the [MATH] formulation since the preserved symmetry in the plane-wave matrix model is [MATH] rather than [MATH].', 'hep-th-0309258-2-51-3': 'In this formulation the [MATH] spinor [MATH] is decomposed as [EQUATION] where [MATH] implies a fundamental [MATH] index and [MATH] is a fundamental [MATH] index.', 'hep-th-0309258-2-51-4': 'According to this decomposition, we may take the expression of [MATH] as [EQUATION]', 'hep-th-0309258-2-51-5': "We also rewrite the [MATH] gamma matrices [MATH]'s in terms of [MATH] and [MATH] ones as follows: [EQUATION] where the [MATH]'s are the standard [MATH] Pauli matrices and six of [MATH] are taken to form a basis of [MATH] anti-symmetric matrices.", 'hep-th-0309258-2-51-6': 'The original [MATH] Clifford algebra is satisfied as long as we take normalizations so that the gamma matrices [MATH] with [MATH] indices satisfy the algebra [EQUATION]', 'hep-th-0309258-2-51-7': 'By introducing the above [MATH] formulation, the Lagrangian for the fermionic fluctuations is rewritten as [EQUATION]', 'hep-th-0309258-2-51-8': 'The diagonalization of the Lagrangian proceeds in the same way as in the previous subsection.', 'hep-th-0309258-2-51-9': 'In the present case, it is achieved by solving the following eigenvalue problem: [EQUATION]', 'hep-th-0309258-2-51-10': 'We first expand the [MATH] fermionic matrix [MATH] in terms of the matrix spherical harmonics as [EQUATION]', 'hep-th-0309258-2-51-11': 'If we plug this expansion into the above eigenvalue equation and use the [MATH] algebra Eq. ([REF]), we see that the eigenvalues are [MATH] and [MATH] [CITATION].', 'hep-th-0309258-2-51-12': 'Let us now introduce the fermionic spherical eigenmodes [MATH] and [MATH] corresponding to [MATH] and [MATH] respectively.', 'hep-th-0309258-2-51-13': 'As for the spinorial structure, [MATH]) carries the (anti) fundamental [MATH] index.', 'hep-th-0309258-2-51-14': 'We note that, from now on, we suppress the [MATH] indices.', 'hep-th-0309258-2-52-0': 'Then, as the eigenstate for [MATH], the matrix [MATH] has the expansion in terms of the eigenmode [MATH] as [EQUATION] where [MATH], [MATH], and the subscripts [MATH] denote the [MATH] indices measured by [MATH].', 'hep-th-0309258-2-53-0': 'On the other hand, for [MATH], we have [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-2-54-0': 'By using the mode-expansions, Eqs. ([REF]) and ([REF]), and the [MATH] algebra ([REF]), the fermionic Lagrangian ([REF]) becomes [EQUATION] where it should be understood that there is the summation over [MATH] with the range [MATH].', 'hep-th-0309258-2-54-1': 'Now, the path integration for this Lagrangian may be evaluated immediately, and gives [EQUATION]', 'hep-th-0309258-2-55-0': '## Ghost fluctuation', 'hep-th-0309258-2-56-0': 'As the final part of the diagonal fluctuations, we consider the Lagrangian for the ghost fluctuations, which is given by [EQUATION]', 'hep-th-0309258-2-56-1': 'By using the [MATH] algebra ([REF]) and the following expansions in terms of the matrix spherical harmonics [EQUATION] with [MATH] and [MATH], we may rewrite the above Lagrangian as [EQUATION] where the sum over [MATH] is implicit.', 'hep-th-0309258-2-56-2': 'Then the result of the path integration for this Lagrangian is [EQUATION]', 'hep-th-0309258-2-57-0': '## One-loop stability', 'hep-th-0309258-2-58-0': 'In the previous subsections, we have evaluated the path integrals for the block diagonal fluctuations of the action [MATH], ([REF]).', 'hep-th-0309258-2-58-1': 'Thus, we may now consider the effective action.', 'hep-th-0309258-2-58-2': 'As can be inferred from Eq. ([REF]), it is enough to consider only the effective action of a given fuzzy sphere.', 'hep-th-0309258-2-58-3': 'If we let [MATH] be the effective action for the [MATH]-th fuzzy sphere, then it is given by [EQUATION]', 'hep-th-0309258-2-58-4': 'The right hand side may be viewed just as the product of determinants of non-interacting quantum mechanical simple harmonic oscillators with various frequencies.', 'hep-th-0309258-2-58-5': 'In fact, as we will see in the next section, this is also the case for the effective action describing the interaction between two fuzzy spheres.', 'hep-th-0309258-2-58-6': 'Thus, it is worthwhile to consider a generic situation, which is useful both in the present and the next section.', 'hep-th-0309258-2-59-0': 'Let us then consider a situation, [EQUATION] where [MATH] is included for the later usage.', 'hep-th-0309258-2-59-1': 'The formal expression of the effective action is read as [EQUATION]', 'hep-th-0309258-2-59-2': 'By using the relation [MATH] for a given matrix [MATH], we may present a prototype calculation of single determinant as [EQUATION] where the momentum integration has appeared by inserting the momentum space identity, [MATH], between bra and ket vectors for time, and the final result has been derived by using the formula [EQUATION]', 'hep-th-0309258-2-59-3': 'If we apply the result of this single determinant calculation to Eq. ([REF]), then the effective action is finally obtained as [EQUATION]', 'hep-th-0309258-2-59-4': 'We now return to the present case.', 'hep-th-0309258-2-59-5': 'First of all, we observe that the ghost part ([REF]) eliminates two determinant factors of the bosonic result ([REF]) which are actually the contributions from the unphysical modes.', 'hep-th-0309258-2-59-6': 'Thus, what we get from the bosonic and the ghost parts is only the determinant factors from the physical bosonic modes.', 'hep-th-0309258-2-59-7': 'By using the generic expression, Eq. ([REF]), it turns out that their contributions to the effective action are [EQUATION]', 'hep-th-0309258-2-59-8': 'On the other hand, the contribution from the fermionic part ([REF]) is obtained as [EQUATION]', 'hep-th-0309258-2-59-9': 'Therefore, there is no net contribution to the effective action and we can conclude that the one loop effective action obtained after integrating out each of the block diagonal fluctuations vanishes; [EQUATION]', 'hep-th-0309258-2-59-10': 'This means that each membrane fuzzy sphere has quantum stability at least at one-loop level.', 'hep-th-0309258-2-59-11': 'We note that, if we take [MATH] in the above two contributions, the result of the previous path integral computation [CITATION] is recovered.', 'hep-th-0309258-2-60-0': '# Interaction between fuzzy spheres', 'hep-th-0309258-2-61-0': 'We turn to the action [MATH] in Eq. ([REF]) for the block off-diagonal fluctuations and compute the one-loop effective potential describing the interaction between two fuzzy spheres.', 'hep-th-0309258-2-61-1': 'The action is given by [EQUATION] where [MATH], [MATH] and [MATH] are the Lagrangians for the off-diagonal bosonic, fermionic and ghost fluctuations of Eq. ([REF]) respectively and their explicit expressions will be presented in due course.', 'hep-th-0309258-2-61-2': 'Since, as in the previous section, there is no mixing between different kinds of fluctuations, each Lagrangian can be considered independently.', 'hep-th-0309258-2-62-0': 'In calculating the effective action, the prescription given by Kabat and Taylor [CITATION] is usually used.', 'hep-th-0309258-2-62-1': 'We note however that, at the present situation, it is more helpful to use the expansion in terms of the matrix spherical harmonics as in the previous section.', 'hep-th-0309258-2-63-0': '## Bosonic fluctuation', 'hep-th-0309258-2-64-0': 'Let us first consider the bosonic Lagrangian and evaluate its path integral.', 'hep-th-0309258-2-64-1': 'The Lagrangian is [EQUATION]', 'hep-th-0309258-2-64-2': 'Here, adopting the notation of [*], we have defined [EQUATION] where [MATH] is the [MATH] matrix which is a block at [MATH]-th row and [MATH]-th column in the blocked form of a given matrix [MATH].', 'hep-th-0309258-2-64-3': 'In the present case, [MATH] and [MATH] take values of [MATH] and [MATH].', 'hep-th-0309258-2-64-4': 'For example, if we look at the [MATH] block matrix form of the gauge field fluctuation [MATH] in Eq. ([REF]), then [MATH], [MATH], and [MATH].', 'hep-th-0309258-2-65-0': 'The matrix fields, [MATH], [MATH], and [MATH] are coupled with each other through the circular motion background.', 'hep-th-0309258-2-65-1': 'Since the time dependent trigonometric functions may make the formulation annoying, we consider the newly defined matrix variables as [EQUATION] where [MATH] may be interpreted as the fluctuation tangential to the circular motion at time [MATH] and [MATH] as the normal fluctuation.', 'hep-th-0309258-2-65-2': 'In terms of these fluctuations, the terms in the Lagrangian ([REF]), which are dependent on [MATH] and [MATH], are rewritten as [EQUATION] where we no longer see the explicit time dependent classical functions.', 'hep-th-0309258-2-66-0': 'We are now in a position to consider the diagonalization of the Lagrangian [MATH].', 'hep-th-0309258-2-66-1': 'We note that, compared to the case in the previous section, we are in a somewhat different situation.', 'hep-th-0309258-2-66-2': 'The fluctuation matrices are [MATH] or [MATH] ones, while those in the last section are rectangular [MATH] matrices.', 'hep-th-0309258-2-66-3': 'This means that, when we regard an [MATH] block off-diagonal matrix as an [MATH]-dimensional reducible representation of [MATH], it has the decomposition into irreducible spin [MATH] representations with the range [MATH], that is, [MATH], and may be expanded as [EQUATION] where [MATH] is the [MATH] matrix spherical harmonics transforming in the irreducible spin [MATH] representation and [MATH] is the corresponding spherical mode.', 'hep-th-0309258-2-66-4': 'The basic operation between [MATH] generators [MATH] and [MATH] is given not by the commutator but by the [MATH] operator ([REF]).', 'hep-th-0309258-2-66-5': 'Thus the algebraic properties for the present situation are given by Eq. ([REF]) where the commutator is replaced by the [MATH] operator.', 'hep-th-0309258-2-67-0': 'Having the expansion and algebraic properties, the diagonalization proceeds in exactly the same way as in the previous section.', 'hep-th-0309258-2-67-1': 'Hence, by noting that one may find a detailed procedure in [CITATION], we will present only the results of diagonalization for the Lagrangian.', 'hep-th-0309258-2-67-2': 'We observe that, because of the background for the circular motion in [MATH]-[MATH] plane, the [MATH] symmetry is broken to [MATH], while the [MATH] symmetry remains intact.', 'hep-th-0309258-2-67-3': 'This fact naturally leads us to break the bosonic Lagrangian ([REF]) into three parts as follows: [EQUATION] where [MATH] is the Lagrangian for [MATH], [MATH] is for [MATH] with [MATH], and [MATH] represents the [MATH] rotational part described by [MATH], [MATH], and the gauge fluctuation [MATH].', 'hep-th-0309258-2-68-0': 'We first consider [MATH] and its path integration.', 'hep-th-0309258-2-68-1': 'Its diagonalized form is obtained by [EQUATION] where the sum of [MATH] over the range [MATH] is implicit and the spherical modes [MATH] are the degrees of freedom for the gauge transformation, the block off-diagonal counterpart of [MATH], ([REF]), in the previous section.', 'hep-th-0309258-2-68-2': '[MATH] and [MATH] are the block off-diagonal counterparts of [MATH] and [MATH], respectively.', 'hep-th-0309258-2-68-3': 'The path integral of this Lagrangian is straightforward and results in [EQUATION]', 'hep-th-0309258-2-68-4': 'As for [MATH], we have as its diagonalized form [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-2-68-5': 'Its path integration leads us to have [EQUATION]', 'hep-th-0309258-2-68-6': 'For the rotational part, the diagonalized Lagrangian is obtained as [EQUATION] where [MATH].', 'hep-th-0309258-2-68-7': 'Since all the coupling coefficients between the modes are time-independent constants, the path integral of this Lagrangian is readily evaluated and gives [EQUATION]', 'hep-th-0309258-2-69-0': '## Fermionic fluctuation', 'hep-th-0309258-2-70-0': 'The fermionic Lagrangian of the block off-diagonal action [MATH], ([REF]), is [EQUATION]', 'hep-th-0309258-2-70-1': 'As was done in Eq. ([REF]), we decompose the fermion [MATH] into [MATH] and [MATH] according to [MATH].', 'hep-th-0309258-2-70-2': 'Then the expression of [MATH] may be taken as [EQUATION] where [MATH].', 'hep-th-0309258-2-70-3': 'It should be noted that, contrary to the block diagonal fermionic matrix [MATH] of ([REF]), [MATH] is the two copy of the [MATH] representation [MATH] as one may see from Eq. ([REF]).', 'hep-th-0309258-2-70-4': 'This means that [MATH] and [MATH] should be treated as independent spinors not related in any way.', 'hep-th-0309258-2-70-5': 'By plugging the decomposition ([REF]) into the Lagrangian ([REF]), we have [EQUATION]', 'hep-th-0309258-2-70-6': 'The Lagrangian has explicit time dependence due to the presence of the circular motion background.', 'hep-th-0309258-2-70-7': 'In order to hide it, we take the fermionic field [MATH] as [EQUATION] where we have introduced a new fermionic field [MATH] which is in the [MATH] of [MATH].', 'hep-th-0309258-2-70-8': 'Then, by using the following identities, [EQUATION] which are proved via the Clifford algebra ([REF]), we may show that the Lagrangian ([REF]) becomes [EQUATION]', 'hep-th-0309258-2-70-9': 'The explicit time dependent classical functions disappear and the term containing [MATH] appears, which originates from the kinetic term of [MATH] in ([REF]).', 'hep-th-0309258-2-71-0': 'With the above Lagrangian ([REF]), the diagonalization proceeds in the same manner with that for the block diagonal fermionic Lagrangian ([REF]), except for some differences pointed out in the previous subsection.', 'hep-th-0309258-2-71-1': 'In the expansion of [MATH] and [MATH] indices are suppressed.)', 'hep-th-0309258-2-71-2': 'in terms of the matrix spherical harmonics like ([REF]), let us denote their spherical modes as [MATH] and [MATH] respectively.', 'hep-th-0309258-2-71-3': 'Then the diagonalization results in [MATH] and [MATH].', 'hep-th-0309258-2-71-4': 'The modes [MATH] and [MATH] have the same mass of [MATH] with [MATH].', 'hep-th-0309258-2-71-5': 'For the modes [MATH] and [MATH], their mass is [MATH] with [MATH].', 'hep-th-0309258-2-71-6': 'All the modes have the same range of [MATH] as [MATH].', 'hep-th-0309258-2-71-7': 'With respect to these diagonalized spherical modes, the Lagrangian ([REF]) is written as [EQUATION] where [MATH] and the [MATH] indices are suppressed.', 'hep-th-0309258-2-72-0': 'The product [MATH] measures the [MATH] chirality in the [MATH]-[MATH] plane where the circular motion takes place.', 'hep-th-0309258-2-72-1': 'Since [MATH], its eigenvalues are [MATH].', 'hep-th-0309258-2-72-2': 'Each spherical mode may split into modes having definite [MATH] eigenvalues as follows: [EQUATION] where the modes on the right hand sides satisfy [EQUATION]', 'hep-th-0309258-2-72-3': 'One may see that the Lagrangian ([REF]) composed of two independent parts.', 'hep-th-0309258-2-72-4': 'One is for [MATH] and [MATH], and the other one for [MATH] and [MATH].', 'hep-th-0309258-2-72-5': 'If we first consider the part for [MATH] and [MATH], then, according to the above splitting of modes, we have [EQUATION] where again [MATH].', 'hep-th-0309258-2-72-6': 'The path integration of this part is straightforward and gives [EQUATION]', 'hep-th-0309258-2-72-7': 'The other part for [MATH] and [MATH] is obtained as [EQUATION] where the sum over [MATH] for [MATH] is implicit.', 'hep-th-0309258-2-72-8': 'The path integration of this part results in [EQUATION]', 'hep-th-0309258-2-73-0': '## Ghost fluctuation', 'hep-th-0309258-2-74-0': 'Finally, we consider the path integration for the ghost part of the action [MATH] ([REF]).', 'hep-th-0309258-2-74-1': 'The block off-diagonal Lagrangian for the ghosts is [EQUATION]', 'hep-th-0309258-2-74-2': 'The diagonalization may be carried out by using the same logic in the previous subsections.', 'hep-th-0309258-2-74-3': 'We expand the ghost fields [MATH] and [MATH] in terms of the matrix spherical harmonics according to ([REF]), and denote their spherical modes as [MATH] and [MATH] respectively.', 'hep-th-0309258-2-74-4': 'Then the diagonalized Lagrangian is obtained as [EQUATION] where the sum over [MATH] for the range [MATH] is implicit.', 'hep-th-0309258-2-75-0': 'The path integral for the above diagonalized Lagrangian is then immediately evaluated as follows: [EQUATION]', 'hep-th-0309258-2-76-0': '## Effective potential', 'hep-th-0309258-2-77-0': 'Having evaluated the path integral for each part of the block off-diagonal action [MATH] ([REF]), the one-loop effective action, [MATH], which describes the interaction between two membrane fuzzy spheres with the classical configuration ([REF]) and ([REF]), is now given by [EQUATION] where the subscripts on the right hand side denote the bosonic, the fermionic, and the ghost contributions.', 'hep-th-0309258-2-77-1': 'We see that the ghost contribution, Eq. ([REF]), eliminate those of unphysical gauge degrees of freedom present in bosonic contributions, Eqs. ([REF]) and ([REF]).', 'hep-th-0309258-2-77-2': 'Thus only the physical degrees of freedom contribute to the effective action, as it should be.', 'hep-th-0309258-2-78-0': 'The explicit expression of the effective action is obtained by consulting the generic result presented from Eq. ([REF]) to ([REF]).', 'hep-th-0309258-2-78-1': 'The effective potential, [MATH], about which we are concerned in this subsection, is then given by [MATH].', 'hep-th-0309258-2-78-2': 'In expressing the effective potential, it is convenient to write [MATH] as [EQUATION] where [MATH]) is the contribution of the physical bosonic (fermionic) degrees of freedom to the effective potential.', 'hep-th-0309258-2-78-3': 'The expression that we obtain for [MATH] is then [EQUATION] while, for the fermionic contribution [MATH], we obtain [EQUATION]', 'hep-th-0309258-2-78-4': 'The above expressions show that [MATH] and [MATH] have the same structure except for the ranges of [MATH].', 'hep-th-0309258-2-78-5': 'This leads us to expect a great amount of cancellation.', 'hep-th-0309258-2-78-6': 'Indeed, what we have found is that they are exactly the same.', 'hep-th-0309258-2-78-7': 'One way to see the cancellation is to adjust all the ranges of the summation parameter [MATH] to the range [MATH].', 'hep-th-0309258-2-78-8': 'Therefore, the one-loop effective potential [MATH], ([REF]), as a function of the distance [MATH] between two fuzzy spheres is just flat potential; [EQUATION]', 'hep-th-0309258-2-79-0': '# Conclusion and discussion', 'hep-th-0309258-2-80-0': 'We have studied the one-loop quantum corrections to a classical background of the plane-wave matrix model in the framework of the path integration.', 'hep-th-0309258-2-80-1': 'The background is composed of two supersymmetric membrane fuzzy spheres, ([REF]) and ([REF]).', 'hep-th-0309258-2-80-2': 'One fuzzy sphere is located at the origin in the [MATH] symmetric space and the other one rotates around it with the distance [MATH].', 'hep-th-0309258-2-81-0': 'Firstly, the quantum stability of each fuzzy sphere has been shown.', 'hep-th-0309258-2-81-1': 'In fact, the stability check already has been done in the operator [CITATION] as well as in the path integral [CITATION] formulation.', 'hep-th-0309258-2-81-2': 'However, while the size [MATH] of the fuzzy sphere has been taken arbitrary in the operator formulation, it has been restricted to the minimal one, that is [MATH], in the path integral formulation.', 'hep-th-0309258-2-81-3': 'Since the fuzzy sphere in this paper has an arbitrary size, our stability check can be regarded as the full generalization of the previous path integral result.', 'hep-th-0309258-2-82-0': 'Secondly, the one-loop effective potential describing the interaction between two fuzzy spheres has been calculated as a function of the distance [MATH].', 'hep-th-0309258-2-82-1': 'Interestingly, the result is that the effective potential [MATH] is flat and thus the fuzzy spheres do not feel any force.', 'hep-th-0309258-2-82-2': 'This implies that the whole configuration of two fuzzy spheres given by Eqs. ([REF]) and ([REF]) is supersymmetric.', 'hep-th-0309258-2-82-3': 'Although the flatness of the effective potential is the one-loop result, we expect that the result holds also for higher loops.', 'hep-th-0309258-2-82-4': 'For the supersymmetric properties of the fuzzy sphere configuration itself, the study of supersymmetry algebra may be more helpful rather than the path integral formulation.', 'hep-th-0309258-2-82-5': 'It would be interesting to investigate the configuration in this paper through the supersymmetry algebra.', 'hep-th-0309258-2-83-0': 'Let us consider the radial distance [MATH] between two fuzzy spheres and discuss about its possible interpretation.', 'hep-th-0309258-2-83-1': 'We first consider the form of the effective action when [MATH] is time dependent.', 'hep-th-0309258-2-83-2': 'Since the circular motion is taken as the background, [MATH] is constant in this paper.', 'hep-th-0309258-2-83-3': 'However, if we slightly deform the circular motion to the elliptic one, [MATH] can be made to have time dependence.', 'hep-th-0309258-2-83-4': 'We can also make the time variation of it, [MATH], arbitrarily small by controlling the degree of deformation.', 'hep-th-0309258-2-83-5': 'In this case, it is expected that the fuzzy spheres begin to interact and the effective action [MATH] may be written as [EQUATION] where the kinetic term for [MATH] is the value of the classical action [MATH] with the rescaling ([REF]), [MATH] is the would-be one-loop contribution to the effective action with the property [MATH], and the term of order [MATH] implies the higher loop corrections.', 'hep-th-0309258-2-84-0': 'If [MATH], the above effective action vanishes and the supersymmetric situation is recovered.', 'hep-th-0309258-2-84-1': 'This is reminiscent of the effective action for graviton-graviton scattering in the flat space matrix model [CITATION].', 'hep-th-0309258-2-84-2': 'The distance between two gravitons comes from the flat directions which are continuous moduli or supersymmetric vacua making the potential of the flat space matrix model vanish.', 'hep-th-0309258-2-84-3': 'In the plane-wave matrix model, it is known that there is no continuous moduli and hence we do not have flat directions.', 'hep-th-0309258-2-84-4': 'However, if we look at the tree level action ([REF]) evaluated for the fuzzy sphere configuration, ([REF]) and ([REF]), we see that it vanishes exactly and does not depend on [MATH] which is continuous from [MATH] to [MATH].', 'hep-th-0309258-2-84-5': 'This means that, as long as the fuzzy sphere dynamics is concerned, the radius of the circular motion may be interpreted as the flat direction.', 'hep-th-0309258-2-85-0': 'In this paper, the circular motion takes place in the [MATH]-[MATH] sub-plane of the [MATH] symmetric space.', 'hep-th-0309258-2-85-1': 'Since the [MATH] symmetric space has two other sub-planes, that is [MATH]-[MATH] and [MATH]-[MATH], and we may embed the circular motion in one of those, there may be three flat directions in total, which are radial directions of three sub-planes.', 'hep-th-0309258-2-85-2': 'However, it is not obvious whether these three directions are connected in a continuous way or not, because of the supersymmetric property of rotating fuzzy sphere; all the points in the supersymmetric moduli are expected to preserve a fixed fraction of supersymmetry.', 'hep-th-0309258-2-85-3': 'While the fuzzy sphere rotating in only one sub-plane is 1/2-BPS object, it is generically 1/4-BPS when it has angular momenta also in the other sub-planes [CITATION].', 'hep-th-0309258-2-85-4': 'If we turn to the [MATH] symmetric space, it seems that we do not have flat directions, since only the fuzzy sphere rotating with fixed radius is supersymmetric [CITATION].', 'hep-th-0309258-2-86-0': 'In view of the gauge/gravity duality, it is interesting to study the same situation in the supergravity side.', 'hep-th-0309258-2-86-1': 'In the large [MATH] limit, the leading order interaction terms obtained from the supergravity side analysis would match with those from the matrix theory analysis.', 'hep-th-0309258-2-86-2': 'We expect that the supergravity side analysis is helpful and provides clearer understanding about the structure of the effective potential.', 'hep-th-0309258-2-87-0': 'The calculation in this paper has been carried out in the Minkowskian time signature.', 'hep-th-0309258-2-87-1': 'The basic reason why we have not taken the Wick rotation for some convenience in the actual calculation is the periodic nature of the circular motion background ([REF]).', 'hep-th-0309258-2-87-2': 'The naive change of the Minkowskian time to the Euclidean one in the process of calculation breaks the periodicity of the background and the reality of the action.', 'hep-th-0309258-2-87-3': 'If one wants to study the time dependent background with the Euclidean time signature, he or she should begin with the Euclidean action at the first setup.', 'hep-th-0309258-2-87-4': 'In the Euclidean case, the time dependent solutions to the equations of motion are given by hyperbolic functions, which give open paths not periodic ones.', 'hep-th-0309258-2-87-5': 'In the sense that there is no classical background solution corresponding to open path in the Minkowskian time plane-wave matrix model, it would be interesting to consider the model in the Euclidean time for studying such a background.', 'hep-th-0309258-2-88-0': 'In the situation where the one-loop effective potential is flat, it is a natural step to consider the case where the radius [MATH] between two fuzzy spheres is time dependent and calculate the form of the interaction.', 'hep-th-0309258-2-88-1': 'We hope to return to this issue in the near future.'}

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[['hep-th-0309258-1-5-3', 'hep-th-0309258-2-5-3'], ['hep-th-0309258-1-23-1', 'hep-th-0309258-2-23-1'], ['hep-th-0309258-1-15-4', 'hep-th-0309258-2-15-4'], ['hep-th-0309258-1-70-4', 'hep-th-0309258-2-71-4'], ['hep-th-0309258-2-64-2', 'hep-th-0309258-3-64-2'], ['hep-th-0309258-2-66-2', 'hep-th-0309258-3-66-2'], ['hep-th-0309258-1-81-1', 'hep-th-0309258-2-82-1'], ['hep-th-0309258-1-81-5', 'hep-th-0309258-2-83-5']]

[]

[['hep-th-0309258-1-10-3', 'hep-th-0309258-2-10-3'], ['hep-th-0309258-1-10-6', 'hep-th-0309258-2-10-8'], ['hep-th-0309258-1-82-1', 'hep-th-0309258-2-86-2'], ['hep-th-0309258-1-84-0', 'hep-th-0309258-2-88-0'], ['hep-th-0309258-2-59-10', 'hep-th-0309258-3-59-10'], ['hep-th-0309258-1-81-2', 'hep-th-0309258-2-83-2']]

[]

['hep-th-0309258-1-1-0', 'hep-th-0309258-1-2-0', 'hep-th-0309258-1-2-1', 'hep-th-0309258-1-2-2', 'hep-th-0309258-1-3-0', 'hep-th-0309258-1-24-0', 'hep-th-0309258-1-25-0', 'hep-th-0309258-1-28-0', 'hep-th-0309258-1-29-0', 'hep-th-0309258-1-30-0', 'hep-th-0309258-1-31-0', 'hep-th-0309258-1-32-0', 'hep-th-0309258-1-33-0', 'hep-th-0309258-1-34-0', 'hep-th-0309258-1-41-5', 'hep-th-0309258-1-53-0', 'hep-th-0309258-2-1-0', 'hep-th-0309258-2-2-0', 'hep-th-0309258-2-2-1', 'hep-th-0309258-2-2-2', 'hep-th-0309258-2-3-0', 'hep-th-0309258-2-24-0', 'hep-th-0309258-2-25-0', 'hep-th-0309258-2-28-0', 'hep-th-0309258-2-29-0', 'hep-th-0309258-2-30-0', 'hep-th-0309258-2-31-0', 'hep-th-0309258-2-32-0', 'hep-th-0309258-2-33-0', 'hep-th-0309258-2-34-0', 'hep-th-0309258-2-41-5', 'hep-th-0309258-2-53-0', 'hep-th-0309258-3-1-0', 'hep-th-0309258-3-2-0', 'hep-th-0309258-3-2-1', 'hep-th-0309258-3-2-2', 'hep-th-0309258-3-3-0', 'hep-th-0309258-3-24-0', 'hep-th-0309258-3-25-0', 'hep-th-0309258-3-28-0', 'hep-th-0309258-3-29-0', 'hep-th-0309258-3-30-0', 'hep-th-0309258-3-31-0', 'hep-th-0309258-3-32-0', 'hep-th-0309258-3-33-0', 'hep-th-0309258-3-34-0', 'hep-th-0309258-3-41-5', 'hep-th-0309258-3-53-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/0309258

{'hep-th-0309258-3-0-0': 'In the plane-wave matrix model, the background configuration of two membrane fuzzy spheres, one of which rotates around the other one in the [MATH] symmetric space, is allowed as a classical solution.', 'hep-th-0309258-3-0-1': 'We study the one-loop quantum corrections to this background in the path integral formulation.', 'hep-th-0309258-3-0-2': 'Firstly, we show that each fuzzy sphere is stable under the quantum correction.', 'hep-th-0309258-3-0-3': 'Secondly, the effective potential describing the interaction between fuzzy spheres is obtained as a function of [MATH], which is the distance between two fuzzy spheres.', 'hep-th-0309258-3-0-4': 'It is shown that the effective potential is flat and hence the fuzzy spheres do not feel any force.', 'hep-th-0309258-3-0-5': 'The possibility on the existence of flat directions is discussed.', 'hep-th-0309258-3-1-0': '[5mm] Keywords : pp-wave, Matrix model, Fuzzy sphere', 'hep-th-0309258-3-2-0': 'PACS numbers : 11.25.-w, 11.27.', 'hep-th-0309258-3-2-1': '+d, 12.60.', 'hep-th-0309258-3-2-2': 'Jv', 'hep-th-0309258-3-3-0': '6.6mm', 'hep-th-0309258-3-4-0': '# Introduction', 'hep-th-0309258-3-5-0': 'The plane-wave matrix model [CITATION] is a microscopic description of the discrete light cone quantized (DLCQ) M-theory in the eleven-dimensional [MATH]-wave or plane-wave background.', 'hep-th-0309258-3-5-1': 'The eleven-dimensional plane-wave [CITATION] is maximally supersymmetric and the limiting case of the eleven-dimensional [MATH] type geometries [CITATION].', 'hep-th-0309258-3-5-2': 'Its explicit form is given by [EQUATION] where [MATH].', 'hep-th-0309258-3-5-3': 'Due to the effect of the [MATH] component of the metric and the presence of the four-form field strength, the plane-wave matrix model has some [MATH] dependent terms, which make the difference between the usual flat space matrix model and the plane-wave one.', 'hep-th-0309258-3-6-0': 'The presence of the [MATH] dependent terms makes the plane-wave matrix model have some peculiar properties.', 'hep-th-0309258-3-6-1': 'One of them is that there are various vacuum structures classified by the [MATH] algebra [CITATION].', 'hep-th-0309258-3-6-2': 'The crucial ingredient for vacua is the membrane fuzzy sphere.', 'hep-th-0309258-3-6-3': 'It preserves the full 16 supersymmetries of the plane-wave matrix model and exists even at finite [MATH] which is the size of matrix.', 'hep-th-0309258-3-7-0': 'After the plane-wave matrix model was proposed and its basic aspects were uncovered, there have been lots of investigations in various directions.', 'hep-th-0309258-3-7-1': 'The structure of vacua has been studied in more detail especially related to the protected multiplet [CITATION].', 'hep-th-0309258-3-7-2': 'The possible BPS objects contained in the plane-wave matrix model have been searched [CITATION].', 'hep-th-0309258-3-7-3': 'The algebraic and structural study of the model itself and the various BPS objects present in it has been performed [CITATION].', 'hep-th-0309258-3-7-4': 'Based on the fact that the low energy description of the M theory is the eleven dimensional supergravity, there also have been supergravity side analysis [CITATION].', 'hep-th-0309258-3-8-0': 'If the M theory is compactified on a circle, then we have ten-dimensional Type IIA string theory.', 'hep-th-0309258-3-8-1': 'Under the circle compactification, the [MATH]-wave geometry ([REF]) becomes the IIA [MATH]-wave background which is not maximally supersymmetric and has 24 supersymmetries [CITATION].', 'hep-th-0309258-3-8-2': 'For the purpose of understanding the plane-wave matrix model as well as the string theory itself, the IIA string theory in the [MATH]-wave background has been also extensively studied, in parallel with the progress in the study of the plane-wave matrix model [CITATION].', 'hep-th-0309258-3-9-0': 'However, despite of quite amount of progress in the study of the plane-wave matrix model, there has been lack of the investigation about the dynamical aspects.', 'hep-th-0309258-3-9-1': 'However, see for example [CITATION].', 'hep-th-0309258-3-9-2': 'In fact, the present status of the plane-wave matrix model enables us to study the dynamics of the model.', 'hep-th-0309258-3-9-3': 'In this paper, we consider the basic objects of the plane-wave matrix model and study their interaction.', 'hep-th-0309258-3-10-0': 'For the interacting objects, we take two membrane fuzzy spheres, both of which are supersymmetric.', 'hep-th-0309258-3-10-1': 'In the [MATH] symmetric subspace which one may see in the [MATH]-wave background ([REF]), two fuzzy spheres are taken to be at the origin.', 'hep-th-0309258-3-10-2': 'In the [MATH] symmetric space, one fuzzy sphere is located at the origin, while the other fuzzy sphere is taken to rotate around the origin with a fixed distance.', 'hep-th-0309258-3-10-3': 'It should be noted that this configuration is allowed as a classical solution of the equations of motion and furthermore the rotating fuzzy sphere itself is supersymmetric.', 'hep-th-0309258-3-10-4': 'We will evaluate the one-loop corrections to the configuration and obtain the effective potential.', 'hep-th-0309258-3-10-5': 'As we will see, the effective potential is flat.', 'hep-th-0309258-3-10-6': 'This implies that the whole configuration of fuzzy spheres is also supersymmetric.', 'hep-th-0309258-3-10-7': 'One may argue that the flat potential is natural since each fuzzy sphere configuration is supersymmetric.', 'hep-th-0309258-3-10-8': 'However, the situation is unconventional from the viewpoint of the flat space matrix model [CITATION] and indicates one of intriguing properties of the plane-wave matrix model.', 'hep-th-0309258-3-10-9': 'Moreover, the flat potential shows us the possibility that the plane-wave matrix model has the flat directions which have not been observed in it.', 'hep-th-0309258-3-11-0': 'The organization of this paper is as follows.', 'hep-th-0309258-3-11-1': 'In the next section, we give the action of the plane-wave matrix model and consider its classical solutions focused on our concern.', 'hep-th-0309258-3-11-2': 'The expansion of the action around a given arbitrary background is given in section [REF].', 'hep-th-0309258-3-11-3': 'In section [REF], we set up the background configuration and consider the fluctuations around it.', 'hep-th-0309258-3-11-4': 'In section [REF], the one-loop stability of each fuzzy sphere is checked for arbitrary size.', 'hep-th-0309258-3-11-5': 'In section [REF], we evaluate the path integration of fluctuations responsible for the interaction between fuzzy spheres.', 'hep-th-0309258-3-11-6': 'It will be shown that the one-loop effective potential is flat.', 'hep-th-0309258-3-11-7': 'Thus, the fuzzy spheres do not feel any force.', 'hep-th-0309258-3-11-8': 'Finally, conclusion and discussion will be given in section [REF].', 'hep-th-0309258-3-11-9': 'We discuss the possibility on the existence of flat directions.', 'hep-th-0309258-3-12-0': '# Plane-wave matrix model and classical solutions', 'hep-th-0309258-3-13-0': 'The plane-wave matrix model is basically composed of two parts.', 'hep-th-0309258-3-13-1': 'One part is the usual matrix model based on eleven-dimensional flat space-time, that is, the flat space matrix model, and another is a set of terms reflecting the structure of the maximally supersymmetric eleven dimensional plane-wave background, Eq. ([REF]).', 'hep-th-0309258-3-13-2': 'Its action is [EQUATION] where each part of the action on the right hand side is given by [EQUATION]', 'hep-th-0309258-3-13-3': 'Here, [MATH] is the radius of circle compactification along [MATH] and [MATH] is the covariant derivative with the gauge field [MATH], [EQUATION]', 'hep-th-0309258-3-13-4': 'For dealing with the problem in this paper, it is convenient to rescale the gauge field and parameters as [EQUATION]', 'hep-th-0309258-3-13-5': 'With this rescaling, the radius parameter [MATH] disappears and the actions in Eq. ([REF]) become [EQUATION]', 'hep-th-0309258-3-13-6': 'The possible backgrounds allowed by the plane-wave matrix model are the classical solutions of the equations of motion for the matrix fields.', 'hep-th-0309258-3-13-7': 'Since the background that we are concerned about is purely bosonic, we concentrate on solutions of the bosonic fields [MATH].', 'hep-th-0309258-3-13-8': 'We would like to note that we will not consider all possible solutions but only those relevant to our interest for the fuzzy sphere interaction.', 'hep-th-0309258-3-13-9': 'Then, from the rescaled action, ([REF]), the bosonic equations of motion are derived as [EQUATION] where the over dot implies the time derivative [MATH].', 'hep-th-0309258-3-14-0': 'Except for the trivial [MATH] solution, the simplest one is the simple harmonic oscillator solution; [EQUATION] where [MATH] and [MATH] are the amplitudes and phases of oscillations respectively, and [MATH] is the [MATH] unit matrix.', 'hep-th-0309258-3-14-1': 'This oscillatory solution is special to the plane-wave matrix model due to the presence of mass terms for [MATH].', 'hep-th-0309258-3-14-2': 'It should be noted that, because of the mass terms, the configuration corresponding to the time dependent straight line motion, say [MATH] with non-zero constants [MATH] and [MATH], is not possible as a solution of ([REF]), that is, a classical background of plane-wave matrix model, contrary to the case of the flat space matrix model.', 'hep-th-0309258-3-14-3': 'As the generalization of the oscillatory solution, Eq. ([REF]), we get the solution of the form of diagonal matrix with each diagonal element having independent amplitude and phase.', 'hep-th-0309258-3-15-0': 'As for the non-trivial constant matrix solution, Eq. ([REF]) allows the following membrane fuzzy sphere or giant graviton solution: [EQUATION] where [MATH] satisfies the [MATH] algebra, [EQUATION]', 'hep-th-0309258-3-15-1': 'The reason why this solution is possible is basically because of the fact that the matrix field [MATH] feels an extra force due to the Myers interaction which may stabilize the oscillatory force.', 'hep-th-0309258-3-15-2': 'The fuzzy sphere solution [MATH] preserves the full 16 dynamical supersymmetries of the plane-wave and hence is 1/2-BPS object.', 'hep-th-0309258-3-15-3': 'We note that actually there is another fuzzy sphere solution of the form [MATH].', 'hep-th-0309258-3-15-4': 'However, it has been shown that such solution does not have quantum stability and is thus non-BPS object [CITATION].', 'hep-th-0309258-3-16-0': '# Matrix model expansion around general background', 'hep-th-0309258-3-17-0': 'In this section, the plane-wave matrix model is expanded around the general bosonic background, which is supposed to satisfy the classical equations of motion, Eq. ([REF]).', 'hep-th-0309258-3-18-0': 'We first split the matrix quantities into as follows: [EQUATION] where [MATH] and [MATH] are the classical background fields while [MATH] and [MATH] are the quantum fluctuations around them.', 'hep-th-0309258-3-18-1': 'The fermionic background [MATH] is taken to vanish from now on, since we will only consider the purely bosonic background.', 'hep-th-0309258-3-18-2': 'The quantum fluctuations are the fields subject to the path integration.', 'hep-th-0309258-3-18-3': 'In taking into account the quantum fluctuations, we should recall that the matrix model itself is a gauge theory.', 'hep-th-0309258-3-18-4': 'This implies that the gauge fixing condition should be specified before proceed further.', 'hep-th-0309258-3-18-5': 'In this paper, we take the background field gauge which is usually chosen in the matrix model calculation as [EQUATION]', 'hep-th-0309258-3-18-6': 'Then the corresponding gauge-fixing [MATH] and Faddeev-Popov ghost [MATH] terms are given by [EQUATION]', 'hep-th-0309258-3-18-7': 'Now by inserting the decomposition of the matrix fields ([REF]) into Eqs. ([REF]) and ([REF]), we get the gauge fixed plane-wave action [MATH] expanded around the background.', 'hep-th-0309258-3-18-8': 'The resulting acting is read as [EQUATION] where [MATH] represents the action of order [MATH] with respect to the quantum fluctuations and, for each [MATH], its expression is [EQUATION]', 'hep-th-0309258-3-18-9': 'Some comments are in order for the background gauge choice, Eq. ([REF]).', 'hep-th-0309258-3-18-10': 'One advantage of this gauge choice is that the quadratic part of the action in terms of fluctuations, that is, the quadratic action, is simplified.', 'hep-th-0309258-3-18-11': 'In more detail, there appears the term [MATH] in the expansion of the potential [MATH], which is canceled exactly by the same term with the opposite sign coming from the gauge fixing term of Eq. ([REF]) and hence absent in [MATH] of Eq. ([REF]).', 'hep-th-0309258-3-18-12': 'This cancellation has given some benefits in the actual flat space matrix model calculation.', 'hep-th-0309258-3-18-13': 'This is also the case in the present plane-wave matrix model, except however for the potential of [MATH].', 'hep-th-0309258-3-18-14': 'As we will see later, [MATH] is responsible for completing the [MATH] potential into the nice square form.', 'hep-th-0309258-3-18-15': 'Thus the same term with the opposite sign from the gauge fixing term remains in the quadratic action.', 'hep-th-0309258-3-18-16': 'At later stage, the presence of this term will have an important implication in taking into account of the unphysical gauge degrees of freedom which are eventually eliminated by those of ghosts.', 'hep-th-0309258-3-19-0': '# Fuzzy sphere configuration and fluctuations', 'hep-th-0309258-3-20-0': 'We now set up the background configuration for the membrane fuzzy spheres.', 'hep-th-0309258-3-20-1': 'Since we will study the interaction of two fuzzy spheres, the matrices representing the background have the [MATH] block diagonal form as [EQUATION] where [MATH] with [MATH] are [MATH] matrices.', 'hep-th-0309258-3-20-2': 'If we take [MATH] as [MATH] matrices, then [MATH].', 'hep-th-0309258-3-21-0': 'The two fuzzy spheres are taken to be static in the space where they span, and hence represented by the classical solution, ([REF]); [EQUATION] where, for each [MATH], [MATH] is in the [MATH]-dimensional irreducible representation of [MATH] and satisfies the [MATH] algebra, Eq. ([REF]).', 'hep-th-0309258-3-21-1': 'In the [MATH] symmetric transverse space, the fuzzy spheres are regarded as point objects, of course, in a sense of ignoring the matrix nature.', 'hep-th-0309258-3-21-2': 'We first let the second fuzzy sphere given by the background of [MATH] be at the origin in the transverse space and stay there.', 'hep-th-0309258-3-21-3': 'As for the first fuzzy sphere, it is made to move around the second sphere in the form of circular motion with the radius [MATH].', 'hep-th-0309258-3-21-4': 'Obviously, this configuration is one of the classical solutions of the equations of motion as one can see from Eq. ([REF]).', 'hep-th-0309258-3-21-5': 'Recalling that the transverse space is [MATH] symmetric, all the possible choices of two-dimensional sub-plane where the circular motion takes place are equivalent.', 'hep-th-0309258-3-21-6': 'Thus, without loss of generality, we can take a certain plane for the circular motion.', 'hep-th-0309258-3-21-7': 'In this paper, the [MATH]-[MATH] plane is chosen.', 'hep-th-0309258-3-21-8': 'Then the configuration in the transverse space is given by [EQUATION]', 'hep-th-0309258-3-21-9': 'Eqs. ([REF]) and ([REF]) compose the background configuration about which we are concerned, and all other elements of matrices [MATH] are set to zero.', 'hep-th-0309258-3-21-10': 'We would like to note that not only the fuzzy sphere at the origin given by [MATH] but also the rotating one, [MATH], is supersymmetric [CITATION].', 'hep-th-0309258-3-21-11': 'A schematic view of the background configuration is presented in Fig. [REF].', 'hep-th-0309258-3-22-0': 'If we evaluate the classical value of the action for this background, it is zero; [EQUATION]', 'hep-th-0309258-3-22-1': 'From now on, we are going to compute the one-loop correction to this action, that is, to the background, ([REF]) and ([REF]), due to the quantum fluctuations via the path integration of the quadratic action [MATH], and obtain the one-loop effective action [MATH] or the effective potential [MATH] as a function of [MATH], the radius of the circular motion.', 'hep-th-0309258-3-23-0': 'For the justification of one-loop computation or the semi-classical analysis, it should be made clear that [MATH] and [MATH] of Eq. ([REF]) can be regarded as perturbations.', 'hep-th-0309258-3-23-1': 'For this purpose, following [CITATION], we rescale the fluctuations and parameters as', 'hep-th-0309258-3-24-0': 'A ^-1/2 A , Y^I ^-1/2 Y^I , C ^-1/2 C , C ^-1/2 C ,', 'hep-th-0309258-3-25-0': 'r r , t ^-1 t .', 'hep-th-0309258-3-26-0': 'Under this rescaling, the action [MATH] in the background ([REF]) and ([REF]) becomes [EQUATION] where [MATH], [MATH] and [MATH] do not have [MATH] dependence.', 'hep-th-0309258-3-26-1': 'Now it is obvious that, in the large [MATH] limit, [MATH] and [MATH] can be treated as perturbations and the one-loop computation gives the sensible result.', 'hep-th-0309258-3-27-0': 'Based on the structure of ([REF]), we now write the quantum fluctuations in the [MATH] block matrix form as follows.', 'hep-th-0309258-3-28-0': 'A = Z_(1)^0 & ^0', 'hep-th-0309258-3-29-0': '^0 & Z_(2)^0 , Y^I = Z_(1)^I & ^I', 'hep-th-0309258-3-30-0': '^I & Z_(2)^I , = _(1) &', 'hep-th-0309258-3-31-0': '^ & _(2) ,', 'hep-th-0309258-3-32-0': 'C = C_(1) & C', 'hep-th-0309258-3-33-0': 'C^ & C_(2) , C = C_(1) & C', 'hep-th-0309258-3-34-0': 'C^& C_(2) .', 'hep-th-0309258-3-35-0': 'Although we denote the block off-diagonal matrices for the ghosts by the same symbols with those of the original ghost matrices, there will be no confusion since [MATH] matrices will never appear in what follows.', 'hep-th-0309258-3-35-1': 'The above form of matrices is convenient, since the block diagonal and block off-diagonal parts decouple from each other in the quadratic action and thus can be taken into account separately at one-loop level; [EQUATION] where [MATH]) implies the action for the block (off-) diagonal fluctuations.', 'hep-th-0309258-3-35-2': 'We note that there is no [MATH] parameter in [MATH] and [MATH] due to the above rescaling ([REF]).', 'hep-th-0309258-3-36-0': '# Stability of fuzzy sphere', 'hep-th-0309258-3-37-0': 'In this section, the path integration of [MATH] is performed.', 'hep-th-0309258-3-37-1': 'The resulting effective action will enable us to check the one-loop stability of the fuzzy sphere configuration, Eqs. ([REF]) and ([REF]).', 'hep-th-0309258-3-37-2': 'In fact, the background encoding the circular motion, Eq. ([REF]), does not contribute to [MATH] basically because of the fact that it is proportional to the identity matrix.', 'hep-th-0309258-3-37-3': 'This leads to the situation that the quantum fluctuations around one fuzzy sphere do not interact with those around another fuzzy sphere, and the effective action is just the sum of that for each fuzzy sphere.', 'hep-th-0309258-3-37-4': 'This can then be stated as [EQUATION] where three Lagrangians, [MATH], [MATH], and [MATH], are those for the bosonic, fermionic, and ghost fluctuations respectively around the [MATH]-th fuzzy sphere.', 'hep-th-0309258-3-37-5': 'We note that, at quadratic level, there is no mixing between these three kinds of fluctuations.', 'hep-th-0309258-3-38-0': '## Bosonic fluctuation', 'hep-th-0309258-3-39-0': 'We first evaluate the path integral of bosonic fluctuations.', 'hep-th-0309258-3-39-1': 'The corresponding Lagrangian is given by [EQUATION] where, as alluded to at the end of section [REF], the quadratic term of [MATH] coming from the gauge fixing term appears because the same term with opposite sign has been used for making the complete square form [MATH].', 'hep-th-0309258-3-40-0': 'In order to perform the actual path integration, it is useful to diagonalize the fluctuation matrices and obtain the mass spectrum.', 'hep-th-0309258-3-40-1': 'By the way, since the diagonalization itself has been already given in [CITATION], we will be brief in its presentation and present only the essential points.', 'hep-th-0309258-3-41-0': 'The starting point is the observation that the mass terms are written in terms of the commutators with [MATH] satisfying Eq. ([REF]), the [MATH] algebra.', 'hep-th-0309258-3-41-1': 'This indicates that we can use the representation theory of [MATH] for the diagonalization.', 'hep-th-0309258-3-41-2': 'We regard an [MATH] matrix as an [MATH]-dimensional reducible representation of [MATH], which decomposes into irreducible spin [MATH] representations with the range of [MATH] from [MATH] to [MATH], that is, [MATH].', 'hep-th-0309258-3-41-3': 'Based on this decomposition, an [MATH] matrix may be expanded as [EQUATION] where the [MATH] matrix [MATH] is the matrix spherical harmonics transforming in the irreducible spin [MATH] representation and [MATH] is the corresponding spherical mode.', 'hep-th-0309258-3-41-4': 'We note that [MATH] satisfies the following reality condition since the fluctuation matrices are Hermitian.', 'hep-th-0309258-3-41-5': '[EQUATION]', 'hep-th-0309258-3-41-6': 'The expansion of matrices in the [MATH] language enables us to use the properties of the [MATH] generators, which are given by [EQUATION] where [MATH].', 'hep-th-0309258-3-41-7': 'For the normalization of the matrix spherical harmonics, we choose [EQUATION]', 'hep-th-0309258-3-41-8': 'Having equipped with the necessary machinery, we now proceed the diagonalization.', 'hep-th-0309258-3-42-0': 'By direct application of Eqs. ([REF]), ([REF]), and ([REF]), the gauge field fluctuation [MATH] and the fluctuations in the [MATH] directions [MATH] are immediately diagonalized.', 'hep-th-0309258-3-42-1': 'The corresponding spherical modes and their masses are [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-3-43-0': 'As for the fluctuations in the [MATH] directions, we should first consider the potential [MATH] and diagonalize it by solving the eigenvalue problem, [EQUATION]', 'hep-th-0309258-3-43-1': 'By defining the combinations of matrices [MATH] and using Eqs. ([REF]) and ([REF]), it turns out that the eigenvalue [MATH] takes the values of [MATH], [MATH], or [MATH].', 'hep-th-0309258-3-43-2': 'Let us now denote the spherical eigenmodes of matrix eigenvectors as [MATH], [MATH], and [MATH] for [MATH], [MATH], and [MATH], respectively.', 'hep-th-0309258-3-43-3': 'The eigenmodes satisfy the reality condition in the form of Eq. ([REF]).', 'hep-th-0309258-3-43-4': 'Then the fluctuation matrices may be expressed with respect to these eigenmodes for each eigenvalue.', 'hep-th-0309258-3-44-0': 'For [MATH], we have the following expressions: [EQUATION] where [MATH], [MATH], and the normalization constant are chosen so that the kinetic term for the mode [MATH] is of the form [MATH].', 'hep-th-0309258-3-44-1': 'Here and in what follows, the summations over [MATH] and [MATH] with specified ranges are implicit.', 'hep-th-0309258-3-45-0': 'For [MATH], we have [EQUATION] where [MATH], [MATH], and the normalization constant are chosen in the same way with the case of [MATH].', 'hep-th-0309258-3-46-0': 'Finally, for [MATH], [EQUATION] with [MATH] and [MATH].', 'hep-th-0309258-3-46-1': 'As pointed out in [CITATION], the modes for [MATH] case correspond to the degrees of freedom for the gauge transformation and are thus unphysical.', 'hep-th-0309258-3-46-2': 'Since the authors of [CITATION] took the physical Weyl gauge, [MATH], and worked in the operator formulation, it was not necessary to consider these unphysical modes seriously.', 'hep-th-0309258-3-46-3': 'However, they should be involved properly in the present context because our gauge choice is the covariant background gauge and we work in the path integral formulation.', 'hep-th-0309258-3-47-0': 'We turn to the remaining potential term in the Lagrangian ([REF]) which is the square of [MATH].', 'hep-th-0309258-3-47-1': 'Interestingly enough, the matrices expanded in terms of the eigenmodes [MATH] and [MATH], Eqs. ([REF]) and ([REF]), do not give any contribution, since we obtain [EQUATION] for [MATH] and [MATH].', 'hep-th-0309258-3-47-2': 'Only the modes corresponding to [MATH] contribute to the potential, which is evaluated as [EQUATION] for each [MATH] and [MATH].', 'hep-th-0309258-3-48-0': 'Having diagonalized the fluctuations [MATH], we get the following list of spherical modes in the [MATH] directions with their masses and the ranges of spin [MATH]: [EQUATION] where [MATH].', 'hep-th-0309258-3-49-0': 'With respect to the spherical modes of Eqs. ([REF]) and ([REF]), the Lagrangian ([REF]) is then written in the diagonalized form as [EQUATION] where the sum over [MATH] with the range [MATH] is understood.', 'hep-th-0309258-3-49-1': 'The Lagrangian is just the sum of various harmonic oscillator Lagrangians, which are non-interacting with each other, and therefore the path integration is now straightforward.', 'hep-th-0309258-3-49-2': 'As a result, what we obtain is [EQUATION]', 'hep-th-0309258-3-50-0': '## Fermionic fluctuation', 'hep-th-0309258-3-51-0': 'We turn to the path integral of fermionic fluctuations.', 'hep-th-0309258-3-51-1': 'The Lagrangian is written as [EQUATION]', 'hep-th-0309258-3-51-2': 'It is convenient for our calculation of fermionic part to introduce the [MATH] formulation since the preserved symmetry in the plane-wave matrix model is [MATH] rather than [MATH].', 'hep-th-0309258-3-51-3': 'In this formulation the [MATH] spinor [MATH] is decomposed as [EQUATION] where [MATH] implies a fundamental [MATH] index and [MATH] is a fundamental [MATH] index.', 'hep-th-0309258-3-51-4': 'According to this decomposition, we may take the expression of [MATH] as [EQUATION]', 'hep-th-0309258-3-51-5': "We also rewrite the [MATH] gamma matrices [MATH]'s in terms of [MATH] and [MATH] ones as follows: [EQUATION] where the [MATH]'s are the standard [MATH] Pauli matrices and six of [MATH] are taken to form a basis of [MATH] anti-symmetric matrices.", 'hep-th-0309258-3-51-6': 'The original [MATH] Clifford algebra is satisfied as long as we take normalizations so that the gamma matrices [MATH] with [MATH] indices satisfy the algebra [EQUATION]', 'hep-th-0309258-3-51-7': 'By introducing the above [MATH] formulation, the Lagrangian for the fermionic fluctuations is rewritten as [EQUATION]', 'hep-th-0309258-3-51-8': 'The diagonalization of the Lagrangian proceeds in the same way as in the previous subsection.', 'hep-th-0309258-3-51-9': 'In the present case, it is achieved by solving the following eigenvalue problem: [EQUATION]', 'hep-th-0309258-3-51-10': 'We first expand the [MATH] fermionic matrix [MATH] in terms of the matrix spherical harmonics as [EQUATION]', 'hep-th-0309258-3-51-11': 'If we plug this expansion into the above eigenvalue equation and use the [MATH] algebra Eq. ([REF]), we see that the eigenvalues are [MATH] and [MATH] [CITATION].', 'hep-th-0309258-3-51-12': 'Let us now introduce the fermionic spherical eigenmodes [MATH] and [MATH] corresponding to [MATH] and [MATH] respectively.', 'hep-th-0309258-3-51-13': 'As for the spinorial structure, [MATH]) carries the (anti) fundamental [MATH] index.', 'hep-th-0309258-3-51-14': 'We note that, from now on, we suppress the [MATH] indices.', 'hep-th-0309258-3-52-0': 'Then, as the eigenstate for [MATH], the matrix [MATH] has the expansion in terms of the eigenmode [MATH] as [EQUATION] where [MATH], [MATH], and the subscripts [MATH] denote the [MATH] indices measured by [MATH].', 'hep-th-0309258-3-53-0': 'On the other hand, for [MATH], we have [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-3-54-0': 'By using the mode-expansions, Eqs. ([REF]) and ([REF]), and the [MATH] algebra ([REF]), the fermionic Lagrangian ([REF]) becomes [EQUATION] where it should be understood that there is the summation over [MATH] with the range [MATH].', 'hep-th-0309258-3-54-1': 'Now, the path integration for this Lagrangian may be evaluated immediately, and gives [EQUATION]', 'hep-th-0309258-3-55-0': '## Ghost fluctuation', 'hep-th-0309258-3-56-0': 'As the final part of the diagonal fluctuations, we consider the Lagrangian for the ghost fluctuations, which is given by [EQUATION]', 'hep-th-0309258-3-56-1': 'By using the [MATH] algebra ([REF]) and the following expansions in terms of the matrix spherical harmonics [EQUATION] with [MATH] and [MATH], we may rewrite the above Lagrangian as [EQUATION] where the sum over [MATH] is implicit.', 'hep-th-0309258-3-56-2': 'Then the result of the path integration for this Lagrangian is [EQUATION]', 'hep-th-0309258-3-57-0': '## One-loop stability', 'hep-th-0309258-3-58-0': 'In the previous subsections, we have evaluated the path integrals for the block diagonal fluctuations of the action [MATH], ([REF]).', 'hep-th-0309258-3-58-1': 'Thus, we may now consider the effective action.', 'hep-th-0309258-3-58-2': 'As can be inferred from Eq. ([REF]), it is enough to consider only the effective action of a given fuzzy sphere.', 'hep-th-0309258-3-58-3': 'If we let [MATH] be the effective action for the [MATH]-th fuzzy sphere, then it is given by [EQUATION]', 'hep-th-0309258-3-58-4': 'The right hand side may be viewed just as the product of determinants of non-interacting quantum mechanical simple harmonic oscillators with various frequencies.', 'hep-th-0309258-3-58-5': 'In fact, as we will see in the next section, this is also the case for the effective action describing the interaction between two fuzzy spheres.', 'hep-th-0309258-3-58-6': 'Thus, it is worthwhile to consider a generic situation, which is useful both in the present and the next section.', 'hep-th-0309258-3-59-0': 'Let us then consider a situation, [EQUATION] where [MATH] is included for the later usage.', 'hep-th-0309258-3-59-1': 'The formal expression of the effective action is read as [EQUATION]', 'hep-th-0309258-3-59-2': 'By using the relation [MATH] for a given matrix [MATH], we may present a prototype calculation of single determinant as [EQUATION] where the momentum integration has appeared by inserting the momentum space identity, [MATH], between bra and ket vectors for time, and the final result has been derived by using the formula [EQUATION]', 'hep-th-0309258-3-59-3': 'If we apply the result of this single determinant calculation to Eq. ([REF]), then the effective action is finally obtained as [EQUATION]', 'hep-th-0309258-3-59-4': 'We now return to the present case.', 'hep-th-0309258-3-59-5': 'First of all, we observe that the ghost part ([REF]) eliminates two determinant factors of the bosonic result ([REF]) which are actually the contributions from the unphysical modes.', 'hep-th-0309258-3-59-6': 'Thus, what we get from the bosonic and the ghost parts is only the determinant factors from the physical bosonic modes.', 'hep-th-0309258-3-59-7': 'By using the generic expression, Eq. ([REF]), it turns out that their contributions to the effective action are [EQUATION]', 'hep-th-0309258-3-59-8': 'On the other hand, the contribution from the fermionic part ([REF]) is obtained as [EQUATION]', 'hep-th-0309258-3-59-9': 'Therefore, there is no net contribution to the effective action and we can conclude that the one loop effective action obtained after integrating out each of the block diagonal fluctuations vanishes; [EQUATION]', 'hep-th-0309258-3-59-10': 'This indicates that each membrane fuzzy sphere has quantum stability at least at one-loop level, by which we mean that the fuzzy sphere does not receive quantum corrections.', 'hep-th-0309258-3-59-11': 'We note that, if we take [MATH] in the above two contributions, the result of the previous path integral computation [CITATION] is recovered.', 'hep-th-0309258-3-60-0': '# Interaction between fuzzy spheres', 'hep-th-0309258-3-61-0': 'We turn to the action [MATH] in Eq. ([REF]) for the block off-diagonal fluctuations and compute the one-loop effective potential describing the interaction between two fuzzy spheres.', 'hep-th-0309258-3-61-1': 'The action is given by [EQUATION] where [MATH], [MATH] and [MATH] are the Lagrangians for the off-diagonal bosonic, fermionic and ghost fluctuations of Eq. ([REF]) respectively and their explicit expressions will be presented in due course.', 'hep-th-0309258-3-61-2': 'Since, as in the previous section, there is no mixing between different kinds of fluctuations, each Lagrangian can be considered independently.', 'hep-th-0309258-3-62-0': 'In calculating the effective action, the prescription given by Kabat and Taylor [CITATION] is usually used.', 'hep-th-0309258-3-62-1': 'We note however that, at the present situation, it is more helpful to use the expansion in terms of the matrix spherical harmonics as in the previous section.', 'hep-th-0309258-3-63-0': '## Bosonic fluctuation', 'hep-th-0309258-3-64-0': 'Let us first consider the bosonic Lagrangian and evaluate its path integral.', 'hep-th-0309258-3-64-1': 'The Lagrangian is [EQUATION]', 'hep-th-0309258-3-64-2': 'Here, adopting the notation of [CITATION], we have defined [EQUATION] where [MATH] is the [MATH] matrix which is a block at [MATH]-th row and [MATH]-th column in the blocked form of a given matrix [MATH].', 'hep-th-0309258-3-64-3': 'In the present case, [MATH] and [MATH] take values of [MATH] and [MATH].', 'hep-th-0309258-3-64-4': 'For example, if we look at the [MATH] block matrix form of the gauge field fluctuation [MATH] in Eq. ([REF]), then [MATH], [MATH], and [MATH].', 'hep-th-0309258-3-65-0': 'The matrix fields, [MATH], [MATH], and [MATH] are coupled with each other through the circular motion background.', 'hep-th-0309258-3-65-1': 'Since the time dependent trigonometric functions may make the formulation annoying, we consider the newly defined matrix variables as [EQUATION] where [MATH] may be interpreted as the fluctuation tangential to the circular motion at time [MATH] and [MATH] as the normal fluctuation.', 'hep-th-0309258-3-65-2': 'In terms of these fluctuations, the terms in the Lagrangian ([REF]), which are dependent on [MATH] and [MATH], are rewritten as [EQUATION] where we no longer see the explicit time dependent classical functions.', 'hep-th-0309258-3-66-0': 'We are now in a position to consider the diagonalization of the Lagrangian [MATH].', 'hep-th-0309258-3-66-1': 'We note that, compared to the case in the previous section, we are in a somewhat different situation.', 'hep-th-0309258-3-66-2': 'The fluctuation matrices are [MATH] or [MATH] ones, while those in the last section are square [MATH] matrices.', 'hep-th-0309258-3-66-3': 'This means that, when we regard an [MATH] block off-diagonal matrix as an [MATH]-dimensional reducible representation of [MATH], it has the decomposition into irreducible spin [MATH] representations with the range [MATH], that is, [MATH], and may be expanded as [EQUATION] where [MATH] is the [MATH] matrix spherical harmonics transforming in the irreducible spin [MATH] representation and [MATH] is the corresponding spherical mode.', 'hep-th-0309258-3-66-4': 'The basic operation between [MATH] generators [MATH] and [MATH] is given not by the commutator but by the [MATH] operator ([REF]).', 'hep-th-0309258-3-66-5': 'Thus the algebraic properties for the present situation are given by Eq. ([REF]) where the commutator is replaced by the [MATH] operator.', 'hep-th-0309258-3-67-0': 'Having the expansion and algebraic properties, the diagonalization proceeds in exactly the same way as in the previous section.', 'hep-th-0309258-3-67-1': 'Hence, by noting that one may find a detailed procedure in [CITATION], we will present only the results of diagonalization for the Lagrangian.', 'hep-th-0309258-3-67-2': 'We observe that, because of the background for the circular motion in [MATH]-[MATH] plane, the [MATH] symmetry is broken to [MATH], while the [MATH] symmetry remains intact.', 'hep-th-0309258-3-67-3': 'This fact naturally leads us to break the bosonic Lagrangian ([REF]) into three parts as follows: [EQUATION] where [MATH] is the Lagrangian for [MATH], [MATH] is for [MATH] with [MATH], and [MATH] represents the [MATH] rotational part described by [MATH], [MATH], and the gauge fluctuation [MATH].', 'hep-th-0309258-3-68-0': 'We first consider [MATH] and its path integration.', 'hep-th-0309258-3-68-1': 'Its diagonalized form is obtained by [EQUATION] where the sum of [MATH] over the range [MATH] is implicit and the spherical modes [MATH] are the degrees of freedom for the gauge transformation, the block off-diagonal counterpart of [MATH], ([REF]), in the previous section.', 'hep-th-0309258-3-68-2': '[MATH] and [MATH] are the block off-diagonal counterparts of [MATH] and [MATH], respectively.', 'hep-th-0309258-3-68-3': 'The path integral of this Lagrangian is straightforward and results in [EQUATION]', 'hep-th-0309258-3-68-4': 'As for [MATH], we have as its diagonalized form [EQUATION] where [MATH] and [MATH].', 'hep-th-0309258-3-68-5': 'Its path integration leads us to have [EQUATION]', 'hep-th-0309258-3-68-6': 'For the rotational part, the diagonalized Lagrangian is obtained as [EQUATION] where [MATH].', 'hep-th-0309258-3-68-7': 'Since all the coupling coefficients between the modes are time-independent constants, the path integral of this Lagrangian is readily evaluated and gives [EQUATION]', 'hep-th-0309258-3-69-0': '## Fermionic fluctuation', 'hep-th-0309258-3-70-0': 'The fermionic Lagrangian of the block off-diagonal action [MATH], ([REF]), is [EQUATION]', 'hep-th-0309258-3-70-1': 'As was done in Eq. ([REF]), we decompose the fermion [MATH] into [MATH] and [MATH] according to [MATH].', 'hep-th-0309258-3-70-2': 'Then the expression of [MATH] may be taken as [EQUATION] where [MATH].', 'hep-th-0309258-3-70-3': 'It should be noted that, contrary to the block diagonal fermionic matrix [MATH] of ([REF]), [MATH] is the two copy of the [MATH] representation [MATH] as one may see from Eq. ([REF]).', 'hep-th-0309258-3-70-4': 'This means that [MATH] and [MATH] should be treated as independent spinors not related in any way.', 'hep-th-0309258-3-70-5': 'By plugging the decomposition ([REF]) into the Lagrangian ([REF]), we have [EQUATION]', 'hep-th-0309258-3-70-6': 'The Lagrangian has explicit time dependence due to the presence of the circular motion background.', 'hep-th-0309258-3-70-7': 'In order to hide it, we take the fermionic field [MATH] as [EQUATION] where we have introduced a new fermionic field [MATH] which is in the [MATH] of [MATH].', 'hep-th-0309258-3-70-8': 'Then, by using the following identities, [EQUATION] which are proved via the Clifford algebra ([REF]), we may show that the Lagrangian ([REF]) becomes [EQUATION]', 'hep-th-0309258-3-70-9': 'The explicit time dependent classical functions disappear and the term containing [MATH] appears, which originates from the kinetic term of [MATH] in ([REF]).', 'hep-th-0309258-3-71-0': 'With the above Lagrangian ([REF]), the diagonalization proceeds in the same manner with that for the block diagonal fermionic Lagrangian ([REF]), except for some differences pointed out in the previous subsection.', 'hep-th-0309258-3-71-1': 'In the expansion of [MATH] and [MATH] indices are suppressed.)', 'hep-th-0309258-3-71-2': 'in terms of the matrix spherical harmonics like ([REF]), let us denote their spherical modes as [MATH] and [MATH] respectively.', 'hep-th-0309258-3-71-3': 'Then the diagonalization results in [MATH] and [MATH].', 'hep-th-0309258-3-71-4': 'The modes [MATH] and [MATH] have the same mass of [MATH] with [MATH].', 'hep-th-0309258-3-71-5': 'For the modes [MATH] and [MATH], their mass is [MATH] with [MATH].', 'hep-th-0309258-3-71-6': 'All the modes have the same range of [MATH] as [MATH].', 'hep-th-0309258-3-71-7': 'With respect to these diagonalized spherical modes, the Lagrangian ([REF]) is written as [EQUATION] where [MATH] and the [MATH] indices are suppressed.', 'hep-th-0309258-3-72-0': 'The product [MATH] measures the [MATH] chirality in the [MATH]-[MATH] plane where the circular motion takes place.', 'hep-th-0309258-3-72-1': 'Since [MATH], its eigenvalues are [MATH].', 'hep-th-0309258-3-72-2': 'Each spherical mode may split into modes having definite [MATH] eigenvalues as follows: [EQUATION] where the modes on the right hand sides satisfy [EQUATION]', 'hep-th-0309258-3-72-3': 'One may see that the Lagrangian ([REF]) composed of two independent parts.', 'hep-th-0309258-3-72-4': 'One is for [MATH] and [MATH], and the other one for [MATH] and [MATH].', 'hep-th-0309258-3-72-5': 'If we first consider the part for [MATH] and [MATH], then, according to the above splitting of modes, we have [EQUATION] where again [MATH].', 'hep-th-0309258-3-72-6': 'The path integration of this part is straightforward and gives [EQUATION]', 'hep-th-0309258-3-72-7': 'The other part for [MATH] and [MATH] is obtained as [EQUATION] where the sum over [MATH] for [MATH] is implicit.', 'hep-th-0309258-3-72-8': 'The path integration of this part results in [EQUATION]', 'hep-th-0309258-3-73-0': '## Ghost fluctuation', 'hep-th-0309258-3-74-0': 'Finally, we consider the path integration for the ghost part of the action [MATH] ([REF]).', 'hep-th-0309258-3-74-1': 'The block off-diagonal Lagrangian for the ghosts is [EQUATION]', 'hep-th-0309258-3-74-2': 'The diagonalization may be carried out by using the same logic in the previous subsections.', 'hep-th-0309258-3-74-3': 'We expand the ghost fields [MATH] and [MATH] in terms of the matrix spherical harmonics according to ([REF]), and denote their spherical modes as [MATH] and [MATH] respectively.', 'hep-th-0309258-3-74-4': 'Then the diagonalized Lagrangian is obtained as [EQUATION] where the sum over [MATH] for the range [MATH] is implicit.', 'hep-th-0309258-3-75-0': 'The path integral for the above diagonalized Lagrangian is then immediately evaluated as follows: [EQUATION]', 'hep-th-0309258-3-76-0': '## Effective potential', 'hep-th-0309258-3-77-0': 'Having evaluated the path integral for each part of the block off-diagonal action [MATH] ([REF]), the one-loop effective action, [MATH], which describes the interaction between two membrane fuzzy spheres with the classical configuration ([REF]) and ([REF]), is now given by [EQUATION] where the subscripts on the right hand side denote the bosonic, the fermionic, and the ghost contributions.', 'hep-th-0309258-3-77-1': 'We see that the ghost contribution, Eq. ([REF]), eliminate those of unphysical gauge degrees of freedom present in bosonic contributions, Eqs. ([REF]) and ([REF]).', 'hep-th-0309258-3-77-2': 'Thus only the physical degrees of freedom contribute to the effective action, as it should be.', 'hep-th-0309258-3-78-0': 'The explicit expression of the effective action is obtained by consulting the generic result presented from Eq. ([REF]) to ([REF]).', 'hep-th-0309258-3-78-1': 'The effective potential, [MATH], about which we are concerned in this subsection, is then given by [MATH].', 'hep-th-0309258-3-78-2': 'In expressing the effective potential, it is convenient to write [MATH] as [EQUATION] where [MATH]) is the contribution of the physical bosonic (fermionic) degrees of freedom to the effective potential.', 'hep-th-0309258-3-78-3': 'The expression that we obtain for [MATH] is then [EQUATION] while, for the fermionic contribution [MATH], we obtain [EQUATION]', 'hep-th-0309258-3-78-4': 'The above expressions show that [MATH] and [MATH] have the same structure except for the ranges of [MATH].', 'hep-th-0309258-3-78-5': 'This leads us to expect a great amount of cancellation.', 'hep-th-0309258-3-78-6': 'Indeed, what we have found is that they are exactly the same.', 'hep-th-0309258-3-78-7': 'One way to see the cancellation is to adjust all the ranges of the summation parameter [MATH] to the range [MATH].', 'hep-th-0309258-3-78-8': 'Therefore, the one-loop effective potential [MATH], ([REF]), as a function of the distance [MATH] between two fuzzy spheres is just flat potential; [EQUATION]', 'hep-th-0309258-3-79-0': '# Conclusion and discussion', 'hep-th-0309258-3-80-0': 'We have studied the one-loop quantum corrections to a classical background of the plane-wave matrix model in the framework of the path integration.', 'hep-th-0309258-3-80-1': 'The background is composed of two supersymmetric membrane fuzzy spheres, ([REF]) and ([REF]).', 'hep-th-0309258-3-80-2': 'One fuzzy sphere is located at the origin in the [MATH] symmetric space and the other one rotates around it with the distance [MATH].', 'hep-th-0309258-3-81-0': 'Firstly, the quantum stability of each fuzzy sphere has been shown.', 'hep-th-0309258-3-81-1': 'In fact, the stability check already has been done in the operator [CITATION] as well as in the path integral [CITATION] formulation.', 'hep-th-0309258-3-81-2': 'However, while the size [MATH] of the fuzzy sphere has been taken arbitrary in the operator formulation, it has been restricted to the minimal one, that is [MATH], in the path integral formulation.', 'hep-th-0309258-3-81-3': 'Since the fuzzy sphere in this paper has an arbitrary size, our stability check can be regarded as the full generalization of the previous path integral result.', 'hep-th-0309258-3-82-0': 'Secondly, the one-loop effective potential describing the interaction between two fuzzy spheres has been calculated as a function of the distance [MATH].', 'hep-th-0309258-3-82-1': 'Interestingly, the result is that the effective potential [MATH] is flat and thus the fuzzy spheres do not feel any force.', 'hep-th-0309258-3-82-2': 'This implies that the whole configuration of two fuzzy spheres given by Eqs. ([REF]) and ([REF]) is supersymmetric.', 'hep-th-0309258-3-82-3': 'Although the flatness of the effective potential is the one-loop result, we expect that the result holds also for higher loops.', 'hep-th-0309258-3-82-4': 'For the supersymmetric properties of the fuzzy sphere configuration itself, the study of supersymmetry algebra may be more helpful rather than the path integral formulation.', 'hep-th-0309258-3-82-5': 'It would be interesting to investigate the configuration in this paper through the supersymmetry algebra.', 'hep-th-0309258-3-83-0': 'Let us consider the radial distance [MATH] between two fuzzy spheres and discuss about its possible interpretation.', 'hep-th-0309258-3-83-1': 'We first consider the form of the effective action when [MATH] is time dependent.', 'hep-th-0309258-3-83-2': 'Since the circular motion is taken as the background, [MATH] is constant in this paper.', 'hep-th-0309258-3-83-3': 'However, if we slightly deform the circular motion to the elliptic one, [MATH] can be made to have time dependence.', 'hep-th-0309258-3-83-4': 'We can also make the time variation of it, [MATH], arbitrarily small by controlling the degree of deformation.', 'hep-th-0309258-3-83-5': 'In this case, it is expected that the fuzzy spheres begin to interact and the effective action [MATH] may be written as [EQUATION] where the kinetic term for [MATH] is the value of the classical action [MATH] with the rescaling ([REF]), [MATH] is the would-be one-loop contribution to the effective action with the property [MATH], and the term of order [MATH] implies the higher loop corrections.', 'hep-th-0309258-3-84-0': 'If [MATH], the above effective action vanishes and the supersymmetric situation is recovered.', 'hep-th-0309258-3-84-1': 'This is reminiscent of the effective action for graviton-graviton scattering in the flat space matrix model [CITATION].', 'hep-th-0309258-3-84-2': 'The distance between two gravitons comes from the flat directions which are continuous moduli or supersymmetric vacua making the potential of the flat space matrix model vanish.', 'hep-th-0309258-3-84-3': 'In the plane-wave matrix model, it is known that there is no continuous moduli and hence we do not have flat directions.', 'hep-th-0309258-3-84-4': 'However, if we look at the tree level action ([REF]) evaluated for the fuzzy sphere configuration, ([REF]) and ([REF]), we see that it vanishes exactly and does not depend on [MATH] which is continuous from [MATH] to [MATH].', 'hep-th-0309258-3-84-5': 'This means that, as long as the fuzzy sphere dynamics is concerned, the radius of the circular motion may be interpreted as the flat direction.', 'hep-th-0309258-3-85-0': 'In this paper, the circular motion takes place in the [MATH]-[MATH] sub-plane of the [MATH] symmetric space.', 'hep-th-0309258-3-85-1': 'Since the [MATH] symmetric space has two other sub-planes, that is [MATH]-[MATH] and [MATH]-[MATH], and we may embed the circular motion in one of those, there may be three flat directions in total, which are radial directions of three sub-planes.', 'hep-th-0309258-3-85-2': 'However, it is not obvious whether these three directions are connected in a continuous way or not, because of the supersymmetric property of rotating fuzzy sphere; all the points in the supersymmetric moduli are expected to preserve a fixed fraction of supersymmetry.', 'hep-th-0309258-3-85-3': 'While the fuzzy sphere rotating in only one sub-plane is 1/2-BPS object, it is generically 1/4-BPS when it has angular momenta also in the other sub-planes [CITATION].', 'hep-th-0309258-3-85-4': 'If we turn to the [MATH] symmetric space, it seems that we do not have flat directions, since only the fuzzy sphere rotating with fixed radius is supersymmetric [CITATION].', 'hep-th-0309258-3-86-0': 'In view of the gauge/gravity duality, it is interesting to study the same situation in the supergravity side.', 'hep-th-0309258-3-86-1': 'In the large [MATH] limit, the leading order interaction terms obtained from the supergravity side analysis would match with those from the matrix theory analysis.', 'hep-th-0309258-3-86-2': 'We expect that the supergravity side analysis is helpful and provides clearer understanding about the structure of the effective potential.', 'hep-th-0309258-3-87-0': 'The calculation in this paper has been carried out in the Minkowskian time signature.', 'hep-th-0309258-3-87-1': 'The basic reason why we have not taken the Wick rotation for some convenience in the actual calculation is the periodic nature of the circular motion background ([REF]).', 'hep-th-0309258-3-87-2': 'The naive change of the Minkowskian time to the Euclidean one in the process of calculation breaks the periodicity of the background and the reality of the action.', 'hep-th-0309258-3-87-3': 'If one wants to study the time dependent background with the Euclidean time signature, he or she should begin with the Euclidean action at the first setup.', 'hep-th-0309258-3-87-4': 'In the Euclidean case, the time dependent solutions to the equations of motion are given by hyperbolic functions, which give open paths not periodic ones.', 'hep-th-0309258-3-87-5': 'In the sense that there is no classical background solution corresponding to open path in the Minkowskian time plane-wave matrix model, it would be interesting to consider the model in the Euclidean time for studying such a background.', 'hep-th-0309258-3-88-0': 'In the situation where the one-loop effective potential is flat, it is a natural step to consider the case where the radius [MATH] between two fuzzy spheres is time dependent and calculate the form of the interaction.', 'hep-th-0309258-3-88-1': 'We hope to return to this issue in the near future.'}

1406.6263

{'1406.6263-1-0-0': 'The nonperturbative regime of electron-positron pair creation by a relativistic proton beam colliding with a highly intense bichromatic laser field is studied.', '1406.6263-1-0-1': 'The laser wave is composed of a strong low-frequency and a weak high-frequency mode, with mutually orthogonal polarization vectors.', '1406.6263-1-0-2': 'We show that the presence of the high-frequency field component can strongly enhance the pair-creation rate.', '1406.6263-1-0-3': 'Besides, a characteristic influence of the high-frequency mode on the angular and energy distributions of the created particles is demonstrated, both in the nuclear rest frame and the laboratory frame.', '1406.6263-1-1-0': '# Introduction', '1406.6263-1-2-0': 'In the presence of very strong electromagnetic fields, the quantum vacuum can decay into electron-positron pairs [CITATION].', '1406.6263-1-2-1': 'Pair creation in a constant electric field was studied in detail by Schwinger [CITATION], employing the then newly established methods of quantum electrodynamics (QED).', '1406.6263-1-2-2': 'The resulting pair-creation rate has the form [MATH], where [MATH] is the applied field and [MATH] the critical field of QED.', '1406.6263-1-2-3': 'Here, [MATH] and [MATH] are the positron charge and mass, respectively, and relativistic units with [MATH] are used.', '1406.6263-1-2-4': 'The Schwinger rate has a manifestly nonperturbative character due to its non-analytic field dependence.', '1406.6263-1-2-5': 'Therefore, it is of fundamental importance for our understanding of nonperturbative quantum field theories.', '1406.6263-1-2-6': 'An experimental verification has so far been prevented, though, by the huge value of [MATH], which is not accessible in the laboratory yet.', '1406.6263-1-3-0': 'The recent progress in high-intensity laser devices has strongly revived the interest in Schwinger pair creation.', '1406.6263-1-4-0': 'In particular, various schemes have been proposed to facilitate its observation.', '1406.6263-1-4-1': 'Among them is a dynamically assisted variant where a rapidly oscillating electric field is superimposed onto a slowly varying electric field [CITATION].', '1406.6263-1-4-2': 'The total rate for pair creation in this field combination was shown to be strongly enhanced, while preserving its nonperturbative character.', '1406.6263-1-4-3': 'This interesting prediction has led to a number of subsequent investigations.', '1406.6263-1-4-4': 'Currently, the momentum distributions of pairs created by the dynamically assisted mechanism are under active scrutiny [CITATION].', '1406.6263-1-5-0': 'Pair-creation phenomena resembling the Schwinger effect can also occur when a beam of charged particles collides with a counterpropagating intense laser beam.', '1406.6263-1-5-1': 'For instance, pair creation has been theoretically predicted in relativistic proton-laser collisions via a strong-field version of the Bethe-Heitler effect (see, e.g., [CITATION]).', '1406.6263-1-5-2': 'When the laser frequency [MATH] is relatively small, so that the dimensionless parameter [MATH] becomes large, [MATH], the total rate for this process adopts the form [MATH] [CITATION].', '1406.6263-1-5-3': 'Here, [MATH] denotes the laser field amplitude transformed to the rest frame of the proton, which is assumed to be sub-critical: [MATH].', '1406.6263-1-5-4': 'The proton Lorentz factor [MATH] and reduced velocity [MATH] have been introduced here.', '1406.6263-1-5-5': 'The similarity with the Schwinger rate is due to the fact that the characteristic length scale of pair creation in this parameter regime is much shorter than the laser wavelength, so that the field appears as quasi-static during the process.', '1406.6263-1-6-0': 'An experimental setup to observe the strong-field Bethe-Heitler process could, in principle, be realized by utilizing the highly relativistic proton beam of the Large Hadron Collider (LHC) at CERN ([MATH]) in conjunction with a counterpropagating high-intensity laser beam ([MATH]V/cm).', '1406.6263-1-6-1': 'In fact, a very similar scheme was employed to observe pair creation by multiphoton absorption at the Stanford Linear Accelerator Center, where a [MATH] electron beam colliding with an intense optical laser pulse triggered the pair creation [CITATION].', '1406.6263-1-6-2': 'This experiment, however, did not probe the Schwinger regime of pair creation because the field intensity was somewhat too low ([MATH]).', '1406.6263-1-6-3': 'Instead, the measured pair-creation rate showed a power-law dependence on the field strength.', '1406.6263-1-7-0': 'Note that, a rate of the form [MATH], with [MATH] denoting the number of absorbed laser photons, follows from a consideration within the [MATH]-th order of perturbation theory, which is valid for [MATH].', '1406.6263-1-8-0': 'Also for the strong-field Bethe-Heitler process, a dynamically assisted variant has been studied recently [CITATION].', '1406.6263-1-8-1': 'There, it has been shown that the total pair-creation rates in relativistic proton-laser collisions may be strongly enhanced when a weak high-frequency component is superimposed onto an intense optical laser beam.', '1406.6263-1-8-2': 'Moreover, similar enhancement effects have been obtained for pair creation by the strong-field Breit-Wheeler process [CITATION] and in spatially localized fields [CITATION].', '1406.6263-1-9-0': 'In this letter, we study strong-field Bethe-Heitler pair creation in a laser field consisting of a strong low-frequency and a weak high-frequency component.', '1406.6263-1-9-1': 'An [MATH]-matrix calculation within the framework of laser-dressed QED employing Dirac-Volkov states is performed.', '1406.6263-1-9-2': 'Besides total pair-creation rates, our method allows us to calculate angular and energy distributions of the created particles that are not accessible to the polarization operator approach used in [CITATION].', '1406.6263-1-9-3': 'The nonperturbative interaction regime with [MATH] is analyzed both in the laboratory and the nuclear rest frame.', '1406.6263-1-9-4': 'We note that Bethe-Heitler pair creation in bichromatic laser fields has been studied recently, with a focus on interference and relative phase effects arising in the case of commensurable frequencies [CITATION].', '1406.6263-1-10-0': 'It is interesting to note that analogies of the Schwinger effect exist in other areas of physics.', '1406.6263-1-10-1': 'They have been demonstrated in various systems such as graphene layers in external electric fields [CITATION], ultracold atom dynamics [CITATION], and light propagation in optical lattices [CITATION].', '1406.6263-1-10-2': 'In these systems, the transition probability between quasiparticle and hole states shows Schwinger-like properties.', '1406.6263-1-11-0': '# Theoretical Framework', '1406.6263-1-12-0': 'Employing the [MATH]-matrix formalism of electron-positron pair creation in combined laser and Coulomb fields, we write the transition amplitude in the nuclear rest frame as [CITATION] [EQUATION]', '1406.6263-1-12-1': 'Electron and positron are created with free momenta [MATH] and spins [MATH], where the subscripted sign denotes their respective charge.', '1406.6263-1-12-2': 'They are described by relativistic Volkov states [MATH] taking their interaction with a plane-wave laser field fully into account.', '1406.6263-1-12-3': 'In our case, this laser field is a superposition of two independent modes: [EQUATION] wherein the phase variables [MATH] are defined via the wave vectors [MATH], where [MATH].', '1406.6263-1-12-4': 'The individual amplitudes [MATH] are chosen to be perpendicular and their absolute value is given by the dimensionless intensity parameters [EQUATION]', '1406.6263-1-12-5': 'The nuclear Coulomb field [MATH], with the nuclear charge number [MATH], is treated in the lowest order of perturbation theory.', '1406.6263-1-13-0': 'The [MATH]-matrix may be evaluated by expanding the occurring periodic functions into Fourier series.', '1406.6263-1-13-1': 'The thereby introduced series summation indices [MATH], with [MATH] or [MATH], may be understood as numbers of photons taken from the respective laser mode [MATH].', '1406.6263-1-13-2': 'This expansion allows to perform the four-dimensional integration from Eq. [REF] analytically by using the Fourier transform of the Coulomb potential and a representation of the [MATH]-function for the integral in space and time, respectively [CITATION]: [EQUATION] where [MATH] is the momentum transfer to the nucleus and [EQUATION] are the laser-dressed momenta.', '1406.6263-1-13-3': 'Note that, by definition of [MATH], the arising [MATH]-function ensures energy conservation.', '1406.6263-1-14-0': 'The fully differential pair-creation rate is obtained by summing the square of the amplitude [MATH] over the final spin states: [EQUATION] where the time interval [MATH] has been introduced.', '1406.6263-1-14-1': 'For incommensurable frequencies, the squared [MATH]-matrix adopts the form [EQUATION] with the matrix element [MATH] containing the Fourier series coefficients.', '1406.6263-1-15-0': 'For the following discussion, it will be particularly convenient to rewrite Eq.[REF] by introducing partial rates distinguished by a single summation index, such that [EQUATION]', '1406.6263-1-15-1': 'The [MATH]-function introduced in Eq.[REF], allows to perform one integration analytically.', '1406.6263-1-15-2': 'In order to gain total rates or rates differential in a single coordinate the necessary remaining integrations are then calculated numerically.', '1406.6263-1-16-0': '# Results', '1406.6263-1-17-0': '## Total Pair-Creation Rates', '1406.6263-1-18-0': 'In the following we shall apply our formalism to pair creation by a relativistic nuclear beam and a bichromatic laser field, which is composed of a high-frequency low-intensity (i.e., weak) mode [MATH], [MATH] and a low-frequency high-intensity (i.e., strong) mode [MATH].', '1406.6263-1-18-1': 'Such a field may be achieved experimentally by the following combination in the laboratory frame (primed): The assisting photon from the weak field with [MATH] may be provided by an extreme-ultraviolet (XUV) source, e.g., one based on High-Harmonic Generation (HHG) [CITATION] or a Free-Electron Laser (FEL) [CITATION].', '1406.6263-1-18-2': 'For the strong field, the [MATH] infrared (IR) light emitted by an Nd:YAG (neodymium-doped yttrium aluminum garnet) laser may be frequency doubled in order to deliver [MATH].', '1406.6263-1-18-3': 'This corresponds to a frequency ratio of [MATH].', '1406.6263-1-18-4': 'In combination with a proton beam of [MATH], as currently provided by the LHC [CITATION], a setup of technology available today is feasible.', '1406.6263-1-18-5': 'Specifically, we will first examine the case of [MATH], with the resulting photon energies in the nuclear rest frame [MATH] and [MATH], as well as [MATH].', '1406.6263-1-18-6': 'Note that, the nuclear rest frame energy of the assisting mode is indicated by the notion of a [MATH]-assisted process.', '1406.6263-1-19-0': 'We shall begin with the comparison of strong-field Bethe-Heitler pair creation with and without the assistance of a single high-energetic photon.', '1406.6263-1-19-1': 'The respective total pair-creation rates [MATH] are shown in Fig. [REF] for the nuclear rest frame.', '1406.6263-1-19-2': 'The corresponding rate in the laboratory frame is given by [MATH].', '1406.6263-1-19-3': 'Even though the depicted intensity range of the strong mode is [MATH], the total rates of both processes show a Schwinger-like dependence on this parameter: [EQUATION]', '1406.6263-1-19-4': 'However, the coefficient [MATH] differs for the two cases.', '1406.6263-1-19-5': 'In the assisted process [MATH] is smaller, leading to an almost flat curve, while in the unassisted case a steep increase is visible.', '1406.6263-1-20-0': 'Taking the absolute height into account, the influence of the assisting photon on the total pair-creation rate becomes apparent: As long as the intensity of the highly intense laser is below [MATH], the assisted process shows a strong enhancement over the unassisted counterpart.', '1406.6263-1-20-1': 'However, from [MATH] onwards the situation inverses and the unassisted process becomes dominant.', '1406.6263-1-21-0': 'Note that, the value of [MATH] where this crossing occurs may be influenced by adjusting [MATH], on which the unassisted process does not depend and the assisted process depends quadratically.', '1406.6263-1-21-1': 'Therefore, a relative scaling between the two graphs is possible.', '1406.6263-1-22-0': 'In a similar scheme, total rates of [MATH]-assisted pair creation have been investigated fully analytically in [CITATION].', '1406.6263-1-22-1': 'Particularly, Eq. (9) therein (here quoted in a form adapted to our notation), [EQUATION] with the parameters [EQUATION] allows for a comparison of the derived dependence on the strong field intensity [MATH] with our numerically integrated result.', '1406.6263-1-22-2': 'This comparison is done by scaling the exponential via a proportionality factor such that the values at [MATH] coincide.', '1406.6263-1-22-3': 'However, we note that Eq.[REF] was obtained under the assumption that [MATH].', '1406.6263-1-22-4': 'Even though this requirement is not met for the results shown in Fig.[REF], where [MATH], the additional graph - for the approximated [MATH] - shows very good agreement.', '1406.6263-1-23-0': 'Besides this requirement of large intensities, there are two additional constraints demanded for the applicability of Eq.[REF]: The two parameters given in Eq.[REF] and Eq.[REF] should be [MATH].', '1406.6263-1-23-1': 'For the parameter set above, their values are both relatively small ([MATH]) indeed.', '1406.6263-1-23-2': 'However, if we keep the laser combination in the laboratory fixed and increase [MATH] to [MATH], leading to photon energies in the nuclear rest frame of [MATH] and [MATH], we end up with [MATH].', '1406.6263-1-23-3': 'This expectedly leads to the significant deviation between our result and that obtained using Eq.[REF] clearly visible in Fig.[REF].', '1406.6263-1-24-0': 'Had we chosen a [MATH] lower than the initial [MATH], the results would also have deviated from the prediction of Eq.[REF], as then the parameter [MATH], indicating the remaining energy gap that has to be overcome by absorption of photons from the strong laser mode [MATH], would have grown to and eventually overcome unity.', '1406.6263-1-24-1': 'For this regime, an empirical extension to Eq.[REF] has been found by rewriting it as [EQUATION] which is identical to Eq.[REF] with the approximated [MATH] for [MATH].', '1406.6263-1-24-2': 'Then we separately identified dependences on the two photon energies via fits to our results from a wide range of parameters, [MATH] and [MATH] in the nuclear rest frame, with according functions extracted from Eq.[REF].', '1406.6263-1-24-3': 'From this we could gain that for [MATH] the exponential scaling works well when [MATH] is not very small.', '1406.6263-1-24-4': 'While this is only a small deviation from the Schwinger-like form of Eq.[REF], it can strongly impact the quantitative value of the predicted rates.', '1406.6263-1-25-0': 'Note that, for the regime where the parameter [MATH] is not very small, i.e., for energies of the assisting photon just below the pair-creation threshold like in Fig.[REF], no constant empirical value for [MATH] could be determined, as the dependence on [MATH] changes rapidly with the precise value of the two parameters.', '1406.6263-1-25-1': 'This regime shall be further investigated in the subsequent section.', '1406.6263-1-26-0': '## Angular- and Energy-Differential Spectra', '1406.6263-1-27-0': 'Besides the total pair-creation rate, further insight into the differences between the assisted and the unassisted process may be found in the arising differential spectra.', '1406.6263-1-27-1': 'In particular, a potential experimental observation would rely on knowing the required acceptance of a detector and the optimal angle under which to measure.', '1406.6263-1-28-0': 'Returning to the above parameter set with [MATH], where the parameter [MATH] (cf. Eq. [REF]) is relatively large, Fig. [REF] shows the pair-creation rate differential in the emission angle [MATH] in the nuclear rest frame and the laboratory frame for the both processes at [MATH], the approximate crossing point found in Fig. [REF].', '1406.6263-1-28-1': 'The two key differences are the shifted peak position and the increased width of the distribution for the assisted process.', '1406.6263-1-29-0': 'It is worth pointing out that, the minimal number of low-frequency photons to overcome the pair-creation threshold for the unassisted process in Fig. [REF] amounts to [MATH], whereas the assisted process requires only [MATH].', '1406.6263-1-29-1': 'The difference between these threshold numbers corresponds to the frequency ratio [MATH].', '1406.6263-1-29-2': 'The main contribution to the total pair-creation rates, though, stems from substantially larger photon numbers.', '1406.6263-1-29-3': 'For the unassisted process, the typical number [MATH] ranges from about [MATH] to [MATH], whereas it ranges from about [MATH] to [MATH] for the assisted process.', '1406.6263-1-29-4': 'Hence, the broader angular distribution arises in the case where the total number of absorbed photons is smaller.', '1406.6263-1-29-5': 'Interestingly, a similar effect of angular broadening for smaller photon numbers has also been found for pair creation by few-photon absorption in the perturbative interaction domain [CITATION].', '1406.6263-1-29-6': 'There, however, a small shift of the emission maximum towards larger angles has been found for larger photon numbers, in contrast to the present situation.', '1406.6263-1-30-0': 'Note that, the Lorentz transformation contracts the full-[MATH] spectrum of the nuclear rest frame (Fig. [REF]) into a small range in the laboratory frame (Fig. [REF]).', '1406.6263-1-30-1': 'Furthermore, it maps small angles to large angles and vice versa.', '1406.6263-1-30-2': 'Therefore, when comparing the distributions in one reference frame with its counterpart in the other, the spectra show a mirrored behavior.', '1406.6263-1-30-3': 'To both spectra, the summed-up differential rates [MATH], with [MATH] as defined in Eq. [REF], have been added.', '1406.6263-1-30-4': 'Note that, the rate of the unassisted process could in principle be measured independently by turning off the assisting light source.', '1406.6263-1-30-5': 'In contrast, the assisted process occurs only simultaneously with its unassisted counterpart.', '1406.6263-1-30-6': 'Nevertheless, the distinction between the two processes in their contribution to the summed-up rate is more illustrative.', '1406.6263-1-31-0': 'The energy distribution is depicted in Fig.[REF] for the same parameters as above.', '1406.6263-1-31-1': 'Similar to Fig.[REF], a broadening for the assisted process is found in both frames of reference.', '1406.6263-1-31-2': 'However, only in the laboratory frame the peak position is shifted.', '1406.6263-1-31-3': 'The latter may be attributed to the interweaving of the emission angle as given in Fig. [REF] (contained in the [MATH]-component of the four-momentum) and the energy as given in Fig. [REF] by the Lorentz transformation to the laboratory frame: [EQUATION]', '1406.6263-1-31-4': 'Hence, the peak position shift in the former manifests itself in the Lorentz-transformed version of the latter.', '1406.6263-1-31-5': 'Again, the summed-up rates [MATH] have been added to both spectra for the same reason as stated above.', '1406.6263-1-32-0': '## Doubly [MATH] Gamma-Assisted Process', '1406.6263-1-33-0': 'Extending our results to a higher-order assisted process may be straightforwardly done by halving [MATH], the frequency of the assisting mode.', '1406.6263-1-33-1': 'This way we may compare the unassisted case (which obviously remains unchanged) to the assisted processes with one or two photons of high energy.', '1406.6263-1-33-2': 'Furthermore, the intensity of the assisting mode is increased to [MATH] to ensure better comparability of the three processes of interest.', '1406.6263-1-33-3': 'Besides, the parameters are identical to the above case of [MATH].', '1406.6263-1-34-0': 'Analogous to Fig. [REF], the total rates for these three processes are shown in Fig. [REF] as function of [MATH].', '1406.6263-1-34-1': 'We note that, in agreement with the above found behavior, for the lowest values of [MATH] the highest number of assisting photons, two, is dominant.', '1406.6263-1-34-2': 'From [MATH] onwards, the singly assisted process overgrows the former, which is in turn overgrown by the unassisted case for the highest depicted intensities [MATH].', '1406.6263-1-35-0': 'It is worth pointing out that here, the singly [MATH]-assisted process with [MATH] represents the aforementioned case where the parameter [MATH] (cf. Eq.[REF]) is larger than unity, as [MATH].', '1406.6263-1-36-0': 'A similar agreement with the behavior found in Fig.[REF] and [REF] is visible for the angular and energy spectra depicted in Fig.[REF] for [MATH], where the three processes are of approximately equal strength.', '1406.6263-1-36-1': 'Evidently, the distributions are further widened and the peak position is shifted for the process with two assisting photons.', '1406.6263-1-37-0': '# Summary and Conclusion', '1406.6263-1-38-0': 'We have studied the strong-field Bethe-Heitler process of electron-positron pair creation in relativistic proton-laser collisions, with the laser field consisting of a strong low-frequency and a weak high-frequency mode.', '1406.6263-1-38-1': 'Focusing on the nonperturbative interaction regime, where the intensity parameter of the strong mode is in the order of unity [MATH], we have shown that the assistance of the high-frequency field component can largely enhance the total pair-creation rate.', '1406.6263-1-38-2': 'The latter still exhibits a manifestly nonperturbative field dependence, similar to the famous Schwinger rate.', '1406.6263-1-38-3': 'This way we extended previous results valid for [MATH] [CITATION] to the intermediate laser-matter coupling regime and showed, moreover, that the field scaling of the Schwinger-like exponent changes significantly when the high laser frequency closely approaches the pair-creation threshold (in the proton rest frame).', '1406.6263-1-39-0': 'Additionally, we have demonstrated that the [MATH]-assisted pair-creation mechanism exerts a characteristic influence on the angular and energy distributions of created particles.', '1406.6263-1-39-1': 'It leads to shifts of the emission maxima and a broadening of the distributions.', '1406.6263-1-39-2': 'Finally, a regime of parameters has been identified where the assistance of two high-frequency photons strongly affects, or even dominates, the pair-creation rate.', '1406.6263-1-39-3': 'In accordance with the above, characteristic changes were also found in the underlying spectra.', '1406.6263-1-40-0': 'Our results can be tested experimentally by combining the LHC proton beam with a counterpropagating laser pulse comprised of a highly intense optical mode and a high-frequency (XUV or soft X-ray) mode of moderate intensity.'}

{'1406.6263-2-0-0': 'The nonperturbative regime of electron-positron pair creation by a relativistic proton beam colliding with a highly intense bichromatic laser field is studied.', '1406.6263-2-0-1': 'The laser wave is composed of a strong low-frequency and a weak high-frequency mode, with mutually orthogonal polarization vectors.', '1406.6263-2-0-2': 'We show that the presence of the high-frequency field component can strongly enhance the pair-creation rate.', '1406.6263-2-0-3': 'Besides, a characteristic influence of the high-frequency mode on the angular and energy distributions of the created particles is demonstrated, both in the nuclear rest frame and the laboratory frame.', '1406.6263-2-1-0': '# Introduction', '1406.6263-2-2-0': 'In the presence of very strong electromagnetic fields, the quantum vacuum can decay into electron-positron pairs [CITATION].', '1406.6263-2-2-1': 'Pair creation in a constant electric field was studied in detail by Schwinger [CITATION], employing the then newly established methods of quantum electrodynamics (QED).', '1406.6263-2-2-2': 'The resulting pair-creation rate has the form [MATH], where [MATH] is the applied field and [MATH] the critical field of QED.', '1406.6263-2-2-3': 'Here, [MATH] and [MATH] are the positron charge and mass, respectively, and relativistic units with [MATH] are used.', '1406.6263-2-2-4': 'The Schwinger rate has a manifestly nonperturbative character due to its non-analytic field dependence.', '1406.6263-2-2-5': 'Therefore, it is of fundamental importance for our understanding of nonperturbative quantum field theories.', '1406.6263-2-2-6': 'An experimental verification has so far been prevented, though, by the huge value of [MATH], which is not accessible in the laboratory yet.', '1406.6263-2-3-0': 'The recent progress in high-intensity laser devices has strongly revived the interest in Schwinger pair creation.', '1406.6263-2-4-0': 'In particular, various schemes have been proposed to facilitate its observation.', '1406.6263-2-4-1': 'Among them is a dynamically assisted variant where a rapidly oscillating electric field is superimposed onto a slowly varying electric field [CITATION].', '1406.6263-2-4-2': 'The total rate for pair creation in this field combination was shown to be strongly enhanced, while preserving its nonperturbative character.', '1406.6263-2-4-3': 'This interesting prediction has led to a number of subsequent investigations.', '1406.6263-2-4-4': 'Currently, the momentum distributions of pairs created by the dynamically assisted mechanism are under active scrutiny [CITATION].', '1406.6263-2-5-0': 'Pair-creation phenomena resembling the Schwinger effect can also occur when a beam of charged particles collides with a counterpropagating intense laser beam.', '1406.6263-2-5-1': 'For instance, pair creation has been theoretically predicted in relativistic proton-laser collisions via a strong-field version of the Bethe-Heitler effect (see, e.g., [CITATION]).', '1406.6263-2-5-2': 'When the laser frequency [MATH] is relatively small, so that the dimensionless parameter [MATH] becomes large, [MATH], the total rate for this process adopts the form [MATH] [CITATION].', '1406.6263-2-5-3': 'Here, [MATH] denotes the laser field amplitude transformed to the rest frame of the proton, which is assumed to be sub-critical: [MATH].', '1406.6263-2-5-4': 'The proton Lorentz factor [MATH] and reduced velocity [MATH] have been introduced here.', '1406.6263-2-5-5': 'The similarity with the Schwinger rate is due to the fact that the characteristic length scale of pair creation in this parameter regime is much shorter than the laser wavelength, so that the field appears as quasi-static during the process.', '1406.6263-2-6-0': 'An experimental setup to observe the strong-field Bethe-Heitler process could, in principle, be realized by utilizing the highly relativistic proton beam of the Large Hadron Collider (LHC) at CERN ([MATH]) in conjunction with a counterpropagating high-intensity laser beam ([MATH]).', '1406.6263-2-6-1': 'In fact, a very similar scheme was employed to observe pair creation by multiphoton absorption at the Stanford Linear Accelerator Center, where a [MATH] electron beam colliding with an intense optical laser pulse triggered the pair creation [CITATION].', '1406.6263-2-6-2': 'This experiment, however, did not probe the Schwinger regime of pair creation because the field intensity was somewhat too low ([MATH]).', '1406.6263-2-6-3': 'Instead, the measured pair-creation rate showed a power-law dependence on the field strength.', '1406.6263-2-7-0': 'Note that, a rate of the form [MATH], with [MATH] denoting the number of absorbed laser photons, follows from a consideration within the [MATH]-th order of perturbation theory, which is valid for [MATH].', '1406.6263-2-8-0': 'Also for the strong-field Bethe-Heitler process, a dynamically assisted variant has been studied recently [CITATION].', '1406.6263-2-8-1': 'There, it has been shown that the total pair-creation rates in relativistic proton-laser collisions may be strongly enhanced when a weak high-frequency component is superimposed onto an intense optical laser beam.', '1406.6263-2-8-2': 'Moreover, similar enhancement effects have been obtained for pair creation by the strong-field Breit-Wheeler process [CITATION] and in spatially localized fields [CITATION].', '1406.6263-2-9-0': 'In this letter, we study strong-field Bethe-Heitler pair creation in a laser field consisting of a strong low-frequency and a weak high-frequency component.', '1406.6263-2-9-1': 'An [MATH]-matrix calculation within the framework of laser-dressed QED employing Dirac-Volkov states is performed.', '1406.6263-2-9-2': 'Besides total pair-creation rates, our method allows us to calculate angular and energy distributions of the created particles that are not accessible to the polarization operator approach used in [CITATION].', '1406.6263-2-9-3': 'The nonperturbative interaction regime with [MATH] is analyzed both in the laboratory and the nuclear rest frame.', '1406.6263-2-9-4': 'We note that Bethe-Heitler pair creation in bichromatic laser fields has been studied recently, with a focus on interference and relative phase effects arising in the case of commensurable frequencies [CITATION].', '1406.6263-2-10-0': 'It is interesting to note that analogies of the Schwinger effect exist in other areas of physics.', '1406.6263-2-10-1': 'They have been demonstrated in various systems such as graphene layers in external electric fields [CITATION], ultracold atom dynamics [CITATION], and light propagation in optical lattices [CITATION].', '1406.6263-2-10-2': 'In these systems, the transition probability between quasiparticle and hole states shows Schwinger-like properties.', '1406.6263-2-11-0': '# Theoretical Framework', '1406.6263-2-12-0': 'Employing the [MATH]-matrix formalism of electron-positron pair creation in combined laser and Coulomb fields, we write the transition amplitude in the nuclear rest frame as [CITATION] [EQUATION]', '1406.6263-2-12-1': 'Electron and positron are created with free momenta [MATH] and spins [MATH], where the subscripted sign denotes their respective charge.', '1406.6263-2-12-2': 'They are described by relativistic Volkov states [MATH] taking their interaction with a plane-wave laser field fully into account.', '1406.6263-2-12-3': 'In our case, this laser field is a superposition of two independent modes with the combined vector potential: [EQUATION] wherein the phase variables [MATH] are defined via the wave vectors [MATH], where [MATH].', '1406.6263-2-12-4': 'The individual amplitude vectors [MATH] are chosen to be perpendicular and their absolute value is given by the dimensionless intensity parameters [EQUATION]', '1406.6263-2-12-5': 'The nuclear Coulomb field [MATH], with the nuclear charge number [MATH], is treated in the lowest order of perturbation theory.', '1406.6263-2-13-0': 'The [MATH]-matrix may be evaluated by expanding the occurring periodic functions into Fourier series.', '1406.6263-2-13-1': 'The thereby introduced series summation indices [MATH], with [MATH] or [MATH], may be understood as numbers of photons taken from the respective laser mode [MATH].', '1406.6263-2-13-2': 'This expansion allows to perform the four-dimensional integration from Eq. [REF] analytically by using the Fourier transform of the Coulomb potential and a representation of the [MATH]-function for the integral in space and time, respectively [CITATION]: [EQUATION] where [MATH] is the momentum transfer to the nucleus and [EQUATION] are the laser-dressed momenta.', '1406.6263-2-13-3': 'Note that, by definition of [MATH], the arising [MATH]-function ensures energy conservation.', '1406.6263-2-14-0': 'The fully differential pair-creation rate is obtained by summing the square of the amplitude [MATH] over the final spin states: [EQUATION] where the time interval [MATH] has been introduced.', '1406.6263-2-14-1': 'For incommensurable frequencies, the squared [MATH]-matrix adopts the form [EQUATION] with the matrix element [MATH] containing the Fourier series coefficients.', '1406.6263-2-15-0': 'For the following discussion, it will be particularly convenient to rewrite Eq.[REF] by introducing partial rates distinguished by a single summation index, such that [EQUATION]', '1406.6263-2-15-1': 'The [MATH]-function introduced in Eq.[REF], allows to perform one integration analytically.', '1406.6263-2-15-2': 'In order to gain total rates or rates differential in a single coordinate the necessary remaining integrations are then calculated numerically.', '1406.6263-2-16-0': '# Results', '1406.6263-2-17-0': '## Total Pair-Creation Rates', '1406.6263-2-18-0': 'In the following we shall apply our formalism to pair creation by a relativistic nuclear beam and a bichromatic laser field, which is composed of a high-frequency low-intensity (i.e., weak) mode [MATH], [MATH] and a low-frequency high-intensity (i.e., strong) mode [MATH].', '1406.6263-2-18-1': 'Such a field may be achieved experimentally by the following combination in the laboratory frame (primed): The assisting photon from the weak field with [MATH] may be provided by an extreme-ultraviolet (XUV) source, e.g., one based on High-Harmonic Generation (HHG) [CITATION] or a Free-Electron Laser (FEL) [CITATION].', '1406.6263-2-18-2': 'For the strong field, the [MATH] infrared (IR) light emitted by an Nd:YAG (neodymium-doped yttrium aluminum garnet) laser may be frequency doubled in order to deliver [MATH].', '1406.6263-2-18-3': 'This corresponds to a frequency ratio of [MATH].', '1406.6263-2-18-4': 'In combination with a proton beam of [MATH], as currently provided by the LHC [CITATION], a setup of technology available today is feasible.', '1406.6263-2-18-5': 'Specifically, we will first examine the case of [MATH], with the resulting photon energies in the nuclear rest frame [MATH] and [MATH], as well as [MATH].', '1406.6263-2-18-6': 'Note that, the nuclear rest frame energy of the assisting mode is indicated by the notion of a [MATH]-assisted process.', '1406.6263-2-18-7': 'Furthermore, the parameters of the assisting mode correspond to an intensity of approximately [MATH] in the laboratory frame.', '1406.6263-2-18-8': 'We point out that this is a moderate value given the fact that present-day HHG and FEL sources may reach field intensities of the order of [MATH] [CITATION] and [MATH] [CITATION], respectively.', '1406.6263-2-19-0': 'We shall begin with the comparison of strong-field Bethe-Heitler pair creation with and without the assistance of a single high-energetic photon.', '1406.6263-2-19-1': 'The respective total pair-creation rates [MATH] are shown in Fig. *fig:assist924 for the nuclear rest frame.', '1406.6263-2-19-2': 'The corresponding rate in the laboratory frame is given by [MATH].', '1406.6263-2-19-3': 'Even though the depicted intensity range of the strong mode is [MATH], the total rates of both processes show a Schwinger-like dependence on this parameter: [EQUATION]', '1406.6263-2-19-4': 'However, the coefficient [MATH] differs for the two cases.', '1406.6263-2-19-5': 'In the assisted process [MATH] is smaller, leading to an almost flat curve, while in the unassisted case a steep increase is visible.', '1406.6263-2-20-0': 'Taking the absolute height into account, the influence of the assisting photon on the total pair-creation rate becomes apparent: As long as the intensity of the highly intense laser is below [MATH], the assisted process shows a strong enhancement over the unassisted counterpart.', '1406.6263-2-20-1': 'However, from [MATH] onwards the situation inverses and the unassisted process becomes dominant.', '1406.6263-2-21-0': 'Note that, the value of [MATH] where this crossing occurs may be influenced by adjusting [MATH], on which the unassisted process does not depend and the assisted process depends quadratically.', '1406.6263-2-21-1': 'Therefore, a relative scaling between the two graphs is possible.', '1406.6263-2-22-0': 'In a similar scheme, total rates of [MATH]-assisted pair creation have been investigated fully analytically in [CITATION].', '1406.6263-2-22-1': 'Particularly, Eq. (9) therein (here quoted in a form adapted to our notation), [EQUATION] with the parameters [EQUATION] allows for a comparison of the derived dependence on the strong field intensity [MATH] with our numerically integrated result.', '1406.6263-2-22-2': 'This comparison is done by scaling the exponential via a proportionality factor such that the values at [MATH] coincide.', '1406.6263-2-22-3': 'However, we note that Eq.[REF] was obtained under the assumption that [MATH].', '1406.6263-2-22-4': 'Even though this requirement is not met for the results shown in Fig.*', '1406.6263-2-22-5': '*fig:assist924, where [MATH], the additional graph - for the approximated [MATH] - shows very good agreement.', '1406.6263-2-23-0': 'Besides this requirement of large intensities, there are two additional constraints demanded for the applicability of Eq.[REF]: The two parameters given in Eq.[REF] and Eq.[REF] should be [MATH].', '1406.6263-2-23-1': 'For the parameter set above, their values are both relatively small ([MATH]) indeed.', '1406.6263-2-23-2': 'However, if we keep the laser combination in the laboratory fixed and increase [MATH] to [MATH], leading to photon energies in the nuclear rest frame of [MATH] and [MATH], we end up with [MATH].', '1406.6263-2-23-3': 'This expectedly leads to the significant deviation between our result and that obtained using Eq.[REF] clearly visible in Fig.*', '1406.6263-2-24-0': 'Had we chosen a [MATH] lower than the initial [MATH], the results would also have deviated from the prediction of Eq.[REF], as then the parameter [MATH], indicating the remaining energy gap that has to be overcome by absorption of photons from the strong laser mode [MATH], would have grown to and eventually overcome unity.', '1406.6263-2-24-1': 'Based on our results, we can establish an empirical extension to Eq.[REF] valid for this regime.', '1406.6263-2-24-2': 'It has been obtained by expressing Eq.[REF] in the form [EQUATION] where we have introduced an empirical fitting parameter [MATH].', '1406.6263-2-24-3': 'Note that, for [MATH] set to unity, Eq.[REF] is identical to Eq.[REF] if the approximated expression for [MATH] from Eq.[REF] is employed.', '1406.6263-2-24-4': 'Then we separately identified dependences on the two photon energies via fits to our results from a wide range of parameters, [MATH] and [MATH] in the nuclear rest frame, with according functions extracted from Eq.[REF].', '1406.6263-2-24-5': 'From this we could gain that for [MATH] the exponential scaling works well when [MATH] is not very small.', '1406.6263-2-24-6': 'While this is only a small deviation from the Schwinger-like form of Eq.[REF], it can strongly impact the quantitative value of the predicted rates.', '1406.6263-2-25-0': 'Note that, for the regime where the parameter [MATH] is not very small, i.e., for energies of the assisting photon just below the pair-creation threshold like in Fig.*', '1406.6263-2-25-1': '*fig:assist980, no constant empirical value for [MATH] could be determined, as the dependence on [MATH] changes rapidly with the precise value of the two parameters.', '1406.6263-2-25-2': 'This regime shall be further investigated in the subsequent section.', '1406.6263-2-26-0': '## Angular- and Energy-Differential Spectra', '1406.6263-2-27-0': 'Besides the total pair-creation rate, further insight into the differences between the assisted and the unassisted process may be found in the arising differential spectra.', '1406.6263-2-27-1': 'In particular, a potential experimental observation would rely on knowing the required acceptance of a detector and the optimal angle under which to measure.', '1406.6263-2-28-0': 'Returning to the above parameter set with [MATH], where the parameter [MATH] (cf. Eq. [REF]) is relatively large, Fig. [REF] shows the pair-creation rate differential in the emission angle [MATH] in the nuclear rest frame and the laboratory frame for the both processes at [MATH], the approximate crossing point found in Fig. *fig:assist980.', '1406.6263-2-28-1': 'The two key differences are the shifted peak position and the increased width of the distribution for the assisted process.', '1406.6263-2-29-0': 'It is worth pointing out that, the minimal number of low-frequency photons to overcome the pair-creation threshold for the unassisted process in Fig. [REF] amounts to [MATH], whereas the assisted process requires only [MATH].', '1406.6263-2-29-1': 'The difference between these threshold numbers corresponds to the frequency ratio [MATH].', '1406.6263-2-29-2': 'The main contribution to the total pair-creation rates, though, stems from substantially larger photon numbers.', '1406.6263-2-29-3': 'For the unassisted process, the typical number [MATH] ranges from about [MATH] to [MATH], whereas it ranges from about [MATH] to [MATH] for the assisted process.', '1406.6263-2-29-4': 'Hence, the broader angular distribution arises in the case where the total number of absorbed photons is smaller.', '1406.6263-2-29-5': 'Interestingly, a similar effect of angular broadening for smaller photon numbers has also been found for pair creation by few-photon absorption in the perturbative interaction domain [CITATION].', '1406.6263-2-29-6': 'There, however, a small shift of the emission maximum towards larger angles has been found for larger photon numbers, in contrast to the present situation.', '1406.6263-2-30-0': 'Note that, the Lorentz transformation contracts the full-[MATH] spectrum of the nuclear rest frame (Fig. *fig:spectra-nuc-theta) into a small range in the laboratory frame (Fig. *fig:spectra-lab-theta).', '1406.6263-2-30-1': 'Furthermore, it maps small angles to large angles and vice versa.', '1406.6263-2-30-2': 'Therefore, when comparing the distributions in one reference frame with its counterpart in the other, the spectra show a mirrored behavior.', '1406.6263-2-30-3': 'To both spectra, the summed-up differential rates [MATH], with [MATH] as defined in Eq. [REF], have been added.', '1406.6263-2-30-4': 'Note that, the rate of the unassisted process could in principle be measured independently by turning off the assisting light source.', '1406.6263-2-30-5': 'In contrast, the assisted process occurs only simultaneously with its unassisted counterpart.', '1406.6263-2-30-6': 'Nevertheless, the distinction between the two processes in their contribution to the summed-up rate is more illustrative.', '1406.6263-2-31-0': 'The energy distribution is depicted in Fig.[REF] for the same parameters as above.', '1406.6263-2-31-1': 'Similar to Fig.[REF], a broadening for the assisted process is found in both frames of reference.', '1406.6263-2-31-2': 'However, only in the laboratory frame the peak position is shifted.', '1406.6263-2-31-3': 'The latter may be attributed to the interweaving of the emission angle as given in Fig. *fig:spectra-nuc-theta (contained in the [MATH]-component of the four-momentum) and the energy as given in Fig. *fig:spectra-nuc-energy by the Lorentz transformation to the laboratory frame: [EQUATION]', '1406.6263-2-31-4': 'Hence, the peak position shift in the former manifests itself in the Lorentz-transformed version of the latter.', '1406.6263-2-31-5': 'Again, the summed-up rates [MATH] have been added to both spectra for the same reason as stated above.', '1406.6263-2-32-0': '## Doubly [MATH] Gamma-Assisted Process', '1406.6263-2-33-0': 'Extending our results to a higher-order assisted process may be straightforwardly done by halving [MATH], the frequency of the assisting mode.', '1406.6263-2-33-1': 'This way we may compare the unassisted case (which obviously remains unchanged) to the assisted processes with one or two photons of high energy.', '1406.6263-2-33-2': 'Furthermore, the intensity of the assisting mode is increased to [MATH] to ensure better comparability of the three processes of interest.', '1406.6263-2-33-3': 'Besides, the parameters are identical to the above case of [MATH].', '1406.6263-2-34-0': 'Analogous to Fig. [REF], the total rates for these three processes are shown in Fig. [REF] as function of [MATH].', '1406.6263-2-34-1': 'We note that, in agreement with the above found behavior, for the lowest values of [MATH] the highest number of assisting photons, two, is dominant.', '1406.6263-2-34-2': 'From [MATH] onwards, the singly assisted process overgrows the former, which is in turn overgrown by the unassisted case for the highest depicted intensities [MATH].', '1406.6263-2-35-0': 'It is worth pointing out that here, the singly [MATH]-assisted process with [MATH] represents the aforementioned case where the parameter [MATH] (cf. Eq.[REF]) is larger than unity, as [MATH].', '1406.6263-2-36-0': 'A similar agreement with the behavior found in Figs. [REF] and [REF] is visible for the angular and energy spectra depicted in Fig.[REF] for [MATH], where the three processes are of approximately equal strength.', '1406.6263-2-36-1': 'Evidently, the distributions are further widened and the peak position is shifted for the process with two assisting photons.', '1406.6263-2-37-0': '# Summary and Conclusion', '1406.6263-2-38-0': 'We have studied the strong-field Bethe-Heitler process of electron-positron pair creation in relativistic proton-laser collisions, with the laser field consisting of a strong low-frequency and a weak high-frequency mode.', '1406.6263-2-38-1': 'Focusing on the nonperturbative interaction regime, where the intensity parameter of the strong mode is in the order of unity [MATH], we have shown that the assistance of the high-frequency field component can largely enhance the total pair-creation rate.', '1406.6263-2-38-2': 'The latter still exhibits a manifestly nonperturbative field dependence, similar to the famous Schwinger rate.', '1406.6263-2-38-3': 'This way we extended previous results valid for [MATH] [CITATION] to the intermediate laser-matter coupling regime and showed, moreover, that the field scaling of the Schwinger-like exponent changes significantly when the high laser frequency closely approaches the pair-creation threshold (in the proton rest frame).', '1406.6263-2-39-0': 'Additionally, we have demonstrated that the [MATH]-assisted pair-creation mechanism exerts a characteristic influence on the angular and energy distributions of created particles.', '1406.6263-2-39-1': 'It leads to shifts of the emission maxima and a broadening of the distributions.', '1406.6263-2-39-2': 'Finally, a regime of parameters has been identified where the assistance of two high-frequency photons strongly affects, or even dominates, the pair-creation rate.', '1406.6263-2-39-3': 'In accordance with the above, characteristic changes were also found in the underlying spectra.', '1406.6263-2-40-0': 'Our results can be tested experimentally by combining the LHC proton beam with a counterpropagating laser pulse comprised of a highly intense optical mode and a high-frequency (XUV or soft X-ray) mode of moderate intensity.'}

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[]

[['1406.6263-1-22-4', '1406.6263-2-22-4'], ['1406.6263-1-22-4', '1406.6263-2-22-5'], ['1406.6263-1-24-1', '1406.6263-2-24-1'], ['1406.6263-1-25-0', '1406.6263-2-25-0'], ['1406.6263-1-25-0', '1406.6263-2-25-1']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1406.6263

1512.07136

{'1512.07136-1-0-0': 'Divided symmetrization of a function [MATH] is symmetrization of the ratio [EQUATION] where the product is taken over the set of edges of some graph [MATH].', '1512.07136-1-0-1': 'We concentrate on the case when [MATH] is a tree and [MATH] is a polynomial of degree [MATH], in this case [MATH] is a constant function.', '1512.07136-1-0-2': 'We give a combinatorial interpretation of the divided symmetrization of monomials for general trees and probabilistic game interpretation for a tree which is a path.', '1512.07136-1-0-3': 'In particular, this implies a result by Postnikov originally proved by computing volumes of special polytopes.', '1512.07136-1-1-0': 'Let [MATH] be set of variables, [MATH], say, [MATH] (but further we need different sets, like [MATH]).', '1512.07136-1-1-1': 'It is convenient to think that [MATH] is well ordered: [MATH].', '1512.07136-1-1-2': 'For a rational function (with coefficients in some field) [MATH] of variables from [MATH], define its symmetrization as [MATH], where summation is taken over all [MATH] permutations [MATH] of variables.', '1512.07136-1-2-0': 'Let [MATH] be polynomial of degree [MATH] in variables from [MATH].', '1512.07136-1-2-1': 'Then, its divided symmetrization [EQUATION] is also polynomial of degree not exceeding [MATH], in particular, it vanishes identically when [MATH].', '1512.07136-1-2-2': 'The reason why [MATH] is polynomial is the following.', '1512.07136-1-2-3': 'Fix variables [MATH] and partition all summands into pairs corresponding to permutations [MATH], where [MATH] is a transposition of [MATH] and [MATH].', '1512.07136-1-2-4': 'We see that in the sum of any pair multiple [MATH] in denominator is canceled.', '1512.07136-1-2-5': 'Thus every multiple is canceled and so we get polynomial.', '1512.07136-1-3-0': 'Let [MATH] be a graph on the set of vertices [MATH].', '1512.07136-1-3-1': 'We view [MATH] as a set of pairs [MATH], [MATH].', '1512.07136-1-3-2': 'We may consider "partial symmetrization in [MATH]", that is, [EQUATION].', '1512.07136-1-3-3': 'Of course this is a polynomial again of degree at most [MATH] due to the obvious formula [EQUATION].', '1512.07136-1-4-0': 'If we restrict [MATH] to polynomials of degree at most [MATH], we get a linear functional.', '1512.07136-1-4-1': 'The kernel [MATH] of this functional is particularly structured.', '1512.07136-1-4-2': 'First of all, all polynomials of degree less then [MATH] lie in [MATH].', '1512.07136-1-4-3': 'Next, if [MATH] has a symmetric factor: [MATH], [MATH] is symmetric and non-constant, then [MATH].', '1512.07136-1-4-4': 'This is true because of the formula [MATH], and the second multiple being equal to 0 since [MATH].', '1512.07136-1-5-0': 'Assume that [MATH] is disconnected.', '1512.07136-1-5-1': 'That is, [MATH], and there are no edges of [MATH] between [MATH] and [MATH]: [MATH], where [MATH] are sets of edges joining vertices of [MATH] respectively.', '1512.07136-1-5-2': 'Denote the corresponding subgraphs of [MATH] by [MATH] and [MATH].', '1512.07136-1-5-3': 'Note that both [MATH] are well ordered sets of variables and thus above the definitions still work for the subgraphs [MATH].', '1512.07136-1-6-0': 'Any polynomial [MATH] may be represented as a sum [MATH], where the polynomials [MATH] depend only on variables from [MATH], while [MATH] depends only on variables from [MATH] (and, of course, the degree [MATH] of each summand does not exceed [MATH]).', '1512.07136-1-6-1': 'Assume that [MATH].', '1512.07136-1-6-2': 'Then [EQUATION] (the binomial factor comes from fixing the sets of variables [MATH] and [MATH].', '1512.07136-1-6-3': 'If [MATH] then the symmetrization [MATH] is just 0, analogously if [MATH].', '1512.07136-1-6-4': 'If [MATH], [MATH], then both [MATH], [MATH] are constants and therefore do not depend on the sets of variables [MATH], [MATH]).', '1512.07136-1-6-5': 'It follows that [MATH] if for any [MATH] either [MATH] or [MATH].', '1512.07136-1-6-6': 'As already noted above, it is so unless [MATH], [MATH].', '1512.07136-1-6-7': 'If [MATH] has a factor symmetric in the variables from [MATH], then [MATH].', '1512.07136-1-7-0': 'Next observation.', '1512.07136-1-7-1': 'If [MATH] and [MATH] that [MATH].', '1512.07136-1-7-2': 'Combining this with our previous argument, we get the following', '1512.07136-1-8-0': 'If [MATH] and [MATH] is a connected component in [MATH], [MATH] is divisible by [MATH], where [MATH] is symmetric in variables from [MATH], then [MATH].', '1512.07136-1-9-0': 'Denoting by [MATH] the set of polynomials [MATH] such that [MATH] provided that [MATH] (it is a sort of ideal, but the set of polynomials with restricted degree is not a ring), we have found some elements in [MATH]: all symmetric polynomials and all polynomials like those in Lemma [REF].', '1512.07136-1-10-0': 'Now we restrict our attention to the following specific situation.', '1512.07136-1-10-1': 'Let [MATH], [MATH], [MATH].', '1512.07136-1-10-2': 'For notational simplicity, denote [MATH], [MATH], [MATH].', '1512.07136-1-10-3': 'Also denote [MATH].', '1512.07136-1-10-4': 'We have [MATH] and [MATH] for [MATH].', '1512.07136-1-11-0': "Let's show how this helps to calculate, say, [MATH].", '1512.07136-1-12-0': '[MATH].', '1512.07136-1-13-0': 'Induction on [MATH].', '1512.07136-1-13-1': 'The base case [MATH] (or even [MATH]) is straightforward.', '1512.07136-1-13-2': 'Denote [MATH].', '1512.07136-1-13-3': 'We have [MATH], since symmetric polynomials lie in [MATH].', '1512.07136-1-13-4': 'Next, denoting [MATH] we get [EQUATION] by [REF] and by the induction assumption.', '1512.07136-1-14-0': 'We obtain [MATH] linear relations on [MATH] which determine them uniquely and also the numbers [MATH] satisfy those relations.', '1512.07136-1-14-1': 'This finishes the induction step.', '1512.07136-1-15-0': 'This lemma and many other nice formulae for [MATH] are obtained recently by T. Amdeberhan in [CITATION].', '1512.07136-1-16-0': 'Now, consider the product [EQUATION]', '1512.07136-1-16-1': 'Proceed as follows.', '1512.07136-1-16-2': 'At first, replace [MATH] by [MATH] in this product.', '1512.07136-1-16-3': 'Next, replace [MATH] by [MATH], and so on.', '1512.07136-1-16-4': 'Finally we get [MATH].', '1512.07136-1-16-5': 'The value of [MATH] does not change after such replacements.', '1512.07136-1-16-6': 'Indeed, when we replace [MATH] by [MATH], we add (to our polynomial) a quantity divisible by [MATH].', '1512.07136-1-16-7': 'This is an element of [MATH], hence what we add lies in [MATH].', '1512.07136-1-16-8': 'For any homogeneous polynomial [MATH] of degree [MATH], we have', '1512.07136-1-17-0': 'It suffices to prove Theorem [REF] for monomials [MATH], [MATH].', '1512.07136-1-17-1': 'We are interested only in cyclic type of [MATH], and for cyclic type [MATH] this follows from [REF] and the fact that [MATH].', '1512.07136-1-17-2': 'Denote by [MATH] the function of a cyclic vector [MATH] which is defined as the difference between two parts of [REF].', '1512.07136-1-17-3': 'It helps to remember that [MATH] contains [MATH] and [MATH] (we may think that indices are taken modulo [MATH], then it is true for all [MATH]).', '1512.07136-1-17-4': 'Thus we have [MATH] and [MATH] if [MATH].', '1512.07136-1-17-5': 'These relations imply that [MATH] is always 0.', '1512.07136-1-17-6': 'Indeed, if, say, [MATH] attains positive values and [MATH] is maximal, and [MATH], then we see that [MATH] and [MATH] are other maximizers.', '1512.07136-1-17-7': 'It is easy to see that by such operations we may come to cyclic type [MATH] and so get a contradiction.', '1512.07136-1-18-0': 'The above proof is indirect in the sense that it does not say explicitly what [MATH] is.', '1512.07136-1-18-1': 'This is partially fixed in the following description in terms of a probabilistic sandpile-type process.', '1512.07136-1-19-0': 'Consider [MATH] coins distributed somehow among the vertices of a regular [MATH]-gon, which are enumerated by [MATH].', '1512.07136-1-19-1': 'Choose any vertex [MATH] containing at least 2 coins and rub it: take 1 coin from the vertex [MATH] and put either to left or right neighbor of [MATH] with equal probability.', '1512.07136-1-19-2': 'Proceed until it is possible.', '1512.07136-1-19-3': 'The number of vertices with at least 1 coin does not decrease.', '1512.07136-1-19-4': 'When it stabilizes, there is an empty vertex [MATH], and its neighbor [MATH] we should rub [MATH] finitely often, else with probability 1 [MATH] becomes non-empty.', '1512.07136-1-19-5': 'Analogously consider neighbor of [MATH] and verify that with probability 1 it must be rubbed finitely often and so on.', '1512.07136-1-19-6': 'So, with probability 1 the process terminates in a finite time.', '1512.07136-1-19-7': 'Next, the ultimate distribution of configurations does not depend on the choice of the rubbed vertex at every turn.', '1512.07136-1-19-8': 'This follows from the facts that operations commute, and any vertex with at least 2 coins must be rubbed at some point before the process terminates.', '1512.07136-1-20-0': 'There are [MATH] possible final configurations (one empty vertex and [MATH] vertices wih 1 coin).', '1512.07136-1-21-0': 'Assume that initially we have [MATH] coins at vertex [MATH].', '1512.07136-1-21-1': 'Then the probability [MATH] that vertex [MATH] is empty in the final configuration equals [MATH].', '1512.07136-1-22-0': 'We have [MATH], [MATH] when [MATH] and for all [MATH] [EQUATION]', '1512.07136-1-22-1': 'The same three properties are shared by the function [MATH].', '1512.07136-1-22-2': 'But they define the function uniquely, as the argument with maximum principle (very much the same as in the proof of Theorem [REF]) shows.', '1512.07136-1-23-0': 'By cyclic symmetry, [MATH] is the probability that in the final configuration vertex [MATH] is empty.', '1512.07136-1-23-1': 'Summing over all [MATH], we get [MATH], where summation indices are cyclic modulo [MATH].', '1512.07136-1-23-2': 'This is another proof of Theorem [REF].', '1512.07136-1-24-0': 'Now we consider arbitrary tree.', '1512.07136-1-25-0': 'Let [MATH] be a tree on a well ordered set [MATH], [MATH].', '1512.07136-1-25-1': 'Let [MATH] be a monomial of degree [MATH], where we call [MATH] a weight of a vertex [MATH].', '1512.07136-1-25-2': 'The total weight of all vertices equals [MATH].', '1512.07136-1-25-3': 'For each edge [MATH], [MATH], consider two connected components of the graph [MATH].', '1512.07136-1-25-4': 'The total weight is negative for exactly for one of them.', '1512.07136-1-25-5': 'If this component contains [MATH], call edge [MATH] regular, else call it inversive.', '1512.07136-1-25-6': 'Define sign [MATH] as [MATH].', '1512.07136-1-25-7': 'Call a permutation [MATH] of the set [MATH]-acceptable if for all edges [MATH], [MATH] if and only if [MATH] is regular.', '1512.07136-1-26-0': 'The partial divided symmetrization [MATH] of the monomial [MATH] equals the number of [MATH]-acceptable permutations times [MATH].', '1512.07136-1-27-0': 'Induction on [MATH].', '1512.07136-1-27-1': 'The base case [MATH] is obvious.', '1512.07136-1-27-2': 'Assume that [MATH] and the assertion is valid for [MATH].', '1512.07136-1-27-3': 'For any monomial [MATH] denote by [MATH] the number of [MATH]-acceptable permutations times [MATH].', '1512.07136-1-27-4': 'We need to check that [MATH] for all [MATH].', '1512.07136-1-27-5': 'To this end, it suffices to verify the following properties of [MATH] and [MATH]:', '1512.07136-1-28-0': '(i) [MATH] does not depend on [MATH];', '1512.07136-1-29-0': '(ii) [MATH].', '1512.07136-1-30-0': 'We start with (i).', '1512.07136-1-30-1': 'In turn, it suffices to prove that [MATH], where [MATH] is a monomial of degree [MATH] and [MATH], [MATH], is an edge of [MATH].', '1512.07136-1-30-2': 'We have [EQUATION].', '1512.07136-1-30-3': 'Denote [MATH], where [MATH], [MATH] are components of [MATH] containing [MATH], [MATH] respectively; [MATH], [MATH] are trees induced by [MATH] on [MATH], [MATH], and [MATH], where [MATH] is a monomial in elements the of [MATH] and [MATH] in the elements of [MATH].', '1512.07136-1-30-4': 'We have [EQUATION].', '1512.07136-1-30-5': 'In the second case we also have [MATH], since any edge [MATH] is either regular for both monomials [MATH], [MATH], or inversive for both monomials.', '1512.07136-1-30-6': 'In the first case edge [MATH] is regular for [MATH] and inversive for [MATH].', '1512.07136-1-30-7': 'It means that s[MATH] and [MATH] have different signs: [MATH].', '1512.07136-1-30-8': 'It follows that [MATH] equals [MATH] times number of permutations [MATH] which are either [MATH]-acceptable or [MATH]-acceptable.', '1512.07136-1-30-9': 'If we fix [MATH] (and thus automatically [MATH]), which may be done in exactly [MATH] ways, this property of a permutation is a combination of independent properties on [MATH] and on [MATH].', '1512.07136-1-30-10': 'Applying the induction hypothesis we get that [MATH] equals [MATH], as desired.', '1512.07136-1-31-0': 'Now we come to (ii).', '1512.07136-1-31-1': 'We have [MATH], since [MATH] is a symmetric polynomial.', '1512.07136-1-31-2': 'So, it suffices to prove that [MATH].', '1512.07136-1-31-3': 'This, however, is an alternating sum of numbers of pointed permutations: [MATH] counts permutations [MATH] pointed by [MATH] satisfying [MATH] for any edge [MATH] so that there exists a path from [MATH] to [MATH] avoiding [MATH].', '1512.07136-1-31-4': 'We prove that it does vanish by providing a sign-changing involution on this set of pointed permutations (sign is understood in the sense of [MATH]).', '1512.07136-1-31-5': 'Let [MATH] be two maximal elements of [MATH].', '1512.07136-1-31-6': 'The involution acts as follows: take permutation [MATH] pointed by [MATH], then [MATH].', '1512.07136-1-31-7': 'Let [MATH], then [MATH] and [MATH] are neighbors.', '1512.07136-1-31-8': 'Replace [MATH] by [MATH] and [MATH] by [MATH], also point a new permutation by [MATH].', '1512.07136-1-32-0': 'I am grateful to Tewodros Amdeberhan for pending my attention to this topic, for fruitful discussions, and for reading and editing the preliminary version of the manuscript.'}

{'1512.07136-2-0-0': 'Divided symmetrization of a function [MATH] is symmetrization of the ratio [EQUATION] where the product is taken over the set of edges of some graph [MATH].', '1512.07136-2-0-1': 'We concentrate on the case when [MATH] is a tree and [MATH] is a polynomial of degree [MATH], in this case [MATH] is a constant function.', '1512.07136-2-0-2': 'We give a combinatorial interpretation of the divided symmetrization of monomials for general trees and probabilistic game interpretation for a tree which is a path.', '1512.07136-2-0-3': 'In particular, this implies a result by Postnikov originally proved by computing volumes of special polytopes, and suggests its generalization.', '1512.07136-2-1-0': '# Introduction', '1512.07136-2-2-0': 'Let [MATH] be a set of variables, [MATH], say, [MATH] (but further, we need and allow sets such as [MATH]).', '1512.07136-2-2-1': 'It is convenient to think that [MATH] is well ordered: [MATH].', '1512.07136-2-2-2': 'For a rational function [MATH], with coefficients in some field, of variables from [MATH], define its symmetrization as [EQUATION] where summation is taken over all [MATH] permutations [MATH] of the variables.', '1512.07136-2-3-0': 'Let [MATH] be polynomial of degree [MATH] in the variables from [MATH].', '1512.07136-2-3-1': 'Then, its divided symmetrization [EQUATION] is also polynomial of degree not exceeding [MATH].', '1512.07136-2-3-2': 'In particular, it vanishes identically when [MATH].', '1512.07136-2-3-3': 'The reason why [MATH] is a polynomial is the following.', '1512.07136-2-3-4': 'Fix variables [MATH] and partition all summands into pairs corresponding to permutations [MATH], where [MATH] is a transposition of [MATH] and [MATH].', '1512.07136-2-3-5': 'We see that in the sum of any pair, the multiple [MATH] in the denominator gets cancelled.', '1512.07136-2-3-6': 'Thus every multiple is cancelled and so we get polynomial.', '1512.07136-2-3-7': "The symmetrization operators have applications, for instance, in the theory of symmetric functions, see Chapter 7 of the A. Lascoux's book [CITATION].", '1512.07136-2-4-0': 'Let [MATH] be a graph on the set of vertices [MATH].', '1512.07136-2-4-1': 'We view [MATH] as a set of pairs [MATH], [MATH].', '1512.07136-2-4-2': 'We may consider partial symmetrization in [MATH], that is, [EQUATION].', '1512.07136-2-4-3': 'Of course this is a polynomial again of degree at most [MATH] due to the obvious formula [EQUATION].', '1512.07136-2-5-0': 'If we restrict [MATH] to polynomials of degree at most [MATH], we get a linear functional.', '1512.07136-2-5-1': 'The kernel [MATH] of this functional is particularly structured.', '1512.07136-2-5-2': 'First of all, all polynomials of degree less then [MATH] lie in [MATH].', '1512.07136-2-5-3': 'Next, if [MATH] has a symmetric factor, i.e., [MATH], where [MATH] is symmetric and non-constant, then [MATH].', '1512.07136-2-5-4': 'This is true because of the formula [MATH], and the second multiple being equal to 0 since [MATH].', '1512.07136-2-6-0': 'Assume that [MATH] is disconnected.', '1512.07136-2-6-1': 'That is, [MATH], and there are no edges of [MATH] between [MATH] and [MATH]: [MATH], where [MATH] are sets of edges joining vertices of [MATH] respectively.', '1512.07136-2-6-2': 'Denote the corresponding subgraphs of [MATH] by [MATH] and [MATH].', '1512.07136-2-6-3': 'Note that both [MATH] are well ordered sets of variables and thus the above definitions still apply to the subgraphs [MATH].', '1512.07136-2-7-0': 'Any polynomial [MATH] may be represented as a sum [MATH], where the polynomials [MATH] depend only on variables from [MATH], while [MATH] depends only on variables from [MATH] (and, of course, the degree [MATH] of each summand does not exceed [MATH]).', '1512.07136-2-7-1': 'Assume that [MATH].', '1512.07136-2-7-2': 'Then [EQUATION] (the binomial factor comes from fixing the sets of variables [MATH] and [MATH].', '1512.07136-2-7-3': 'If [MATH] then the symmetrization [MATH] is just 0, analogously if [MATH].', '1512.07136-2-7-4': 'If [MATH], [MATH], then both [MATH], [MATH] are constants and therefore do not depend on the sets of variables [MATH], [MATH]).', '1512.07136-2-7-5': 'It follows that [MATH] if for any [MATH] either [MATH] or [MATH].', '1512.07136-2-7-6': 'As already noted above, it is so unless [MATH], [MATH].', '1512.07136-2-7-7': 'If [MATH] has a factor symmetric in the variables from [MATH], then [MATH].', '1512.07136-2-8-0': 'Next observation.', '1512.07136-2-8-1': 'If [MATH] and [MATH] then [MATH].', '1512.07136-2-8-2': 'Combining this with our previous argument, we get the following', '1512.07136-2-9-0': 'If [MATH] and [MATH] is a connected component in [MATH], [MATH] is divisible by [MATH], where [MATH] is symmetric in variables from [MATH], then [MATH].', '1512.07136-2-10-0': 'Denoting by [MATH] the set of polynomials [MATH] such that [MATH] provided that [MATH] (it is sort of an ideal, but the set of polynomials with restricted degree is not a ring), we have found some elements in [MATH]: all symmetric polynomials and all polynomials like those in Lemma [REF].', '1512.07136-2-11-0': 'Next, we consider the case of partial divided symmetrization w.r.t. tree [MATH] on [MATH] vertices of a polynomial [MATH], [MATH].', '1512.07136-2-11-1': 'This is a linear functional and we give combinatorial formulae for its values in a natural monomial base.', '1512.07136-2-12-0': '# Tree', '1512.07136-2-13-0': 'Let [MATH] be a tree on a well ordered set [MATH], [MATH].', '1512.07136-2-13-1': 'Let [MATH] be a monomial of degree [MATH], where we call [MATH] a weight of a vertex [MATH].', '1512.07136-2-13-2': 'The total weight of all vertices equals [MATH].', '1512.07136-2-13-3': 'For each edge [MATH], [MATH], consider two connected components of the graph [MATH].', '1512.07136-2-13-4': 'The total weight is negative for exactly one of them.', '1512.07136-2-13-5': 'If this component contains [MATH], call edge [MATH] regular, else call it inversive.', '1512.07136-2-13-6': 'Define sign [MATH] as [MATH].', '1512.07136-2-13-7': 'Call a permutation [MATH] of the set [MATH] to be [MATH]-acceptable if for all edges [MATH], [MATH] if and only if [MATH] is regular.', '1512.07136-2-14-0': 'The partial divided symmetrization [MATH] of the monomial [MATH] equals the number of [MATH]-acceptable permutations times [MATH].', '1512.07136-2-15-0': 'Induction on [MATH].', '1512.07136-2-15-1': 'The base case [MATH] is obvious.', '1512.07136-2-15-2': 'Assume that [MATH] and the assertion is valid for [MATH].', '1512.07136-2-15-3': 'For any monomial [MATH] denote by [MATH] the number of [MATH]-acceptable permutations times [MATH].', '1512.07136-2-15-4': 'We need to check that [MATH] for all [MATH].', '1512.07136-2-15-5': 'To this end, it suffices to verify the following properties of [MATH] and [MATH]:', '1512.07136-2-16-0': '(i) [MATH] does not depend on [MATH];', '1512.07136-2-17-0': '(ii) [MATH].', '1512.07136-2-18-0': 'We start with (i).', '1512.07136-2-18-1': 'In turn, it suffices to prove that [MATH], where [MATH] is a monomial of degree [MATH] and [MATH], [MATH], is an edge of [MATH].', '1512.07136-2-18-2': 'We have [EQUATION].', '1512.07136-2-18-3': 'Denote [MATH], where [MATH], [MATH] are components of [MATH] containing [MATH], [MATH] respectively; [MATH], [MATH] are trees induced by [MATH] on [MATH], [MATH], and [MATH], where [MATH] is a monomial in elements the of [MATH] and [MATH] in the elements of [MATH].', '1512.07136-2-18-4': 'We have [EQUATION].', '1512.07136-2-18-5': 'In the second case we also have [MATH], since any edge [MATH] is either regular for both monomials [MATH], [MATH], or inversive for both monomials.', '1512.07136-2-18-6': 'In the first case edge [MATH] is regular for [MATH] and inversive for [MATH].', '1512.07136-2-18-7': 'It means that [MATH] and [MATH] have different signs: [MATH].', '1512.07136-2-18-8': 'It follows that [MATH] equals [MATH] times number of permutations [MATH] which are either [MATH]-acceptable or [MATH]-acceptable.', '1512.07136-2-18-9': 'If we fix [MATH] (and thus automatically [MATH]), which may be done in exactly [MATH] ways, this property of a permutation is a combination of independent properties on [MATH] and on [MATH].', '1512.07136-2-18-10': 'Applying the induction hypothesis we get that [MATH] equals [MATH], as desired.', '1512.07136-2-19-0': 'Now we come to (ii).', '1512.07136-2-19-1': 'We have [MATH], since [MATH] is a symmetric polynomial.', '1512.07136-2-19-2': 'So, it suffices to prove that [MATH].', '1512.07136-2-19-3': 'This, however, is an alternating sum of numbers of pointed permutations: [MATH] counts permutations [MATH] pointed by [MATH] satisfying [MATH] for any edge [MATH] so that there exists a path from [MATH] to [MATH] avoiding [MATH].', '1512.07136-2-19-4': 'We prove that it does vanish by providing a sign-changing involution on this set of pointed permutations (sign is understood in the sense of [MATH]).', '1512.07136-2-19-5': 'Let [MATH] be two maximal elements of [MATH].', '1512.07136-2-19-6': 'The involution acts as follows: take permutation [MATH] pointed by [MATH], then [MATH].', '1512.07136-2-19-7': 'Let [MATH], then [MATH] and [MATH] are neighbours.', '1512.07136-2-19-8': 'Replace [MATH] by [MATH] and [MATH] by [MATH], also point a new permutation by [MATH].', '1512.07136-2-20-0': '# Path', '1512.07136-2-21-0': 'Now we restrict our attention to the following specific situation.', '1512.07136-2-21-1': 'Let [MATH], [MATH], [MATH].', '1512.07136-2-21-2': 'For notational simplicity, denote [MATH], [MATH], [MATH].', '1512.07136-2-21-3': 'Also denote [MATH].', '1512.07136-2-21-4': 'We have [MATH] and [MATH] for [MATH].', '1512.07136-2-22-0': "Let's show how this helps to calculate, say, [MATH].", '1512.07136-2-23-0': '[MATH].', '1512.07136-2-24-0': 'Induction on [MATH].', '1512.07136-2-24-1': 'The base case [MATH] (or even [MATH]) is straightforward.', '1512.07136-2-24-2': 'Denote [MATH].', '1512.07136-2-24-3': 'We have [MATH], since symmetric polynomials lie in [MATH].', '1512.07136-2-24-4': 'Next, denoting [MATH] we get [EQUATION] by [REF] and by the induction assumption.', '1512.07136-2-25-0': 'We obtain [MATH] linear relations on [MATH] which determine them uniquely and also the numbers [MATH] satisfy those relations.', '1512.07136-2-25-1': 'This finishes the induction step.', '1512.07136-2-26-0': 'This lemma and many other nice formulae for [MATH] are obtained recently by T. Amdeberhan in [CITATION].', '1512.07136-2-27-0': 'Now, consider the product [EQUATION] and proceed as follows.', '1512.07136-2-27-1': 'At first, replace [MATH] by [MATH] in this product.', '1512.07136-2-27-2': 'Next, replace [MATH] by [MATH], and so on.', '1512.07136-2-27-3': 'Finally we get [MATH].', '1512.07136-2-27-4': 'The value of [MATH] does not change after such replacements.', '1512.07136-2-27-5': 'Indeed, when we replace [MATH] by [MATH], we add (to our polynomial) a quantity divisible by [MATH].', '1512.07136-2-27-6': 'This is an element of [MATH], hence what we add lies in [MATH].', '1512.07136-2-27-7': 'For any homogeneous polynomial [MATH] of degree [MATH], we have', '1512.07136-2-28-0': 'It suffices to prove Theorem [REF] for monomials [MATH], [MATH].', '1512.07136-2-28-1': 'We are interested only in cyclic type of [MATH], and for cyclic type [MATH] this follows from [REF] and the fact that [MATH].', '1512.07136-2-28-2': 'Denote by [MATH] the function of a cyclic vector [MATH] which is defined as the difference between two parts of [REF].', '1512.07136-2-28-3': 'It helps to remember that [MATH] contains [MATH] and [MATH] (we may think that indices are taken modulo [MATH], then it is true for all [MATH]).', '1512.07136-2-28-4': 'Thus we have [MATH] and [MATH] if [MATH].', '1512.07136-2-28-5': 'These relations imply that [MATH] is always 0 by a standard maximum principle argument.', '1512.07136-2-28-6': 'Indeed, if, say, [MATH] attains positive values and [MATH] is maximal, and [MATH], then we see that [MATH] and [MATH] are other maximizers.', '1512.07136-2-28-7': 'It is easy to see that by such operations we may come to the cyclic type [MATH] and so get a contradiction.', '1512.07136-2-29-0': 'The above proof is indirect in the sense that it does not say explicitly what [MATH] is.', '1512.07136-2-29-1': 'This is partially fixed in the following description in terms of a probabilistic sandpile-type process.', '1512.07136-2-30-0': 'Consider [MATH] coins distributed somehow among the vertices of a regular [MATH]-gon, which are enumerated by [MATH].', '1512.07136-2-30-1': 'Choose any vertex [MATH] containing at least 2 coins and rob it as follows: take 1 coin from the vertex [MATH] and put it either to the left or right neighbour of [MATH] with equal probability.', '1512.07136-2-30-2': 'Proceed as long as it is possible, i.e., until there remain no vertices containing at least 2 coins.', '1512.07136-2-30-3': 'Assume the process does not terminate for infinitely long.', '1512.07136-2-30-4': 'The number of vertices with at least 1 coin does not decrease during our process.', '1512.07136-2-30-5': 'When it stabilizes, there is an empty vertex [MATH], and its neighbour [MATH] must be robbed finitely often, else with probability 1 [MATH] becomes non-empty.', '1512.07136-2-30-6': 'Analogously consider the other (different from [MATH]) neighbour of [MATH] and verify that with probability 1 it must be robbed finitely often and so on.', '1512.07136-2-30-7': 'So, with probability 1 the process terminates in a finite time.', '1512.07136-2-30-8': 'Next, the ultimate distribution of configurations does not depend on the choice of the robbed vertex at every turn.', '1512.07136-2-30-9': 'This follows from the facts that operations commute, and any vertex with at least 2 coins must be robbed at some point before the process terminates.', '1512.07136-2-31-0': 'There are [MATH] possible final configurations (one empty vertex and [MATH] vertices with 1 coin).', '1512.07136-2-32-0': 'Assume that initially we have [MATH] coins at vertex [MATH].', '1512.07136-2-32-1': 'Then the probability [MATH] that vertex [MATH] is empty in the final configuration equals [MATH].', '1512.07136-2-33-0': 'We have [MATH], [MATH] when [MATH] and [EQUATION] whenever [MATH].', '1512.07136-2-33-1': 'The same three properties are shared by the function [MATH].', '1512.07136-2-33-2': 'But they define the function uniquely, as the argument with maximum principle (very much the same as in the proof of Theorem [REF]) shows.', '1512.07136-2-34-0': 'By cyclic symmetry, [MATH] is the probability that in the final configuration vertex [MATH] is empty.', '1512.07136-2-34-1': 'Summing over all [MATH], we get [MATH], where summation indices are cyclic modulo [MATH].', '1512.07136-2-34-2': 'This is another proof of Theorem [REF].', '1512.07136-2-35-0': 'We may consider a slightly more general problem.', '1512.07136-2-35-1': 'Namely, let [MATH] be a cycle with [MATH] vertices [MATH] counted counter-clockwise, [MATH] are the corresponding variables.', '1512.07136-2-35-2': 'Indices are taken modulo [MATH].', '1512.07136-2-35-3': 'Put [MATH] coins in the vertices, [MATH] coins in a vertex [MATH], [MATH].', '1512.07136-2-35-4': 'Do the same robbing process as above until we get exactly [MATH] empty vertices, and 1 coin in each of the others.', '1512.07136-2-35-5': 'The sum of probabilities over all [MATH] possible final configurations equals 1, and this identity may be rewritten as a divided symmetrization identity due to Theorem [REF].', '1512.07136-2-36-0': 'Namely, let [MATH] be a set of [MATH] empty vertices in the final configuration.', '1512.07136-2-36-1': 'Define the weight [MATH] as a product of sizes of [MATH] groups onto which the vertices from [MATH] divide the cycle (each vertex from [MATH] belongs to exactly one group, so the sum of sizes of these groups equals [MATH]).', '1512.07136-2-36-2': 'For any vertex [MATH] define [MATH] as a sum of variables between [MATH] and the clockwise-next to [MATH] vertex [MATH] from [MATH] included, [MATH] not included).', '1512.07136-2-36-3': 'Define also [MATH] if [MATH].', '1512.07136-2-36-4': 'Then the probability that [MATH] is empty in the final configuration is [EQUATION]', '1512.07136-2-36-5': 'Indeed, this is clear if [MATH] for some [MATH], both the probability that [MATH] is empty and the divided symmetrization [REF] are equal to 0.', '1512.07136-2-36-6': 'If [MATH] for all [MATH], the events in the [MATH] groups between elements of [MATH] are independent.', '1512.07136-2-36-7': 'The symmetrizations are also independent due to the multiples [MATH], thus we apply [REF] (strictly speaking, a generalization of [REF] for [MATH] groups of independent variables.)', '1512.07136-2-36-8': 'The following identity generalizes Theorem [REF] (which corresponds to the case [MATH]):', '1512.07136-2-37-0': 'In above notations any polynomial [MATH] of degree [MATH] in [MATH] variables satisfies the following identity [EQUATION]', '1512.07136-2-37-1': 'The proof is the same as in Theorem [REF]: it suffices to consider [MATH] a monomial, in this case the result follows from [REF] and the fact that with probability 1 our process with coins should terminate.'}

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[]

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[]

['1512.07136-1-6-1', '1512.07136-1-7-0', '1512.07136-1-7-1', '1512.07136-1-7-2', '1512.07136-1-8-0', '1512.07136-1-10-1', '1512.07136-1-10-3', '1512.07136-1-11-0', '1512.07136-1-12-0', '1512.07136-1-13-0', '1512.07136-1-13-2', '1512.07136-1-16-1', '1512.07136-1-22-0', '1512.07136-1-24-0', '1512.07136-1-27-0', '1512.07136-1-27-5', '1512.07136-1-28-0', '1512.07136-1-29-0', '1512.07136-1-30-2', '1512.07136-1-30-4', '1512.07136-2-7-1', '1512.07136-2-8-0', '1512.07136-2-8-1', '1512.07136-2-8-2', '1512.07136-2-9-0', '1512.07136-2-15-0', '1512.07136-2-15-5', '1512.07136-2-16-0', '1512.07136-2-17-0', '1512.07136-2-18-2', '1512.07136-2-18-4', '1512.07136-2-21-1', '1512.07136-2-21-3', '1512.07136-2-22-0', '1512.07136-2-23-0', '1512.07136-2-24-0', '1512.07136-2-24-2', '1512.07136-2-33-0', '1512.07136-2-36-8']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1512.07136

1311.1095

{'1311.1095-1-0-0': 'Phenomena inherent to quantum theory on curved space-time, such as Hawking radiation [CITATION], are typically assumed to be only relevant at extreme physical conditions: at high energies and in strong gravitational fields.', '1311.1095-1-0-1': 'Here we consider low-energy quantum mechanics in the presence of weak gravitational time dilation and show that the latter leads to universal decoherence of quantum superpositions.', '1311.1095-1-0-2': "Time dilation induces a universal coupling between internal degrees-of-freedom and the centre-of-mass of a composite particle and we show that the resulting entanglement causes the particle's position to decohere.", '1311.1095-1-0-3': 'We derive the decoherence timescale and show that the weak time dilation on Earth is already sufficient to decohere micro-scale objects.', '1311.1095-1-0-4': 'No coupling to an external environment is necessary, thus even completely isolated composite systems will decohere on curved space-time.', '1311.1095-1-0-5': 'In contrast to gravitational collapse models [CITATION], no modification of quantum theory is assumed.', '1311.1095-1-0-6': 'General relativity therefore can account for the emergence of classicality and the effect can in principle be tested in future matter wave experiments with large molecules [CITATION] or with trapped microspheres [CITATION].', '1311.1095-1-1-0': 'The superposition principle is a major cornerstone of quantum mechanics.', '1311.1095-1-1-1': 'Quantum interference experiments have demonstrated superpositions of neutrons [CITATION], atoms [CITATION] and even large molecules [CITATION], however, quantum superpositions are not accessible on human scales.', '1311.1095-1-1-2': 'The origin of the quantum-to-classical transition is still an active field of research and a prominent role in this transition is commonly attributed to decoherence [CITATION]: due to interaction with an external environment, a particle loses its quantum coherence.', '1311.1095-1-1-3': 'Many specific models have been studied in which a particle interacts with its environment, such as for example scattering with surrounding phonons [CITATION], photons [CITATION] and gravitational waves [CITATION].', '1311.1095-1-1-4': 'A different approach in explaining classicality is taken in so-called "collapse models", in which macroscopic superpositions are prohibited without any external environment [CITATION].', '1311.1095-1-1-5': 'Such models are often inspired by general relativity but they all rely on a postulated breakdown of quantum mechanics.', '1311.1095-1-1-6': 'In contrast, here we show the existence of decoherence due to general relativistic time dilation without any breakdown of quantum mechanics and which takes place even for isolated composite systems.', '1311.1095-1-1-7': 'We show that even the weak time dilation on earth is already sufficient to decohere the position of micro-scale composite quantum systems.', '1311.1095-1-2-0': 'To derive the decoherence we consider standard quantum mechanics in the presence of general-relativistic time dilation, which causes clocks to run slower near a massive object.', '1311.1095-1-2-1': 'A derivation of the quantum dynamics of a composite system on curved space-time can be found in Appendix A.', '1311.1095-1-2-2': 'We consider the dynamics in weak gravitational fields (i.e. to lowest order in [MATH]), in which case the results already follow from the mass-energy equivalence in a gravitational field [CITATION]: any internal energy contributes to the total weight of a system and gravity couples to the total mass [MATH], where the Hamiltonian [MATH] generates the time evolution of the internal degrees of freedom of this particle and [MATH] is the remaining static rest mass.', '1311.1095-1-2-3': 'The interaction with the gravitational field is therefore [MATH], where [MATH].', '1311.1095-1-2-4': 'For example, if the particle is a simple harmonic oscillator with frequency [MATH], the above interaction with gravity effectively changes the frequency according to [MATH].', '1311.1095-1-2-5': 'This is the well-known gravitational redshift to lowest order in [MATH].', '1311.1095-1-2-6': 'Classically, the time-dilation-induced interaction [MATH] yields only this frequency shift.', '1311.1095-1-2-7': 'On the other hand, in quantum mechanics the internal energy [MATH] and the position [MATH] are quantized operators, thus time dilation causes an additional, purely quantum mechanical effect: entanglement between the internal degrees of freedom and the centre-of-mass position of the particle[CITATION].', '1311.1095-1-2-8': 'However, gravitational time dilation on Earth is very weak and the resulting entanglement has typically no effect on small quantum systems.', '1311.1095-1-2-9': 'For composite quantum systems, on the other hand, this effect can become significant, as we will show below.', '1311.1095-1-3-0': 'We consider composite particles subject to gravitational time dilation and model them by having [MATH] constituents that are independent three-dimensional harmonic oscillators.', '1311.1095-1-3-1': 'Such a model equivalently describes [MATH] internal harmonic modes of the particle.', '1311.1095-1-3-2': 'The internal Hamiltonian for this system is [MATH], where [MATH] are the number operators for the [MATH]-th mode with frequency [MATH].', '1311.1095-1-3-3': 'The centre-of-mass (with [MATH] and [MATH] being its vertical position and momentum, respectively) of the whole system is subject to the gravitational potential [MATH].', '1311.1095-1-3-4': 'For a homogeneous gravitational field in the x-direction we can approximate [MATH], where [MATH] m/s[MATH] is the gravitational acceleration on earth.', '1311.1095-1-3-5': 'The total Hamiltonian is therefore [MATH], where [MATH] is the Hamiltonian for the centre-of-mass of the particle and the time-dilation-induced interaction (to lowest order in [MATH]) between position and internal energy is [EQUATION]', '1311.1095-1-3-6': 'We now consider the particle to be at rest in superposition of two vertically distinct positions [MATH] and [MATH], with the height difference [MATH], such that the centre-of-mass is in the state [MATH].', '1311.1095-1-3-7': 'The internal degrees of freedom are in thermal equilibrium at temperature T, thus the [MATH]-th constituent is in the state [MATH], where we used the coherent state representation, [MATH] is the average excitation and [MATH] is the Boltzmann constant.', '1311.1095-1-3-8': 'The total initial state is thus given by [MATH].', '1311.1095-1-3-9': 'General relativistic time dilation now couples the centre-of-mass position of the system to the internal degrees of freedom [MATH] via the Hamiltonian in Eq. [REF].', '1311.1095-1-3-10': 'The off-diagonal elements [MATH], responsible for quantum interference, therefore evolve in time as [EQUATION] where [MATH] and [MATH].', '1311.1095-1-3-11': 'The individual internal states evolve at different frequencies according to general relativistic time dilation (Fig. 1).', '1311.1095-1-3-12': 'To see decoherence of the centre-of-mass position, we trace out the internal degrees of freedom.', '1311.1095-1-3-13': 'The quantum coherence, quantified by the visibility [MATH], becomes [MATH].', '1311.1095-1-3-14': 'This expression can be simplified by noting that typically [MATH].', '1311.1095-1-3-15': 'In the high temperature limit we also have [MATH], so that the frequency-dependence completely drops out from the visibility.', '1311.1095-1-3-16': 'In this case, the reduction of quantum interference is given by [EQUATION] where we used [MATH] and defined the decoherence time [EQUATION]', '1311.1095-1-3-17': 'The above equation shows that gravitational time dilation causes superpositions of composite systems to decohere.', '1311.1095-1-3-18': 'The decoherence rate derived here scales linearly with the superposition size [MATH], in contrast to other decoherence mechanisms that typically show a quadratic scaling [CITATION].', '1311.1095-1-3-19': 'Also, decoherence due to gravitational time dilation depends on the number of oscillating internal states of the system, [MATH], since the internal structure of the system causes decoherence via time dilation.', '1311.1095-1-3-20': 'Thus the suppression of quantum effects takes place even for completely isolated systems, provided that the superposition amplitudes acquire a proper-time difference.', '1311.1095-1-3-21': 'In the high-temperature limit the frequencies of the internal oscillations drop out entirely from the final expression, therefore it is not necessary to have fast-evolving internal states.', '1311.1095-1-3-22': 'Note that the decoherence derived here depends on the constants [MATH], [MATH], [MATH] and the gravitational constant [MATH]: it can therefore be considered a general relativistic, thermodynamic and quantum mechanical effect.', '1311.1095-1-4-0': 'The arguments presented above can be made more rigorous by considering the full time-evolution in the presence of relativistic time dilation.', '1311.1095-1-4-1': 'To this end, we derive in the Methods Section a master equation that describes the quantum dynamics of a composite system under the influence of time dilation to lowest non-vanishing order in [MATH].', '1311.1095-1-4-2': 'The derivation is fully general and captures special and general relativistic time dilation for any space-time and any internal Hamiltonian [MATH].', '1311.1095-1-4-3': 'For the model considered here the master equation becomes [EQUATION]', '1311.1095-1-4-4': 'This equation effectively replaces the Schrodinger equation for the time evolution of centre-of-mass of a composite particle in the presence of a gravitational field.', '1311.1095-1-4-5': 'The first term describes the unitary evolution of the centre-of-mass due to an arbitrary Hamiltonian [MATH] which can include general relativistic corrections.', '1311.1095-1-4-6': 'For clarity we separated the gravitational potential, which couples to an effective total mass [MATH] that includes the average internal energy [MATH].', '1311.1095-1-4-7': 'This is in accordance with the notion of heat in general relativity (in Einstein\'s words [CITATION]: "a piece of iron weighs more when red-hot than when cool").', '1311.1095-1-4-8': 'The second term causes the suppression of off-diagonal elements of the density matrix and is responsible for the decoherence.', '1311.1095-1-4-9': 'The integral captures the fact that decoherence depends on the overall acquired proper-time difference along the different paths.', '1311.1095-1-4-10': 'If the centre-of-mass Hamiltonian [MATH] does not induce significant changes to the off-diagonal elements on the decoherence time-scale, the above euqation can be approximated by [EQUATION]', '1311.1095-1-4-11': 'This equation is approximately valid if one can neglect the accumulation of proper time difference during the initial build-up of the superposition.', '1311.1095-1-4-12': 'The decoherence time-scale is found from the solution to eq. [REF], which for the off-diagonal terms [MATH] is approximately (to order [MATH]): [MATH], with [MATH] as in eq. [REF].', '1311.1095-1-4-13': 'The reduction of visibility is Gaussian and agrees with eq. [REF].', '1311.1095-1-4-14': 'The master equation due to time dilation, eq. [REF], is similar in form to other master equations typically studied in the field of decoherence [CITATION] but does not include any dissipative term.', '1311.1095-1-4-15': 'Thus time dilation provides naturally an "ideal" master equation for decoherence that affects only off-diagonal terms in the position basis.', '1311.1095-1-4-16': 'The master equation for gravitational time dilation differs from other decoherence mechanisms mainly by having a Gaussian decay instead of an exponential decay and in the particular form and parameter-dependence of the decoherence time-scale, eq. [REF].', '1311.1095-1-5-0': 'To estimate the strength of the decoherence due to time dilation, we consider a human-scale macroscopic system at room temperature.', '1311.1095-1-5-1': "Assuming that the system has Avogadro's number of constituent particles which oscillate, we set [MATH], which amounts to a gram-scale system.", '1311.1095-1-5-2': 'For a superposition size of [MATH] m, the decoherence time [REF] becomes [EQUATION]', '1311.1095-1-5-3': 'Remarkably, even though the gravitational time dilation is very weak, its resulting decoherence is already substantial on human scales and not just for astrophysical objects.', '1311.1095-1-5-4': 'Macroscopic objects completely decohere on earth on a short timescale due to gravitational time dilation and no vertical superpositions of gram-scale composite systems can be sustained.', '1311.1095-1-5-5': 'In contrast to other decoherence mechanisms, this effect cannot be shielded or overcome without going to a space-time region with no time dilation.', '1311.1095-1-5-6': 'It therefore poses a fundamental limit on ground-based quantum interference experiments with complex systems, unless trajectories with exactly vanishing proper time difference are chosen.', '1311.1095-1-5-7': 'In astrophysical settings, the decoherence can even be substantially stronger: the time scale [REF] can be rewritten in terms of the Schwarzschild-radius [MATH] of the background space-time as [MATH], where [MATH] is the distance between the particle and the centre of the gravitating object (since the nature of the effect is gravitational, quantum mechanical and thermodynamic, the decoherence time may also be written in terms of the Hawking-temperature [CITATION] [MATH] of the body with mass [MATH], but the decoherence is not related to any horizon and the appearance of the Hawking temperature is solely a reformulation of the fundamental constants involved).', '1311.1095-1-5-8': 'The decoherence is stronger for high masses [MATH] and for small distances [MATH] to the mass, i.e. for stronger time dilation.', '1311.1095-1-5-9': 'At the horizon of a black hole with 5 solar masses, a nm-size superposition of a gram-scale object at [MATH]K would decohere after about [MATH]ns.', '1311.1095-1-6-0': 'The effect predicted here is completely general and affects arbitrary quantum states on all curved space-time metrics: a composite quantum system in superposition between two trajectories with proper time difference [MATH] will decohere.', '1311.1095-1-6-1': 'For example, for a particle in superposition of semi-classical (i.e. well-localised) trajectories, the state of the centre-of-mass can easily be calculated as in the derivation of eq. [REF] above: the visibility of the off diagonal elements will drop according to [MATH], where the mean is taken with respect to all internal states [MATH].', '1311.1095-1-6-2': 'This holds for arbitrary internal states and proper time-differences [MATH], so long as the individual trajectories are well-localised.', '1311.1095-1-6-3': 'Thus time dilation universally affects all composite quantum systems and causes their transition to classicality, unless all internal degrees-of-freedom are frozen out.', '1311.1095-1-7-0': 'We stress that the decoherence effect described here is not related to previously considered general relativistic models of decoherence [CITATION].', '1311.1095-1-7-1': 'The mechanism here is time dilation, which arises already in stationary space-times and decoheres composite systems into the position basis, even if they are isolated from any external environment.', '1311.1095-1-7-2': 'This is conceptually different than previously considered scattering of gravitational waves [CITATION], which causes decoherene into the energy basis.', '1311.1095-1-7-3': 'Our treatment, based on a Hamiltonian formulation of low-energy quantum systems, allows for a direct description of composite quantum systems in weak, static gravitational fields and requires neither the ad-hoc modelling of the matter field distribution nor of a gravitational wave background.', '1311.1095-1-7-4': 'We also stress that the time-dilation-induced decoherence is entirely within the framework of quantum mechanics and classical general relativity.', '1311.1095-1-7-5': 'This is in stark contrast to hypothetical models where gravity leads to spontaneous collapse of the wave function and that require a breakdown of unitarity [CITATION] or include stochastic fluctuations of the metric [CITATION].', '1311.1095-1-7-6': 'Thus we show that classical general relativity can account for the suppression of quantum behavior for macroscopic objects without introducing any modifications to quantum mechanics or to general relativity.', '1311.1095-1-8-0': 'We now discuss a possible direct experimental verification of the derived decoherence mechanism.', '1311.1095-1-8-1': 'The gravitational time dilation is well-tested in classical physics but the quantized Hamiltonian has not yet been studied experimentally.', '1311.1095-1-8-2': 'In particular, an experiment to study the induced quantum entanglement between internal degrees of freedom with the centre-of-mass mode, first proposed in Ref. [CITATION], has not yet been realised.', '1311.1095-1-8-3': 'To confirm this quantum mechanical interaction one can use controllable internal states in matter wave interferometry [CITATION], or use Shapiro-delay in single photon interference [CITATION].', '1311.1095-1-8-4': 'Such an experimental verification of the quantum Hamiltonian [REF] would be a strong indication for the presence of the decoherence described here.', '1311.1095-1-8-5': 'To test the decoherence due to time dilation directly it is necessary to bring relatively complex systems into superposition.', '1311.1095-1-8-6': 'This can in principle be achieved with molecule interferometry [CITATION], trapped microspheres [CITATION] or with micro-mechanical mirrors [CITATION].', '1311.1095-1-8-7': 'The latter, however, is currently restricted to very small separations only [CITATION] (on the order of 1 pm) and is therefore less suitable.', '1311.1095-1-8-8': 'To see decoherence caused by time dilation, other decoherence mechanisms will need to be suppressed: The scattering with surrounding molecules and with thermal radiation requires such an experiment to be performed at liquid Helium temperatures and in ultra-high vacuum [CITATION], which has no direct effect on the decoherence due to time dilation.', '1311.1095-1-8-9': 'However, the emission of thermal radiation by the system will be a competing decoherence source.', '1311.1095-1-8-10': 'The decoherence-time due to photon emission is found to be [CITATION] [MATH], where [MATH] is the mode density of the wave vectors [MATH] and [MATH] the effective scattering cross section.', '1311.1095-1-8-11': 'To see the time-dilation-induced decoherence, we require that the decoherence due to emission of radiation is weaker than due to time dilation, i.e. [MATH].', '1311.1095-1-8-12': 'In Fig. 2 we show the parameter regime where time-dilation-induced decoherence can in principle be observed, focussing on micro-scale particles at cryogenic temperatures (sapphire was chosen due to its low microwave emission at low temperatures [CITATION]).', '1311.1095-1-8-13': 'The emission of radiation can be further suppressed if the mode density is reduced, which can ease the restrictions on temperature.', '1311.1095-1-8-14': 'Although an experiment to measure decoherence due to proper time is challenging, the rapid developments in controlling large quantum systems [CITATION] for quantum metrology and for testing collapse-theories [CITATION] will inevitably come to the regime where the time-dilation-induced decoherence predicted here will be of importance.', '1311.1095-1-8-15': 'In the long run, decoherence due to gravitational time dilation poses a fundamental limit on the ability to sustain superpositions on earth.', '1311.1095-1-9-0': 'As a final remark, we note that the presented estimate for the time-dilation-induced decoherence is likely to be an underestimation of the actual effect.', '1311.1095-1-9-1': 'The internal structure of the constituents is not taken into account and additional mechanisms, such as nuclear processes, may contribute to the decoherence effect.', '1311.1095-1-9-2': 'Additionally, special relativistic time dilation and the decoherence during the build-up of the superposition were neglected.', '1311.1095-1-9-3': 'We therefore expect that the actual decoherence due to time dilation is even stronger than predicted here.', '1311.1095-1-10-0': 'We thank M. Arndt and M. Aspelmeyer for discussions and S. Eibenberger for providing us with the illustration of the TPPF20 molecule.', '1311.1095-1-10-1': 'This work was supported by the the Austrian Science Fund (FWF) through the doctoral program Complex Quantum Systems (CoQuS), the SFB FoQuS and the Individual Project 24621, by the Foundational Questions Institute (FQXi), the John Templeton Foundation and by the COST Action MP1209.', '1311.1095-1-11-0': '# Appendix', '1311.1095-1-12-0': '## Hamiltonian for gravitational time dilation', '1311.1095-1-13-0': 'We consider the time evolution of a composite quantum state in the low-energy limit on curved space-time, described by an arbitrary static metric [MATH] with signature (-+++).', '1311.1095-1-13-1': 'In the treatment here we assume small velocities ([MATH]) and neglect special relativistic corrections.', '1311.1095-1-13-2': 'The time evolution of a system in its rest frame, typically given by the Schrodinger equation in this limit, needs to be replaced by the covariant derivative with respect to proper time: [EQUATION]', '1311.1095-1-13-3': 'The derivative can be expressed in terms of the 4-velocity as [MATH].', '1311.1095-1-13-4': 'For a static observer the 4-velocity is proportional to the time translation Killing vector and has therefore only the component [MATH] on a static metric.', '1311.1095-1-13-5': 'The above equation becomes [MATH].', '1311.1095-1-13-6': 'The 4-velocity can now be expressed in terms of the metric via [MATH], such that for a static observer we have [MATH].', '1311.1095-1-13-7': 'Thus the time evolution for a quantum state on curved space-time is given by [EQUATION]', '1311.1095-1-13-8': 'This expression neglects velocity and special relativistic terms, but captures gravitational time dilation through the redshift factor [MATH].', '1311.1095-1-13-9': 'We now use the Schwarzschild-metric in the weak-field limit: [MATH].', '1311.1095-1-13-10': 'Using the full rest Hamiltonian that includes the internal evolution, [MATH], and expanding the resulting expression to lowest order in [MATH], we get an effective Schrodinger equation that includes general relativistic corrections due to time dilation: [EQUATION] which gives the coupling term between the potential [MATH] and the internal Hamiltonian [MATH] as in the main text.', '1311.1095-1-13-11': 'The approach presented here is valid only in the low-energy limit in which the Schrodinger equation can be used.', '1311.1095-1-13-12': 'It agrees with previous results for this limit [CITATION] and additionally allows the study of composite low-energy quantum systems in curved space-time.', '1311.1095-1-14-0': '## Master equation due to gravitational time dilation', '1311.1095-1-15-0': 'Here we derive an equation of motion for a composite quantum system in the presence of time dilation.', '1311.1095-1-15-1': 'We keep the composition and the relativistic time dilation completely general.', '1311.1095-1-15-2': 'The overall Hamiltonian of the system is [MATH], where [MATH] governs the time evolution of the centre-of-mass of the particle, [MATH] governs the evolution of the [MATH] internal constituents and [MATH] captures the time-dilation-induced coupling between internal degrees of freedom and the centre-of-mass to lowest order in [MATH].', '1311.1095-1-15-3': '[MATH] is a function of the centre-of-mass position [MATH] and momentum [MATH] to which the internal states couple due to special relativistic and general relativistic time dilation (for the Schwarzschild metric in the weak-field limit we have [CITATION] [MATH]).', '1311.1095-1-15-4': 'We start with the von Neumann equation for the full state [MATH] and write [MATH], where [MATH].', '1311.1095-1-15-5': 'We change frame to primed coordinates, which we define through [MATH], where [MATH] with the average internal energy [MATH].', '1311.1095-1-15-6': 'The resulting von-Neumann equation is [EQUATION] where [MATH].', '1311.1095-1-15-7': 'The formal solution [MATH] is used in the equation above, which yields the integro-differential equation for the density matrix [EQUATION]', '1311.1095-1-15-8': 'We can now trace over the internal degrees of freedom.', '1311.1095-1-15-9': 'The state is initially uncorrelated [MATH] and we take the Born approximation, keeping only terms to second order in [MATH].', '1311.1095-1-15-10': 'In this case [MATH] can be replaced under the integral by [MATH] and the master equation for the centre-of-mass becomes [EQUATION]', '1311.1095-1-15-11': 'Here we used the notation [MATH] for the energy fluctuations of the internal states and [MATH].', '1311.1095-1-15-12': 'Changing back to the original picture, and introducing [MATH] we obtain the integro-differential equation: [EQUATION] where [MATH].', '1311.1095-1-15-13': 'This is the general equation of motion for a composite particle of arbitrary composition that undergoes time dilation.', '1311.1095-1-15-14': 'The decoherence of its off-diagonal elements is governed by its internal energy spread [MATH] and by the metric-dependent coupling [MATH].', '1311.1095-1-15-15': 'The former is [MATH] in the high-temperature limit for [MATH] non-interacting internal harmonic oscillators and the latter is [MATH] in the homogeneous weak-field limit of the Schwarzschild metric when neglecting special relativistic contributions.'}

{'1311.1095-2-0-0': 'The physics of low-energy quantum systems is usually studied without explicit consideration of the background spacetime.', '1311.1095-2-0-1': 'Phenomena inherent to quantum theory on curved space-time, such as Hawking radiation, are typically assumed to be only relevant at extreme physical conditions: at high energies and in strong gravitational fields.', '1311.1095-2-0-2': 'Here we consider low-energy quantum mechanics in the presence of gravitational time dilation and show that the latter leads to decoherence of quantum superpositions.', '1311.1095-2-0-3': 'Time dilation induces a universal coupling between internal degrees of freedom and the centre-of-mass of a composite particle.', '1311.1095-2-0-4': "The resulting correlations cause decoherence of the particle's position, even without any external environment.", '1311.1095-2-0-5': 'We also show that the weak time dilation on Earth is already sufficient to decohere micron scale objects.', '1311.1095-2-0-6': 'Gravity therefore can account for the emergence of classicality and the effect can in principle be tested in future matter wave experiments.', '1311.1095-2-1-0': 'One of the most striking features of quantum theory is the quantum superposition principle.', '1311.1095-2-1-1': 'It has been demonstrated in numerous experiments with diverse systems, such as neutrons [CITATION], atoms [CITATION] and even large molecules [CITATION].', '1311.1095-2-1-2': 'However, quantum superpositions are not observed on everyday, macroscopic scales.', '1311.1095-2-1-3': 'The origin of the quantum-to-classical transition is still an active field of research.', '1311.1095-2-1-4': 'A prominent role in this transition is commonly attributed to decoherence [CITATION]: due to interaction with an external environment, a particle gets entangled with its environment and loses its quantum coherence.', '1311.1095-2-1-5': 'Many specific models have been studied in which a particle interacts with its surrounding, such as a bath of phonons [CITATION], photons [CITATION], spins [CITATION] and gravitational waves [CITATION].', '1311.1095-2-1-6': 'An alternative route to explain classicality is taken in so-called wave function collapse models, which postulate an inherent breakdown of the superposition principle at some scale without any external environment [CITATION].', '1311.1095-2-1-7': 'Such models are often inspired by general relativity but they rely on a fundamental modification of quantum theory.', '1311.1095-2-1-8': 'In contrast, here we derive the existence of decoherence due to time dilation without any modification of quantum mechanics and which takes place even for isolated composite systems.', '1311.1095-2-1-9': 'We show that even the weak time dilation on Earth is already sufficient to decohere micro-scale quantum systems.', '1311.1095-2-2-0': 'We consider standard quantum mechanics in the presence of time dilation, with the focus on gravitational time dilation which causes clocks to run slower near a massive object.', '1311.1095-2-2-1': 'In Appendix A, we derive the Hamiltonian governing the quantum dynamics of a composite system on an arbitrary, static background space-time (and show that the same result is obtained as a limit of a quantum field model).', '1311.1095-2-2-2': 'Since we consider slowly moving particles and weak gravitational fields (i.e. to lowest order in [MATH]), the results can also be obtained directly from the mass-energy equivalence [CITATION]: any internal energy contributes to the total weight of a system and thus also couples to gravity.', '1311.1095-2-2-3': 'Given any particle of mass [MATH] and an arbitrary Hamiltonian [MATH] that generates the time evolution of its internal degrees of freedom, gravity couples to the total rest mass [MATH], i.e. gravity also couples to internal energy.', '1311.1095-2-2-4': 'The interaction with the gravitational potential [MATH] is therefore [MATH], where [MATH].', '1311.1095-2-2-5': 'This interaction term is just another formulation of gravitational time dilation (the same argument applies to inertial mass as well, so one recovers both the special relativistic and gravitational time dilation governed by [MATH], with [MATH]).', '1311.1095-2-2-6': 'For example, if the particle is a simple harmonic oscillator with frequency [MATH], the above interaction with gravity effectively changes the frequency according to [MATH].', '1311.1095-2-2-7': 'This is the well-tested [CITATION] gravitational redshift to lowest order in [MATH].', '1311.1095-2-2-8': 'When the energy is treated as a classical variable the time-dilation-induced interaction [MATH] yields only this frequency shift.', '1311.1095-2-2-9': 'However, in quantum mechanics the internal energy [MATH] and the position [MATH] are quantized operators, thus time dilation causes an additional, purely quantum mechanical effect: entanglement between the internal degrees of freedom and the centre-of-mass position of the particle [CITATION].', '1311.1095-2-2-10': 'Even though the time dilation on Earth is very weak, it leads to a significant effect for composite quantum systems, as we will show below.', '1311.1095-2-3-0': 'Before deriving in full generality the time evolution of the centre-of-mass of an arbitrary composite system subject to time dilation, we consider a simplified model where a particle has [MATH] constituents that are independent three-dimensional harmonic oscillators.', '1311.1095-2-3-1': 'Such a model equivalently describes [MATH] internal harmonic modes of the particle.', '1311.1095-2-3-2': 'The internal Hamiltonian for this system is [MATH], where [MATH] are the number operators for the [MATH]-th mode with frequency [MATH].', '1311.1095-2-3-3': 'The centre-of-mass (with [MATH] and [MATH] being its vertical position and momentum, respectively) of the whole system is subject to the gravitational potential [MATH].', '1311.1095-2-3-4': 'For a homogeneous gravitational field in the x-direction we can approximate [MATH], where [MATH] m/s[MATH] is the gravitational acceleration on Earth.', '1311.1095-2-3-5': 'The total Hamiltonian of the system is therefore [MATH], where [MATH] is some Hamiltonian for the centre-of-mass of the particle and the gravitational time-dilation-induced interaction (to lowest order in [MATH]) between position and internal energy is [EQUATION]', '1311.1095-2-3-6': 'To demonstrate decoherence, we first consider the case when the gravitational contribution to time dilation is dominant such that the velocity contributions can be neglected.', '1311.1095-2-3-7': 'A typical such case is a particle at rest in superposition of two vertically distinct positions [MATH] and [MATH] and a height difference [MATH].', '1311.1095-2-3-8': 'The centre-of-mass is in the state [MATH].', '1311.1095-2-3-9': 'The internal degrees of freedom are in thermal equilibrium at local temperature [MATH], thus each [MATH]-th constituent is described by the thermal density matrix [MATH], where we used the coherent state representation with the average excitation [MATH] and where [MATH] is the Boltzmann constant.', '1311.1095-2-3-10': 'The total initial state is thus given by [MATH].', '1311.1095-2-3-11': 'Gravitational time dilation now couples the centre-of-mass position of the system to the internal degrees of freedom [MATH] via the Hamiltonian in eq. [REF].', '1311.1095-2-3-12': 'The off-diagonal elements [MATH], which are responsible for quantum interference, evolve to [MATH], where [MATH].', '1311.1095-2-3-13': 'The frequencies of the internal oscillators depend on the position in the gravitational field, in accordance with gravitational time dilation (see also fig. 1).', '1311.1095-2-3-14': 'To see decoherence of the centre-of-mass, we trace out the internal degrees of freedom.', '1311.1095-2-3-15': 'The quantum coherence can be quantified by the interferometric visibility [MATH], which becomes [MATH].', '1311.1095-2-3-16': 'This expression can be simplified for the typical case [MATH].', '1311.1095-2-3-17': 'In the high temperature limit we also have [MATH], so that the frequency-dependence completely drops out from the visibility.', '1311.1095-2-3-18': 'In this case, the reduction of quantum interference is given by [MATH].', '1311.1095-2-3-19': 'For times [MATH] this can be written as [EQUATION] where we defined the decoherence time [EQUATION]', '1311.1095-2-3-20': 'The above equation shows that gravitational time dilation causes superpositions of composite systems to decohere.', '1311.1095-2-3-21': 'The decoherence rate derived here scales linearly with the superposition size [MATH], in contrast to other decoherence mechanisms that typically show a quadratic scaling [CITATION].', '1311.1095-2-3-22': 'Also, decoherence due to gravitational time dilation depends on the number of oscillating internal states of the system, [MATH].', '1311.1095-2-3-23': 'The suppression of quantum effects takes place even for completely isolated systems, provided that the superposition amplitudes acquire a sufficient proper time difference.', '1311.1095-2-3-24': 'In the high temperature limit the frequencies of the internal oscillations drop out entirely from the final expression, therefore it is not necessary to have fast-evolving internal states.', '1311.1095-2-3-25': 'Note that the decoherence derived here depends on the constants [MATH], [MATH], [MATH] and the gravitational acceleration [MATH]: it can therefore be considered a relativistic, thermodynamic and quantum mechanical effect.', '1311.1095-2-4-0': 'The effect is very general and originates from the total proper time difference between superposed world lines.', '1311.1095-2-4-1': 'Quantum systems with internal degrees-of-freedom are affected on arbitrary space-time metrics, as long as a proper time difference is accumulated.', '1311.1095-2-4-2': 'To highlight this, consider a particle moving in superposition along two arbitrary world lines with proper time difference [MATH] (see fig. 2).', '1311.1095-2-4-3': 'The two superposed amplitudes can interfere when the world lines meet.', '1311.1095-2-4-4': 'Due to time dilation, the internal energy is effectively altered by [MATH], with [MATH] (see also Appendix A).', '1311.1095-2-4-5': 'Each amplitude therefore evolves with [MATH] along the respective world lines.', '1311.1095-2-4-6': '[MATH] does not depend on [MATH], [MATH] and [MATH] and for clarity, we restrict the analysis to semi-classical paths, i.e. constrained to have coordinates [MATH], [MATH] and [MATH], [MATH] along the two world lines, respectively (as in fig. 2c).', '1311.1095-2-4-7': 'The interference visibility is then [MATH].', '1311.1095-2-4-8': 'Since [MATH], the interference visibility is simply [EQUATION] where [MATH] is the proper time difference between the two world lines and the expectation value is taken with respect to the initial state.', '1311.1095-2-4-9': 'This result is manifestly coordinate invariant and shows that decoherence occurs if a proper time difference is present, and if the internal states are not eigenstates of internal energy.', '1311.1095-2-4-10': 'Eq. [REF] is recovered as a special case of the general formula above (expanding to lowest order in [MATH], assuming N internal harmonic oscillators such that [MATH] and neglecting the [MATH]-dependent term yields eq. [REF]).', '1311.1095-2-4-11': 'For the special case of a pure two-level system, eq. [REF] also reproduces the effect discussed in ref. [CITATION], which can be interpreted as due to the which-way information acquired by a clock.', '1311.1095-2-4-12': 'In contrast, which-way information is never available for thermal states (a more detailed discussion can be found in Appendix C).', '1311.1095-2-4-13': 'Eq. [REF] shows, however, that time dilation affects any state that is not an eigenstate of [MATH].', '1311.1095-2-4-14': 'The coupling is universal, which follows directly from the universality of time dilation.', '1311.1095-2-4-15': 'Thus the decoherence is as universal as time dilation itself, in the sense that all composite quantum systems are affected, independently of the nature and kind of their internal energy [MATH].', '1311.1095-2-5-0': 'We now consider the full time evolution in the presence of time dilation that takes any arbitrary internal Hamiltonian [MATH] and centre-of-mass Hamiltonian [MATH] into account.', '1311.1095-2-5-1': 'To this end, we derive a master equation that describes the quantum dynamics of a composite system on a background space-time to lowest non-vanishing order in [MATH] (see Appendix B for details).', '1311.1095-2-5-2': 'The resulting time evolution of the centre-of-mass in the presence of special relativistic and gravitational time dilation is [EQUATION] where [MATH].', '1311.1095-2-5-3': 'The first term describes the unitary evolution of the centre-of-mass due to an arbitrary Hamiltonian [MATH], which is completely general and can also include external interactions (as for example those necessary for keeping the particle in superposition or realizing an interference experiment) as well as relativistic corrections to the centre-of-mass dynamics.', '1311.1095-2-5-4': 'The correction dependent upon [MATH] stems from the relativistic contribution to the mass.', '1311.1095-2-5-5': 'The second term causes the suppression of off-diagonal elements of the density matrix and is responsible for the decoherence.', '1311.1095-2-5-6': 'It is proportional to [MATH], the fluctuations in internal energy, or equivalently the heat capacity [MATH] (the high temperature limit of the model that we used previously corresponds to the Einstein solid model).', '1311.1095-2-5-7': "The integral captures the fact that decoherence depends on the overall acquired proper time difference during a particle's evolution.", '1311.1095-2-5-8': 'For a stationary particles, and if the centre-of-mass Hamiltonian [MATH] does not induce significant changes to the off-diagonal elements on the decoherence time scale, the master equation becomes approximately [EQUATION]', '1311.1095-2-5-9': 'In the unitary part we have separated for clarity the Newtonian gravitational potential (i.e. [MATH]): It is evident that the potential couples to an effective total mass [MATH] that includes the average internal energy, which becomes [MATH] for the previously considered model with [MATH] thermal internal harmonic oscillators.', '1311.1095-2-5-10': 'This is in accordance with the notion of heat in general relativity (in Einstein\'s words [CITATION]: "a piece of iron weighs more when red-hot than when cool"), the relation to the Tolman effect is briefly discussed in Appendix C.', '1311.1095-2-5-11': 'The non-unitary part now depends only on the stationary [MATH]-contributions (see also fig. 2a).', '1311.1095-2-5-12': 'The decoherence time scale is found from the solution to eq. [REF], which for the off-diagonal terms [MATH] is approximately (to order [MATH]): [MATH], with [MATH].', '1311.1095-2-5-13': 'The loss of visibility thus agrees with eq. [REF] and the decoherence time scale reduces to eq. [REF] for the specific model used previously.', '1311.1095-2-5-14': 'The master equation due to gravitational time dilation, eq. [REF], is similar in form to other master equations typically studied in the field of decoherence [CITATION] but does not include any dissipative term.', '1311.1095-2-5-15': 'Thus time dilation provides naturally an "ideal" master equation for decoherence that suppresses off-diagonal terms in the position basis for stationary particles.', '1311.1095-2-5-16': 'For non-stationary systems, decoherence is governed by eq. [REF] and the pointer basis derives from a combination of [MATH] and [MATH].', '1311.1095-2-5-17': 'Position and momentum are therefore naturally driven into becoming classical variables.', '1311.1095-2-5-18': 'The evolution in the presence of gravitational time dilation is inherently non-Markovian, since the overall acquired proper time difference is crucial.', '1311.1095-2-5-19': 'This results in a Gaussian decay (rather than an exponential decay as in Markovian models) of the off-diagonal elements and the decoherence time directly depends on the fluctuations in internal energy.', '1311.1095-2-5-20': 'This again highlights the interplay between thermodynamics, relativity and quantum theory that is relevant for this effect.', '1311.1095-2-6-0': 'To estimate the strength of the decoherence due to time dilation, we make use of eq. [REF] and consider a human-scale macroscopic system at room temperature.', '1311.1095-2-6-1': "Assuming that the system has Avogadro's number of constituent particles which oscillate, we set [MATH], which amounts to a gram-scale system.", '1311.1095-2-6-2': 'For a superposition size of [MATH] m, the decoherence time [REF] becomes [EQUATION]', '1311.1095-2-6-3': 'Remarkably, even though the gravitational time dilation is very weak, its resulting decoherence is already substantial on human scales and not just for astrophysical objects.', '1311.1095-2-6-4': 'Macroscopic objects completely decohere on Earth on a short time scale due to gravitational time dilation.', '1311.1095-2-6-5': 'In contrast to other decoherence mechanisms, this effect cannot be shielded and decoherence will occur whenever there is time dilation between superposed amplitudes.', '1311.1095-2-6-6': 'But as any other decoherence in quantum theory, the effect is in principle reversible: as is apparent from eq. [REF], "revivals" of coherence will occur after a sufficiently large proper time difference is accumulated, dependent on the frequencies of the internal degrees of freedom.', '1311.1095-2-6-7': 'However, the corresponding recurrence times typically scale exponentially with the size of the system [CITATION], thus time dilation can cause decoherence which is irreversible "for all practical purposes" if interfering paths have a proper time difference.', '1311.1095-2-6-8': 'This will be increasingly difficult to control in experiments with large, composite systems, and is simply unavoidable for systems not under experimental control.', '1311.1095-2-7-0': 'From the perspective of quantum theory, decoherence due to time dilation is fully analogous to any other decoherence source.', '1311.1095-2-7-1': 'The loss of coherence takes place due to correlations with degrees of freedom that are not accessible, here internal degrees of freedom of the composite system.', '1311.1095-2-7-2': 'The unique aspect of the effect described here is that the correlations are induced by relativistic time dilation, and would not take place in Newtonian gravity.', '1311.1095-2-7-3': "Thus, to understand this effect it is necessary to invoke quantum theory and time dilation stemming from Earth's gravitational field.", '1311.1095-2-7-4': 'The phenomenon arises already in weak, stationary space-times and decoheres composite systems into the position basis, even if they are isolated from any external environment.', '1311.1095-2-7-5': 'It is thus of different nature than decoherence due to scattering with gravitational waves [CITATION].', '1311.1095-2-7-6': 'Importantly, the time-dilation-induced decoherence is entirely within the framework of quantum mechanics and classical general relativity.', '1311.1095-2-7-7': 'No free "model parameters" enter and unitarity is preserved on a fundamental level.', '1311.1095-2-7-8': 'This is in stark contrast to hypothetical models where gravity leads to spontaneous collapse of the wave function and that require a breakdown of unitarity [CITATION] or include stochastic fluctuations of the metric [CITATION].', '1311.1095-2-7-9': 'Our results show that general relativity can account for the suppression of quantum behavior for macroscopic objects without introducing any modifications to quantum mechanics or to general relativity.', '1311.1095-2-8-0': 'In astrophysical settings, the decoherence can even be substantially stronger: the time scale [REF] can be rewritten in terms of the Schwarzschild-radius [MATH] of the background space-time as [MATH], where [MATH] is the distance between the particle and the centre of the gravitating object (since the nature of the effect is gravitational, quantum mechanical and thermodynamic, the decoherence time may also be written in terms of the Hawking-temperature [MATH] of the body with mass [MATH], but the decoherence is not related to any horizon and the appearance of the Hawking temperature is solely a reformulation of the fundamental constants involved).', '1311.1095-2-8-1': 'The decoherence is stronger for high masses [MATH] and for small distances [MATH] to the mass, i.e. for stronger time dilation.', '1311.1095-2-8-2': 'At the horizon of a black hole with 5 solar masses, a nm-size superposition of a gram-scale object at [MATH]K would decohere after about [MATH]ns.', '1311.1095-2-9-0': 'We now discuss a possible direct experimental verification of the derived decoherence mechanism.', '1311.1095-2-9-1': 'The gravitational time dilation is well tested in classical physics [CITATION] but the quantized Hamiltonian [REF] has not yet been studied experimentally.', '1311.1095-2-9-2': 'In particular, an experiment to study the induced quantum entanglement of internal degrees of freedom with the centre-of-mass mode, first proposed in ref. [CITATION], has not yet been realised.', '1311.1095-2-9-3': 'To confirm this quantum mechanical interaction one can use controllable internal states in matter wave interferometry [CITATION], or use Shapiro-delay in single photon interference [CITATION] (a related effect for entangled photon pairs was also discussed in ref. [CITATION]).', '1311.1095-2-9-4': 'Such an experimental verification of the quantum Hamiltonian [REF] would be a strong indication for the presence of the decoherence described here.', '1311.1095-2-9-5': 'To test directly the decoherence due to time dilation it is necessary to bring relatively complex systems into superposition.', '1311.1095-2-9-6': 'This can in principle be achieved with molecule interferometry [CITATION], cooled microspheres [CITATION] or with micro-mechanical mirrors [CITATION].', '1311.1095-2-9-7': 'The latter, however, is expected to be limited to very small separations only (on the order of 1 pm) and is therefore less suitable.', '1311.1095-2-9-8': 'To see decoherence caused by time dilation, other decoherence mechanisms will need to be suppressed: The scattering with surrounding molecules and with thermal radiation requires such an experiment to be performed at liquid Helium temperatures and in ultra-high vacuum [CITATION].', '1311.1095-2-9-9': 'Additionally, the emission and absorption of thermal radiation by the system [CITATION] will be a competing decoherence source.', '1311.1095-2-9-10': 'For the parameter regime studied here, emission of radiation is expected to be the dominant decoherence effect, with the decoherence time [CITATION] [MATH], where [MATH] is the mode density of the wave vectors [MATH] and [MATH] the effective scattering cross section.', '1311.1095-2-9-11': 'To see the time-dilation-induced decoherence, we require that the decoherence due to emission of radiation is weaker than due to time dilation, i.e. [MATH].', '1311.1095-2-9-12': 'To get quantitative estimates, we rely on the harmonic oscillator model introduced previously and show in fig. 3 the parameter regime where time-dilation-induced decoherence can in principle be distinguished from decoherence due to thermal emission, focussing on micro-scale particles at cryogenic temperatures (sapphire was chosen due to its low microwave emission at low temperatures [CITATION]).', '1311.1095-2-9-13': 'The emission of radiation can be further suppressed if the mode density is reduced, which can ease the restrictions on temperature.', '1311.1095-2-9-14': 'However, we note that the simple model for the composition of the system, necessary to estimate the time dilation decoherence rate, is very crude and at low temperatures we expect the model to break down.', '1311.1095-2-9-15': 'Given a specific system, the time dilation decoherence can be estimated more accurately by measurement of the internal energy fluctuations through the heat capacity.', '1311.1095-2-9-16': 'Although an experiment to measure decoherence due to proper time is very challenging, the rapid developments in controlling large quantum systems [CITATION] for quantum metrology and for testing wave function collapse models [CITATION] will inevitably come to the regime where the time-dilation-induced decoherence predicted here will be of importance.', '1311.1095-2-9-17': 'In the long run, experiments on Earth will have to be specifically designed to avoid this gravitational effect on quantum coherence.', '1311.1095-2-9-18': 'As a final remark, we note that due to the universality of time dilation, all dynamical processes contribute to this decoherence, even those that are typically experimentally inaccessible (such as nuclear dynamics, which has not been taken into account in our treatment).', '1311.1095-2-9-19': 'Thus time dilation decoherence could also serve as a tool to indirectly study dynamical processes within composite systems.', '1311.1095-2-10-0': 'We thank M. Arndt, M. Aspelmeyer, L. Diosi and M. Vanner for discussions and S. Eibenberger for providing us with the illustration of the TPPF20 molecule.', '1311.1095-2-10-1': 'This work was supported by the the Austrian Science Fund (FWF) through the doctoral program Complex Quantum Systems (CoQuS), the Vienna Center for Quantum Science and Technology (VCQ), the SFB FoQuS and the Individual Project 24621, by the Foundational Questions Institute (FQXi), the John Templeton Foundation, the Australian Research Council Centre of Excellence for Engineered Quantum Systems (grant number CE110001013), the European Commission through RAQUEL (No. 323970) and the COST Action MP1209.', '1311.1095-2-11-0': '# Appendix', '1311.1095-2-12-0': '## Hamiltonian for gravitational time dilation', '1311.1095-2-13-0': 'We present a Hamiltonian formalism useful to describe the dynamics of low energy quantum systems with internal degrees of freedom subject to time dilation.', '1311.1095-2-13-1': 'We further show an explicit example of a quantum field in curved space-time whose dynamics reduces to the same Hamiltonian in the appropriate limit.', '1311.1095-2-14-0': 'We consider the time evolution of a composite quantum system in the low-energy limit on a generic space-time, described by a metric [MATH] with signature ([MATH]).', '1311.1095-2-14-1': 'We restrict the treatment to static metrics with [MATH] and [MATH], where Latin indices refer to the spatial 3-components.', '1311.1095-2-14-2': 'The typical systems we consider are low energy quantum systems (such as atoms, molecules, nanospheres, etc.) with internal energy levels (such as electronic, rotational, vibrational) in a weak gravitational field and under small accelerations.', '1311.1095-2-14-3': 'For such systems, one can assume that the relative distances between their constituents are sufficiently small, such that variations of the metric over their extension can be neglected.', '1311.1095-2-14-4': 'In this case one can assign a single position degree of freedom to the centre-of-mass of the system, which in the classical limit describes a single world line.', '1311.1095-2-14-5': 'In other words, we consider the limit in which the system can be effectively considered as "point-like" with internal degrees of freedom.', '1311.1095-2-14-6': 'This is directly analogous to the notion of ideal clocks in relativity, which measure time along a well-localized world line.', '1311.1095-2-15-0': 'The rest energy [MATH] of the system is defined as the invariant quantity [EQUATION] it corresponds to the total mass-energy as measured by a co-moving observer.', '1311.1095-2-15-1': "Here [MATH] is the system's total 4-momentum in arbitrary coordinates (we restrict to coordinates which keep the stated assumptions for the metric).", '1311.1095-2-15-2': 'The dynamics of the system can be described in terms of the evolution with respect to an arbitrary time coordinate [MATH].', '1311.1095-2-15-3': 'The generator of the coordinate-time translations follows from eq. [REF] and is given by the Hamiltonian [EQUATION]', '1311.1095-2-15-4': 'If the particle is at rest with respect to a static observer, the energy is [MATH].', '1311.1095-2-15-5': 'This "redshift factor" is also sometimes expressed in terms of the time-like killing vector [MATH] as [MATH].', '1311.1095-2-15-6': 'The rest energy [MATH] is the total Hamiltonian of the system in its local comoving frame.', '1311.1095-2-15-7': 'The static rest mass contribution [MATH] can be explicitly separated, and the remaining part is just the Hamiltonian of the internal degrees of freedom, which we denote by [MATH].', '1311.1095-2-15-8': 'The full dynamics of internal and external degrees of freedom is thus governed by the total Hamiltonian [REF], with [EQUATION]', '1311.1095-2-15-9': 'Relativistically, there is no distinction between "rest mass" and "rest energy".', '1311.1095-2-15-10': 'In fact, the largest contribution to the rest mass [MATH] of the systems we consider, e.g. a molecule, is already given by binding energies between atoms, nucleons, quarks and all other constituents (at an even more fundamental level, masses of fundamental particles are reducible to interaction energies with the Higgs field, according to the standard model of particle physics).', '1311.1095-2-15-11': 'The natural choice for the split [REF] is dictated by the energy scale: if some degrees of freedom are "frozen", their contribution to the rest energy can be incorporated in the mass term.', '1311.1095-2-15-12': 'The split between mass and internal energy [REF] is thus merely conventional, and amounts to a choice of the zero-energy state of the internal degrees of freedom.', '1311.1095-2-15-13': 'The general expression [REF] can also be derived from the action of a particle in the comoving frame, in which the time coordinate coincides with the proper time: [MATH], where [MATH] is the Lagrangian describing the internal degrees of freedom with coordinates [MATH] and [MATH].', '1311.1095-2-15-14': 'Changing to the lab frame this expression becomes [MATH], with [MATH] and [MATH].', '1311.1095-2-15-15': 'The Legendre transform yields the Hamiltonian for the system, which gives exactly the expression [REF] with [EQUATION]', '1311.1095-2-15-16': 'If the internal dynamics is irrelevant, we simply have [MATH] and [MATH].', '1311.1095-2-15-17': 'But in general any arbitrary internal dynamics governed by [MATH] gives rise to the total energy as in eq. [REF] with [MATH] as in eq. [REF].', '1311.1095-2-15-18': 'For example, two masses on a spring (with total mass [MATH], reduced mass [MATH] and spring constant [MATH]) in the comoving frame of the centre-of-mass are described by [MATH], where [MATH] is the relative degree of freedom of the two masses.', '1311.1095-2-15-19': 'This gives [MATH], where [MATH] describes the dynamical part of the internal degrees of freedom.', '1311.1095-2-16-0': 'To obtain the quantum equations of motion one can replace the 4-momenta in the full expression [REF] with covariant derivatives, which leads to a modified Klein-Gordon equation with [MATH] as the invariant total mass (a similar derivation can be applied to the Dirac equation to describe particles with spin).', '1311.1095-2-16-1': 'For energies small compared to [MATH] (such that particle creation and other quantum field effects are negligible), the Klein-Gordon field is treated as a particle in first quantization and the low-energy Schrodinger evolution is obtained.', '1311.1095-2-16-2': 'Specifically, the Hamiltonian for the dynamics of a point particle on a post-Newtonian background metric with [MATH] and [MATH] is obtained following ref. [CITATION] (with the addition of internal degrees of freedom in [MATH]), resulting in [EQUATION] where [MATH] acts only on the potential and we introduced [MATH] to keep the expansion to order [MATH] explicit (in deriving the Hamiltonian there is an ambiguity in the ordering of the [MATH] terms, but which does not affect the results for time dilation).', '1311.1095-2-16-3': 'For an eigenstate [MATH] of the internal Hamiltonian [MATH] the rest energy in the Hamiltonian [REF] can be treated as a parameter [MATH].', '1311.1095-2-16-4': 'For arbitrary internal states and due to the linearity of quantum mechanics, [MATH] has to be treated as an operator acting on the internal degrees of freedom according to [REF].', '1311.1095-2-16-5': 'The Hamiltonian [REF] thus describes the full quantum dynamics of the system, including internal and external degrees of freedom.', '1311.1095-2-16-6': 'Expanding the result to first order in [MATH], we find [EQUATION] where [MATH] includes all terms acting on the centre-of-mass to this order of approximation (and can also include any other interaction, such as the electromagnetic interaction [CITATION]).', '1311.1095-2-16-7': 'The term [MATH] captures the time dilation, which effectively shifts the internal energy.', '1311.1095-2-16-8': 'An alternative route to get eq. [REF] is to directly expand eq. [REF] in orders of [MATH] and canonically quantize the result, which yields the same Schrodinger equation with relativistic corrections.', '1311.1095-2-16-9': 'The term proportional to [MATH] stems from [MATH], while the [MATH]-term stems from the spatial [MATH] components of the metric.', '1311.1095-2-16-10': 'The former captures gravitational time dilation and the latter is the velocity-dependent special relativistic time dilation.', '1311.1095-2-16-11': 'In total, eq. [REF] describes the special and general relativistic corrections to the dynamics of a quantum system with internal degrees of freedom, to lowest order in [MATH].', '1311.1095-2-16-12': 'Note that the coupling between internal and external degrees of freedom is completely independent of the nature and kinds of interactions involved in the internal dynamics [MATH].', '1311.1095-2-16-13': 'This is a consequence of the universality of time dilation, which affects all kinds of clocks, irrespectively of their specific construction.', '1311.1095-2-17-0': 'The description above is well-suited for low-energy particles on a background space-time.', '1311.1095-2-17-1': 'The same results can also be obtained from a field theory description and to highlight this, we consider an explicit example in which the effective Hamiltonian is derived starting from a quantum field.', '1311.1095-2-17-2': 'We consider a Klein-Gordon field of mass [MATH], in a state approximately localized in some region of space.', '1311.1095-2-17-3': 'The position of the region of space represents the "centre-of-mass", while the field describes the "internal degrees of freedom" of our system.', '1311.1095-2-17-4': 'The dynamics of the field is given by the action [EQUATION] where [MATH] is the determinant of [MATH] and we use units with [MATH].', '1311.1095-2-17-5': 'To simplify the discussion, we consider a system at a fixed height above Earth.', '1311.1095-2-17-6': 'The effect already appears by expanding the metric at the first order: [MATH], [MATH], where [MATH] is the Newtonian potential.', '1311.1095-2-17-7': 'In this case the determinant is [MATH] and [MATH], so we can rewrite eq. [REF] as [EQUATION]', '1311.1095-2-17-8': 'Once the coordinates are fixed, we can write [MATH] and single out the Lagrangian [EQUATION]', '1311.1095-2-17-9': 'In order to pass to the Hamiltonian picture, we need the conjugate momenta [EQUATION]', '1311.1095-2-17-10': 'The Hamiltonian in these coordinates is given by [MATH].', '1311.1095-2-17-11': 'Substituting [MATH], we get [EQUATION]', '1311.1095-2-17-12': 'We are in particular interested in a system of small size sitting at a fixed space coordinate [MATH] above Earth.', '1311.1095-2-17-13': 'This can be modelled by confining the field to a small box of volume [MATH] around [MATH], so that the space integral [REF] can be restricted to that volume.', '1311.1095-2-17-14': 'If the potential is approximately constant within the volume, [MATH] for [MATH], we can take it out of the integral and obtain [EQUATION] where the rest Hamiltonian is given by [MATH], which is just the usual Klein-Gordon Hamiltonian in a finite volume in Minkowski space-time.', '1311.1095-2-17-15': 'The factor in front of [MATH] in eq. [REF] is responsible for the gravitational red-shift: all energies, when measured according to coordinate time, are rescaled with respect to those measured locally at the position of the system.', '1311.1095-2-17-16': 'We can see this explicitly by considering a cubic box of side [MATH], for which the rest Hamiltonian can be diagonalized as [EQUATION] where, restoring units, [MATH], [MATH] for [MATH], [MATH] and [MATH]) creates (annihilates) a boson with momentum [MATH] (neglecting the constant vacuum energy).', '1311.1095-2-17-17': 'For a particle at a distance [MATH] above Earth, we can set [MATH], where [MATH] is the gravitational acceleration.', '1311.1095-2-17-18': 'The Hamiltonian [REF] thus becomes [EQUATION]', '1311.1095-2-17-19': 'Equations [REF] shows the same coupling between position and internal energy as described in the main text (eq. [REF]).', '1311.1095-2-17-20': 'It is valid for a system at rest in the chosen coordinate system, i.e. with vanishing external momentum.', '1311.1095-2-17-21': 'The contribution of the external momentum is recovered by moving to an arbitrary coordinate system, yielding the coupling [REF].', '1311.1095-2-18-0': '## Master equation due to gravitational time dilation', '1311.1095-2-19-0': 'Here we derive an equation of motion for the centre-of-mass of a composite quantum system in the presence of time dilation.', '1311.1095-2-19-1': 'We keep the composition, the centre-of-mass Hamiltonian [MATH] and the relativistic time dilation completely general.', '1311.1095-2-19-2': 'The overall Hamiltonian of the system is [MATH], where [MATH] governs the evolution of the internal constituents and [MATH] captures the time-dilation-induced coupling between internal degrees of freedom and the centre-of-mass to lowest order in [MATH].', '1311.1095-2-19-3': '[MATH] is a function of the centre-of-mass position [MATH] and momentum [MATH] to which the internal states couple due to special relativistic and general relativistic time dilation (for the Schwarzschild metric in the weak-field limit we have [MATH]).', '1311.1095-2-19-4': 'We start with the von Neumann equation for the full state [MATH] and write [MATH], where [MATH].', '1311.1095-2-19-5': 'We change frame to primed coordinates, which we define through [MATH], where [MATH] with the average internal energy [MATH].', '1311.1095-2-19-6': 'The resulting von-Neumann equation is [EQUATION] where [MATH].', '1311.1095-2-19-7': 'The formal solution [MATH] is used in the equation above, which yields the integro-differential equation [EQUATION]', '1311.1095-2-19-8': 'We can now trace over the internal degrees of freedom.', '1311.1095-2-19-9': 'The state is initially uncorrelated [MATH] and we take the Born approximation, keeping only terms to second order in [MATH].', '1311.1095-2-19-10': 'In this case [MATH] can be replaced under the integral by [MATH] and the master equation for the centre-of-mass becomes [EQUATION]', '1311.1095-2-19-11': 'Here we used the notation [MATH] for the energy fluctuations of the internal states and [MATH].', '1311.1095-2-19-12': 'Changing back to the Schrodinger picture, and introducing [MATH] we obtain the integro-differential equation: [EQUATION] where [MATH].', '1311.1095-2-19-13': 'This is the general equation of motion for a composite particle of arbitrary composition that undergoes time dilation.', '1311.1095-2-19-14': 'The decoherence of its off-diagonal elements is governed by its internal energy spread [MATH] and by the metric-dependent coupling [MATH].', '1311.1095-2-19-15': 'The former is [MATH] in the high-temperature limit for [MATH] non-interacting internal harmonic oscillators and the latter is [MATH] for stationary particles in the homogeneous weak-field limit of the Schwarzschild metric.', '1311.1095-2-20-0': '## Effect of time dilation on clocks and thermal states', '1311.1095-2-21-0': 'The time-dilation-induced coupling between internal degrees of freedom and the centre of mass causes decoherence of the latter.', '1311.1095-2-21-1': 'The complementarity between the visibility [MATH] of interference and the which-path information [MATH] is given by the inequality [MATH] [CITATION].', '1311.1095-2-21-2': 'The equal sign holds for pure states, i.e. for well-defined clocks as considered in ref. [CITATION] in the main text.', '1311.1095-2-21-3': 'For mixed states, it is possible to have loss of visibility with no accessible which-path information, as is the case here (as well as in most other decoherence models for which the Born approximation is used).', '1311.1095-2-21-4': 'The thermal state of the internal degrees of freedom can be seen as a mixture of clock-states each measuring the proper time along its respective path.', '1311.1095-2-21-5': 'To highlight this, consider for example a particle with a single 2-level internal degree of freedom which is in a clock-state, i.e. in a superposition of the ground and excited state with transition frequency [MATH] and an arbitrary relative phase [MATH]: [MATH].', '1311.1095-2-21-6': 'If the particle moves along a world line with overall proper time [MATH], the internal state will evolve to [MATH].', '1311.1095-2-21-7': 'For the particle in superposition along two paths with proper time difference [MATH], the internal clock state will therefore acquire which-path information, thus leading to a loss in visibility given by [MATH], independent of the phase [MATH].', '1311.1095-2-21-8': 'For a fully mixed internal state (analogous to a thermal state), [EQUATION] the relative phase between [MATH] and [MATH] is unknown and thus no which-path information is available, but it still results in a drop in visibility.', '1311.1095-2-21-9': 'The above state can be equivalently written in the basis [EQUATION]', '1311.1095-2-21-10': 'Since [REF] and [REF] represent the same state, they cannot be discriminated and will result in the same loss of visibility for the centre of mass.', '1311.1095-2-21-11': 'The states [MATH] and [MATH] individually, however, are not clock-states, thus no time dilation can be read out directly.', '1311.1095-2-21-12': 'The interpretation of the visibility drop in this representation is the phase scrambling due to the red shift, since a system with internal states [MATH] has different weight than if the states were [MATH] and thus acquires a different phase shift.', '1311.1095-2-21-13': 'Irrespective of the state representation, time dilation couples the internal states to the centre of mass and thus cause decoherence of composite particles with internal degrees of freedom.', '1311.1095-2-22-0': 'To highlight that time dilation can physically affect thermal states, we re-derive in our formalism the well known classical, general relativistic and thermodynamical effect first considered by Tolman [CITATION].', '1311.1095-2-22-1': 'By definition, the dynamics of internal degrees of freedom is governed by the Hamiltonian [MATH] in the local rest-frame of the system.', '1311.1095-2-22-2': 'If the system is thermal, its state is therefore [MATH], where the subscript [MATH] highlights that the quantities are with respect to the local rest-frame (which is the temperature used in the main text).', '1311.1095-2-22-3': 'However, from the point of view of a stationary observer (laboratory frame) at some distance, the internal dynamics of the system at height [MATH] in a gravitational potential [MATH] is governed by the Hamiltonian [MATH].', '1311.1095-2-22-4': 'Thus the state in that frame is given by [MATH].', '1311.1095-2-22-5': 'But since the density matrix assigned to a system does not depend on the reference frame (it can be interpreted in terms of occupation numbers of its eigenstates, irrespective of the energy labels assigned to them), we have [MATH], from which it follows that local and laboratory temperatures are related by [MATH].', '1311.1095-2-22-6': 'The condition of thermal equilibrium for systems with different Hamiltonians is simply that they have the same temperature, thus two systems at heights [MATH], [MATH] are in thermal equilibrium for an outside observer if [MATH].', '1311.1095-2-22-7': 'Hence, in terms of the locally measured temperatures, the condition for thermal equilibrium between the two systems is [MATH], which means that [MATH] and [MATH] are different for systems in different gravitational potentials.', '1311.1095-2-22-8': 'For a single extended system in thermal equilibrium in a gravitational field the locally measured temperatures satisfy [MATH] constant, which is the Tolman law.', '1311.1095-2-23-0': 'This classical effect can be seen as caused by the "weight of heat" or gravitational redshift - as in Tolman\'s own work [CITATION] - or, equivalently, as caused by time dilation between parts of the system at different gravitational potentials: the local internal frequencies [MATH] are time dilated with respect to the laboratory reference frame, in which they become [MATH].', '1311.1095-2-23-1': 'For the system to be in equilibrium the radiation coming from any part of the system must have the same frequency spectrum.', '1311.1095-2-23-2': 'The local quantities [MATH] must therefore differ with height and satisfy [MATH] constant.', '1311.1095-2-23-3': 'The Tolman effect and the decoherence effect discussed in this work have the same origin: time dilation affecting thermal states in the presence gravity.'}

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[]

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[['1311.1095-1-10-0', '1311.1095-2-10-0'], ['1311.1095-1-10-1', '1311.1095-2-10-1'], ['1311.1095-1-13-0', '1311.1095-2-14-0']]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1311.1095

astro-ph-0512020

{'astro-ph-0512020-1-0-0': 'We studied the WMAP temperature anisotropy data using two different methods.', 'astro-ph-0512020-1-0-1': 'The derived signal gradient maps show regions with low mean gradients in structures near the ecliptic poles and higher gradient values in the wide ecliptic equatorial zone, being the result of non-uniform observational time sky coverage.', 'astro-ph-0512020-1-0-2': 'We show that the distinct observational time pattern present in the raw (cleaned) data leaves also its imprints on the composite CMB maps.', 'astro-ph-0512020-1-0-3': 'Next, studying distribution of the signal dispersion we show that the north-south asymmetry of the WMAP signal diminishes with galactic altitude, confirming the earlier conclusions that it possibly reveals galactic foreground effects.', 'astro-ph-0512020-1-0-4': 'As based on these results, one can suspect that the instrumental noise sky distribution and non-removed foregrounds can have affected some of the analyses of the CMB signal.', 'astro-ph-0512020-1-0-5': 'We show that actually the different characteristic axes of the CMB sky distribution derived by numerous authors are preferentially oriented towards some distinguished regions on the sky, defined by the observational time pattern and the galactic plane orientation.', 'astro-ph-0512020-1-1-0': '# Introduction', 'astro-ph-0512020-1-2-0': 'The first-year data from the Wilkinson Microwave Anisotropy Probe (WMAP) (Bennett at al. 2003a,b) became a basis for testing cosmological scenarios, substantial improving of evaluation accuracy of basic cosmological parameters related to the geometry of our Universe, dynamics of its expansion, matter and energy contents.', 'astro-ph-0512020-1-2-1': 'Therefore any studies of reliability of the provided data are of great interest and importance.', 'astro-ph-0512020-1-2-2': 'In particular, application of non-standard methods provide additional independent tests of data quality and consistency, as well as the correctness of involved interpretations.', 'astro-ph-0512020-1-3-0': 'All the sky CMB anisotropy maps give possibility to test with high accuracy the fundamental principles of current cosmology, such as the isotropy of our Universe and the Gaussianity of primordial cosmological perturbations.', 'astro-ph-0512020-1-3-1': 'The first analysis of WMAP data for Gaussianity was performed by WMAP team [CITATION].', 'astro-ph-0512020-1-3-2': 'They found that the WMAP data are consistent with the Gaussian primordial fluctuations and provided limits to the amplitude of non-Gaussian ones.', 'astro-ph-0512020-1-3-3': 'Other authors, however, applying advanced or sophisticated statistical methods to analyse the foreground-subtracted WMAP maps pointed out non-Gaussian signatures or north-south asymmetry of galactic hemispheres (Chiang et al. 2003, Park 2004, Eriksen et al. 2004a, Hansen et al. 2004a,b,c, Eriksen et al. 2005a, Cayon et al. 2005).', 'astro-ph-0512020-1-3-4': 'Some authors suppose that it may be caused by residual foreground contamination from unknown galactic or extra-galactic sources and find their possible locations, angular scales, and amplitudes (Park 2004, Eriksen et al. 2004a, Hansen et al. 2004b, Chiang Naselsky 2004, Patanchon et al. 2004, Naselsky et al. 2005, Tojeiro et al. 2005, Land Magueijo 2005).', 'astro-ph-0512020-1-3-5': 'For example, Tojeiro et al. (2005) argued that the evidence for non-Gaussianity on large scales is associated with cold spots of unsubtracted foregrounds.', 'astro-ph-0512020-1-3-6': 'Some part of non-Gaussian statistics of [MATH] fluctuations can be due to the ring of pixels of radius of 5 degrees centered at ([MATH]) [CITATION], close to the spot detected in Vielva et al. (2004) and Cruz et al. (2005).', 'astro-ph-0512020-1-3-7': 'Wibig Wolfendale (2005) performed the correlation analysis of CMB temperature and gamma-ray whole-sky maps and detected positive correlation, which may suggest an actual contamination of the WMAP data by, if not cosmic rays directly, some component accompanying them in interstellar space with a sufficiently flat emission spectrum.', 'astro-ph-0512020-1-4-0': 'All these studies indicate that WMAP Q-, V-, and W-band maps contain information on unidentified extended low-amplitude foreground structures that contaminate the relic CMB signal.', 'astro-ph-0512020-1-4-1': 'The effective extraction of these structures from the WMAP maps would improve the current and future CMB analyses, as well as reveal possible new foreground objects or effects (cf. Davies et al. 2005, Eriksen et al. 2005b).', 'astro-ph-0512020-1-5-0': 'Our principal goal in the present paper is to understand the physical meaning of structures within WMAP measurements in order to detect or limit possible non-cosmological factors affecting the data used for determining cosmological parameters.', 'astro-ph-0512020-1-5-1': 'We apply two simple analytical methods to study possible foreground or observational effects in the WMAP first-year data, not recognized and/or p removed by the cleaning methods used by the WMAP team (Bennett et al. 2003b).', 'astro-ph-0512020-1-5-2': 'In the paper, first we shortly provide the basic information about the data analysed, and next describe our method of generating gradient maps from the foreground-cleaned WMAP maps and interpret the large-scale gradient structures by the observational strategy of the WMAP satellite.', 'astro-ph-0512020-1-5-3': 'We demonstrate that the scanning effects present in the raw signal data can leave clear imprints on the gradients maps from all frequency channels, retained even in coadded, smoothed, and ILC maps.', 'astro-ph-0512020-1-5-4': 'Then, we investigate distribution of signal dispersion in the WMAP maps to find a significant asymmetry between the northern and southern galactic hemispheres.', 'astro-ph-0512020-1-5-5': 'Following some previous authors, we argue that a natural explanation of this asymmetry is residual galactic contamination.', 'astro-ph-0512020-1-5-6': 'Finally, the inevitable influence of the above-mentioned effects on the analysis of the large-scale CMB structure is shortly considered.', 'astro-ph-0512020-1-6-0': '# The observational data', 'astro-ph-0512020-1-7-0': 'In this paper we study the publicly available first-year WMAP maps in 5 frequency channels, K (22.8 GHz), Ka (33.0 GHz), Q (40.7 GHz), V (60.8 GHz) and W (93.5 GHz) downloaded from LAMBDA website.', 'astro-ph-0512020-1-7-1': 'The maps provide distributions of differential temperature measured in micro-Kelvin units, [MATH], over the entire sky.', 'astro-ph-0512020-1-7-2': 'The signal in each individual map, after subtracting the kinematic dipole component, foreground point sources, and diffuse galactic emission, is expected to consist primarily of cosmic microwave background (see Bennett et al. 2003a,b for details).', 'astro-ph-0512020-1-7-3': 'We studied mainly individual differential assemblies maps with the full available resolution (NSIDE=512 in the HEALPix pixelization).', 'astro-ph-0512020-1-7-4': 'Most analyses presented in the paper use the V2 channel map, showing the lowest galactic contamination and small instrumental noise.', 'astro-ph-0512020-1-8-0': '# Gradient analysis of the WMAP CMB anisotropy maps', 'astro-ph-0512020-1-9-0': '## Derivation of gradient maps', 'astro-ph-0512020-1-10-0': 'We study the variations of measured temperature fluctuations in the WMAP sky maps, including CMB signal and instrumental noise, by analysing the gradient structure of the data.', 'astro-ph-0512020-1-10-1': 'We applied two methods in order to derive the gradients.', 'astro-ph-0512020-1-10-2': 'In the first approach, we evaluate the maximum gradient value at a given location in the map (i.e. in each pixel/cell) calculated between the reference cell and the surrounding cells using: [EQUATION] where the index "[MATH]" indicates the reference cell. "[', 'astro-ph-0512020-1-10-3': '[MATH]" runs through all adjacent cells and [MATH] is the distance between the cells considered .', 'astro-ph-0512020-1-10-4': 'The second approach involved deriving a gradient map with the use of the Sobol operator (filter) known in the image analysis as the edge detector (Huang et al. 2003).', 'astro-ph-0512020-1-10-5': 'This method evaluated the mean gradient at a given pixel using the relative values of the signal, [MATH], in its 8 adjacent pixels.', 'astro-ph-0512020-1-10-6': 'In this case, the obtained gradient maps reveal the same pattern as in the maximum gradient method, but the derived mean gradient values are lower.', 'astro-ph-0512020-1-10-7': 'Below we present the maximum gradient maps derived with the simpler method using eq. (3.1), which is more efficient in visualizing gradient structures.', 'astro-ph-0512020-1-11-0': 'The map of [MATH] derived for the WMAP V2 channel is presented in Fig. 1b.', 'astro-ph-0512020-1-11-1': 'In the figure one can clearly see regions of low gradients at both the ecliptic poles and within two extended circular regions around the poles at the radius [MATH].', 'astro-ph-0512020-1-11-2': 'The extended regions of enhanced gradient occupy a wide stripe along the ecliptic equator.', 'astro-ph-0512020-1-11-3': 'The largest gradients occur at the galactic plane due to some residual galactic component.', 'astro-ph-0512020-1-11-4': 'Relatively large gradient values are also found in places where data were partly discarded because of the planets appearing in the WMAP field of view.', 'astro-ph-0512020-1-11-5': 'The comparison of the gradient pattern with the map of observational time ([MATH] the effective number of observations in the WMAP team nomenclature, [MATH]) (Fig.1d) proves that most of the observed extended gradient structures represent variations of instrumental noise within the data.', 'astro-ph-0512020-1-11-6': 'The noise distribution corresponds to the varying observational time in different areas of the sky due to the WMAP sky scanning pattern.', 'astro-ph-0512020-1-11-7': 'It also leads to a distinct anti-correlation between the observational time and the signal gradient.', 'astro-ph-0512020-1-12-0': '## Analysis of gradient pattern', 'astro-ph-0512020-1-13-0': 'Using the above method, we derived the gradient maps for all the WMAP channels (the maps can be obtained upon request from K.Ch.)', 'astro-ph-0512020-1-13-1': 'with analogous characteristic gradient patterns, spots at galactic poles, and rings around them.', 'astro-ph-0512020-1-13-2': 'These characteristic features in the gradient distribution are more distinct after smoothing the maps (Fig. 1c).', 'astro-ph-0512020-1-13-3': 'In order to show it quantitatively, we smoothed both the V2 CMB and the number of observation maps with increasing Gaussian beams, and calculated the correlation coefficient.', 'astro-ph-0512020-1-13-4': 'Without smoothing, the correlation is -0.16, for 1 smoothing it is -0.26, and for 5, 10, and 15 smoothing it becomes -0.69, -0.83, and -0.89, respectively.', 'astro-ph-0512020-1-14-0': 'We also obtained the gradient maps from the coadded maps for individual assemblies in V, Q and W bands, with the position-dependent, or band-dependent inverse noise-weighting.', 'astro-ph-0512020-1-14-1': 'We got gradient patterns much similar to those obtained from the V2 channel alone.', 'astro-ph-0512020-1-14-2': 'Next we applied the gradient analysis to the ILC map.', 'astro-ph-0512020-1-14-3': 'In this case, the characteristic gradient pattern is weaker, because of the convolved to 1 scale component maps of the ILC combined map.', 'astro-ph-0512020-1-14-4': 'However, the distinct pattern appears also in this case, as we demonstrate at Fig. 2b by convolving the gradient map with the 15 Gaussian beam.', 'astro-ph-0512020-1-14-5': 'In the figure, the observational time pattern analogous to that in Fig. 1d is clearly reproduced, while significantly perturbed by the effects of CMB or galactic foreground signal structures.', 'astro-ph-0512020-1-15-0': 'The performed experiments confirm that the systematic pattern revealed in the gradient maps is firmly established and cannot be removed by simple filtering procedures, like convolving, or adding data maps.', 'astro-ph-0512020-1-15-1': 'Thus it can affect the analysis and interpretation of the large-scale distribution of the signal in the WMAP maps.', 'astro-ph-0512020-1-16-0': '# Asymmetric noise distributions of galactic hemispheres', 'astro-ph-0512020-1-17-0': 'Apart from the above analysis of signal gradients, we studied also the large-scale asymmetries within the WMAP data by deriving diagrams of the measured signal variance across the sky versus the effective number of observations, N.', 'astro-ph-0512020-1-17-1': 'Such diagrams, presented by Jarosik et al. (2003) for the whole sky (see their Fig.10), we constructed independently for the northern and southern galactic hemispheres, in a couple of galactic polar-cap regions formed by removing the galactic equatorial strip with [MATH], [MATH] and [MATH] (Fig. 3).', 'astro-ph-0512020-1-17-2': 'We used these growing galactic cuts to check if possible galactic foregrounds may affect the data.', 'astro-ph-0512020-1-17-3': 'To derive the sample signal variance for a consecutive bin "[MATH]" centered at the number of observations [MATH] we used data from all strip cells with [MATH] in the respective range ([MATH], [MATH]).', 'astro-ph-0512020-1-17-4': 'There were also much smaller numbers of cells with very small or very large [MATH], which can explain growing scatter at limiting parts of the plot presented in the upper-left panel of Fig 3.', 'astro-ph-0512020-1-17-5': 'In the other three panels in this figure the zoomed distributions are presented, excluding the range with low [MATH], where the linear form of distribution is lost.', 'astro-ph-0512020-1-17-6': 'In the figure one can easily see the south-north asymmetry in the variance distribution.', 'astro-ph-0512020-1-17-7': 'The difference is preserved in a range of regions in the sky with varying number of observations.', 'astro-ph-0512020-1-17-8': 'Moreover, there is a distinct tendency for decreasing the difference between hemispheres at higher galactic latitudes.', 'astro-ph-0512020-1-18-0': 'We performed another analysis of the signal dispersion in the channel V2.', 'astro-ph-0512020-1-18-1': 'We considered 4 separate galactic altitude stripes ([MATH] [[MATH], [MATH]], [[MATH], [MATH]], [[MATH], [MATH]], [[MATH], [MATH]]) and 4 ranges of the observational time ([MATH], [MATH], [MATH], [MATH]), separately in both hemispheres.', 'astro-ph-0512020-1-18-2': 'The individual dispersion values were derived in the following way.', 'astro-ph-0512020-1-18-3': 'We assumed a strict relation to hold between dispersions of the cosmic signal (dominated by the CMB) fluctuations, [MATH], and of the instrumental noise, [MATH], constituting the measured dispersion [MATH] of temperature fluctuations ([MATH]) on the map, [EQUATION]', 'astro-ph-0512020-1-18-4': 'The noise constant for the analysed V2 channel is given by the WMAP team as [MATH] (Bennett et al. 2003a).', 'astro-ph-0512020-1-18-5': 'Our operational procedure to derive the individual dispersion value involves data from n ([MATH]) consecutive cells from the V2 sky map (for the given stripe and the observation number range): [EQUATION]', 'astro-ph-0512020-1-18-6': 'One may note that the presented algorithm, while not supposed to yield strict statistical quantities, provides a uniform measure of fluctuation amplitude, allowing for comparing the sky signal in both hemispheres.', 'astro-ph-0512020-1-19-0': 'The normalized distributions of the above dispersion values [MATH] are presented at Fig. 4.', 'astro-ph-0512020-1-19-1': 'A distinct systematic trend can be observed in the successive panels, with the south-hemisphere distributions shifted to higher dispersion values.', 'astro-ph-0512020-1-19-2': 'The observed difference is larger in sky stripes closer to the galactic plane and within these stripes is more pronounced for panels with larger [MATH] (the longer observational times).', 'astro-ph-0512020-1-19-3': 'However, in regions near the galactic poles such weak but regular shift is also observed.', 'astro-ph-0512020-1-20-0': '# Summary and Conclusions', 'astro-ph-0512020-1-21-0': 'The gradient analysis is a very efficient method in recognition of weak signal non-uniformities in the sky maps.', 'astro-ph-0512020-1-21-1': 'By analysing the gradient maps we demonstrate the WMAP noise pattern resulting from non-uniformity of the observational time distribution.', 'astro-ph-0512020-1-21-2': 'The pattern involves regions of low noise (small gradients) near the ecliptic poles and along the circles at the angular distance of [MATH] from the ecliptic poles, and non-uniform extended regions of larger noise amplitudes (larger gradients) along the ecliptic equator.', 'astro-ph-0512020-1-21-3': 'The pattern introduced by local variations in the observational time is much alike at all available frequency channels and must inevitably affect the resulting CMB maps, independently of the applied approach to constructing such maps from the original data (e.g. Bennett at al. 2003b, Eriksen et al. 2004b, Tegmark et al. 2003).', 'astro-ph-0512020-1-21-4': 'The apparent gradient structure can "propagate" through the noise filtering procedures used in attempt to reconstruct the CMB signal maps, as it is proved in section 3 by showing its imprints in the frequently presented and discussed ILC map.', 'astro-ph-0512020-1-21-5': 'It suggests that analyses of the ILC maps, or similar maps, must be affected - besides statistical flaws due to the signal generation procedure and non-removed foreground effects - by the WMAP observational time pattern (cf. in this context e.g. Fig. 1 in Hansen et al. 2004c, Fig. 4 in Jaffe et al. 2005a and Jaffe et al. 2005b).', 'astro-ph-0512020-1-21-6': 'One may also note that for the CMB study based on the WMAP data (and the future Planck mission) the best regions to study are those with the lowest gradients around the ecliptic poles, and the lowest instrumental noise.', 'astro-ph-0512020-1-22-0': 'Independently, the possible galactic foreground effects were studied by simple methods of analysing distributions of signal fluctuations versus the observational time at the WMAP V2 band.', 'astro-ph-0512020-1-22-1': 'We confirm existence of the south-north asymmetry between hemispheres, with the larger fluctuation amplitude in the south hemisphere (cf. Eriksen et al. 2004a, Hansen et al. 2004a,b).', 'astro-ph-0512020-1-22-2': 'We show that the asymmetry extends throughout a large part of the sky and diminishes at higher galactic altitudes.', 'astro-ph-0512020-1-22-3': 'It suggests that the foreground effects may be due to some galactic component concentrating along the galactic plane.', 'astro-ph-0512020-1-22-4': 'Furthermore, higher amplitude of signal fluctuations in the southern hemisphere can be, in our opinion, explained by the known fact that our Sun is situated approximately 20 pc north off this plane.', 'astro-ph-0512020-1-22-5': 'Let us also note that regularity of observed effects with growing galactic altitude, also in the separate ranges of [MATH] at Fig. 4, does not support the hypothesis suggesting an extragalactic origin of the asymmetry.', 'astro-ph-0512020-1-23-0': 'The findings suggest that the analysis of large-scale patterns of the CMB signal based on the WMAP data must be affected by observational time non-uniformities, which are not possible to be quite removed.', 'astro-ph-0512020-1-23-1': 'Thus we expect that analysis results of the global CMB sky distribution can be also influenced by this fact, introducing the ecliptic symmetry to the data, and by the foreground effects, leading to some symmetry/asymmetry with respect to the galactic plane.', 'astro-ph-0512020-1-23-2': 'Thus let us look at examples of characteristic orientations/axes of the universal CMB distribution discovered by the other authors (e.g. in discussion of non-trivial universe topology by de Oliveira-Costa et al. 2004, Weeks et al. 2004, Roukema et al. 2004).', 'astro-ph-0512020-1-23-3': 'Such 10 directions imposed onto the gradient map in Fig. 5 include: 1.)', 'astro-ph-0512020-1-23-4': '([MATH], [MATH]) Vielva et al. (2004); 2) (222[MATH], -62[MATH]) Jaffe et al. (2005a); 3) (-100[MATH], 60[MATH]) Land et al. (2005); 4) (-110[MATH], 60[MATH]) de Oliveira et al. (2004); 5) (252[MATH] , 65[MATH]); 6) (51[MATH], 51[MATH]); 7) (144[MATH], 38[MATH]); 8) (271[MATH], 3[MATH]); 9) (207[MATH], 10[MATH]); 10) (332[MATH], 25[MATH]) Roukema et al. (2005).', 'astro-ph-0512020-1-23-5': 'The presented directions show more or less preferential orientation with respect to four circles: the ecliptic equator and its meridian perpendicular to the galactic plane, plus two small circles in the distance of [MATH] around the ecliptic poles, the sites corresponding to the local maxima of the observational time distribution.', 'astro-ph-0512020-1-23-6': 'Also the discussed unexplained quadrupole and octopole orientations in respect to the ecliptic frame (cf. de Oliveira-Costa et al. 2004, Land Magueijo 2005, Jaffe et al. 2005) can be possibly explained in a natural way, if their detections are affected by the sky pattern due to the observational time, retained in the cleaned maps.', 'astro-ph-0512020-1-24-0': 'In the present work we were not able to verify the supposition that several detected structures in the CMB maps result from the discussed instrumental or foreground effects, but the distinct grouping of the observed structure orientations within some regions of the ecliptic and galactic reference frames suggests that such possible effects should be properly considered.', 'astro-ph-0512020-1-25-0': 'This work made use of the WMAP data archive and the HEALPIX software package.', 'astro-ph-0512020-1-25-1': 'We are grateful to Reinhard Schlickeiser and Andrzej Woszczyna for stimulating discussions.', 'astro-ph-0512020-1-25-2': 'The work was performed within the research program of the Jagiellonian Center of Astrophysics.'}

{'astro-ph-0512020-2-0-0': 'We studied the WMAP temperature anisotropy data using two different methods.', 'astro-ph-0512020-2-0-1': 'The derived signal gradient maps show regions with low mean gradients in structures near the ecliptic poles and higher gradient values in the wide ecliptic equatorial zone, being the result of non-uniform observational time sky coverage.', 'astro-ph-0512020-2-0-2': 'We show that the distinct observational time pattern present in the raw (cleaned) data leaves also its imprints on the composite CMB maps.', 'astro-ph-0512020-2-0-3': 'Next, studying distribution of the signal dispersion we show that the north-south asymmetry of the WMAP signal diminishes with galactic altitude, confirming the earlier conclusions that it possibly reveals galactic foreground effects.', 'astro-ph-0512020-2-0-4': 'As based on these results, one can suspect that the instrumental noise sky distribution and non-removed foregrounds can have affected some of the analyses of the CMB signal.', 'astro-ph-0512020-2-0-5': 'We show that actually the different characteristic axes of the CMB sky distribution derived by numerous authors are preferentially oriented towards some distinguished regions on the sky, defined by the observational time pattern and the galactic plane orientation.', 'astro-ph-0512020-2-1-0': '# Introduction', 'astro-ph-0512020-2-2-0': 'The first-year data from the Wilkinson Microwave Anisotropy Probe (WMAP) (Bennett at al. 2003a,b) became a basis for testing cosmological scenarios, substantial improving of evaluation accuracy of basic cosmological parameters related to the geometry of our Universe, dynamics of its expansion, matter and energy contents.', 'astro-ph-0512020-2-2-1': 'Therefore any studies of reliability of the provided data are of great interest and importance.', 'astro-ph-0512020-2-2-2': 'In particular, application of non-standard methods provide additional independent tests of data quality and consistency, as well as the correctness of involved interpretations.', 'astro-ph-0512020-2-3-0': 'All the sky CMB anisotropy maps give possibility to test with high accuracy the fundamental principles of current cosmology, such as the isotropy of our Universe and the Gaussianity of primordial cosmological perturbations.', 'astro-ph-0512020-2-3-1': 'The first analysis of WMAP data for Gaussianity was performed by WMAP team [CITATION].', 'astro-ph-0512020-2-3-2': 'They found that the WMAP data are consistent with the Gaussian primordial fluctuations and provided limits to the amplitude of non-Gaussian ones.', 'astro-ph-0512020-2-3-3': 'Other authors, however, applying advanced or sophisticated statistical methods to analyse the foreground-subtracted WMAP maps pointed out non-Gaussian signatures or north-south asymmetry of galactic hemispheres (Chiang et al. 2003, Park 2004, Eriksen et al. 2004a, Hansen et al. 2004a,b,c, Eriksen et al. 2005a, Cayon et al. 2005).', 'astro-ph-0512020-2-3-4': 'Some authors suppose that it may be caused by residual foreground contamination from unknown galactic or extra-galactic sources and find their possible locations, angular scales, and amplitudes (Park 2004, Eriksen et al. 2004a, Hansen et al. 2004b, Chiang Naselsky 2004, Patanchon et al. 2004, Naselsky et al. 2005, Tojeiro et al. 2005, Land Magueijo 2005).', 'astro-ph-0512020-2-3-5': 'For example, Tojeiro et al. (2005) argued that the evidence for non-Gaussianity on large scales is associated with cold spots of unsubtracted foregrounds.', 'astro-ph-0512020-2-3-6': 'Some part of non-Gaussian statistics of [MATH] fluctuations can be due to the ring of pixels of radius of 5 degrees centered at ([MATH]) [CITATION], close to the spot detected in Vielva et al. (2004) and Cruz et al. (2005).', 'astro-ph-0512020-2-3-7': 'Wibig Wolfendale (2005) performed the correlation analysis of CMB temperature and gamma-ray whole-sky maps and detected positive correlation, which may suggest an actual contamination of the WMAP data by, if not cosmic rays directly, some component accompanying them in interstellar space with a sufficiently flat emission spectrum.', 'astro-ph-0512020-2-4-0': 'All these studies indicate that WMAP Q-, V-, and W-band maps contain information on unidentified extended low-amplitude foreground structures that contaminate the relic CMB signal.', 'astro-ph-0512020-2-4-1': 'The effective extraction of these structures from the WMAP maps would improve the current and future CMB analyses, as well as reveal possible new foreground objects or effects (cf. Davies et al. 2005, Eriksen et al. 2005b).', 'astro-ph-0512020-2-5-0': 'Our principal goal in the present paper is to understand the physical meaning of structures within WMAP measurements in order to detect or limit possible non-cosmological factors affecting the data used for determining cosmological parameters.', 'astro-ph-0512020-2-5-1': 'We apply two simple analytical methods to study possible foreground or observational effects in the WMAP first-year data, not recognized and/or p removed by the cleaning methods used by the WMAP team (Bennett et al. 2003b).', 'astro-ph-0512020-2-5-2': 'In the paper, first we shortly provide the basic information about the data analysed, and next describe our method of generating gradient maps from the foreground-cleaned WMAP maps and interpret the large-scale gradient structures by the observational strategy of the WMAP satellite.', 'astro-ph-0512020-2-5-3': 'We demonstrate that the scanning effects present in the raw signal data can leave clear imprints on the gradients maps from all frequency channels, retained even in coadded, smoothed, and ILC maps.', 'astro-ph-0512020-2-5-4': 'Then, we investigate distribution of signal dispersion in the WMAP maps to find a significant asymmetry between the northern and southern galactic hemispheres.', 'astro-ph-0512020-2-5-5': 'Following some previous authors, we argue that a natural explanation of this asymmetry is residual galactic contamination.', 'astro-ph-0512020-2-5-6': 'Finally, the inevitable influence of the above-mentioned effects on the analysis of the large-scale CMB structure is shortly considered.', 'astro-ph-0512020-2-6-0': '# The observational data', 'astro-ph-0512020-2-7-0': 'In this paper we study the publicly available first-year WMAP maps in 5 frequency channels, K (22.8 GHz), Ka (33.0 GHz), Q (40.7 GHz), V (60.8 GHz) and W (93.5 GHz) downloaded from LAMBDA website.', 'astro-ph-0512020-2-7-1': 'The maps provide distributions of differential temperature measured in micro-Kelvin units, [MATH], over the entire sky.', 'astro-ph-0512020-2-7-2': 'The signal in each individual map, after subtracting the kinematic dipole component, foreground point sources, and diffuse galactic emission, is expected to consist primarily of cosmic microwave background (see Bennett et al. 2003a,b for details).', 'astro-ph-0512020-2-7-3': 'We studied mainly individual differential assemblies maps with the full available resolution (NSIDE=512 in the HEALPix pixelization).', 'astro-ph-0512020-2-7-4': 'Most analyses presented in the paper use the V2 channel map, showing the lowest galactic contamination and small instrumental noise.', 'astro-ph-0512020-2-8-0': '# Gradient analysis of the WMAP CMB anisotropy maps', 'astro-ph-0512020-2-9-0': '## Derivation of gradient maps', 'astro-ph-0512020-2-10-0': 'We study the variations of measured temperature fluctuations in the WMAP sky maps, including CMB signal and instrumental noise, by analysing the gradient structure of the data.', 'astro-ph-0512020-2-10-1': 'We applied two methods in order to derive the gradients.', 'astro-ph-0512020-2-10-2': 'In the first approach, we evaluate the maximum gradient value at a given location in the map (i.e. in each pixel/cell) calculated between the reference cell and the surrounding cells using: [EQUATION] where the index "[MATH]" indicates the reference cell. "[', 'astro-ph-0512020-2-10-3': '[MATH]" runs through all adjacent cells and [MATH] is the distance between the cells considered .', 'astro-ph-0512020-2-10-4': 'The second approach involved deriving a gradient map with the use of the Sobol operator (filter) known in the image analysis as the edge detector (Huang et al. 2003).', 'astro-ph-0512020-2-10-5': 'This method evaluated the mean gradient at a given pixel using the relative values of the signal, [MATH], in its 8 adjacent pixels.', 'astro-ph-0512020-2-10-6': 'In this case, the obtained gradient maps reveal the same pattern as in the maximum gradient method, but the derived mean gradient values are lower.', 'astro-ph-0512020-2-10-7': 'Below we present the maximum gradient maps derived with the simpler method using eq. (3.1), which is more efficient in visualizing gradient structures.', 'astro-ph-0512020-2-11-0': 'The map of [MATH] derived for the WMAP V2 channel is presented in Fig. 1b.', 'astro-ph-0512020-2-11-1': 'In the figure one can clearly see regions of low gradients at both the ecliptic poles and within two extended circular regions around the poles at the radius [MATH].', 'astro-ph-0512020-2-11-2': 'The extended regions of enhanced gradient occupy a wide stripe along the ecliptic equator.', 'astro-ph-0512020-2-11-3': 'The largest gradients occur at the galactic plane due to some residual galactic component.', 'astro-ph-0512020-2-11-4': 'Relatively large gradient values are also found in places where data were partly discarded because of the planets appearing in the WMAP field of view.', 'astro-ph-0512020-2-11-5': 'The comparison of the gradient pattern with the map of observational time ([MATH] the effective number of observations in the WMAP team nomenclature, [MATH]) (Fig.1d) proves that most of the observed extended gradient structures represent variations of instrumental noise within the data.', 'astro-ph-0512020-2-11-6': 'The noise distribution corresponds to the varying observational time in different areas of the sky due to the WMAP sky scanning pattern.', 'astro-ph-0512020-2-11-7': 'It also leads to a distinct anti-correlation between the observational time and the signal gradient.', 'astro-ph-0512020-2-12-0': '## Analysis of gradient pattern', 'astro-ph-0512020-2-13-0': 'Using the above method, we derived the gradient maps for all the WMAP channels (the maps can be obtained upon request from K.Ch.)', 'astro-ph-0512020-2-13-1': 'with analogous characteristic gradient patterns, spots at galactic poles, and rings around them.', 'astro-ph-0512020-2-13-2': 'These characteristic features in the gradient distribution are more distinct after smoothing the maps (Fig. 1c).', 'astro-ph-0512020-2-13-3': 'In order to show it quantitatively, we smoothed both the V2 CMB and the number of observation maps with increasing Gaussian beams, and calculated the correlation coefficient.', 'astro-ph-0512020-2-13-4': 'Without smoothing, the correlation is -0.16, for 1 smoothing it is -0.26, and for 5, 10, and 15 smoothing it becomes -0.69, -0.83, and -0.89, respectively.', 'astro-ph-0512020-2-14-0': 'We also obtained the gradient maps from the coadded maps for individual assemblies in V, Q and W bands, with the position-dependent, or band-dependent inverse noise-weighting.', 'astro-ph-0512020-2-14-1': 'We got gradient patterns much similar to those obtained from the V2 channel alone.', 'astro-ph-0512020-2-14-2': 'Next we applied the gradient analysis to the ILC map.', 'astro-ph-0512020-2-14-3': 'In this case, the characteristic gradient pattern is weaker, because of the convolved to 1 scale component maps of the ILC combined map.', 'astro-ph-0512020-2-14-4': 'However, the distinct pattern appears also in this case, as we demonstrate at Fig. 2b by convolving the gradient map with the 15 Gaussian beam.', 'astro-ph-0512020-2-14-5': 'In the figure, the observational time pattern analogous to that in Fig. 1d is clearly reproduced, while significantly perturbed by the effects of CMB or galactic foreground signal structures.', 'astro-ph-0512020-2-15-0': 'The performed experiments confirm that the systematic pattern revealed in the gradient maps is firmly established and cannot be removed by simple filtering procedures, like convolving, or adding data maps.', 'astro-ph-0512020-2-15-1': 'Thus it can affect the analysis and interpretation of the large-scale distribution of the signal in the WMAP maps.', 'astro-ph-0512020-2-16-0': '# Asymmetric noise distributions of galactic hemispheres', 'astro-ph-0512020-2-17-0': 'Apart from the above analysis of signal gradients, we studied also the large-scale asymmetries within the WMAP data by deriving diagrams of the measured signal variance across the sky versus the effective number of observations, N.', 'astro-ph-0512020-2-17-1': 'Such diagrams, presented by Jarosik et al. (2003) for the whole sky (see their Fig.10), we constructed independently for the northern and southern galactic hemispheres, in a couple of galactic polar-cap regions formed by removing the galactic equatorial strip with [MATH], [MATH] and [MATH] (Fig. 3).', 'astro-ph-0512020-2-17-2': 'We used these growing galactic cuts to check if possible galactic foregrounds may affect the data.', 'astro-ph-0512020-2-17-3': 'To derive the sample signal variance for a consecutive bin "[MATH]" centered at the number of observations [MATH] we used data from all strip cells with [MATH] in the respective range ([MATH], [MATH]).', 'astro-ph-0512020-2-17-4': 'There were also much smaller numbers of cells with very small or very large [MATH], which can explain growing scatter at limiting parts of the plot presented in the upper-left panel of Fig 3.', 'astro-ph-0512020-2-17-5': 'In the other three panels in this figure the zoomed distributions are presented, excluding the range with low [MATH], where the linear form of distribution is lost.', 'astro-ph-0512020-2-17-6': 'In the figure one can easily see the south-north asymmetry in the variance distribution.', 'astro-ph-0512020-2-17-7': 'The difference is preserved in a range of regions in the sky with varying number of observations.', 'astro-ph-0512020-2-17-8': 'Moreover, there is a distinct tendency for decreasing the difference between hemispheres at higher galactic latitudes.', 'astro-ph-0512020-2-18-0': 'We performed another analysis of the signal dispersion in the channel V2.', 'astro-ph-0512020-2-18-1': 'We considered 4 separate galactic altitude stripes ([MATH] [[MATH], [MATH]], [[MATH], [MATH]], [[MATH], [MATH]], [[MATH], [MATH]]) and 4 ranges of the observational time ([MATH], [MATH], [MATH], [MATH]), separately in both hemispheres.', 'astro-ph-0512020-2-18-2': 'The individual dispersion values were derived in the following way.', 'astro-ph-0512020-2-18-3': 'We assumed a strict relation to hold between dispersions of the cosmic signal (dominated by the CMB) fluctuations, [MATH], and of the instrumental noise, [MATH], constituting the measured dispersion [MATH] of temperature fluctuations ([MATH]) on the map, [EQUATION]', 'astro-ph-0512020-2-18-4': 'The noise constant for the analysed V2 channel is given by the WMAP team as [MATH] (Bennett et al. 2003a).', 'astro-ph-0512020-2-18-5': 'Our operational procedure to derive the individual dispersion value involves data from n ([MATH]) consecutive cells from the V2 sky map (for the given stripe and the observation number range): [EQUATION]', 'astro-ph-0512020-2-18-6': 'One may note that the presented algorithm, while not supposed to yield strict statistical quantities, provides a uniform measure of fluctuation amplitude, allowing for comparing the sky signal in both hemispheres.', 'astro-ph-0512020-2-19-0': 'The normalized distributions of the above dispersion values [MATH] are presented at Fig. 4.', 'astro-ph-0512020-2-19-1': 'A distinct systematic trend can be observed in the successive panels, with the south-hemisphere distributions shifted to higher dispersion values.', 'astro-ph-0512020-2-19-2': 'The observed difference is larger in sky stripes closer to the galactic plane and within these stripes is more pronounced for panels with larger [MATH] (the longer observational times).', 'astro-ph-0512020-2-19-3': 'However, in regions near the galactic poles such weak but regular shift is also observed.', 'astro-ph-0512020-2-20-0': '# Summary and Conclusions', 'astro-ph-0512020-2-21-0': 'The gradient analysis is a very efficient method in recognition of weak signal non-uniformities in the sky maps.', 'astro-ph-0512020-2-21-1': 'By analysing the gradient maps we demonstrate the WMAP noise pattern resulting from non-uniformity of the observational time distribution.', 'astro-ph-0512020-2-21-2': 'The pattern involves regions of low noise (small gradients) near the ecliptic poles and along the circles at the angular distance of [MATH] from the ecliptic poles, and non-uniform extended regions of larger noise amplitudes (larger gradients) along the ecliptic equator.', 'astro-ph-0512020-2-21-3': 'The pattern introduced by local variations in the observational time is much alike at all available frequency channels and must inevitably affect the resulting CMB maps, independently of the applied approach to constructing such maps from the original data (e.g. Bennett at al. 2003b, Eriksen et al. 2004b, Tegmark et al. 2003).', 'astro-ph-0512020-2-21-4': 'The apparent gradient structure can "propagate" through the noise filtering procedures used in attempt to reconstruct the CMB signal maps, as it is proved in section 3 by showing its imprints in the frequently presented and discussed ILC map.', 'astro-ph-0512020-2-21-5': 'It suggests that analyses of the ILC maps, or similar maps, must be affected - besides statistical flaws due to the signal generation procedure and non-removed foreground effects - by the WMAP observational time pattern (cf. in this context e.g. Fig. 1 in Hansen et al. 2004c, Fig. 4 in Jaffe et al. 2005a and Jaffe et al. 2005b).', 'astro-ph-0512020-2-21-6': 'One may also note that for the CMB study based on the WMAP data (and the future Planck mission) the best regions to study are those with the lowest gradients around the ecliptic poles, and the lowest instrumental noise.', 'astro-ph-0512020-2-22-0': 'Independently, the possible galactic foreground effects were studied by simple methods of analysing distributions of signal fluctuations versus the observational time at the WMAP V2 band.', 'astro-ph-0512020-2-22-1': 'We confirm existence of the south-north asymmetry between hemispheres, with the larger fluctuation amplitude in the south hemisphere (cf. Eriksen et al. 2004a, Hansen et al. 2004a,b).', 'astro-ph-0512020-2-22-2': 'We show that the asymmetry extends throughout a large part of the sky and diminishes at higher galactic altitudes.', 'astro-ph-0512020-2-22-3': 'It suggests that the foreground effects may be due to some galactic component concentrating along the galactic plane.', 'astro-ph-0512020-2-22-4': 'Furthermore, higher amplitude of signal fluctuations in the southern hemisphere can be, in our opinion, explained by the known fact that our Sun is situated approximately 20 pc north off this plane.', 'astro-ph-0512020-2-22-5': 'Let us also note that regularity of observed effects with growing galactic altitude, also in the separate ranges of [MATH] at Fig. 4, does not support the hypothesis suggesting an extragalactic origin of the asymmetry.', 'astro-ph-0512020-2-23-0': 'The findings suggest that the analysis of large-scale patterns of the CMB signal based on the WMAP data must be affected by observational time non-uniformities, which are not possible to be quite removed.', 'astro-ph-0512020-2-23-1': 'Thus we expect that analysis results of the global CMB sky distribution can be also influenced by this fact, introducing the ecliptic symmetry to the data, and by the foreground effects, leading to some symmetry/asymmetry with respect to the galactic plane.', 'astro-ph-0512020-2-23-2': 'Thus let us look at examples of characteristic orientations/axes of the universal CMB distribution discovered by the other authors (e.g. in discussion of non-trivial universe topology by de Oliveira-Costa et al. 2004, Weeks et al. 2004, Roukema et al. 2004).', 'astro-ph-0512020-2-23-3': 'Such 10 directions imposed onto the gradient map in Fig. 5 include: 1.)', 'astro-ph-0512020-2-23-4': '([MATH], [MATH]) Vielva et al. (2004); 2) (222[MATH], -62[MATH]) Jaffe et al. (2005a); 3) (-100[MATH], 60[MATH]) Land et al. (2005); 4) (-110[MATH], 60[MATH]) de Oliveira et al. (2004); 5) (252[MATH] , 65[MATH]); 6) (51[MATH], 51[MATH]); 7) (144[MATH], 38[MATH]); 8) (271[MATH], 3[MATH]); 9) (207[MATH], 10[MATH]); 10) (332[MATH], 25[MATH]) Roukema et al. (2005).', 'astro-ph-0512020-2-23-5': 'The presented directions show more or less preferential orientation with respect to four circles: the ecliptic equator and its meridian perpendicular to the galactic plane, plus two small circles in the distance of [MATH] around the ecliptic poles, the sites corresponding to the local maxima of the observational time distribution.', 'astro-ph-0512020-2-23-6': 'Also the discussed unexplained quadrupole and octopole orientations in respect to the ecliptic frame (cf. de Oliveira-Costa et al. 2004, Land Magueijo 2005, Jaffe et al. 2005) can be possibly explained in a natural way, if their detections are affected by the sky pattern due to the observational time, retained in the cleaned maps.', 'astro-ph-0512020-2-24-0': 'In the present work we were not able to verify the supposition that several detected structures in the CMB maps result from the discussed instrumental or foreground effects, but the distinct grouping of the observed structure orientations within some regions of the ecliptic and galactic reference frames suggests that such possible effects should be properly considered.', 'astro-ph-0512020-2-25-0': 'This work made use of the WMAP data archive and the HEALPIX software package.', 'astro-ph-0512020-2-25-1': 'We are grateful to Reinhard Schlickeiser and Andrzej Woszczyna for stimulating discussions.', 'astro-ph-0512020-2-25-2': 'The work was performed within the research program of the Jagiellonian Center of Astrophysics.'}

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[]

[]

[]

[]

['astro-ph-0512020-1-13-4', 'astro-ph-0512020-1-23-4', 'astro-ph-0512020-2-13-4', 'astro-ph-0512020-2-23-4']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0512020

1710.02468

{'1710.02468-1-0-0': 'In this paper we consider the evaluation of the Araki-Sucher correction for arbitrary many-electron atomic and molecular systems.', '1710.02468-1-0-1': 'This contribution appears in the leading order quantum electrodynamics corrections to the energy of a bound state.', '1710.02468-1-0-2': 'The conventional one-electron basis set of Gaussian-type orbitals (GTOs) is adopted; this leads to two-electron matrix elements which are evaluated with help of generalised the McMurchie-Davidson scheme.', '1710.02468-1-0-3': 'We also consider the convergence of the results towards the complete basis set.', '1710.02468-1-0-4': 'A rigorous analytic result for the convergence rate is obtained and verified by comparing with independent numerical values for the helium atom.', '1710.02468-1-0-5': 'Finally, we present a selection of numerical examples and compare our results with the available reference data for small systems.', '1710.02468-1-0-6': 'In contrast with other methods used for the evaluation of the Araki-Sucher correction, our method is not restricted to few-electron atoms or molecules.', '1710.02468-1-0-7': 'This is illustrated by calculations for several many-electron atoms and molecules.', '1710.02468-1-1-0': '# Introduction', '1710.02468-1-2-0': 'In the past few decades there has been remarkable a progress in the many-body electronic structure theory.', '1710.02468-1-2-1': 'This has allowed to treat large systems of chemical or biological significance containing hundreds of electrons and, at the same time, obtain very accurate results for small systems which are intensively studied spectroscopically.', '1710.02468-1-2-2': 'Introduction of general explicitly correlated methods [CITATION], reliable extrapolation techniques [CITATION], general coupled cluster theories [CITATION], and new or improved one-electron basis sets [CITATION] made the so-called spectroscopic accuracy (few cm[MATH] or less) achievable for many small molecules.', '1710.02468-1-3-0': 'However, as the accuracy standards of routine calculations are tightened up one encounters new challenges.', '1710.02468-1-3-1': 'One of these challenges is the necessity to include corrections due to finite mass of the nuclei [CITATION], relativistic, and quantum electrodynamic (QED) effects [CITATION], and possibly finite nuclear size [CITATION].', '1710.02468-1-3-2': 'The former two have been subjects to many studies in the past decades, see Refs. [CITATION] and references therein.', '1710.02468-1-3-3': 'However, systematic studies of importance of the QED effects in atoms and molecules are scarce and have begun relatively recently [CITATION].', '1710.02468-1-3-4': 'High accuracy of the ab initio calculations and reliability of the theoretical predictions is of prime importance, e.g., in the field of ultracold molecules.', '1710.02468-1-3-5': 'This is best illustrated by the papers of McGuyer et al. [CITATION] devoted to observation of the subradiant states of Sr[MATH] or by McDonald and collaborators [CITATION] where photodissociation of ultracold molecules was studied.', '1710.02468-1-3-6': 'Notably, the importance of QED effects has also been realised in the first-principles studies of He[MATH] for the purposes of metrology [CITATION].', '1710.02468-1-4-0': 'QED is definitely one of the most successful theories in physics, with calculation of the anomalous magnetic moment of the electron being the prime example [CITATION].', '1710.02468-1-4-1': 'However, applications to the bound states, e.g., with a strong Coulomb field are marred with problems.', '1710.02468-1-4-2': 'Two physical phenomena, electron self-energy and vacuum polarisation, giving rise to the Lamb shift [CITATION] are difficult to include in the standard many-body theories.', '1710.02468-1-4-3': 'For moderate and large [MATH] approaches based on the Uehling potential [CITATION] with optional corrections [CITATION], scaling of the hydrogen-like values [CITATION], effective potentials of Shabaev et al. [CITATION], multiple commutator approach by Labzowsky and Goidenko [CITATION], and effective Hamiltonians of Flambaum and Ginges [CITATION] were used with a considerable success.', '1710.02468-1-5-0': 'However, for small and moderate [MATH] the most theoretically consistent approach is the nonrelativistic QED theory (NRQED) proposed by Caswell and Lepage [CITATION] and further developed and extended by Pachucki [CITATION].', '1710.02468-1-5-1': 'This method relies on the expansion of the exact energy in power series of the fine structure constant, [MATH].', '1710.02468-1-5-2': 'The coefficients of the expansion are evaluated as expectation values of an effective Hamiltonian with the nonrelativistic wavefunction.', '1710.02468-1-5-3': 'Thus, the zeroth-order term is simply the nonrelativistic energy, the first-order term is zero, the second-order contributions are expectation values of the Breit-Pauli Hamiltonian (the relativistic corrections).', '1710.02468-1-6-0': 'NRQED has been successfully applied to numerous few-electron atomic and molecular systems.', '1710.02468-1-6-1': 'Obvious applications are the one-electron systems such as hydrogen-like atoms (see Ref. [CITATION] for a comprehensive review)', '1710.02468-1-7-0': 'and hydrogen molecular ion [CITATION].', '1710.02468-1-7-1': 'Beyond that, very accurate results are available for the helium atom [CITATION], hydrogen molecule [CITATION], and their isotopomers.', '1710.02468-1-7-2': 'Remarkably, corrections of the order [MATH] have been derived and evaluated recently [CITATION].', '1710.02468-1-7-3': 'Other examples are lithium [CITATION] and beryllium atoms [CITATION] with the corresponding ions [CITATION], and helium dimer [CITATION].', '1710.02468-1-7-4': 'In all these examples very accurate agreement with the experimental data has been obtained which confirms validity and applicability of NRQED to light molecular and atomic systems.', '1710.02468-1-7-5': 'However, all presently available rigorous methods for calculation of the NRQED corrections are inherently limited to few-body systems and cannot be straightforwardly extended to larger ones.', '1710.02468-1-8-0': 'In the framework of NRQED, the leading (pure) QED corrections are of the order [MATH] and [MATH].', '1710.02468-1-8-1': 'For a singlet atomic or molecular state one has [CITATION] [EQUATION] where [MATH] is the so-called Bethe logarithm [CITATION]; [MATH] and [MATH] are the one- and two-electron Darwin terms [EQUATION] where [MATH] are the nuclear charges, and [MATH] is the three-dimensional Dirac delta distribution.', '1710.02468-1-8-2': 'Throughout the paper, we use letters [MATH] and [MATH] to denote summations over electrons and nuclei, respectively.', '1710.02468-1-8-3': 'The last term in Eq. ([REF]) is the Araki-Sucher correction [EQUATION] where the regularised distribution in the brackets is defined by the following formulae [EQUATION] and [EQUATION] where [MATH] is the Euler–Mascheroni constant, and [MATH] is the Heaviside step function in the usual convention.', '1710.02468-1-9-0': 'In evaluation of the QED corrections for many-electron atoms and molecules, two quantities present in Eq. ([REF]) are the major source of difficulties.', '1710.02468-1-9-1': 'The first one is the Bethe logarithm and the second is the Araki-Sucher correction.', '1710.02468-1-9-2': 'In this paper we are concerned with the latter quantity; evaluation of the Bethe logarithm will be considered in subsequent papers.', '1710.02468-1-10-0': 'From the point of view of the many-body electronic structure theory Eq. ([REF]) is an ordinary expectation value of a two-electron operator.', '1710.02468-1-10-1': 'Provided that the corresponding matrix elements are available, evaluation of such expectation values by using the coupled cluster (CC) [CITATION] or configuration interaction (CI) wavefunctions is a standard task [CITATION].', '1710.02468-1-10-2': 'Therefore, in this work we are concerned with evaluation of the matrix elements (i.e., two-electron integrals) of the Araki-Sucher distribution in the Gaussian-type orbitals (GTOs) basis [CITATION].', '1710.02468-1-10-3': 'Importantly, the proposed method can be applied to an arbitrary molecule and is not limited to few-electron systems.', '1710.02468-1-11-0': 'Throughout the paper, we follow Ref. [CITATION] in definitions of all special and elementary functions.', '1710.02468-1-12-0': '# Calculation of the matrix elements', '1710.02468-1-13-0': 'In this section we consider evaluation of matrix elements necessary to calculate the Araki-Sucher correction for many-electron atomic and molecular systems.', '1710.02468-1-13-1': 'We adopt the usual Gaussian-type orbitals (GTOs) in the Cartesian representation [CITATION] as one-electron basis set [EQUATION] where [MATH] is a vector specifying location of the orbital, [MATH] and similarly for the remaining coordinates.', '1710.02468-1-13-2': 'For brevity, we omit the normalisation constant in the definition ([REF]).', '1710.02468-1-13-3': 'However, normalised orbitals are used in all calculations described further in the paper.', '1710.02468-1-14-0': 'Evaluation of the Araki-Sucher correction from the many-electron coupled cluster wavefunction within the basis set ([REF]) requires the following two-electron matrix elements [EQUATION]', '1710.02468-1-14-1': 'The scheme presented further in the paper relies on the McMurchie-Davidson method [CITATION].', '1710.02468-1-14-2': 'This method was first introduced in the context of the standard two-electron repulsion integrals and various one-electron integrals necessary for calculation of the molecular properties.', '1710.02468-1-14-3': 'Later, it was extended to handle integral derivatives and more involved two-electron integrals found in the so-called explicitly correlated methods [CITATION].', '1710.02468-1-14-4': 'While some other methods of calculation of the usual electron repulsion integrals are more computationally efficient (cf. Ref. [CITATION]) than the McMurchie-Davidson scheme, the latter is much simpler to implement and extend to more complicated integrals.', '1710.02468-1-14-5': 'This was the main motivation for its use in the present context.', '1710.02468-1-15-0': '## Generalised McMurchie-Davidson scheme', '1710.02468-1-16-0': 'The backbone of the McMurchie-Davidson scheme is the so-called Gauss-Hermite function, [MATH], defined formally as [EQUATION]', '1710.02468-1-16-1': 'Clearly, the functions [MATH] are closely related (by scaling) with the well-known Hermite polynomials.', '1710.02468-1-16-2': 'It is also straightforward to prove the following relation [EQUATION] where [MATH] and [MATH].', '1710.02468-1-16-3': 'The coefficients [MATH] can be calculated with convenient recursion relations [CITATION].', '1710.02468-1-17-0': 'With the help of Eqs. ([REF]) and ([REF]) one can show that the product of two off-centred GTOs can be written as [EQUATION]', '1710.02468-1-17-1': 'Returning to the initial integrals ([REF]) and use Eq. ([REF]) for both orbital products.', '1710.02468-1-17-2': 'This leads to [EQUATION] where [EQUATION] and [MATH] is the so-called basic integral defined as [MATH] with [EQUATION]', '1710.02468-1-17-3': 'Note that differentiation with respect to the coordinates of [MATH] in Eq. ([REF]) has been replaced by differentiation with respect to the corresponding components of [MATH].', '1710.02468-1-17-4': 'This is valid because the basic integral is dependent only on the length of [MATH] but not on the individual components.', '1710.02468-1-18-0': 'The biggest inconvenience connected with Eq. ([REF]) is the necessity to differentiate with respect to Cartesian coordinates.', '1710.02468-1-18-1': 'In the original treatment of McMurchie and Davidson (concerning the standard electron repulsion integrals) a four-dimensional recursion relation was introduced to resolve this issue [CITATION].', '1710.02468-1-18-2': 'This approach is difficult to generalise to other basic integrals and typically requires a separate treatment in each case.', '1710.02468-1-18-3': 'In a recent paper we proposed a different strategy based on the following expression [CITATION] [EQUATION] where [MATH], relating Cartesian coordinates with the real solid spherical harmonics, [MATH] (note that the Racah normalisation is not adopted here).', '1710.02468-1-18-4': 'The numerical coefficients [MATH] can be precalculated and stored in memory as a look-up table (cf. the work of Schlegel [CITATION]).', '1710.02468-1-18-5': 'In analogy, the differentials present in Eq. ([REF]) are rewritten as [EQUATION] where [MATH] is the gradient operator and [MATH] are the (real) spherical harmonic gradient operators [CITATION].', '1710.02468-1-18-6': 'Heuristically, they are obtained by taking an explicit expression for [MATH] and replacing all Cartesian coordinates with the corresponding differentials.', '1710.02468-1-19-0': 'By the virtue of the Hobson theorem [CITATION] one has [EQUATION] where [MATH], for an arbitrary function [MATH] dependent only on the length of the vector, [MATH].', '1710.02468-1-19-1': 'With help of Eqs. ([REF]) and ([REF]) one can write [EQUATION]', '1710.02468-1-19-2': 'Note that the quantity in the subscript, [MATH], is always even (otherwise the coefficients [MATH] vanish).', '1710.02468-1-19-3': 'Therefore, the last step amounts to repeated action of the Laplacian on the terms in the square brackets.', '1710.02468-1-19-4': 'The final result can be obtained by noting that the solid harmonics are eigenfunctions of the Laplace operator and by using the obvious relationship [MATH] for the radial part of the integrations [EQUATION] where [MATH].', '1710.02468-1-19-5': 'The auxiliary coefficients, [MATH], are calculated recursively [EQUATION] starting with [MATH]; the last term of the recursion is neglected for [MATH].', '1710.02468-1-19-6': 'Note that the coefficients [MATH] can also be stored as a look-up table.', '1710.02468-1-20-0': 'To sum up, by means of Eq. ([REF]) all integrals [MATH] are expressed through the derivatives of the basic integral, [MATH].', '1710.02468-1-20-1': 'We consider evaluation of these quantities in the next section.', '1710.02468-1-20-2': 'Let us also note in passing that to achieve an optimal efficiency during the evaluation of Eq. ([REF]) the summations need to be carried out stepwise, paying attention to the order of the individual sums.', '1710.02468-1-21-0': '## Basic integral and derivatives', '1710.02468-1-22-0': 'Calculation of the basic integral, given formally by the limit of Eq. ([REF]), is hampered by the troublesome form of the Araki-Sucher distribution.', '1710.02468-1-22-1': 'It was shown in Ref. [CITATION] that an equivalent general formula for the basic integral reads [EQUATION] where [MATH].', '1710.02468-1-22-2': 'In the present case this expression naturally splits into two parts, [MATH] [EQUATION] where the second formula follows directly from the properties of the Dirac delta distribution.', '1710.02468-1-22-3': 'The first integral can be simplified by changing the coordinates to [MATH], [MATH], [MATH] and three arbitrary angles.', '1710.02468-1-22-4': 'Integration over all variables apart from [MATH] is elementary [EQUATION] where [MATH].', '1710.02468-1-22-5': 'The next step is to expand the [MATH]-dependent part of the integrand into power series [EQUATION] where [MATH].', '1710.02468-1-22-6': 'The first term of the series (corresponding to [MATH]) must be extracted and treated separately, but the remaining integrals are straightforward.', '1710.02468-1-22-7': 'Importantly, to simplify the integration process we drop all higher-order terms in [MATH] which do not contribute to the final result (once the [MATH] limit is taken).', '1710.02468-1-22-8': 'After integration and some rearrangements one obtains [EQUATION]', '1710.02468-1-22-9': 'Let us return to the second part of the basic integral, [MATH].', '1710.02468-1-22-10': 'Fortunately, this integration is elementary [EQUATION]', '1710.02468-1-22-11': 'Let us now add both contributions and take the limit [MATH].', '1710.02468-1-22-12': 'The logarithmic singularities present in [MATH] and [MATH] cancel out, and the result reads [EQUATION]', '1710.02468-1-22-13': 'One can easily prove that the infinite series present in the above expression is convergent for an arbitrary real [MATH].', '1710.02468-1-22-14': 'Therefore, this formula constitutes an exact analytical result.', '1710.02468-1-22-15': 'However, the rate of convergence of this series can be expected to be very slow for large values of the parameter, greatly increasing the cost of the calculations.', '1710.02468-1-22-16': 'Moreover, this representation does not allow for a straightforward calculation of the derivatives, [MATH].', '1710.02468-1-22-17': 'Therefore, it is desirable to bring this expression into a more computationally convenient form.', '1710.02468-1-23-0': 'For this sake, the series in Eq. ([REF]) is summed analytically giving the following integral representation [EQUATION]', '1710.02468-1-23-1': 'Validity of this formula can easily be verified by expanding the integrand into power series in [MATH] and integrating term-by-term.', '1710.02468-1-23-2': 'Guided by Eq. ([REF]) one can introduce a more general family of functions [EQUATION]', '1710.02468-1-23-3': 'Note that [MATH] directly corresponds to the result of the summation in Eq. ([REF]) and [MATH].', '1710.02468-1-23-4': 'With the help of the newly introduced quantities the basic integral is rewritten as [EQUATION]', '1710.02468-1-23-5': 'Within this particular representation of the basic integral it becomes disarmingly simple to perform the required differentiation.', '1710.02468-1-23-6': 'In fact, one can show that [EQUATION] which completes the present section.', '1710.02468-1-23-7': 'Parenthetically, we note that in the above formula the argument of [MATH] is always positive, i.e., [MATH], despite the fact that the formal definition of these integrals given by Eq. ([REF]) is valid for an arbitrary complex-valued [MATH].', '1710.02468-1-24-0': 'At this point we would like to compare our results with some other expressions published in the literature.', '1710.02468-1-24-1': 'An integral closely related to the basic integral [MATH] was considered in Ref. [CITATION].', '1710.02468-1-24-2': 'In fact, one can verify that by setting [MATH] in Eq. (15) of Ref. [CITATION] one obtains Eq. ([REF]) of the present work (after taking the [MATH] limit).', '1710.02468-1-24-3': 'However, no results for the derivatives [MATH] were provided as they do not appear in the explicitly correlated Gaussian calculations.', '1710.02468-1-24-4': 'Interestingly, an alternative integral representation of [MATH] was given in Ref. [CITATION].', '1710.02468-1-24-5': 'In our notation [EQUATION] where [MATH] is the error function.', '1710.02468-1-24-6': 'One can verify that the definitions ([REF]) and ([REF]) coincide by exchanging the variables and working out the inner integral.', '1710.02468-1-24-7': 'We have not found the above representation particularly useful in the present context, but it provides an additional verification that our final result is correct.', '1710.02468-1-25-0': '## Auxiliary integrals [MATH]', '1710.02468-1-26-0': 'The only missing building block of the present theory is the calculation of the integrals [MATH].', '1710.02468-1-26-1': 'First, let us specify the range of parameters ([MATH] and [MATH]) which are of interest.', '1710.02468-1-26-2': 'The maximal value of [MATH] is set to [MATH] in our program.', '1710.02468-1-26-3': 'This allows to compute the integrals ([REF]) with the maximal value [MATH] in the one-electron basis set, see Eq. ([REF]).', '1710.02468-1-26-4': 'This corresponds to the maximal value of the angular momentum [MATH]-type functions) in a purely spherical representation.', '1710.02468-1-26-5': 'The are no limitations on the value of [MATH], i.e., the code is open-ended with respect to positive values of [MATH].', '1710.02468-1-26-6': 'Below, we provide a set of procedures based mostly on the recursive relations which allow to calculate the integrals [MATH] with accuracy of at least 12 significant digits over the whole range of parameters specified above.', '1710.02468-1-27-0': 'First, for [MATH] the integrals [MATH] take a particularly simple analytic form [EQUATION] where [MATH] are the harmonic numbers.', '1710.02468-1-27-1': 'This expression can be rewritten as a convenient recursion [MATH], starting with [MATH].', '1710.02468-1-27-2': 'The values of [MATH] constitute an important special case corresponding to the atomic integrals, but they appear in large numbers also in molecular calculations.', '1710.02468-1-28-0': 'For any value of [MATH] the integrals [MATH] obey the following recursion relation [EQUATION] where [EQUATION]', '1710.02468-1-28-1': 'The latter quantity is nothing but the famous Boys function [CITATION] considered countless number of times in the quantum chemistry literature (see, for example, Refs. [CITATION] and references therein).', '1710.02468-1-28-2': 'Accurate and efficient methods for calculation of [MATH] are available and there is no reason for us to elaborate on this issue.', '1710.02468-1-29-0': 'Returning to the recursion relation ([REF]), its direct use is hampered by a peculiar behaviour of the integrals [MATH].', '1710.02468-1-29-1': 'Let us temporarily consider [MATH] to be a continuous variable.', '1710.02468-1-29-2': 'Then, for any fixed [MATH] the integrals [MATH] have a root as a function of [MATH].', '1710.02468-1-29-3': 'For brevity, let us call the exact position of the root (as a function of [MATH]) the critical line, [MATH].', '1710.02468-1-29-4': 'The exact location of the root cannot be obtained with elementary methods, but we found that a simple linear function [EQUATION] provides a reasonably faithful picture.', '1710.02468-1-29-5': 'If the recursion ([REF]) is carried out and the critical line is crossed, one can expect an unacceptable loss of significant digits due to the cancellations.', '1710.02468-1-29-6': 'Therefore, this simple approach is inherently numerically unstable, independently of whether the recursion is carried out upward or downward.', '1710.02468-1-30-0': 'One of the possible solutions to this problem is to assert that the critical line is never crossed during the recursive process.', '1710.02468-1-30-1': 'This can be achieved as follows.', '1710.02468-1-30-2': 'For an interval of [MATH] of approximately unit length we find the smallest value of [MATH] above the critical line ([MATH]) and the largest value of [MATH] below the critical line ([MATH]).', '1710.02468-1-30-3': 'Starting from the value at [MATH] the upward recursion is initiated and carried out up to the maximal desired value of [MATH].', '1710.02468-1-30-4': 'Similarly, the downward recursion is initiated at [MATH] and stopped at [MATH].', '1710.02468-1-30-5': 'This guarantees that the integrals [MATH] do not change sign in both sub-recursions and the whole process is completely numerically stable since the integrals [MATH] are always positive.', '1710.02468-1-31-0': 'The remaining problem is to evaluate the integrals [MATH] at [MATH] and [MATH] for a given [MATH].', '1710.02468-1-31-1': 'This is achieved by fitting [MATH] and [MATH] for each interval of [MATH].', '1710.02468-1-31-2': 'Since the length of each interval is only about unity the ordinary exponential-polynomial [CITATION] fitting is sufficient, i.e. [EQUATION]', '1710.02468-1-31-3': 'The length of the expansion was chosen to be [MATH] in each interval both for [MATH] and [MATH].', '1710.02468-1-32-0': 'For [MATH] the method described above needs to be slightly modified.', '1710.02468-1-32-1': 'This is the point where the critical line crosses [MATH].', '1710.02468-1-32-2': 'Therefore, for [MATH] all [MATH] with [MATH] are positive, and it is sufficient to evaluate [MATH] by fitting and carry out the recursion ([REF]) downward.', '1710.02468-1-32-3': 'We used the following fitting function [EQUATION] with [MATH].', '1710.02468-1-32-4': 'The prefactor in the second term of this expression comes from the asymptotic expansion of [MATH] which will be introduced in the next paragraph.', '1710.02468-1-33-0': 'Finally, for [MATH] we use large-[MATH] asymptotic expansion of the [MATH] functions.', '1710.02468-1-33-1': 'For [MATH] the necessary expression was given in Ref. [CITATION] [EQUATION] and for larger [MATH] the corresponding formulae can be obtained by noting that [MATH].', '1710.02468-1-33-2': 'The value of [MATH] was set to [MATH] after some numerical experimentation.', '1710.02468-1-33-3': 'Under these conditions the summation converges to the machine precision after at most [MATH] terms.', '1710.02468-1-33-4': 'In general, the rate of convergence improves with increasing [MATH] and thus the expansion ([REF]) is able to handle arbitrarily large values of [MATH].', '1710.02468-1-33-5': 'Moreover, all terms in Eq. ([REF]) are positive and thus no loss of digits in the summation is possible.', '1710.02468-1-33-6': 'This observation remains valid for [MATH].', '1710.02468-1-34-0': 'To sum up, the integrals [MATH] are calculated with a union of three algorithms, involving polynomial fitting, recursion relations and asymptotic expansion.', '1710.02468-1-34-1': 'We note that the efficiency of the resulting code is only somewhat worse than for the aforementioned Boys function.', '1710.02468-1-34-2': 'A C++ implementation of the methods described in this section can be obtained upon request.', '1710.02468-1-35-0': '# Basis set convergence issue', '1710.02468-1-36-0': 'Most of the ab initio methods used nowadays in the electronic structure theory rely on a basis set for expansion of the exact wavefunction.', '1710.02468-1-36-1': 'Consequently, observables obtained with a (necessarily finite) basis set suffer from the basis set incompleteness error.', '1710.02468-1-36-2': 'To allow for a meaningful comparison with the experimental data this error should be estimated and minimised, if possible.', '1710.02468-1-37-0': 'One of the prominent techniques applied to remove a bulk fraction of the basis set incompleteness error is the extrapolation towards the exact theoretical value.', '1710.02468-1-37-1': 'However, to ensure that such a procedure is reliable one typically requires some information on how the calculated values converge towards the exact result as a function of the basis set size.', '1710.02468-1-37-2': 'For example, it was shown by Hill [CITATION] that the nonrelativistic energy converges as [MATH], where [MATH] is the largest angular momentum present in the basis set.', '1710.02468-1-37-3': 'It can be shown that some relativistic corrections converge even slower, as [MATH].', '1710.02468-1-37-4': 'This was numerically observed in Refs. [CITATION] and later proved by Kutzelnigg [CITATION].', '1710.02468-1-37-5': 'In this case the values calculated with a finite basis set can be in error of tens of percents and extrapolation is necessary to arrive at a reliable result.', '1710.02468-1-38-0': 'Concerning the Araki-Sucher correction, it has never been assessed thus far how the results obtained with finite basis sets converge as a function of the largest angular momentum included.', '1710.02468-1-38-1': 'To answer this question we consider the ground state of the helium atom as a model system where a strict asymptotic result can be obtained.', '1710.02468-1-38-2': 'Further in the paper we show numerically that the main conclusions are valid also for many-electron many-centre systems.', '1710.02468-1-38-3': 'This allows for a reliable extrapolation towards the complete basis set limit, dramatically improving the final results.', '1710.02468-1-39-0': '## Definitions and notation', '1710.02468-1-40-0': 'We consider the ground 1[MATH]S state of the helium atom with the exact wavefunction given by [MATH], where [MATH] are the positions of the electrons and [MATH].', '1710.02468-1-40-1': 'The corresponding wavefunction can be represented as [EQUATION] where [MATH], [MATH] are the Legendre polynomials, and [MATH] is the angle between vectors [MATH], [MATH].', '1710.02468-1-40-2': 'The above expression is dubbed the partial wave expansion (PWE) by many authors and we shall follow this nomenclature for the wavefunctions and operators.', '1710.02468-1-41-0': 'It is natural to define a family of approximants to the exact wavefunction by truncating Eq. ([REF]) at a given [MATH], i.e. [EQUATION]', '1710.02468-1-41-1': 'The Araki-Sucher correction can be than approximated as the [MATH] limit of the following expectation values [EQUATION]', '1710.02468-1-41-2': 'Obviously, in the infinite [MATH] limit this series converges to the exact value, [MATH].', '1710.02468-1-41-3': 'Therefore, we may consider the error [EQUATION] as a function of [MATH] and ask what is the asymptotic form of [MATH] at large [MATH].', '1710.02468-1-41-4': 'After taking the [MATH] limit one recovers the actual result for the Araki-Sucher correction.', '1710.02468-1-41-5': 'This is only a precise mathematical restatement of the intuitive picture presented at the beginning of this section.', '1710.02468-1-42-0': 'All derivations presented further rely on the seminal work of Hill and the methods introduced therein [CITATION].', '1710.02468-1-42-1': 'The original presentation of Hill relies on a chain of postulates which are extremely difficult to prove strictly but are nonetheless very physically sound and hard to deny (especially in the face of ample numerical evidence).', '1710.02468-1-42-2': 'First, the denominator in Eq. ([REF]) can be replaced by unity as it converges much faster than the numerator and does not contribute in the leading order.', '1710.02468-1-42-3': 'Second, for large [MATH] the dominant contribution to the integral in Eq. ([REF]) comes from the region around the electrons coalescence points.', '1710.02468-1-42-4': 'The famous Kato cusp condition [CITATION] teaches us that in this regime the exact wavefunction behaves as [EQUATION] where [MATH] is the value of the exact wavefunction at [MATH].', '1710.02468-1-42-5': 'By using these assumptions modulus square of the wavefunction present is rewritten as [EQUATION]', '1710.02468-1-42-6': 'Finally, let us recall PWE for [MATH] [EQUATION] where [MATH], [MATH].', '1710.02468-1-42-7': 'This expression is closely related to the well-known Laplace expansion of the potential.', '1710.02468-1-43-0': '## PWE for the Araki-Sucher distribution', '1710.02468-1-44-0': 'Throughout the presentation we shall need PWE for the distribution of Eq. ([REF]), [EQUATION] where the radial coefficients are defined formally through the expression [EQUATION]', '1710.02468-1-44-1': 'Derivation of the explicit expression for [MATH] is fairly straightforward and relies solely on Eq. ([REF]).', '1710.02468-1-44-2': 'However, it requires some tedious technical algebra.', '1710.02468-1-44-3': 'In order to shorten the main article, we decided to move the entire derivation to the Supplemental Material [CITATION].', '1710.02468-1-44-4': 'Herein, we present only the final result [EQUATION] where [EQUATION] and [EQUATION]', '1710.02468-1-44-5': 'Note that only the leading-order terms in [MATH] have been retained in the above formulae.', '1710.02468-1-44-6': 'This is justified because the actual Araki-Sucher correction of Eq. ([REF]) involves the [MATH] limit and all higher-order contributions in [MATH] vanish.', '1710.02468-1-45-0': '## Large-[MATH] asymptotic formula for [MATH]', '1710.02468-1-46-0': 'Let us insert Eqs. ([REF]), ([REF]) and ([REF]) into Eq. ([REF]) and change the variables to [MATH], [MATH], [MATH].', '1710.02468-1-46-1': 'Integration over the remaining 3 variables gives [MATH] and integration over [MATH] is trivial due to the orthogonality of the Legendre polynomials.', '1710.02468-1-46-2': 'The error is given by [EQUATION] where the factor [MATH] comes from the volume element.', '1710.02468-1-46-3': 'Let us define the following quantities [EQUATION] which are natural constituents of Eq. ([REF]).', '1710.02468-1-46-4': 'Analogous definitions hold for the doubly primed and triply primed quantities in accordance with Eqs. ([REF])-([REF]) and for the sum of the three (without the prime).', '1710.02468-1-47-0': 'Starting with Eq. ([REF]), we change the variables to [MATH] and [MATH], insert the explicit form of Eqs. ([REF]) and ([REF]), and execute the Heaviside theta to arrive at the result [EQUATION]', '1710.02468-1-47-1': 'The inner integral can be brought into a closed form, but it is much simpler to obtain the leading-order expression in [MATH] from the integration by parts.', '1710.02468-1-47-2': 'This leads to [EQUATION] where additionally some higher-order terms in [MATH] have been neglected.', '1710.02468-1-48-0': 'Passing to the doubly primed quantities, we insert Eq. ([REF]) into Eq. ([REF]) and after elementary rearrangements and a change of variable, we obtain [EQUATION]', '1710.02468-1-48-1': 'By the virtues of the [MATH] function the integral can be rewritten to the form [EQUATION]', '1710.02468-1-48-2': 'It turns out that in our case the first integral gives zero contribution (in the small [MATH] limit).', '1710.02468-1-48-3': 'The inner integral of the second component can be expanded as a powers series in [MATH].', '1710.02468-1-48-4': 'The first term vanishes and only the second (i.e., proportional to [MATH]) has to be retained giving [EQUATION]', '1710.02468-1-48-5': 'Upon reinserting into Eq. ([REF]) and rearranging one obtains [EQUATION]', '1710.02468-1-48-6': 'The last integral [MATH] is the simplest to evaluate.', '1710.02468-1-48-7': 'One inserts Eq. ([REF]) into Eq. ([REF]) and executes the Dirac delta to arrive at [EQUATION] without invoking any approximations.', '1710.02468-1-48-8': 'Finally, we add up the three integrals evaluated above [EQUATION] where [EQUATION]', '1710.02468-1-48-9': 'One can see that in the final expression all logarithmic singularities cancel out.', '1710.02468-1-48-10': 'Therefore, we can now take the limit [MATH] removing all higher-order terms in [MATH].', '1710.02468-1-49-0': 'Let us now return to the formula for the error, Eq. ([REF]) at [MATH].', '1710.02468-1-49-1': 'Making use of Eq. ([REF]) and after some algebra the result can be written as [EQUATION]', '1710.02468-1-49-2': 'The first infinite sum is nontrivial to evaluate, but we can utilise the Euler-Maclaurin resummation formula to get the large-[MATH] asymptotics.', '1710.02468-1-49-3': 'This gives the leading-order expressions and their error estimates [EQUATION]', '1710.02468-1-49-4': 'Finally, we rewrite the error formula as [EQUATION] which indicates a very slow, i.e., logarithmic convergence of the Araki-Sucher correction towards the exact value.', '1710.02468-1-49-5': 'In fact, the convergence rate is even slower than for the aforementioned relativistic corrections [CITATION].', '1710.02468-1-49-6': 'Nevertheless, the above formula gives precise information on how the values from the finite basis sets should be extrapolated.', '1710.02468-1-50-0': '# Numerical results', '1710.02468-1-51-0': '## Benchmark calculations', '1710.02468-1-52-0': 'To verify that the method of calculation of the matrix elements of the Araki-Sucher distribution and the extrapolation scheme ([REF]) are both valid we performed calculations for several systems where reference values of this quantity are known to a sufficient accuracy.', '1710.02468-1-52-1': 'The includes the helium atom (He), lithium atom (Li) and its cation (Li[MATH]), beryllium atom (Be) and its cation (Be[MATH]), and the hydrogen molecule (H[MATH]).', '1710.02468-1-52-2': 'Expectation values of the Araki-Sucher distribution were computed by using the finite-field approach.', '1710.02468-1-52-3': 'Suitable values of the displacement parameter were found individually for each system by trial-and-error.', '1710.02468-1-52-4': 'Typically, a value of about [MATH] was optimal.', '1710.02468-1-52-5': 'For the two- and three-electron systems (He, Li[MATH], H[MATH], Li, Be[MATH]) we used the full CI method to solve the electronic Schrodinger equation (this method is exact in the complete basis set limit).', '1710.02468-1-52-6': 'For larger systems we employed the CCSD(T) method [CITATION].', '1710.02468-1-52-7': 'All electronic structure calculations reported in this work were performed with help of a locally modified version of the Gamess program package [CITATION].', '1710.02468-1-52-8': 'For the helium atom we used the customised basis sets developed by Cencek et al. [CITATION].', '1710.02468-1-52-9': 'For the hydrogen molecule, lithium and beryllium (both neutral atoms and cations) the standard basis sets developed in Refs. [CITATION] were employed.', '1710.02468-1-53-0': 'In Table [REF] we show results for the calculations for the helium atom.', '1710.02468-1-53-1': 'One can see a very slow convergence of the results with the size of the basis set.', '1710.02468-1-53-2': 'To overcome this difficulty we applied a two-point extrapolation formula, Eq. ([REF]).', '1710.02468-1-53-3': 'Note that in the present case we do not extrapolate with respect to the maximal angular momentum present in the basis ([MATH]) but rather with respect to the so-called cardinal number ([MATH]) [CITATION].', '1710.02468-1-53-4': 'This does not change the asymptotic formula ([REF]) but changes values of the numerical coefficients in the expansion.', '1710.02468-1-53-5': 'Therefore, we do not attempt to compare the values obtained by fitting with the analytic results given by Eqs. ([REF]) and ([REF]).', '1710.02468-1-53-6': 'Nonetheless, the quality of the extrapolation is very good.', '1710.02468-1-53-7': 'Extrapolation from the basis sets [MATH] reduces the error from about 30% to less than 1.5% (cf. Table [REF]).', '1710.02468-1-53-8': 'One can safely say that the extrapolation is mandatory to obtain results of any reasonable quality.', '1710.02468-1-54-0': 'Let us now pass to the calculations for the hydrogen molecule, H[MATH].', '1710.02468-1-54-1': 'We calculated the Araki-Sucher correction for several internuclear distances and compared them with more accurate values given by Piszczatowski et al. [CITATION] and by Stanke et al. [CITATION] obtained with the explicitly correlated Gaussian wavefunctions.', '1710.02468-1-54-2': 'Our extrapolated results are given in Table [REF] and compared with the two sets of reference values.', '1710.02468-1-54-3': 'One can see a reasonable agreement between the present results and Refs. [CITATION].', '1710.02468-1-54-4': 'The biggest absolute deviation from the values of Piszczatowski et al. [CITATION] is about 4%.', '1710.02468-1-54-5': 'This is only slightly larger than for the helium atom.', '1710.02468-1-54-6': 'This error increase can be (at least partially) attributed to the fact that a larger d7Z basis set was used for helium atom while calculations for the hydrogen molecule were restricted to the d6Z basis.', '1710.02468-1-55-0': 'Finally, in Table [REF] we show results of the calculations for the lithium and beryllium atoms as well as the corresponding cations.', '1710.02468-1-55-1': 'Our extrapolated values are compared with the reference data taken from the papers of Frolov et al. [CITATION] and Pachucki et al. [CITATION].', '1710.02468-1-55-2': 'The errors are consistently within the range of 1[MATH]2.', '1710.02468-1-55-3': 'Only for the lithium cation the accuracy is slightly worse ([MATH]3) but this is probably accidental.', '1710.02468-1-55-4': 'We can also check how well relative differences are reproduced in our method.', '1710.02468-1-55-5': 'To this end, we calculate contributions of the Araki-Sucher term to the ionisation energies of the lithium and beryllium atoms and compare the results with Refs. [CITATION].', '1710.02468-1-55-6': 'In both cases we find a remarkable agreement within approx. 1% of the total value.', '1710.02468-1-56-0': 'To sum up, the method of calculating the Araki-Sucher correction proposed here is fundamentally valid and useful in practice.', '1710.02468-1-56-1': 'By comparing our results with the reference data available in the literature for several few-body systems we conclude that it is capable of reaching an accuracy of a few percents or better.', '1710.02468-1-56-2': 'This is true provided that sufficiently large basis sets and accurate electronic structure methods are employed.', '1710.02468-1-56-3': 'Moreover, extrapolation to the complete basis set limit must be performed in every case.', '1710.02468-1-56-4': 'The theoretically derived leading-order formula ([REF]) is very efficient in this respect.', '1710.02468-1-57-0': '## Results for many-electron systems', '1710.02468-1-58-0': 'The biggest advantage of the method proposed here is that it can be applied to systems much larger than studied previously.', '1710.02468-1-58-1': 'This includes not only many-electron atoms, but also diatomic and even polyatomic molecules.', '1710.02468-1-58-2': 'To illustrate this we performed calculations for several many-body systems - the magnesium atom (Mg) and its ion (Mg[MATH]) and the argon atom (Ar) and its dimer (Ar[MATH]).', '1710.02468-1-58-3': 'In the case of Mg and Mg[MATH] we employed the IP-EOM-CCSD-3A method [CITATION] and the basis sets "aug-cc-pwCVX" reported in Ref. [CITATION].', '1710.02468-1-58-4': 'For the Ar and Ar[MATH] systems we used the CCSD(T) method and the basis sets "disp-XZ+2/AE" developed by Patkowski and Szalewicz [CITATION] specifically for accurate description of the argon dimer.', '1710.02468-1-58-5': 'The results are shown in Table [REF].', '1710.02468-1-58-6': 'Overall, the rate of convergence of the values obtained in finite basis sets is similar as for the helium atom which validates the extrapolation formula ([REF]) for many-electron systems.', '1710.02468-1-58-7': 'We can estimate that the accuracy of the results shown in Table [REF] is not worse than 5%.', '1710.02468-1-59-0': 'With the help of the results from Table [REF] one can also calculate the contribution of the Araki-Sucher correction to the ionisation energy of the magnesium atom and interaction energy the argon dimer.', '1710.02468-1-59-1': 'The former quantity is approximately equal to [MATH]0.02 cm[MATH] (the negative sign indicates that this correction decreases the ionisation energy).', '1710.02468-1-59-2': 'While this value seems to be very small we note that it is of the same order of magnitude as the present-day experimental uncertainty in the measurement of the ionisation energy of the magnesium atom, 0.03 cm[MATH] [CITATION].', '1710.02468-1-59-3': 'For the argon dimer we calculate that the contribution to the interaction energy of the Araki-Sucher term is equal to 0.02 cm[MATH].', '1710.02468-1-59-4': 'Again, this value has to be put into context.', '1710.02468-1-59-5': 'The total interaction energy of the argon dimer is approximately 99 cm[MATH].', '1710.02468-1-59-6': 'Therefore, while the Araki-Sucher contribution is small on the absolute scale, it becomes non-negligible in relation to other subtle effects.', '1710.02468-1-59-7': 'Moreover, the theoretical accuracy attainable for the argon dimer at present [CITATION] is already quite close to the level where the QED effects come into play.', '1710.02468-1-60-0': '# Conclusions', '1710.02468-1-61-0': 'In the present work we have put forward a general scheme to calculate the Araki-Sucher correction for many-electron systems.', '1710.02468-1-61-1': 'Several obstacles had to be removed to accomplish this goal.', '1710.02468-1-61-2': 'First, the complicated two-electron integrals involving the Araki-Sucher distribution have been solved with help of the McMurchie-Davidson technique (within the Gaussian-type orbitals basis set).', '1710.02468-1-61-3': 'It has been shown that they can be expressed through a family of one-dimensional integrals.', '1710.02468-1-61-4': 'Recursive and numerically stable computation of the latter integrals has been discussed in details.', '1710.02468-1-62-0': 'Second, the issue of convergence of the results with respect to the size of the basis set has been considered.', '1710.02468-1-62-1': 'We have demonstrated a slow convergence pattern ([MATH] in the leading order) towards the complete basis set limit.', '1710.02468-1-62-2': 'This result has been verified by comparing with reference data for the helium atom.', '1710.02468-1-62-3': 'With the analytic information about the convergence at hand, extrapolations have been used to improve the accuracy of the results.', '1710.02468-1-62-4': 'The accuracy of about 1% has been achieved in this case.', '1710.02468-1-63-0': 'To confirm the validity of the proposed approach we have performed calculations for several few- and many-electron systems.', '1710.02468-1-63-1': 'First, we have concentrated on small systems (e.g., few-electron atoms, hydrogen molecule) for which accurate reference values are available in the literature.', '1710.02468-1-63-2': 'A consistent accuracy of few percents have been obtained and the molecular results are only slightly less accurate than the atoms.', '1710.02468-1-63-3': 'Next, we have moved on to many-electron systems.', '1710.02468-1-63-4': 'We have estimated the contribution of the Araki-Sucher correction to the ionisation energy of the magnesium atom and interaction energy of the argon dimer.', '1710.02468-1-63-5': 'The final values of the Araki-Sucher correction are comparable to the present day experimental uncertainties of the measurements.', '1710.02468-1-64-0': 'This work was supported by the Polish National Science Centre through the project 2016/21/B/ST4/03877.', '1710.02468-1-64-1': 'The authors would like to thank Bogumi Jeziorski for reading and commenting on the manuscript.'}

{'1710.02468-2-0-0': 'In this paper we consider the evaluation of the Araki-Sucher correction for arbitrary many-electron atomic and molecular systems.', '1710.02468-2-0-1': 'This contribution appears in the leading order quantum electrodynamics corrections to the energy of a bound state.', '1710.02468-2-0-2': 'The conventional one-electron basis set of Gaussian-type orbitals (GTOs) is adopted; this leads to two-electron matrix elements which are evaluated with help of generalised the McMurchie-Davidson scheme.', '1710.02468-2-0-3': 'We also consider the convergence of the results towards the complete basis set.', '1710.02468-2-0-4': 'A rigorous analytic result for the convergence rate is obtained and verified by comparing with independent numerical values for the helium atom.', '1710.02468-2-0-5': 'Finally, we present a selection of numerical examples and compare our results with the available reference data for small systems.', '1710.02468-2-0-6': 'In contrast with other methods used for the evaluation of the Araki-Sucher correction, our method is not restricted to few-electron atoms or molecules.', '1710.02468-2-0-7': 'This is illustrated by calculations for several many-electron atoms and molecules.', '1710.02468-2-1-0': '# Introduction', '1710.02468-2-2-0': 'In the past few decades there has been remarkable a progress in the many-body electronic structure theory.', '1710.02468-2-2-1': 'This has allowed to treat large systems of chemical or biological significance containing hundreds of electrons and, at the same time, obtain very accurate results for small systems which are intensively studied spectroscopically.', '1710.02468-2-2-2': 'Introduction of general explicitly correlated methods [CITATION], reliable extrapolation techniques [CITATION], general coupled cluster theories [CITATION], and new or improved one-electron basis sets [CITATION] made the so-called spectroscopic accuracy (few cm[MATH] or less) achievable for many small molecules.', '1710.02468-2-3-0': 'However, as the accuracy standards of routine calculations are tightened up one encounters new challenges.', '1710.02468-2-3-1': 'One of these challenges is the necessity to include corrections due to finite mass of the nuclei [CITATION], relativistic, and quantum electrodynamic (QED) effects [CITATION], and possibly finite nuclear size [CITATION].', '1710.02468-2-3-2': 'The former two have been subjects to many studies in the past decades, see Refs. [CITATION] and references therein.', '1710.02468-2-3-3': 'However, systematic studies of importance of the QED effects in atoms and molecules are scarce and have begun relatively recently [CITATION].', '1710.02468-2-3-4': 'High accuracy of the ab initio calculations and reliability of the theoretical predictions is of prime importance, e.g., in the field of ultracold molecules.', '1710.02468-2-3-5': 'This is best illustrated by the papers of McGuyer et al. [CITATION] devoted to observation of the subradiant states of Sr[MATH] or by McDonald and collaborators [CITATION] where photodissociation of ultracold molecules was studied.', '1710.02468-2-3-6': 'Notably, the importance of QED effects has also been realised in the first-principles studies of He[MATH] for the purposes of metrology [CITATION].', '1710.02468-2-4-0': 'QED is definitely one of the most successful theories in physics, with calculation of the anomalous magnetic moment of the electron being the prime example [CITATION].', '1710.02468-2-4-1': 'However, applications to the bound states, e.g., with a strong Coulomb field are marred with problems.', '1710.02468-2-4-2': 'Two physical phenomena, electron self-energy and vacuum polarisation, giving rise to the Lamb shift [CITATION] are difficult to include in the standard many-body theories.', '1710.02468-2-4-3': 'For moderate and large [MATH] approaches based on the Uehling potential [CITATION] with optional corrections [CITATION], scaling of the hydrogen-like values [CITATION], effective potentials of Shabaev et al. [CITATION], multiple commutator approach by Labzowsky and Goidenko [CITATION], and effective Hamiltonians of Flambaum and Ginges [CITATION] were used with a considerable success.', '1710.02468-2-5-0': 'However, for small and moderate [MATH] the most theoretically consistent approach is the nonrelativistic QED theory (NRQED) proposed by Caswell and Lepage [CITATION] and further developed and extended by Pachucki [CITATION].', '1710.02468-2-5-1': 'This method relies on the expansion of the exact energy in power series of the fine structure constant, [MATH].', '1710.02468-2-5-2': 'The coefficients of the expansion are evaluated as expectation values of an effective Hamiltonian with the nonrelativistic wavefunction.', '1710.02468-2-5-3': 'Thus, the zeroth-order term is simply the nonrelativistic energy, the first-order term is zero, the second-order contributions are expectation values of the Breit-Pauli Hamiltonian (the relativistic corrections).', '1710.02468-2-6-0': 'NRQED has been successfully applied to numerous few-electron atomic and molecular systems.', '1710.02468-2-6-1': 'Obvious applications are the one-electron systems such as hydrogen-like atoms (see Ref. [CITATION] for a comprehensive review)', '1710.02468-2-7-0': 'and hydrogen molecular ion [CITATION].', '1710.02468-2-7-1': 'Beyond that, very accurate results are available for the helium atom [CITATION], hydrogen molecule [CITATION], and their isotopomers.', '1710.02468-2-7-2': 'Remarkably, corrections of the order [MATH] have been derived and evaluated recently [CITATION].', '1710.02468-2-7-3': 'Other examples are lithium [CITATION] and beryllium atoms [CITATION] with the corresponding ions [CITATION], and helium dimer [CITATION].', '1710.02468-2-7-4': 'In all these examples very accurate agreement with the experimental data has been obtained which confirms validity and applicability of NRQED to light molecular and atomic systems.', '1710.02468-2-7-5': 'However, all presently available rigorous methods for calculation of the NRQED corrections are inherently limited to few-body systems and cannot be straightforwardly extended to larger ones.', '1710.02468-2-8-0': 'In the framework of NRQED, the leading (pure) QED corrections are of the order [MATH] and [MATH].', '1710.02468-2-8-1': 'For a singlet atomic or molecular state one has [CITATION] [EQUATION] where [MATH] is the so-called Bethe logarithm [CITATION]; [MATH] and [MATH] are the one- and two-electron Darwin terms [EQUATION] where [MATH] are the nuclear charges, and [MATH] is the three-dimensional Dirac delta distribution.', '1710.02468-2-8-2': 'Throughout the paper, we use letters [MATH] and [MATH] to denote summations over electrons and nuclei, respectively.', '1710.02468-2-8-3': 'The last term in Eq. ([REF]) is the Araki-Sucher correction [EQUATION] where the regularised distribution in the brackets is defined by the following formulae [EQUATION] and [EQUATION] where [MATH] is the Euler–Mascheroni constant, and [MATH] is the Heaviside step function in the usual convention.', '1710.02468-2-9-0': 'In evaluation of the QED corrections for many-electron atoms and molecules, two quantities present in Eq. ([REF]) are the major source of difficulties.', '1710.02468-2-9-1': 'The first one is the Bethe logarithm and the second is the Araki-Sucher correction.', '1710.02468-2-9-2': 'In this paper we are concerned with the latter quantity; evaluation of the Bethe logarithm will be considered in subsequent papers.', '1710.02468-2-9-3': 'Let us point out that the Araki-Sucher term is not necessarily the largest of the [MATH] QED corrections.', '1710.02468-2-9-4': 'In fact, the one-electron terms typically dominate in Eq. ([REF]).', '1710.02468-2-9-5': 'However, the relative importance of the Araki-Sucher correction is expected to increase for heavier atoms similarly as for the two-electron terms of the Breit-Pauli Hamiltonian.', '1710.02468-2-9-6': 'Moreover, for polyatomic systems the Araki-Sucher correction possesses an usual [MATH] asymptotics for large interatomic distances, [MATH].', '1710.02468-2-9-7': 'As a result, it decays much less rapidly than the other components of the potential energy curve [CITATION] and its importance is substantial for large [MATH].', '1710.02468-2-10-0': 'From the point of view of the many-body electronic structure theory Eq. ([REF]) is an ordinary expectation value of a two-electron operator.', '1710.02468-2-10-1': 'Provided that the corresponding matrix elements are available, evaluation of such expectation values by using the coupled cluster (CC) [CITATION] or configuration interaction (CI) wavefunctions is a standard task [CITATION].', '1710.02468-2-10-2': 'Therefore, in this work we are concerned with evaluation of the matrix elements (i.e., two-electron integrals) of the Araki-Sucher distribution in the Gaussian-type orbitals (GTOs) basis [CITATION].', '1710.02468-2-10-3': 'Importantly, the proposed method can be applied to an arbitrary molecule and is not limited to few-electron systems.', '1710.02468-2-11-0': 'Throughout the paper, we follow Ref. [CITATION] in definitions of all special and elementary functions.', '1710.02468-2-12-0': '# Calculation of the matrix elements', '1710.02468-2-13-0': 'In this section we consider evaluation of matrix elements necessary to calculate the Araki-Sucher correction for many-electron atomic and molecular systems.', '1710.02468-2-13-1': 'We adopt the usual Gaussian-type orbitals (GTOs) in the Cartesian representation [CITATION] as one-electron basis set [EQUATION] where [MATH] is a vector specifying location of the orbital, [MATH] and similarly for the remaining coordinates.', '1710.02468-2-13-2': 'For brevity, we omit the normalisation constant in the definition ([REF]).', '1710.02468-2-13-3': 'However, normalised orbitals are used in all calculations described further in the paper.', '1710.02468-2-14-0': 'Evaluation of the Araki-Sucher correction from the many-electron coupled cluster wavefunction within the basis set ([REF]) requires the following two-electron matrix elements [EQUATION]', '1710.02468-2-14-1': 'The scheme presented further in the paper relies on the McMurchie-Davidson method [CITATION].', '1710.02468-2-14-2': 'This method was first introduced in the context of the standard two-electron repulsion integrals and various one-electron integrals necessary for calculation of the molecular properties.', '1710.02468-2-14-3': 'Later, it was extended to handle integral derivatives and more involved two-electron integrals found in the so-called explicitly correlated methods [CITATION].', '1710.02468-2-14-4': 'While some other methods of calculation of the usual electron repulsion integrals are more computationally efficient (cf. Ref. [CITATION]) than the McMurchie-Davidson scheme, the latter is much simpler to implement and extend to more complicated integrals.', '1710.02468-2-14-5': 'This was the main motivation for its use in the present context.', '1710.02468-2-15-0': '## Generalised McMurchie-Davidson scheme', '1710.02468-2-16-0': 'The backbone of the McMurchie-Davidson scheme is the so-called Gauss-Hermite function, [MATH], defined formally as [EQUATION]', '1710.02468-2-16-1': 'Clearly, the functions [MATH] are closely related (by scaling) with the well-known Hermite polynomials.', '1710.02468-2-16-2': 'It is also straightforward to prove the following relation [EQUATION] where [MATH] and [MATH].', '1710.02468-2-16-3': 'The coefficients [MATH] can be calculated with convenient recursion relations [CITATION].', '1710.02468-2-17-0': 'With the help of Eqs. ([REF]) and ([REF]) one can show that the product of two off-centred GTOs can be written as [EQUATION]', '1710.02468-2-17-1': 'Returning to the initial integrals ([REF]) and use Eq. ([REF]) for both orbital products.', '1710.02468-2-17-2': 'This leads to [EQUATION] where [EQUATION] and [MATH] is the so-called basic integral defined as [MATH] with [EQUATION]', '1710.02468-2-17-3': 'Note that differentiation with respect to the coordinates of [MATH] in Eq. ([REF]) has been replaced by differentiation with respect to the corresponding components of [MATH].', '1710.02468-2-17-4': 'This is valid because the basic integral is dependent only on the length of [MATH] but not on the individual components.', '1710.02468-2-18-0': 'The biggest inconvenience connected with Eq. ([REF]) is the necessity to differentiate with respect to Cartesian coordinates.', '1710.02468-2-18-1': 'In the original treatment of McMurchie and Davidson (concerning the standard electron repulsion integrals) a four-dimensional recursion relation was introduced to resolve this issue [CITATION].', '1710.02468-2-18-2': 'This approach is difficult to generalise to other basic integrals and typically requires a separate treatment in each case.', '1710.02468-2-18-3': 'In a recent paper we proposed a different strategy based on the following expression [CITATION] [EQUATION] where [MATH], relating Cartesian coordinates with the real solid spherical harmonics, [MATH] (note that the Racah normalisation is not adopted here).', '1710.02468-2-18-4': 'The numerical coefficients [MATH] can be precalculated and stored in memory as a look-up table (cf. the work of Schlegel [CITATION]).', '1710.02468-2-18-5': 'In analogy, the differentials present in Eq. ([REF]) are rewritten as [EQUATION] where [MATH] is the gradient operator and [MATH] are the (real) spherical harmonic gradient operators [CITATION].', '1710.02468-2-18-6': 'Heuristically, they are obtained by taking an explicit expression for [MATH] and replacing all Cartesian coordinates with the corresponding differentials.', '1710.02468-2-19-0': 'By the virtue of the Hobson theorem [CITATION] one has [EQUATION] where [MATH], for an arbitrary function [MATH] dependent only on the length of the vector, [MATH].', '1710.02468-2-19-1': 'With help of Eqs. ([REF]) and ([REF]) one can write [EQUATION]', '1710.02468-2-19-2': 'Note that the quantity in the subscript, [MATH], is always even (otherwise the coefficients [MATH] vanish).', '1710.02468-2-19-3': 'Therefore, the last step amounts to repeated action of the Laplacian on the terms in the square brackets.', '1710.02468-2-19-4': 'The final result can be obtained by noting that the solid harmonics are eigenfunctions of the Laplace operator and by using the obvious relationship [MATH] for the radial part of the integrations [EQUATION] where [MATH].', '1710.02468-2-19-5': 'The auxiliary coefficients, [MATH], are calculated recursively [EQUATION] starting with [MATH]; the last term of the recursion is neglected for [MATH].', '1710.02468-2-19-6': 'Note that the coefficients [MATH] can also be stored as a look-up table.', '1710.02468-2-20-0': 'To sum up, by means of Eq. ([REF]) all integrals [MATH] are expressed through the derivatives of the basic integral, [MATH].', '1710.02468-2-20-1': 'We consider evaluation of these quantities in the next section.', '1710.02468-2-20-2': 'Let us also note in passing that to achieve an optimal efficiency during the evaluation of Eq. ([REF]) the summations need to be carried out stepwise, paying attention to the order of the individual sums.', '1710.02468-2-21-0': '## Basic integral and derivatives', '1710.02468-2-22-0': 'Calculation of the basic integral, given formally by the limit of Eq. ([REF]), is hampered by the troublesome form of the Araki-Sucher distribution.', '1710.02468-2-22-1': 'It was shown in Ref. [CITATION] that an equivalent general formula for the basic integral reads [EQUATION] where [MATH].', '1710.02468-2-22-2': 'In the present case this expression naturally splits into two parts, [MATH] [EQUATION] where the second formula follows directly from the properties of the Dirac delta distribution.', '1710.02468-2-22-3': 'The first integral can be simplified by changing the coordinates to [MATH], [MATH], [MATH] and three arbitrary angles.', '1710.02468-2-22-4': 'Integration over all variables apart from [MATH] is elementary [EQUATION] where [MATH].', '1710.02468-2-22-5': 'The next step is to expand the [MATH]-dependent part of the integrand into power series [EQUATION] where [MATH].', '1710.02468-2-22-6': 'The first term of the series (corresponding to [MATH]) must be extracted and treated separately, but the remaining integrals are straightforward.', '1710.02468-2-22-7': 'Importantly, to simplify the integration process we drop all higher-order terms in [MATH] which do not contribute to the final result (once the [MATH] limit is taken).', '1710.02468-2-22-8': 'After integration and some rearrangements one obtains [EQUATION]', '1710.02468-2-22-9': 'Let us return to the second part of the basic integral, [MATH].', '1710.02468-2-22-10': 'Fortunately, this integration is elementary [EQUATION]', '1710.02468-2-22-11': 'Let us now add both contributions and take the limit [MATH].', '1710.02468-2-22-12': 'The logarithmic singularities present in [MATH] and [MATH] cancel out, and the result reads [EQUATION]', '1710.02468-2-22-13': 'One can easily prove that the infinite series present in the above expression is convergent for an arbitrary real [MATH].', '1710.02468-2-22-14': 'Therefore, this formula constitutes an exact analytical result.', '1710.02468-2-22-15': 'However, the rate of convergence of this series can be expected to be very slow for large values of the parameter, greatly increasing the cost of the calculations.', '1710.02468-2-22-16': 'Moreover, this representation does not allow for a straightforward calculation of the derivatives, [MATH].', '1710.02468-2-22-17': 'Therefore, it is desirable to bring this expression into a more computationally convenient form.', '1710.02468-2-23-0': 'For this sake, the series in Eq. ([REF]) is summed analytically giving the following integral representation [EQUATION]', '1710.02468-2-23-1': 'Validity of this formula can easily be verified by expanding the integrand into power series in [MATH] and integrating term-by-term.', '1710.02468-2-23-2': 'Guided by Eq. ([REF]) one can introduce a more general family of functions [EQUATION]', '1710.02468-2-23-3': 'Note that [MATH] directly corresponds to the result of the summation in Eq. ([REF]) and [MATH].', '1710.02468-2-23-4': 'With the help of the newly introduced quantities the basic integral is rewritten as [EQUATION]', '1710.02468-2-23-5': 'Within this particular representation of the basic integral it becomes disarmingly simple to perform the required differentiation.', '1710.02468-2-23-6': 'In fact, one can show that [EQUATION] which completes the present section.', '1710.02468-2-23-7': 'Parenthetically, we note that in the above formula the argument of [MATH] is always positive, i.e., [MATH], despite the fact that the formal definition of these integrals given by Eq. ([REF]) is valid for an arbitrary complex-valued [MATH].', '1710.02468-2-24-0': 'At this point we would like to compare our results with some other expressions published in the literature.', '1710.02468-2-24-1': 'An integral closely related to the basic integral [MATH] was considered in Ref. [CITATION].', '1710.02468-2-24-2': 'In fact, one can verify that by setting [MATH] in Eq. (15) of Ref. [CITATION] one obtains Eq. ([REF]) of the present work (after taking the [MATH] limit).', '1710.02468-2-24-3': 'However, no results for the derivatives [MATH] were provided as they do not appear in the explicitly correlated Gaussian calculations.', '1710.02468-2-24-4': 'Interestingly, an alternative integral representation of [MATH] was given in Ref. [CITATION].', '1710.02468-2-24-5': 'In our notation [EQUATION] where [MATH] is the error function.', '1710.02468-2-24-6': 'One can verify that the definitions ([REF]) and ([REF]) coincide by exchanging the variables and working out the inner integral.', '1710.02468-2-24-7': 'We have not found the above representation particularly useful in the present context, but it provides an additional verification that our final result is correct.', '1710.02468-2-25-0': '## Auxiliary integrals [MATH]', '1710.02468-2-26-0': 'The only missing building block of the present theory is the calculation of the integrals [MATH].', '1710.02468-2-26-1': 'First, let us specify the range of parameters ([MATH] and [MATH]) which are of interest.', '1710.02468-2-26-2': 'The maximal value of [MATH] is set to [MATH] in our program.', '1710.02468-2-26-3': 'This allows to compute the integrals ([REF]) with the maximal value [MATH] in the one-electron basis set, see Eq. ([REF]).', '1710.02468-2-26-4': 'This corresponds to the maximal value of the angular momentum [MATH]-type functions) in a purely spherical representation.', '1710.02468-2-26-5': 'The are no limitations on the value of [MATH], i.e., the code is open-ended with respect to positive values of [MATH].', '1710.02468-2-26-6': 'Below, we provide a set of procedures based mostly on the recursive relations which allow to calculate the integrals [MATH] with accuracy of at least 12 significant digits over the whole range of parameters specified above.', '1710.02468-2-27-0': 'First, for [MATH] the integrals [MATH] take a particularly simple analytic form [EQUATION] where [MATH] are the harmonic numbers.', '1710.02468-2-27-1': 'This expression can be rewritten as a convenient recursion [MATH], starting with [MATH].', '1710.02468-2-27-2': 'The values of [MATH] constitute an important special case corresponding to the atomic integrals, but they appear in large numbers also in molecular calculations.', '1710.02468-2-28-0': 'For any value of [MATH] the integrals [MATH] obey the following recursion relation [EQUATION] where [EQUATION]', '1710.02468-2-28-1': 'The latter quantity is nothing but the famous Boys function [CITATION] considered countless number of times in the quantum chemistry literature (see, for example, Refs. [CITATION] and references therein).', '1710.02468-2-28-2': 'Accurate and efficient methods for calculation of [MATH] are available and there is no reason for us to elaborate on this issue.', '1710.02468-2-29-0': 'Returning to the recursion relation ([REF]), its direct use is hampered by a peculiar behaviour of the integrals [MATH].', '1710.02468-2-29-1': 'Let us temporarily consider [MATH] to be a continuous variable.', '1710.02468-2-29-2': 'Then, for any fixed [MATH] the integrals [MATH] have a root as a function of [MATH].', '1710.02468-2-29-3': 'For brevity, let us call the exact position of the root (as a function of [MATH]) the critical line, [MATH].', '1710.02468-2-29-4': 'The exact location of the root cannot be obtained with elementary methods, but we found that a simple linear function [EQUATION] provides a reasonably faithful picture.', '1710.02468-2-29-5': 'If the recursion ([REF]) is carried out and the critical line is crossed, one can expect an unacceptable loss of significant digits due to the cancellations.', '1710.02468-2-29-6': 'Therefore, this simple approach is inherently numerically unstable, independently of whether the recursion is carried out upward or downward.', '1710.02468-2-30-0': 'One of the possible solutions to this problem is to assert that the critical line is never crossed during the recursive process.', '1710.02468-2-30-1': 'This can be achieved as follows.', '1710.02468-2-30-2': 'For an interval of [MATH] of approximately unit length we find the smallest value of [MATH] above the critical line ([MATH]) and the largest value of [MATH] below the critical line ([MATH]).', '1710.02468-2-30-3': 'Starting from the value at [MATH] the upward recursion is initiated and carried out up to the maximal desired value of [MATH].', '1710.02468-2-30-4': 'Similarly, the downward recursion is initiated at [MATH] and stopped at [MATH].', '1710.02468-2-30-5': 'This guarantees that the integrals [MATH] do not change sign in both sub-recursions and the whole process is completely numerically stable since the integrals [MATH] are always positive.', '1710.02468-2-31-0': 'The remaining problem is to evaluate the integrals [MATH] at [MATH] and [MATH] for a given [MATH].', '1710.02468-2-31-1': 'This is achieved by fitting [MATH] and [MATH] for each interval of [MATH].', '1710.02468-2-31-2': 'Since the length of each interval is only about unity the ordinary exponential-polynomial [CITATION] fitting is sufficient, i.e. [EQUATION]', '1710.02468-2-31-3': 'The length of the expansion was chosen to be [MATH] in each interval both for [MATH] and [MATH].', '1710.02468-2-32-0': 'For [MATH] the method described above needs to be slightly modified.', '1710.02468-2-32-1': 'This is the point where the critical line crosses [MATH].', '1710.02468-2-32-2': 'Therefore, for [MATH] all [MATH] with [MATH] are positive, and it is sufficient to evaluate [MATH] by fitting and carry out the recursion ([REF]) downward.', '1710.02468-2-32-3': 'We used the following fitting function [EQUATION] with [MATH].', '1710.02468-2-32-4': 'The prefactor in the second term of this expression comes from the asymptotic expansion of [MATH] which will be introduced in the next paragraph.', '1710.02468-2-33-0': 'Finally, for [MATH] we use large-[MATH] asymptotic expansion of the [MATH] functions.', '1710.02468-2-33-1': 'For [MATH] the necessary expression was given in Ref. [CITATION] [EQUATION] and for larger [MATH] the corresponding formulae can be obtained by noting that [MATH].', '1710.02468-2-33-2': 'The value of [MATH] was set to [MATH] after some numerical experimentation.', '1710.02468-2-33-3': 'Under these conditions the summation converges to the machine precision after at most [MATH] terms.', '1710.02468-2-33-4': 'In general, the rate of convergence improves with increasing [MATH] and thus the expansion ([REF]) is able to handle arbitrarily large values of [MATH].', '1710.02468-2-33-5': 'Moreover, all terms in Eq. ([REF]) are positive and thus no loss of digits in the summation is possible.', '1710.02468-2-33-6': 'This observation remains valid for [MATH].', '1710.02468-2-34-0': 'To sum up, the integrals [MATH] are calculated with a union of three algorithms, involving polynomial fitting, recursion relations and asymptotic expansion.', '1710.02468-2-34-1': 'We note that the efficiency of the resulting code is only somewhat worse than for the aforementioned Boys function.', '1710.02468-2-34-2': 'A C++ implementation of the methods described in this section can be obtained upon request.', '1710.02468-2-35-0': '# Basis set convergence issue', '1710.02468-2-36-0': 'Most of the ab initio methods used nowadays in the electronic structure theory rely on a basis set for expansion of the exact wavefunction.', '1710.02468-2-36-1': 'Consequently, observables obtained with a (necessarily finite) basis set suffer from the basis set incompleteness error.', '1710.02468-2-36-2': 'To allow for a meaningful comparison with the experimental data this error should be estimated and minimised, if possible.', '1710.02468-2-37-0': 'One of the prominent techniques applied to remove a bulk fraction of the basis set incompleteness error is the extrapolation towards the exact theoretical value.', '1710.02468-2-37-1': 'However, to ensure that such a procedure is reliable one typically requires some information on how the calculated values converge towards the exact result as a function of the basis set size.', '1710.02468-2-37-2': 'For example, it was shown by Hill [CITATION] that the nonrelativistic energy converges as [MATH], where [MATH] is the largest angular momentum present in the basis set.', '1710.02468-2-37-3': 'It can be shown that some relativistic corrections converge even slower, as [MATH].', '1710.02468-2-37-4': 'This was numerically observed in Refs. [CITATION] and later proved by Kutzelnigg [CITATION].', '1710.02468-2-37-5': 'In this case the values calculated with a finite basis set can be in error of tens of percents and extrapolation is necessary to arrive at a reliable result.', '1710.02468-2-38-0': 'Concerning the Araki-Sucher correction, it has never been assessed thus far how the results obtained with finite basis sets converge as a function of the largest angular momentum included.', '1710.02468-2-38-1': 'To answer this question we consider the ground state of the helium atom as a model system where a strict asymptotic result can be obtained.', '1710.02468-2-38-2': 'Further in the paper we show numerically that the main conclusions are valid also for many-electron many-centre systems.', '1710.02468-2-38-3': 'This allows for a reliable extrapolation towards the complete basis set limit, dramatically improving the final results.', '1710.02468-2-39-0': '## Definitions and notation', '1710.02468-2-40-0': 'We consider the ground 1[MATH]S state of the helium atom with the exact wavefunction given by [MATH], where [MATH] are the positions of the electrons and [MATH].', '1710.02468-2-40-1': 'The corresponding wavefunction can be represented as [EQUATION] where [MATH], [MATH] are the Legendre polynomials, and [MATH] is the angle between vectors [MATH], [MATH].', '1710.02468-2-40-2': 'The above expression is dubbed the partial wave expansion (PWE) by many authors and we shall follow this nomenclature for the wavefunctions and operators.', '1710.02468-2-41-0': 'It is natural to define a family of approximants to the exact wavefunction by truncating Eq. ([REF]) at a given [MATH], i.e. [EQUATION]', '1710.02468-2-41-1': 'The Araki-Sucher correction can be than approximated as the [MATH] limit of the following expectation values [EQUATION]', '1710.02468-2-41-2': 'Obviously, in the infinite [MATH] limit this series converges to the exact value, [MATH].', '1710.02468-2-41-3': 'Therefore, we may consider the error [EQUATION] as a function of [MATH] and ask what is the asymptotic form of [MATH] at large [MATH].', '1710.02468-2-41-4': 'After taking the [MATH] limit one recovers the actual result for the Araki-Sucher correction.', '1710.02468-2-41-5': 'This is only a precise mathematical restatement of the intuitive picture presented at the beginning of this section.', '1710.02468-2-42-0': 'All derivations presented further rely on the seminal work of Hill and the methods introduced therein [CITATION].', '1710.02468-2-42-1': 'The original presentation of Hill relies on a chain of postulates which are extremely difficult to prove strictly but are nonetheless very physically sound and hard to deny (especially in the face of ample numerical evidence).', '1710.02468-2-42-2': 'First, the denominator in Eq. ([REF]) can be replaced by unity as it converges much faster than the numerator and does not contribute in the leading order.', '1710.02468-2-42-3': 'Second, for large [MATH] the dominant contribution to the integral in Eq. ([REF]) comes from the region around the electrons coalescence points.', '1710.02468-2-42-4': 'The famous Kato cusp condition [CITATION] teaches us that in this regime the exact wavefunction behaves as [EQUATION] where [MATH] is the value of the exact wavefunction at [MATH].', '1710.02468-2-42-5': 'By using these assumptions modulus square of the wavefunction present is rewritten as [EQUATION]', '1710.02468-2-42-6': 'Finally, let us recall PWE for [MATH] [EQUATION] where [MATH], [MATH].', '1710.02468-2-42-7': 'This expression is closely related to the well-known Laplace expansion of the potential.', '1710.02468-2-43-0': '## PWE for the Araki-Sucher distribution', '1710.02468-2-44-0': 'Throughout the presentation we shall need PWE for the distribution of Eq. ([REF]), [EQUATION] where the radial coefficients are defined formally through the expression [EQUATION]', '1710.02468-2-44-1': 'Derivation of the explicit expression for [MATH] is fairly straightforward and relies solely on Eq. ([REF]).', '1710.02468-2-44-2': 'However, it requires some tedious technical algebra.', '1710.02468-2-44-3': 'In order to shorten the main article, we decided to move the entire derivation to Appendix [REF].', '1710.02468-2-44-4': 'Herein, we present only the final result [EQUATION] where [EQUATION] and [EQUATION]', '1710.02468-2-44-5': 'Note that only the leading-order terms in [MATH] have been retained in the above formulae.', '1710.02468-2-44-6': 'This is justified because the actual Araki-Sucher correction of Eq. ([REF]) involves the [MATH] limit and all higher-order contributions in [MATH] vanish.', '1710.02468-2-45-0': '## Large-[MATH] asymptotic formula for [MATH]', '1710.02468-2-46-0': 'Let us insert Eqs. ([REF]), ([REF]) and ([REF]) into Eq. ([REF]) and change the variables to [MATH], [MATH], [MATH].', '1710.02468-2-46-1': 'Integration over the remaining 3 variables gives [MATH] and integration over [MATH] is trivial due to the orthogonality of the Legendre polynomials.', '1710.02468-2-46-2': 'The error is given by [EQUATION] where the factor [MATH] comes from the volume element.', '1710.02468-2-46-3': 'Let us define the following quantities [EQUATION] which are natural constituents of Eq. ([REF]).', '1710.02468-2-46-4': 'Analogous definitions hold for the doubly primed and triply primed quantities in accordance with Eqs. ([REF])-([REF]) and for the sum of the three (without the prime).', '1710.02468-2-47-0': 'Starting with Eq. ([REF]), we change the variables to [MATH] and [MATH], insert the explicit form of Eqs. ([REF]) and ([REF]), and execute the Heaviside theta to arrive at the result [EQUATION]', '1710.02468-2-47-1': 'The inner integral can be brought into a closed form, but it is much simpler to obtain the leading-order expression in [MATH] from the integration by parts.', '1710.02468-2-47-2': 'This leads to [EQUATION] where additionally some higher-order terms in [MATH] have been neglected.', '1710.02468-2-48-0': 'Passing to the doubly primed quantities, we insert Eq. ([REF]) into Eq. ([REF]) and after elementary rearrangements and a change of variable, we obtain [EQUATION]', '1710.02468-2-48-1': 'By the virtues of the [MATH] function the integral can be rewritten to the form [EQUATION]', '1710.02468-2-48-2': 'It turns out that in our case the first integral gives zero contribution (in the small [MATH] limit).', '1710.02468-2-48-3': 'The inner integral of the second component can be expanded as a powers series in [MATH].', '1710.02468-2-48-4': 'The first term vanishes and only the second (i.e., proportional to [MATH]) has to be retained giving [EQUATION]', '1710.02468-2-48-5': 'Upon reinserting into Eq. ([REF]) and rearranging one obtains [EQUATION]', '1710.02468-2-48-6': 'The last integral [MATH] is the simplest to evaluate.', '1710.02468-2-48-7': 'One inserts Eq. ([REF]) into Eq. ([REF]) and executes the Dirac delta to arrive at [EQUATION] without invoking any approximations.', '1710.02468-2-48-8': 'Finally, we add up the three integrals evaluated above [EQUATION] where [EQUATION]', '1710.02468-2-48-9': 'One can see that in the final expression all logarithmic singularities cancel out.', '1710.02468-2-48-10': 'Therefore, we can now take the limit [MATH] removing all higher-order terms in [MATH].', '1710.02468-2-49-0': 'Let us now return to the formula for the error, Eq. ([REF]) at [MATH].', '1710.02468-2-49-1': 'Making use of Eq. ([REF]) and after some algebra the result can be written as [EQUATION]', '1710.02468-2-49-2': 'The first infinite sum is nontrivial to evaluate, but we can utilise the Euler-Maclaurin resummation formula to get the large-[MATH] asymptotics.', '1710.02468-2-49-3': 'This gives the leading-order expressions and their error estimates [EQUATION]', '1710.02468-2-49-4': 'Finally, we rewrite the error formula as [EQUATION] which indicates a very slow, i.e., logarithmic convergence of the Araki-Sucher correction towards the exact value.', '1710.02468-2-49-5': 'In fact, the convergence rate is even slower than for the aforementioned relativistic corrections [CITATION].', '1710.02468-2-49-6': 'Nevertheless, the above formula gives precise information on how the values from the finite basis sets should be extrapolated.', '1710.02468-2-50-0': '# Numerical results', '1710.02468-2-51-0': '## Benchmark calculations', '1710.02468-2-52-0': 'To verify that the method of calculation of the matrix elements of the Araki-Sucher distribution and the extrapolation scheme ([REF]) are both valid we performed calculations for several systems where reference values of this quantity are known to a sufficient accuracy.', '1710.02468-2-52-1': 'The includes the helium atom (He), lithium atom (Li) and its cation (Li[MATH]), beryllium atom (Be) and its cation (Be[MATH]), and the hydrogen molecule (H[MATH]).', '1710.02468-2-52-2': 'Expectation values of the Araki-Sucher distribution were computed by using the finite-field approach.', '1710.02468-2-52-3': 'Suitable values of the displacement parameter were found individually for each system by trial-and-error.', '1710.02468-2-52-4': 'Typically, a value of about [MATH] was optimal.', '1710.02468-2-52-5': 'For the two- and three-electron systems (He, Li[MATH], H[MATH], Li, Be[MATH]) we used the full CI method to solve the electronic Schrodinger equation (this method is exact in the complete basis set limit).', '1710.02468-2-52-6': 'For larger systems we employed the CCSD(T) method [CITATION].', '1710.02468-2-52-7': 'All electronic structure calculations reported in this work were performed with help of a locally modified version of the Gamess program package [CITATION].', '1710.02468-2-52-8': 'For the helium atom we used the customised basis sets developed by Cencek et al. [CITATION].', '1710.02468-2-52-9': 'For the hydrogen molecule, lithium and beryllium (both neutral atoms and cations) the standard basis sets developed in Refs. [CITATION] were employed.', '1710.02468-2-53-0': 'In Table [REF] we show results for the calculations for the helium atom.', '1710.02468-2-53-1': 'One can see a very slow convergence of the results with the size of the basis set.', '1710.02468-2-53-2': 'To overcome this difficulty we applied a two-point extrapolation formula, Eq. ([REF]).', '1710.02468-2-53-3': 'Note that in the present case we do not extrapolate with respect to the maximal angular momentum present in the basis ([MATH]) but rather with respect to the so-called cardinal number ([MATH]) [CITATION].', '1710.02468-2-53-4': 'This does not change the asymptotic formula ([REF]) but changes values of the numerical coefficients in the expansion.', '1710.02468-2-53-5': 'Therefore, we do not attempt to compare the values obtained by fitting with the analytic results given by Eqs. ([REF]) and ([REF]).', '1710.02468-2-53-6': 'Nonetheless, the quality of the extrapolation is very good.', '1710.02468-2-53-7': 'Extrapolation from the basis sets [MATH] reduces the error from about 30% to less than 1.5% (cf. Table [REF]).', '1710.02468-2-53-8': 'One can safely say that the extrapolation is mandatory to obtain results of any reasonable quality.', '1710.02468-2-54-0': 'Let us now pass to the calculations for the hydrogen molecule, H[MATH].', '1710.02468-2-54-1': 'We calculated the Araki-Sucher correction for several internuclear distances and compared them with more accurate values given by Piszczatowski et al. [CITATION] and by Stanke et al. [CITATION] obtained with the explicitly correlated Gaussian wavefunctions.', '1710.02468-2-54-2': 'Our extrapolated results are given in Table [REF] and compared with the two sets of reference values.', '1710.02468-2-54-3': 'One can see a reasonable agreement between the present results and Refs. [CITATION].', '1710.02468-2-54-4': 'The biggest absolute deviation from the values of Piszczatowski et al. [CITATION] is about 4%.', '1710.02468-2-54-5': 'This is only slightly larger than for the helium atom.', '1710.02468-2-54-6': 'This error increase can be (at least partially) attributed to the fact that a larger d7Z basis set was used for helium atom while calculations for the hydrogen molecule were restricted to the d6Z basis.', '1710.02468-2-55-0': 'Finally, in Table [REF] we show results of the calculations for the lithium and beryllium atoms as well as the corresponding cations.', '1710.02468-2-55-1': 'Our extrapolated values are compared with the reference data taken from the papers of Frolov et al. [CITATION] and Pachucki et al. [CITATION].', '1710.02468-2-55-2': 'The errors are consistently within the range of 1[MATH]2.', '1710.02468-2-55-3': 'Only for the lithium cation the accuracy is slightly worse ([MATH]3) but this is probably accidental.', '1710.02468-2-55-4': 'We can also check how well relative differences are reproduced in our method.', '1710.02468-2-55-5': 'To this end, we calculate contributions of the Araki-Sucher term to the ionisation energies of the lithium and beryllium atoms and compare the results with Refs. [CITATION].', '1710.02468-2-55-6': 'In both cases we find a remarkable agreement within approx. 1% of the total value.', '1710.02468-2-56-0': 'To sum up, the method of calculating the Araki-Sucher correction proposed here is fundamentally valid and useful in practice.', '1710.02468-2-56-1': 'By comparing our results with the reference data available in the literature for several few-body systems we conclude that it is capable of reaching an accuracy of a few percents or better.', '1710.02468-2-56-2': 'This is true provided that sufficiently large basis sets and accurate electronic structure methods are employed.', '1710.02468-2-56-3': 'Moreover, extrapolation to the complete basis set limit must be performed in every case.', '1710.02468-2-56-4': 'The theoretically derived leading-order formula ([REF]) is very efficient in this respect.', '1710.02468-2-57-0': '## Results for many-electron systems', '1710.02468-2-58-0': 'The biggest advantage of the method proposed here is that it can be applied to systems much larger than studied previously.', '1710.02468-2-58-1': 'This includes not only many-electron atoms, but also diatomic and even polyatomic molecules.', '1710.02468-2-58-2': 'To illustrate this we performed calculations for several many-body systems - the magnesium atom (Mg) and its ion (Mg[MATH]) and the argon atom (Ar) and its dimer (Ar[MATH]).', '1710.02468-2-58-3': 'In the case of Mg and Mg[MATH] we employed the IP-EOM-CCSD-3A method [CITATION] and the basis sets "aug-cc-pwCVX" reported in Ref. [CITATION].', '1710.02468-2-58-4': 'For the Ar and Ar[MATH] systems we used the CCSD(T) method and the basis sets "disp-XZ+2/AE" developed by Patkowski and Szalewicz [CITATION] specifically for accurate description of the argon dimer.', '1710.02468-2-58-5': 'The results are shown in Table [REF].', '1710.02468-2-58-6': 'Overall, the rate of convergence of the values obtained in finite basis sets is similar as for the helium atom which validates the extrapolation formula ([REF]) for many-electron systems.', '1710.02468-2-58-7': 'We can estimate that the accuracy of the results shown in Table [REF] is not worse than 5%.', '1710.02468-2-59-0': 'With the help of the results from Table [REF] one can also calculate the contribution of the Araki-Sucher correction to the ionisation energy of the magnesium atom and interaction energy the argon dimer.', '1710.02468-2-59-1': 'The former quantity is approximately equal to [MATH]0.02 cm[MATH] (the negative sign indicates that this correction decreases the ionisation energy).', '1710.02468-2-59-2': 'While this value seems to be very small we note that it is of the same order of magnitude as the present-day experimental uncertainty in the measurement of the ionisation energy of the magnesium atom, 0.03 cm[MATH] [CITATION].', '1710.02468-2-59-3': 'For the argon dimer we calculate that the contribution to the interaction energy of the Araki-Sucher term is equal to 0.02 cm[MATH].', '1710.02468-2-59-4': 'Again, this value has to be put into context.', '1710.02468-2-59-5': 'The total interaction energy of the argon dimer is approximately 99 cm[MATH].', '1710.02468-2-59-6': 'Therefore, while the Araki-Sucher contribution is small on the absolute scale, it becomes non-negligible in relation to other subtle effects.', '1710.02468-2-59-7': 'Moreover, the theoretical accuracy attainable for the argon dimer at present [CITATION] is already quite close to the level where the QED effects come into play.', '1710.02468-2-60-0': '# Conclusions', '1710.02468-2-61-0': 'In the present work we have put forward a general scheme to calculate the Araki-Sucher correction for many-electron systems.', '1710.02468-2-61-1': 'Several obstacles had to be removed to accomplish this goal.', '1710.02468-2-61-2': 'First, the complicated two-electron integrals involving the Araki-Sucher distribution have been solved with help of the McMurchie-Davidson technique (within the Gaussian-type orbitals basis set).', '1710.02468-2-61-3': 'It has been shown that they can be expressed through a family of one-dimensional integrals.', '1710.02468-2-61-4': 'Recursive and numerically stable computation of the latter integrals has been discussed in details.', '1710.02468-2-62-0': 'Second, the issue of convergence of the results with respect to the size of the basis set has been considered.', '1710.02468-2-62-1': 'We have demonstrated a slow convergence pattern ([MATH] in the leading order) towards the complete basis set limit.', '1710.02468-2-62-2': 'This result has been verified by comparing with reference data for the helium atom.', '1710.02468-2-62-3': 'With the analytic information about the convergence at hand, extrapolations have been used to improve the accuracy of the results.', '1710.02468-2-62-4': 'The accuracy of about 1% has been achieved in this case.', '1710.02468-2-63-0': 'To confirm the validity of the proposed approach we have performed calculations for several few- and many-electron systems.', '1710.02468-2-63-1': 'First, we have concentrated on small systems (e.g., few-electron atoms, hydrogen molecule) for which accurate reference values are available in the literature.', '1710.02468-2-63-2': 'A consistent accuracy of few percents have been obtained and the molecular results are only slightly less accurate than the atoms.', '1710.02468-2-63-3': 'Next, we have moved on to many-electron systems.', '1710.02468-2-63-4': 'We have estimated the contribution of the Araki-Sucher correction to the ionisation energy of the magnesium atom and interaction energy of the argon dimer.', '1710.02468-2-63-5': 'The final values of the Araki-Sucher correction are comparable to the present day experimental uncertainties of the measurements.', '1710.02468-2-64-0': 'This work was supported by the Polish National Science Centre through the project 2016/21/B/ST4/03877.', '1710.02468-2-64-1': 'The authors would like to thank Bogumi Jeziorski for reading and commenting on the manuscript.', '1710.02468-2-65-0': '# Partial wave expansion of the Araki-Sucher distribution', '1710.02468-2-66-0': 'In this section we present details of the derivation of Eqs. ([REF])-([REF]).', '1710.02468-2-66-1': 'Let us start with the definition ([REF]) and split it into two parts, [MATH], with [EQUATION] and [EQUATION]', '1710.02468-2-66-2': 'The second of these integrals is straightforward to evaluate because in the present context [EQUATION]', '1710.02468-2-66-3': 'Upon inserting back into Eq. ([REF]) the integration over the angle becomes straightforward and with help of the expression [MATH] one arrives at [EQUATION]', '1710.02468-2-66-4': 'Evaluation of the first term [MATH] is much more complicated.', '1710.02468-2-66-5': 'Changing the integration variable in Eq. ([REF]) to [MATH] gives [EQUATION]', '1710.02468-2-66-6': 'To get rid of the theta function under the integral sign we need to distinguish three possible (and disjoint) cases.', '1710.02468-2-66-7': 'First, assuming that [MATH] the integration range remains unchanged because [MATH] is equal to the unity there.', '1710.02468-2-66-8': 'The second case is [MATH] - the integrand vanishes whenever [MATH] so that the lower integration limit has to be shifted to [MATH].', '1710.02468-2-66-9': 'The third case is [MATH] - the result is zero because the integrand vanishes here.', '1710.02468-2-66-10': 'With this reasoning the integral can be rewritten as [EQUATION] after the change of variables to [MATH] and with the shorthand notation [MATH].', '1710.02468-2-66-11': 'The first integral is evaluated with elementary methods [EQUATION]', '1710.02468-2-66-12': 'The second integral is more complicated because of the function in the upper integration limit.', '1710.02468-2-66-13': 'It is probably quite difficult to derive the explicit expression for this integral but fortunately we require only the leading-order term in [MATH].', '1710.02468-2-66-14': 'The higher-order terms vanish in the final result due to the [MATH] limit.', '1710.02468-2-66-15': 'To extract the leading-order contribution we return to the original variable and integrate by parts once [EQUATION]', '1710.02468-2-66-16': 'The first and the second terms are of the order [MATH] and [MATH], respectively.', '1710.02468-2-66-17': 'By integrating by parts again one can show that the last term is also of the order [MATH].', '1710.02468-2-66-18': 'Therefore, we can write [EQUATION] which is sufficient for the present purposes.', '1710.02468-2-66-19': 'Finally, to arrive at Eqs. ([REF])-([REF]) from the main text one has to gather Eqs. ([REF]), ([REF]), ([REF]), ([REF]), and ([REF]) and rearrange.'}

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'1710.02468-2-17-4'], ['1710.02468-1-59-0', '1710.02468-2-59-0'], ['1710.02468-1-59-1', '1710.02468-2-59-1'], ['1710.02468-1-59-2', '1710.02468-2-59-2'], ['1710.02468-1-59-3', '1710.02468-2-59-3'], ['1710.02468-1-59-4', '1710.02468-2-59-4'], ['1710.02468-1-59-5', '1710.02468-2-59-5'], ['1710.02468-1-59-6', '1710.02468-2-59-6'], ['1710.02468-1-59-7', '1710.02468-2-59-7'], ['1710.02468-1-20-0', '1710.02468-2-20-0'], ['1710.02468-1-20-1', '1710.02468-2-20-1'], ['1710.02468-1-20-2', '1710.02468-2-20-2'], ['1710.02468-1-32-0', '1710.02468-2-32-0'], ['1710.02468-1-32-1', '1710.02468-2-32-1'], ['1710.02468-1-32-2', '1710.02468-2-32-2'], ['1710.02468-1-32-3', '1710.02468-2-32-3'], ['1710.02468-1-32-4', '1710.02468-2-32-4'], ['1710.02468-1-62-0', '1710.02468-2-62-0'], ['1710.02468-1-62-1', '1710.02468-2-62-1'], ['1710.02468-1-62-2', '1710.02468-2-62-2'], ['1710.02468-1-62-3', '1710.02468-2-62-3'], ['1710.02468-1-62-4', '1710.02468-2-62-4'], ['1710.02468-1-64-0', '1710.02468-2-64-0'], 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[['1710.02468-1-44-3', '1710.02468-2-44-3']]

[]

[]

[]

['1710.02468-1-46-0', '1710.02468-2-46-0', '1710.02468-2-66-19']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1710.02468

0904.4189

{'0904.4189-1-0-0': 'We construct a polynomial planar vector field of degree two with one invariant algebraic curves of genus one.', '0904.4189-1-0-1': 'We exhibit an explicit quadratic vector fields which invariant curves of degree nine and twelve .', '0904.4189-1-1-0': '# Introduction', '0904.4189-1-2-0': 'In the paper [CITATION] the authors present for the first time examples of algebraic limit cycles of degree greater than 4 for planar quadratic vector fields.', '0904.4189-1-2-1': 'They also give an example of an invariant algebraic curve of degree 12 and genus one for which the quadratic system has no Darboux integrating factors or first integrals.', '0904.4189-1-3-0': 'One of the first example of quadratic planar system with invariant algebraic curve of genus one is the Filipstov differential system [CITATION] [EQUATION] which possesses the irreducible invariant algebraic curve of degree four and genus one [EQUATION].', '0904.4189-1-4-0': 'Another example we can find in the paper [CITATION].', '0904.4189-1-5-0': 'This diferential system [EQUATION] a has the invariant algebraic curve of degree four and genus one [EQUATION]', '0904.4189-1-5-1': 'In the both case we have one algebraic limit cycles.', '0904.4189-1-6-0': 'In [CITATION] we can find a surprisingly simple quadratic system', '0904.4189-1-7-0': '[EQUATION] which has invariant algebraic curve of degree 12 and genus one', '0904.4189-1-8-0': '[EQUATION].', '0904.4189-1-9-0': 'The main result of this paper is to construct a set of quadratic planar vector field with an invariant curves of large degree and genus one.', '0904.4189-1-10-0': '# Quadratic vector fields with a given invariant algebraic curves of degree nine', '0904.4189-1-11-0': 'In this section we determine three quadratic vector field with invariant curves of degree nine and genus one.', '0904.4189-1-12-0': 'The following quadratic differential systems have the invariant curve [MATH] of degree nine and genus one and cofactor [MATH]', '0904.4189-1-13-0': '# Quadratic vector fields with a given invariant algebraic curves of degree twelve', '0904.4189-1-14-0': 'In this section we determine five quadratic vector field with invariant curves of degree twelve and genus one.', '0904.4189-1-15-0': 'The following quadratic differential systems have the invariant curve [MATH]of degree nine and genus one with cofactor [MATH].', '0904.4189-1-16-0': 'The quadratic vector fields with invariant algebraic curve of genus one admits at most two algebraic limit cycles.', '0904.4189-1-17-0': 'The proof follow from the fact that if the algebraic curve has genus [MATH] then admits at most [MATH] ovals.'}

{'0904.4189-2-0-0': 'We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree.', '0904.4189-2-0-1': 'We exhibit an explicit quadratic vector fields which invariant curves of degree nine and twelve and fifteen degree.', '0904.4189-2-1-0': '# Introduction', '0904.4189-2-2-0': 'In the paper [CITATION] the authors present for the first time examples of algebraic limit cycles of degree greater than 4 for planar quadratic vector fields.', '0904.4189-2-2-1': 'They also give an example of an invariant algebraic curve of degree 12 and genus one for which the quadratic system has no Darboux integrating factors or first integrals.', '0904.4189-2-3-0': 'One of the first example of quadratic planar system with invariant algebraic curve of genus one is the Filipstov differential system [CITATION] [EQUATION] which possesses the irreducible invariant algebraic curve of degree four and genus one [EQUATION].', '0904.4189-2-4-0': 'Another example we can find in the paper [CITATION].', '0904.4189-2-5-0': 'This diferential system [EQUATION] a has the invariant algebraic curve of degree four and genus one [EQUATION]', '0904.4189-2-5-1': 'In the both case we have one algebraic limit cycles.', '0904.4189-2-6-0': 'In [CITATION] we can find a surprisingly simple quadratic system', '0904.4189-2-7-0': '[EQUATION] which has invariant algebraic curve of degree 12 and genus one', '0904.4189-2-8-0': '[EQUATION].', '0904.4189-2-9-0': 'The main result of this paper is to construct a set of quadratic planar vector field with an invariant curves of large degree and genus greater o equal to one.', '0904.4189-2-10-0': '# Quadratic vector fields with a given invariant algebraic curves of degree nine', '0904.4189-2-11-0': 'In this section we determine three quadratic vector field with invariant curves of degree nine and genus one.', '0904.4189-2-12-0': 'The following quadratic differential systems have the invariant curve [MATH] of degree nine and genus one and cofactor [MATH]', '0904.4189-2-13-0': '# Quadratic vector fields with a given invariant algebraic curves of degree twelve', '0904.4189-2-14-0': 'In this section we determine five quadratic vector field with invariant curves of degree twelve and genus one.', '0904.4189-2-15-0': 'The following quadratic differential systems have the invariant curve [MATH]of degree nine and genus one with cofactor [MATH].', '0904.4189-2-16-0': 'The quadratic vector fields with invariant algebraic curve of genus one admits at most two algebraic limit cycles.', '0904.4189-2-17-0': 'The proof follow from the fact that if the algebraic curve has genus [MATH] then admits at most [MATH] ovals.', '0904.4189-2-18-0': '# Quadratic vector fields with a given invariant algebraic curves of degree fifteen', '0904.4189-2-19-0': 'We construct four quadratic vector field with invariant curve of degree fifteen.', '0904.4189-2-19-1': 'In this section we give two of theses quadratic vector fields.', '0904.4189-2-20-0': 'The following quadratic differential systems have the invariant curve [MATH] of degree fifteen with cofactor [MATH] .', '0904.4189-2-21-0': 'The quadratic vector fields with invariant curve of degree fifteen are Darboux integrable.'}

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[['0904.4189-1-9-0', '0904.4189-2-9-0'], ['0904.4189-1-0-0', '0904.4189-2-0-0'], ['0904.4189-1-0-1', '0904.4189-2-0-1'], ['0904.4189-2-2-1', '0904.4189-3-2-1'], ['0904.4189-2-20-0', '0904.4189-3-20-0'], ['0904.4189-3-0-1', '0904.4189-4-0-1']]

[]

[['0904.4189-2-19-0', '0904.4189-3-19-0'], ['0904.4189-2-21-0', '0904.4189-3-19-0']]

[]

['0904.4189-1-4-0', '0904.4189-1-6-0', '0904.4189-1-8-0', '0904.4189-2-4-0', '0904.4189-2-6-0', '0904.4189-2-8-0', '0904.4189-3-4-0', '0904.4189-3-6-0', '0904.4189-3-8-0', '0904.4189-4-4-0', '0904.4189-4-6-0', '0904.4189-4-8-0', '0904.4189-4-23-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0904.4189

{'0904.4189-3-0-0': 'We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree.', '0904.4189-3-0-1': 'We exhibit an explicit quadratic vector fields which invariant curves of degree nine and twelve and fifteen degree.', '0904.4189-3-1-0': '# Introduction', '0904.4189-3-2-0': 'In the paper [CITATION] the authors present for the first time examples of algebraic limit cycles of degree greater than 4 for planar quadratic vector fields.', '0904.4189-3-2-1': 'They also give an example of an invariant algebraic curve of degree 12 and genus one for which the quadratic system has no Darboux integrating factors or first integrals [CITATION].', '0904.4189-3-3-0': 'One of the first example of quadratic planar system with invariant algebraic curve of genus one is the Filipstov differential system [CITATION] [EQUATION] which possesses the irreducible invariant algebraic curve of degree four and genus one [EQUATION].', '0904.4189-3-4-0': 'Another example we can find in the paper [CITATION].', '0904.4189-3-5-0': 'This diferential system [EQUATION] a has the invariant algebraic curve of degree four and genus one [EQUATION]', '0904.4189-3-5-1': 'In the both case we have one algebraic limit cycles.', '0904.4189-3-6-0': 'In [CITATION] we can find a surprisingly simple quadratic system', '0904.4189-3-7-0': '[EQUATION] which has invariant algebraic curve of degree 12 and genus one', '0904.4189-3-8-0': '[EQUATION].', '0904.4189-3-9-0': 'The main result of this paper is to construct a set of quadratic planar vector field with an invariant curves of large degree and genus greater o equal to one.', '0904.4189-3-10-0': '# Quadratic vector fields with a given invariant algebraic curves of degree nine', '0904.4189-3-11-0': 'In this section we determine three quadratic vector field with invariant curves of degree nine and genus one.', '0904.4189-3-12-0': 'The following quadratic differential systems have the invariant curve [MATH] of degree nine and genus one and cofactor [MATH]', '0904.4189-3-13-0': '# Quadratic vector fields with a given invariant algebraic curves of degree twelve', '0904.4189-3-14-0': 'In this section we determine five quadratic vector field with invariant curves of degree twelve and genus one.', '0904.4189-3-15-0': 'The following quadratic differential systems have the invariant curve [MATH]of degree nine and genus one with cofactor [MATH].', '0904.4189-3-16-0': 'The quadratic vector fields with invariant algebraic curve of genus one admits at most two algebraic limit cycles.', '0904.4189-3-17-0': 'The proof follow from the fact that if the algebraic curve has genus [MATH] then admits at most [MATH] ovals.', '0904.4189-3-18-0': '# Quadratic vector fields with a given invariant algebraic curves of degree fifteen', '0904.4189-3-19-0': 'We construct three quadratic vector field with invariant curve of degree fifteen and genus two.', '0904.4189-3-19-1': 'In this section we give two of theses quadratic vector fields.', '0904.4189-3-20-0': 'The following quadratic differential systems have the invariant curve [MATH] of degree fifteen with cofactor [MATH] and genus two.'}

{'0904.4189-4-0-0': 'We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree.', '0904.4189-4-0-1': 'We exhibit an explicit quadratic vector fields which invariant curves of degree nine , twelve , fifteen and eighteen degree.', '0904.4189-4-1-0': '# Introduction', '0904.4189-4-2-0': 'In the paper [CITATION] the authors present for the first time examples of algebraic limit cycles of degree greater than 4 for planar quadratic vector fields.', '0904.4189-4-2-1': 'They also give an example of an invariant algebraic curve of degree 12 and genus one for which the quadratic system has no Darboux integrating factors or first integrals [CITATION].', '0904.4189-4-3-0': 'One of the first example of quadratic planar system with invariant algebraic curve of genus one is the Filipstov differential system [CITATION] [EQUATION] which possesses the irreducible invariant algebraic curve of degree four and genus one [EQUATION].', '0904.4189-4-4-0': 'Another example we can find in the paper [CITATION].', '0904.4189-4-5-0': 'This diferential system [EQUATION] a has the invariant algebraic curve of degree four and genus one [EQUATION]', '0904.4189-4-5-1': 'In the both case we have one algebraic limit cycles.', '0904.4189-4-6-0': 'In [CITATION] we can find a surprisingly simple quadratic system', '0904.4189-4-7-0': '[EQUATION] which has invariant algebraic curve of degree 12 and genus one', '0904.4189-4-8-0': '[EQUATION].', '0904.4189-4-9-0': 'The main result of this paper is to construct a set of quadratic planar vector field with an invariant curves of large degree and genus greater o equal to one.', '0904.4189-4-9-1': 'We apply the normal form of the quadratic fields deduced by J.Llibre [CITATION].', '0904.4189-4-10-0': '# Quadratic vector fields with a given invariant algebraic curves of degree nine', '0904.4189-4-11-0': 'In this section we determine three quadratic vector field with invariant curves of degree nine and genus one.', '0904.4189-4-12-0': 'The following quadratic differential systems have the invariant curve [MATH] of degree nine and genus one and cofactor [MATH]', '0904.4189-4-13-0': '# Quadratic vector fields with a given invariant algebraic curves of degree twelve', '0904.4189-4-14-0': 'In this section we determine five quadratic vector field with invariant curves of degree twelve and genus one.', '0904.4189-4-15-0': 'The following quadratic differential systems have the invariant curve [MATH]of degree nine and genus one with cofactor [MATH].', '0904.4189-4-16-0': 'The quadratic vector fields with invariant algebraic curve of genus one admits at most two algebraic limit cycles.', '0904.4189-4-17-0': 'The proof follow from the fact that if the algebraic curve has genus [MATH] then admits at most [MATH] ovals.', '0904.4189-4-18-0': '# Quadratic vector fields with a given invariant algebraic curves of degree fifteen', '0904.4189-4-19-0': 'We construct three quadratic vector field with invariant curve of degree fifteen and genus two.', '0904.4189-4-19-1': 'In this section we give two of theses quadratic vector fields.', '0904.4189-4-20-0': 'The following quadratic differential systems have the invariant curve [MATH] of degree fifteen with cofactor [MATH] and genus two.', '0904.4189-4-21-0': '# Quadratic vector fields with a given invariant algebraic curve of degree eighteen', '0904.4189-4-22-0': 'We construct a quadratic vector field with invariant curve of degree eighteen and genus two.', '0904.4189-4-23-0': 'The following quadratic differential systems have the invariant curve [MATH] of degree eighteen with cofactor [MATH] and genus two.', '0904.4189-4-23-1': '[EQUATION]'}

1908.09347

{'1908.09347-1-0-0': 'The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus [MATH].', '1908.09347-1-0-1': 'These flows are weakly mixing by the Avila-Forni theorem.', '1908.09347-1-0-2': 'In genus 2, the Holder property for the spectral measures of these flows was established in [CITATION].', '1908.09347-1-0-3': 'Recently Forni [CITATION], building on [CITATION], obtained Holder estimates for spectral measures in the case of surfaces of arbitrary genus.', '1908.09347-1-0-4': "Here we combine Forni's idea with the symbolic approach of [CITATION] and prove Holder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.", '1908.09347-1-1-0': '# Introduction', '1908.09347-1-2-0': '## Formulation of the main result', '1908.09347-1-3-0': 'Let [MATH] be a compact orientable surface.', '1908.09347-1-3-1': 'To a holomorphic one-form [MATH] on [MATH] one can assign the corresponding vertical flow [MATH] on [MATH], i.e., the flow at unit speed along the leaves of the foliation [MATH].', '1908.09347-1-3-2': 'The vertical flow preserves the measure [MATH] the area form induced by [MATH].', '1908.09347-1-3-3': 'By a theorem of Katok [CITATION], the flow [MATH] is never mixing.', '1908.09347-1-3-4': 'The moduli space of abelian differentials carries a natural volume measure, called the Masur-Veech measure [CITATION], [CITATION].', '1908.09347-1-3-5': 'For almost every Abelian differential with respect to the Masur-Veech measure, Masur [CITATION] and Veech [CITATION] independently and simultaneously proved that the flow [MATH] is uniquely ergodic.', '1908.09347-1-3-6': 'Weak mixing for almost all translation flows has been established by Veech in [CITATION] under additional assumptions on the combinatorics of the abelian differentials and by Avila and Forni [CITATION] in full generality.', '1908.09347-1-3-7': 'The spectrum of translation flows is therefore almost surely continuous and always has a singular component.', '1908.09347-1-4-0': 'Sinai [personal communication] raised the question: to find the local asymptotics for the spectral measures of translation flows.', '1908.09347-1-4-1': 'In [CITATION] we developed an approach to this problem and succeeded in obtaining Holder estimates for spectral measures in the case of surfaces of genus 2.', '1908.09347-1-4-2': 'The proof proceeds via uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markov compacta and arguments of Diophantine nature in the spirit of Salem, Erdos and Kahane.', '1908.09347-1-4-3': 'Recently Forni [CITATION] obtained Holder estimates for spectral measures in the case of surfaces of arbitrary genus.', '1908.09347-1-4-4': 'While Forni does not use the symbolic formalism, the main idea of his approach can also be formulated in symbolic terms: namely, that instead of the scalar estimates of [CITATION], we can use the Erdos-Kahane argument in vector form, cf. [REF] et seq. Following the idea of Forni and directly using the vector form of the Erdos-Kahane argument yields a considerable simplification of our initial proof and allows us to prove the Holder property for a general class of random Markov compacta, cf. [CITATION], and, in particular, for almost all translation flows on surfaces of arbitrary genus.', '1908.09347-1-5-0': 'Let [MATH] be a stratum of abelian differentials on a surface of genus [MATH].', '1908.09347-1-5-1': 'The natural smooth Masur-Veech measure on the stratum [MATH] is denoted by [MATH].', '1908.09347-1-5-2': 'Our main result is that for almost all abelian differentials in [MATH], the spectral measures of Lipschitz functions with respect to the corresponding translation flows have the Holder property.', '1908.09347-1-5-3': 'Recall that for a square-integrable test function [MATH], the spectral measure [MATH] is defined by [EQUATION] see Section [REF].', '1908.09347-1-5-4': 'A point mass for the spectral measure corresponds to an eigenvalue, so Holder estimates for spectral measures quantify weak mixing for our systems.', '1908.09347-1-6-0': 'There exists [MATH] such that for [MATH]-almost every abelian differential [MATH] the following holds.', '1908.09347-1-6-1': 'For any [MATH] there exist constants [MATH] and [MATH] such that for any Lipschitz function [MATH] on [MATH], for all [MATH] we have [EQUATION]', '1908.09347-1-6-2': 'This theorem is analogous to Forni [CITATION].', '1908.09347-1-7-0': "Our argument, as well as Forni's, see [CITATION], remains valid for almost every translation flow under a more general class of measures.", '1908.09347-1-7-1': 'Let [MATH] be a Borel probability measure invariant and ergodic under the Teichmuller flow.', '1908.09347-1-7-2': 'Let [MATH] be the number of positive Lyapunov exponents for the Kontsevich-Zorich cocycle under the measure [MATH].', '1908.09347-1-7-3': 'To formulate our condition precisely, recall that, by the Hubbard-Masur theorem on the existence of cohomological coordinates, the moduli space of abelian differentials with prescribed singularities can be locally identified with the space [MATH] of relative cohomology, with complex coefficients, of the underlying surface with respect to the singularities.', '1908.09347-1-7-4': 'Consider the subspace [MATH] corresponding to the absolute cohomology, the corresponding fibration of [MATH] into translates of [MATH] and its image, a fibration [MATH] on the moduli space of abelian differentials with prescribed singularities.', '1908.09347-1-7-5': 'Each fibre is locally isomorphic to [MATH] and thus has dimension equal to [MATH], where [MATH] is the genus of the underlying surface.', '1908.09347-1-7-6': 'We now restrict ourselves to the subspace of abelian differentials of area [MATH], and let the fibration [MATH] be the restriction of the fibration [MATH]; the dimension of each fibre of the fibration [MATH] is equal to [MATH].', '1908.09347-1-7-7': 'Almost every fibre of [MATH] carries a conditional measure, defined up to multiplication by a constant, of the measure [MATH].', '1908.09347-1-7-8': 'If there exists [MATH] such that the Hausdorff dimension of the conditional measure of [MATH] on almost every fibre of [MATH] has Hausdorff dimension at least [MATH], then Theorem [REF] holds for [MATH]-almost every abelian differential.', '1908.09347-1-8-0': 'In the case of the Masur-Veech measure [MATH], it is well-known that the conditional measure on almost every fibre is mutually absolutely continuous with the Lebesgue measure, hence has Hausdorff dimension [MATH].', '1908.09347-1-8-1': 'By the celebrated result of Forni [CITATION], there are [MATH] positive Lyapunov exponents for the Kontsevich-Zorich cocycle under the measure [MATH], so Theorem [REF] will follow by taking any [MATH].', '1908.09347-1-9-0': 'The proof of the Holder property for spectral measures proceeds via upper bounds on the growth of twisted Birkhoff integrals [EQUATION].', '1908.09347-1-10-0': 'There exists [MATH] such that for [MATH]-almost every abelian differential [MATH] and any [MATH] there exist [MATH] and [MATH] such that for any Lipschitz function [MATH] on [MATH], for all [MATH] and all [MATH], [EQUATION].', '1908.09347-1-11-0': 'This theorem is analogous to Forni [CITATION].', '1908.09347-1-11-1': 'The derivation of Theorem [REF] from Theorem [REF] is standard, with [MATH]; see Lemma [REF].', '1908.09347-1-11-2': 'In fact, in order to obtain ([REF]), [MATH]-estimates (with respect to the area measure on [MATH]) of [MATH] suffice; we obtain bounds that are uniform in [MATH], which is of independent interest.', '1908.09347-1-12-0': '## Quantitative weak mixing There is a close relation between Holder regularity of spectral measures and quantitative rates of weak mixing, see [CITATION].', '1908.09347-1-12-1': "One can also note a connection of our arguments with the proofs of weak mixing, via Veech's criterion [CITATION].", '1908.09347-1-12-2': 'A translation flow can be represented, by considering its return map to a transverse interval, as a special flow over an interval exchange transformation (IET) with a roof function constant on each sub-interval.', '1908.09347-1-12-3': 'The roof function is determined by a vector [MATH] with positive coordinates, where [MATH] is the number of sub-intervals of the IET and [MATH] is a subspace of dimension [MATH] corresponding to the space of absolute cohomology from Remark 1.2.', '1908.09347-1-12-4': 'Let [MATH] be the Zorich acceleration of the Masur-Veech cocycle on [MATH], corresponding to returns to a "good set", where [MATH] encodes the IET.', '1908.09347-1-12-5': "Veech's criterion [CITATION] says that if [EQUATION] then the translation flow corresponding to [MATH] is weakly mixing.", '1908.09347-1-12-6': 'This was used by Avila and Forni [CITATION] to show that for almost every [MATH] the set of [MATH], such that the special flow is not weakly mixing, has Hausdorff dimension at most [MATH].', '1908.09347-1-12-7': 'On the other hand, our Proposition [REF] says that if the set [EQUATION] has positive lower density in [MATH] for some [MATH] uniformly in [MATH] bounded away from zero and from infinity, then spectral measures corresponding to Lipschitz functions have the Holder property.', '1908.09347-1-12-8': 'The Erdos-Kahane argument is used to estimate the dimension of those [MATH] for which this fails.', '1908.09347-1-12-9': "In recent Forni's work, a version of the weak stable space for the Kontsevich-Zorich cocycle, denoted [MATH] and defined in [CITATION], seems analogous to our exceptional set [MATH] defined in ([REF]).", '1908.09347-1-12-10': 'The Hausdorff dimension of this set is estimated in [CITATION], with the help of [CITATION], which plays a role similar to our Erdos-Kahane argument.', '1908.09347-1-13-0': '## Organization of the paper, comparison with [CITATION] and [CITATION] A large part of the paper [CITATION] was written in complete generality, without the genus 2 assumptions, and is directly used here; for example, we do not reproduce the estimates of twisted Birkhoff integrals using generalized matrix Riesz products, but refer the reader to [CITATION] instead.', '1908.09347-1-14-0': 'Sharper estimates of matrix Riesz products were obtained in [CITATION], where the matrix Riesz products products are interpreted in terms of the spectral cocycle.', '1908.09347-1-14-1': 'In particular, we established a formula relating the local upper Lyapunov exponents of the cocycle with the pointwise lower dimension of spectral measures.', '1908.09347-1-14-2': 'Note nonetheless that the cocycle structure is not used in this paper.', '1908.09347-1-14-3': 'Section 3, parallel to [CITATION], contains the main result on Holder regularity of spectral measures in the setting of random [MATH]-adic, or, equivalently, random Bratteli-Verhik systems.', '1908.09347-1-14-4': 'The main novelty is that here we only require that the second Lyapunov exponent [MATH] of the Kontsevich-Zorich cocycle be positive, while in [CITATION] the assumption was that [MATH] are the only non-negative exponents.', '1908.09347-1-14-5': 'The preliminary Section 4 closely follows [CITATION].', '1908.09347-1-14-6': 'The crucial changes occurs in Sections 5 and 6, where, in contrast with [CITATION], Diophantine approximation is established in vector form, cf. Lemma [REF].', '1908.09347-1-14-7': 'The exceptional set is defined in ([REF]), and the Hausdorff dimension of the exceptional set is estimated in Proposition [REF].', '1908.09347-1-14-8': 'Although the general strategy of the "Erdos-Kahane argument" remains, the implementation is now significantly simplified.', '1908.09347-1-14-9': 'In [CITATION] we worked with scalar parameters, the coordinates of the vector of heights with respect to the Oseledets basis, but here we simply consider the vector parameter and work with the projection of the vector of heights to the strong unstable subspace.', '1908.09347-1-15-0': 'In particular, the cumbersome estimates of [CITATION] are no longer needed.', '1908.09347-1-15-1': 'Section 7, devoted to the derivation of the main theorem on translation flows from its symbolic counterpart, parallels [CITATION] with some changes.', '1908.09347-1-15-2': 'The most significant one is that we require a stronger property of the good returns, which is achieved in Lemma [REF].', '1908.09347-1-15-3': 'On the other hand, the large deviation estimate for the Teichmuller flow required in Theorem [REF] remains unchanged, and we directly use [CITATION].'}

{'1908.09347-2-0-0': 'The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus [MATH].', '1908.09347-2-0-1': 'These flows are weakly mixing by the Avila-Forni theorem.', '1908.09347-2-0-2': 'In genus 2, the Holder property for the spectral measures of these flows was established in [CITATION].', '1908.09347-2-0-3': 'Recently Forni [CITATION], motivated by [CITATION], obtained Holder estimates for spectral measures in the case of surfaces of arbitrary genus.', '1908.09347-2-0-4': "Here we combine Forni's idea with the symbolic approach of [CITATION] and prove Holder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.", '1908.09347-2-1-0': '# Introduction', '1908.09347-2-2-0': '## Formulation of the main result', '1908.09347-2-3-0': 'Let [MATH] be a compact orientable surface.', '1908.09347-2-3-1': 'To a holomorphic one-form [MATH] on [MATH] one can assign the corresponding vertical flow [MATH] on [MATH], i.e., the flow at unit speed along the leaves of the foliation [MATH].', '1908.09347-2-3-2': 'The vertical flow preserves the measure [MATH] the area form induced by [MATH].', '1908.09347-2-3-3': 'By a theorem of Katok [CITATION], the flow [MATH] is never mixing.', '1908.09347-2-3-4': 'The moduli space of abelian differentials carries a natural volume measure, called the Masur-Veech measure [CITATION], [CITATION].', '1908.09347-2-3-5': 'For almost every Abelian differential with respect to the Masur-Veech measure, Masur [CITATION] and Veech [CITATION] independently and simultaneously proved that the flow [MATH] is uniquely ergodic.', '1908.09347-2-3-6': 'Weak mixing for almost all translation flows has been established by Veech in [CITATION] under additional assumptions on the combinatorics of the abelian differentials and by Avila and Forni [CITATION] in full generality.', '1908.09347-2-3-7': 'The spectrum of translation flows is therefore almost surely continuous and always has a singular component.', '1908.09347-2-4-0': 'Sinai [personal communication] raised the question: to find the local asymptotics for the spectral measures of translation flows.', '1908.09347-2-4-1': 'In [CITATION] we developed an approach to this problem and succeeded in obtaining Holder estimates for spectral measures in the case of surfaces of genus 2.', '1908.09347-2-4-2': 'The proof proceeds via uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markov compacta and arguments of Diophantine nature in the spirit of Salem, Erdos and Kahane.', '1908.09347-2-4-3': 'Recently Forni [CITATION] obtained Holder estimates for spectral measures in the case of surfaces of arbitrary genus.', '1908.09347-2-4-4': 'While Forni does not use the symbolic formalism, the main idea of his approach can also be formulated in symbolic terms: namely, that instead of the scalar estimates of [CITATION], we can use the Erdos-Kahane argument in vector form, cf. [REF] et seq. Following the idea of Forni and directly using the vector form of the Erdos-Kahane argument yields a considerable simplification of our initial proof and allows us to prove the Holder property for a general class of random Markov compacta, cf. [CITATION], and, in particular, for almost all translation flows on surfaces of arbitrary genus.', '1908.09347-2-5-0': 'Let [MATH] be a stratum of abelian differentials on a surface of genus [MATH].', '1908.09347-2-5-1': 'The natural smooth Masur-Veech measure on the stratum [MATH] is denoted by [MATH].', '1908.09347-2-5-2': 'Our main result is that for almost all abelian differentials in [MATH], the spectral measures of Lipschitz functions with respect to the corresponding translation flows have the Holder property.', '1908.09347-2-5-3': 'Recall that for a square-integrable test function [MATH], the spectral measure [MATH] is defined by [EQUATION] see Section [REF].', '1908.09347-2-5-4': 'A point mass for the spectral measure corresponds to an eigenvalue, so Holder estimates for spectral measures quantify weak mixing for our systems.', '1908.09347-2-6-0': 'There exists [MATH] such that for [MATH]-almost every abelian differential [MATH] the following holds.', '1908.09347-2-6-1': 'For any [MATH] there exist constants [MATH] and [MATH] such that for any Lipschitz function [MATH] on [MATH], for all [MATH] we have [EQUATION]', '1908.09347-2-6-2': 'This theorem is analogous to Forni [CITATION].', '1908.09347-2-7-0': "Our argument, as well as Forni's, see [CITATION], remains valid for almost every translation flow under a more general class of measures.", '1908.09347-2-7-1': 'Let [MATH] be a Borel probability measure invariant and ergodic under the Teichmuller flow.', '1908.09347-2-7-2': 'Let [MATH] be the number of positive Lyapunov exponents for the Kontsevich-Zorich cocycle under the measure [MATH].', '1908.09347-2-7-3': 'To formulate our condition precisely, recall that, by the Hubbard-Masur theorem on the existence of cohomological coordinates, the moduli space of abelian differentials with prescribed singularities can be locally identified with the space [MATH] of relative cohomology, with complex coefficients, of the underlying surface with respect to the singularities.', '1908.09347-2-7-4': 'Consider the subspace [MATH] corresponding to the absolute cohomology, the corresponding fibration of [MATH] into translates of [MATH] and its image, a fibration [MATH] on the moduli space of abelian differentials with prescribed singularities.', '1908.09347-2-7-5': 'Each fibre is locally isomorphic to [MATH] and thus has dimension equal to [MATH], where [MATH] is the genus of the underlying surface.', '1908.09347-2-7-6': 'We now restrict ourselves to the subspace of abelian differentials of area [MATH], and let the fibration [MATH] be the restriction of the fibration [MATH]; the dimension of each fibre of the fibration [MATH] is equal to [MATH].', '1908.09347-2-7-7': 'Almost every fibre of [MATH] carries a conditional measure, defined up to multiplication by a constant, of the measure [MATH].', '1908.09347-2-7-8': 'If there exists [MATH] such that the Hausdorff dimension of the conditional measure of [MATH] on almost every fibre of [MATH] has Hausdorff dimension at least [MATH], then Theorem [REF] holds for [MATH]-almost every abelian differential.', '1908.09347-2-8-0': 'In the case of the Masur-Veech measure [MATH], it is well-known that the conditional measure on almost every fibre is mutually absolutely continuous with the Lebesgue measure, hence has Hausdorff dimension [MATH].', '1908.09347-2-8-1': 'By the celebrated result of Forni [CITATION], there are [MATH] positive Lyapunov exponents for the Kontsevich-Zorich cocycle under the measure [MATH], so Theorem [REF] will follow by taking any [MATH].', '1908.09347-2-9-0': 'The proof of the Holder property for spectral measures proceeds via upper bounds on the growth of twisted Birkhoff integrals [EQUATION].', '1908.09347-2-10-0': 'There exists [MATH] such that for [MATH]-almost every abelian differential [MATH] and any [MATH] there exist [MATH] and [MATH] such that for any Lipschitz function [MATH] on [MATH], for all [MATH] and all [MATH], [EQUATION].', '1908.09347-2-11-0': 'This theorem is analogous to Forni [CITATION].', '1908.09347-2-11-1': 'The derivation of Theorem [REF] from Theorem [REF] is standard, with [MATH]; see Lemma [REF].', '1908.09347-2-11-2': 'In fact, in order to obtain ([REF]), [MATH]-estimates (with respect to the area measure on [MATH]) of [MATH] suffice; we obtain bounds that are uniform in [MATH], which is of independent interest.', '1908.09347-2-12-0': '## Quantitative weak mixing There is a close relation between Holder regularity of spectral measures and quantitative rates of weak mixing, see [CITATION].', '1908.09347-2-12-1': "One can also note a connection of our arguments with the proofs of weak mixing, via Veech's criterion [CITATION].", '1908.09347-2-12-2': 'A translation flow can be represented, by considering its return map to a transverse interval, as a special flow over an interval exchange transformation (IET) with a roof function constant on each sub-interval.', '1908.09347-2-12-3': 'The roof function is determined by a vector [MATH] with positive coordinates, where [MATH] is the number of sub-intervals of the IET and [MATH] is a subspace of dimension [MATH] corresponding to the space of absolute cohomology from Remark 1.2.', '1908.09347-2-12-4': 'Let [MATH] be the Zorich acceleration of the Masur-Veech cocycle on [MATH], corresponding to returns to a "good set", where [MATH] encodes the IET.', '1908.09347-2-12-5': "Veech's criterion [CITATION] says that if [EQUATION] then the translation flow corresponding to [MATH] is weakly mixing.", '1908.09347-2-12-6': 'This was used by Avila and Forni [CITATION] to show that for almost every [MATH] the set of [MATH], such that the special flow is not weakly mixing, has Hausdorff dimension at most [MATH].', '1908.09347-2-12-7': 'On the other hand, our Proposition [REF] says that if the set [EQUATION] has positive lower density in [MATH] for some [MATH] uniformly in [MATH] bounded away from zero and from infinity, then spectral measures corresponding to Lipschitz functions have the Holder property.', '1908.09347-2-12-8': 'The Erdos-Kahane argument is used to estimate the dimension of those [MATH] for which this fails.', '1908.09347-2-12-9': "In recent Forni's work, a version of the weak stable space for the Kontsevich-Zorich cocycle, denoted [MATH] and defined in [CITATION], seems analogous to our exceptional set [MATH] defined in ([REF]).", '1908.09347-2-12-10': 'The Hausdorff dimension of this set is estimated in [CITATION], with the help of [CITATION], which plays a role similar to our Erdos-Kahane argument.', '1908.09347-2-13-0': '## Organization of the paper, comparison with [CITATION] and [CITATION] A large part of the paper [CITATION] was written in complete generality, without the genus 2 assumptions, and is directly used here; for example, we do not reproduce the estimates of twisted Birkhoff integrals using generalized matrix Riesz products, but refer the reader to [CITATION] instead.', '1908.09347-2-14-0': 'Sharper estimates of matrix Riesz products were obtained in [CITATION], where the matrix Riesz products products are interpreted in terms of the spectral cocycle.', '1908.09347-2-14-1': 'In particular, we established a formula relating the local upper Lyapunov exponents of the cocycle with the pointwise lower dimension of spectral measures.', '1908.09347-2-14-2': 'Note nonetheless that the cocycle structure is not used in this paper.', '1908.09347-2-14-3': 'Section 3, parallel to [CITATION], contains the main result on Holder regularity of spectral measures in the setting of random [MATH]-adic, or, equivalently, random Bratteli-Verhik systems.', '1908.09347-2-14-4': 'The main novelty is that here we only require that the second Lyapunov exponent [MATH] of the Kontsevich-Zorich cocycle be positive, while in [CITATION] the assumption was that [MATH] are the only non-negative exponents.', '1908.09347-2-14-5': 'The preliminary Section 4 closely follows [CITATION].', '1908.09347-2-14-6': 'The crucial changes occurs in Sections 5 and 6, where, in contrast with [CITATION], Diophantine approximation is established in vector form, cf. Lemma [REF].', '1908.09347-2-14-7': 'The exceptional set is defined in ([REF]), and the Hausdorff dimension of the exceptional set is estimated in Proposition [REF].', '1908.09347-2-14-8': 'Although the general strategy of the "Erdos-Kahane argument" remains, the implementation is now significantly simplified.', '1908.09347-2-14-9': 'In [CITATION] we worked with scalar parameters, the coordinates of the vector of heights with respect to the Oseledets basis, but here we simply consider the vector parameter and work with the projection of the vector of heights to the strong unstable subspace.', '1908.09347-2-15-0': 'In particular, the cumbersome estimates of [CITATION] are no longer needed.', '1908.09347-2-15-1': 'Section 7, devoted to the derivation of the main theorem on translation flows from its symbolic counterpart, parallels [CITATION] with some changes.', '1908.09347-2-15-2': 'The most significant one is that we require a stronger property of the good returns, which is achieved in Lemma [REF].', '1908.09347-2-15-3': 'On the other hand, the large deviation estimate for the Teichmuller flow required in Theorem [REF] remains unchanged, and we directly use [CITATION].'}

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[['1908.09347-1-0-3', '1908.09347-2-0-3']]

[]

[]

[]

['1908.09347-1-10-0', '1908.09347-2-10-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1908.09347

1404.4238

{'1404.4238-1-0-0': 'It is shown that the optically bright excitonic transitions near the Dirac points of quasi-two dimensional systems with honeycomb lattice structure have a [MATH]-like symmetry whereas the [MATH]-like states are dipole forbidden.', '1404.4238-1-0-1': 'For the example of [MATH], excellent agreement of the computed excitonic resonance energies with recent experiments is demonstrated.', '1404.4238-1-0-2': 'The deduced large binding energy and the long radiative lifetime of the energetically lowest [MATH]-exciton state yields ideal conditions for Bose Einstein condensation, with a condensate that is stable against radiative decay.', '1404.4238-1-1-0': 'Since its first isolation by Novosolev and Geim 2004, graphene and graphene based systems have attracted lots of interest, both for fundamental reasons and technological applications.', '1404.4238-1-1-1': 'Meanwhile, a variety of graphene-analogues materials like h-BN, silicene or transition-metal dichalcogenides have been fabricated[CITATION].', '1404.4238-1-1-2': 'Similar to graphene, these novel material systems display exciting new physical properties, distinct from their bulk counterparts[CITATION].', '1404.4238-1-2-0': 'Specifically, transition metal dichalgonides (TMD) show a cross-over from an indirect to a direct gap semiconductor if the thickness is reduced to a single layer, with a gap in the visible range[CITATION].', '1404.4238-1-2-1': 'Moreover, the optical spectra are dominated by excitonic effects, making these systems particularly interesting for optoelectronic applications[CITATION].', '1404.4238-1-3-0': 'The key to understand the fascinating optoelectronic properties of this novel material class is the basic single-particle band structure.', '1404.4238-1-3-1': 'Owing to the symmetry of the honeycomb lattice, the band edges are located at two nonequivalent corners of the Brillioun zone, known as valleys[CITATION].', '1404.4238-1-3-2': 'In each of these valleys, the carriers are theoretically described as massive Dirac Fermions[CITATION].', '1404.4238-1-3-3': 'At the band edges, the valley index can be considered as discrete degree of freedom similar to the spin, which has lead to the fascinating concept of valleytronics[CITATION].', '1404.4238-1-4-0': 'The valley degree of freedom and the resulting of valley and frequency dependent optical selection rules in the presence of a strong spin-orbit coupling have been discussed and demonstrated in the literature[CITATION].', '1404.4238-1-4-1': 'However, it has been widely ignored that the associated optical transitions are governed by selection rules that differ fundamentally from those in conventional semiconductors.', '1404.4238-1-5-0': 'In conventional semiconductors with near-parabolic [MATH]-like conduction and a [MATH]-like valence bands, the near-bandgap optical properties are best described by the so-called Elliot formula, expressing the energetic position and oscillator strength of the excitonic transitions in terms of the excitonic wave functions.', '1404.4238-1-5-1': 'Simple selection rules show that only [MATH]-like excitonic states couple to the light field, resulting in the well-known excitonic Rydberg series.', '1404.4238-1-6-0': 'In this letter, we show that the optically active states in quasi-two dimensional (2D) systems with honeycomb-lattice structure have a [MATH]-like symmetry.', '1404.4238-1-6-1': 'Hence, the [MATH]-resonance is the lowest dipole allowed transition whereas the [MATH]-excitonic ground state is optically dark.', '1404.4238-1-6-2': 'To support our result, we compare our predictions with recent reflection measurements performed on monolayer WS[MATH] grown on a SiO[MATH] substrate[CITATION].', '1404.4238-1-7-0': 'Dictated by the symmetry properties of the hexagonal lattice, the lowest order [MATH] Hamiltonian has the form[CITATION] [EQUATION] where [MATH] is the valley index and [MATH] is the tensor product of the electron spin state and a two-component quasi-spinor.', '1404.4238-1-7-1': 'The Pauli matrices [MATH] and [MATH] act in the pseudo-spin space and [MATH] in the real spin space.', '1404.4238-1-7-2': 'The basis functions for the pseudo-spinor are a linear combination of the relevant atomic orbitals that contribute to the valence and conduction bands and depend on the specific material system under consideration.', '1404.4238-1-7-3': 'Within a tight-binding model, the parameters [MATH], [MATH] and [MATH] correspond to the bias in on-site energies, the effective hopping matrix element, and the lattice constant, respectively, and [MATH] is the spin-splitting of the valence band due to the intra-atomic spin-orbit coupling (SOC).', '1404.4238-1-7-4': 'The Hamiltonian ([REF]) is valid for the entire class of monolayer hexagonal structures including graphene with a zero gap and negligible SOC, h-BN with a large gap and also negligible SOC, and the variety of TMD with a gap in the optical range and strong SOC.', '1404.4238-1-8-0': 'Independent of the explicit expressions for the speudo-spinor basis functions, the light-matter interaction can be found by the minimal substitution [MATH].', '1404.4238-1-8-1': 'Using [MATH], one obtains [EQUATION]', '1404.4238-1-8-2': 'Using the eigenstates of [MATH], the Hamiltonian for the light-matter (LM) interaction can be written as [EQUATION]', '1404.4238-1-8-3': 'Here, [MATH]) creates an electron with spin [MATH] and valley index [MATH] in the conduction (valence) band, [MATH] is the spin and valley dependent gap, [MATH] is the relativistic dispersion of a quasi-particle with rest energy [MATH], [MATH], and [MATH] is the angle in [MATH]-space defined by [MATH].', '1404.4238-1-9-0': 'In Eq. [REF], the first line describes the intraband and the second line the interband transitions, respectively.', '1404.4238-1-9-1': 'Contributions proportional to [MATH] correspond to the absorption of a photon with circular polarization [MATH].', '1404.4238-1-9-2': 'In zero-gap materials like graphene or silicene, the transition matrix elements for both intra- and interband absorption become equal in magnitude and differences in the transition probabilities exclusively result from the different occupation numbers in the initial state.', '1404.4238-1-9-3': 'In contrast, in a wide gap system, the transition amplitudes [MATH] and [MATH] vanish, while [MATH], giving the much simpler expression [EQUATION]', '1404.4238-1-9-4': 'From this simplified LM Hamiltonian, one recognizes clearly that the valence-to-conduction-band excitations at the [MATH] valley require the absorption of a [MATH] polarized photon, as has been shown by Xiao et al.[CITATION].', '1404.4238-1-9-5': 'Moreover, only collective states with a [MATH]-like symmetry contribute to the macroscopic current [EQUATION]', '1404.4238-1-9-6': 'The microcopic transition amplitudes [MATH] can be computed from the semiconductor Bloch equations (SBE).', '1404.4238-1-9-7': 'Using Eq. [REF] together with the Coulomb-interaction Hamitonian, the linearized polarization equation of the SBE is given by [EQUATION]', '1404.4238-1-9-8': 'Here, [MATH] denotes the Coulomb renormalization of the band-gap and the source term [MATH] is anisotropic due to the phase factor in the LM Hamiltonian.', '1404.4238-1-9-9': 'Expansion into eigenstates of the Wannier equation [EQUATION] gives the modified Elliot formula [EQUATION] with the oscillator strength [EQUATION]', '1404.4238-1-9-10': 'As usual, the resonances of the reflection and absorption spectra occur at the poles of the Elliot formula.', '1404.4238-1-9-11': 'As a consequence of the phase factors in Eq. [REF], only [MATH]-like states have a finite oscillator strength and couple to the optical field.', '1404.4238-1-9-12': 'This result differs fundamentally from the common knowledge that bright excitations always have [MATH]- like symmetry.', '1404.4238-1-10-0': 'To test our general theory, we analyze the recently reported experimental observation of five bright excitonic states corresponding to the [MATH]-exciton in WS[MATH] on SiO[MATH] [CITATION].', '1404.4238-1-10-1': 'Assuming an isotropic Coulomb interaction, the [MATH]- and [MATH]- states are degenerate for [MATH], while the lowest lying [MATH]-excitonic state is nondegenerate and dark.', '1404.4238-1-10-2': 'As far as relativistic corrections to the dispersion may be neglected and the Coulomb potential can be considered as strictly 2D, the binding energy of the [MATH]th exciton is given by the Rydberg series [EQUATION] with a gap [MATH] containing the self energy of the filled valence band, and the binding of the lowest (dark) exciton [EQUATION]', '1404.4238-1-10-3': 'Here, [MATH] is the reduced mass of the electron-hole pair, [MATH] is the effective fine structure constant and [MATH] is the effective dielectric constant of the substrate/environment system.', '1404.4238-1-10-4': 'While the knowledge of [MATH] suffices to compute the energetic separation between any two bright states, the prediction of the absolute spectral position requires the additional knowledge of the Coulombic band gap renormalization [MATH].', '1404.4238-1-10-5': 'Whereas [MATH] diverges for an ideal, strictly 2D Coulomb potential, it is finite for any real system where the Coulomb matrix elements must be evaluated with the atomic orbitals used to represent the pseudo spinor space in Eq. [REF].', '1404.4238-1-10-6': 'This leads to a quasi 2D Coulomb interaction [EQUATION] where [MATH] the normalization area and [MATH] is a monotonically decreasing formfactor with [MATH].', '1404.4238-1-10-7': 'The form factor leads to a regularization of the [MATH] singularity at small distances, and the parameter [MATH] can be interpreted as the effective thickness of the monolayer.', '1404.4238-1-10-8': 'Since its explicit expression depends on the specific atomic orbitals contributing to the valence and conduction band, we do not calculate the band gap but use it as a fit parameter instead.', '1404.4238-1-11-0': 'With the material parameters for WS[MATH] eV, [MATH]eV, [MATH] eV, [MATH]A given in Ref. [CITATION], and an effective dielectric constant [MATH] for the SiO[MATH] substrate, we find an [MATH]-exciton binding energy [MATH] for the [MATH]-exciton.', '1404.4238-1-11-1': 'Using [MATH] eV, we show in Fig. [REF] (blue solid line) the computed result for the spectral position of the lowest five bright exciton transitions.', '1404.4238-1-11-2': 'The red dots show the experimental data points taken from Ref. [CITATION].', '1404.4238-1-11-3': 'For comparison, the green dashed line shows the predicted values assuming [MATH]-like symmetry of the bright excitons.', '1404.4238-1-11-4': 'Our computed results show remarkably good agreement with the experimental observations given the fact that only the effective band gap was used as a single fit paratmeter.', '1404.4238-1-11-5': 'In contrast, the results based on the assumption that the series has [MATH]-type character and starts with the [MATH]-resonance are completely off the scale, in particularfor the energetically lowest states.', '1404.4238-1-12-0': 'In conclusion, we have shown that excitonic transitions in layered hexagonal material have a [MATH]-like symmetry of the excitonic wave functions.', '1404.4238-1-12-1': 'This is fundamentally different from the situation in conventional semiconductors, where bright excitons always have [MATH]-like symmetry.', '1404.4238-1-13-0': 'Our results have far reaching consequences, the most fascinating among them being the existence of a dark excitonic [MATH]-state energetically below the lowest optical active state.', '1404.4238-1-13-1': 'If the energy of this state is well above the ground state level, it should be observable, e.,g., by two photon transitions, such as two-photon absorption to the [MATH] state or stimulated emission from the [MATH] to the [MATH] state.', '1404.4238-1-13-2': 'Similarly, in the presence of a strong spin orbit coupling, a two photon absorption exciting the [MATH] of the [MATH] exciton via the [MATH] state of the [MATH] exciton should be observable.', '1404.4238-1-13-3': 'Moreover, both the lowest lying [MATH] and [MATH] excitons, and the [MATH] exciton can be populated via excitation of a higher [MATH]-level and subsequent relaxation.', '1404.4238-1-13-4': 'Since these low lying [MATH]- states are dark, they do not suffer from radiative decay and thus provide optimal condictions to observe a Bose-Einstein condensation.'}

{'1404.4238-2-0-0': 'It is shown that the optically bright excitonic transitions near the Dirac points of quasi-two dimensional systems with honeycomb lattice structure have a [MATH]-like symmetry whereas the [MATH]-like states are dipole forbidden.', '1404.4238-2-0-1': 'For the example of WS[MATH], excellent agreement of the computed excitonic resonance energies with recent experiments is demonstrated.', '1404.4238-2-0-2': 'The deduced large binding energy and the long radiative lifetime of the energetically lowest [MATH]-exciton state yields ideal conditions for Bose Einstein condensation, with a condensate that is stable against radiative decay.', '1404.4238-2-1-0': 'Since its first isolation by Novosolev and Geim 2004, graphene and graphene based systems have attracted lots of interest, both for fundamental reasons and technological applications.', '1404.4238-2-1-1': 'Meanwhile, a variety of graphene-analogues materials like h-BN, silicene or transition-metal dichalcogenides have been fabricated[CITATION].', '1404.4238-2-1-2': 'Similar to graphene, these novel material systems display exciting new physical properties, distinct from their bulk counterparts[CITATION].', '1404.4238-2-2-0': 'Specifically, transition metal dichalgonides (TMD) show a cross-over from an indirect to a direct gap semiconductor if the thickness is reduced to a single layer, with a gap in the visible range[CITATION].', '1404.4238-2-2-1': 'Moreover, the optical spectra are dominated by excitonic effects, making these systems particularly interesting for optoelectronic applications[CITATION].', '1404.4238-2-3-0': 'The key to understand the fascinating optoelectronic properties of this novel material class is the basic single-particle band structure.', '1404.4238-2-3-1': 'Owing to the symmetry of the honeycomb lattice, the band edges are located at two nonequivalent corners of the Brillioun zone, known as valleys[CITATION].', '1404.4238-2-3-2': 'In each of these valleys, the carriers are theoretically described as massive Dirac Fermions[CITATION].', '1404.4238-2-3-3': 'At the band edges, the valley index can be considered as discrete degree of freedom similar to the spin, which has lead to the fascinating concept of valleytronics[CITATION].', '1404.4238-2-4-0': 'The valley degree of freedom and the resulting of valley and frequency dependent optical selection rules in the presence of a strong spin-orbit coupling have been discussed and demonstrated in the literature[CITATION].', '1404.4238-2-4-1': 'However, it has been widely ignored that the associated optical transitions are governed by selection rules that differ fundamentally from those in conventional semiconductors.', '1404.4238-2-5-0': 'In conventional semiconductors with near-parabolic [MATH]-like conduction and a [MATH]-like valence bands, the near-bandgap optical properties are best described by the so-called Elliot formula, expressing the energetic position and oscillator strength of the excitonic transitions in terms of the excitonic wave functions.', '1404.4238-2-5-1': 'Simple selection rules show that only [MATH]-like excitonic states couple to the light field, resulting in the well-known excitonic Rydberg series.', '1404.4238-2-6-0': 'In this letter, we show that the optically active states in quasi-two dimensional (2D) systems with honeycomb-lattice structure have a [MATH]-like symmetry.', '1404.4238-2-6-1': 'Hence, the [MATH]-resonance is the lowest dipole allowed transition whereas the [MATH]-excitonic ground state is optically dark.', '1404.4238-2-6-2': 'To support our result, we compare our predictions with recent reflection measurements performed on monolayer WS[MATH] grown on a SiO[MATH] substrate[CITATION].', '1404.4238-2-7-0': 'Dictated by the symmetry properties of the hexagonal lattice, the lowest order [MATH] Hamiltonian has the form[CITATION] [EQUATION] where [MATH] is the valley index and [MATH] is the tensor product of the electron spin state and a two-component quasi-spinor.', '1404.4238-2-7-1': 'The Pauli matrices [MATH] and [MATH] act in the pseudo-spin space and [MATH] in the real spin space.', '1404.4238-2-7-2': 'The basis functions for the pseudo-spinor are a linear combination of the relevant atomic orbitals that contribute to the valence and conduction bands and depend on the specific material system under consideration.', '1404.4238-2-7-3': 'Within a tight-binding model, the parameters [MATH], [MATH] and [MATH] correspond to the bias in on-site energies, the effective hopping matrix element, and the lattice constant, respectively, and [MATH] is the spin-splitting of the valence band due to the intra-atomic spin-orbit coupling (SOC).', '1404.4238-2-7-4': 'The Hamiltonian ([REF]) is valid for the entire class of monolayer hexagonal structures including graphene with a zero gap and negligible SOC, h-BN with a large gap and also negligible SOC, and the variety of TMD with a gap in the optical range and strong SOC.', '1404.4238-2-8-0': 'Independent of the explicit expressions for the speudo-spinor basis functions, the light-matter interaction can be found by the minimal substitution [MATH].', '1404.4238-2-8-1': 'Using [MATH], one obtains [EQUATION]', '1404.4238-2-8-2': 'Using the eigenstates of [MATH], the Hamiltonian for the light-matter (LM) interaction can be written as [EQUATION]', '1404.4238-2-8-3': 'Here, [MATH]) creates an electron with spin [MATH] and valley index [MATH] in the conduction (valence) band, [MATH] is the spin and valley dependent gap, [MATH] is the relativistic dispersion of a quasi-particle with rest energy [MATH], [MATH], and [MATH] is the angle in [MATH]-space defined by [MATH].', '1404.4238-2-9-0': 'In Eq. [REF], the first line describes the intraband and the second line the interband transitions, respectively.', '1404.4238-2-9-1': 'Contributions proportional to [MATH] correspond to the absorption of a photon with circular polarization [MATH].', '1404.4238-2-9-2': 'In zero-gap materials like graphene or silicene, the transition matrix elements for both intra- and interband absorption become equal in magnitude and differences in the transition probabilities exclusively result from the different occupation numbers in the initial state.', '1404.4238-2-9-3': 'In contrast, in a wide gap system, the transition amplitudes [MATH] and [MATH] vanish, while [MATH], giving the much simpler expression [EQUATION]', '1404.4238-2-9-4': 'From this simplified LM Hamiltonian, one recognizes clearly that the valence-to-conduction-band excitations at the [MATH] valley require the absorption of a [MATH] polarized photon, as has been shown by Xiao et al.[CITATION].', '1404.4238-2-9-5': 'Moreover, only collective states with a [MATH]-like symmetry contribute to the macroscopic current [EQUATION]', '1404.4238-2-9-6': 'The microscopic transition amplitudes [MATH] can be computed from the semiconductor Bloch equations (SBE).', '1404.4238-2-9-7': 'Using Eq. [REF] together with the Coulomb-interaction Hamiltonian, the linearized polarization equation of the SBE is given by [EQUATION]', '1404.4238-2-9-8': 'Here, [MATH] denotes the Coulomb renormalization of the band-gap and the source term [MATH] is anisotropic due to the phase factor in the LM Hamiltonian.', '1404.4238-2-9-9': 'Expansion into eigenstates of the Wannier equation [EQUATION] gives the modified Elliot formula [EQUATION] with the oscillator strength [EQUATION]', '1404.4238-2-9-10': 'As usual, the resonances of the reflection and absorption spectra occur at the poles of the Elliot formula.', '1404.4238-2-9-11': 'As a consequence of the phase factors in Eq. [REF], only [MATH]-like states have a finite oscillator strength and couple to the optical field.', '1404.4238-2-9-12': 'This result differs fundamentally from the common knowledge that bright excitations always have [MATH]- like symmetry.', '1404.4238-2-10-0': 'To test our general theory, we analyze the recently reported experimental observation of five bright excitonic states corresponding to the [MATH]-exciton in WS[MATH] on SiO[MATH] [CITATION].', '1404.4238-2-10-1': 'Assuming an isotropic Coulomb interaction, the [MATH]- and [MATH]- states are degenerate for [MATH], while the lowest lying [MATH]-excitonic state is nondegenerate and dark.', '1404.4238-2-10-2': 'As far as relativistic corrections to the dispersion may be neglected and the Coulomb potential can be considered as strictly 2D, the binding energy of the [MATH]th exciton is given by the Rydberg series [EQUATION] with a gap [MATH] containing the self energy of the filled valence band, and the binding of the lowest (dark) exciton [EQUATION]', '1404.4238-2-10-3': 'Here, [MATH] is the reduced mass of the electron-hole pair, [MATH] is the effective fine structure constant and [MATH] is the effective dielectric constant of the substrate/environment system.', '1404.4238-2-10-4': 'While the knowledge of [MATH] suffices to compute the energetic separation between any two bright states, the prediction of the absolute spectral position requires the additional knowledge of the Coulombic band gap renormalization [MATH].', '1404.4238-2-10-5': 'Whereas [MATH] diverges for an ideal, strictly 2D Coulomb potential, it is finite for any real system where the Coulomb matrix elements must be evaluated with the atomic orbitals used to represent the pseudo spinor space in Eq. [REF].', '1404.4238-2-10-6': 'This leads to a quasi 2D Coulomb interaction [EQUATION] where [MATH] the normalization area and [MATH] is a monotonically decreasing form factor with [MATH].', '1404.4238-2-10-7': 'The form factor leads to a regularization of the [MATH] singularity at small distances, and the parameter [MATH] can be interpreted as the effective thickness of the monolayer.', '1404.4238-2-10-8': 'Since its explicit expression depends on the specific atomic orbitals contributing to the valence and conduction band, we do not calculate the band gap but use it as a fit parameter instead.', '1404.4238-2-11-0': 'With the material parameters for WS[MATH] eV, [MATH]eV, [MATH] eV, [MATH]A given in Ref. [CITATION], and an effective dielectric constant [MATH] for the SiO[MATH] substrate, we find an [MATH]-exciton binding energy [MATH] for the [MATH]-exciton.', '1404.4238-2-11-1': 'Using [MATH] eV, we show in Fig. [REF] (blue solid line) the computed result for the spectral position of the lowest five bright exciton transitions.', '1404.4238-2-11-2': 'The red dots show the experimental data points taken from Ref. [CITATION].', '1404.4238-2-11-3': 'For comparison, the green dashed line shows the predicted values assuming [MATH]-like symmetry of the bright excitons.', '1404.4238-2-11-4': 'Our computed results show remarkably good agreement with the experimental observations given the fact that only the effective band gap was used as a single fit parameter.', '1404.4238-2-11-5': 'In contrast, the results based on the assumption that the series has [MATH]-type character and starts with the [MATH]-resonance are completely off the scale, in particular for the energetically lowest states.', '1404.4238-2-12-0': 'In conclusion, we have shown that excitonic transitions in layered hexagonal material have a [MATH]-like symmetry of the excitonic wave functions.', '1404.4238-2-12-1': 'This is fundamentally different from the situation in conventional semiconductors, where bright excitons always have [MATH]-like symmetry.', '1404.4238-2-13-0': 'Our results have far reaching consequences, the most fascinating among them being the existence of a dark excitonic [MATH]-state energetically below the lowest optical active state.', '1404.4238-2-13-1': 'If the energy of this state is well above the ground state level, it should be observable, e.g., by two photon transitions, such as two-photon absorption to the [MATH] state or stimulated emission from the [MATH] to the [MATH] state.', '1404.4238-2-13-2': 'Similarly, in the presence of a strong spin orbit coupling, a two photon absorption exciting the [MATH] of the [MATH] exciton via the [MATH] state of the [MATH] exciton should be observable.', '1404.4238-2-13-3': 'Moreover, both the lowest lying [MATH] and [MATH] excitons, and the [MATH] exciton can be populated via excitation of a higher [MATH]-level and subsequent relaxation.', '1404.4238-2-13-4': 'Since these low lying [MATH]- states are dark, they do not suffer from radiative decay and thus provide optimal condictions to observe a Bose-Einstein condensation.'}

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['1404.4238-2-9-10', '1404.4238-3-9-4'], ['1404.4238-2-11-0', '1404.4238-3-13-0'], ['1404.4238-2-11-1', '1404.4238-3-13-1'], ['1404.4238-2-11-2', '1404.4238-3-13-2'], ['1404.4238-2-11-3', '1404.4238-3-13-3'], ['1404.4238-2-11-4', '1404.4238-3-13-5'], ['1404.4238-2-11-5', '1404.4238-3-13-6']]

[['1404.4238-1-11-0', '1404.4238-2-11-0'], ['1404.4238-1-11-1', '1404.4238-2-11-1'], ['1404.4238-1-11-2', '1404.4238-2-11-2'], ['1404.4238-1-11-3', '1404.4238-2-11-3'], ['1404.4238-1-9-0', '1404.4238-2-9-0'], ['1404.4238-1-9-1', '1404.4238-2-9-1'], ['1404.4238-1-9-2', '1404.4238-2-9-2'], ['1404.4238-1-9-3', '1404.4238-2-9-3'], ['1404.4238-1-9-4', '1404.4238-2-9-4'], ['1404.4238-1-9-5', '1404.4238-2-9-5'], ['1404.4238-1-9-8', '1404.4238-2-9-8'], ['1404.4238-1-9-9', '1404.4238-2-9-9'], ['1404.4238-1-9-10', '1404.4238-2-9-10'], ['1404.4238-1-9-11', '1404.4238-2-9-11'], ['1404.4238-1-9-12', '1404.4238-2-9-12'], ['1404.4238-1-3-0', '1404.4238-2-3-0'], ['1404.4238-1-3-1', '1404.4238-2-3-1'], ['1404.4238-1-3-2', '1404.4238-2-3-2'], ['1404.4238-1-3-3', '1404.4238-2-3-3'], ['1404.4238-1-1-0', '1404.4238-2-1-0'], ['1404.4238-1-1-1', '1404.4238-2-1-1'], ['1404.4238-1-1-2', '1404.4238-2-1-2'], ['1404.4238-1-13-0', '1404.4238-2-13-0'], ['1404.4238-1-13-2', '1404.4238-2-13-2'], ['1404.4238-1-13-3', '1404.4238-2-13-3'], ['1404.4238-1-13-4', '1404.4238-2-13-4'], ['1404.4238-1-7-0', '1404.4238-2-7-0'], ['1404.4238-1-7-1', '1404.4238-2-7-1'], ['1404.4238-1-7-2', '1404.4238-2-7-2'], ['1404.4238-1-7-3', '1404.4238-2-7-3'], ['1404.4238-1-7-4', '1404.4238-2-7-4'], ['1404.4238-1-4-0', '1404.4238-2-4-0'], ['1404.4238-1-4-1', '1404.4238-2-4-1'], ['1404.4238-1-8-0', '1404.4238-2-8-0'], ['1404.4238-1-8-1', '1404.4238-2-8-1'], ['1404.4238-1-8-2', '1404.4238-2-8-2'], ['1404.4238-1-8-3', '1404.4238-2-8-3'], ['1404.4238-1-12-0', '1404.4238-2-12-0'], ['1404.4238-1-12-1', '1404.4238-2-12-1'], ['1404.4238-1-10-0', '1404.4238-2-10-0'], ['1404.4238-1-10-1', '1404.4238-2-10-1'], ['1404.4238-1-10-2', '1404.4238-2-10-2'], ['1404.4238-1-10-3', '1404.4238-2-10-3'], ['1404.4238-1-10-4', '1404.4238-2-10-4'], ['1404.4238-1-10-5', '1404.4238-2-10-5'], ['1404.4238-1-10-7', '1404.4238-2-10-7'], ['1404.4238-1-10-8', '1404.4238-2-10-8'], ['1404.4238-1-6-0', '1404.4238-2-6-0'], ['1404.4238-1-6-1', '1404.4238-2-6-1'], ['1404.4238-1-6-2', '1404.4238-2-6-2'], ['1404.4238-1-0-0', '1404.4238-2-0-0'], ['1404.4238-1-0-2', '1404.4238-2-0-2'], ['1404.4238-1-2-0', '1404.4238-2-2-0'], ['1404.4238-1-2-1', '1404.4238-2-2-1'], ['1404.4238-1-5-0', '1404.4238-2-5-0'], ['1404.4238-1-5-1', '1404.4238-2-5-1'], ['1404.4238-2-7-0', '1404.4238-3-6-0'], ['1404.4238-2-7-2', '1404.4238-3-6-2'], ['1404.4238-2-7-4', '1404.4238-3-6-5'], ['1404.4238-2-10-7', '1404.4238-3-12-6'], ['1404.4238-2-1-2', '1404.4238-3-1-2'], ['1404.4238-3-22-2', '1404.4238-4-49-2'], ['1404.4238-3-22-3', '1404.4238-4-49-3'], ['1404.4238-3-22-4', '1404.4238-4-49-4'], ['1404.4238-3-13-1', '1404.4238-4-29-2'], ['1404.4238-3-13-2', '1404.4238-4-29-3'], ['1404.4238-3-13-4', '1404.4238-4-29-5'], ['1404.4238-3-13-6', '1404.4238-4-29-7'], ['1404.4238-3-18-1', '1404.4238-4-34-2'], ['1404.4238-3-18-3', '1404.4238-4-34-5'], ['1404.4238-3-7-1', '1404.4238-4-8-1'], ['1404.4238-3-7-3', '1404.4238-4-8-3'], ['1404.4238-3-8-1', '1404.4238-4-9-1'], ['1404.4238-3-8-2', '1404.4238-4-9-2'], ['1404.4238-3-8-3', '1404.4238-4-9-3'], ['1404.4238-3-0-3', '1404.4238-4-0-3'], ['1404.4238-3-6-1', '1404.4238-4-7-2'], ['1404.4238-3-6-3', '1404.4238-4-7-4'], ['1404.4238-3-6-4', '1404.4238-4-7-5'], ['1404.4238-3-14-5', '1404.4238-4-30-2'], ['1404.4238-3-14-6', '1404.4238-4-30-3'], ['1404.4238-3-17-1', '1404.4238-4-33-1'], ['1404.4238-3-17-3', '1404.4238-4-33-3'], ['1404.4238-3-17-4', '1404.4238-4-33-4'], ['1404.4238-3-17-5', '1404.4238-4-33-5'], ['1404.4238-3-17-8', '1404.4238-4-33-9'], ['1404.4238-3-17-9', '1404.4238-4-33-10'], ['1404.4238-3-19-0', '1404.4238-4-35-0'], ['1404.4238-3-19-2', '1404.4238-4-35-2'], ['1404.4238-3-19-4', '1404.4238-4-35-4'], ['1404.4238-3-12-0', '1404.4238-4-26-0'], ['1404.4238-3-12-2', '1404.4238-4-26-1'], ['1404.4238-3-15-0', '1404.4238-4-32-0'], ['1404.4238-3-16-0', '1404.4238-4-32-1'], ['1404.4238-3-16-1', '1404.4238-4-32-2']]

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[]

[['1404.4238-2-12-0', '1404.4238-3-22-0'], ['1404.4238-2-7-3', '1404.4238-3-6-3'], ['1404.4238-2-7-3', '1404.4238-3-6-4'], ['1404.4238-2-0-0', '1404.4238-3-0-0'], ['1404.4238-2-10-0', '1404.4238-3-12-0'], ['1404.4238-2-10-0', '1404.4238-3-12-1'], ['1404.4238-3-21-1', '1404.4238-4-37-1'], ['1404.4238-3-22-0', '1404.4238-4-49-0'], ['1404.4238-3-18-0', '1404.4238-4-34-0'], ['1404.4238-3-18-0', '1404.4238-4-34-1'], ['1404.4238-3-7-2', '1404.4238-4-8-2'], ['1404.4238-3-8-4', '1404.4238-4-9-4'], ['1404.4238-3-11-0', '1404.4238-4-21-2'], ['1404.4238-3-11-2', '1404.4238-4-21-3'], ['1404.4238-3-3-0', '1404.4238-4-3-2'], ['1404.4238-3-3-0', '1404.4238-4-3-3'], ['1404.4238-3-3-1', '1404.4238-4-3-5'], ['1404.4238-3-0-0', '1404.4238-4-0-1'], ['1404.4238-3-4-0', '1404.4238-4-4-0'], ['1404.4238-3-14-0', '1404.4238-4-30-0'], ['1404.4238-3-14-4', '1404.4238-4-30-1'], ['1404.4238-3-17-6', '1404.4238-4-33-6'], ['1404.4238-3-17-6', '1404.4238-4-33-7'], ['1404.4238-3-12-3', '1404.4238-4-27-0'], ['1404.4238-3-9-3', '1404.4238-4-13-2'], ['1404.4238-3-9-4', '1404.4238-4-16-1'], ['1404.4238-3-9-5', '1404.4238-4-17-1'], ['1404.4238-3-9-5', '1404.4238-4-17-2'], ['1404.4238-3-9-7', '1404.4238-4-16-2'], ['1404.4238-3-10-4', '1404.4238-4-15-0'], ['1404.4238-3-10-5', '1404.4238-4-15-1'], ['1404.4238-3-10-8', '1404.4238-4-20-1'], ['1404.4238-3-2-1', '1404.4238-4-2-2'], ['1404.4238-3-2-1', '1404.4238-4-2-3'], ['1404.4238-3-2-4', '1404.4238-4-2-5'], ['1404.4238-3-2-5', '1404.4238-4-2-7']]

[['1404.4238-2-5-0', '1404.4238-3-3-0'], ['1404.4238-2-5-1', '1404.4238-3-3-1'], ['1404.4238-2-8-0', '1404.4238-3-7-0'], ['1404.4238-2-8-2', '1404.4238-3-7-2'], ['1404.4238-2-8-3', '1404.4238-3-7-3'], ['1404.4238-2-9-0', '1404.4238-3-8-0'], ['1404.4238-2-9-1', '1404.4238-3-8-1'], ['1404.4238-2-9-2', '1404.4238-3-8-2'], ['1404.4238-2-9-2', '1404.4238-3-8-3'], ['1404.4238-2-9-3', '1404.4238-3-8-4'], ['1404.4238-2-9-4', '1404.4238-3-8-5'], ['1404.4238-2-9-6', '1404.4238-3-9-0'], ['1404.4238-2-9-7', '1404.4238-3-9-1'], ['1404.4238-2-9-9', '1404.4238-3-9-3'], ['1404.4238-2-9-10', '1404.4238-3-9-4'], ['1404.4238-2-11-0', '1404.4238-3-13-0'], ['1404.4238-2-11-1', '1404.4238-3-13-1'], ['1404.4238-2-11-2', '1404.4238-3-13-2'], ['1404.4238-2-11-3', '1404.4238-3-13-3'], ['1404.4238-2-11-4', '1404.4238-3-13-5'], ['1404.4238-2-11-5', '1404.4238-3-13-6']]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1404.4238

{'1404.4238-3-0-0': 'It is shown that the optically bright excitonic transitions near the Dirac points of suspended semiconducting transition metal dichalcogenides have a [MATH]-like symmetry whereas the [MATH]-like states are dipole forbidden.', '1404.4238-3-0-1': 'The unusual optical selection rules result from the strong Coulomb coupling that induces an excitonic ground state from which the dipole-allowed optical excitations correspond to intra-excitonic transitions.', '1404.4238-3-0-2': 'The huge intrinsic coupling constant seems to be a generic property of the semiconducting transition metal dichalcogenides and strong Coulomb-coupling signatures in the form of the optical selection rules can be observed even in samples grown on typical substrates like SiO[MATH].', '1404.4238-3-0-3': 'For the examples of WS[MATH] and WSe[MATH], excellent agreement of the computed excitonic resonance energies with recent experiments is demonstrated.', '1404.4238-3-1-0': 'Since its first isolation by Novosolev and Geim[CITATION], graphene and graphene based systems have attracted lots of interest, both for fundamental reasons and technological applications.', '1404.4238-3-1-1': 'Meanwhile, a variety of graphene-analogues materials like h-BN, silicene or transition-metal dichalcogenides (TMDs) have been fabricated[CITATION].', '1404.4238-3-1-2': 'Similar to graphene, these novel material systems display exciting new physical properties, distinct from their bulk counterparts[CITATION].', '1404.4238-3-2-0': 'Particularly interesting among this class of materials are monolayers of TMDs, exhibiting unique linear and nonlinear optical properties.', '1404.4238-3-2-1': 'The linear optical spectra are dominated by strong resonances, with a peak absorption of more than 10%, and the breaking of inversion symmetry allows for an efficient second harmonic generation (SHG).', '1404.4238-3-2-2': 'Though initially assigned to free particle transitions, meanwhile the excitonic nature of the strong optical response is widely accepted.', '1404.4238-3-2-3': 'Nevertheless, the precise role of the Coulomb interaction in these structures remains unclear and is still under intensive investigation.', '1404.4238-3-2-4': 'Recent linear and nonlinear optical experimental studies on WS2 [1-3] report large binding energies for the lowest bright exciton transition ranging from 0.32-0.71 eV and strong deviations from the usual hydrogenic Rydberg series.', '1404.4238-3-2-5': 'However, due to the difficulty to observe the band gap directly, these values have been extracted either from the energetic separation between different resonances, or by comparison with ab initio predictions, and thus depend critically on the correct assignment of optically active states.', '1404.4238-3-2-6': 'The widely accepted assignment of the lowest observed resonance to the 1s-exciton ground state not only leads to a large uncertainty in the estimated exciton binding energy, but also leads to mutually contradicting conclusions drawn from different experimental techniques on nominally equivalent samples.', '1404.4238-3-3-0': 'In conventional direct-gap GaAs-type semiconductors with near-parabolic [MATH]-like conduction and [MATH]-like valence bands, the near-bandgap optical properties are best described by the so-called Elliot formula[CITATION], expressing the energetic position and oscillator strength of the excitonic transitions in terms of the excitonic wave functions.', '1404.4238-3-3-1': 'Simple selection rules show that only [MATH]-like excitonic states couple to the light field, resulting in the well-known excitonic Rydberg series, and the assignment of the lowest optical active state to the 1s exciton resonance is based on these selection rules.', '1404.4238-3-4-0': 'However, the s-type optical selection rules follow from the implicit assumption that the system is excited from the noninteracting ground state.', '1404.4238-3-4-1': 'In a strongly interacting system, the ground state itself may be excitonic in which case the optical transitions correspond to intra-excitonic transitions, that are governed by fundamentally different optical selection rules.', '1404.4238-3-5-0': 'In this letter, we present strong evidence that not only for suspended TMDs, but also for TMDs on a SiO[MATH] substrate, the optically active states correspond to the [MATH]-type excited states of a 2D hydrogenic Rydberg series, while the lowest lying [MATH]-exciton is merged with the ground state level.', '1404.4238-3-5-1': 'To support our theory, we compare its predictions with recent experiments on WS[MATH] [CITATION] and WSe[MATH][CITATION], demonstrating that the assumption of bright [MATH]-states allows for a simple interpretation of all the different experimental results.', '1404.4238-3-6-0': 'Dictated by the symmetry properties of the hexagonal lattice, the lowest order [MATH] Hamiltonian has the form[CITATION] [EQUATION] where [MATH] is the valley index and [MATH] is the tensor product of the electron spin state and a two-component quasi-spinor.', '1404.4238-3-6-1': 'The Pauli matrices [MATH] and [MATH] act in the pseudo-spin space and [MATH] in the real spin space, respectively.', '1404.4238-3-6-2': 'The basis functions for the pseudo-spinor are a linear combination of the relevant atomic orbitals that contribute to the valence and conduction bands and depend on the specific material system under consideration.', '1404.4238-3-6-3': 'Within a tight-binding model, the parameters [MATH], [MATH] and [MATH] correspond to the bias in on-site energies, the effective hopping matrix element, and the lattice constant, respectively.', '1404.4238-3-6-4': 'Furthermore, [MATH] is the spin-splitting of the valence band due to the intra-atomic spin-orbit coupling (SOC).', '1404.4238-3-6-5': 'The Hamiltonian ([REF]) is valid for the entire class of monolayer hexagonal structures including graphene with a zero gap and negligible SOC, h-BN with a large gap and also negligible SOC, and the variety of TMD with a gap in the optical range and strong SOC.', '1404.4238-3-7-0': 'Independent of the explicit expressions for the pseudo-spinor basis functions, the light-matter interaction can be found by the minimal substitution [MATH].', '1404.4238-3-7-1': 'Defining the Fermi velocity by [MATH], one obtains [EQUATION]', '1404.4238-3-7-2': 'Using the eigenstates of [MATH], the Hamiltonian for the light-matter (LM) interaction can be written as [EQUATION]', '1404.4238-3-7-3': 'Here, [MATH]) creates an electron with spin [MATH] and valley index [MATH] in the conduction (valence) band, [MATH] is the spin and valley dependent gap, [MATH] is the relativistic dispersion of a quasi-particle with rest energy [MATH], [MATH], and [MATH] is the angle in [MATH]-space defined by [MATH].', '1404.4238-3-8-0': 'In Eq. [REF], the first line describes the intraband and the second line the interband transitions, respectively.', '1404.4238-3-8-1': 'Contributions proportional to [MATH] correspond to the absorption of a photon with circular polarization [MATH].', '1404.4238-3-8-2': 'In zero-gap materials like graphene or silicene, the transition matrix elements for both intra- and interband absorption are equal in magnitude.', '1404.4238-3-8-3': 'Differences in the transition probabilities exclusively result from the different occupation numbers in the initial state.', '1404.4238-3-8-4': 'In contrast, in a wide gap system, the transition amplitudes [MATH] and [MATH] vanish, while [MATH], giving the much simpler expression [EQUATION]', '1404.4238-3-8-5': 'From this simplified LM Hamiltonian, one recognizes clearly that the valence-to-conduction-band excitations at the [MATH] valley require the absorption of a [MATH] polarized photon, as has been shown by Xiao et al.[CITATION].', '1404.4238-3-9-0': 'The microscopic transition amplitudes [MATH] can be computed from the semiconductor Bloch equations (SBE)[CITATION].', '1404.4238-3-9-1': 'Using Eq. [REF] together with the standard Coulomb-interaction Hamiltonian and assuming that the initial state is the noninteracting ground state, in linear response the SBE for the polarization is given by [EQUATION]', '1404.4238-3-9-2': 'Here, [MATH] denotes renormalized single particle energy and includes the Coulomb renormalization of the band-gap.', '1404.4238-3-9-3': 'The expansion into the eigenstates of the relativistic Wannier equation [EQUATION] yields the Elliot formula [EQUATION] with the oscillator strength [MATH].', '1404.4238-3-9-4': 'As usual, the resonances of the reflection and absorption spectra occur at the poles of the Elliot formula.', '1404.4238-3-9-5': 'Equation [REF] shows that the resonant contributions for a given circular polarization component [MATH] stem from the [MATH] valley, while the nonresonant contributions stem from the [MATH] valley, opening the possibility of a valley selective excitation with circularly polarized light.', '1404.4238-3-9-6': 'Moreover, the dependence of the resonance energies [MATH] on the product of the spin and valley index couples the valley dynamics to the spin dynamics[CITATION].', '1404.4238-3-9-7': 'The oscillator strength is a measure for the transition probability from the noninteracting ground state to excitonic state with quantum number [MATH] and is nonvanishing for exciton wave functions with [MATH]-type orbital angular momentum.', '1404.4238-3-10-0': 'The relativistic Wannier equation is the [MATH]-space representation of the Dirac Coulomb problem.', '1404.4238-3-10-1': 'In real space, the wave functions are two-component spinors[CITATION] [EQUATION] where [MATH] is the eigenvalue of the total angular momentum [MATH] and [MATH] is the principle quantum number.', '1404.4238-3-10-2': 'The absorption or emission of a circularly polarized photon corresponds to a spin flip in pseudospin space.', '1404.4238-3-10-3': 'Clearly, assuming a noninteracting ground state, only spinors with [MATH] couple to the optical field.', '1404.4238-3-10-4': 'As far as the Coulomb interaction can be considered as strictly 2D, the eigenvalues the relativistic hydrogen problem are given by[CITATION] [EQUATION].', '1404.4238-3-10-5': 'Here, [MATH] is the coupling constant, [MATH] is the effective dielectric constant of the environment, and the real spectrum contains al values for [MATH] with [MATH].', '1404.4238-3-10-6': 'For small values of [MATH], the spectrum reduces to that of the nonrelativistic Rydberg series [MATH].', '1404.4238-3-10-7': 'Interestingly, the bright exciton series with [MATH] vanishes in the strong coupling regime, which is characterized by [MATH].', '1404.4238-3-10-8': 'Using the material parameters given in Ref. [CITATION], the nominal values for [MATH] are given by [MATH] for WS[MATH], [MATH] for WSe[MATH] and [MATH] for MoS[MATH] respectively.', '1404.4238-3-10-9': 'In free standing monolayers of TMDs, these values are very large and even on top of a typical substrate like SiO[MATH] the effective coupling exceeds the critical value.', '1404.4238-3-11-0': 'In this regime of strong Coulomb coupling, the system has an excitonic ground state characterized by a static interband polarization [MATH] and static populations [MATH] that can be obtained as solutions of the gap equations[CITATION].', '1404.4238-3-11-1': 'The static polarization obeys the equation [EQUATION] while the linear polarization, Eq. [REF], is modified by the Pauli blocking of the ground-state populations: [EQUATION]', '1404.4238-3-11-2': 'Defining [MATH] and using [MATH], where [MATH] is the dispersion of the Bogoliubov bands, the ground state is the lowest eigenstate of the generalized Wannier equation [EQUATION] with [MATH].', '1404.4238-3-11-3': 'Here, [MATH] is the Coulomb interaction weakened by the phase space filling of the ground state populations.', '1404.4238-3-11-4': 'The optical polarization can be expanded into the eigenfunctions orthogonal to the ground state.', '1404.4238-3-11-5': 'Thus, in the strong Coulomb-interacting regime, optical transitions connect the excitonic ground state [MATH] with the excited excitonic states such that the dipole allowed transitions with [MATH] have the total angular momentum [MATH], corresponding to [MATH] like states.', '1404.4238-3-12-0': 'To test our general theory, we compare its predictions with the observations of several recent experiments on WS[MATH] [CITATION] and WSe[MATH][CITATION].', '1404.4238-3-12-1': 'We start with the analysis of the bright excitonic states in WS[MATH] on SiO[MATH] corresponding to the lowest exciton with spin and valley combination [MATH][CITATION].', '1404.4238-3-12-2': 'As commonly adopted, we refer to A and B excitons for the lower and higher direct exciton transitions with a spin and valley index combination [MATH] respectively.', '1404.4238-3-12-3': 'While the knowledge of the tight binding parameters and the dielectric screening of the substrate suffices to compute the energetic separation between any two bright states, the prediction of the absolute spectral position requires the additional knowledge of the Coulombic band gap renormalization [MATH].', '1404.4238-3-12-4': 'Whereas [MATH] diverges for an ideal, strictly 2D Coulomb potential, it is finite for any real system where the Coulomb matrix elements must be evaluated from the gap equations with the atomic orbitals used to represent the pseudo spinor space in Eq. [REF].', '1404.4238-3-12-5': 'This leads to a quasi 2D Coulomb interaction [EQUATION] where [MATH] is the normalization area and [MATH] is a monotonically decreasing form factor with [MATH] and [MATH].', '1404.4238-3-12-6': 'The form factor leads to a regularization of the [MATH] singularity at small distances, and the parameter [MATH] can be interpreted as the effective thickness of the monolayer.', '1404.4238-3-12-7': 'Since its explicit expression depends on the specific atomic orbitals contributing to the valence and conduction band, we do not calculate the band gap but rather use it as a fit parameter.', '1404.4238-3-13-0': 'With the material parameters for WS[MATH] eV, [MATH]eV, [MATH] eV, [MATH]A given in Ref. [CITATION], and an effective dielectric constant [MATH] for the SiO[MATH] substrate, we find an exciton binding energy a binding of [MATH] eV for the lowest bright exciton transition with [MATH].', '1404.4238-3-13-1': 'Using [MATH] eV, we present in Fig. [REF] the computed result for the spectral position of the lowest five bright exciton transitions.', '1404.4238-3-13-2': 'The red diamonds show the experimental data points taken from Ref. [CITATION].', '1404.4238-3-13-3': 'For comparison, we also show the predicted values assuming a nonrelativistic bright [MATH] states.', '1404.4238-3-13-4': '(Note that the [MATH] series gives complex eigenvalues of the relativistic wave equation and thus cannot be used for comparison.)', '1404.4238-3-13-5': 'Our computed results show remarkably good agreement with the experimental observations given the fact that only the effective band gap was used as a single fit parameter.', '1404.4238-3-13-6': 'In contrast, the results based on the assumption that the series has [MATH]-type character are completely off the scale, in particular for the energetically lowest states.', '1404.4238-3-14-0': 'Taking the renormalized band gap extracted from the experimental data, we can also compute the spectral positions of the B-excitons.', '1404.4238-3-14-1': 'Though the Coulomb renormalization contains some excitonic contributions, it is dominated by the renormalization of the single particle energy of the filled valence band.', '1404.4238-3-14-2': 'This part depends on the effective monolayer thickness and the background dielectric screening.', '1404.4238-3-14-3': 'Thus, it is reasonable to assume that the band gap renormalization is equal for the [MATH] and [MATH] excitons in WS[MATH], giving a gap [MATH] for the B-exciton.', '1404.4238-3-14-4': 'The ratio of the binding energy is fixed by the ratio of the single particle gaps, i.e. [MATH], giving a binding of the lowest bright B exciton of [MATH] eV.', '1404.4238-3-14-5': 'Thus the [MATH] resonance is predicted to occur at [MATH] eV, which is also in very good agreement with the experimental findings[CITATION].', '1404.4238-3-14-6': 'In table [REF], a list of the lowest theoretically predicted resonance positions of both A and B exciton is given.', '1404.4238-3-15-0': 'To provide further evidence for the theoretically predicted selection rules, we also analyze recent experiments probing dark states by two-photon spectroscopy which can only access excitonic states with a parity opposite to those excited by a single photon process[CITATION].', '1404.4238-3-16-0': 'In Ref. [CITATION], linear reflection and TP-PLE have been performed on WS[MATH] flakes having nominally the same material parameters as those in Ref. [CITATION].', '1404.4238-3-16-1': 'The resulting experimental data are shown in Fig. [REF].', '1404.4238-3-17-0': 'In Ref. Zhu2014, linear reflection has been measured for different temperatures, showing two clearly recognazable peaks at [MATH] eV and [MATH] eV at [MATH]K.', '1404.4238-3-17-1': 'These peaks agree well with the dominant peaks in the reflection spectrum of Ref. Chernikov2014 and can be assigned to the A and B exciton respectively.', '1404.4238-3-17-2': 'At room temperature, both peaks experience a red shift of approximately [MATH] eV, most likely resulting from a temperature dependent band gap renormalization.', '1404.4238-3-17-3': 'The TP-PLE measurements have been performed at room temperature and cover the spectral range between [MATH] and [MATH] eV.', '1404.4238-3-17-4': 'Within this spectral range two peaks in the TP-PLE spectrum are observed at approximately [MATH] and [MATH] eV, and a significant gap at approximately [MATH] eV.', '1404.4238-3-17-5': 'In Ref. Ye2014, the A and B excitonic resonances occur at [MATH] and [MATH]eV in the linear reflection spectrum respectively.', '1404.4238-3-17-6': 'Thus, the energetic separation of the two dominant peaks is good agreement with Refs. Chernikov2014 and Zhu2014, and deviations in the absolute values are most likely due to different experimental conditions.', '1404.4238-3-17-7': 'The TP-PLE spectrum covers the range between [MATH] and [MATH] eV and thus compliments the measurements of Ref. Zhu2014.', '1404.4238-3-17-8': 'Within this energy range, the TP-PLE spectrum exhibits pronounced maxima between [MATH] eV and at [MATH] eV.', '1404.4238-3-17-9': 'Remarkably, both experiments observe a large TP-PLE signal resonant with the lowest bright transition of the B exciton, whereas the TP-PLE in Ref. Ye2014 shows no significant signal at the lowest bright A exciton.', '1404.4238-3-18-0': 'In Ref. Wang2014, TP-PLE measurements have been performed in conjunction with SHG experiments on WSe[MATH] which has very similar properties as WS[MATH], showing the same symmetries and similar spin-splitting of the valence bands.', '1404.4238-3-18-1': 'Using PLE, the resonance position of the A exciton is determined as [MATH] eV, whereas in TP-PLE, the dominant peak occurs at approximately 1.9 eV.', '1404.4238-3-18-2': 'In agreement with the observation in Ref. [CITATION], the TP-PLE spectrum shows no significant signal at the transition energy of the lowest bright A exciton.', '1404.4238-3-18-3': 'In contrast, the SHG spectrum exhibits two dominant peaks at [MATH] eV, i.e, at the lowest bright resonance, and at [MATH] eV, and less prominent peaks at [MATH], [MATH] and [MATH] eV.', '1404.4238-3-18-4': 'The SHG spectrum extracted from the data in Ref. [CITATION] is shown in Fig. [REF].', '1404.4238-3-19-0': 'In Refs. [CITATION], the experiments have been analyzed under the assumption of bright [MATH]-excitons.', '1404.4238-3-19-1': 'This assumption is seemingly supported by the absence of a TP-PLE signal at the resonance frequency of the lowest bright A exciton transition.', '1404.4238-3-19-2': 'Thus, the resonances observed in TP-PLE spectra are assigned to [MATH]-like states.', '1404.4238-3-19-3': 'Since the lowest [MATH]-state should be energetically well above the lowest [MATH]- state, all resonances observed in the TP-PLE spectra of WS[MATH] are assigned to excited states of the A exciton, including those resonant with the lowest bright B exciton and above.', '1404.4238-3-19-4': 'This assignement directly contradicts the Rydberg series observed by Chernikov et al.[CITATION], where all excited states and the onset of the band gap of the A-exciton are found below the B- exciton.', '1404.4238-3-19-5': 'Furthermore, the assumption of bright [MATH]-excitons cannot explain the large SHG signal resonant with the lowest A exciton transition in the experiment of Wang et al.[CITATION].', '1404.4238-3-20-0': 'Remarkably, all observations can be explained quite naturally by the selection rules predicted by our theory.', '1404.4238-3-20-1': 'Since all [MATH]-states in a 2D system are accompanied by energetically degenerate states with opposite parity, one- and two-photon transitions are simultaneously possible such that all resonances observed in a linear optical experiment should also appear in SHG, in full agreement with the findings of Ref. [CITATION].', '1404.4238-3-20-2': 'Since a TP-PLE signal not only requires a two-photon-absorption transition but also a fast relaxation to the lowest bright state, the absence of light emmision at the [MATH]-resonance after a two-photon excitation of the [MATH] -transiton is most likely explained by the lack of an efficient relaxation channel between the two energetically degenerate states.', '1404.4238-3-21-0': 'In Table [REF],we summarize the main findings and compare our theoretically predicted resonance positions of WS[MATH] and WSe[MATH] with the experimentally available data.', '1404.4238-3-21-1': 'The theoretical values have been obtained using the material parameters of Ref. [CITATION] and [MATH] which is sufficient to fix the spacings between all A and B resonance positions.', '1404.4238-3-21-2': 'To fix the overall energy scale, we need to fit one additional value per series, marked by an asterisk in table [REF].', '1404.4238-3-21-3': 'Note, that we assigned quantum numbers to the respective resonance positions that result from our theoretical assignment and do not correpond to the labeling in the respective original publications.', '1404.4238-3-21-4': "Resonances that could not be resolved by the specific experiment remain as empty spaces, while those resonances that could in principle occur in the experiment but are not observed are indicated by an 'x'.", '1404.4238-3-21-5': 'As can be recognized from the table, not only the agreement between theory and experiment is remarkable, but also the data obtained by the different experimental techniques on different (but equivalent) samples is brought into almost perfect mutual agreement as soon as one adopts our interpretation in terms of optically active [MATH]-states.', '1404.4238-3-22-0': 'In conclusion, we presented a general theory showing that optically active excitonic transitions in recently investigated TMDs have a [MATH]-like symmetry of the excitonic wave functions.', '1404.4238-3-22-1': 'Our theory is supported by recently published experimental data, including the observation of a series of resonances in the linear reflectrum spectrum of WS[MATH][CITATION], two photon absorption in WS[MATH][CITATION] and WSe[MATH][CITATION], as well as second harmonic generation in WSe[MATH][CITATION].', '1404.4238-3-22-2': 'This observation of optically active [MATH]-transitions is not only of crucial importance for the correct interpretation of experimental data, but impressively demonstrates that in the TMDs investigated here, the Coulomb interaction is strong enough to induce an excitonic ground state.', '1404.4238-3-22-3': 'The dipole allowed excited states of such an excitonic insulator correspond to intra-excitonic transitions and are governed by the corresponding optical selection rules.', '1404.4238-3-22-4': 'Even for systems in the weakly interacting regime without an excitonic ground state, the lowest excitonic transition should be in the THz or infrared regime and the observed optical resonances correspond to excited exciton levels with main quantum number [MATH].'}

{'1404.4238-4-0-0': 'It is shown that the strong Coulomb coupling in intrinsic suspended semiconducting transition metal dichalcogenides can exceed the critical value needed for an excitonic ground state.', '1404.4238-4-0-1': 'The dipole-allowed optical excitations then correspond to intra-excitonic transitions such that the optically bright excitonic transitions near the Dirac points have a [MATH]-like symmetry whereas the [MATH]-like states are dipole forbidden.', '1404.4238-4-0-2': 'The large intrinsic coupling strength seems to be a generic property of the semiconducting transition metal dichalcogenides and strong Coulomb-coupling signatures in the form of the optical selection rules can be observed even in samples grown on typical substrates like SiO[MATH].', '1404.4238-4-0-3': 'For the examples of WS[MATH] and WSe[MATH], excellent agreement of the computed excitonic resonance energies with recent experiments is demonstrated.', '1404.4238-4-1-0': '# Introduction', '1404.4238-4-2-0': 'Monolayers of transition metal dichalcogenides (TMDs) are novel two-dimensional (2D) semiconductors.', '1404.4238-4-2-1': 'Similar to graphene, these materials display exciting new physical properties, distinct from their bulk counterparts[CITATION].', '1404.4238-4-2-2': 'The linear optical spectra are dominated by strong resonances with peak absorption values of more than 10% of the incoming light.', '1404.4238-4-2-3': 'Furthermore, the absence of inversion symmetry allows for efficient second harmonic generation (SHG).', '1404.4238-4-2-4': 'Though initially assigned to free-particle transitions, meanwhile the excitonic nature of the strong optical response is widely accepted.', '1404.4238-4-2-5': 'On the basis of recent linear and nonlinear optical experiments on WS[MATH] [1-3], the respective authors report large binding energies for the energetically lowest, optically bright excitons ranging from 0.32-0.71 eV.', '1404.4238-4-2-6': 'At the same time, strong deviations from the usual hydrogenic Rydberg series are observed.', '1404.4238-4-2-7': 'Due to the difficulty to directly measure the single-particle bandgap energy, the binding energies have been deduced either from the energetic separation between different resonances or by the comparison with theoretical predictions.', '1404.4238-4-3-0': 'In general, the proper analysis of the measured spectra depends critically on the correct identification of the optically active states.', '1404.4238-4-3-1': 'For this purpose, one derives optical selection rules that result from the conservation of the total angular momentum in the excitation and emission processes.', '1404.4238-4-3-2': 'The [MATH]-point transitions in conventional direct-gap GaAs-type semiconductors occur between the [MATH]-hybridized near-parabolic [MATH]-like conduction and [MATH]-like valence bands.', '1404.4238-4-3-3': 'The near-bandgap optical properties are well described by the so-called Elliot formula[CITATION], expressing the energetic position and oscillator strength of the excitonic transitions in terms of the excitonic wave functions.', '1404.4238-4-3-4': 'As a result of the simple dipole selection rules, only [MATH]-like excitonic states couple to the light field.', '1404.4238-4-3-5': 'This leads to the well-known excitonic Rydberg series where the lowest optical active state is the 1s exciton resonance.', '1404.4238-4-4-0': 'The derivation of the Elliot formula[CITATION] is based on the implicit assumption that the system is excited from the noninteracting ground state.', '1404.4238-4-4-1': 'In contrast, if one considers an excitonic insulator[CITATION], the ground state itself is excitonic and light absorption or emission involves intra-excitonic transitions that are governed by fundamentally different optical selection rules.', '1404.4238-4-4-2': 'The energetically lowest transition from an [MATH]-type excitonic ground state will lead to optically excited [MATH]-type excitons.', '1404.4238-4-5-0': 'In this paper, we present strong evidence that not only for suspended TMDs, but also for TMDs on a SiO[MATH] substrate, the optically active states correspond to the [MATH]-type excited states of a two-dimensional (2D) hydrogenic Rydberg series, while the lowest lying [MATH]-exciton is merged with the ground-state level.', '1404.4238-4-5-1': 'Our predictions are based on the analysis of the Hamiltonian for 2D massive Dirac Fermions with Coulomb interaction.', '1404.4238-4-5-2': 'To support our theory, we compare its predictions with recent experiments on WS[MATH] [CITATION] and WSe[MATH][CITATION], demonstrating that the assumption of bright [MATH]-states allows for a simple interpretation of a wide range of different experimental results.', '1404.4238-4-6-0': '# Microscopic Theory', '1404.4238-4-7-0': 'Starting point of our analysis is the low-energy Hamiltonian for the bandstructure in the vincinity of the Dirac-points.', '1404.4238-4-7-1': 'Dictated by the symmetry properties of the hexagonal lattice, the lowest order [MATH] Hamiltonian has the form[CITATION] [EQUATION] where [MATH] is the so-called valley index identifying the two non-equivalent Dirac points, and [MATH] is the tensor product of the electron spin state and a two-component quasi-spinor.', '1404.4238-4-7-2': 'The Pauli matrices [MATH] and [MATH] act in the pseudo-spin space and [MATH] in the real spin space, respectively.', '1404.4238-4-7-3': 'The basis functions for the pseudo-spinors are a linear combination of the relevant atomic orbitals that contribute to the valence and conduction bands and depend on the specific material system under consideration.', '1404.4238-4-7-4': 'Within a tight-binding model, the parameters [MATH], [MATH] and [MATH] correspond to the bias in on-site energies, the effective hopping matrix element, and the lattice constant, respectively.', '1404.4238-4-7-5': 'Furthermore, [MATH] is the spin-splitting of the valence band due to the intra-atomic spin-orbit coupling (SOC).', '1404.4238-4-7-6': 'The Hamiltonian ([REF]) is valid for the entire class of monolayer hexagonal structures including graphene with a zero gap and negligible SOC, h-BN with a large gap and also negligible SOC, and the variety of TMDs with a gap in the optical range and strong SOC.', '1404.4238-4-8-0': 'Independent of the explicit expressions for the pseudo-spinor basis functions, the Hamiltonian describing the light-matter (LM) interaction can be found by the minimal substitution [MATH].', '1404.4238-4-8-1': 'Defining the Fermi velocity by [MATH], one obtains [EQUATION]', '1404.4238-4-8-2': 'Using the eigenstates of [MATH], the LM Hamiltonian can be written as [EQUATION]', '1404.4238-4-8-3': 'Here, [MATH]) creates an electron with spin [MATH] and valley index [MATH] in the conduction (valence) band, [MATH] is the spin and valley dependent gap, [MATH] is the relativistic dispersion of a quasi-particle with rest energy [MATH], [MATH], and [MATH] is the angle in [MATH]-space defined by [MATH].', '1404.4238-4-9-0': 'In Eq. ([REF]), the first line describes the intraband and the second line the interband transitions, respectively.', '1404.4238-4-9-1': 'Contributions proportional to [MATH] correspond to the absorption of a photon with circular polarization [MATH].', '1404.4238-4-9-2': 'In zero-gap materials like graphene or silicene, the transition matrix elements for both intra- and interband absorption are equal in magnitude.', '1404.4238-4-9-3': 'Differences in the transition probabilities exclusively result from the different occupation numbers in the initial state.', '1404.4238-4-9-4': 'In contrast, in a wide gap system, the leading orders of the transition matrix elements are given by [MATH], [MATH], and [MATH].', '1404.4238-4-9-5': 'Keeping only contributions up to first order in [MATH] gives [EQUATION] where the intraband contributions have been rewritten with the aid of the intraband current matrix element [MATH].', '1404.4238-4-9-6': 'From this simplified LM Hamiltonian, Eq. ([REF]), one recognizes clearly that the valence-to-conduction-band excitations at the [MATH] valley require the absorption of a [MATH] polarized photon, as has been shown by Xiao et al.[CITATION].', '1404.4238-4-10-0': 'The optically induced current can be obtained from the LM Hamiltonian via [MATH] and contains both intra- and interband transitions.', '1404.4238-4-10-1': 'From Eq. ([REF]) it is clear that a finite macroscopic interband current requires an even part of the microscopic transition amplitudes [MATH], while intraband transitions contribute to the macroscopic current only if [MATH] contains an odd part.', '1404.4238-4-11-0': 'The microscopic transition amplitudes [MATH] and the Pauli blocking factor [MATH] can be computed from the Heisenberg equations of motion yielding the semiconductor Bloch equations (SBE)[CITATION].', '1404.4238-4-11-1': 'Using Eq. ([REF]) and the standard Coulomb-interaction Hamiltonian, one obtains [EQUATION] where [MATH] denotes the renormalized single particle energy that includes the Coulomb renormalization of the band-gap.', '1404.4238-4-12-0': '## Regime of weak Coulomb coupling', '1404.4238-4-13-0': 'Assuming the noninteracting ground state as the initial state before optical excitation, the linear response SBE for the polarization is given by [EQUATION]', '1404.4238-4-13-1': 'Since the populations are at least second order in the exciting field, we use [MATH].', '1404.4238-4-13-2': 'The expansion into the eigenstates of the Wannier equation with the relativistic single-particle dispersion [EQUATION] yields [EQUATION] for the linear susceptibility [MATH], describing the optical response via [EQUATION]', '1404.4238-4-13-3': 'Here, the oscillator strength is given as [MATH].', '1404.4238-4-14-0': 'The fully relativistic Wannier equation ([REF]) corresponds to the [MATH]-space representation of the Dirac Coulomb problem.', '1404.4238-4-14-1': 'In real space, the wave functions are two-component spinors[CITATION] [EQUATION] obeying the wave equation [EQUATION]', '1404.4238-4-14-2': 'Here [MATH] is the eigenvalue of the total angular momentum [MATH], and [MATH] is the principle quantum number.', '1404.4238-4-14-3': 'We can identify the set of quantum numbers [MATH] such that, e.g., an [MATH] -type wave function has [MATH].', '1404.4238-4-14-4': 'The eigenfunctions [MATH] of Eq. ([REF]) correspond to the solutions of Eq. ([REF]) with positive energy and are given by [EQUATION] while the solutions corresponding to negative energies are of no physical relevance here.', '1404.4238-4-15-0': 'For the 2D Coulomb interaction potential, the eigenvalues of the relativistic hydrogen problem are given by[CITATION] [EQUATION] with [MATH] and [MATH].', '1404.4238-4-15-1': 'Here, [MATH] is the coupling constant and [MATH] is the effective dielectric constant of the environment.', '1404.4238-4-15-2': 'The eigenvalues are real for those [MATH]-values where [MATH].', '1404.4238-4-15-3': 'For small values of [MATH], the eigenvalue spectrum reduces to that of the nonrelativistic Rydberg series [EQUATION] with [MATH].', '1404.4238-4-16-0': 'From the expression ([REF]) for the linear susceptibility, one can derive the exciton spectrum and optical selection rules for dipole allowed optical transitions.', '1404.4238-4-16-1': 'At the poles of Eq. ([REF]), the reflection and absorption spectra display resonances.', '1404.4238-4-16-2': 'The oscillator strength is a measure for the transition probability from the noninteracting groundstate to the excitonic state with quantum number [MATH].', '1404.4238-4-17-0': 'Thus, we see that only excitonic transitions with [MATH]-type orbital angular momentum are dipole allowed.', '1404.4238-4-17-1': 'Furthermore, the resonant contributions for a given circular polarization component [MATH] stem from the [MATH] valley, while the nonresonant contributions stem from the [MATH] valley.', '1404.4238-4-17-2': 'This opens the possibility of a valley selective excitation with circularly polarized light.', '1404.4238-4-17-3': 'Moreover, the dependence of the resonance energies [MATH] on the product of the spin and valley index couples the valley dynamics to the spin dynamics.', '1404.4238-4-17-4': 'Both effects have been predicted in [CITATION] and demonstrated experimentally for various monolayers of TMDs.', '1404.4238-4-18-0': '## Regime of Strong Coulomb Coupling', '1404.4238-4-19-0': 'Interestingly, in the regime of strong Coulomb coupling characterized by [MATH], the optical spectrum contains no [MATH]-type wave functions with [MATH].', '1404.4238-4-19-1': 'In this case, total angular momentum conservation cannot be fulfilled for optical transitions from the noninteracting groundstate such that no one-photon transitions should be possible.', '1404.4238-4-20-0': 'To check if any of the TMD systems can be in the regime of strong Coulomb coupling, we use the material parameters given in Ref. [CITATION].', '1404.4238-4-20-1': 'Thus, we obtain the nominal values [MATH] for WS[MATH], [MATH] for WSe[MATH] and [MATH] for MoS[MATH] respectively.', '1404.4238-4-20-2': 'In free standing monolayers of TMDs, these values are very large and even on top of a typical substrate like SiO[MATH] with [MATH], the effective coupling exceeds the critical value.', '1404.4238-4-20-3': 'Thus, the assumption of a noninteracting ground state predicts the absence of the one-photon optical transitions for all of these systems, clearly contradicting the experimental findings.', '1404.4238-4-21-0': 'However, in the regime of strong Coulomb coupling, the real part of the lowest exciton energy [MATH] vanishes, i.e., it is degenerate with the noninteracting groundstate and the system transitions into an excitonic insulator state.', '1404.4238-4-21-1': 'This excitonic insulator state is a BEC-like condensate of excitons, i.e. a coherent superposition of the noninteracting groundstate and all exciton states with [MATH][CITATION].', '1404.4238-4-21-2': 'This state exhibits a static interband polarization [MATH] and static blocking factor [MATH] that can be obtained as nontrivial solutions of the gap equations[CITATION] [EQUATION]', '1404.4238-4-21-3': 'Here, [MATH] is the dispersion of the Bogoliubov bands.', '1404.4238-4-22-0': 'The static groundstate polarizations and populations induce a coupling between the linearized equations for the optical polarization and populations and add an additional source term to the equation of motion for the optical interband transition amplitude.', '1404.4238-4-22-1': 'Explicitely, we have [EQUATION]', '1404.4238-4-22-2': 'As the additional source term is proportional to the intraband current matrixelement with odd parity, it mixes states with different parity.', '1404.4238-4-22-3': 'Though the odd terms of the linear interband polarization do not couple to the optical field directly, they serve as source terms for the optically induced intraband current.', '1404.4238-4-22-4': 'Eqs. ([REF]) and ([REF]) can be decoupled by the transformation [EQUATION] yielding [EQUATION]', '1404.4238-4-22-5': 'Hence, the solutions of Eq. ([REF]) can be expanded in terms of the eigenfunctions of the Bogoliubov-Wannier equation [EQUATION] where the dispersion of the noninteracting bands has been replaced by the Bogoliubov quasi-particle dispersion.', '1404.4238-4-22-6': 'The groundstate polarization obeys [MATH] such that it can be expanded into the eigenfunctions of the Bogoliubov-Wannier equation with the eigenvalue [MATH].', '1404.4238-4-22-7': 'Hence, we make the ansatz [EQUATION]', '1404.4238-4-22-8': 'Inserting the ansatz into Eq. ([REF]) yields [EQUATION]', '1404.4238-4-22-9': 'Here, [MATH] is the coupling strength of the macroscopic interband polarization to the optical field.', '1404.4238-4-22-10': 'For an [MATH]-type groundstate ([MATH]), the populations also have spherical symmetry and the coupling strength vanishes for all non [MATH]-type excition states, i.e. for all real exciton resonances.', '1404.4238-4-22-11': 'Generally, if [MATH], the groundstate populations are composed of exciton states with [MATH], while real exciton resonances require [MATH].', '1404.4238-4-22-12': 'Thus, in the strong Coulomb coupling regime, the first term on the RHS of Eq. ([REF]) vanishes.', '1404.4238-4-23-0': 'Reversing the transformation [REF], we can compute the optical susceptibility for the excitonic insulator state [EQUATION]', '1404.4238-4-23-1': 'Here, all contributions have intraband character.', '1404.4238-4-24-0': 'As can be recognized from Eq. ([REF]), the poles of the linear susceptibility for the excitonic insulator state occur at the spectrum of the exciton Hamiltonian with Bogoliubov dispersion.', '1404.4238-4-24-1': 'The dominant effect of the exciton condensation on the single quasi-particle dispersion is a renormalization of the gap.', '1404.4238-4-24-2': 'Hence, the spectrum can be obtained in very good approximation from Eq. [REF], provided one uses the appropriate renormalized gap.', '1404.4238-4-24-3': 'The major difference as compared to the optical susceptibility, Eq. ([REF]), for the uncorrelated groundstate is the oscillator strength, which is now given by the intraband current matrix element between the excited exciton state and the excitonic groundstate [EQUATION].', '1404.4238-4-24-4': 'This matrix element connects that part of the groundstate with largest [MATH] to the excited states with [MATH] such that dipole allowed transitions increase (decrease) the total angular momentum by one.', '1404.4238-4-24-5': 'Explicitly, in the regime with [MATH], optical transitions connect the [MATH]-type excitonic groundstate [MATH] with the excited excitonic states [MATH], corresponding to [MATH]-like states.', '1404.4238-4-24-6': 'Thus, combining the results of the weakly and strongly interacting regimes, the modified optical selection rules can be summarized by the condition [EQUATION] for the optically active states.', '1404.4238-4-25-0': '# Analysis of Experimental Observations', '1404.4238-4-26-0': 'To test our general theory, we compare its predictions with the observations of several recent experiments on WS[MATH] [CITATION] and WSe[MATH][CITATION].', '1404.4238-4-26-1': 'As commonly adopted, we refer to A and B excitons for the lower and higher direct exciton transitions with a spin and valley index combination [MATH] respectively.', '1404.4238-4-27-0': 'While the effective fine structure constant can be extracted from the tight binding parameters and the dielectric screening, to compute the energetic positions of the bright states requires the additional knowledge of the Coulombic bandgap renormalization.', '1404.4238-4-27-1': 'Whereas the bandgap renormalization diverges for an ideal, strictly 2D Coulomb potential, it is finite for any real system where the Coulomb matrix elements must be evaluated from the gap equations with the atomic orbitals used to represent the pseudo spinor space in Eq. ([REF]).', '1404.4238-4-27-2': 'This leads to a quasi-2D Coulomb interaction [EQUATION] where [MATH] is the normalization area and [MATH] is a monotonically decreasing form factor with [MATH] and [MATH].', '1404.4238-4-27-3': 'The form factor leads to a regularization of the [MATH] singularity at small distances and the parameter [MATH] can be interpreted as the effective thickness of the monolayer.', '1404.4238-4-27-4': 'Since its explicit expression depends on the specific atomic orbitals contributing to the valence and conduction band, we do not calculate the bandgap but rather use it as a fit parameter.', '1404.4238-4-28-0': '## Linear Reflection experiments', '1404.4238-4-29-0': 'We start with the analysis of the bright excitonic states in WS[MATH] on SiO[MATH] corresponding to the lowest A exciton with spin and valley combination [MATH][CITATION].', '1404.4238-4-29-1': 'With the material parameters for WS[MATH]eV, [MATH] eV, [MATH]A given in Ref. [CITATION], an effective dielectric constant [MATH] for the SiO[MATH] substrate, and using [MATH] eV, we find a binding energy of [MATH] eV for the lowest bright exciton transition with [MATH].', '1404.4238-4-29-2': 'Using [MATH] eV, we present in Fig. [REF] the computed result for the spectral position of the lowest five bright exciton transitions.', '1404.4238-4-29-3': 'The red diamonds show the experimental data points taken from Ref. [CITATION].', '1404.4238-4-29-4': 'For comparison, we also show the predicted values assuming nonrelativistic bright [MATH]-states.', '1404.4238-4-29-5': '(Note that the [MATH] series gives complex eigenvalues of the relativistic wave equation and thus cannot be used for comparison.)', '1404.4238-4-29-6': 'Our computed results show remarkably good agreement with the experimental observations given the fact that only the effective bandgap was used as a single fit parameter.', '1404.4238-4-29-7': 'In contrast, the results based on the assumption that the series has [MATH]-type character are completely off the scale, in particular for the energetically lowest states.', '1404.4238-4-30-0': 'Assuming that the observed spin splitting corresponds to the difference in the renormalized bandgaps, we obtain [MATH] which can be used to compute the spectral positions of the B-excitons.', '1404.4238-4-30-1': 'The ratio of the binding energy is fixed by the ratio of the renormalized gaps, i.e. [MATH], giving a binding energy of [MATH] eV for the lowest bright B exciton.', '1404.4238-4-30-2': 'Thus the [MATH] resonance is predicted to occur at [MATH] eV, which is also in very good agreement with the experimental findings[CITATION].', '1404.4238-4-30-3': 'In table [REF], a list of the lowest theoretically predicted resonance positions of both A and B exciton is given.', '1404.4238-4-31-0': '## Two Photon Experiments', '1404.4238-4-32-0': 'To provide further evidence for the theoretically predicted selection rules, we also analyze recent experiments probing dark states by two-photon spectroscopy which can only access excitonic states with a parity opposite to those excited by a single photon process[CITATION].', '1404.4238-4-32-1': 'In Ref. [CITATION], linear reflection and TP-PLE have been performed on WS[MATH] flakes having nominally the same material parameters as those in Ref. [CITATION].', '1404.4238-4-32-2': 'The resulting experimental data are shown in Fig. [REF].', '1404.4238-4-33-0': 'In Ref. Zhu2014, the linear reflection has been measured for different temperatures, showing two clearly recognizable peaks for [MATH]K at [MATH] eV and [MATH] eV, respectively.', '1404.4238-4-33-1': 'These peaks agree well with the dominant peaks in the reflection spectrum of Ref. Chernikov2014 and can be assigned to the A and B exciton respectively.', '1404.4238-4-33-2': 'At room temperature, both peaks experience a red shift of approximately [MATH] eV, most likely resulting from a temperature dependent bandgap renormalization.', '1404.4238-4-33-3': 'The TP-PLE measurements have been performed at room temperature and cover the spectral range between [MATH] and [MATH] eV.', '1404.4238-4-33-4': 'Within this spectral range two peaks in the TP-PLE spectrum are observed at approximately [MATH] and [MATH] eV, and a significant gap at approximately [MATH] eV.', '1404.4238-4-33-5': 'In Ref. Ye2014, the A and B excitonic resonances occur at [MATH] and [MATH]eV in the linear reflection spectrum respectively.', '1404.4238-4-33-6': 'Thus, the energetic separation of the two dominant peaks is in good agreement with Refs. Chernikov2014 and Zhu2014.', '1404.4238-4-33-7': 'The small deviations in the absolute values are most likely due to different experimental conditions.', '1404.4238-4-33-8': 'The TP-PLE spectrum covers the range between [MATH] and [MATH] eV and thus complements the measurements of Ref. [CITATION].', '1404.4238-4-33-9': 'Within this energy range, the TP-PLE spectrum exhibits pronounced maxima between [MATH] eV and at [MATH] eV.', '1404.4238-4-33-10': 'Remarkably, both experiments observe a large TP-PLE signal resonant with the lowest bright transition of the B exciton, whereas the TP-PLE in Ref. Ye2014 shows no significant signal at the lowest bright A exciton.', '1404.4238-4-34-0': 'In Ref. Wang2014, TP-PLE measurements have been performed in conjunction with SHG experiments on WSe[MATH].', '1404.4238-4-34-1': 'This material system has very similar properties as WS[MATH], in particular it has the same symmetries and similar spin-splitting of the valence bands.', '1404.4238-4-34-2': 'Using PLE, the resonance position of the A exciton is determined as [MATH] eV, whereas in TP-PLE, the dominant peak occurs at approximately 1.9 eV.', '1404.4238-4-34-3': 'In agreement with the observation in Ref. Ye2014, the TP-PLE spectrum shows no significant signal at the transition energy of the lowest bright A exciton.', '1404.4238-4-34-4': 'In contradiction to the results of Ref. Ye2014, the TP-PLE spectrum does not show a significant signal at the transition energy of the lowest bright B exciton.', '1404.4238-4-34-5': 'In contrast, the SHG spectrum exhibits two dominant peaks at [MATH] eV, i.e, at the lowest bright resonance, and at [MATH] eV, and less prominent peaks at [MATH], [MATH] and [MATH] eV.', '1404.4238-4-35-0': 'In Refs. [CITATION], the experiments have been analyzed under the assumption of bright [MATH]-excitons.', '1404.4238-4-35-1': 'This assumption is seemingly supported by the absence of a TP-PLE signal at the resonance frequency of the lowest bright A exciton transition[CITATION], and at the lowest A and B exciton[CITATION] respectively.', '1404.4238-4-35-2': 'Thus, the resonances observed in TP-PLE spectra are assigned to [MATH]-like states.', '1404.4238-4-35-3': 'Since the lowest [MATH]-state should be energetically well above the lowest [MATH]- state, all resonances observed in the TP-PLE spectra of WS[MATH] are therefore assigned to excited states of the A exciton, including those resonant with the lowest bright B exciton and above.', '1404.4238-4-35-4': 'This assignement directly contradicts the Rydberg series observed by Chernikov et al.[CITATION], where all excited states and the onset of the band gap of the A-exciton are found below the B- exciton.', '1404.4238-4-36-0': 'Remarkably, all the above mentioned experimental observations can be explained quite naturally by the selection rules predicted by our theory.', '1404.4238-4-36-1': 'In the strongly interacting regime, the lowest bright exciton state is [MATH]-type and nondegerate, as [MATH]-type wave functions do not exist.', '1404.4238-4-36-2': 'The higher states are at least two-fold degenerate, and a two-photon absorption excites a [MATH]-type excited state.', '1404.4238-4-36-3': 'Thus, if the groundstate is excitonic, the lowest excited state should be dark in a two-photon experiment, exactly as is the case for the lowest [MATH]-exciton in a weakly interacting system.', '1404.4238-4-36-4': 'However, if the coupling constant is slightly below the critical value, the [MATH] states become bright and the state with [MATH] becomes (nearly) degenerate with the [MATH] state, which is the first excited [MATH]-type state.', '1404.4238-4-36-5': 'Simultaneously, the [MATH]-state can only be reached via a two-photon transition, while the corresponding degenerate [MATH]-state would be bright.', '1404.4238-4-36-6': 'Thus, the observation of a TP-PLE signal at the resonance position of the lowest bright B-exciton in the WS[MATH] sample seems to indicate that the WS[MATH] sample is at the edge of an excitonic insulator, with the lowest B[MATH] exciton transition somewhere in the (far) infrared.', '1404.4238-4-37-0': 'In Table [REF], we summarize the reported observations and compare our theoretically predicted resonance positions of WS[MATH] and WSe[MATH] with the experimentally available data.', '1404.4238-4-37-1': 'The theoretical values have been obtained using the material parameters of Ref. [CITATION], [MATH] with the band gap as a single fit parameter.', '1404.4238-4-37-2': 'Note, that we assigned quantum numbers to the respective resonance positions that result from our theoretical assignment and do not necessarily correspond to the labeling in the respective original publications.', '1404.4238-4-37-3': "Resonances that could not be resolved by the specific experiment remain as empty spaces in Table [REF] , while those resonances that could in principle occur in the experiment but are not observed are indicated by an 'x'.", '1404.4238-4-37-4': 'As can be recognized from the table, not only the agreement between theory and experiment is remarkable, but also the data obtained by the different experimental techniques on different samples is brought into almost perfect mutual agreement as soon as one adopts our interpretation in terms of optically active [MATH]-states.', '1404.4238-4-38-0': '# Discussion', '1404.4238-4-39-0': 'The results discussed in the previous sections are based on the model Hamiltonian ([REF]) treating the electrons and holes in the vicinity of the [MATH]-points as massive Dirac Fermions.', '1404.4238-4-39-1': 'The strength of this model Hamiltonian is its simplicity, allowing for an analytical solution of the exciton problem and thus providing insight into the optical properties of Coulomb-interacting chiral quasi-particles that are difficult to obtain numerically.', '1404.4238-4-39-2': 'The obvious disadvantage is that it contains several simplifications that might restrict the application of the model to real material systems.', '1404.4238-4-39-3': 'In the following, we give a brief discussion of the approximations underlying our analysis, and estimate its validity and restrictions.', '1404.4238-4-40-0': 'Inherent to the massive Dirac model is a single-particle dispersion that has a perfect electron-hole symmetry.', '1404.4238-4-40-1': 'The effect of an eventual electron-hole asymmetry is twofold: It modifies the single-particle dispersion and therewith the homogeneous part of the Wannier equation and it alters the dipole matrix element, i.e. the light-matter interaction.', '1404.4238-4-40-2': 'The latter effect may weaken the optical selection rules, i.e. unlike for the case of a perfect electron-hole symmetry, one- and two-photon processes can address the same states, while the first effect yields a modified exciton spectrum.', '1404.4238-4-41-0': 'For the exciton equation, the relevant quantity is the energy difference between the conduction and valence band.', '1404.4238-4-41-1': 'In a conventional semiconductor with parabolic bands, this leads to an exciton dispersion and binding energy that depend on the reduced mass of the electron-hole pair only.', '1404.4238-4-41-2': 'Adding a term to the single-particle Hamiltonian of the form [EQUATION] that has been proposed in Refs. [CITATION] to account for different electron and hole masses within the chiral two-band model, it is straightforward to show that, up to corrections of the order [MATH], the exciton dispersion corresponds to that of a relativistic, symmetric electron-hole pair with a Fermi velocity [MATH] and effective mass [MATH].', '1404.4238-4-41-3': 'Here, [MATH] and [MATH] is the reduced mass of the electron-hole pair.', '1404.4238-4-41-4': 'It is worth to note that the so defined effective mass is not directly related to the electron and hole masses but describes the pair properties only.', '1404.4238-4-41-5': 'Using the parameters for WS[MATH] given in [CITATION], this gives [MATH] and a nominal coupling constant [MATH] which is still well above the critical value.', '1404.4238-4-41-6': 'Thus, although a possible electron-hole asymmetry may lead to a violation of the optical selection rules, the massive Fermion model is well suited to describe the exciton spectrum even for asymmetric electrons and holes, provided one uses the effective Fermi-velocity.', '1404.4238-4-42-0': 'To obtain the spectral position of the excitons, we utilize the solutions of the resulting relativistic Wannier equation where we include only the substrate background screening but neglect all other screening effects.', '1404.4238-4-42-1': 'In general, screening effects arise both from filled valence bands that are not considered explicitely in the model Hamiltonian, and screening of the conduction and valence band under consideration.', '1404.4238-4-42-2': 'The Coulomb matrix elements used to set up the interacting model Hamiltonian should contain only the first type of contributions which can be considered to lead to a constant background dielectric constant.', '1404.4238-4-42-3': 'For a film of vanishing thickness [MATH], this contribution can be neglected[CITATION].', '1404.4238-4-42-4': 'The second type of contributions should then be computed selfconsistently, using the Coulomb matrix elements screened by the background dielectric constant.', '1404.4238-4-43-0': 'For the massive Dirac-Hamiltonian ([REF]), the RPA dielectric function of the TMD monolayer is given by [CITATION], [EQUATION] where [MATH] is the effective dielectric screening of the substrate and [MATH] is the [MATH] exciton Bohr radius.', '1404.4238-4-43-1': 'An analytical expression for [MATH] is given in Ref. [CITATION].', '1404.4238-4-43-2': 'In the nonrelativistic limit [MATH], [MATH], yielding [MATH].', '1404.4238-4-43-3': 'In this regime, the intrinsic screening corresponds to the anti-screening proposed by[CITATION] with an anti-screening length [MATH].', '1404.4238-4-43-4': 'Using the nominal values of [MATH] in suspended TMDs, the anti screening length is typically [MATH] , which corresponds roughly to the physical thickness of the monolayer.', '1404.4238-4-43-5': 'For supported samples, the anti-screening length is reduced correspondingly.', '1404.4238-4-44-0': 'The anti-screening is typical for a truly 2D-dielectric material and and weakens the strength of the Coulomb interaction on a length scale small compared to the anti-screening length.', '1404.4238-4-44-1': 'Thus, for the weakly bound excitons with a spatial extension large compared to the 2D-exciton Bohr radius, we expect the anti-screening effect to be of minor importance.', '1404.4238-4-44-2': 'This expectation is indirectly confirmed by the excellent agreement of the experimentally observed resonance positions and our theoretical predictions.', '1404.4238-4-45-0': 'At this point, we mention that the anti-screening model has also been employed to understand the observed non-hydrogenic series in TMDs based on an analysis in terms of optically bright [MATH]-states, using the anti-screening length as fit parameter[CITATION].', '1404.4238-4-45-1': 'For WS[MATH] on SiO[MATH], reasonable agreement with the experimental observations could be obtained using [MATH] , which is much larger than the physical thickness.', '1404.4238-4-46-0': 'In the ultrarelativistic regime, [MATH], [MATH] and the dielectric function [MATH] reduces to a constant, as in graphene.', '1404.4238-4-46-1': 'For large [MATH]-values, intrinsic screening therefore may play a significant role and eventually prevent the phase transition into the excitonic state.', '1404.4238-4-47-0': 'Nevertheless, the weakly bound states are not affected by this screening, nor by other deviations of the full electron spectrum beyond the quadatic approximation.', '1404.4238-4-47-1': 'Thus, the excited states still have [MATH].', '1404.4238-4-47-2': 'Hence, assuming a noninteracting groundstate and a nominal value of [MATH], the excited exciton states should be dark and the only bright resonance in a linear optical experiment should correspond to the [MATH] exciton.', '1404.4238-4-47-3': 'Since the experiments do not observe this feature but rather show a whole series of excitonic resonances, this serves as a clear indication that the groundstate is indeed excitonic and the bright exciton resonances correspond to [MATH]-type wave functions with [MATH].', '1404.4238-4-48-0': '# Conclusion', '1404.4238-4-49-0': 'In conclusion, we presented a general theory which assigns a [MATH]-like symmetry to the wave functions of the optically active excitonic transitions in recently investigated TMDs.', '1404.4238-4-49-1': 'This theory is supported by recently published experimental data, including the observation of a series of resonances in the linear reflectrum spectrum of WS[MATH][CITATION], two photon absorption in WS[MATH][CITATION] and WSe[MATH][CITATION].', '1404.4238-4-49-2': 'This observation of optically active [MATH]-transitions is not only of crucial importance for the correct interpretation of experimental data, but impressively demonstrates that in the TMDs investigated here, the Coulomb interaction is strong enough to induce an excitonic ground state.', '1404.4238-4-49-3': 'The dipole allowed excited states of such an excitonic insulator correspond to intra-excitonic transitions and are governed by the corresponding optical selection rules.', '1404.4238-4-49-4': 'Even for systems in the weakly interacting regime without an excitonic ground state, the lowest excitonic transition should be in the THz or infrared regime and the observed optical resonances correspond to excited exciton levels with main quantum number [MATH].'}

astro-ph-9704018

{'astro-ph-9704018-1-0-0': 'We calculate the flux and spectrum of [MATH]-rays emitted by a two-temperature advection-dominated accretion flow (ADAF) around a black hole.', 'astro-ph-9704018-1-0-1': 'The [MATH]-rays are from the decay of neutral pions produced through proton-proton collisions.', 'astro-ph-9704018-1-0-2': 'We discuss both thermal and power-law distributions of proton energies and show that the [MATH]-ray spectra in the two cases are very different.', 'astro-ph-9704018-1-0-3': 'We apply the calculations to the [MATH]-ray source, 2EG J1746-2852, detected by EGRET from the direction of the Galactic Center.', 'astro-ph-9704018-1-0-4': 'We show that the flux and spectrum of this source are consistent with emission from an ADAF around the supermassive accreting black hole Sgr A[MATH] if the proton distribution is a power-law.', 'astro-ph-9704018-1-0-5': 'The model uses accretion parameters within the range made likely by other considerations.', 'astro-ph-9704018-1-0-6': 'If this model is correct, it provides evidence for the presence of a two temperature plasma in Sgr A[MATH], and predicts [MATH]-ray fluxes from other accreting black holes which could be observed with more sensitive detectors.', 'astro-ph-9704018-1-1-0': '# Introduction', 'astro-ph-9704018-1-2-0': 'Advection-dominated accretion flows (ADAFs) are optically thin hot accretion flows with low radiative efficiency (Narayan Yi 1994, 1995 a,b; Abramowicz et al. 1995).', 'astro-ph-9704018-1-2-1': 'Unlike standard thin disks (see Frank et al. 1992) in which the viscously generated energy is thermalized and radiated locally, ADAFs store most of the viscous energy and advect it into the central star.', 'astro-ph-9704018-1-2-2': 'The gas in ADAFs has a two temperature structure (Shapiro, Lightman, Eardley 1976; Rees et al. 1982), with the ions being hotter than the electrons.', 'astro-ph-9704018-1-2-3': 'The viscous heating affects mainly the ions, the more massive species, while the radiation is produced primarily by the electrons.', 'astro-ph-9704018-1-2-4': 'Since the ions transfer only a small fraction of their energy to the electrons via Coulomb scattering, the energy which is radiated is much less than the total energy released during accretion (Rees et al. 1982).', 'astro-ph-9704018-1-3-0': 'The emission spectrum of an ADAF is mainly determined by the cooling processes of the electrons, viz. synchrotron, bremsstrahlung, and Compton processes (see eg.', 'astro-ph-9704018-1-3-1': 'Narayan Yi 1995b; Mahadevan 1997).', 'astro-ph-9704018-1-3-2': 'Detailed calculations (e.g. Narayan 1996) show that ADAFs around accreting black holes have a characteristic spectrum ranging from radio frequencies [MATH] Hz up to hard X-ray frequencies [MATH] Hz, whose shape is a function primarily of the mass accretion rate, and to some extent the mass of the accreting star.', 'astro-ph-9704018-1-3-3': 'A number of accreting black hole systems have been shown to contain ADAFs (e.g. Narayan, Yi, Mahadevan 1995; Narayan, McClintock, Yi 1996; Lasota et al. 1996, Fabian Rees 1995, Mahadevan 1997, Narayan, Barret, McClintock 1997, Reynolds et al. 1997).', 'astro-ph-9704018-1-4-0': 'In the work done so far, on ADAFs, only the cooling of the electrons has been considered in calculating the spectra.', 'astro-ph-9704018-1-4-1': 'However, since ADAFs are two temperature plasmas with very high ion temperatures, e.g. [MATH] K close to a black hole (Narayan Yi 1995b), one wonders if there might be radiative processes associated directly with the ions.', 'astro-ph-9704018-1-4-2': 'In particular, at such high temperatures, collisions between protons can lead to substantial production of neutral pions, [MATH], which would decay into high energy [MATH]MeV [MATH]-rays.', 'astro-ph-9704018-1-4-3': 'Charged pions can also be produced by proton-proton collisions, but the final decay products are neutrinos and electrons (and their anti-particles), so that most of the energy leaves the plasma either in neutrinos or lower energy photons.', 'astro-ph-9704018-1-5-0': 'Gamma-ray emission from a two temperature gas accreting onto a black hole has been computed previously (see e.g. Dahlbacka, Chapline, Weaver 1974; Colpi, Maraschi, Treves 1986, Berezinsky Dokuchaev 1990).', 'astro-ph-9704018-1-5-1': 'Most of the previous work considered a thermal distribution of protons, and the predicted [MATH]-ray fluxes depend sensitively on the particular accretion scenario being considered.', 'astro-ph-9704018-1-5-2': 'In this paper we consider both thermal and non-thermal distributions of protons and focus specifically on the ADAF paradigm.', 'astro-ph-9704018-1-5-3': 'Using the methods described by Dermer (1986ab) we calculate the flux and spectrum of [MATH]-ray emission from ADAFs and show that these flows produce interesting levels of high energy [MATH]-rays.', 'astro-ph-9704018-1-5-4': 'In the case of the source Sagittarius A[MATH] (Sgr A[MATH]) at the Galactic Center, we predict a [MATH]-ray flux which is above the sensitivity limit of the EGRET detector on the Compton Gamma-Ray Observatory, and we compare the prediction with a possible detection of this source.', 'astro-ph-9704018-1-6-0': 'The outline of the paper is as follows.', 'astro-ph-9704018-1-6-1': 'In 2 we present the reaction rate for pion production and describe the structure and basic equations of ADAFs.', 'astro-ph-9704018-1-6-2': 'In 3 we calculate the [MATH]-ray spectra from thermal and non-thermal distributions of protons in ADAFs, and in 4 we apply these results to EGRET observations of the Galactic Center.', 'astro-ph-9704018-1-6-3': 'In 5 we discuss other applications of the results.', 'astro-ph-9704018-1-7-0': '# General Relations', 'astro-ph-9704018-1-8-0': '## Pion Production', 'astro-ph-9704018-1-9-0': 'The production of [MATH]-rays from [MATH] collisions is a two step process, and has been worked out in detail by Stecker (1971, see also Stephens Badhwar 1981; Dermer 1986ab).', 'astro-ph-9704018-1-9-1': 'The colliding protons produce an intermediate particle, a neutral pion, [MATH], which then decays into two [MATH]-ray photons.', 'astro-ph-9704018-1-9-2': 'The reaction is [EQUATION] where the photons, [MATH], in general have different energies in the gas frame.', 'astro-ph-9704018-1-9-3': 'The number of neutral pions produced is calculated by considering two protons with four momenta [MATH] and [MATH], moving towards each other with relative velocity [MATH].', 'astro-ph-9704018-1-9-4': 'The number of collisions that occur in a volume [MATH], for a time [MATH], is a frame invariant quantity, which in an arbitrary reference frame can be written as (see Landau Lifshitz 1975, 12) [EQUATION]', 'astro-ph-9704018-1-9-5': 'Here, [MATH], [MATH] are the number densities of the protons, [MATH] are their energies, [MATH] is the total cross-section for pion production, and [MATH] are the respective velocity parameters of the two particles.', 'astro-ph-9704018-1-9-6': 'For a distribution of particle velocities, the total number of pions produced per second is just the integral of [MATH] over volume and velocities, which in spherical co-ordinates is [EQUATION] where [MATH] are the respective Lorentz factors of the two protons, [MATH], and [MATH] is the radius with respect to the accreting star.', 'astro-ph-9704018-1-9-7': 'The total number of [MATH]-rays produced is [MATH].', 'astro-ph-9704018-1-10-0': 'To obtain the [MATH]-ray spectrum, we first calculate the spectrum of pion energies, [MATH], in the frame of the observer.', 'astro-ph-9704018-1-10-1': 'The observed [MATH]-ray spectrum is then obtained through the relation (Stecker 1971) [EQUATION] where [MATH] GeV[MATH] is the mass of the pion, and [MATH] is the minimum pion energy required to produce a [MATH]-ray with energy [MATH], [EQUATION]', 'astro-ph-9704018-1-10-2': 'The pion spectrum, [MATH] is obtained by substituting the differential cross-section [MATH] for [MATH] in equation ([REF]).', 'astro-ph-9704018-1-10-3': 'Two models are generally used to determine the differential cross-section for pion production: the isobar model and the scaling model.', 'astro-ph-9704018-1-10-4': 'In the isobar model, pions are produced by a two step process.', 'astro-ph-9704018-1-10-5': 'Colliding protons produce an isobar, with rest mass 1.238 GeV, which subsequently decays into a proton and a pion (Lindenbaum Sternheimer 1957; Stecker 1971).', 'astro-ph-9704018-1-10-6': 'Dermer (1986b) has shown that the isobar model agrees with the experimental data quite well for proton energies near threshold: [MATH] GeV.', 'astro-ph-9704018-1-10-7': 'On the other hand, for proton energies [MATH] GeV, the scaling model of Stephens and Badhwar (1981) represents the experimental data better (Dermer 1986b).', 'astro-ph-9704018-1-10-8': 'In our calculations, we therefore use the two models in their respective ranges of validity and interpolate smoothly (cubic interpolation) in the intermediate regime.', 'astro-ph-9704018-1-11-0': '## ADAF Equations', 'astro-ph-9704018-1-12-0': 'To evaluate equations ([REF]) and ([REF]) we need the number density of protons as a function of energy in the accretion flow, [MATH].', 'astro-ph-9704018-1-12-1': 'We write [MATH] as the product of a normalized velocity distribution, which depends on the temperature at a given radius, and the total number density of protons, [EQUATION] with [EQUATION]', 'astro-ph-9704018-1-12-2': 'Here, [MATH] is the dimensionless ion temperature, [MATH], at radius [MATH].', 'astro-ph-9704018-1-13-0': 'An ADAF is characterized by four parameters: the viscosity parameter, [MATH] (Shakura Sunyaev 1973), the ratio of gas pressure to total pressure, [MATH], the mass of the central black hole, [MATH], and the mass accretion rate, [MATH].', 'astro-ph-9704018-1-13-1': 'Given these parameters, the global structure and dynamics of ADAFs can be calculated (Narayan, Kato, Honma 1997; Chen, Abramowicz, Lasota 1997).', 'astro-ph-9704018-1-13-2': 'Generally, ADAFs appear to have large values of [MATH] (Narayan 1996), and the uncertainty in the value is a factor of a few.', 'astro-ph-9704018-1-13-3': 'Many published ADAF models in the literature use [MATH] (e.g. Narayan et al. 1996, 1997).', 'astro-ph-9704018-1-13-4': 'If there is equipartition between magnetic and gas pressure, we expect [MATH]; this is the value we favor.', 'astro-ph-9704018-1-13-5': 'However, for completeness we also consider the more extreme case [MATH] = 0.95, where the magnetic pressure contributes negligibly ([MATH]) to the total pressure.', 'astro-ph-9704018-1-14-0': 'The radial structure of ADAFs is well approximated by a series of concentric spherical shells, with the properties of the gas varying as a function of radius (Narayan Yi 1995a).', 'astro-ph-9704018-1-14-1': 'The continuity equation gives [MATH], where [MATH] is the radial velocity of the gas, and [MATH] is the mass density.', 'astro-ph-9704018-1-14-2': 'Writing the radius in units of the Schwarzschild radius, [MATH], where [MATH] cm and [MATH] is the mass of the central star in solar mass units, and assuming that the mass fraction of hydrogen is [MATH], we have [EQUATION]', 'astro-ph-9704018-1-14-3': 'Here [MATH] is the accretion rate in Eddington units, with [MATH] g s[MATH], and [MATH] is defined so as to scale out the dependence of [MATH] on [MATH] and [MATH].', 'astro-ph-9704018-1-14-4': 'Given [MATH], [MATH], [MATH] and [MATH], equation ([REF]) allows us to calculate [MATH] provided we have [MATH].', 'astro-ph-9704018-1-14-5': 'We obtain [MATH] from the global ADAF solution of Narayan et al. (1997).', 'astro-ph-9704018-1-15-0': 'The ion temperature [MATH] is obtained from the total gas pressure [MATH] (Narayan Yi 1995b), [EQUATION] where [MATH] is the isothermal sound speed, and [MATH] are the effective molecular weights of the ions and electrons respectively.', 'astro-ph-9704018-1-15-1': 'Since [MATH] we neglect the second term to obtain [EQUATION] where [MATH] is the velocity of light.', 'astro-ph-9704018-1-15-2': 'We obtain [MATH] again from the global solution of the ADAF (Narayan et al. 1997).', 'astro-ph-9704018-1-16-0': 'Given [MATH] and [MATH] and an assumed functional form for the proton energy distribution [MATH], we can substitute equation ([REF]) in equations ([REF]) and ([REF]) to calculate the [MATH]-ray spectrum.', 'astro-ph-9704018-1-17-0': '# Results', 'astro-ph-9704018-1-18-0': 'At present our understanding of viscous heating is poor, and we do not know whether the process leads to a thermal or power-law distribution of proton energies, or perhaps some combination of the two.', 'astro-ph-9704018-1-18-1': 'In this section we calculate the pion spectra by considering two different proton energy distributions: a relativistic Maxwell-Boltzmann distribution and a power-law distribution.', 'astro-ph-9704018-1-19-0': '## Thermal Distribution', 'astro-ph-9704018-1-20-0': 'The proton temperature in an ADAF close to a black hole is marginally relativistic, [MATH].', 'astro-ph-9704018-1-20-1': 'At such temperatures, pions are produced primarily by protons in the tail of the Maxwell-Boltzmann distribution, since only these particles have energies above the threshold needed for pion production.', 'astro-ph-9704018-1-20-2': 'Thus, the pion production is very sensitive to the proton temperature, and since [MATH] scales approximately as [MATH] (Narayan Yi 1995b; Mahadevan 1997), most of the production is from the innermost radii of the ADAF, [MATH].', 'astro-ph-9704018-1-21-0': 'The protons at these temperatures have momenta near the threshold for pion production, and do not exceed [MATH] 1 GeV/[MATH].', 'astro-ph-9704018-1-21-1': 'At such energies the isobar model (Lindenbaum Sternheimer 1957; Stecker 1971; Dermer 1986a) is more accurate and we have therefore used it to calculate the cross-sections and energy spectra.', 'astro-ph-9704018-1-21-2': 'In this model, the isobars move along the initial directions of the colliding protons in the center of momentum (CM) system.', 'astro-ph-9704018-1-21-3': 'It is therefore convenient to transform variables in equation ([REF]) via a Lorentz boost to the CM reference frame, followed by a rotation so that the colliding protons are moving along the z-axis (Dermer 1984, 1986a).', 'astro-ph-9704018-1-21-4': 'Using equation ([REF]) with equation ([REF]), and rewriting in terms of the new variables (see Dermer 1984, 1986a for details), we obtain [EQUATION]', 'astro-ph-9704018-1-21-5': "Here [MATH] is the pion energy in the observer's frame, [MATH] is the relative Lorentz factor of the two colliding protons, [MATH], [MATH] is the differential cross-section for the production of a pion with Lorentz factor [MATH] in the CM frame for a given [MATH], [MATH] where [MATH] characterizes the strength of the collision, [MATH] is the modified Bessel function of order 2, and we have used a Maxwell-Boltzmann energy distribution for the protons.", 'astro-ph-9704018-1-21-6': 'We take [MATH] to be [MATH]; however, since the proton temperature decreases steeply with radius ([MATH]) the contribution to the pion spectrum from the gas at [MATH] is negligible.', 'astro-ph-9704018-1-22-0': 'The calculation in equation ([REF]) gives the pion spectrum, [MATH].', 'astro-ph-9704018-1-22-1': 'Using equation ([REF]), we then obtain the [MATH]-ray spectrum.', 'astro-ph-9704018-1-22-2': 'The results are shown in Fig. 1.', 'astro-ph-9704018-1-22-3': 'The plot corresponds to two values of the viscosity parameter [MATH], 0.1 and 0.3, and two values of [MATH], 0.5 and 0.95.', 'astro-ph-9704018-1-22-4': 'For a given [MATH], increasing [MATH] leads to increasing gas pressure, and therefore to increasing ion temperature (see eq. [REF]).', 'astro-ph-9704018-1-22-5': 'Changing [MATH] from 0.5 to 0.95 causes the temperature to go up by a factor [MATH], and since the pion production is extremely sensitive to temperature, the [MATH]-ray luminosity increases by nearly two orders in magnitude.', 'astro-ph-9704018-1-22-6': 'Changes in [MATH] also affect the [MATH]-ray luminosity, though less sensitively.', 'astro-ph-9704018-1-22-7': 'Since the total luminosity is proportional to [MATH] which varies roughly as [MATH] (Narayan Yi 1995b), we might expect the luminosity to increase with decreasing [MATH].', 'astro-ph-9704018-1-22-8': 'This effect is counterbalanced however by small changes in the temperature, and the luminosity in fact decreases slightly when [MATH] goes down from [MATH] to [MATH].', 'astro-ph-9704018-1-23-0': '## Non-Thermal Distribution', 'astro-ph-9704018-1-24-0': 'In this section we consider a power-law distribution of proton energies described by a spectral index [MATH].', 'astro-ph-9704018-1-24-1': 'Equation ([REF]) continues to hold, but [MATH] is no longer a Maxwell-Boltzmann distribution.', 'astro-ph-9704018-1-24-2': 'To determine the form of [MATH], two requirements have to be satisfied (see eq. [REF]), namely the normalization condition and the requirement that the average kinetic energy of the power-law distribution of protons, at each radius, be equal to the average energy of the protons as given by the local ion temperature.', 'astro-ph-9704018-1-25-0': 'We model the energy distribution of protons as [EQUATION] where a fraction [MATH] of the protons have [MATH], and a fraction [MATH] are in a power-law tail with index [MATH].', 'astro-ph-9704018-1-25-1': 'The distribution is properly normalized.', 'astro-ph-9704018-1-25-2': 'The fraction [MATH] is fixed by the energy requirement: [EQUATION]', 'astro-ph-9704018-1-25-3': 'Since the exact energy distribution for the fraction [MATH] of protons with energy below threshold is not important for the present calculation, we have simplified the distribution to a [MATH] function in equation ([REF]).', 'astro-ph-9704018-1-26-0': 'To calculate the total pion luminosity, [MATH], we substitute equation ([REF]) in equation ([REF]).', 'astro-ph-9704018-1-26-1': 'We then obtain three different terms proportional to [MATH], [MATH], and [MATH], respectively.', 'astro-ph-9704018-1-26-2': 'The [MATH] term gives no contribution since it corresponds to protons with [MATH] colliding with one another.', 'astro-ph-9704018-1-26-3': 'These protons do not have enough energy to produce pions.', 'astro-ph-9704018-1-26-4': 'The [MATH] term corresponds to protons with [MATH] colliding with protons with [MATH], and we denote the contribution by [MATH].', 'astro-ph-9704018-1-26-5': 'Similarly, the [MATH] term corresponds to protons with [MATH] colliding with protons with [MATH], and we denote it as [MATH].', 'astro-ph-9704018-1-26-6': 'We have [EQUATION] and [EQUATION] where [MATH] is the cross-section in units of millibarns, and [EQUATION]', 'astro-ph-9704018-1-26-7': 'In an ADAF, we generally have [MATH] so that most of the protons have [MATH] (cf. eq. [REF]).', 'astro-ph-9704018-1-26-8': 'We therefore expect most of the pion production and [MATH]-ray luminosity to come from [MATH].', 'astro-ph-9704018-1-26-9': 'Table [REF] gives numerical results and we see that [MATH] is indeed negligible compared with [MATH].', 'astro-ph-9704018-1-27-0': 'In calculating the pion spectrum, we consider only the contribution from [MATH] and write [EQUATION]', 'astro-ph-9704018-1-27-1': 'Unlike in the case of a thermal distribution where most of the protons have energies either below or just above threshold, in a power-law distribution the protons have a wide range of energy and some protons are well above threshold.', 'astro-ph-9704018-1-27-2': 'We therefore use both the isobar and scaling models, as described in [REF]', 'astro-ph-9704018-1-28-0': 'Equation ([REF]) reveals the dependence of the pion flux and energy spectrum on the parameters: (1) The flux is proportional to [MATH].', 'astro-ph-9704018-1-28-1': '(2) The flux depends on [MATH], [MATH] and [MATH] through the function [MATH] defined in equation ([REF]).', 'astro-ph-9704018-1-28-2': '(3) The shape of the spectrum depends only on [MATH] and is given by the integral in equation ([REF]).', 'astro-ph-9704018-1-29-0': 'Tables [REF] and [REF] give [MATH] for different values of its parameters.', 'astro-ph-9704018-1-29-1': 'For fixed [MATH], increasing [MATH] leads to increasing [MATH] since the temperature in the flow increases.', 'astro-ph-9704018-1-29-2': 'However, unlike the thermal case, the luminosity is not excessively sensitive to [MATH] since the number of protons above threshold is directly proportional to [MATH] (cf. eq. [REF]); therefore, a change in [MATH] by a factor of two changes the total luminosity only by [MATH].', 'astro-ph-9704018-1-29-3': 'For fixed [MATH] in a self-similar solution (Narayan Yi 1994, 1995b), the density varies in proportion to [MATH], so the luminosity scales as [MATH].', 'astro-ph-9704018-1-29-4': 'However, in the more accurate global solutions employed in this paper, the dependence of density-and hence luminosity-on [MATH] is not strictly a power-law, and the overall strength of the dependence is somewhat weaker than in the self-similar solutions.', 'astro-ph-9704018-1-29-5': 'To illustrate this dependence, we list the numerical values of [MATH] for three different choices of [MATH] in Tables [REF] and [REF].', 'astro-ph-9704018-1-30-0': 'Figure 2 shows [MATH]-ray spectra for different values of the proton energy spectral index [MATH] (we fix [MATH], [MATH]).', 'astro-ph-9704018-1-30-1': 'The [MATH]-ray spectral index at high energies [MATH] GeV is the same as the energy spectral index of the protons [MATH], as shown by Dermer (1986b).', 'astro-ph-9704018-1-30-2': 'For this reason, we investigated values of [MATH] in the range 2.3 - 3.3.', 'astro-ph-9704018-1-30-3': 'The spectrum turns over at [MATH] MeV (approximately half the pion rest mass) and falls for lower photon energies.', 'astro-ph-9704018-1-30-4': 'Comparing Figures 1 and 2, we see that the total [MATH]-ray luminosity from a power-law energy distribution of protons is comparable to that obtained from a thermal distribution when [MATH], but is very much higher for [MATH].', 'astro-ph-9704018-1-30-5': 'In our view [MATH] is the natural choice since it corresponds to equipartition between gas and magnetic pressure.', 'astro-ph-9704018-1-30-6': 'The [MATH]-ray spectrum extends to much higher photon energies when the protons have a power-law distribution compared to the thermal case.', 'astro-ph-9704018-1-31-0': '# Application to Sgr A[MATH]', 'astro-ph-9704018-1-32-0': 'The black hole candidate at the center of our Galaxy, Sagittarius A[MATH] (Sgr A[MATH]), appears to be a scaled down version of an AGN.', 'astro-ph-9704018-1-32-1': 'The source is believed to consist of a black hole of mass [MATH] (Eckart Genzel 1996a, 1996b) accreting gas from its surroundings.', 'astro-ph-9704018-1-33-0': 'The mass accretion rate in Sgr A[MATH] has been estimated by various methods (see Genzel et al. 1994 for a summary).', 'astro-ph-9704018-1-33-1': 'The accretion of wind gas from neighboring stars (principally IRS 16) seems to be the most likely scenario.', 'astro-ph-9704018-1-33-2': 'The accretion rate is determined by considering the fraction of the wind that comes within the accretion radius of the central black hole, but the estimate depends sensitively on the assumed wind velocity.', 'astro-ph-9704018-1-33-3': 'Genzel et al. (1994) obtain an accretion rate of [MATH] yr[MATH] which corresponds to [MATH] for [MATH], assuming a wind velocity of 1000 km s[MATH] and mass loss rate from winds of [MATH] yr[MATH].', 'astro-ph-9704018-1-33-4': 'Melia (1992) uses a wind velocity of [MATH] km s[MATH] and obtains an accretion rate of [MATH] yr[MATH]).', 'astro-ph-9704018-1-33-5': 'The true accretion rate probably lies between these two estimates, and we will require our models to have [MATH].', 'astro-ph-9704018-1-33-6': 'Note that Lacy et al. (1982; see also Rees 1982) estimated a much higher mass accretion rate of [MATH] yr[MATH] from stellar disruptions at the Galactic Center, but argued that the accretion due to this probably has a low duty cycle.', 'astro-ph-9704018-1-34-0': 'Sgr A[MATH] has an extremely low luminosity, [MATH], which corresponds to [MATH] if the accretion flow has a standard radiative efficiency of 10%.', 'astro-ph-9704018-1-34-1': 'Such a low [MATH] is in serious conflict with the estimate of [MATH] given in the previous paragraph.', 'astro-ph-9704018-1-34-2': 'Narayan, Yi Mahadevan (1995) suggested that Sgr A[MATH] does accrete at a rate near the one estimated, but in an advection-dominated mode.', 'astro-ph-9704018-1-34-3': 'Using a self-similar ADAF model, they were able to reconcile the low luminosity of the source with a relatively large [MATH]: [MATH].', 'astro-ph-9704018-1-34-4': 'If [MATH], the accretion rate inferred on the basis of the ADAF model would be in agreement with that estimated on the basis of gas supply.', 'astro-ph-9704018-1-34-5': 'Furthermore, they obtained a reasonable fit to the observed spectrum from radio to hard X-ray frequencies.', 'astro-ph-9704018-1-34-6': 'In this section, we calculate the [MATH]-ray luminosity and spectrum of Sgr A[MATH] due to pion production, and compare these predictions with the flux seen by EGRET from the direction of the Galactic Center.', 'astro-ph-9704018-1-35-0': 'Among several unidentified sources detected by EGRET in the Galactic plane (Merck et al. 1996), the source 2EG 1746-2852 is of particular interest since it is spatially coincident with the Galactic Center.', 'astro-ph-9704018-1-35-1': 'There is no firm identification of this source, but it is point-like to within the resolution of the instrument ([MATH]), it is [MATH] above the local diffuse emission, and its spectrum differs significantly from the spectra of other unidentified EGRET sources.', 'astro-ph-9704018-1-35-2': 'Out of the 32 sources whose spectra are reported by Merck et al. (1996), 27 have spectral slopes [MATH] (photon index), 4 have spectral slopes [MATH] and one has a very hard spectrum with a spectral slope of 1.7.', 'astro-ph-9704018-1-35-3': 'This last source is 2EG 1746-2852, and it is located exactly at the Galactic Center.', 'astro-ph-9704018-1-35-4': 'Following Merck et al. (1996), we make the reasonable assumption that 2EG 1746-2852 corresponds to Sgr A[MATH].', 'astro-ph-9704018-1-35-5': 'Figure 3 shows the spectrum of the source as measured by EGRET.', 'astro-ph-9704018-1-35-6': 'The dashed line is the best-fit power-law obtained by Merck et al. (1996) over the photon energy range 100-4000 MeV.', 'astro-ph-9704018-1-35-7': 'Their fitted spectral slope is [MATH].', 'astro-ph-9704018-1-35-8': 'At higher energies, the data suggest a roll-over in the spectrum which would give a softer spectral index at higher energies [MATH] GeV).', 'astro-ph-9704018-1-36-0': '## Thermal Distribution', 'astro-ph-9704018-1-37-0': 'Figure 3 compares the [MATH]-ray spectrum from the thermal model with the EGRET data.', 'astro-ph-9704018-1-37-1': 'The solid line corresponds to [MATH], [MATH], and the dotted line to [MATH], [MATH].', 'astro-ph-9704018-1-37-2': 'The black hole mass is taken to be [MATH], and the accretion rates in the two models have been varied such that the predicted spectra agree with the data point at [MATH]MeV; the accretion rates are indicated on the plot.', 'astro-ph-9704018-1-38-0': 'It is clear that the thermal model predicts a spectrum with the wrong shape and fails to explain the EGRET detection at [MATH] GeV.', 'astro-ph-9704018-1-38-1': 'Moreover, for the preferred value of [MATH], the model requires a rather high [MATH], which is much larger than the range we consider reasonable (see above).', 'astro-ph-9704018-1-38-2': 'We conclude that the thermal model is inconsistent with the observations.', 'astro-ph-9704018-1-39-0': '## Non-Thermal Distribution', 'astro-ph-9704018-1-40-0': 'Fig. 4 compares [MATH]-ray spectra from various power-law models ([MATH]) with the EGRET data.', 'astro-ph-9704018-1-40-1': 'The different curves correspond to different values of the proton energy index [MATH].', 'astro-ph-9704018-1-40-2': 'For each [MATH], we have varied the accretion rate so as to obatin the closest agreement with the observed spectrum.', 'astro-ph-9704018-1-40-3': 'The results ([MATH]), which are only weakly dependent on [MATH], are given in Table [REF].', 'astro-ph-9704018-1-40-4': 'The accretion rates so inferred fall well within the range of accretion rates estimated on the basis of the properties of nearby gas ([MATH], Genzel et al. 1994), and are within factors of a few of the accretion rate estimated on the basis of fitting the radio to X-ray spectrum with the somewhat less accurate self-similar version of the ADAF model ([MATH], Narayan, Yi, Mahadevan 1995).', 'astro-ph-9704018-1-41-0': 'Reproducing the power-law portion of the [MATH]-ray spectrum is a by-product of postulating a power-law proton energy distribution.', 'astro-ph-9704018-1-41-1': 'However, it is striking in this context that the best-fit slope, [MATH], is identical to the slope of the low-energy cosmic ray distribution.', 'astro-ph-9704018-1-41-2': 'The models predict roll-overs at both the high energy (few GeV) and low energy ([MATH]MeV) ends of the spectrum.', 'astro-ph-9704018-1-41-3': 'At the high-energy end there is just a hint of such a roll-over in the data, but at the low-energy end, the evidence seems somewhat stronger.', 'astro-ph-9704018-1-42-0': 'The similarity of the best-fit spectral shape to the diffuse Galactic [MATH]-ray emission has two possible interpretations.', 'astro-ph-9704018-1-42-1': 'First, since the diffuse emission is generally interpreted as being due to cosmic ray protons striking thermal protons in the interstellar medium, a mechanism similar to what occurs in an ADAF, the EGRET detection of 2EG 1746-2852 could be interpreted as cosmic ray protons colliding with a compact dense cloud of gas that is spatially coincident with the Galactic Center.', 'astro-ph-9704018-1-42-2': 'This interpretation would require such a cloud to have a considerably greater gas density (and possibly also cosmic ray density) than in the interstellar medium only slightly farther away from the Galactic Center.', 'astro-ph-9704018-1-42-3': 'If virtually all the Galactic Center region [MATH]-ray flux were due to such a cloud, the power-law proton distribution function version of the ADAF model would be put seriously in doubt.', 'astro-ph-9704018-1-42-4': 'Thermal proton distribution function versions of the ADAF model (or entirely different models) would not be seriously constrained if the [MATH]-rays coming from the Galactic Center prove not to have their source in Sgr A[MATH].', 'astro-ph-9704018-1-43-0': 'The alternative is that the observed [MATH]-ray spectrum is in fact due to Sgr A[MATH].', 'astro-ph-9704018-1-43-1': 'In this case, not only is the deduced accretion rate roughly consistent with previous estimates, but we also have the interesting result that whatever process determines the cosmic proton energy distribution (shock acceleration ?)', 'astro-ph-9704018-1-43-2': 'is also at work in ADAFs.', 'astro-ph-9704018-1-43-3': 'We find the numerical coincidence between the prediction of the power-law version of the ADAF model and the observed [MATH]-ray flux striking enough to justify pursuing this interpretation.', 'astro-ph-9704018-1-44-0': 'To gauge just how strong the quantitative agreement is, we must consider how much freedom the model is given by adjustable parameters.', 'astro-ph-9704018-1-44-1': 'Three free parameters are significant.', 'astro-ph-9704018-1-44-2': 'One ([MATH]) is fixed to within an order of magnitude or so by other considerations (the amount of gas available to accrete, and fitting the radio-X-ray spectrum).', 'astro-ph-9704018-1-44-3': 'Another ([MATH]), while formally unconstrained over a range of several orders of magnitude, is often supposed to have a value quite near the one which we infer (i.e. between 0.1 and 1, cf. Narayan 1996).', 'astro-ph-9704018-1-44-4': 'Only the third ([MATH]) is chosen almost entirely on the basis of fitting the observed [MATH]-ray spectrum.', 'astro-ph-9704018-1-44-5': 'Because the [MATH]-ray luminosity predicted by the ADAF model (assuming a power-law proton energy distribution) approximately scales with [MATH] (the actual scaling with [MATH] is only very roughly described by a power-law), the combined a priori uncertainty in [MATH] and [MATH] may be regarded as giving the [MATH]-ray luminosity predicted by the model a possible range of [MATH].', 'astro-ph-9704018-1-45-0': '# Discussion Conclusions', 'astro-ph-9704018-1-46-0': 'Since it is not understood whether viscous heating produces a thermal or power-law distribution of proton energies, we have calculated the [MATH]-ray emission of ADAFs corresponding to both types of distributions.', 'astro-ph-9704018-1-46-1': 'We expect that the true energy distribution in any given source will be bracketed by these two extremes.', 'astro-ph-9704018-1-46-2': 'Spectra corresponding to intermediate models can be easily calculated by taking a weighted sum of the thermal and power-law models.', 'astro-ph-9704018-1-47-0': 'If the proton distribution is thermal, the spectrum has a characteristic shape with a peak at [MATH] MeV and very little emission at either lower or higher photon energies.', 'astro-ph-9704018-1-47-1': 'For the kinds of proton temperatures expected in an ADAF, only a small fraction of the protons (in the tail of the Maxwellian distribution) have sufficient energy to produce pions.', 'astro-ph-9704018-1-47-2': 'Consequently, even a minor change in the temperature, say by a factor of [MATH], can modify the [MATH]-ray luminosity by orders of magnitude (cf. Figure 1).', 'astro-ph-9704018-1-47-3': 'Since the flux is very sensitive to the gas temperature, a detailed understanding of the physics of the ADAF in the region [MATH] (where most of the [MATH]-ray luminosity originates) is necessary in order to make testable predictions.', 'astro-ph-9704018-1-47-4': 'The present study does not have the necessary accuracy for this.', 'astro-ph-9704018-1-47-5': 'General relativistic effects become important at these radii and need to be included consistently.', 'astro-ph-9704018-1-47-6': 'In principle, if these effects are included, the [MATH]-ray spectrum could be used as a sensitive probe of the proton temperature.', 'astro-ph-9704018-1-47-7': 'One might even hope to distinguish between rotating and non-rotating black holes, since the physical properties of the flow at [MATH] will be different for the two cases.', 'astro-ph-9704018-1-48-0': 'If the protons have a power-law distribution of energies, the [MATH]-ray spectrum again peaks at [MATH] MeV, but there is significant emission at higher photon energies.', 'astro-ph-9704018-1-48-1': 'Indeed, the spectrum asymptotically has a spectral slope [MATH] which is equal to the power-law index [MATH] of the proton distribution (Dermer 1986b).', 'astro-ph-9704018-1-48-2': 'Thus, the detection of a power-law [MATH]-ray spectrum not only indicates the nonthermal nature of the protons but also helps determine the energy index.', 'astro-ph-9704018-1-49-0': 'In contrast to the thermal case, only half the [MATH]-ray luminosity due to a power-law distribution of proton energies originates from [MATH], while the other half is emitted from [MATH].', 'astro-ph-9704018-1-49-1': 'Further, the luminosity is not very sensitive to changes in the temperature.', 'astro-ph-9704018-1-49-2': 'Therefore, our present understanding of the physics of ADAFs is probably adequate for a reasonable estimate of the [MATH]-ray flux, and we can thus test this version of the ADAF paradigm usefully against observations.', 'astro-ph-9704018-1-50-0': 'The Galactic Center source, Sgr A[MATH] presents an excellent opportunity to test the model, since there is a good case for the presence of an ADAF in this source (Narayan et al. 1995) and the predicted [MATH]-ray flux is large enough to be detectable with current instruments.', 'astro-ph-9704018-1-50-1': 'Indeed, EGRET has detected an unresolved source in the Galactic Center region, 2EG J1746-2852 (Merck et al. 1996).', 'astro-ph-9704018-1-50-2': 'It is possible that this source corresponds to a dense compact gas cloud interacting with cosmic rays.', 'astro-ph-9704018-1-50-3': 'However, the source could equally well be associated with Sgr A[MATH].', 'astro-ph-9704018-1-50-4': 'The [MATH]-ray flux of Sgr A[MATH] which we calculate with our model using a power-law distribution of proton energies is in good agreement with the observed flux of 2EG J1746-2852 given parameters which are almost determined on independent grounds.', 'astro-ph-9704018-1-50-5': 'In order to fit the flux, we need an Eddington-scaled mass accretion rate of [MATH]-[MATH] (for [MATH]), close to the estimate [MATH] obtained by Narayan et al. (1995) from fitting the lower energy spectrum due to the electrons.', 'astro-ph-9704018-1-50-6': 'It is also compatible with the range of [MATH] quoted by Genzel et al. (1994) on the basis of gas motions in the vicinity of Sgr A[MATH]: [MATH].', 'astro-ph-9704018-1-50-7': 'In addition, the pion-decay model for the Galactic Center [MATH]-ray spectrum predicts a roll over at energies below [MATH] MeV that may be present in the observed spectrum.', 'astro-ph-9704018-1-51-0': 'An exciting aspect of the present study is that it provides for the first time a direct probe of the protons in hot accretion flows.', 'astro-ph-9704018-1-51-1': 'A fundamental assumption in ADAF models is that the plasma is two-temperature.', 'astro-ph-9704018-1-51-2': 'However, until now there has been no direct test of this assumption since all previous investigations of the emission from ADAFs dealt only with the electrons.', 'astro-ph-9704018-1-51-3': 'The [MATH]-ray emission we have considered in this paper is due entirely to the protons.', 'astro-ph-9704018-1-51-4': 'Moreover, it requires that the protons have nearly virial energies in order to be able to exceed the energy threshold for pion production.', 'astro-ph-9704018-1-51-5': 'The encouraging results we have obtained in the case of Sgr A[MATH] indicate that this source might have a two-temperature plasma exactly as postulated in the models.', 'astro-ph-9704018-1-51-6': 'More detailed study of Sgr A[MATH] coupled with future detections of other sources (see below), could strengthen the case for two-temperature accretion flows significantly.', 'astro-ph-9704018-1-52-0': 'Another exciting aspect of this study is that the [MATH]-ray spectrum from pion decay provides a direct window to the energy distribution of the protons.', 'astro-ph-9704018-1-52-1': 'In the ADAF paradigm, viscous heat energy goes almost entirely into the protons.', 'astro-ph-9704018-1-52-2': 'Furthermore, it is easy to show that, at least at the low [MATH] expected in quiescent systems, the protons have no energetically important interactions either among themselves or with the electrons.', 'astro-ph-9704018-1-52-3': 'Therefore, each proton retains memory of all the heating events it has undergone during the accretion, and so the energy distribution of the protons directly reflects the heating processes present in the plasma.', 'astro-ph-9704018-1-52-4': 'In principle, with sufficiently sensitive observations, one could use [MATH]-ray measurements as a direct probe of viscous heating.', 'astro-ph-9704018-1-52-5': 'This would be invaluable for the theory of hot accretion flows.', 'astro-ph-9704018-1-53-0': 'The ADAF model has been applied successfully to several other low-luminosity black holes in addition to Sgr A[MATH].', 'astro-ph-9704018-1-53-1': 'The sources studied include two X-ray binaries in quiescence, A0620-00 and V404 Cyg (Narayan, McClintock, Yi 1996, Narayan, Barret McClintock 1997), and several quiescent AGNs, viz. the LINER galaxy NGC 4258 (Lasota et al. 1996, Herrnstein et al. 1996) and the nearby elliptical galaxies NGC 4472, NGC 4486, NGC 4649, and NGC 4636 (Mahadevan 1997; Reynolds et al. 1997).', 'astro-ph-9704018-1-53-2': 'Table [REF] gives the values of [MATH], [MATH], [MATH] and [MATH] of several of these systems (taken from the references indicated) and presents the expected [MATH]-ray fluxes according to our model.', 'astro-ph-9704018-1-53-3': 'Among the nearby ellipticals, we have included only NGC4486 since it is the only system with a reliable mass estimate (Ford et al. 1995; Harms et al. 1995).', 'astro-ph-9704018-1-53-4': 'The [MATH]-ray flux estimates in Table [REF] assume that the protons have a power-law energy distribution with [MATH].', 'astro-ph-9704018-1-53-5': 'We see that, only in the case of Sgr A[MATH] does the detection threshold of EGRET ([MATH]) permit a test of the predictions, although some of the other sources might be not far below the EGRET threshold.', 'astro-ph-9704018-1-53-6': 'These sources are potentially detectable with future instruments such as the Gamma Ray Large Area Space Telescope (GLAST), which is designed to be 100 times more sensitive than EGRET.', 'astro-ph-9704018-1-54-0': 'Narayan (1996) has argued that X-ray binaries in the "hard state" (also called the "low state") may also contain ADAFs with [MATH].', 'astro-ph-9704018-1-54-1': 'If the proton energy distribution in these sources is a power-law of the sort we suggest exists in Sgr A[MATH], several of them might be expected to produce detectable [MATH]-ray fluxes.', 'astro-ph-9704018-1-54-2': 'However, the accretion rates in these objects (e.g. Cyg X-1) is likely to be close to the maximum rate permitting an ADAF (Narayan Yi 1995b).', 'astro-ph-9704018-1-54-3': 'If this is the case, particle interactions might be rapid enough that maintenance of a true power-law distribution function would be questionable.', 'astro-ph-9704018-1-54-4': 'Accurate predictions of the [MATH]-ray flux would then require a more elaborate calculation than we have performed for Sgr A[MATH].', 'astro-ph-9704018-1-55-0': 'Finally, we note that in determining the [MATH]-ray spectra, we have neglected gravitational redshift effects which become important at [MATH].', 'astro-ph-9704018-1-55-1': 'We have also neglected the Doppler blueshift associated with the large radial and orbital velocities of the gas at these radii (Narayan Yi 1995b).', 'astro-ph-9704018-1-55-2': 'Only a detailed calculation can tell which of the two effects predominates.', 'astro-ph-9704018-1-55-3': 'We note, however, that these effects are not likely to modify the results we have presented for a power-law distribution of protons because more than half the emission in this case occurs at [MATH].'}

{'astro-ph-9704018-2-0-0': 'We calculate the flux and spectrum of [MATH]-rays emitted by a two-temperature advection-dominated accretion flow (ADAF) around a black hole.', 'astro-ph-9704018-2-0-1': 'The [MATH]-rays are from the decay of neutral pions produced through proton-proton collisions.', 'astro-ph-9704018-2-0-2': 'We discuss both thermal and power-law distributions of proton energies and show that the [MATH]-ray spectra in the two cases are very different.', 'astro-ph-9704018-2-0-3': 'We apply the calculations to the [MATH]-ray source, 2EG J1746-2852, detected by EGRET from the direction of the Galactic Center.', 'astro-ph-9704018-2-0-4': 'We show that the flux and spectrum of this source are consistent with emission from an ADAF around the supermassive accreting black hole Sgr A[MATH] if the proton distribution is a power-law.', 'astro-ph-9704018-2-0-5': 'The model uses accretion parameters within the range made likely by other considerations.', 'astro-ph-9704018-2-0-6': 'If this model is correct, it provides evidence for the presence of a two temperature plasma in Sgr A[MATH], and predicts [MATH]-ray fluxes from other accreting black holes which could be observed with more sensitive detectors.', 'astro-ph-9704018-2-1-0': '# Introduction', 'astro-ph-9704018-2-2-0': 'Advection-dominated accretion flows (ADAFs) are optically thin hot accretion flows with low radiative efficiency (Narayan Yi 1994, 1995 a,b; Abramowicz et al. 1995).', 'astro-ph-9704018-2-2-1': 'Unlike standard thin disks (see Frank et al. 1992) in which the viscously generated energy is thermalized and radiated locally, ADAFs store most of the viscous energy and advect it into the central star.', 'astro-ph-9704018-2-2-2': 'The gas in ADAFs has a two temperature structure (Shapiro, Lightman, Eardley 1976; Rees et al. 1982), with the ions being hotter than the electrons.', 'astro-ph-9704018-2-2-3': 'The viscous heating affects mainly the ions, the more massive species, while the radiation is produced primarily by the electrons.', 'astro-ph-9704018-2-2-4': 'Since the ions transfer only a small fraction of their energy to the electrons via Coulomb scattering, the energy which is radiated is much less than the total energy released during accretion (Rees et al. 1982).', 'astro-ph-9704018-2-3-0': 'The emission spectrum of an ADAF is mainly determined by the cooling processes of the electrons, viz. synchrotron, bremsstrahlung, and Compton processes (see eg.', 'astro-ph-9704018-2-3-1': 'Narayan Yi 1995b; Mahadevan 1997).', 'astro-ph-9704018-2-3-2': 'Detailed calculations (e.g. Narayan 1996) show that ADAFs around accreting black holes have a characteristic spectrum ranging from radio frequencies [MATH] Hz up to hard X-ray frequencies [MATH] Hz, whose shape is a function primarily of the mass accretion rate, and to some extent the mass of the accreting star.', 'astro-ph-9704018-2-3-3': 'A number of accreting black hole systems have been shown to contain ADAFs (e.g. Narayan, Yi, Mahadevan 1995; Narayan, McClintock, Yi 1996; Lasota et al. 1996, Fabian Rees 1995, Mahadevan 1997, Narayan, Barret, McClintock 1997, Reynolds et al. 1997).', 'astro-ph-9704018-2-4-0': 'In the work done so far, on ADAFs, only the cooling of the electrons has been considered in calculating the spectra.', 'astro-ph-9704018-2-4-1': 'However, since ADAFs are two temperature plasmas with very high ion temperatures, e.g. [MATH] K close to a black hole (Narayan Yi 1995b), one wonders if there might be radiative processes associated directly with the ions.', 'astro-ph-9704018-2-4-2': 'In particular, at such high temperatures, collisions between protons can lead to substantial production of neutral pions, [MATH], which would decay into high energy [MATH]MeV [MATH]-rays.', 'astro-ph-9704018-2-4-3': 'Charged pions can also be produced by proton-proton collisions, but the final decay products are neutrinos and electrons (and their anti-particles), so that most of the energy leaves the plasma either in neutrinos or lower energy photons.', 'astro-ph-9704018-2-5-0': 'Gamma-ray emission from a two temperature gas accreting onto a black hole has been computed previously (see e.g. Dahlbacka, Chapline, Weaver 1974; Colpi, Maraschi, Treves 1986, Berezinsky Dokuchaev 1990).', 'astro-ph-9704018-2-5-1': 'Most of the previous work considered a thermal distribution of protons, and the predicted [MATH]-ray fluxes depend sensitively on the particular accretion scenario being considered.', 'astro-ph-9704018-2-5-2': 'In this paper we consider both thermal and non-thermal distributions of protons and focus specifically on the ADAF paradigm.', 'astro-ph-9704018-2-5-3': 'Using the methods described by Dermer (1986ab) we calculate the flux and spectrum of [MATH]-ray emission from ADAFs and show that these flows produce interesting levels of high energy [MATH]-rays.', 'astro-ph-9704018-2-5-4': 'In the case of the source Sagittarius A[MATH] (Sgr A[MATH]) at the Galactic Center, we predict a [MATH]-ray flux which is above the sensitivity limit of the EGRET detector on the Compton Gamma-Ray Observatory, and we compare the prediction with a possible detection of this source.', 'astro-ph-9704018-2-6-0': 'The outline of the paper is as follows.', 'astro-ph-9704018-2-6-1': 'In 2 we present the reaction rate for pion production and describe the structure and basic equations of ADAFs.', 'astro-ph-9704018-2-6-2': 'In 3 we calculate the [MATH]-ray spectra from thermal and non-thermal distributions of protons in ADAFs, and in 4 we apply these results to EGRET observations of the Galactic Center.', 'astro-ph-9704018-2-6-3': 'In 5 we discuss other applications of the results.', 'astro-ph-9704018-2-7-0': '# General Relations', 'astro-ph-9704018-2-8-0': '## Pion Production', 'astro-ph-9704018-2-9-0': 'The production of [MATH]-rays from [MATH] collisions is a two step process, and has been worked out in detail by Stecker (1971, see also Stephens Badhwar 1981; Dermer 1986ab).', 'astro-ph-9704018-2-9-1': 'The colliding protons produce an intermediate particle, a neutral pion, [MATH], which then decays into two [MATH]-ray photons.', 'astro-ph-9704018-2-9-2': 'The reaction is [EQUATION] where the photons, [MATH], in general have different energies in the gas frame.', 'astro-ph-9704018-2-9-3': 'The number of neutral pions produced is calculated by considering two protons with four momenta [MATH] and [MATH], moving towards each other with relative velocity [MATH].', 'astro-ph-9704018-2-9-4': 'The number of collisions that occur in a volume [MATH], for a time [MATH], is a frame invariant quantity, which in an arbitrary reference frame can be written as (see Landau Lifshitz 1975, 12) [EQUATION]', 'astro-ph-9704018-2-9-5': 'Here, [MATH], [MATH] are the number densities of the protons, [MATH] are their energies, [MATH] is the total cross-section for pion production, and [MATH] are the respective velocity parameters of the two particles.', 'astro-ph-9704018-2-9-6': 'For a distribution of particle velocities, the total number of pions produced per second is just the integral of [MATH] over volume and velocities, which in spherical co-ordinates is [EQUATION] where [MATH] are the respective Lorentz factors of the two protons, [MATH], and [MATH] is the radius with respect to the accreting star.', 'astro-ph-9704018-2-9-7': 'The total number of [MATH]-rays produced is [MATH].', 'astro-ph-9704018-2-10-0': 'To obtain the [MATH]-ray spectrum, we first calculate the spectrum of pion energies, [MATH], in the frame of the observer.', 'astro-ph-9704018-2-10-1': 'The observed [MATH]-ray spectrum is then obtained through the relation (Stecker 1971) [EQUATION] where [MATH] GeV[MATH] is the mass of the pion, and [MATH] is the minimum pion energy required to produce a [MATH]-ray with energy [MATH], [EQUATION]', 'astro-ph-9704018-2-10-2': 'The pion spectrum, [MATH] is obtained by substituting the differential cross-section [MATH] for [MATH] in equation ([REF]).', 'astro-ph-9704018-2-10-3': 'Two models are generally used to determine the differential cross-section for pion production: the isobar model and the scaling model.', 'astro-ph-9704018-2-10-4': 'In the isobar model, pions are produced by a two step process.', 'astro-ph-9704018-2-10-5': 'Colliding protons produce an isobar, with rest mass 1.238 GeV, which subsequently decays into a proton and a pion (Lindenbaum Sternheimer 1957; Stecker 1971).', 'astro-ph-9704018-2-10-6': 'Dermer (1986b) has shown that the isobar model agrees with the experimental data quite well for proton energies near threshold: [MATH] GeV.', 'astro-ph-9704018-2-10-7': 'On the other hand, for proton energies [MATH] GeV, the scaling model of Stephens and Badhwar (1981) represents the experimental data better (Dermer 1986b).', 'astro-ph-9704018-2-10-8': 'In our calculations, we therefore use the two models in their respective ranges of validity and interpolate smoothly (cubic interpolation) in the intermediate regime.', 'astro-ph-9704018-2-11-0': '## ADAF Equations', 'astro-ph-9704018-2-12-0': 'To evaluate equations ([REF]) and ([REF]) we need the number density of protons as a function of energy in the accretion flow, [MATH].', 'astro-ph-9704018-2-12-1': 'We write [MATH] as the product of a normalized velocity distribution, which depends on the temperature at a given radius, and the total number density of protons, [EQUATION] with [EQUATION]', 'astro-ph-9704018-2-12-2': 'Here, [MATH] is the dimensionless ion temperature, [MATH], at radius [MATH].', 'astro-ph-9704018-2-13-0': 'An ADAF is characterized by four parameters: the viscosity parameter, [MATH] (Shakura Sunyaev 1973), the ratio of gas pressure to total pressure, [MATH], the mass of the central black hole, [MATH], and the mass accretion rate, [MATH].', 'astro-ph-9704018-2-13-1': 'Given these parameters, the global structure and dynamics of ADAFs can be calculated (Narayan, Kato, Honma 1997; Chen, Abramowicz, Lasota 1997).', 'astro-ph-9704018-2-13-2': 'Generally, ADAFs appear to have large values of [MATH] (Narayan 1996), and the uncertainty in the value is a factor of a few.', 'astro-ph-9704018-2-13-3': 'Many published ADAF models in the literature use [MATH] (e.g. Narayan et al. 1996, 1997).', 'astro-ph-9704018-2-13-4': 'If there is equipartition between magnetic and gas pressure, we expect [MATH]; this is the value we favor.', 'astro-ph-9704018-2-13-5': 'However, for completeness we also consider the more extreme case [MATH] = 0.95, where the magnetic pressure contributes negligibly ([MATH]) to the total pressure.', 'astro-ph-9704018-2-14-0': 'The radial structure of ADAFs is well approximated by a series of concentric spherical shells, with the properties of the gas varying as a function of radius (Narayan Yi 1995a).', 'astro-ph-9704018-2-14-1': 'The continuity equation gives [MATH], where [MATH] is the radial velocity of the gas, and [MATH] is the mass density.', 'astro-ph-9704018-2-14-2': 'Writing the radius in units of the Schwarzschild radius, [MATH], where [MATH] cm and [MATH] is the mass of the central star in solar mass units, and assuming that the mass fraction of hydrogen is [MATH], we have [EQUATION]', 'astro-ph-9704018-2-14-3': 'Here [MATH] is the accretion rate in Eddington units, with [MATH] g s[MATH], and [MATH] is defined so as to scale out the dependence of [MATH] on [MATH] and [MATH].', 'astro-ph-9704018-2-14-4': 'Given [MATH], [MATH], [MATH] and [MATH], equation ([REF]) allows us to calculate [MATH] provided we have [MATH].', 'astro-ph-9704018-2-14-5': 'We obtain [MATH] from the global ADAF solution of Narayan et al. (1997).', 'astro-ph-9704018-2-15-0': 'The ion temperature [MATH] is obtained from the total gas pressure [MATH] (Narayan Yi 1995b), [EQUATION] where [MATH] is the isothermal sound speed, and [MATH] are the effective molecular weights of the ions and electrons respectively.', 'astro-ph-9704018-2-15-1': 'Since [MATH] we neglect the second term to obtain [EQUATION] where [MATH] is the velocity of light.', 'astro-ph-9704018-2-15-2': 'We obtain [MATH] again from the global solution of the ADAF (Narayan et al. 1997).', 'astro-ph-9704018-2-16-0': 'Given [MATH] and [MATH] and an assumed functional form for the proton energy distribution [MATH], we can substitute equation ([REF]) in equations ([REF]) and ([REF]) to calculate the [MATH]-ray spectrum.', 'astro-ph-9704018-2-17-0': '# Results', 'astro-ph-9704018-2-18-0': 'At present our understanding of viscous heating is poor, and we do not know whether the process leads to a thermal or power-law distribution of proton energies, or perhaps some combination of the two.', 'astro-ph-9704018-2-18-1': 'In this section we calculate the pion spectra by considering two different proton energy distributions: a relativistic Maxwell-Boltzmann distribution and a power-law distribution.', 'astro-ph-9704018-2-19-0': '## Thermal Distribution', 'astro-ph-9704018-2-20-0': 'The proton temperature in an ADAF close to a black hole is marginally relativistic, [MATH].', 'astro-ph-9704018-2-20-1': 'At such temperatures, pions are produced primarily by protons in the tail of the Maxwell-Boltzmann distribution, since only these particles have energies above the threshold needed for pion production.', 'astro-ph-9704018-2-20-2': 'Thus, the pion production is very sensitive to the proton temperature, and since [MATH] scales approximately as [MATH] (Narayan Yi 1995b; Mahadevan 1997), most of the production is from the innermost radii of the ADAF, [MATH].', 'astro-ph-9704018-2-21-0': 'The protons at these temperatures have momenta near the threshold for pion production, and do not exceed [MATH] 1 GeV/[MATH].', 'astro-ph-9704018-2-21-1': 'At such energies the isobar model (Lindenbaum Sternheimer 1957; Stecker 1971; Dermer 1986a) is more accurate and we have therefore used it to calculate the cross-sections and energy spectra.', 'astro-ph-9704018-2-21-2': 'In this model, the isobars move along the initial directions of the colliding protons in the center of momentum (CM) system.', 'astro-ph-9704018-2-21-3': 'It is therefore convenient to transform variables in equation ([REF]) via a Lorentz boost to the CM reference frame, followed by a rotation so that the colliding protons are moving along the z-axis (Dermer 1984, 1986a).', 'astro-ph-9704018-2-21-4': 'Using equation ([REF]) with equation ([REF]), and rewriting in terms of the new variables (see Dermer 1984, 1986a for details), we obtain [EQUATION]', 'astro-ph-9704018-2-21-5': "Here [MATH] is the pion energy in the observer's frame, [MATH] is the relative Lorentz factor of the two colliding protons, [MATH], [MATH] is the differential cross-section for the production of a pion with Lorentz factor [MATH] in the CM frame for a given [MATH], [MATH] where [MATH] characterizes the strength of the collision, [MATH] is the modified Bessel function of order 2, and we have used a Maxwell-Boltzmann energy distribution for the protons.", 'astro-ph-9704018-2-21-6': 'We take [MATH] to be [MATH]; however, since the proton temperature decreases steeply with radius ([MATH]) the contribution to the pion spectrum from the gas at [MATH] is negligible.', 'astro-ph-9704018-2-22-0': 'The calculation in equation ([REF]) gives the pion spectrum, [MATH].', 'astro-ph-9704018-2-22-1': 'Using equation ([REF]), we then obtain the [MATH]-ray spectrum.', 'astro-ph-9704018-2-22-2': 'The results are shown in Fig. 1.', 'astro-ph-9704018-2-22-3': 'The plot corresponds to two values of the viscosity parameter [MATH], 0.1 and 0.3, and two values of [MATH], 0.5 and 0.95.', 'astro-ph-9704018-2-22-4': 'For a given [MATH], increasing [MATH] leads to increasing gas pressure, and therefore to increasing ion temperature (see eq. [REF]).', 'astro-ph-9704018-2-22-5': 'Changing [MATH] from 0.5 to 0.95 causes the temperature to go up by a factor [MATH], and since the pion production is extremely sensitive to temperature, the [MATH]-ray luminosity increases by nearly two orders in magnitude.', 'astro-ph-9704018-2-22-6': 'Changes in [MATH] also affect the [MATH]-ray luminosity, though less sensitively.', 'astro-ph-9704018-2-22-7': 'Since the total luminosity is proportional to [MATH] which varies roughly as [MATH] (Narayan Yi 1995b), we might expect the luminosity to increase with decreasing [MATH].', 'astro-ph-9704018-2-22-8': 'This effect is counterbalanced however by small changes in the temperature, and the luminosity in fact decreases slightly when [MATH] goes down from [MATH] to [MATH].', 'astro-ph-9704018-2-23-0': '## Non-Thermal Distribution', 'astro-ph-9704018-2-24-0': 'In this section we consider a power-law distribution of proton energies described by a spectral index [MATH].', 'astro-ph-9704018-2-24-1': 'Equation ([REF]) continues to hold, but [MATH] is no longer a Maxwell-Boltzmann distribution.', 'astro-ph-9704018-2-24-2': 'To determine the form of [MATH], two requirements have to be satisfied (see eq. [REF]), namely the normalization condition and the requirement that the average kinetic energy of the power-law distribution of protons, at each radius, be equal to the average energy of the protons as given by the local ion temperature.', 'astro-ph-9704018-2-25-0': 'We model the energy distribution of protons as [EQUATION] where a fraction [MATH] of the protons have [MATH], and a fraction [MATH] are in a power-law tail with index [MATH].', 'astro-ph-9704018-2-25-1': 'The distribution is properly normalized.', 'astro-ph-9704018-2-25-2': 'The fraction [MATH] is fixed by the energy requirement: [EQUATION]', 'astro-ph-9704018-2-25-3': 'Since the exact energy distribution for the fraction [MATH] of protons with energy below threshold is not important for the present calculation, we have simplified the distribution to a [MATH] function in equation ([REF]).', 'astro-ph-9704018-2-26-0': 'To calculate the total pion luminosity, [MATH], we substitute equation ([REF]) in equation ([REF]).', 'astro-ph-9704018-2-26-1': 'We then obtain three different terms proportional to [MATH], [MATH], and [MATH], respectively.', 'astro-ph-9704018-2-26-2': 'The [MATH] term gives no contribution since it corresponds to protons with [MATH] colliding with one another.', 'astro-ph-9704018-2-26-3': 'These protons do not have enough energy to produce pions.', 'astro-ph-9704018-2-26-4': 'The [MATH] term corresponds to protons with [MATH] colliding with protons with [MATH], and we denote the contribution by [MATH].', 'astro-ph-9704018-2-26-5': 'Similarly, the [MATH] term corresponds to protons with [MATH] colliding with protons with [MATH], and we denote it as [MATH].', 'astro-ph-9704018-2-26-6': 'We have [EQUATION] and [EQUATION] where [MATH] is the cross-section in units of millibarns, and [EQUATION]', 'astro-ph-9704018-2-26-7': 'In an ADAF, we generally have [MATH] so that most of the protons have [MATH] (cf. eq. [REF]).', 'astro-ph-9704018-2-26-8': 'We therefore expect most of the pion production and [MATH]-ray luminosity to come from [MATH].', 'astro-ph-9704018-2-26-9': 'Table [REF] gives numerical results and we see that [MATH] is indeed negligible compared with [MATH].', 'astro-ph-9704018-2-27-0': 'In calculating the pion spectrum, we consider only the contribution from [MATH] and write [EQUATION]', 'astro-ph-9704018-2-27-1': 'Unlike in the case of a thermal distribution where most of the protons have energies either below or just above threshold, in a power-law distribution the protons have a wide range of energy and some protons are well above threshold.', 'astro-ph-9704018-2-27-2': 'We therefore use both the isobar and scaling models, as described in [REF]', 'astro-ph-9704018-2-28-0': 'Equation ([REF]) reveals the dependence of the pion flux and energy spectrum on the parameters: (1) The flux is proportional to [MATH].', 'astro-ph-9704018-2-28-1': '(2) The flux depends on [MATH], [MATH] and [MATH] through the function [MATH] defined in equation ([REF]).', 'astro-ph-9704018-2-28-2': '(3) The shape of the spectrum depends only on [MATH] and is given by the integral in equation ([REF]).', 'astro-ph-9704018-2-29-0': 'Tables [REF] and [REF] give [MATH] for different values of its parameters.', 'astro-ph-9704018-2-29-1': 'For fixed [MATH], increasing [MATH] leads to increasing [MATH] since the temperature in the flow increases.', 'astro-ph-9704018-2-29-2': 'However, unlike the thermal case, the luminosity is not excessively sensitive to [MATH] since the number of protons above threshold is directly proportional to [MATH] (cf. eq. [REF]); therefore, a change in [MATH] by a factor of two changes the total luminosity only by [MATH].', 'astro-ph-9704018-2-29-3': 'For fixed [MATH] in a self-similar solution (Narayan Yi 1994, 1995b), the density varies in proportion to [MATH], so the luminosity scales as [MATH].', 'astro-ph-9704018-2-29-4': 'However, in the more accurate global solutions employed in this paper, the dependence of density-and hence luminosity-on [MATH] is not strictly a power-law, and the overall strength of the dependence is somewhat weaker than in the self-similar solutions.', 'astro-ph-9704018-2-29-5': 'To illustrate this dependence, we list the numerical values of [MATH] for three different choices of [MATH] in Tables [REF] and [REF].', 'astro-ph-9704018-2-30-0': 'Figure 2 shows [MATH]-ray spectra for different values of the proton energy spectral index [MATH] (we fix [MATH], [MATH]).', 'astro-ph-9704018-2-30-1': 'The [MATH]-ray spectral index at high energies [MATH] GeV is the same as the energy spectral index of the protons [MATH], as shown by Dermer (1986b).', 'astro-ph-9704018-2-30-2': 'For this reason, we investigated values of [MATH] in the range 2.3 - 3.3.', 'astro-ph-9704018-2-30-3': 'The spectrum turns over at [MATH] MeV (approximately half the pion rest mass) and falls for lower photon energies.', 'astro-ph-9704018-2-30-4': 'Comparing Figures 1 and 2, we see that the total [MATH]-ray luminosity from a power-law energy distribution of protons is comparable to that obtained from a thermal distribution when [MATH], but is very much higher for [MATH].', 'astro-ph-9704018-2-30-5': 'In our view [MATH] is the natural choice since it corresponds to equipartition between gas and magnetic pressure.', 'astro-ph-9704018-2-30-6': 'The [MATH]-ray spectrum extends to much higher photon energies when the protons have a power-law distribution compared to the thermal case.', 'astro-ph-9704018-2-31-0': '# Application to Sgr A[MATH]', 'astro-ph-9704018-2-32-0': 'The black hole candidate at the center of our Galaxy, Sagittarius A[MATH] (Sgr A[MATH]), appears to be a scaled down version of an AGN.', 'astro-ph-9704018-2-32-1': 'The source is believed to consist of a black hole of mass [MATH] (Eckart Genzel 1996a, 1996b) accreting gas from its surroundings.', 'astro-ph-9704018-2-33-0': 'The mass accretion rate in Sgr A[MATH] has been estimated by various methods (see Genzel et al. 1994 for a summary).', 'astro-ph-9704018-2-33-1': 'The accretion of wind gas from neighboring stars (principally IRS 16) seems to be the most likely scenario.', 'astro-ph-9704018-2-33-2': 'The accretion rate is determined by considering the fraction of the wind that comes within the accretion radius of the central black hole, but the estimate depends sensitively on the assumed wind velocity.', 'astro-ph-9704018-2-33-3': 'Genzel et al. (1994) obtain an accretion rate of [MATH] yr[MATH] which corresponds to [MATH] for [MATH], assuming a wind velocity of 1000 km s[MATH] and mass loss rate from winds of [MATH] yr[MATH].', 'astro-ph-9704018-2-33-4': 'Melia (1992) uses a wind velocity of [MATH] km s[MATH] and obtains an accretion rate of [MATH] yr[MATH]).', 'astro-ph-9704018-2-33-5': 'The true accretion rate probably lies between these two estimates, and we will require our models to have [MATH].', 'astro-ph-9704018-2-33-6': 'Note that Lacy et al. (1982; see also Rees 1982) estimated a much higher mass accretion rate of [MATH] yr[MATH] from stellar disruptions at the Galactic Center, but argued that the accretion due to this probably has a low duty cycle.', 'astro-ph-9704018-2-34-0': 'Sgr A[MATH] has an extremely low luminosity, [MATH], which corresponds to [MATH] if the accretion flow has a standard radiative efficiency of 10%.', 'astro-ph-9704018-2-34-1': 'Such a low [MATH] is in serious conflict with the estimate of [MATH] given in the previous paragraph.', 'astro-ph-9704018-2-34-2': 'Narayan, Yi Mahadevan (1995) suggested that Sgr A[MATH] does accrete at a rate near the one estimated, but in an advection-dominated mode.', 'astro-ph-9704018-2-34-3': 'Using a self-similar ADAF model, they were able to reconcile the low luminosity of the source with a relatively large [MATH]: [MATH].', 'astro-ph-9704018-2-34-4': 'If [MATH], the accretion rate inferred on the basis of the ADAF model would be in agreement with that estimated on the basis of gas supply.', 'astro-ph-9704018-2-34-5': 'Furthermore, they obtained a reasonable fit to the observed spectrum from radio to hard X-ray frequencies.', 'astro-ph-9704018-2-34-6': 'In this section, we calculate the [MATH]-ray luminosity and spectrum of Sgr A[MATH] due to pion production, and compare these predictions with the flux seen by EGRET from the direction of the Galactic Center.', 'astro-ph-9704018-2-35-0': 'Among several unidentified sources detected by EGRET in the Galactic plane (Merck et al. 1996), the source 2EG 1746-2852 is of particular interest since it is spatially coincident with the Galactic Center.', 'astro-ph-9704018-2-35-1': 'There is no firm identification of this source, but it is point-like to within the resolution of the instrument ([MATH]), it is [MATH] above the local diffuse emission, and its spectrum differs significantly from the spectra of other unidentified EGRET sources.', 'astro-ph-9704018-2-35-2': 'Out of the 32 sources whose spectra are reported by Merck et al. (1996), 27 have spectral slopes [MATH] (photon index), 4 have spectral slopes [MATH] and one has a very hard spectrum with a spectral slope of 1.7.', 'astro-ph-9704018-2-35-3': 'This last source is 2EG 1746-2852, and it is located exactly at the Galactic Center.', 'astro-ph-9704018-2-35-4': 'Following Merck et al. (1996), we make the reasonable assumption that 2EG 1746-2852 corresponds to Sgr A[MATH].', 'astro-ph-9704018-2-35-5': 'Figure 3 shows the spectrum of the source as measured by EGRET.', 'astro-ph-9704018-2-35-6': 'The dashed line is the best-fit power-law obtained by Merck et al. (1996) over the photon energy range 100-4000 MeV.', 'astro-ph-9704018-2-35-7': 'Their fitted spectral slope is [MATH].', 'astro-ph-9704018-2-35-8': 'At higher energies, the data suggest a roll-over in the spectrum which would give a softer spectral index at higher energies [MATH] GeV).', 'astro-ph-9704018-2-36-0': '## Thermal Distribution', 'astro-ph-9704018-2-37-0': 'Figure 3 compares the [MATH]-ray spectrum from the thermal model with the EGRET data.', 'astro-ph-9704018-2-37-1': 'The solid line corresponds to [MATH], [MATH], and the dotted line to [MATH], [MATH].', 'astro-ph-9704018-2-37-2': 'The black hole mass is taken to be [MATH], and the accretion rates in the two models have been varied such that the predicted spectra agree with the data point at [MATH]MeV; the accretion rates are indicated on the plot.', 'astro-ph-9704018-2-38-0': 'It is clear that the thermal model predicts a spectrum with the wrong shape and fails to explain the EGRET detection at [MATH] GeV.', 'astro-ph-9704018-2-38-1': 'Moreover, for the preferred value of [MATH], the model requires a rather high [MATH], which is much larger than the range we consider reasonable (see above).', 'astro-ph-9704018-2-38-2': 'We conclude that the thermal model is inconsistent with the observations.', 'astro-ph-9704018-2-39-0': '## Non-Thermal Distribution', 'astro-ph-9704018-2-40-0': 'Fig. 4 compares [MATH]-ray spectra from various power-law models ([MATH]) with the EGRET data.', 'astro-ph-9704018-2-40-1': 'The different curves correspond to different values of the proton energy index [MATH].', 'astro-ph-9704018-2-40-2': 'For each [MATH], we have varied the accretion rate so as to obatin the closest agreement with the observed spectrum.', 'astro-ph-9704018-2-40-3': 'The results ([MATH]), which are only weakly dependent on [MATH], are given in Table [REF].', 'astro-ph-9704018-2-40-4': 'The accretion rates so inferred fall well within the range of accretion rates estimated on the basis of the properties of nearby gas ([MATH], Genzel et al. 1994), and are within factors of a few of the accretion rate estimated on the basis of fitting the radio to X-ray spectrum with the somewhat less accurate self-similar version of the ADAF model ([MATH], Narayan, Yi, Mahadevan 1995).', 'astro-ph-9704018-2-41-0': 'Reproducing the power-law portion of the [MATH]-ray spectrum is a by-product of postulating a power-law proton energy distribution.', 'astro-ph-9704018-2-41-1': 'However, it is striking in this context that the best-fit slope, [MATH], is identical to the slope of the low-energy cosmic ray distribution.', 'astro-ph-9704018-2-41-2': 'The models predict roll-overs at both the high energy (few GeV) and low energy ([MATH]MeV) ends of the spectrum.', 'astro-ph-9704018-2-41-3': 'At the high-energy end there is just a hint of such a roll-over in the data, but at the low-energy end, the evidence seems somewhat stronger.', 'astro-ph-9704018-2-42-0': 'The similarity of the best-fit spectral shape to the diffuse Galactic [MATH]-ray emission has two possible interpretations.', 'astro-ph-9704018-2-42-1': 'First, since the diffuse emission is generally interpreted as being due to cosmic ray protons striking thermal protons in the interstellar medium, a mechanism similar to what occurs in an ADAF, the EGRET detection of 2EG 1746-2852 could be interpreted as cosmic ray protons colliding with a compact dense cloud of gas that is spatially coincident with the Galactic Center.', 'astro-ph-9704018-2-42-2': 'This interpretation would require such a cloud to have a considerably greater gas density (and possibly also cosmic ray density) than in the interstellar medium only slightly farther away from the Galactic Center.', 'astro-ph-9704018-2-42-3': 'If virtually all the Galactic Center region [MATH]-ray flux were due to such a cloud, the power-law proton distribution function version of the ADAF model would be put seriously in doubt.', 'astro-ph-9704018-2-42-4': 'Thermal proton distribution function versions of the ADAF model (or entirely different models) would not be seriously constrained if the [MATH]-rays coming from the Galactic Center prove not to have their source in Sgr A[MATH].', 'astro-ph-9704018-2-43-0': 'The alternative is that the observed [MATH]-ray spectrum is in fact due to Sgr A[MATH].', 'astro-ph-9704018-2-43-1': 'In this case, not only is the deduced accretion rate roughly consistent with previous estimates, but we also have the interesting result that whatever process determines the cosmic proton energy distribution (shock acceleration ?)', 'astro-ph-9704018-2-43-2': 'is also at work in ADAFs.', 'astro-ph-9704018-2-43-3': 'We find the numerical coincidence between the prediction of the power-law version of the ADAF model and the observed [MATH]-ray flux striking enough to justify pursuing this interpretation.', 'astro-ph-9704018-2-44-0': 'To gauge just how strong the quantitative agreement is, we must consider how much freedom the model is given by adjustable parameters.', 'astro-ph-9704018-2-44-1': 'Three free parameters are significant.', 'astro-ph-9704018-2-44-2': 'One ([MATH]) is fixed to within an order of magnitude or so by other considerations (the amount of gas available to accrete, and fitting the radio-X-ray spectrum).', 'astro-ph-9704018-2-44-3': 'Another ([MATH]), while formally unconstrained over a range of several orders of magnitude, is often supposed to have a value quite near the one which we infer (i.e. between 0.1 and 1, cf. Narayan 1996).', 'astro-ph-9704018-2-44-4': 'Only the third ([MATH]) is chosen almost entirely on the basis of fitting the observed [MATH]-ray spectrum.', 'astro-ph-9704018-2-44-5': 'Because the [MATH]-ray luminosity predicted by the ADAF model (assuming a power-law proton energy distribution) approximately scales with [MATH] (the actual scaling with [MATH] is only very roughly described by a power-law), the combined a priori uncertainty in [MATH] and [MATH] may be regarded as giving the [MATH]-ray luminosity predicted by the model a possible range of [MATH].', 'astro-ph-9704018-2-45-0': '# Discussion Conclusions', 'astro-ph-9704018-2-46-0': 'Since it is not understood whether viscous heating produces a thermal or power-law distribution of proton energies, we have calculated the [MATH]-ray emission of ADAFs corresponding to both types of distributions.', 'astro-ph-9704018-2-46-1': 'We expect that the true energy distribution in any given source will be bracketed by these two extremes.', 'astro-ph-9704018-2-46-2': 'Spectra corresponding to intermediate models can be easily calculated by taking a weighted sum of the thermal and power-law models.', 'astro-ph-9704018-2-47-0': 'If the proton distribution is thermal, the spectrum has a characteristic shape with a peak at [MATH] MeV and very little emission at either lower or higher photon energies.', 'astro-ph-9704018-2-47-1': 'For the kinds of proton temperatures expected in an ADAF, only a small fraction of the protons (in the tail of the Maxwellian distribution) have sufficient energy to produce pions.', 'astro-ph-9704018-2-47-2': 'Consequently, even a minor change in the temperature, say by a factor of [MATH], can modify the [MATH]-ray luminosity by orders of magnitude (cf. Figure 1).', 'astro-ph-9704018-2-47-3': 'Since the flux is very sensitive to the gas temperature, a detailed understanding of the physics of the ADAF in the region [MATH] (where most of the [MATH]-ray luminosity originates) is necessary in order to make testable predictions.', 'astro-ph-9704018-2-47-4': 'The present study does not have the necessary accuracy for this.', 'astro-ph-9704018-2-47-5': 'General relativistic effects become important at these radii and need to be included consistently.', 'astro-ph-9704018-2-47-6': 'In principle, if these effects are included, the [MATH]-ray spectrum could be used as a sensitive probe of the proton temperature.', 'astro-ph-9704018-2-47-7': 'One might even hope to distinguish between rotating and non-rotating black holes, since the physical properties of the flow at [MATH] will be different for the two cases.', 'astro-ph-9704018-2-48-0': 'If the protons have a power-law distribution of energies, the [MATH]-ray spectrum again peaks at [MATH] MeV, but there is significant emission at higher photon energies.', 'astro-ph-9704018-2-48-1': 'Indeed, the spectrum asymptotically has a spectral slope [MATH] which is equal to the power-law index [MATH] of the proton distribution (Dermer 1986b).', 'astro-ph-9704018-2-48-2': 'Thus, the detection of a power-law [MATH]-ray spectrum not only indicates the nonthermal nature of the protons but also helps determine the energy index.', 'astro-ph-9704018-2-49-0': 'In contrast to the thermal case, only half the [MATH]-ray luminosity due to a power-law distribution of proton energies originates from [MATH], while the other half is emitted from [MATH].', 'astro-ph-9704018-2-49-1': 'Further, the luminosity is not very sensitive to changes in the temperature.', 'astro-ph-9704018-2-49-2': 'Therefore, our present understanding of the physics of ADAFs is probably adequate for a reasonable estimate of the [MATH]-ray flux, and we can thus test this version of the ADAF paradigm usefully against observations.', 'astro-ph-9704018-2-50-0': 'The Galactic Center source, Sgr A[MATH] presents an excellent opportunity to test the model, since there is a good case for the presence of an ADAF in this source (Narayan et al. 1995) and the predicted [MATH]-ray flux is large enough to be detectable with current instruments.', 'astro-ph-9704018-2-50-1': 'Indeed, EGRET has detected an unresolved source in the Galactic Center region, 2EG J1746-2852 (Merck et al. 1996).', 'astro-ph-9704018-2-50-2': 'It is possible that this source corresponds to a dense compact gas cloud interacting with cosmic rays.', 'astro-ph-9704018-2-50-3': 'However, the source could equally well be associated with Sgr A[MATH].', 'astro-ph-9704018-2-50-4': 'The [MATH]-ray flux of Sgr A[MATH] which we calculate with our model using a power-law distribution of proton energies is in good agreement with the observed flux of 2EG J1746-2852 given parameters which are almost determined on independent grounds.', 'astro-ph-9704018-2-50-5': 'In order to fit the flux, we need an Eddington-scaled mass accretion rate of [MATH]-[MATH] (for [MATH]), close to the estimate [MATH] obtained by Narayan et al. (1995) from fitting the lower energy spectrum due to the electrons.', 'astro-ph-9704018-2-50-6': 'It is also compatible with the range of [MATH] quoted by Genzel et al. (1994) on the basis of gas motions in the vicinity of Sgr A[MATH]: [MATH].', 'astro-ph-9704018-2-50-7': 'In addition, the pion-decay model for the Galactic Center [MATH]-ray spectrum predicts a roll over at energies below [MATH] MeV that may be present in the observed spectrum.', 'astro-ph-9704018-2-51-0': 'An exciting aspect of the present study is that it provides for the first time a direct probe of the protons in hot accretion flows.', 'astro-ph-9704018-2-51-1': 'A fundamental assumption in ADAF models is that the plasma is two-temperature.', 'astro-ph-9704018-2-51-2': 'However, until now there has been no direct test of this assumption since all previous investigations of the emission from ADAFs dealt only with the electrons.', 'astro-ph-9704018-2-51-3': 'The [MATH]-ray emission we have considered in this paper is due entirely to the protons.', 'astro-ph-9704018-2-51-4': 'Moreover, it requires that the protons have nearly virial energies in order to be able to exceed the energy threshold for pion production.', 'astro-ph-9704018-2-51-5': 'The encouraging results we have obtained in the case of Sgr A[MATH] indicate that this source might have a two-temperature plasma exactly as postulated in the models.', 'astro-ph-9704018-2-51-6': 'More detailed study of Sgr A[MATH] coupled with future detections of other sources (see below), could strengthen the case for two-temperature accretion flows significantly.', 'astro-ph-9704018-2-52-0': 'Another exciting aspect of this study is that the [MATH]-ray spectrum from pion decay provides a direct window to the energy distribution of the protons.', 'astro-ph-9704018-2-52-1': 'In the ADAF paradigm, viscous heat energy goes almost entirely into the protons.', 'astro-ph-9704018-2-52-2': 'Furthermore, it is easy to show that, at least at the low [MATH] expected in quiescent systems, the protons have no energetically important interactions either among themselves or with the electrons.', 'astro-ph-9704018-2-52-3': 'Therefore, each proton retains memory of all the heating events it has undergone during the accretion, and so the energy distribution of the protons directly reflects the heating processes present in the plasma.', 'astro-ph-9704018-2-52-4': 'In principle, with sufficiently sensitive observations, one could use [MATH]-ray measurements as a direct probe of viscous heating.', 'astro-ph-9704018-2-52-5': 'This would be invaluable for the theory of hot accretion flows.', 'astro-ph-9704018-2-53-0': 'The ADAF model has been applied successfully to several other low-luminosity black holes in addition to Sgr A[MATH].', 'astro-ph-9704018-2-53-1': 'The sources studied include two X-ray binaries in quiescence, A0620-00 and V404 Cyg (Narayan, McClintock, Yi 1996, Narayan, Barret McClintock 1997), and several quiescent AGNs, viz. the LINER galaxy NGC 4258 (Lasota et al. 1996, Herrnstein et al. 1996) and the nearby elliptical galaxies NGC 4472, NGC 4486, NGC 4649, and NGC 4636 (Mahadevan 1997; Reynolds et al. 1997).', 'astro-ph-9704018-2-53-2': 'Table [REF] gives the values of [MATH], [MATH], [MATH] and [MATH] of several of these systems (taken from the references indicated) and presents the expected [MATH]-ray fluxes according to our model.', 'astro-ph-9704018-2-53-3': 'Among the nearby ellipticals, we have included only NGC4486 since it is the only system with a reliable mass estimate (Ford et al. 1995; Harms et al. 1995).', 'astro-ph-9704018-2-53-4': 'The [MATH]-ray flux estimates in Table [REF] assume that the protons have a power-law energy distribution with [MATH].', 'astro-ph-9704018-2-53-5': 'We see that, only in the case of Sgr A[MATH] does the detection threshold of EGRET ([MATH]) permit a test of the predictions, although some of the other sources might be not far below the EGRET threshold.', 'astro-ph-9704018-2-53-6': 'These sources are potentially detectable with future instruments such as the Gamma Ray Large Area Space Telescope (GLAST), which is designed to be 100 times more sensitive than EGRET.', 'astro-ph-9704018-2-54-0': 'Narayan (1996) has argued that X-ray binaries in the "hard state" (also called the "low state") may also contain ADAFs with [MATH].', 'astro-ph-9704018-2-54-1': 'If the proton energy distribution in these sources is a power-law of the sort we suggest exists in Sgr A[MATH], several of them might be expected to produce detectable [MATH]-ray fluxes.', 'astro-ph-9704018-2-54-2': 'However, the accretion rates in these objects (e.g. Cyg X-1) is likely to be close to the maximum rate permitting an ADAF (Narayan Yi 1995b).', 'astro-ph-9704018-2-54-3': 'If this is the case, particle interactions might be rapid enough that maintenance of a true power-law distribution function would be questionable.', 'astro-ph-9704018-2-54-4': 'Accurate predictions of the [MATH]-ray flux would then require a more elaborate calculation than we have performed for Sgr A[MATH].', 'astro-ph-9704018-2-55-0': 'Finally, we note that in determining the [MATH]-ray spectra, we have neglected gravitational redshift effects which become important at [MATH].', 'astro-ph-9704018-2-55-1': 'We have also neglected the Doppler blueshift associated with the large radial and orbital velocities of the gas at these radii (Narayan Yi 1995b).', 'astro-ph-9704018-2-55-2': 'Only a detailed calculation can tell which of the two effects predominates.', 'astro-ph-9704018-2-55-3': 'We note, however, that these effects are not likely to modify the results we have presented for a power-law distribution of protons because more than half the emission in this case occurs at [MATH].'}

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['astro-ph-9704018-1-33-6', 'astro-ph-9704018-2-33-6'], ['astro-ph-9704018-1-4-0', 'astro-ph-9704018-2-4-0'], ['astro-ph-9704018-1-4-1', 'astro-ph-9704018-2-4-1'], ['astro-ph-9704018-1-4-2', 'astro-ph-9704018-2-4-2'], ['astro-ph-9704018-1-4-3', 'astro-ph-9704018-2-4-3'], ['astro-ph-9704018-1-34-0', 'astro-ph-9704018-2-34-0'], ['astro-ph-9704018-1-34-1', 'astro-ph-9704018-2-34-1'], ['astro-ph-9704018-1-34-2', 'astro-ph-9704018-2-34-2'], ['astro-ph-9704018-1-34-3', 'astro-ph-9704018-2-34-3'], ['astro-ph-9704018-1-34-4', 'astro-ph-9704018-2-34-4'], ['astro-ph-9704018-1-34-5', 'astro-ph-9704018-2-34-5'], ['astro-ph-9704018-1-34-6', 'astro-ph-9704018-2-34-6'], ['astro-ph-9704018-1-20-0', 'astro-ph-9704018-2-20-0'], ['astro-ph-9704018-1-20-1', 'astro-ph-9704018-2-20-1'], ['astro-ph-9704018-1-20-2', 'astro-ph-9704018-2-20-2'], ['astro-ph-9704018-1-12-0', 'astro-ph-9704018-2-12-0'], ['astro-ph-9704018-1-12-1', 'astro-ph-9704018-2-12-1'], ['astro-ph-9704018-1-12-2', 'astro-ph-9704018-2-12-2'], ['astro-ph-9704018-1-54-0', 'astro-ph-9704018-2-54-0'], ['astro-ph-9704018-1-54-1', 'astro-ph-9704018-2-54-1'], ['astro-ph-9704018-1-54-2', 'astro-ph-9704018-2-54-2'], ['astro-ph-9704018-1-54-3', 'astro-ph-9704018-2-54-3'], ['astro-ph-9704018-1-54-4', 'astro-ph-9704018-2-54-4'], ['astro-ph-9704018-1-44-0', 'astro-ph-9704018-2-44-0'], ['astro-ph-9704018-1-44-1', 'astro-ph-9704018-2-44-1'], ['astro-ph-9704018-1-44-2', 'astro-ph-9704018-2-44-2'], ['astro-ph-9704018-1-44-3', 'astro-ph-9704018-2-44-3'], ['astro-ph-9704018-1-44-4', 'astro-ph-9704018-2-44-4'], ['astro-ph-9704018-1-44-5', 'astro-ph-9704018-2-44-5'], ['astro-ph-9704018-1-30-0', 'astro-ph-9704018-2-30-0'], ['astro-ph-9704018-1-30-1', 'astro-ph-9704018-2-30-1'], ['astro-ph-9704018-1-30-2', 'astro-ph-9704018-2-30-2'], ['astro-ph-9704018-1-30-3', 'astro-ph-9704018-2-30-3'], ['astro-ph-9704018-1-30-4', 'astro-ph-9704018-2-30-4'], ['astro-ph-9704018-1-30-5', 'astro-ph-9704018-2-30-5'], ['astro-ph-9704018-1-30-6', 'astro-ph-9704018-2-30-6'], ['astro-ph-9704018-1-5-0', 'astro-ph-9704018-2-5-0'], ['astro-ph-9704018-1-5-1', 'astro-ph-9704018-2-5-1'], ['astro-ph-9704018-1-5-2', 'astro-ph-9704018-2-5-2'], ['astro-ph-9704018-1-5-3', 'astro-ph-9704018-2-5-3'], ['astro-ph-9704018-1-5-4', 'astro-ph-9704018-2-5-4'], ['astro-ph-9704018-1-55-0', 'astro-ph-9704018-2-55-0'], ['astro-ph-9704018-1-55-1', 'astro-ph-9704018-2-55-1'], ['astro-ph-9704018-1-55-2', 'astro-ph-9704018-2-55-2'], ['astro-ph-9704018-1-55-3', 'astro-ph-9704018-2-55-3'], ['astro-ph-9704018-1-51-0', 'astro-ph-9704018-2-51-0'], ['astro-ph-9704018-1-51-1', 'astro-ph-9704018-2-51-1'], ['astro-ph-9704018-1-51-2', 'astro-ph-9704018-2-51-2'], ['astro-ph-9704018-1-51-3', 'astro-ph-9704018-2-51-3'], ['astro-ph-9704018-1-51-4', 'astro-ph-9704018-2-51-4'], ['astro-ph-9704018-1-51-5', 'astro-ph-9704018-2-51-5'], ['astro-ph-9704018-1-51-6', 'astro-ph-9704018-2-51-6'], ['astro-ph-9704018-1-32-0', 'astro-ph-9704018-2-32-0'], ['astro-ph-9704018-1-32-1', 'astro-ph-9704018-2-32-1'], ['astro-ph-9704018-1-22-0', 'astro-ph-9704018-2-22-0'], ['astro-ph-9704018-1-22-1', 'astro-ph-9704018-2-22-1'], ['astro-ph-9704018-1-22-2', 'astro-ph-9704018-2-22-2'], ['astro-ph-9704018-1-22-3', 'astro-ph-9704018-2-22-3'], ['astro-ph-9704018-1-22-4', 'astro-ph-9704018-2-22-4'], ['astro-ph-9704018-1-22-5', 'astro-ph-9704018-2-22-5'], ['astro-ph-9704018-1-22-6', 'astro-ph-9704018-2-22-6'], ['astro-ph-9704018-1-22-7', 'astro-ph-9704018-2-22-7'], ['astro-ph-9704018-1-22-8', 'astro-ph-9704018-2-22-8'], ['astro-ph-9704018-1-47-0', 'astro-ph-9704018-2-47-0'], ['astro-ph-9704018-1-47-1', 'astro-ph-9704018-2-47-1'], ['astro-ph-9704018-1-47-2', 'astro-ph-9704018-2-47-2'], ['astro-ph-9704018-1-47-3', 'astro-ph-9704018-2-47-3'], ['astro-ph-9704018-1-47-4', 'astro-ph-9704018-2-47-4'], ['astro-ph-9704018-1-47-5', 'astro-ph-9704018-2-47-5'], ['astro-ph-9704018-1-47-6', 'astro-ph-9704018-2-47-6'], ['astro-ph-9704018-1-47-7', 'astro-ph-9704018-2-47-7']]

[]

[]

[]

[]

['astro-ph-9704018-1-3-1', 'astro-ph-9704018-2-3-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/9704018

1907.02985

{'1907.02985-1-0-0': 'In Vision-and-Language Navigation (VLN), an embodied agent needs to reach a target destination with the only guidance of a natural language instruction.', '1907.02985-1-0-1': 'To explore the environment and progress towards the target location, the agent must perform a series of low-level actions, such as rotate, before stepping ahead.', '1907.02985-1-0-2': 'In this paper, we propose to exploit dynamic convolutional filters to encode the visual information and the lingual description in an efficient way.', '1907.02985-1-0-3': 'Differently from some previous works that abstract from the agent perspective and use high-level navigation spaces, we design a policy which decodes the information provided by dynamic convolution into a series of low-level, agent friendly actions.', '1907.02985-1-0-4': 'Results show that our model exploiting dynamic filters performs better than other architectures with traditional convolution, being the new state of the art for embodied VLN in the low-level action space.', '1907.02985-1-0-5': 'Additionally, we attempt to categorize recent work on VLN depending on their architectural choices and distinguish two main groups: we call them low-level actions and high-level actions models.', '1907.02985-1-0-6': 'To the best of our knowledge, we are the first to propose this analysis and categorization for VLN.', '1907.02985-1-1-0': '# Introduction', '1907.02985-1-2-0': 'Imagine finding yourself in a large conference hall, with an assistant giving you instructions on how to reach the room for your talk.', '1907.02985-1-2-1': 'You are likely to hear something like: turn right at the end of the corridor, head upstairs and reach the third floor: your room is immediately on the left.', '1907.02985-1-2-2': 'Succeeding in the task of finding your target location is rather nontrivial because of the length of the instruction and its sequential nature: the flow of actions must be coordinated with a series of visual examinations - like recognizing the end of the corridor or the floor number.', '1907.02985-1-2-3': 'Furthermore, navigation complexity dramatically increases if the environment is unknown, and no prior knowledge, such as a map, is available.', '1907.02985-1-3-0': 'Vision-and-Language Navigation (VLN) [CITATION] is a challenging task that demands an embodied agent to reach a target location by navigating unseen environments, with a natural language instruction as its only clue.', '1907.02985-1-3-1': 'Similarly to the previous example, the agent must assess different sub-tasks to succeed.', '1907.02985-1-3-2': 'First, a fine-grained comprehension of the given instruction is needed.', '1907.02985-1-3-3': 'Then, the agent must be able to map parts of the description into the visual perception.', '1907.02985-1-3-4': 'For example, walking past the piano requires to find and focus on the piano, rather than considering other objects in the scene.', '1907.02985-1-3-5': 'Finally, the agent needs to understand when the navigation has been completed and send a stop signal.', '1907.02985-1-4-0': 'VLN has been first proposed by Anderson et al [CITATION], with the aim of connecting the research efforts on vision-and-language understanding [CITATION] with the raising area of embodied AI [CITATION].', '1907.02985-1-4-1': 'This is particularly challenging, as embodied agents must deal with a series of issues that do not belong to traditional vision and language tasks [CITATION], like contextual decision-making and planning.', '1907.02985-1-4-2': 'Recent works on VLN [CITATION] integrate the agent with a simplified action space in which it "only needs to make high-level decisions as to which navigable direction to go next" [CITATION].', '1907.02985-1-4-3': 'In this scenario, the agent does not need to infer the sequence of actions to progress in the environment (e.g., turn right 30 degrees, then move forward) but it exploits a navigation graph to teleport itself to an adjacent location.', '1907.02985-1-4-4': 'The adoption of this high-level action space allowed for a significant boost in success rates, while partly depriving the task of its embodied nature, and leaving space for little more than pure visual and language understanding.', '1907.02985-1-4-5': 'We claim that this type of approach is inconvenient, as it strongly relies on prior knowledge on the environment.', '1907.02985-1-4-6': 'Depending on information such as the position and the availability of navigable directions, it reduces the task to a pure graph navigation.', '1907.02985-1-4-7': 'Moreover, it ignores the active role of the agent, as it only perceives the surrounding scene and selects the next target viewpoint from a limited set.', '1907.02985-1-4-8': 'We claim instead that the agent should be the principal component of embodied VLN [CITATION].', '1907.02985-1-4-9': 'Consequently, the output space should match with the low-level set of movements that the agent can perform.', '1907.02985-1-5-0': 'In this paper, we propose a novel architecture for embodied VLN which employs dynamic convolutional filters [CITATION] to identify the next target direction, without getting any information about the navigable viewpoints from the simulator.', '1907.02985-1-5-1': 'Convolutional filters are produced via an attention mechanism which follows the given instruction, and are in turn used to attend relevant directions of the scene towards which the agent should move.', '1907.02985-1-5-2': 'We then rely on a policy network to predict the sequence of low-level actions.', '1907.02985-1-6-0': 'Dynamic convolutional filters, proposed by Li et al [CITATION], were first conceived to identify and track people by a natural language specification.', '1907.02985-1-6-1': 'They were then successfully employed in other computer vision tasks, such as actor and action video segmentation from a sentence [CITATION].', '1907.02985-1-6-2': 'Nonetheless, these works considered mainly short descriptions, while we deal with complex sentences and long-term dependencies.', '1907.02985-1-6-3': 'We generate dynamic filters according to the given instruction, to extract only the relevant information from the visual context.', '1907.02985-1-6-4': 'In this way, the same observation can lead to different feature maps, depending on the part of the instruction that the agents must complete (Fig. [REF]).', '1907.02985-1-7-0': 'The proposed method is competitive with prior work that performs high-level navigation exploiting information about the reachable viewpoints (i.ethe navigation graph).', '1907.02985-1-7-1': 'Additionally, our approach is fully compliant with recent recommendations for embodied navigation [CITATION].', '1907.02985-1-7-2': 'When compared with models that are compliant with the VLN setup, we overcome the current state of the art by a significant margin.', '1907.02985-1-8-0': 'To sum up, our contributions are as follows:', '1907.02985-1-9-0': '# Method', '1907.02985-1-10-0': 'We propose an encoder-decoder architecture for vision-and-language navigation.', '1907.02985-1-10-1': 'Our work employs dynamic convolutional filters conditioned on the current instruction to extract the relevant piece of information from the visual perception, which is in turn used to feed a policy network which controls the actions performed by the agent.', '1907.02985-1-10-2': 'The output of our model is a probability distribution over a low-level action space [MATH], which comprises the following actions: turn 30 left, turn 30 right, raise elevation, lower elevation, go ahead, <end episode>.', '1907.02985-1-10-3': 'The output probability distribution at a given step, [MATH], depends on a natural language instruction [MATH], the current visual observation [MATH], and on the policy hidden state at time step [MATH].', '1907.02985-1-10-4': 'Our architecture is depicted in Fig. [REF] and detailed next.', '1907.02985-1-11-0': '## Encoder', '1907.02985-1-12-0': 'To represent the two inputs of the architecture, i.ethe instruction and the visual input at time [MATH], we devise an instruction and a visual encoder.', '1907.02985-1-12-1': 'The instruction encoder provides a representation of the navigation instructions that is employed to guide the whole navigation episode.', '1907.02985-1-12-2': 'On the other hand, the visual encoding module operates at every single step, building a representation of the current observation which depends on the agent position.', '1907.02985-1-13-0': 'Instruction Encoding.', '1907.02985-1-13-1': 'The given natural language instruction is split into single words via tokenization, and stop words are filtered out to obtain a shorter description.', '1907.02985-1-13-2': 'Differently from previous works that train word embeddings from scratch, we rely on word embeddings obtained from a large corpus of documents.', '1907.02985-1-13-3': 'Beside providing semantic information which could not be learned purely from VLN instructions, this also let us handle words that are not present in the training set (see Sec. [REF] for a discussion).', '1907.02985-1-13-4': 'Given an instruction with length [MATH], we denote its embeddings sequence as [MATH], where [MATH] indicates the embedding for the [MATH]-th word.', '1907.02985-1-13-5': 'Then, we adopt a Long Short-Term Memory (LSTM) network to provide a timewise contextual representation of the instruction: [EQUATION] where each [MATH] denotes the hidden state of the LSTM at time [MATH], thus leading to a final representation with shape [MATH], where [MATH] is the size of the LSTM hidden state.', '1907.02985-1-14-0': 'Visual Features Encoding.', '1907.02985-1-14-1': 'As visual input, we employ the panoramic 360 view of the agent, and discretize the resulting equirectangular image in a [MATH] grid, consisting of three different elevation levels and 30 heading shift from each other.', '1907.02985-1-14-2': 'Each location of the grid is then encoded via the 2048-dimensional features extracted from a ResNet-152 [CITATION] pre-trained on ImageNet [CITATION].', '1907.02985-1-14-3': 'We also append to each cell vector a set of coordinates relative to the current agent heading and elevation: [EQUATION] where [MATH] and [MATH] are the heading and elevation angles w.r.t. the agent position.', '1907.02985-1-14-4': 'By adding [MATH] to the image feature map, we encode information related to concepts such as right, left, above, below into the agent observation.', '1907.02985-1-15-0': '## Decoder', '1907.02985-1-16-0': 'Given the instruction embedding [MATH] for the whole episode, we use an attention mechanism to select the next part of the sentence that the agent has to fulfill.', '1907.02985-1-16-1': 'We denote this encoded piece of instruction as [MATH].', '1907.02985-1-16-2': 'We detail our attentive module in the next section.', '1907.02985-1-17-0': 'Dynamic Convolutional Filters.', '1907.02985-1-17-1': 'Dynamic filters are different from traditional, fixed filters typically used in CNN, as they depend on an input rather than being purely learnable parameters.', '1907.02985-1-17-2': 'In our case, we can think about them as specialized feature extractors reflecting the semantics of the natural language specification.', '1907.02985-1-17-3': 'For example, starting from an instruction like "head towards the red chair" our model can learn specific filters to focus on concepts such as red and chair.', '1907.02985-1-17-4': 'In this way, our model can rely on a large ensemble of specialized kernels and apply only the most suitable ones, depending on the current goal.', '1907.02985-1-17-5': 'Naturally, this approach is more efficient and flexible than learning a fixed set of filters for all the navigation steps.', '1907.02985-1-17-6': 'We use the representation of the current piece of instruction [MATH] to generate multiple [MATH] dynamic convolutional kernels, according to the following equation: [EQUATION] where [MATH] indicates L2 normalization, and [MATH] is a tensor of filters reshaped to have the same number of channels as the image feature map.', '1907.02985-1-17-7': 'We then perform the dynamic convolution over the image features [MATH], thus obtaining a response map for the current timestep as follows: [EQUATION]', '1907.02985-1-17-8': 'As the aforementioned operation is equivalent to a dot product, we can conceive the dynamic convolution as a specialized form of dot-product attention, in which [MATH] acts as key and the filters in [MATH] act as time-varying queries.', '1907.02985-1-17-9': 'Following this interpretation, we rescale [MATH] by [MATH], where [MATH] is the dynamic filter size [CITATION] to maintain dot products smaller in magnitude.', '1907.02985-1-18-0': 'Action Selection.', '1907.02985-1-18-1': 'We use the response maps dynamically generated as input for the policy network.', '1907.02985-1-18-2': 'We implement it with an LSTM whose hidden state at time step [MATH] is employed to obtain the action scores.', '1907.02985-1-18-3': 'Formally, [EQUATION] where [MATH] indicates concatenation, [MATH] is the one-hot encoding of the action performed at the previous timestep, and [MATH] is the flattened tensor obtained from [MATH].', '1907.02985-1-18-4': 'To select the next action [MATH], we sample from a multinomial distribution parametrized with the output probability distribution during training, and select [MATH] during the test.', '1907.02985-1-18-5': 'In line with previous work, we find out that sampling during the training phase encourages exploration and improves overall performances.', '1907.02985-1-19-0': 'Note that, as previously stated, we do not employ a high-level action space, where the agent selects the next viewpoint in the image feature map, but instead make the agent responsible of learning the sequence of low-level actions needed to perform the navigation.', '1907.02985-1-19-1': 'The agent can additionally send a specific stop signal when it considers the goal reached, as suggested by recent standardization attempts [CITATION].', '1907.02985-1-20-0': '## Encoder-Decoder Attention', '1907.02985-1-21-0': 'The navigation instructions are very complex, as they involve not only different actions but also temporal dependencies between them.', '1907.02985-1-21-1': 'Moreover, their high average length represents an additional challenge for traditional embedding methods.', '1907.02985-1-21-2': 'For these reasons, we enrich our architecture with a mechanism to attend different locations of the sentence representation, as the navigation moves towards the goal.', '1907.02985-1-21-3': 'In line with previous work on VLN [CITATION], we employ an attention mechanism to identify the most relevant parts of the navigation instruction.', '1907.02985-1-21-4': 'We employ the hidden state of our policy LSTM to get the information about our progress in the navigation episode and extract a time-varying query [MATH].', '1907.02985-1-21-5': 'We then project our sentence embedding into a lower dimensional space to obtain key vectors, and perform a scaled dot-product attention [CITATION] among them.', '1907.02985-1-21-6': '[EQUATION]', '1907.02985-1-21-7': 'After a softmax layer, we obtain the current instruction embedding [MATH] by matrix multiplication between the initial sentence embedding and the softmax scores.', '1907.02985-1-21-8': '[EQUATION]', '1907.02985-1-21-9': 'At each timestep of the navigation process [MATH] is obtained by attending the instruction embedding at different locations.', '1907.02985-1-21-10': 'The same vector is in turn used to obtain a time-varying query for attending spatial locations in the visual input.', '1907.02985-1-22-0': '## Training', '1907.02985-1-23-0': 'Our training sample consists of a batch of navigation instructions and the corresponding ground truth paths coming from the R2R (Room-to-Room) dataset [CITATION] (described in section [REF]).', '1907.02985-1-23-1': 'The path denotes a list of discretized viewpoints that the agent has to traverse to progress towards the goal.', '1907.02985-1-23-2': 'The agent spawns in the first viewpoint, and its goal is to reach the last viewpoint in the ground truth list.', '1907.02985-1-23-3': 'At each step, the simulator is responsible for providing the next ground truth action in the low-level action space that enables the agent to progress.', '1907.02985-1-23-4': 'Specifically, the ground truth action is computed by comparing the coordinates of the next target node in the navigation graph with the agent position and orientation.', '1907.02985-1-23-5': 'At each time step [MATH], we minimize the following objective function: [EQUATION] where [MATH] is the output of our network, and [MATH] is the ground truth low-level action provided by the simulator at time step [MATH].', '1907.02985-1-23-6': 'We train our network with a batch size of 128 and use Adam optimizer [CITATION] with a learning rate of [MATH].', '1907.02985-1-23-7': 'We adopt early stopping to terminate the training if the mean success rate does not improve for 10 epochs.', '1907.02985-1-24-0': '# Experiments', '1907.02985-1-25-0': '## Experimental Settings', '1907.02985-1-26-0': 'For our experiments, we employ the R2R (Room-to-Room) dataset [CITATION].', '1907.02985-1-26-1': 'This challenging benchmark builds upon Matterport3D dataset of spaces [CITATION] and contains [MATH] different navigation paths in [MATH] different scenes.', '1907.02985-1-26-2': 'For each route, the dataset provides 3 natural language instructions, for a total of 21,567 instructions with an average length of 29 words.', '1907.02985-1-26-3': 'The R2R dataset is split into 4 partitions: training, validation on seen environments, validation on unseen scenes, and test on unseen environments.', '1907.02985-1-27-0': 'Evaluation Metrics.', '1907.02985-1-27-1': 'We adopt the same evaluation metrics employed by previous work on the R2R dataset: navigation error (NE), oracle success rate (OSR), success rate (SR), and success rate weighted by path length (SPL).', '1907.02985-1-27-2': 'NE is the mean distance in meters between the final position and the goal.', '1907.02985-1-27-3': 'SR is fraction of episodes terminated within no more than 3 meters from the goal position.', '1907.02985-1-27-4': 'OSR is the success rate that the agent would have achieved if it received an oracle stop signal in the closest point to the goal along its navigation.', '1907.02985-1-27-5': 'SPL is the success rate weighted by normalized inverse path length and penalizes overlong navigations.', '1907.02985-1-28-0': 'Implementation Details.', '1907.02985-1-28-1': 'For each LSTM, we set the hidden size to 512.', '1907.02985-1-28-2': 'Word embeddings are obtained with GloVe [CITATION].', '1907.02985-1-28-3': 'In our visual encoder, we apply a bottleneck layer to reduce the dimension of the image feature map to 512.', '1907.02985-1-28-4': 'We generate dynamic filters with 512 channels using a linear layer with dropout [CITATION] ([MATH]).', '1907.02985-1-28-5': 'In our attention module, [MATH] and [MATH] have 128 channels and we apply a ReLU non-linearity after the linear transformation.', '1907.02985-1-28-6': 'For our action selection, we apply dropout with [MATH] to the policy hidden state before feeding it to the linear layer.', '1907.02985-1-29-0': '## Ablation Study', '1907.02985-1-30-0': 'In our ablation study, we test the influence of our implementation choices on VLN.', '1907.02985-1-30-1': 'As a first step, we discuss the impact of dynamic convolution by comparing our model with a similar seq2seq architecture that employs fixed convolutional filters.', '1907.02985-1-30-2': 'We then detail the importance of using an attention mechanism to extract the current piece of instruction to be fulfilled.', '1907.02985-1-30-3': 'Finally, we compare the results obtained using a pre-trained word embedding instead of learning the word representation from scratch.', '1907.02985-1-30-4': 'Results are reported in Table [REF].', '1907.02985-1-31-0': 'Static Filters Vs. Dynamic Convolution.', '1907.02985-1-31-1': 'As results show, dynamic convolutional filters surpass traditional fixed filters for VLN.', '1907.02985-1-31-2': 'This because they can easily adapt to new instructions and reflect the variability of the task.', '1907.02985-1-31-3': 'When compared to a baseline model that employs traditional convolution [CITATION], our method performs [MATH] and [MATH] better, in terms of success rate, on the val-seen and val-unseen splits respectively.', '1907.02985-1-32-0': 'Fixed Instruction Representation Vs. Attention.', '1907.02985-1-32-1': 'The navigation instructions are very complex and rich.', '1907.02985-1-32-2': 'When removing the attention module from our architecture, we keep the last hidden state [MATH] as instruction representation for the whole episode.', '1907.02985-1-32-3': 'Even with this limitation, dynamic filters achieve better results than static convolution, as the success rate is higher for both of the validation splits.', '1907.02985-1-32-4': 'However, our attention module further increases the success rate by [MATH] and [MATH].', '1907.02985-1-33-0': 'Word Embedding from Scratch Vs. Pre-trained Embedding.', '1907.02985-1-33-1': 'Learning a meaningful word embedding is nontrivial and requires a large corpus of natural language descriptions.', '1907.02985-1-33-2': 'For this reason, we adopt a pre-trained word embedding to encode single words in our instructions.', '1907.02985-1-33-3': 'We then run the same model while trying to learn the word embedding from scratch.', '1907.02985-1-33-4': 'We discover that a pre-trained word embedding significantly eases VLN.', '1907.02985-1-33-5': 'Our model with GloVe [CITATION] obtains [MATH] and [MATH] more on the val-seen and val-unseen splits respectively, in terms of success rate.', '1907.02985-1-34-0': '## Multi-headed Dynamic Convolution', '1907.02985-1-35-0': 'In this experiment, we test the impact of using a different number of dynamically-generated filters.', '1907.02985-1-35-1': 'We test our architecture when using 1, 2, 4, 8, and 16 dynamic filters.', '1907.02985-1-35-2': 'We find out that the best setup corresponds to the use of 4 different convolutional filters.', '1907.02985-1-35-3': 'Results in Fig. [REF] show that the success rate and the SPL increase linearly with the number of dynamic kernels for a small number of filters, reaching a maximum at 4.', '1907.02985-1-35-4': 'The metrics then decrease when adding new parameters to the network.', '1907.02985-1-35-5': 'This suggests that a low number of dynamic filters can represent a wide variety of natural language specifications.', '1907.02985-1-35-6': 'However, as the number of dynamic filters increase, the representation provided by the convolution becomes less efficient.', '1907.02985-1-36-0': '## Comparison with the State-of-the-art', '1907.02985-1-37-0': 'Finally, we compare our architecture with the state-of-the-art methods for VLN.', '1907.02985-1-37-1': 'Results are reported in Table [REF].', '1907.02985-1-37-2': 'We distinguish two main categories of models, depending on their output space: the first, to which our approach belongs, predicts the next atomic action (e.gturn right, go ahead).', '1907.02985-1-37-3': 'We call architectures in this category low-level actions methods.', '1907.02985-1-37-4': 'The second, instead, searches in the visual space to match the current instruction with the most suitable navigable viewpoint.', '1907.02985-1-37-5': 'In these models, atomic actions are not considered, as the agent displacements are done with a teleport system, using the next viewpoint identifier as target destination.', '1907.02985-1-37-6': 'Hence, we refer to these works as high-level actions methods.', '1907.02985-1-37-7': 'While the latter achieve better results, they make strong assumptions on the underlying simulating platform and on the navigation graph.', '1907.02985-1-37-8': 'Our method, exploiting dynamic convolutional filters and predicting atomic actions, outperforms comparable architectures and achieves state of the art results for low-level actions VLN.', '1907.02985-1-37-9': 'Our final implementation takes advantage of the synthetic data provided by Fried et al [CITATION] and overcomes comparable methods [CITATION] by [MATH] and [MATH] success rate points on the R2R test set.', '1907.02985-1-37-10': 'Additionally, we note that our method is competitive with some high-level actions models, especially in terms of SPL.', '1907.02985-1-37-11': 'When considering the test set, we notice in fact that our model outperforms Speaker-Follower [CITATION] by [MATH], while performing only [MATH] worse than [CITATION].', '1907.02985-1-38-0': 'Low-level Action Space or High-level Navigation Space?', '1907.02985-1-38-1': 'While previous work on VLN never considered this important difference, we claim that it is imperative to categorize navigation architectures depending on their output space.', '1907.02985-1-38-2': 'In our opinion, ignoring this aspect would lead to inappropriate comparisons and wrong conclusions.', '1907.02985-1-38-3': 'Considering the results in Table [REF], we separate the two classes of work and highlight the best results for each category.', '1907.02985-1-38-4': 'Please note that the random baseline was initially provided by [CITATION] and belongs to low-level actions architectures (a random high-level actions agent was never provided by previous work).', '1907.02985-1-38-5': 'We immediately notice that, with this new categorization, intra-class results have less variance and are much more aligned to each other.', '1907.02985-1-38-6': 'We believe that future work on VLN should consider this new taxonomy in order to provide meaningful and fair comparisons.', '1907.02985-1-39-0': '## Qualitative Results', '1907.02985-1-40-0': 'Fig. [REF] shows two navigation episodes from the R2R validation set.', '1907.02985-1-40-1': 'We display the predicted action in a green box on the bottom-right corner of each image.', '1907.02985-1-40-2': 'Both examples are successful.', '1907.02985-1-41-0': '# Conclusion'}

{'1907.02985-2-0-0': 'In Vision-and-Language Navigation (VLN), an embodied agent needs to reach a target destination with the only guidance of a natural language instruction.', '1907.02985-2-0-1': 'To explore the environment and progress towards the target location, the agent must perform a series of low-level actions, such as rotate, before stepping ahead.', '1907.02985-2-0-2': 'In this paper, we propose to exploit dynamic convolutional filters to encode the visual information and the lingual description in an efficient way.', '1907.02985-2-0-3': 'Differently from some previous works that abstract from the agent perspective and use high-level navigation spaces, we design a policy which decodes the information provided by dynamic convolution into a series of low-level, agent friendly actions.', '1907.02985-2-0-4': 'Results show that our model exploiting dynamic filters performs better than other architectures with traditional convolution, being the new state of the art for embodied VLN in the low-level action space.', '1907.02985-2-0-5': 'Additionally, we attempt to categorize recent work on VLN depending on their architectural choices and distinguish two main groups: we call them low-level actions and high-level actions models.', '1907.02985-2-0-6': 'To the best of our knowledge, we are the first to propose this analysis and categorization for VLN.', '1907.02985-2-0-7': 'Code is available at https://github.com/aimagelab/DynamicConv-agent.', '1907.02985-2-1-0': '# Introduction', '1907.02985-2-2-0': 'Imagine finding yourself in a large conference hall, with an assistant giving you instructions on how to reach the room for your talk.', '1907.02985-2-2-1': 'You are likely to hear something like: turn right at the end of the corridor, head upstairs and reach the third floor: your room is immediately on the left.', '1907.02985-2-2-2': 'Succeeding in the task of finding your target location is rather nontrivial because of the length of the instruction and its sequential nature: the flow of actions must be coordinated with a series of visual examinations - like recognizing the end of the corridor or the floor number.', '1907.02985-2-2-3': 'Furthermore, navigation complexity dramatically increases if the environment is unknown, and no prior knowledge, such as a map, is available.', '1907.02985-2-3-0': 'Vision-and-Language Navigation (VLN) [CITATION] is a challenging task that demands an embodied agent to reach a target location by navigating unseen environments, with a natural language instruction as its only clue.', '1907.02985-2-3-1': 'Similarly to the previous example, the agent must assess different sub-tasks to succeed.', '1907.02985-2-3-2': 'First, a fine-grained comprehension of the given instruction is needed.', '1907.02985-2-3-3': 'Then, the agent must be able to map parts of the description into the visual perception.', '1907.02985-2-3-4': 'For example, walking past the piano requires to find and focus on the piano, rather than considering other objects in the scene.', '1907.02985-2-3-5': 'Finally, the agent needs to understand when the navigation has been completed and send a stop signal.', '1907.02985-2-4-0': 'VLN has been first proposed by Anderson et al [CITATION], with the aim of connecting the research efforts on vision-and-language understanding [CITATION] with the raising area of embodied AI [CITATION].', '1907.02985-2-4-1': 'This is particularly challenging, as embodied agents must deal with a series of issues that do not belong to traditional vision and language tasks [CITATION], like contextual decision-making and planning.', '1907.02985-2-4-2': 'Recent works on VLN [CITATION] integrate the agent with a simplified action space in which it "only needs to make high-level decisions as to which navigable direction to go next" [CITATION].', '1907.02985-2-4-3': 'In this scenario, the agent does not need to infer the sequence of actions to progress in the environment (e.g., turn right 30 degrees, then move forward) but it exploits a navigation graph to teleport itself to an adjacent location.', '1907.02985-2-4-4': 'The adoption of this high-level action space allowed for a significant boost in success rates, while partly depriving the task of its embodied nature, and leaving space for little more than pure visual and language understanding.', '1907.02985-2-4-5': 'We claim that this type of approach is inconvenient, as it strongly relies on prior knowledge on the environment.', '1907.02985-2-4-6': 'Depending on information such as the position and the availability of navigable directions, it reduces the task to a pure graph navigation.', '1907.02985-2-4-7': 'Moreover, it ignores the active role of the agent, as it only perceives the surrounding scene and selects the next target viewpoint from a limited set.', '1907.02985-2-4-8': 'We claim instead that the agent should be the principal component of embodied VLN [CITATION].', '1907.02985-2-4-9': 'Consequently, the output space should match with the low-level set of movements that the agent can perform.', '1907.02985-2-5-0': 'In this paper, we propose a novel architecture for embodied VLN which employs dynamic convolutional filters [CITATION] to identify the next target direction, without getting any information about the navigable viewpoints from the simulator.', '1907.02985-2-5-1': 'Convolutional filters are produced via an attention mechanism which follows the given instruction, and are in turn used to attend relevant directions of the scene towards which the agent should move.', '1907.02985-2-5-2': 'We then rely on a policy network to predict the sequence of low-level actions.', '1907.02985-2-6-0': 'Dynamic convolutional filters, proposed by Li et al [CITATION], were first conceived to identify and track people by a natural language specification.', '1907.02985-2-6-1': 'They were then successfully employed in other computer vision tasks, such as actor and action video segmentation from a sentence [CITATION].', '1907.02985-2-6-2': 'Nonetheless, these works considered mainly short descriptions, while we deal with complex sentences and long-term dependencies.', '1907.02985-2-6-3': 'We generate dynamic filters according to the given instruction, to extract only the relevant information from the visual context.', '1907.02985-2-6-4': 'In this way, the same observation can lead to different feature maps, depending on the part of the instruction that the agents must complete (Fig. [REF]).', '1907.02985-2-7-0': 'The proposed method is competitive with prior work that performs high-level navigation exploiting information about the reachable viewpoints (i.ethe navigation graph).', '1907.02985-2-7-1': 'Additionally, our approach is fully compliant with recent recommendations for embodied navigation [CITATION].', '1907.02985-2-7-2': 'When compared with models that are compliant with the VLN setup, we overcome the current state of the art by a significant margin.', '1907.02985-2-8-0': 'To sum up, our contributions are as follows:', '1907.02985-2-9-0': '# Method', '1907.02985-2-10-0': 'We propose an encoder-decoder architecture for vision-and-language navigation.', '1907.02985-2-10-1': 'Our work employs dynamic convolutional filters conditioned on the current instruction to extract the relevant piece of information from the visual perception, which is in turn used to feed a policy network which controls the actions performed by the agent.', '1907.02985-2-10-2': 'The output of our model is a probability distribution over a low-level action space [MATH], which comprises the following actions: turn 30 left, turn 30 right, raise elevation, lower elevation, go ahead, <end episode>.', '1907.02985-2-10-3': 'The output probability distribution at a given step, [MATH], depends on a natural language instruction [MATH], the current visual observation [MATH], and on the policy hidden state at time step [MATH].', '1907.02985-2-10-4': 'Our architecture is depicted in Fig. [REF] and detailed next.', '1907.02985-2-11-0': '## Encoder', '1907.02985-2-12-0': 'To represent the two inputs of the architecture, i.ethe instruction and the visual input at time [MATH], we devise an instruction and a visual encoder.', '1907.02985-2-12-1': 'The instruction encoder provides a representation of the navigation instructions that is employed to guide the whole navigation episode.', '1907.02985-2-12-2': 'On the other hand, the visual encoding module operates at every single step, building a representation of the current observation which depends on the agent position.', '1907.02985-2-13-0': 'Instruction Encoding.', '1907.02985-2-13-1': 'The given natural language instruction is split into single words via tokenization, and stop words are filtered out to obtain a shorter description.', '1907.02985-2-13-2': 'Differently from previous works that train word embeddings from scratch, we rely on word embeddings obtained from a large corpus of documents.', '1907.02985-2-13-3': 'Beside providing semantic information which could not be learned purely from VLN instructions, this also let us handle words that are not present in the training set (see Sec. [REF] for a discussion).', '1907.02985-2-13-4': 'Given an instruction with length [MATH], we denote its embeddings sequence as [MATH], where [MATH] indicates the embedding for the [MATH]-th word.', '1907.02985-2-13-5': 'Then, we adopt a Long Short-Term Memory (LSTM) network to provide a timewise contextual representation of the instruction: [EQUATION] where each [MATH] denotes the hidden state of the LSTM at time [MATH], thus leading to a final representation with shape [MATH], where [MATH] is the size of the LSTM hidden state.', '1907.02985-2-14-0': 'Visual Features Encoding.', '1907.02985-2-14-1': 'As visual input, we employ the panoramic 360 view of the agent, and discretize the resulting equirectangular image in a [MATH] grid, consisting of three different elevation levels and 30 heading shift from each other.', '1907.02985-2-14-2': 'Each location of the grid is then encoded via the 2048-dimensional features extracted from a ResNet-152 [CITATION] pre-trained on ImageNet [CITATION].', '1907.02985-2-14-3': 'We also append to each cell vector a set of coordinates relative to the current agent heading and elevation: [EQUATION] where [MATH] and [MATH] are the heading and elevation angles w.r.t. the agent position.', '1907.02985-2-14-4': 'By adding [MATH] to the image feature map, we encode information related to concepts such as right, left, above, below into the agent observation.', '1907.02985-2-15-0': '## Decoder', '1907.02985-2-16-0': 'Given the instruction embedding [MATH] for the whole episode, we use an attention mechanism to select the next part of the sentence that the agent has to fulfill.', '1907.02985-2-16-1': 'We denote this encoded piece of instruction as [MATH].', '1907.02985-2-16-2': 'We detail our attentive module in the next section.', '1907.02985-2-17-0': 'Dynamic Convolutional Filters.', '1907.02985-2-17-1': 'Dynamic filters are different from traditional, fixed filters typically used in CNN, as they depend on an input rather than being purely learnable parameters.', '1907.02985-2-17-2': 'In our case, we can think about them as specialized feature extractors reflecting the semantics of the natural language specification.', '1907.02985-2-17-3': 'For example, starting from an instruction like "head towards the red chair" our model can learn specific filters to focus on concepts such as red and chair.', '1907.02985-2-17-4': 'In this way, our model can rely on a large ensemble of specialized kernels and apply only the most suitable ones, depending on the current goal.', '1907.02985-2-17-5': 'Naturally, this approach is more efficient and flexible than learning a fixed set of filters for all the navigation steps.', '1907.02985-2-17-6': 'We use the representation of the current piece of instruction [MATH] to generate multiple [MATH] dynamic convolutional kernels, according to the following equation: [EQUATION] where [MATH] indicates L2 normalization, and [MATH] is a tensor of filters reshaped to have the same number of channels as the image feature map.', '1907.02985-2-17-7': 'We then perform the dynamic convolution over the image features [MATH], thus obtaining a response map for the current timestep as follows: [EQUATION]', '1907.02985-2-17-8': 'As the aforementioned operation is equivalent to a dot product, we can conceive the dynamic convolution as a specialized form of dot-product attention, in which [MATH] acts as key and the filters in [MATH] act as time-varying queries.', '1907.02985-2-17-9': 'Following this interpretation, we rescale [MATH] by [MATH], where [MATH] is the dynamic filter size [CITATION] to maintain dot products smaller in magnitude.', '1907.02985-2-18-0': 'Action Selection.', '1907.02985-2-18-1': 'We use the response maps dynamically generated as input for the policy network.', '1907.02985-2-18-2': 'We implement it with an LSTM whose hidden state at time step [MATH] is employed to obtain the action scores.', '1907.02985-2-18-3': 'Formally, [EQUATION] where [MATH] indicates concatenation, [MATH] is the one-hot encoding of the action performed at the previous timestep, and [MATH] is the flattened tensor obtained from [MATH].', '1907.02985-2-18-4': 'To select the next action [MATH], we sample from a multinomial distribution parametrized with the output probability distribution during training, and select [MATH] during the test.', '1907.02985-2-18-5': 'In line with previous work, we find out that sampling during the training phase encourages exploration and improves overall performances.', '1907.02985-2-19-0': 'Note that, as previously stated, we do not employ a high-level action space, where the agent selects the next viewpoint in the image feature map, but instead make the agent responsible of learning the sequence of low-level actions needed to perform the navigation.', '1907.02985-2-19-1': 'The agent can additionally send a specific stop signal when it considers the goal reached, as suggested by recent standardization attempts [CITATION].', '1907.02985-2-20-0': '## Encoder-Decoder Attention', '1907.02985-2-21-0': 'The navigation instructions are very complex, as they involve not only different actions but also temporal dependencies between them.', '1907.02985-2-21-1': 'Moreover, their high average length represents an additional challenge for traditional embedding methods.', '1907.02985-2-21-2': 'For these reasons, we enrich our architecture with a mechanism to attend different locations of the sentence representation, as the navigation moves towards the goal.', '1907.02985-2-21-3': 'In line with previous work on VLN [CITATION], we employ an attention mechanism to identify the most relevant parts of the navigation instruction.', '1907.02985-2-21-4': 'We employ the hidden state of our policy LSTM to get the information about our progress in the navigation episode and extract a time-varying query [MATH].', '1907.02985-2-21-5': 'We then project our sentence embedding into a lower dimensional space to obtain key vectors, and perform a scaled dot-product attention [CITATION] among them.', '1907.02985-2-21-6': '[EQUATION]', '1907.02985-2-21-7': 'After a softmax layer, we obtain the current instruction embedding [MATH] by matrix multiplication between the initial sentence embedding and the softmax scores.', '1907.02985-2-21-8': '[EQUATION]', '1907.02985-2-21-9': 'At each timestep of the navigation process [MATH] is obtained by attending the instruction embedding at different locations.', '1907.02985-2-21-10': 'The same vector is in turn used to obtain a time-varying query for attending spatial locations in the visual input.', '1907.02985-2-22-0': '## Training', '1907.02985-2-23-0': 'Our training sample consists of a batch of navigation instructions and the corresponding ground truth paths coming from the R2R (Room-to-Room) dataset [CITATION] (described in section [REF]).', '1907.02985-2-23-1': 'The path denotes a list of discretized viewpoints that the agent has to traverse to progress towards the goal.', '1907.02985-2-23-2': 'The agent spawns in the first viewpoint, and its goal is to reach the last viewpoint in the ground truth list.', '1907.02985-2-23-3': 'At each step, the simulator is responsible for providing the next ground truth action in the low-level action space that enables the agent to progress.', '1907.02985-2-23-4': 'Specifically, the ground truth action is computed by comparing the coordinates of the next target node in the navigation graph with the agent position and orientation.', '1907.02985-2-23-5': 'At each time step [MATH], we minimize the following objective function: [EQUATION] where [MATH] is the output of our network, and [MATH] is the ground truth low-level action provided by the simulator at time step [MATH].', '1907.02985-2-23-6': 'We train our network with a batch size of 128 and use Adam optimizer [CITATION] with a learning rate of [MATH].', '1907.02985-2-23-7': 'We adopt early stopping to terminate the training if the mean success rate does not improve for 10 epochs.', '1907.02985-2-24-0': '# Experiments', '1907.02985-2-25-0': '## Experimental Settings', '1907.02985-2-26-0': 'For our experiments, we employ the R2R (Room-to-Room) dataset [CITATION].', '1907.02985-2-26-1': 'This challenging benchmark builds upon Matterport3D dataset of spaces [CITATION] and contains [MATH] different navigation paths in [MATH] different scenes.', '1907.02985-2-26-2': 'For each route, the dataset provides 3 natural language instructions, for a total of 21,567 instructions with an average length of 29 words.', '1907.02985-2-26-3': 'The R2R dataset is split into 4 partitions: training, validation on seen environments, validation on unseen scenes, and test on unseen environments.', '1907.02985-2-27-0': 'Evaluation Metrics.', '1907.02985-2-27-1': 'We adopt the same evaluation metrics employed by previous work on the R2R dataset: navigation error (NE), oracle success rate (OSR), success rate (SR), and success rate weighted by path length (SPL).', '1907.02985-2-27-2': 'NE is the mean distance in meters between the final position and the goal.', '1907.02985-2-27-3': 'SR is fraction of episodes terminated within no more than 3 meters from the goal position.', '1907.02985-2-27-4': 'OSR is the success rate that the agent would have achieved if it received an oracle stop signal in the closest point to the goal along its navigation.', '1907.02985-2-27-5': 'SPL is the success rate weighted by normalized inverse path length and penalizes overlong navigations.', '1907.02985-2-28-0': 'Implementation Details.', '1907.02985-2-28-1': 'For each LSTM, we set the hidden size to 512.', '1907.02985-2-28-2': 'Word embeddings are obtained with GloVe [CITATION].', '1907.02985-2-28-3': 'In our visual encoder, we apply a bottleneck layer to reduce the dimension of the image feature map to 512.', '1907.02985-2-28-4': 'We generate dynamic filters with 512 channels using a linear layer with dropout [CITATION] ([MATH]).', '1907.02985-2-28-5': 'In our attention module, [MATH] and [MATH] have 128 channels and we apply a ReLU non-linearity after the linear transformation.', '1907.02985-2-28-6': 'For our action selection, we apply dropout with [MATH] to the policy hidden state before feeding it to the linear layer.', '1907.02985-2-29-0': '## Ablation Study', '1907.02985-2-30-0': 'In our ablation study, we test the influence of our implementation choices on VLN.', '1907.02985-2-30-1': 'As a first step, we discuss the impact of dynamic convolution by comparing our model with a similar seq2seq architecture that employs fixed convolutional filters.', '1907.02985-2-30-2': 'We then detail the importance of using an attention mechanism to extract the current piece of instruction to be fulfilled.', '1907.02985-2-30-3': 'Finally, we compare the results obtained using a pre-trained word embedding instead of learning the word representation from scratch.', '1907.02985-2-30-4': 'Results are reported in Table [REF].', '1907.02985-2-31-0': 'Static Filters Vs. Dynamic Convolution.', '1907.02985-2-31-1': 'As results show, dynamic convolutional filters surpass traditional fixed filters for VLN.', '1907.02985-2-31-2': 'This because they can easily adapt to new instructions and reflect the variability of the task.', '1907.02985-2-31-3': 'When compared to a baseline model that employs traditional convolution [CITATION], our method performs [MATH] and [MATH] better, in terms of success rate, on the val-seen and val-unseen splits respectively.', '1907.02985-2-32-0': 'Fixed Instruction Representation Vs. Attention.', '1907.02985-2-32-1': 'The navigation instructions are very complex and rich.', '1907.02985-2-32-2': 'When removing the attention module from our architecture, we keep the last hidden state [MATH] as instruction representation for the whole episode.', '1907.02985-2-32-3': 'Even with this limitation, dynamic filters achieve better results than static convolution, as the success rate is higher for both of the validation splits.', '1907.02985-2-32-4': 'However, our attention module further increases the success rate by [MATH] and [MATH].', '1907.02985-2-33-0': 'Word Embedding from Scratch Vs. Pre-trained Embedding.', '1907.02985-2-33-1': 'Learning a meaningful word embedding is nontrivial and requires a large corpus of natural language descriptions.', '1907.02985-2-33-2': 'For this reason, we adopt a pre-trained word embedding to encode single words in our instructions.', '1907.02985-2-33-3': 'We then run the same model while trying to learn the word embedding from scratch.', '1907.02985-2-33-4': 'We discover that a pre-trained word embedding significantly eases VLN.', '1907.02985-2-33-5': 'Our model with GloVe [CITATION] obtains [MATH] and [MATH] more on the val-seen and val-unseen splits respectively, in terms of success rate.', '1907.02985-2-34-0': '## Multi-headed Dynamic Convolution', '1907.02985-2-35-0': 'In this experiment, we test the impact of using a different number of dynamically-generated filters.', '1907.02985-2-35-1': 'We test our architecture when using 1, 2, 4, 8, and 16 dynamic filters.', '1907.02985-2-35-2': 'We find out that the best setup corresponds to the use of 4 different convolutional filters.', '1907.02985-2-35-3': 'Results in Fig. [REF] show that the success rate and the SPL increase linearly with the number of dynamic kernels for a small number of filters, reaching a maximum at 4.', '1907.02985-2-35-4': 'The metrics then decrease when adding new parameters to the network.', '1907.02985-2-35-5': 'This suggests that a low number of dynamic filters can represent a wide variety of natural language specifications.', '1907.02985-2-35-6': 'However, as the number of dynamic filters increase, the representation provided by the convolution becomes less efficient.', '1907.02985-2-36-0': '## Comparison with the State-of-the-art', '1907.02985-2-37-0': 'Finally, we compare our architecture with the state-of-the-art methods for VLN.', '1907.02985-2-37-1': 'Results are reported in Table [REF].', '1907.02985-2-37-2': 'We distinguish two main categories of models, depending on their output space: the first, to which our approach belongs, predicts the next atomic action (e.gturn right, go ahead).', '1907.02985-2-37-3': 'We call architectures in this category low-level actions methods.', '1907.02985-2-37-4': 'The second, instead, searches in the visual space to match the current instruction with the most suitable navigable viewpoint.', '1907.02985-2-37-5': 'In these models, atomic actions are not considered, as the agent displacements are done with a teleport system, using the next viewpoint identifier as target destination.', '1907.02985-2-37-6': 'Hence, we refer to these works as high-level actions methods.', '1907.02985-2-37-7': 'While the latter achieve better results, they make strong assumptions on the underlying simulating platform and on the navigation graph.', '1907.02985-2-37-8': 'Our method, exploiting dynamic convolutional filters and predicting atomic actions, outperforms comparable architectures and achieves state of the art results for low-level actions VLN.', '1907.02985-2-37-9': 'Our final implementation takes advantage of the synthetic data provided by Fried et al [CITATION] and overcomes comparable methods [CITATION] by [MATH] and [MATH] success rate points on the R2R test set.', '1907.02985-2-37-10': 'Additionally, we note that our method is competitive with some high-level actions models, especially in terms of SPL.', '1907.02985-2-37-11': 'When considering the test set, we notice in fact that our model outperforms Speaker-Follower [CITATION] by [MATH], while performing only [MATH] worse than [CITATION].', '1907.02985-2-38-0': 'Low-level Action Space or High-level Navigation Space?', '1907.02985-2-38-1': 'While previous work on VLN never considered this important difference, we claim that it is imperative to categorize navigation architectures depending on their output space.', '1907.02985-2-38-2': 'In our opinion, ignoring this aspect would lead to inappropriate comparisons and wrong conclusions.', '1907.02985-2-38-3': 'Considering the results in Table [REF], we separate the two classes of work and highlight the best results for each category.', '1907.02985-2-38-4': 'Please note that the random baseline was initially provided by [CITATION] and belongs to low-level actions architectures (a random high-level actions agent was never provided by previous work).', '1907.02985-2-38-5': 'We immediately notice that, with this new categorization, intra-class results have less variance and are much more aligned to each other.', '1907.02985-2-38-6': 'We believe that future work on VLN should consider this new taxonomy in order to provide meaningful and fair comparisons.', '1907.02985-2-39-0': '## Qualitative Results', '1907.02985-2-40-0': 'Fig. [REF] shows two navigation episodes from the R2R validation set.', '1907.02985-2-40-1': 'We display the predicted action in a green box on the bottom-right corner of each image.', '1907.02985-2-40-2': 'Both examples are successful.', '1907.02985-2-41-0': '# Conclusion'}

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['1907.02985-1-35-0', '1907.02985-2-35-0'], ['1907.02985-1-35-1', '1907.02985-2-35-1'], ['1907.02985-1-35-2', '1907.02985-2-35-2'], ['1907.02985-1-35-3', '1907.02985-2-35-3'], ['1907.02985-1-35-4', '1907.02985-2-35-4'], ['1907.02985-1-35-5', '1907.02985-2-35-5'], ['1907.02985-1-35-6', '1907.02985-2-35-6'], ['1907.02985-1-4-0', '1907.02985-2-4-0'], ['1907.02985-1-4-1', '1907.02985-2-4-1'], ['1907.02985-1-4-2', '1907.02985-2-4-2'], ['1907.02985-1-4-3', '1907.02985-2-4-3'], ['1907.02985-1-4-4', '1907.02985-2-4-4'], ['1907.02985-1-4-5', '1907.02985-2-4-5'], ['1907.02985-1-4-6', '1907.02985-2-4-6'], ['1907.02985-1-4-7', '1907.02985-2-4-7'], ['1907.02985-1-4-8', '1907.02985-2-4-8'], ['1907.02985-1-4-9', '1907.02985-2-4-9'], ['1907.02985-1-5-0', '1907.02985-2-5-0'], ['1907.02985-1-5-1', '1907.02985-2-5-1'], ['1907.02985-1-5-2', '1907.02985-2-5-2'], ['1907.02985-1-16-0', '1907.02985-2-16-0'], ['1907.02985-1-16-1', '1907.02985-2-16-1'], ['1907.02985-1-16-2', '1907.02985-2-16-2'], ['1907.02985-1-27-1', '1907.02985-2-27-1'], ['1907.02985-1-27-2', '1907.02985-2-27-2'], ['1907.02985-1-27-3', '1907.02985-2-27-3'], ['1907.02985-1-27-4', '1907.02985-2-27-4'], ['1907.02985-1-27-5', '1907.02985-2-27-5'], ['1907.02985-1-18-1', '1907.02985-2-18-1'], ['1907.02985-1-18-2', '1907.02985-2-18-2'], ['1907.02985-1-18-3', '1907.02985-2-18-3'], ['1907.02985-1-18-4', '1907.02985-2-18-4'], ['1907.02985-1-18-5', '1907.02985-2-18-5'], ['1907.02985-1-19-0', '1907.02985-2-19-0'], ['1907.02985-1-19-1', '1907.02985-2-19-1'], ['1907.02985-1-21-0', '1907.02985-2-21-0'], ['1907.02985-1-21-1', '1907.02985-2-21-1'], ['1907.02985-1-21-2', '1907.02985-2-21-2'], ['1907.02985-1-21-3', '1907.02985-2-21-3'], ['1907.02985-1-21-4', '1907.02985-2-21-4'], ['1907.02985-1-21-5', '1907.02985-2-21-5'], ['1907.02985-1-21-7', '1907.02985-2-21-7'], ['1907.02985-1-21-9', '1907.02985-2-21-9'], ['1907.02985-1-21-10', '1907.02985-2-21-10'], ['1907.02985-1-23-0', '1907.02985-2-23-0'], ['1907.02985-1-23-1', '1907.02985-2-23-1'], ['1907.02985-1-23-2', '1907.02985-2-23-2'], ['1907.02985-1-23-3', '1907.02985-2-23-3'], ['1907.02985-1-23-4', '1907.02985-2-23-4'], ['1907.02985-1-23-5', '1907.02985-2-23-5'], ['1907.02985-1-23-6', '1907.02985-2-23-6'], ['1907.02985-1-23-7', '1907.02985-2-23-7'], ['1907.02985-1-2-0', '1907.02985-2-2-0'], ['1907.02985-1-2-1', '1907.02985-2-2-1'], ['1907.02985-1-2-2', '1907.02985-2-2-2'], ['1907.02985-1-2-3', '1907.02985-2-2-3'], ['1907.02985-1-32-0', '1907.02985-2-32-0'], ['1907.02985-1-32-1', '1907.02985-2-32-1'], ['1907.02985-1-32-2', '1907.02985-2-32-2'], ['1907.02985-1-32-3', '1907.02985-2-32-3'], ['1907.02985-1-32-4', '1907.02985-2-32-4'], ['1907.02985-1-7-0', '1907.02985-2-7-0'], ['1907.02985-1-7-1', '1907.02985-2-7-1'], ['1907.02985-1-7-2', '1907.02985-2-7-2'], ['1907.02985-1-3-0', '1907.02985-2-3-0'], ['1907.02985-1-3-1', '1907.02985-2-3-1'], ['1907.02985-1-3-2', '1907.02985-2-3-2'], ['1907.02985-1-3-3', '1907.02985-2-3-3'], ['1907.02985-1-3-4', '1907.02985-2-3-4'], ['1907.02985-1-3-5', '1907.02985-2-3-5'], ['1907.02985-1-14-1', '1907.02985-2-14-1'], ['1907.02985-1-14-2', '1907.02985-2-14-2'], ['1907.02985-1-14-3', '1907.02985-2-14-3'], ['1907.02985-1-14-4', '1907.02985-2-14-4'], ['1907.02985-1-10-0', '1907.02985-2-10-0'], ['1907.02985-1-10-1', '1907.02985-2-10-1'], ['1907.02985-1-10-2', '1907.02985-2-10-2'], ['1907.02985-1-10-3', '1907.02985-2-10-3'], ['1907.02985-1-10-4', '1907.02985-2-10-4'], ['1907.02985-1-13-1', '1907.02985-2-13-1'], ['1907.02985-1-13-2', '1907.02985-2-13-2'], ['1907.02985-1-13-3', '1907.02985-2-13-3'], ['1907.02985-1-13-4', '1907.02985-2-13-4'], ['1907.02985-1-13-5', '1907.02985-2-13-5'], ['1907.02985-1-40-0', '1907.02985-2-40-0'], ['1907.02985-1-40-1', '1907.02985-2-40-1'], ['1907.02985-1-40-2', '1907.02985-2-40-2']]

[]

[]

[]

[]

['1907.02985-1-8-0', '1907.02985-1-13-0', '1907.02985-1-14-0', '1907.02985-1-17-0', '1907.02985-1-18-0', '1907.02985-1-21-6', '1907.02985-1-21-8', '1907.02985-1-27-0', '1907.02985-1-28-0', '1907.02985-2-8-0', '1907.02985-2-13-0', '1907.02985-2-14-0', '1907.02985-2-17-0', '1907.02985-2-18-0', '1907.02985-2-21-6', '1907.02985-2-21-8', '1907.02985-2-27-0', '1907.02985-2-28-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1907.02985

1310.6369

{'1310.6369-1-0-0': '# Introduction', '1310.6369-1-1-0': 'The last two years have been remarkably exciting for both experimentalists and theorists working in high energy physics due to the discovery of the Higgs particle [CITATION].', '1310.6369-1-1-1': 'However, a Higgs mass of [MATH] poses intriguing questions, for theorists, about the naturalness of minimal supersymmetric models [CITATION].', '1310.6369-1-1-2': 'Due to the fact that, at tree level, the mass of the Higgs is bounded by the [MATH] boson mass, a large contribution to [MATH] must come from radiative corrections, which are dominated by stop fields.', '1310.6369-1-1-3': 'Thus, it would require to have very heavy stop masses in order to generate such mass value.', '1310.6369-1-1-4': 'However, large stop masses make the soft mass [MATH] large as well, which is straining the electroweak symmetry breaking condition [EQUATION] implying a large amount of fine tuning to achieve the proper cancellation between [MATH] and [MATH].', '1310.6369-1-1-5': 'This is known in the literature as the "little hierarchy problem".', '1310.6369-1-2-0': 'There have been several approaches to raising the mass of the Higgs in the literature.', '1310.6369-1-2-1': 'In one strategy, keeping SUSY minimal, the use of large trilinear [MATH]terms has been studied as a way to avoid the requirement of very heavy stops.', '1310.6369-1-2-2': 'This is, however, somewhat difficult to achieve in the standard gauge mediation of supersymmetry breaking scenario (GMSB) [CITATION], and would require very high messenger masses or a modification of the mediation mechanism.', '1310.6369-1-2-3': 'For instance, GMSB has been modified in models where mixing between the low energy degrees of freedom and the messenger fields has been proposed as a way to generate non-zero [MATH]terms at the messenger scale [CITATION].', '1310.6369-1-3-0': 'In a different approach, extensions of the minimal supersymmetric standard model (MSSM) have been proposed in order to alleviate the little hierarchy problem.', '1310.6369-1-3-1': 'A clear example of such attempts is the next-to-MSSM (see [CITATION] for a review).', '1310.6369-1-3-2': 'Another possibility is the addition of new degrees of freedom at the low energy scale.', '1310.6369-1-3-3': 'Concretely, adding a set of vector-like fields that couple to the Higgs multiplets raises the mass of the Higgs without requiring very heavy superpartners [CITATION].', '1310.6369-1-3-4': 'This type of extensions of the low energy matter content provides a spectrum with low masses that can be tested in the near future at the LHC.', '1310.6369-1-4-0': 'In this work, we explore a model that combines these two approaches.', '1310.6369-1-4-1': 'In this model GMSB is modified to include Yukawa couplings between the MSSM and the messenger sector and, at the same time, there are additional vector-like fields at the TeV scale.', '1310.6369-1-4-2': 'A similar approach was used several decades ago by Dine and Fischler [CITATION] to generate electroweak symmetry breaking; however in their approach, the vector matter was much heavier.', '1310.6369-1-4-3': 'In our case, we provide a microscopic completion for a type of models presented in [CITATION] in the context of gauge mediated superymmetry breaking, with the additional feature of having large [MATH]terms in the effective low energy Lagrangian, which leads to a Higgs mass consistent with the observed values at the LHC.', '1310.6369-1-4-4': 'We explore the parameter space and find a set of regions that yield a 126 GeV Higgs mass with stop masses under 2 TeV.', '1310.6369-1-4-5': 'We argue that the fine tune problem is substantially improved in this kind of scenario to GMSB as there is no need for large stop masses and where [MATH] gets a smaller radiative contribution from stops and vector-like sfermions.', '1310.6369-1-5-0': 'This article is structured as follows: In section [REF], we introduce the model with the new fields.', '1310.6369-1-5-1': 'We also present the soft terms calculated at the messenger scale.', '1310.6369-1-5-2': 'In section [REF] we show the correction the the Higgs mass due to the new fields and the results obtained in the numerical analysis.', '1310.6369-1-5-3': 'We close with some conclusions and we include two appendices with the RGEs of the model and the complete expressions for the soft masses.', '1310.6369-1-6-0': '# The model', '1310.6369-1-7-0': 'In the same spirit of [CITATION], we introduce a set of new vector-like chiral superfields that are charged under the Standard Model (SM) gauge symmetry group.', '1310.6369-1-7-1': 'In such work, new superheavy superfields were added in order to achieve the breaking of [MATH] symmetry by driving [MATH] negative in the soft Lagrangian.', '1310.6369-1-7-2': 'In the present case, we consider similar new superfields but with vector-like masses somewhere between 500 GeV and 1.4 TeV.', '1310.6369-1-7-3': 'The addition of these new superfields is expected to lift the mass of the Higgs through radiative corrections in such a way that no multi-TeV squark masses are necessary.', '1310.6369-1-7-4': 'Furthermore, as done in [CITATION], we allow the MSSM and new vector-like matter content to interact directly with the messenger sector through Yukawa couplings.', '1310.6369-1-7-5': 'This mixing with the messengers will generate large trilinear terms ([MATH]terms) that contribute to the Higgs mass enhancement.', '1310.6369-1-8-0': 'The new vector-like superfields can be arranged in complete representations of [MATH].', '1310.6369-1-8-1': 'Here, we choose to use a pair of 10 dimensional representations [MATH].', '1310.6369-1-8-2': '[EQUATION]', '1310.6369-1-8-3': 'On the other hand, we use a pair of [MATH] representations in the messenger sector, [EQUATION]', '1310.6369-1-8-4': 'These messengers couple to the spurion [MATH], which generates the messenger mass and breaks SUSY through its expectation value [MATH], in the superpotential [EQUATION]', '1310.6369-1-8-5': 'We consider a superpotential that connects three sectors: the MSSM, the new vector-like superfields, and the messengers fields.', '1310.6369-1-8-6': '[EQUATION] where the [MATH] and [MATH] indices have been contracted in the usual way; i.e. [MATH] and [MATH], with [MATH] and [MATH].', '1310.6369-1-8-7': 'We assign [MATH]parity [MATH] to the new vector-like fields, this prevents the low energy vector-like fields from mixing with the MSSM quarks or leptons in the superpotential.', '1310.6369-1-8-8': 'It is worth mentioning that, in this work, we are allowing all the couplings in Equation ([REF]) to be non-zero; this is different from previous works, where only one non-zero coupling was consider at a time.', '1310.6369-1-9-0': '## Effective mass terms', '1310.6369-1-10-0': 'Supersymmetry is broken due to the non-zero F-term [MATH].', '1310.6369-1-10-1': 'This generates soft mases for the MSSM fields as well as for the new vector-like fields.', '1310.6369-1-10-2': 'Besides the soft masses generated through the usual gauge mediation mechanism (GMSB), there is an additional contribution to the masses of the sfermions due to the Yukawa couplings to the messenger fields.', '1310.6369-1-10-3': 'The calculation of these soft masses follow the same methodology presented in [CITATION].', '1310.6369-1-11-0': 'This modified gauge mediation mechanism results in the soft Lagrangian [EQUATION] where [MATH], for [MATH].', '1310.6369-1-11-1': 'These soft parameters are calculated at the messenger scale, [MATH], and evolved down to the electroweak scale through the running of the renormalization group.', '1310.6369-1-11-2': 'For simplicity, we write here just the leading contributions to the soft masses of the vector-like field [MATH] coming from the gauge and Yukawa interactions with the messengers.', '1310.6369-1-11-3': 'In Appendix [REF], we present a general formula to compute these soft terms.', '1310.6369-1-11-4': 'Also, we make [MATH].', '1310.6369-1-11-5': '[EQUATION]', '1310.6369-1-11-6': 'At the messenger scale, the [MATH]terms are non-zero due to the Yukawa couplings to the messengers.', '1310.6369-1-11-7': 'For instance, [MATH] takes the value [EQUATION]', '1310.6369-1-11-8': 'This implies that, at low energies, there is a significant mixing between the scalar superpartners.', '1310.6369-1-12-0': 'After calculating the soft masses and [MATH]terms, we run down the renormalization group from the messenger scale to the low energies where electroweak symmetry breaking is computed and the Higgs mass is obtained, as shown in the next section.', '1310.6369-1-12-1': 'The RGEs corresponding to this model are presented in the appendix .', '1310.6369-1-13-0': 'In the effective theory, we have a set of electrically neutral fermionic fields that couple to the neutral components of the Higgs multiplets and with mass matrix [EQUATION]', '1310.6369-1-13-1': 'On the other hand, the mass matrix for the electrically neutral scalars coupled to the neutral Higgs bosons is given by [EQUATION] where [MATH].', '1310.6369-1-14-0': '# The lighest CP-even Higgs mass', '1310.6369-1-15-0': 'At tree level, the scalar potential for the Higgs fields is the same as in the MSSM [CITATION]: [EQUATION]', '1310.6369-1-15-1': 'In the mass spectrum, we find a charged pair of Higg fields [MATH], a CP-odd neutral scalar [MATH] and two CP-even neutral scalars [MATH] and [MATH].', '1310.6369-1-15-2': 'As it is well known, at tree level, the mass of the lightest neutral Higgs is bounded from above by the mass of the [MATH] boson, .', '1310.6369-1-15-3': 'i.e. [MATH].', '1310.6369-1-15-4': 'In the so-called "decoupling limit", [MATH], [MATH] saturates this bound.', '1310.6369-1-15-5': 'This is the limit that we use in this work.', '1310.6369-1-16-0': 'In the MSSM scenario, at one-loop level, the top and stop fields contribute to the mass of the lightest Higgs.', '1310.6369-1-16-1': 'The effective potential due to these fields is [EQUATION]', '1310.6369-1-16-2': 'The one-loop corrected mass of the lightest CP-even Higgs is, then, given by [CITATION] [EQUATION] with [MATH] and [MATH].', '1310.6369-1-16-3': 'Attaining a Higgs mass [MATH] is somewhat challenging for minimal GMSB unless the top squarks have masses larger than [MATH], which, as already stated, generates some conflict with the naturalness of the MSSM.', '1310.6369-1-16-4': 'Additionally, in GMSB, the [MATH]terms are zero at the messenger scale, although they are non-zero at the electroweak scale due to the running of their RGEs.', '1310.6369-1-16-5': 'However, [MATH] is still smaller than 1, which does not favor the enhancement of the Higgs mass coming from stop mixing; instead, the radiative corrections to [MATH] are dominated by the logarithmic term in Equation ([REF])[CITATION].', '1310.6369-1-17-0': 'When we introduce the new vector-like fields, there is a similar effective one-loop potential from the vector-like fields, [EQUATION] where [MATH] runs over [MATH] and [MATH] are the eigenvalues of the mass matrices ([REF], [REF]) respectively.', '1310.6369-1-17-1': 'The correction to the Higgs mass is given by [CITATION] [EQUATION]', '1310.6369-1-17-2': 'The correction to the Higgs mass is, then, given by [EQUATION]', '1310.6369-1-18-0': '## Parameter scan', '1310.6369-1-19-0': 'In order to analyze the implications of this correction to the Higgs mass, we numerically scan the parameter space over the ranges shown in Table [REF].', '1310.6369-1-19-1': 'We use these input parameters to compute the soft terms at the messenger scale and, then, we use the one-loop RG equations to extract the value of the different masses and couplings at the electroweak scale.', '1310.6369-1-19-2': 'For each set of input parameters we calculate the Higgs mass with and without the extra vector-like fields.', '1310.6369-1-20-0': 'In our parameter scan, we vary [MATH] and [MATH] such that the model is perturbative below the unification scale .', '1310.6369-1-20-1': '[MATH] does not play an important role in raising [MATH], hence we assume [MATH].', '1310.6369-1-20-2': 'The best results for [MATH] are obtained for [MATH].', '1310.6369-1-21-0': 'For simplicity, we have set the mass of the vector-like fermions to be the same at the GUT scale.', '1310.6369-1-21-1': 'At the electroweak scale, this fermions have vector-like masses between 700 GeV and 2 TeV.', '1310.6369-1-21-2': 'Figure [REF] shows the values of [MATH] vs [MATH] when the vector-like fields are included.', '1310.6369-1-21-3': 'It is important to note that 126 GeV are easily achieved in this model even for stop masses under 1 TeV.', '1310.6369-1-21-4': 'In the analyzed parameter space, the correction to the Higgs mass, given in equation ([REF]), can be as small as [MATH] and and as large as 35 GeV.', '1310.6369-1-21-5': 'However, when we just look at the region where the corrected mass of the Higgs is [MATH] (the blue dots in Figure [REF]), we obtain that [MATH] takes values between [MATH] and [MATH].', '1310.6369-1-21-6': 'This enhancement is more notable for vector-like masses around [MATH].', '1310.6369-1-22-0': 'As for the SUSY scales [MATH] that correspond to [MATH] and [MATH] TeV, we obtained values as low as [MATH] GeV and as high as [MATH] GeV with messengers masses between [MATH] and [MATH] GeV.', '1310.6369-1-22-1': 'Notice that this implies a lower scaler for [MATH] than in the GMSB scenario, where [MATH] GeV [CITATION].', '1310.6369-1-23-0': 'Now, we take a look at the values of the [MATH]terms when we include the vector-like fields.', '1310.6369-1-23-1': 'Figure [REF] shows the values for [MATH] for different values of [MATH].', '1310.6369-1-23-2': 'It can be seen that, indeed, it is possible to obtain trilinear terms such that [MATH], unlike the MSSM case with minimal GMSB.', '1310.6369-1-23-3': 'Thus, this feature of the model aides the addition of the vector-like fields in the lifting of the Higgs mass.', '1310.6369-1-23-4': 'To gain some insight about how having larger [MATH] terms helps in raising [MATH], we compare, in Figure [REF], the regions in the [MATH] vs [MATH] plane that are compatible with a Higgs mass of [MATH] GeV for two cases: our model with messenger mixing (green area) and a model without mixing (yellow area, see [CITATION]).', '1310.6369-1-23-5': 'This is shown for [MATH], [MATH], [MATH] GeV and [MATH] .', '1310.6369-1-23-6': 'It can be noticed that, even if the yellow region shows that a realistic mass for the Higgs can be obtained with low masses for the vector-like fermions and the gaugino, the addition of the couplings to the messengers improve the possibilities.', '1310.6369-1-24-0': 'Considering the fact that the Higgs mass is easily lifted in this type of models and the mass of the stops is not required to be larger than 2 TeV, one could think naively that such a proposal is more "natural" than the MSSM with GMSB.', '1310.6369-1-24-1': 'However, as pointed out in [CITATION], the new vector-like fields will contribute to the soft term [MATH], in the same way than stops, through the radiative term [EQUATION]', '1310.6369-1-24-2': 'This, apparently, would make the little hierarchy problem even worse.', '1310.6369-1-24-3': 'However, the effect of this term, and the stop contribution, in making [MATH] large is smaller in this case (where the stops and the vector sfermions have masses just above 1 TeV) compared to the case where 5 TeV stops are required.', '1310.6369-1-24-4': 'Concretely, [EQUATION]', '1310.6369-1-24-5': 'Thus, this effect together with the correction to the Higgs mass lead us to argue that there is some alleviation of the fine tuning problem existing in the MSSM.', '1310.6369-1-24-6': 'In fact, if we consider the measure [CITATION] [EQUATION] the fine tuning in this model is improved by a factor of 10 respect to the MSSM.', '1310.6369-1-25-0': '## Some phenomenology comments', '1310.6369-1-26-0': 'For a more general discussion about the LHC signatures of this type of models with extra vector-like matter, we refer the reader to [CITATION].', '1310.6369-1-26-1': 'In our model, the lightest neutralino has a mass [MATH] GeV and [MATH] GeV, in which case the limits set by the LHC searches using simplified models and CMSSM/mSUGRA are applicable to our analysis [CITATION].', '1310.6369-1-26-2': 'This implies a lower bound on the gluino mass around 1.3 TeV and 500 GeV for the stau mass.', '1310.6369-1-27-0': 'Since the MSSM matter does not mix, at tree level, with the vector-like fermions, the lightest of these, [MATH] (a combination of [MATH] and [MATH]), is long-lived.', '1310.6369-1-27-1': 'For [MATH] TeV, its lifetime is [MATH] s.', '1310.6369-1-27-2': 'There exist cosmological bounds on this type of relics, coming from photodissociation of light elements[CITATION].', '1310.6369-1-27-3': 'After computing the number density of these colored particles [CITATION], we find that [MATH], where [MATH] is the entropy density and [MATH] is the relic number density.', '1310.6369-1-27-4': 'This is consistent with the cosmological constraints.', '1310.6369-1-28-0': 'On the collider side, searches for a charged massive stable particle have been done at the LHC at collision energies of [MATH] and [MATH] TeV[CITATION], where these particles can be identified by their anomalous energy loss or a long time-of-flight to the outer detectors.', '1310.6369-1-28-1': 'The limit obtained in [CITATION] for the mass of this fermion is [MATH] GeV.', '1310.6369-1-29-0': 'Finally, for [MATH] GeV, cosmological gravitino constraints set some bounds on the mass of the messengers and the reheating temperature after inflation.', '1310.6369-1-29-1': 'In order to avoid an overclosure of the universe by the thermal relic abundance of the gravitino, the reheat temperature should be lower than [MATH] GeV.', '1310.6369-1-29-2': 'The next constraint comes from big bang nucleosynthesis (BBN), which sets a bound on the gravitino mass such that the NLSP decays to a photon and a gravitino before [MATH].', '1310.6369-1-29-3': 'This implies [MATH] MeV, which translates into [MATH] GeV.', '1310.6369-1-30-0': '# Conclusions', '1310.6369-1-31-0': 'We have presented a model that yields a Higgs mass around [MATH] with stop masses under 2 TeV.', '1310.6369-1-31-1': 'In order to do so, we have synthesized two approaches used in the literature to tackle the little hierarchy problem.', '1310.6369-1-31-2': 'On one side, we have added a new set of fields to the effective theory.', '1310.6369-1-31-3': 'These fields possess vector-like masses around 1 TeV and contribute to raise the mass of the Higgs field through radiative corrections.', '1310.6369-1-31-4': 'On the other hand, we have included in the superpotential several marginal terms that couple the low energy degrees of freedom to the messenger sector.', '1310.6369-1-31-5': 'This provides a mechanism to enhance the trilinear [MATH] terms in the soft Lagrangian.', '1310.6369-1-31-6': 'This effect complements the enhancement of the Higgs mass by increasing the one-loop corrections through the stop (and vector-like scalar) mixing.', '1310.6369-1-31-7': 'This way, we obtained a wider region in the parameter space for which the measured mass of the Higgs can be achieved, while keeping the sfermions with relatively small masses.', '1310.6369-1-31-8': 'This, together with the fact that the soft term [MATH] is smaller for this typer of models, allow us to argue that the fine tuning problem in the MSSM is improved by this construction.', '1310.6369-1-32-0': 'So far, no signature of new vector matter or SUSY has been observed at the LHC.', '1310.6369-1-32-1': 'However, since this type of models contain stops and vector-like sfermions with masses just above the TeV scale, it presents a spectrum that can be searched for in the near future.', '1310.6369-1-33-0': 'We are grateful to Michael Dine for providing very useful comments and for reminding W.F. about their work done a few decades ago.', '1310.6369-1-33-1': 'W.T. also thanks Jianghao Yu, Can Kilic, Alejandro de la Puente and Simon Knapen for insightful discussions.', '1310.6369-1-33-2': 'This material is based upon work supported by the National Science Foundation under Grant Numbers PHY-1316033 and PHY-0969020 and by the Texas Cosmology Center.', '1310.6369-1-34-0': '# Soft terms', '1310.6369-1-35-0': 'We calculate the soft masses at the messenger scale based on [CITATION], using the superpotential [REF].', '1310.6369-1-35-1': 'In such work, the formulas for the soft parameters are derived from wave function renormalization taking into account the discontinuity of the wave functions at the messenger threshold.', '1310.6369-1-36-0': 'In addition to the soft mass contribution from the standard gauge mediation [EQUATION] there is a two-loop contribution due to the Yukawa couplings between the messenger sector and the effective theory.', '1310.6369-1-36-1': '[EQUATION] where [MATH] is a multiplicity factor, [MATH] sums the quadratic Casimirs of the fields, and the sum over [MATH] includes only the MSSM matter.', '1310.6369-1-36-2': 'There is also a subleading one-loop contribution [EQUATION]', '1310.6369-1-36-3': 'On the other hand, the [MATH]-terms are given by [EQUATION]', '1310.6369-1-37-0': '# Renormalization group equations', '1310.6369-1-38-0': '[EQUATION]'}

{'1310.6369-2-0-0': '# Introduction', '1310.6369-2-1-0': 'The last two years have been remarkably exciting for both experimentalists and theorists working in high energy physics due to the discovery of the Higgs particle [CITATION].', '1310.6369-2-1-1': 'However, a Higgs mass of [MATH] poses intriguing questions, for theorists, about the naturalness of minimal supersymmetric models [CITATION].', '1310.6369-2-1-2': 'Due to the fact that, at tree level, the mass of the Higgs is bounded by the [MATH] boson mass, a large contribution to [MATH] must come from radiative corrections, which are dominated by stop fields.', '1310.6369-2-1-3': 'Thus, it would require to have very heavy stop masses in order to generate such mass value.', '1310.6369-2-1-4': 'However, large stop masses make the soft mass [MATH] large as well, which is straining the electroweak symmetry breaking condition [EQUATION] implying a large amount of fine tuning to achieve the proper cancellation between [MATH] and [MATH].', '1310.6369-2-1-5': 'This is known in the literature as the "little hierarchy problem".', '1310.6369-2-2-0': 'There have been several approaches to raising the mass of the Higgs in the literature.', '1310.6369-2-2-1': 'In one strategy, keeping SUSY minimal, the use of large trilinear [MATH]terms has been studied as a way to avoid the requirement of very heavy stops.', '1310.6369-2-2-2': 'This is, however, somewhat difficult to achieve in the standard gauge mediation of supersymmetry breaking scenario (GMSB) [CITATION], and would require very high messenger masses or a modification of the mediation mechanism.', '1310.6369-2-2-3': 'For instance, GMSB has been modified in models where mixing between the low energy degrees of freedom and the messenger fields has been proposed as a way to generate non-zero [MATH]terms at the messenger scale [CITATION] .', '1310.6369-2-3-0': 'In a different approach, extensions of the minimal supersymmetric standard model (MSSM) have been proposed in order to alleviate the little hierarchy problem.', '1310.6369-2-3-1': 'A clear example of such attempts is the next-to-MSSM (see [CITATION] for a review).', '1310.6369-2-3-2': 'Another possibility is the addition of new degrees of freedom at the low energy scale.', '1310.6369-2-3-3': 'Concretely, adding a set of vector-like fields that couple to the Higgs multiplets raises the mass of the Higgs without requiring very heavy superpartners [CITATION].', '1310.6369-2-3-4': 'This type of extensions of the low energy matter content provides a spectrum with low masses that can be tested in the near future at the LHC.', '1310.6369-2-4-0': 'In this work, we explore a model that combines these two approaches.', '1310.6369-2-4-1': 'In this model GMSB is modified to include Yukawa couplings between the MSSM and the messenger sector and, at the same time, there are additional vector-like fields at the TeV scale.', '1310.6369-2-4-2': 'A similar approach was used several decades ago by Dine and Fischler [CITATION] to generate electroweak symmetry breaking; however in their approach, the vector matter was much heavier.', '1310.6369-2-4-3': 'In our case, we provide a microscopic completion for a type of models presented in [CITATION] in the context of gauge mediated superymmetry breaking, with the additional feature of having large [MATH]terms in the effective low energy Lagrangian, which leads to a Higgs mass consistent with the observed values at the LHC.', '1310.6369-2-4-4': 'We explore the parameter space and find a set of regions that yield a 126 GeV Higgs mass with stop masses under 2 TeV.', '1310.6369-2-4-5': 'We argue that the fine tune problem is substantially improved in this kind of scenario to GMSB as there is no need for large stop masses and where [MATH] gets a smaller radiative contribution from stops and vector-like sfermions.', '1310.6369-2-5-0': 'This article is structured as follows: In section [REF], we introduce the model with the new fields.', '1310.6369-2-5-1': 'We also present the soft terms calculated at the messenger scale.', '1310.6369-2-5-2': 'In section [REF] we show the correction the the Higgs mass due to the new fields and the results obtained in the numerical analysis.', '1310.6369-2-5-3': 'We close with some conclusions and we include two appendices with the RGEs of the model and the complete expressions for the soft masses.', '1310.6369-2-6-0': '# The model', '1310.6369-2-7-0': 'In the same spirit of [CITATION], we introduce a set of new vector-like chiral superfields that are charged under the Standard Model (SM) gauge symmetry group.', '1310.6369-2-7-1': 'In such work, new superheavy superfields were added in order to achieve the breaking of [MATH] symmetry by driving [MATH] negative in the soft Lagrangian.', '1310.6369-2-7-2': 'In the present case, we consider similar new superfields but with vector-like masses somewhere between 500 GeV and 1.4 TeV.', '1310.6369-2-7-3': 'The addition of these new superfields is expected to lift the mass of the Higgs through radiative corrections in such a way that no multi-TeV squark masses are necessary.', '1310.6369-2-7-4': 'Furthermore, as done in [CITATION], we allow the MSSM and new vector-like matter content to interact directly with the messenger sector through Yukawa couplings.', '1310.6369-2-7-5': 'This mixing with the messengers will generate large trilinear terms ([MATH]terms) that contribute to the Higgs mass enhancement.', '1310.6369-2-8-0': 'The new vector-like superfields can be arranged in complete representations of [MATH].', '1310.6369-2-8-1': 'Here, we choose to use a pair of 10 dimensional representations [MATH].', '1310.6369-2-8-2': '[EQUATION]', '1310.6369-2-8-3': 'On the other hand, we use a pair of [MATH] representations in the messenger sector, [EQUATION]', '1310.6369-2-8-4': 'These messengers couple to the spurion [MATH], which generates the messenger mass and breaks SUSY through its expectation value [MATH], in the superpotential [EQUATION]', '1310.6369-2-8-5': 'We consider a superpotential that connects three sectors: the MSSM, the new vector-like superfields, and the messengers fields.', '1310.6369-2-8-6': '[EQUATION] where the [MATH] and [MATH] indices have been contracted in the usual way; i.e. [MATH], with [MATH] and [MATH].', '1310.6369-2-8-7': 'We assign [MATH]parity [MATH] to the new vector-like fields, this prevents the low energy vector-like fields from mixing with the MSSM quarks or leptons in the superpotential.', '1310.6369-2-8-8': 'It is worth mentioning that, in this work, we are allowing all the couplings in Equation ([REF]) to be non-zero; this is different from previous works, where only one non-zero coupling was consider at a time.', '1310.6369-2-9-0': '## Effective mass terms', '1310.6369-2-10-0': 'Supersymmetry is broken due to the non-zero F-term [MATH].', '1310.6369-2-10-1': 'This generates soft mases for the MSSM fields as well as for the new vector-like fields.', '1310.6369-2-10-2': 'Besides the soft masses generated through the usual gauge mediation mechanism (GMSB), there is an additional contribution to the masses of the sfermions due to the Yukawa couplings to the messenger fields.', '1310.6369-2-10-3': 'The calculation of these soft masses follow the same methodology presented in [CITATION].', '1310.6369-2-11-0': 'This modified gauge mediation mechanism results in the soft Lagrangian [EQUATION] where [MATH], for [MATH].', '1310.6369-2-11-1': 'These soft parameters are calculated at the messenger scale, [MATH], and evolved down to the electroweak scale through the running of the renormalization group.', '1310.6369-2-11-2': 'For simplicity, we write here just the leading contributions to the soft masses of the vector-like field [MATH] coming from the gauge and Yukawa interactions with the messengers.', '1310.6369-2-11-3': 'In Appendix [REF], we present a general formula to compute these soft terms.', '1310.6369-2-11-4': 'Also, we make [MATH].', '1310.6369-2-11-5': '[EQUATION]', '1310.6369-2-11-6': 'At the messenger scale, the [MATH]terms are non-zero due to the Yukawa couplings to the messengers.', '1310.6369-2-11-7': 'For instance, [MATH] takes the value [EQUATION]', '1310.6369-2-11-8': 'This implies that, at low energies, there is a significant mixing between the scalar superpartners.', '1310.6369-2-12-0': 'After calculating the soft masses and [MATH]terms, we run down the renormalization group from the messenger scale to the low energies where electroweak symmetry breaking is computed and the Higgs mass is obtained, as shown in the next section.', '1310.6369-2-12-1': 'The RGEs corresponding to this model are presented in the appendix .', '1310.6369-2-13-0': 'In the effective theory, we have a set of electrically neutral fermionic fields that couple to the neutral components of the Higgs multiplets and with mass matrix [EQUATION]', '1310.6369-2-13-1': 'On the other hand, the mass matrix for the electrically neutral scalars coupled to the neutral Higgs bosons is given by [EQUATION] where [MATH].', '1310.6369-2-14-0': '# The lighest CP-even Higgs mass', '1310.6369-2-15-0': 'At tree level, the scalar potential for the Higgs fields is the same as in the MSSM [CITATION]: [EQUATION]', '1310.6369-2-15-1': 'In the mass spectrum, we find a charged pair of Higg fields [MATH], a CP-odd neutral scalar [MATH] and two CP-even neutral scalars [MATH] and [MATH].', '1310.6369-2-15-2': 'As it is well known, at tree level, the mass of the lightest neutral Higgs is bounded from above by the mass of the [MATH] boson, .', '1310.6369-2-15-3': 'i.e. [MATH].', '1310.6369-2-15-4': 'In the so-called "decoupling limit", [MATH], [MATH] saturates this bound.', '1310.6369-2-15-5': 'This is the limit that we use in this work.', '1310.6369-2-16-0': 'In the MSSM scenario, at one-loop level, the top and stop fields contribute to the mass of the lightest Higgs.', '1310.6369-2-16-1': 'The effective potential due to these fields is [EQUATION]', '1310.6369-2-16-2': 'The one-loop corrected mass of the lightest CP-even Higgs is, then, given by [CITATION] [EQUATION] with [MATH] and [MATH].', '1310.6369-2-16-3': 'Attaining a Higgs mass [MATH] is somewhat challenging for minimal GMSB unless the top squarks have masses larger than [MATH], which, as already stated, generates some conflict with the naturalness of the MSSM.', '1310.6369-2-16-4': 'Additionally, in GMSB, the [MATH]terms are zero at the messenger scale, although they are non-zero at the electroweak scale due to the running of their RGEs.', '1310.6369-2-16-5': 'However, [MATH] is still smaller than 1, which does not favor the enhancement of the Higgs mass coming from stop mixing; instead, the radiative corrections to [MATH] are dominated by the logarithmic term in Equation ([REF])[CITATION].', '1310.6369-2-17-0': 'When we introduce the new vector-like fields, there is a similar effective one-loop potential from the vector-like fields, [EQUATION] where [MATH] runs over [MATH] and [MATH] are the eigenvalues of the mass matrices ([REF], [REF]) respectively.', '1310.6369-2-17-1': 'The correction to the Higgs mass is given by [CITATION] [EQUATION]', '1310.6369-2-17-2': 'The correction to the Higgs mass is, then, given by [EQUATION]', '1310.6369-2-18-0': '## Parameter scan', '1310.6369-2-19-0': 'In order to analyze the implications of this correction to the Higgs mass, we numerically scan the parameter space over the ranges shown in Table [REF].', '1310.6369-2-19-1': 'We use these input parameters to compute the soft terms at the messenger scale and, then, we use the one-loop RG equations to extract the value of the different masses and couplings at the electroweak scale.', '1310.6369-2-19-2': 'For each set of input parameters we calculate the Higgs mass with and without the extra vector-like fields.', '1310.6369-2-20-0': 'In our parameter scan, we vary [MATH] and [MATH] such that the model is perturbative below the unification scale .', '1310.6369-2-20-1': '[MATH] does not play an important role in raising [MATH], hence we assume [MATH].', '1310.6369-2-20-2': 'The best results for [MATH] are obtained for [MATH].', '1310.6369-2-21-0': 'For simplicity, we have set the mass of the vector-like fermions to be the same at the GUT scale.', '1310.6369-2-21-1': 'At the electroweak scale, this fermions have vector-like masses between 700 GeV and 2 TeV.', '1310.6369-2-21-2': 'Figure [REF] shows the values of [MATH] vs [MATH] when the vector-like fields are included.', '1310.6369-2-21-3': 'It is important to note that 126 GeV are easily achieved in this model even for stop masses under 1 TeV.', '1310.6369-2-21-4': 'In the analyzed parameter space, the correction to the Higgs mass, given in equation ([REF]), can be as small as [MATH] and and as large as 35 GeV.', '1310.6369-2-21-5': 'However, when we just look at the region where the corrected mass of the Higgs is [MATH] (the blue dots in Figure [REF]), we obtain that [MATH] takes values between [MATH] and [MATH].', '1310.6369-2-21-6': 'This enhancement is more notable for vector-like masses around [MATH].', '1310.6369-2-22-0': 'As for the SUSY scales [MATH] that correspond to [MATH] and [MATH] TeV, we obtained values as low as [MATH] GeV and as high as [MATH] GeV with messengers masses between [MATH] and [MATH] GeV.', '1310.6369-2-22-1': 'Notice that this implies a lower scaler for [MATH] than in the GMSB scenario, where [MATH] GeV [CITATION].', '1310.6369-2-23-0': 'Now, we take a look at the values of the [MATH]terms when we include the vector-like fields.', '1310.6369-2-23-1': 'Figure [REF] shows the values for [MATH] for different values of [MATH].', '1310.6369-2-23-2': 'It can be seen that, indeed, it is possible to obtain trilinear terms such that [MATH], unlike the MSSM case with minimal GMSB.', '1310.6369-2-23-3': 'Thus, this feature of the model aides the addition of the vector-like fields in the lifting of the Higgs mass.', '1310.6369-2-23-4': 'To gain some insight about how having larger [MATH] terms helps in raising [MATH], we compare, in Figure [REF], the regions in the [MATH] vs [MATH] plane that are compatible with a Higgs mass of [MATH] GeV for two cases: our model with messenger mixing (green area) and a model without mixing (yellow area, see [CITATION]).', '1310.6369-2-23-5': 'This is shown for [MATH], [MATH], [MATH] GeV and [MATH] .', '1310.6369-2-23-6': 'It can be noticed that, even if the yellow region shows that a realistic mass for the Higgs can be obtained with low masses for the vector-like fermions and the gaugino, the addition of the couplings to the messengers improve the possibilities.', '1310.6369-2-24-0': 'Considering the fact that the Higgs mass is easily lifted in this type of models and the mass of the stops is not required to be larger than 2 TeV, one could think naively that such a proposal is more "natural" than the MSSM with GMSB.', '1310.6369-2-24-1': 'However, as pointed out in [CITATION], the new vector-like fields will contribute to the soft term [MATH], in the same way than stops, through the radiative term [EQUATION]', '1310.6369-2-24-2': 'This, apparently, would make the little hierarchy problem even worse.', '1310.6369-2-24-3': 'However, the effect of this term, and the stop contribution, in making [MATH] large is smaller in this case (where the stops and the vector sfermions have masses just above 1 TeV) compared to the case where 5 TeV stops are required.', '1310.6369-2-24-4': 'Concretely, [EQUATION]', '1310.6369-2-24-5': 'Thus, this effect together with the correction to the Higgs mass lead us to argue that there is some alleviation of the fine tuning problem existing in the MSSM.', '1310.6369-2-24-6': 'In fact, if we consider the measure [CITATION] [EQUATION] the fine tuning in this model is improved by a factor of 10 respect to the MSSM.', '1310.6369-2-25-0': '## Some phenomenology comments', '1310.6369-2-26-0': 'For a more general discussion about the LHC signatures of this type of models with extra vector-like matter, we refer the reader to [CITATION].', '1310.6369-2-26-1': 'In our model, the lightest neutralino has a mass [MATH] GeV and [MATH] GeV, in which case the limits set by the LHC searches using simplified models and CMSSM/mSUGRA are applicable to our analysis [CITATION].', '1310.6369-2-26-2': 'This implies a lower bound on the gluino mass around 1.3 TeV and 500 GeV for the stau mass.', '1310.6369-2-27-0': 'Since the MSSM matter does not mix, at tree level, with the vector-like fermions, the lightest of these, [MATH] (a combination of [MATH] and [MATH] fermions), is long-lived.', '1310.6369-2-27-1': 'For [MATH] TeV, its lifetime is [MATH] s.', '1310.6369-2-27-2': 'There exist cosmological bounds on this type of relics, coming from photodissociation of light elements[CITATION].', '1310.6369-2-27-3': 'After computing the number density of these colored particles [CITATION], we find that [MATH], where [MATH] is the entropy density and [MATH] is the relic number density.', '1310.6369-2-27-4': 'This is consistent with the cosmological constraints.', '1310.6369-2-28-0': 'On the collider side, searches for a charged massive stable particle have been done at the LHC at collision energies of [MATH] and [MATH] TeV[CITATION], where these particles can be identified by their anomalous energy loss or a long time-of-flight to the outer detectors.', '1310.6369-2-28-1': 'The limit obtained in [CITATION] for the mass of this fermion is [MATH] GeV.', '1310.6369-2-29-0': 'Finally, for [MATH] GeV, cosmological gravitino constraints set some bounds on the mass of the messengers and the reheating temperature after inflation.', '1310.6369-2-29-1': 'In order to avoid an overclosure of the universe by the thermal relic abundance of the gravitino, the reheat temperature should be lower than [MATH] GeV.', '1310.6369-2-29-2': 'The next constraint comes from big bang nucleosynthesis (BBN), which sets a bound on the gravitino mass such that the NLSP decays to a photon and a gravitino before [MATH].', '1310.6369-2-29-3': 'This implies [MATH] MeV, which translates into [MATH] GeV.', '1310.6369-2-30-0': '# Conclusions', '1310.6369-2-31-0': 'We have presented a model that yields a Higgs mass around [MATH] with stop masses under 2 TeV.', '1310.6369-2-31-1': 'In order to do so, we have synthesized two approaches used in the literature to tackle the little hierarchy problem.', '1310.6369-2-31-2': 'On one side, we have added a new set of fields to the effective theory.', '1310.6369-2-31-3': 'These fields possess vector-like masses around 1 TeV and contribute to raise the mass of the Higgs field through radiative corrections.', '1310.6369-2-31-4': 'On the other hand, we have included in the superpotential several marginal terms that couple the low energy degrees of freedom to the messenger sector.', '1310.6369-2-31-5': 'This provides a mechanism to enhance the trilinear [MATH] terms in the soft Lagrangian.', '1310.6369-2-31-6': 'This effect complements the enhancement of the Higgs mass by increasing the one-loop corrections through the stop (and vector-like scalar) mixing.', '1310.6369-2-31-7': 'This way, we obtained a wider region in the parameter space for which the measured mass of the Higgs can be achieved, while keeping the sfermions with relatively small masses.', '1310.6369-2-31-8': 'This, together with the fact that the soft term [MATH] is smaller for this typer of models, allow us to argue that the fine tuning problem in the MSSM is improved by this construction.', '1310.6369-2-32-0': 'So far, no signature of new vector matter or SUSY has been observed at the LHC.', '1310.6369-2-32-1': 'However, since this type of models contain stops and vector-like sfermions with masses just above the TeV scale, it presents a spectrum that can be searched for in the near future.', '1310.6369-2-33-0': 'We are grateful to Michael Dine for providing very useful comments and for reminding W.F. about their work done a few decades ago.', '1310.6369-2-33-1': 'W.T. also thanks Jianghao Yu, Can Kilic, Alejandro de la Puente and Simon Knapen for insightful discussions.', '1310.6369-2-33-2': 'We also thank Renato Fonseca for pointing out a couple of issues in the original version of this article.', '1310.6369-2-33-3': 'This material is based upon work supported by the National Science Foundation under Grant Numbers PHY-1316033 and PHY-0969020 and by the Texas Cosmology Center.', '1310.6369-2-34-0': '# Soft terms', '1310.6369-2-35-0': 'We calculate the soft masses at the messenger scale based on [CITATION], using the superpotential [REF].', '1310.6369-2-35-1': 'In such work, the formulas for the soft parameters are derived from wave function renormalization taking into account the discontinuity of the wave functions at the messenger threshold.', '1310.6369-2-36-0': 'In addition to the soft mass contribution from the standard gauge mediation [EQUATION] there is a two-loop contribution due to the Yukawa couplings between the messenger sector and the effective theory.', '1310.6369-2-36-1': '[EQUATION] where [MATH] is a multiplicity factor, [MATH] sums the quadratic Casimirs of the fields, and the sum over [MATH] includes only the MSSM matter.', '1310.6369-2-36-2': 'There is also a subleading one-loop contribution [EQUATION]', '1310.6369-2-36-3': 'On the other hand, the [MATH]-terms are given by [EQUATION]', '1310.6369-2-37-0': '# Renormalization group equations', '1310.6369-2-38-0': '[EQUATION]'}

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'1310.6369-2-8-5'], ['1310.6369-1-8-7', '1310.6369-2-8-7'], ['1310.6369-1-8-8', '1310.6369-2-8-8'], ['1310.6369-1-7-0', '1310.6369-2-7-0'], ['1310.6369-1-7-1', '1310.6369-2-7-1'], ['1310.6369-1-7-2', '1310.6369-2-7-2'], ['1310.6369-1-7-3', '1310.6369-2-7-3'], ['1310.6369-1-7-4', '1310.6369-2-7-4'], ['1310.6369-1-7-5', '1310.6369-2-7-5'], ['1310.6369-2-13-0', '1310.6369-3-13-0'], ['1310.6369-2-13-1', '1310.6369-3-13-1'], ['1310.6369-2-22-0', '1310.6369-3-22-0'], ['1310.6369-2-22-1', '1310.6369-3-22-1'], ['1310.6369-2-23-0', '1310.6369-3-23-0'], ['1310.6369-2-23-1', '1310.6369-3-23-1'], ['1310.6369-2-23-2', '1310.6369-3-23-2'], ['1310.6369-2-23-3', '1310.6369-3-23-3'], ['1310.6369-2-23-4', '1310.6369-3-23-4'], ['1310.6369-2-23-5', '1310.6369-3-23-5'], ['1310.6369-2-23-6', '1310.6369-3-23-6'], ['1310.6369-2-24-0', '1310.6369-3-24-0'], ['1310.6369-2-24-1', '1310.6369-3-24-1'], ['1310.6369-2-24-2', '1310.6369-3-24-2'], ['1310.6369-2-24-3', '1310.6369-3-24-3'], ['1310.6369-2-24-5', '1310.6369-3-24-5'], ['1310.6369-2-24-6', '1310.6369-3-24-6'], ['1310.6369-2-20-0', '1310.6369-3-20-0'], ['1310.6369-2-20-1', '1310.6369-3-20-1'], ['1310.6369-2-20-2', '1310.6369-3-20-2'], ['1310.6369-2-19-0', '1310.6369-3-19-0'], ['1310.6369-2-19-1', '1310.6369-3-19-1'], ['1310.6369-2-19-2', '1310.6369-3-19-2'], ['1310.6369-2-21-0', '1310.6369-3-21-0'], ['1310.6369-2-21-1', '1310.6369-3-21-1'], ['1310.6369-2-21-2', '1310.6369-3-21-2'], ['1310.6369-2-21-3', '1310.6369-3-21-3'], ['1310.6369-2-21-4', '1310.6369-3-21-4'], ['1310.6369-2-21-5', '1310.6369-3-21-5'], ['1310.6369-2-21-6', '1310.6369-3-21-6'], ['1310.6369-2-29-0', '1310.6369-3-29-0'], ['1310.6369-2-29-1', '1310.6369-3-29-1'], ['1310.6369-2-29-2', '1310.6369-3-29-2'], ['1310.6369-2-29-3', '1310.6369-3-29-3'], ['1310.6369-2-1-0', '1310.6369-3-1-0'], ['1310.6369-2-1-1', '1310.6369-3-1-1'], ['1310.6369-2-1-2', '1310.6369-3-1-2'], ['1310.6369-2-1-3', '1310.6369-3-1-3'], ['1310.6369-2-1-4', '1310.6369-3-1-4'], ['1310.6369-2-1-5', '1310.6369-3-1-5'], ['1310.6369-2-2-0', '1310.6369-3-2-0'], ['1310.6369-2-2-1', '1310.6369-3-2-1'], ['1310.6369-2-2-2', '1310.6369-3-2-2'], ['1310.6369-2-2-3', '1310.6369-3-2-3'], ['1310.6369-2-32-0', '1310.6369-3-32-0'], ['1310.6369-2-32-1', '1310.6369-3-32-1'], ['1310.6369-2-3-0', '1310.6369-3-3-0'], ['1310.6369-2-3-1', '1310.6369-3-3-1'], ['1310.6369-2-3-2', '1310.6369-3-3-2'], ['1310.6369-2-3-3', '1310.6369-3-3-3'], ['1310.6369-2-3-4', '1310.6369-3-3-4'], ['1310.6369-2-15-0', '1310.6369-3-15-0'], ['1310.6369-2-15-1', '1310.6369-3-15-1'], ['1310.6369-2-15-2', '1310.6369-3-15-2'], ['1310.6369-2-15-4', '1310.6369-3-15-4'], ['1310.6369-2-15-5', '1310.6369-3-15-5'], ['1310.6369-2-4-0', '1310.6369-3-4-0'], ['1310.6369-2-4-1', '1310.6369-3-4-1'], ['1310.6369-2-4-2', '1310.6369-3-4-2'], ['1310.6369-2-4-3', '1310.6369-3-4-3'], ['1310.6369-2-4-4', '1310.6369-3-4-4'], ['1310.6369-2-4-5', '1310.6369-3-4-5'], ['1310.6369-2-5-0', '1310.6369-3-5-0'], ['1310.6369-2-5-1', '1310.6369-3-5-1'], ['1310.6369-2-5-2', '1310.6369-3-5-2'], ['1310.6369-2-5-3', '1310.6369-3-5-3'], ['1310.6369-2-12-0', '1310.6369-3-12-0'], ['1310.6369-2-12-1', '1310.6369-3-12-1'], ['1310.6369-2-11-0', '1310.6369-3-11-0'], ['1310.6369-2-11-1', '1310.6369-3-11-1'], ['1310.6369-2-11-2', '1310.6369-3-11-2'], ['1310.6369-2-11-3', '1310.6369-3-11-3'], ['1310.6369-2-11-4', '1310.6369-3-11-4'], ['1310.6369-2-11-6', '1310.6369-3-11-6'], ['1310.6369-2-11-7', '1310.6369-3-11-7'], ['1310.6369-2-11-8', '1310.6369-3-11-8'], ['1310.6369-2-17-0', '1310.6369-3-17-0'], ['1310.6369-2-17-1', '1310.6369-3-17-1'], ['1310.6369-2-17-2', '1310.6369-3-17-2'], ['1310.6369-2-8-0', '1310.6369-3-8-0'], ['1310.6369-2-8-1', '1310.6369-3-8-1'], ['1310.6369-2-8-3', '1310.6369-3-8-3'], ['1310.6369-2-8-4', '1310.6369-3-8-4'], ['1310.6369-2-8-5', '1310.6369-3-8-5'], ['1310.6369-2-8-6', '1310.6369-3-8-6'], ['1310.6369-2-8-7', '1310.6369-3-8-7'], ['1310.6369-2-8-8', '1310.6369-3-8-8'], ['1310.6369-2-27-0', '1310.6369-3-27-0'], 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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1310.6369

{'1310.6369-3-0-0': '# Introduction', '1310.6369-3-1-0': 'The last two years have been remarkably exciting for both experimentalists and theorists working in high energy physics due to the discovery of the Higgs particle [CITATION].', '1310.6369-3-1-1': 'However, a Higgs mass of [MATH] poses intriguing questions, for theorists, about the naturalness of minimal supersymmetric models [CITATION].', '1310.6369-3-1-2': 'Due to the fact that, at tree level, the mass of the Higgs is bounded by the [MATH] boson mass, a large contribution to [MATH] must come from radiative corrections, which are dominated by stop fields.', '1310.6369-3-1-3': 'Thus, it would require to have very heavy stop masses in order to generate such mass value.', '1310.6369-3-1-4': 'However, large stop masses make the soft mass [MATH] large as well, which is straining the electroweak symmetry breaking condition [EQUATION] implying a large amount of fine tuning to achieve the proper cancellation between [MATH] and [MATH].', '1310.6369-3-1-5': 'This is known in the literature as the "little hierarchy problem".', '1310.6369-3-2-0': 'There have been several approaches to raising the mass of the Higgs in the literature.', '1310.6369-3-2-1': 'In one strategy, keeping SUSY minimal, the use of large trilinear [MATH]terms has been studied as a way to avoid the requirement of very heavy stops.', '1310.6369-3-2-2': 'This is, however, somewhat difficult to achieve in the standard gauge mediation of supersymmetry breaking scenario (GMSB) [CITATION], and would require very high messenger masses or a modification of the mediation mechanism.', '1310.6369-3-2-3': 'For instance, GMSB has been modified in models where mixing between the low energy degrees of freedom and the messenger fields has been proposed as a way to generate non-zero [MATH]terms at the messenger scale [CITATION] .', '1310.6369-3-3-0': 'In a different approach, extensions of the minimal supersymmetric standard model (MSSM) have been proposed in order to alleviate the little hierarchy problem.', '1310.6369-3-3-1': 'A clear example of such attempts is the next-to-MSSM (see [CITATION] for a review).', '1310.6369-3-3-2': 'Another possibility is the addition of new degrees of freedom at the low energy scale.', '1310.6369-3-3-3': 'Concretely, adding a set of vector-like fields that couple to the Higgs multiplets raises the mass of the Higgs without requiring very heavy superpartners [CITATION].', '1310.6369-3-3-4': 'This type of extensions of the low energy matter content provides a spectrum with low masses that can be tested in the near future at the LHC.', '1310.6369-3-4-0': 'In this work, we explore a model that combines these two approaches.', '1310.6369-3-4-1': 'In this model GMSB is modified to include Yukawa couplings between the MSSM and the messenger sector and, at the same time, there are additional vector-like fields at the TeV scale.', '1310.6369-3-4-2': 'A similar approach was used several decades ago by Dine and Fischler [CITATION] to generate electroweak symmetry breaking; however in their approach, the vector matter was much heavier.', '1310.6369-3-4-3': 'In our case, we provide a microscopic completion for a type of models presented in [CITATION] in the context of gauge mediated superymmetry breaking, with the additional feature of having large [MATH]terms in the effective low energy Lagrangian, which leads to a Higgs mass consistent with the observed values at the LHC.', '1310.6369-3-4-4': 'We explore the parameter space and find a set of regions that yield a 126 GeV Higgs mass with stop masses under 2 TeV.', '1310.6369-3-4-5': 'We argue that the fine tune problem is substantially improved in this kind of scenario to GMSB as there is no need for large stop masses and where [MATH] gets a smaller radiative contribution from stops and vector-like sfermions.', '1310.6369-3-5-0': 'This article is structured as follows: In section [REF], we introduce the model with the new fields.', '1310.6369-3-5-1': 'We also present the soft terms calculated at the messenger scale.', '1310.6369-3-5-2': 'In section [REF] we show the correction the the Higgs mass due to the new fields and the results obtained in the numerical analysis.', '1310.6369-3-5-3': 'We close with some conclusions and we include two appendices with the RGEs of the model and the complete expressions for the soft masses.', '1310.6369-3-6-0': '# The model', '1310.6369-3-7-0': 'In the same spirit of [CITATION], we introduce a set of new vector-like chiral superfields that are charged under the Standard Model (SM) gauge symmetry group.', '1310.6369-3-7-1': 'In such work, new superheavy superfields were added in order to achieve the breaking of [MATH] symmetry by driving [MATH] negative in the soft Lagrangian.', '1310.6369-3-7-2': 'In the present case, we consider similar new superfields but with vector-like masses somewhere between 500 GeV and 1.4 TeV.', '1310.6369-3-7-3': 'The addition of these new superfields is expected to lift the mass of the Higgs through radiative corrections in such a way that no multi-TeV squark masses are necessary.', '1310.6369-3-7-4': 'Furthermore, as done in [CITATION], we allow the MSSM and new vector-like matter content to interact directly with the messenger sector through Yukawa couplings.', '1310.6369-3-7-5': 'This mixing with the messengers will generate large trilinear terms ([MATH]terms) that contribute to the Higgs mass enhancement.', '1310.6369-3-8-0': 'The new vector-like superfields can be arranged in complete representations of [MATH].', '1310.6369-3-8-1': 'Here, we choose to use a pair of 10 dimensional representations [MATH].', '1310.6369-3-8-2': '[EQUATION]', '1310.6369-3-8-3': 'On the other hand, we use a pair of [MATH] representations in the messenger sector, [EQUATION]', '1310.6369-3-8-4': 'These messengers couple to the spurion [MATH], which generates the messenger mass and breaks SUSY through its expectation value [MATH], in the superpotential [EQUATION]', '1310.6369-3-8-5': 'We consider a superpotential that connects three sectors: the MSSM, the new vector-like superfields, and the messengers fields.', '1310.6369-3-8-6': '[EQUATION] where the [MATH] and [MATH] indices have been contracted in the usual way; i.e. [MATH], with [MATH] and [MATH].', '1310.6369-3-8-7': 'We assign [MATH]parity [MATH] to the new vector-like fields, this prevents the low energy vector-like fields from mixing with the MSSM quarks or leptons in the superpotential.', '1310.6369-3-8-8': 'It is worth mentioning that, in this work, we are allowing all the couplings in Equation ([REF]) to be non-zero; this is different from previous works, where only one non-zero coupling was consider at a time.', '1310.6369-3-9-0': '## Effective mass terms', '1310.6369-3-10-0': 'Supersymmetry is broken due to the non-zero F-term [MATH].', '1310.6369-3-10-1': 'This generates soft mases for the MSSM fields as well as for the new vector-like fields.', '1310.6369-3-10-2': 'Besides the soft masses generated through the usual gauge mediation mechanism (GMSB), there is an additional contribution to the masses of the sfermions due to the Yukawa couplings to the messenger fields.', '1310.6369-3-10-3': 'The calculation of these soft masses follow the same methodology presented in [CITATION].', '1310.6369-3-11-0': 'This modified gauge mediation mechanism results in the soft Lagrangian [EQUATION] where [MATH], for [MATH].', '1310.6369-3-11-1': 'These soft parameters are calculated at the messenger scale, [MATH], and evolved down to the electroweak scale through the running of the renormalization group.', '1310.6369-3-11-2': 'For simplicity, we write here just the leading contributions to the soft masses of the vector-like field [MATH] coming from the gauge and Yukawa interactions with the messengers.', '1310.6369-3-11-3': 'In Appendix [REF], we present a general formula to compute these soft terms.', '1310.6369-3-11-4': 'Also, we make [MATH].', '1310.6369-3-11-5': '[EQUATION]', '1310.6369-3-11-6': 'At the messenger scale, the [MATH]terms are non-zero due to the Yukawa couplings to the messengers.', '1310.6369-3-11-7': 'For instance, [MATH] takes the value [EQUATION]', '1310.6369-3-11-8': 'This implies that, at low energies, there is a significant mixing between the scalar superpartners.', '1310.6369-3-12-0': 'After calculating the soft masses and [MATH]terms, we run down the renormalization group from the messenger scale to the low energies where electroweak symmetry breaking is computed and the Higgs mass is obtained, as shown in the next section.', '1310.6369-3-12-1': 'The RGEs corresponding to this model are presented in the appendix .', '1310.6369-3-13-0': 'In the effective theory, we have a set of electrically neutral fermionic fields that couple to the neutral components of the Higgs multiplets and with mass matrix [EQUATION]', '1310.6369-3-13-1': 'On the other hand, the mass matrix for the electrically neutral scalars coupled to the neutral Higgs bosons is given by [EQUATION] where [MATH].', '1310.6369-3-14-0': '# The lighest CP-even Higgs mass', '1310.6369-3-15-0': 'At tree level, the scalar potential for the Higgs fields is the same as in the MSSM [CITATION]: [EQUATION]', '1310.6369-3-15-1': 'In the mass spectrum, we find a charged pair of Higg fields [MATH], a CP-odd neutral scalar [MATH] and two CP-even neutral scalars [MATH] and [MATH].', '1310.6369-3-15-2': 'As it is well known, at tree level, the mass of the lightest neutral Higgs is bounded from above by the mass of the [MATH] boson, .', '1310.6369-3-15-3': 'i.e. [MATH].', '1310.6369-3-15-4': 'In the so-called "decoupling limit", [MATH], [MATH] saturates this bound.', '1310.6369-3-15-5': 'This is the limit that we use in this work.', '1310.6369-3-16-0': 'In the MSSM scenario, at one-loop level, the top and stop fields contribute to the mass of the lightest Higgs.', '1310.6369-3-16-1': 'The effective potential due to these fields is [EQUATION]', '1310.6369-3-16-2': 'The one-loop corrected mass of the lightest CP-even Higgs is, then, given by [CITATION] [EQUATION] with [MATH] and [MATH].', '1310.6369-3-16-3': 'Attaining a Higgs mass [MATH] is somewhat challenging for minimal GMSB unless the top squarks have masses larger than [MATH], which, as already stated, generates some conflict with the naturalness of the MSSM.', '1310.6369-3-16-4': 'Additionally, in GMSB, the [MATH]terms are zero at the messenger scale, although they are non-zero at the electroweak scale due to the running of their RGEs.', '1310.6369-3-16-5': 'However, [MATH] is still smaller than 1, which does not favor the enhancement of the Higgs mass coming from stop mixing; instead, the radiative corrections to [MATH] are dominated by the logarithmic term in Equation ([REF])[CITATION].', '1310.6369-3-17-0': 'When we introduce the new vector-like fields, there is a similar effective one-loop potential from the vector-like fields, [EQUATION] where [MATH] runs over [MATH] and [MATH] are the eigenvalues of the mass matrices ([REF], [REF]) respectively.', '1310.6369-3-17-1': 'The correction to the Higgs mass is given by [CITATION] [EQUATION]', '1310.6369-3-17-2': 'The correction to the Higgs mass is, then, given by [EQUATION]', '1310.6369-3-18-0': '## Parameter scan', '1310.6369-3-19-0': 'In order to analyze the implications of this correction to the Higgs mass, we numerically scan the parameter space over the ranges shown in Table [REF].', '1310.6369-3-19-1': 'We use these input parameters to compute the soft terms at the messenger scale and, then, we use the one-loop RG equations to extract the value of the different masses and couplings at the electroweak scale.', '1310.6369-3-19-2': 'For each set of input parameters we calculate the Higgs mass with and without the extra vector-like fields.', '1310.6369-3-20-0': 'In our parameter scan, we vary [MATH] and [MATH] such that the model is perturbative below the unification scale .', '1310.6369-3-20-1': '[MATH] does not play an important role in raising [MATH], hence we assume [MATH].', '1310.6369-3-20-2': 'The best results for [MATH] are obtained for [MATH].', '1310.6369-3-21-0': 'For simplicity, we have set the mass of the vector-like fermions to be the same at the GUT scale.', '1310.6369-3-21-1': 'At the electroweak scale, this fermions have vector-like masses between 700 GeV and 2 TeV.', '1310.6369-3-21-2': 'Figure [REF] shows the values of [MATH] vs [MATH] when the vector-like fields are included.', '1310.6369-3-21-3': 'It is important to note that 126 GeV are easily achieved in this model even for stop masses under 1 TeV.', '1310.6369-3-21-4': 'In the analyzed parameter space, the correction to the Higgs mass, given in equation ([REF]), can be as small as [MATH] and and as large as 35 GeV.', '1310.6369-3-21-5': 'However, when we just look at the region where the corrected mass of the Higgs is [MATH] (the blue dots in Figure [REF]), we obtain that [MATH] takes values between [MATH] and [MATH].', '1310.6369-3-21-6': 'This enhancement is more notable for vector-like masses around [MATH].', '1310.6369-3-22-0': 'As for the SUSY scales [MATH] that correspond to [MATH] and [MATH] TeV, we obtained values as low as [MATH] GeV and as high as [MATH] GeV with messengers masses between [MATH] and [MATH] GeV.', '1310.6369-3-22-1': 'Notice that this implies a lower scaler for [MATH] than in the GMSB scenario, where [MATH] GeV [CITATION].', '1310.6369-3-23-0': 'Now, we take a look at the values of the [MATH]terms when we include the vector-like fields.', '1310.6369-3-23-1': 'Figure [REF] shows the values for [MATH] for different values of [MATH].', '1310.6369-3-23-2': 'It can be seen that, indeed, it is possible to obtain trilinear terms such that [MATH], unlike the MSSM case with minimal GMSB.', '1310.6369-3-23-3': 'Thus, this feature of the model aides the addition of the vector-like fields in the lifting of the Higgs mass.', '1310.6369-3-23-4': 'To gain some insight about how having larger [MATH] terms helps in raising [MATH], we compare, in Figure [REF], the regions in the [MATH] vs [MATH] plane that are compatible with a Higgs mass of [MATH] GeV for two cases: our model with messenger mixing (green area) and a model without mixing (yellow area, see [CITATION]).', '1310.6369-3-23-5': 'This is shown for [MATH], [MATH], [MATH] GeV and [MATH] .', '1310.6369-3-23-6': 'It can be noticed that, even if the yellow region shows that a realistic mass for the Higgs can be obtained with low masses for the vector-like fermions and the gaugino, the addition of the couplings to the messengers improve the possibilities.', '1310.6369-3-24-0': 'Considering the fact that the Higgs mass is easily lifted in this type of models and the mass of the stops is not required to be larger than 2 TeV, one could think naively that such a proposal is more "natural" than the MSSM with GMSB.', '1310.6369-3-24-1': 'However, as pointed out in [CITATION], the new vector-like fields will contribute to the soft term [MATH], in the same way than stops, through the radiative term [EQUATION]', '1310.6369-3-24-2': 'This, apparently, would make the little hierarchy problem even worse.', '1310.6369-3-24-3': 'However, the effect of this term, and the stop contribution, in making [MATH] large is smaller in this case (where the stops and the vector sfermions have masses just above 1 TeV) compared to the case where 5 TeV stops are required.', '1310.6369-3-24-4': 'Concretely, [EQUATION]', '1310.6369-3-24-5': 'Thus, this effect together with the correction to the Higgs mass lead us to argue that there is some alleviation of the fine tuning problem existing in the MSSM.', '1310.6369-3-24-6': 'In fact, if we consider the measure [CITATION] [EQUATION] the fine tuning in this model is improved by a factor of 10 respect to the MSSM.', '1310.6369-3-25-0': '## Some phenomenology comments', '1310.6369-3-26-0': 'For a more general discussion about the LHC signatures of this type of models with extra vector-like matter, we refer the reader to [CITATION].', '1310.6369-3-26-1': 'In our model, the lightest neutralino has a mass [MATH] GeV and [MATH] GeV, in which case the limits set by the LHC searches using simplified models and CMSSM/mSUGRA are applicable to our analysis [CITATION].', '1310.6369-3-26-2': 'This implies a lower bound on the gluino mass around 1.3 TeV and 500 GeV for the stau mass.', '1310.6369-3-27-0': 'Since the MSSM matter does not mix, at tree level, with the vector-like fermions, the lightest of these, [MATH] (a combination of [MATH] and [MATH] fermions), is long-lived.', '1310.6369-3-27-1': 'For [MATH] TeV, its lifetime is [MATH] s.', '1310.6369-3-27-2': 'There exist cosmological bounds on this type of relics, coming from photodissociation of light elements[CITATION].', '1310.6369-3-27-3': 'After computing the number density of these colored particles [CITATION], we find that [MATH], where [MATH] is the entropy density and [MATH] is the relic number density.', '1310.6369-3-27-4': 'This is consistent with the cosmological constraints.', '1310.6369-3-28-0': 'On the collider side, searches for a charged massive stable particle have been done at the LHC at collision energies of [MATH] and [MATH] TeV[CITATION], where these particles can be identified by their anomalous energy loss or a long time-of-flight to the outer detectors.', '1310.6369-3-28-1': 'The limit obtained in [CITATION] for the mass of this fermion is [MATH] GeV.', '1310.6369-3-29-0': 'Finally, for [MATH] GeV, cosmological gravitino constraints set some bounds on the mass of the messengers and the reheating temperature after inflation.', '1310.6369-3-29-1': 'In order to avoid an overclosure of the universe by the thermal relic abundance of the gravitino, the reheat temperature should be lower than [MATH] GeV.', '1310.6369-3-29-2': 'The next constraint comes from big bang nucleosynthesis (BBN), which sets a bound on the gravitino mass such that the NLSP decays to a photon and a gravitino before [MATH].', '1310.6369-3-29-3': 'This implies [MATH] MeV, which translates into [MATH] GeV.', '1310.6369-3-30-0': '# Conclusions', '1310.6369-3-31-0': 'We have presented a model that yields a Higgs mass around [MATH] with stop masses under 2 TeV.', '1310.6369-3-31-1': 'In order to do so, we have synthesized two approaches used in the literature to tackle the little hierarchy problem.', '1310.6369-3-31-2': 'On one side, we have added a new set of fields to the effective theory.', '1310.6369-3-31-3': 'These fields possess vector-like masses around 1 TeV and contribute to raise the mass of the Higgs field through radiative corrections.', '1310.6369-3-31-4': 'On the other hand, we have included in the superpotential several marginal terms that couple the low energy degrees of freedom to the messenger sector.', '1310.6369-3-31-5': 'This provides a mechanism to enhance the trilinear [MATH] terms in the soft Lagrangian.', '1310.6369-3-31-6': 'This effect complements the enhancement of the Higgs mass by increasing the one-loop corrections through the stop (and vector-like scalar) mixing.', '1310.6369-3-31-7': 'This way, we obtained a wider region in the parameter space for which the measured mass of the Higgs can be achieved, while keeping the sfermions with relatively small masses.', '1310.6369-3-31-8': 'This, together with the fact that the soft term [MATH] is smaller for this typer of models, allow us to argue that the fine tuning problem in the MSSM is improved by this construction.', '1310.6369-3-32-0': 'So far, no signature of new vector matter or SUSY has been observed at the LHC.', '1310.6369-3-32-1': 'However, since this type of models contain stops and vector-like sfermions with masses just above the TeV scale, it presents a spectrum that can be searched for in the near future.', '1310.6369-3-33-0': 'We are grateful to Michael Dine for providing very useful comments and for reminding W.F. about their work done a few decades ago.', '1310.6369-3-33-1': 'W.T. also thanks Jianghao Yu, Can Kilic, Alejandro de la Puente and Simon Knapen for insightful discussions.', '1310.6369-3-33-2': 'We also thank Renato Fonseca for pointing out a couple of issues in the original version of this article.', '1310.6369-3-33-3': 'This material is based upon work supported by the National Science Foundation under Grant Numbers PHY-1316033 and PHY-0969020 and by the Texas Cosmology Center.', '1310.6369-3-34-0': '# Soft terms', '1310.6369-3-35-0': 'We calculate the soft masses at the messenger scale based on [CITATION], using the superpotential [REF].', '1310.6369-3-35-1': 'In such work, the formulas for the soft parameters are derived from wave function renormalization taking into account the discontinuity of the wave functions at the messenger threshold.', '1310.6369-3-36-0': 'In addition to the soft mass contribution from the standard gauge mediation [EQUATION] there is a two-loop contribution due to the Yukawa couplings between the messenger sector and the effective theory.', '1310.6369-3-36-1': '[EQUATION] where [MATH] is a multiplicity factor, [MATH] sums the quadratic Casimirs of the fields, and the sum over [MATH] includes only the MSSM matter.', '1310.6369-3-36-2': 'There is also a subleading one-loop contribution [EQUATION]', '1310.6369-3-36-3': 'On the other hand, the [MATH]-terms are given by [EQUATION]', '1310.6369-3-37-0': '# Renormalization group equations', '1310.6369-3-38-0': '[EQUATION]'}

{'1310.6369-4-0-0': '# Introduction', '1310.6369-4-1-0': 'The last two years have been remarkably exciting for both experimentalists and theorists working in high energy physics due to the discovery of the Higgs particle [CITATION].', '1310.6369-4-1-1': 'However, a Higgs mass of [MATH] poses intriguing questions, for theorists, about the naturalness of minimal supersymmetric models [CITATION].', '1310.6369-4-1-2': 'Due to the fact that, at tree level, the mass of the Higgs is bounded by the [MATH] boson mass, a large contribution to [MATH] must come from radiative corrections, which are dominated by the stop fields.', '1310.6369-4-1-3': 'Thus, it would require to have very heavy stop masses in order to generate such mass value.', '1310.6369-4-1-4': 'However, large stop masses make the soft parameter [MATH] large as well, which is straining the electroweak symmetry breaking condition [EQUATION] implying a large amount of fine tuning to achieve the proper cancellation between [MATH] and [MATH].', '1310.6369-4-1-5': 'This is known in the literature as the "little hierarchy problem".', '1310.6369-4-2-0': 'There have been several approaches to raising the mass of the Higgs in the literature.', '1310.6369-4-2-1': 'In one strategy, keeping SUSY minimal, the use of large trilinear [MATH]terms has been studied as a way to avoid the requirement of very heavy stops.', '1310.6369-4-2-2': 'This is, however, somewhat difficult to achieve in the standard gauge mediation of supersymmetry breaking scenario (GMSB) [CITATION], and would require very high messenger masses or a modification of the mediation mechanism.', '1310.6369-4-2-3': 'For instance, GMSB has been modified in models where mixing between the low energy degrees of freedom and the messenger fields is proposed proposed as a way to generate non-zero [MATH]terms at the messenger scale [CITATION] .', '1310.6369-4-3-0': 'In a different approach, extensions of the minimal supersymmetric standard model (MSSM) have been introduced in order to alleviate the little hierarchy problem.', '1310.6369-4-3-1': 'A clear example of such attempts is the next-to-MSSM (NMSSM, see [CITATION] for a review).', '1310.6369-4-3-2': 'Another proposal that falls in this category is the addition of a set of vector-like fields that couple to the Higgs multiplets.', '1310.6369-4-3-3': 'This raises the mass of the Higgs without requiring very heavy superpartners [CITATION].', '1310.6369-4-3-4': 'This type of extensions of the low energy matter content provides a spectrum with low masses that can be tested in the near future at the LHC.', '1310.6369-4-4-0': 'In this work, we explore a model that combines these two approaches.', '1310.6369-4-4-1': 'Minimal GMSB is modified to include Yukawa couplings between the MSSM and the messenger sector and, at the same time, there are additional vector-like fields at the TeV scale.', '1310.6369-4-4-2': 'A similar approach was used several decades ago by Dine and Fischler [CITATION] to generate electroweak symmetry breaking; however in their approach, the vector matter was much heavier.', '1310.6369-4-4-3': 'In our case, we provide a microscopic completion for a type of models presented in [CITATION] in the context of gauge mediated supersymmetry breaking, with the additional feature of having large [MATH]terms in the effective low energy Lagrangian, which leads to a Higgs mass consistent with the observed values at the LHC.', '1310.6369-4-4-4': 'We explore the parameter space and find a set of regions that yield a 126 GeV Higgs mass with stop masses under 2 TeV.', '1310.6369-4-4-5': 'We argue that the fine tune problem is substantially improved in this kind of scenario, compared to minimal GMSB, as there is no need for large stop masses and [MATH] gets a smaller radiative contribution from stops and vector-like sfermions.', '1310.6369-4-5-0': 'This article is structured as follows: In section [REF], we introduce the model with the new fields.', '1310.6369-4-5-1': 'We also present the soft terms calculated at the messenger scale.', '1310.6369-4-5-2': 'In section [REF] we show the correction the the Higgs mass due to the new fields and the results obtained in the numerical analysis.', '1310.6369-4-5-3': 'We close with some conclusions and we include two appendices with the RGEs of the model and the complete expressions for the soft masses.', '1310.6369-4-6-0': '# The model', '1310.6369-4-7-0': 'In this paper, we introduce a set of new vector-like chiral superfields that are charged under the Standard Model (SM) gauge symmetry group.', '1310.6369-4-7-1': 'This is motivated by the work presented in [CITATION].', '1310.6369-4-7-2': 'There, new superheavy superfields were added in order to achieve the breaking of the gauge symmetry [MATH] by generating negative values of [MATH] in the Higgs potential.', '1310.6369-4-7-3': 'In the present case, we consider similar new superfields with vector-like masses somewhere between 500 GeV and 1.4 TeV.', '1310.6369-4-7-4': 'The addition of these new superfields is expected to lift the mass of the Higgs through radiative corrections in such a way that very large stop masses are unnecessary.', '1310.6369-4-7-5': 'Furthermore, as done in [CITATION], we allow the MSSM fields and the new vector-like matter to interact directly with the messenger sector through Yukawa couplings.', '1310.6369-4-7-6': 'This mixing with the messengers will generate large trilinear terms ([MATH]terms) that contribute to the Higgs mass enhancement.', '1310.6369-4-8-0': 'The new vector-like superfields can be arranged in complete representations of [MATH].', '1310.6369-4-8-1': 'Here, we choose to use a pair of 10 dimensional representations [MATH], [EQUATION]', '1310.6369-4-8-2': 'On the other hand, we use a pair of [MATH] representations in the messenger sector, [EQUATION]', '1310.6369-4-8-3': 'These messengers couple to the spurion [MATH], which generates the messenger mass and breaks SUSY through its expectation value, [MATH], in the superpotential [EQUATION]', '1310.6369-4-8-4': 'We consider a superpotential that connects three sectors: the MSSM, the new vector-like superfields, and the messengers fields.', '1310.6369-4-8-5': '[EQUATION] where the [MATH] and [MATH] indices have been contracted in the usual way, i.e. [MATH], with [MATH] and [MATH].', '1310.6369-4-8-6': 'The negative sign in front of [MATH] is not necessary; however, it facilitates our analysis in analogy to [MATH] in the MSSM.', '1310.6369-4-8-7': 'We assign [MATH]parity [MATH] to the new vector-like fields to prevent the low energy vector-like fields from mixing with the MSSM quarks or leptons in the superpotential.', '1310.6369-4-8-8': 'It is worth mentioning that, in this work, we are allowing all the couplings in Equation (2.4) to be non-zero; this is different from previous works, where only one non-zero coupling was consider at a time.', '1310.6369-4-9-0': '## Effective mass terms', '1310.6369-4-10-0': 'Supersymmetry is broken due to the non-zero F-term [MATH].', '1310.6369-4-10-1': 'This generates soft mases for the MSSM fields as well as for the new vector-like fields.', '1310.6369-4-10-2': 'Besides the soft masses generated through the usual gauge mediation mechanism (GMSB), there is an additional contribution to the masses of the sfermions due to the Yukawa couplings to the messenger fields.', '1310.6369-4-10-3': 'The calculation of these soft masses follow the same methodology presented in [CITATION].', '1310.6369-4-11-0': 'This modified gauge mediation mechanism results in the soft Lagrangian [EQUATION] where [MATH], for [MATH].', '1310.6369-4-11-1': 'These soft parameters are calculated at the messenger scale, [MATH], and evolved down to the electroweak scale through the running of the renormalization group.', '1310.6369-4-11-2': 'For simplicity, we write here just the leading contributions to the soft masses of the vector-like field [MATH] coming from the gauge and Yukawa interactions with the messengers.', '1310.6369-4-11-3': 'In Appendix [REF], we present a general formula to compute these soft terms.', '1310.6369-4-11-4': 'Also, for simplicity, we make [MATH].', '1310.6369-4-11-5': '[EQUATION]', '1310.6369-4-11-6': 'At the messenger scale, the [MATH]terms are non-zero due to the Yukawa couplings to the messengers.', '1310.6369-4-11-7': 'For instance, [MATH] takes the value [EQUATION]', '1310.6369-4-11-8': 'This implies that, at low energies, there is a significant mixing between the scalar superpartners.', '1310.6369-4-12-0': 'After calculating the soft masses and [MATH]terms, we run down the renormalization group from the messenger scale to the low energies where electroweak symmetry breaking is computed and the Higgs mass is obtained, as shown in the next section.', '1310.6369-4-12-1': 'The RGEs corresponding to this model are presented in the appendix .', '1310.6369-4-13-0': 'In the effective theory, we have a set of electrically neutral fermionic fields that couple to the neutral components of the Higgs multiplets and with mass matrix [EQUATION]', '1310.6369-4-13-1': 'On the other hand, the mass matrix for the electrically neutral scalars coupled to the neutral Higgs bosons is given by [EQUATION] where [MATH].', '1310.6369-4-14-0': 'In the stop sector, the scalar masses are given by [EQUATION] where [MATH] and [MATH] are the [MATH] and [MATH] soft masses, which at the messenger scale are given by [EQUATION] while the trilinear parameter is given by [EQUATION]', '1310.6369-4-14-1': 'It is noteworthy the fact that the effect of having large values for [MATH] on Equation ([REF]) is an increment in the splitting of the stop masses.', '1310.6369-4-14-2': 'However [MATH] contributes directly to [MATH] and [MATH], that implies that we can benefit from having relatively large [MATH]-terms as long as they are not much larger than the squark soft masses.', '1310.6369-4-14-3': 'A numerical depiction of [MATH] can be seen in Figure [REF] in the next section.', '1310.6369-4-15-0': '# The lightest CP-even Higgs mass', '1310.6369-4-16-0': 'At tree level, the scalar potential for the Higgs fields is the same as in the MSSM [CITATION]: [EQUATION]', '1310.6369-4-16-1': 'In the mass spectrum, we find a charged pair of Higgs fields [MATH], a CP-odd neutral scalar [MATH] and two CP-even neutral scalars [MATH] and [MATH].', '1310.6369-4-16-2': 'As it is well known, at tree level, the mass of the lightest neutral Higgs is bounded from above by the mass of the [MATH] boson, .', '1310.6369-4-16-3': 'i.e. [MATH].', '1310.6369-4-16-4': 'In the so-called "decoupling limit", [MATH], [MATH] saturates this bound.', '1310.6369-4-16-5': 'This is the limit that we use in this work.', '1310.6369-4-17-0': 'In the MSSM scenario, at one-loop level, the top and stop fields contribute to the mass of the lightest Higgs.', '1310.6369-4-17-1': 'The effective potential due to these fields is [EQUATION]', '1310.6369-4-17-2': 'The one-loop corrected mass of the lightest CP-even Higgs is, then, given by [CITATION] [EQUATION] with [MATH].', '1310.6369-4-17-3': 'The two-loop correction to this expression is [CITATION] [EQUATION]', '1310.6369-4-17-4': 'For the numerical calculations in the following section, we will use both 1-loop and 2 -loop contributions to the Higgs mass, [MATH].', '1310.6369-4-18-0': 'Attaining a Higgs mass [MATH] is somewhat challenging for minimal GMSB unless the top squarks have masses larger than [MATH], which, as already stated, generates some conflict with the naturalness of the MSSM.', '1310.6369-4-18-1': 'Additionally, in GMSB, the [MATH]terms are zero at the messenger scale, although they are non-zero at the electroweak scale due to the running of their RGEs.', '1310.6369-4-18-2': 'However, in most minimal GMSB models, [MATH] is still smaller than 1, which does not favor the enhancement of the Higgs mass coming from stop mixing; instead, the radiative corrections to [MATH] are dominated by the logarithmic term in Equation ([REF])[CITATION].', '1310.6369-4-18-3': 'There are some scenarios in GMSB where stops lighter than [MATH] and [MATH] are possible, by having heavy gauginos or a very large messenger scale [CITATION].', '1310.6369-4-19-0': 'When we introduce the new vector-like fields, there is a similar effective one-loop potential from the vector-like fields, [EQUATION] where [MATH] runs over [MATH] and [MATH] are the eigenvalues of the mass matrices ([REF], [REF]) respectively.', '1310.6369-4-19-1': 'The correction to the Higgs mass is given by [CITATION] [EQUATION] where [MATH] and [MATH].', '1310.6369-4-20-0': 'The correction to the Higgs mass is, then, given by [EQUATION]', '1310.6369-4-21-0': '## Parameter scan', '1310.6369-4-22-0': 'In order to analyze the implications of this correction to the Higgs mass, we numerically scan the parameter space over the ranges shown in Table [REF].', '1310.6369-4-22-1': 'We use these input parameters to compute the soft terms at the messenger scale and, then, we use the one-loop RG equations to extract the value of the different masses and couplings at the electroweak scale.', '1310.6369-4-22-2': 'For each set of input parameters we calculate the Higgs mass with and without the extra vector-like fields.', '1310.6369-4-23-0': 'In our parameter scan, we vary [MATH] and [MATH] such that the model is perturbative below the unification scale .', '1310.6369-4-23-1': '[MATH] does not play an important role in raising [MATH], hence we assume [MATH].', '1310.6369-4-23-2': 'The best results for [MATH] are obtained for [MATH].', '1310.6369-4-24-0': 'For simplicity, we have set the mass of the vector-like fermions to be the same at the GUT scale.', '1310.6369-4-24-1': 'At the electroweak scale, this fermions have vector-like masses between 700 GeV and 2 TeV.', '1310.6369-4-24-2': 'Figure [REF] shows the values of [MATH] vs [MATH] when the vector-like fields are included.', '1310.6369-4-24-3': 'It is important to note that [MATH] GeV is easily achieved in this model even for stop masses under 1 TeV.', '1310.6369-4-24-4': 'In the analyzed parameter space, the correction to the Higgs mass, given in equation ([REF]), can be as small as [MATH] and and as large as 35 GeV.', '1310.6369-4-24-5': 'However, when we just look at the region where the corrected mass of the Higgs is [MATH] (the blue dots in Figure [REF]), we find that [MATH] takes values between [MATH] and [MATH].', '1310.6369-4-24-6': 'This enhancement is more notable for vector-like masses around [MATH].', '1310.6369-4-25-0': 'As for the SUSY scale [MATH] that corresponds to [MATH] and [MATH] TeV, we obtained values as low as [MATH] GeV and as high as [MATH] GeV with messengers masses between [MATH] and [MATH] GeV.', '1310.6369-4-25-1': 'Notice that this implies a lower scaler for [MATH] than in the GMSB scenario, where [MATH] GeV [CITATION].', '1310.6369-4-25-2': 'A comparison of these scales is depicted in Figure [REF].', '1310.6369-4-26-0': 'Now, we take a look at the values of the [MATH]terms when we include the vector-like fields.', '1310.6369-4-26-1': 'Figure [REF] shows the values for [MATH] for different values of [MATH].', '1310.6369-4-26-2': 'It can be seen that, indeed, it is possible to obtain trilinear terms such that [MATH], unlike the MSSM case with minimal GMSB.', '1310.6369-4-26-3': 'Thus, this feature of the model aides the addition of the vector-like fields in the lifting of the Higgs mass.', '1310.6369-4-26-4': 'It is worth quantifying the effect of the messenger-matter mixing to the values of [MATH] compared to the mGMSB scenario.', '1310.6369-4-26-5': 'Defining [MATH] at the low scale, we find that [EQUATION]', '1310.6369-4-26-6': 'To gain further insight about how having larger [MATH] terms helps in raising [MATH], we compare, in Figure [REF], the regions in the vector-like fermion mass [MATH] vs [MATH] plane that are compatible with a Higgs mass of [MATH] GeV for two cases: our model with messenger mixing (green area) and a model without mixing (yellow area, see [CITATION]).', '1310.6369-4-26-7': 'This is shown for [MATH], [MATH], [MATH] GeV and [MATH] .', '1310.6369-4-26-8': 'It can be noticed that, even if the yellow region shows that a realistic mass for the Higgs can be obtained with low masses for the vector-like fermions and the gaugino, the addition of the couplings to the messengers improves the possibilities.', '1310.6369-4-27-0': 'Considering the fact that the Higgs mass is easily lifted in this type of models and the mass of the stops is not required to be larger than 2 TeV, one could think naively that such a proposal is more "natural" than the MSSM with mGMSB.', '1310.6369-4-27-1': 'However, as pointed out in [CITATION], the new vector-like fields will contribute to the soft term [MATH], in the same way than stops, through the radiative term [EQUATION] where [MATH].', '1310.6369-4-27-2': 'This, apparently, would make the little hierarchy problem even worse.', '1310.6369-4-27-3': 'However, the effect of this term, and the stop contribution, in making [MATH] large is smaller in this case (where the stops and the vector sfermions have masses just above 1 TeV) compared to the case where 5 TeV stops are required.', '1310.6369-4-27-4': 'Concretely, [EQUATION]', '1310.6369-4-27-5': 'Thus, this effect together with the correction to the Higgs mass lead us to argue that there is some alleviation of the fine tuning problem existing in the MSSM with mGMSB.', '1310.6369-4-27-6': 'In fact, if we consider the measure [CITATION] [EQUATION] where [MATH], this model produces values for [MATH] between 300 and 500 for the range of parameters studied in this section and corresponding to the green region in Figure [REF].', '1310.6369-4-27-7': 'This turns out to be an order of magnitude smaller than the values of [MATH] corresponding to mGMSB in the blue region of the same figure.', '1310.6369-4-27-8': 'It is interesting to remark that for one of the least tuned points in the parameter space, which is depicted in Figure [REF], the contribution to the Higgs mass enhancement due to the presence of the vector-like fields is dominant respect to the fact of having a large [MATH] term.', '1310.6369-4-28-0': '## Some phenomenology comments', '1310.6369-4-29-0': 'For a more general discussion about the LHC signatures of this type of models with extra vector-like matter, we refer the reader to [CITATION].', '1310.6369-4-29-1': 'For the sake of clarity, we present a sample spectrum in Figure [REF], where the new charged quarks (combinations of the [MATH] and [MATH] fermions) are denoted as [MATH] and their spartners are [MATH] with [MATH].', '1310.6369-4-29-2': 'Likewise, the charged vectorlike lepton is denoted as [MATH] and its spartners [MATH].', '1310.6369-4-30-0': 'In our model, the lightest neutralino has a mass [MATH] GeV and [MATH] GeV, in which case the limits set by the LHC searches using simplified models and CMSSM/mSUGRA are applicable to our analysis [CITATION].', '1310.6369-4-30-1': 'This implies a lower bound on the gluino mass around 1.3 TeV (see Figure [REF]) and 500 GeV for the stau mass.', '1310.6369-4-31-0': 'Since the MSSM matter does not mix, at tree level, with the vector-like fermions, the lightest of these, [MATH] (a combination of [MATH] and [MATH] fermions), is long-lived.', '1310.6369-4-31-1': 'For [MATH] TeV, its lifetime is [MATH] s.', '1310.6369-4-31-2': 'There exist cosmological bounds on this type of relics, coming from photodissociation of light elements[CITATION].', '1310.6369-4-31-3': 'After computing the number density of these colored particles [CITATION], we find that [MATH], where [MATH] is the entropy density and [MATH] is the relic number density.', '1310.6369-4-31-4': 'This is consistent with the cosmological constraints.', '1310.6369-4-32-0': 'On the collider side, searches for a charged massive stable particle have been done at the LHC at collision energies of [MATH] and [MATH] TeV[CITATION], where these particles can be identified by their anomalous energy loss or a long time-of-flight to the outer detectors.', '1310.6369-4-32-1': 'The limit obtained in [CITATION] for the mass of this fermion is [MATH] GeV, which constraints the bottom part of Figure [REF], but still leaves a large region in the parameter space where the vector-like fermion mass is above 1 TeV.', '1310.6369-4-33-0': 'Finally, for [MATH] GeV, cosmological gravitino constraints set some bounds on the mass of the messengers and the reheating temperature after inflation.', '1310.6369-4-33-1': 'In order to avoid overclosure of the universe by the thermal relic abundance of the gravitino, the reheat temperature should be lower than [MATH] GeV.', '1310.6369-4-33-2': 'The next constraint comes from big bang nucleosynthesis (BBN), which sets a bound on the gravitino mass such that the NLSP decays to a photon and a gravitino before [MATH] (see [CITATION] and references therein).', '1310.6369-4-33-3': 'This implies [MATH] MeV, which translates into [MATH] GeV.', '1310.6369-4-34-0': '# Conclusions', '1310.6369-4-35-0': 'We have presented a model that yields a Higgs mass around [MATH] with stop masses under 2 TeV.', '1310.6369-4-35-1': 'In order to do so, we have synthesized two approaches used in the literature to tackle the little hierarchy problem.', '1310.6369-4-35-2': 'First, we have added a new set of fields to the effective theory.', '1310.6369-4-35-3': 'These fields possess vector-like masses around 1 TeV and contribute to raise the mass of the Higgs field through radiative corrections.', '1310.6369-4-35-4': 'On the other hand, we have included in the superpotential several marginal terms that couple the low energy degrees of freedom to the messenger sector.', '1310.6369-4-35-5': 'This provides a mechanism to enhance the trilinear [MATH] terms in the soft Lagrangian.', '1310.6369-4-35-6': 'This effect complements the enhancement of the Higgs mass by increasing the one-loop corrections through the stop (and vector-like scalar) mixing.', '1310.6369-4-35-7': 'This way, we obtained a wider region in the parameter space for which the measured mass of the Higgs can be achieved, while keeping the sfermions with relatively small masses.', '1310.6369-4-35-8': 'In addition, the soft term [MATH] is smaller for this type of models.', '1310.6369-4-35-9': 'These facts allow us to argue that the fine tuning problem in the MSSM is improved by this construction.', '1310.6369-4-36-0': 'So far, no signature of new vector matter or SUSY has been observed at the LHC.', '1310.6369-4-36-1': 'However, since this sort of models contains stops and vector-like sfermions with masses just above the TeV scale, it presents a spectrum that can be searched for in the near future.', '1310.6369-4-37-0': 'We are grateful to Michael Dine for providing very useful comments and for reminding W.F. about their work done a few decades ago.', '1310.6369-4-37-1': 'W.T. also thanks Jianghao Yu, Can Kilic, Alejandro de la Puente and Simon Knapen for insightful discussions.', '1310.6369-4-37-2': 'We also thank Renato Fonseca for pointing out a couple of issues in the original version of this article.', '1310.6369-4-37-3': 'This material is based upon work supported by the National Science Foundation under Grant Numbers PHY-1316033 and PHY-0969020 and by the Texas Cosmology Center.', '1310.6369-4-38-0': '# Soft terms', '1310.6369-4-39-0': 'We calculate the soft masses at the messenger scale based on [CITATION], using the superpotential [REF].', '1310.6369-4-39-1': 'In such work, the formulas for the soft parameters are derived from wave function renormalization taking into account the discontinuity of the wave functions at the messenger threshold.', '1310.6369-4-40-0': 'In addition to the soft mass contribution from the standard gauge mediation [EQUATION] there is a two-loop contribution due to the Yukawa couplings between the messenger sector and the effective theory.', '1310.6369-4-40-1': '[EQUATION] where [MATH] is a multiplicity factor, [MATH] sums the quadratic Casimirs of the fields, and the sum over [MATH] includes only the MSSM matter.', '1310.6369-4-40-2': 'There is also a subleading one-loop contribution [EQUATION]', '1310.6369-4-40-3': 'On the other hand, the [MATH]-terms are given by [EQUATION]', '1310.6369-4-41-0': '# Renormalization group equations', '1310.6369-4-42-0': '[EQUATION]'}

1710.11529

{'1710.11529-1-0-0': 'The 4D-Var method for filtering partially observed nonlinear chaotic dynamical systems [CITATION] consists of finding the maximum a-posteriori (MAP) estimator of the initial condition of the system given observations over a time window, and propagating it forward to the current time via the model dynamics.', '1710.11529-1-0-1': 'This method forms the basis of most currently operational weather forecasting systems (see [CITATION]).', '1710.11529-1-0-2': 'In practice the optimization becomes infeasible if the time window is too long due to the multimodality of the likelihood [CITATION] and the effect of model errors [CITATION].', '1710.11529-1-0-3': 'Hence the window has to be kept sufficiently short, and the observations in the previous windows can be taken into account via a Gaussian background (prior) distribution.', '1710.11529-1-0-4': 'The choice of the background covariance matrix is an important question that has received much attention in the literature, see e.g.[CITATION].', '1710.11529-1-0-5': 'In this paper, we define the background covariances in a principled manner, based on observations in the previous [MATH] assimilation windows, for a parameter [MATH].', '1710.11529-1-0-6': 'The method is at most [MATH] times more computationally expensive than using fixed background covariances, requires little tuning, and greatly improves the accuracy of 4D-Var.', '1710.11529-1-0-7': 'As a concrete example, we focus on the shallow-water equations.', '1710.11529-1-0-8': 'The proposed method is compared against state-of-the-art approaches in data assimilation and is shown to perform very favourably on simulated data.', '1710.11529-1-0-9': 'We also illustrate our approach on data from the recent tsunami of 2011 in Fukushima, Japan.', '1710.11529-1-1-0': 'Key words: Filtering; Smoothing; Gauss-Newton Method; Shallow-Water Equations.', '1710.11529-1-2-0': '# Introduction', '1710.11529-1-3-0': 'Filtering, or data assimilation, is a field of core importance in a wide variety of real applications, such as numerical weather forecasting, climate modelling and finance; see e.g. [CITATION] for an introduction.', '1710.11529-1-3-1': 'Informally, one is interested in carrying out inference about an unobserved signal process conditionally upon noisy observations.', '1710.11529-1-3-2': 'The type of unobserved process considered in this paper is that of a nonlinear chaotic dynamical system, with unknown initial condition.', '1710.11529-1-3-3': 'We focus on the case where the unobserved dynamics correspond to the discretized version of the shallow-water equations; see e.g. [CITATION].', '1710.11529-1-3-4': 'These latter equations are of great practical importance, generating realistic approximations of real world phenomena, useful in tsunami and flood modelling (see e.g. [CITATION]).', '1710.11529-1-4-0': 'For systems of the type considered in this article, the filtering problem is notoriously challenging.', '1710.11529-1-4-1': 'Firstly, the filter is seldom available in analytic form due to the non-linearity.', '1710.11529-1-4-2': 'Secondly, even if the given system were solvable, the associated dimension of the object to be filtered is very high (of order of [MATH] or greater) thus posing great computational challenges, e.g. with regards to the storage of covariance matrices.', '1710.11529-1-4-3': 'Thirdly, the algorithmic cost cannot grow prohibitively with the (real) time of the system.', '1710.11529-1-5-0': 'Despite the challenges, due to the practical interest in weather prediction, several techniques, able to process massive datasets, have been devised and implemented operationally in weather forecasting centres worldwide.', '1710.11529-1-5-1': 'One of the most successful methods is the so-called 4D-Var algorithm [CITATION]: it consists of optimising a loss-functional so that under Gaussian noise it is equivalent to finding the maximum a-posteriori (MAP) estimator of the initial condition.', '1710.11529-1-5-2': 'Since its introduction, a lot of further developments in the 4D-Var methodology have appeared in the literature; for an overview of some recent advances, we refer the reader to [CITATION].', '1710.11529-1-5-3': 'The main focus of this article is to consider principled improvements of the 4D-Var algorithm.', '1710.11529-1-6-0': 'While it is well understood that the MAP estimator is optimal for linear systems, in [CITATION] it was also shown to be asymptotically optimal for a class of non-linear systems - in an appropriate sense - under a small-noise/high-frequency regime for a fixed observation window.', '1710.11529-1-6-1': 'Based on this theoretical result, we expect the 4D-Var method to perform well in the small-noise/high-frequency setting, when the dynamical system and the observations are not too non-linear.', '1710.11529-1-6-2': 'In this setting, based on the results of [CITATION], one expects the posterior distribution to be approximately Gaussian.', '1710.11529-1-6-3': 'If the conditions are not met, as in the case of highly non-linear systems and observations, high observation noise, low observation frequency, or long assimilation window, it is likely that the Gaussian approximation of the smoothing distribution is no longer accurate and the MAP is no longer necessarily close to optimal in mean square error.', '1710.11529-1-6-4': 'Nevertheless, as long as the smoothing distribution has most of its mass contained in a single mode, we expect the typical error of the MAP estimator to still be of the order of the observation noise, giving 4D-Var methodology a certain robustness that is not necessarily present in filters that are directly based on Gaussian approximations in the update (such as the extended or the ensemble Kalman filter).', '1710.11529-1-7-0': 'An important practical issue of the 4D-Var method is that, due to chaotic nature of the systems encountered in weather prediction, the likelihood can become highly multimodal if the assimilation window is too long.', '1710.11529-1-7-1': 'The observations in the previous window are taken into account via the background (prior) distribution, which is a Gaussian whose mean is the estimate of the current position based on the previous windows, and has a certain covariance matrix.', '1710.11529-1-7-2': 'The choice of this background covariance matrix is an important and difficult problem that has attracted much research.', '1710.11529-1-7-3': '[CITATION] states that in operational weather forecasting systems up to 85% of the information in the smoothing distribution comes from the background (prior) distribution.', '1710.11529-1-7-4': 'The main contribution of this paper is the improvement of the 4D-Var methodology by a principled definition of this matrix in a flow-dependent way.', '1710.11529-1-7-5': 'This is based on the observations in the previous [MATH] assimilation windows (for a parameter [MATH]).', '1710.11529-1-7-6': 'Our method is computationally efficient, leads to improved precision, and requires less tuning than using a fixed background covariance matrix.', '1710.11529-1-8-0': 'There exist mathematically rigorous techniques to obtain the filter with precision and use the mean of the posterior distribution as the estimate, based upon sequential Monte Carlo methods (e.g. [CITATION]) which can provably work in high-dimensional systems [CITATION].', '1710.11529-1-8-1': 'While these are optimal in mean square error, and are probably considerably more accurate than the methods we work with in this paper, nonetheless such methodology can be practically overly expensive.', '1710.11529-1-8-2': 'As a result, one may have to resort to less accurate but more computationally efficient methodologies.', '1710.11529-1-8-3': 'These include 3D-Var, 4D-Var and the ensemble Kalman filter; see [CITATION] for a review.', '1710.11529-1-8-4': 'There are some relatively recent applications of particle filtering methods to high dimensional data assimilation problems, see e.g. [CITATION].', '1710.11529-1-8-5': 'While these algorithms seem to be promising for certain highly non-linear problems, their theoretical understanding is limited at the moment due to the bias they introduce via various approximations.', '1710.11529-1-9-0': 'The structure of the paper is as follows.', '1710.11529-1-9-1': 'In Section [REF] the modelling framework for the shallow-water equations is described in detail.', '1710.11529-1-9-2': 'In Section [REF] we introduce our 4D-Var method with flow-dependent background covariance.', '1710.11529-1-9-3': 'In particular, Section [REF] compares our method with other choices of flow-dependent background covariances in the literature.', '1710.11529-1-9-4': 'Section [REF] reviews some existing alternative methods for data assimilation.', '1710.11529-1-9-5': 'In Section [REF] we present some simulation results and compare the performance of our method with the 3D-Var and ensemble Kalman Filter methods.', '1710.11529-1-9-6': 'Finally, in Section [REF] we state some conclusions for this paper.', '1710.11529-1-10-0': '# Model and Multimodality', '1710.11529-1-11-0': '## The Model', '1710.11529-1-12-0': 'We consider the shallow-water equations, e.g. as in [CITATION], but with added diffusion and bottom friction terms, i.e. [EQUATION]', '1710.11529-1-12-1': 'Here, [MATH] and [MATH] are the velocity fields in the [MATH] and [MATH] directions respectively, and [MATH] the field for the height of the wave.', '1710.11529-1-12-2': 'Also, [MATH] is the depth of the ocean, [MATH] the gravity constant, [MATH] the Coriolis parameter, [MATH] the bottom friction coefficient and [MATH] the viscosity coefficient.', '1710.11529-1-12-3': 'Parameters [MATH], [MATH], [MATH] and [MATH] are assumed to be constant in time but in general depend on the location.', '1710.11529-1-12-4': 'The total height of the water column is the sum [MATH].', '1710.11529-1-13-0': 'For square grids, under periodic boundary conditions, the equations are discretised as [EQUATION] where [MATH], for a typically large [MATH], with the indices understood modulo [MATH], and some space-step [MATH].', '1710.11529-1-13-1': 'Summing up [REF] over [MATH], one can see that the discretisation preserves the total mass [MATH].', '1710.11529-1-13-2': 'If we assume that the viscosity and bottom friction are negligible, i.e. [MATH], then the total energy [MATH] is also preserved.', '1710.11529-1-13-3': 'When the coefficients [MATH] and [MATH] are not zero, the bottom friction term always decreases the total energy (the sum of the kinetic and potential energy), while the diffusion term tends to smooth the velocity profile.', '1710.11529-1-13-4': 'We denote the solution of equations [REF]-[REF] at time [MATH] as [EQUATION]', '1710.11529-1-13-5': 'The unknown and random initial condition is denoted by [MATH].', '1710.11529-1-13-6': 'One can show by standard methods (see [CITATION]) that the solution of [REF]-[REF] exists up to some time [MATH].', '1710.11529-1-13-7': 'In order to represent the components of [MATH], we introduce a vector index notation.', '1710.11529-1-13-8': 'The set [MATH] denotes the possible indices, with the first component referring to one of [MATH], [MATH], [MATH], the second component to coordinate [MATH], and the third to [MATH].', '1710.11529-1-13-9': 'A vector index in [MATH] will usually be denoted as [EQUATION]-^2q_(m-1)k(U(t_(m-1)k))+D_m,[EQUATION] from the observed ones, and choose the coefficients [MATH] as the least square estimator for these samples (i.e. such that the difference between the sum of the squares for all the samples of the linear estimator of the [MATH]th derivative of the chosen component [EQUATION] and [MATH] corresponding different components [MATH], [MATH] or [MATH] (see Section [REF] for the notation [MATH]), while equals [EQUATION] for [MATH] corresponding to component [MATH] (see equation (A10) of [CITATION]), and similarly for the other components with constants [MATH] and [MATH].', '1710.11529-1-13-10': 'For one dimensional grids, this covariance matrix is known to have spectral response (see equation (A11) of [CITATION]) [EQUATION] corresponding to damping the high frequency terms.', '1710.11529-1-13-11': 'Due to the high dimensionality of the system, it is not practical to operate with this covariance matrix and its inverse directly.', '1710.11529-1-13-12': 'Instead there are approximate ways to apply them on a vector that leads to similar spectral response, see [CITATION].', '1710.11529-1-13-13': 'An alternative way of obtaining a spectral response similar to [REF] for our model is via the two dimensional discrete cosine transform (DCT), see [CITATION] for an overview.', '1710.11529-1-13-14': 'Based on this, the effect of the covariance matrix on a vector is computed by first transforming the vector by DCT (separately in components [MATH], [MATH] and [MATH]), then applying the spectral response [REF] (which is equivalent to multiplying with a diagonal matrix), and finally transforming it back (see Section 7.1.2 of [CITATION] for the description of such an approach).', '1710.11529-1-13-15': 'The effect of the precision matrix is analogous except that the inverse of [REF] is applied in the second step.', '1710.11529-1-14-0': '## Literature on Flow-Dependent Covariances for 4D-Var', '1710.11529-1-15-0': 'Flow-dependent background covariances have been used in practice since [CITATION] that proposed the so-called NMC method based on a comparison of 24 and 48 hour forecast differences.', '1710.11529-1-15-1': 'This method was refined in [CITATION].', '1710.11529-1-15-2': '[CITATION] proposed the use of wavelets for forming background covariances; these retain the computational advantages of spectral methods, while also allow for spatial inhomogeneity.', '1710.11529-1-15-3': 'The background covariances are made flow-dependent via a suitable modification of the NMC approach.', '1710.11529-1-15-4': 'The paper [CITATION] reviews some of the practical aspects of modelling 4D-Var error covariances, while [CITATION] makes a comparison between 4D-Var for long assimilation windows and the Extended Kalman Filter.', '1710.11529-1-15-5': 'As we have noted previously, long windows are not always applicable due to the presence of model errors and multimodalities in the likelihood.', '1710.11529-1-16-0': 'More recently, there have been several methods proposing the use of ensembles for modelling the covariances, see e.g. [CITATION].', '1710.11529-1-16-1': 'However, it is not clear how well can a small ensemble approximate the covariances, which are localised only over short assimilation intervals, but become full matrices over longer intervals.', '1710.11529-1-16-2': 'Our method uses a factorised form of the Hessian and the background precision matrix described in Sections [REF]-[REF], thus we only need to store the Jacobians between observation times in a sparse form.', '1710.11529-1-16-3': 'This allows us to compute the effect of these matrices on a vector efficiently, without needing to store all the elements of the background precision matrix, which would require too much memory.', '1710.11529-1-17-0': 'Although in this paper we have assumed that the model is perfect, there have been efforts to account for model error in the literature, see [CITATION].', '1710.11529-1-17-1': 'The effect of nonlinearities in the dynamics and the observations can be in some cases so strong that the Gaussian approximations are no longer reasonable, see [CITATION] for some examples and suggestions for overcoming these problems.', '1710.11529-1-18-0': '# Alternative Data Assimilation Methods', '1710.11529-1-19-0': 'Consider a discrete-time deterministic dynamical system in [MATH], started from an initial point [MATH] and evolving according to [EQUATION] for some [MATH].', '1710.11529-1-19-1': 'Assume that we have linear observations of the form [EQUATION] with [MATH] and [MATH].', '1710.11529-1-19-2': 'We briefly review some commonly used filtering techniques that can approximate [MATH] given observations [MATH], see e.g. [CITATION] where it is shown that all of relevant methods can be interpreted as solutions of minimization problems delivering a balance between model prediction, [MATH], and corresponding observation, [MATH].', '1710.11529-1-20-0': '## Kalman Filter', '1710.11529-1-21-0': 'In the linear case [MATH], and a Gaussian prior for [MATH], the laws [MATH] are Gaussian with means [MATH] and covariances [MATH] obtained via the Kalman filter update equations [EQUATION]', '1710.11529-1-21-1': 'To extend such updates to a non-linear system, [MATH] can be replaced by the linearisation of the system dynamics at the current estimate.', '1710.11529-1-21-2': 'A slight modification of this idea gives the extended Kalman filter, which propagates the covariances according to the linearisation of the dynamics and the means according to the non-linear dynamics (see e.g. Section 4.2.2 of [CITATION]).', '1710.11529-1-21-3': 'For linear or weakly non-linear systems, such methods can be accurate.', '1710.11529-1-21-4': 'However, to update the mean, in general one needs to invert a [MATH] matrix, with [MATH] computational costs and [MATH] memory requirements, thus these methods, in their standard form, can be overly expensive for high-dimensional systems.', '1710.11529-1-21-5': 'Besides the computational cost, the Kalman filter and the extended Kalman filter are known to diverge in some cases of highly non-linear systems.', '1710.11529-1-21-6': 'To remedy such issues, two approximate filters have been widely used in the literature, and are described below.', '1710.11529-1-22-0': '## 3D-Var', '1710.11529-1-23-0': 'This algorithm starts with an initial estimator [MATH], fixes a covariance [MATH] for all steps, and applies the iterations [EQUATION]', '1710.11529-1-23-1': 'The covariance matrix being kept fixed, this method can have low computational cost (the exact order depends on the choice of [MATH]).', '1710.11529-1-23-2': '[CITATION] proves some theoretical results about a version of 3D-Var (e.g., under conditions, the mean square error of the filter is [MATH] - assuming [MATH] and the time between observations is [MATH]).', '1710.11529-1-24-0': '## Ensemble Kalman Filter', '1710.11529-1-25-0': 'Ensemble Kalman filter (EnKF) uses an ensemble of particles to implicitly store the uncertainty of the filtering laws, and updates them using the Kalman approach based on ensemble means and covariances.', '1710.11529-1-25-1': 'There are various ways to execute this algorithm, so we describe the so-called perturbed observation EnKF.', '1710.11529-1-26-0': 'One starts with an ensemble of [MATH] particles [MATH] that are i.i.d. draws from some [MATH].', '1710.11529-1-26-1': 'At each step [MATH], there is the prediction,', '1710.11529-1-27-0': '_l+1^(n)=(_l^(n)),n=1,,N,', '1710.11529-1-28-0': "m_l+1=1N_n=1^N _l+1^(n), C_l+1=1N-1_n=1^N(_l+1^(n)-m_l+1)(_l+1^(n)-m_l+1)',", '1710.11529-1-29-0': 'followed by the analysis,', '1710.11529-1-30-0': '_l+1^(n)=(I-K_l+1Hr)_l+1^(n)+K_l+1Y_l+1^(n), n=1,,N;', '1710.11529-1-31-0': "S_l+1:=HhatC_l+1H'+, K_l+1:=C_l+1H'S_l+1^-1, Y_l+1^(n)=Y_l+1+Z_l+1^(n),", '1710.11529-1-32-0': 'where [MATH] are i.i.d draws from [MATH].', '1710.11529-1-32-1': 'This filter has the property that as [MATH], then for linear dynamics it converges to the exact Kalman filter.', '1710.11529-1-32-2': 'One typically chooses [MATH] (otherwise, the computational cost can be no lower than that of the extended Kalman filter).', '1710.11529-1-32-3': 'In such situations, storing the empirical covariance matrix [MATH] and inverting the matrix [MATH] cost [MATH].', '1710.11529-1-32-4': 'There are two frequently used methods to improve upon these costs.', '1710.11529-1-32-5': 'The first is based on the Sherman-Morrison-Woodbury formula (see e.g. [CITATION]) which for appropriate matrices states [EQUATION]', '1710.11529-1-32-6': 'Denoting by [MATH] the [MATH] matrix with [MATH]th column [MATH], then [MATH], thus applying ([REF]) for [MATH] and [MATH] gives [EQUATION] which only requires the inversion of [MATH] and of the [MATH] matrix [MATH].', '1710.11529-1-32-7': 'Implementation of EnKF based on this formula needs [MATH] memory.', '1710.11529-1-32-8': 'The second method is based on localizing the covariance [MATH], i.e. setting covariances for pairs of gridpoints that are far apart equal to zero.', '1710.11529-1-32-9': 'This allows matrices [MATH] and [MATH] to be stored in a sparse format, thus also greatly speeding up the calculation of the updates [MATH].', '1710.11529-1-32-10': 'If one does not store the inverse but uses [MATH] to solve the linear equations for the updates, the memory requirement for fixed localization distance can be [MATH].', '1710.11529-1-33-0': '# Simulations', '1710.11529-1-34-0': 'Firstly, we compare the performance of various methods using synthetic data.', '1710.11529-1-34-1': 'The shallow water equations were solved on the torus [MATH] with [MATH].', '1710.11529-1-34-2': 'The initial condition [MATH], ocean depth [MATH] and other ODE parameters were chosen as follows,', '1710.11529-1-35-0': 'u(0)=0.5+0.5 ( 2(x+y)L), v(0)=0.5-0.5 ( 2(x-y)L), h(0)=2(2xL)(2yL);', '1710.11529-1-36-0': 'H=100+100(1+0.5(2xL))(1+0.5(2yL));', '1710.11529-1-37-0': '-3,c_b=10^-5,g=9.81, f=10^-4.', '1710.11529-1-38-0': 'The discretised versions of the initial condition and the ocean depth were obtained under the choices [MATH], [MATH].', '1710.11529-1-38-1': 'In the second observation scenario (see Section [REF]) we have chosen the spatial frequency of the velocity observations as [MATH] (giving [MATH] velocity observations).', '1710.11529-1-38-2': 'Observations are made every 10 s.', '1710.11529-1-38-3': 'The background (prior) precision matrix [MATH] was chosen as the inverse of the SOAR covariance matrix (see [REF] in Section [REF]).', '1710.11529-1-38-4': 'The assimilation window [MATH] was chosen as 3 hours, which offered the best performance for fixed background covariance.', '1710.11529-1-38-5': 'Fig. [REF] illustrates the performance of the various methods in the two observation scenarios of Section [REF].', '1710.11529-1-38-6': 'In this case, 4D-Var-based methods offer the best performance, and using observations in earlier assimilation windows to update the background covariance matrix in a flow-dependent way is very beneficial, with relative errors reduced by as much as 72% compared to using a fixed background matrix.', '1710.11529-1-39-0': 'The shallow water equations are applied in tsunami modelling.', '1710.11529-1-39-1': '[CITATION] estimate the initial distribution of the tsunami waves after the 2011 Japan earthquake.', '1710.11529-1-39-2': 'They use data from 17 locations in the ocean, where the wave heights were observed continuously in time.', '1710.11529-1-39-3': 'We have used these estimates as our initial condition for the heights, and set the initial velocities to zero (as they are unknown).', '1710.11529-1-39-4': 'Using publicly available bathymetry data for [MATH], and the above described initial condition, we have run a simulation of 40 minutes for our model, see Fig. [REF].', '1710.11529-1-39-5': 'We have tested the efficiency of the data assimilation methods also on this simulated dataset, considering a time interval from 10 to 40 minutes (thus the initial condition corresponds to the value of the model after 10 minutes and is shown in Fig. [REF]).', '1710.11529-1-39-6': 'Due to the somewhat rough nature of the tsunami waves, in this example we have found that setting the background (prior) precision matrix [MATH] as zero offered the best performance for the 4-DVar based methods, while diagonal covariance matrices offered the best performance for the 3DVAR and ENKF methods.', '1710.11529-1-39-7': 'Fig. [REF] compares the performance of the methods for this synthetic dataset implemented for grid size [MATH], in the two observation scenarios of this paper.', '1710.11529-1-39-8': 'In this case the dimension on the dynamical system is [MATH].', '1710.11529-1-39-9': 'As in the previous synthetic example, the 4D-Var based methods offer best performance.', '1710.11529-1-39-10': 'We have also tried the localised EnKF but it was not stable and quickly diverged in this case.', '1710.11529-1-40-0': 'We stress that this comparison is based on synthetic data and cannot be considered as testing the accuracy and predictive performance of the data assimilation methods on "real data" (due to the fact that available real observations are very sparse and they are quite uninformative about some model parameters so comparison based on them would be very sensitive to the tuning parameters).', '1710.11529-1-40-1': 'Such comparisons are beyond the scope of this paper.', '1710.11529-1-41-0': '# Conclusion', '1710.11529-1-42-0': 'In this work we have presented a new method for updating the background covariances in 4D-Var filtering, and applied it to the shallow-water equations.', '1710.11529-1-42-1': "Our method finds the MAP estimator of the initial position using the Gauss-Newton's method with the Hessian matrix stored and the background covariances obtained in a factorised form.", '1710.11529-1-42-2': 'We also use a mathematically justified initialisation to overcome multimodality problems.', '1710.11529-1-42-3': 'Our method is computationally efficient and has memory and computational costs that scale nearly linearly with the size of the grid.', '1710.11529-1-42-4': 'In comparisons on synthetic datasets, we have found that out method outperforms competing one (3D-Var and EnKF) by a significant margin.', '1710.11529-1-42-5': 'Our approach also allows for uncertainty quantification via Gaussian approximations.'}

{'1710.11529-2-0-0': 'The 4D-Var method for filtering partially observed nonlinear chaotic dynamical systems consists of finding the maximum a-posteriori (MAP) estimator of the initial condition of the system given observations over a time window, and propagating it forward to the current time via the model dynamics.', '1710.11529-2-0-1': 'This method forms the basis of most currently operational weather forecasting systems.', '1710.11529-2-0-2': 'In practice the optimization becomes infeasible if the time window is too long due to the multimodality of the likelihood and the effect of model errors.', '1710.11529-2-0-3': 'Hence the window has to be kept sufficiently short, and the observations in the previous windows can be taken into account via a Gaussian background (prior) distribution.', '1710.11529-2-0-4': 'The choice of the background covariance matrix is an important question that has received much attention in the literature.', '1710.11529-2-0-5': 'In this paper, we define the background covariances in a principled manner, based on observations in the previous [MATH] assimilation windows, for a parameter [MATH].', '1710.11529-2-0-6': 'The method is at most [MATH] times more computationally expensive than using fixed background covariances, requires little tuning, and greatly improves the accuracy of 4D-Var.', '1710.11529-2-0-7': 'As a concrete example, we focus on the shallow-water equations.', '1710.11529-2-0-8': 'The proposed method is compared against state-of-the-art approaches in data assimilation and is shown to perform very favourably on simulated data.', '1710.11529-2-0-9': 'We also illustrate our approach on data from the recent tsunami of 2011 in Fukushima, Japan.', '1710.11529-2-1-0': 'Key words: Filtering; Smoothing; Data assimilation; Gauss-Newton Method; Shallow-Water Equations.', '1710.11529-2-2-0': '# Introduction', '1710.11529-2-3-0': 'Filtering, or data assimilation, is a field of core importance in a wide variety of real applications, such as numerical weather forecasting, climate modelling and finance; see e.g. [CITATION] for an introduction.', '1710.11529-2-3-1': 'Informally, one is interested in carrying out inference about an unobserved signal process conditionally upon noisy observations.', '1710.11529-2-3-2': 'The type of unobserved process considered in this paper is that of a nonlinear chaotic dynamical system, with unknown initial condition.', '1710.11529-2-3-3': 'We focus on the case where the unobserved dynamics correspond to the discretised version of the shallow-water equations; see e.g. [CITATION].', '1710.11529-2-3-4': 'These latter equations are of great practical importance, generating realistic approximations of real world phenomena, useful in tsunami and flood modelling (see e.g. [CITATION]).', '1710.11529-2-4-0': 'For systems of the type considered in this article, the filtering problem is notoriously challenging.', '1710.11529-2-4-1': 'Firstly, the filter is seldom available in analytic form due to the non-linearity.', '1710.11529-2-4-2': 'Secondly, even if the given system were solvable, the associated dimension of the object to be filtered is very high (of order of [MATH] or greater) thus posing great computational challenges, e.g. with regards to the storage of covariance matrices.', '1710.11529-2-4-3': 'Thirdly, the algorithmic cost cannot grow prohibitively with the (real) time of the system.', '1710.11529-2-5-0': 'Despite the challenges, due to the practical interest in weather prediction, several techniques, able to process massive datasets, have been devised and implemented operationally in weather forecasting centers worldwide.', '1710.11529-2-5-1': 'One of the most successful methods is the so-called 4D-Var algorithm [CITATION]: it consists of optimizing a loss-functional so that under Gaussian noise it is equivalent to finding the maximum a-posteriori (MAP) estimator of the initial condition.', '1710.11529-2-5-2': 'Since its introduction, a lot of further developments in the 4D-Var methodology have appeared in the literature; for an overview of some recent advances, we refer the reader to [CITATION].', '1710.11529-2-5-3': 'The main focus of this article is to consider principled improvements of the 4D-Var algorithm.', '1710.11529-2-6-0': 'While it is well understood that the MAP estimator is optimal for linear systems, in [CITATION] it was also shown to be asymptotically optimal for a class of non-linear systems - in an appropriate sense - under a small-noise/high-frequency regime for a fixed observation window.', '1710.11529-2-6-1': 'Based on this theoretical result, we expect the 4D-Var method to perform well in the small-noise/high-frequency setting, when the dynamical system and the observations are not too non-linear.', '1710.11529-2-6-2': 'In this setting, based on the results of [CITATION], one expects the posterior distribution to be approximately Gaussian.', '1710.11529-2-6-3': 'If the conditions are not met, as in the case of highly non-linear systems and observations, high observation noise, low observation frequency, or long assimilation window, it is likely that the Gaussian approximation of the smoothing distribution is no longer accurate and the MAP is no longer necessarily close to optimal in mean square error.', '1710.11529-2-6-4': 'Nevertheless, as long as the smoothing distribution has most of its mass contained in a single mode, we expect the typical error of the MAP estimator to still be of the order of the observation noise, giving 4D-Var methodology a certain robustness that is not necessarily present in filters that are directly based on Gaussian approximations in the update (such as the extended or the ensemble Kalman filter).', '1710.11529-2-7-0': 'An important practical issue of the 4D-Var method is that, due to chaotic nature of the systems encountered in weather prediction, the likelihood can become highly multimodal if the assimilation window is too long.', '1710.11529-2-7-1': 'The observations in the previous window are taken into account via the background (prior) distribution, which is a Gaussian whose mean is the estimate of the current position based on the previous windows, and has a certain covariance matrix.', '1710.11529-2-7-2': 'The choice of this background covariance matrix is an important and difficult problem that has attracted much research.', '1710.11529-2-7-3': '[CITATION] states that in operational weather forecasting systems up to 85% of the information in the smoothing distribution comes from the background (prior) distribution.', '1710.11529-2-7-4': 'The main contribution of this paper is the improvement of the 4D-Var methodology by a principled definition of this matrix in a flow-dependent way.', '1710.11529-2-7-5': 'This is based on the observations in the previous [MATH] assimilation windows (for a parameter [MATH]).', '1710.11529-2-7-6': 'Our method is computationally efficient, leads to improved precision, and requires less tuning than using a fixed background covariance matrix.', '1710.11529-2-8-0': 'There exist mathematically rigorous techniques to obtain the filter with precision and use the mean of the posterior distribution as the estimate, based upon sequential Monte Carlo methods (e.g. [CITATION]) which can provably work in high-dimensional systems [CITATION].', '1710.11529-2-8-1': 'While these are optimal in mean square error, and are probably considerably more accurate than the methods we work with in this paper, nonetheless such methodology can be practically overly expensive.', '1710.11529-2-8-2': 'As a result, one may have to resort to less accurate but more computationally efficient methodologies.', '1710.11529-2-8-3': 'These include 3D-Var, 4D-Var and the ensemble Kalman filter; see [CITATION] for a review.', '1710.11529-2-8-4': 'There are some relatively recent applications of particle filtering methods to high dimensional data assimilation problems, see e.g. [CITATION].', '1710.11529-2-8-5': 'While these algorithms seem to be promising for certain highly non-linear problems, their theoretical understanding is limited at the moment due to the bias they introduce via various approximations.', '1710.11529-2-9-0': 'The structure of the paper is as follows.', '1710.11529-2-9-1': 'In Section [REF] the modelling framework for the shallow-water equations is described in detail.', '1710.11529-2-9-2': 'In Section [REF] we introduce our 4D-Var method with flow-dependent background covariance.', '1710.11529-2-9-3': 'In particular, Section [REF] compares our method with other choices of flow-dependent background covariances in the literature.', '1710.11529-2-9-4': 'In Section [REF] we present some simulation results and compare the performance of our method with the 3D-Var and ensemble Kalman Filter methods.', '1710.11529-2-9-5': 'Finally, in Section [REF] we state some conclusions for this paper.', '1710.11529-2-10-0': '# Notations, Model and Multimodality', '1710.11529-2-11-0': '## Notations', '1710.11529-2-12-0': 'In this paper, we will be generally using the unified notations for data assimilation introduced in [CITATION].', '1710.11529-2-12-1': 'In this section we briefly review the required notations for the 4D-Var data assimilation method.', '1710.11529-2-13-0': 'The state vector at time [MATH] will be denoted by [MATH], and it is assumed that it has dimension [MATH].', '1710.11529-2-13-1': 'The evolution of the system from time [MATH] to time [MATH] will be governed by the equation [EQUATION] where [MATH] is the model evolution operator from time [MATH] to time [MATH].', '1710.11529-2-13-2': 'In practice this finite dimensional model is usually obtained by discretisation of the full partial differential equations governing the flow of the system.', '1710.11529-2-14-0': 'Observations are made at times [MATH], and they are of the form [EQUATION] where [MATH] is the observation operator, and [MATH] is the random noise.', '1710.11529-2-14-1': 'We will denote the dimension [MATH] by [MATH], and assume that [MATH] are independent normally distributed random vectors with mean 0 and covariance matrix [MATH].', '1710.11529-2-14-2': 'The Jacobian matrix (i.e. linearization) of the operator [MATH] at position [MATH] will be denoted by [MATH], and the Jacobian of [MATH] at [MATH] will be denoted by [MATH].', '1710.11529-2-14-3': 'The inverse, and transpose of a matrix will be denoted by [MATH] and [MATH], respectively.', '1710.11529-2-15-0': 'The 4D-Var method for assimilating the observations in the time interval [MATH] consists of minimising the cost functional [EQUATION] where [MATH], and [MATH] denotes the background covariance matrix, and [MATH] denotes the background mean.', '1710.11529-2-15-1': 'minimising this functional is equivalent to maximizing the likelihood of the smoothing distribution for [MATH] given [MATH] and normally distributed prior with mean [MATH] and covariance [MATH].', '1710.11529-2-15-2': 'Note that the cost function [REF] corresponds to the so-called strong constraint 4D-Var (i.e. no noise is allowed in the dynamics), there are also weak constraint alternatives that account for possible model errors by allowing noise in the dynamics (see e.g. [CITATION]).', '1710.11529-2-16-0': '## The Model', '1710.11529-2-17-0': 'We consider the shallow-water equations, e.g. as in [CITATION], but with added diffusion and bottom friction terms, i.e. [EQUATION]', '1710.11529-2-17-1': 'Here, [MATH] and [MATH] are the velocity fields in the [MATH] and [MATH] directions respectively, and [MATH] the field for the height of the wave.', '1710.11529-2-17-2': 'Also, [MATH] is the depth of the ocean, [MATH] the gravity constant, [MATH] the Coriolis parameter, [MATH] the bottom friction coefficient and [MATH] the viscosity coefficient.', '1710.11529-2-17-3': 'Parameters [MATH], [MATH], [MATH] and [MATH] are assumed to be constant in time but in general depend on the location.', '1710.11529-2-17-4': 'The total height of the water column is the sum [MATH].', '1710.11529-2-18-0': 'For square grids, under periodic boundary conditions, the equations are discretised as [EQUATION] where [MATH], for a typically large [MATH], with the indices understood modulo [MATH], and some space-step [MATH].', '1710.11529-2-18-1': 'Summing up [REF] over [MATH], one can see that the discretisation preserves the total mass [MATH].', '1710.11529-2-18-2': 'If we assume that the viscosity and bottom friction are negligible, i.e. [MATH], then the total energy [MATH] is also preserved.', '1710.11529-2-18-3': 'When the coefficients [MATH] and [MATH] are not zero, the bottom friction term always decreases the total energy (the sum of the kinetic and potential energy), while the diffusion term tends to smooth the velocity profile.', '1710.11529-2-18-4': 'We denote the solution of equations [REF]-[REF] at time [MATH] as [EQUATION]', '1710.11529-2-18-5': 'The unknown and random initial condition is denoted by [MATH].', '1710.11529-2-18-6': 'One can show by standard methods (see [CITATION]) that the solution of [REF]-[REF] exists up to some time [MATH].', '1710.11529-2-18-7': 'In order to represent the components of [MATH], we introduce a vector index notation.', '1710.11529-2-18-8': 'The set [MATH] denotes the possible indices, with the first component referring to one of [MATH], [MATH], [MATH], the second component to coordinate [MATH], and the third to [MATH].', '1710.11529-2-18-9': 'A vector index in [MATH] will usually be denoted as [EQUATION]', '1710.11529-2-18-10': 'B_(m-1)k^-1+D_(m-1)k:mk-1, [EQUATION] from the observed ones, and choose the coefficients [MATH] as the least square estimator for these samples (i.e. such that the difference between the sum of the squares for all the samples of the linear estimator of the [MATH]th derivative of the chosen component [EQUATION] and [MATH] corresponding different components [MATH], [MATH] or [MATH] (see Section [REF] for the notation [MATH]), while equals [EQUATION] for [MATH] corresponding to component [MATH] (see equation (A10) of [CITATION]), and similarly for the other components with constants [MATH] and [MATH].', '1710.11529-2-18-11': 'For one dimensional grids, this covariance matrix is known to have spectral response (see equation (A11) of [CITATION]) [EQUATION] corresponding to damping the high frequency terms.', '1710.11529-2-18-12': 'Due to the high dimensionality of the system, it is not practical to operate with this covariance matrix and its inverse directly.', '1710.11529-2-18-13': 'Instead there are approximate ways to apply them on a vector that leads to similar spectral response, see [CITATION].', '1710.11529-2-18-14': 'An alternative way of obtaining a spectral response similar to [REF] for our model is via the two dimensional discrete cosine transform (DCT), see [CITATION] for an overview.', '1710.11529-2-18-15': 'Based on this, the effect of the covariance matrix on a vector is computed by first transforming the vector by DCT (separately in components [MATH], [MATH] and [MATH]), then applying the spectral response [REF] (which is equivalent to multiplying with a diagonal matrix), and finally transforming it back (see Section 7.1.2 of [CITATION] for the description of such an approach).', '1710.11529-2-18-16': 'The effect of the precision matrix is analogous except that the inverse of [REF] is applied in the second step.', '1710.11529-2-19-0': '## Literature on Flow-Dependent Covariances for 4D-Var', '1710.11529-2-20-0': 'Flow-dependent background covariances have been used in practice since [CITATION] proposed the so-called NMC method based on a comparison of 24 and 48 hour forecast differences.', '1710.11529-2-20-1': 'This method was refined in [CITATION].', '1710.11529-2-20-2': '[CITATION] proposed the use of wavelets for forming background covariances; these retain the computational advantages of spectral methods, while also allow for spatial inhomogeneity.', '1710.11529-2-20-3': 'The background covariances are made flow-dependent via a suitable modification of the NMC approach.', '1710.11529-2-20-4': 'The paper [CITATION] reviews some of the practical aspects of modelling 4D-Var error covariances, while [CITATION] makes a comparison between 4D-Var for long assimilation windows and the Extended Kalman Filter.', '1710.11529-2-20-5': 'As we have noted previously, long windows are not always applicable due to the presence of model errors and multimodalities in the likelihood.', '1710.11529-2-21-0': 'More recently, there have been several methods proposing the use of ensembles for modelling the covariances, see e.g. [CITATION].', '1710.11529-2-21-1': 'However, it is not clear how well a small ensemble can approximate the covariances, which are localised only over short assimilation intervals, but become full matrices over longer intervals.', '1710.11529-2-21-2': 'Our method uses a factorised form of the Hessian and the background precision matrix described in Sections [REF]-[REF], thus we only need to store the Jacobians between observation times in a sparse form.', '1710.11529-2-21-3': 'This allows us to compute the effect of these matrices on a vector efficiently, without needing to store all the elements of the background precision matrix, which would require too much memory.', '1710.11529-2-22-0': 'Although in this paper we have assumed that the model is perfect, there have been efforts to account for model error in the literature, see [CITATION].', '1710.11529-2-22-1': 'The effect of nonlinearities in the dynamics and the observations can be in some cases so strong that the Gaussian approximations are no longer reasonable, see [CITATION] for some examples and suggestions for overcoming these problems.', '1710.11529-2-23-0': '# Simulations', '1710.11529-2-24-0': 'In this section, we are going to illustrate the performance of our proposed method through simulation results.', '1710.11529-2-24-1': 'As a comparison, we also apply the 3D-Var and ENKF methods on the same datasets.', '1710.11529-2-24-2': 'Section 5 of [CITATION] and Sections 7-8 of [CITATION] offer excellent introductions to standard data assimilation methods such as 3D-Var, 4D-Var and EnKF and its variants.', '1710.11529-2-24-3': 'Note that the literature in data assimilation is large, and many variants of 3D-Var, 4D-Var, ENKF and hybrid methods have been proposed (see [CITATION], [CITATION] and [CITATION]).', '1710.11529-2-25-0': 'Firstly, we compare the performance of various methods using synthetic data.', '1710.11529-2-25-1': 'The shallow water equations were solved on the torus [MATH] with [MATH].', '1710.11529-2-25-2': 'The initial condition [MATH], ocean depth [MATH] and other ODE parameters were chosen as follows,', '1710.11529-2-26-0': 'u(0)=0.5+0.5 ( 2(x+y)L), v(0)=0.5-0.5 ( 2(x-y)L), h(0)=2(2xL)(2yL);', '1710.11529-2-27-0': 'H=100+100(1+0.5(2xL))(1+0.5(2yL));', '1710.11529-2-28-0': '-3,c_b=10^-5,g=9.81, f=10^-4.', '1710.11529-2-29-0': 'The discretised versions of the initial condition and the ocean depth were obtained under the choices [MATH], [MATH].', '1710.11529-2-29-1': 'In the second observation scenario (see Section [REF]) we have chosen the spatial frequency of the velocity observations as [MATH] (giving [MATH] velocity observations).', '1710.11529-2-29-2': 'Observations are made every 10 s, and the observation errors are i.i.d. [MATH] random variables for [MATH].', '1710.11529-2-29-3': 'The background (prior) covariance matrix [MATH] was chosen as the SOAR covariance matrix (see [REF] in Section [REF]).', '1710.11529-2-29-4': 'The assimilation window [MATH] was chosen as 3 hours, which offered the best performance for fixed background covariance.', '1710.11529-2-29-5': 'Fig. [REF] illustrates the performance of the various methods in the two observation scenarios of Section [REF].', '1710.11529-2-29-6': 'The 3D-Var method was implemented as described in Section 5.5 of [CITATION], with background covariances chosen as the SOAR covariance matrix [MATH].', '1710.11529-2-29-7': 'Note that due to the linearity of the observation operator, the minimisation problem in this case only requires solving a linear equation, which can be done efficiently by the preconditioned conjugate gradient (PCG) method.', '1710.11529-2-29-8': 'For the ENKF, we have tried the non-localised and localised versions, as described in Sections 7.1 and 8.3 of [CITATION].', '1710.11529-2-29-9': 'The filter function in the localisation was done according to equation (8.29) of [CITATION], and the localisation radius was tuned for optimal performance.', '1710.11529-2-29-10': 'For both versions of the ENKF, we have also used multiplicative ensemble inflation, as described in Section 8.2 of [CITATION], which was also tuned for optimal performance.', '1710.11529-2-29-11': 'The 4D-Var method was optimised based on the Gauss-Newton method with preconditioned conjugate gradient (PCG) based linear solver.', '1710.11529-2-29-12': 'We did not use any preconditioner, and the maximum number of iterations per PCG step was set to 100 (which was sufficient for reducing the relative residual below [MATH] in most cases).', '1710.11529-2-29-13': 'All of the methods were implemented in Matlab and ran on an ordinary desktop computer with 4 core 2.5Ghz Intel i5 CPU.', '1710.11529-2-29-14': 'The measure of performance is the relative error of the unobserved component at a certain time [MATH], i.e. if [MATH] denotes the true value of the unobserved component, and [MATH] is the estimator, then [MATH] is the relative error ([MATH] refers to the Euclidean norm in [MATH]).', '1710.11529-2-30-0': 'As we can see on Figure [REF], for this synthetic dataset, 4D-Var-based methods offer the best performance, and using observations in earlier assimilation windows to update the background covariance matrix in a flow-dependent way is very beneficial, with relative errors reduced by as much as 72% compared to using a fixed background matrix.', '1710.11529-2-31-0': 'The shallow water equations are applied in tsunami modelling.', '1710.11529-2-31-1': '[CITATION] estimate the initial distribution of the tsunami waves after the 2011 Japan earthquake.', '1710.11529-2-31-2': 'They use data from 17 locations in the ocean, where the wave heights were observed continuously in time.', '1710.11529-2-31-3': 'We have used these estimates as our initial condition for the heights, and set the initial velocities to zero (as they are unknown).', '1710.11529-2-31-4': 'Using publicly available bathymetry data for [MATH], and the above described initial condition, we have run a simulation of 40 minutes for our model, see Fig. [REF].', '1710.11529-2-31-5': 'We have tested the efficiency of the data assimilation methods also on this simulated dataset, considering a time interval from 10 to 40 minutes (thus the initial condition corresponds to the value of the model after 10 minutes and is shown in Fig. [REF]).', '1710.11529-2-31-6': 'Due to the somewhat rough nature of the tsunami waves, in this example we have found that setting the background precision (inverse covariance) matrix [MATH] as zero offered the best performance for the 4D-Var based methods, while diagonal covariance matrices offered the best performance for the 3DVAR and ENKF methods (for ENKF, this matrix is only used for sampling the initial ensemble).', '1710.11529-2-31-7': 'The ENKF and localised ENKF were implemented with optimally tuned ensemble inflation and localisation as described in Sections 8.2 and 8.3 of [CITATION].', '1710.11529-2-31-8': 'The 4D-Var method was optimised based on the Gauss-Newton method with preconditioned conjugate gradient (PCG) based linear solver without any preconditioner, and the maximum number of iterations per PCG step was set to 200.', '1710.11529-2-32-0': 'Fig. [REF] compares the performance of the methods for this synthetic dataset implemented for grid size [MATH] ( so the dimension on the dynamical system is [MATH]), in the two observation scenarios of this paper.', '1710.11529-2-32-1': 'For the second scenario, the spatial frequency of the velocity observations was chosen as [MATH] (i.e. [MATH] velocity observations in total).', '1710.11529-2-33-0': 'As in the previous synthetic example, the 4D-Var based methods offer best performance, and they require less tuning than the ENKF based methods (which are sensitive to the variance inflation and localisation parameters in this example).', '1710.11529-2-33-1': 'We believe that the relatively poor performance of the ENKF and localized ENKF methods on Figure [REF] are due to the instability of the ODE, and the sparsity of the velocity observations, which require very accurate covariance modelling.', '1710.11529-2-34-0': 'We believe that the difference in performance is due to the better modelling of forecast error covariances by the 4D-Var method, which are not necessarily well-localised for this particular model when assimilating observations over longer periods.', '1710.11529-2-34-1': 'This is especially evident in the first synthetic example on Figure [REF], where we have ran the model for 24 hours.', '1710.11529-2-35-0': 'We stress that this comparison is based on synthetic data and cannot be considered as testing the accuracy and predictive performance of the data assimilation methods on "real data" (due to the fact that available real observations are very sparse and they are quite uninformative about some model parameters so comparison based on them would be very sensitive to the tuning parameters).', '1710.11529-2-35-1': 'Moreover, in real data problems, model errors can also cause significant complications.', '1710.11529-2-35-2': 'Such comparisons are beyond the scope of this paper.', '1710.11529-2-36-0': '# Conclusion', '1710.11529-2-37-0': 'In this work we have presented a new method for updating the background covariances in 4D-Var filtering, and applied it to the shallow-water equations.', '1710.11529-2-37-1': "Our method finds the MAP estimator of the initial position using the Gauss-Newton's method with the Hessian matrix stored and the background covariances obtained in a factorised form.", '1710.11529-2-37-2': 'We also use a mathematically justified initialisation to overcome multimodality problems.', '1710.11529-2-37-3': 'Our method is computationally efficient and has memory and computational costs that scale nearly linearly with the size of the grid.', '1710.11529-2-37-4': 'In comparisons on synthetic datasets, we have found that out method outperforms competing one (3D-Var and EnKF) by a significant margin.', '1710.11529-2-37-5': 'Our approach also allows for uncertainty quantification via Gaussian approximations.'}

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'1710.11529-2-21-0'], ['1710.11529-1-16-2', '1710.11529-2-21-2'], ['1710.11529-1-16-3', '1710.11529-2-21-3'], ['1710.11529-1-5-2', '1710.11529-2-5-2'], ['1710.11529-1-5-3', '1710.11529-2-5-3'], ['1710.11529-1-7-0', '1710.11529-2-7-0'], ['1710.11529-1-7-1', '1710.11529-2-7-1'], ['1710.11529-1-7-2', '1710.11529-2-7-2'], ['1710.11529-1-7-3', '1710.11529-2-7-3'], ['1710.11529-1-7-4', '1710.11529-2-7-4'], ['1710.11529-1-7-5', '1710.11529-2-7-5'], ['1710.11529-1-7-6', '1710.11529-2-7-6'], ['1710.11529-1-9-0', '1710.11529-2-9-0'], ['1710.11529-1-9-1', '1710.11529-2-9-1'], ['1710.11529-1-9-2', '1710.11529-2-9-2'], ['1710.11529-1-9-3', '1710.11529-2-9-3'], ['1710.11529-1-9-5', '1710.11529-2-9-4'], ['1710.11529-1-9-6', '1710.11529-2-9-5'], ['1710.11529-1-40-0', '1710.11529-2-35-0'], ['1710.11529-1-40-1', '1710.11529-2-35-2'], ['1710.11529-1-12-0', '1710.11529-2-17-0'], ['1710.11529-1-12-1', '1710.11529-2-17-1'], ['1710.11529-1-12-2', '1710.11529-2-17-2'], ['1710.11529-1-12-3', 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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1710.11529

{'1710.11529-3-0-0': 'The 4D-Var method for filtering partially observed nonlinear chaotic dynamical systems consists of finding the maximum a-posteriori (MAP) estimator of the initial condition of the system given observations over a time window, and propagating it forward to the current time via the model dynamics.', '1710.11529-3-0-1': 'This method forms the basis of most currently operational weather forecasting systems.', '1710.11529-3-0-2': 'In practice the optimization becomes infeasible if the time window is too long due to the non-convexity of the cost function, the effect of model errors, and the limited precision of the ODE solvers.', '1710.11529-3-0-3': 'Hence the window has to be kept sufficiently short, and the observations in the previous windows can be taken into account via a Gaussian background (prior) distribution.', '1710.11529-3-0-4': 'The choice of the background covariance matrix is an important question that has received much attention in the literature.', '1710.11529-3-0-5': 'In this paper, we define the background covariances in a principled manner, based on observations in the previous [MATH] assimilation windows, for a parameter [MATH].', '1710.11529-3-0-6': 'The method is at most [MATH] times more computationally expensive than using fixed background covariances, requires little tuning, and greatly improves the accuracy of 4D-Var.', '1710.11529-3-0-7': 'As a concrete example, we focus on the shallow-water equations.', '1710.11529-3-0-8': 'The proposed method is compared against state-of-the-art approaches in data assimilation and is shown to perform favourably on simulated data.', '1710.11529-3-0-9': 'We also illustrate our approach on data from the recent tsunami of 2011 in Fukushima, Japan.', '1710.11529-3-1-0': 'Key words: Filtering; Smoothing; Data assimilation; Gauss-Newton Method; Shallow-Water Equations.', '1710.11529-3-2-0': '# Introduction', '1710.11529-3-3-0': 'Filtering, or data assimilation, is a field of core importance in a wide variety of real applications, such as numerical weather forecasting, climate modelling and finance; see e.g. [CITATION] for an introduction.', '1710.11529-3-3-1': 'Informally, one is interested in carrying out inference about an unobserved signal process conditionally upon noisy observations.', '1710.11529-3-3-2': 'The type of unobserved process considered in this paper is that of a nonlinear chaotic dynamical system, with unknown initial condition.', '1710.11529-3-3-3': 'As an application in this paper we consider the case where the unobserved dynamics correspond to the discretised version of the shallow-water equations; see e.g. [CITATION].', '1710.11529-3-3-4': 'These latter equations are of great practical importance, generating realistic approximations of real world phenomena, useful in tsunami and flood modelling (see e.g. [CITATION]).', '1710.11529-3-4-0': 'For systems of this type, the filtering problem is notoriously challenging.', '1710.11529-3-4-1': 'Firstly, the filter is seldom available in analytic form due to the non-linearity.', '1710.11529-3-4-2': 'Secondly, even if the given system were solvable, the associated dimension of the object to be filtered is very high (of order of [MATH] or greater) thus posing great computational challenges.', '1710.11529-3-5-0': 'One of the most successful methods capable of handling such high dimensional datasets is the so-called 4D-Var algorithm [CITATION]: it consists of optimizing a loss-functional so that under Gaussian noise it is equivalent to finding the maximum a-posteriori (MAP) estimator of the initial condition.', '1710.11529-3-5-1': 'Since its introduction, a lot of further developments in the 4D-Var methodology have appeared in the literature; for an overview of some recent advances, we refer the reader to [CITATION].', '1710.11529-3-5-2': 'The main focus of this article is to consider principled improvements of the 4D-Var algorithm.', '1710.11529-3-6-0': 'It is well understood that the MAP estimator is optimal in mean square error (MSE) for linear systems (since it equals to the posterior mean for such systems, and the posterior mean is always the optimal estimator in mean square error, see page 136 of [CITATION] for a proof).', '1710.11529-3-6-1': 'In [CITATION], the MAP was also shown to be asymptotically optimal in MSE for a class of non-linear systems under small obervation noise or high observation frequency scenarios for a fixed observation window length.', '1710.11529-3-6-2': 'The reason for this is that in these situations, the posterior distribution near the true initial position can be well approximated by a suitably chosen Gaussian distribution, and hence the MAP estimator is close to the posterior mean, which is optimal in MSE.', '1710.11529-3-7-0': 'Based on this theoretical result, we expect the 4D-Var method to perform well in the small-noise/high-frequency setting, when the dynamical system and the observations are not too non-linear.', '1710.11529-3-7-1': 'In this setting, based on the results of [CITATION], one expects the posterior distribution to be approximately Gaussian.', '1710.11529-3-7-2': 'If the conditions are not met, it is likely that the Gaussian approximation of the smoothing distribution is no longer accurate and the MAP is no longer necessarily close to optimal in MSE.', '1710.11529-3-8-0': 'An important practical issue of the 4D-Var method is that, due to chaotic nature of the systems encountered in weather prediction, the negative log-likelihood (cost function) can become highly non-convex if the assimilation window is too long.', '1710.11529-3-8-1': 'The reason for this is that for deterministic dynamical systems, as the assimilation window grows, the smoothing distribution gets more and more concentrated on the stable manifold of the initial position, which is a complicated lower dimensional set (see [CITATION], [CITATION] for more details).', '1710.11529-3-8-2': 'On one hand, this means that it becomes very difficult to find the MAP estimator.', '1710.11529-3-8-3': 'On the other hand, due to the highly non-Gaussian nature of the posterior in this setting, the MAP might be far away from the posterior mean, and have a large mean square error.', '1710.11529-3-8-4': 'Moreover, for longer windows, the precision of the tangent linear model/adjoint solvers might decrease.', '1710.11529-3-8-5': 'Due to these facts, the performance of 4D-Var deteriorates for many models when the observation window becomes too long (see [CITATION]).', '1710.11529-3-9-0': 'The observations in the previous window are taken into account via the background (prior) distribution, which is a Gaussian whose mean is the estimate of the current position based on the previous windows, and has a certain covariance matrix.', '1710.11529-3-9-1': 'The choice of this background covariance matrix is an important and difficult problem that has attracted much research.', '1710.11529-3-9-2': '[CITATION] states that in operational weather forecasting systems up to 85% of the information in the smoothing distribution comes from the background (prior) distribution.', '1710.11529-3-9-3': 'The main contribution of this paper is an improvement of the 4D-Var methodology by a principled definition of this matrix in a flow-dependent way.', '1710.11529-3-9-4': 'This is based on the observations in the previous [MATH] assimilation windows (for a parameter [MATH]).', '1710.11529-3-9-5': 'Via simulations on the shallow-water model, we show that our method compares favourably in precision and computational time with the state-of-the-art methods, Ensemble Kalman filter (ENKF) and En4D-Var (a 4D-Var method which uses background covariances generated by an ENKF).', '1710.11529-3-9-6': 'The main reason for the improved precision is that we do not rely on localisation, hence we can capture long range dependencies in the covariances that arise over long assimilation windows.', '1710.11529-3-10-0': 'The structure of the paper is as follows.', '1710.11529-3-10-1': 'In the rest of this section, we briefly review the literature on 4D-Var background covariances.', '1710.11529-3-10-2': 'In Section [REF] the modelling framework for the shallow-water equations is described in detail.', '1710.11529-3-10-3': 'In Section [REF] we introduce our 4D-Var method with flow-dependent background covariance.', '1710.11529-3-10-4': 'In particular, Section [REF] compares our method with other choices of flow-dependent background covariances in the literature.', '1710.11529-3-10-5': 'In Section [REF] we present some simulation results and compare the performance of our method with the ENKF and En4D-Var methods.', '1710.11529-3-10-6': 'Finally, in Section [REF] we state some conclusions for this paper.', '1710.11529-3-11-0': '## Comparison with the literature', '1710.11529-3-12-0': 'There exist mathematically rigorous techniques to obtain the filter with precision and use the mean of the posterior distribution as the estimate, based upon sequential Monte Carlo methods (e.g. [CITATION]) which can provably work in high-dimensional systems [CITATION].', '1710.11529-3-12-1': 'While these approximate the posterior means and hence are optimal in mean square error, and are possibly considerably more accurate than optimization based methods, nonetheless such methodology can be practically overly expensive.', '1710.11529-3-12-2': 'As a result, one may have to resort to less accurate but more computationally efficient methodologies (see [CITATION] for a review).', '1710.11529-3-12-3': 'There are some relatively recent applications of particle filtering methods to high dimensional data assimilation problems, see e.g. [CITATION].', '1710.11529-3-12-4': 'While these algorithms seem to be promising for certain highly non-linear problems, their theoretical understanding is limited at the moment due to the bias they introduce via various approximations.', '1710.11529-3-13-0': 'Despite the difficulty of solving the non-linear filtering problem exactly, due to the practical interest in weather prediction, several techniques have been devised and implemented operationally in weather forecasting centers worldwide.', '1710.11529-3-13-1': 'These techniques are based on optimization methods, hence they scale well to high dimensions and are able to process massive datasets.', '1710.11529-3-13-2': 'Although initially such methods were lacking in mathematical foundation, the books [CITATION] and [CITATION] are among the first to open up the field of data assimilation to mathematics.', '1710.11529-3-13-3': 'Among the earlier works devoted to the efforts of bringing together data assimilation and mathematics, we also mention [CITATION] and [CITATION], where a comparison between the Kalman filter (sequential-estimation) and variational methods is presented.', '1710.11529-3-14-0': 'The performance of 4D-Var methods depends very strongly on the choice of background covariances.', '1710.11529-3-14-1': 'One of the first principled ways of choosing 4D-Var background covariances was introduced by [CITATION].', '1710.11529-3-14-2': 'They have proposed the so-called NMC method for choosing climatological prior error covariances based on a comparison of 24 and 48 hour forecast differences.', '1710.11529-3-14-3': 'This method was refined in [CITATION].', '1710.11529-3-14-4': '[CITATION] proposed the use of wavelets for forming background covariances; these retain the computational advantages of spectral methods, while also allow for spatial inhomogeneity.', '1710.11529-3-14-5': 'The background covariances are made flow-dependent via a suitable modification of the NMC approach.', '1710.11529-3-14-6': '[CITATION] reviews some of the practical aspects of modelling 4D-Var error covariances, while [CITATION] makes a comparison between 4D-Var for long assimilation windows and the Extended Kalman Filter.', '1710.11529-3-14-7': 'As we have noted previously, long windows are not always applicable due to the presence of model errors and the non-convexity of the likelihood.', '1710.11529-3-15-0': 'More recently, there have been several methods proposing the use of ensembles combined with localization methods for modelling the covariances, see e.g. [CITATION].', '1710.11529-3-16-0': 'Localization eliminates spurious correlations between elements of the covariance matrix that are far away from each other and hence they have little correlation.', '1710.11529-3-16-1': 'This means that these long range correlations are set to zero, which allows the sparse storage of the covariance matrix and efficient computations of the matrix-vector products.', '1710.11529-3-16-2': 'Bishop et al 2011 proposes an efficient implementation of localization by introducing some further approximations, using the product structure of the grid, and doing part of the calculations on lower resolution grid points.', '1710.11529-3-16-3': 'Such efficient implementations have allowed localized ENKF based background covariance modelling to provide the state-of-the-art performance in data assimilation, and they form the core of most operational NWP systems at the moment.', '1710.11529-3-17-0': 'Over longer time periods, given sufficient data available, most of the variables become correlated, and imposing a localized structure over them leads to some loss of information vs the benefit of computational efficiency.', '1710.11529-3-17-1': 'Our method does not impose such a structure as it writes the precision matrix in a factorized form.', '1710.11529-3-17-2': 'Moreover, the localization structure is assumed to be fixed in time, so even with a considerable amount of tuning for a certain time period of data it is not guaranteed that the same localization structure will be optimal in the future.', '1710.11529-3-17-3': 'Our method does not make such a constant localization assumption and hence it is able to adapt to different correlation structures automatically.', '1710.11529-3-18-0': 'We use an implicit factorised form of the Hessian and the background precision matrix described in Sections [REF]-[REF], thus we only need to store the positions of the system started from the 4D-Var estimate of the previous [MATH] windows at the observation times.', '1710.11529-3-18-1': 'This allows us to compute the effect of these matrices on a vector efficiently, without needing to store all the elements of the background precision matrix, which would require too much memory.', '1710.11529-3-19-0': 'Although in this paper we have assumed that the model is perfect, there have been efforts to account for model error in the literature, see [CITATION].', '1710.11529-3-19-1': 'The effect of nonlinearities in the dynamics and the observations can be in some cases so strong that the Gaussian approximations are no longer reasonable, see [CITATION] for some examples and suggestions for overcoming these problems.', '1710.11529-3-20-0': '# Notations and Model', '1710.11529-3-21-0': '## Notations', '1710.11529-3-22-0': 'In this paper, we will be generally using the unified notations for data assimilation introduced in [CITATION].', '1710.11529-3-22-1': 'In this section we briefly review the required notations for the 4D-Var data assimilation method.', '1710.11529-3-23-0': 'The state vector at time [MATH] will be denoted by [MATH], and it is assumed that it has dimension [MATH].', '1710.11529-3-23-1': 'The evolution of the system from time [MATH] to time [MATH] will be governed by the equation [EQUATION] where [MATH] is the model evolution operator from time [MATH] to time [MATH].', '1710.11529-3-23-2': 'In practice, this finite dimensional model is usually obtained by discretisation of the full partial differential equations governing the flow of the system.', '1710.11529-3-24-0': 'Observations are made at times [MATH], and they are of the form [EQUATION] where [MATH] is the observation operator, and [MATH] is the random noise.', '1710.11529-3-24-1': 'We will denote the dimension [MATH] by [MATH], and assume that [MATH] are independent normally distributed random vectors with mean 0 and covariance matrix [MATH].', '1710.11529-3-24-2': 'The Jacobian matrix (i.e. linearization) of the operator [MATH] at position [MATH] will be denoted by [MATH], and the Jacobian of [MATH] at [MATH] will be denoted by [MATH].', '1710.11529-3-24-3': 'The inverse and transpose of a matrix will be denoted by [MATH] and [MATH], respectively.', '1710.11529-3-25-0': 'The 4D-Var method for assimilating the observations in the time interval [MATH] consists of minimizing the cost functional [EQUATION] where [MATH], and [MATH] denotes the background covariance matrix, and [MATH] denotes the background mean.', '1710.11529-3-25-1': 'Minimizing this functional is equivalent to maximizing the likelihood of the smoothing distribution for [MATH] given [MATH] and normally distributed prior with mean [MATH] and covariance [MATH].', '1710.11529-3-25-2': 'Note that the cost function [REF] corresponds to the so-called strong constraint 4D-Var (i.e. no noise is allowed in the dynamics), there are also weak constraint alternatives that account for possible model errors by allowing noise in the dynamics (see e.g. [CITATION]).', '1710.11529-3-26-0': '## The Model', '1710.11529-3-27-0': 'We consider the shallow-water equations, e.g. as in [CITATION], but with added diffusion and bottom friction terms, i.e. [EQUATION]', '1710.11529-3-27-1': 'Here, [MATH] and [MATH] are the velocity fields in the [MATH] and [MATH] directions respectively, and [MATH] the field for the height of the wave.', '1710.11529-3-27-2': 'Also, [MATH] is the depth of the ocean, [MATH] the gravity constant, [MATH] the Coriolis parameter, [MATH] the bottom friction coefficient and [MATH] the viscosity coefficient.', '1710.11529-3-27-3': 'Parameters [MATH], [MATH], [MATH] and [MATH] are assumed to be constant in time but in general depend on the location.', '1710.11529-3-27-4': 'The total height of the water column is the sum [MATH].', '1710.11529-3-28-0': 'For square grids, under periodic boundary conditions, the equations are discretised as [EQUATION] where [MATH], for a typically large [MATH], with the indices understood modulo [MATH] (hence the domain is a torus), and some space-step [MATH].', '1710.11529-3-28-1': 'Summing up [REF] over [MATH], one can see that the discretisation preserves the total mass [MATH].', '1710.11529-3-28-2': 'If we assume that the viscosity and bottom friction are negligible, i.e. [MATH], then the total energy [MATH] is also preserved.', '1710.11529-3-28-3': 'When the coefficients [MATH] and [MATH] are not zero, the bottom friction term always decreases the total energy (the sum of the kinetic and potential energy), while the diffusion term tends to smooth the velocity profile.', '1710.11529-3-28-4': 'We denote the solution of equations [REF]-[REF] at time [MATH] as [EQUATION]', '1710.11529-3-28-5': 'The unknown and random initial condition is denoted by [MATH].', '1710.11529-3-28-6': 'One can show by standard methods (see [CITATION]) that the solution of [REF]-[REF] exists up to some time [MATH].', '1710.11529-3-28-7': 'In order to represent the components of [MATH], we introduce a vector index notation.', '1710.11529-3-28-8': 'The set [MATH] denotes the possible indices, with the first component referring to one of [MATH], [MATH], [MATH], the second component to coordinate [MATH], and the third to [MATH].', '1710.11529-3-28-9': 'A vector index in [MATH] will usually be denoted as [EQUATION]', '1710.11529-3-28-10': "B_(m-1)k^-1+D_(m-1)k:mk-1, [MATH]D_(m-1)k:mk-1[MATH]J[MATH]x(t_(m-1)k)[MATH]D_(m-1)k:mk-1[MATH]Zsim N(m, P^-1)[MATH]R^n[MATH]:R^nR^n[MATH](Z)[MATH]Z[MATH][MATH]b[MATH]j=1,, b[MATH]B_mk:=B_mk^m[MATH](1+)[MATH]>0[MATH]B_(m-b)k^m[MATH]B_mk^-1[MATH]win R^n[MATH]B_mk^-1w=B_-1[MATH]M_1,, M_k[MATH]M_1^-1,, M_k^-1[MATH]b[MATH]D_(m-l)k:(m-l+1)k-1[MATH][REF][MATH]M_j [MATH]M_j' [MATH]M_j^-1[MATH](M_j^-1)'[MATH]M_j[MATH]b[MATH]O(b)[MATH]T[MATH](b+1)T[MATH](b+1)T[MATH]T[MATH]b[MATH]b=[MATH]b[MATH]O(d^2)[MATH]O(d^3)[MATH]O(bd)[MATH]O(bd)[MATH]Jx(t_0)[MATH]B_0^-1[x(t_0)-x^b(t_0)][MATH]Jx(t_0)[MATH]B_0^-1[x(t_0)-x^b(t_0)][MATH]h[MATH]H_i[MATH]Hin R^n^n[MATH]u[MATH]v[MATH]1i,jd[MATH]N(0,^2)[MATH]R_l=R=^2 I_n^[MATH]l0[MATH]h[MATH]1i,jd[MATH]u[MATH]v[MATH]r[MATH]r[MATH]N(0,^2)[MATH]h[MATH]r[MATH]r[MATH]N(0,^2)[MATH][0,L]^2[MATH]L=210km[MATH]U(0):=(u(0),v(0),h(0))[MATH]H[MATH]d=21[MATH]km[MATH]r=3[MATH]N(0,^2)[MATH]-2[MATH]B_0[MATH]T[MATH]0.01[MATH]t[MATH]w(t)R^n-n^[MATH]w(t)R^n-n^[MATH]w(t)- w(t)/w(t)[MATH]R^n-n^[MATH]r=3[MATH]d=21[MATH]km[MATH]-2[MATH]T=3h[MATH]6h[MATH]9h[MATH]12h[MATH]18h[MATH]12h[MATH]18h[MATH]T[MATH]T=12h[MATH]T=9h[MATH]T=18h[MATH]T=9h[MATH]b[MATH]T=12h[MATH]b=3[MATH]b=4[MATH]b=5[MATH]b=1, 2, 3[MATH]b[MATH]h[MATH]B_0^-1[MATH]d=336[MATH]n=3d^2=338,688[MATH]r=48[MATH]b=1[MATH]b=2[MATH]b[MATH]M_i[MATH]1ik[MATH](M_i)_1ik[MATH]M_i v[MATH]M_i^T v[MATH]dxdt=-Ax-B(x,x)+f[MATH]A[MATH]nn[MATH]B[MATH]nnn[MATH]f[MATH]R^n[MATH]f=0[MATH]M(t,s)[x(s)][MATH]l_>0[MATH]l_[MATH]t-s[MATH]M(t,s)[x(s)][MATH]O(n)[MATH]M_1,, M_k[MATH](M(t,s)[x(s)])^-1=M(s,t)[x(t)][MATH](s-t)^l[MATH](t-s)^l[MATH]x(t)[MATH]x(s)[MATH]M(t_l,t_0)[MATH]ln[MATH]M_1,, M_l[MATH]O(n l)[MATH]M_l M_l-1M_1[MATH]O(n l)[MATH]M_lM_1[MATH]O(nl^2)[MATH]O(n l^2)[MATH]O(n l^3)[MATH]r>1[MATH]M_l[MATH]"}

1805.01605

{'1805.01605-1-0-0': 'Magnetorelaxometry imaging is a novel tool for quantitative determination of the spatial distribution of magnetic nanoparticles inside an organism.', '1805.01605-1-0-1': 'While the use of multiple excitation patterns has been demonstrated to significantly improve spatial resolution, the use of several activation patterns is considerably more time consuming as several experiments have to be performed in a sequential manner.', '1805.01605-1-0-2': 'In this paper, we use compressed sensing in combination with sparse recovery, for reducing scanning time while keeping improved spatial resolution.', '1805.01605-1-0-3': 'We investigate single-stage and two-stage approaches for image reconstructing and sparse recovery algorithms based on Douglas Rachford splitting.', '1805.01605-1-0-4': 'Our numerical experiments demonstrate that the single-stage approach clearly outperforms the two-stage approach.', '1805.01605-1-0-5': 'Further, the proposed single-stage algorithm is shown to be capable to recover the magnetic nanoparticles with high spatial resolution from a small number of activation patterns.', '1805.01605-1-1-0': 'Keywords: Compressed sensing; magnetorelaxometry; image reconstruction; magnetic nanoparticles; multiple excitation; Douglas Rachford splitting; sparse recovery.', '1805.01605-1-2-0': '# Introduction', '1805.01605-1-3-0': 'Magnetic nanoparticles (MNP) offer a variety of promising biomedical applications.', '1805.01605-1-3-1': 'For example, they can be used as agents for drug delivery or hyperthermia, where the aim is to heat up specific regions inside a biological specimen [CITATION].', '1805.01605-1-3-2': 'These applications require quantitative knowledge of the magnetic nanoparticles distribution for safety and efficacy monitoring.', '1805.01605-1-3-3': 'In this paper, we consider magnetorelaxometry (MRX) imaging, which is a novel and promising non-invasive technique to spatially resolve the location and concentration of magnetic particles in vivo [CITATION], and beneficially combines a highly sensitive magnetic measurement technology for MNP imaging with a broad range of parameters and the potential to image particle distributions in a comparably large volume [CITATION].', '1805.01605-1-4-0': 'In MRX, the magnetic moments of the magnetic nanoparticles are aligned by a magnetic field generated by excitation coils [CITATION].', '1805.01605-1-4-1': 'Therewith, a magnetization can be measured from the MNP.', '1805.01605-1-4-2': 'After switching off the excitation field, the decay of magnetization (relaxation) is recorded, typically by SQUID sensors, yielding information about the particle concentration and other properties.', '1805.01605-1-4-3': 'If the relaxation is measured by a sensor array, the particle distribution can be reconstructed by inverse imaging methods [CITATION].', '1805.01605-1-5-0': 'In multiple excitation MRX (ME-MRX), several inhomogeneous activation fields generated by an array of excitation coils are applied.', '1805.01605-1-5-1': 'With such a coil array, a variety of inhomogeneous excitation fields can be generated, for example, by switching on the coils in a sequential manner [CITATION].', '1805.01605-1-5-2': 'First experimental realization of ME-MRX with sequential activation and least squares estimation for imaging has been obtained in [CITATION].', '1805.01605-1-6-0': 'ME-MRX using sequential coil activation requires a large number of measurement cycles and thus a considerable time for data acquisition.', '1805.01605-1-6-1': 'Recently, different approaches have been proposed for advanced excitation schemes [CITATION].', '1805.01605-1-7-0': "A possibility to maintain improved resolution while reducing measurement time is via compressed sensing (CS), a new sensing paradigm [CITATION] that allows to capture a high resolution image (or signal) by using much fewer measurements than predicted by Shannon's sampling theorem.", '1805.01605-1-7-1': 'CS replaces point-measurements by general linear measurements, where each measurement consists of a linear combination of the entries of the image of interest.', '1805.01605-1-7-2': 'Recovering the original image is highly under-determined.', '1805.01605-1-7-3': 'The standard theory of CS predicts that under suitable assumptions on the image of interest (sparsity) and the measurement matrix (incoherence), stable image reconstruction is nevertheless possible.', '1805.01605-1-7-4': 'CS has led to several new sampling strategies in medical imaging, for example, for speeding up MRI data acquisition [CITATION], accelerating photoacoustic tomography [CITATION], or completing under-sampled CT images [CITATION].', '1805.01605-1-7-5': 'While these applications involve relatively mildly ill-conditioned problems, ME-MRX constitutes a severely ill-conditioned or ill-posed problem.', '1805.01605-1-7-6': 'No standard approaches for image reconstruction combining compressive measurements and such severely ill-posed problems exist.', '1805.01605-1-8-0': 'In this work, we investigate CS techniques for accelerating ME-MRX.', '1805.01605-1-8-1': 'For that purpose, we consider random activations of the coils as well as sparse deterministic subsampling schemes.', '1805.01605-1-8-2': 'For reconstructing the magnetic particle distribution from the CS data we develop two-stage and single-stage reconstruction approaches.', '1805.01605-1-8-3': 'In the two-stage approach, the CS data are used to recover the complete data (corresponding to a full sequential activation pattern) in a first step, from which the magnetic particle distribution is recovered in a second stage.', '1805.01605-1-8-4': 'In the single-stage approach, the magnetic particle distribution is directly recovered from the CS data.', '1805.01605-1-8-5': 'For both approaches, we develop sparse reconstruction algorithms based on Douglas Rachford splitting [CITATION].', '1805.01605-1-8-6': 'Note that due to the ill-posed nature of the MRX inverse problem, the standard uniform CS recovery theory [CITATION] cannot be applied to the single-stage approach.', '1805.01605-1-8-7': 'Nevertheless, our numerical results demonstrate that the single-stage approach works well, and significantly outperforms the two-stage approach in terms of reconstruction quality.', '1805.01605-1-8-8': 'We therefore recommend the single-stage approach for further investigations in CS ME-MRX imaging.', '1805.01605-1-9-0': '## Outline', '1805.01605-1-10-0': 'This article is organized as follows.', '1805.01605-1-10-1': 'In Section [REF] we describe the forward problem of ME-MRX imaging and CS.', '1805.01605-1-10-2': 'The proposed single-stage and two-stage algorithms for ME-MRX image reconstruction are developed in Section [REF].', '1805.01605-1-10-3': 'Details on the numerical implementation together with numerical results and a discussion are presented in Section [REF].', '1805.01605-1-10-4': 'The paper ends with a conclusion and outlook presented in Section [REF].', '1805.01605-1-11-0': '# The forward model in ME-MRX imaging', '1805.01605-1-12-0': 'This section gives a precise formulation of the forward problem of ME-MRX and CS.', '1805.01605-1-12-1': 'For the sake of clarity, throughout we work with a discrete model [CITATION].', '1805.01605-1-12-2': 'A continuous model has recently been presented in [CITATION].', '1805.01605-1-13-0': '## Signal generation', '1805.01605-1-14-0': 'In MRX, an activation field is applied to a region of interest containing magnetic particles.', '1805.01605-1-14-1': 'We assume this region to be divided into a number of [MATH] quadratic voxels, each represented by its midpoint located at [MATH] and containing a concentration of magnetic particles [MATH] (compare Fig. [REF]).', '1805.01605-1-14-2': 'Due to the presence of the applied field, the magnetic particles align up and, after the activation field is switched off, produce a relaxation signal.', '1805.01605-1-14-3': 'According to Biot-Savart law, the contribution of the magnetic particles concentration in the [MATH]th voxel to the measured signal at sensor location [MATH] is given by [CITATION] [EQUATION]', '1805.01605-1-14-4': 'Here [MATH] is the activation field, [MATH] is the vector joining [MATH] and [MATH], [MATH] is the normal vector of the sensor at location [MATH], and [MATH] and [MATH] are used to denote the tensor and scalar product, respectively.', '1805.01605-1-15-0': 'Assuming the concentrations [MATH] and the activation field [MATH] to be known, the measured data can be computed by [REF].', '1805.01605-1-15-1': 'Collecting the individual measurements in a vector [MATH], we obtain the linear equation [EQUATION]', '1805.01605-1-15-2': 'Here [MATH] is the vector of magnetic particle concentrations, and [MATH] is the system matrix having entries [EQUATION] as derived from relation [REF].', '1805.01605-1-15-3': 'The matrix [MATH] is called Lead field matrix corresponding to the excitation field [MATH].', '1805.01605-1-15-4': 'Eq. [REF] constitutes the standard discrete forward model of MRX using a single activation field.', '1805.01605-1-16-0': '## Full coil activation', '1805.01605-1-17-0': 'In ME-MRX, the volume of interest is (partially) surrounded by an array of excitation coils and sensor arrays (see Fig. [REF]).', '1805.01605-1-17-1': 'The coils can be controlled to generate different types of activation fields.', '1805.01605-1-17-2': 'In the following we denote by [MATH] the magnetic fields induced by individual activation of the coils.', '1805.01605-1-17-3': 'Further, we write [MATH] and [MATH] for the data and the Lead field matrix according to [REF] and [REF], corresponding to the activation field [MATH].', '1805.01605-1-17-4': 'The measurements from all activations can be combined to a single equation of the form [EQUATION] where [EQUATION]', '1805.01605-1-17-5': 'We will refer to [MATH] in [REF] as full activation data.', '1805.01605-1-17-6': 'Evaluating [REF] constitutes the forward model of ME-MRX.', '1805.01605-1-17-7': 'The corresponding inverse problem consists in determining the magnetic particle distribution [MATH] from the data vector [MATH] that is additionally corrupted by noise.', '1805.01605-1-17-8': 'Note that Eq. [REF] is known to be severely ill-conditioned as its singular values are rapidly decreasing (see [CITATION]; an analysis in the infinite-dimensional setting has been done in [CITATION]).', '1805.01605-1-18-0': 'The multiple coil setup has been been realized in [CITATION].', '1805.01605-1-18-1': 'In order to collect the required data, any coil is switched on individually and data are collected by the sensor array.', '1805.01605-1-18-2': 'This process is repeated in a sequential manner until each coil in the array has been activated.', '1805.01605-1-18-3': 'The use of multiple excitation patters has been shown to significantly improve the spatial resolution compared to single activation [CITATION].', '1805.01605-1-18-4': 'However, the consecutive activation leads to a more time consuming measurement process.', '1805.01605-1-18-5': 'In order to accelerate data acquisition while keeping the advantages of multiple activation coils, in this paper we use CS techniques where we use random as well as deterministic coil activation patterns.', '1805.01605-1-18-6': 'Additionally, we develop sparse reconstruction algorithms for the arising CS ME-MRX inverse problems.', '1805.01605-1-19-0': '## Compressive coil activations', '1805.01605-1-20-0': 'The basic idea of employing CS for ME-MRX is to use [MATH] specific coil activations with [MATH] instead of activating all [MATH] coils in a sequential manner.', '1805.01605-1-20-1': 'See Fig. [REF] for a possible random activation pattern compared to sequential activation.', '1805.01605-1-20-2': 'Because of linearity, the data corresponding to the [MATH]th random activation pattern is given by [EQUATION]', '1805.01605-1-20-3': 'Here [MATH] models the error in the data and [MATH] is the contribution of coil [MATH] to the [MATH]th activation pattern.', '1805.01605-1-20-4': 'The measurement matrix [EQUATION] represents all activation patters.', '1805.01605-1-20-5': 'Typical choices for [MATH] are Bernoulli or Gaussian matrices, since they are known to guarantee stable recovery of sparse signals [CITATION].', '1805.01605-1-20-6': 'Such kind of measurements can be realized in natural manner in ME-MRX, by simultaneous activation of several coils.', '1805.01605-1-20-7': 'The corresponding image reconstruction problem consists in recovering the magnetic particle distribution [MATH] from the data in [REF].', '1805.01605-1-21-0': '## Matrix formulation of the reconstruction problem', '1805.01605-1-22-0': 'In order to write the reconstruction problem in a compact form we introduce some additional notation.', '1805.01605-1-22-1': 'First, we define the CS measurement vector as [EQUATION]', '1805.01605-1-22-2': 'Second, we introduce the vectorization or reshaping operator [MATH], that takes a matrix to a vector whose block entries are equal to the transposes of the rows of the matrix: [EQUATION]', '1805.01605-1-22-3': 'We denote by [MATH] the inverse reshaping operation that maps a vector in [MATH] to a matrix in [MATH].', '1805.01605-1-22-4': 'Further, we write [MATH] and [MATH].', '1805.01605-1-23-0': 'With the above notations, the CS data in [REF] can be written in the compact form [EQUATION] where [MATH] is the reshaped Lead field matrix defined by the property [MATH].', '1805.01605-1-23-1': 'With the Kronecker (or tensor) product [MATH] between the CS activation matrix [MATH] and the identity matrix [MATH], equation [REF] can further be rewritten in the form [EQUATION]', '1805.01605-1-23-2': 'The image reconstruction task in CS ME-MRX now consists in recovering the magnetic particle distribution [MATH] from equation [REF] or, equivalently, from equation [REF].', '1805.01605-1-24-0': '# Sparse reconstruction algorithms for CS ME-MRX imaging', '1805.01605-1-25-0': 'In order to recover the magnetic particle distribution [MATH] from equation [REF] (or [REF]) we devise new sparse recovery algorithms.', '1805.01605-1-25-1': 'These methods will either follow a two-stage or a single-stage strategy.', '1805.01605-1-25-2': 'In the two-stage approach, the full measurement data [MATH] is recovered from the CS data as an intermediate result, from which the magnetic particle distribution is recovered in the second step.', '1805.01605-1-25-3': 'In the single-stage approach, the magnetic particle distribution is directly recovered from the CS data [REF].', '1805.01605-1-25-4': 'We note that no standard approach for image reconstruction combining compressive measurements and severely ill-posed problems exist.', '1805.01605-1-25-5': 'This section is therefore also of interest from a general perspective on CS in inverse problems.', '1805.01605-1-26-0': '## Single-stage reconstruction', '1805.01605-1-27-0': 'Probably the most straightforward way to address CS ME-MRX is to view [REF] as a single inverse problem with system matrix [MATH].', '1805.01605-1-27-1': 'To address instability and non-uniqueness and to account for prior knowledge, we approach this by sparse regularization.', '1805.01605-1-27-2': 'For that purpose we minimize the generalized Tikhonov functional [CITATION] [EQUATION]', '1805.01605-1-27-3': 'Here [MATH] is a transform that sparsifies the magnetic particle concentration, [MATH] is an additional regularizer that incorporates further a priori knowledge about the magnetic particle distribution (such as positivity and other convex constraints), and [MATH] is the standard [MATH]-norm defined by [EQUATION]', '1805.01605-1-27-4': 'We call [MATH] a sparsifying transform if [MATH] can well be approximated by [MATH]-sparse vectors [MATH], defined by the property that [MATH] has at most [MATH] elements.', '1805.01605-1-28-0': '[Recovery theory for [REF]] The [MATH]-RIP constant of [MATH] (after appropriate scaling) is defined as the smallest number [MATH] such that for all [MATH]-sparse vectors [MATH] we have [EQUATION]', '1805.01605-1-28-1': 'Roughly spoken, standard CS theory [CITATION] predicts uniform stable recovery with [REF] (using [MATH] and orthogonal [MATH]), in the sense that it stably recovers any [MATH]-sparse vector provided that the [MATH]-RIP constant of [MATH] is sufficiently small.', '1805.01605-1-28-2': 'Due to the severe ill-posedness of [MATH], the [MATH]-RIP is expected to be at most satisfied for very small problem size.', '1805.01605-1-28-3': 'Anyway, even in the case of full measurement data, where [MATH] is the identity matrix, uniform reconstruction seems hardly possible.', '1805.01605-1-28-4': 'To obtain quantitative error estimates for [REF], the stable recovery results for individual elements [CITATION] are a promising alternative.', '1805.01605-1-28-5': 'Such theoretical investigations are beyond the scope of this paper and an interesting line of future research.', '1805.01605-1-29-0': 'In order to minimize [REF], we propose using the Douglas Rachford minimization algorithm, which is an backward-backward type splitting method for minimizing the sum of two functionals [MATH] and [MATH].', '1805.01605-1-29-1': 'For our purpose we take [EQUATION]', '1805.01605-1-29-2': 'The Douglas Rachford algorithm for minimizing [REF] generates a sequence [MATH] of estimated magnetic particle distributions and auxiliary sequences [MATH], [MATH] as described in Algorithm [REF].', '1805.01605-1-30-0': 'Proposed single-stage reconstruction sparse recovery algorithm for CS ME-MRX.', '1805.01605-1-30-1': '[1] Select [MATH], and [MATH] Initialize [MATH]', '1805.01605-1-31-0': 'Under the reasonable assumptions that the regularizer [MATH] is lower semicontinuous and convex, and that the sparse Tikhonov functional [MATH] is coercive, the sequence [MATH] generated by Algorithm [REF] is known to converge to a minimizer of [REF]; see [CITATION].', '1805.01605-1-31-1': 'Note that Algorithm [REF] performs implicit steps with respect to the residual functional [MATH], which we found to have much faster convergence than forward-backward type splitting algorithms that suffer from the ill-conditioned nature of [MATH].', '1805.01605-1-32-0': 'For our numerical experiments we take [MATH] as the discrete gradient operator as appropriate sparsifying transform for piecewise smooth magnetic particle distributions.', '1805.01605-1-32-1': 'The additional regularizer is taken as the indicator function [MATH] of the convex set [EQUATION] defined by [MATH] if [MATH] and [MATH].', '1805.01605-1-32-2': 'It guarantees non-negativity and boundedness, and [REF] reduces to total variation regularization with box constraints.', '1805.01605-1-32-3': 'The minimization of [MATH] required for implementing Algorithm [REF], is again performed by Douglas Rachford splitting.', '1805.01605-1-33-0': '## Two stage reconstruction approach', '1805.01605-1-34-0': 'An alternative reconstruction approach is based on a two-stage procedure.', '1805.01605-1-34-1': 'Thereby, the CS measurements are used to first recover the full activation data from which the magnetic particle distribution is recovered in a second step.', '1805.01605-1-34-2': 'In order to recover the full activation data, note that the columns of [MATH], corresponding to CS data collected by individual sensors, satisfy [EQUATION]', '1805.01605-1-34-3': 'Eq. [REF] is a family of [MATH] separate reconstruction problems involving the same measurement matrix [MATH], and known as the multiple measurement problem in the CS literature.', '1805.01605-1-34-4': 'In order to solve [REF], we use sparse recovery.', '1805.01605-1-34-5': 'We determine an approximation to [MATH] by minimizing [EQUATION] where we [MATH] is the Frobenius norm, [MATH] is a sparsifying transform for [MATH] defined by an orthogonal basis [MATH], and [MATH] the second derivative with respect to the coil variable.', '1805.01605-1-34-6': 'The additional regularization term [MATH] accounts for the fact that the forward operator is smoothing and therefore [MATH] is expected to be smooth.', '1805.01605-1-35-0': 'The functional [REF] is again approached by Douglas Rachford splitting.', '1805.01605-1-35-1': 'The resulting algorithm is summarized in Algorithm [REF], where [MATH] denotes the soft-thresholding operation in the basis [MATH], [EQUATION]', '1805.01605-1-35-2': 'Because the sparse Tikhonov functional [MATH] defined in [REF] is coercive, the sequence [MATH] generated by Algorithm [REF] converges to a minimizer of the [MATH].', '1805.01605-1-36-0': 'Proposed sparse data recovery algorithm for CS ME-MRX.', '1805.01605-1-36-1': '[1] Select [MATH], and [MATH] Initialize [MATH]', '1805.01605-1-37-0': 'Having recovered an approximation of the complete data [MATH] by Algorithm [REF], the magnetic particle distribution can be recovered by solving the full measurement data problem.', '1805.01605-1-37-1': 'For that purpose we use Algorithm [REF] with [MATH].', '1805.01605-1-38-0': '# Numerical results', '1805.01605-1-39-0': 'For the following numerical simulations, we use a data setup similar to the realization in [CITATION].', '1805.01605-1-39-1': 'For simplification, we consider a two-dimensional setup representing one voxel plane, containing two parallel arrays of detector elements (one measuring the horizontal [MATH] and one measuring the vertical [MATH] component) located outside a quadratic region of interest.', '1805.01605-1-39-2': 'Circular shaped activation coils are arranged in U-form around the region of interest all having the same normal vector [MATH].', '1805.01605-1-39-3': 'The used arrangement of sensor and coil locations for the full measurement data setup is shown in Fig. [REF].', '1805.01605-1-39-4': 'For our simulations we choose a discretization of imaging space into [MATH] voxels covering a the region of interest of [MATH] cm^2.', '1805.01605-1-39-5': 'The data are generated for [MATH] positions and full measurement data corresponds to [MATH] activation coils outside region of interest.', '1805.01605-1-40-0': '## Forward computations', '1805.01605-1-41-0': 'To set up the forward model [REF], [REF], [REF], we first have to compute the magnetic fields generated by each coil.', '1805.01605-1-41-1': 'For that purpose we follow the approach of [CITATION], where any circular activation coil is approximated by [MATH] short line segments, each having the same current [MATH].', '1805.01605-1-41-2': 'Using this approximation, the induced magnetic field at the voxel center [MATH] can be computed by (see [CITATION]) [EQUATION] where [MATH] and [MATH] are the distance vectors between the voxel center [MATH] and the beginning and end points of the [MATH]th line segment, respectively.', '1805.01605-1-41-3': 'For the presented numerical computations we use [MATH] line segments with a diameter of [MATH], illustrating an almost point like coil.', '1805.01605-1-41-4': 'Having computed the activation field [MATH], we compute the entries of the lead field matrix according to [REF].', '1805.01605-1-41-5': 'By activating the coils sequentially, we obtain the full measurement data Lead field matrix.', '1805.01605-1-42-0': 'For the presented results we use three different magnetic particle distributions which, together with the corresponding measurements full data, are shown in Fig. [REF].', '1805.01605-1-42-1': 'Any column in the measurement data (in Fig. [REF] and below) corresponds to a single activation pattern and contains the data of all detectors.', '1805.01605-1-42-2': 'The phantoms are rescaled to have maximum value 1 and the forward operator [MATH] is rescaled to have matrix norm [MATH].', '1805.01605-1-42-3': 'To all data we have added additive Gaussian noise amounting to an SNR of 80dB.', '1805.01605-1-42-4': 'The corresponding reconstruction [MATH] with standard quadratic Tikhonov regularization (penalized least squares) using regularization parameter [MATH] are shown in the bottom row of Fig. [REF].', '1805.01605-1-42-5': 'They can be seen as benchmark for the more sophisticated CS reconstructions applied to less data presented below.', '1805.01605-1-43-0': 'The CS forward matrix [MATH] is computed by multiplying the full Lead field matrix with [MATH].', '1805.01605-1-43-1': 'CS measurements have been generated in two random ways and one deterministic way.', '1805.01605-1-43-2': 'In the random case, [MATH] is taken either as Bernoulli matrix having entries [MATH] with equal probability, or a Gaussian matrix consisting of i.i.d. [MATH]-Gaussian random variables in each entry.', '1805.01605-1-43-3': 'The deterministic subsampling is performed by choosing [MATH] equispaced coil activations.', '1805.01605-1-43-4': 'The CS data [MATH] for [MATH] coil activations using the three sampling schemes and the three phantoms are shown in Fig. [REF].', '1805.01605-1-44-0': '## Results for the two-stage approach', '1805.01605-1-45-0': 'In the two-stage reconstruction approach, the complete data [MATH] are recovered with Algorithm [REF] in the first step.', '1805.01605-1-45-1': 'The results are shown in Fig. [REF].', '1805.01605-1-45-2': 'We see that any data set is recovered well from the CS data.', '1805.01605-1-45-3': 'For the sparsifying transform [MATH] we use the wavelet basis derived from Daubechies least asymmetric wavelet having 8 vanishing moments.', '1805.01605-1-45-4': 'The parameters are selected as [MATH], [MATH] in the regularization functional, and [MATH], [MATH], [MATH] for the numerical minimization.', '1805.01605-1-45-5': 'The relative mean squared errors [MATH] are [MATH] for the CS-phantom, [MATH] for the Smiley-phantom and [MATH] for the tumor-phantom.', '1805.01605-1-45-6': 'In particular, for the CS-phantom and the tumor-phantom the error is smallest for the deterministic subsampling scheme, whereas for the Smiley-phantom, the Bernoulli measurements lead to the smallest error.', '1805.01605-1-46-0': 'Using the completed data from the first step, we recover the magnetic particle distribution using the proposed Algorithm [REF] with [MATH].', '1805.01605-1-46-1': 'The results of the two-stage procedure are shown in Fig. [REF].', '1805.01605-1-46-2': 'We used [MATH], [MATH], [MATH] and [MATH].', '1805.01605-1-46-3': 'The relative mean squared errors for the reconstructed particle distributions are [MATH] for the CS-phantom, [MATH] for the Smiley-phantom and [MATH] for the tumor-phantom.', '1805.01605-1-47-0': '## Results for the single-stage approach', '1805.01605-1-48-0': 'In the single-stage approach, the phantoms are directly recovered from the CS data using the proposed Algorithm [REF].', '1805.01605-1-48-1': 'We have rescaled [MATH] to have unit matrix norm and use the same parameter setting [MATH], [MATH], [MATH], [MATH] and [MATH].', '1805.01605-1-48-2': 'The reconstructed magnetic particle distributions using the single-stage procedure from 40 coil activations are shown in Fig. [REF].', '1805.01605-1-48-3': 'Each reconstruction with Algorithm [REF] takes about 50 seconds in Matlab R 2017a on a MacBook Pro (2016) with 2.9 GHz Intel Core i7 processor.', '1805.01605-1-48-4': 'The relative mean squared errors are [MATH] for the CS-phantom, [MATH] for the Smiley-phantom and [MATH] for the tumor-phantom.', '1805.01605-1-48-5': 'These numbers as well as visual inspection show that the single-stage approach significantly outperforms the two-stage procedure in terms of reconstruction quality.', '1805.01605-1-48-6': 'The reconstruction results are even much better than for full data using quadratic Tikhonov regularization.', '1805.01605-1-48-7': 'All sampling schemes yields comparable performance; however the regular subsampling scheme slightly outperforms the random activation schemes.', '1805.01605-1-49-0': 'To further explore the capabilities of Algorithm [REF] we also perform experiments using different numbers of CS measurements.', '1805.01605-1-49-1': 'For that purpose we repeated the above simulations for [MATH] coils activations using the same parameters for image reconstruction.', '1805.01605-1-49-2': 'The results are shown in [REF].', '1805.01605-1-49-3': 'One observes that for the CS-phantom and the tumor-phantom, accurate results are obtained even for very few coil activations.', '1805.01605-1-49-4': 'For the more complex Smiley-phantom, the quality significantly decreases with decreasing number of measurements.', '1805.01605-1-50-0': 'Finally, Figure [REF] compares the reconstruction results for a small number of [MATH] activation patterns.', '1805.01605-1-50-1': 'The relative mean squared errors are [MATH] for the CS-phantom, [MATH] for the Smiley-phantom and [MATH] for the Tumor-phantom.', '1805.01605-1-50-2': 'While the average reconstruction quality for the CS-phantom over the the whole region of interest is the best for the deterministic sampling scheme, some aspects might be better recovered with the random sampling schemes.', '1805.01605-1-50-3': 'For the other two phantoms the random schemes now outperform the Deterministic activation pattern schemes.', '1805.01605-1-51-0': '## Discussion', '1805.01605-1-52-0': 'Our results clearly demonstrate, that the single-stage approach yields significantly better results than the two-stage approach.', '1805.01605-1-52-1': 'While the full measurement data are completed well by Algorithm [REF] (see Fig. [REF]) in the first step of the two-stage approach, recovering the particle distribution using Algorithm [REF], performs much better with the original CS data than with the recovered full measurement.', '1805.01605-1-52-2': 'One the one hand, this indicates that the data contains key features allowing high quality reconstruction of the magnetic particle distributions.', '1805.01605-1-52-3': 'On the other hand, the data completion procedure seemingly fails to recover these key features in a sufficient manner.', '1805.01605-1-53-0': 'For a large numbers of coil activations shown in Fig. [REF] and Fig. [REF], the deterministic subsampling scheme leads to better results than the random activation.', '1805.01605-1-53-1': 'We address this due to the strong smoothing effect of the forward operator, which removes most high frequency components in the data.', '1805.01605-1-53-2': 'The regular sparse sampling pattern might therefore yield to the largest information gain given a certain number of linear measurements.', '1805.01605-1-53-3': 'For the small number of coil activations shown (see Fig. [REF]), the random sampling patterns outperforms the deterministic scheme for the Smiley-phantom and the tumor-phantom.', '1805.01605-1-54-0': 'We point out, that hardly any recovery results for compressed sensing applied to severely ill-conditioned matrices [MATH] are available, as [MATH] cannot be expected to satisfy the RIP or a similar uniform sparse recovery condition.', '1805.01605-1-54-1': 'The numerical results presented in the paper indicate that for the type of considered phantoms (especially the CS-shaped and the tumor-shaped phantom) stable recovery is possible from a small number of coil activations.', '1805.01605-1-54-2': 'One might derive such estimates using the recovery result of individual vectors (see Remark [REF]).', '1805.01605-1-54-3': 'Such investigations are an interesting line of future research.', '1805.01605-1-55-0': '# Conclusion', '1805.01605-1-56-0': 'The use of multiple coil activation patterns in magnetorelaxometry imaging is time consuming and requires performing several consecutive measurements.', '1805.01605-1-56-1': 'It is therefore desirable to make the number of coils activations as small as possible, while keeping high spatial resolution.', '1805.01605-1-56-2': 'For that purpose we investigated CS strategies in this paper.', '1805.01605-1-56-3': 'We compared Gaussian random activation, Bernoulli activations and deterministic subsampling schemes.', '1805.01605-1-56-4': 'Further, we established and compared single-stage and two-stage reconstruction approaches.', '1805.01605-1-56-5': 'In the two-stage approach, the full activation data is computed as an intermediate result, whereas in the single stage approach, the magnetic particle distributions is directly recovered from the CS data.', '1805.01605-1-57-0': 'Some conclusions that can be drawn from our numerical experiments are as follows.', '1805.01605-1-57-1': 'First, the single-stage approach significantly outperforms the two-stage approach.', '1805.01605-1-57-2': 'We therefore propose to use the single-stage approach for ME-MRX image reconstruction.', '1805.01605-1-57-3': 'Second, in the experiments with a large number of activations, the deterministic sampling scheme performed better than the random sampling patterns.', '1805.01605-1-57-4': 'For a small number of coil activations (see Fig. [REF]) the random schemes outperform the deterministic scheme for the Smiley-phantom and tumor phantom.', '1805.01605-1-57-5': 'Finally, for simple phantoms, a small number of activation patterns seem to be sufficient for accurately estimating the magnetic particle distribution.', '1805.01605-1-57-6': 'For actual image reconstruction, we developed an algorithm based on Douglas Rachford splitting.', '1805.01605-1-57-7': 'The single-stage algorithm includes a TV penalty as well as accounts for positivity and other known prior.', '1805.01605-1-57-8': 'The algorithm performs well for the considered setup and recovers the particle distribution in less that one minute on a standard Mac Book (2016).', '1805.01605-1-58-0': 'Several interesting research directions following this work are possible.', '1805.01605-1-58-1': 'First, we can replace the inner TV minimization in the single-stage approach by a different algorithm which should accelerate Algorithm [REF].', '1805.01605-1-58-2': 'Second, the derivation of theoretical error estimates for the single-stage approach is of significant interest.', '1805.01605-1-58-3': 'Results in that direction can advice which type of MRX measurements are best to obtain accurate results for certain phantom classes.', '1805.01605-1-58-4': 'Moreover, the derivation of adaptive compressed sensing strategies for online monitoring is of significant interest.', '1805.01605-1-58-5': 'Advising optimal coil activations given previous activations is a difficult problem that will benefit from theoretical error estimates, numerical simulations as well as a real-world experiments that are currently been implemented.', '1805.01605-1-58-6': 'In this paper we investigated standard random compressed sensing schemes (using Gaussian and Bernoulli activation patterns).', '1805.01605-1-58-7': 'Using more problem adapted and task oriented measurement design we will expect to derive improved coil activation schemes.'}

{'1805.01605-2-0-0': 'Magnetorelaxometry imaging is a novel tool for quantitative determination of the spatial distribution of magnetic nanoparticles inside an organism.', '1805.01605-2-0-1': 'The use of multiple excitation patterns has been demonstrated to significantly improve spatial resolution.', '1805.01605-2-0-2': 'However, increasing the number of excitation patterns is considerably more time consuming, because several sequential measurements have to be performed.', '1805.01605-2-0-3': 'In this paper, we use compressed sensing in combination with sparse recovery to reduce the total measurement time and to improve spatial resolution.', '1805.01605-2-0-4': 'For image reconstruction, we propose using the Douglas-Rachford splitting algorithm applied to the sparse Tikhonov functional including a positivity constraint.', '1805.01605-2-0-5': 'Our numerical experiments demonstrate that the resulting algorithm is capable to accurately recover the magnetic nanoparticle distribution from a small number of activation patterns.', '1805.01605-2-0-6': 'For example, our algorithm applied with 10 activations yields half the reconstruction error of quadratic Tikhonov regularization applied with 50 activations, for a tumor-like phantom.', '1805.01605-2-1-0': 'Keywords: Compressed sensing; magnetorelaxometry; image reconstruction; magnetic nanoparticles; multiple excitation; Douglas-Rachford splitting; sparse recovery.', '1805.01605-2-2-0': '# Introduction', '1805.01605-2-3-0': 'Magnetic nanoparticles (MNP) offer a variety of promising biomedical applications.', '1805.01605-2-3-1': 'For example, they can be used as agents for drug delivery or hyperthermia, where the aim is to heat up specific regions inside a biological specimen [CITATION].', '1805.01605-2-3-2': 'These applications require quantitative knowledge of the magnetic nanoparticles distribution for safety and efficacy monitoring.', '1805.01605-2-3-3': 'In this paper, we consider magnetorelaxometry (MRX) imaging, which is a novel and promising non-invasive technique to spatially resolve the location and concentration of MNPs in vivo [CITATION].', '1805.01605-2-3-4': 'It beneficially combines a highly sensitive magnetic measurement technology for MNP imaging with a broad range of parameters and has the potential to image particle distributions in a comparably large volume [CITATION].', '1805.01605-2-4-0': 'In MRX, the magnetic moments of the magnetic nanoparticles are aligned by a magnetic field generated by excitation coils [CITATION].', '1805.01605-2-4-1': 'Therewith, a magnetization can be measured from the MNP.', '1805.01605-2-4-2': 'After switching off the excitation field, the decay of magnetization (relaxation) is recorded, typically by SQUIDs (superconducting quantum interference device), yielding information about the particle concentration and related properties.', '1805.01605-2-4-3': 'If the relaxation is measured by a sensor array, the particle distribution can be reconstructed by inverse imaging methods [CITATION].', '1805.01605-2-5-0': 'In multiple excitation MRX (ME-MRX), several inhomogeneous activation fields generated by an array of excitation coils are applied.', '1805.01605-2-5-1': 'With such a coil array, a variety of inhomogeneous excitation fields can be generated, for example, by switching on the coils in a sequential manner [CITATION].', '1805.01605-2-5-2': 'First experimental realizations of ME-MRX with sequential activation and least squares estimation for imaging have been obtained in [CITATION].', '1805.01605-2-5-3': 'ME-MRX using sequential coil activation and standard image reconstruction techniques requires a large number of measurement cycles and thus a considerable time for data acquisition.', '1805.01605-2-5-4': 'Recently, different approaches have been proposed using advanced excitation schemes [CITATION].', '1805.01605-2-6-0': "A strategy to maintain high resolution while reducing measurement time is via compressed sensing (CS), a new sensing paradigm [CITATION] that allows to capture a high-resolution image (or signal) by using fewer measurements than predicted by Shannon's sampling theorem.", '1805.01605-2-6-1': 'CS replaces point-measurements by general linear measurements, where each measurement consists of a linear combination of the entries of the image of interest.', '1805.01605-2-6-2': 'Recovering the original image is highly under-determined.', '1805.01605-2-6-3': 'The standard theory of CS predicts that under suitable assumptions on the image (sparsity) and the measurement matrix (incoherence), stable image reconstruction is nevertheless possible.', '1805.01605-2-6-4': 'CS has led to several new sampling strategies in medical imaging, for example, for speeding up MRI data acquisition [CITATION], accelerating photoacoustic tomography [CITATION], or completing under-sampled CT images [CITATION].', '1805.01605-2-6-5': 'While these applications involve relatively mildly ill-conditioned problems, ME-MRX constitutes a severely ill-conditioned or ill-posed problem [CITATION].', '1805.01605-2-6-6': 'No standard approaches for image reconstruction combining compressive measurements and such severely ill-posed problems exist.', '1805.01605-2-7-0': 'In this work, we investigate CS techniques for accelerating ME-MRX.', '1805.01605-2-7-1': 'For that purpose, we consider random activations of the coils as well as sparse deterministic activation schemes.', '1805.01605-2-7-2': 'In order to image the MNP distribution from the CS data we propose a sparse reconstruction framework.', '1805.01605-2-7-3': 'We use [MATH]-Tikhonov regularization together with a positivity constraint on the set of reconstructed MNP distributions, and use the Douglas-Rachford splitting algorithm [CITATION] for its minimization.', '1805.01605-2-7-4': 'We evaluate the performance of the resulting sparse recovery algorithm using different phantoms in dependence of the number of activation patterns.', '1805.01605-2-7-5': 'In any of the cases, the sparse recovery algorithm reduces the root mean square error (RSME) by more than [MATH].', '1805.01605-2-7-6': 'For two of the three phantoms (CS and smiley), the deterministic scheme outperforms the random pattern by around [MATH] in terms of reduced RSME.', '1805.01605-2-7-7': 'For the third phantom (tumor), all tested sampling patterns roughly give the same results.', '1805.01605-2-7-8': 'We therefore can propose the sparse sampling pattern among the tested schemes, in combination with the proposed sparse recovery framework.', '1805.01605-2-8-0': '# The forward model in ME-MRX imaging', '1805.01605-2-9-0': 'This section gives a precise formulation of the forward problem of ME-MRX and CS.', '1805.01605-2-9-1': 'Throughout this text we work with a discrete model [CITATION].', '1805.01605-2-9-2': 'A continuous model has recently been presented in [CITATION].', '1805.01605-2-10-0': '## Signal generation', '1805.01605-2-11-0': 'In MRX, an activation field is applied to a region of interest containing MNPs.', '1805.01605-2-11-1': 'We assume this region to be divided into a number of [MATH] quadratic voxels, each represented by its midpoint located at [MATH] and containing a concentration [MATH] of MNPs (compare Figure [REF]).', '1805.01605-2-11-2': 'Due to the presence of the applied field, the MNPs align up and, after the activation field is switched off, produce a relaxation signal.', '1805.01605-2-11-3': 'According to the Biot-Savart law, the overall contribution of the MNP concentrations in the individual voxels to the measured signal at sensor location [MATH] is given by [CITATION] [EQUATION]', '1805.01605-2-11-4': 'Here [MATH] is the activation field, [MATH] is the vector joining [MATH] and [MATH], [MATH] is the normal vector of the sensor at location [MATH], and [MATH] and [MATH] are used to denote the tensor and scalar product, respectively.', '1805.01605-2-12-0': 'Assuming the concentrations [MATH] and the activation field [MATH] to be known, the measured data can be computed by [REF].', '1805.01605-2-12-1': 'Collecting the individual measurements in a vector [MATH], we obtain the linear equation [EQUATION]', '1805.01605-2-12-2': 'Here [MATH] is the vector of MNP concentrations, and [MATH] is the system matrix having entries [EQUATION] as derived from relation [REF].', '1805.01605-2-12-3': 'The matrix [MATH] is called Lead field matrix corresponding to the excitation field [MATH].', '1805.01605-2-12-4': 'Equations [REF], [REF] constitute the standard discrete forward model of MRX using a single activation field.', '1805.01605-2-13-0': '## Sequential coil activation', '1805.01605-2-14-0': 'In ME-MRX, the volume is (partially) surrounded by an array of excitation coils and sensor arrays (see Figure [REF]).', '1805.01605-2-14-1': 'The coils can be steered to generate different types of activation fields.', '1805.01605-2-14-2': 'In the following we denote by [MATH] the magnetic fields induced by individual activation of the coils.', '1805.01605-2-14-3': 'Further, we write [MATH] and [MATH] for the data and the Lead field matrix according to [REF] and [REF], corresponding to the activation field [MATH].', '1805.01605-2-14-4': 'The measurements from all activations can be combined to a single equation of the form [EQUATION] where [EQUATION]', '1805.01605-2-14-5': 'We will refer to [MATH] in [REF] as full activation data.', '1805.01605-2-14-6': 'Evaluating [REF] constitutes the forward model of ME-MRX.', '1805.01605-2-14-7': 'The corresponding inverse problem consists in determining the MNP distribution [MATH] from the data vector [MATH] that is additionally corrupted by noise.', '1805.01605-2-14-8': 'Note that Eq. [REF] is known to be severely ill-conditioned as its singular values are rapidly decreasing (see [CITATION]; an analysis in the infinite-dimensional setting has been performed in [CITATION]).', '1805.01605-2-14-9': 'In Figure [REF] we plot the singular values of the forward matrix for full data.', '1805.01605-2-14-10': 'The singular values decay rapidly.', '1805.01605-2-14-11': 'This reflects the severe ill-conditioning and implies that only a small number of singular components can be robustly reconstructed.', '1805.01605-2-15-0': 'The multiple coil setup has been realized in [CITATION] and shown to significantly improve the spatial resolution compared to a single coil activation [CITATION].', '1805.01605-2-15-1': 'However, consecutive activations lead to a more time consuming measurement process.', '1805.01605-2-15-2': 'In order to accelerate data acquisition and to improve imaging resolution, we use CS techniques and develop a Douglas-Rachford based sparse reconstruction algorithm.', '1805.01605-2-16-0': '## Compressive coil activations', '1805.01605-2-17-0': 'The basic idea to employ CS for ME-MRX is to use [MATH] specific coil activations, with [MATH], instead of activating all [MATH] coils in a sequential manner.', '1805.01605-2-17-1': 'Because of linearity, the data corresponding to the [MATH]th random activation pattern are given by [EQUATION]', '1805.01605-2-17-2': 'Here [MATH] models the error in the data and [MATH] is the contribution of coil [MATH] to the [MATH]th activation pattern.', '1805.01605-2-17-3': 'The measurement matrix [EQUATION] represents all activation patters.', '1805.01605-2-17-4': 'Typical choices for [MATH] are Bernoulli or Gaussian matrices, since they are known to guarantee stable recovery of sparse signals [CITATION].', '1805.01605-2-17-5': 'Such kind of measurements can be naturally realized for ME-MRX, by simultaneous activation of several coils.', '1805.01605-2-17-6': 'The corresponding image reconstruction problem consists in recovering the MNP distribution [MATH] from the data in [REF].', '1805.01605-2-18-0': '## Matrix formulation of the reconstruction problem', '1805.01605-2-19-0': 'In order to write the reconstruction problem in a compact form we introduce some additional notation.', '1805.01605-2-19-1': 'First, we define the CS measurement vector as [EQUATION]', '1805.01605-2-19-2': 'Second, we introduce the vectorization or reshaping operator [MATH], that takes a matrix to a vector whose block entries are equal to the transposes of the rows of the matrix: [EQUATION]', '1805.01605-2-19-3': 'We denote by [MATH] the inverse reshaping operation that maps a vector in [MATH] to a matrix in [MATH].', '1805.01605-2-19-4': 'Further, we write [MATH] and [MATH].', '1805.01605-2-20-0': 'With the above notations, the CS data in [REF] can be written in the compact form [EQUATION] where [MATH] is the reshaped Lead field matrix defined by the property [MATH] and [MATH] denotes the noise matrix.', '1805.01605-2-20-1': 'With the Kronecker (or tensor) product [MATH] between the CS activation matrix [MATH] and the identity matrix [MATH], equation [REF] can further be rewritten in the form [EQUATION]', '1805.01605-2-20-2': 'The image reconstruction task in CS ME-MRX consists in recovering the MNP distribution [MATH] from equation [REF] or, equivalently, from equation [REF].', '1805.01605-2-21-0': '# Douglas-Rachford algorithm for CS MRX imaging', '1805.01605-2-22-0': 'In order to recover the MNP distribution, the inverse problem [REF] (or [REF]) has to be solved which is known to be severely ill-posed, see Figure [REF].', '1805.01605-2-22-1': 'We note that no standard approach for image reconstruction combining compressive measurements and severely ill-posed problems exist.', '1805.01605-2-22-2': 'This section is also of interest from a general perspective on CS for inverse problems.', '1805.01605-2-23-0': '## Sparse Tikhonov regularization', '1805.01605-2-24-0': 'We consider [REF] as a single inverse problem with system matrix [MATH].', '1805.01605-2-24-1': 'To address instability and non-uniqueness and to incorporate prior knowledge, we use a sparse regularization approach.', '1805.01605-2-24-2': 'For that purpose, we minimize the generalized Tikhonov functional [CITATION] [EQUATION]', '1805.01605-2-24-3': 'Here [MATH] is a transform that sparsifies the MNP concentration (with the [MATH] the dimension of the transformed domain), [MATH] is an additional regularizer that incorporates additional prior knowledge about the MNP distribution (such as positivity and other convex constraints) and [MATH] is the standard [MATH]-norm defined by [EQUATION] where [MATH] is the number of components of the vector [MATH].', '1805.01605-2-24-4': 'The non-negative parameters [MATH] and [MATH] allow to balance between the data consistency term [MATH], the sparsity prior [MATH] and the additional regularizer [MATH].', '1805.01605-2-25-0': 'We call [MATH] a sparsifying transform if [MATH] can well be approximated by [MATH]-sparse vectors [MATH], defined by the property that [MATH] has at most [MATH] elements.', '1805.01605-2-25-1': 'This approximation be can be quantified using the [MATH]-term approximation error, [EQUATION]', '1805.01605-2-25-2': 'The [MATH]-term approximation error [MATH] appears as additive term in standard error estimates in compressed sensing theory.', '1805.01605-2-25-3': 'For a mathematically precise discussion of the [MATH]-term approximation error we refer to [CITATION].', '1805.01605-2-26-0': '[Recovery theory for [REF]] The [MATH]-restricted isometry property ([MATH]-RIP) of [MATH] (after appropriate scaling) is defined as the smallest number [MATH] such that for all [MATH]-sparse vectors [MATH] we have [EQUATION]', '1805.01605-2-26-1': 'Roughly spoken, CS theory [CITATION] predicts uniform stable recovery with [REF] for [MATH] and in the sense that it stably recovers any [MATH]-sparse vector provided that the [MATH]-RIP constant of [MATH] is sufficiently small.', '1805.01605-2-27-0': 'Due to the severe ill-conditioning of the Lead field matrix [MATH], the [MATH]-RIP is expected to be at most satisfied when the number [MATH] of voxels is small.', '1805.01605-2-27-1': 'Figure [REF] shows that the singular values of [MATH] with [MATH] are rapidly decaying and therefore the estimate [MATH] for reasonable [MATH] can only be satisfied for signals mainly formed by the first singular vectors.', '1805.01605-2-27-2': 'This implies that even in the case of full measurement data, where [MATH] is the identity matrix, uniform stable reconstruction of all sparse vectors is impossible.', '1805.01605-2-27-3': 'To obtain quantitative error estimates for [REF], stable recovery results for individual elements [CITATION] are a promising alternative.', '1805.01605-2-27-4': 'Such theoretical investigations are beyond the scope of this paper and an interesting line of future research.', '1805.01605-2-28-0': 'Standard CS theory requires orthogonality of the sparsifying transform, which means that [MATH] equals the identity matrix.', '1805.01605-2-28-1': 'Total variation (TV)-regularization [CITATION] uses the gradient as sparsifying transform which is non-orthogonal.', '1805.01605-2-28-2': 'As in other compressed sensing imaging applications [CITATION] we observed TV to outperform orthogonal sparsifying transforms.', '1805.01605-2-28-3': 'We will therefore focus on TV and, due to space limitations, no results for orthogonal bases like wavelets are included.', '1805.01605-2-29-0': '## Douglas-Rachford algorithm', '1805.01605-2-30-0': 'In order to minimize [REF], we propose using the Douglas-Rachford minimization algorithm, which is a backward-backward type splitting method for minimizing the sum [MATH] of two functionals [MATH] and [MATH].', '1805.01605-2-30-1': 'For our purpose we take [EQUATION]', '1805.01605-2-30-2': 'Minimizing [REF] by the Douglas-Rachford algorithm, generates a sequence [MATH] of estimated MNP distributions and auxiliary sequences [MATH], [MATH] as described in Algorithm [REF].', '1805.01605-2-31-0': 'Proposed sparse reconstruction algorithm for CS ME-MRX.', '1805.01605-2-31-1': '[1] Select [MATH], and [MATH] Initialize [MATH]', '1805.01605-2-32-0': 'The Douglas-Rachford algorithm is a splitting type algorithm for minimizing [MATH] which alternately performs updates according for [MATH] (in our case, the residual functional or data fitting term) and [MATH] (in our case, the regularizer).', '1805.01605-2-32-1': 'Because the regularizing term is non-smooth, standard methods such as Newton type methods cannot be applied.', '1805.01605-2-32-2': 'Splitting type methods are a natural choice in this case, and the implicit step in line 5 in Algorithm [REF] accounts for the non-smoothness of [MATH].', '1805.01605-2-32-3': 'Other splitting algorithms that are well suited to treat the non-smooth regularizer are the forward-backward splitting algorithm [CITATION] and the Chambolle-Pock algorithm [CITATION].', '1805.01605-2-32-4': 'These algorithms use explicit gradient updates for the data fitting term requiring a matrix inversion.', '1805.01605-2-32-5': 'Therefore, they are not applicable to ME-MRX imaging.', '1805.01605-2-32-6': 'The Douglas-Rachford algorithm performs an implicit update for the data fitting term (line 4 Algorithm [REF]), which is recognized as quadratic Tikhonov regularization step for the given equation and potentially accelerates the iteration.', '1805.01605-2-33-0': 'For our numerical experiments, we take [MATH] as the discrete gradient operator as appropriate sparsifying transform for piecewise smooth MNP distributions.', '1805.01605-2-33-1': 'The additional regularizer is taken as the indicator function [MATH] of the convex set [EQUATION] defined by [MATH] if [MATH] and [MATH].', '1805.01605-2-33-2': 'It guarantees non-negativity and boundedness.', '1805.01605-2-33-3': 'With the above choices, the Tikhonov functional [REF] reduces to total variation (TV) regularization with positivity constraint.', '1805.01605-2-33-4': 'The minimization of [MATH] required for implementing Algorithm [REF] is a TV denoising step and again performed by the Douglas-Rachford algorithm using the decomposition in [MATH] and [MATH].', '1805.01605-2-34-0': 'Under the reasonable assumptions that the regularizer [MATH] is lower semicontinuous and convex, and that the sparse Tikhonov functional [MATH] is coercive, the sequence [MATH] generated by Algorithm [REF] is known to converge to a minimizer of [REF]; see [CITATION].', '1805.01605-2-34-1': 'Recall that the functional [MATH] is called lower semicontinuous if [MATH] for any [MATH] and any sequence [MATH] converging to [MATH].', '1805.01605-2-34-2': 'Note that Algorithm [REF] performs implicit steps with respect to the residual functional [MATH], which we found to have much faster convergence than forward-backward splitting algorithm [CITATION] [EQUATION] where [MATH] are auxiliary quantities and [MATH] the reconstructed MNPs.', '1805.01605-2-35-0': '# Numerical results', '1805.01605-2-36-0': 'For the following numerical simulations, we use a data setup similar to the realization in [CITATION].', '1805.01605-2-36-1': 'For simplification, we consider a two-dimensional setup representing one voxel plane, containing two parallel arrays of detector elements (one measuring the horizontal [MATH] and one measuring the vertical [MATH] component) located outside a quadratic region of interest.', '1805.01605-2-36-2': 'Circular shaped activation coils are arranged in U-form around the region of interest, all having the same normal vector [MATH].', '1805.01605-2-36-3': 'The used arrangement of sensor and coil locations for the full measurement data setup is shown in Figure [REF].', '1805.01605-2-36-4': 'For our simulations, we choose a discretization of imaging space into [MATH] voxels covering a region of interest of [MATH] cm^2.', '1805.01605-2-36-5': 'The data are generated for [MATH] positions and full measurement data correspond to [MATH] activation coils outside the region of interest.', '1805.01605-2-37-0': '## Forward computations', '1805.01605-2-38-0': 'To set up the forward model [REF], [REF], [REF], we first have to compute the magnetic fields generated by each coil.', '1805.01605-2-38-1': 'For that purpose, we follow the approach of [CITATION], where any circular activation coil is approximated by short line segments, each carrying the same current [MATH].', '1805.01605-2-38-2': 'Every coil is modeled by 45 line segments for numerical calculations of the magnetic field.', '1805.01605-2-38-3': 'This is considered to be a sufficiently accurate approximation since the deviation of only 36 line segments from the analytical solution was shown to be already below 1 in [CITATION].', '1805.01605-2-38-4': 'Using this approximation, the induced magnetic field at the voxel center [MATH] can be computed by (see [CITATION]) [EQUATION] where [MATH] and [MATH] are the distance vectors between the voxel center [MATH] and the beginning and end points of the [MATH]th line segment, respectively.', '1805.01605-2-38-5': 'For the presented numerical computations, we use a coil diameter of [MATH], illustrating an almost point like coil.', '1805.01605-2-38-6': 'Having computed the activation field [MATH], we compute the entries of the Lead field matrix according to [REF].', '1805.01605-2-38-7': 'By activating the coils sequentially, we obtain the full measurement data Lead field matrix.', '1805.01605-2-39-0': 'For the presented results, we use three different magnetic particle distributions which, together with the corresponding measurements full data, are shown in Figure [REF].', '1805.01605-2-39-1': 'Any column in the measurement data (in Figure [REF] and below) corresponds to a single activation pattern and contains the data of all detectors.', '1805.01605-2-39-2': 'The phantoms are rescaled to have maximum value 1 and the forward operator [MATH] is rescaled to have matrix norm [MATH].', '1805.01605-2-39-3': 'To all data we have added additive Gaussian noise amounting to a signal-to-noise (SNR) of 80dB.', '1805.01605-2-39-4': 'The corresponding reconstruction [EQUATION] with standard quadratic Tikhonov regularization (penalized least squares) using regularization parameter [MATH] are shown in the bottom row of Figure [REF].', '1805.01605-2-39-5': 'They can be seen as benchmark for the more sophisticated CS reconstructions applied to less data presented below.', '1805.01605-2-39-6': 'The regularization parameter has been chosen empirically as the trade-off between stability and smoothing.', '1805.01605-2-39-7': 'Taking a smaller regularization parameter would not suppress the noise (amplification) well enough, whereas a larger regularization parameter yields to overshooting.', '1805.01605-2-39-8': 'There are many strategies for selecting the regularization parameter analytically [CITATION].', '1805.01605-2-39-9': 'An example is the L-curve method, where the resi+dual [MATH] is plotted against the solution norm [MATH] both on a logarithmic scale.', '1805.01605-2-39-10': 'The L-curve method predicts the graph to be an L-shaped curve and the regularization parameter should be taken close to the corner of the L-shaped curve.', '1805.01605-2-39-11': 'The L-curves for the considered phantoms are plotted in Figure [REF].', '1805.01605-2-39-12': 'Indeed, the selected [MATH] are close to the corner in each case.', '1805.01605-2-39-13': 'Strictly taken, the L-curve method predicts a larger regularization which we however found to yield to a slight over-smoothing and a larger reconstruction error.', '1805.01605-2-40-0': 'The CS forward matrix [MATH] is computed by multiplying the full Lead field matrix with [MATH].', '1805.01605-2-40-1': 'CS measurements have been generated in two random ways and one deterministic way.', '1805.01605-2-40-2': 'In the random case, [MATH] is taken either as Bernoulli matrix having entries [MATH] appearing with equal probability, or a Gaussian matrix consisting of i.i.d. [MATH]-Gaussian random variables in each entry.', '1805.01605-2-40-3': 'The deterministic sparse sampling is performed by choosing [MATH] equispaced coil activations.', '1805.01605-2-41-0': '## Reconstruction results', '1805.01605-2-42-0': 'The system matrix [MATH] is rescaled to have matrix norm 1 and we use the parameter setting [MATH], [MATH], [MATH], [MATH] and [MATH].', '1805.01605-2-42-1': 'Note that the parameter [MATH] has a similar role for each iterative step as in quadratic Tikhonov regularization [REF].', '1805.01605-2-42-2': 'Choosing it too small causes noise amplification.', '1805.01605-2-42-3': 'Choosing it too large causes oversmoothing, which however will be reduced during the iteration.', '1805.01605-2-42-4': 'Therefore, taking it somewhat larger than the value found in Tikhonov case seems reasonable.', '1805.01605-2-42-5': 'The regularization parameter [MATH] has been selected empirically to yield good numerical performance.', '1805.01605-2-43-0': 'We observed that after 50 iterations the reconstructed MNP stagnates which indicates that the iterates are close to the minimizer of the Tikhonov functional.', '1805.01605-2-43-1': 'The reconstructed MNP distributions using Algorithm [REF] from 40 coil activations are shown in Figure [REF].', '1805.01605-2-43-2': 'Each reconstruction takes about 50 seconds in Matlab R 2017a on a MacBook Pro (2016) with 2.9 GHz Intel Core i7 processor.', '1805.01605-2-44-0': 'To quantitatively evaluate the reconstruction results, we compute the relative root mean squared error (RMSE) [MATH], the SNR [MATH], and the Pearson correlation coefficient [MATH] with [MATH] denoting the covariance and [MATH] the variance.', '1805.01605-2-44-1': 'The results are shown in Table [REF].', '1805.01605-2-44-2': 'For all phantoms and for any evaluation metric, the proposed algorithm clearly outperforms quadratic Tikhonov regularization.', '1805.01605-2-44-3': 'More specifically, the RSME for the proposed algorithm applied with 40 activations yields half of the RSME compared to quadratic Tikhonov regularization with 120 activations.', '1805.01605-2-44-4': 'For the CS-phantom and the smiley-phantom, the deterministic scheme performs best and decreases the reconstruction error by more than [MATH] compared to the random schemes.', '1805.01605-2-44-5': 'For the tumor phantom, all sampling schemes yield comparable performance.', '1805.01605-2-45-0': 'Figure [REF] shows a convergence study (for the correlation and the RMSE) in dependence of the number of activations.', '1805.01605-2-45-1': 'We see that the Douglas-Rachford algorithm constantly outperforms Tikhonov regularization and significantly reduces the RSME and increases the correlation.', '1805.01605-2-45-2': 'For the CS and the smiley phantom, the deterministic sampling pattern has a significantly smaller RMSE than the random schemes.', '1805.01605-2-45-3': 'We address this behavior to the strong smoothing effect of the forward operator, which removes most high frequency components in the data.', '1805.01605-2-45-4': 'In the full forward matrix [MATH], the incoherence contained in the measurements matrix [MATH] is annihilated by the smoothing effect of [MATH].', '1805.01605-2-45-5': 'Finding particular activation patterns that outperform the sparse sampling scheme is a nontrivial issue and will be investigated in future work.', '1805.01605-2-46-0': '# Conclusion', '1805.01605-2-47-0': 'The use of multiple coil activation patterns in magnetorelaxometry imaging is time consuming and requires performing several consecutive measurements.', '1805.01605-2-47-1': 'It is therefore desirable to make the number of coil activations as small as possible, while keeping high spatial resolution.', '1805.01605-2-47-2': 'For that purpose, we investigated CS strategies in this paper.', '1805.01605-2-47-3': 'We compared Gaussian random activations, Bernoulli activations and deterministic sparse sampling schemes.', '1805.01605-2-47-4': 'For actual image reconstruction, we applied Douglas-Rachford splitting to the sparse Tikhonov functional; see Algorithm [REF].', '1805.01605-2-47-5': 'For a small number of coil activations the random schemes slightly outperform the deterministic scheme for the smiley-phantom and tumor phantom.', '1805.01605-2-47-6': 'For a large number of activations, the deterministic sampling scheme clearly performed better than the random sampling patterns.', '1805.01605-2-47-7': 'We explain this behavior by the severe ill-conditioning of the Lead field matrix, which is more severe for a large number of activations.', '1805.01605-2-47-8': 'Only for a small number of activations, where the resolution is poor anyway, the incoherence at the low frequencies is not destroyed by the smoothing effect of the Lead field matrix.', '1805.01605-2-47-9': 'The reconstruction results clearly demonstrate the proposed Algorithm [REF] significantly outperforms standard algorithms such as Tikhonov regularization.', '1805.01605-2-47-10': 'Depending on the complexity of the phantom between 6 (tumor) and 30 activations (CS) are sufficient for the used framework and further increasing the number of activations only decreases the RMSE by a few percent; see Figure [REF].', '1805.01605-2-48-0': 'Several interesting research directions following this work are possible.', '1805.01605-2-48-1': 'First, we can replace the inner TV minimization in the single-stage approach by a different algorithm which should accelerate Algorithm [REF].', '1805.01605-2-48-2': 'Second, the derivation of theoretical error estimates for the single-stage approach is of significant interest.', '1805.01605-2-48-3': 'Results in that direction can advice which type of MRX measurements are best to obtain accurate results for certain phantom classes.', '1805.01605-2-48-4': 'Moreover, the derivation of adaptive compressed sensing strategies for online monitoring is of significant interest.', '1805.01605-2-48-5': 'Advising optimal coil activations given previous activations is a practically important challenge that will benefit from theoretical error estimates, numerical simulations as well as real-world experiments.', '1805.01605-2-48-6': 'Such issues will be addressed in future work.', '1805.01605-2-48-7': 'In this paper, we investigated standard random compressed sensing schemes (using Gaussian and Bernoulli activation patterns and sparse sampling).', '1805.01605-2-48-8': 'Using more problem adapted and task oriented measurement design we will expect to derive improved coil activation schemes.'}

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[]

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[['1805.01605-1-48-1', '1805.01605-2-42-0']]

['1805.01605-1-46-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1805.01605

{'1805.01605-3-0-0': 'Magnetorelaxometry imaging is a novel tool for quantitative determination of the spatial distribution of magnetic nanoparticle inside an organism.', '1805.01605-3-0-1': 'The use of multiple excitation patterns has been demonstrated to significantly improve spatial resolution.', '1805.01605-3-0-2': 'However, increasing the number of excitation patterns is considerably more time consuming, because several sequential measurements have to be performed.', '1805.01605-3-1-0': 'In this paper, we use compressed sensing in combination with sparse recovery to reduce the total measurement time and to improve spatial resolution.', '1805.01605-3-1-1': 'For image reconstruction, we propose using the Douglas-Rachford splitting algorithm applied to the sparse Tikhonov functional including a positivity constraint.', '1805.01605-3-1-2': 'Our numerical experiments demonstrate that the resulting algorithm is capable to accurately recover the magnetic nanoparticle distribution from a small number of activation patterns.', '1805.01605-3-1-3': 'For example, our algorithm applied with 10 activations yields half the reconstruction error of quadratic Tikhonov regularization applied with 50 activations, for a tumor-like phantom.', '1805.01605-3-2-0': 'Keywords: Compressed sensing; magnetorelaxometry; image reconstruction; magnetic nanoparticles; multiple excitation; Douglas-Rachford splitting; sparse recovery.', '1805.01605-3-3-0': '# Introduction', '1805.01605-3-4-0': 'Magnetic nanoparticles (MNP) offer a variety of promising biomedical applications.', '1805.01605-3-4-1': 'For example, they can be used as agents for drug delivery or hyperthermia, where the aim is to heat up specific regions inside a biological specimen [CITATION].', '1805.01605-3-4-2': 'These applications require quantitative knowledge of the magnetic nanoparticles distribution for safety and efficacy monitoring.', '1805.01605-3-4-3': 'In this paper, we consider magnetorelaxometry (MRX) imaging, which is a novel and promising non-invasive technique to spatially resolve the location and concentration of MNPs in vivo [CITATION].', '1805.01605-3-4-4': 'It beneficially combines a highly sensitive magnetic measurement technology for MNP imaging with a broad range of parameters and has the potential to image particle distributions in a comparably large volume [CITATION].', '1805.01605-3-5-0': 'In MRX, the magnetic moments of the magnetic nanoparticles are aligned by a magnetic field generated by excitation coils [CITATION].', '1805.01605-3-5-1': 'Therewith, a magnetization can be measured from the MNP.', '1805.01605-3-5-2': 'After switching off the excitation field, the decay of magnetization (relaxation) is recorded, typically by SQUIDs (superconducting quantum interference device), yielding information about the particle concentration and related properties.', '1805.01605-3-5-3': 'If the relaxation is measured by a sensor array, the particle distribution can be reconstructed by inverse imaging methods [CITATION].', '1805.01605-3-6-0': 'In multiple excitation MRX (ME-MRX), several inhomogeneous activation fields generated by an array of excitation coils are applied.', '1805.01605-3-6-1': 'With such a coil array, a variety of inhomogeneous excitation fields can be generated, for example, by switching on the coils in a sequential manner [CITATION].', '1805.01605-3-6-2': 'First experimental realizations of ME-MRX with sequential activation and least squares estimation for imaging have been obtained in [CITATION].', '1805.01605-3-6-3': 'ME-MRX using sequential coil activation and standard image reconstruction techniques requires a large number of measurement cycles and thus a considerable time for data acquisition.', '1805.01605-3-6-4': 'Recently, different approaches have been proposed using advanced excitation schemes [CITATION].', '1805.01605-3-7-0': "A strategy to maintain high resolution while reducing measurement time is via compressed sensing (CS), a new sensing paradigm [CITATION] that allows to capture a high-resolution image (or signal) by using fewer measurements than predicted by Shannon's sampling theorem.", '1805.01605-3-7-1': 'CS replaces point-measurements by general linear measurements, where each measurement consists of a linear combination of the entries of the image of interest.', '1805.01605-3-7-2': 'Recovering the original image is highly under-determined.', '1805.01605-3-7-3': 'The standard theory of CS predicts that under suitable assumptions on the image (sparsity) and the measurement matrix (incoherence), stable image reconstruction is nevertheless possible.', '1805.01605-3-7-4': 'CS has led to several new sampling strategies in medical imaging, for example, for speeding up MRI data acquisition [CITATION], accelerating photoacoustic tomography [CITATION], or completing under-sampled CT images [CITATION].', '1805.01605-3-7-5': 'While these applications involve relatively mildly ill-conditioned problems, ME-MRX constitutes a severely ill-conditioned or ill-posed problem [CITATION].', '1805.01605-3-7-6': 'No standard approaches for image reconstruction combining compressive measurements and such severely ill-posed problems exist.', '1805.01605-3-8-0': 'In this work, we investigate CS techniques for accelerating ME-MRX.', '1805.01605-3-8-1': 'For that purpose, we consider random activations of the coils as well as sparse deterministic activation schemes.', '1805.01605-3-8-2': 'In order to image the MNP distribution from the CS data we propose a sparse reconstruction framework.', '1805.01605-3-8-3': 'We use [MATH]-Tikhonov regularization together with a positivity constraint on the set of reconstructed MNP distributions, and use the Douglas-Rachford splitting algorithm [CITATION] for its minimization.', '1805.01605-3-8-4': 'We evaluate the performance of the resulting sparse recovery algorithm using different phantoms in dependence of the number of activation patterns.', '1805.01605-3-8-5': 'In any of the cases, the sparse recovery algorithm reduces the root mean square error (RSME) by more than [MATH].', '1805.01605-3-8-6': 'For two of the three phantoms (CS and smiley), the deterministic scheme outperforms the random pattern by around [MATH] in terms of reduced RSME.', '1805.01605-3-8-7': 'For the third phantom (tumor), all tested sampling patterns roughly give the same results.', '1805.01605-3-8-8': 'We therefore can propose the sparse sampling pattern among the tested schemes, in combination with the proposed sparse recovery framework.', '1805.01605-3-9-0': '# The forward model in ME-MRX imaging', '1805.01605-3-10-0': 'This section gives a precise formulation of the forward problem of ME-MRX and CS.', '1805.01605-3-10-1': 'Throughout this text we work with a discrete model [CITATION].', '1805.01605-3-10-2': 'A continuous model has recently been presented in [CITATION].', '1805.01605-3-11-0': '## Signal generation', '1805.01605-3-12-0': 'In MRX, an activation field is applied to a region of interest containing MNPs.', '1805.01605-3-12-1': 'We assume this region to be divided into a number of [MATH] quadratic voxels, each represented by its midpoint located at [MATH] and containing a concentration [MATH] of MNPs (compare Figure [REF]).', '1805.01605-3-12-2': 'Due to the presence of the applied field, the MNPs align up and, after the activation field is switched off, produce a relaxation signal.', '1805.01605-3-12-3': 'According to the Biot-Savart law, the overall contribution of the MNP concentrations in the individual voxels to the measured signal at sensor location [MATH] is given by [CITATION] [EQUATION]', '1805.01605-3-12-4': 'Here [MATH] is the activation field, [MATH] is the vector joining [MATH] and [MATH], [MATH] is the normal vector of the sensor at location [MATH], and [MATH] and [MATH] are used to denote the tensor and inner product, respectively.', '1805.01605-3-12-5': 'Recall that the tensor product of two column vectors [MATH] is defined by [MATH] which results in a matrix, whereas the inner product [MATH] results in a scalar.', '1805.01605-3-13-0': 'Assuming the concentrations [MATH] and the activation field [MATH] to be known, the measured data can be computed by [REF].', '1805.01605-3-13-1': 'Collecting the individual measurements in a vector [MATH], we obtain the linear equation [EQUATION]', '1805.01605-3-13-2': 'Here [MATH] is the vector of MNP concentrations, and [MATH] is the system matrix having entries [EQUATION] as derived from relation [REF].', '1805.01605-3-13-3': 'The matrix [MATH] is called Lead field matrix corresponding to the excitation field [MATH].', '1805.01605-3-13-4': 'Equations [REF], [REF] constitute the standard discrete forward model of MRX using a single activation field.', '1805.01605-3-14-0': '## Sequential coil activation', '1805.01605-3-15-0': 'In ME-MRX, the volume is (partially) surrounded by an array of excitation coils and sensor arrays (see Figure [REF]).', '1805.01605-3-15-1': 'The coils can be steered to generate different types of activation fields.', '1805.01605-3-15-2': 'In the following we denote by [MATH] the magnetic fields induced by individual activation of the coils.', '1805.01605-3-15-3': 'Further, we write [MATH] and [MATH] for the data and the Lead field matrix according to [REF] and [REF], corresponding to the activation field [MATH].', '1805.01605-3-15-4': 'The measurements from all activations can be combined to a single equation of the form [EQUATION] where [EQUATION]', '1805.01605-3-15-5': 'We will refer to [MATH] in [REF] as full activation data.', '1805.01605-3-15-6': 'Evaluating [REF] constitutes the forward model of ME-MRX.', '1805.01605-3-15-7': 'The corresponding inverse problem consists in determining the MNP distribution [MATH] from the data vector [MATH] that is additionally corrupted by noise.', '1805.01605-3-15-8': 'Note that Eq. [REF] is known to be severely ill-conditioned as its singular values are rapidly decreasing (see [CITATION]; an analysis in the infinite-dimensional setting has been performed in [CITATION]).', '1805.01605-3-15-9': 'In Figure [REF] we plot the singular values of the forward matrix for full data.', '1805.01605-3-15-10': 'The singular values decay rapidly.', '1805.01605-3-15-11': 'This reflects the severe ill-conditioning and implies that only a small number of singular components can be robustly reconstructed.', '1805.01605-3-16-0': 'The multiple coil setup has been realized in [CITATION] and shown to significantly improve the spatial resolution compared to a single coil activation [CITATION].', '1805.01605-3-16-1': 'However, consecutive activations lead to a more time consuming measurement process.', '1805.01605-3-16-2': 'In order to accelerate data acquisition and to improve imaging resolution, we use CS techniques and develop a Douglas-Rachford based sparse reconstruction algorithm.', '1805.01605-3-17-0': '## Compressive coil activations', '1805.01605-3-18-0': 'The basic idea to employ CS for ME-MRX is to use [MATH] specific coil activations, with [MATH], instead of activating all [MATH] coils in a sequential manner.', '1805.01605-3-18-1': 'Because of linearity, the data corresponding to the [MATH]th random activation pattern are given by [EQUATION]', '1805.01605-3-18-2': 'Here [MATH] models the error in the data and [MATH] is the contribution of coil [MATH] to the [MATH]th activation pattern.', '1805.01605-3-18-3': 'The measurement matrix [EQUATION] represents all activation patters.', '1805.01605-3-18-4': 'Typical choices for [MATH] are Bernoulli or Gaussian matrices, since they are known to guarantee stable recovery of sparse signals [CITATION].', '1805.01605-3-18-5': 'Such kind of measurements can be naturally realized for ME-MRX, by simultaneous activation of several coils.', '1805.01605-3-18-6': 'The corresponding image reconstruction problem consists in recovering the MNP distribution [MATH] from the data in [REF].', '1805.01605-3-19-0': '## Matrix formulation of the reconstruction problem', '1805.01605-3-20-0': 'In order to write the reconstruction problem in a compact form we introduce some additional notation.', '1805.01605-3-20-1': 'First, we define the CS measurement vector as [EQUATION]', '1805.01605-3-20-2': 'Second, we introduce the vectorization or reshaping operator [MATH], that takes a matrix to a vector whose block entries are equal to the transposes of the rows of the matrix: [EQUATION]', '1805.01605-3-20-3': 'We denote by [MATH] the inverse reshaping operation that maps a vector in [MATH] to a matrix in [MATH].', '1805.01605-3-20-4': 'Further, we write [MATH] and [MATH].', '1805.01605-3-21-0': 'With the above notations, the CS data in [REF] can be written in the compact form [EQUATION] where [MATH] is the reshaped Lead field matrix defined by the property [MATH] and [MATH] denotes the noise matrix.', '1805.01605-3-21-1': 'With the Kronecker (or tensor) product [MATH] between the CS activation matrix [MATH] and the identity matrix [MATH], equation [REF] can further be rewritten in the form [EQUATION]', '1805.01605-3-21-2': 'The image reconstruction task in CS ME-MRX consists in recovering the MNP distribution [MATH] from equation [REF] or, equivalently, from equation [REF].', '1805.01605-3-22-0': '# Douglas-Rachford algorithm for CS MRX imaging', '1805.01605-3-23-0': 'In order to recover the MNP distribution, the inverse problem [REF] (or [REF]) has to be solved which is known to be severely ill-posed, see Figure [REF].', '1805.01605-3-23-1': 'We note that no standard approach for image reconstruction combining compressive measurements and severely ill-posed problems exist.', '1805.01605-3-23-2': 'This section is also of interest from a general perspective on CS for inverse problems.', '1805.01605-3-24-0': '## Sparse Tikhonov regularization', '1805.01605-3-25-0': 'We consider [REF] as a single inverse problem with system matrix [MATH].', '1805.01605-3-25-1': 'To address instability and non-uniqueness and to incorporate prior knowledge, we use a sparse regularization approach.', '1805.01605-3-25-2': 'For that purpose, we minimize the generalized Tikhonov functional [CITATION] [EQUATION]', '1805.01605-3-25-3': 'Here [MATH] is a transform that sparsifies the MNP concentration (with the [MATH] the dimension of the transformed domain), [MATH] is an additional regularizer that incorporates additional prior knowledge about the MNP distribution (such as positivity and other convex constraints) and [MATH] is the standard [MATH]-norm defined by [EQUATION]', '1805.01605-3-25-4': 'The non-negative parameters [MATH] and [MATH] allow to balance between the data consistency term [MATH], the sparsity prior [MATH] and the additional regularizer [MATH].', '1805.01605-3-26-0': 'We call [MATH] a sparsifying transform if [MATH] can well be approximated by [MATH]-sparse vectors [MATH], defined by the property that [MATH] has at most [MATH] elements.', '1805.01605-3-26-1': 'This approximation be can be quantified using the [MATH]-term approximation error, [EQUATION]', '1805.01605-3-26-2': 'The [MATH]-term approximation error [MATH] appears as additive term in standard error estimates in compressed sensing theory.', '1805.01605-3-26-3': 'For a mathematically precise discussion of the [MATH]-term approximation error we refer to [CITATION].', '1805.01605-3-27-0': '[Recovery theory for [REF]] The [MATH]-restricted isometry property ([MATH]-RIP) of [MATH] (after appropriate scaling) is defined as the smallest number [MATH] such that for all [MATH]-sparse vectors [MATH] we have [EQUATION]', '1805.01605-3-27-1': 'Roughly spoken, CS theory [CITATION] predicts uniform stable recovery with [REF] for [MATH] and in the sense that it stably recovers any [MATH]-sparse vector provided that the [MATH]-RIP constant of [MATH] is sufficiently small.', '1805.01605-3-28-0': 'Due to the severe ill-conditioning of the Lead field matrix [MATH], the [MATH]-RIP is expected to be at most satisfied when the number [MATH] of voxels is small.', '1805.01605-3-28-1': 'Figure [REF] shows that the singular values of [MATH] with [MATH] are rapidly decaying and therefore the estimate [MATH] for reasonable [MATH] can only be satisfied for signals mainly formed by the first singular vectors.', '1805.01605-3-28-2': 'This implies that even in the case of full measurement data, where [MATH] is the identity matrix, uniform stable reconstruction of all sparse vectors is impossible.', '1805.01605-3-28-3': 'To obtain quantitative error estimates for [REF], stable recovery results for individual elements [CITATION] are a promising alternative.', '1805.01605-3-28-4': 'Such theoretical investigations are beyond the scope of this paper and an interesting line of future research.', '1805.01605-3-29-0': 'Standard CS theory requires orthogonality of the sparsifying transform, which means that [MATH] equals the identity matrix.', '1805.01605-3-29-1': 'Total variation (TV)-regularization [CITATION] uses the gradient as sparsifying transform which is non-orthogonal.', '1805.01605-3-29-2': 'As in other compressed sensing imaging applications [CITATION] we observed TV to outperform orthogonal sparsifying transforms.', '1805.01605-3-29-3': 'We will therefore focus on TV and, due to space limitations, no results for orthogonal bases like wavelets are included.', '1805.01605-3-30-0': '## Douglas-Rachford algorithm', '1805.01605-3-31-0': 'In order to minimize [REF], we propose using the Douglas-Rachford minimization algorithm, which is a backward-backward type splitting method for minimizing the sum [MATH] of two functionals [MATH] and [MATH].', '1805.01605-3-31-1': 'For our purpose we take [EQUATION]', '1805.01605-3-31-2': 'Minimizing [REF] by the Douglas-Rachford algorithm, generates a sequence [MATH] of estimated MNP distributions and auxiliary sequences [MATH], [MATH] as described in Algorithm [REF].', '1805.01605-3-31-3': 'See Section [REF], for a discussion of the role of the parameter [MATH].', '1805.01605-3-32-0': 'Proposed sparse reconstruction algorithm for CS ME-MRX.', '1805.01605-3-32-1': '[1] Select [MATH], and [MATH] Initialize [MATH]', '1805.01605-3-33-0': 'The Douglas-Rachford algorithm is a splitting type algorithm for minimizing [MATH] which alternately performs updates according for [MATH] (in our case, the residual functional or data fitting term) and [MATH] (in our case, the regularizer).', '1805.01605-3-33-1': 'Because the regularizing term is non-smooth, standard methods such as Newton type methods cannot be applied.', '1805.01605-3-33-2': 'Splitting type methods are a natural choice in this case, and the implicit step in line 5 in Algorithm [REF] accounts for the non-smoothness of [MATH].', '1805.01605-3-33-3': 'Other splitting algorithms that are well suited to treat the non-smooth regularizer are the forward-backward splitting algorithm [CITATION] and the Chambolle-Pock algorithm [CITATION].', '1805.01605-3-33-4': 'These algorithms use explicit gradient updates for the data fitting term requiring a matrix inversion.', '1805.01605-3-33-5': 'Therefore, they are not applicable to ME-MRX imaging.', '1805.01605-3-33-6': 'The Douglas-Rachford algorithm performs an implicit update for the data fitting term (line 4 Algorithm [REF]), which is recognized as quadratic Tikhonov regularization step for the given equation and potentially accelerates the iteration.', '1805.01605-3-34-0': 'For our numerical experiments, we take [MATH] as the discrete gradient operator as appropriate sparsifying transform for piecewise smooth MNP distributions.', '1805.01605-3-34-1': 'The additional regularizer is taken as the indicator function [MATH] of the convex set [EQUATION] defined by [MATH] if [MATH] and [MATH].', '1805.01605-3-34-2': 'It guarantees non-negativity and boundedness.', '1805.01605-3-34-3': 'With the above choices, the Tikhonov functional [REF] reduces to total variation (TV) regularization with positivity constraint.', '1805.01605-3-34-4': 'The minimization of [MATH] required for implementing Algorithm [REF] is a TV denoising step and again performed by the Douglas-Rachford algorithm using the decomposition in [MATH] and [MATH].', '1805.01605-3-35-0': 'Under the reasonable assumptions that the regularizer [MATH] is lower semicontinuous and convex, and that the sparse Tikhonov functional [MATH] is coercive, the sequence [MATH] generated by Algorithm [REF] is known to converge to a minimizer of [REF]; see [CITATION].', '1805.01605-3-35-1': 'Recall that the functional [MATH] is called lower semicontinuous if [MATH] for any [MATH] and any sequence [MATH] converging to [MATH].', '1805.01605-3-35-2': 'Note that Algorithm [REF] performs implicit steps with respect to the residual functional [MATH], which we found to have much faster convergence than forward-backward splitting algorithm [CITATION] [EQUATION] where [MATH] are auxiliary quantities and [MATH] the reconstructed MNPs.', '1805.01605-3-36-0': '# Numerical results', '1805.01605-3-37-0': 'For the following numerical simulations, we use a data setup similar to the realization in [CITATION].', '1805.01605-3-37-1': 'For simplification, we consider a two-dimensional setup representing one voxel plane, containing two parallel arrays of detector elements (one measuring the horizontal [MATH] and one measuring the vertical [MATH] component) located outside a quadratic region of interest.', '1805.01605-3-37-2': 'Circular shaped activation coils are arranged in U-form around the region of interest, all having the same normal vector [MATH].', '1805.01605-3-37-3': 'The used arrangement of sensor and coil locations for the full measurement data setup is shown in Figure [REF].', '1805.01605-3-37-4': 'For our simulations, we choose a discretization of imaging space into [MATH] voxels covering a region of interest of [MATH] cm^2.', '1805.01605-3-37-5': 'The data are generated for [MATH] positions and full measurement data correspond to [MATH] activation coils outside the region of interest.', '1805.01605-3-38-0': '## Forward computations', '1805.01605-3-39-0': 'To set up the forward model [REF], [REF], [REF], we first have to compute the magnetic fields generated by each coil.', '1805.01605-3-39-1': 'For that purpose, we follow the approach of [CITATION], where any circular activation coil is approximated by short line segments, each carrying the same current [MATH].', '1805.01605-3-39-2': 'Every coil is modeled by 45 line segments for numerical calculations of the magnetic field.', '1805.01605-3-39-3': 'This is considered to be a sufficiently accurate approximation since the deviation of only 36 line segments from the analytical solution was shown to be already below 1 in [CITATION].', '1805.01605-3-39-4': 'Using this approximation, the induced magnetic field at the voxel center [MATH] can be computed by (see [CITATION]) [EQUATION] where [MATH] and [MATH] are the distance vectors between the voxel center [MATH] and the beginning and end points of the [MATH]th line segment, respectively.', '1805.01605-3-39-5': 'For the presented numerical computations, we use a coil diameter of [MATH], illustrating an almost point like coil.', '1805.01605-3-39-6': 'Having computed the activation field [MATH], we compute the entries of the Lead field matrix according to [REF].', '1805.01605-3-39-7': 'By activating the coils sequentially, we obtain the full measurement data Lead field matrix.', '1805.01605-3-40-0': 'For the presented results, we use three different magnetic particle distributions which, together with the corresponding measurements full data, are shown in Figure [REF].', '1805.01605-3-40-1': 'Any column in the measurement data (in Figure [REF] and below) corresponds to a single activation pattern and contains the data of all detectors.', '1805.01605-3-40-2': 'The phantoms are rescaled to have maximum value 1 and the forward operator [MATH] is rescaled to have matrix norm [MATH].', '1805.01605-3-40-3': 'To all data we have added additive Gaussian noise amounting to a signal-to-noise (SNR) of 80dB.', '1805.01605-3-40-4': 'The corresponding reconstructions [EQUATION] with standard quadratic Tikhonov regularization (penalized least squares) using regularization parameter [MATH] are shown in the bottom row of Figure [REF].', '1805.01605-3-40-5': 'Recall that [MATH] is the full Lead field matrix and [MATH] the full activation data.', '1805.01605-3-40-6': 'Reconstructions with [REF] can be seen as benchmark for the more sophisticated CS reconstructions applied to less data presented below.', '1805.01605-3-40-7': 'The regularization parameter has been chosen empirically as the trade-off between stability and smoothing.', '1805.01605-3-40-8': 'Taking a smaller regularization parameter would not suppress the noise (amplification) well enough, whereas a larger regularization parameter yields to overshooting.', '1805.01605-3-40-9': 'There are many strategies for selecting the regularization parameter analytically [CITATION].', '1805.01605-3-40-10': 'An example is the L-curve method, where the residual [MATH] is plotted against the solution norm [MATH] both on a logarithmic scale.', '1805.01605-3-40-11': 'The L-curve method predicts the graph to be an L-shaped curve and the regularization parameter should be taken close to the corner of the L-shaped curve.', '1805.01605-3-40-12': 'The L-curves for the considered phantoms are plotted in Figure [REF].', '1805.01605-3-40-13': 'Indeed, the selected [MATH] are close to the corner in each case.', '1805.01605-3-40-14': 'Strictly taken, the L-curve method predicts a larger regularization which we however found to yield to a slight over-smoothing and a larger reconstruction error.', '1805.01605-3-41-0': 'The CS forward matrix [MATH] is computed by multiplying the full Lead field matrix with [MATH].', '1805.01605-3-41-1': 'CS measurements have been generated in two random ways and one deterministic way.', '1805.01605-3-41-2': 'In the random case, [MATH] is taken either as Bernoulli matrix having entries [MATH] appearing with equal probability, or a Gaussian matrix consisting of i.i.d. [MATH]-Gaussian random variables in each entry.', '1805.01605-3-41-3': 'The deterministic sparse sampling is performed by choosing [MATH] equispaced coil activations.', '1805.01605-3-42-0': '## Reconstruction results', '1805.01605-3-43-0': 'The system matrix [MATH] is rescaled to have matrix norm 1 and we use the parameter setting [MATH], [MATH], [MATH], [MATH] and [MATH].', '1805.01605-3-43-1': 'Note that the parameter [MATH] has a similar role for each iterative step as in quadratic Tikhonov regularization [REF].', '1805.01605-3-43-2': 'Choosing it too small causes noise amplification.', '1805.01605-3-43-3': 'Choosing it too large causes oversmoothing, which however will be reduced during the iteration.', '1805.01605-3-43-4': 'Therefore, taking it somewhat larger than the value found in Tikhonov case seems reasonable.', '1805.01605-3-43-5': 'The regularization parameter [MATH] has been selected empirically to yield good numerical performance.', '1805.01605-3-44-0': 'We observed that after 50 iterations the reconstructed MNP stagnates which indicates that the iterates are close to the minimizer of the Tikhonov functional.', '1805.01605-3-44-1': 'The reconstructed MNP distributions using Algorithm [REF] from 40 coil activations are shown in Figure [REF].', '1805.01605-3-44-2': 'Each reconstruction takes about 50 seconds in Matlab R 2017a on a MacBook Pro (2016) with 2.9 GHz Intel Core i7 processor.', '1805.01605-3-45-0': 'To quantitatively evaluate the reconstruction results, we compute the relative root mean squared error (RMSE) [MATH], the SNR [MATH], and the Pearson correlation coefficient [MATH] with [MATH] denoting the covariance and [MATH] the variance.', '1805.01605-3-45-1': 'The results are shown in Table [REF].', '1805.01605-3-45-2': 'For all phantoms and for any evaluation metric, the proposed algorithm clearly outperforms quadratic Tikhonov regularization.', '1805.01605-3-45-3': 'More specifically, the RSME for the proposed algorithm applied with 40 activations yields half of the RSME compared to quadratic Tikhonov regularization with 120 activations.', '1805.01605-3-45-4': 'For the CS-phantom and the smiley-phantom, the deterministic scheme performs best and decreases the reconstruction error by more than [MATH] compared to the random schemes.', '1805.01605-3-45-5': 'For the tumor phantom, all sampling schemes yield comparable performance.', '1805.01605-3-46-0': 'Figure [REF] shows a convergence study (for the correlation and the RMSE) in dependence of the number of activations.', '1805.01605-3-46-1': 'We see that the Douglas-Rachford algorithm constantly outperforms Tikhonov regularization and significantly reduces the RSME and increases the correlation.', '1805.01605-3-46-2': 'For the CS and the smiley phantom, the deterministic sampling pattern has a significantly smaller RMSE than the random schemes.', '1805.01605-3-46-3': 'We address this behavior to the strong smoothing effect of the forward operator, which removes most high frequency components in the data.', '1805.01605-3-46-4': 'In the full forward matrix [MATH], the incoherence contained in the measurements matrix [MATH] is annihilated by the smoothing effect of [MATH].', '1805.01605-3-46-5': 'Finding particular activation patterns that outperform the sparse sampling scheme is a nontrivial issue and will be investigated in future work.', '1805.01605-3-47-0': '# Conclusion', '1805.01605-3-48-0': 'The use of multiple coil activation patterns in magnetorelaxometry imaging is time consuming and requires performing several consecutive measurements.', '1805.01605-3-48-1': 'It is therefore desirable to make the number of coil activations as small as possible, while keeping high spatial resolution.', '1805.01605-3-48-2': 'For that purpose, we investigated CS strategies in this paper.', '1805.01605-3-48-3': 'We compared Gaussian random activations, Bernoulli activations and deterministic sparse sampling schemes.', '1805.01605-3-48-4': 'For actual image reconstruction, we applied Douglas-Rachford splitting to the sparse Tikhonov functional; see Algorithm [REF].', '1805.01605-3-48-5': 'For a small number of coil activations, the random schemes slightly outperform the deterministic scheme for the smiley-phantom and tumor phantom.', '1805.01605-3-48-6': 'For a large number of activations, the deterministic sampling scheme clearly performed better than the random sampling patterns.', '1805.01605-3-48-7': 'We explain this behavior by the severe ill-conditioning of the Lead field matrix, which is more severe for a large number of activations.', '1805.01605-3-48-8': 'Only for a small number of activations, where the resolution is poor anyway, the incoherence at the low frequencies is not destroyed by the smoothing effect of the Lead field matrix.', '1805.01605-3-48-9': 'The reconstruction results clearly demonstrate the proposed Algorithm [REF] significantly outperforms standard algorithms such as Tikhonov regularization.', '1805.01605-3-48-10': 'Depending on the complexity of the phantom between 6 (tumor) and 30 activations (CS) are sufficient for the used framework and further increasing the number of activations only decreases the RMSE by a few percent; see Figure [REF].', '1805.01605-3-49-0': 'Several interesting research directions following this work are possible.', '1805.01605-3-49-1': 'First, we can replace the inner TV minimization in the single-stage approach by a different algorithm which should accelerate Algorithm [REF].', '1805.01605-3-49-2': 'Second, the derivation of theoretical error estimates for the single-stage approach is of significant interest.', '1805.01605-3-49-3': 'Results in that direction can advise which type of MRX measurements are best to obtain accurate results for certain phantom classes.', '1805.01605-3-49-4': 'Moreover, the derivation of adaptive compressed sensing strategies for online monitoring is of significant interest.', '1805.01605-3-49-5': 'Advising optimal coil activations given previous activations is a practically important challenge that will benefit from theoretical error estimates, numerical simulations as well as real-world experiments.', '1805.01605-3-49-6': 'Such issues will be addressed in future work.', '1805.01605-3-49-7': 'In this paper, we investigated standard random compressed sensing schemes (using Gaussian and Bernoulli activation patterns and sparse sampling).', '1805.01605-3-49-8': 'Using more problem adapted and task oriented measurement design we expect to derive improved coil activation schemes.'}

0905.1005

{'0905.1005-1-0-0': '15pt An efficient numerical algorithm to evaluate one-loop amplitudes using tensor integrals is presented.', '0905.1005-1-0-1': 'In particular, it is shown by explicit calculations that for ordered QCD amplitudes with a number of external legs up to 10, its performance is competitive with other methods.', '0905.1005-1-1-0': '# Introduction', '0905.1005-1-2-0': 'In order to deal with the data from the experiments at LHC for the study of elementary particles, signals and potential backgrounds for new physics have to be under control at sufficient accuracy [CITATION].', '0905.1005-1-2-1': 'In particular, hard processes with high multiplicities, involving many particles or partons, cannot be neglected.', '0905.1005-1-2-2': 'On top of that, such processes have to be dealt with at the next-to-leading order (NLO) level to, for example, reduce the scale dependence of observables and to have a better description of the shape of their distributions.', '0905.1005-1-3-0': 'An important part of a NLO calculation concerns the one-loop amplitude.', '0905.1005-1-3-1': 'Recently, impressive results have been published for one-loop QCD amplitudes for very high numbers of partons [CITATION].', '0905.1005-1-3-2': 'They were obtained with the so-called unitarity-approach.', '0905.1005-1-3-3': 'Originally restricted to analytical calculations [CITATION], the potential of this method in a numerical approach became, after the crucial input from [CITATION], clear with the work of [CITATION] and [CITATION].', '0905.1005-1-3-4': 'It is considered an alternative to the "traditional" approach involving tensor integrals.', '0905.1005-1-3-5': 'Both approaches expand the one-loop amplitude in terms of a basis set of one-loop functions.', '0905.1005-1-3-6': 'In the unitarity-approach, this set consists of scalar-integrals up to [MATH]-point or [MATH]-point functions, and it aims at determining the coefficients directly.', '0905.1005-1-3-7': 'In the "tensor-approach", the basis set is larger and consists of tensor integrals or their coefficients functions when expanded in terms of Lorentz-covariant objects [CITATION].', '0905.1005-1-3-8': 'Also these basis-functions are eventually calculated by expressing them in terms of a smaller set of scalar-integrals, but this happens in a, for the particular method, universal way, independent of the amplitude.', '0905.1005-1-4-0': 'Multiplicities with up to [MATH] partons as achieved in [CITATION] are unattainable in the tensor-approach because of the asymptotic computational complexity of the latter.', '0905.1005-1-4-1': 'It arises because the basis set contains [MATH]-point functions where [MATH] goes up to the total number of external legs of the amplitude.', '0905.1005-1-4-2': 'Let us make a crude comparison between the unitarity-approach and the tensor-approach of [CITATION] in which the basis set consists of "normal" tensor integrals carrying explicit Lorentz-indices.', '0905.1005-1-4-3': 'A first step in the analysis of the computational complexity of the two methods is the determination of the number of coefficients to be evaluated in case ordered amplitudes have to be calculated.', '0905.1005-1-4-4': 'For the unitarity-approach which determines coefficients up to [MATH]-point functions, it is given by [EQUATION]', '0905.1005-1-4-5': 'The number of tensor integrals is, using the fact that only symmetric tensors have to be considered of a rank not higher than the multiplicity, [EQUATION]', '0905.1005-1-4-6': 'The number of tensor integrals is obviously much larger than the number of scalar functions.', '0905.1005-1-4-7': 'The asymptotic behavior of [MATH] for the tensor integrals is particularly disastrous.', '0905.1005-1-4-8': 'It is a result of the expansion in terms of [MATH]-point integrals.', '0905.1005-1-4-9': 'The accompanying [MATH]-behavior stems from the symmetric tensor components.', '0905.1005-1-4-10': 'Of course this is not the whole story.', '0905.1005-1-4-11': 'Also the operations to be performed in order to determine the coefficients have to be taken into account.', '0905.1005-1-4-12': 'For the unitarity-approach as presented in [CITATION] for example, this leads to a final computational complexity of [MATH].', '0905.1005-1-4-13': 'In this write-up, we will see that, by using recursion on both the tensor integrals and their coefficients, the complexityonly changes from [MATH] of Eq. ([REF]) to [MATH].', '0905.1005-1-4-14': 'The asymptotic complexity is still exponential as opposed to the polynomial complexity of the unitarity-approach, but for moderate values of [MATH] may be competitive, and in fact, we will see that in practice it is up to [MATH].', '0905.1005-1-5-0': 'Besides the computational complexity, also the numerical stability is an important issue concerning one-loop calculations.', '0905.1005-1-5-1': 'It is typically related to inverse Gram-determinants approaching zero.', '0905.1005-1-5-2': 'When using tensor integrals, this issue is isolated to the calculation of the tensor integrals themselves.', '0905.1005-1-5-3': 'In that sense, it allows for universal solutions, and several methods to achieve this exist.', '0905.1005-1-5-4': 'As a last resort, the calculation of the tensor integrals can be performed at higher precision level for phase-space points at which numerical instabilities occur, and the decision to do so can be made at relatively low cost.', '0905.1005-1-5-5': 'In the unitarity-approach, numerical instabilities can show up in the computation of the coefficients for the scalar functions.', '0905.1005-1-5-6': 'At the moment, there is no better cure known than to increase the precision level for the full calculation for phase-space points at which numerical instabilities occur.', '0905.1005-1-5-7': 'Fortunately, also in this case the decision is relatively cheap.', '0905.1005-1-6-0': 'A final issue worth mentioning is the potential towards automation of the method.', '0905.1005-1-6-1': 'The unitarity-approach proofs to be successful in this respect because it optimally allows for the use of existing tree-level machinery related to the calculation of off-shell currents or "sub-amplitudes".', '0905.1005-1-6-2': 'We will see that the method presented here allows for the same.', '0905.1005-1-6-3': 'In particular, it is completely numerical and no computer algebra is involved.', '0905.1005-1-7-0': 'The outline of the paper is as follows.', '0905.1005-1-7-1': 'In Section [REF] the tensor reduction is addressed, and in Section [REF] tensor symmetrization, which is crucial for the efficiency of the presented algorithm.', '0905.1005-1-7-2': 'Recursive relations for one-loop amplitudes are presented in Section [REF], and in Section [REF] results can be found obtained with the help of an explicit implementation of the algorithms in the foregoing sections.', '0905.1005-1-7-3': 'The conclusions in Section [REF] finally close the paper.', '0905.1005-1-8-0': '# Tensor reduction', '0905.1005-1-9-0': 'Tensor integrals are usually calculated using recursive equations relating high-multiplicity and high-rank tensor integrals to lower-multiplicity and lower-rank ones.', '0905.1005-1-9-1': 'Tensor reduction is this formal process, in practice the opposite process, tensor building, is performed.', '0905.1005-1-9-2': 'The multiplicity [MATH] and and rank [MATH] are defined with the formula [EQUATION]', '0905.1005-1-9-3': 'As the formula suggests, we consider tensor integrals defined in [MATH] dimensions, but only with [MATH]-dimensional components of the integration momentum in the numerator.', '0905.1005-1-9-4': 'This will lead to a calculation of the one-loop amplitude within the scheme of [CITATION], which is equivalent to calculations in the unitarity-approach neglecting the so-called [MATH]-term [CITATION].', '0905.1005-1-9-5': "Stated differently, the [MATH] term of a one-loop amplitude, which is shown how to be determined in [CITATION], is exactly the difference between the scheme of [CITATION] and other schemes like 't Hooft-Veltman or FDH.", '0905.1005-1-10-0': 'The computational complexity given in Eq. ([REF]) does not increase considerably when the operations needed to calculate the tensor integrals are taken into account, because for high-[MATH] each integral can be obtained using a fixed number of lower integrals, independent of [MATH] or [MATH].', '0905.1005-1-10-1': 'This can be easily understood as follows.', '0905.1005-1-10-2': 'Using the fact that we can write [EQUATION] we have [EQUATION] where [MATH] is obtained from [MATH] by removing the [MATH]-th denominator.', '0905.1005-1-11-0': 'Choosing [MATH] different vectors [MATH] appearing in the denominators, we get [MATH] relations, enough to determine the [MATH] integrals [MATH] with the first [MATH] Lorentz indices fixed.', '0905.1005-1-11-1': 'So [MATH] tensor integrals can be determined using [MATH] lower integrals.', '0905.1005-1-11-2': 'Although very straightforward, this is numerically not necessarily the best method to calculate tensor integrals, since it involves the inversion of a [MATH]-matrix.', '0905.1005-1-11-3': 'The method presented in [CITATION] involves the square-root of an inverse Gram determinant of only [MATH] vectors.', '0905.1005-1-11-4': 'Also these can be chosen out of [MATH] denominators, and for high [MATH] the probability that a phase-space point is such that all combinations lead to small Gram determinants is rather low.', '0905.1005-1-12-0': 'For low-[MATH] integrals, i.e. for [MATH], the previous statement is obviously not true, but several recipes and their implementations to deal with numerical instabilities exist.', '0905.1005-1-12-1': 'Notice that, in renormalizable gauges, [MATH], so that for low [MATH] also the cost, for example, of converting Passarino-Veltman functions calculated following [CITATION] to tensor integrals like above is acceptable.', '0905.1005-1-12-2': 'In fact, to obtain the results in this write-up, the "Alternative Passarino-Veltman-like reduction" from [CITATION] was used for the [MATH]-point integrals.', '0905.1005-1-12-3': 'It can easily be predicted when this method fails, in which case the method from [CITATION] was used.', '0905.1005-1-12-4': 'For the [MATH]-point functions, conventional Passarino-Veltman reduction was used.', '0905.1005-1-13-0': 'The end-points of the tensor reduction are scalar integrals.', '0905.1005-1-13-1': 'Also these can recursively be reduced further, and eventually be expressed in terms of [MATH]-point scalar functions.', '0905.1005-1-13-2': 'For the application in this write-up, the unitarity-approach as presented in [CITATION] was used to express scalar functions into [MATH]-point functions directly.', '0905.1005-1-13-3': 'This choice, of course, is not in correspondence with the "recursivity philosophy", and in fact it strictly speaking increases the computational complexity to [MATH].', '0905.1005-1-14-0': 'In practice, however, it appears to be rather numerically stable and efficient, in particular because the formulations of [CITATION] and [CITATION] are compatible to large extend, avoiding the re-calculation of some overhead.', '0905.1005-1-14-1': 'Furthermore, for the scalar-functions, no numerator functions have to be evaluated, and only the coefficients for the [MATH]-point functions have to be calculated, avoiding the computationally more challenging issues coming with the method of [CITATION] in general.', '0905.1005-1-15-0': 'The scalar one-loop [MATH]-point, [MATH]-point, [MATH]-point and [MATH]-point functions, finally, were evaluated with OneLOop [CITATION].', '0905.1005-1-15-1': 'Also the tensor [MATH]-point functions were evaluated with this program.', '0905.1005-1-16-0': '# Tensor symmetrization', '0905.1005-1-17-0': 'Also for high-rank tensors, one has to deal with tensor-contractions in the end.', '0905.1005-1-17-1': 'Contracting rank-[MATH] tensors with [MATH] tensor components seems hopeless, and the solution to this problem is tensor symmetrizations.', '0905.1005-1-17-2': 'Here, we use the fact that the tensor integrals Eq. ([REF]) are symmetric, and a tensor integral of rank [MATH] has only [EQUATION] independent components.', '0905.1005-1-17-3': 'So for symmetric tensors, the issue of contraction does not look hopeless at all.', '0905.1005-1-17-4': 'We only have to make sure we can calculate the symmetrized coefficients to be contracted with the tensor integrals directly.', '0905.1005-1-17-5': 'By symmetrization we mean adding tensor-components which are multiplied by the same tensor integral together, so [EQUATION] etc..', '0905.1005-1-17-6': 'In general, a symmetric tensor [MATH] of rank [MATH] with [MATH]-dimensional indices can be represented as [EQUATION] where [MATH] is the number of indices referring to dimension [MATH].', '0905.1005-1-17-7': 'These numbers satisfy [MATH].', '0905.1005-1-17-8': 'Now suppose we have a linear recursive relation between tensors of the type [EQUATION] with [MATH].', '0905.1005-1-17-9': 'The solution is a product of the components of the vectors [MATH] to [MATH].', '0905.1005-1-17-10': 'To calculate the symmetrized product, we can cast the relation in the form [EQUATION] with the convention that [MATH] is identically zero whenever any of the indices is negative.', '0905.1005-1-17-11': 'If we use this relation to calculate all components of the symmetrized product of [MATH] vectors, we need to perform a number of multiplications which is [MATH] times the total number of symmetric tensor components showing up during the calculation, starting from rank [MATH], and is given by [EQUATION]', '0905.1005-1-17-12': 'Notice that we included multiplications with components which are actually [MATH] here.', '0905.1005-1-17-13': 'So the number of multiplications to be performed grows like [MATH].', '0905.1005-1-17-14': 'Comparing this with the number [MATH] of components only for general, non-symmetrized, tensors makes the importance of calculating symmetric tensors directly clear.', '0905.1005-1-18-0': 'The relation expressed by Eq.([REF]) seems rather trivial, but we will see that the tensor components we have to calculate satisfy very similar relations.', '0905.1005-1-18-1': 'Only the simple multiplication is replaced by a more complicated contraction, i.e. the number [MATH] on the l.h.s. of Eq.([REF]) will be replaced by a larger number, but essentially still a constant.', '0905.1005-1-18-2': 'The really crucial point is that, in the above example, we only calculated strictly ordered products of the vectors, i.e. only [MATH], [MATH], [MATH] etc., and not [MATH] for example.', '0905.1005-1-18-3': 'In the following, we will show how to calculate ordered one-loop QCD amplitudes, and we will see that because of the ordering we will only need to calculate strictly ordered tensor components related to given denominator structures.', '0905.1005-1-18-4': 'As a result, the computational complexity mentionened in Section [REF] is only raised from [MATH] in Eq. ([REF]) to [MATH].', '0905.1005-1-18-5': 'The equivalents of the vectors [MATH] above will essentially consist of tree-level off-shell currents, which are computed at a cost of [MATH].', '0905.1005-1-19-0': '# Ordered gluon one-loop amplitudes', '0905.1005-1-20-0': 'In the following, we wil derive recursive relations for the tensors to be contracted with the tensor integrals in order to arrive at one-loop amplitudes.', '0905.1005-1-20-1': 'First we repeat the known tree-level relations to introduce some notation.', '0905.1005-1-21-0': '## Recursive relations for ordered gluon tree-level amplitudes', '0905.1005-1-22-0': 'The recursive relations for tree-level gluon off-shell currents are given by [CITATION] [EQUATION] with [EQUATION] and [EQUATION]', '0905.1005-1-22-1': 'The starting points [MATH] of these recursive equations are the polarizations vectors of the external gluons, and we denote [EQUATION] where [MATH] is the momentum of gluon [MATH].', '0905.1005-1-22-2': 'For [MATH] we define [MATH].', '0905.1005-1-22-3': 'If [MATH], then [EQUATION] is the tree-level color-ordered amplitude for gluon [MATH] to [MATH].', '0905.1005-1-22-4': 'The full tree-level amplitude for the [MATH] gluons is then given by [CITATION] [EQUATION] where [MATH] are the color indices of the gluons and [MATH] are the generators of [MATH].', '0905.1005-1-22-5': 'The sum is over all permutations of the gluons except the cyclic permutations.', '0905.1005-1-22-6': 'The off-shell currents satisfy [MATH] , and the three-point vertex can be reduced to [EQUATION]', '0905.1005-1-22-7': 'The recursive equation may be represented diagrammatically by [EQUATION]', '0905.1005-1-22-8': 'Even for a diagrammatic representation this formula is rather rudimentary, but it encodes enough information for our purpose.', '0905.1005-1-22-9': 'For a more detailed description of the recursive relation, we prefer to refer to Eq. ([REF]) instead of dressing up the diagrammatic representation.', '0905.1005-1-23-0': '## Recursive relations for ordered gluon one-loop amplitudes', '0905.1005-1-24-0': 'A so called color decomposition as in Eq. ([REF]) also exists for one-loop amplitudes [CITATION], and is given by [EQUATION] where [MATH] is the subset of [MATH] that leaves the corresponding double trace structure invariant.', '0905.1005-1-24-1': 'The objects [MATH] are called primitive amplitudes.', '0905.1005-1-24-2': 'They only receive contributions from diagrams with a particular ordering of the gluons.', '0905.1005-1-24-3': 'The partial amplitudes [MATH] can be calculated as linear combinations of permutations of the primitive amplitudes.', '0905.1005-1-25-0': 'Given the definition of the primitive amplitudes, one can write down a recursive relation for off-shell currents from which the primitive amplitudes can be constructed following a relation like Eq.([REF]).', '0905.1005-1-25-1': 'The blobs in the diagrammatic equation Eq.([REF]) represent off-shell currents consisting of sums of tree-level diagrams.', '0905.1005-1-25-2': 'We represent off-shell currents consisting of diagrams containing exactly [MATH] loop by a blob with a hole, and we have', '0905.1005-1-26-0': 'Concerning the first line, it is clear that, since the result may only consist of one-loop diagrams, exactly one blob with a hole must be connected to a vertex.', '0905.1005-1-26-1': 'Since we are considering ordered amplitudes, all distributions of the blob with a hole over the different legs of a vertex have to be represented separately.', '0905.1005-1-27-0': 'The actual loops are generated in the second line of Eq. ([REF]).', '0905.1005-1-27-1': 'Both the second and the third term on this line have to be added explicitly because of the ordering.', '0905.1005-1-27-2': 'These loops are constructed from tree-level off-shell currents with one auxiliary gluon with momentum and polarization vector, say, [MATH] and [MATH] respectively.', '0905.1005-1-27-3': 'The index [MATH] runs from [MATH] to [MATH] such that [EQUATION]', '0905.1005-1-27-4': 'This gluon is supposed to be virtual, and thus off-shell.', '0905.1005-1-27-5': 'The polarization vector has no real physical meaning, and just plays the role of the end-point of the gluonic line.', '0905.1005-1-27-6': 'We will now introduce objects [MATH] including this auxiliary gluon through the formal relations [EQUATION]', '0905.1005-1-27-7': 'The seemingly superfluous momentum shift [MATH] in the first line is to make sure that only inverse denominators of the form [MATH] appear in the calculation, and not for example [MATH].', '0905.1005-1-27-8': 'This also means that the auxiliary external gluon in [MATH] is carrying momentum [MATH] instead of [MATH].', '0905.1005-1-27-9': 'At this point, the question is how to assign a meaning to the relations above, and we will explain this in the following.', '0905.1005-1-28-0': 'Since we are interested only in the contribution of ordered one-loop diagrams, the auxiliary gluon with momentum [MATH] must be the first one', '0905.1005-1-29-0': 'for every off-shell current [MATH], so these off-shell currents satisfy [EQUATION]', '0905.1005-1-29-1': 'More explicitly, the relation is [EQUATION]', '0905.1005-1-30-0': 'Notice that the sum over [MATH] starts with [MATH]: the case that [MATH] does not contain any of the gluons [MATH] to [MATH] and for which it is given by [EQUATION] for every [MATH].', '0905.1005-1-30-1': 'Introducing the symbol [EQUATION] we can separate the [MATH]-dependent part of the [MATH]-point vertex and write [EQUATION]', '0905.1005-1-31-0': 'From these recursive equations, we can see that [MATH] can be expressed as follows [EQUATION] where [MATH] is the number of elements in [MATH], which is a subset of the set [MATH] containing at least [MATH] and [MATH].', '0905.1005-1-31-1': 'The tensors [MATH] do not depend on [MATH].', '0905.1005-1-31-2': 'As an explicit example, we can write', '0905.1005-1-32-0': 'With this observation, we can assign a meaning to the relations of Eq.([REF]) as follows.', '0905.1005-1-32-1': 'Given the tensor integrals [EQUATION] for which the numerator only contains [MATH]-dimensional components of [MATH], we define the object [EQUATION] which does not depend on [MATH], and assign [EQUATION]', '0905.1005-1-32-2': 'This will lead to a calculation of the one-loop amplitude within the scheme of [CITATION].', '0905.1005-1-32-3': 'We cannot use the tensors [MATH] directly to define the first line of Eq. ([REF]), and we will discuss this below.', '0905.1005-1-33-0': 'First, however, we need to answer the question how to calculate the tensors [MATH].', '0905.1005-1-33-1': 'Obviously, from Eq. ([REF]) we can derive recursive equations for them.', '0905.1005-1-33-2': 'Writing [MATH], so the largest two elements of [MATH] are [MATH], we find [EQUATION] where the third line is absent for the case [MATH], and the second line is absent for the case [MATH].', '0905.1005-1-33-3': 'As the starting points of the relations we define [EQUATION] for any [MATH].', '0905.1005-1-34-0': 'Let us address the discussion about the computational complexity in Section [REF] and compare Eq.([REF]) with Eq.([REF]).', '0905.1005-1-34-1': 'The first difference is that Eq.([REF]) has tensors of rank [MATH] also on the r.h.s..', '0905.1005-1-34-2': 'Secondly, Eq.([REF]) involves the contraction with index [MATH] instead of a simple multiplication.', '0905.1005-1-34-3': 'These differences cause the constant with value [MATH] in Eq.([REF]) for the computational complexity to increase.', '0905.1005-1-34-4': 'Most important for the argument stated in Section [REF], however, is the similarity between the two equations regarding the strict ordering.', '0905.1005-1-34-5': 'Only tensors following the strict ordering of [MATH] are involved in Eq.([REF]).', '0905.1005-1-35-0': 'In order to deal with the first line of Eq.([REF]), we introduce the objects [EQUATION]', '0905.1005-1-35-1': 'With the help of the symbol [EQUATION] we can separate the [MATH]-dependent part of the [MATH]-point vertex again and write [EQUATION] and express [EQUATION] with [EQUATION]', '0905.1005-1-35-2': 'The second term on the r.h.s. is absent for the case [MATH] and the first one is absent for the case [MATH].', '0905.1005-1-35-3': 'Now we simply assign [EQUATION]', '0905.1005-1-35-4': 'We can now write down the diagrammatic relation Eq.([REF]) explicitly.', '0905.1005-1-35-5': 'Denoting a one-loop off-shell current by [MATH], we have', '0905.1005-1-36-0': 'So the program to calculate these one-loop off-shell currents is to', '0905.1005-1-37-0': 'calculate the tree-level off-shell currents [MATH] with recursive equation Eq. ([REF]); calculate the tensor integrals [MATH] defined in Eq. ([REF]); calculate the tensors [MATH] using Eq. ([REF]) and the tensors [MATH] using Eq. ([REF]); calculate the objects [MATH] following Eq. ([REF]) and [MATH] following Eq. ([REF]); solve Eq. ([REF]) recursively.', '0905.1005-1-38-0': 'The one-loop amplitude finally is given by [EQUATION]', '0905.1005-1-39-0': '## Including ghost- and fermion-loops', '0905.1005-1-40-0': 'In order to arrive at gauge-invariant amplitudes, ghost contributions have to be included.', '0905.1005-1-40-1': 'Also, one could want to include fermion-loops.', '0905.1005-1-40-2': 'Then, Eq. ([REF]) has to be extended to [EQUATION]', '0905.1005-1-41-0': 'The factor [MATH] for the ghost contribution is included instead of a separation between the ghost-and anti-ghost contribution.', '0905.1005-1-41-1': 'For the fermion loops something similar can be done.', '0905.1005-1-41-2': "Instead of including both orientations, one may, according to Furry's theorem, multiply loops with even number of propagators with [MATH] and discard loops with odd number of propagators.", '0905.1005-1-41-3': 'Because we use tensor integrals, which are explicitly labelled by the propagators, this can actually be easily accommodated for.', '0905.1005-1-42-0': 'The calculation of the ghost-tensors is rather trivial once the calculation of the gluonic ones are understood.', '0905.1005-1-42-1': 'We prefer to focus the attention on the fermion-loops.', '0905.1005-1-42-2': 'The main difference is that for these, the [MATH]-dependence of the numerators in the off-shell currents with an auxiliary fermion comes from the fermion propagators instead of the vertices.', '0905.1005-1-42-3': 'The off-shell currents, or "off-shell spinors", with an auxiliary fermion satisfy [EQUATION] where [MATH] denote the spinor-indices.', '0905.1005-1-42-4': 'More explicitly, the relation can be written as [EQUATION] where implicit summation also over spinor-indices is understood.', '0905.1005-1-42-5': 'The [MATH]-dependent part on the r.h.s. can be isolated even more straightforwardly than in the gluon case.', '0905.1005-1-42-6': 'We write [EQUATION] and find that [EQUATION]', '0905.1005-1-42-7': 'The contribution from the fermion loops to the integrands of the one-loop gluon off-shell currents are given by [EQUATION]', '0905.1005-1-42-8': 'Remember that [MATH] already contains the denominator of the propagator factor on the r.h.s..', '0905.1005-1-42-9': 'Now we write [EQUATION] with [EQUATION]', '0905.1005-1-42-10': 'The integrated contributions from the fermion loops to the one-loop gluon off-shell currents are then given by [EQUATION]', '0905.1005-1-43-0': '## The [MATH]-term', '0905.1005-1-44-0': 'Finally, to obtain gauge-invariant results, the [MATH]-term [CITATION] has to be included.', '0905.1005-1-44-1': "The necessary extra vertices are given by [EQUATION] where [MATH] is then number of fermions, and [MATH] in the 't Hooft-Veltman scheme and [MATH] in the FDH scheme.", '0905.1005-1-44-2': 'The off-shell currents [MATH] containing graphs with exactly [MATH] such vertex satisfy the recursive equation', '0905.1005-1-45-0': '# Results', '0905.1005-1-46-0': 'The presented algorithm has been implemented in a Fortran77 program which, first of all, reproduces all the numeric results given in [CITATION] for multi-gluon amplitudes up to [MATH] gluons to at least [MATH] decimals precision (Appendix [REF]).', '0905.1005-1-47-0': 'Secondly an analysis of the accuracy like in [CITATION] and [CITATION] has been performed, which makes use of the existence of a simple formula for the divergent part of color-ordered one-loop gluon amplitudes within dimensional regularization [CITATION].', '0905.1005-1-47-1': 'It is given in [CITATION] as [EQUATION] where [MATH] and [MATH] is the dimension, [MATH] is the dimensional scale.', '0905.1005-1-47-2': 'Since the divergent part of the one-loop amplitude should also be obtained by using the divergent parts of the initial scalar functions as the starting points in the calculation of the tensor integrals, there is the opportunity to compare the two and assess the accuracy of the latter.', '0905.1005-1-47-3': 'The left of Fig. [REF] gives the distribution of the quantity [EQUATION] where [MATH] refers to the coefficient of [MATH] in Eq. ([REF]), and [MATH] refers to this coefficient calculated with the program presented in this write-up.', '0905.1005-1-47-4': 'The distribution is obtained from a large sample of uniformly distributed phase-space points with the same kinematical cuts as in [CITATION] being [EQUATION] where [MATH] is the rapidity of gluon [MATH], [MATH] its transverse momentum and [MATH] its azimuthal angle, all with respect to the axis of the incoming gluons.', '0905.1005-1-47-5': 'Also helicity configurations where sampled uniformly distributed, only avoiding configurations for which the tree-level amplitude vanishes.', '0905.1005-1-47-6': 'The distributions for a total number of [MATH], [MATH] and [MATH] gluons are shown for calculations at the double precision level.', '0905.1005-1-47-7': 'We see a behavior compatible with [CITATION] at the double precision level, and slightly better than [CITATION] at double precision level.', '0905.1005-1-48-0': 'Another hint at the accuracy of the program can be given by the extend at which gauge invariance is satisfied.', '0905.1005-1-48-1': 'A one-loop amplitude should vanish whenever the polarization vector of any of the external gluons is replaced by the momentum of that gluon.', '0905.1005-1-48-2': 'The quantity [EQUATION] may serve as a measure of the number of decimals being eliminated by performing such a replacement in a numerical calculation, which then again may give an estimate of the accuracy.', '0905.1005-1-48-3': 'The right of Fig. [REF] presents the distribution of (the finite part of) this quantity, obtained from the same sample of phase-space points as before, now however including helicity configurations for which the tree-level amplitude vanishes.', '0905.1005-1-48-4': 'The distributions are compatible with the ones on the left.', '0905.1005-1-49-0': 'In Table [REF] the typical cpu-times [MATH] are given needed for [MATH] evaluation of the one-loop amplitude for a number of gluons from [MATH] to [MATH] on a 2.80GHz Intel Xeon processor.', '0905.1005-1-49-1': 'They are determined by taking the average over the evaluations for a large number of different phase-space points.', '0905.1005-1-49-2': 'The numbers are roughly comparable with those in [CITATION], but do show a worse behavior as function of the number of external particles.', '0905.1005-1-49-3': 'The numbers for [MATH] seem to converge, supporting the statements in Section [REF] about the computational complexity being [MATH].', '0905.1005-1-49-4': 'The numbers for [MATH], where [MATH] is the cpu-time needed for [MATH] evaluation of the tree-level amplitude, give a machine-independent measure of the computational cost.', '0905.1005-1-49-5': 'Notice that they increase as [MATH], consistent with the computational complexities of [MATH] for the one-loop amplitude, and [MATH] for the tree-level amplitude.', '0905.1005-1-50-0': '# Conclusion'}

{'0905.1005-2-0-0': '15pt An efficient numerical algorithm to evaluate one-loop amplitudes using tensor integrals is presented.', '0905.1005-2-0-1': 'In particular, it is shown by explicit calculations that for ordered QCD amplitudes with a number of external legs up to 10, its performance is competitive with other methods.', '0905.1005-2-1-0': '# Introduction', '0905.1005-2-2-0': 'In order to deal with the data from the experiments at LHC for the study of elementary particles, signals and potential backgrounds for new physics have to be under control at sufficient accuracy [CITATION].', '0905.1005-2-2-1': 'In particular, hard processes with high multiplicities, involving many particles or partons, cannot be neglected.', '0905.1005-2-2-2': 'On top of that, such processes have to be dealt with at the next-to-leading order (NLO) level to, for example, reduce the scale dependence of observables and to have a better description of the shape of their distributions.', '0905.1005-2-3-0': 'An important part of a NLO calculation concerns the one-loop amplitude.', '0905.1005-2-3-1': 'Recently, impressive results have been published for one-loop QCD amplitudes for very high numbers of partons [CITATION].', '0905.1005-2-3-2': 'They were obtained with the so-called unitarity-approach.', '0905.1005-2-3-3': 'Originally restricted to analytical calculations [CITATION], the potential of this method in a numerical approach became, after the crucial input from [CITATION], clear with the work of [CITATION] and [CITATION].', '0905.1005-2-3-4': 'It is considered an alternative to the "traditional" approach involving tensor integrals.', '0905.1005-2-3-5': 'Both approaches expand the one-loop amplitude in terms of a basis set of one-loop functions.', '0905.1005-2-3-6': 'In the unitarity-approach, this set consists of scalar-integrals up to [MATH]-point or [MATH]-point functions, and it aims at determining the coefficients directly.', '0905.1005-2-3-7': 'In the "tensor-approach", the basis set is larger and consists of tensor integrals or their coefficients functions when expanded in terms of Lorentz-covariant objects [CITATION].', '0905.1005-2-3-8': 'Also these basis-functions are eventually calculated by expressing them in terms of a smaller set of scalar-integrals, but this happens in a, for the particular method, universal way, independent of the amplitude.', '0905.1005-2-4-0': 'Multiplicities with up to [MATH] partons as achieved in [CITATION] are unattainable in the tensor-approach because of the asymptotic computational complexity of the latter.', '0905.1005-2-4-1': 'It arises because the basis set contains [MATH]-point functions where [MATH] goes up to the total number of external legs of the amplitude.', '0905.1005-2-4-2': 'Let us make a crude comparison between the unitarity-approach and the tensor-approach of [CITATION] in which the basis set consists of "normal" tensor integrals carrying explicit Lorentz-indices.', '0905.1005-2-4-3': 'A first step in the analysis of the computational complexity of the two methods is the determination of the number of coefficients to be evaluated in case ordered amplitudes have to be calculated.', '0905.1005-2-4-4': 'For the unitarity-approach which determines coefficients up to [MATH]-point functions, it is given by [EQUATION]', '0905.1005-2-4-5': 'The number of tensor integrals is, using the fact that only symmetric tensors have to be considered of a rank not higher than the multiplicity, [EQUATION]', '0905.1005-2-4-6': 'Here, all tensors up to the maximal ranks have been included.', '0905.1005-2-4-7': 'The number is obviously much larger than the number of scalar functions.', '0905.1005-2-4-8': 'The asymptotic behavior of [MATH] for the tensor integrals is particularly disastrous.', '0905.1005-2-4-9': 'It is a result of the expansion in terms of [MATH]-point integrals.', '0905.1005-2-4-10': 'The accompanying [MATH]-behavior stems from the symmetric tensor components.', '0905.1005-2-4-11': 'Of course this is not the whole story.', '0905.1005-2-4-12': 'Also the operations to be performed in order to determine the coefficients have to be taken into account.', '0905.1005-2-4-13': 'For the unitarity-approach as presented in [CITATION] for example, this leads to a final computational complexity of [MATH].', '0905.1005-2-4-14': 'In this write-up, we will see that, by using recursion on both the tensor integrals and their coefficients, the complexity as given in Eq. ([REF]) does not change.', '0905.1005-2-4-15': 'Although the asymptotic complexity is exponential as opposed to the polynomial complexity of the unitarity-approach, it may be competitive for moderate values of [MATH], and in fact, we will see that in practice it is up to [MATH].', '0905.1005-2-5-0': 'Besides the computational complexity, also the numerical stability is an important issue concerning one-loop calculations.', '0905.1005-2-5-1': 'It is typically related to inverse Gram-determinants approaching zero.', '0905.1005-2-5-2': 'When using tensor integrals, this issue is isolated to the calculation of the tensor integrals themselves.', '0905.1005-2-5-3': 'In that sense, it allows for universal solutions, and several methods to achieve this exist.', '0905.1005-2-5-4': 'As a last resort, the calculation of the tensor integrals can be performed at higher precision level for phase-space points at which numerical instabilities occur, and the decision to do so can be made at relatively low cost.', '0905.1005-2-5-5': 'In the unitarity-approach, numerical instabilities can show up in the computation of the coefficients for the scalar functions.', '0905.1005-2-5-6': 'At the moment, there is no better cure known than to increase the precision level for the full calculation for phase-space points at which numerical instabilities occur.', '0905.1005-2-5-7': 'Fortunately, also in this case the decision is relatively cheap.', '0905.1005-2-6-0': 'A final issue worth mentioning is the potential towards automation of the method.', '0905.1005-2-6-1': 'The unitarity-approach proofs to be successful in this respect because it optimally allows for the use of existing tree-level machinery related to the calculation of off-shell currents or "sub-amplitudes".', '0905.1005-2-6-2': 'We will see that the method presented here allows for the same.', '0905.1005-2-6-3': 'In particular, it is completely numerical and no computer algebra is involved.', '0905.1005-2-7-0': 'The outline of the paper is as follows.', '0905.1005-2-7-1': 'In Section [REF] the tensor reduction is addressed, and in Section [REF] tensor symmetrization, which is crucial for the efficiency of the presented algorithm.', '0905.1005-2-7-2': 'Recursive relations for one-loop amplitudes are presented in Section [REF], and in Section [REF] results can be found obtained with the help of an explicit implementation of the algorithms in the foregoing sections.', '0905.1005-2-7-3': 'The conclusions in Section [REF] finally close the paper.', '0905.1005-2-8-0': '# Tensor reduction', '0905.1005-2-9-0': 'Tensor integrals are usually calculated using recursive equations relating high-multiplicity and high-rank tensor integrals to lower-multiplicity and lower-rank ones.', '0905.1005-2-9-1': 'Tensor reduction is this formal process, in practice the opposite process, tensor building, is performed.', '0905.1005-2-9-2': 'The multiplicity [MATH] and and rank [MATH] are defined with the formula [EQUATION]', '0905.1005-2-9-3': 'As the formula suggests, we consider tensor integrals defined in [MATH] dimensions, but only with [MATH]-dimensional components of the integration momentum in the numerator.', '0905.1005-2-9-4': "This will lead to a calculation of the one-loop amplitude within the scheme of [CITATION], which asks for a finite counterterm in order to arrive at gauge-invariant results and to cast the result into other schemes like 't Hooft-Veltman or FDH.", '0905.1005-2-9-5': 'This finite counterterm is exactly given by the so-called [MATH]-term, showing up explicitly in the OPP unitarity-approach as part of the rational terms [CITATION], and which is shown how to be determined in [CITATION].', '0905.1005-2-10-0': 'The asymptotic computational complexity given in Eq. ([REF]) does not increase when the operations needed to calculate the tensor integrals are taken into account, because for high-[MATH] each integral can be obtained using a fixed number of lower integrals, independent of [MATH] or [MATH].', '0905.1005-2-10-1': 'This can be easily understood as follows.', '0905.1005-2-10-2': 'Using the fact that we can write [EQUATION] we have [EQUATION] where [MATH] is obtained from [MATH] by removing the [MATH]-th denominator.', '0905.1005-2-11-0': 'Choosing [MATH] different vectors [MATH] appearing in the denominators, we get [MATH] relations, enough to determine the [MATH] integrals [MATH] with the first [MATH] Lorentz indices fixed.', '0905.1005-2-11-1': 'So [MATH] tensor integrals can be determined using [MATH] lower integrals.', '0905.1005-2-11-2': 'Although very straightforward, this is numerically not necessarily the best method to calculate tensor integrals, since it involves the inversion of a [MATH]-matrix.', '0905.1005-2-11-3': 'The method presented in [CITATION] involves the square-root of an inverse Gram determinant of only [MATH] vectors.', '0905.1005-2-11-4': 'Also these can be chosen out of [MATH] denominators, and for high [MATH] the probability that a phase-space point is such that all combinations lead to small Gram determinants is rather low.', '0905.1005-2-12-0': 'For low-[MATH] integrals, i.e. for [MATH], the previous statement is obviously not true, but several recipes and their implementations to deal with numerical instabilities exist.', '0905.1005-2-12-1': 'Notice that, in renormalizable gauges, [MATH], so that for low [MATH] also the cost, for example, of converting Passarino-Veltman functions calculated following [CITATION] to tensor integrals like above is acceptable.', '0905.1005-2-12-2': 'In fact, to obtain the results in this write-up, the "Alternative Passarino-Veltman-like reduction" from [CITATION] was used for the [MATH]-point integrals.', '0905.1005-2-12-3': 'It can easily be predicted when this method fails, in which case the method from [CITATION] was used.', '0905.1005-2-12-4': 'For the [MATH]-point functions, conventional Passarino-Veltman reduction was used.', '0905.1005-2-13-0': 'The end-points of the tensor reduction are scalar integrals.', '0905.1005-2-13-1': 'Also these can recursively be reduced further, and eventually be expressed in terms of [MATH]-point scalar functions.', '0905.1005-2-13-2': 'For the application in this write-up, the unitarity-approach as presented in [CITATION] was used to express scalar functions into [MATH]-point functions directly.', '0905.1005-2-13-3': 'This choice, of course, is not in correspondence with the "recursivity philosophy", and in fact it strictly speaking increases the asymptotic computational complexity to [MATH].', '0905.1005-2-14-0': 'In practice, however, it appears to be rather numerically stable and efficient, in particular because the formulations of [CITATION] and [CITATION] are compatible to large extend, avoiding the re-calculation of some overhead.', '0905.1005-2-14-1': 'Furthermore, for the scalar-functions, no numerator functions have to be evaluated, and only the coefficients for the [MATH]-point functions have to be calculated, avoiding the computationally more challenging issues coming with the method of [CITATION] in general.', '0905.1005-2-15-0': 'The scalar one-loop [MATH]-point, [MATH]-point, [MATH]-point and [MATH]-point functions, finally, were evaluated with OneLOop [CITATION].', '0905.1005-2-15-1': 'Also the tensor [MATH]-point functions were evaluated with this program.', '0905.1005-2-16-0': '# Tensor symmetrization', '0905.1005-2-17-0': 'Also for high-rank tensors, one has to deal with tensor-contractions in the end.', '0905.1005-2-17-1': 'Contracting rank-[MATH] tensors with [MATH] tensor components seems hopeless, and the solution to this problem is tensor symmetrization.', '0905.1005-2-17-2': 'Here, we use the fact that the tensor integrals Eq. ([REF]) are symmetric, and a tensor integral of rank [MATH] has only [EQUATION] independent components.', '0905.1005-2-17-3': 'So for symmetric tensors, the issue of contraction does not look hopeless at all.', '0905.1005-2-17-4': 'We only have to make sure we can calculate the symmetrized coefficients to be contracted with the tensor integrals directly.', '0905.1005-2-17-5': 'By symmetrization we mean adding tensor-components which are multiplied by the same tensor integral together, so [EQUATION] etc..', '0905.1005-2-17-6': 'In general, a symmetric tensor [MATH] of rank [MATH] with [MATH]-dimensional indices can be represented as [EQUATION] where [MATH] is the number of indices referring to dimension [MATH].', '0905.1005-2-17-7': 'These numbers satisfy [MATH].', '0905.1005-2-17-8': 'Now suppose we have a linear recursive relation between tensors of the type [EQUATION] with [MATH].', '0905.1005-2-17-9': 'The solution is a product of the components of the vectors [MATH] to [MATH].', '0905.1005-2-17-10': 'To calculate the symmetrized product, we can cast the relation in the form [EQUATION] with the convention that [MATH] is identically zero whenever any of the indices is negative.', '0905.1005-2-18-0': 'The relation expressed by Eq. ([REF]) seems rather trivial, but we will see that the tensor components we have to calculate satisfy very similar relations.', '0905.1005-2-18-1': 'The main difference will be that the simple multiplications on the r.h.s. will be replaced by more complicated contractions.', '0905.1005-2-18-2': 'This does not have any influence on the possibility to calculate symmetrized components directly, nor on the asymptotic computational complexity.', '0905.1005-2-18-3': 'In fact, like the calculation of the tensor integrals, also the calculation of the tensor components does not increase the asymptotic complexity given in Eq. ([REF]).', '0905.1005-2-18-4': 'This stems from the facts that the number of operations to be performed to calculate a tensor given the lower tensors is constant, and that the number of tensors entering the recursive equation is equal to the number of tensor integrals, i.e., no intermediate "auxiliary" tensors have to be calculated.', '0905.1005-2-18-5': 'The equivalents of the vectors [MATH] above will essentially consist of tree-level off-shell currents, which are computed at a cost of [MATH].', '0905.1005-2-19-0': '# Ordered gluon one-loop amplitudes', '0905.1005-2-20-0': 'In the following, we wil derive recursive relations for the tensors to be contracted with the tensor integrals in order to arrive at one-loop amplitudes.', '0905.1005-2-20-1': 'First we repeat the known tree-level relations to introduce some notation.', '0905.1005-2-21-0': '## Recursive relations for ordered gluon tree-level amplitudes', '0905.1005-2-22-0': 'The recursive relations for tree-level gluon off-shell currents are given by [CITATION] [EQUATION] with [EQUATION] and [EQUATION]', '0905.1005-2-22-1': 'The starting points [MATH] of these recursive equations are the polarizations vectors of the external gluons, and we denote [EQUATION] where [MATH] is the momentum of gluon [MATH].', '0905.1005-2-22-2': 'For [MATH] we define [MATH].', '0905.1005-2-22-3': 'If [MATH], then [EQUATION] is the tree-level color-ordered amplitude for gluon [MATH] to [MATH].', '0905.1005-2-22-4': 'The full tree-level amplitude for the [MATH] gluons is then given by [CITATION] [EQUATION] where [MATH] are the color indices of the gluons and [MATH] are the generators of [MATH].', '0905.1005-2-22-5': 'The sum is over all permutations of the gluons except the cyclic permutations.', '0905.1005-2-22-6': 'The off-shell currents satisfy [MATH] , and the three-point vertex can be reduced to [EQUATION]', '0905.1005-2-22-7': 'The recursive equation may be represented diagrammatically by [EQUATION]', '0905.1005-2-22-8': 'Even for a diagrammatic representation this formula is rather rudimentary, but it encodes enough information for our purpose.', '0905.1005-2-22-9': 'For a more detailed description of the recursive relation, we prefer to refer to Eq. ([REF]) instead of dressing up the diagrammatic representation.', '0905.1005-2-23-0': '## Recursive relations for ordered gluon one-loop amplitudes', '0905.1005-2-24-0': 'A so called color decomposition as in Eq. ([REF]) also exists for one-loop amplitudes [CITATION], and is given by [EQUATION] where [MATH] is the subset of [MATH] that leaves the corresponding double trace structure invariant.', '0905.1005-2-24-1': 'The objects [MATH] are called primitive amplitudes.', '0905.1005-2-24-2': 'They only receive contributions from diagrams with a particular ordering of the gluons.', '0905.1005-2-24-3': 'The partial amplitudes [MATH] can be calculated as linear combinations of permutations of the primitive amplitudes.', '0905.1005-2-25-0': 'Given the definition of the primitive amplitudes, one can write down a recursive relation for off-shell currents from which the primitive amplitudes can be constructed following a relation like Eq.([REF]).', '0905.1005-2-25-1': 'The blobs in the diagrammatic equation Eq.([REF]) represent off-shell currents consisting of sums of tree-level diagrams.', '0905.1005-2-25-2': 'We represent off-shell currents consisting of diagrams containing exactly [MATH] loop by a blob with a hole, and we have', '0905.1005-2-26-0': 'Concerning the first line, it is clear that, since the result may only consist of one-loop diagrams, exactly one blob with a hole must be connected to a vertex.', '0905.1005-2-26-1': 'Since we are considering ordered amplitudes, all distributions of the blob with a hole over the different legs of a vertex have to be represented separately.', '0905.1005-2-27-0': 'The actual loops are generated in the second line of Eq. ([REF]).', '0905.1005-2-27-1': 'Both the second and the third term on this line have to be added explicitly because of the ordering.', '0905.1005-2-27-2': 'These loops are constructed from tree-level off-shell currents with one auxiliary gluon with momentum and polarization vector, say, [MATH] and [MATH] respectively.', '0905.1005-2-27-3': 'The index [MATH] runs from [MATH] to [MATH] such that [EQUATION]', '0905.1005-2-27-4': 'This gluon is supposed to be virtual, and thus off-shell.', '0905.1005-2-27-5': 'The polarization vector has no real physical meaning, and just plays the role of the end-point of the gluonic line.', '0905.1005-2-27-6': 'We will now introduce objects [MATH] including this auxiliary gluon through the formal relations [EQUATION]', '0905.1005-2-27-7': 'The seemingly superfluous momentum shift [MATH] in the first line is to make sure that only inverse denominators of the form [MATH] appear in the calculation, and not for example [MATH].', '0905.1005-2-27-8': 'This also means that the auxiliary external gluon in [MATH] is carrying momentum [MATH] instead of [MATH].', '0905.1005-2-27-9': 'At this point, the question is how to assign a meaning to the relations above, and we will explain this in the following.', '0905.1005-2-28-0': 'Since we are interested only in the contribution of ordered one-loop diagrams, the auxiliary gluon with momentum [MATH] must be the first one', '0905.1005-2-29-0': 'for every off-shell current [MATH], so these off-shell currents satisfy [EQUATION]', '0905.1005-2-29-1': 'More explicitly, the relation is [EQUATION]', '0905.1005-2-30-0': 'Notice that the sum over [MATH] starts with [MATH]: the case that [MATH] does not contain any of the gluons [MATH] to [MATH] and for which it is given by [EQUATION] for every [MATH].', '0905.1005-2-30-1': 'Introducing the symbol [EQUATION] we can separate the [MATH]-dependent part of the [MATH]-point vertex and write [EQUATION]', '0905.1005-2-31-0': 'From these recursive equations, we can see that [MATH] can be expressed as follows [EQUATION] where [MATH] is the number of elements in [MATH], which is a subset of the set [MATH] containing at least [MATH] and [MATH].', '0905.1005-2-31-1': 'The tensors [MATH] do not depend on [MATH].', '0905.1005-2-31-2': 'As an explicit example, we can write', '0905.1005-2-32-0': 'With this observation, we can assign a meaning to the relations of Eq.([REF]) as follows.', '0905.1005-2-32-1': 'Given the tensor integrals [EQUATION] for which the numerator only contains [MATH]-dimensional components of [MATH], we define the object [EQUATION] which does not depend on [MATH], and assign [EQUATION]', '0905.1005-2-32-2': 'This will lead to a calculation of the one-loop amplitude within the scheme of [CITATION].', '0905.1005-2-32-3': 'We cannot use the tensors [MATH] directly to define the first line of Eq. ([REF]), and we will discuss this below.', '0905.1005-2-33-0': 'First, however, we need to answer the question how to calculate the tensors [MATH].', '0905.1005-2-33-1': 'Obviously, from Eq. ([REF]) we can derive recursive equations for them.', '0905.1005-2-33-2': 'Writing [MATH], so the largest two elements of [MATH] are [MATH], we find [EQUATION] where the third line is absent for the case [MATH], and the second line is absent for the case [MATH].', '0905.1005-2-33-3': 'As the starting points of the relations we define [EQUATION] for any [MATH].', '0905.1005-2-34-0': 'Let us address the discussion about the computational complexity in Section [REF] and compare Eq.([REF]) with Eq.([REF]).', '0905.1005-2-34-1': 'The first difference is that Eq.([REF]) has tensors of rank [MATH] also on the r.h.s..', '0905.1005-2-34-2': 'Secondly, Eq.([REF]) involves the contraction with index [MATH] instead of a simple multiplication.', '0905.1005-2-34-3': 'These differences do not influence the asymptotic computational complexity.', '0905.1005-2-34-4': 'Finally, all objects calculated using the recursive relation are needed in Eq.([REF]), and no auxiliary tensors show up whose calculation could influence the asymptotic computational complexity.', '0905.1005-2-35-0': 'In order to deal with the first line of Eq.([REF]), we introduce the objects [EQUATION]', '0905.1005-2-35-1': 'With the help of the symbol [EQUATION] we can separate the [MATH]-dependent part of the [MATH]-point vertex again and write [EQUATION] and express [EQUATION] with [EQUATION]', '0905.1005-2-35-2': 'The second term on the r.h.s. is absent for the case [MATH] and the first one is absent for the case [MATH].', '0905.1005-2-35-3': 'Now we simply assign [EQUATION]', '0905.1005-2-35-4': 'We can now write down the diagrammatic relation Eq.([REF]) explicitly.', '0905.1005-2-35-5': 'Denoting a one-loop off-shell current by [MATH], we have', '0905.1005-2-36-0': 'So the program to calculate these one-loop off-shell currents is to', '0905.1005-2-37-0': 'calculate the tree-level off-shell currents [MATH] with recursive equation Eq. ([REF]); calculate the tensor integrals [MATH] defined in Eq. ([REF]); calculate the tensors [MATH] using Eq. ([REF]) and the tensors [MATH] using Eq. ([REF]); calculate the objects [MATH] following Eq. ([REF]) and [MATH] following Eq. ([REF]); solve Eq. ([REF]) recursively.', '0905.1005-2-38-0': 'The one-loop amplitude finally is given by [EQUATION]', '0905.1005-2-39-0': '## Including ghost and quark loops', '0905.1005-2-40-0': 'In order to arrive at gauge-invariant amplitudes, ghost contributions have to be included.', '0905.1005-2-40-1': 'Also, one could want to include quark loops.', '0905.1005-2-40-2': 'Then, Eq. ([REF]) has to be extended to [EQUATION]', '0905.1005-2-41-0': 'The factor [MATH] for the ghost loop is needed to arrive at gauge-invariant ordered amplitudes.', '0905.1005-2-41-1': 'The calculation of the ghost-tensors is rather trivial once the calculation of the gluonic ones are understood.', '0905.1005-2-41-2': 'We prefer to focus the attention on the quark loops.', '0905.1005-2-41-3': 'The main difference is that for these, the [MATH]-dependence of the numerators in the off-shell currents with an auxiliary quark comes from the quark propagators instead of the vertices.', '0905.1005-2-41-4': 'The off-shell currents, or "off-shell spinors", with an auxiliary quark satisfy [EQUATION] where [MATH] denote the spinor-indices.', '0905.1005-2-41-5': 'More explicitly, the relation can be written as [EQUATION] where implicit summation also over spinor-indices is understood.', '0905.1005-2-41-6': 'Here, we introduced the matrix [EQUATION]', '0905.1005-2-41-7': 'The [MATH]-dependent part on the r.h.s. can be isolated even more straightforwardly than in the gluon case.', '0905.1005-2-41-8': 'We write [EQUATION] and find that [EQUATION]', '0905.1005-2-41-9': 'The contribution from the quark loops to the integrands of the one-loop gluon off-shell currents, without the the factor [MATH], are given by [EQUATION]', '0905.1005-2-41-10': 'Remember that [MATH] already contains the denominator of the propagator factor on the r.h.s..', '0905.1005-2-41-11': 'Now we write [EQUATION] with [EQUATION]', '0905.1005-2-41-12': 'The integrated contributions from the quark loops to the one-loop gluon off-shell currents are then given by [EQUATION]', '0905.1005-2-42-0': '## The [MATH]-term', '0905.1005-2-43-0': 'Finally, to obtain gauge-invariant results, the [MATH]-term [CITATION] has to be included.', '0905.1005-2-43-1': "The necessary extra vertices are given by [EQUATION] where [MATH] is then number of quarks, and [MATH] in the 't Hooft-Veltman scheme and [MATH] in the FDH scheme.", '0905.1005-2-43-2': 'The off-shell currents [MATH] containing graphs with exactly [MATH] such vertex satisfy the recursive equation', '0905.1005-2-44-0': '# Results', '0905.1005-2-45-0': 'The presented algorithm has been implemented in a Fortran77 program which, first of all, reproduces all the numeric results given in [CITATION] for multi-gluon amplitudes up to [MATH] gluons to at least [MATH] decimals precision (Appendix [REF]).', '0905.1005-2-45-1': 'Results involving one massless quark-loop are presented in Appendix [REF].', '0905.1005-2-46-0': 'Secondly an analysis of the accuracy like in [CITATION] and [CITATION] has been performed, which makes use of the existence of a simple formula for the divergent part of color-ordered one-loop gluon amplitudes within dimensional regularization [CITATION].', '0905.1005-2-46-1': 'It is given in [CITATION] as [EQUATION] where [MATH] and [MATH] is the dimension, [MATH] is the dimensional scale.', '0905.1005-2-46-2': 'Since the divergent part of the one-loop amplitude should also be obtained by using the divergent parts of the initial scalar functions as the starting points in the calculation of the tensor integrals, there is the opportunity to compare the two and assess the accuracy of the latter.', '0905.1005-2-46-3': 'The left of Fig. [REF] gives the distribution of the quantity [EQUATION] where [MATH] refers to the coefficient of [MATH] in Eq. ([REF]), and [MATH] refers to this coefficient calculated with the program presented in this write-up.', '0905.1005-2-46-4': 'The distribution is obtained from a large sample of uniformly distributed phase-space points with the same kinematical cuts as in [CITATION] being [EQUATION] where [MATH] is the rapidity of gluon [MATH], [MATH] its transverse momentum and [MATH] its azimuthal angle, all with respect to the axis of the incoming gluons.', '0905.1005-2-46-5': 'Also helicity configurations where sampled uniformly distributed, only avoiding configurations for which the tree-level amplitude vanishes.', '0905.1005-2-46-6': 'The distributions for a total number of [MATH], [MATH] and [MATH] gluons are shown for calculations at the double precision level.', '0905.1005-2-46-7': 'We see a behavior compatible with [CITATION] at the double precision level, and slightly better than [CITATION] at double precision level.', '0905.1005-2-46-8': 'The lower graph at the left of Fig. [REF] shows the same distributions in a log scale for the y-axis in order to highlight the right tail.', '0905.1005-2-47-0': 'Obviously, for a small part of the phase space points the accuracy becomes unacceptably bad.', '0905.1005-2-47-1': 'Results of re-evaluation for these at higher precision are given on the left of Fig. [REF].', '0905.1005-2-47-2': 'Presented are the right tail of the distribution for [MATH] starting from [MATH], and distributions obtained with the same set of phase space points for different options of applying quadruple precision arithmetic.', '0905.1005-2-47-3': 'Evaluating the tensor integrals only (but including the scalar functions) at quadruple precision, the tail is shifted to the left to a large extend, but still phase space points may show up leading to an unacceptably bad accuracy.', '0905.1005-2-47-4': 'The more expensive option of evaluating everything at the quadruple precision level, however, moves the whole tail below [MATH].', '0905.1005-2-48-0': 'Another hint at the accuracy of the program can be given by the extend at which gauge invariance is satisfied.', '0905.1005-2-48-1': 'A one-loop amplitude should vanish whenever the polarization vector of any of the external gluons is replaced by the momentum of that gluon.', '0905.1005-2-48-2': 'The quantity [EQUATION] may serve as a measure of the number of decimals being eliminated by performing such a replacement in a numerical calculation, which then again may give an estimate of the accuracy.', '0905.1005-2-48-3': 'The right of Fig. [REF] presents the distribution of (the finite part of) this quantity, obtained from the same sample of phase-space points as before, now however including helicity configurations for which the tree-level amplitude vanishes.', '0905.1005-2-48-4': 'The distributions are compatible with the ones on the left.', '0905.1005-2-48-5': 'The lower graph at the rights show the same distributions again in a log scale for the y-axis.', '0905.1005-2-48-6': 'The right tails behave similarly to the ones left, and the same holds for the distributions in Fig. [REF].', '0905.1005-2-49-0': 'In Table [REF] the typical cpu-times [MATH] are given needed for [MATH] evaluation of the one-loop amplitude for a number of gluons from [MATH] to [MATH] on a 2.80GHz Intel Xeon processor.', '0905.1005-2-49-1': 'They are determined by taking the average over the evaluations for a large number of different phase-space points.', '0905.1005-2-49-2': 'The numbers are roughly comparable with those in [CITATION], but do show a worse behavior as function of the number of external particles.', '0905.1005-2-49-3': 'The numbers for [MATH] seem to converge, supporting the statements in Section [REF] about the computational complexity being [MATH].', '0905.1005-2-49-4': 'The numbers for [MATH], where [MATH] is the cpu-time needed for [MATH] evaluation of the tree-level amplitude, give a machine-independent measure of the computational cost.', '0905.1005-2-49-5': 'Notice that they increase as [MATH], consistent with the computational complexities of [MATH] for the one-loop amplitude, and [MATH] for the tree-level amplitude.', '0905.1005-2-50-0': '# Conclusion'}

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['0905.1005-1-24-1', '0905.1005-2-24-1'], ['0905.1005-1-24-2', '0905.1005-2-24-2'], ['0905.1005-1-24-3', '0905.1005-2-24-3'], ['0905.1005-1-40-0', '0905.1005-2-40-0'], ['0905.1005-1-40-2', '0905.1005-2-40-2'], ['0905.1005-1-10-2', '0905.1005-2-10-2'], ['0905.1005-1-11-0', '0905.1005-2-11-0'], ['0905.1005-1-11-1', '0905.1005-2-11-1'], ['0905.1005-1-11-2', '0905.1005-2-11-2'], ['0905.1005-1-11-3', '0905.1005-2-11-3'], ['0905.1005-1-11-4', '0905.1005-2-11-4'], ['0905.1005-1-4-0', '0905.1005-2-4-0'], ['0905.1005-1-4-1', '0905.1005-2-4-1'], ['0905.1005-1-4-2', '0905.1005-2-4-2'], ['0905.1005-1-4-3', '0905.1005-2-4-3'], ['0905.1005-1-4-4', '0905.1005-2-4-4'], ['0905.1005-1-4-5', '0905.1005-2-4-5'], ['0905.1005-1-4-7', '0905.1005-2-4-8'], ['0905.1005-1-4-8', '0905.1005-2-4-9'], ['0905.1005-1-4-9', '0905.1005-2-4-10'], ['0905.1005-1-4-10', '0905.1005-2-4-11'], ['0905.1005-1-4-11', '0905.1005-2-4-12'], ['0905.1005-1-4-12', '0905.1005-2-4-13'], ['0905.1005-1-29-0', '0905.1005-2-29-0'], ['0905.1005-1-29-1', '0905.1005-2-29-1'], ['0905.1005-1-6-0', '0905.1005-2-6-0'], ['0905.1005-1-6-1', '0905.1005-2-6-1'], ['0905.1005-1-6-2', '0905.1005-2-6-2'], ['0905.1005-1-6-3', '0905.1005-2-6-3'], ['0905.1005-1-30-0', '0905.1005-2-30-0'], ['0905.1005-1-30-1', '0905.1005-2-30-1'], ['0905.1005-1-13-0', '0905.1005-2-13-0'], ['0905.1005-1-13-1', '0905.1005-2-13-1'], ['0905.1005-1-13-2', '0905.1005-2-13-2'], ['0905.1005-1-47-0', '0905.1005-2-46-0'], ['0905.1005-1-47-1', '0905.1005-2-46-1'], ['0905.1005-1-47-2', '0905.1005-2-46-2'], ['0905.1005-1-47-3', '0905.1005-2-46-3'], ['0905.1005-1-47-4', '0905.1005-2-46-4'], ['0905.1005-1-47-5', '0905.1005-2-46-5'], ['0905.1005-1-47-6', '0905.1005-2-46-6'], ['0905.1005-1-47-7', '0905.1005-2-46-7'], ['0905.1005-1-26-0', '0905.1005-2-26-0'], ['0905.1005-1-26-1', '0905.1005-2-26-1'], ['0905.1005-1-25-0', '0905.1005-2-25-0'], ['0905.1005-1-25-1', '0905.1005-2-25-1'], ['0905.1005-1-25-2', '0905.1005-2-25-2'], ['0905.1005-1-3-0', '0905.1005-2-3-0'], ['0905.1005-1-3-1', '0905.1005-2-3-1'], ['0905.1005-1-3-2', '0905.1005-2-3-2'], ['0905.1005-1-3-3', '0905.1005-2-3-3'], ['0905.1005-1-3-4', '0905.1005-2-3-4'], ['0905.1005-1-3-5', '0905.1005-2-3-5'], ['0905.1005-1-3-6', '0905.1005-2-3-6'], ['0905.1005-1-3-7', '0905.1005-2-3-7'], ['0905.1005-1-3-8', '0905.1005-2-3-8'], ['0905.1005-1-20-0', '0905.1005-2-20-0'], ['0905.1005-1-20-1', '0905.1005-2-20-1'], ['0905.1005-1-17-0', '0905.1005-2-17-0'], ['0905.1005-1-17-2', '0905.1005-2-17-2'], ['0905.1005-1-17-3', '0905.1005-2-17-3'], ['0905.1005-1-17-4', '0905.1005-2-17-4'], ['0905.1005-1-17-5', '0905.1005-2-17-5'], ['0905.1005-1-17-6', '0905.1005-2-17-6'], ['0905.1005-1-17-7', '0905.1005-2-17-7'], ['0905.1005-1-17-8', '0905.1005-2-17-8'], ['0905.1005-1-17-9', '0905.1005-2-17-9'], ['0905.1005-1-17-10', '0905.1005-2-17-10'], ['0905.1005-1-22-0', '0905.1005-2-22-0'], ['0905.1005-1-22-1', '0905.1005-2-22-1'], ['0905.1005-1-22-2', '0905.1005-2-22-2'], ['0905.1005-1-22-3', '0905.1005-2-22-3'], ['0905.1005-1-22-4', '0905.1005-2-22-4'], ['0905.1005-1-22-5', '0905.1005-2-22-5'], ['0905.1005-1-22-6', '0905.1005-2-22-6'], ['0905.1005-1-22-7', '0905.1005-2-22-7'], ['0905.1005-1-22-8', '0905.1005-2-22-8'], ['0905.1005-1-22-9', '0905.1005-2-22-9'], ['0905.1005-1-32-0', '0905.1005-2-32-0'], ['0905.1005-1-32-1', '0905.1005-2-32-1'], ['0905.1005-1-32-2', '0905.1005-2-32-2'], ['0905.1005-1-32-3', '0905.1005-2-32-3'], ['0905.1005-1-36-0', '0905.1005-2-36-0'], ['0905.1005-1-48-0', '0905.1005-2-48-0'], ['0905.1005-1-48-1', '0905.1005-2-48-1'], ['0905.1005-1-48-2', '0905.1005-2-48-2'], ['0905.1005-1-48-3', '0905.1005-2-48-3'], ['0905.1005-1-48-4', '0905.1005-2-48-4'], ['0905.1005-1-33-0', '0905.1005-2-33-0'], ['0905.1005-1-33-1', '0905.1005-2-33-1'], ['0905.1005-1-33-2', '0905.1005-2-33-2'], ['0905.1005-1-33-3', '0905.1005-2-33-3'], ['0905.1005-1-31-0', '0905.1005-2-31-0'], ['0905.1005-1-31-1', '0905.1005-2-31-1'], ['0905.1005-1-31-2', '0905.1005-2-31-2'], ['0905.1005-1-2-0', '0905.1005-2-2-0'], ['0905.1005-1-2-1', '0905.1005-2-2-1'], ['0905.1005-1-2-2', '0905.1005-2-2-2'], ['0905.1005-1-9-0', '0905.1005-2-9-0'], ['0905.1005-1-9-1', '0905.1005-2-9-1'], ['0905.1005-1-9-2', '0905.1005-2-9-2'], ['0905.1005-1-9-3', '0905.1005-2-9-3'], ['0905.1005-1-14-0', '0905.1005-2-14-0'], ['0905.1005-1-14-1', '0905.1005-2-14-1'], ['0905.1005-1-15-0', '0905.1005-2-15-0'], ['0905.1005-1-15-1', '0905.1005-2-15-1'], ['0905.1005-1-0-0', '0905.1005-2-0-0'], ['0905.1005-1-0-1', '0905.1005-2-0-1'], ['0905.1005-1-5-0', '0905.1005-2-5-0'], ['0905.1005-1-5-1', '0905.1005-2-5-1'], ['0905.1005-1-5-2', '0905.1005-2-5-2'], ['0905.1005-1-5-3', '0905.1005-2-5-3'], ['0905.1005-1-5-4', '0905.1005-2-5-4'], ['0905.1005-1-5-5', '0905.1005-2-5-5'], ['0905.1005-1-5-6', '0905.1005-2-5-6'], ['0905.1005-1-5-7', '0905.1005-2-5-7'], ['0905.1005-1-49-0', '0905.1005-2-49-0'], ['0905.1005-1-49-1', '0905.1005-2-49-1'], ['0905.1005-1-49-2', '0905.1005-2-49-2'], ['0905.1005-1-49-3', '0905.1005-2-49-3'], ['0905.1005-1-49-4', '0905.1005-2-49-4'], ['0905.1005-1-49-5', '0905.1005-2-49-5'], ['0905.1005-1-44-0', '0905.1005-2-43-0'], ['0905.1005-1-44-2', '0905.1005-2-43-2'], ['0905.1005-1-18-0', '0905.1005-2-18-0'], ['0905.1005-1-18-5', '0905.1005-2-18-5'], ['0905.1005-1-35-0', '0905.1005-2-35-0'], ['0905.1005-1-35-1', '0905.1005-2-35-1'], ['0905.1005-1-35-2', '0905.1005-2-35-2'], ['0905.1005-1-35-3', '0905.1005-2-35-3'], ['0905.1005-1-35-4', '0905.1005-2-35-4'], ['0905.1005-1-35-5', '0905.1005-2-35-5'], ['0905.1005-1-10-1', '0905.1005-2-10-1'], ['0905.1005-1-42-0', '0905.1005-2-41-1'], ['0905.1005-1-42-4', '0905.1005-2-41-5'], ['0905.1005-1-42-5', '0905.1005-2-41-7'], ['0905.1005-1-42-6', '0905.1005-2-41-8'], ['0905.1005-1-42-8', '0905.1005-2-41-10'], ['0905.1005-1-42-9', '0905.1005-2-41-11'], ['0905.1005-1-46-0', '0905.1005-2-45-0'], ['0905.1005-1-40-1', '0905.1005-2-40-1'], ['0905.1005-1-4-6', '0905.1005-2-4-7'], ['0905.1005-1-4-14', '0905.1005-2-4-15'], ['0905.1005-1-13-3', 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[]

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[]

['0905.1005-1-37-0', '0905.1005-1-38-0', '0905.1005-2-37-0', '0905.1005-2-38-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0905.1005

1511.03842

{'1511.03842-1-0-0': 'We consider a [MATH] extended supersymmetric model with a right handed neutrino superfield which can generate light neutrino mass by Type I seesaw mechanism.', '1511.03842-1-0-1': 'The lighter part of superpartner of the right-handed neutrino cloud be the candidate dark matter.', '1511.03842-1-0-2': 'Right-handed snuetrinos can come from the decay of [MATH], superpartner of the extra gauge boson [MATH].', '1511.03842-1-0-3': 'Left-right handed snuetrino mixing in this case can give rise to displaced lepton flavour violating signatures.', '1511.03842-1-0-4': 'Lighter right-handed snutrinos along with a wino-like chargino NLSP (next to lightest supersymmetric particle) creates an interesting decay topology.', '1511.03842-1-0-5': 'We investigate such displaced multi-leptonic final states with lepton flavour violation from the third generation supersymmetric cascade decays at the LHC.', '1511.03842-1-1-0': 'The recent discovery of a Higgs boson around [MATH] GeV has established the role of at least one scalar in the electro-weak symmetry breaking [CITATION].', '1511.03842-1-1-1': 'However, experimental evidences for tiny neutrino masses and indirect evidence of dark matter candidate, and a theoretical requirement for naturalness of the electroweak scale needs a theory beyond Standard Model(SM).', '1511.03842-1-1-2': 'A supersymmetric seesaw model [CITATION] with an additional [MATH] [CITATION] gauge symmetry can address these issues consistently.', '1511.03842-1-2-0': 'In supersymmetric theories with R-parity, the lightest supersymmetric particle (LSP) is stable and thus a neutral LSP, typically a linear combination of neutral gauginos and Higgsinos, becomes a good thermal dark matter (DM) candidate if supersymmetry (SUSY) is broken around the TeV scale [CITATION].', '1511.03842-1-2-1': 'Similarly [MATH] breaking can also be indued via quantum effects at [MATH](TeV) [CITATION] which then generates the seesaw scale around TeV testable at LHC.', '1511.03842-1-2-2': 'The additional [MATH] if comes from any grand unified gauge group which requires the presence of right-handed neutrinos (RHNs) for the anomaly-free condition, and cloud be at TeV-scale.', '1511.03842-1-3-0': 'A right-handed sneutrino (RHsN) can be the LSP and a good dark matter candidate whose thermal relic density is in the right range if the [MATH] gaugino [MATH], the superpartner of the [MATH] gauge boson [MATH], is relatively light [CITATION].', '1511.03842-1-3-1': 'LHC recently put bounds on the lighter electro-weak chargino-neutralino production cross-sections [CITATION].', '1511.03842-1-3-2': 'If the lighter neutralino states are made of RHsNs which are SM gauge singlets, such production cross-sections are automatically suppressed.', '1511.03842-1-3-3': 'The same reason also prompts the non-standard decays of lighter chargions and neutralino which further evade the recent bounds.', '1511.03842-1-3-4': 'We consider the [MATH] extended supersymmetric scenario with an right handed sneutrino-superfiled.', '1511.03842-1-3-5': 'The [MATH] under question can be [MATH] or [MATH] or any other kind of such extensions.', '1511.03842-1-3-6': 'For the current study we focus on the [MATH] extended supersymmetric scenarios.', '1511.03842-1-4-0': 'In this work we explore the possible LHC signatures of displaced lepton flavour violating decays of RHsN caused by left-right handed sneutrino mixings.', '1511.03842-1-4-1': 'Being SM gauge singlet such RHsNs will mostly come from [MATH] decay.', '1511.03842-1-4-2': 'The pair production cross-section of such [MATH] is not so promising [CITATION] but its production from SUSY cascade could still be encouraging, in particular from the lighter third generation squark ([MATH]) decays.', '1511.03842-1-4-3': 'When the heavier RHsN is (Next to next LSP) NNLSP and the lighter one is LSP along with chargino NLSP (Next to LSP), the situation can lead to multi-leptonic final states along with lepton flavour violating displaced charged track.', '1511.03842-1-5-0': 'Among various possibilities of an extra gauge symmetry [MATH] and the presence of the associated right-handed neutrinos [CITATION], we will take the [MATH] model for our explicit analysis as in [CITATION].', '1511.03842-1-5-1': 'The particle content of our [MATH] model is as follows: [EQUATION] where [MATH] representations and [MATH] charges of the SM fermions ([MATH]), Higgs bosons ([MATH]), and additional singlet fields ([MATH]) are shown.', '1511.03842-1-5-2': 'Here [MATH] denotes the right-handed neutrino, [MATH] is an additional singlet field fit into the 27 representation of [MATH], and we introduced more singlets [MATH], vector-like under [MATH], to break [MATH] and generate the Majorana mass term of [MATH] [CITATION].', '1511.03842-1-5-3': 'Note that the right-handed neutrinos carry the largest charge under [MATH] and thus the corresponding [MATH] decays dominantly to right-handed neutrinos.', '1511.03842-1-5-4': 'The [MATH] breaking can be generated radiatively in the minimal Higgs sector [CITATION] alternatively the additional singlet field [MATH] is neutral under [MATH] so that it can be used to generate a mass to the [MATH] Higgsinos in non-minimal Higgs sectors [CITATION].', '1511.03842-1-5-5': 'Here we do not assume any grand unification theory as the origin of our model and the grand unification structure is just used for a convenient guide to a theoretically consistent model guaranteeing the anomaly free condition.', '1511.03842-1-6-0': 'The gauge invariant superpotential in the seesaw sector is given by [EQUATION] where [MATH] and [MATH] denote the lepton and Higgs doublet superfields, respectively.', '1511.03842-1-6-1': 'Given the vacuum expectation values [MATH], the [MATH] mass is generated as [MATH], which gets [MATH]>[MATH][MATH] TeV bound depending on [MATH] types [CITATION].', '1511.03842-1-7-0': 'After the [MATH] breaking, the right-handed neutrinos (RHNs) obtain the mass [MATH] and induce the seesaw mass for the light neutrinos: [EQUATION]', '1511.03842-1-7-1': 'The same vev also induces the mixing between the RHN [MATH] and light neutrino [MATH] and corresponding mixing angle is given by [EQUATION] which leads to [MATH] decay to SM gauge bosons.', '1511.03842-1-7-2': 'The real and imaginary components of the scalar part of [MATH] superfiled, [MATH], get a mass splitting proportional to the lepton number violating (Majorana) mass term [MATH][CITATION].', '1511.03842-1-7-3': 'The mixings also occur between the real and imaginary components of the left- and right- sneutrinos proportional to the vev [MATH] as given by Eq. [REF].', '1511.03842-1-7-4': 'Here [MATH] are masses for the real and imaginary part of the left-handed sneutrino, [MATH] is the tri-linear soft term corresponding to the leptonic doublet, [MATH] is the up and down type Higgs mixing parameter and [MATH] is the ratio of their vevs as explained in [CITATION].', '1511.03842-1-7-5': '[EQUATION]', '1511.03842-1-7-6': 'These mixing angles can be naturally small due to [MATH] for [MATH] eV and [MATH] GeV and the given the cancelation among the parameters.', '1511.03842-1-8-0': 'An interesting scenario can happen if [MATH] is NNLSP and [MATH] is the LSP with a wino like chargino [MATH] as NLSP as shown in Figure [REF].', '1511.03842-1-8-1': 'If a relatively lighter [MATH] produced, preferably by SUSY cascade decays, it decays into a RHN and RHsNs ([MATH]).', '1511.03842-1-8-2': 'Such [MATH] then undergoes a lepton flavour violating decay to an electron and chargino via the mixing [MATH], with a decay width given in Eq. [REF], where [MATH] is one of the chargino mixing matrices, [MATH] is the [MATH] gauge coupling and [MATH] is the lighter chargino mass.', '1511.03842-1-8-3': 'This decay-width can be very small, [MATH] GeV or less, thus gives rise to displaced charged tracks of length cm to m which can be observed at inner tracker and calorimeters at CMS and ATLAS detectors of the the LHC.', '1511.03842-1-9-0': '[EQUATION]', '1511.03842-1-9-1': 'Such long lived lighter chargino then causes another lepton flavour violating decay to [MATH] for [MATH] via snutrino mixing ([MATH]).', '1511.03842-1-9-2': 'The decay in this case can even be smaller as given in Eq. [REF].', '1511.03842-1-9-3': 'If the lighter chargino is [MATH] type, then [MATH] decays to [MATH] would be proportional to the [MATH].', '1511.03842-1-10-0': 'Left-right-handed snutrino mixings when go to very small values [MATH], can result in displaced decays for both [MATH] and [MATH].', '1511.03842-1-10-1': 'Due to the fact [MATH] is neutral scalar compared to [MATH] which is a charged fermion, the former decay width is four times more given everything else are the same.', '1511.03842-1-10-2': 'For a given [MATH] and [MATH] the cancelation happens either for [MATH] or for [MATH].', '1511.03842-1-11-0': 'Figure [REF] shows such decay lengths for [MATH] and [MATH] in cm for a scanned parameter space, where [MATH], [MATH], [MATH], [MATH] in GeV, [MATH] and [MATH] for [MATH] eV.', '1511.03842-1-11-1': 'We can see that depending on the parameter space, chargino produced from the first displaced decays of [MATH] can travel few cm to m before gets another recoil from its decay.', '1511.03842-1-12-0': 'The RHN [MATH] can be produced from the [MATH] decay, [MATH], and then can go through a flavour violating decay, [MATH].', '1511.03842-1-12-1': 'However, recent bound of [MATH]>[MATH][MATH] TeV from LHC [CITATION], reduces such production cross-section drastically[CITATION].', '1511.03842-1-12-2': 'The other source of RHNs could be from the decays of its super-partner [MATH], [MATH].', '1511.03842-1-12-3': 'Due to suppressed pair production of [MATH], it is encouraged to look for [MATH] is supersymmetric (SUSY) cascade decays [CITATION].', '1511.03842-1-12-4': 'From [CITATION] (Figure 13), we can see that the right-handed down type squark mostly decays to [MATH] for the [MATH] model.', '1511.03842-1-13-0': 'Third generation squarks, in particular lighter sbottom ([MATH]) cascade decay to [MATH] is our immediate interest.', '1511.03842-1-13-1': 'Once produced, the [MATH] can go through one or two displaced decays as shown in Figure [REF].', '1511.03842-1-13-2': 'Thus [MATH] can give rise to [MATH], where the two pairs of electron come from the displaced lepton flavour violating decays of [MATH] and [MATH] respectively.', '1511.03842-1-13-3': 'The RHNs [MATH] produced from [MATH] decay are not displaced but add to the lepton flavour violating decays [MATH] along with [MATH] and [MATH] permitted by the phase space.', '1511.03842-1-13-4': 'Thus final state can be constituted of [MATH], among which four of the electrons are displaced.', '1511.03842-1-13-5': 'Now the two [MATH]s can decay both muons and electrons with the same ratios.', '1511.03842-1-13-6': 'Checking the final state muon and electron numbers tagged with two [MATH]-jets among which 4 electrons are displaced cloud be a clear signature for such cascade decay.', '1511.03842-1-13-7': 'Along with [MATH] contributions, [MATH] adds to the final states with more jets or leptons from the top decay.', '1511.03842-1-14-0': 'For the collider study we select few benchamark points given in Table [REF] and we mainly follow Ref [CITATION] for fixing the Higgs sector with decoupled first two generations of squark and gluino.', '1511.03842-1-14-1': 'The recent bounds from LHC on Higgs data [CITATION], bounds on third generation SUSY masses [CITATION] and recent bounds on lighter charginos and neutralino production cross-sections [CITATION] are taken into account.', '1511.03842-1-15-0': 'In this collider study we mainly focus on the multi-leptonic final states tagged with at least two [MATH]-jets ([MATH]) along with lepton flavour number, i.e. [MATH] and [MATH].', '1511.03842-1-15-1': 'Four of the electrons coming from [MATH] and [MATH] are displaced with average decay length of 795.8 and 404.6, 736.9 and 69.9, and 50.5 and 53.6 in cm for BP1, BP2 and BP3 respectively.', '1511.03842-1-15-2': 'Table [REF] shows the relevant decay branching fractions involved in SUSY cascade decays.', '1511.03842-1-15-3': 'For BP1 and BP2 the RHN is 100 GeV so decays to [MATH] and [MATH] with branching fraction of 88%, 12% respectively.', '1511.03842-1-15-4': 'However for BP3 as [MATH] GeV it decays to [MATH] as well with branching fraction of [MATH].', '1511.03842-1-16-0': 'For this purpose we simulate the events coming from [MATH]s and [MATH]s via CalcHEP-PYTHIA[CITATION] interface for the final states.', '1511.03842-1-16-1': 'The jet formation has been performed using the Fastjet-3.0.3 [CITATION] with the CAMBRIDGE AACHEN algorithm with jet size [MATH] where the calorimeter coverage is [MATH].', '1511.03842-1-16-2': 'The threshold [MATH] cuts on the jets and the leptons are chosen to be 20 GeV and 10 GeV respectively as the leptons in such cascade decays cloud be soft.', '1511.03842-1-16-3': '[MATH] and [MATH] are imposed to isolated the leptons from other leptons and/or jets.', '1511.03842-1-17-0': 'Figure [REF] (Upper) shows the electron and muon number multiplicity distribution from [MATH] for BP1 and it is evident that due to lepton flavour violating decays, electron numbers are more than muons.', '1511.03842-1-17-1': 'Figure [REF] (lower) shows the [MATH] distributions of such electrons coming from [MATH] and [MATH] and they cloud be on softer side due to smaller mass gaps in the corresponding lepton flavour violating decays.', '1511.03842-1-17-2': 'Figure [REF] shows the transverse decay lengths of [MATH] and [MATH] for BP1 at the LHC for ECM of 14 TeV.', '1511.03842-1-18-0': 'Finally in Table [REF] we present the number of events for such final states at an integrated luminosity of 100 fb[MATH] for the center of mass energy of 13 (14) TeV at the LHC.', '1511.03842-1-18-1': 'This includes [MATH], [MATH], [MATH] and [MATH] inclusive or/and with two tagged [MATH]-jets with single [MATH]-jet tagged efficiency of 0.5 [CITATION].', '1511.03842-1-18-2': 'These final states has 4 displaced electrons for the lepton flavour violating signatures.', '1511.03842-1-18-3': 'If the lepton flavour is conserved then we expect both displaced electrons and muons in the final sates.', '1511.03842-1-18-4': 'The almost background free final states can have some contributions for [MATH]-jet final state from SM [MATH], [MATH] but without any displaced charged leptons.', '1511.03842-1-18-5': 'Thus the signals remain clean with almost no SM backgrounds.', '1511.03842-1-19-0': 'However depending on the benchmark points the cascade decay branching may affect the final state numbers.', '1511.03842-1-19-1': 'For example in BP2, though [MATH]-jtes and [MATH] can be observed with few hundreds of fb[MATH] of data, but higher leptonic final states require higher luminosity.', '1511.03842-1-19-2': 'BP1 and BP3 cases are favourable points to probe the full decay chain including the electrons coming from the RHNs and electrons and muons from the [MATH] bosons, which are not displaced via [MATH] and [MATH] final states.', '1511.03842-1-19-3': 'Lighter RHsNs can be probed via their mixings with the left-handed ones through the displaced multi-leptonic final sates at the LHC.', '1511.03842-1-19-4': 'Lepton flavour violation if any can be measured by checking the flavour multiplicities in the final states.', '1511.03842-1-19-5': 'In case of supersymmetric Type III seesaw a pair of charged RHsN makes the scenario very interesting [CITATION].', '1511.03842-1-20-0': 'PB wants to acknowledge Prof. Eung Jin Chun for the useful discussions.'}

{'1511.03842-2-0-0': '# Introduction', '1511.03842-2-1-0': 'The recent discovery of a Higgs boson around [MATH] GeV has established the role of at least one scalar in the electro-weak symmetry breaking [CITATION].', '1511.03842-2-1-1': 'However, experimental evidences for tiny neutrino masses and indirect evidence of dark matter candidate, and a theoretical requirement for naturalness of the electro-weak scale need a theory beyond the Standard Model (SM).', '1511.03842-2-1-2': 'A supersymmetric seesaw model [CITATION] with an additional [MATH] [CITATION] gauge symmetry can address these issues consistently.', '1511.03842-2-2-0': 'In supersymmetric theories with R-parity, the lightest supersymmetric particle (LSP) is stable and thus a neutral LSP, typically a linear combination of neutral gauginos and Higgsinos, becomes a good thermal dark matter (DM) candidate if supersymmetry (SUSY) is broken around the TeV scale [CITATION].', '1511.03842-2-2-1': 'Similarly, [MATH] breaking can also be induced via quantum effects at [MATH](TeV) [CITATION] which then generates the seesaw scale around TeV, testable at LHC.', '1511.03842-2-2-2': 'The additional [MATH], arising from any grand unified gauge group, requires the presence of right-handed neutrinos (RHNs) for the anomaly-free condition.', '1511.03842-2-3-0': 'A right-handed sneutrino (RHsN) can be a good scalar dark matter candidate with correct thermal relic if the [MATH] gaugino [MATH], the superpartner of the [MATH] gauge boson [MATH], is relatively light [CITATION].', '1511.03842-2-3-1': 'The scenario gets very interesting when one of the RHsNs is LSP with a chargino next to LSP (NLPS) and the other one is next to next LSP (NNLSP).', '1511.03842-2-3-2': 'The RHsNs being SM gauge singlets can evade the recent bounds on NLSP-LSP [CITATION].', '1511.03842-2-3-3': 'The same reason leads to the unusual decays of lighter chargino and RHsNs which are discussed in this article.', '1511.03842-2-3-4': 'We consider the [MATH] extended supersymmetric scenario with an right handed sneutrino-superfield.', '1511.03842-2-3-5': 'The characteristics of such decays are rather general and though we consider [MATH] extended supersymmetric scenario for the analysis, such observations are feasible in other [MATH] extensions [CITATION].', '1511.03842-2-4-0': 'In this article we show the possible LHC signatures of the displaced "lepton flavour violating" decays of RHsN caused by left-right handed sneutrino mixings.', '1511.03842-2-4-1': 'Being SM gauge singlet and only charged under [MATH], such RHsNs will mostly come from [MATH] decay.', '1511.03842-2-4-2': 'The pair production cross-section of such [MATH] is not so promising [CITATION] but its production from SUSY cascade could still be encouraging, in particular from the lighter third generation squark ([MATH]) decays.', '1511.03842-2-4-3': 'When the heavier RHsN ([MATH]) is NNLSP and the lighter one ([MATH]) is LSP along with wino-like chargino NLSP, the situation can lead to multi-leptonic final states along with "lepton flavour violating" displaced charged track.', '1511.03842-2-5-0': 'In section [REF] we briefly discuss the model and in section [REF] we describe the mixings between left and right handed sneutrinos.', '1511.03842-2-5-1': 'We discuss the decay phenomenology of [MATH] in section [REF] and perform a collider study for some benchmark points in section [REF].', '1511.03842-2-5-2': 'Finally, we conclude in section [REF].', '1511.03842-2-6-0': '# The Model', '1511.03842-2-7-0': 'We consider the [MATH] model for our explicit analysis as in [CITATION] and the particle content is as follows: [EQUATION] where [MATH] representations and [MATH] charges of the SM fermions ([MATH]), Higgs bosons ([MATH]), and additional singlet fields ([MATH]) are shown.', '1511.03842-2-7-1': 'Here [MATH] denotes the right-handed neutrino, [MATH] is an additional singlet field fit into the 27 representation of [MATH], and we introduced more singlets [MATH], vector-like under [MATH], to break [MATH] and generate the Majorana mass term of [MATH] [CITATION].', '1511.03842-2-7-2': 'Note that the right-handed neutrinos carry the largest charge under [MATH] and thus the corresponding [MATH] decays dominantly to right-handed neutrinos.', '1511.03842-2-7-3': 'The [MATH] breaking can be generated radiatively in the minimal Higgs sector [CITATION] alternatively the additional singlet field [MATH] is neutral under [MATH] so that it can be used to generate a mass for the [MATH] Higgsinos in non-minimal Higgs sectors [CITATION].', '1511.03842-2-7-4': 'Here we do not assume any grand unification theory as the origin of our model and the grand unification structure is just used for a convenient guide to a theoretically consistent model guaranteeing the anomaly free condition.', '1511.03842-2-8-0': 'The gauge invariant superpotential in the seesaw sector is given by [EQUATION] where [MATH] and [MATH] denote the lepton, Higgs doublet superfields and [MATH], [MATH] are the dimensionless couplings respectively.', '1511.03842-2-8-1': 'Given the vacuum expectation values [MATH], the [MATH] mass is generated as [MATH], which gets [MATH]>[MATH][MATH] TeV bound depending on [MATH] types [CITATION].', '1511.03842-2-9-0': '# Sneutrino mixings and displaced decay', '1511.03842-2-10-0': 'The [MATH] symmetry is broken via [MATH] acquiring vev and this also generates the Majorana mass terms for the right-handed neutrinos (RHNs), i.e., [MATH].', '1511.03842-2-10-1': 'The Majorana mass terms along with electro-weak vev of the up-type Higgs doublet [MATH], induce the masses for the light neutrinos via Type-I seesaw mechanism as shown below: [EQUATION]', '1511.03842-2-10-2': 'The same vev also instigates the mixing between the right-handed neutrino [MATH] and left-handed neutrino [MATH] and corresponding mixing angle is given by [EQUATION] which leads to [MATH] decay to SM gauge bosons and leptons.', '1511.03842-2-10-3': 'The real and imaginary components of the scalar part of [MATH] superfield, i.e., [MATH], get a mass splitting proportional to the lepton number violating (Majorana) mass term [MATH][CITATION].', '1511.03842-2-10-4': 'The mixings also occur between the real and imaginary components of the left- and right-sneutrinos proportional to the vev [MATH] as given by Eq. [REF] [EQUATION] where [MATH] are the masses for the real and imaginary part of the left-handed sneutrino, [MATH] is the tri-linear soft term corresponding to the leptonic doublet, [MATH] is the up and down type Higgs mixing parameter in the superpotential ([MATH]) and [MATH] is the ratio of their vevs as explained in [CITATION].', '1511.03842-2-11-0': 'The mixing angles can be naturally small due to small Yukawa coupling, [MATH] which corresponds to a light neutrino mass scale of [MATH] eV for [MATH] GeV.', '1511.03842-2-11-1': 'The other possibility of very small mixing angles, comes from the cancelation among the parameters in the numerator of Eq. [REF].', '1511.03842-2-11-2': 'We now see how these mixing angles affect the decays of lighter chargino and sneutrinos.', '1511.03842-2-12-0': 'Figure [REF] shows a typical mass spectrum, where lighter stop and sbottom can decay to [MATH] which further decays to RHN, RHsNs ([MATH]).', '1511.03842-2-12-1': 'The lighter sneutrino, [MATH] is the LSP and a dark matter candidate [CITATION].', '1511.03842-2-12-2': 'The wino-like chargino [MATH] is the NLSP and the heavier sneutrino [MATH] stays as NNLSP.', '1511.03842-2-12-3': '[MATH] can be produced from the decays of [MATH] and [MATH] [CITATION].', '1511.03842-2-13-0': 'The choice of electro-weak scale mass of [MATH] is required to have the correct DM relic in a scenario where the lighter RHsN [MATH] is DM candidate [CITATION].', '1511.03842-2-13-1': 'Both the right-handed superfield and DM are charged under the [MATH] as can be seen from Eq. [REF] and otherwise both are SM gauge singlets.', '1511.03842-2-13-2': 'The lightest superpartner of [MATH], i.e., [MATH] is a real scalar field which does not have any [MATH]-channel annihilation diagram via [MATH] and [MATH] (See couplings in Table1 of [CITATION]).', '1511.03842-2-13-3': 'These leave no choice other than a [MATH]-channel annihilation of [MATH] via [MATH] [CITATION].', '1511.03842-2-13-4': 'In this process, the decays and inverse decays of the right- handed neutrino, [MATH] through the small Yukawa coupling play an important role in keeping the right-handed neutrino in thermal equilibrium and thus controlling the dark matter relic density, which is a distinguishable feature of the thermal history of the right-handed sneutrino dark matter compared with the conventional neutralino LSP dark matter.', '1511.03842-2-13-5': 'The [MATH]-channel annihilation of [MATH] via [MATH] and [MATH] decays make the masses of [MATH] correlated.', '1511.03842-2-13-6': 'This results into very light electro-weak scale [MATH] masses for a given electro-weak scale masses [MATH] and [MATH] [CITATION].', '1511.03842-2-13-7': 'Thus the mass spectrum and their phenomenology studied in this context is very specific to the choice of the dark matter candidate, in our case which is the lightest right-handed sneutrino [MATH].', '1511.03842-2-14-0': 'The Majorana mass term in Eq. [REF] violates the lepton number.', '1511.03842-2-14-1': 'However the decay [MATH] maintains the lepton number.', '1511.03842-2-14-2': 'For the current collider analysis we have taken only one [MATH] superfield, where [MATH] decays into [MATH] but not to [MATH] and [MATH].', '1511.03842-2-14-3': "We have chosen only one RHN, with the lepton flavour number of 'e'.", '1511.03842-2-14-4': 'This preference of one flavour over another can be seen in collider searches as flavour violating decays.', '1511.03842-2-14-5': 'From now on the "lepton flavour violation" phrase has been used in their collider search context only.', '1511.03842-2-15-0': 'The effect of kinetic mixing of [MATH] and [MATH] is not considered here as their effects are not potential threat to change the mass hierarchy.', '1511.03842-2-15-1': 'This is due to the fact that the right-handed neutrino superfield is singlet under SM gauge groups thus does not couple to [MATH] and only couples to [MATH].', '1511.03842-2-15-2': 'Such kinetic mixing must appear twice in the vacuum polarisation diagram which affects the loop mass proportional to the quadratic power of the mixing angle.', '1511.03842-2-15-3': 'Those vacuum polarisation diagrams are further suppressed by the three gauge boson (gaugino) and one RHN (RHsN) propagators.', '1511.03842-2-15-4': 'The bounds on the [MATH] kinetic mixing and its phenomenology can be found out [CITATION].', '1511.03842-2-16-0': 'Figure [REF] depicts the decay of [MATH] which leads to the scenario we are interested in (shown in Figure [REF]).', '1511.03842-2-16-1': 'Once a relatively lighter [MATH] produced, preferably by SUSY cascade decays, it decays into a RHN and RHsNs ([MATH]).', '1511.03842-2-16-2': '[MATH] then undergoes a "lepton flavour violating" decay to an electron and chargino via the mixing [MATH].', '1511.03842-2-16-3': 'Such chargino undergoes another "lepton flavour violating" two-body decay to [MATH] via sneutrino mixing [MATH].', '1511.03842-2-17-0': 'The decay width of [MATH] and [MATH] are shown below by Eq. [REF] and Eq. [REF] respectively, [EQUATION] where [MATH] is one of the chargino mixing matrices, [MATH] is the [MATH] gauge coupling, [MATH] is the lighter chargino mass and [MATH] can be calculated from Eq. [REF].', '1511.03842-2-17-1': 'We can see from Eq. [REF] that for some parameter space, either [MATH] or [MATH] can go to zero.', '1511.03842-2-17-2': 'Small mixing angles, i.e., [MATH] can be expected for certain parameter space.', '1511.03842-2-17-3': 'Such small mixing angles induce decay widths for [MATH] and [MATH] as low as [MATH] GeV or less, that lead to displaced charged tracks.', '1511.03842-2-17-4': 'These tracks can be of length cm to meter that can be observed at inner tracker and calorimeters at CMS and ATLAS detectors of the LHC.', '1511.03842-2-18-0': 'Figure [REF] shows such decay lengths for [MATH] and [MATH] in cm for a scanned parameter space given below in Eq. [REF] [EQUATION]', '1511.03842-2-18-1': 'The parameter space is scanned for the displaced decays of lighter chargino and NLSP where the right-handed neutrino mass [MATH], the right-handed sneutrino masses [MATH] and the left-handed sneutrino mass [MATH], the lighter chargino [MATH] and [MATH] neutralino masses are varied according to Eq. [REF] for [MATH].', '1511.03842-2-18-2': 'The scan is not performed with the random variations of the parameters within the limit given in Eq. [REF] but are varied by [MATH] GeV, [MATH] GeV and [MATH] GeV for the senutrinos masses([MATH]), right-handed neutrino mass ([MATH]) and [MATH] (See Eq. [REF]) respectively.', '1511.03842-2-18-3': 'From Figure [REF] we can see that depending on the parameter space, chargino produced from the first displaced decays of [MATH] can travel few cms to meters before its own displaced decay.', '1511.03842-2-18-4': 'The little parabolic behaviour between the two decay lengths are originating due to the choice of the parameter scan [MATH] (See Eq. [REF] and Eq. [REF]).', '1511.03842-2-19-0': 'The left-handed sneutrino and sleptons are heavy enough, such that [MATH] decays are not kinematically allowed.', '1511.03842-2-19-1': 'In principle the lighter chargino can go through three-body decays, but the fact that the lighter RHsN ([MATH]) is the LSP, such decays will be further suppressed due to the following reasons.', '1511.03842-2-19-2': 'Given the scenario, the lighter chargino ([MATH]) is wino-type and also NLSP, the possible three-body decays are possible via off-shell selectron ([MATH]), i.e., [MATH], via off-shell [MATH] or higgsino i.e., [MATH], via charged higgsino ([MATH]) i.e., [MATH] and via off-shell left-handed sneutrino ([MATH]), i.e., [MATH].', '1511.03842-2-19-3': 'All the cases the three-body decay widths are proportional to the square of mixing angle of left-right handed sneutrino ([MATH] ) and again it is further suppressed by the mass of off-shell selectron ([MATH]) or left-handed sneutrino ([MATH]) propagators.', '1511.03842-2-19-4': 'However for the given scenario, the three-body decays with final states involving [MATH] are not kinematically allowed as [MATH].', '1511.03842-2-19-5': 'Another possibility is that of four-body decays when such [MATH] are off-shell and lead to [MATH] and [MATH].', '1511.03842-2-19-6': 'Thus the wino characteristics of the chargino makes such three-body and four-body decays more suppressed than the two-body decays.', '1511.03842-2-20-0': '# [MATH] and [MATH] phenomenology', '1511.03842-2-21-0': 'Another kind of lepton-flavour violating decay happens when the RHN [MATH] decays to [MATH].', '1511.03842-2-21-1': 'This RHN can be produced from the [MATH].', '1511.03842-2-21-2': 'However, such production rate is suppressed [CITATION] given the recent bound on [MATH]>[MATH][MATH] TeV from LHC [CITATION].', '1511.03842-2-21-3': 'The decays of [MATH] could be an alternative source of such RHN.', '1511.03842-2-21-4': 'Being electro-weak in nature, pair production of [MATH] is suppressed and it is encouraged to look for [MATH] is supersymmetric (SUSY) cascade decays [CITATION].', '1511.03842-2-21-5': 'From Figure 13 of [CITATION], we can see that the right-handed down type squark mostly decays to [MATH] for the [MATH] model, which is a feature of [MATH] model.', '1511.03842-2-21-6': 'This prompts us to follow the decays of third generation squarks specially [MATH] and [MATH], which are expected to be lighter and can be produced at the LHC earlier than others as a signature of the supersymmetry.', '1511.03842-2-21-7': 'In this model the third generation squarks, in particular lighter sbottom ([MATH]) cascade decay to [MATH] which then follow the decay topology of Figure [REF].', '1511.03842-2-21-8': '[EQUATION]', '1511.03842-2-21-9': 'Eq. [REF] shows that the sbottom pair production and their following decay topologies that lead to [MATH] final states, where the two pairs of electron come from the displaced "lepton flavour violating" decays of [MATH] and [MATH] respectively.', '1511.03842-2-21-10': 'The RHNs [MATH] produced from [MATH] decay are not displaced but add to the "lepton flavour violating" decays [MATH] along with [MATH] and [MATH] permitted by the phase space.', '1511.03842-2-21-11': 'Thus final state can be constituted of [MATH], among which four of the electrons are displaced.', '1511.03842-2-21-12': 'Now the two [MATH]s can decay to both muons and electrons with the same ratios.', '1511.03842-2-21-13': 'Checking the final state muon and electron numbers tagged with two [MATH]-jets among which 4 electrons are displaced could be a clear signature for such cascade decay.', '1511.03842-2-21-14': 'Along with [MATH] contributions, [MATH] adds to the final states with more jets or leptons from the top decay.', '1511.03842-2-21-15': 'In the case where there is no "lepton flavour violation", we can still look for displaced charged leptons ([MATH], [MATH]) tagged with prompt leptons and [MATH]-jets can probe such decay modes.', '1511.03842-2-22-0': 'For the collider study we select few benchmark points given in Table [REF] where we mainly follow Ref [CITATION] for fixing the Higgs sector with decoupled first two generations of squark and gluino.', '1511.03842-2-22-1': 'The recent bounds from LHC on Higgs data [CITATION], bounds on third generation SUSY masses [CITATION] and recent bounds on lighter charginos and neutralinos production cross-sections [CITATION] are taken into account.', '1511.03842-2-22-2': 'The mass spectrum in the benchmark points generally follows the hierarchy shown in Figure [REF], except for BP3 where the right-handed neutrino [MATH] decay to [MATH] is kinematically allowed unlike the other two benchmark points.', '1511.03842-2-23-0': 'In this collider study we mainly focus on the multi-leptonic final states tagged with at least two [MATH]-jets ([MATH]) along with lepton flavour, i.e. [MATH] and [MATH].', '1511.03842-2-23-1': 'Four of the electrons coming from [MATH] and [MATH] are displaced and their average decay lengths are shown in Table [REF].', '1511.03842-2-23-2': 'Table [REF] shows the relevant decay branching fractions involved in SUSY cascade decays.', '1511.03842-2-23-3': 'For BP1 and BP2 the RHN is 100 GeV so decays to [MATH] and [MATH] with branching fraction of 88%, 12% respectively.', '1511.03842-2-23-4': 'However, for BP3 as [MATH] GeV it decays to [MATH] as well with branching fraction of [MATH].', '1511.03842-2-24-0': '# Collider simulation', '1511.03842-2-25-0': 'For this purpose we simulate the events coming from stops and sbottoms via CalcHEP-PYTHIA[CITATION] interface for the final states.', '1511.03842-2-25-1': 'The jet formation has been performed using the Fastjet-3.0.3 [CITATION] with the CAMBRIDGE AACHEN algorithm.', '1511.03842-2-25-2': 'We have selected a jet size [MATH] for the jet formation, with the following criteria:', '1511.03842-2-26-0': 'Figure [REF] (a) shows the electron and muon number multiplicity distribution from [MATH] for BP1 and it is evident that due to "lepton flavour violating" decays, electron numbers are more than muons.', '1511.03842-2-26-1': 'Figure [REF] (b) shows the [MATH] distributions of such electrons coming from [MATH] and [MATH] and they could be on softer side due to smaller mass gaps in the corresponding "lepton flavour violating" decays.', '1511.03842-2-26-2': 'Figure [REF] shows the transverse decay lengths of [MATH] and [MATH] for BP1 at the LHC for ECM of 14 TeV.', '1511.03842-2-27-0': 'Table [REF] presents the number of events for the benchmark points for the final states [MATH]-jets, [MATH]-jets and [MATH]-jets at an integrated luminosity of 100 fb[MATH] for the center of mass energy of 14 TeV at the LHC.', '1511.03842-2-27-1': 'Here [MATH] includes [MATH] and [MATH] and we take the single [MATH]-jet tagged efficiency of 0.5 [CITATION].', '1511.03842-2-27-2': 'We can see that that [MATH]-jets has greater numbers of events compared to [MATH]-jets.', '1511.03842-2-27-3': 'This is due to the "lepton flavour violating" decays of [MATH] and [MATH], from where four electrons are coming from.', '1511.03842-2-27-4': 'Whereas the muons are coming from the [MATH]s which are produced from the decays of RHNs ([MATH]).', '1511.03842-2-27-5': 'The difference between electron and muon numbers obviously gives the "lepton flavour violating" signature.', '1511.03842-2-27-6': 'On top of that, four of the electrons from [MATH] and [MATH] decays are displaced ones.', '1511.03842-2-27-7': 'If the lepton flavour is conserved then we expect same number of displaced electrons and muons in the final sates.', '1511.03842-2-27-8': 'The existence of displaced charged leptonic final states make the final state topologies almost background free.', '1511.03842-2-27-9': 'In terms of charged lepton numbers and [MATH]-jets, there can be some contributions for [MATH]-jet final state from SM [MATH], [MATH] but without any displaced charged leptons.', '1511.03842-2-27-10': 'However, [MATH] background fails to contribute after the selection cuts and due to low cross-section.', '1511.03842-2-27-11': 'Table [REF] presents the corresponding [MATH] number for 14 TeV at the LHC, where all the charged leptons are prompt ones.', '1511.03842-2-27-12': 'The signals remain clean if we consider the displaced charged leptons due to no SM backgrounds.', '1511.03842-2-27-13': 'However, considering all the charged leptons are prompt we can calculate the signal significance for the benchmark points where the cascade decay branching fractions affect the final state numbers.', '1511.03842-2-27-14': 'For [MATH]-jet final state final state gets signal significance of 49.6[MATH], 8.6[MATH], and 44.7[MATH] respectively for center of mass energy of 14 TeV at the LHC.', '1511.03842-2-27-15': 'A [MATH] discovery can be achieved with very earlier data of few fb[MATH] for BP1 and BP3 to 34 fb[MATH] for BP2 for 14 TeV at the LHC.', '1511.03842-2-27-16': '[MATH]-jet final state get significance of 42.1[MATH], 7.8[MATH] and 36.7[MATH] respectively for BP1, BP2 and BP3 at the LHC with ECM=14 TeV.', '1511.03842-2-27-17': 'Due to "lepton flavour violating" decays [MATH]-jet final state will have greater significance than [MATH]-jet case, which can be realised from Table [REF] and Table [REF].', '1511.03842-2-28-0': 'Table [REF] presents [MATH] final states event numbers for the benchmark points at the LHC with ECM=14 TeV at an integrated luminosity of 100 fb[MATH], which also shows [MATH] and [MATH] final states.', '1511.03842-2-28-1': 'The final states are almost background free specially with the displaced charged electrons.', '1511.03842-2-28-2': 'For [MATH] final state final state gets signal significance of 29.4[MATH], 4.2[MATH], and 25.1[MATH] respectively for BP1, BP2 and BP3 for center of mass energy of 14 TeV at the LHC at an integrated luminosity of 100 fb[MATH].', '1511.03842-2-28-3': 'The required luminosity for [MATH] significance can vary from few fb[MATH] to [MATH] fb[MATH].', '1511.03842-2-28-4': '[MATH]-jets final state signal significance numbers are 12.8[MATH], 2.1[MATH], and 10.1[MATH] respectively for BP1, BP2 and BP3 for center of mass energy of 14 TeV at the LHC at an integrated luminosity of 100 fb[MATH].', '1511.03842-2-28-5': 'Similarly, we can see that for [MATH]-jets the corresponding significances are 1.0[MATH], 1.7[MATH], and 10.1[MATH] respectively for BP1, BP2 and BP3 for center of mass energy of 14 TeV at the LHC at an integrated luminosity of 100 fb[MATH].', '1511.03842-2-29-0': 'BP1 and BP3 cases are favourable points to probe the full decay chain including the electrons coming from the RHNs and electrons and muons from the [MATH] bosons along with the four displaced electrons from chargino and RHsN decays.', '1511.03842-2-29-1': 'When both the [MATH]s decay leptonically we get final states with [MATH] and [MATH].', '1511.03842-2-29-2': 'Finally in Table [REF] we present the number of events for the benchmark points for the final states at an integrated luminosity of 100 fb[MATH] for the center of mass energy of 14 TeV at the LHC respectively.', '1511.03842-2-29-3': 'Though for higher charged leptonic final states signal numbers suffer a lot but finding such multi-leptonic final state is still a possibility which can probe this particular decay chain.', '1511.03842-2-29-4': 'If such a decay chain with displaced charged lepton can be found then we can measure the mixings between the lighter RHsNs with the left-handed at the LHC.', '1511.03842-2-30-0': '# Discussion and conclusion', '1511.03842-2-31-0': 'We have seen in this study that displaced leptonic signatures can probe the left-right handed sneutrino mixings as well as the possible "lepton flavour violation" at the LHC.', '1511.03842-2-31-1': 'Here we analysed multi-leptonic final state with some displaced charged leptonic signature.', '1511.03842-2-31-2': 'The choice of parameter space is responsible for such displaced decays.', '1511.03842-2-31-3': 'However, these multi-leptonic final states can address other parameter space without having a dispalced decay.', '1511.03842-2-31-4': 'In a [MATH] NLSP scenario, [MATH] decay produces more prompt electron than muon in the final states and such "lepton flavour violation" if any can be measured by checking the flavour multiplicities in the final states [CITATION] which can also give rise to needed same sign di-leptonic excess recently found at the LHC [CITATION].', '1511.03842-2-31-5': 'The nature of the compact mass spectrum as well as the nature of NLSP changes the scenario quite a bit which we discuss below.', '1511.03842-2-32-0': 'In the present case compact mass spectrum is responsible for the final state resulting two displaced charged leptons from the decays of [MATH] NNLSP and of wino-like chargino NLSP.', '1511.03842-2-32-1': 'The fact is that [MATH] only mix via [MATH] (see Eq. [REF]) and couple to wino-like chargino.', '1511.03842-2-32-2': 'As long as [MATH] is NNLSP, wino-like chargino is NLSP and [MATH] is LSP, the above signature is normally expected.', '1511.03842-2-32-3': 'Even if the mass gap between chargino and LSP is greater than [MATH], [MATH] is not allowed due to the absence of direct coupling.', '1511.03842-2-32-4': 'However in such situation, the three-body displaced decays, i.e., [MATH] open up via off-shell charged slepton, [MATH] and higgsino respectively.', '1511.03842-2-32-5': 'The displaced [MATH] then decays promptly into charged leptons and neutrinos or di-jet.', '1511.03842-2-32-6': 'For an increased mass gap between [MATH] and [MATH], i.e., [MATH], a displaced Higgs production may be visible via the three-body decay of [MATH].', '1511.03842-2-32-7': 'Similar argument also holds for the displaced decay [MATH].', '1511.03842-2-32-8': 'As long as the NLSP is wino-like chargino, no other kind of two-body decay is allowed even if the mass gap is more than [MATH].', '1511.03842-2-32-9': 'However if the mass gap is more than the Higgs([MATH]) mass, [MATH] is open due to the existence of [MATH] couplings, which leads to displaced Higgs production along with a displaced leptonic charged track.', '1511.03842-2-33-0': 'If we allow charged slepton ([MATH] instead of wino-like chargino as NLSP, then also we can have displaced charged lepton or [MATH] in one or both the vertices depending on the mass spectrum.', '1511.03842-2-33-1': 'For a [MATH] like NLSP scenario, such displaced decays of both [MATH] and [MATH] will give rise to neutrinos and [MATH], i.e. only the missing particles and thus not recognisable at the collider.', '1511.03842-2-33-2': 'The signature of displaced charged lepton or [MATH] changes drastically if [MATH] is NLSP and decays to [MATH], which gives rise to non-displaced "lepton flavour violating" signature at the collider [CITATION].', '1511.03842-2-33-3': 'If the lighter chargino is [MATH] type, then [MATH] decay width is proportional to the [MATH] which may also lead to displaced decay of [MATH].', '1511.03842-2-33-4': "However, given [MATH] is LSP, such higgsino like chargino will decay to [MATH] via [MATH] and travel a few cms to meters before encountering a recoil in it's charged track.", '1511.03842-2-34-0': 'These final states are still interesting even when [MATH] and [MATH] do not have "lepton flavour violating" final states, i.e. they decay into the electron, muon and tau channels equally.', '1511.03842-2-34-1': 'Thus displaced electron and muon number difference is a good observable to probe such scenarios.', '1511.03842-2-34-2': 'In case of supersymmetric Type III seesaw, a pair of charged RHsN have similar interesting phenomenology[CITATION].', '1511.03842-2-34-3': 'There are some other studies where right-handed sneutrino is considered as cold dark matter [CITATION] and similar long lived charged particles are predicted from supersymmetric cascade decays [CITATION].', '1511.03842-2-35-0': 'The displaced charged leptonic signature can appear in various supersymmetric models, viz. [MATH]-parity violation [CITATION], gauge mediation [CITATION], etc and non-supersymmetric models, viz. Type-I seesaw [CITATION], Hyperpions [CITATION], Leptoquark [CITATION], etc.', '1511.03842-2-35-1': 'Thus there is a wide range of models that can mimic the generic signature of displaced decays and the corresponding charged tracks.', '1511.03842-2-35-2': 'However, we see in this article that the presence of one of right-handed sneutrino as dark matter makes the production of the corresponding right-handed neutrino dominant over other two right-handed neutrino flavours.', '1511.03842-2-35-3': 'This brings not only displaced charged tracks, but also lepton-flavour violating signatures.', '1511.03842-2-35-4': 'In that respect this is a "smoking gun" signature which can distinguish such model from others with generic displaced leptonic charged track.', '1511.03842-2-36-0': 'PB wants to acknowledge Eung Jin Chun for the useful discussions.'}

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[]

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[['1511.03842-1-9-3', '1511.03842-2-33-3']]

['1511.03842-1-7-5', '1511.03842-1-9-0', '1511.03842-2-21-8', '1511.03842-2-25-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1511.03842

gr-qc-0601086

{'gr-qc-0601086-1-0-0': 'The introduction of an interaction for dark energy to the standard cosmology offers a potential solution to the cosmic coincidence problem.', 'gr-qc-0601086-1-0-1': 'We examine the conditions on the dark energy density that must be satisfied for this scenario to be realized.', 'gr-qc-0601086-1-0-2': 'Under some general conditions we find a stable attractor for the evolution of the Universe in the future.', 'gr-qc-0601086-1-0-3': 'Holographic conjectures for the dark energy offer some specific examples of models with the desired properties.', 'gr-qc-0601086-1-1-0': '[MATH]', 'gr-qc-0601086-1-2-0': '# Introduction', 'gr-qc-0601086-1-3-0': 'The observation that the Universe appears to be accelerating has been one of the biggest surprises in both the astronomy and particle physics communities.', 'gr-qc-0601086-1-3-1': "This acceleration can be accounted for by adding a cosmological constant in Einstein's equations.", 'gr-qc-0601086-1-3-2': 'While particle physicists never had a definite reason for setting the cosmological constant to zero, there was always the view that like all small or zero quantities, there must be a symmetry enforcing it.', 'gr-qc-0601086-1-3-3': 'The observation that the size of the cosmological constant is neither zero nor at one of the natural scales like the Planck length or the scale of supersymmetry breaking has focused attention on explaining the dark energy component.', 'gr-qc-0601086-1-4-0': 'There are at least two requirements for any model of dark energy: (1) one must obtain in a natural way the observed size of the energy density, [MATH] eV[MATH].', 'gr-qc-0601086-1-4-1': 'Many models attempt to relate this to the observed coincidence of this scale with the scale [MATH] where [MATH] is the Hubble constant at the current epoch and [MATH] is the Planck mass.', 'gr-qc-0601086-1-4-2': '(2) One must also obtain an equation of state [MATH] which is similar to that of a cosmological constant at least in the recent past, namely [MATH].', 'gr-qc-0601086-1-4-3': 'In fact, the most recent data from the WMAP satellite and supernova and sky surveys have constrained the equation of state [MATH] at the 95% confidence level for dark energy with a constant [MATH].', 'gr-qc-0601086-1-5-0': 'An interesting approach to the dark matter problem that has arisen from recent advances in understanding string theory and black hole physics is to employ a holographic principle.', 'gr-qc-0601086-1-5-1': 'Here the motivation is the recognition that the observed size of the energy density at the present epoch seems to be consistent with taking the geometric average of the two scales [MATH] and [MATH].', 'gr-qc-0601086-1-5-2': 'Early attempts to relate the dark energy to the Hubble scale or the particle horizon did not give acceptable solutions in detail that were consistent with the observations.', 'gr-qc-0601086-1-6-0': 'The evolution of the dark energy density also depends on its precise nature.', 'gr-qc-0601086-1-6-1': 'If the dark energy is like a cosmological constant with equation of state [MATH], the dark energy density is constant and within the expanding Universe ultimately the dark energy density comes to dominate over the matter density.', 'gr-qc-0601086-1-6-2': 'In this scenario the transition from matter domination to dark energy domination is rapid, and the fact that experimental observations seem to be consistent with sizeable amounts of both matter and dark energy indicates that we are currently residing in this transition period.', 'gr-qc-0601086-1-6-3': 'The cosmic coincidence problem is the statement that it is unlikely that the current epoch coincides with this rapid transition period.', 'gr-qc-0601086-1-7-0': 'A possible solution is to assume that there is some mechanism that is converting dark energy into matter.', 'gr-qc-0601086-1-7-1': 'Cosmic antifriction forces were introduced in Ref. [CITATION] as a possible solution to the cosmic coincidence problem.', 'gr-qc-0601086-1-7-2': 'If these forces are present one can obtain a fixed ratio of matter density to dark energy density as the final state of the Universe and hence solve the coincidence problem.', 'gr-qc-0601086-1-7-3': 'The resulting cosmology is described by sourced Friedmann equations.', 'gr-qc-0601086-1-7-4': 'The basic scenario is quite simple: the natural tendency of a cosmological component with a more negative equation of state to dominate at large times the energy density of the Universe is compensated by the decay of the dark energy component into the other component(s).', 'gr-qc-0601086-1-7-5': 'At large times an equilibrium solution can develop for which the various components can coexist.', 'gr-qc-0601086-1-7-6': 'The interpretation is then that we are living in or close to this equilibrium thus eliminating the cosmic coincidence problem.', 'gr-qc-0601086-1-8-0': 'Frictional and antifrictional forces we envision here were introduced for a single component in Refs. [CITATION].', 'gr-qc-0601086-1-8-1': 'The particular case where the dark energy component is assumed to have a constant equation of state was treated in Ref. [CITATION].', 'gr-qc-0601086-1-8-2': 'Some attempts have been made to explain the size of the dark energy density on the basis of holographic ideas[CITATION], and lead to the consideration of the Hubble horizon as the holographic scale[CITATION].', 'gr-qc-0601086-1-8-3': 'A holographic model based on the future event horizon was examined in Ref. [CITATION].', 'gr-qc-0601086-1-8-4': 'Danielsson studied the effects of a transplanckian backreaction as a particular source for the holographic dark energy[CITATION].', 'gr-qc-0601086-1-8-5': 'More recenlty an emphasis on the case of interacting dark energy has been discussed in Refs. [CITATION].', 'gr-qc-0601086-1-9-0': 'Ultimately one needs to appeal to the observational data to constrain these ideas.', 'gr-qc-0601086-1-9-1': 'In this direction Wang, et al. have begun a more detailed comparison of the predictions of a holographic universe with interaction with the observational data[CITATION].', 'gr-qc-0601086-1-9-2': 'In addition to the overall development of the densities of the cosmological components there are other observables which are sensitive to the modified scenarios[CITATION].', 'gr-qc-0601086-1-10-0': '# Framework of Sourced Friedmann Equations', 'gr-qc-0601086-1-11-0': 'We assume two component equations for dark energy and matter, [EQUATION]', 'gr-qc-0601086-1-11-1': 'The equality of [MATH] in the two equations guarantees the overall conservation of the energy-momentum tensor, and positive [MATH] can be interpreted as a transfer from the dark energy component to the matter component.', 'gr-qc-0601086-1-11-2': 'Presumably this arises from some microscopic mechanism, but we do not specify one here.', 'gr-qc-0601086-1-11-3': 'The quantity [MATH] is the matter component for which one usually takes [MATH], but we temporarily retain a nonzero equation of state so as to retain the generality of an two interacting components.', 'gr-qc-0601086-1-12-0': 'One can define effective equations of state, [EQUATION]', 'gr-qc-0601086-1-12-1': 'Define the ratio [MATH] and the rate [MATH], then the effective equations of state are expressed in terms of the native equations of state and a ratio of rates as [EQUATION]', 'gr-qc-0601086-1-12-2': 'The time evolution of the ratio [MATH] is then [EQUATION]', 'gr-qc-0601086-1-12-3': 'The general behavior of solutions can be ascertained from this equation.', 'gr-qc-0601086-1-12-4': 'The stable points in the evolution are obtained when either [MATH] or when [MATH].', 'gr-qc-0601086-1-12-5': 'If [MATH] then the vanishing dark energy density ([MATH]) solution will apply in the infinite past while a constant nonzero ratio of dark energy to matter will apply in the infinite future with equal effective equations of state.', 'gr-qc-0601086-1-13-0': 'Using the definitions [EQUATION] which satisfy a Friedmann equation [EQUATION] one can convert to the physical parameters [EQUATION]', 'gr-qc-0601086-1-13-1': 'One can write the differential equation in the suggestive form using [MATH], [EQUATION] where [MATH].', 'gr-qc-0601086-1-13-2': 'The utility of the evolution equation in this form is that it allows one to understand in a general way the possible solutions for [MATH].', 'gr-qc-0601086-1-13-3': 'Firstly if one turns off the interaction ([MATH]) one recovers the usual evolution equation with the effective equations of state replaced with the usual equations of state [MATH] and [MATH] (we will call these the native equations of state).', 'gr-qc-0601086-1-13-4': 'For constant [MATH] and [MATH] and [MATH] then the dark energy evolution proceeds from the fixed points of the differential equation (zeros of the right hand side) in the usual way from [MATH] to [MATH] in the usual way.', 'gr-qc-0601086-1-13-5': 'When an interaction is present a more interesting solution may occur where the evolution of [MATH] approaches a condition where instead [MATH] at which [MATH] approaches a fixed asymptotic value for large time less than 1.', 'gr-qc-0601086-1-13-6': 'For a properly chosen interaction this might constitue a solution to the cosmic concidence problem.', 'gr-qc-0601086-1-14-0': 'For illustration one can also consider the (redundant) differential equation for the matter density, [EQUATION]', 'gr-qc-0601086-1-14-1': 'An initial condition [MATH] is inevitably driven to an asymptotic solution for which [MATH] and [MATH] and [MATH] approach their limiting (equal) value.', 'gr-qc-0601086-1-14-2': 'This behavior is displayed for a specific solution[CITATION] shown in Fig. 1.', 'gr-qc-0601086-1-15-0': 'Two physical assumptions must be provided to determine the evolution of the parameters of the universe.', 'gr-qc-0601086-1-15-1': 'A holographic condition on the dark energy [MATH] will determine by itself the effective equation of state [MATH].', 'gr-qc-0601086-1-15-2': 'A specific specification for the source [MATH] will determine the effective matter equation of state [MATH].', 'gr-qc-0601086-1-15-3': 'Alternatively one of these conditions can be replaced by the assumption that [MATH] is constant.', 'gr-qc-0601086-1-16-0': 'One typically obtains an equation for the evolution of the dark energy density of the form [EQUATION]', 'gr-qc-0601086-1-16-1': 'The evolution of [MATH] with respect to [MATH] or [MATH] is qualitatively determined by the fixed points of the differential equation in Eq. ([REF]) which are determined by the condition [MATH].', 'gr-qc-0601086-1-16-2': 'Examining the equations in Eqs. ([REF]), it would appear that this condition would require that [MATH] and then that [MATH] via Eq. [REF].', 'gr-qc-0601086-1-16-3': 'However, as shown in Ref. [CITATION] if one of the fixed points occurs for the physically interesting value of [MATH], then one can have the Universe with close to vanishing dark energy simultaneously with [MATH].', 'gr-qc-0601086-1-16-4': 'In the remainder of this paper we explore the general conditions that give rise to this type of equilibrium solution.', 'gr-qc-0601086-1-17-0': '# Dimensional analysis', 'gr-qc-0601086-1-18-0': 'First define a length scale [MATH] via [EQUATION] where the constant [MATH] represents an order one constant.', 'gr-qc-0601086-1-18-1': 'Specific assumptions might associate the length scale with a physical length such as the Hubble horizon, the particle horizon, the future event horizon, or perhaps some other parameter.', 'gr-qc-0601086-1-18-2': 'The common feature of any realistic choice is that the length scale at the present epoch should be of order the size of the Universe so as to obtain the appropriate size of the dark energy density consistent with experimental observations.', 'gr-qc-0601086-1-18-3': 'One has [EQUATION]', 'gr-qc-0601086-1-18-4': 'There need to be two constraints imposed to solve the set of equations.', 'gr-qc-0601086-1-18-5': 'We have three choices: 1) holographic principle on [MATH], 2) suppose a form for [MATH] or equivalently [MATH], and 3) suppose a form for the equation of state [MATH].', 'gr-qc-0601086-1-18-6': 'The equation linking them is obtained from Eq. ([REF]), [EQUATION]', 'gr-qc-0601086-1-18-7': 'This implies the generic scale of the decay rate [MATH] is the same order as [MATH].', 'gr-qc-0601086-1-18-8': 'In the particular case [MATH] Eq. ([REF]) reduces to [EQUATION] which connects the sign of [MATH] to the sign of [MATH].', 'gr-qc-0601086-1-18-9': 'In the case that the interaction vanishes, [MATH], one returns to the condition of a cosmological constant with constant energy density (and therefore constant [MATH]).', 'gr-qc-0601086-1-18-10': 'On the other hand the interaction can either increase or decrease [MATH] depending on the sign of the interaction.', 'gr-qc-0601086-1-18-11': 'Henceforth we will refer to the rate [MATH] as the holographic rate, but it should be understood that it is an equivalent description for the time development of the dark energy density independent of any assumption based on the holographic principle.', 'gr-qc-0601086-1-19-0': 'Since the dark energy density scales as [MATH], one has [EQUATION]', 'gr-qc-0601086-1-19-1': 'Comparing to Eq. ([REF]) one obtains the relationship between the effective equation of state and the length scale [EQUATION]', 'gr-qc-0601086-1-19-2': 'This equation, which is just a consequence of dimensional analysis, via Eq. ([REF]), merely recasts the scaling exhibited by the dark energy as the ratio of the two rates, [MATH] and [MATH].', 'gr-qc-0601086-1-19-3': 'When there is no interaction the same equation applies for the native equation of state [MATH].', 'gr-qc-0601086-1-19-4': 'The familiar case of a constant dark energy density simply corresponds to the statement that the length scale does not change, [MATH].', 'gr-qc-0601086-1-19-5': 'More generally Eq. ([REF]) shows that an assumption for [MATH] based on some holographic principle determines the form of the effective equation of state even in the presence of an interaction.', 'gr-qc-0601086-1-19-6': 'If one applies Eq. ([REF]) one has [EQUATION]', 'gr-qc-0601086-1-19-7': 'Inserting this into Eq. ([REF]) and using the definition for the deceleration parameter [MATH] and [MATH], one obtains [EQUATION]', 'gr-qc-0601086-1-19-8': 'Asymptotically (i.e. for large [MATH]) the last term vanishes, so that [EQUATION]', 'gr-qc-0601086-1-19-9': 'Finally using Eq. ([REF]) one obtains [EQUATION] which indicates that the differential equation (right hand side) has a zero for [MATH] provided the term in brackets does not vanish like [MATH].', 'gr-qc-0601086-1-20-0': 'The resulting interpretation of the possible evolutions of the Universe becomes straightforward using Eq. ([REF]).', 'gr-qc-0601086-1-20-1': 'The derivative of [MATH] has fixed points at [MATH] and at some larger value where the condition [EQUATION] is satisfied.', 'gr-qc-0601086-1-20-2': 'For large times the Universe approaches this solution where the effective equations of states for matter and the dark energy become equal.', 'gr-qc-0601086-1-20-3': 'At early times the dark energy is repelled from the [MATH] and grows monitonically to a positive value.', 'gr-qc-0601086-1-20-4': 'For the case [MATH] one observes that [MATH] must be negative for asympotically large times.', 'gr-qc-0601086-1-21-0': '# Examples', 'gr-qc-0601086-1-22-0': 'A holographic condition based on relating [MATH] to the Hubble scale or to the particle horizon leads to conflict with observation.', 'gr-qc-0601086-1-22-1': 'In the former case the ratio of the dark energy density to matter density is constant.', 'gr-qc-0601086-1-22-2': 'In the latter case the equation of state of the dark energy is greater than [MATH].', 'gr-qc-0601086-1-22-3': 'Taking the length scale to be the Hubble horizon [MATH] which implies [EQUATION] so that [MATH] is a constant.', 'gr-qc-0601086-1-22-4': 'This yields [EQUATION] so that the asymptotic solution obtained in Eq. ([REF]) is identically satisfied.', 'gr-qc-0601086-1-23-0': 'Setting the holographic length scale equal to the particle horizon gives [EQUATION] while setting it equal to the future event horizon[CITATION] yields [EQUATION]', 'gr-qc-0601086-1-23-1': 'If the length scale is set equal to the Hubble horizon, the particle horizon, or the future horizon, then the rates go like [EQUATION] respectively.', 'gr-qc-0601086-1-23-2': 'Finally for a cosmological constant one has [MATH].', 'gr-qc-0601086-1-23-3': 'These conditions yield the following expressions for the effective dark energy equation of state [EQUATION] and [MATH] for a cosmological constant.', 'gr-qc-0601086-1-24-0': 'The case of the Hubble horizon is particularly simple: From Eq. ([REF]) one has that [MATH], so that the ratio of dark energy to matter is constant[CITATION].', 'gr-qc-0601086-1-24-1': 'This is independent of the existence of a interaction [MATH].', 'gr-qc-0601086-1-24-2': 'If the interaction is absent, then [MATH] and [MATH] identically.', 'gr-qc-0601086-1-24-3': 'Including an interaction these generalize to [EQUATION]', 'gr-qc-0601086-1-24-4': 'One particular mechanism for determining the holographic condition is to assume the dark energy density is created out of the vacuum as the Universe expands[CITATION].', 'gr-qc-0601086-1-24-5': 'The component is created via the Bogolubov modes which result from the expansion of the Universe and the variation of the vacuum in such a situation.', 'gr-qc-0601086-1-24-6': 'One has [EQUATION] where [MATH] is some mass of order the Planck scale.', 'gr-qc-0601086-1-24-7': 'Translating this into the varying scale [MATH], one obtains [EQUATION] where an overall constant that arises in this definition is absorbed into the definition of the order one constant coefficient [MATH].', 'gr-qc-0601086-1-24-8': 'One calculates [EQUATION]', 'gr-qc-0601086-1-24-9': 'The effective equation of state then follows from Eq. ([REF]).', 'gr-qc-0601086-1-24-10': 'In this formulation the behavior near [MATH] results from lower momentum modes which were created at earlier times when [MATH] was larger.', 'gr-qc-0601086-1-25-0': 'The asymptotic value for the effective equation of state should approach something near that of a cosmological constant to be consistent with oberservations.', 'gr-qc-0601086-1-25-1': 'Having accumulated the results for the previous examples it is easy to conclude that the future horizon[CITATION] yields an acceptable effective equation of state near [MATH], while the particle horizon has an effective equation of state which is too large.', 'gr-qc-0601086-1-25-2': 'Generally one wants a length scale rate change that becomes small compared to the Hubble expansion [MATH] so that the effective equation of state approaches [MATH] as in Eq. ([REF]).', 'gr-qc-0601086-1-25-3': 'The future horizon and other holographic conditions that result in a decreasing value for [MATH] as a function of increasing [MATH] have the desired properties for an accelerating Universe.', 'gr-qc-0601086-1-26-0': '# Interactions', 'gr-qc-0601086-1-27-0': 'The coupled equations in Eq. ([REF]) defines an interaction between the dark energy and matter components.', 'gr-qc-0601086-1-27-1': 'Given a specific rate [MATH] as a function of [MATH] and [MATH], one can study the behavior of the evolution equations with a specific behavior on the energy density.', 'gr-qc-0601086-1-27-2': 'The ratio of rates is taken to be a function of [MATH] [EQUATION] where the dimensionless constant [MATH] is included to facilitate comparison with previous works.', 'gr-qc-0601086-1-27-3': 'In particular Ref. [CITATION] assumed an interaction of this form with [MATH].', 'gr-qc-0601086-1-27-4': 'If [MATH] is the only time scale that enters into the definition, then dimensional analysis dictates this form (both sides are dimensionless).', 'gr-qc-0601086-1-27-5': 'One then obtains (we assume henceforth that [MATH]) [EQUATION]', 'gr-qc-0601086-1-27-6': 'So the effective equation of state [MATH] for the matter component depends on the interaction only.', 'gr-qc-0601086-1-27-7': 'Together with the fact that the effective equation of state [MATH] depends only on the assumption for the physical condition setting the scale [MATH], allows for a more general analysis of possible cases that give the desired equilibrium solution.', 'gr-qc-0601086-1-27-8': 'Using Eqs. ([REF]) and ([REF]) one obtains [EQUATION]', 'gr-qc-0601086-1-27-9': 'Using Eq. ([REF]) to convert this into an equation involving only [MATH], one obtains [EQUATION]', 'gr-qc-0601086-1-27-10': 'This expression is quite general, so it facilitates a more general understanding of the physical conditions required for the holographic conditions determining [MATH] and the interaction [MATH].', 'gr-qc-0601086-1-27-11': 'Furthermore the differential equation involves the effective equations of state in a simple way.', 'gr-qc-0601086-1-27-12': 'The first two terms in brackets are [MATH] while the last term is [MATH].', 'gr-qc-0601086-1-27-13': 'Together with Eq. ([REF]) it determines the acceleration of the Universe through [MATH].', 'gr-qc-0601086-1-27-14': 'One has [EQUATION]', 'gr-qc-0601086-1-27-15': 'A complete specification of the problems involves the specification of two ratios of rates as in Eq. ([REF]).', 'gr-qc-0601086-1-27-16': 'A generic choice of definition for [MATH] and [MATH] will determine the native equation of state [MATH] which will then vary with time.', 'gr-qc-0601086-1-27-17': 'Alternatively one can choose a constant [MATH] and [MATH] in which case the interaction rate is determined.', 'gr-qc-0601086-1-27-18': 'We examine these cases in more detail in the remainder of this section.', 'gr-qc-0601086-1-28-0': '## Constant equation of state', 'gr-qc-0601086-1-29-0': 'The first case we shall examine is the one where the native equation of state for dark energy is constant.', 'gr-qc-0601086-1-29-1': 'For this purpose take the equation involving the three rates, Eq. ([REF]), and eliminate the interaction term in Eq. ([REF]) in favor of the other two rates.', 'gr-qc-0601086-1-29-2': 'One obtains [EQUATION]', 'gr-qc-0601086-1-29-3': 'This equation is valid irregardless of whether [MATH] is constant in time, but we are presently interested in the constant case.', 'gr-qc-0601086-1-29-4': 'Provided the holographic rate does not have a singularity for [MATH], the right hand side has the required zero at [MATH], and for [MATH] the first two terms in brackets is positive provided [MATH].', 'gr-qc-0601086-1-29-5': 'For both the case of the holographic rate based on the particle horizon and future event horizon, there is another fixed point at positive [MATH] (see Eq. ([REF])).', 'gr-qc-0601086-1-30-0': 'Using Eqs. ([REF]) and ([REF]) one obtains the simple result [EQUATION] which shows that the fixed point occurs when [MATH].', 'gr-qc-0601086-1-30-1': 'As examples take [MATH] and [MATH].', 'gr-qc-0601086-1-30-2': 'Then the attractive fixed point occurs at [MATH] for the particle horizon (PH) case and at [MATH] for the future event horizon (FH) case, and from Eq. ([REF]) these correspond to [MATH] and [MATH] respectively.', 'gr-qc-0601086-1-30-3': 'More generally Eq. ([REF]) implies that [MATH].', 'gr-qc-0601086-1-31-0': 'In the case the scale [MATH] is identified with the Hubble scale [MATH], the differential equation vanishes and one has the relation [EQUATION]', 'gr-qc-0601086-1-31-1': 'If one further assumes a constant [MATH] then the interaction [MATH] is determined, and [MATH] is simply proportional to the constant [MATH].', 'gr-qc-0601086-1-31-2': 'The interaction rate is then proportional to the [MATH], [EQUATION]', 'gr-qc-0601086-1-31-3': 'So this holographic condition corresponds to the one in which the differential equation is identically zero and equilibrium between the dark energy density and matter density holds for all times.', 'gr-qc-0601086-1-32-0': '## Assumptions about the holographic and interaction rates', 'gr-qc-0601086-1-33-0': 'The inclusion of an iteraction term for the dark energy component can be accomodated most easily by defining an effective equation of state [MATH] which differs from the native equation of state [MATH].', 'gr-qc-0601086-1-33-1': 'By virtue of its definition it satisfies the same equation that the native equation of state satisfies in the noninteraction case, namely Eq. ([REF]).', 'gr-qc-0601086-1-33-2': 'If [MATH] then [MATH] and the universe accelerates.', 'gr-qc-0601086-1-33-3': 'The effective equation of state must lie between that of a cosmological constant ([MATH]) and the curvature component ([MATH]).', 'gr-qc-0601086-1-34-0': 'While there is a certain appeal in taking the future event horizon as the length scale for a holographic prinicple[CITATION], it is clear from the evolution equation for the dark energy that an equilibrium fixed point solution can occur for a wider variety of assumptions.', 'gr-qc-0601086-1-35-0': 'In order to examine the conditions on the interaction rate consider the cases [EQUATION] for some exponent [MATH].', 'gr-qc-0601086-1-35-1': 'One then obtains [EQUATION]', 'gr-qc-0601086-1-35-2': 'Figure 2 shows the functional dependence of [MATH] and [MATH] versus [MATH] for various choices of the holographic condition and interaction.', 'gr-qc-0601086-1-35-3': 'An intial condition of [MATH] will evolve to the right until [MATH] which respresents the asymptotic solution.', 'gr-qc-0601086-1-35-4': 'It is clear that an interaction of the form [MATH] with [MATH] can give an acceptable result with a equilibrium balance between [MATH] and [MATH] provided the holographic condition involves the future event horizon.', 'gr-qc-0601086-1-36-0': 'For [MATH] the effective equation of state [MATH] diverges for small [MATH], and in fact is less than [MATH] so does not yield an acceptable solution.', 'gr-qc-0601086-1-36-1': 'We note that for [MATH] one has [MATH] for [MATH] which may be needed for better agreement with the observational data.', 'gr-qc-0601086-1-36-2': 'An explicit example is plotted in Fig. 3 for which the iteraction is assumed to be of the form in Eq. ([REF]) with [MATH].', 'gr-qc-0601086-1-36-3': 'In fact this is the typical behavior for all [MATH].', 'gr-qc-0601086-1-37-0': '# Multicomponent Generalization', 'gr-qc-0601086-1-38-0': 'An obvious generalization of the model considered here is the multicomponent case [EQUATION] subject to the constraint [MATH].', 'gr-qc-0601086-1-38-1': 'The evolution equation for the dark energy component is [EQUATION] where [EQUATION] are quantities defined for each component other than the dark energy.', 'gr-qc-0601086-1-38-2': 'The generic solution that occurs for typical cases where the effective equations of state are monotonic functions of [MATH] is that two of the components come to equilibrium with the remaining components being diluted away by the expansion of the Univerese.', 'gr-qc-0601086-1-38-3': 'This case is of coursed realized if one adds a radiation component for example.', 'gr-qc-0601086-1-39-0': '# Summary', 'gr-qc-0601086-1-40-0': 'We have described the general conditions such that a dark energy component can arise in a coupled framework and lead to an asymptotic state where the effective equations of state of the matter component and the dark energy component are equal.', 'gr-qc-0601086-1-40-1': 'The solutions are characterized by the condition [MATH] for which the late time solution is [MATH].', 'gr-qc-0601086-1-40-2': 'This inequality can be satisfied by a variety of choices for the dark energy length scale and for the dark energy-matter interaction.', 'gr-qc-0601086-1-40-3': 'The resulting equilibrium between dark energy and matter offers a possible solution to the cosmic coincidence problem.', 'gr-qc-0601086-1-41-0': 'The dependence of the dark energy effective equation of state has been related to a general holographic condition, while the dependence of the matter effective equation of state has been related to the form of the interaction.', 'gr-qc-0601086-1-41-1': 'The effective equations of state and the general analysis of the differential equation that governs the evolution of the dark energy density should be useful for analyzing the conditions for an acceptable cosmological solution in the future.'}

{'gr-qc-0601086-2-0-0': 'The introduction of an interaction for dark energy to the standard cosmology offers a potential solution to the cosmic coincidence problem.', 'gr-qc-0601086-2-0-1': 'We examine the conditions on the dark energy density that must be satisfied for this scenario to be realized.', 'gr-qc-0601086-2-0-2': 'Under some general conditions we find a stable attractor for the evolution of the Universe in the future.', 'gr-qc-0601086-2-0-3': 'Holographic conjectures for the dark energy offer some specific examples of models with the desired properties.', 'gr-qc-0601086-2-1-0': '[MATH]', 'gr-qc-0601086-2-2-0': '# Introduction', 'gr-qc-0601086-2-3-0': 'The observation that the Universe appears to be accelerating has been one of the biggest surprises in both the astronomy and particle physics communities.', 'gr-qc-0601086-2-3-1': "This acceleration can be accounted for by adding a cosmological constant in Einstein's equations.", 'gr-qc-0601086-2-3-2': 'While particle physicists never had a definite reason for setting the cosmological constant to zero, there was always the view that like all small or zero quantities, there must be a symmetry enforcing it.', 'gr-qc-0601086-2-3-3': 'The observation that the size of the cosmological constant is neither zero nor at one of the natural scales like the Planck length or the scale of supersymmetry breaking has focused attention on explaining the dark energy component.', 'gr-qc-0601086-2-4-0': 'There are at least two requirements for any model of dark energy: (1) one must obtain in a natural way the observed size of the energy density, [MATH] eV[MATH].', 'gr-qc-0601086-2-4-1': 'Many models attempt to relate this to the observed coincidence of this scale with the scale [MATH] where [MATH] is the Hubble constant at the current epoch and [MATH] is the Planck mass.', 'gr-qc-0601086-2-4-2': '(2) One must also obtain an equation of state [MATH] which is similar to that of a cosmological constant at least in the recent past, namely [MATH].', 'gr-qc-0601086-2-4-3': 'In fact, the most recent data from the WMAP satellite and supernova and sky surveys have constrained the equation of state [MATH] at the 95% confidence level for dark energy with a constant [MATH].', 'gr-qc-0601086-2-5-0': 'An interesting approach to the dark matter problem that has arisen from recent advances in understanding string theory and black hole physics is to employ a holographic principle.', 'gr-qc-0601086-2-5-1': 'Here the motivation is the recognition that the observed size of the energy density at the present epoch seems to be consistent with taking the geometric average of the two scales [MATH] and [MATH].', 'gr-qc-0601086-2-5-2': 'Early attempts to relate the dark energy to the Hubble scale or the particle horizon did not give acceptable solutions in detail that were consistent with the observations.', 'gr-qc-0601086-2-6-0': 'The evolution of the dark energy density also depends on its precise nature.', 'gr-qc-0601086-2-6-1': 'If the dark energy is like a cosmological constant with equation of state [MATH], the dark energy density is constant and within the expanding Universe ultimately the dark energy density comes to dominate over the matter density.', 'gr-qc-0601086-2-6-2': 'In this scenario the transition from matter domination to dark energy domination is rapid, and the fact that experimental observations seem to be consistent with sizeable amounts of both matter and dark energy indicates that we are currently residing in this transition period.', 'gr-qc-0601086-2-6-3': 'The cosmic coincidence problem is the statement that it is unlikely that the current epoch coincides with this rapid transition period.', 'gr-qc-0601086-2-7-0': 'A possible solution is to assume that there is some mechanism that is converting dark energy into matter.', 'gr-qc-0601086-2-7-1': 'Cosmic antifriction forces were introduced in Ref. [CITATION] as a possible solution to the cosmic coincidence problem.', 'gr-qc-0601086-2-7-2': 'If these forces are present one can obtain a fixed ratio of matter density to dark energy density as the final state of the Universe and hence solve the coincidence problem.', 'gr-qc-0601086-2-7-3': 'The resulting cosmology is described by sourced Friedmann equations.', 'gr-qc-0601086-2-7-4': 'The basic scenario is quite simple: the natural tendency of a cosmological component with a more negative equation of state to dominate at large times the energy density of the Universe is compensated by the decay of the dark energy component into the other component(s).', 'gr-qc-0601086-2-7-5': 'At large times an equilibrium solution can develop for which the various components can coexist.', 'gr-qc-0601086-2-7-6': 'The interpretation is then that we are living in or close to this equilibrium thus eliminating the cosmic coincidence problem.', 'gr-qc-0601086-2-8-0': 'Frictional and antifrictional forces we envision here were introduced for a single component in Refs. [CITATION].', 'gr-qc-0601086-2-8-1': 'The particular case where the dark energy component is assumed to have a constant equation of state was treated in Ref. [CITATION].', 'gr-qc-0601086-2-8-2': 'These efforts were preceded by searches for attractor solutions involving scalar fields[CITATION].', 'gr-qc-0601086-2-8-3': 'Recently these kind of scenarios have been employed in models with brane-bulk energy exchange[CITATION].', 'gr-qc-0601086-2-8-4': 'Some attempts have been made to explain the size of the dark energy density on the basis of holographic ideas[CITATION], and lead to the consideration of the Hubble horizon as the holographic scale[CITATION].', 'gr-qc-0601086-2-8-5': 'A holographic model based on the future event horizon was examined in Ref. [CITATION].', 'gr-qc-0601086-2-8-6': 'Danielsson studied the effects of a transplanckian backreaction as a particular source for the holographic dark energy[CITATION].', 'gr-qc-0601086-2-8-7': 'More recenlty an emphasis on the case of interacting dark energy has been discussed in Refs. [CITATION].', 'gr-qc-0601086-2-9-0': 'Ultimately one needs to appeal to the observational data to constrain these ideas.', 'gr-qc-0601086-2-9-1': 'In this direction Wang, et al. have begun a more detailed comparison of the predictions of a holographic universe with interaction with the observational data[CITATION].', 'gr-qc-0601086-2-9-2': 'In addition to the overall development of the densities of the cosmological components there are other observables which are sensitive to the modified scenarios[CITATION].', 'gr-qc-0601086-2-10-0': '# Framework of Sourced Friedmann Equations', 'gr-qc-0601086-2-11-0': 'We assume two component equations for dark energy and matter, [EQUATION]', 'gr-qc-0601086-2-11-1': 'The equality of [MATH] in the two equations guarantees the overall conservation of the energy-momentum tensor, and positive [MATH] can be interpreted as a transfer from the dark energy component to the matter component.', 'gr-qc-0601086-2-11-2': 'Presumably this arises from some microscopic mechanism, but we do not specify one here.', 'gr-qc-0601086-2-11-3': 'The quantity [MATH] is the matter component for which one usually takes [MATH], but we temporarily retain a nonzero equation of state so as to retain the generality of an two interacting components.', 'gr-qc-0601086-2-12-0': 'One can define effective equations of state, [EQUATION]', 'gr-qc-0601086-2-12-1': 'Define the ratio [MATH] and the rate [MATH], then the effective equations of state are expressed in terms of the native equations of state and a ratio of rates as [EQUATION]', 'gr-qc-0601086-2-12-2': 'The time evolution of the ratio [MATH] is then [EQUATION]', 'gr-qc-0601086-2-12-3': 'The general behavior of solutions can be ascertained from this equation.', 'gr-qc-0601086-2-12-4': 'The stable points in the evolution are obtained when either [MATH] or when [MATH].', 'gr-qc-0601086-2-12-5': 'If [MATH] then the vanishing dark energy density ([MATH]) solution will apply in the infinite past while a constant nonzero ratio of dark energy to matter will apply in the infinite future with equal effective equations of state.', 'gr-qc-0601086-2-13-0': 'Using the definitions [EQUATION] which satisfy a Friedmann equation [EQUATION] one can convert to the physical parameters [EQUATION]', 'gr-qc-0601086-2-13-1': 'One can write the differential equation in the suggestive form using [MATH], [EQUATION] where [MATH].', 'gr-qc-0601086-2-13-2': 'The utility of the evolution equation in this form is that it allows one to understand in a general way the possible solutions for [MATH].', 'gr-qc-0601086-2-13-3': 'Firstly if one turns off the interaction ([MATH]) one recovers the usual evolution equation with the effective equations of state replaced with the usual equations of state [MATH] and [MATH] (we will call these the native equations of state).', 'gr-qc-0601086-2-13-4': 'For constant [MATH] and [MATH] and [MATH] then the dark energy evolution proceeds from the fixed points of the differential equation (zeros of the right hand side) in the usual way from [MATH] to [MATH] in the usual way.', 'gr-qc-0601086-2-13-5': 'When an interaction is present a more interesting solution may occur where the evolution of [MATH] approaches a condition where instead [MATH] at which [MATH] approaches a fixed asymptotic value for large time less than 1.', 'gr-qc-0601086-2-13-6': 'For a properly chosen interaction this might constitue a solution to the cosmic concidence problem.', 'gr-qc-0601086-2-14-0': 'For illustration one can also consider the (redundant) differential equation for the matter density, [EQUATION]', 'gr-qc-0601086-2-14-1': 'An initial condition [MATH] is inevitably driven to an asymptotic solution for which [MATH] and [MATH] and [MATH] approach their limiting (equal) value.', 'gr-qc-0601086-2-14-2': 'This behavior is displayed for a specific solution[CITATION] shown in Fig. 1.', 'gr-qc-0601086-2-15-0': 'Two physical assumptions must be provided to determine the evolution of the parameters of the universe.', 'gr-qc-0601086-2-15-1': 'A holographic condition on the dark energy [MATH] will determine by itself the effective equation of state [MATH].', 'gr-qc-0601086-2-15-2': 'A specific specification for the source [MATH] will determine the effective matter equation of state [MATH].', 'gr-qc-0601086-2-15-3': 'Alternatively one of these conditions can be replaced by the assumption that [MATH] is constant.', 'gr-qc-0601086-2-16-0': 'One typically obtains an equation for the evolution of the dark energy density of the form [EQUATION]', 'gr-qc-0601086-2-16-1': 'The evolution of [MATH] with respect to [MATH] or [MATH] is qualitatively determined by the fixed points of the differential equation in Eq. ([REF]) which are determined by the condition [MATH].', 'gr-qc-0601086-2-16-2': 'Examining the equations in Eqs. ([REF]), it would appear that this condition would require that [MATH] and then that [MATH] via Eq. [REF].', 'gr-qc-0601086-2-16-3': 'However, as shown in Ref. [CITATION] if one of the fixed points occurs for the physically interesting value of [MATH], then one can have the Universe with close to vanishing dark energy simultaneously with [MATH].', 'gr-qc-0601086-2-16-4': 'In the remainder of this paper we explore the general conditions that give rise to this type of equilibrium solution.', 'gr-qc-0601086-2-17-0': '# Dimensional analysis', 'gr-qc-0601086-2-18-0': 'First define a length scale [MATH] via [EQUATION] where the constant [MATH] represents an order one constant.', 'gr-qc-0601086-2-18-1': 'Specific assumptions might associate the length scale with a physical length such as the Hubble horizon, the particle horizon, the future event horizon, or perhaps some other parameter.', 'gr-qc-0601086-2-18-2': 'The common feature of any realistic choice is that the length scale at the present epoch should be of order the size of the Universe so as to obtain the appropriate size of the dark energy density consistent with experimental observations.', 'gr-qc-0601086-2-18-3': 'One has [EQUATION]', 'gr-qc-0601086-2-18-4': 'There need to be two constraints imposed to solve the set of equations.', 'gr-qc-0601086-2-18-5': 'We have three choices: 1) holographic principle on [MATH], 2) suppose a form for [MATH] or equivalently [MATH], and 3) suppose a form for the equation of state [MATH].', 'gr-qc-0601086-2-18-6': 'The equation linking them is obtained from Eq. ([REF]), [EQUATION]', 'gr-qc-0601086-2-18-7': 'This implies the generic scale of the decay rate [MATH] is the same order as [MATH].', 'gr-qc-0601086-2-18-8': 'In the particular case [MATH] Eq. ([REF]) reduces to [EQUATION] which connects the sign of [MATH] to the sign of [MATH].', 'gr-qc-0601086-2-18-9': 'In the case that the interaction vanishes, [MATH], one returns to the condition of a cosmological constant with constant energy density (and therefore constant [MATH]).', 'gr-qc-0601086-2-18-10': 'On the other hand the interaction can either increase or decrease [MATH] depending on the sign of the interaction.', 'gr-qc-0601086-2-18-11': 'Henceforth we will refer to the rate [MATH] as the holographic rate, but it should be understood that it is an equivalent description for the time development of the dark energy density independent of any assumption based on the holographic principle.', 'gr-qc-0601086-2-19-0': 'Since the dark energy density scales as [MATH], one has [EQUATION]', 'gr-qc-0601086-2-19-1': 'Comparing to Eq. ([REF]) one obtains the relationship between the effective equation of state and the length scale [EQUATION]', 'gr-qc-0601086-2-19-2': 'This equation, which is just a consequence of dimensional analysis, via Eq. ([REF]), merely recasts the scaling exhibited by the dark energy as the ratio of the two rates, [MATH] and [MATH].', 'gr-qc-0601086-2-19-3': 'When there is no interaction the same equation applies for the native equation of state [MATH].', 'gr-qc-0601086-2-19-4': 'The familiar case of a constant dark energy density simply corresponds to the statement that the length scale does not change, [MATH].', 'gr-qc-0601086-2-19-5': 'More generally Eq. ([REF]) shows that an assumption for [MATH] based on some holographic principle determines the form of the effective equation of state even in the presence of an interaction.', 'gr-qc-0601086-2-19-6': 'If one applies Eq. ([REF]) one has [EQUATION]', 'gr-qc-0601086-2-19-7': 'Inserting this into Eq. ([REF]) and using the definition for the deceleration parameter [MATH] and [MATH], one obtains [EQUATION]', 'gr-qc-0601086-2-19-8': 'Asymptotically (i.e. for large [MATH]) the last term vanishes, so that [EQUATION]', 'gr-qc-0601086-2-19-9': 'Finally using Eq. ([REF]) one obtains [EQUATION] which indicates that the differential equation (right hand side) has a zero for [MATH] provided the term in brackets does not vanish like [MATH].', 'gr-qc-0601086-2-20-0': 'The resulting interpretation of the possible evolutions of the Universe becomes straightforward using Eq. ([REF]).', 'gr-qc-0601086-2-20-1': 'The derivative of [MATH] has fixed points at [MATH] and at some larger value where the condition [EQUATION] is satisfied.', 'gr-qc-0601086-2-20-2': 'For large times the Universe approaches this solution where the effective equations of states for matter and the dark energy become equal.', 'gr-qc-0601086-2-20-3': 'At early times the dark energy is repelled from the [MATH] and grows monitonically to a positive value.', 'gr-qc-0601086-2-20-4': 'For the case [MATH] one observes that [MATH] must be negative for asympotically large times.', 'gr-qc-0601086-2-21-0': '# Examples', 'gr-qc-0601086-2-22-0': 'A holographic condition based on relating [MATH] to the Hubble scale or to the particle horizon leads to conflict with observation.', 'gr-qc-0601086-2-22-1': 'In the former case the ratio of the dark energy density to matter density is constant.', 'gr-qc-0601086-2-22-2': 'In the latter case the equation of state of the dark energy is greater than [MATH].', 'gr-qc-0601086-2-22-3': 'Taking the length scale to be the Hubble horizon [MATH] which implies [EQUATION] so that [MATH] is a constant.', 'gr-qc-0601086-2-22-4': 'This yields [EQUATION] so that the asymptotic solution obtained in Eq. ([REF]) is identically satisfied.', 'gr-qc-0601086-2-23-0': 'Setting the holographic length scale equal to the particle horizon gives [EQUATION] while setting it equal to the future event horizon[CITATION] yields [EQUATION]', 'gr-qc-0601086-2-23-1': 'If the length scale is set equal to the Hubble horizon, the particle horizon, or the future horizon, then the rates go like [EQUATION] respectively.', 'gr-qc-0601086-2-23-2': 'Finally for a cosmological constant one has [MATH].', 'gr-qc-0601086-2-23-3': 'These conditions yield the following expressions for the effective dark energy equation of state [EQUATION] and [MATH] for a cosmological constant.', 'gr-qc-0601086-2-24-0': 'The case of the Hubble horizon is particularly simple: From Eq. ([REF]) one has that [MATH], so that the ratio of dark energy to matter is constant[CITATION].', 'gr-qc-0601086-2-24-1': 'This is independent of the existence of a interaction [MATH].', 'gr-qc-0601086-2-24-2': 'If the interaction is absent, then [MATH] and [MATH] identically.', 'gr-qc-0601086-2-24-3': 'Including an interaction these generalize to [EQUATION]', 'gr-qc-0601086-2-24-4': 'One particular mechanism for determining the holographic condition is to assume the dark energy density is created out of the vacuum as the Universe expands[CITATION].', 'gr-qc-0601086-2-24-5': 'The component is created via the Bogolubov modes which result from the expansion of the Universe and the variation of the vacuum in such a situation.', 'gr-qc-0601086-2-24-6': 'One has [EQUATION] where [MATH] is some mass of order the Planck scale.', 'gr-qc-0601086-2-24-7': 'Translating this into the varying scale [MATH], one obtains [EQUATION] where an overall constant that arises in this definition is absorbed into the definition of the order one constant coefficient [MATH].', 'gr-qc-0601086-2-24-8': 'One calculates [EQUATION]', 'gr-qc-0601086-2-24-9': 'The effective equation of state then follows from Eq. ([REF]).', 'gr-qc-0601086-2-24-10': 'In this formulation the behavior near [MATH] results from lower momentum modes which were created at earlier times when [MATH] was larger.', 'gr-qc-0601086-2-25-0': 'The asymptotic value for the effective equation of state should approach something near that of a cosmological constant to be consistent with oberservations.', 'gr-qc-0601086-2-25-1': 'Having accumulated the results for the previous examples it is easy to conclude that the future horizon[CITATION] yields an acceptable effective equation of state near [MATH], while the particle horizon has an effective equation of state which is too large.', 'gr-qc-0601086-2-25-2': 'Generally one wants a length scale rate change that becomes small compared to the Hubble expansion [MATH] so that the effective equation of state approaches [MATH] as in Eq. ([REF]).', 'gr-qc-0601086-2-25-3': 'The future horizon and other holographic conditions that result in a decreasing value for [MATH] as a function of increasing [MATH] have the desired properties for an accelerating Universe.', 'gr-qc-0601086-2-26-0': '# Interactions', 'gr-qc-0601086-2-27-0': 'The coupled equations in Eq. ([REF]) defines an interaction between the dark energy and matter components.', 'gr-qc-0601086-2-27-1': 'Given a specific rate [MATH] as a function of [MATH] and [MATH], one can study the behavior of the evolution equations with a specific behavior on the energy density.', 'gr-qc-0601086-2-27-2': 'The ratio of rates is taken to be a function of [MATH] [EQUATION] where the dimensionless constant [MATH] is included to facilitate comparison with previous works.', 'gr-qc-0601086-2-27-3': 'In particular Ref. [CITATION] assumed an interaction of this form with [MATH].', 'gr-qc-0601086-2-27-4': 'If [MATH] is the only time scale that enters into the definition, then dimensional analysis dictates this form (both sides are dimensionless).', 'gr-qc-0601086-2-27-5': 'One then obtains (we assume henceforth that [MATH]) [EQUATION]', 'gr-qc-0601086-2-27-6': 'So the effective equation of state [MATH] for the matter component depends on the interaction only.', 'gr-qc-0601086-2-27-7': 'Together with the fact that the effective equation of state [MATH] depends only on the assumption for the physical condition setting the scale [MATH], allows for a more general analysis of possible cases that give the desired equilibrium solution.', 'gr-qc-0601086-2-27-8': 'Using Eqs. ([REF]) and ([REF]) one obtains [EQUATION]', 'gr-qc-0601086-2-27-9': 'Using Eq. ([REF]) to convert this into an equation involving only [MATH], one obtains [EQUATION]', 'gr-qc-0601086-2-27-10': 'This expression is quite general, so it facilitates a more general understanding of the physical conditions required for the holographic conditions determining [MATH] and the interaction [MATH].', 'gr-qc-0601086-2-27-11': 'Furthermore the differential equation involves the effective equations of state in a simple way.', 'gr-qc-0601086-2-27-12': 'The first two terms in brackets are [MATH] while the last term is [MATH].', 'gr-qc-0601086-2-27-13': 'Together with Eq. ([REF]) it determines the acceleration of the Universe through [MATH].', 'gr-qc-0601086-2-27-14': 'One has [EQUATION]', 'gr-qc-0601086-2-27-15': 'A complete specification of the problems involves the specification of two ratios of rates as in Eq. ([REF]).', 'gr-qc-0601086-2-27-16': 'A generic choice of definition for [MATH] and [MATH] will determine the native equation of state [MATH] which will then vary with time.', 'gr-qc-0601086-2-27-17': 'Alternatively one can choose a constant [MATH] and [MATH] in which case the interaction rate is determined.', 'gr-qc-0601086-2-27-18': 'We examine these cases in more detail in the remainder of this section.', 'gr-qc-0601086-2-28-0': '## Constant equation of state', 'gr-qc-0601086-2-29-0': 'The first case we shall examine is the one where the native equation of state for dark energy is constant.', 'gr-qc-0601086-2-29-1': 'For this purpose take the equation involving the three rates, Eq. ([REF]), and eliminate the interaction term in Eq. ([REF]) in favor of the other two rates.', 'gr-qc-0601086-2-29-2': 'One obtains [EQUATION]', 'gr-qc-0601086-2-29-3': 'This equation is valid irregardless of whether [MATH] is constant in time, but we are presently interested in the constant case.', 'gr-qc-0601086-2-29-4': 'Provided the holographic rate does not have a singularity for [MATH], the right hand side has the required zero at [MATH], and for [MATH] the first two terms in brackets is positive provided [MATH].', 'gr-qc-0601086-2-29-5': 'For both the case of the holographic rate based on the particle horizon and future event horizon, there is another fixed point at positive [MATH] (see Eq. ([REF])).', 'gr-qc-0601086-2-30-0': 'Using Eqs. ([REF]) and ([REF]) one obtains the simple result [EQUATION] which shows that the fixed point occurs when [MATH].', 'gr-qc-0601086-2-30-1': 'As examples take [MATH] and [MATH].', 'gr-qc-0601086-2-30-2': 'Then the attractive fixed point occurs at [MATH] for the particle horizon (PH) case and at [MATH] for the future event horizon (FH) case, and from Eq. ([REF]) these correspond to [MATH] and [MATH] respectively.', 'gr-qc-0601086-2-30-3': 'More generally Eq. ([REF]) implies that [MATH].', 'gr-qc-0601086-2-31-0': 'In the case the scale [MATH] is identified with the Hubble scale [MATH], the differential equation vanishes and one has the relation [EQUATION]', 'gr-qc-0601086-2-31-1': 'If one further assumes a constant [MATH] then the interaction [MATH] is determined, and [MATH] is simply proportional to the constant [MATH].', 'gr-qc-0601086-2-31-2': 'The interaction rate is then proportional to the [MATH], [EQUATION]', 'gr-qc-0601086-2-31-3': 'So this holographic condition corresponds to the one in which the differential equation is identically zero and equilibrium between the dark energy density and matter density holds for all times.', 'gr-qc-0601086-2-32-0': '## Assumptions about the holographic and interaction rates', 'gr-qc-0601086-2-33-0': 'The inclusion of an iteraction term for the dark energy component can be accomodated most easily by defining an effective equation of state [MATH] which differs from the native equation of state [MATH].', 'gr-qc-0601086-2-33-1': 'By virtue of its definition it satisfies the same equation that the native equation of state satisfies in the noninteraction case, namely Eq. ([REF]).', 'gr-qc-0601086-2-33-2': 'If [MATH] then [MATH] and the universe accelerates.', 'gr-qc-0601086-2-33-3': 'The effective equation of state must lie between that of a cosmological constant ([MATH]) and the curvature component ([MATH]).', 'gr-qc-0601086-2-34-0': 'While there is a certain appeal in taking the future event horizon as the length scale for a holographic prinicple[CITATION], it is clear from the evolution equation for the dark energy that an equilibrium fixed point solution can occur for a wider variety of assumptions.', 'gr-qc-0601086-2-35-0': 'In order to examine the conditions on the interaction rate consider the cases [EQUATION] for some exponent [MATH].', 'gr-qc-0601086-2-35-1': 'One then obtains [EQUATION]', 'gr-qc-0601086-2-35-2': 'Figure 2 shows the functional dependence of [MATH] and [MATH] versus [MATH] for various choices of the holographic condition and interaction.', 'gr-qc-0601086-2-35-3': 'An intial condition of [MATH] will evolve to the right until [MATH] which respresents the asymptotic solution.', 'gr-qc-0601086-2-35-4': 'It is clear that an interaction of the form [MATH] with [MATH] can give an acceptable result with a equilibrium balance between [MATH] and [MATH] provided the holographic condition involves the future event horizon.', 'gr-qc-0601086-2-36-0': 'For [MATH] the effective equation of state [MATH] diverges for small [MATH], and in fact is less than [MATH] so does not yield an acceptable solution.', 'gr-qc-0601086-2-36-1': 'We note that for [MATH] one has [MATH] for [MATH] which may be needed for better agreement with the observational data.', 'gr-qc-0601086-2-36-2': 'An explicit example is plotted in Fig. 3 for which the iteraction is assumed to be of the form in Eq. ([REF]) with [MATH].', 'gr-qc-0601086-2-36-3': 'In fact this is the typical behavior for all [MATH].', 'gr-qc-0601086-2-37-0': '# Multicomponent Generalization', 'gr-qc-0601086-2-38-0': 'An obvious generalization of the model considered here is the multicomponent case [EQUATION] subject to the constraint [MATH].', 'gr-qc-0601086-2-38-1': 'The evolution equation for the dark energy component is [EQUATION] where [EQUATION] are quantities defined for each component other than the dark energy.', 'gr-qc-0601086-2-38-2': 'The generic solution that occurs for typical cases where the effective equations of state are monotonic functions of [MATH] is that two of the components come to equilibrium with the remaining components being diluted away by the expansion of the Univerese.', 'gr-qc-0601086-2-38-3': 'This case is of coursed realized if one adds a radiation component for example.', 'gr-qc-0601086-2-39-0': '# Summary', 'gr-qc-0601086-2-40-0': 'We have described the general conditions such that a dark energy component can arise in a coupled framework and lead to an asymptotic state where the effective equations of state of the matter component and the dark energy component are equal.', 'gr-qc-0601086-2-40-1': 'The solutions are characterized by the condition [MATH] for which the late time solution is [MATH].', 'gr-qc-0601086-2-40-2': 'This inequality can be satisfied by a variety of choices for the dark energy length scale and for the dark energy-matter interaction.', 'gr-qc-0601086-2-40-3': 'The resulting equilibrium between dark energy and matter offers a possible solution to the cosmic coincidence problem.', 'gr-qc-0601086-2-41-0': 'The dependence of the dark energy effective equation of state has been related to a general holographic condition, while the dependence of the matter effective equation of state has been related to the form of the interaction.', 'gr-qc-0601086-2-41-1': 'The effective equations of state and the general analysis of the differential equation that governs the evolution of the dark energy density should be useful for analyzing the conditions for an acceptable cosmological solution in the future.'}

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['gr-qc-0601086-1-11-3', 'gr-qc-0601086-2-11-3'], ['gr-qc-0601086-1-18-0', 'gr-qc-0601086-2-18-0'], ['gr-qc-0601086-1-18-1', 'gr-qc-0601086-2-18-1'], ['gr-qc-0601086-1-18-2', 'gr-qc-0601086-2-18-2'], ['gr-qc-0601086-1-18-4', 'gr-qc-0601086-2-18-4'], ['gr-qc-0601086-1-18-5', 'gr-qc-0601086-2-18-5'], ['gr-qc-0601086-1-18-6', 'gr-qc-0601086-2-18-6'], ['gr-qc-0601086-1-18-7', 'gr-qc-0601086-2-18-7'], ['gr-qc-0601086-1-18-8', 'gr-qc-0601086-2-18-8'], ['gr-qc-0601086-1-18-9', 'gr-qc-0601086-2-18-9'], ['gr-qc-0601086-1-18-10', 'gr-qc-0601086-2-18-10'], ['gr-qc-0601086-1-18-11', 'gr-qc-0601086-2-18-11'], ['gr-qc-0601086-1-36-0', 'gr-qc-0601086-2-36-0'], ['gr-qc-0601086-1-36-1', 'gr-qc-0601086-2-36-1'], ['gr-qc-0601086-1-36-2', 'gr-qc-0601086-2-36-2'], ['gr-qc-0601086-1-36-3', 'gr-qc-0601086-2-36-3'], ['gr-qc-0601086-1-40-0', 'gr-qc-0601086-2-40-0'], ['gr-qc-0601086-1-40-1', 'gr-qc-0601086-2-40-1'], ['gr-qc-0601086-1-40-2', 'gr-qc-0601086-2-40-2'], ['gr-qc-0601086-1-40-3', 'gr-qc-0601086-2-40-3'], ['gr-qc-0601086-1-6-0', 'gr-qc-0601086-2-6-0'], ['gr-qc-0601086-1-6-1', 'gr-qc-0601086-2-6-1'], ['gr-qc-0601086-1-6-2', 'gr-qc-0601086-2-6-2'], ['gr-qc-0601086-1-6-3', 'gr-qc-0601086-2-6-3'], ['gr-qc-0601086-1-9-0', 'gr-qc-0601086-2-9-0'], ['gr-qc-0601086-1-9-1', 'gr-qc-0601086-2-9-1'], ['gr-qc-0601086-1-9-2', 'gr-qc-0601086-2-9-2'], ['gr-qc-0601086-1-15-0', 'gr-qc-0601086-2-15-0'], ['gr-qc-0601086-1-15-1', 'gr-qc-0601086-2-15-1'], ['gr-qc-0601086-1-15-2', 'gr-qc-0601086-2-15-2'], ['gr-qc-0601086-1-15-3', 'gr-qc-0601086-2-15-3'], ['gr-qc-0601086-1-33-0', 'gr-qc-0601086-2-33-0'], ['gr-qc-0601086-1-33-1', 'gr-qc-0601086-2-33-1'], ['gr-qc-0601086-1-33-2', 'gr-qc-0601086-2-33-2'], ['gr-qc-0601086-1-33-3', 'gr-qc-0601086-2-33-3'], ['gr-qc-0601086-1-35-0', 'gr-qc-0601086-2-35-0'], ['gr-qc-0601086-1-35-1', 'gr-qc-0601086-2-35-1'], ['gr-qc-0601086-1-35-2', 'gr-qc-0601086-2-35-2'], ['gr-qc-0601086-1-35-3', 'gr-qc-0601086-2-35-3'], ['gr-qc-0601086-1-35-4', 'gr-qc-0601086-2-35-4'], ['gr-qc-0601086-1-31-0', 'gr-qc-0601086-2-31-0'], ['gr-qc-0601086-1-31-1', 'gr-qc-0601086-2-31-1'], ['gr-qc-0601086-1-31-2', 'gr-qc-0601086-2-31-2'], ['gr-qc-0601086-1-31-3', 'gr-qc-0601086-2-31-3'], ['gr-qc-0601086-1-13-0', 'gr-qc-0601086-2-13-0'], ['gr-qc-0601086-1-13-1', 'gr-qc-0601086-2-13-1'], ['gr-qc-0601086-1-13-2', 'gr-qc-0601086-2-13-2'], ['gr-qc-0601086-1-13-3', 'gr-qc-0601086-2-13-3'], ['gr-qc-0601086-1-13-4', 'gr-qc-0601086-2-13-4'], ['gr-qc-0601086-1-13-5', 'gr-qc-0601086-2-13-5'], ['gr-qc-0601086-1-13-6', 'gr-qc-0601086-2-13-6'], ['gr-qc-0601086-1-34-0', 'gr-qc-0601086-2-34-0'], ['gr-qc-0601086-1-3-0', 'gr-qc-0601086-2-3-0'], ['gr-qc-0601086-1-3-1', 'gr-qc-0601086-2-3-1'], ['gr-qc-0601086-1-3-2', 'gr-qc-0601086-2-3-2'], ['gr-qc-0601086-1-3-3', 'gr-qc-0601086-2-3-3'], ['gr-qc-0601086-1-7-0', 'gr-qc-0601086-2-7-0'], ['gr-qc-0601086-1-7-1', 'gr-qc-0601086-2-7-1'], ['gr-qc-0601086-1-7-2', 'gr-qc-0601086-2-7-2'], ['gr-qc-0601086-1-7-3', 'gr-qc-0601086-2-7-3'], ['gr-qc-0601086-1-7-4', 'gr-qc-0601086-2-7-4'], ['gr-qc-0601086-1-7-5', 'gr-qc-0601086-2-7-5'], ['gr-qc-0601086-1-7-6', 'gr-qc-0601086-2-7-6'], ['gr-qc-0601086-1-22-0', 'gr-qc-0601086-2-22-0'], ['gr-qc-0601086-1-22-1', 'gr-qc-0601086-2-22-1'], ['gr-qc-0601086-1-22-2', 'gr-qc-0601086-2-22-2'], ['gr-qc-0601086-1-22-3', 'gr-qc-0601086-2-22-3'], ['gr-qc-0601086-1-22-4', 'gr-qc-0601086-2-22-4'], ['gr-qc-0601086-1-16-0', 'gr-qc-0601086-2-16-0'], ['gr-qc-0601086-1-16-1', 'gr-qc-0601086-2-16-1'], ['gr-qc-0601086-1-16-2', 'gr-qc-0601086-2-16-2'], ['gr-qc-0601086-1-16-3', 'gr-qc-0601086-2-16-3'], ['gr-qc-0601086-1-16-4', 'gr-qc-0601086-2-16-4'], ['gr-qc-0601086-1-20-0', 'gr-qc-0601086-2-20-0'], ['gr-qc-0601086-1-20-1', 'gr-qc-0601086-2-20-1'], ['gr-qc-0601086-1-20-2', 'gr-qc-0601086-2-20-2'], ['gr-qc-0601086-1-20-3', 'gr-qc-0601086-2-20-3'], ['gr-qc-0601086-1-20-4', 'gr-qc-0601086-2-20-4'], ['gr-qc-0601086-1-29-0', 'gr-qc-0601086-2-29-0'], ['gr-qc-0601086-1-29-1', 'gr-qc-0601086-2-29-1'], ['gr-qc-0601086-1-29-3', 'gr-qc-0601086-2-29-3'], ['gr-qc-0601086-1-29-4', 'gr-qc-0601086-2-29-4'], ['gr-qc-0601086-1-29-5', 'gr-qc-0601086-2-29-5'], ['gr-qc-0601086-1-8-0', 'gr-qc-0601086-2-8-0'], ['gr-qc-0601086-1-8-1', 'gr-qc-0601086-2-8-1'], ['gr-qc-0601086-1-8-2', 'gr-qc-0601086-2-8-4'], ['gr-qc-0601086-1-8-3', 'gr-qc-0601086-2-8-5'], ['gr-qc-0601086-1-8-4', 'gr-qc-0601086-2-8-6'], ['gr-qc-0601086-1-8-5', 'gr-qc-0601086-2-8-7'], ['gr-qc-0601086-1-4-0', 'gr-qc-0601086-2-4-0'], ['gr-qc-0601086-1-4-1', 'gr-qc-0601086-2-4-1'], ['gr-qc-0601086-1-4-2', 'gr-qc-0601086-2-4-2'], ['gr-qc-0601086-1-4-3', 'gr-qc-0601086-2-4-3'], ['gr-qc-0601086-1-30-0', 'gr-qc-0601086-2-30-0'], ['gr-qc-0601086-1-30-1', 'gr-qc-0601086-2-30-1'], ['gr-qc-0601086-1-30-2', 'gr-qc-0601086-2-30-2'], ['gr-qc-0601086-1-30-3', 'gr-qc-0601086-2-30-3'], ['gr-qc-0601086-1-27-0', 'gr-qc-0601086-2-27-0'], ['gr-qc-0601086-1-27-1', 'gr-qc-0601086-2-27-1'], ['gr-qc-0601086-1-27-2', 'gr-qc-0601086-2-27-2'], ['gr-qc-0601086-1-27-3', 'gr-qc-0601086-2-27-3'], ['gr-qc-0601086-1-27-4', 'gr-qc-0601086-2-27-4'], ['gr-qc-0601086-1-27-5', 'gr-qc-0601086-2-27-5'], ['gr-qc-0601086-1-27-6', 'gr-qc-0601086-2-27-6'], ['gr-qc-0601086-1-27-7', 'gr-qc-0601086-2-27-7'], ['gr-qc-0601086-1-27-8', 'gr-qc-0601086-2-27-8'], ['gr-qc-0601086-1-27-9', 'gr-qc-0601086-2-27-9'], ['gr-qc-0601086-1-27-10', 'gr-qc-0601086-2-27-10'], ['gr-qc-0601086-1-27-11', 'gr-qc-0601086-2-27-11'], ['gr-qc-0601086-1-27-12', 'gr-qc-0601086-2-27-12'], ['gr-qc-0601086-1-27-13', 'gr-qc-0601086-2-27-13'], ['gr-qc-0601086-1-27-15', 'gr-qc-0601086-2-27-15'], ['gr-qc-0601086-1-27-16', 'gr-qc-0601086-2-27-16'], ['gr-qc-0601086-1-27-17', 'gr-qc-0601086-2-27-17'], ['gr-qc-0601086-1-27-18', 'gr-qc-0601086-2-27-18'], ['gr-qc-0601086-1-41-0', 'gr-qc-0601086-2-41-0'], ['gr-qc-0601086-1-41-1', 'gr-qc-0601086-2-41-1'], ['gr-qc-0601086-1-25-0', 'gr-qc-0601086-2-25-0'], ['gr-qc-0601086-1-25-1', 'gr-qc-0601086-2-25-1'], ['gr-qc-0601086-1-25-2', 'gr-qc-0601086-2-25-2'], ['gr-qc-0601086-1-25-3', 'gr-qc-0601086-2-25-3'], ['gr-qc-0601086-1-5-0', 'gr-qc-0601086-2-5-0'], ['gr-qc-0601086-1-5-1', 'gr-qc-0601086-2-5-1'], ['gr-qc-0601086-1-5-2', 'gr-qc-0601086-2-5-2'], ['gr-qc-0601086-1-23-0', 'gr-qc-0601086-2-23-0'], ['gr-qc-0601086-1-23-1', 'gr-qc-0601086-2-23-1'], ['gr-qc-0601086-1-23-2', 'gr-qc-0601086-2-23-2'], ['gr-qc-0601086-1-23-3', 'gr-qc-0601086-2-23-3'], ['gr-qc-0601086-1-24-0', 'gr-qc-0601086-2-24-0'], ['gr-qc-0601086-1-24-1', 'gr-qc-0601086-2-24-1'], ['gr-qc-0601086-1-24-2', 'gr-qc-0601086-2-24-2'], ['gr-qc-0601086-1-24-3', 'gr-qc-0601086-2-24-3'], ['gr-qc-0601086-1-24-4', 'gr-qc-0601086-2-24-4'], ['gr-qc-0601086-1-24-5', 'gr-qc-0601086-2-24-5'], ['gr-qc-0601086-1-24-6', 'gr-qc-0601086-2-24-6'], ['gr-qc-0601086-1-24-7', 'gr-qc-0601086-2-24-7'], ['gr-qc-0601086-1-24-9', 'gr-qc-0601086-2-24-9'], ['gr-qc-0601086-1-24-10', 'gr-qc-0601086-2-24-10'], ['gr-qc-0601086-1-19-0', 'gr-qc-0601086-2-19-0'], ['gr-qc-0601086-1-19-1', 'gr-qc-0601086-2-19-1'], ['gr-qc-0601086-1-19-2', 'gr-qc-0601086-2-19-2'], ['gr-qc-0601086-1-19-3', 'gr-qc-0601086-2-19-3'], ['gr-qc-0601086-1-19-4', 'gr-qc-0601086-2-19-4'], ['gr-qc-0601086-1-19-5', 'gr-qc-0601086-2-19-5'], ['gr-qc-0601086-1-19-6', 'gr-qc-0601086-2-19-6'], ['gr-qc-0601086-1-19-7', 'gr-qc-0601086-2-19-7'], ['gr-qc-0601086-1-19-8', 'gr-qc-0601086-2-19-8'], ['gr-qc-0601086-1-19-9', 'gr-qc-0601086-2-19-9'], ['gr-qc-0601086-2-12-0', 'gr-qc-0601086-3-12-0'], ['gr-qc-0601086-2-12-1', 'gr-qc-0601086-3-12-1'], ['gr-qc-0601086-2-12-2', 'gr-qc-0601086-3-12-2'], ['gr-qc-0601086-2-12-3', 'gr-qc-0601086-3-12-3'], ['gr-qc-0601086-2-12-4', 'gr-qc-0601086-3-12-4'], ['gr-qc-0601086-2-12-5', 'gr-qc-0601086-3-12-5'], ['gr-qc-0601086-2-24-0', 'gr-qc-0601086-3-24-0'], ['gr-qc-0601086-2-24-1', 'gr-qc-0601086-3-24-1'], ['gr-qc-0601086-2-24-2', 'gr-qc-0601086-3-24-2'], ['gr-qc-0601086-2-24-3', 'gr-qc-0601086-3-24-3'], ['gr-qc-0601086-2-24-4', 'gr-qc-0601086-3-24-4'], ['gr-qc-0601086-2-24-5', 'gr-qc-0601086-3-24-5'], ['gr-qc-0601086-2-24-6', 'gr-qc-0601086-3-24-6'], ['gr-qc-0601086-2-24-7', 'gr-qc-0601086-3-24-7'], ['gr-qc-0601086-2-24-9', 'gr-qc-0601086-3-24-9'], ['gr-qc-0601086-2-24-10', 'gr-qc-0601086-3-24-10'], ['gr-qc-0601086-2-3-0', 'gr-qc-0601086-3-3-0'], ['gr-qc-0601086-2-3-1', 'gr-qc-0601086-3-3-1'], ['gr-qc-0601086-2-3-2', 'gr-qc-0601086-3-3-2'], ['gr-qc-0601086-2-3-3', 'gr-qc-0601086-3-3-3'], ['gr-qc-0601086-2-16-0', 'gr-qc-0601086-3-16-0'], ['gr-qc-0601086-2-16-1', 'gr-qc-0601086-3-16-1'], ['gr-qc-0601086-2-16-2', 'gr-qc-0601086-3-16-2'], ['gr-qc-0601086-2-16-3', 'gr-qc-0601086-3-16-3'], ['gr-qc-0601086-2-16-4', 'gr-qc-0601086-3-16-4'], ['gr-qc-0601086-2-9-0', 'gr-qc-0601086-3-9-0'], ['gr-qc-0601086-2-9-1', 'gr-qc-0601086-3-9-1'], ['gr-qc-0601086-2-9-2', 'gr-qc-0601086-3-9-2'], ['gr-qc-0601086-2-23-0', 'gr-qc-0601086-3-23-0'], ['gr-qc-0601086-2-23-1', 'gr-qc-0601086-3-23-1'], ['gr-qc-0601086-2-23-2', 'gr-qc-0601086-3-23-2'], ['gr-qc-0601086-2-23-3', 'gr-qc-0601086-3-23-3'], ['gr-qc-0601086-2-31-0', 'gr-qc-0601086-3-31-0'], ['gr-qc-0601086-2-31-1', 'gr-qc-0601086-3-31-1'], ['gr-qc-0601086-2-31-3', 'gr-qc-0601086-3-31-3'], ['gr-qc-0601086-2-14-0', 'gr-qc-0601086-3-14-0'], ['gr-qc-0601086-2-14-1', 'gr-qc-0601086-3-14-1'], ['gr-qc-0601086-2-14-2', 'gr-qc-0601086-3-14-2'], ['gr-qc-0601086-2-6-0', 'gr-qc-0601086-3-6-0'], ['gr-qc-0601086-2-6-1', 'gr-qc-0601086-3-6-1'], ['gr-qc-0601086-2-6-2', 'gr-qc-0601086-3-6-2'], ['gr-qc-0601086-2-6-3', 'gr-qc-0601086-3-6-3'], ['gr-qc-0601086-2-33-0', 'gr-qc-0601086-3-33-0'], ['gr-qc-0601086-2-33-1', 'gr-qc-0601086-3-33-1'], ['gr-qc-0601086-2-33-2', 'gr-qc-0601086-3-33-2'], ['gr-qc-0601086-2-33-3', 'gr-qc-0601086-3-33-3'], ['gr-qc-0601086-2-11-0', 'gr-qc-0601086-3-11-0'], ['gr-qc-0601086-2-11-1', 'gr-qc-0601086-3-11-1'], ['gr-qc-0601086-2-11-2', 'gr-qc-0601086-3-11-2'], ['gr-qc-0601086-2-11-3', 'gr-qc-0601086-3-11-3'], ['gr-qc-0601086-2-29-0', 'gr-qc-0601086-3-29-0'], 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['gr-qc-0601086-2-4-1', 'gr-qc-0601086-3-4-1'], ['gr-qc-0601086-2-4-2', 'gr-qc-0601086-3-4-2'], ['gr-qc-0601086-2-13-0', 'gr-qc-0601086-3-13-0'], ['gr-qc-0601086-2-13-1', 'gr-qc-0601086-3-13-1'], ['gr-qc-0601086-2-13-2', 'gr-qc-0601086-3-13-2'], ['gr-qc-0601086-2-13-3', 'gr-qc-0601086-3-13-3'], ['gr-qc-0601086-2-13-4', 'gr-qc-0601086-3-13-4'], ['gr-qc-0601086-2-13-5', 'gr-qc-0601086-3-13-5'], ['gr-qc-0601086-2-13-6', 'gr-qc-0601086-3-13-6'], ['gr-qc-0601086-2-5-0', 'gr-qc-0601086-3-5-0'], ['gr-qc-0601086-2-5-1', 'gr-qc-0601086-3-5-1'], ['gr-qc-0601086-2-5-2', 'gr-qc-0601086-3-5-2'], ['gr-qc-0601086-2-20-0', 'gr-qc-0601086-3-20-0'], ['gr-qc-0601086-2-20-1', 'gr-qc-0601086-3-20-1'], ['gr-qc-0601086-2-20-2', 'gr-qc-0601086-3-20-2'], ['gr-qc-0601086-2-20-3', 'gr-qc-0601086-3-20-3'], ['gr-qc-0601086-2-20-4', 'gr-qc-0601086-3-20-4'], ['gr-qc-0601086-2-0-0', 'gr-qc-0601086-3-0-0'], ['gr-qc-0601086-2-0-1', 'gr-qc-0601086-3-0-1'], ['gr-qc-0601086-2-0-2', 'gr-qc-0601086-3-0-2'], ['gr-qc-0601086-2-0-3', 'gr-qc-0601086-3-0-3'], ['gr-qc-0601086-2-7-0', 'gr-qc-0601086-3-7-0'], ['gr-qc-0601086-2-7-1', 'gr-qc-0601086-3-7-1'], ['gr-qc-0601086-2-7-2', 'gr-qc-0601086-3-7-2'], ['gr-qc-0601086-2-7-4', 'gr-qc-0601086-3-7-4'], ['gr-qc-0601086-2-7-5', 'gr-qc-0601086-3-7-5'], ['gr-qc-0601086-2-7-6', 'gr-qc-0601086-3-7-6'], ['gr-qc-0601086-2-40-3', 'gr-qc-0601086-3-41-0']]

[['gr-qc-0601086-2-18-0', 'gr-qc-0601086-3-18-0'], ['gr-qc-0601086-2-18-2', 'gr-qc-0601086-3-18-2'], ['gr-qc-0601086-2-4-3', 'gr-qc-0601086-3-4-3'], ['gr-qc-0601086-2-7-3', 'gr-qc-0601086-3-7-3'], ['gr-qc-0601086-2-40-1', 'gr-qc-0601086-3-40-7']]

[]

[['gr-qc-0601086-2-31-2', 'gr-qc-0601086-3-31-2'], ['gr-qc-0601086-2-19-2', 'gr-qc-0601086-3-19-2'], ['gr-qc-0601086-2-40-2', 'gr-qc-0601086-3-40-1']]

[]

['gr-qc-0601086-1-1-0', 'gr-qc-0601086-1-18-3', 'gr-qc-0601086-1-24-8', 'gr-qc-0601086-1-27-14', 'gr-qc-0601086-1-29-2', 'gr-qc-0601086-2-1-0', 'gr-qc-0601086-2-18-3', 'gr-qc-0601086-2-24-8', 'gr-qc-0601086-2-27-14', 'gr-qc-0601086-2-29-2', 'gr-qc-0601086-3-1-0', 'gr-qc-0601086-3-18-3', 'gr-qc-0601086-3-24-8', 'gr-qc-0601086-3-27-15', 'gr-qc-0601086-3-29-2']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/gr-qc/0601086

{'gr-qc-0601086-3-0-0': 'The introduction of an interaction for dark energy to the standard cosmology offers a potential solution to the cosmic coincidence problem.', 'gr-qc-0601086-3-0-1': 'We examine the conditions on the dark energy density that must be satisfied for this scenario to be realized.', 'gr-qc-0601086-3-0-2': 'Under some general conditions we find a stable attractor for the evolution of the Universe in the future.', 'gr-qc-0601086-3-0-3': 'Holographic conjectures for the dark energy offer some specific examples of models with the desired properties.', 'gr-qc-0601086-3-1-0': 'defet al.', 'gr-qc-0601086-3-2-0': '# Introduction', 'gr-qc-0601086-3-3-0': 'The observation that the Universe appears to be accelerating has been one of the biggest surprises in both the astronomy and particle physics communities.', 'gr-qc-0601086-3-3-1': "This acceleration can be accounted for by adding a cosmological constant in Einstein's equations.", 'gr-qc-0601086-3-3-2': 'While particle physicists never had a definite reason for setting the cosmological constant to zero, there was always the view that like all small or zero quantities, there must be a symmetry enforcing it.', 'gr-qc-0601086-3-3-3': 'The observation that the size of the cosmological constant is neither zero nor at one of the natural scales like the Planck length or the scale of supersymmetry breaking has focused attention on explaining the dark energy component.', 'gr-qc-0601086-3-4-0': 'There are at least two requirements for any model of dark energy: (1) one must obtain in a natural way the observed size of the energy density, [MATH] eV[MATH].', 'gr-qc-0601086-3-4-1': 'Many models attempt to relate this to the observed coincidence of this scale with the scale [MATH] where [MATH] is the Hubble constant at the current epoch and [MATH] is the Planck mass.', 'gr-qc-0601086-3-4-2': '(2) One must also obtain an equation of state [MATH] which is similar to that of a cosmological constant at least in the recent past, namely [MATH].', 'gr-qc-0601086-3-4-3': 'In fact, the most recent data from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite and supernova and sky surveys have constrained the equation of state [MATH] at the 95% confidence level for dark energy with a constant [MATH].', 'gr-qc-0601086-3-5-0': 'An interesting approach to the dark matter problem that has arisen from recent advances in understanding string theory and black hole physics is to employ a holographic principle.', 'gr-qc-0601086-3-5-1': 'Here the motivation is the recognition that the observed size of the energy density at the present epoch seems to be consistent with taking the geometric average of the two scales [MATH] and [MATH].', 'gr-qc-0601086-3-5-2': 'Early attempts to relate the dark energy to the Hubble scale or the particle horizon did not give acceptable solutions in detail that were consistent with the observations.', 'gr-qc-0601086-3-6-0': 'The evolution of the dark energy density also depends on its precise nature.', 'gr-qc-0601086-3-6-1': 'If the dark energy is like a cosmological constant with equation of state [MATH], the dark energy density is constant and within the expanding Universe ultimately the dark energy density comes to dominate over the matter density.', 'gr-qc-0601086-3-6-2': 'In this scenario the transition from matter domination to dark energy domination is rapid, and the fact that experimental observations seem to be consistent with sizeable amounts of both matter and dark energy indicates that we are currently residing in this transition period.', 'gr-qc-0601086-3-6-3': 'The cosmic coincidence problem is the statement that it is unlikely that the current epoch coincides with this rapid transition period.', 'gr-qc-0601086-3-7-0': 'A possible solution is to assume that there is some mechanism that is converting dark energy into matter.', 'gr-qc-0601086-3-7-1': 'Cosmic antifriction forces were introduced in Ref. [CITATION] as a possible solution to the cosmic coincidence problem.', 'gr-qc-0601086-3-7-2': 'If these forces are present one can obtain a fixed ratio of matter density to dark energy density as the final state of the Universe and hence solve the coincidence problem.', 'gr-qc-0601086-3-7-3': 'The resulting cosmology is described by Friedmann equations.', 'gr-qc-0601086-3-7-4': 'The basic scenario is quite simple: the natural tendency of a cosmological component with a more negative equation of state to dominate at large times the energy density of the Universe is compensated by the decay of the dark energy component into the other component(s).', 'gr-qc-0601086-3-7-5': 'At large times an equilibrium solution can develop for which the various components can coexist.', 'gr-qc-0601086-3-7-6': 'The interpretation is then that we are living in or close to this equilibrium thus eliminating the cosmic coincidence problem.', 'gr-qc-0601086-3-8-0': 'Frictional and antifrictional forces we envision here were introduced for a single component in Refs. [CITATION].', 'gr-qc-0601086-3-8-1': 'The particular case where the dark energy component is assumed to have a constant equation of state was treated in Ref. [CITATION].', 'gr-qc-0601086-3-8-2': 'These efforts were preceded by searches for attractor solutions involving scalar fields[CITATION].', 'gr-qc-0601086-3-8-3': 'Recently these kind of scenarios have been employed in models with brane-bulk energy exchange[CITATION].', 'gr-qc-0601086-3-8-4': 'Some attempts have been made to explain the size of the dark energy density on the basis of holographic ideas[CITATION], and lead to the consideration of the Hubble horizon as the holographic scale[CITATION].', 'gr-qc-0601086-3-8-5': 'A holographic model based on the future event horizon was examined in Ref. [CITATION].', 'gr-qc-0601086-3-8-6': 'Danielsson studied the effects of a transplanckian backreaction as a particular source for the holographic dark energy[CITATION].', 'gr-qc-0601086-3-8-7': 'More recenlty an emphasis on the case of interacting dark energy has been discussed in Refs. [CITATION].', 'gr-qc-0601086-3-9-0': 'Ultimately one needs to appeal to the observational data to constrain these ideas.', 'gr-qc-0601086-3-9-1': 'In this direction Wang, et al. have begun a more detailed comparison of the predictions of a holographic universe with interaction with the observational data[CITATION].', 'gr-qc-0601086-3-9-2': 'In addition to the overall development of the densities of the cosmological components there are other observables which are sensitive to the modified scenarios[CITATION].', 'gr-qc-0601086-3-10-0': '# Framework of Interacting Dark Energy', 'gr-qc-0601086-3-11-0': 'We assume two component equations for dark energy and matter, [EQUATION]', 'gr-qc-0601086-3-11-1': 'The equality of [MATH] in the two equations guarantees the overall conservation of the energy-momentum tensor, and positive [MATH] can be interpreted as a transfer from the dark energy component to the matter component.', 'gr-qc-0601086-3-11-2': 'Presumably this arises from some microscopic mechanism, but we do not specify one here.', 'gr-qc-0601086-3-11-3': 'The quantity [MATH] is the matter component for which one usually takes [MATH], but we temporarily retain a nonzero equation of state so as to retain the generality of an two interacting components.', 'gr-qc-0601086-3-12-0': 'One can define effective equations of state, [EQUATION]', 'gr-qc-0601086-3-12-1': 'Define the ratio [MATH] and the rate [MATH], then the effective equations of state are expressed in terms of the native equations of state and a ratio of rates as [EQUATION]', 'gr-qc-0601086-3-12-2': 'The time evolution of the ratio [MATH] is then [EQUATION]', 'gr-qc-0601086-3-12-3': 'The general behavior of solutions can be ascertained from this equation.', 'gr-qc-0601086-3-12-4': 'The stable points in the evolution are obtained when either [MATH] or when [MATH].', 'gr-qc-0601086-3-12-5': 'If [MATH] then the vanishing dark energy density ([MATH]) solution will apply in the infinite past while a constant nonzero ratio of dark energy to matter will apply in the infinite future with equal effective equations of state.', 'gr-qc-0601086-3-13-0': 'Using the definitions [EQUATION] which satisfy a Friedmann equation [EQUATION] one can convert to the physical parameters [EQUATION]', 'gr-qc-0601086-3-13-1': 'One can write the differential equation in the suggestive form using [MATH], [EQUATION] where [MATH].', 'gr-qc-0601086-3-13-2': 'The utility of the evolution equation in this form is that it allows one to understand in a general way the possible solutions for [MATH].', 'gr-qc-0601086-3-13-3': 'Firstly if one turns off the interaction ([MATH]) one recovers the usual evolution equation with the effective equations of state replaced with the usual equations of state [MATH] and [MATH] (we will call these the native equations of state).', 'gr-qc-0601086-3-13-4': 'For constant [MATH] and [MATH] and [MATH] then the dark energy evolution proceeds from the fixed points of the differential equation (zeros of the right hand side) in the usual way from [MATH] to [MATH] in the usual way.', 'gr-qc-0601086-3-13-5': 'When an interaction is present a more interesting solution may occur where the evolution of [MATH] approaches a condition where instead [MATH] at which [MATH] approaches a fixed asymptotic value for large time less than 1.', 'gr-qc-0601086-3-13-6': 'For a properly chosen interaction this might constitue a solution to the cosmic concidence problem.', 'gr-qc-0601086-3-14-0': 'For illustration one can also consider the (redundant) differential equation for the matter density, [EQUATION]', 'gr-qc-0601086-3-14-1': 'An initial condition [MATH] is inevitably driven to an asymptotic solution for which [MATH] and [MATH] and [MATH] approach their limiting (equal) value.', 'gr-qc-0601086-3-14-2': 'This behavior is displayed for a specific solution[CITATION] shown in Fig. 1.', 'gr-qc-0601086-3-15-0': 'Two physical assumptions must be provided to determine the evolution of the parameters of the universe.', 'gr-qc-0601086-3-15-1': 'A holographic condition on the dark energy [MATH] will determine by itself the effective equation of state [MATH].', 'gr-qc-0601086-3-15-2': 'A specific specification for the source [MATH] will determine the effective matter equation of state [MATH].', 'gr-qc-0601086-3-15-3': 'Alternatively one of these conditions can be replaced by the assumption that [MATH] is constant.', 'gr-qc-0601086-3-16-0': 'One typically obtains an equation for the evolution of the dark energy density of the form [EQUATION]', 'gr-qc-0601086-3-16-1': 'The evolution of [MATH] with respect to [MATH] or [MATH] is qualitatively determined by the fixed points of the differential equation in Eq. ([REF]) which are determined by the condition [MATH].', 'gr-qc-0601086-3-16-2': 'Examining the equations in Eqs. ([REF]), it would appear that this condition would require that [MATH] and then that [MATH] via Eq. [REF].', 'gr-qc-0601086-3-16-3': 'However, as shown in Ref. [CITATION] if one of the fixed points occurs for the physically interesting value of [MATH], then one can have the Universe with close to vanishing dark energy simultaneously with [MATH].', 'gr-qc-0601086-3-16-4': 'In the remainder of this paper we explore the general conditions that give rise to this type of equilibrium solution.', 'gr-qc-0601086-3-17-0': '# Dimensional analysis', 'gr-qc-0601086-3-18-0': 'First define a length scale [MATH] via [EQUATION] where the constant [MATH] represents an order one constant[CITATION].', 'gr-qc-0601086-3-18-1': 'Specific assumptions might associate the length scale with a physical length such as the Hubble horizon, the particle horizon, the future event horizon, or perhaps some other parameter.', 'gr-qc-0601086-3-18-2': 'The common feature of any realistic choice is that the length scale at the present epoch should be of order the size of the observable Universe so as to obtain the appropriate size of the dark energy density consistent with experimental observations.', 'gr-qc-0601086-3-18-3': 'One has [EQUATION]', 'gr-qc-0601086-3-18-4': 'There need to be two constraints imposed to solve the set of equations.', 'gr-qc-0601086-3-18-5': 'We have three choices: 1) holographic principle on [MATH], 2) suppose a form for [MATH] or equivalently [MATH], and 3) suppose a form for the equation of state [MATH].', 'gr-qc-0601086-3-18-6': 'The equation linking them is obtained from Eq. ([REF]), [EQUATION]', 'gr-qc-0601086-3-18-7': 'This implies the generic scale of the decay rate [MATH] is the same order as [MATH].', 'gr-qc-0601086-3-18-8': 'In the particular case [MATH] Eq. ([REF]) reduces to [EQUATION] which connects the sign of [MATH] to the sign of [MATH].', 'gr-qc-0601086-3-18-9': 'In the case that the interaction vanishes, [MATH], one returns to the condition of a cosmological constant with constant energy density (and therefore constant [MATH]).', 'gr-qc-0601086-3-18-10': 'On the other hand the interaction can either increase or decrease [MATH] depending on the sign of the interaction.', 'gr-qc-0601086-3-18-11': 'Henceforth we will refer to the rate [MATH] as the holographic rate, but it should be understood that it is an equivalent description for the time development of the dark energy density independent of any assumption based on the holographic principle.', 'gr-qc-0601086-3-19-0': 'Since the dark energy density scales as [MATH], one has [EQUATION]', 'gr-qc-0601086-3-19-1': 'Comparing to Eq. ([REF]) one obtains the relationship between the effective equation of state and the length scale [EQUATION]', 'gr-qc-0601086-3-19-2': 'This equation recasts the scaling exhibited by the dark energy in terms of the ratio of the two rates, [MATH] and [MATH].', 'gr-qc-0601086-3-19-3': 'When there is no interaction the same equation applies for the native equation of state [MATH].', 'gr-qc-0601086-3-19-4': 'The familiar case of a constant dark energy density simply corresponds to the statement that the length scale does not change, [MATH].', 'gr-qc-0601086-3-19-5': 'More generally Eq. ([REF]) shows that an assumption for [MATH] based on some holographic principle determines the form of the effective equation of state even in the presence of an interaction.', 'gr-qc-0601086-3-19-6': 'If one applies Eq. ([REF]) one has [EQUATION]', 'gr-qc-0601086-3-19-7': 'Inserting this into Eq. ([REF]) and using the definition for the deceleration parameter [MATH] and [MATH], one obtains [EQUATION]', 'gr-qc-0601086-3-19-8': 'Asymptotically (i.e. for large [MATH]) the last term vanishes, so that [EQUATION]', 'gr-qc-0601086-3-19-9': 'Finally using Eq. ([REF]) one obtains [EQUATION] which indicates that the differential equation (right hand side) has a zero for [MATH] provided the term in brackets does not vanish like [MATH].', 'gr-qc-0601086-3-20-0': 'The resulting interpretation of the possible evolutions of the Universe becomes straightforward using Eq. ([REF]).', 'gr-qc-0601086-3-20-1': 'The derivative of [MATH] has fixed points at [MATH] and at some larger value where the condition [EQUATION] is satisfied.', 'gr-qc-0601086-3-20-2': 'For large times the Universe approaches this solution where the effective equations of states for matter and the dark energy become equal.', 'gr-qc-0601086-3-20-3': 'At early times the dark energy is repelled from the [MATH] and grows monitonically to a positive value.', 'gr-qc-0601086-3-20-4': 'For the case [MATH] one observes that [MATH] must be negative for asympotically large times.', 'gr-qc-0601086-3-21-0': '# Examples', 'gr-qc-0601086-3-22-0': 'A holographic condition based on relating [MATH] to the Hubble scale or to the particle horizon leads to conflict with observation.', 'gr-qc-0601086-3-22-1': 'In the former case the ratio of the dark energy density to matter density is constant.', 'gr-qc-0601086-3-22-2': 'In the latter case the equation of state of the dark energy is greater than [MATH].', 'gr-qc-0601086-3-22-3': 'Taking the length scale to be the Hubble horizon [MATH] which implies [EQUATION] so that [MATH] is a constant.', 'gr-qc-0601086-3-22-4': 'This yields [EQUATION] so that the asymptotic solution obtained in Eq. ([REF]) is identically satisfied.', 'gr-qc-0601086-3-23-0': 'Setting the holographic length scale equal to the particle horizon gives [EQUATION] while setting it equal to the future event horizon[CITATION] yields [EQUATION]', 'gr-qc-0601086-3-23-1': 'If the length scale is set equal to the Hubble horizon, the particle horizon, or the future horizon, then the rates go like [EQUATION] respectively.', 'gr-qc-0601086-3-23-2': 'Finally for a cosmological constant one has [MATH].', 'gr-qc-0601086-3-23-3': 'These conditions yield the following expressions for the effective dark energy equation of state [EQUATION] and [MATH] for a cosmological constant.', 'gr-qc-0601086-3-24-0': 'The case of the Hubble horizon is particularly simple: From Eq. ([REF]) one has that [MATH], so that the ratio of dark energy to matter is constant[CITATION].', 'gr-qc-0601086-3-24-1': 'This is independent of the existence of a interaction [MATH].', 'gr-qc-0601086-3-24-2': 'If the interaction is absent, then [MATH] and [MATH] identically.', 'gr-qc-0601086-3-24-3': 'Including an interaction these generalize to [EQUATION]', 'gr-qc-0601086-3-24-4': 'One particular mechanism for determining the holographic condition is to assume the dark energy density is created out of the vacuum as the Universe expands[CITATION].', 'gr-qc-0601086-3-24-5': 'The component is created via the Bogolubov modes which result from the expansion of the Universe and the variation of the vacuum in such a situation.', 'gr-qc-0601086-3-24-6': 'One has [EQUATION] where [MATH] is some mass of order the Planck scale.', 'gr-qc-0601086-3-24-7': 'Translating this into the varying scale [MATH], one obtains [EQUATION] where an overall constant that arises in this definition is absorbed into the definition of the order one constant coefficient [MATH].', 'gr-qc-0601086-3-24-8': 'One calculates [EQUATION]', 'gr-qc-0601086-3-24-9': 'The effective equation of state then follows from Eq. ([REF]).', 'gr-qc-0601086-3-24-10': 'In this formulation the behavior near [MATH] results from lower momentum modes which were created at earlier times when [MATH] was larger.', 'gr-qc-0601086-3-25-0': 'The asymptotic value for the effective equation of state should approach something near that of a cosmological constant to be consistent with oberservations.', 'gr-qc-0601086-3-25-1': 'Having accumulated the results for the previous examples it is easy to conclude that the future horizon[CITATION] yields an acceptable effective equation of state near [MATH], while the particle horizon has an effective equation of state which is too large.', 'gr-qc-0601086-3-25-2': 'Generally one wants a length scale rate change that becomes small compared to the Hubble expansion [MATH] so that the effective equation of state approaches [MATH] as in Eq. ([REF]).', 'gr-qc-0601086-3-25-3': 'The future horizon and other holographic conditions that result in a decreasing value for [MATH] as a function of increasing [MATH] have the desired properties for an accelerating Universe.', 'gr-qc-0601086-3-26-0': '# Interactions', 'gr-qc-0601086-3-27-0': 'The coupled equations in Eq. ([REF]) defines an interaction between the dark energy and matter components.', 'gr-qc-0601086-3-27-1': 'Given a specific rate [MATH] as a function of [MATH] and [MATH], one can study the behavior of the evolution equations with a specific behavior on the energy density.', 'gr-qc-0601086-3-27-2': 'The ratio of rates is taken to be a function of [MATH] [EQUATION] where the dimensionless constant [MATH] is included to facilitate comparison with previous works.', 'gr-qc-0601086-3-27-3': 'In particular Ref. [CITATION] assumed an interaction of this form with [MATH].', 'gr-qc-0601086-3-27-4': 'One then obtains (we assume henceforth that [MATH]) [EQUATION]', 'gr-qc-0601086-3-27-5': 'So the effective equation of state [MATH] for the matter component depends on the interaction only.', 'gr-qc-0601086-3-27-6': 'Together with the fact that the effective equation of state [MATH] depends only on the assumption for the physical condition setting the scale [MATH], allows for a more general analysis of possible cases that give the desired equilibrium solution.', 'gr-qc-0601086-3-27-7': 'Using Eqs. ([REF]) and ([REF]) one obtains [EQUATION]', 'gr-qc-0601086-3-27-8': 'Using Eq. ([REF]) to convert this into an equation involving only [MATH], one obtains [EQUATION]', 'gr-qc-0601086-3-27-9': 'This expression is quite general, so it facilitates a more general understanding of the physical conditions required for the holographic conditions determining [MATH] and the interaction [MATH].', 'gr-qc-0601086-3-27-10': 'If these conditions have a functional expression in terms of [MATH], then Eq. ([REF]) represents a differential equation that can be solved.', 'gr-qc-0601086-3-27-11': 'This is the case for the holographic conditions arising from the particle and future horizons as well as the form of the interactions assumed in this paper.', 'gr-qc-0601086-3-27-12': 'Furthermore the differential equation involves the effective equations of state in a simple way.', 'gr-qc-0601086-3-27-13': 'The first two terms in brackets are [MATH] while the last term is [MATH].', 'gr-qc-0601086-3-27-14': 'Together with Eq. ([REF]) it determines the acceleration of the Universe through [MATH].', 'gr-qc-0601086-3-27-15': 'One has [EQUATION]', 'gr-qc-0601086-3-27-16': 'A complete specification of the problems involves the specification of two ratios of rates as in Eq. ([REF]).', 'gr-qc-0601086-3-27-17': 'A generic choice of definition for [MATH] and [MATH] will determine the native equation of state [MATH] which will then vary with time.', 'gr-qc-0601086-3-27-18': 'Alternatively one can choose a constant [MATH] and [MATH] in which case the interaction rate is determined.', 'gr-qc-0601086-3-27-19': 'We examine these cases in more detail in the remainder of this section.', 'gr-qc-0601086-3-28-0': '## Constant equation of state', 'gr-qc-0601086-3-29-0': 'The first case we shall examine is the one where the native equation of state for dark energy is constant.', 'gr-qc-0601086-3-29-1': 'For this purpose take the equation involving the three rates, Eq. ([REF]), and eliminate the interaction term in Eq. ([REF]) in favor of the other two rates.', 'gr-qc-0601086-3-29-2': 'One obtains [EQUATION]', 'gr-qc-0601086-3-29-3': 'This equation is valid irregardless of whether [MATH] is constant in time, but we are presently interested in the constant case.', 'gr-qc-0601086-3-29-4': 'Provided the holographic rate does not have a singularity for [MATH], the right hand side has the required zero at [MATH], and for [MATH] the first two terms in brackets is positive provided [MATH].', 'gr-qc-0601086-3-29-5': 'For both the case of the holographic rate based on the particle horizon and future event horizon, there is another fixed point at positive [MATH] (see Eq. ([REF])).', 'gr-qc-0601086-3-30-0': 'Using Eqs. ([REF]) and ([REF]) one obtains the simple result [EQUATION] which shows that the fixed point occurs when [MATH].', 'gr-qc-0601086-3-30-1': 'As examples take [MATH] and [MATH].', 'gr-qc-0601086-3-30-2': 'Then the attractive fixed point occurs at [MATH] for the particle horizon (PH) case and at [MATH] for the future event horizon (FH) case, and from Eq. ([REF]) these correspond to [MATH] and [MATH] respectively.', 'gr-qc-0601086-3-30-3': 'More generally Eq. ([REF]) implies that [MATH].', 'gr-qc-0601086-3-31-0': 'In the case the scale [MATH] is identified with the Hubble scale [MATH], the differential equation vanishes and one has the relation [EQUATION]', 'gr-qc-0601086-3-31-1': 'If one further assumes a constant [MATH] then the interaction [MATH] is determined, and [MATH] is simply proportional to the constant [MATH].', 'gr-qc-0601086-3-31-2': 'The interaction rate is then proportional to [MATH] as shown in Ref. [CITATION], [EQUATION]', 'gr-qc-0601086-3-31-3': 'So this holographic condition corresponds to the one in which the differential equation is identically zero and equilibrium between the dark energy density and matter density holds for all times.', 'gr-qc-0601086-3-32-0': '## Assumptions about the holographic and interaction rates', 'gr-qc-0601086-3-33-0': 'The inclusion of an iteraction term for the dark energy component can be accomodated most easily by defining an effective equation of state [MATH] which differs from the native equation of state [MATH].', 'gr-qc-0601086-3-33-1': 'By virtue of its definition it satisfies the same equation that the native equation of state satisfies in the noninteraction case, namely Eq. ([REF]).', 'gr-qc-0601086-3-33-2': 'If [MATH] then [MATH] and the universe accelerates.', 'gr-qc-0601086-3-33-3': 'The effective equation of state must lie between that of a cosmological constant ([MATH]) and the curvature component ([MATH]).', 'gr-qc-0601086-3-34-0': 'While there is a certain appeal in taking the future event horizon as the length scale for a holographic prinicple[CITATION], it is clear from the evolution equation for the dark energy that an equilibrium fixed point solution can occur for a wider variety of assumptions.', 'gr-qc-0601086-3-35-0': 'In order to examine the conditions on the interaction rate consider the cases [EQUATION] for some exponent [MATH].', 'gr-qc-0601086-3-35-1': 'One then obtains [EQUATION]', 'gr-qc-0601086-3-35-2': 'Figure 2 shows the functional dependence of [MATH] and [MATH] versus [MATH] for various choices of the holographic condition and interaction.', 'gr-qc-0601086-3-35-3': 'An intial condition of [MATH] will evolve to the right until [MATH] which respresents the asymptotic solution.', 'gr-qc-0601086-3-35-4': 'It is clear that an interaction of the form [MATH] with [MATH] can give an acceptable result with a equilibrium balance between [MATH] and [MATH] provided the holographic condition involves the future event horizon.', 'gr-qc-0601086-3-36-0': 'For [MATH] the effective equation of state [MATH] diverges for small [MATH], and in fact is less than [MATH] so does not yield an acceptable solution.', 'gr-qc-0601086-3-36-1': 'We note that for [MATH] one has [MATH] for [MATH] which may be needed for better agreement with the observational data.', 'gr-qc-0601086-3-36-2': 'An explicit example is plotted in Fig. 3 for which the iteraction is assumed to be of the form in Eq. ([REF]) with [MATH].', 'gr-qc-0601086-3-36-3': 'In fact this is the typical behavior for all [MATH].', 'gr-qc-0601086-3-37-0': '# Multicomponent Generalization', 'gr-qc-0601086-3-38-0': 'An obvious generalization of the model considered here is the multicomponent case [EQUATION] subject to the constraint [MATH].', 'gr-qc-0601086-3-38-1': 'The evolution equation for the dark energy component is [EQUATION] where [EQUATION] are quantities defined for each component other than the dark energy.', 'gr-qc-0601086-3-38-2': 'The generic solution that occurs for typical cases where the effective equations of state are monotonic functions of [MATH] is that two of the components come to equilibrium with the remaining components being diluted away by the expansion of the Univerese.', 'gr-qc-0601086-3-38-3': 'This case is of coursed realized if one adds a radiation component for example.', 'gr-qc-0601086-3-39-0': '# Summary', 'gr-qc-0601086-3-40-0': 'This work presents a unifying and general treatment of the physical assumptions that have been made for models with an interacting dark energy component in the Universe.', 'gr-qc-0601086-3-40-1': 'The discussion in the literature has typically involved specifications for a holographic principle, a choice for an interaction between dark matter and dark energy, or the assumption of constant equations of state.', 'gr-qc-0601086-3-40-2': 'A specification for any two of the three choices determines the third.', 'gr-qc-0601086-3-40-3': 'In particular a generic specification of a holographic principle and an interaction gives an effective equation of state that is not constant.', 'gr-qc-0601086-3-40-4': 'The existence of a future fixed point for the dark energy of the Universe can be easily established without a numerical calculation by analyzing Eq. ([REF]) with appropriate assumptions for the relative size of the rate for the Universe expansion [MATH], the rate for the evolution of the dark energy density [MATH], and the interaction rate [MATH].', 'gr-qc-0601086-3-40-5': 'Furthermore if the Universe in the present epoch is at or near this fixed point, a condition on the parameters involved in these physical assumptions can be derived by the vanishing of this differential equation.', 'gr-qc-0601086-3-40-6': 'The evolution of a Universe with vanishing dark energy can be understood in a qualitative fashion if the interaction rate and the holographic rate or plotted versus as shown in Fig. 2, and the asymptotic future state of the Universe is identified as the balance between two competing factors: the natural tendency for dark energy to dominate over matter as the Universe expands versus the decay of dark energy into matter.', 'gr-qc-0601086-3-40-7': 'The solutions are characterized by the condition [MATH] for which the late time (fixed point) solution is [MATH].', 'gr-qc-0601086-3-41-0': 'The resulting equilibrium between dark energy and matter offers a possible solution to the cosmic coincidence problem.', 'gr-qc-0601086-3-41-1': 'For example, the viability of holographic models using the future horizon rather than the particle horizon is clearly a result of the effective equation of state decreasing with [MATH].', 'gr-qc-0601086-3-41-2': 'However it should be clear that the existence of an equilibrium solution is more general than the holographic principle in terms of the future horizon.', 'gr-qc-0601086-3-41-3': 'Detailed quantitative comparison of interacting models with present and future observational data should be able to further delineate between them.', 'gr-qc-0601086-3-42-0': 'The interacting models have been generalized to the case involving more than two interacting components.', 'gr-qc-0601086-3-42-1': 'The resulting evolution equation were shown to have a universal form in terms of the effective equations of state.'}

1803.01455

{'1803.01455-1-0-0': 'Recently, the Cubic Galileon Gravity (CGG) model has been suggested as an alternative gravity theory to General Relativity.', '1803.01455-1-0-1': "The model consists of an extra field potential term which can serve as the 'fifth-force'.", '1803.01455-1-0-2': 'In this article, we examine the possibility whether this extra force term can explain the missing mass problem in galaxies without the help of dark matter.', '1803.01455-1-0-3': 'By using the Milky Way rotation curve and the Spitzer Photomery Accurate Rotation Curves (SPARC) data, we show that this CGG model can satisfactorily explain the shapes of these rotation curves without dark matter.', '1803.01455-1-0-4': 'The CGG model can be regarded as a new alternative theory to challenge the existing dark matter paradigm.', '1803.01455-1-1-0': '# Introduction', '1803.01455-1-2-0': "Recent observations indicate some deviation of the observed galactic rotational curves from the curves predicted by visible mass distribution under Kepler's laws.", '1803.01455-1-2-1': "With the mass distribution extracted from the luminosity of spiral galaxies, a typical galactic rotation curve should have a decreasing trend moving away from the center of the galaxy under Kepler's laws, while observations in recent decades show that a flattened rotation curve is the common case .", '1803.01455-1-2-2': 'Such discrepancy naturally leads to the conjecture on the existence of dark matter which cannot be probed from luminosity.', '1803.01455-1-2-3': 'Besides, observations of hot gas in galaxy clusters also reveal the existence of dark matter .', '1803.01455-1-2-4': 'It is commonly believed that dark matter consists of some unknown massive particles, such as sterile neutrinos or weakly interacting massive particles (WIMPs) .', '1803.01455-1-2-5': 'However, recent experiments for dark matter search show negative results.', '1803.01455-1-2-6': 'No dark matter particle has been discovered for a wide range of mass and energy .', '1803.01455-1-2-7': 'The null result of direct detection might indicate that the assumption of the existence of particle dark matter is wrong.', '1803.01455-1-3-0': 'Besides the null detection problem, when we compare the rotational speed from the Newtonian gravity theory and the observations, one may reckon that the disagreement between the rotational speed values occurs mainly at large distance from the center of the galaxies .', '1803.01455-1-3-1': 'Furthermore, the entire shapes of rotation curves of many galaxies trace their baryonic mass distributions.', '1803.01455-1-3-2': 'It is also surprising that the combination of the contributions of dark matter and baryons results in a nearly flat rotation curve.', '1803.01455-1-3-3': 'The transition from baryon to dark matter domination is very smooth.', '1803.01455-1-3-4': 'This problem is now known as the halo-disk conspiracy (for spiral galaxies) or the dark-spheroid conspiracy (for elliptical galaxies) .', '1803.01455-1-4-0': "Therefore, some suggest that the missing mass in galaxies and galaxy clusters can be explained by alternative theories of Newton's law or gravitational law so that no dark matter exists.", '1803.01455-1-4-1': 'The earliest version is the Modified Newtonian Dynamics (MOND) .', '1803.01455-1-4-2': 'Although this theory can explain some observations successfully, it works very poor in galaxy clusters .', '1803.01455-1-4-3': 'Besides, some studies suggest that MOND is equivalent to some particular form of dark matter profile which suggests that the success of MOND is just a delusion .', '1803.01455-1-4-4': 'On the other hand, some suggest that the missing higher order terms in the current gravity theory makes itself only applicable to local space, including extra terms may explain why the galactic rotational curves usually flattens out.', '1803.01455-1-4-5': 'For instance, the Modified Gravity (MOG) theory suggests that some extra terms in the gravitational law can mimic the effect of dark matter .', '1803.01455-1-5-0': 'In this article, we test a new alternative gravity theory proposed by [CITATION] with the Milky Way rotation curve and the Spitzer Photomery Accurate Rotation Curves (SPARC) data .', '1803.01455-1-5-1': 'We show that this model can also provide a viable solution to the missing mass problem.', '1803.01455-1-5-2': 'The Cubic Galileon Gravity (CGG) model includes an extra interaction term, which can be regarded as an additional force.', '1803.01455-1-5-3': 'By tuning the scaling parameter in the model, it provides a possible way to account for the significant boost in the galactic rotational speed at large galactocentric radius of the galaxies in our universe and the halo-disk (or dark-spheroid) conspiracy problem.', '1803.01455-1-5-4': 'Some previous studies have applied this model to galaxy clusters and dwarf stars .', '1803.01455-1-6-0': '# The Cubic Galileon Gravity (CGG) Model', '1803.01455-1-7-0': 'There have been some new gravity theory alternatives to General Relativity arose among the community and one of them includes an additional field potential which results in the presence of a fifth-force as a supplement to the four fundamental forces .', '1803.01455-1-7-1': 'The theories with a fifth-force are considered to possibly explain the anomalies of our universe observed in recent decades that do not agree with our existing theories.', '1803.01455-1-7-2': '[CITATION] introduce an additional field [MATH] to the Newtonian gravitational potential [MATH] such that a fifth-force [MATH] comes into play, where [MATH] is a dimensionless coupling parameter which plays an important role in determining the magnitude of the fifth-force.', '1803.01455-1-8-0': 'Assume that the potential of the new field behaves the same way as the Newtonian potential, namely [MATH].', '1803.01455-1-8-1': 'In order for the fifth-force to exhibit the Vainshtein Mechanism (screening effect at small galactocentric radius), a new derivative interaction term is added to the equation.', '1803.01455-1-8-2': 'By imposing spherical symmetry, the Poisson equation for the fifth-force potential field becomes [EQUATION] where [MATH] is a mass scale being of the order of the Hubble constant [MATH].', '1803.01455-1-8-3': 'Integrating both sides and substituting the expressions of the fifth-force and the Newtonian gravitational force, one may obtain an equation for the ratio between the two forces [MATH] : [EQUATION] where [EQUATION] is the Vainshtein Radius and generally has a dependence on [MATH].', '1803.01455-1-8-4': 'One may realise that when the galactocentric radius [MATH] is much greater than the Vainshtein Radius, [MATH] and the fifth-force is unscreened.', '1803.01455-1-8-5': 'When [MATH] is comparatively much smaller, the ratio [MATH] (the strength of the fifth-force) is suppressed by a factor of [MATH].', '1803.01455-1-8-6': 'Such screening effect is called the Vainshtein Mechanism .', '1803.01455-1-8-7': 'The resultant force now has the magnitude of [EQUATION] where the positive solution of [MATH] is taken: [EQUATION] with [MATH].', '1803.01455-1-9-0': '# Consequence on the Galactic Rotational Speed', '1803.01455-1-10-0': 'The corresponding rotational speed is larger for a larger centripetal force.', '1803.01455-1-10-1': 'Within the screened regime, i.e. [MATH], the effect of the fifth-force is not significant enough to make the rotational speed deviate from that under the Newtonian gravitational force.', '1803.01455-1-10-2': 'However, in the unscreened regime, i.e. [MATH], the extra force term helps in boosting the rotational speed such that a flattened or even increasing rotational speed is possible at large galactocentric radius even without dark matter.', '1803.01455-1-10-3': 'The rotational speed can generally be given from the relation [EQUATION]', '1803.01455-1-10-4': 'Note that [MATH], the ratio between the fifth-force and the Newtonian force, is a function in [MATH].', '1803.01455-1-10-5': 'Hence, adjusting the value of [MATH] essentially changes the shape of rotation curve.', '1803.01455-1-10-6': 'In the CGG model, since [MATH] is generally considered to be constant , [MATH] is the only parameter that is adjustable.', '1803.01455-1-10-7': 'With a suitable value of [MATH], the gravity model provides a possible candidate for the underlying functional form of gravity in order to account for the anomalous behavior of the galactic rotational curves.', '1803.01455-1-11-0': '# Curve-fitting for the Galactic Rotational Speed', '1803.01455-1-12-0': '## Milky Way rotation curve', '1803.01455-1-13-0': 'First, we apply the CGG model to fit the Milky Way rotation curve data which can be obtained in [CITATION].', '1803.01455-1-13-1': 'The baryonic components can be modeled by the SLFC model suggested in [CITATION].', '1803.01455-1-13-2': 'The SLFC model includes a potential to mimic the effect of dark matter in order to fit the data of the observed rotational speed.', '1803.01455-1-13-3': 'By taking away the potential term for dark matter, one may obtain the Newtonian prediction of the rotational speed for Milky Way using the visible mass profile in the SLFC model.', '1803.01455-1-13-4': 'Given the visible mass profile, one may also use the CGG model to boost the rotational speed.', '1803.01455-1-13-5': 'Our result shows that the CGG model can be an alternative theory of dark matter to describe the dynamics of Milky Way (see Fig. 1).', '1803.01455-1-13-6': 'Here, we assume [MATH] km s[MATH] Mpc[MATH] .', '1803.01455-1-13-7': 'The best-fit parameter is [MATH], which does not violate the observational constraint for our Solar system.', '1803.01455-1-13-8': 'As discussed in [CITATION], the value of [MATH] for our Solar system is [MATH] pc.', '1803.01455-1-13-9': 'Therefore, compared with the size of our Solar system ([MATH] AU or [MATH] pc), the largest correction in the Newtonian gravitational force (see Eq. (5)) within our Solar system is less than 1% ([MATH]), which cannot be ruled out based on current observational data (for the Solar system constraints, see [CITATION]).', '1803.01455-1-14-0': '## SPARC rotation curves', '1803.01455-1-15-0': 'Next, we apply the CGG model to other galaxies.', '1803.01455-1-15-1': 'We test our model with the SPARC (Spitzer Photometry Accurate Rotation Curves) obtained by [CITATION].', '1803.01455-1-15-2': 'Combining these data with the CGG model, we can give a further test for the possible range of [MATH].', '1803.01455-1-16-0': 'The baryonic mass profiles for different galaxies can be obtained by their corresponding luminosity profiles.', '1803.01455-1-16-1': 'In the database of SPARC, the luminosity contributed from the disk component and the bulge component are separated and presented as the velocity contributions [MATH] and [MATH] correspondingly by assuming the mass-to-luminosity ratio to be the Solar ratio, i.e. [MATH].', '1803.01455-1-16-2': 'The mass inscribed up to galactocentric radius [MATH] is given by [EQUATION] where [MATH] and [MATH] are the mass-to-luminosity ratios for the bulge and disk component respectively.', '1803.01455-1-16-3': 'Both [MATH] and [MATH] are taken to be of the value [MATH] respectively.', '1803.01455-1-17-0': 'We are not going to use all data of SPARC (175 spiral galaxies) to test the CGG model.', '1803.01455-1-17-1': 'In particular, we choose the best candidates which satisfy the following 3 criteria: 1.', '1803.01455-1-17-2': 'Type S0-Sc (Hubble stage [MATH]) galaxy (i.e. ruled out Scd-Im); 2.', '1803.01455-1-17-3': 'Distance to the galaxy [MATH] Mpc; 3.', '1803.01455-1-17-4': 'Small uncertainty of the distance to the galaxy ([MATH]% ).', '1803.01455-1-17-5': 'The use of the above criteria can be justified by the following arguments.', '1803.01455-1-17-6': "Generally speaking, type Scd-Im (Hubble stage [MATH]) galaxies have 'broken-arm' and diffuse features which made up of individual stellar clusters and nebulae .", '1803.01455-1-17-7': 'Therefore, they usually have irregular shapes so that symmetry is difficult to apply in these galaxies.', '1803.01455-1-17-8': 'For distant galaxies ([MATH] Mpc), the uncertainties in observations are usually quite large, especially in luminosity and rotational velocity determination.', '1803.01455-1-17-9': 'Lastly, there are some galaxies in the database of SPARC with large uncertainty in distance estimation (e.g. UGC 08699).', '1803.01455-1-17-10': 'The criteria [MATH] Mpc and [MATH]% are reasonable cutoffs to minimize the uncertainties.', '1803.01455-1-17-11': 'Based on these reasons, we rule out galaxies which do not satisfy all of the above criteria 1-3 and finally have 28 galaxies for testing.', '1803.01455-1-18-0': 'By using the mass profile in Eq. (7), the rotational speed of the CGG model is calculated from Eq. (6) and is compared with the observations.', '1803.01455-1-18-1': 'Obviously, different galaxies should have slightly different mass-to-luminosity values.', '1803.01455-1-18-2': 'Therefore, we have 3 parameters for fitting: [MATH], [MATH] and [MATH].', '1803.01455-1-18-3': 'For those galaxies without bulge, we set [MATH].', '1803.01455-1-18-4': "Similar adjustment of mass-to-luminosity ratio has also been found in MOND's framework .", '1803.01455-1-18-5': 'The best-fit values of the parameters are obtained when the reduced [MATH] value is minimized.', '1803.01455-1-18-6': 'The reduced [MATH] is defined as [MATH], where [MATH] is the degrees of freedom, [MATH] are the calculated rotational speed in the CGG model, [MATH] are the observed rotational speed and [MATH] are the uncertainties of the observed rotational speed.', '1803.01455-1-18-7': 'We adapt the following criteria in tunning these three parameters: [MATH], [MATH] and [MATH] .', '1803.01455-1-19-0': "Generally speaking, the value of [MATH] basically controls the magnitude of the rotational speed at large radius, lifting and lowering down the 'tail' of the rotation curve, while the values of [MATH] and [MATH] enlarge and diminish the contribution of the corresponding velocity component according to Eq. (7).", '1803.01455-1-19-1': 'The relative values between [MATH] and [MATH] also affect the shape of the graph.', '1803.01455-1-19-2': 'Galaxies generally have a bulge-dominant behavior when the value of [MATH] is sufficiently larger than [MATH], and vice versa.', '1803.01455-1-19-3': 'The best-fit results are shown in Figs. 2-8.', '1803.01455-1-20-0': 'From the figures, we notice that the general shapes of the resultant rotation curves trace their baryonic distributions.', '1803.01455-1-20-1': 'This can provide an explanation for the halo-disk conspiracy problem.', '1803.01455-1-20-2': 'Also, the outer parts of the resultant rotation curves are generally flat which provide good agreement with the observational data.', '1803.01455-1-20-3': 'The best-fit values of [MATH] fall within a small range [MATH] which suggests that [MATH] is likely to be a universal constant.', '1803.01455-1-20-4': 'This is consistent with the CGG model.', '1803.01455-1-21-0': 'Note that some of the fits generate relatively large reduced [MATH] values, especially for NGC5055.', '1803.01455-1-21-1': 'However, it does not mean that the CGG model works poor in these galaxies.', '1803.01455-1-21-2': 'In fact, the systematic uncertainties in obtaining the rotation curves are not completely negligible.', '1803.01455-1-21-3': 'For example, many rotation curve data in [CITATION] are obtained by combining the H[MATH] data in the inner regions with HI data in the outer parts.', '1803.01455-1-21-4': 'It can give good quality of rotation curves because the H[MATH] data can trace the kinematics at high spatial resolutions so that the beam-smearing effects are minimal .', '1803.01455-1-21-5': 'However, for NGC5055, its H[MATH] rotation curve has not been taken into account .', '1803.01455-1-21-6': 'The resultant data points rely on HI data only, which may generate some systematic errors.', '1803.01455-1-21-7': 'Besides, the uncertainties due to the inclination of galaxies have not been considered either .', '1803.01455-1-21-8': 'In particular, the uncertainty of the inclination of NGC5055 is greater than 10.', '1803.01455-1-21-9': 'Since the overall systematic error is relatively large while the observational uncertainty is small (less than 5 for most of the data points) for those galaxies, the overall reduced [MATH] values are somewhat overestimated.', '1803.01455-1-21-10': 'In fact, by comparing the rotation curves with the data points, we can see that the overall fits of rotation curves are not bad.', '1803.01455-1-22-0': '# Discussion', '1803.01455-1-23-0': 'In this article, we test the CGG model by the Milky Way rotation curve and the SPARC data.', '1803.01455-1-23-1': 'We show that the CGG model can provide a satisfactory explanation to the missing mass problem in galaxies.', '1803.01455-1-23-2': "The extra term arising from the CGG model can provide enough extra 'gravity' to flatten the outer rotation curves without any dark matter.", '1803.01455-1-23-3': 'This model can also provide a solution to the disk-halo conspiracy problem.', '1803.01455-1-23-4': 'The only free parameter in this model is constrained to [MATH].', '1803.01455-1-23-5': 'This value does not violate any observational constraint based on our Solar system.', '1803.01455-1-24-0': 'However, recent studies with cosmological data of the Integrated Sachs-Wolfe (ISW) effect and bounds on gravitational wave speed seem to rule out the CGG as a cosmological viable theory.', '1803.01455-1-24-1': 'In order to match the cosmological constraints (e.g. the cosmic microwave background data), the evolution of the Galileon field must satisfy a so-called tracker solution before the energy density of the Galileon field starts to contribute non-negligibly to the total energy density of our universe .', '1803.01455-1-24-2': 'The ISW data and the bounds on gravitational wave speed suggest difficulties with the CGG based on the tracker solution.', '1803.01455-1-24-3': 'Nevertheless, our focus in this article is not CGG as an alternative model in cosmology without dark energy, but an alternative to dark matter.', '1803.01455-1-24-4': 'The Galileon field is in general a free parameter in the CGG theory .', '1803.01455-1-24-5': 'Here, the field used in this analysis satisfies the Poisson equation instead of the cosmological tracker solution.', '1803.01455-1-24-6': 'Besides, although the Vainshtein Radius [MATH] has a term [MATH], its physical meaning is not related to the dark energy.', '1803.01455-1-24-7': 'We only take it with the order of the Hubble constant [MATH].', '1803.01455-1-24-8': 'Therefore, it is possible that CGG as an alternative to dark energy is wrong but CGG as an alternative to dark matter is correct.', '1803.01455-1-24-9': 'Dark energy may still exist within the CGG model.', '1803.01455-1-24-10': 'Moreover, the cosmological evidence can sometimes be ambiguous.', '1803.01455-1-24-11': 'It is too early to conclude that CGG is ruled out by these studies (even as an alternative to dark energy).', '1803.01455-1-24-12': 'For example, a recent study shows that the problem of the tracker solution in CGG theory due to the bounds on gravitational wave speed can be avoided by the existence of a quadratic k-essence Lagrangian .', '1803.01455-1-24-13': 'Also, some recent studies point out that the mass of neutrinos might affect the cosmological ISW data, though it cannot completely rescue the CGG theory .', '1803.01455-1-25-0': 'On the other hand, some studies point out that the observational data of the Bullet Cluster 1E0657-558 may give some challenge to the alternative theory of gravity, especially for MOND .', '1803.01455-1-25-1': 'The spatial offset of the center of the total mass from the center of the baryonic mass is difficult to be explained with the alternative theory of gravity.', '1803.01455-1-25-2': 'Nevertheless, [CITATION] show that the observed cluster thermal profile gives good agreement with the MOG.', '1803.01455-1-25-3': 'The theory of MOG gives some extra terms in Newtonian gravity which demonstrate similar effects with the CGG model.', '1803.01455-1-25-4': 'Although we have not applied the CGG model to the Bullet Cluster, we believe that it is not a big problem to the CGG model.', '1803.01455-1-25-5': 'In fact, it is also a controversial issue whether the standard [MATH]CDM model is consistent with the data of the Bullet Cluster collision .', '1803.01455-1-25-6': 'Therefore, the case of the Bullet Cluster should not be treated as a smoking gun to rule out any alternative theory of gravity.', '1803.01455-1-26-0': 'As mentioned above, the null detection of dark matter in direct detection experiments might indicate that the assumption of the existence of particle dark matter is wrong.', '1803.01455-1-26-1': 'Therefore, alternative theories of gravity can provide another way to address the missing mass problem in galaxies, galaxy clusters and our entire universe.', '1803.01455-1-26-2': 'Not only MOG can achieve this purpose, the CGG model can also achieve the same goal.', '1803.01455-1-26-3': 'Further observational tests for dwarf galaxies might be able to differentiate which alternative theories of gravity can provide the best explanation to the missing mass problem.', '1803.01455-1-27-0': "This work is supported by the Dean's Research Fund from The Education University of Hong Kong (Project No.:SFRS9 2017)."}

{'1803.01455-2-0-0': 'Recently, the Cubic Galileon Gravity (CGG) model has been suggested as an alternative gravity theory to General Relativity.', '1803.01455-2-0-1': "The model consists of an extra field potential term which can serve as the 'fifth-force'.", '1803.01455-2-0-2': 'In this article, we examine the possibility whether this extra force term can explain the missing mass problem in galaxies without the help of dark matter.', '1803.01455-2-0-3': 'By using the Milky Way rotation curve and the Spitzer Photomery Accurate Rotation Curves (SPARC) data, we show that this CGG model can satisfactorily explain the shapes of these rotation curves without dark matter.', '1803.01455-2-0-4': 'The CGG model can be regarded as a new alternative theory to challenge the existing dark matter paradigm.', '1803.01455-2-1-0': '# Introduction', '1803.01455-2-2-0': "Recent observations indicate some deviation of the observed galactic rotational curves from the curves predicted by visible mass distribution under Kepler's laws.", '1803.01455-2-2-1': "With the mass distribution extracted from the luminosity of spiral galaxies, a typical galactic rotation curve should have a decreasing trend moving away from the center of the galaxy under Kepler's laws, while observations in recent decades show that a flattened rotation curve is the common case .", '1803.01455-2-2-2': 'Such discrepancy naturally leads to the conjecture on the existence of dark matter which cannot be probed from luminosity.', '1803.01455-2-2-3': 'Besides, observations of hot gas in galaxy clusters also reveal the existence of dark matter .', '1803.01455-2-2-4': 'It is commonly believed that dark matter consists of some unknown massive particles, such as sterile neutrinos or weakly interacting massive particles (WIMPs) .', '1803.01455-2-2-5': 'However, recent experiments for dark matter search show negative results.', '1803.01455-2-2-6': 'No dark matter particle has been discovered for a wide range of mass and energy .', '1803.01455-2-2-7': 'The null result of direct detection might indicate that the assumption of the existence of particle dark matter is wrong.', '1803.01455-2-3-0': 'Besides the null detection problem, when we compare the rotational speed from the Newtonian gravity theory and the observations, one may reckon that the disagreement between the rotational speed values occurs mainly at large distance from the center of the galaxies .', '1803.01455-2-3-1': 'Furthermore, the entire shapes of rotation curves of many galaxies trace their baryonic mass distributions.', '1803.01455-2-3-2': 'It is also surprising that the combination of the contributions of dark matter and baryons results in a nearly flat rotation curve.', '1803.01455-2-3-3': 'The transition from baryon to dark matter domination is very smooth.', '1803.01455-2-3-4': 'This problem is now known as the halo-disk conspiracy (for spiral galaxies) or the dark-spheroid conspiracy (for elliptical galaxies) .', '1803.01455-2-4-0': "Therefore, some suggest that the missing mass in galaxies and galaxy clusters can be explained by alternative theories of Newton's law or gravitational law so that no dark matter exists.", '1803.01455-2-4-1': 'The earliest version is the Modified Newtonian Dynamics (MOND) .', '1803.01455-2-4-2': 'Although this theory can explain some observations successfully, it works very poor in galaxy clusters .', '1803.01455-2-4-3': 'Besides, some studies suggest that MOND is equivalent to some particular form of dark matter profile which suggests that the success of MOND is just a delusion .', '1803.01455-2-4-4': 'On the other hand, some suggest that the missing higher order terms in the current gravity theory makes itself only applicable to local space, including extra terms may explain why the galactic rotational curves usually flattens out.', '1803.01455-2-4-5': 'For instance, the Modified Gravity (MOG) theory suggests that some extra terms in the gravitational law can mimic the effect of dark matter .', '1803.01455-2-5-0': 'In this article, we test a new alternative gravity theory proposed by [CITATION] with the Milky Way rotation curve and the Spitzer Photomery Accurate Rotation Curves (SPARC) data .', '1803.01455-2-5-1': 'We show that this model can also provide a viable solution to the missing mass problem.', '1803.01455-2-5-2': 'The Cubic Galileon Gravity (CGG) model includes an extra interaction term, which can be regarded as an additional force.', '1803.01455-2-5-3': 'By tuning the scaling parameter in the model, it provides a possible way to account for the significant boost in the galactic rotational speed at large galactocentric radius of the galaxies in our universe and the halo-disk (or dark-spheroid) conspiracy problem.', '1803.01455-2-5-4': 'Some previous studies have applied this model to galaxy clusters and dwarf stars .', '1803.01455-2-6-0': '# The Cubic Galileon Gravity (CGG) Model', '1803.01455-2-7-0': 'There have been some new gravity theory alternatives to General Relativity arose among the community and one of them includes an additional field potential which results in the presence of a fifth-force as a supplement to the four fundamental forces .', '1803.01455-2-7-1': 'The theories with a fifth-force are considered to possibly explain the anomalies of our universe observed in recent decades that do not agree with our existing theories.', '1803.01455-2-7-2': '[CITATION] introduce an additional field [MATH] to the Newtonian gravitational potential [MATH] such that a fifth-force [MATH] comes into play, where [MATH] is a dimensionless coupling parameter which plays an important role in determining the magnitude of the fifth-force.', '1803.01455-2-8-0': 'Assume that the potential of the new field behaves the same way as the Newtonian potential, namely [MATH].', '1803.01455-2-8-1': 'In order for the fifth-force to exhibit the Vainshtein Mechanism (screening effect at small galactocentric radius), a new derivative interaction term is added to the equation.', '1803.01455-2-8-2': 'By imposing spherical symmetry, the Poisson equation for the fifth-force potential field becomes [EQUATION] where [MATH] is a mass scale being of the order of the Hubble constant [MATH].', '1803.01455-2-8-3': 'Integrating both sides and substituting the expressions of the fifth-force and the Newtonian gravitational force, one may obtain an equation for the ratio between the two forces [MATH] : [EQUATION] where [EQUATION] is the Vainshtein Radius and generally has a dependence on [MATH].', '1803.01455-2-8-4': 'One may realise that when the galactocentric radius [MATH] is much greater than the Vainshtein Radius, [MATH] and the fifth-force is unscreened.', '1803.01455-2-8-5': 'When [MATH] is comparatively much smaller, the ratio [MATH] (the strength of the fifth-force) is suppressed by a factor of [MATH].', '1803.01455-2-8-6': 'Such screening effect is called the Vainshtein Mechanism .', '1803.01455-2-8-7': 'The resultant force now has the magnitude of [EQUATION] where the positive solution of [MATH] is taken: [EQUATION] with [MATH].', '1803.01455-2-9-0': '# Consequence on the Galactic Rotational Speed', '1803.01455-2-10-0': 'The corresponding rotational speed is larger for a larger centripetal force.', '1803.01455-2-10-1': 'Within the screened regime, i.e. [MATH], the effect of the fifth-force is not significant enough to make the rotational speed deviate from that under the Newtonian gravitational force.', '1803.01455-2-10-2': 'However, in the unscreened regime, i.e. [MATH], the extra force term helps in boosting the rotational speed such that a flattened or even increasing rotational speed is possible at large galactocentric radius even without dark matter.', '1803.01455-2-10-3': 'The rotational speed can generally be given from the relation [EQUATION]', '1803.01455-2-10-4': 'Note that [MATH], the ratio between the fifth-force and the Newtonian force, is a function in [MATH].', '1803.01455-2-10-5': 'Hence, adjusting the value of [MATH] essentially changes the shape of rotation curve.', '1803.01455-2-10-6': 'In the CGG model, since [MATH] is generally considered to be constant , [MATH] is the only parameter that is adjustable.', '1803.01455-2-10-7': 'With a suitable value of [MATH], the gravity model provides a possible candidate for the underlying functional form of gravity in order to account for the anomalous behavior of the galactic rotational curves.', '1803.01455-2-11-0': '# Curve-fitting for the Galactic Rotational Speed', '1803.01455-2-12-0': '## Milky Way rotation curve', '1803.01455-2-13-0': 'First, we apply the CGG model to fit the Milky Way rotation curve data which can be obtained in [CITATION].', '1803.01455-2-13-1': 'The baryonic components can be modeled by the SLFC model suggested in [CITATION].', '1803.01455-2-13-2': 'The SLFC model includes a potential to mimic the effect of dark matter in order to fit the data of the observed rotational speed.', '1803.01455-2-13-3': 'By taking away the potential term for dark matter, one may obtain the Newtonian prediction of the rotational speed for Milky Way using the visible mass profile in the SLFC model.', '1803.01455-2-13-4': 'Given the visible mass profile, one may also use the CGG model to boost the rotational speed.', '1803.01455-2-13-5': 'Our result shows that the CGG model can be an alternative theory of dark matter to describe the dynamics of Milky Way (see Fig. 1).', '1803.01455-2-13-6': 'Here, we assume [MATH] km s[MATH] Mpc[MATH] .', '1803.01455-2-13-7': 'The best-fit parameter is [MATH], which does not violate the observational constraint for our Solar system.', '1803.01455-2-13-8': 'As discussed in [CITATION], the value of [MATH] for our Solar system is [MATH] pc.', '1803.01455-2-13-9': 'Therefore, compared with the size of our Solar system ([MATH] AU or [MATH] pc), the largest correction in the Newtonian gravitational force (see Eq. (5)) within our Solar system is less than 1% ([MATH]), which cannot be ruled out based on current observational data (for the Solar system constraints, see [CITATION]).', '1803.01455-2-14-0': '## SPARC rotation curves', '1803.01455-2-15-0': 'Next, we apply the CGG model to other galaxies.', '1803.01455-2-15-1': 'We test our model with the SPARC (Spitzer Photometry Accurate Rotation Curves) obtained by [CITATION].', '1803.01455-2-15-2': 'Combining these data with the CGG model, we can give a further test for the possible range of [MATH].', '1803.01455-2-16-0': 'The baryonic mass profiles for different galaxies can be obtained by their corresponding luminosity profiles.', '1803.01455-2-16-1': 'In the database of SPARC, the luminosity contributed from the disk component and the bulge component are separated and presented as the velocity contributions [MATH] and [MATH] correspondingly by assuming the mass-to-luminosity ratio to be the Solar ratio, i.e. [MATH].', '1803.01455-2-16-2': 'The mass inscribed up to galactocentric radius [MATH] is given by [EQUATION] where [MATH] and [MATH] are the mass-to-luminosity ratios for the bulge and disk component respectively.', '1803.01455-2-16-3': 'Both [MATH] and [MATH] are taken to be of the value [MATH] respectively.', '1803.01455-2-17-0': 'We are not going to use all data of SPARC (175 spiral galaxies) to test the CGG model.', '1803.01455-2-17-1': 'In particular, we choose the best candidates which satisfy the following 3 criteria: 1.', '1803.01455-2-17-2': 'Type S0-Sc (Hubble stage [MATH]) galaxy (i.e. ruled out Scd-Im); 2.', '1803.01455-2-17-3': 'Distance to the galaxy [MATH] Mpc; 3.', '1803.01455-2-17-4': 'Small uncertainty of the distance to the galaxy ([MATH]% ).', '1803.01455-2-17-5': 'The use of the above criteria can be justified by the following arguments.', '1803.01455-2-17-6': "Generally speaking, type Scd-Im (Hubble stage [MATH]) galaxies have 'broken-arm' and diffuse features which made up of individual stellar clusters and nebulae .", '1803.01455-2-17-7': 'Therefore, they usually have irregular shapes so that symmetry is difficult to apply in these galaxies.', '1803.01455-2-17-8': 'For distant galaxies ([MATH] Mpc), the uncertainties in observations are usually quite large, especially in luminosity and rotational velocity determination.', '1803.01455-2-17-9': 'Lastly, there are some galaxies in the database of SPARC with large uncertainty in distance estimation (e.g. UGC 08699).', '1803.01455-2-17-10': 'The criteria [MATH] Mpc and [MATH]% are reasonable cutoffs to minimize the uncertainties.', '1803.01455-2-17-11': 'Based on these reasons, we rule out galaxies which do not satisfy all of the above criteria 1-3 and finally have 28 galaxies for testing.', '1803.01455-2-18-0': 'By using the mass profile in Eq. (7), the rotational speed of the CGG model is calculated from Eq. (6) and is compared with the observations.', '1803.01455-2-18-1': 'Obviously, different galaxies should have slightly different mass-to-luminosity values.', '1803.01455-2-18-2': 'Therefore, we have 3 parameters for fitting: [MATH], [MATH] and [MATH].', '1803.01455-2-18-3': 'For those galaxies without bulge, we set [MATH].', '1803.01455-2-18-4': "Similar adjustment of mass-to-luminosity ratio has also been found in MOND's framework .", '1803.01455-2-18-5': 'The best-fit values of the parameters are obtained when the reduced [MATH] value is minimized.', '1803.01455-2-18-6': 'The reduced [MATH] is defined as [MATH], where [MATH] is the degrees of freedom, [MATH] are the calculated rotational speed in the CGG model, [MATH] are the observed rotational speed and [MATH] are the uncertainties of the observed rotational speed.', '1803.01455-2-18-7': 'We adapt the following criteria in tunning these three parameters: [MATH], [MATH] and [MATH] .', '1803.01455-2-19-0': "Generally speaking, the value of [MATH] basically controls the magnitude of the rotational speed at large radius, lifting and lowering down the 'tail' of the rotation curve, while the values of [MATH] and [MATH] enlarge and diminish the contribution of the corresponding velocity component according to Eq. (7).", '1803.01455-2-19-1': 'The relative values between [MATH] and [MATH] also affect the shape of the graph.', '1803.01455-2-19-2': 'Galaxies generally have a bulge-dominant behavior when the value of [MATH] is sufficiently larger than [MATH], and vice versa.', '1803.01455-2-19-3': 'The best-fit results are shown in Figs. 2-8.', '1803.01455-2-20-0': 'From the figures, we notice that the general shapes of the resultant rotation curves trace their baryonic distributions.', '1803.01455-2-20-1': 'This can provide an explanation for the halo-disk conspiracy problem.', '1803.01455-2-20-2': 'Also, the outer parts of the resultant rotation curves are generally flat which provide good agreement with the observational data.', '1803.01455-2-20-3': 'The best-fit values of [MATH] fall within a small range [MATH] which suggests that [MATH] is likely to be a universal constant.', '1803.01455-2-20-4': 'This is consistent with the CGG model.', '1803.01455-2-21-0': 'Note that some of the fits generate relatively large reduced [MATH] values, especially for NGC5055.', '1803.01455-2-21-1': 'However, it does not mean that the CGG model works poor in these galaxies.', '1803.01455-2-21-2': 'In fact, the systematic uncertainties in obtaining the rotation curves are not completely negligible.', '1803.01455-2-21-3': 'For example, many rotation curve data in [CITATION] are obtained by combining the H[MATH] data in the inner regions with HI data in the outer parts.', '1803.01455-2-21-4': 'It can give good quality of rotation curves because the H[MATH] data can trace the kinematics at high spatial resolutions so that the beam-smearing effects are minimal .', '1803.01455-2-21-5': 'However, for NGC5055, its H[MATH] rotation curve has not been taken into account .', '1803.01455-2-21-6': 'The resultant data points rely on HI data only, which may generate some systematic errors.', '1803.01455-2-21-7': 'Besides, the uncertainties due to the inclination of galaxies have not been considered either .', '1803.01455-2-21-8': 'In particular, the uncertainty of the inclination of NGC5055 is greater than 10.', '1803.01455-2-21-9': 'Since the overall systematic error is relatively large while the observational uncertainty is small (less than 5 for most of the data points) for those galaxies, the overall reduced [MATH] values are somewhat overestimated.', '1803.01455-2-21-10': 'In fact, by comparing the rotation curves with the data points, we can see that the overall fits of rotation curves are not bad.', '1803.01455-2-22-0': '# Discussion', '1803.01455-2-23-0': 'In this article, we test the CGG model by the Milky Way rotation curve and the SPARC data.', '1803.01455-2-23-1': 'We show that the CGG model can provide a satisfactory explanation to the missing mass problem in galaxies.', '1803.01455-2-23-2': "The extra term arising from the CGG model can provide enough extra 'gravity' to flatten the outer rotation curves without any dark matter.", '1803.01455-2-23-3': 'This model can also provide a solution to the disk-halo conspiracy problem.', '1803.01455-2-23-4': 'The only free parameter in this model is constrained to [MATH].', '1803.01455-2-23-5': 'This value does not violate any observational constraint based on our Solar system.', '1803.01455-2-24-0': 'However, recent studies with cosmological data of the Integrated Sachs-Wolfe (ISW) effect and bounds on gravitational wave speed seem to rule out the CGG as a cosmological viable theory.', '1803.01455-2-24-1': 'In order to match the cosmological constraints (e.g. the cosmic microwave background data), the evolution of the Galileon field must satisfy a so-called tracker solution before the energy density of the Galileon field starts to contribute non-negligibly to the total energy density of our universe .', '1803.01455-2-24-2': 'The ISW data and the bounds on gravitational wave speed suggest difficulties with the CGG based on the tracker solution.', '1803.01455-2-24-3': 'Nevertheless, our focus in this article is not CGG as an alternative model in cosmology without dark energy, but an alternative to dark matter.', '1803.01455-2-24-4': 'The Galileon field is in general a free parameter in the CGG theory .', '1803.01455-2-24-5': 'Here, the field used in this analysis satisfies the Poisson equation instead of the cosmological tracker solution.', '1803.01455-2-24-6': 'Besides, although the Vainshtein Radius [MATH] has a term [MATH], its physical meaning is not related to the dark energy.', '1803.01455-2-24-7': 'We only take it with the order of the Hubble constant [MATH].', '1803.01455-2-24-8': 'Therefore, it is possible that CGG as an alternative to dark energy is wrong but CGG as an alternative to dark matter is correct.', '1803.01455-2-24-9': 'Dark energy may still exist within the CGG model.', '1803.01455-2-24-10': 'Moreover, the cosmological evidence can sometimes be ambiguous.', '1803.01455-2-24-11': 'It is too early to conclude that CGG is ruled out by these studies (even as an alternative to dark energy).', '1803.01455-2-24-12': 'For example, a recent study shows that the problem of the tracker solution in CGG theory due to the bounds on gravitational wave speed can be avoided by the existence of a quadratic k-essence Lagrangian .', '1803.01455-2-24-13': 'Also, some recent studies point out that the mass of neutrinos might affect the cosmological ISW data, though it cannot completely rescue the CGG theory .', '1803.01455-2-25-0': 'On the other hand, some studies point out that the observational data of the Bullet Cluster 1E0657-558 may give some challenge to the alternative theory of gravity, especially for MOND .', '1803.01455-2-25-1': 'The spatial offset of the center of the total mass from the center of the baryonic mass is difficult to be explained with the alternative theory of gravity.', '1803.01455-2-25-2': 'Nevertheless, [CITATION] show that the observed cluster thermal profile gives good agreement with the MOG.', '1803.01455-2-25-3': 'The theory of MOG gives some extra terms in Newtonian gravity which demonstrate similar effects with the CGG model.', '1803.01455-2-25-4': 'Although we have not applied the CGG model to the Bullet Cluster, we believe that it is not a big problem to the CGG model.', '1803.01455-2-25-5': 'In fact, it is also a controversial issue whether the standard [MATH]CDM model is consistent with the data of the Bullet Cluster collision .', '1803.01455-2-25-6': 'Therefore, the case of the Bullet Cluster should not be treated as a smoking gun to rule out any alternative theory of gravity.', '1803.01455-2-26-0': 'As mentioned above, the null detection of dark matter in direct detection experiments might indicate that the assumption of the existence of particle dark matter is wrong.', '1803.01455-2-26-1': 'Therefore, alternative theories of gravity can provide another way to address the missing mass problem in galaxies, galaxy clusters and our entire universe.', '1803.01455-2-26-2': 'Not only MOG can achieve this purpose, the CGG model can also achieve the same goal.', '1803.01455-2-26-3': 'Further observational tests for dwarf galaxies might be able to differentiate which alternative theories of gravity can provide the best explanation to the missing mass problem.', '1803.01455-2-27-0': "This work is supported by the Dean's Research Fund from The Education University of Hong Kong (Project No.:SFRS9 2017)."}

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['1803.01455-1-8-6', '1803.01455-2-8-6'], ['1803.01455-1-8-7', '1803.01455-2-8-7'], ['1803.01455-1-17-0', '1803.01455-2-17-0'], ['1803.01455-1-17-1', '1803.01455-2-17-1'], ['1803.01455-1-17-2', '1803.01455-2-17-2'], ['1803.01455-1-17-3', '1803.01455-2-17-3'], ['1803.01455-1-17-4', '1803.01455-2-17-4'], ['1803.01455-1-17-5', '1803.01455-2-17-5'], ['1803.01455-1-17-6', '1803.01455-2-17-6'], ['1803.01455-1-17-7', '1803.01455-2-17-7'], ['1803.01455-1-17-8', '1803.01455-2-17-8'], ['1803.01455-1-17-9', '1803.01455-2-17-9'], ['1803.01455-1-17-10', '1803.01455-2-17-10'], ['1803.01455-1-17-11', '1803.01455-2-17-11'], ['1803.01455-1-24-0', '1803.01455-2-24-0'], ['1803.01455-1-24-1', '1803.01455-2-24-1'], ['1803.01455-1-24-2', '1803.01455-2-24-2'], ['1803.01455-1-24-3', '1803.01455-2-24-3'], ['1803.01455-1-24-4', '1803.01455-2-24-4'], ['1803.01455-1-24-5', '1803.01455-2-24-5'], ['1803.01455-1-24-6', '1803.01455-2-24-6'], ['1803.01455-1-24-7', '1803.01455-2-24-7'], ['1803.01455-1-24-8', '1803.01455-2-24-8'], ['1803.01455-1-24-9', '1803.01455-2-24-9'], ['1803.01455-1-24-10', '1803.01455-2-24-10'], ['1803.01455-1-24-11', '1803.01455-2-24-11'], ['1803.01455-1-24-12', '1803.01455-2-24-12'], ['1803.01455-1-24-13', '1803.01455-2-24-13'], ['1803.01455-1-19-0', '1803.01455-2-19-0'], ['1803.01455-1-19-1', '1803.01455-2-19-1'], ['1803.01455-1-19-2', '1803.01455-2-19-2'], ['1803.01455-1-19-3', '1803.01455-2-19-3'], ['1803.01455-1-2-0', '1803.01455-2-2-0'], ['1803.01455-1-2-1', '1803.01455-2-2-1'], ['1803.01455-1-2-2', '1803.01455-2-2-2'], ['1803.01455-1-2-3', '1803.01455-2-2-3'], ['1803.01455-1-2-4', '1803.01455-2-2-4'], ['1803.01455-1-2-5', '1803.01455-2-2-5'], ['1803.01455-1-2-6', '1803.01455-2-2-6'], ['1803.01455-1-2-7', '1803.01455-2-2-7'], ['1803.01455-1-27-0', '1803.01455-2-27-0'], ['1803.01455-1-0-0', '1803.01455-2-0-0'], ['1803.01455-1-0-1', '1803.01455-2-0-1'], ['1803.01455-1-0-2', '1803.01455-2-0-2'], ['1803.01455-1-0-3', '1803.01455-2-0-3'], ['1803.01455-1-0-4', '1803.01455-2-0-4']]

[]

[]

[]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1803.01455

astro-ph-0011556

{'astro-ph-0011556-1-0-0': 'Deep long-slit spectra of the diffuse ionized gas halos of the edge-on spiral galaxies NGC 4302 and UGC 10288 are presented.', 'astro-ph-0011556-1-0-1': 'Emission lines are detected up to about [MATH] kpc in NGC 4302, and to nearly [MATH] kpc on the north side of UGC 10288.', 'astro-ph-0011556-1-0-2': 'For both galaxies, the line ratios [N2]/H[MATH] and [S2]/H[MATH] increase with [MATH] in accordance with dilute photo-ionization models.', 'astro-ph-0011556-1-0-3': 'Runs of [S2]/[N2], and for UGC 10288, the run of [O3]/H[MATH], however, are not explained by the models.', 'astro-ph-0011556-1-0-4': 'Scale height determinations of their DIG halos are generally lower than those of galaxies with more prominent extraplanar DIG features.', 'astro-ph-0011556-1-1-0': 'These data, along with previously presented data for NGC 5775 and NGC 891, are used to address the issue of how DIG halos are energized.', 'astro-ph-0011556-1-1-1': 'Composite photo-ionization/shock models are generally better at explaining runs of line ratios in these galaxies than photoionization models alone.', 'astro-ph-0011556-1-1-2': 'Models of line ratios in NGC 5775 require a greater contribution from shocks for filamentary regions than for non-filamentary regions to explain the run of [O3]/H[MATH].', 'astro-ph-0011556-1-1-3': 'In either case, the [S2]/[N2] ratio is not well fit by the models.', 'astro-ph-0011556-1-1-4': "Composite models for UGC 10288 are successful at reproducing the run of [S2]/[N2] for all but the the highest values of [N2]/H[MATH]; however, the run of [O3]/H[MATH] vs. [N2]/H[MATH] does not show any discernible trend, making it difficult to determine whether or not shocks contribute to the layer's maintenance.", 'astro-ph-0011556-1-2-0': 'We also examine whether the data can be explained simply by an increase in temperature with [MATH] in a pure photo-ionization model without a secondary source of ionization.', 'astro-ph-0011556-1-2-1': 'Runs of [S2]/H[MATH], [N2]/H[MATH], and [S2]/[N2] in each of the four galaxies are consistent with such an increase.', 'astro-ph-0011556-1-2-2': 'However, the runs of [O3]/H[MATH] vs. [MATH] in NGC 5775 and UGC 10288 require unusually high ionization fractions of O[MATH] that can not be explained without invoking a secondary ionization source or at the very least a much higher temperature for the [O3]-emitting component than for the [S2]- and [N2]-emitting component.', 'astro-ph-0011556-1-2-3': 'An increase in temperature with [MATH] is generally more successful at explaining the [O3]/H[MATH] run in NGC 891, with the ionization fraction of O[MATH] remaining relatively low and nearly constant with [MATH].', 'astro-ph-0011556-1-3-0': '# Introduction', 'astro-ph-0011556-1-4-0': 'A vertically extended layer known as the Reynolds Layer or the Warm Ionized Medium (WIM) of the Milky Way contains the majority of the ionized gas in the interstellar medium.', 'astro-ph-0011556-1-4-1': 'The layer fills about 20 of the ISM volume with a local midplane density of [MATH]0.1 cm[MATH] and a scale height of [MATH]1 kpc (Reynolds 1993).', 'astro-ph-0011556-1-4-2': "Energetic arguments favor photo-ionization by massive stars in the disk for the layer's maintenance, as roughly 15 their ionizing output is sufficient to maintain the layer in the solar neighborhood (Reynolds 1993).", 'astro-ph-0011556-1-4-3': 'The Wisconsin H[MATH] Mapper (WHAM; Reynolds et.', 'astro-ph-0011556-1-4-4': 'al. 1998b) has been instrumental in recent years in characterizing the WIM.', 'astro-ph-0011556-1-4-5': 'In external galaxies, where the layer is more commonly referred to as Diffuse Ionized Gas (DIG), analogous information has been obtained through the use of narrow-band emission-line imaging, long-slit spectroscopy, and Fabry-Perot observations [e.g. Rand, Kulkarni, Hester 1990 (hereafter RKH); Rand 1996, 1997, 1998; Golla, Dettmar, Domgorgen 1996; Ferguson, Wyse, Gallagher 1996; Wang, Heckman, Lehnert 1997; Hoopes, Walterbos, Rand 1999; Collins et.', 'astro-ph-0011556-1-4-6': 'al. 2000 (hereafter CRDW)].', 'astro-ph-0011556-1-4-7': "Estimates of the diffuse fraction in DIG layers put the ionization requirement closer to 40% of the massive stars' ionizing output (e.g. Hoopes et.", 'astro-ph-0011556-1-4-8': 'al. 1996; Ferguson et.', 'astro-ph-0011556-1-4-9': 'al. 1996).', 'astro-ph-0011556-1-4-10': 'The advantage in observing external edge-on galaxies is the ability to obtain an overview of the entire halo, so that brightness, spatial distribution, and emission line ratios can be better characterized as they vary with [MATH].', 'astro-ph-0011556-1-5-0': 'One of the most important issues regarding these layers is their ionization.', 'astro-ph-0011556-1-5-1': 'In the WIM, as well as in external DIG layers, [S2] [MATH]H[MATH] and [N2] [MATH]H[MATH] are enhanced relative to [H2] regions [e.g. Haffner, Reynolds, Tufte 1999 (hereafter HRT); Ferguson, Wyse, Gallagher 1996; Rand 1997].', 'astro-ph-0011556-1-5-2': 'Such a scenario is predicted by photo-ionization models in which the dilution of the radiation field, measured by the ionization parameter, [MATH], increases as photons propagate away from H2 regions [Domgorgen Mathis 1994; Sokolowski 1994 (hereafter S94); Bland-Hawthorn et.', 'astro-ph-0011556-1-5-3': 'al. 1997].', 'astro-ph-0011556-1-5-4': 'Species such as S which are predominantly doubly ionized in H2 regions are able to recombine to a singly ionized state as the field becomes increasingly dilute.', 'astro-ph-0011556-1-5-5': 'Such an effect should not be as significant for N or O, which have considerably higher second ionization potentials and thus may not be fully doubly ionized in H2 regions.', 'astro-ph-0011556-1-6-0': 'The photo-ionization scenario is not without problems, however.', 'astro-ph-0011556-1-6-1': 'Firstly, in an increasingly dilute field, photoionization models predict that the ratio [S2]/[N2] should increase with [MATH] due to differing second ionization potentials of S and N. However, [S2]/[N2] vs. [MATH] has been found to remain nearly constant in the WIM (HRT) and NGC 891 (Rand 1998), and in the case of NGC 4631, the variations with [MATH] are not as pronounced as those predicted by the models (Golla et al. 1996).', 'astro-ph-0011556-1-6-2': 'In addition, the ratio [O3] [MATH]H[MATH] should decrease with [MATH] in a photo-ionization scenario as the field becomes increasingly dilute.', 'astro-ph-0011556-1-6-3': 'The opposite trend, however, is seen in NGC 891 where [O3]/H[MATH] increases significantly from the midplane to [MATH] kpc (Rand 1998).', 'astro-ph-0011556-1-6-4': '[O3]/H[MATH] enhancements in DIG relative to H2 regions are also seen in a number of more face-on galaxies, including M51 (Wang et.', 'astro-ph-0011556-1-6-5': 'al. 1997).', 'astro-ph-0011556-1-6-6': 'Finally, photoionization models have trouble explaining values of [N2]/H[MATH] and [S2]/H[MATH] that are often [MATH] (Wang et.', 'astro-ph-0011556-1-6-7': 'al. 1997; Rand 1998; HRT).', 'astro-ph-0011556-1-6-8': 'Models of S94, who took observed line ratios in NGC 891 into consideration, can only explain such ratios by assuming an upper mass cutoff for the stellar IMF of [MATH], hardening of the radiation field as it propagates, and a depletion of important gas-phase coolants onto dust grains.', 'astro-ph-0011556-1-6-9': 'Observations of the weak He1 [MATH] line, however, in the Reynolds Layer (Reynolds Tufte 1995) and in NGC 891 (Rand 1997) each imply a significantly softer spectrum than indicated by the forbidden lines.', 'astro-ph-0011556-1-7-0': 'One possibility that has been forwarded to explain some of this line ratio behavior is to assume a secondary ionizing source that contributes a small but increasing fraction of the H[MATH] emission with [MATH].', 'astro-ph-0011556-1-7-1': 'Such secondary sources include shocks (Shull McKee 1979) or turbulent mixing layers (TMLs; Slavin, Shull, Begelman 1993), each of which may produce large amounts of [O3] emission.', 'astro-ph-0011556-1-7-2': 'Other, less well-studied ionization sources may be relevant in halos, such as ionization from soft X-rays or cosmic rays.', 'astro-ph-0011556-1-8-0': 'Another approach that has been taken is to assume, instead of the ionization parameter or state changing with [MATH], that the line ratio behavior can be explained by the gas temperature increasing with [MATH].', 'astro-ph-0011556-1-8-1': 'This requires a source of additional heating, such as photo-electric heating from dust grains (Reynolds Cox 1992) or dissipation of interstellar turbulence (Minter Balser 1997), that becomes more significant at lower gas densities, but may not require any additional source of ionization.', 'astro-ph-0011556-1-8-2': 'A temperature increase with [MATH] is successful in explaining enhancements in [N2]/H[MATH] and [S2]/H[MATH] with [MATH], as well the constancy in [S2]/[N2] in the Reynolds Layer up to [MATH] kpc (HRT; Reynolds, Haffner, Tufte 1999).', 'astro-ph-0011556-1-8-3': 'Such a rise in temperature could also explain an increase in [O3]/H[MATH], though data on the run of [O3]/H[MATH] vs. [MATH] in the WIM are currently lacking.', 'astro-ph-0011556-1-9-0': 'The goal of this research is to attempt to better understand how DIG halos are energized.', 'astro-ph-0011556-1-9-1': 'We attempt to determine whether measured runs of line ratios versus [MATH] can be explained by: (a) photo-ionization with a contribution from shock ionization that increases with height above the midplane, and/or (b) only photo-ionization, but with an increase in gas temperature with [MATH].', 'astro-ph-0011556-1-9-2': 'We examine our data in the context of each of these scenarios, realizing that elements of both could be important for understanding the ionization of DIG halos in general.', 'astro-ph-0011556-1-9-3': 'To this end, we have obtained data for two galaxies: NGC 4302 and UGC 10288.', 'astro-ph-0011556-1-9-4': 'We also employ data for NGC 5775, already presented by Rand (2000).', 'astro-ph-0011556-1-9-5': 'For the purpose of assessing the rising temperature scenario, we use data for NGC 891, which has been compared with composite shock/photo-ionization models previously (Rand 1998).', 'astro-ph-0011556-1-9-6': 'The observations of these galaxies are discussed in 2.', 'astro-ph-0011556-1-9-7': 'Line ratio data are presented in 3 along with comparisons to composite shock/photo-ionization and rising temperature models.', 'astro-ph-0011556-1-9-8': 'We briefly summarize our results in 4.', 'astro-ph-0011556-1-10-0': '# Observations', 'astro-ph-0011556-1-11-0': 'Observational details for the NGC 891 (Rand 1998) and NGC 5775 (Rand 2000) data have been discussed in previous papers, though some modifications have been made which will be discussed in this section.', 'astro-ph-0011556-1-11-1': 'Spectra for UGC 10288 and NGC 4302 were obtained at the KPNO 4-m telescope on 1999 June 10-13 and 1998 February 23-26, respectively.', 'astro-ph-0011556-1-11-2': 'Slits for all four galaxies run perpendicular to the plane.', 'astro-ph-0011556-1-11-3': 'The slit position for NGC 4302, covering the most prominent region of extraplanar DIG, is shown in Figure 1, overlaid on the H[MATH] image.', 'astro-ph-0011556-1-11-4': 'The H[MATH] image of UGC 10288, showing little extraplanar H[MATH] emission, is shown in Figure 2 with slit position overlays.', 'astro-ph-0011556-1-11-5': "The slit positions for UGC 10288 were chosen to be roughly 30 east and west of the galaxy's center.", 'astro-ph-0011556-1-11-6': 'Due to the presence of a foreground star on the axis perpendicular the galactic disk 30 west of the radio continuum center, slit 2 was placed at a position 35 west of center.', 'astro-ph-0011556-1-11-7': 'A summary of the observations is given in Table 1.', 'astro-ph-0011556-1-12-0': 't1', 'astro-ph-0011556-1-13-0': 'The observations of NGC 4302 utilized the KPC-24 grating in conjunction with the T2KB 2048x2048 CCD, providing a dispersion of 0.51 per pixel, a resolution of 1.3 and a useful coverage of about 800.', 'astro-ph-0011556-1-13-1': 'The slit length is 5\', the slit width was set at 3", and the spatial scale is 0.69" per pixel.', 'astro-ph-0011556-1-13-2': 'For red line observations, the grating was tilted to give an approximate central wavelength of 6600, allowing H[MATH], [N2] [MATH]6548, 6583, [S2] [MATH]6716, 6731, and [O1] [MATH]6300 to be observed.', 'astro-ph-0011556-1-13-3': 'Due to poor weather during the 1998 observing run, spectra containing [O3] [MATH]5007 could not be obtained.', 'astro-ph-0011556-1-14-0': 'For observations of UGC 10288, the KPC-007 grating was used with the T2KB 2048x2048 CCD, providing a dispersion of 1.42 per pixel, a resolution of 3.5 and a useful coverage of about 2100 encompassing the spectral lines H[MATH], [N2] [MATH]6548, 6583, [N2] [MATH]5755, [S2] [MATH]6716, 6731, He1 [MATH]5876, [O1] [MATH]6300, [O3] [MATH]5007, and H[MATH].', 'astro-ph-0011556-1-14-1': 'The slit length is 5\', the slit width was set at 2", and the spatial scale is 0.69" per pixel.', 'astro-ph-0011556-1-15-0': 'Data reductions were performed with the IRAFpackage.', 'astro-ph-0011556-1-15-1': 'Projector flats were used to remove small-scale variations in response, while sky flats were used to determine the slit illumination correction.', 'astro-ph-0011556-1-15-2': 'Arc lamp exposures were used to calibrate the wavelength scale as a function of location along the slit, and the spectral response function was determined with standard stars.', 'astro-ph-0011556-1-15-3': 'Regions of the spectra free of emission from the galaxies were used to form sky-subtracted spectra.', 'astro-ph-0011556-1-15-4': 'In order to increase sensitivity for the determination of line parameters, blocks of 5 pixels were averaged in the spatial direction.', 'astro-ph-0011556-1-15-5': 'Since Gaussians appear to accurately represent the line profiles, Gaussian fits were used to determine line properties.', 'astro-ph-0011556-1-15-6': 'Uncertainties in line parameters for NGC 4302 were determined with the IRAF task splot.', 'astro-ph-0011556-1-15-7': 'The use of this method for NGC 5775 and UGC 10288 yielded error bars that were far greater than uncertainties suggested by the data.', 'astro-ph-0011556-1-15-8': 'In this case error bars were estimated by determining flux variations along the slit for sky lines of a range of intensities, from which a relation between these variances and sky line intensity could be established.', 'astro-ph-0011556-1-15-9': 'This relation was then used to determine flux uncertainties for the various emission lines.', 'astro-ph-0011556-1-15-10': 'Uncertainties for H[MATH] line strengths in NGC 5775 also take into account the coincidence of the H[MATH] line with a blended pair of sky lines.', 'astro-ph-0011556-1-15-11': 'This factor was not considered in the presentation of NGC 5775 line data in Rand (2000).', 'astro-ph-0011556-1-16-0': 'In order to compare [O3] [MATH]5007 emission to that of H[MATH] it was necessary to make a reddening correction to the [O3] line in NGC 5775, UGC 10288, and NGC 891.', 'astro-ph-0011556-1-16-1': 'This correction was not performed for the presentation of the [O3]/H[MATH] ratio in Rand (2000) for NGC 5775.', 'astro-ph-0011556-1-16-2': 'In all three galaxies, H[MATH] could be measured for all points where [OIII][MATH] could be measured, thus allowing extinction corrections to be made for the full range in [MATH].', 'astro-ph-0011556-1-16-3': 'The correction was determined by assuming a uniform foreground dust screen and comparing measured values of the H[MATH]/H[MATH] ratio to the value calculated by Osterbrock (1989) for a 10[MATH] K gas.', 'astro-ph-0011556-1-16-4': 'The assumption of a 10[MATH] K gas is not contradictory to a scenario including a rising temperature with [MATH] in that the H[MATH]/H[MATH] ratio is not nearly as temperature sensitive as the ratios of forbidden lines to Balmer lines: from 5,000 to 10,000 K, H[MATH]/H[MATH] increases by [MATH], whereas the ratios of [N2], [S2], and [O3] to H[MATH] increase by an order of magnitude.', 'astro-ph-0011556-1-16-5': 'In NGC 5775, the correction to the [OIII] flux decreases from a factor of 2.9 in the midplane to no measurable extinction by [MATH] kpc.', 'astro-ph-0011556-1-16-6': 'The correction is more variable for UGC 10288, with a maximum correction of 3.6 at [MATH] pc on the south side.', 'astro-ph-0011556-1-16-7': 'In NGC 891, the correction decreases from 2.2 in the midplane to 1.5 at [MATH] kpc.', 'astro-ph-0011556-1-16-8': 'In both NGC 5775 and NGC 891, the H[MATH] line is superposed on a weak stellar absorption line close to the midplane.', 'astro-ph-0011556-1-16-9': 'However, the absorption line is much broader than the emission line and thus can be accounted for by the background fitting process in the line fitting routine.', 'astro-ph-0011556-1-17-0': '# Results and Discussion', 'astro-ph-0011556-1-18-0': '## Line Data and Line Ratios', 'astro-ph-0011556-1-19-0': 'The H[MATH], [N2] [MATH]6583, and [S2] [MATH]6716 lines are detected up to nearly [MATH] kpc on either side of the plane of NGC 4302.', 'astro-ph-0011556-1-19-1': 'The [S2] [MATH]6716 line was not detected at [MATH] kpc on the west side, though the fainter line at 6731 was detected, due to confusion with noise in that part of the spectra.', 'astro-ph-0011556-1-19-2': 'The flux of the 6716 line at that part of the spectra was inferred from that of the 6731 line assuming the low-density limit of [S2] [MATH].', 'astro-ph-0011556-1-19-3': 'The vertical runs of [N2] [MATH]H[MATH], [S2] [MATH]H[MATH], and [S2] [MATH][N2] [MATH] are shown in Figure 3.', 'astro-ph-0011556-1-19-4': 'The data are averaged over 5 spatial pixels.', 'astro-ph-0011556-1-19-5': 'The [N2]/H[MATH] and [S2]/H[MATH] each show a similar rise with [MATH], with the [N2]/H[MATH] rising from about 0.4 in the midplane to 1.0 at [MATH] kpc on the east side, and to nearly 1.4 at [MATH] kpc on the west side.', 'astro-ph-0011556-1-19-6': 'The [S2]/[N2] remains fairly constant at about 0.6 for the full range of [MATH], with slight increases in the midplane and beyond [MATH] kpc.', 'astro-ph-0011556-1-20-0': 'In UGC 10288, the H[MATH], [N2] [MATH]6583, and [S2] [MATH]6716 lines are all detected well above the midplane, with the [N2] [MATH]6583 and H[MATH] lines being detected up to nearly [MATH] kpc on the north side for both slit positions.', 'astro-ph-0011556-1-20-1': 'Detections of the [O3] [MATH]5007 and [O1] [MATH]6300 lines are limited to [MATH] kpc, except for the south side of slit 1 where both lines are detected up to [MATH] kpc.', 'astro-ph-0011556-1-20-2': 'The vertical runs of [N2] [MATH]H[MATH], [S2] [MATH]H[MATH], [S2] [MATH][N2] [MATH], [O3] [MATH]H[MATH], and [O1] [MATH]H[MATH] for both slits are shown in Figure 4.', 'astro-ph-0011556-1-20-3': 'The data are averaged over 5 spatial pixels.', 'astro-ph-0011556-1-20-4': 'For slit 1, [N2]/H[MATH] rises from 0.3 at the midplane to 1.6 at [MATH] kpc on the north side and 1.3 at [MATH] kpc on the south, while for slit 2 the ratio remains somewhat lower, rising to 1.2 for [MATH] kpc on either side of the disk.', 'astro-ph-0011556-1-20-5': 'The [S2]/[N2] ratio shows a similar range of values for each slit, varying from 0.8 to 1.4 with the lower values generally occurring in the midplane.', 'astro-ph-0011556-1-20-6': 'These values are significantly higher than those of the other three galaxies.', 'astro-ph-0011556-1-20-7': 'The [O3]/H[MATH] ratio shows some evidence for an increase with [MATH] on the south side of either slit, reaching values as high as 0.75 in Slit 1 and 0.4 in Slit 2.', 'astro-ph-0011556-1-20-8': 'On the north side of each slit, values reach no higher than 0.5 in Slit 1 and 0.4 in Slit 2.', 'astro-ph-0011556-1-20-9': 'Such values of [O3]/H[MATH] are consistent with those observed at comparable [MATH]-heights in NGC 5775.', 'astro-ph-0011556-1-21-0': 'Line ratio data for NGC 5775, except for the [O1] [MATH]H[MATH] and extinction corrected [O3] [MATH]H[MATH] ratios, have been presented previously (Rand 2000).', 'astro-ph-0011556-1-21-1': 'These, along with [N2] [MATH]H[MATH], [S2] [MATH]H[MATH], and [S2] [MATH][N2] [MATH] ratios are shown in Figure 12.', 'astro-ph-0011556-1-21-2': 'The data are averaged over 10 spatial pixels.', 'astro-ph-0011556-1-21-3': 'An increase with [MATH] in the extinction corrected [O3]/H[MATH] for the NE side of Slit 1 and the SW side of Slit 2 is still indicated by the data, where values range from 0.2-0.4 in the disk to 1.1 in the halo.', 'astro-ph-0011556-1-21-4': 'The [O1/H[MATH] ratio ranges from about 0.02 in the midplane to 0.06 at [MATH] kpc in Slit 2, while the [O1] [MATH] line was not detected for Slit 1.', 'astro-ph-0011556-1-21-5': 'Because of the 85 (Irwin 1994) inclination of the galaxy, points within 10 of the midplane ([MATH] pc in the figures) on the spatial axis reflect in-plane, highly inclined disk structure rather than true vertical structure.', 'astro-ph-0011556-1-21-6': 'In these lines of sight, H2 regions and areas between them have a greater effect on line ratio variations than extraplanar diffuse gas.', 'astro-ph-0011556-1-21-7': 'This fact explains why some line ratio minima are not at [MATH] kpc.', 'astro-ph-0011556-1-22-0': 'Line ratio data for NGC 891, previously presented by Rand (1998), can be seen in Figure 13.', 'astro-ph-0011556-1-23-0': '## DIG scale heights', 'astro-ph-0011556-1-24-0': "In this section we attempt to determine the H[MATH] emission scale height, [MATH], of the galaxies' DIG halos at the various slit locations.", 'astro-ph-0011556-1-24-1': 'We attempt to model only an exponential halo component, avoiding any disk component where contamination by bright H2 regions could complicate the analysis.', 'astro-ph-0011556-1-24-2': 'We plot the logarithm of H[MATH] intensity versus [MATH] for each slit, then fit an exponential component with a least-squares fit except in cases where an exponential is obviously a bad description.', 'astro-ph-0011556-1-24-3': 'Plots of logarithmic H[MATH] intensity vs. [MATH], along with exponential fits, for NGC 5775, NGC 4302, and UGC 10288 are shown in Figures 5, 6, and 7, respectively.', 'astro-ph-0011556-1-24-4': 'Results of the fitting procedure are summarized in Table 2.', 'astro-ph-0011556-1-25-0': 'Position 1 of NGC 5775 covers the NE filament, DIG emission from which has been modeled previously (CRDW).', 'astro-ph-0011556-1-25-1': 'The value [MATH] pc is somewhat consistent with the value of [MATH] kpc determined from image data.', 'astro-ph-0011556-1-25-2': 'The image data however does include some [N2] contamination.', 'astro-ph-0011556-1-25-3': 'The previously determined value assumes an [N2]/H[MATH] ratio that does not vary with [MATH].', 'astro-ph-0011556-1-25-4': 'Since our data shows that this ratio in fact rises with [MATH] on the NE filament, from 0.5 at [MATH] kpc to 0.75 at [MATH] kpc, the value [MATH] kpc becomes an upper limit on the actual scale height.', 'astro-ph-0011556-1-26-0': 'The DIG halos of NGC 4302 and UGC 10288 have been modeled previously as well (Rand 1996).', 'astro-ph-0011556-1-26-1': 'The slit position for NGC 4302 coincides with one of the previously modeled regions, the scale height of which was determined to be [MATH] pc (the image is not contaminated by [N2] emission).', 'astro-ph-0011556-1-26-2': 'The values determined from spectroscopic data, [MATH] pc and [MATH] pc for the east and west sides, are close to those determined from image data.', 'astro-ph-0011556-1-26-3': 'The scale height determined from the image, however, is an average over a 3 kpc radial extent of the disk and thus should not be in exact agreement with the spectroscopically determined value.', 'astro-ph-0011556-1-26-4': 'Image data of UGC 10288 revealed very little extraplanar H[MATH] emission, and thus profiles were not well modeled with exponentials.', 'astro-ph-0011556-1-26-5': 'Increased sensitivity of the spectroscopic data have allowed the detection of H[MATH] emission to greater [MATH]-heights.', 'astro-ph-0011556-1-26-6': 'The resulting H[MATH] profiles seem to be well modeled by exponentials with small scale heights on the south side, while more complicated structure is present on the north side.', 'astro-ph-0011556-1-27-0': 't2', 'astro-ph-0011556-1-28-0': 'NGC 5775, having the most prominent extraplanar DIG emission, also has the DIG halo with the greatest scale height.', 'astro-ph-0011556-1-28-1': 'In contrast, UGC 10288, showing very little extraplanar H[MATH] emission, has a DIG halo of relatively small scale height.', 'astro-ph-0011556-1-28-2': 'NGC 4302, having a number of extraplanar plumes of H[MATH] emission, has a DIG scale height intermediate between that of NGC 5775 and UGC 10288.', 'astro-ph-0011556-1-28-3': 'The halo of NGC 891 has been modeled previously (RKH), the scale height of which has been measured to be [MATH] kpc on the west side and [MATH] kpc on the east side from imaging data.', 'astro-ph-0011556-1-28-4': 'These results suggest that not only do galaxies such as NGC 5775 and NGC 891 have brighter DIG halos, where sensitivity limits allow faint emission to be detected at much greater heights than for a galaxy with a lower H[MATH] surface brightness, but the halos also have much larger scale heights.', 'astro-ph-0011556-1-28-5': 'This adds further evidence to the notion of a dynamic halo where higher levels of star formation in the disk result in material driven further into the halo (e.g. Rand 1996; Hoopes et.', 'astro-ph-0011556-1-28-6': 'al. 1999).', 'astro-ph-0011556-1-29-0': '## Shocks as a Secondary Source of Ionization', 'astro-ph-0011556-1-30-0': 'Some of the line ratio behavior observed in NGC 5775 and UGC 10288, namely the rise in [N2]/H[MATH] and [S2]/H[MATH] with [MATH], is predicted by the photo-ionization models (e.g. S94; Bland-Hawthorn 1997).', 'astro-ph-0011556-1-30-1': 'As previously stated, rises in these ratios are expected in such models as the radiation field becomes more dilute with [MATH].', 'astro-ph-0011556-1-30-2': 'If the field is in fact becoming more dilute (low values of the ionization parameter, [MATH], which measures the "diluteness\'\'of the radiation field and is proportional to the ratio of ionizing photon number density to gas density) with [MATH], then the models predict that the [O3]/H[MATH] ratio should decrease away from the midplane.', 'astro-ph-0011556-1-30-3': 'However, in NGC 5775, and to a lesser extent UGC 10288, we see a definite incease in [O3]/H[MATH] with [MATH], indicative of higher excitation conditions above the disk in such models (high-[MATH]).', 'astro-ph-0011556-1-30-4': 'It is then possible that the [O3] emission originates from a different DIG component than that of the [N2] and [S2].', 'astro-ph-0011556-1-30-5': 'Consequently, a secondary mechanism, such as shock ionization, could account for the bright [O3] emission above the plane.', 'astro-ph-0011556-1-31-0': 'In this section we explore the possibility that shock ionization is responsible for some of the DIG emission in these galaxies.', 'astro-ph-0011556-1-31-1': 'One could also consider TMLs, since they also produce large amounts of [O3] emission.', 'astro-ph-0011556-1-31-2': 'In fact in NGC 891, TMLs do just as well as shocks in fitting the observed data (Rand 1998).', 'astro-ph-0011556-1-31-3': 'Our approach is that of Rand (1998), with some minor changes.', 'astro-ph-0011556-1-31-4': 'We adopt the S94 matter-bounded photoionization model with the lowest terminal hydrogen column density, [MATH] cm[MATH], of individual gas clumps.', 'astro-ph-0011556-1-31-5': 'The S94 models feature a hard stellar spectrum (an upper mass cutoff of [MATH]), hardening of the radiation field as it propagates through the ISM, and depletion of gas-phase coolants onto dust grains.', 'astro-ph-0011556-1-31-6': 'We do not consider the S94 radiation-bounded model as this model predicts extremely bright [O3] emission.', 'astro-ph-0011556-1-31-7': 'Any composite shock model utilizing radiation-bounded models requires a very unusual secondary source of H[MATH] emission.', 'astro-ph-0011556-1-31-8': 'In the case of NGC 5775, such a scenario would require a secondary source that accounts for nearly half of the H[MATH] emission at the midplane (assuming log [MATH] at [MATH] kpc), and produces virtually no [O3] or [N2] emission.', 'astro-ph-0011556-1-31-9': 'In addition, such a source must produce significant [O3] and [S2] emission at high-[MATH], and still contribute at least 50% of the total H[MATH] emission.', 'astro-ph-0011556-1-31-10': 'Such emission characteristics could be explained with low-speed shocks near the midplane with a shock velocity increasing with [MATH].', 'astro-ph-0011556-1-31-11': 'However, the problem is further complicated by the fact that low-speed shocks produce copious amounts of [O1] emission relative to [N2].', 'astro-ph-0011556-1-31-12': "In addition, basic energetic arguments render such a scenario, where the secondary source contributes half of the layer's ionization, highly unlikely as the ionizing flux from massive stars exceeds supernovae power output, presumably the initiator of these shocks, by a factor of 6 or 7 (Reynolds 1984, 1992).", 'astro-ph-0011556-1-31-13': 'In any case, parameters describing these shocks require considerable tweaking to account for the observed run of line ratios.', 'astro-ph-0011556-1-31-14': 'Matter-bounded models on the other hand, better match the data before composite modeling, and require a less complicated parameter space describing the secondary source.', 'astro-ph-0011556-1-32-0': 'To obtain line ratios due to shock ionization, we use the models of Shull McKee (1979; hereafter SM79).', 'astro-ph-0011556-1-32-1': 'Line ratios in these models, especially [O3]/H[MATH], are highly dependent on shock velocity.', 'astro-ph-0011556-1-32-2': 'Other variables affecting line ratios include the preshock gas density and ionization state, abundances, and transverse magnetic field strength.', 'astro-ph-0011556-1-32-3': 'These models assume the gas to be initially neutral at [MATH] cm[MATH], then penetrated by a precursor ionization front.', 'astro-ph-0011556-1-32-4': 'More appropriate to the case of a partially photo-ionized DIG layer, would be a initial density of order [MATH] cm[MATH], and a high initial ionization fraction.', 'astro-ph-0011556-1-32-5': 'We mainly consider the models with shock velocities of [MATH] km s[MATH] and [MATH] km s[MATH] in that these higher velocity models are best able to produce large amounts of [O3] emission, while keeping [O1] low relative to [N2].', 'astro-ph-0011556-1-32-6': 'SM79 also calculate a single model containing depleted abundances with [MATH] km s[MATH], which we also consider in the modeling to allow a comparison between shock and photo-ionization models with depleted abundances.', 'astro-ph-0011556-1-32-7': 'In general, though, we do not consider this analysis as providing a definitive measure of the shock speed in these composite models given the simplicity of the model and the many unrealistic parameters which describe the shocks.', 'astro-ph-0011556-1-33-0': 'For the composite modeling, we assume that some fraction of H[MATH] emission arises from shock ionization, with that fraction possibly changing with [MATH].', 'astro-ph-0011556-1-33-1': 'We still assume a stellar radiation field distinguished by a decrease in ionization parameter, [MATH], with [MATH].', 'astro-ph-0011556-1-33-2': 'To simplify matters we assume a single shock velocity for each composite model.', 'astro-ph-0011556-1-33-3': 'We attempt to fit the composite model to the data by varying the percent contribution at a given value of [MATH] (and thus [MATH]).', 'astro-ph-0011556-1-33-4': 'For NGC 5775 the modeling worked best by allowing shock contributions to begin at log [MATH], while models with shocks beginning at log [MATH] worked best for UGC 10288.', 'astro-ph-0011556-1-33-5': 'Such values are somewhat lower than would be expected for a low-[MATH] radiation field.', 'astro-ph-0011556-1-33-6': 'However, kinematics of edge-on galaxies indicate that at low-[MATH], the line emission we observe is mostly associated with the outer disk regions due to an absorbing dust layer.', 'astro-ph-0011556-1-33-7': 'Since star formation is typically concentrated in the inner disk, the radiation field in the outer disk may be relatively dilute, in which case, the value of [MATH] at low-[MATH] may be quite low (Rand 1998).', 'astro-ph-0011556-1-34-0': 'We do not carry out a statistical test of the goodness of fit since our goal is to qualitatively assess the feasibility of a secondary source of ionization being able to account for some of the line ratios.', 'astro-ph-0011556-1-34-1': 'Although no model reproduces the line ratios to within the errors, we do find that the composite models better reproduce the run of line ratios than photo-ionization models alone.', 'astro-ph-0011556-1-34-2': 'Rand (1998) finds a similar conclusion for NGC 891, where composite modeling reproduces the run of [O3]/H[MATH] vs. [N2]/H[MATH] and [S2]/H[MATH], yet is unable to duplicate the [S2]/[N2] ratio at low-[MATH].', 'astro-ph-0011556-1-34-3': 'NGC 4302 is not considered due to the lack of critical [O3] [MATH]5007 data.', 'astro-ph-0011556-1-35-0': '### NGC 5775', 'astro-ph-0011556-1-36-0': 'For the composite modeling, we consider separately the cases of filamentary and non-filamentary emission.', 'astro-ph-0011556-1-36-1': 'The NE portion of Slit 1 crosses the prominent NE H[MATH] filament, previously identified by CRDW as being associated with a shell-like H1 structure and possibly being involved in vigorous disk-halo interactions.', 'astro-ph-0011556-1-36-2': 'In addition, the SW portion of Slit 2 crosses a bright plume of H[MATH] emission immediately adjacent to the very prominent SW H[MATH] filament, also a region where intense disk-halo activity is suspected.', 'astro-ph-0011556-1-36-3': 'Since these two regions appear associated with current cycling of material from disk to halo, it follows that shock ionization may play a greater role in energizing DIG in these areas.', 'astro-ph-0011556-1-36-4': 'The SW portion of Slit 1 and the NE portion of Slit 2 each cross regions with relatively less extended DIG emission, where disk-halo interactions appear far less vigorous than regions with more filamentary features.', 'astro-ph-0011556-1-37-0': 'The runs of [O3]/H[MATH], [S2]/H[MATH], and [O1]/H[MATH] vs. [N2]/H[MATH] are shown in Figures 8, 9, and 10, respectively.', 'astro-ph-0011556-1-37-1': 'Figures 8 and 9 also present filamentary and non-filamentary regions separately.', 'astro-ph-0011556-1-37-2': 'The runs of these ratios in the matter-bounded S94 models, with various values of [MATH] labeled, are plotted along with predicted line ratios from the SM79 models for [MATH] km s[MATH], [MATH] km s[MATH], and [MATH] km s[MATH] with depletions.', 'astro-ph-0011556-1-37-3': 'We find the [MATH] km s[MATH] model is best able to reproduce the observed run of line ratios as a secondary source to photoionization.', 'astro-ph-0011556-1-37-4': 'The [MATH] km s[MATH] model has difficulty reproducing the run of [S2]/H[MATH] vs. [N2]/H[MATH], and the depleted [MATH] km s[MATH] model, though able to duplicate the run of [S2]/H[MATH] vs. [N2]/H[MATH], is unable to reproduce the run of [O3]/H[MATH] vs. [N2]/H[MATH] at high-[MATH].', 'astro-ph-0011556-1-38-0': 'We find that the run of [O3]/H[MATH] vs. [N2]/H[MATH] seen in Figure 8a for the filamentary regions is distinctly different from the trend seen in Figure 8b for the non-filamentary regions.', 'astro-ph-0011556-1-38-1': 'Clearly, for a given value of [N2]/H[MATH], the ratio [O3]/H[MATH] is much greater in DIG with a more filamentary morphology.', 'astro-ph-0011556-1-38-2': 'The non-filamentary regions show a run that is very similar to the run observed in NGC 891 (Rand 1998).', 'astro-ph-0011556-1-38-3': 'The run of [S2]/H[MATH] vs. [N2]/H[MATH] on the other hand, does not seem to depend strongly on the presence of filamentary DIG structure.', 'astro-ph-0011556-1-38-4': "Non-filamentary regions do seem to have greater values of [N2]/H[MATH] and [S2]/H[MATH], in keeping with these ratios' anti-correlation with relative H[MATH] surface brightness (see also Figure 12).", 'astro-ph-0011556-1-38-5': 'All data for the ratio [O1]/H[MATH], being detected only for Slit 2 to [MATH] kpc, are plotted in Figure 10.', 'astro-ph-0011556-1-38-6': 'The lack of [O1]/H[MATH] data for higher values of [N2]/H[MATH] makes a comparison between various DIG morphologies impossible.', 'astro-ph-0011556-1-39-0': 'Composite models are plotted in Figures 8, 9, and 10 as dashed lines joining open hexagons, which mark various values of log [MATH].', 'astro-ph-0011556-1-39-1': 'For the filamentary regions, 20% of the H[MATH] emission at log [MATH] arises from shock ionization, increasing to 85% at log [MATH].', 'astro-ph-0011556-1-39-2': 'This does not mean the majority of extraplanar DIG is shock ionized.', 'astro-ph-0011556-1-39-3': 'It does indicate however that at very high-[MATH] (log [MATH] corresponds to a height of about [MATH] kpc for the filamentary regions and [MATH] kpc for the non-filamentary), it is possible that most of the emission within the filaments arises from shock ionized gas.', 'astro-ph-0011556-1-39-4': 'In the non-filamentary case, shocks contribute 16% of the H[MATH] emission at log [MATH], rising to 27% at log [MATH].', 'astro-ph-0011556-1-39-5': 'For the [O1]/H[MATH] vs. [N2]/H[MATH] data of Figure 10, the filamentary composite model is plotted over all data points for both filamentary and non-filamentary regions.', 'astro-ph-0011556-1-39-6': 'This model and the non-filamentary model are somewhat successful in explaining the [O1]/H[MATH] vs. [N2]/H[MATH] run at low-[MATH], in the sense of matching the excess of [O1]/H[MATH] for a given [N2]/H[MATH] compared to the S94 model alone.', 'astro-ph-0011556-1-40-0': 'The composite models both match the runs of [O3]/H[MATH] vs. [N2]/H[MATH].', 'astro-ph-0011556-1-40-1': 'The main deficiency in the models is their inability to reproduce the run of [S2]/[N2] at low-[MATH].', 'astro-ph-0011556-1-40-2': 'The models do not predict enough [S2] emission for a given value of [N2].', 'astro-ph-0011556-1-40-3': 'The models perform somewhat better at high-[MATH], though the spread in data values at high-[MATH] make a trend somewhat difficult to discern.', 'astro-ph-0011556-1-40-4': 'Again, though models with depleted abundances are better able to produce an excess of [S2] emission relative to [N2], they do not produce enough [O3] emission.', 'astro-ph-0011556-1-41-0': 'In any case, it is clear that the filamentary regions require a greater contribution from shocks to explain the observed line ratios.', 'astro-ph-0011556-1-41-1': 'The correlation of these features with H1 shells and enhanced 20-cm radio continuum emission is also suggestive of a larger percent contribution to the observed emission in these regions from shock ionization.', 'astro-ph-0011556-1-41-2': 'Enhanced [O3] emission can also be produced by other sources of ionization such as TMLs.', 'astro-ph-0011556-1-41-3': 'Rand (1998) found that composite photo-ionization/TML models were just as successful in reproducing the runs of observed line ratios in NGC 891 as composite models featuring shocks.', 'astro-ph-0011556-1-41-4': 'TMLs are expected to occur at interfaces of gas at different temperatures in the ISM, such as in shell walls.', 'astro-ph-0011556-1-41-5': 'It is likely then for NGC 5775, where H1 shells are associated with the more prominent H[MATH] filaments, that TMLs could also account for the enhanced line ratios.', 'astro-ph-0011556-1-41-6': 'We have performed cursory modeling of the line ratios for NGC 5775 with composite photo-ionization/TML models.', 'astro-ph-0011556-1-41-7': 'Using the TML model of Slavin, Shull, Begelman (1993) with depleted abundances in the mixing layers of hot and warm gas, a hot gas mixing speed of [MATH] km s[MATH], and a mixed gas temperature of log [MATH], we adopt the same approach as the composite photo-ionization/shock modeling.', 'astro-ph-0011556-1-41-8': 'We find that the photo-ionization/TML models work nearly as well as the photo-ionization/shock models, though they are unable to reproduce the run of ratios for the highest values of [O3]/H[MATH].', 'astro-ph-0011556-1-42-0': '### UGC 10288', 'astro-ph-0011556-1-43-0': 'A preliminary analysis of the diagnostic diagrams for UGC 10288 has revealed no discernible trend in the run of [O3]/H[MATH] vs. [N2]/H[MATH].', 'astro-ph-0011556-1-43-1': 'To attempt to better reveal any possible trend, line ratios on either side of the disk have been averaged for each slit to establish a run of each ratio vs. [MATH].', 'astro-ph-0011556-1-43-2': 'These averaged runs of [O3]/H[MATH], [S2]/H[MATH], and [O1]/H[MATH] vs. [N2]/H[MATH] are shown in Figure 11.', 'astro-ph-0011556-1-43-3': 'The matter-bounded S94 models, with various values of [MATH] labeled, are plotted along with predicted line ratios from the SM79 models for [MATH] km s[MATH], [MATH] km s[MATH], and [MATH] km s[MATH] with depletions.', 'astro-ph-0011556-1-43-4': "Even with the averaging, the [O3]/H[MATH] vs. [N2]/H[MATH] run doesn't show an obvious trend: Slit 1 seems to show a slight trend of an increase in [O3]/H[MATH] with [N2]/H[MATH], while Slit 2 data is clustered near the photo-ionization model.", 'astro-ph-0011556-1-43-5': 'For this reason, we attempt to fit the run of [S2]/[N2] instead of [O3]/H[MATH] vs. [N2]/H[MATH].', 'astro-ph-0011556-1-43-6': 'We adopt the [MATH] km s[MATH] model with depleted abundances as the other two models do not produce enough [S2] emission relative to [N2].', 'astro-ph-0011556-1-44-0': 'Composite models are plotted in Figure 11 as dashed lines joining open hexagons, which mark various values of log [MATH].', 'astro-ph-0011556-1-44-1': 'We find that the composite models best agree with the data when shocks begin contributing to emission at log [MATH].', 'astro-ph-0011556-1-44-2': 'For the composite model, 20% of the H[MATH] emission at log [MATH] arises from a 100 km s[MATH] shock with depleted abundances, rising to 45% at log [MATH] (log [MATH] corresponds to a height of about [MATH] pc).', 'astro-ph-0011556-1-45-0': 'The composite model successfully replicates the run of [S2]/[N2] at all but very high-[MATH] where matter bounded photo-ionization models cannot explain values of [N2]/H[MATH] and [S2]/H[MATH] greater than unity.', 'astro-ph-0011556-1-45-1': 'Radiation bounded photo-ionization models feature greater values of these line ratios; however, predicted runs of [O3]/H[MATH] cannot be reconciled with observations.', 'astro-ph-0011556-1-45-2': 'The run of [O1]/H[MATH] vs. [N2]/H[MATH] is also well explained by the composite model.', 'astro-ph-0011556-1-45-3': 'The lack of any obvious trend in the run of [O3]/H[MATH] vs. [N2]/H[MATH] makes it difficult to assess whether shocks are a viable contributor to the H[MATH] emission.', 'astro-ph-0011556-1-45-4': 'However, the high values of [O3]/H[MATH] in Slit 1 do suggest a large contribution from shocks in those regions.', 'astro-ph-0011556-1-45-5': 'It should be noted, though, that the galaxy has a fairly quiescent appearance in the H[MATH] image with little extraplanar emission, calling into question the notion that shocks are permeating the ISM.', 'astro-ph-0011556-1-45-6': 'It would be useful to obtain more [O3] data for this galaxy to better diagnose the possible contributions from secondary ionization sources.', 'astro-ph-0011556-1-45-7': 'Nevertheless, a clear departure from a pure photoionization model is indicated by the data.', 'astro-ph-0011556-1-46-0': '## DIG Temperature as the Main Cause of Line Ratio Variations', 'astro-ph-0011556-1-47-0': 'In this section, following the approach of HRT, we attempt to determine whether variations in line ratios with [MATH] can be explained by a change in gas temperature as one moves off the midplane.', 'astro-ph-0011556-1-47-1': 'A temperature increase with [MATH] provides a simple explanation of why all ratios of forbidden lines to Balmer lines are generally seen to increase with [MATH].', 'astro-ph-0011556-1-47-2': 'In fact, HRT find for the Reynolds Layer that the increase in [S2]/H[MATH] and [N2]/H[MATH], and the constancy of [S2]/[N2] with [MATH], up to [MATH] kpc, can be accounted for by an increase in gas temperature alone, without having to invoke a secondary source of ionization.', 'astro-ph-0011556-1-47-3': 'Such a scenario requires additional non-ionizing heating for low-density, high-[MATH] gas, such as photo-electric heating from dust grains (Reynolds Cox 1992) or dissipation of interstellar turbulence (Minter Balser 1997).', 'astro-ph-0011556-1-47-4': 'Using line ratio data versus [MATH] for NGC 4302, NGC 5775, UGC 10288, and NGC 891, we attempt to test this scenario by determining gas temperatures as well as ionization fractions of constituent elements.', 'astro-ph-0011556-1-48-0': 'For collisionally excited ions, the equation for intensity of emission lines is given by Osterbrock (1989), [EQUATION] where [MATH] and [MATH] are the collision strength and energy of the transition, [MATH] is the statistical weight of the ground level, [MATH] is the temperature in units of 10[MATH] K, and [MATH] is the fraction of downward transitions that produce the emission line.', 'astro-ph-0011556-1-49-0': 'To simplify matters, we assume no change in physical conditions, namely temperature, density, ionization fraction, and gas abundance, along a given line-of-sight through the DIG layer.', 'astro-ph-0011556-1-49-1': 'Such uniformity of physical parameters within the ISM is clearly not accurate when lines-of-sight cross small scale structures such as H2 regions or shocks, where temperature or ionization conditions may change dramatically.', 'astro-ph-0011556-1-49-2': 'The assumption is valid, however, if these parameters are thought of as averages along a given line of sight instead of describing localized conditions within the ISM.', 'astro-ph-0011556-1-50-0': 'This assumption, along with the use of the collision strengths provided by Aller (1984), allows the intensity of [S2], [N2], and [O3] emission to be represented, in units of rayleighs, by: [EQUATION] and [EQUATION]', 'astro-ph-0011556-1-50-1': 'The intensity of H[MATH] emission is given by, [EQUATION] where [MATH] is the emission measure in units of pc cm[MATH].', 'astro-ph-0011556-1-51-0': 'A further simplification is to assume N[MATH] and H[MATH] have approximately equal ionization fractions.', 'astro-ph-0011556-1-51-1': 'This assumption is justified for two reasons.', 'astro-ph-0011556-1-51-2': 'First, photoionization models of S94 predict that N[MATH]/N[MATH] tracks H[MATH]/H[MATH] due to similar first ionization potentials (14.5 eV versus 13.6 eV) and a weak charge exchange reaction.', 'astro-ph-0011556-1-51-3': 'Second, N is not likely to ionize higher than N[MATH] in a pure photoionization scenario due to a second ionization potential of 29.6 eV (HRT).', 'astro-ph-0011556-1-51-4': 'A high ionization fraction of O[MATH], however, would imply a significant fraction in the N[MATH] state as the second ionization potential of oxygen is 35.1 eV.', 'astro-ph-0011556-1-51-5': 'Thus we will be able to check the validity of this assumption with the [O3]/H[MATH] data.', 'astro-ph-0011556-1-51-6': 'Due to this simplification, of the line ratios available from this data, [N2]/H[MATH] is the best probe of gas temperature.', 'astro-ph-0011556-1-52-0': 'The forbidden lines are all dependent on their abundance relative to atomic hydrogen.', 'astro-ph-0011556-1-52-1': 'The issue of abundances is a complicated one, the understanding of which is critical for interpreting the line ratios.', 'astro-ph-0011556-1-52-2': 'Current evidence suggests that abundances in halo gas may be quite different from gas in the disk.', 'astro-ph-0011556-1-52-3': 'Savage and Sembach (1996) find increasing gas-phase abundances in the Milky Way from disk to halo, possibly as a result of partial grain destruction due to shocks.', 'astro-ph-0011556-1-52-4': 'However, models by Sembach et.', 'astro-ph-0011556-1-52-5': 'al. (2000; hereafter SHRK) predict that the ratios of forbidden lines to Balmer lines tend to rise as abundances fall due to decreased cooling efficiency.', 'astro-ph-0011556-1-52-6': 'For this reason, it is unlikely that abundance variations can explain the increase in ratios of forbidden lines to Balmer lines.', 'astro-ph-0011556-1-52-7': 'However, observations of the [O2] [MATH]3727 line, to complement [O3] [MATH]5007 and [O1] [MATH]6300 measurements, would be very useful to completely decouple line ratios (which depend on excitation and temperature) from variations in abundance.', 'astro-ph-0011556-1-52-8': 'To simplify matters, we adopt abundances that do not vary with [MATH].', 'astro-ph-0011556-1-52-9': 'Using galactic gas-phase abundances of S,N, and O of (S/H)[MATH] (Anders Grevesse 1989), (N/H)[MATH] (Meyer, Cardelli, Sofia 1997), and (O/H)[MATH] (Cardelli Meyer 1997), along with the assumption of equal fractions of N[MATH] and H[MATH], we arrive at expressions relating line ratios to temperature and ionization fraction: [EQUATION] and [EQUATION]', 'astro-ph-0011556-1-52-10': 'The ionization fractions of oxygen and hydrogen are coupled through a charge exchange reaction that allows the ionization state of Hydrogen to be determined from the [O1]/H[MATH] ratio.', 'astro-ph-0011556-1-52-11': 'We use the ratio of emissivities of the [O1] [MATH]6300 line to the H[MATH] line provided by Reynolds et.', 'astro-ph-0011556-1-52-12': 'al. (1998a) to obtain, [EQUATION]', 'astro-ph-0011556-1-52-13': 'Gas temperatures are derived from the [N2]/H[MATH] ratio.', 'astro-ph-0011556-1-52-14': 'Assuming this ratio accurately reflects temperatures within the medium, line ratios involving the S and O lines are then calculated for a variety of ionization fractions.', 'astro-ph-0011556-1-52-15': 'We can then see whether all the ionization fractions are reasonable and consistent with our initial assumptions.', 'astro-ph-0011556-1-52-16': 'Plots of temperature versus [MATH] and calculated ratios overlayed on measured ratios are shown in Figures 3, 4, 12, and 13 for NGC 4302, UGC 10288, NGC 5775, and NGC 891, respectively.', 'astro-ph-0011556-1-52-17': 'Resulting ionization fractions from the temperature modeling are shown in Table 3.', 'astro-ph-0011556-1-53-0': 't3', 'astro-ph-0011556-1-54-0': 'To first order, a rise in DIG temperature can account for the observed rises in line intensities relative to H[MATH] emission, as well as the near constancy in the [S2]/[N2] ratio.', 'astro-ph-0011556-1-54-1': 'We find, except in the case of UGC 10288, the ionization fraction of S[MATH] relative to H[MATH] remains very close to 0.5 independent of [MATH].', 'astro-ph-0011556-1-54-2': 'This is similar to the result obtained by HRT for the WIM.', 'astro-ph-0011556-1-54-3': 'Photo-ionization models performed by SHRK predict a comparable S[MATH] ionization fraction for low-density H2 regions.', 'astro-ph-0011556-1-54-4': 'UGC 10288 on the other hand, shows a considerably higher fraction of S[MATH] relative to H[MATH] as well as some evidence for an increase with [MATH].', 'astro-ph-0011556-1-54-5': 'However, the ionization fraction of H[MATH], determined from points relatively close to the midplane, is significantly lower than in the case of NGC 5775 or NGC 891.', 'astro-ph-0011556-1-54-6': 'As a result, the range in ionization fraction of S[MATH] in UGC 10288 is nearly equal to the range in values for the other galaxies.', 'astro-ph-0011556-1-54-7': 'This does not explain the observed rise with [MATH], however, and the lack of [O1] data at high-[MATH] makes it impossible to determine whether the ionization state of sulfur changes with [MATH].', 'astro-ph-0011556-1-55-0': 'The [O3]/H[MATH] data, in addition to revealing information about the ionization state of O, is also an excellent test of the validity of some of the assumptions used for this temperature modeling.', 'astro-ph-0011556-1-55-1': 'The data for NGC 5775 and UGC 10288 suggest that a significant fraction of oxygen is in the O[MATH] state; at least 15% and in some lines-of-sight nearly 100% relative to H[MATH].', 'astro-ph-0011556-1-55-2': 'The second ionization potential of oxygen is 35.1 eV, while that of nitrogen is 29.6 eV.', 'astro-ph-0011556-1-55-3': 'Thus one would expect the ionization fractions of the doubly-ionized species of these two elements to roughly track one another in a warm medium.', 'astro-ph-0011556-1-55-4': 'The determined values of ionization fraction of O[MATH] for these two galaxies imply that a significant percentage of nitrogen atoms are in the N[MATH] state, particularly at high-[MATH] for NGC 5775.', 'astro-ph-0011556-1-55-5': 'The likely presence of significant amounts of N[MATH] as well as the possibility of its fraction changing with [MATH] complicates the use of [N2]/H[MATH] as a temperature indicator.', 'astro-ph-0011556-1-55-6': 'In any event, the high values of O[MATH] ionization fraction suggest that a temperature increase alone is not able to fully explain the runs of line ratios observed in NGC 5775 and UGC 10288 since pure photo-ionization models predict very low O[MATH] fractions.', 'astro-ph-0011556-1-55-7': "One possible explanation for the large ionization fractions of doubly-ionized species is that a secondary ionizing source with emission characteristics similar to shocks or TMLs contributes to the layer's ionization.", 'astro-ph-0011556-1-55-8': 'Alternatively, the [O3] emitting gas could represent a component that is at a significantly higher temperature than the component responsible for the [N2] emission.', 'astro-ph-0011556-1-56-0': 'NGC 891 on the other hand, shows considerably lower values for the ionization fraction of O[MATH] relative to H[MATH]: 13% on either side of the disk and just above 30% in the midplane.', 'astro-ph-0011556-1-56-1': 'The slit runs near a bright filament that is not as prominent as those observed in NGC 5775.', 'astro-ph-0011556-1-56-2': 'These fractions are slightly larger than those predicted by the SHRK model for low-density H2 regions, though still more than an order of magnitude higher than that of the diffuse gas model of SHRK.', 'astro-ph-0011556-1-56-3': 'Thus it seems the run of line-ratios in NGC 891 is better explained by an increase in gas temperature with [MATH] than for NGC 5775.', 'astro-ph-0011556-1-56-4': 'A slightly warmer [O3] emitting component than indicated by the [N2]/H[MATH] ratio could explain measured values of [O3]/H[MATH] while keeping the ionization fraction of O[MATH] at levels predicted by the models.', 'astro-ph-0011556-1-56-5': 'If this is the case, a secondary ionizing source would not be necessary to explain the runs of line ratios in NGC 891.', 'astro-ph-0011556-1-57-0': 'The run of [SII]/H[MATH] in NGC 4302 is well fit by the increase in temperature reflected by the [NII]/H[MATH] ratio.', 'astro-ph-0011556-1-57-1': 'It would be interesting to obtain [OIII][MATH]5007 data for this galaxy to better assess whether line ratio variations can be explained by DIG temperature variations.', 'astro-ph-0011556-1-58-0': 'Other factors may influence the line ratios such as afore-mentioned abundance gradients (Savage Sembach 1996) as well as scattered light due to extraplanar dust.', 'astro-ph-0011556-1-58-1': 'The effects of dust absorption on extraplanar emission have been shown to be significant for a number of DIG halos (Howk Savage 1999).', 'astro-ph-0011556-1-58-2': 'Modeling of the Milky Way (Wood Reynolds 1999) and NGC 891 (Ferrara et.', 'astro-ph-0011556-1-58-3': 'al. 1996) suggest that scattered light makes a minor contribution to diffuse halo H[MATH] emission that decreases with [MATH].', 'astro-ph-0011556-1-58-4': 'In NGC 891, the contribution to DIG emission from scattered light falls to 10% at [MATH] pc (Ferrara et.', 'astro-ph-0011556-1-58-5': 'al. 1996).', 'astro-ph-0011556-1-58-6': 'In any case, dust-scattered light may be an important consideration for the interpretation of these line ratios.', 'astro-ph-0011556-1-59-0': '# Conclusions', 'astro-ph-0011556-1-60-0': 'Deep spectroscopy of four edge-on spiral galaxies has allowed physical conditions within their DIG halos to be probed, and has provided information on possible sources of ionization of the gas.', 'astro-ph-0011556-1-60-1': 'This study has revealed the following conclusions:', 'astro-ph-0011556-1-61-0': '1.', 'astro-ph-0011556-1-61-1': 'The general trend for the runs of line ratios for NGC 4302 and UGC 10288 is very similar to that of NGC 5775 (Rand 2000) and NGC 891 (Rand 1998): an increase in [N2]/H[MATH] and [S2]/H[MATH] with [S2]/[N2] remaining nearly constant with [MATH], though one slit position for UGC 10288 does show evidence for an increase in [S2]/[N2] with [MATH].', 'astro-ph-0011556-1-61-2': 'Determinations of halo emission scale height have revealed that the more prominent DIG halos also have greater scale height.', 'astro-ph-0011556-1-61-3': 'This is consistent with models of dynamic halos where greater levels of star formation in the disk can push material farther off the midplane.', 'astro-ph-0011556-1-62-0': '2.', 'astro-ph-0011556-1-62-1': 'Pure photo-ionization models do not explain the runs of line ratios in NGC 5775 and UGC 10288.', 'astro-ph-0011556-1-62-2': 'Composite photo-ionization/shock models can replicate the run of [O3]/H[MATH] vs. [N2]/H[MATH] in NGC 5775, though the run of [S2]/H[MATH] vs. [N2]/H[MATH] is still problematic, especially at low-[MATH].', 'astro-ph-0011556-1-62-3': 'In addition, we find that the composite models require a greater percent contribution from shocks in filamentary than non-filamentary regions to match the run of [O3]/H[MATH].', 'astro-ph-0011556-1-62-4': 'A lack of a discernible trend for the [O3]/H[MATH] vs. [N2]/H[MATH] run in UGC 10288 makes it difficult to determine whether shocks contribute to line emission, though the high values of [O3]/H[MATH] in Slit 1 indicate that a shock contribution may be necessary.', 'astro-ph-0011556-1-62-5': 'A composite model, though, can account for the run of [S2]/H[MATH] vs. [N2]/H[MATH].', 'astro-ph-0011556-1-62-6': "However, if it follows the trend seen in NGC 5775 of more disturbed regions requiring a larger shock contribution, then the lack of a bright, filamentary halo in UGC 10288 may imply that shocks do not play as significant a role in layer's ionization.", 'astro-ph-0011556-1-63-0': 'The [O3]/H[MATH] vs. [N2]/H[MATH] run seems to be a key discriminant for shocks.', 'astro-ph-0011556-1-63-1': 'Fabry-Perot observations of these 3 lines for brighter halos, such as NGC 5775, are now feasible and would provide much more information on the importance of secondary ionizing sources with local morphology and height.', 'astro-ph-0011556-1-63-2': 'It is important to emphasize that shocks are not the only possible secondary source.', 'astro-ph-0011556-1-63-3': 'Other sources such as TMLs, X-rays, and ionization by cosmic rays may contribute as well.', 'astro-ph-0011556-1-64-0': '3.', 'astro-ph-0011556-1-64-1': 'Another scenario that we consider is a pure photoionization model with an increasing temperature with [MATH].', 'astro-ph-0011556-1-64-2': 'By making a few reasonably valid assumptions, the run of gas temperature vs. [MATH] has been determined in each of the four galaxies.', 'astro-ph-0011556-1-64-3': 'We find that an increase in temperature with [MATH] can explain the general trend of an increase in the ratios of forbidden lines to Balmer lines with [MATH], including an increase in [O3]/H[MATH] and the constancy of [S2]/[N2].', 'astro-ph-0011556-1-64-4': 'By establishing the run of gas temperature, constraints on ionization fractions of Sulfur, Oxygen, and Hydrogen could then be established.', 'astro-ph-0011556-1-64-5': 'The [O3]/H[MATH] ratios in NGC 5775 and UGC 10288 generally indicate an exceptionally high ionization fraction of O[MATH], that invalidates the assumption that N[MATH] and H[MATH] have equal ionization fractions, indicating that the line ratios cannot be explained by a temperature increase alone.', 'astro-ph-0011556-1-64-6': 'The runs of line ratios in NGC 891, where the ionization fraction of O[MATH] is still somewhat high, and NGC 4302 are better explained by an increase in temperature with [MATH], though the absence of [O3][MATH]5007 data for NGC 4302 makes a proper assessment difficult.', 'astro-ph-0011556-1-65-0': 'Finally, [O1][MATH]6300, [O2][MATH]3727, and [O3][MATH]5007 data for a vertically oriented slit would provide additional information on ionization and excitation conditions in these DIG halos.', 'astro-ph-0011556-1-66-0': 'We thank the KPNO staff for their help in obtaining the data.', 'astro-ph-0011556-1-66-1': 'We also thank L. M. Haffner, R. Reynolds, and R. Benjamin for comments in the preparation of this paper.', 'astro-ph-0011556-1-66-2': 'This research has made use of the NASA/IPAC Extragalactic database (NED).', 'astro-ph-0011556-1-66-3': 'This work was partially supported by NSF grant AST-9986113.', 'astro-ph-0011556-1-67-0': 'Aller, L. H. 1984, Physics of Thermal Gaseous Nebulae (Dordrecht: Reidel) Anders, E., Grevesse, N. 1989, Geochim.', 'astro-ph-0011556-1-67-1': '278, Back to the Galaxy , ed.', 'astro-ph-0011556-1-68-0': 'Galaxy & 2cGalaxy Centera & Adopted & Offset along & Hours of & Noisec', 'astro-ph-0011556-1-69-0': 'and Slit & R.A. & Dec. & Distanceb & Major axis & Integration &', 'astro-ph-0011556-1-70-0': '& (J2000) & (J2000) & (Mpc) & & & NGC 5775 (Slit 1) & 14[MATH]6 & 03[MATH] & 24.8 & 32 NW & 6.0 & 2.2 NGC 5775 (Slit 2) & & & & 20 SE & 5.0 & 2.4 NGC 4302 & 12 21 42.5 & 14 36 05 & 16.8 & 24 S & 4.0 & 9.7 UGC 10288 (Slit 1) & 16 14 25.1 & -00 12 27 & 31.5 & 30 E & 6.17 & 2.5 UGC 10288 (Slit 2) & & & & 35 W & 5.33 & 3.6 NGC 891 & 02 22 33.1 & 42 20 48 & 9.6 & 100 N & 1.5 & 16.1 aThe H1 kinematic center of NGC 5775 was determined by Irwin (1994).', 'astro-ph-0011556-1-70-1': 'Central positions of NGC 4302, UGC 10288, and NGC 891 are from de Vaucouleurs et al. (1991).', 'astro-ph-0011556-1-70-2': 'bAdopted distance of NGC 5775 is from Irwin (1994).', 'astro-ph-0011556-1-70-3': 'Distances for NGC 4302, UGC 10288, and NGC 891 are from Tully (1988).', 'astro-ph-0011556-1-70-4': 'cBefore spatial averaging.', 'astro-ph-0011556-1-70-5': 'Noise units are 10[MATH] erg cm[MATH] s[MATH] arcsec[MATH].', 'astro-ph-0011556-1-71-0': 'lcc 3 0pc DIG SCALE HEIGHTS & Profile & [MATH] Galaxy & Description & (pc) NGC 5775 & Slit 1 (NE) & 2300 NGC 5775 & Slit 1 (SW) & 2300 NGC 5775 & Slit 2 (NE) & 2500 NGC 5775 & Slit 2 (SW) & 1800 NGC 4302 & East & 700 NGC 4302 & West & 550 UGC 10288 & Slit 1 (S) & 320 UGC 10288 & Slit 2 (S) & 460', 'astro-ph-0011556-1-72-0': 'lcccc 5 0pc CONSTRAINTS ON IONIZATION FRACTIONSa', 'astro-ph-0011556-1-73-0': 'Galaxy & Gas Temperature & [MATH]^+[MATH]^+[MATH] & [MATH]^++[MATH]^+[MATH] & [MATH]^+[MATH]', 'astro-ph-0011556-1-74-0': 'and Slit & (10[MATH] K) & & & NGC 5775 (Slit 1) & 0.64-0.95 & 0.30-0.60 & 0.15-0.70b & n/a NGC 5775 (Slit 2) & 0.62-0.95 & 0.35-0.80 & 0.30-0.75b & 0.65-0.85 UGC 10288 (Slit 1) & 0.60-1.05 & 0.55-1.00 & 0.50-0.80b & 0.45-0.65 UGC 10288 (Slit 2) & 0.60-0.95 & 0.50-1.00 & 0.50-0.80b & 0.30-0.60 NGC 4302 & 0.66-1.00 & 0.35-0.60 & n/a & n/a NGC 891 & 0.65-1.05 & 0.40-0.65 & 0.13-0.30 & 0.55-0.80 aValues of ionization fractions are valid only for the case of a pure photoionization model with an increasing temperature with [MATH].', 'astro-ph-0011556-1-74-1': 'See text for other assumptions used to determine these values.', 'astro-ph-0011556-1-74-2': 'bNearly all data points fall within this range for NGC 5775 and UGC 10288.', 'astro-ph-0011556-1-74-3': 'However, some measured values indicate an ionization fraction of O[MATH] relative to H[MATH] that is nearly 1.0.'}

{'astro-ph-0011556-2-0-0': 'Deep long-slit spectra of the diffuse ionized gas halos of the edge-on spiral galaxies NGC 4302 and UGC 10288 are presented.', 'astro-ph-0011556-2-0-1': 'Emission lines are detected up to about [MATH] kpc in NGC 4302, and to nearly [MATH] kpc on the north side of UGC 10288.', 'astro-ph-0011556-2-0-2': 'For both galaxies, the line ratios [N2]/H[MATH] and [S2]/H[MATH] increase with [MATH] in accordance with dilute photo-ionization models.', 'astro-ph-0011556-2-0-3': 'Runs of [S2]/[N2], and for UGC 10288, the run of [O3]/H[MATH], however, are not explained by the models.', 'astro-ph-0011556-2-0-4': 'Scale height determinations of their DIG halos are generally lower than those of galaxies with more prominent extraplanar DIG features.', 'astro-ph-0011556-2-1-0': 'These data, along with previously presented data for NGC 5775 and NGC 891, are used to address the issue of how DIG halos are energized.', 'astro-ph-0011556-2-1-1': 'Composite photo-ionization/shock models are generally better at explaining runs of line ratios in these galaxies than photoionization models alone.', 'astro-ph-0011556-2-1-2': 'Models of line ratios in NGC 5775 require a greater contribution from shocks for filamentary regions than for non-filamentary regions to explain the run of [O3]/H[MATH].', 'astro-ph-0011556-2-1-3': 'In either case, the [S2]/[N2] ratio is not well fit by the models.', 'astro-ph-0011556-2-1-4': "Composite models for UGC 10288 are successful at reproducing the run of [S2]/[N2] for all but the the highest values of [N2]/H[MATH]; however, the run of [O3]/H[MATH] vs. [N2]/H[MATH] does not show any discernible trend, making it difficult to determine whether or not shocks contribute to the layer's maintenance.", 'astro-ph-0011556-2-2-0': 'We also examine whether the data can be explained simply by an increase in temperature with [MATH] in a pure photo-ionization model without a secondary source of ionization.', 'astro-ph-0011556-2-2-1': 'Runs of [S2]/H[MATH], [N2]/H[MATH], and [S2]/[N2] in each of the four galaxies are consistent with such an increase.', 'astro-ph-0011556-2-2-2': 'However, the runs of [O3]/H[MATH] vs. [MATH] in NGC 5775 and UGC 10288 require unusually high ionization fractions of O[MATH] that can not be explained without invoking a secondary ionization source or at the very least a much higher temperature for the [O3]-emitting component than for the [S2]- and [N2]-emitting component.', 'astro-ph-0011556-2-2-3': 'An increase in temperature with [MATH] is generally more successful at explaining the [O3]/H[MATH] run in NGC 891, with the ionization fraction of O[MATH] remaining relatively low and nearly constant with [MATH].', 'astro-ph-0011556-2-3-0': '# Introduction', 'astro-ph-0011556-2-4-0': 'A vertically extended layer known as the Reynolds Layer or the Warm Ionized Medium (WIM) of the Milky Way contains the majority of the ionized gas in the interstellar medium.', 'astro-ph-0011556-2-4-1': 'The layer fills about 20 of the ISM volume with a local midplane density of [MATH]0.1 cm[MATH] and a scale height of [MATH]1 kpc (Reynolds 1993).', 'astro-ph-0011556-2-4-2': "Energetic arguments favor photo-ionization by massive stars in the disk for the layer's maintenance, as roughly 15 their ionizing output is sufficient to maintain the layer in the solar neighborhood (Reynolds 1993).", 'astro-ph-0011556-2-4-3': 'The Wisconsin H[MATH] Mapper (WHAM; Reynolds et.', 'astro-ph-0011556-2-4-4': 'al. 1998b) has been instrumental in recent years in characterizing the WIM.', 'astro-ph-0011556-2-4-5': 'In external galaxies, where the layer is more commonly referred to as Diffuse Ionized Gas (DIG), analogous information has been obtained through the use of narrow-band emission-line imaging, long-slit spectroscopy, and Fabry-Perot observations [e.g. Rand, Kulkarni, Hester 1990 (hereafter RKH); Rand 1996, 1997, 1998; Golla, Dettmar, Domgorgen 1996; Ferguson, Wyse, Gallagher 1996; Wang, Heckman, Lehnert 1997; Hoopes, Walterbos, Rand 1999; Collins et.', 'astro-ph-0011556-2-4-6': 'al. 2000 (hereafter CRDW)].', 'astro-ph-0011556-2-4-7': "Estimates of the diffuse fraction in DIG layers put the ionization requirement closer to 40% of the massive stars' ionizing output (e.g. Hoopes et.", 'astro-ph-0011556-2-4-8': 'al. 1996; Ferguson et.', 'astro-ph-0011556-2-4-9': 'al. 1996).', 'astro-ph-0011556-2-4-10': 'The advantage in observing external edge-on galaxies is the ability to obtain an overview of the entire halo, so that brightness, spatial distribution, and emission line ratios can be better characterized as they vary with [MATH].', 'astro-ph-0011556-2-5-0': 'One of the most important issues regarding these layers is their ionization.', 'astro-ph-0011556-2-5-1': 'In the WIM, as well as in external DIG layers, [S2] [MATH]H[MATH] and [N2] [MATH]H[MATH] are enhanced relative to [H2] regions [e.g. Haffner, Reynolds, Tufte 1999 (hereafter HRT); Ferguson, Wyse, Gallagher 1996; Rand 1997].', 'astro-ph-0011556-2-5-2': 'Such a scenario is predicted by photo-ionization models in which the dilution of the radiation field, measured by the ionization parameter, [MATH], increases as photons propagate away from H2 regions [Domgorgen Mathis 1994; Sokolowski 1994 (hereafter S94); Bland-Hawthorn et.', 'astro-ph-0011556-2-5-3': 'al. 1997].', 'astro-ph-0011556-2-5-4': 'Species such as S which are predominantly doubly ionized in H2 regions are able to recombine to a singly ionized state as the field becomes increasingly dilute.', 'astro-ph-0011556-2-5-5': 'Such an effect should not be as significant for N or O, which have considerably higher second ionization potentials and thus may not be fully doubly ionized in H2 regions.', 'astro-ph-0011556-2-6-0': 'The photo-ionization scenario is not without problems, however.', 'astro-ph-0011556-2-6-1': 'Firstly, in an increasingly dilute field, photoionization models predict that the ratio [S2]/[N2] should increase with [MATH] due to differing second ionization potentials of S and N. However, [S2]/[N2] vs. [MATH] has been found to remain nearly constant in the WIM (HRT) and NGC 891 (Rand 1998), and in the case of NGC 4631, the variations with [MATH] are not as pronounced as those predicted by the models (Golla et al. 1996).', 'astro-ph-0011556-2-6-2': 'In addition, the ratio [O3] [MATH]H[MATH] should decrease with [MATH] in a photo-ionization scenario as the field becomes increasingly dilute.', 'astro-ph-0011556-2-6-3': 'The opposite trend, however, is seen in NGC 891 (Rand 1998), where [O3]/H[MATH] increases significantly from the midplane to [MATH] kpc, as well as in NGC 5775 (Rand 2000; Tullmann et.', 'astro-ph-0011556-2-6-4': 'al. 2000).', 'astro-ph-0011556-2-6-5': '[O3]/H[MATH] enhancements in DIG relative to H2 regions are also seen in a number of more face-on galaxies, including M51 (Wang et.', 'astro-ph-0011556-2-6-6': 'al. 1997).', 'astro-ph-0011556-2-6-7': 'Finally, photoionization models have trouble explaining values of [N2]/H[MATH] and [S2]/H[MATH] that are often [MATH] (Wang et.', 'astro-ph-0011556-2-6-8': 'al. 1997; Rand 1998; HRT).', 'astro-ph-0011556-2-6-9': 'Models of S94, who took observed line ratios in NGC 891 into consideration, can only explain such ratios by assuming an upper mass cutoff for the stellar IMF of [MATH], hardening of the radiation field as it propagates, and a depletion of important gas-phase coolants onto dust grains.', 'astro-ph-0011556-2-6-10': 'Observations of the weak He1 [MATH] line, however, in the Reynolds Layer (Reynolds Tufte 1995) and in NGC 891 (Rand 1997) each imply a significantly softer spectrum than indicated by the forbidden lines.', 'astro-ph-0011556-2-7-0': 'One possibility that has been forwarded to explain some of this line ratio behavior is to assume a secondary ionizing source that contributes a small but increasing fraction of the H[MATH] emission with [MATH].', 'astro-ph-0011556-2-7-1': 'Such secondary sources include shocks (Shull McKee 1979) or turbulent mixing layers (TMLs; Slavin, Shull, Begelman 1993), each of which may produce large amounts of [O3] emission.', 'astro-ph-0011556-2-7-2': 'Other, less well-studied ionization sources may be relevant in halos, such as ionization from soft X-rays or cosmic rays.', 'astro-ph-0011556-2-8-0': 'Another approach that has been taken is to assume, instead of the ionization parameter or state changing with [MATH], that the line ratio behavior can be explained by the gas temperature increasing with [MATH].', 'astro-ph-0011556-2-8-1': 'This requires a source of additional heating, such as photo-electric heating from dust grains (Reynolds Cox 1992) or dissipation of interstellar turbulence (Minter Balser 1997), that becomes more significant at lower gas densities, but may not require any additional source of ionization.', 'astro-ph-0011556-2-8-2': 'A temperature increase with [MATH] is successful in explaining enhancements in [N2]/H[MATH] and [S2]/H[MATH] with [MATH], as well the constancy in [S2]/[N2] in the Reynolds Layer up to [MATH] kpc (HRT; Reynolds, Haffner, Tufte 1999).', 'astro-ph-0011556-2-8-3': 'Such a rise in temperature could also explain an increase in [O3]/H[MATH], though data on the run of [O3]/H[MATH] vs. [MATH] in the WIM are currently lacking.', 'astro-ph-0011556-2-9-0': 'The goal of this research is to attempt to better understand how DIG halos are energized.', 'astro-ph-0011556-2-9-1': 'We attempt to determine whether measured runs of line ratios versus [MATH] can be explained by: (a) photo-ionization with a contribution from shock ionization that increases with height above the midplane, and/or (b) only photo-ionization, but with an increase in gas temperature with [MATH].', 'astro-ph-0011556-2-9-2': 'We examine our data in the context of each of these scenarios, realizing that elements of both could be important for understanding the ionization of DIG halos in general.', 'astro-ph-0011556-2-9-3': 'To this end, we have obtained data for two galaxies: NGC 4302 and UGC 10288.', 'astro-ph-0011556-2-9-4': 'We also employ data for NGC 5775, already presented by Rand (2000).', 'astro-ph-0011556-2-9-5': 'For the purpose of assessing the rising temperature scenario, we use data for NGC 891, which has been compared with composite shock/photo-ionization models previously (Rand 1998).', 'astro-ph-0011556-2-9-6': 'The observations of these galaxies are discussed in 2.', 'astro-ph-0011556-2-9-7': 'Line ratio data are presented in 3 along with comparisons to composite shock/photo-ionization and rising temperature models.', 'astro-ph-0011556-2-9-8': 'We briefly summarize our results in 4.', 'astro-ph-0011556-2-10-0': '# Observations', 'astro-ph-0011556-2-11-0': 'Observational details for the NGC 891 (Rand 1998) and NGC 5775 (Rand 2000) data have been discussed in previous papers, though some modifications have been made which will be discussed in this section.', 'astro-ph-0011556-2-11-1': 'Spectra for UGC 10288 and NGC 4302 were obtained at the KPNO 4-m telescope on 1999 June 10-13 and 1998 February 23-26, respectively.', 'astro-ph-0011556-2-11-2': 'Slits for all four galaxies run perpendicular to the plane.', 'astro-ph-0011556-2-11-3': 'The slit position for NGC 4302, covering the most prominent region of extraplanar DIG, is shown in Figure 1, overlaid on the H[MATH] image.', 'astro-ph-0011556-2-11-4': 'The H[MATH] image of UGC 10288, showing little extraplanar H[MATH] emission, is shown in Figure 2 with slit position overlays.', 'astro-ph-0011556-2-11-5': "The slit positions for UGC 10288 were chosen to be roughly 30 east and west of the galaxy's center.", 'astro-ph-0011556-2-11-6': 'Due to the presence of a foreground star on the axis perpendicular the galactic disk 30 west of the radio continuum center, slit 2 was placed at a position 35 west of center.', 'astro-ph-0011556-2-11-7': 'A summary of the observations is given in Table 1.', 'astro-ph-0011556-2-12-0': 't1', 'astro-ph-0011556-2-13-0': 'The observations of NGC 4302 utilized the KPC-24 grating in conjunction with the T2KB 2048x2048 CCD, providing a dispersion of 0.51 per pixel, a resolution of 1.3 and a useful coverage of about 800.', 'astro-ph-0011556-2-13-1': 'The slit length is 5\', the slit width was set at 3", and the spatial scale is 0.69" per pixel.', 'astro-ph-0011556-2-13-2': 'For red line observations, the grating was tilted to give an approximate central wavelength of 6600, allowing H[MATH], [N2] [MATH]6548, 6583, [S2] [MATH]6716, 6731, and [O1] [MATH]6300 to be observed.', 'astro-ph-0011556-2-13-3': 'Due to poor weather during the 1998 observing run, spectra containing [O3] [MATH]5007 could not be obtained.', 'astro-ph-0011556-2-14-0': 'For observations of UGC 10288, the KPC-007 grating was used with the T2KB 2048x2048 CCD, providing a dispersion of 1.42 per pixel, a resolution of 3.5 and a useful coverage of about 2100 encompassing the spectral lines H[MATH], [N2] [MATH]6548, 6583, [N2] [MATH]5755, [S2] [MATH]6716, 6731, He1 [MATH]5876, [O1] [MATH]6300, [O3] [MATH]5007, and H[MATH].', 'astro-ph-0011556-2-14-1': 'The slit length is 5\', the slit width was set at 2", and the spatial scale is 0.69" per pixel.', 'astro-ph-0011556-2-15-0': 'Data reductions were performed with the IRAFpackage.', 'astro-ph-0011556-2-15-1': 'Projector flats were used to remove small-scale variations in response, while sky flats were used to determine the slit illumination correction.', 'astro-ph-0011556-2-15-2': 'Arc lamp exposures were used to calibrate the wavelength scale as a function of location along the slit, and the spectral response function was determined with standard stars.', 'astro-ph-0011556-2-15-3': 'Regions of the spectra free of emission from the galaxies were used to form sky-subtracted spectra.', 'astro-ph-0011556-2-15-4': 'In order to increase sensitivity for the determination of line parameters, blocks of 5 pixels were averaged in the spatial direction.', 'astro-ph-0011556-2-15-5': 'Since Gaussians appear to accurately represent the line profiles, Gaussian fits were used to determine line properties.', 'astro-ph-0011556-2-15-6': 'Uncertainties in line parameters for NGC 4302 were determined with the IRAF task splot.', 'astro-ph-0011556-2-15-7': 'The use of this method for NGC 5775 and UGC 10288 yielded error bars that were far greater than uncertainties suggested by the data.', 'astro-ph-0011556-2-15-8': 'In this case error bars were estimated by determining flux variations along the slit for sky lines of a range of intensities, from which a relation between these variances and sky line intensity could be established.', 'astro-ph-0011556-2-15-9': 'This relation was then used to determine flux uncertainties for the various emission lines.', 'astro-ph-0011556-2-15-10': 'Uncertainties for H[MATH] line strengths in NGC 5775 also take into account the coincidence of the H[MATH] line with a blended pair of sky lines.', 'astro-ph-0011556-2-15-11': 'This factor was not considered in the presentation of NGC 5775 line data in Rand (2000).', 'astro-ph-0011556-2-16-0': 'In order to compare [O3] [MATH]5007 emission to that of H[MATH] it was necessary to make a reddening correction to the [O3] line in NGC 5775, UGC 10288, and NGC 891.', 'astro-ph-0011556-2-16-1': 'This correction was not performed for the presentation of the [O3]/H[MATH] ratio in Rand (2000) for NGC 5775.', 'astro-ph-0011556-2-16-2': 'In all three galaxies, H[MATH] could be measured for all points where [OIII][MATH] could be measured, thus allowing extinction corrections to be made for the full range in [MATH].', 'astro-ph-0011556-2-16-3': 'The correction was determined by assuming a uniform foreground dust screen and comparing measured values of the H[MATH]/H[MATH] ratio to the value calculated by Osterbrock (1989) for a 10[MATH] K gas.', 'astro-ph-0011556-2-16-4': 'The assumption of a 10[MATH] K gas is not contradictory to a scenario including a rising temperature with [MATH] in that the H[MATH]/H[MATH] ratio is not nearly as temperature sensitive as the ratios of forbidden lines to Balmer lines: from 5,000 to 10,000 K, H[MATH]/H[MATH] increases by [MATH], whereas the ratios of [N2], [S2], and [O3] to H[MATH] increase by an order of magnitude.', 'astro-ph-0011556-2-16-5': 'In NGC 5775, the correction to the [OIII] flux decreases from a factor of 2.9 in the midplane to no measurable extinction by [MATH] kpc.', 'astro-ph-0011556-2-16-6': 'The correction is more variable for UGC 10288, with a maximum correction of 3.6 at [MATH] pc on the south side.', 'astro-ph-0011556-2-16-7': 'In NGC 891, the correction decreases from 2.2 in the midplane to 1.5 at [MATH] kpc.', 'astro-ph-0011556-2-16-8': 'In both NGC 5775 and NGC 891, the H[MATH] line is superposed on a weak stellar absorption line close to the midplane.', 'astro-ph-0011556-2-16-9': 'However, the absorption line is much broader than the emission line and thus can be accounted for by the background fitting process in the line fitting routine.', 'astro-ph-0011556-2-17-0': '# Results and Discussion', 'astro-ph-0011556-2-18-0': '## Line Data and Line Ratios', 'astro-ph-0011556-2-19-0': 'The H[MATH], [N2] [MATH]6583, and [S2] [MATH]6716 lines are detected up to nearly [MATH] kpc on either side of the plane of NGC 4302.', 'astro-ph-0011556-2-19-1': 'The [S2] [MATH]6716 line was not detected at [MATH] kpc on the west side, though the fainter line at 6731 was detected, due to confusion with noise in that part of the spectra.', 'astro-ph-0011556-2-19-2': 'The flux of the 6716 line at that part of the spectra was inferred from that of the 6731 line assuming the low-density limit of [S2] [MATH].', 'astro-ph-0011556-2-19-3': 'The vertical runs of [N2] [MATH]H[MATH], [S2] [MATH]H[MATH], and [S2] [MATH][N2] [MATH] are shown in Figure 3.', 'astro-ph-0011556-2-19-4': 'The data are averaged over 5 spatial pixels.', 'astro-ph-0011556-2-19-5': 'The [N2]/H[MATH] and [S2]/H[MATH] each show a similar rise with [MATH], with the [N2]/H[MATH] rising from about 0.4 in the midplane to 1.0 at [MATH] kpc on the east side, and to nearly 1.4 at [MATH] kpc on the west side.', 'astro-ph-0011556-2-19-6': 'The [S2]/[N2] remains fairly constant at about 0.6 for the full range of [MATH], with slight increases in the midplane and beyond [MATH] kpc.', 'astro-ph-0011556-2-20-0': 'In UGC 10288, the H[MATH], [N2] [MATH]6583, and [S2] [MATH]6716 lines are all detected well above the midplane, with the [N2] [MATH]6583 and H[MATH] lines being detected up to nearly [MATH] kpc on the north side for both slit positions.', 'astro-ph-0011556-2-20-1': 'Detections of the [O3] [MATH]5007 and [O1] [MATH]6300 lines are limited to [MATH] kpc, except for the south side of slit 1 where both lines are detected up to [MATH] kpc.', 'astro-ph-0011556-2-20-2': 'The vertical runs of [N2] [MATH]H[MATH], [S2] [MATH]H[MATH], [S2] [MATH][N2] [MATH], [O3] [MATH]H[MATH], and [O1] [MATH]H[MATH] for both slits are shown in Figure 4.', 'astro-ph-0011556-2-20-3': 'The data are averaged over 5 spatial pixels.', 'astro-ph-0011556-2-20-4': 'For slit 1, [N2]/H[MATH] rises from 0.3 at the midplane to 1.6 at [MATH] kpc on the north side and 1.3 at [MATH] kpc on the south, while for slit 2 the ratio remains somewhat lower, rising to 1.2 for [MATH] kpc on either side of the disk.', 'astro-ph-0011556-2-20-5': 'The [S2]/[N2] ratio shows a similar range of values for each slit, varying from 0.8 to 1.4 with the lower values generally occurring in the midplane.', 'astro-ph-0011556-2-20-6': 'These values are significantly higher than those of the other three galaxies.', 'astro-ph-0011556-2-20-7': 'The [O3]/H[MATH] ratio shows some evidence for an increase with [MATH] on the south side of either slit, reaching values as high as 0.75 in Slit 1 and 0.4 in Slit 2.', 'astro-ph-0011556-2-20-8': 'On the north side of each slit, values reach no higher than 0.5 in Slit 1 and 0.4 in Slit 2.', 'astro-ph-0011556-2-20-9': 'Such values of [O3]/H[MATH] are consistent with those observed at comparable [MATH]-heights in NGC 5775.', 'astro-ph-0011556-2-21-0': 'Line ratio data for NGC 5775, except for the [O1] [MATH]H[MATH] and extinction corrected [O3] [MATH]H[MATH] ratios, have been presented previously (Rand 2000).', 'astro-ph-0011556-2-21-1': 'These, along with [N2] [MATH]H[MATH], [S2] [MATH]H[MATH], and [S2] [MATH][N2] [MATH] ratios are shown in Figure 12.', 'astro-ph-0011556-2-21-2': 'The data are averaged over 10 spatial pixels.', 'astro-ph-0011556-2-21-3': 'An increase with [MATH] in the extinction corrected [O3]/H[MATH] for the NE side of Slit 1 and the SW side of Slit 2 is still indicated by the data, where values range from 0.2-0.4 in the disk to 1.1 in the halo.', 'astro-ph-0011556-2-21-4': 'The [O1/H[MATH] ratio ranges from about 0.02 in the midplane to 0.06 at [MATH] kpc in Slit 2, while the [O1] [MATH] line was not detected for Slit 1.', 'astro-ph-0011556-2-21-5': 'Because of the 85 (Irwin 1994) inclination of the galaxy, points within 10 of the midplane ([MATH] pc in the figures) on the spatial axis reflect in-plane, highly inclined disk structure rather than true vertical structure.', 'astro-ph-0011556-2-21-6': 'In these lines of sight, H2 regions and areas between them have a greater effect on line ratio variations than extraplanar diffuse gas.', 'astro-ph-0011556-2-21-7': 'This fact explains why some line ratio minima are not at [MATH] kpc.', 'astro-ph-0011556-2-22-0': 'Line ratio data for NGC 891, previously presented by Rand (1998), can be seen in Figure 13.', 'astro-ph-0011556-2-23-0': '## DIG scale heights', 'astro-ph-0011556-2-24-0': "In this section we attempt to determine the H[MATH] emission scale height, [MATH], of the galaxies' DIG halos at the various slit locations.", 'astro-ph-0011556-2-24-1': 'We attempt to model only an exponential halo component, avoiding any disk component where contamination by bright H2 regions could complicate the analysis.', 'astro-ph-0011556-2-24-2': 'We plot the logarithm of H[MATH] intensity versus [MATH] for each slit, then fit an exponential component with a least-squares fit except in cases where an exponential is obviously a bad description.', 'astro-ph-0011556-2-24-3': 'Plots of logarithmic H[MATH] intensity vs. [MATH], along with exponential fits, for NGC 5775, NGC 4302, and UGC 10288 are shown in Figures 5, 6, and 7, respectively.', 'astro-ph-0011556-2-24-4': 'Results of the fitting procedure are summarized in Table 2.', 'astro-ph-0011556-2-25-0': 'Position 1 of NGC 5775 covers the NE filament, DIG emission from which has been modeled previously (CRDW).', 'astro-ph-0011556-2-25-1': 'The value [MATH] pc is somewhat consistent with the value of [MATH] kpc determined from image data.', 'astro-ph-0011556-2-25-2': 'The image data however does include some [N2] contamination.', 'astro-ph-0011556-2-25-3': 'The previously determined value assumes an [N2]/H[MATH] ratio that does not vary with [MATH].', 'astro-ph-0011556-2-25-4': 'Since our data shows that this ratio in fact rises with [MATH] on the NE filament, from 0.5 at [MATH] kpc to 0.75 at [MATH] kpc, the value [MATH] kpc becomes an upper limit on the actual scale height.', 'astro-ph-0011556-2-26-0': 'The DIG halos of NGC 4302 and UGC 10288 have been modeled previously as well (Rand 1996).', 'astro-ph-0011556-2-26-1': 'The slit position for NGC 4302 coincides with one of the previously modeled regions, the scale height of which was determined to be [MATH] pc (the image is not contaminated by [N2] emission).', 'astro-ph-0011556-2-26-2': 'The values determined from spectroscopic data, [MATH] pc and [MATH] pc for the east and west sides, are close to those determined from image data.', 'astro-ph-0011556-2-26-3': 'The scale height determined from the image, however, is an average over a 3 kpc radial extent of the disk and thus should not be in exact agreement with the spectroscopically determined value.', 'astro-ph-0011556-2-26-4': 'Image data of UGC 10288 revealed very little extraplanar H[MATH] emission, and thus profiles were not well modeled with exponentials.', 'astro-ph-0011556-2-26-5': 'Increased sensitivity of the spectroscopic data have allowed the detection of H[MATH] emission to greater [MATH]-heights.', 'astro-ph-0011556-2-26-6': 'The resulting H[MATH] profiles seem to be well modeled by exponentials with small scale heights on the south side, while more complicated structure is present on the north side.', 'astro-ph-0011556-2-27-0': 't2', 'astro-ph-0011556-2-28-0': 'NGC 5775, having the most prominent extraplanar DIG emission, also has the DIG halo with the greatest scale height.', 'astro-ph-0011556-2-28-1': 'In contrast, UGC 10288, showing very little extraplanar H[MATH] emission, has a DIG halo of relatively small scale height.', 'astro-ph-0011556-2-28-2': 'NGC 4302, having a number of extraplanar plumes of H[MATH] emission, has a DIG scale height intermediate between that of NGC 5775 and UGC 10288.', 'astro-ph-0011556-2-28-3': 'The halo of NGC 891 has been modeled previously (RKH), the scale height of which has been measured to be [MATH] kpc on the west side and [MATH] kpc on the east side from imaging data.', 'astro-ph-0011556-2-28-4': 'These results suggest that not only do galaxies such as NGC 5775 and NGC 891 have brighter DIG halos, where sensitivity limits allow faint emission to be detected at much greater heights than for a galaxy with a lower H[MATH] surface brightness, but the halos also have much larger scale heights.', 'astro-ph-0011556-2-28-5': 'This adds further evidence to the notion of a dynamic halo where higher levels of star formation in the disk result in material driven further into the halo (e.g. Rand 1996; Hoopes et.', 'astro-ph-0011556-2-28-6': 'al. 1999).', 'astro-ph-0011556-2-29-0': '## Shocks as a Secondary Source of Ionization', 'astro-ph-0011556-2-30-0': 'Some of the line ratio behavior observed in NGC 5775 and UGC 10288, namely the rise in [N2]/H[MATH] and [S2]/H[MATH] with [MATH], is predicted by the photo-ionization models (e.g. S94; Bland-Hawthorn 1997).', 'astro-ph-0011556-2-30-1': 'As previously stated, rises in these ratios are expected in such models as the radiation field becomes more dilute with [MATH].', 'astro-ph-0011556-2-30-2': 'If the field is in fact becoming more dilute (low values of the ionization parameter, [MATH], which measures the "diluteness\'\'of the radiation field and is proportional to the ratio of ionizing photon number density to gas density) with [MATH], then the models predict that the [O3]/H[MATH] ratio should decrease away from the midplane.', 'astro-ph-0011556-2-30-3': 'However, in NGC 5775, and to a lesser extent UGC 10288, we see a definite incease in [O3]/H[MATH] with [MATH], indicative of higher excitation conditions above the disk in such models (high-[MATH]).', 'astro-ph-0011556-2-30-4': 'Recent spectroscopy of NGC 5775 by Tullmann et.', 'astro-ph-0011556-2-30-5': 'al. (2000) also reveals a significant rise in [O3]/H[MATH] with [MATH], though these data are not extinction corrected.', 'astro-ph-0011556-2-30-6': 'It is then possible that the [O3] emission originates from a different DIG component than that of the [N2] and [S2].', 'astro-ph-0011556-2-30-7': 'Consequently, a secondary mechanism, such as shock ionization, could account for the bright [O3] emission above the plane.', 'astro-ph-0011556-2-31-0': 'In this section we explore the possibility that shock ionization is responsible for some of the DIG emission in these galaxies.', 'astro-ph-0011556-2-31-1': 'One could also consider TMLs, since they also produce large amounts of [O3] emission.', 'astro-ph-0011556-2-31-2': 'In fact in NGC 891, TMLs do just as well as shocks in fitting the observed data (Rand 1998).', 'astro-ph-0011556-2-31-3': 'Our approach is that of Rand (1998), with some minor changes.', 'astro-ph-0011556-2-31-4': 'We adopt the S94 matter-bounded photoionization model with the lowest terminal hydrogen column density, [MATH] cm[MATH], of individual gas clumps.', 'astro-ph-0011556-2-31-5': 'The S94 models feature a hard stellar spectrum (an upper mass cutoff of [MATH]), hardening of the radiation field as it propagates through the ISM, and depletion of gas-phase coolants onto dust grains.', 'astro-ph-0011556-2-31-6': 'We do not consider the S94 radiation-bounded model as this model predicts extremely bright [O3] emission.', 'astro-ph-0011556-2-31-7': 'Any composite shock model utilizing radiation-bounded models requires a very unusual secondary source of H[MATH] emission.', 'astro-ph-0011556-2-31-8': 'In the case of NGC 5775, such a scenario would require a secondary source that accounts for nearly half of the H[MATH] emission at the midplane (assuming log [MATH] at [MATH] kpc), and produces virtually no [O3] or [N2] emission.', 'astro-ph-0011556-2-31-9': 'In addition, such a source must produce significant [O3] and [S2] emission at high-[MATH], and still contribute at least 50% of the total H[MATH] emission.', 'astro-ph-0011556-2-31-10': 'Such emission characteristics could be explained with low-speed shocks near the midplane with a shock velocity increasing with [MATH].', 'astro-ph-0011556-2-31-11': 'However, the problem is further complicated by the fact that low-speed shocks produce copious amounts of [O1] emission relative to [N2].', 'astro-ph-0011556-2-31-12': "In addition, basic energetic arguments render such a scenario, where the secondary source contributes half of the layer's ionization, highly unlikely as the ionizing flux from massive stars exceeds supernovae power output, presumably the initiator of these shocks, by a factor of 6 or 7 (Reynolds 1984, 1992).", 'astro-ph-0011556-2-31-13': 'In any case, parameters describing these shocks require considerable tweaking to account for the observed run of line ratios.', 'astro-ph-0011556-2-31-14': 'Matter-bounded models on the other hand, better match the data before composite modeling, and require a less complicated parameter space describing the secondary source.', 'astro-ph-0011556-2-32-0': 'To obtain line ratios due to shock ionization, we use the models of Shull McKee (1979; hereafter SM79).', 'astro-ph-0011556-2-32-1': 'Line ratios in these models, especially [O3]/H[MATH], are highly dependent on shock velocity.', 'astro-ph-0011556-2-32-2': 'Other variables affecting line ratios include the preshock gas density and ionization state, abundances, and transverse magnetic field strength.', 'astro-ph-0011556-2-32-3': 'These models assume the gas to be initially neutral at [MATH] cm[MATH], then penetrated by a precursor ionization front.', 'astro-ph-0011556-2-32-4': 'More appropriate to the case of a partially photo-ionized DIG layer, would be a initial density of order [MATH] cm[MATH], and a high initial ionization fraction.', 'astro-ph-0011556-2-32-5': 'We mainly consider the models with shock velocities of [MATH] km s[MATH] and [MATH] km s[MATH] in that these higher velocity models are best able to produce large amounts of [O3] emission, while keeping [O1] low relative to [N2].', 'astro-ph-0011556-2-32-6': 'SM79 also calculate a single model containing depleted abundances with [MATH] km s[MATH], which we also consider in the modeling to allow a comparison between shock and photo-ionization models with depleted abundances.', 'astro-ph-0011556-2-32-7': 'In general, though, we do not consider this analysis as providing a definitive measure of the shock speed in these composite models given the simplicity of the model and the many unrealistic parameters which describe the shocks.', 'astro-ph-0011556-2-33-0': 'For the composite modeling, we assume that some fraction of H[MATH] emission arises from shock ionization, with that fraction possibly changing with [MATH].', 'astro-ph-0011556-2-33-1': 'We still assume a stellar radiation field distinguished by a decrease in ionization parameter, [MATH], with [MATH].', 'astro-ph-0011556-2-33-2': 'To simplify matters we assume a single shock velocity for each composite model.', 'astro-ph-0011556-2-33-3': 'We attempt to fit the composite model to the data by varying the percent contribution at a given value of [MATH] (and thus [MATH]).', 'astro-ph-0011556-2-33-4': 'For NGC 5775 the modeling worked best by allowing shock contributions to begin at log [MATH], while models with shocks beginning at log [MATH] worked best for UGC 10288.', 'astro-ph-0011556-2-33-5': 'Such values are somewhat lower than would be expected for a low-[MATH] radiation field.', 'astro-ph-0011556-2-33-6': 'However, kinematics of edge-on galaxies indicate that at low-[MATH], the line emission we observe is mostly associated with the outer disk regions due to an absorbing dust layer.', 'astro-ph-0011556-2-33-7': 'Since star formation is typically concentrated in the inner disk, the radiation field in the outer disk may be relatively dilute, in which case, the value of [MATH] at low-[MATH] may be quite low (Rand 1998).', 'astro-ph-0011556-2-34-0': 'We do not carry out a statistical test of the goodness of fit since our goal is to qualitatively assess the feasibility of a secondary source of ionization being able to account for some of the line ratios.', 'astro-ph-0011556-2-34-1': 'Although no model reproduces the line ratios to within the errors, we do find that the composite models better reproduce the run of line ratios than photo-ionization models alone.', 'astro-ph-0011556-2-34-2': 'Rand (1998) finds a similar conclusion for NGC 891, where composite modeling reproduces the run of [O3]/H[MATH] vs. [N2]/H[MATH] and [S2]/H[MATH], yet is unable to duplicate the [S2]/[N2] ratio at low-[MATH].', 'astro-ph-0011556-2-34-3': 'NGC 4302 is not considered due to the lack of critical [O3] [MATH]5007 data.', 'astro-ph-0011556-2-35-0': '### NGC 5775', 'astro-ph-0011556-2-36-0': 'For the composite modeling, we consider separately the cases of filamentary and non-filamentary emission.', 'astro-ph-0011556-2-36-1': 'The NE portion of Slit 1 crosses the prominent NE H[MATH] filament, previously identified by CRDW as being associated with a shell-like H1 structure and possibly being involved in vigorous disk-halo interactions.', 'astro-ph-0011556-2-36-2': 'In addition, the SW portion of Slit 2 crosses a bright plume of H[MATH] emission immediately adjacent to the very prominent SW H[MATH] filament, also a region where intense disk-halo activity is suspected.', 'astro-ph-0011556-2-36-3': 'Since these two regions appear associated with current cycling of material from disk to halo, it follows that shock ionization may play a greater role in energizing DIG in these areas.', 'astro-ph-0011556-2-36-4': 'The SW portion of Slit 1 and the NE portion of Slit 2 each cross regions with relatively less extended DIG emission, where disk-halo interactions appear far less vigorous than regions with more filamentary features.', 'astro-ph-0011556-2-37-0': 'The runs of [O3]/H[MATH], [S2]/H[MATH], and [O1]/H[MATH] vs. [N2]/H[MATH] are shown in Figures 8, 9, and 10, respectively.', 'astro-ph-0011556-2-37-1': 'Figures 8 and 9 also present filamentary and non-filamentary regions separately.', 'astro-ph-0011556-2-37-2': 'The runs of these ratios in the matter-bounded S94 models, with various values of [MATH] labeled, are plotted along with predicted line ratios from the SM79 models for [MATH] km s[MATH], [MATH] km s[MATH], and [MATH] km s[MATH] with depletions.', 'astro-ph-0011556-2-37-3': 'We find the [MATH] km s[MATH] model is best able to reproduce the observed run of line ratios as a secondary source to photoionization.', 'astro-ph-0011556-2-37-4': 'The [MATH] km s[MATH] model has difficulty reproducing the run of [S2]/H[MATH] vs. [N2]/H[MATH], and the depleted [MATH] km s[MATH] model, though able to duplicate the run of [S2]/H[MATH] vs. [N2]/H[MATH], is unable to reproduce the run of [O3]/H[MATH] vs. [N2]/H[MATH] at high-[MATH].', 'astro-ph-0011556-2-38-0': 'We find that the run of [O3]/H[MATH] vs. [N2]/H[MATH] seen in Figure 8a for the filamentary regions is distinctly different from the trend seen in Figure 8b for the non-filamentary regions.', 'astro-ph-0011556-2-38-1': 'Clearly, for a given value of [N2]/H[MATH], the ratio [O3]/H[MATH] is much greater in DIG with a more filamentary morphology.', 'astro-ph-0011556-2-38-2': 'The non-filamentary regions show a run that is very similar to the run observed in NGC 891 (Rand 1998).', 'astro-ph-0011556-2-38-3': 'The run of [S2]/H[MATH] vs. [N2]/H[MATH] on the other hand, does not seem to depend strongly on the presence of filamentary DIG structure.', 'astro-ph-0011556-2-38-4': "Non-filamentary regions do seem to have greater values of [N2]/H[MATH] and [S2]/H[MATH], in keeping with these ratios' anti-correlation with relative H[MATH] surface brightness (see also Figure 12).", 'astro-ph-0011556-2-38-5': 'All data for the ratio [O1]/H[MATH], being detected only for Slit 2 to [MATH] kpc, are plotted in Figure 10.', 'astro-ph-0011556-2-38-6': 'The lack of [O1]/H[MATH] data for higher values of [N2]/H[MATH] makes a comparison between various DIG morphologies impossible.', 'astro-ph-0011556-2-39-0': 'Composite models are plotted in Figures 8, 9, and 10 as dashed lines joining open hexagons, which mark various values of log [MATH].', 'astro-ph-0011556-2-39-1': 'For the filamentary regions, 20% of the H[MATH] emission at log [MATH] arises from shock ionization, increasing to 85% at log [MATH].', 'astro-ph-0011556-2-39-2': 'This does not mean the majority of extraplanar DIG is shock ionized.', 'astro-ph-0011556-2-39-3': 'It does indicate however that at very high-[MATH] (log [MATH] corresponds to a height of about [MATH] kpc for the filamentary regions and [MATH] kpc for the non-filamentary), it is possible that most of the emission within the filaments arises from shock ionized gas.', 'astro-ph-0011556-2-39-4': 'In the non-filamentary case, shocks contribute 16% of the H[MATH] emission at log [MATH], rising to 27% at log [MATH].', 'astro-ph-0011556-2-39-5': 'For the [O1]/H[MATH] vs. [N2]/H[MATH] data of Figure 10, the filamentary composite model is plotted over all data points for both filamentary and non-filamentary regions.', 'astro-ph-0011556-2-39-6': 'This model and the non-filamentary model are somewhat successful in explaining the [O1]/H[MATH] vs. [N2]/H[MATH] run at low-[MATH], in the sense of matching the excess of [O1]/H[MATH] for a given [N2]/H[MATH] compared to the S94 model alone.', 'astro-ph-0011556-2-40-0': 'The composite models both match the runs of [O3]/H[MATH] vs. [N2]/H[MATH].', 'astro-ph-0011556-2-40-1': 'The main deficiency in the models is their inability to reproduce the run of [S2]/[N2] at low-[MATH].', 'astro-ph-0011556-2-40-2': 'The models do not predict enough [S2] emission for a given value of [N2].', 'astro-ph-0011556-2-40-3': 'The models perform somewhat better at high-[MATH], though the spread in data values at high-[MATH] make a trend somewhat difficult to discern.', 'astro-ph-0011556-2-40-4': 'Again, though models with depleted abundances are better able to produce an excess of [S2] emission relative to [N2], they do not produce enough [O3] emission.', 'astro-ph-0011556-2-41-0': 'In any case, it is clear that the filamentary regions require a greater contribution from shocks to explain the observed line ratios.', 'astro-ph-0011556-2-41-1': 'The correlation of these features with H1 shells and enhanced 20-cm radio continuum emission is also suggestive of a larger percent contribution to the observed emission in these regions from shock ionization.', 'astro-ph-0011556-2-41-2': 'Enhanced [O3] emission can also be produced by other sources of ionization such as TMLs.', 'astro-ph-0011556-2-41-3': 'Rand (1998) found that composite photo-ionization/TML models were just as successful in reproducing the runs of observed line ratios in NGC 891 as composite models featuring shocks.', 'astro-ph-0011556-2-41-4': 'TMLs are expected to occur at interfaces of gas at different temperatures in the ISM, such as in shell walls.', 'astro-ph-0011556-2-41-5': 'It is likely then for NGC 5775, where H1 shells are associated with the more prominent H[MATH] filaments, that TMLs could also account for the enhanced line ratios.', 'astro-ph-0011556-2-41-6': 'We have performed cursory modeling of the line ratios for NGC 5775 with composite photo-ionization/TML models.', 'astro-ph-0011556-2-41-7': 'Using the TML model of Slavin, Shull, Begelman (1993) with depleted abundances in the mixing layers of hot and warm gas, a hot gas mixing speed of [MATH] km s[MATH], and a mixed gas temperature of log [MATH], we adopt the same approach as the composite photo-ionization/shock modeling.', 'astro-ph-0011556-2-41-8': 'We find that the photo-ionization/TML models work nearly as well as the photo-ionization/shock models, though they are unable to reproduce the run of ratios for the highest values of [O3]/H[MATH].', 'astro-ph-0011556-2-42-0': '### UGC 10288', 'astro-ph-0011556-2-43-0': 'A preliminary analysis of the diagnostic diagrams for UGC 10288 has revealed no discernible trend in the run of [O3]/H[MATH] vs. [N2]/H[MATH].', 'astro-ph-0011556-2-43-1': 'To attempt to better reveal any possible trend, line ratios on either side of the disk have been averaged for each slit to establish a run of each ratio vs. [MATH].', 'astro-ph-0011556-2-43-2': 'These averaged runs of [O3]/H[MATH], [S2]/H[MATH], and [O1]/H[MATH] vs. [N2]/H[MATH] are shown in Figure 11.', 'astro-ph-0011556-2-43-3': 'The matter-bounded S94 models, with various values of [MATH] labeled, are plotted along with predicted line ratios from the SM79 models for [MATH] km s[MATH], [MATH] km s[MATH], and [MATH] km s[MATH] with depletions.', 'astro-ph-0011556-2-43-4': "Even with the averaging, the [O3]/H[MATH] vs. [N2]/H[MATH] run doesn't show an obvious trend: Slit 1 seems to show a slight trend of an increase in [O3]/H[MATH] with [N2]/H[MATH], while Slit 2 data is clustered near the photo-ionization model.", 'astro-ph-0011556-2-43-5': 'For this reason, we attempt to fit the run of [S2]/[N2] instead of [O3]/H[MATH] vs. [N2]/H[MATH].', 'astro-ph-0011556-2-43-6': 'We adopt the [MATH] km s[MATH] model with depleted abundances as the other two models do not produce enough [S2] emission relative to [N2].', 'astro-ph-0011556-2-44-0': 'Composite models are plotted in Figure 11 as dashed lines joining open hexagons, which mark various values of log [MATH].', 'astro-ph-0011556-2-44-1': 'We find that the composite models best agree with the data when shocks begin contributing to emission at log [MATH].', 'astro-ph-0011556-2-44-2': 'For the composite model, 20% of the H[MATH] emission at log [MATH] arises from a 100 km s[MATH] shock with depleted abundances, rising to 45% at log [MATH] (log [MATH] corresponds to a height of about [MATH] pc).', 'astro-ph-0011556-2-45-0': 'The composite model successfully replicates the run of [S2]/[N2] at all but very high-[MATH] where matter bounded photo-ionization models cannot explain values of [N2]/H[MATH] and [S2]/H[MATH] greater than unity.', 'astro-ph-0011556-2-45-1': 'Radiation bounded photo-ionization models feature greater values of these line ratios; however, predicted runs of [O3]/H[MATH] cannot be reconciled with observations.', 'astro-ph-0011556-2-45-2': 'The run of [O1]/H[MATH] vs. [N2]/H[MATH] is also well explained by the composite model.', 'astro-ph-0011556-2-45-3': 'The lack of any obvious trend in the run of [O3]/H[MATH] vs. [N2]/H[MATH] makes it difficult to assess whether shocks are a viable contributor to the H[MATH] emission.', 'astro-ph-0011556-2-45-4': 'However, the high values of [O3]/H[MATH] in Slit 1 do suggest a large contribution from shocks in those regions.', 'astro-ph-0011556-2-45-5': 'It should be noted, though, that the galaxy has a fairly quiescent appearance in the H[MATH] image with little extraplanar emission, calling into question the notion that shocks are permeating the ISM.', 'astro-ph-0011556-2-45-6': 'It would be useful to obtain more [O3] data for this galaxy to better diagnose the possible contributions from secondary ionization sources.', 'astro-ph-0011556-2-45-7': 'Nevertheless, a clear departure from a pure photoionization model is indicated by the data.', 'astro-ph-0011556-2-46-0': '## DIG Temperature as the Main Cause of Line Ratio Variations', 'astro-ph-0011556-2-47-0': 'In this section, following the approach of HRT, we attempt to determine whether variations in line ratios with [MATH] can be explained by a change in gas temperature as one moves off the midplane.', 'astro-ph-0011556-2-47-1': 'A temperature increase with [MATH] provides a simple explanation of why all ratios of forbidden lines to Balmer lines are generally seen to increase with [MATH].', 'astro-ph-0011556-2-47-2': 'In fact, HRT find for the Reynolds Layer that the increase in [S2]/H[MATH] and [N2]/H[MATH], and the constancy of [S2]/[N2] with [MATH], up to [MATH] kpc, can be accounted for by an increase in gas temperature alone, without having to invoke a secondary source of ionization.', 'astro-ph-0011556-2-47-3': 'Such a scenario requires additional non-ionizing heating for low-density, high-[MATH] gas, such as photo-electric heating from dust grains (Reynolds Cox 1992) or dissipation of interstellar turbulence (Minter Balser 1997).', 'astro-ph-0011556-2-47-4': 'Using line ratio data versus [MATH] for NGC 4302, NGC 5775, UGC 10288, and NGC 891, we attempt to test this scenario by determining gas temperatures as well as ionization fractions of constituent elements.', 'astro-ph-0011556-2-48-0': 'For collisionally excited ions, the equation for intensity of emission lines is given by Osterbrock (1989), [EQUATION] where [MATH] and [MATH] are the collision strength and energy of the transition, [MATH] is the statistical weight of the ground level, [MATH] is the temperature in units of 10[MATH] K, and [MATH] is the fraction of downward transitions that produce the emission line.', 'astro-ph-0011556-2-49-0': 'To simplify matters, we assume no change in physical conditions, namely temperature, density, ionization fraction, and gas abundance, along a given line-of-sight through the DIG layer.', 'astro-ph-0011556-2-49-1': 'Such uniformity of physical parameters within the ISM is clearly not accurate when lines-of-sight cross small scale structures such as H2 regions or shocks, where temperature or ionization conditions may change dramatically.', 'astro-ph-0011556-2-49-2': 'The assumption is valid, however, if these parameters are thought of as averages along a given line of sight instead of describing localized conditions within the ISM.', 'astro-ph-0011556-2-50-0': 'This assumption, along with the use of the collision strengths provided by Aller (1984), allows the intensity of [S2], [N2], and [O3] emission to be represented, in units of rayleighs, by: [EQUATION] and [EQUATION]', 'astro-ph-0011556-2-50-1': 'The intensity of H[MATH] emission is given by, [EQUATION] where [MATH] is the emission measure in units of pc cm[MATH].', 'astro-ph-0011556-2-51-0': 'A further simplification is to assume N[MATH] and H[MATH] have approximately equal ionization fractions.', 'astro-ph-0011556-2-51-1': 'This assumption is justified for two reasons.', 'astro-ph-0011556-2-51-2': 'First, photoionization models of S94 predict that N[MATH]/N[MATH] tracks H[MATH]/H[MATH] due to similar first ionization potentials (14.5 eV versus 13.6 eV) and a weak charge exchange reaction.', 'astro-ph-0011556-2-51-3': 'Second, N is not likely to ionize higher than N[MATH] in a pure photoionization scenario due to a second ionization potential of 29.6 eV (HRT).', 'astro-ph-0011556-2-51-4': 'A high ionization fraction of O[MATH], however, would imply a significant fraction in the N[MATH] state as the second ionization potential of oxygen is 35.1 eV.', 'astro-ph-0011556-2-51-5': 'Thus we will be able to check the validity of this assumption with the [O3]/H[MATH] data.', 'astro-ph-0011556-2-51-6': 'Due to this simplification, of the line ratios available from this data, [N2]/H[MATH] is the best probe of gas temperature.', 'astro-ph-0011556-2-52-0': 'The forbidden lines are all dependent on their abundance relative to atomic hydrogen.', 'astro-ph-0011556-2-52-1': 'The issue of abundances is a complicated one, the understanding of which is critical for interpreting the line ratios.', 'astro-ph-0011556-2-52-2': 'Current evidence suggests that abundances in halo gas may be quite different from gas in the disk.', 'astro-ph-0011556-2-52-3': 'Savage and Sembach (1996) find increasing gas-phase abundances in the Milky Way from disk to halo, possibly as a result of partial grain destruction due to shocks.', 'astro-ph-0011556-2-52-4': 'However, models by Sembach et.', 'astro-ph-0011556-2-52-5': 'al. (2000; hereafter SHRK) predict that the ratios of forbidden lines to Balmer lines tend to rise as abundances fall due to decreased cooling efficiency.', 'astro-ph-0011556-2-52-6': 'For this reason, it is unlikely that abundance variations can explain the increase in ratios of forbidden lines to Balmer lines.', 'astro-ph-0011556-2-52-7': 'However, observations of the [O2] [MATH]3727 line, to complement [O3] [MATH]5007 and [O1] [MATH]6300 measurements, would be very useful to completely decouple line ratios (which depend on excitation and temperature) from variations in abundance.', 'astro-ph-0011556-2-52-8': 'To simplify matters, we adopt abundances that do not vary with [MATH].', 'astro-ph-0011556-2-52-9': 'Using galactic gas-phase abundances of S,N, and O of (S/H)[MATH] (Anders Grevesse 1989), (N/H)[MATH] (Meyer, Cardelli, Sofia 1997), and (O/H)[MATH] (Cardelli Meyer 1997), along with the assumption of equal fractions of N[MATH] and H[MATH], we arrive at expressions relating line ratios to temperature and ionization fraction: [EQUATION] and [EQUATION]', 'astro-ph-0011556-2-52-10': 'The ionization fractions of oxygen and hydrogen are coupled through a charge exchange reaction that allows the ionization state of Hydrogen to be determined from the [O1]/H[MATH] ratio.', 'astro-ph-0011556-2-52-11': 'We use the ratio of emissivities of the [O1] [MATH]6300 line to the H[MATH] line provided by Reynolds et.', 'astro-ph-0011556-2-52-12': 'al. (1998a) to obtain, [EQUATION]', 'astro-ph-0011556-2-52-13': 'Gas temperatures are derived from the [N2]/H[MATH] ratio.', 'astro-ph-0011556-2-52-14': 'Assuming this ratio accurately reflects temperatures within the medium, line ratios involving the S and O lines are then calculated for a variety of ionization fractions.', 'astro-ph-0011556-2-52-15': 'We can then see whether all the ionization fractions are reasonable and consistent with our initial assumptions.', 'astro-ph-0011556-2-52-16': 'Plots of temperature versus [MATH] and calculated ratios overlayed on measured ratios are shown in Figures 3, 4, 12, and 13 for NGC 4302, UGC 10288, NGC 5775, and NGC 891, respectively.', 'astro-ph-0011556-2-52-17': 'Resulting ionization fractions from the temperature modeling are shown in Table 3.', 'astro-ph-0011556-2-53-0': 't3', 'astro-ph-0011556-2-54-0': 'To first order, a rise in DIG temperature can account for the observed rises in line intensities relative to H[MATH] emission, as well as the near constancy in the [S2]/[N2] ratio.', 'astro-ph-0011556-2-54-1': 'We find, except in the case of UGC 10288, the ionization fraction of S[MATH] relative to H[MATH] remains very close to 0.5 independent of [MATH].', 'astro-ph-0011556-2-54-2': 'This is similar to the result obtained by HRT for the WIM.', 'astro-ph-0011556-2-54-3': 'Photo-ionization models performed by SHRK predict a comparable S[MATH] ionization fraction for low-density H2 regions.', 'astro-ph-0011556-2-54-4': 'UGC 10288 on the other hand, shows a considerably higher fraction of S[MATH] relative to H[MATH] as well as some evidence for an increase with [MATH].', 'astro-ph-0011556-2-54-5': 'However, the ionization fraction of H[MATH], determined from points relatively close to the midplane, is significantly lower than in the case of NGC 5775 or NGC 891.', 'astro-ph-0011556-2-54-6': 'As a result, the range in ionization fraction of S[MATH] in UGC 10288 is nearly equal to the range in values for the other galaxies.', 'astro-ph-0011556-2-54-7': 'This does not explain the observed rise with [MATH], however, and the lack of [O1] data at high-[MATH] makes it impossible to determine whether the ionization state of sulfur changes with [MATH].', 'astro-ph-0011556-2-55-0': 'The [O3]/H[MATH] data, in addition to revealing information about the ionization state of O, is also an excellent test of the validity of some of the assumptions used for this temperature modeling.', 'astro-ph-0011556-2-55-1': 'The data for NGC 5775 and UGC 10288 suggest that a significant fraction of oxygen is in the O[MATH] state; at least 15% and in some lines-of-sight nearly 100% relative to H[MATH].', 'astro-ph-0011556-2-55-2': 'The second ionization potential of oxygen is 35.1 eV, while that of nitrogen is 29.6 eV.', 'astro-ph-0011556-2-55-3': 'Thus one would expect the ionization fractions of the doubly-ionized species of these two elements to roughly track one another in a warm medium.', 'astro-ph-0011556-2-55-4': 'The determined values of ionization fraction of O[MATH] for these two galaxies imply that a significant percentage of nitrogen atoms are in the N[MATH] state, particularly at high-[MATH] for NGC 5775.', 'astro-ph-0011556-2-55-5': 'The likely presence of significant amounts of N[MATH] as well as the possibility of its fraction changing with [MATH] complicates the use of [N2]/H[MATH] as a temperature indicator.', 'astro-ph-0011556-2-55-6': 'In any event, the high values of O[MATH] ionization fraction suggest that a temperature increase alone is not able to fully explain the runs of line ratios observed in NGC 5775 and UGC 10288 since pure photo-ionization models predict very low O[MATH] fractions.', 'astro-ph-0011556-2-55-7': "One possible explanation for the large ionization fractions of doubly-ionized species is that a secondary ionizing source with emission characteristics similar to shocks or TMLs contributes to the layer's ionization.", 'astro-ph-0011556-2-55-8': 'Alternatively, the [O3] emitting gas could represent a component that is at a significantly higher temperature than the component responsible for the [N2] emission.', 'astro-ph-0011556-2-56-0': 'NGC 891 on the other hand, shows considerably lower values for the ionization fraction of O[MATH] relative to H[MATH]: 13% on either side of the disk and just above 30% in the midplane.', 'astro-ph-0011556-2-56-1': 'The slit runs near a bright filament that is not as prominent as those observed in NGC 5775.', 'astro-ph-0011556-2-56-2': 'These fractions are slightly larger than those predicted by the SHRK model for low-density H2 regions, though still more than an order of magnitude higher than that of the diffuse gas model of SHRK.', 'astro-ph-0011556-2-56-3': 'Thus it seems the run of line-ratios in NGC 891 is better explained by an increase in gas temperature with [MATH] than for NGC 5775.', 'astro-ph-0011556-2-56-4': 'A slightly warmer [O3] emitting component than indicated by the [N2]/H[MATH] ratio could explain measured values of [O3]/H[MATH] while keeping the ionization fraction of O[MATH] at levels predicted by the models.', 'astro-ph-0011556-2-56-5': 'If this is the case, a secondary ionizing source would not be necessary to explain the runs of line ratios in NGC 891.', 'astro-ph-0011556-2-57-0': 'The run of [SII]/H[MATH] in NGC 4302 is well fit by the increase in temperature reflected by the [NII]/H[MATH] ratio.', 'astro-ph-0011556-2-57-1': 'It would be interesting to obtain [OIII][MATH]5007 data for this galaxy to better assess whether line ratio variations can be explained by DIG temperature variations.', 'astro-ph-0011556-2-58-0': 'Other factors may influence the line ratios such as afore-mentioned abundance gradients (Savage Sembach 1996) as well as scattered light due to extraplanar dust.', 'astro-ph-0011556-2-58-1': 'The effects of dust absorption on extraplanar emission have been shown to be significant for a number of DIG halos (Howk Savage 1999).', 'astro-ph-0011556-2-58-2': 'Modeling of the Milky Way (Wood Reynolds 1999) and NGC 891 (Ferrara et.', 'astro-ph-0011556-2-58-3': 'al. 1996) suggest that scattered light makes a minor contribution to diffuse halo H[MATH] emission that decreases with [MATH].', 'astro-ph-0011556-2-58-4': 'In NGC 891, the contribution to DIG emission from scattered light falls to 10% at [MATH] pc (Ferrara et.', 'astro-ph-0011556-2-58-5': 'al. 1996).', 'astro-ph-0011556-2-58-6': 'In any case, dust-scattered light may be an important consideration for the interpretation of these line ratios.', 'astro-ph-0011556-2-59-0': '# Conclusions', 'astro-ph-0011556-2-60-0': 'Deep spectroscopy of four edge-on spiral galaxies has allowed physical conditions within their DIG halos to be probed, and has provided information on possible sources of ionization of the gas.', 'astro-ph-0011556-2-60-1': 'This study has revealed the following conclusions:', 'astro-ph-0011556-2-61-0': '1.', 'astro-ph-0011556-2-61-1': 'The general trend for the runs of line ratios for NGC 4302 and UGC 10288 is very similar to that of NGC 5775 (Rand 2000) and NGC 891 (Rand 1998): an increase in [N2]/H[MATH] and [S2]/H[MATH] with [S2]/[N2] remaining nearly constant with [MATH], though one slit position for UGC 10288 does show evidence for an increase in [S2]/[N2] with [MATH].', 'astro-ph-0011556-2-61-2': 'Determinations of halo emission scale height have revealed that the more prominent DIG halos also have greater scale height.', 'astro-ph-0011556-2-61-3': 'This is consistent with models of dynamic halos where greater levels of star formation in the disk can push material farther off the midplane.', 'astro-ph-0011556-2-62-0': '2.', 'astro-ph-0011556-2-62-1': 'Pure photo-ionization models do not explain the runs of line ratios in NGC 5775 and UGC 10288.', 'astro-ph-0011556-2-62-2': 'Composite photo-ionization/shock models can replicate the run of [O3]/H[MATH] vs. [N2]/H[MATH] in NGC 5775, though the run of [S2]/H[MATH] vs. [N2]/H[MATH] is still problematic, especially at low-[MATH].', 'astro-ph-0011556-2-62-3': 'In addition, we find that the composite models require a greater percent contribution from shocks in filamentary than non-filamentary regions to match the run of [O3]/H[MATH].', 'astro-ph-0011556-2-62-4': 'A lack of a discernible trend for the [O3]/H[MATH] vs. [N2]/H[MATH] run in UGC 10288 makes it difficult to determine whether shocks contribute to line emission, though the high values of [O3]/H[MATH] in Slit 1 indicate that a shock contribution may be necessary.', 'astro-ph-0011556-2-62-5': 'A composite model, though, can account for the run of [S2]/H[MATH] vs. [N2]/H[MATH].', 'astro-ph-0011556-2-62-6': "However, if it follows the trend seen in NGC 5775 of more disturbed regions requiring a larger shock contribution, then the lack of a bright, filamentary halo in UGC 10288 may imply that shocks do not play as significant a role in layer's ionization.", 'astro-ph-0011556-2-63-0': 'The [O3]/H[MATH] vs. [N2]/H[MATH] run seems to be a key discriminant for shocks.', 'astro-ph-0011556-2-63-1': 'Fabry-Perot observations of these 3 lines for brighter halos, such as NGC 5775, are now feasible and would provide much more information on the importance of secondary ionizing sources with local morphology and height.', 'astro-ph-0011556-2-63-2': 'It is important to emphasize that shocks are not the only possible secondary source.', 'astro-ph-0011556-2-63-3': 'Other sources such as TMLs, X-rays, and ionization by cosmic rays may contribute as well.', 'astro-ph-0011556-2-64-0': '3.', 'astro-ph-0011556-2-64-1': 'Another scenario that we consider is a pure photoionization model with an increasing temperature with [MATH].', 'astro-ph-0011556-2-64-2': 'By making a few reasonably valid assumptions, the run of gas temperature vs. [MATH] has been determined in each of the four galaxies.', 'astro-ph-0011556-2-64-3': 'We find that an increase in temperature with [MATH] can explain the general trend of an increase in the ratios of forbidden lines to Balmer lines with [MATH], including an increase in [O3]/H[MATH] and the constancy of [S2]/[N2].', 'astro-ph-0011556-2-64-4': 'By establishing the run of gas temperature, constraints on ionization fractions of Sulfur, Oxygen, and Hydrogen could then be established.', 'astro-ph-0011556-2-64-5': 'The [O3]/H[MATH] ratios in NGC 5775 and UGC 10288 generally indicate an exceptionally high ionization fraction of O[MATH], that invalidates the assumption that N[MATH] and H[MATH] have equal ionization fractions, indicating that the line ratios cannot be explained by a temperature increase alone.', 'astro-ph-0011556-2-64-6': 'The runs of line ratios in NGC 891, where the ionization fraction of O[MATH] is still somewhat high, and NGC 4302 are better explained by an increase in temperature with [MATH], though the absence of [O3][MATH]5007 data for NGC 4302 makes a proper assessment difficult.', 'astro-ph-0011556-2-65-0': 'Finally, [O1][MATH]6300, [O2][MATH]3727, and [O3][MATH]5007 data for a vertically oriented slit would provide additional information on ionization and excitation conditions in these DIG halos.', 'astro-ph-0011556-2-66-0': 'We thank the KPNO staff for their help in obtaining the data.', 'astro-ph-0011556-2-66-1': 'We also thank L. M. Haffner, R. Reynolds, and R. Benjamin for comments in the preparation of this paper.', 'astro-ph-0011556-2-66-2': 'This research has made use of the NASA/IPAC Extragalactic database (NED).', 'astro-ph-0011556-2-66-3': 'This work was partially supported by NSF grant AST-9986113.', 'astro-ph-0011556-2-67-0': 'Aller, L. H. 1984, Physics of Thermal Gaseous Nebulae (Dordrecht: Reidel) Anders, E., Grevesse, N. 1989, Geochim.', 'astro-ph-0011556-2-67-1': '278, Back to the Galaxy , ed.', 'astro-ph-0011556-2-68-0': 'Galaxy & 2cGalaxy Centera & Adopted & Offset along & Hours of & Noisec', 'astro-ph-0011556-2-69-0': 'and Slit & R.A. & Dec. & Distanceb & Major axis & Integration &', 'astro-ph-0011556-2-70-0': '& (J2000) & (J2000) & (Mpc) & & & NGC 5775 (Slit 1) & 14[MATH]6 & 03[MATH] & 24.8 & 32 NW & 6.0 & 2.2 NGC 5775 (Slit 2) & & & & 20 SE & 5.0 & 2.4 NGC 4302 & 12 21 42.5 & 14 36 05 & 16.8 & 24 S & 4.0 & 9.7 UGC 10288 (Slit 1) & 16 14 25.1 & -00 12 27 & 31.5 & 30 E & 6.17 & 2.5 UGC 10288 (Slit 2) & & & & 35 W & 5.33 & 3.6 NGC 891 & 02 22 33.1 & 42 20 48 & 9.6 & 100 N & 1.5 & 16.1 aThe H1 kinematic center of NGC 5775 was determined by Irwin (1994).', 'astro-ph-0011556-2-70-1': 'Central positions of NGC 4302, UGC 10288, and NGC 891 are from de Vaucouleurs et al. (1991).', 'astro-ph-0011556-2-70-2': 'bAdopted distance of NGC 5775 is from Irwin (1994).', 'astro-ph-0011556-2-70-3': 'Distances for NGC 4302, UGC 10288, and NGC 891 are from Tully (1988).', 'astro-ph-0011556-2-70-4': 'cBefore spatial averaging.', 'astro-ph-0011556-2-70-5': 'Noise units are 10[MATH] erg cm[MATH] s[MATH] arcsec[MATH].', 'astro-ph-0011556-2-71-0': 'lcc 3 0pc DIG SCALE HEIGHTS & Profile & [MATH] Galaxy & Description & (pc) NGC 5775 & Slit 1 (NE) & 2300 NGC 5775 & Slit 1 (SW) & 2300 NGC 5775 & Slit 2 (NE) & 2500 NGC 5775 & Slit 2 (SW) & 1800 NGC 4302 & East & 700 NGC 4302 & West & 550 UGC 10288 & Slit 1 (S) & 320 UGC 10288 & Slit 2 (S) & 460', 'astro-ph-0011556-2-72-0': 'lcccc 5 0pc CONSTRAINTS ON IONIZATION FRACTIONSa', 'astro-ph-0011556-2-73-0': 'Galaxy & Gas Temperature & [MATH]^+[MATH]^+[MATH] & [MATH]^++[MATH]^+[MATH] & [MATH]^+[MATH]', 'astro-ph-0011556-2-74-0': 'and Slit & (10[MATH] K) & & & NGC 5775 (Slit 1) & 0.64-0.95 & 0.30-0.60 & 0.15-0.70b & n/a NGC 5775 (Slit 2) & 0.62-0.95 & 0.35-0.80 & 0.30-0.75b & 0.65-0.85 UGC 10288 (Slit 1) & 0.60-1.05 & 0.55-1.00 & 0.50-0.80b & 0.45-0.65 UGC 10288 (Slit 2) & 0.60-0.95 & 0.50-1.00 & 0.50-0.80b & 0.30-0.60 NGC 4302 & 0.66-1.00 & 0.35-0.60 & n/a & n/a NGC 891 & 0.65-1.05 & 0.40-0.65 & 0.13-0.30 & 0.55-0.80 aValues of ionization fractions are valid only for the case of a pure photoionization model with an increasing temperature with [MATH].', 'astro-ph-0011556-2-74-1': 'See text for other assumptions used to determine these values.', 'astro-ph-0011556-2-74-2': 'bNearly all data points fall within this range for NGC 5775 and UGC 10288.', 'astro-ph-0011556-2-74-3': 'However, some measured values indicate an ionization fraction of O[MATH] relative to H[MATH] that is nearly 1.0.'}

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'astro-ph-0011556-2-61-3'], ['astro-ph-0011556-1-13-0', 'astro-ph-0011556-2-13-0'], ['astro-ph-0011556-1-13-1', 'astro-ph-0011556-2-13-1'], ['astro-ph-0011556-1-13-2', 'astro-ph-0011556-2-13-2'], ['astro-ph-0011556-1-13-3', 'astro-ph-0011556-2-13-3'], ['astro-ph-0011556-1-36-0', 'astro-ph-0011556-2-36-0'], ['astro-ph-0011556-1-36-1', 'astro-ph-0011556-2-36-1'], ['astro-ph-0011556-1-36-2', 'astro-ph-0011556-2-36-2'], ['astro-ph-0011556-1-36-3', 'astro-ph-0011556-2-36-3'], ['astro-ph-0011556-1-36-4', 'astro-ph-0011556-2-36-4'], ['astro-ph-0011556-1-20-0', 'astro-ph-0011556-2-20-0'], ['astro-ph-0011556-1-20-1', 'astro-ph-0011556-2-20-1'], ['astro-ph-0011556-1-20-3', 'astro-ph-0011556-2-20-3'], ['astro-ph-0011556-1-20-4', 'astro-ph-0011556-2-20-4'], ['astro-ph-0011556-1-20-5', 'astro-ph-0011556-2-20-5'], ['astro-ph-0011556-1-20-6', 'astro-ph-0011556-2-20-6'], ['astro-ph-0011556-1-20-7', 'astro-ph-0011556-2-20-7'], ['astro-ph-0011556-1-20-8', 'astro-ph-0011556-2-20-8'], ['astro-ph-0011556-1-20-9', 'astro-ph-0011556-2-20-9'], ['astro-ph-0011556-1-34-0', 'astro-ph-0011556-2-34-0'], ['astro-ph-0011556-1-34-1', 'astro-ph-0011556-2-34-1'], ['astro-ph-0011556-1-34-2', 'astro-ph-0011556-2-34-2'], ['astro-ph-0011556-1-34-3', 'astro-ph-0011556-2-34-3'], ['astro-ph-0011556-1-14-1', 'astro-ph-0011556-2-14-1'], ['astro-ph-0011556-1-70-1', 'astro-ph-0011556-2-70-1'], ['astro-ph-0011556-1-70-2', 'astro-ph-0011556-2-70-2'], ['astro-ph-0011556-1-70-5', 'astro-ph-0011556-2-70-5']]

[]

[]

[['astro-ph-0011556-1-6-3', 'astro-ph-0011556-2-6-3']]

[]

['astro-ph-0011556-1-4-6', 'astro-ph-0011556-1-4-8', 'astro-ph-0011556-1-4-9', 'astro-ph-0011556-1-5-3', 'astro-ph-0011556-1-6-5', 'astro-ph-0011556-1-6-7', 'astro-ph-0011556-1-12-0', 'astro-ph-0011556-1-14-0', 'astro-ph-0011556-1-19-3', 'astro-ph-0011556-1-20-2', 'astro-ph-0011556-1-21-1', 'astro-ph-0011556-1-27-0', 'astro-ph-0011556-1-28-6', 'astro-ph-0011556-1-37-0', 'astro-ph-0011556-1-43-2', 'astro-ph-0011556-1-52-12', 'astro-ph-0011556-1-53-0', 'astro-ph-0011556-1-58-5', 'astro-ph-0011556-1-60-1', 'astro-ph-0011556-1-61-0', 'astro-ph-0011556-1-62-0', 'astro-ph-0011556-1-64-0', 'astro-ph-0011556-1-70-0', 'astro-ph-0011556-1-70-3', 'astro-ph-0011556-1-70-4', 'astro-ph-0011556-1-71-0', 'astro-ph-0011556-1-72-0', 'astro-ph-0011556-1-73-0', 'astro-ph-0011556-1-74-0', 'astro-ph-0011556-2-4-6', 'astro-ph-0011556-2-4-8', 'astro-ph-0011556-2-4-9', 'astro-ph-0011556-2-5-3', 'astro-ph-0011556-2-6-4', 'astro-ph-0011556-2-6-6', 'astro-ph-0011556-2-6-8', 'astro-ph-0011556-2-12-0', 'astro-ph-0011556-2-14-0', 'astro-ph-0011556-2-19-3', 'astro-ph-0011556-2-20-2', 'astro-ph-0011556-2-21-1', 'astro-ph-0011556-2-27-0', 'astro-ph-0011556-2-28-6', 'astro-ph-0011556-2-37-0', 'astro-ph-0011556-2-43-2', 'astro-ph-0011556-2-52-12', 'astro-ph-0011556-2-53-0', 'astro-ph-0011556-2-58-5', 'astro-ph-0011556-2-60-1', 'astro-ph-0011556-2-61-0', 'astro-ph-0011556-2-62-0', 'astro-ph-0011556-2-64-0', 'astro-ph-0011556-2-70-0', 'astro-ph-0011556-2-70-3', 'astro-ph-0011556-2-70-4', 'astro-ph-0011556-2-71-0', 'astro-ph-0011556-2-72-0', 'astro-ph-0011556-2-73-0', 'astro-ph-0011556-2-74-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0011556

astro-ph-0103192

{'astro-ph-0103192-1-0-0': 'The main evidence of the internal-external scenario within the fireball model of GRBs is their high variability.', 'astro-ph-0103192-1-0-1': 'External shocks cannot produce variable bursts and therefore, according to this model, the [MATH] emission produced by internal shocks while the afterglow produced by the following external shock.', 'astro-ph-0103192-1-0-2': 'So far the variability was shown only in long bursts (BATSE [MATH] .', 'astro-ph-0103192-1-0-3': 'It is not known whether short bursts, which form a different class, are produced in the same way.', 'astro-ph-0103192-1-0-4': 'We have developed an algorithm which is sensitive enough to analyze the temporal structure of short bursts.', 'astro-ph-0103192-1-0-5': 'We analyze a sample of bright short bursts from the BATSE 4B-catalog and find that many short bursts are highly variable ( [MATH] , where [MATH] is the shortest pulse).', 'astro-ph-0103192-1-0-6': 'This indicates that short bursts, like long ones, are produced by internal shocks.', 'astro-ph-0103192-1-0-7': 'We also use the same algorithm to analyze the high resolution (TTE) data of long bursts (only the first 1-2 seconds of each burst).', 'astro-ph-0103192-1-0-8': 'We find that 10ms time-scales are common in long bursts.', 'astro-ph-0103192-1-0-9': 'This result show that long bursts are even more variable then it was thought before ( [MATH] ).', 'astro-ph-0103192-1-1-0': '# Introduction', 'astro-ph-0103192-1-2-0': 'From the early days of the GRB studies many efforts were invested in classifying the large variety of GRBs into different classes.', 'astro-ph-0103192-1-2-1': 'Until today only one classification is wildly accepted.', 'astro-ph-0103192-1-2-2': 'This is the long/short classification (Kouveliotou et al. 1993, Mao, Narayan Piran 1994 ).', 'astro-ph-0103192-1-2-3': "The main argument for this classification is the bimodal duration histogram of BATSE's archives, but other parameters are supporting this classification as well (Katz Canel 1996, Kouveliotou et al. 1993).", 'astro-ph-0103192-1-2-4': 'The GRB classes are the long bursts ( [MATH] ) and the short bursts ( [MATH] ).', 'astro-ph-0103192-1-2-5': 'The temporal features of the long bursts were wildly investigated (e.g. Norris et al. 1996, Norris 1995, Lee et.', 'astro-ph-0103192-1-2-6': 'al. 1995 ), while there are only few works about short bursts temporal structure.', 'astro-ph-0103192-1-2-7': 'The main reason for this is that short bursts are much harder to analyze (much lower signal to noise ratio in the relevant resolution).', 'astro-ph-0103192-1-3-0': 'The observations of GRBs afterglow led to a great progress in our understanding of the phenomenon.', 'astro-ph-0103192-1-3-1': 'But there is no observation of an afterglow that followed a short burst.', 'astro-ph-0103192-1-3-2': "Therefore almost all the deductions from the afterglows' properties are limited to the long bursts.", 'astro-ph-0103192-1-3-3': "At present the lack of short bursts afterglows can't teach us much as it could be simply due to instrumental limitations.", 'astro-ph-0103192-1-4-0': 'According to the internal-external GRB model (review Piran 1999) an internal engine ejects a highly relativistic modulated wind.', 'astro-ph-0103192-1-4-1': 'The observed [MATH] -ray emission is a result of internal shocks (collisions between different parts within the wind) (Rees Meszaros 1994).', 'astro-ph-0103192-1-4-2': 'After the internal collisions the wind still have a relativistic bulk motion that is converted to radiation through an external shock (a collision between the wind and the surrounding matter), this radiation is the observed afterglow (Rees Meszaros 1992).', 'astro-ph-0103192-1-5-0': 'The evidences for this scenario is based only on long bursts.', 'astro-ph-0103192-1-5-1': 'The main argument supporting this scenario is the high variability of long bursts.', 'astro-ph-0103192-1-5-2': 'In most long bursts the shortest time scale in the burst (a single pulse duration) is about [MATH] of the longest time scale (the whole [MATH] -ray emission duration).', 'astro-ph-0103192-1-5-3': 'External shocks cannot produce such variability efficiently (Sari Piran 1997), while internal shocks can (Kobayashi, Piran Sari 1997, Beloborodov 2000).', 'astro-ph-0103192-1-5-4': 'As no variability analysis was made for short bursts, it is not clear if this argument is valid in short bursts as well.', 'astro-ph-0103192-1-5-5': 'An additional argument in favor of the internal-external model is the afterglow and specifically the lack of simple scaling between the GRB and the afterglow.', 'astro-ph-0103192-1-5-6': 'As no short burst afterglow was detected, this argument is not valid for short bursts as well.', 'astro-ph-0103192-1-6-0': 'We examine here the variability of short bursts and we compare the shortest time scales of long and short bursts.', 'astro-ph-0103192-1-6-1': 'In order to do so we have developed an algorithm that is sensitive enough to find pulses in short bursts.', 'astro-ph-0103192-1-6-2': 'Our algorithm finds the pulses and determine their duration ( [MATH] ).', 'astro-ph-0103192-1-6-3': 'The burst duration, [MATH] , is taken from the beginning of the first pulse till the end of the last one.', 'astro-ph-0103192-1-6-4': 'Using this algorithm we analyze the variability of short bursts.', 'astro-ph-0103192-1-6-5': 'We also analyze the high resolution (TTE) data of long bursts.', 'astro-ph-0103192-1-6-6': 'Unfortunately, TTE data is available only for the first 1-2 seconds of the bursts, so we have only a small fraction of any whole long burst.', 'astro-ph-0103192-1-7-0': 'We find first, that most short bursts (although not all of them) show a high variability.', 'astro-ph-0103192-1-7-1': 'This result indicates that these short bursts, as well as the long ones, are produced by internal shocks.', 'astro-ph-0103192-1-7-2': 'Second, the shortest time scales in long bursts are similar to these in short bursts.', 'astro-ph-0103192-1-7-3': 'This result is limited to the highest resolution in which we analyzed the long bursts - 5ms.', 'astro-ph-0103192-1-7-4': 'Such time scales were already observed in long bursts (Lee, Bloom Petrosian 2000), but we show that these time scales are common in long bursts.', 'astro-ph-0103192-1-8-0': 'In section 2 we describe the data sets we considered for our analysis.', 'astro-ph-0103192-1-8-1': 'The results are brought in section 3 and the conclusions in 4.', 'astro-ph-0103192-1-8-2': 'Our algorithm is described in the appendix.', 'astro-ph-0103192-1-9-0': '# The data sets', 'astro-ph-0103192-1-10-0': 'The two BATSE data formats we use are the 64ms concatenate data and the TTE data (see Scargle (1998) for a detailed review).', 'astro-ph-0103192-1-10-1': 'The 64ms concatenate data includes the photon counts of a burst, in a 64ms time bins, from a few seconds before the burst trigger till a few hundred of seconds after the trigger.', 'astro-ph-0103192-1-10-2': 'The concatenate data contains also a very early and a very late data of the burst in a 1024ms resolution.', 'astro-ph-0103192-1-10-3': 'We used only the data in the 64ms resolution.', 'astro-ph-0103192-1-10-4': 'The TTE (Time Tagged Events) data includes the arrival time of each photon in a resolution of [MATH] .', 'astro-ph-0103192-1-10-5': 'This data contains records of only the first 1-2 seconds of the burst.', 'astro-ph-0103192-1-10-6': 'Hence it is usually used only for analyzing short bursts.', 'astro-ph-0103192-1-10-7': 'Both data formats have four energy channels.', 'astro-ph-0103192-1-10-8': 'We used the sum of all the channels (in both formats) that is E [MATH] 25Kev.', 'astro-ph-0103192-1-11-0': 'We have several data sets that we consider in this work.', 'astro-ph-0103192-1-11-1': "In order to find [MATH] in short bursts we consider a sample of short bursts (called and a comparable set of long bursts (called However, the properties of the set can't be compared directly with these of the set.", 'astro-ph-0103192-1-11-2': "The long bursts' data is binned in longer time bins then the short bursts data.", 'astro-ph-0103192-1-11-3': 'Therefore two equally intense bursts (one short and one long) would have a different signal-to-noise ratio (S/N).', 'astro-ph-0103192-1-11-4': 'Hence, we need to add noise to the set so that it would be comparable to the set.', 'astro-ph-0103192-1-11-5': "The set of long bursts with added noise is called long'.", 'astro-ph-0103192-1-11-6': 'Finally we want to find out the shortest time scale in long bursts.', 'astro-ph-0103192-1-11-7': 'In order to do so we consider a set of long bursts with a good TTE coverage of the first second.', 'astro-ph-0103192-1-11-8': "This set is called res long'.", 'astro-ph-0103192-1-12-0': '## The data set', 'astro-ph-0103192-1-13-0': 'In the BATSE 4B-catalog there are about 400 records of short bursts, only part of them have good TTE coverage.', 'astro-ph-0103192-1-13-1': 'This is a large statistical sample.', 'astro-ph-0103192-1-13-2': 'But most of these bursts are faint and it is impossible to retrieve their temporal features.', 'astro-ph-0103192-1-13-3': 'There is a trade-off between the sample size, the resolution and the signal to noise ratio.', 'astro-ph-0103192-1-13-4': 'We decided to consider a sample of the brightest 33 short bursts (peak flux in 64ms[MATH]4.37 [MATH] ) with a good TTE data coverage.', 'astro-ph-0103192-1-13-5': 'In order to get a reasonable signal to noise ratio we have binned this data into 2ms time bins.', 'astro-ph-0103192-1-13-6': 'In this resolution the S/N of the brightest peak in the faintest burst (from our sample) is 4.7.', 'astro-ph-0103192-1-13-7': 'As described in the appendix a peak is significant only if it is more then 4 [MATH] above the background, hence this is the largest sample we could consider under the circumstances.', 'astro-ph-0103192-1-13-8': 'The minimal recognized pulse width with this resolution is 4ms.', 'astro-ph-0103192-1-14-0': '## The data sets', 'astro-ph-0103192-1-15-0': 'We need a set of long bursts that could be compared to the set.', 'astro-ph-0103192-1-15-1': 'The first sample we considered is a sample of 34 long bursts (called set) with the same 64ms peak fluxes (one to one) as the bursts in the set.', 'astro-ph-0103192-1-15-2': 'This way we prevent differences that arise from different brightness.', 'astro-ph-0103192-1-15-3': "But, this set couldn't be compared directly with the set.", 'astro-ph-0103192-1-15-4': 'Since the highest resolution for these bursts is 64ms, the signal to noise ratio of this set is larger by a factor of [MATH] (assuming the same background level) then the S/N of the set (binned into 2ms time bins).', 'astro-ph-0103192-1-15-5': "To obtain a comparable set we produced another data set (called long' set) out of the set by treating this set as if the basic time bin was 2ms and adding a Poisson noise accordingly.", 'astro-ph-0103192-1-15-6': 'This procedure reduces the signal to noise ratio and made this set comparable to the set.', 'astro-ph-0103192-1-16-0': "We used the long' set also as a test to our algorithm.", 'astro-ph-0103192-1-16-1': "By analyzing the long' set we can learn how good is our algorithm in retrieving the attributes of the set .", 'astro-ph-0103192-1-16-2': 'We also can learn how does the noise influence the analyzed temporal structure.', 'astro-ph-0103192-1-16-3': 'In this way we can try to estimate what was the original temporal structure of the set.', 'astro-ph-0103192-1-17-0': '## High-resolution long bursts', 'astro-ph-0103192-1-18-0': 'The comparison between the shortest time scales in long and short burst requires the analysis of high-resolution long bursts.', 'astro-ph-0103192-1-18-1': 'The only data type with high enough resolution is TTE data, but this data is available only for the first 1-2sec of the bursts.', 'astro-ph-0103192-1-18-2': 'We have searched for long bursts that begin with a bright enough pulse during the first two seconds.', 'astro-ph-0103192-1-18-3': "We also demanded that the counts would drop back to the background level during this time, so the beginning of the light curve will not be dominated by a pulse longer then 2sec. We found 15 such bursts with good data called res long' set.", 'astro-ph-0103192-1-18-4': 'We compared the first 1-2sec of these bursts with 15 short bursts with comparable peak fluxes (taken out of the set).', 'astro-ph-0103192-1-18-5': 'The analysis of both groups is done in 5ms time bins.', 'astro-ph-0103192-1-19-0': '# Results', 'astro-ph-0103192-1-20-0': '## Long bursts pulses attributes', 'astro-ph-0103192-1-21-0': 'We analyzed the set and found the duration, [MATH] , and the shortest pulse duration, [MATH] , of the bursts.', 'astro-ph-0103192-1-21-1': 'Figure [REF] show [MATH] and [MATH] as a function of [MATH] .', 'astro-ph-0103192-1-21-2': 'Figure [REF]a shows that [MATH] and [MATH] are not correlated.', 'astro-ph-0103192-1-21-3': 'Consequently, [MATH] is smaller for longer durations.', 'astro-ph-0103192-1-21-4': 'The value of [MATH] for the longer bursts are [MATH] .', 'astro-ph-0103192-1-21-5': 'The grey areas are restricted because of the resolution.', 'astro-ph-0103192-1-21-6': 'We see that [MATH] is limited by the resolution, suggesting even higher variability.', 'astro-ph-0103192-1-22-0': "## Long' VS Long bursts", 'astro-ph-0103192-1-23-0': 'Before analyzing the short bursts we studied the effect of noise on the time profile.', 'astro-ph-0103192-1-23-1': "We do so by comparing the attributes of the set with these of the long' set (see [REF]).", 'astro-ph-0103192-1-23-2': 'This procedure also tests our algorithm.', 'astro-ph-0103192-1-23-3': "Since we know the original signal (in the data set) we can find out the efficiency of the algorithm in retrieving the set attributes out of the long' set.", 'astro-ph-0103192-1-24-0': "Figure [REF]a represent all the pulses found in the and long' samples.", 'astro-ph-0103192-1-24-1': 'We can see that the algorithm retrieve the main features of the bursts out of the noisy sample, but there are some effects of the noise.', 'astro-ph-0103192-1-24-2': "One effect is that many pulses are 'lost' because of the noise (only 30% of the pulses are found in the set).", 'astro-ph-0103192-1-24-3': 'This is as a result of two effects: a) some pulses are too weak to be distinguished from the amplified noise.', 'astro-ph-0103192-1-24-4': 'b) Some pulses merge with other pulses because the gap between them is of the same order of the noise.', 'astro-ph-0103192-1-24-5': "The first effect shouldn't change the width of the found pulses, but the second one causes pulse widening.", 'astro-ph-0103192-1-24-6': 'So we expect fewer pulses and a shift toward wide pulses in the noisy sample.', 'astro-ph-0103192-1-24-7': 'Both effects are seen clearly in Figure [REF].', 'astro-ph-0103192-1-24-8': 'In the noisy sample there are 203 pulses with an average width of 1.62sec while in the sample there are 695 pulses with an average width of 1.39sec.', 'astro-ph-0103192-1-25-0': 'Another important issue is the effect of the noise on the parameter [MATH] .', 'astro-ph-0103192-1-25-1': 'This parameter is affected by the pulse widening as well as by the reduction in the bursts duration.', 'astro-ph-0103192-1-25-2': 'The burst duration becomes shorter, because of the noise, if the first or the last pulses of the burst are lost.', 'astro-ph-0103192-1-25-3': 'Figure [REF]b show the effect of the noise on [MATH] .', 'astro-ph-0103192-1-25-4': 'We can see that the noise increases it by a factor of 10 in average.', 'astro-ph-0103192-1-26-0': "## Pulses attributes of short bursts (compared with the long' )", 'astro-ph-0103192-1-27-0': 'As already mentioned the same algorithm was applied to the data set.', 'astro-ph-0103192-1-27-1': 'In Fig [REF]a all the pulses widths are shown as a function of the burst duration.', 'astro-ph-0103192-1-27-2': 'The gray areas are not allowed because of the resolution ( [MATH] ) or the parameters definition [MATH] ).', 'astro-ph-0103192-1-27-3': 'Fig [REF]b is the distribution of the pulses width ( [MATH] ).', 'astro-ph-0103192-1-27-4': 'One can see that [MATH] is about 50-100 ms without a strong correlation with T.', 'astro-ph-0103192-1-28-0': 'These pulse widths are only upper limits.', 'astro-ph-0103192-1-28-1': 'First, the noise causes the pulses to seem wider (see [REF]) by a factor of few.', 'astro-ph-0103192-1-28-2': 'Second, the resolution is limited.', 'astro-ph-0103192-1-28-3': 'It is likely that the shortest pulses in the set are shorter then the best highest resolution of our data.', 'astro-ph-0103192-1-28-4': 'Fig [REF]b shows a histogram that begins with pulses at 12ms (except for one very bright and short pulse of 6ms), which is only three times of best resolution.', 'astro-ph-0103192-1-28-5': "This is also the case with the long' set - the shortest pulse in this set is about 400ms which is three times the best resolution, while the shortest pulse of the set begin at the resolution limit (128ms).", 'astro-ph-0103192-1-28-6': 'Indication for very short time scales in short bursts was already found (Scargle, Norris Bonnel 1997).', 'astro-ph-0103192-1-28-7': '25% of the bursts in the set contain time scales of less then 20ms.', 'astro-ph-0103192-1-29-0': "Fig [REF] show [MATH] and [MATH] in both groups and long'.", 'astro-ph-0103192-1-29-1': 'In the set the median [MATH] is 0.25 while 35% of bursts have [MATH] and 35% of the bursts show a smooth structure ( [MATH] ).', 'astro-ph-0103192-1-29-2': 'This result could mislead us to the conclusion that a significant fraction of the short bursts have a smooth time profile.', 'astro-ph-0103192-1-29-3': "But a look at the long' results show that also in this group more than 20% of the bursts are single pulsed while there were no such bursts in the original set.", 'astro-ph-0103192-1-29-4': 'Naturally, the bursts that loose the fine structure because of the noise are bursts with fewer original pulses.', 'astro-ph-0103192-1-29-5': 'It is clear that short bursts have less pulses then long ones so they are more \'"vulnerable" to the noise.', 'astro-ph-0103192-1-29-6': 'Hence we must conclude that the majority of the short bursts are very variable ( [MATH] ).', 'astro-ph-0103192-1-30-0': '## High resolution long bursts vs. short bursts', 'astro-ph-0103192-1-31-0': 'We compared the time profile of the first seconds of 15 long bursts with the time profiles of 15 short bursts.', 'astro-ph-0103192-1-31-1': 'Figure [REF] shows the light curves of a short burst and the first second of a long burst.', 'astro-ph-0103192-1-31-2': 'The time scales of both bursts are quite similar.', 'astro-ph-0103192-1-32-0': 'Figure [REF] shows the pulse widths histogram of the beginning of long bursts vs. the same histogram of the short bursts.', 'astro-ph-0103192-1-32-1': 'The time scales in both samples are quite similar in the range of 10-200ms, while the long bursts have additional pulses in the range of 0.2-1sec. Long bursts contain, of course, longer pulses, but in the sample we considered we demanded that the counts would fall back to the background level within the first second.', 'astro-ph-0103192-1-32-2': 'In this way we have limited the pulses width of the long bursts.', 'astro-ph-0103192-1-32-3': 'Both histograms begin at 10-20ms, which is at the limit of the pulse width resolution (10ms).', 'astro-ph-0103192-1-32-4': 'It is very likely that both samples contain shorter time scales that cannot be resolved.', 'astro-ph-0103192-1-32-5': 'Walker, Schaefer Fenimore (2000) performed similar analysis of TTE data in 14 long bursts.', 'astro-ph-0103192-1-32-6': 'They found only one long burst with very short time scales.', 'astro-ph-0103192-1-32-7': 'The difference between the results is because of the different samples considered.', 'astro-ph-0103192-1-32-8': 'Walker et al. (2000) considered the bursts with the maximal total photon counts within the TTE burst recored, while we demanded that the counts will return close to the background level within the TTE recored.', 'astro-ph-0103192-1-32-9': 'Most of the bursts in Walker et al. (2000) sample are dominated by a long and bright pulse.', 'astro-ph-0103192-1-32-10': 'Walker et al. (2000) also make a wavelet analysis (in which they are not looking for separate pulses but for intrinsic time scales), and in this analysis their results are in agreement with ours.', 'astro-ph-0103192-1-33-0': 'Finally we have to remember that a sample of 15 bursts is rather small but it is enough to show that time scales of a few tens of msec are common even in long bursts.', 'astro-ph-0103192-1-33-1': 'The part of the long bursts that we have analyzed is very similar to a complete short burst.', 'astro-ph-0103192-1-33-2': 'Even though the sample is small and the bursts were not chosen randomly, this similarity is not a result of the way the sample was selected.', 'astro-ph-0103192-1-33-3': 'This is because the way we selected the sample gave no restriction on the structure of the light curves in time scales shorter then 1sec.', 'astro-ph-0103192-1-34-0': "This similarity raises the question whether it's possible that short bursts are actually only a small fraction, which is above the background noise, of long bursts.", 'astro-ph-0103192-1-34-1': 'We have already seen that the noise cause us to loose pulses.', 'astro-ph-0103192-1-34-2': 'Is it possible that a long burst with a single dominant, very intense pulse, (or a group of very close and intense pulses) will loose all its structure, apart for this intense pulse, due to noise and become a short burst.', 'astro-ph-0103192-1-34-3': 'Figure [REF] rule out this option.', 'astro-ph-0103192-1-34-4': 'It depicts the counts ratio between the most intense and the second most intense pulses within long bursts.', 'astro-ph-0103192-1-34-5': 'The graph shows that almost in all bursts (around 80%) the two most intense peaks are at the same level.', 'astro-ph-0103192-1-34-6': 'It means that there is no way that noise can cause one pulse to disappear without the other.', 'astro-ph-0103192-1-34-7': 'As these two peaks in each burst are well separated in time, there is no way that noise can convert significant fraction of long bursts into short ones.', 'astro-ph-0103192-1-34-8': 'In very few cases we managed to add noise and convert a long burst into a short one.', 'astro-ph-0103192-1-34-9': 'It means that noise could have some effect on the duration histogram but it certainly cannot produce the observed bimodality.', 'astro-ph-0103192-1-35-0': '# Conclusions', 'astro-ph-0103192-1-36-0': 'Our main result is that [MATH] for most short bursts as well as for long bursts.', 'astro-ph-0103192-1-36-1': 'As external shocks cannot produce such a temporal structure, this result indicates that most short bursts are produced in internal shocks.', 'astro-ph-0103192-1-36-2': 'In 30% of the short bursts the observed [MATH] is the same as the observed duration - [MATH] .', 'astro-ph-0103192-1-36-3': "However, a comparison with the long' set, shows that this feature is quite likely to be an artifact of the noise.", 'astro-ph-0103192-1-37-0': 'As the short bursts are produced in internal shocks, the expanding wind still have a high Lorentz factor at the end of the internal shocks.', 'astro-ph-0103192-1-37-1': "This bulk motion should slow down by the surrounding medium (if it's dense enough); this deceleration would emit an afterglow.", 'astro-ph-0103192-1-37-2': 'Therefore we predict that afterglows would be observed in the future following short bursts.', 'astro-ph-0103192-1-37-3': 'These afterglows should have no simple extrapolation to the [MATH] -ray emission.', 'astro-ph-0103192-1-38-0': 'A second result is that short time scales (10ms or less) are common in long bursts.', 'astro-ph-0103192-1-38-1': 'Hence, [MATH] in many long bursts.', 'astro-ph-0103192-1-38-2': 'This result make it very difficult to explain the high variability in any other way but the internal shocks (e.g. high inhomogeneity in the ISM).', 'astro-ph-0103192-1-39-0': '# Appendix -The algorithm', 'astro-ph-0103192-1-40-0': 'Our algorithm finds the peaks of the bursts.', 'astro-ph-0103192-1-40-1': 'Each peak corresponds to a single pulse; a pulse is the basic event of the light curve.', 'astro-ph-0103192-1-40-2': 'The algorithm is based on the algorithm suggested by Li Fenimore (1996).', 'astro-ph-0103192-1-40-3': 'Li Fenimore define a time bin [MATH] (with count [MATH] ) as a peak if there are two time bins [MATH] (with counts [MATH] respectively) which satisfies a) [MATH] and b) [MATH] is the maximal count between [MATH] and [MATH] .', 'astro-ph-0103192-1-40-4': '[MATH] is a parameter that determines the significance of the peak.', 'astro-ph-0103192-1-41-0': 'There are two problems with this algorithm.', 'astro-ph-0103192-1-41-1': 'First, this algorithm analyzes only data in a single time resolution (fixed time bin size).', 'astro-ph-0103192-1-41-2': 'Therefore the algorithm looses long and faint pulses.', 'astro-ph-0103192-1-41-3': 'A peak that does not satisfy the criterion described above in the raw data resolution could satisfy the criterion if the data resolution is lower (longer time-bins).', 'astro-ph-0103192-1-41-4': 'This algorithm would miss such a pulse.', 'astro-ph-0103192-1-41-5': 'Second, [MATH] determines the trade-off between sensitivity and false peaks identification rate.', 'astro-ph-0103192-1-41-6': 'When [MATH] is low the algorithm finds false peaks as a result of the Poisson noise.', 'astro-ph-0103192-1-41-7': 'When [MATH] is high the algorithm misses real peaks.', 'astro-ph-0103192-1-41-8': 'Finding false peaks is a severe problem during long periods of constant level Poisson noise, like the background (as will be explained shortly).', 'astro-ph-0103192-1-41-9': 'Long bursts contain such periods (periods of only background noise).', 'astro-ph-0103192-1-41-10': 'These periods are called quiescent times.', 'astro-ph-0103192-1-41-11': 'In order to avoid false peaks in long bursts [MATH] must be large( [MATH] ), which means an insensitive algorithm.', 'astro-ph-0103192-1-41-12': 'Short bursts contain less quiescent times, and of course shorter ones.', 'astro-ph-0103192-1-41-13': 'But, short bursts contain much less pulses then long burst (three to four compared to an average of more than thirty) and much smaller S/N.', 'astro-ph-0103192-1-41-14': 'Finding even one false peak could change the features of the burst drastically.', 'astro-ph-0103192-1-41-15': 'Too insensitive algorithm could loose all the burst structure.', 'astro-ph-0103192-1-41-16': 'In order to avoid false peaks in short bursts [MATH] must be at least as large as 5.', 'astro-ph-0103192-1-41-17': 'As described in section [REF], the S/N in some of the bright short bursts is smaller then 5.', 'astro-ph-0103192-1-41-18': 'Such [MATH] will prevent the algorithm from finding even one peak in these bursts.', 'astro-ph-0103192-1-42-0': 'We solved the first problem by analyzing the data in different resolutions.', 'astro-ph-0103192-1-42-1': 'The results of the algorithm in different resolutions are merged into a single set of peaks.', 'astro-ph-0103192-1-42-2': "We solved the second problem by restricting the search for peaks only to 'Active Periods'.", 'astro-ph-0103192-1-42-3': 'Active periods are periods with counts that correspond to source activity (we will define it later on).', 'astro-ph-0103192-1-43-0': 'There are few advantages for analyzing only active periods.', 'astro-ph-0103192-1-43-1': 'The main one is that a lower [MATH] can be used during these periods with smaller risk of finding false peaks.', 'astro-ph-0103192-1-43-2': 'The risk of finding false peaks due to a Poisson noise depends on the time scale in which the original signal (i.e. without the noise) changes its counts rate in the same order as the Poisson noise level.', 'astro-ph-0103192-1-43-3': 'If the signal is constant (no real pulses), then this time scale is as long as the signal.', 'astro-ph-0103192-1-43-4': 'If the constant signal is longer, it contains more time bins with the same level of Poisson noise counts.', 'astro-ph-0103192-1-43-5': 'Hence, there is a larger chance of finding within these time-bins three time-bins, [MATH] , [MATH] and [MATH] , that satisfy the criterion in Li Fenimore algorithm.', 'astro-ph-0103192-1-43-6': 'Then [MATH] would become a false peak.', 'astro-ph-0103192-1-43-7': 'On the other hand, if the signal is changing monotonically then the length of the signal is irrelevant.', 'astro-ph-0103192-1-43-8': '[MATH] , [MATH] and [MATH] must be within the period in which the signal changes at the same order as the Poisson noise level; there is no chance of finding [MATH] with [MATH] significantly below [MATH] (if the signal is rising) out of this period.', 'astro-ph-0103192-1-43-9': 'During the active periods the signal is changing rapidly (usually on time scales of seconds or less), and the Poisson noise is superimposed on steep slopes.', 'astro-ph-0103192-1-43-10': "In this case a false peak could only be found during the period in which the signal didn't change compared to the noise level.", 'astro-ph-0103192-1-43-11': 'There are much less time bins during this period and hence there are much less chance of finding false peaks.', 'astro-ph-0103192-1-44-0': 'The second advantage is that when an active period is found we almost certain that it is a part of the burst.', 'astro-ph-0103192-1-44-1': "This is important since one false peak in the 'wrong' place (for example hundred of seconds after the burst ended) can change the burst properties drastically.", 'astro-ph-0103192-1-44-2': 'By analyzing only active periods we can use smaller [MATH] (=4) and get a more sensitive and accurate algorithm.', 'astro-ph-0103192-1-45-0': 'Our algorithm works in several steps.', 'astro-ph-0103192-1-45-1': 'First, it determines the background level of the signal (as a function of time).', 'astro-ph-0103192-1-45-2': "Then it finds an activity level, demanding a probability of 0.9 (per burst) that all the time bins with counts above this level (called 'active bins') correspond to source activity and not of the background Poisson noise (we demand that on every ten bursts there is, on average, a single false 'active bin').", 'astro-ph-0103192-1-45-3': 'The Activity level depends on the background and its value is between 4 [MATH] to 5 [MATH] above the background.', 'astro-ph-0103192-1-45-4': 'From each active bin we search to the right and to the left until the count level drops to the background level on both sides.', 'astro-ph-0103192-1-45-5': 'We call all these bins together an active period (from the time bin that the counts are above the background until the time bin that the counts reaches the background level again).', 'astro-ph-0103192-1-45-6': 'In most cases a single active period includes many active bins and a burst may contain more then a single active period (see Fig [REF]).', 'astro-ph-0103192-1-45-7': 'Note that if the algorithm misses an active period in one resolution, it can still find it in a different (lower) resolution, in which the noise level is lower.', 'astro-ph-0103192-1-46-0': 'Once the active periods of a given burst have been determined we apply the Li Fenimore algorithm to the active periods (using Nvar=4) and determine the peaks.', 'astro-ph-0103192-1-46-1': 'We repeat this procedure (finding the active periods and the corresponding peaks), several times for different time resolution.', 'astro-ph-0103192-1-46-2': 'To obtain lower resolution data we convolve the original signal (in the basic time bins) with a Gaussian, whose width determines the resolution.', 'astro-ph-0103192-1-46-3': 'Finally, after finding the peaks in different resolutions we merge those sets of peaks to a single set (requiring that a peak must appear in at least two different resolutions).', 'astro-ph-0103192-1-46-4': 'The merge is done by merging the highest resolution set with the second highest one and then taking this merged set and merging it with the third highest resolution set and so on.', 'astro-ph-0103192-1-46-5': 'On different resolutions the same peak could be found on different time bins.', 'astro-ph-0103192-1-46-6': 'In each case two peaks on different resolutions are considered as a single one if the peak in one resolution falls between [MATH] and [MATH] of the peak in the other resolution.', 'astro-ph-0103192-1-47-0': 'Each peak corresponds, of course, to a pulse.', 'astro-ph-0103192-1-47-1': 'The pulse width ( [MATH] ) is defined by two points (on each side of the peak) that are higher than the background by 1/4 of the peaks height or by the minimum between two neighboring peaks (if the latter is higher).', 'astro-ph-0103192-1-47-2': 'The duration ( [MATH] ) of the burst is the time elapsed from the beginning of the first pulse till the end of the last pulse (so in single pulsed burst [MATH] ).'}

{'astro-ph-0103192-2-0-0': 'We analyze a sample of bright short bursts from the BATSE 4B-catalog and find that many short bursts are highly variable ( [MATH] , where [MATH] is the shortest pulse duration and [MATH] is the burst duration).', 'astro-ph-0103192-2-0-1': 'This indicates that it is unlikely that short bursts are produced by external shocks.', 'astro-ph-0103192-2-0-2': 'We also analyze the available (first 1-2 seconds) high resolution (TTE) data of some of the long bursts.', 'astro-ph-0103192-2-0-3': 'We find that variability on a 10ms time-scale is common in long bursts.', 'astro-ph-0103192-2-0-4': 'This result shows that some long bursts are even more variable than it was thought before ( [MATH] ).', 'astro-ph-0103192-2-1-0': '# Introduction', 'astro-ph-0103192-2-2-0': 'The temporal features of GRBs are among the more interesting clues on their origin.', 'astro-ph-0103192-2-2-1': 'The temporal features of long bursts ( [MATH] ) were widely investigated (e.g. Norris 1995; Norris et al. 1996; Lee, Bloom Scargle 1995; Beloborodov Stern Svensson, 2000), while only a few works (Scargle, Norris Bonnel 1997; Cline, Matthey Otwinowski 1999) discuss the temporal structure of short ( [MATH] ) bursts.', 'astro-ph-0103192-2-2-2': 'This is not surprising.', 'astro-ph-0103192-2-2-3': 'Short bursts are much harder to analyze because of their significantly lower signal to noise ratios.', 'astro-ph-0103192-2-2-4': 'We developed a new algorithm that is sensitive enough to identify pulses in short bursts.', 'astro-ph-0103192-2-2-5': 'Using this algorithm we search here for subpulses in short bursts and determine their duration ( [MATH] ).', 'astro-ph-0103192-2-2-6': 'This enables us to set upper limits on the shortest time scale seen in the bursts to set limits on the variability of short bursts.', 'astro-ph-0103192-2-3-0': 'Sari Piran (1997) and Fenimore, Madras, Nayakshin (1996) have shown that angular spreading would smooth any variability produced by external shocks (unless the GRB production is very inefficient with the efficiency of the order of [MATH] defined below (Sari Piran, 1997)).', 'astro-ph-0103192-2-3-1': 'The critical parameter in this analysis is the ratio between the shortest time scale, on which the burst varies significantly, and the longest time scale in the burst.', 'astro-ph-0103192-2-3-2': 'This motivates us to focus here on the ratio [MATH] , where [MATH] is the minimal observed duration of an individual pulse and [MATH] is the duration of the burst.', 'astro-ph-0103192-2-3-3': 'High variability means low [MATH] ) values while for smooth bursts [MATH] .', 'astro-ph-0103192-2-3-4': 'Our aim is to explore whether short bursts can be produced by external shocks.', 'astro-ph-0103192-2-3-5': 'However, our analysis is not directed by this motivation and the results concerning the variability of short bursts are valid independently of this motivation.', 'astro-ph-0103192-2-4-0': 'In the second part of this paper we compare the shortest time scales of long and short bursts.', 'astro-ph-0103192-2-4-1': 'We analyze (using the same algorithm) the high resolution (TTE) data of long bursts.', 'astro-ph-0103192-2-4-2': 'Unfortunately, this data is available only for the first 1-2 seconds of each burst, so we can analyze only a small fraction of each long burst.', 'astro-ph-0103192-2-4-3': 'still we are able to demonstrate that very short time scales (10ms or less) are common.', 'astro-ph-0103192-2-5-0': 'Our analysis deals with statistically significant individual pulses.', 'astro-ph-0103192-2-5-1': 'We define an observed peak in the light curve as the highest count rate within a series of counts that is statistically significant (more than 4 [MATH]) above the counts at some time before and some time after it.', 'astro-ph-0103192-2-5-2': 'Each peak corresponds to a pulse.', 'astro-ph-0103192-2-5-3': 'The width of the pulse is determined by the width at a quarter of the maximum, or, rarely, by the minima between neighboring peaks if the counts do not drop below quarter of the maximum (see the Appendix for a detailed definition and for a discussion of the algorithm).', 'astro-ph-0103192-2-5-4': 'An elementary pulse in the observed light curve must not necessarily correspond directly to an elementary emission event in the source.', 'astro-ph-0103192-2-5-5': 'A single pulse in the light curve could, in principle, be composed of numerous emission events.', 'astro-ph-0103192-2-5-6': 'In this case the emission process is even more variable then the observed light curve and the results we obtain here should be considered only as upper limits to the intrinsic variability of the sources.', 'astro-ph-0103192-2-6-0': 'We find that most short bursts (although not all of them) are highly variable ( [MATH] ) .', 'astro-ph-0103192-2-6-1': 'When analyzing the high resolution data of long burst we find that the shortest time scales seen in long bursts are similar to those in short bursts.', 'astro-ph-0103192-2-6-2': 'This result is limited to the highest resolution in which we analyzed the long bursts - 5ms.', 'astro-ph-0103192-2-6-3': 'Such time scales were already observed in at least one long burst (Lee, Bloom Petrosian 2000).', 'astro-ph-0103192-2-6-4': 'We show here that these time scales are common.', 'astro-ph-0103192-2-7-0': 'In section 2 we describe the data samples considered in our analysis.', 'astro-ph-0103192-2-7-1': 'We describe the results in section 3 and we discuss their implications in section 4.', 'astro-ph-0103192-2-7-2': 'Our algorithm is described in the appendix.', 'astro-ph-0103192-2-8-0': '# The data samples', 'astro-ph-0103192-2-9-0': 'We examine bright short and long bursts from BATSE 4B-catalog.', 'astro-ph-0103192-2-9-1': 'We use two BATSE data formats: the 64ms concatenate data and the TTE data (see Scargle (1998) for a detailed review).', 'astro-ph-0103192-2-9-2': 'The 64ms concatenate data includes the photon counts of each burst, in a 64ms time bins, from a few seconds before the burst trigger till a few hundred of seconds after the trigger.', 'astro-ph-0103192-2-9-3': 'The concatenate data includes also very early and very late data of the burst in a 1024ms resolution.', 'astro-ph-0103192-2-9-4': 'We use only the 64ms resolution data.', 'astro-ph-0103192-2-9-5': 'The TTE (Time Tagged Events) data includes the arrival time of each photon in a [MATH] resolution.', 'astro-ph-0103192-2-9-6': 'This data contains only records of the first 1-2 seconds of each burst.', 'astro-ph-0103192-2-9-7': 'Hence it contains whole short bursts, but only a fraction of long bursts.', 'astro-ph-0103192-2-9-8': 'Both data formats have four energy channels.', 'astro-ph-0103192-2-9-9': 'We use the sum of all the channels (in both formats), that is photon energy E [MATH] 25Kev.', 'astro-ph-0103192-2-10-0': 'We consider several samples.', 'astro-ph-0103192-2-10-1': "We consider a sample of short bursts (denoted 'short') and a comparable sample of long bursts (denoted 'long').", 'astro-ph-0103192-2-10-2': "However, the properties of the 'long' sample cannot be compared directly with those of the 'short' sample.", 'astro-ph-0103192-2-10-3': 'The long bursts data is binned in longer time bins then the short bursts data.', 'astro-ph-0103192-2-10-4': 'Therefore two equally intense bursts (one short and one long) would have a different signal-to-noise ratio (S/N).', 'astro-ph-0103192-2-10-5': "Hence, we generate a third sample denoted 'noisy long' by adding noise to the 'long' sample so that the S/N of the bursts in this sample would be comparable to the S/N of the 'short' sample.", 'astro-ph-0103192-2-10-6': 'Finally, in order to determine the shortest time scale in long bursts, we consider a sample of long bursts with a good TTE coverage of the first second.', 'astro-ph-0103192-2-10-7': "This sample is called 'high res long'.", 'astro-ph-0103192-2-11-0': "## The 'short' data sample", 'astro-ph-0103192-2-12-0': 'There are about 400 records of short bursts in the BATSE 4B-catalog.', 'astro-ph-0103192-2-12-1': 'However most of these bursts are too faint and it is impossible to retrieve their temporal features.', 'astro-ph-0103192-2-12-2': 'Furthermore, not all short bursts have a good TTE coverage.', 'astro-ph-0103192-2-12-3': 'There is a trade-off between the sample size, the resolution and the signal to noise ratio.', 'astro-ph-0103192-2-12-4': 'We consider, here, a sample of the brightest 33 short bursts (peak flux in 64ms [MATH] 4.37 [MATH] ) with a good TTE data coverage.', 'astro-ph-0103192-2-12-5': 'In order to get a reasonable signal to noise ratio we have binned this data into 2ms time bins.', 'astro-ph-0103192-2-12-6': 'In this resolution the S/N of the brightest peak in the faintest burst (from our sample) is 4.7.', 'astro-ph-0103192-2-12-7': 'As described in the appendix we consider a peak as statically significant only if it is more then 4 [MATH] above the background.', 'astro-ph-0103192-2-12-8': 'Hence this is the largest sample we could consider.', 'astro-ph-0103192-2-12-9': 'The minimal recognized pulse width with this resolution is 4ms.', 'astro-ph-0103192-2-13-0': "## The 'long' data samples", 'astro-ph-0103192-2-14-0': "We need a sample of long bursts that could be compared to the 'short' sample.", 'astro-ph-0103192-2-14-1': "The first sample we considered is a sample of 34 long bursts (called 'long' sample) with the same 64ms peak fluxes (one to one) as the bursts in the 'short' sample.", 'astro-ph-0103192-2-14-2': 'This sample contain fairly bright long bursts, but not the brightest long bursts.', 'astro-ph-0103192-2-14-3': 'This way we prevent differences that arise from different brightness.', 'astro-ph-0103192-2-15-0': "However, the 'long' sample cannot be compared directly with the 'short' sample.", 'astro-ph-0103192-2-15-1': "The 'long' sample is binned in 64ms bins while the 'short' sample is binned into 2ms time bins.", 'astro-ph-0103192-2-15-2': "Therefore, assuming the same background noise level, the S/N of the 'long' sample is larger by a factor of [MATH] then the S/N of the 'short' sample.", 'astro-ph-0103192-2-15-3': "To obtain a comparable sample we produced another data sample (denoted 'noisy long' sample).", 'astro-ph-0103192-2-15-4': "This data set is produced by adding noise to the 'long' sample.", 'astro-ph-0103192-2-15-5': 'We treat this sample as if the basic time bin is 2ms and add a Poisson noise accordingly.', 'astro-ph-0103192-2-15-6': 'For a given long burst with counts [MATH] we generate a noisy signal, [MATH] , using the following simple procedure.', 'astro-ph-0103192-2-15-7': 'The noisy signal at time [MATH], [MATH] , is a Poisson variable ( [MATH] ) with [MATH] .', 'astro-ph-0103192-2-15-8': 'This noisy signal is smaller by a factor of [MATH] than the original signal and its standard deviation is correspondingly smaller by a factor of [MATH] then the original standard deviation.', 'astro-ph-0103192-2-15-9': 'There is a minor caveat in this procedure.', 'astro-ph-0103192-2-15-10': 'The original signal contains its own noise, but since this original noise is smaller by a factor of [MATH] then the added noise, it is negligible.', 'astro-ph-0103192-2-15-11': "The S/N ratio of the new 'noisy' sample is comparable to the S/N ratio of the 'short' sample.", 'astro-ph-0103192-2-16-0': "We use the 'noisy long' sample to investigate the influence of the noise on the analyzed temporal properties.", 'astro-ph-0103192-2-16-1': "We do so by comparing the temporal properties of the 'long' sample with the temporal properties of the 'noisy long' sample.", 'astro-ph-0103192-2-16-2': "In this way we can estimate what was the original temporal structure of the 'short' sample.", 'astro-ph-0103192-2-17-0': '## High-resolution long bursts', 'astro-ph-0103192-2-18-0': 'The comparison between the shortest time scales in long and short bursts requires the analysis of high-resolution long bursts.', 'astro-ph-0103192-2-18-1': 'The only data with high enough resolution is the TTE data, which is available only for the first 1-2sec of the bursts.', 'astro-ph-0103192-2-18-2': 'We have searched for long bursts that begin with a bright pulse during the first two seconds, also demanding that the counts would drop back to the background level during this time.', 'astro-ph-0103192-2-18-3': "This way the beginning of the light curve is not be dominated by a pulse longer then 2sec. We found 15 such bursts which we denoted as the 'high res long' sample.", 'astro-ph-0103192-2-18-4': "We compared the first 1-2sec of these bursts with 15 short bursts with comparable peak fluxes (taken out of the 'short' sample).", 'astro-ph-0103192-2-18-5': 'The analysis of both groups is done in 5ms time bins.', 'astro-ph-0103192-2-18-6': 'This sample is rather small and not randomly chosen, but this is the best sample one could get within the data limitation.', 'astro-ph-0103192-2-19-0': '# Results', 'astro-ph-0103192-2-20-0': '## Attributes of long bursts pulses', 'astro-ph-0103192-2-21-0': "We begin by estimating the duration, [MATH] , and the shortest pulse duration, [MATH] , of the 'long' bursts.", 'astro-ph-0103192-2-21-1': 'Fig. [REF] shows [MATH] and [MATH] as a function of [MATH] .', 'astro-ph-0103192-2-21-2': 'Fig. [REF]a shows that [MATH] and [MATH] are not correlated.', 'astro-ph-0103192-2-21-3': 'Consequently, [MATH] is smaller for longer bursts.', 'astro-ph-0103192-2-21-4': 'The value of [MATH] for the longer bursts are [MATH] .', 'astro-ph-0103192-2-21-5': 'The gray areas are restricted because of the resolution.', 'astro-ph-0103192-2-21-6': '[MATH] is limited by the resolution, suggesting that the bursts are variable even on shorter time scales and therefore [MATH] is even smaller.', 'astro-ph-0103192-2-21-7': 'This suggestion is confirmed later when we discuss the high resolution data (see sec. [REF]).', 'astro-ph-0103192-2-22-0': '## The effects of noise on the temporal structure', 'astro-ph-0103192-2-23-0': 'We turn now to the effect of noise on the time profile.', 'astro-ph-0103192-2-23-1': "We do so by comparing the attributes of the 'long' sample with these of the 'noisy long' sample (see [REF]).", 'astro-ph-0103192-2-23-2': 'This procedure also tests our algorithm.', 'astro-ph-0103192-2-23-3': "Since we know the original signal (in the 'long' sample) we can find out the efficiency of the algorithm in retrieving the attributes of the 'long' sample out of the 'noisy long' sample.", 'astro-ph-0103192-2-24-0': "Fig. [REF]a represents all the pulses in the 'long' and 'noisy long' samples.", 'astro-ph-0103192-2-24-1': 'The algorithm retrieves the basic features of the bursts out of the noisy sample.', 'astro-ph-0103192-2-24-2': "However, many pulses are 'lost' because of the noise (only 30% of the 'long' pulses are found in the 'noisy' sample): (i) Some pulses are too weak to be distinguished within the amplified noise.", 'astro-ph-0103192-2-24-3': '(ii) Some pulses merge with others as the minimum between them are not statistically significant with the increased noise.', 'astro-ph-0103192-2-24-4': 'The first effect does not affect the width of the pulses.', 'astro-ph-0103192-2-24-5': 'However, the second one causes pulse widening.', 'astro-ph-0103192-2-24-6': 'Therefore we expect fewer and wider pulses in the noisy sample.', 'astro-ph-0103192-2-24-7': 'Both effects are seen clearly in Fig. [REF].', 'astro-ph-0103192-2-24-8': "In the 'noisy' sample there are 203 pulses with an average width of 1.62sec while in the original 'long' sample there are 695 pulses with an average width of 1.39sec. The burst duration is affected by the noise as well, it becomes shorter.", 'astro-ph-0103192-2-24-9': 'This happens when the first or the last pulses of the burst are lost.', 'astro-ph-0103192-2-25-0': 'These two effects (pulse widening and shorter burst duration) tend to increase the value of [MATH] .', 'astro-ph-0103192-2-25-1': 'Fig. [REF]b show that [MATH] increases by a factor of 10 in average, due to the noise.', 'astro-ph-0103192-2-25-2': 'Thus the ratio [MATH] obtained from a noisy data can be considered as an upper limit to the real ratio.', 'astro-ph-0103192-2-26-0': '## Attributes of short bursts pulses', 'astro-ph-0103192-2-27-0': "We have applied the same algorithm to the 'short' data sample.", 'astro-ph-0103192-2-27-1': 'The pulses widths are shown as a function of the bursts duration in Fig [REF] .', 'astro-ph-0103192-2-27-2': 'The gray areas are not allowed because of the resolution ( [MATH] ) or simply by ( [MATH] ).', 'astro-ph-0103192-2-27-3': 'Fig [REF]b depicts the distribution of the pulses width, [MATH] .', 'astro-ph-0103192-2-27-4': 'One can see that typical values of [MATH] are 50-100 ms with no significant correlation with T (provided we delete the smooth single peaked bursts with [MATH]).', 'astro-ph-0103192-2-28-0': 'The pulse width found in this analysis is influenced by several effects.', 'astro-ph-0103192-2-28-1': 'Some of this effects are due to our algorithm: First, few relatively close pulses could be seen by the algorithm as a single wide pulse.', 'astro-ph-0103192-2-28-2': 'Second, the width of two pulses that are not well separated is determined by the minimum between the pulses.', 'astro-ph-0103192-2-28-3': 'In this case the measured width of both pulses is shorter then their actual width.', 'astro-ph-0103192-2-29-0': 'There are also observational effects that influence the pulse width: First, as indicated in section [REF], the pulses become wider by a factor of few because of the noise.', 'astro-ph-0103192-2-29-1': 'Second, the resolution is limited.', 'astro-ph-0103192-2-29-2': "It is likely that the shortest pulses in the 'short' sample are shorter then the best resolution of our data.", 'astro-ph-0103192-2-29-3': 'Time scales shorter then the data resolution were already found in short bursts (Scargle, Norris Bonnel 1997).', 'astro-ph-0103192-2-30-0': 'All the effects described above except one cause pulse widening.', 'astro-ph-0103192-2-30-1': 'The exception is when two pulses overlap leading to a shortening of the estimated widths of both pulses.', 'astro-ph-0103192-2-30-2': 'However, this effect rarely happens in our short bursts analysis.', 'astro-ph-0103192-2-30-3': 'The S/N in this sample is very low, and the significance level we demand (4 [MATH] ) is almost at the signal height (see sec. [REF]).', 'astro-ph-0103192-2-30-4': 'Hence, a pulse determined by the algorithm is almost always well separated (otherwise the minimum between the pulse and its neighbor would be insignificant).', 'astro-ph-0103192-2-30-5': 'Specifically 61 pulses out of the 65 pulses found in our sample are well separated: the minimum between pulses, on both sides, is lower than half of the maximum of the pulse.', 'astro-ph-0103192-2-30-6': 'Thus, pulse widths are almost never underestimated.', 'astro-ph-0103192-2-30-7': 'The combination of the other effects causes pulse widening.', 'astro-ph-0103192-2-30-8': 'Hence our estimate of [MATH] , is only an upper limit.', 'astro-ph-0103192-2-31-0': "Fig. [REF] shows [MATH] and [MATH] for both groups 'short' and 'noisy long'.", 'astro-ph-0103192-2-31-1': "In the 'short' sample the median [MATH] is 0.25 while 35% of bursts have [MATH] and 35% of the bursts show a smooth structure ( [MATH] ).", 'astro-ph-0103192-2-31-2': 'This result could mislead us to the conclusion that a significant fraction of the short bursts have a smooth time profile.', 'astro-ph-0103192-2-31-3': "But a look at the 'noisy long' results show that also in this group more than 20% of the bursts are single pulsed, while there were no such bursts in the original 'long' sample.", 'astro-ph-0103192-2-31-4': 'Naturally, the bursts that loose the fine structure because of the noise are bursts with fewer original pulses.', 'astro-ph-0103192-2-31-5': 'It is clear that short bursts have less pulses then long ones and therefore they are more "vulnerable" to this effect.', 'astro-ph-0103192-2-31-6': 'Hence we conclude that at least one third of short bursts are highly variable ( [MATH] ) and it is very likely that another third (those bursts with [MATH] )is variable as well.', 'astro-ph-0103192-2-31-7': 'We cannot tell whether the smooth structure ( [MATH] ) seen in a third of the short bursts is intrinsic or whether it arises due to the noise.', 'astro-ph-0103192-2-32-0': 'While we are mostly interested in this analysis in the pulse width, it is worth mentioning that the amplitude of the variations is large.', 'astro-ph-0103192-2-32-1': 'In most (61 out of 65) cases, when there is no overlap of nearby pulses the number of counts drops to less than half of the maximal counts.', 'astro-ph-0103192-2-32-2': 'Thus the pulse we describe are not only statistically significant, they also correspond to a significant (factor of 2) variation in the output of the source.', 'astro-ph-0103192-2-33-0': '## High resolution analysis of long bursts', 'astro-ph-0103192-2-34-0': 'We compare the time profile of the first seconds of 15 long bursts with the time profiles of 15 short bursts.', 'astro-ph-0103192-2-34-1': 'Fig. [REF] shows the light curves of a short burst and the first second of a long burst.', 'astro-ph-0103192-2-34-2': 'The time scales of both bursts are quite similar.', 'astro-ph-0103192-2-34-3': 'It is difficult to decide, on the base of these light curves alone, which one belongs to a short burst and which one is a fraction of a long one.', 'astro-ph-0103192-2-35-0': 'Fig. [REF] shows the pulse widths histograms of the beginning of long bursts and of the short bursts.', 'astro-ph-0103192-2-35-1': 'The time scales in both samples are quite similar in the range of 10-200ms.', 'astro-ph-0103192-2-35-2': 'The long bursts have additional pulses in the range of 0.2-1sec. Long bursts contain, of course, longer pulses, but in the sample we considered we demanded that the counts would fall back to the background level within the TTE data.', 'astro-ph-0103192-2-35-3': 'In this way we have limited the pulses width of the long bursts.', 'astro-ph-0103192-2-35-4': 'Both histograms begin at 10-20ms, which is at the limit of the pulse width resolution (10ms).', 'astro-ph-0103192-2-35-5': 'It is likely that both samples contain shorter time scales that cannot be resolved.', 'astro-ph-0103192-2-36-0': 'The counts variations within the pulses observed in the high resolution long bursts is high.', 'astro-ph-0103192-2-36-1': 'In 41 out of 46 pulses, the photon counts drops to less than half of the maximal counts of the pulse on both sides.', 'astro-ph-0103192-2-36-2': 'Therefore, like in short bursts, the statistically significant variations in the bursts reflects also a significant variation in the output of the source.', 'astro-ph-0103192-2-37-0': 'Walker, Schaefer Fenimore (2000) performed a similar analysis of TTE data of 14 long bursts.', 'astro-ph-0103192-2-37-1': 'They found only one long burst with very short time scales.', 'astro-ph-0103192-2-37-2': 'The difference between the results arises from the different samples considered.', 'astro-ph-0103192-2-37-3': 'Walker et al. (2000) considered the bursts with maximal total photon counts within the TTE burst record.', 'astro-ph-0103192-2-37-4': 'We demanded that the counts will return close to the background level within the TTE record.', 'astro-ph-0103192-2-37-5': 'Walker et al. (2000) criterion favors bursts that are active during the whole TTE record and therefore most of the bursts in their sample are dominated by a long and bright pulse.', 'astro-ph-0103192-2-38-0': 'This similarity between the short bursts and the fraction of long bursts raises the question whether it is possible that short bursts are actually only a small fraction, which is above the background noise, of long bursts.', 'astro-ph-0103192-2-38-1': 'We have already seen that the noise cause us to loose pulses.', 'astro-ph-0103192-2-38-2': 'Is it possible that a long burst with a single dominant, very intense pulse, (or a group of very close and intense pulses) will loose all its structure, apart for this intense pulse, due to noise and become a short burst.', 'astro-ph-0103192-2-38-3': 'Fig. [REF] rules out this option.', 'astro-ph-0103192-2-38-4': 'It depicts the counts ratio between the most intense pulse and the second most intense pulse within long bursts.', 'astro-ph-0103192-2-38-5': 'The graph shows that in 32 out of 34 bursts the second highest peak is more then third of the most intense peak and in 27 bursts the second peak is more than two thirds of the first.', 'astro-ph-0103192-2-38-6': 'The noise cannot cause one pulse to disappear without the other.', 'astro-ph-0103192-2-38-7': 'As these two peaks are usually separated by more than two seconds, the noise cannot convert a significant fraction of long bursts to short ones.', 'astro-ph-0103192-2-38-8': 'Clearly the noise has some effect on the duration histogram and in a few cases the added noise converted long bursts into a short ones, but it certainly cannot produce the observed bimodality.', 'astro-ph-0103192-2-39-0': '# Discussion', 'astro-ph-0103192-2-40-0': 'We have shown that most short bursts as well as most long bursts are multi-peaked and highly variable with [MATH] .', 'astro-ph-0103192-2-40-1': 'These statistically significant variations involve generally a change of more than a factor of two in the count rate.', 'astro-ph-0103192-2-40-2': '30% of the short bursts have a single pulse for which [MATH] is the same as the observed duration - [MATH] .', 'astro-ph-0103192-2-40-3': 'These are smooth bursts.', 'astro-ph-0103192-2-40-4': "However, a comparison with the 'noisy long' sample, shows that this might be an artifact of the low S/N in the short bursts sample.", 'astro-ph-0103192-2-40-5': 'We also find significant variability on very short time scale in long GRBs.', 'astro-ph-0103192-2-40-6': 'This reduces the values of the variability parameter [MATH] to [MATH] in long bursts.', 'astro-ph-0103192-2-41-0': 'External shocks (at least simple models of external shocks) cannot produce variable bursts (Sari Piran, 1997; Fenimore, Madras Nayakshin, 1996).', 'astro-ph-0103192-2-41-1': 'Our result suggests that most short bursts are produced via internal shocks.', 'astro-ph-0103192-2-41-2': '30% of the short bursts are smooth.', 'astro-ph-0103192-2-41-3': 'We cannot rule out the possibility that these bursts are produced by external shock.', 'astro-ph-0103192-2-41-4': 'The observed very short time scales in long GRBs strengthen further the argument in favor of internal shocks in these bursts and requires more contrived and fine tuned external shocks models for variability.', 'astro-ph-0103192-2-42-0': 'Kobayashi, Piran Sari (1997) have shown that internal shocks cannot convert all the relativistic kinetic energy to gamma-rays.', 'astro-ph-0103192-2-42-1': 'The remaining kinetic energy is dissipated latter when the relativistic ejecta is slowed down by the surrounding medium.', 'astro-ph-0103192-2-42-2': 'The resulting external shocks produce the afterglow.', 'astro-ph-0103192-2-42-3': 'An essential feature of this internal-external shocks scenario is that the afterglow is not a direct extrapolation of the initial [MATH]-ray emission.', 'astro-ph-0103192-2-42-4': 'This model suggests (Sari, 1997) for long bursts an overlap between the GRB and initial phase of afterglow.', 'astro-ph-0103192-2-42-5': 'This have been indeed observed in several cases as a build up of a softer component during the GRB and a corresponding transition in the spectrum of long GRBs to a soft x-ray dominated stage towards the end of the burst.', 'astro-ph-0103192-2-43-0': 'For short GRBs the internal-external scenario suggests that the afterglow will begin few dozen seconds after the end of a short burst that was produced by internal shocks (Sari, 1997).', 'astro-ph-0103192-2-43-1': 'It also suggest that this afterglow will not be a direct extrapolation of the GRB (as the two are produced by different mechanisms).', 'astro-ph-0103192-2-43-2': 'Thus we predict that for most short bursts there will be a gap between the burst and the beginning of the afterglow.', 'astro-ph-0103192-2-43-3': 'We have already remarked that at this stage one cannot tell whether the remaining 30% smooth short bursts are produced by external or by internal shocks.', 'astro-ph-0103192-2-43-4': 'This could be tested in the distant future with better data.', 'astro-ph-0103192-2-43-5': 'However, there are clear predictions concerning the afterglow that could distinguish between the two possibilities.', 'astro-ph-0103192-2-43-6': 'If these short GRBs are produced by internal shocks then we expect a clear gap between the end of the GRB and the onset of the x-ray afterglow.', 'astro-ph-0103192-2-43-7': 'If on the other hand these short bursts are produced by external shocks then we predict that the afterglow would continue immediately with no interpretation after the GRB and its features would be a direct extrapolation of the properties of the GRB.', 'astro-ph-0103192-2-44-0': '# Appendix -The algorithm', 'astro-ph-0103192-2-45-0': 'Our algorithm finds the peaks of the bursts.', 'astro-ph-0103192-2-45-1': 'Each peak corresponds to a single pulse; a pulse is the basic event of the light curve.', 'astro-ph-0103192-2-45-2': 'The algorithm is based on the algorithm suggested by Li Fenimore (1996).', 'astro-ph-0103192-2-45-3': 'Li Fenimore define a time bin [MATH] (with count [MATH] ) as a peak if there are two time bins [MATH] (with counts [MATH] respectively) which satisfies a) [MATH] and b) [MATH] is the maximal count between [MATH] and [MATH] .', 'astro-ph-0103192-2-45-4': '[MATH] is a parameter that determines the significance of the peak.', 'astro-ph-0103192-2-46-0': 'There are two problems with this algorithm.', 'astro-ph-0103192-2-46-1': 'First, this algorithm analyzes only data in a single time resolution (fixed time bin size).', 'astro-ph-0103192-2-46-2': 'Therefore the algorithm looses long and faint pulses.', 'astro-ph-0103192-2-46-3': 'A peak that does not satisfy the criterion described above in the raw data resolution could satisfy the criterion if the data resolution is lower (longer time-bins).', 'astro-ph-0103192-2-46-4': 'This algorithm would miss such a pulse.', 'astro-ph-0103192-2-46-5': 'Second, [MATH] determines the trade-off between sensitivity and false peaks identification rate.', 'astro-ph-0103192-2-46-6': 'When [MATH] is low the algorithm finds false peaks as a result of the Poisson noise.', 'astro-ph-0103192-2-46-7': 'When [MATH] is high the algorithm misses real peaks.', 'astro-ph-0103192-2-46-8': 'Finding false peaks is a severe problem during long periods of constant level Poisson noise, like the background (as will be explained shortly).', 'astro-ph-0103192-2-46-9': 'Long bursts contain such periods (periods of only background noise).', 'astro-ph-0103192-2-46-10': 'These periods are called quiescent times.', 'astro-ph-0103192-2-46-11': 'In order to avoid false peaks in long bursts [MATH] must be large( [MATH] ), which means an insensitive algorithm.', 'astro-ph-0103192-2-46-12': 'Short bursts contain less quiescent times, and of course shorter ones.', 'astro-ph-0103192-2-46-13': 'But, short bursts contain much less pulses then long burst (three to four compared to an average of more than thirty) and much smaller S/N.', 'astro-ph-0103192-2-46-14': 'Finding even one false peak could change the features of the burst drastically.', 'astro-ph-0103192-2-46-15': 'Too insensitive algorithm could loose all the burst structure.', 'astro-ph-0103192-2-46-16': 'In order to avoid false peaks in short bursts [MATH] must be at least as large as 5.', 'astro-ph-0103192-2-46-17': 'As described in section [REF], the S/N in some of the bright short bursts is smaller than 5.', 'astro-ph-0103192-2-46-18': 'Such [MATH] will prevent the algorithm from finding even one peak in these bursts.', 'astro-ph-0103192-2-47-0': 'We solved the first problem by analyzing the data in different resolutions.', 'astro-ph-0103192-2-47-1': 'The results of the algorithm in different resolutions are merged into a single sample of peaks.', 'astro-ph-0103192-2-47-2': "We solved the second problem by restricting the search for peaks only to 'Active Periods'.", 'astro-ph-0103192-2-47-3': 'Active periods are periods with counts that correspond to source activity (we will define it later on).', 'astro-ph-0103192-2-48-0': 'There are few advantages for analyzing only active periods.', 'astro-ph-0103192-2-48-1': 'The main one is that a lower [MATH] can be used during these periods with smaller risk of finding false peaks.', 'astro-ph-0103192-2-48-2': 'The risk of finding false peaks due to a Poisson noise depends on the time scale in which the original signal (i.e. without the noise) changes its count rate in the same order as the Poisson noise level.', 'astro-ph-0103192-2-48-3': 'If the signal is constant (no real pulses), then this time scale is as long as the signal.', 'astro-ph-0103192-2-48-4': 'If the constant signal is longer, it contains more time bins with the same level of Poisson noise counts.', 'astro-ph-0103192-2-48-5': 'Hence, there is a larger chance of finding within these time-bins three time-bins, [MATH] , [MATH] and [MATH] , that satisfy the criterion in Li Fenimore algorithm.', 'astro-ph-0103192-2-48-6': 'Then [MATH] would become a false peak.', 'astro-ph-0103192-2-48-7': 'On the other hand, if the signal is changing monotonically then the length of the signal is irrelevant.', 'astro-ph-0103192-2-48-8': '[MATH] , [MATH] and [MATH] must be within the period in which the signal changes at the same order as the Poisson noise level; there is no chance of finding [MATH] with [MATH] significantly below [MATH] (if the signal is rising) out of this period.', 'astro-ph-0103192-2-48-9': 'During the active periods the signal is changing rapidly (usually on time scales of seconds or less), and the Poisson noise is superimposed on steep slopes.', 'astro-ph-0103192-2-48-10': "In this case a false peak could only be found during the period in which the signal didn't change compared to the noise level.", 'astro-ph-0103192-2-48-11': 'There are much less time bins during this period and hence there are much less chance of finding false peaks.', 'astro-ph-0103192-2-49-0': 'The second advantage is that when an active period is found we almost certain that it is a part of the burst.', 'astro-ph-0103192-2-49-1': "This is important since one false peak in the 'wrong' place (for example hundred of seconds after the burst ended) can change the burst properties drastically.", 'astro-ph-0103192-2-49-2': 'By analyzing only active periods we can use smaller [MATH] (=4) and get a more sensitive and accurate algorithm.', 'astro-ph-0103192-2-50-0': 'Our algorithm works in several steps.', 'astro-ph-0103192-2-50-1': 'First, it determines the background level of the signal (as a function of time).', 'astro-ph-0103192-2-50-2': "Then it finds an activity level, demanding a probability of 0.9 (per burst) that all the time bins with counts above this level (called 'active bins') correspond to source activity and not of the background Poisson noise (we demand that on every ten bursts there is, on average, a single false 'active bin').", 'astro-ph-0103192-2-50-3': 'The activity level depends on the background and its value is between 4 [MATH] to 5 [MATH] above the background.', 'astro-ph-0103192-2-50-4': 'From each active bin we search to the right and to the left until the count level drops to the background level on both sides.', 'astro-ph-0103192-2-50-5': 'We call all these bins together an active period (from the time bin that the counts are above the background until the time bin that the counts reaches the background level again).', 'astro-ph-0103192-2-50-6': 'In most cases a single active period includes many active bins and a burst may contain more then a single active period (see Fig [REF]).', 'astro-ph-0103192-2-50-7': 'Note that if the algorithm misses an active period in one resolution, it can still find it in a different (lower) resolution, in which the noise level is lower.', 'astro-ph-0103192-2-51-0': 'Once the active periods of a given burst have been determined we apply the Li Fenimore algorithm to the active periods (using Nvar=4) and determine the peaks.', 'astro-ph-0103192-2-51-1': 'We repeat this procedure (finding the active periods and the corresponding peaks), several times for different time resolution.', 'astro-ph-0103192-2-51-2': 'To obtain lower resolution data we convolve the original signal (in the basic time bins) with a Gaussian, whose width determines the resolution.', 'astro-ph-0103192-2-51-3': 'Finally, after finding the peaks in different resolutions we merge those samples of peaks to a single sample (requiring that a peak must appear in at least two different resolutions).', 'astro-ph-0103192-2-51-4': 'The merge is done by merging the highest resolution sample with the second highest one and then taking this merged sample and merging it with the third highest resolution sample and so on.', 'astro-ph-0103192-2-51-5': 'On different resolutions the same peak could be found on different time bins.', 'astro-ph-0103192-2-51-6': 'In each case two peaks on different resolutions are considered as a single one if the peak in one resolution falls between [MATH] and [MATH] of the peak in the other resolution.', 'astro-ph-0103192-2-52-0': 'Each peak corresponds, of course, to a pulse.', 'astro-ph-0103192-2-52-1': 'The pulse width ( [MATH] ) is defined by two points (on each side of the peak) that are higher than the background by 1/4 of the peaks height or by the minimum between two neighboring peaks (if the latter is higher).', 'astro-ph-0103192-2-52-2': 'The duration ( [MATH] ) of the burst is the time elapsed from the beginning of the first pulse till the end of the last pulse (so in single pulsed burst [MATH] ).'}

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[]

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[['astro-ph-0103192-1-25-2', 'astro-ph-0103192-2-24-9']]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0103192

1105.4605

{'1105.4605-1-0-0': 'We use the IRAM HERACLES survey to study CO emission from 33 nearby spiral galaxies down to very low intensities.', '1105.4605-1-0-1': 'Using 21-cm line atomic hydrogen (H[MATH]i) data, mostly from THINGS, we predict the local mean CO velocity based on the mean H[MATH]i velocity.', '1105.4605-1-0-2': 'By re-normalizing the CO velocity axis so that zero corresponds to the local mean H[MATH]i velocity we are able to stack spectra coherently over large regions.', '1105.4605-1-0-3': 'This enables us to measure CO intensities with high significance as low as [MATH] K km s[MATH] M[MATH] pc[MATH]), an improvement of about one order of magnitude over previous studies.', '1105.4605-1-0-4': 'We detect CO out to galactocentric radii [MATH] and find the CO radial profile to follow a remarkably uniform exponential decline with scale length of [MATH] km s[MATH] (Figure [REF] upper right panel).', '1105.4605-1-0-5': 'However, in the central regions of some galaxies the line can be very broad with a flattened or double-horned peak (Figure [REF] upper left panel).', '1105.4605-1-0-6': 'These profiles often coincide with central enhancements like bars or molecular rings and are poorly parametrized by a single Gaussian.', '1105.4605-1-0-7': 'Instead we fit a double-horn profile, a Gaussian scaled by a symmetric second-order polynomial .', '1105.4605-1-0-8': 'Based on "by eye" inspection, the asymmetry in these profiles is small enough that a symmetric function is sufficient.', '1105.4605-1-1-0': 'We derive the best fit profile via a non-linear least squares fit.', '1105.4605-1-1-1': 'We constrain the fit parameters so that the center of the profile lies within [MATH]50 km s[MATH] of zero velocity after shifting, the FWHM is larger than 15 km s[MATH] (to avoid the fit latching onto individual channels), and the amplitude is positive.', '1105.4605-1-1-2': 'We always carry out a Gaussian fit first and in those cases where the FWHM exceeds 60 km s[MATH] we switch to a double-horn profile.', '1105.4605-1-1-3': 'We verify by eye that this yields sensible results.', '1105.4605-1-2-0': 'The integral of the fitted profile gives us the integrated CO line intensity.', '1105.4605-1-2-1': 'We derive the uncertainty in this quantity from the noise, estimated from the signal-free part of the spectrum, and the width of the profile.', '1105.4605-1-2-2': 'It proved useful to define a quality scale for the fit.', '1105.4605-1-2-3': 'The quality is "high" where the peak intensity is larger than [MATH] and its integrated intensity is larger than [MATH] times its uncertainty.', '1105.4605-1-2-4': 'In cases where the peak intensity is less than [MATH] or the integrated intensity is less than [MATH] times its uncertainty, we do not trust the fit and instead determine an upper limit.', '1105.4605-1-2-5': 'We label cases that fall between these regimes as "marginal".', '1105.4605-1-2-6': 'Figure [REF] shows examples of our quality measures and line profile fits.', '1105.4605-1-3-0': 'We derive our upper limits integrating over a Gaussian line profile with FWHM set to 18 km s[MATH] , the typical FWHM found for high SNR spectra at [MATH], and fixed amplitude of 3[MATH].', '1105.4605-1-4-0': '## Stacking as a Function of Radius', '1105.4605-1-5-0': 'We present our stacking technique applied to radial bins.', '1105.4605-1-5-1': 'In principle this method allows us to stack spectra across any region.', '1105.4605-1-5-2': 'For example, we could define regions by total gas column, infrared intensity, or features such as spiral arms and bars.', '1105.4605-1-5-3': 'In practice, radius makes an excellent ordinate.', '1105.4605-1-5-4': 'We wish to study the underlying relationship between CO, H[MATH]i, IR, FUV, and H[MATH] intensity.', '1105.4605-1-5-5': 'Stacking with one of these quantities as the ordinate would require carefully modeling the biases involved to measure the underlying relationships.', '1105.4605-1-5-6': 'Galactocentric radius is a well-determined, independent quantity that is also highly covariant with these other intensities.', '1105.4605-1-5-7': 'This yields a dataset with large dynamic range that is easy to interpret.', '1105.4605-1-6-0': 'Therefore we focus our analysis on data stacked in bins of galactocentric radius.', '1105.4605-1-6-1': 'We average over tilted rings 15 wide, comparable to the angular resolution of our data.', '1105.4605-1-6-2': 'This width corresponds to [MATH] pc for our most distant targets (25 Mpc).', '1105.4605-1-6-3': 'We construct the rings assuming that each galaxy is a thin disk with the inclination and position angle given in Table [REF].', '1105.4605-1-6-4': 'To measure CO with highest sensitivity we construct stacked CO spectra using the procedure described above and fit those to determine the integrated CO line intensities.', '1105.4605-1-6-5': 'For the other observables - H[MATH]i, IR, FUV, and H[MATH] intensities - we use two-dimensional maps of intensity and determine the mean intensity for each tilted ring.', '1105.4605-1-7-0': 'Error bars on the H[MATH]i, IR, FUV, and H[MATH] intensities show the [MATH] scatter within that tilted ring, capturing both statistical noise and deviations from axial symmetry.', '1105.4605-1-7-1': 'We estimate the [MATH] scatter for our CO measurements by integrating the CO cube over a velocity window that is adjusted for each line of sight such that it includes all channels of significant CO emission but at least all channels with velocities within 25 km s[MATH] of the local mean H[MATH]i velocity.', '1105.4605-1-7-2': 'Note that our [MATH] values reflect the scatter in (integrated) intensities of individual lines of sight inside a ring.', '1105.4605-1-7-3': 'They should not be confused with the uncertainty in the determination of the mean intensity inside a ring which is typically much smaller.', '1105.4605-1-8-0': '# Results', '1105.4605-1-9-0': 'In Figure [REF] for NGC 628 and in the Appendix for the rest of the sample, we present stacked radial profiles of integrated CO intensity along with profiles of H[MATH]i , infrared intensity at 24 and 70 , FUV, and H[MATH] intensity.', '1105.4605-1-9-1': 'Following Equations [REF] we combine H[MATH] and 24 intensities to estimate the star formation rate surface density SFR(H[MATH][MATH]24) and compare those to SFR(FUV[MATH]24) derived from FUV and 24 intensities using Equation [REF].', '1105.4605-1-10-0': 'For each galaxy there are two plots: The left panel shows H[MATH]i , H[MATH] (from CO), and SFR for both FUV[MATH]24 and H[MATH][MATH]24.', '1105.4605-1-10-1': 'The right panel shows our SFR tracers - H[MATH] , FUV, 24 and 70 emission.', '1105.4605-1-10-2': 'We present the profiles in both observed intensity (left hand [MATH]-axis) and units of surface density (right hand [MATH]-axis of the left panel) - [MATH], [MATH], and [MATH].', '1105.4605-1-10-3': 'Note that we have projected (only) the surface densities of H[MATH]i , H[MATH] , and SFR to face-on values (i.e., we corrected for inclination).', '1105.4605-1-10-4': 'The observed surface brightnesses are not corrected for the effect of inclination.', '1105.4605-1-10-5': 'Such a correction will just move each galaxy up and down in lockstep and we find it more useful to report the observed values.', '1105.4605-1-10-6': 'The color of a point in the CO profile indicates the significance of the fit to the stacked spectrum: green for high significance, orange for marginal significance, and red for upper limits, these correspond to [MATH] upper limits on the fitted intensity (see Section [REF]).', '1105.4605-1-11-0': '## CO and Star Formation', '1105.4605-1-12-0': 'With these azimuthally averaged data we are able to compare CO to tracers of recent star formation across a large range of H[MATH]-to-H[MATH]i ratios.', '1105.4605-1-12-1': 'In this subsection, we make empirical comparisons between measured intensities, examine the relative roles of H[MATH] and H[MATH]i in the "star formation law," and investigate the origin of the scatter in these relations.', '1105.4605-1-13-0': '### Scaling Relations between CO and IR, FUV, and H[MATH]', '1105.4605-1-14-0': 'Figures [REF] [REF] show scaling relations between observed intensities of CO ([MATH]-axis) and different tracers of recent star formation ([MATH]-axis).', '1105.4605-1-14-1': 'Figure [REF] shows infrared intensities at 24 (top panels) and 70 (bottom panels) and Figure [REF] shows intensities of FUV (top panels) and H[MATH] (bottom panels).', '1105.4605-1-14-2': 'The left hand panels show the relations for all galaxies and all radii.', '1105.4605-1-14-3': 'The panels on the right hand side show only radii [MATH].', '1105.4605-1-14-4': 'H[MATH] and H[MATH]i make up roughly equal parts of the ISM near this radius (see the left panel of Figure [REF]), so most of the points in the right hand panels are H[MATH]i -dominated.', '1105.4605-1-14-5': 'A dotted vertical line shows an integrated CO intensity of [MATH] K km s[MATH] , which corresponds to [MATH] M[MATH] pc[MATH] (assuming [MATH]), which is about the surface density at which H[MATH] and H[MATH]i make up equal parts of the ISM.', '1105.4605-1-14-6': 'A dashed vertical line at [MATH] K km s[MATH] M[MATH] pc[MATH] for [MATH]) shows a conservative sensitivity limit for the whole sample.', '1105.4605-1-14-7': 'Typically our upper limits, which are not displayed in these plots, lie to the left of this line.', '1105.4605-1-14-8': 'They will be systematically higher at large radii when radial rings partially exceed the coverage of our CO maps.', '1105.4605-1-15-0': 'CO emission correlates tightly with IR emission at 24 and 70 (Figure [REF]) (rank correlation coefficient [MATH]).', '1105.4605-1-15-1': 'The correlation extends over three orders of magnitude in CO and IR intensities and crosses the H[MATH]i -to-H[MATH] transition without substantial change in slope or normalization.', '1105.4605-1-15-2': 'Comparing the left panels (all radii) and the right panels ([MATH]) does not reveal any significant radial dependence.', '1105.4605-1-16-0': 'CO emission exhibits a weaker correlation with FUV and H[MATH] (Figure [REF]) ([MATH]) than with IR emission, i.e., both the CO-FUV and CO-H[MATH] relation show much larger scatter than the CO-IR relations.', '1105.4605-1-16-1': 'The CO-FUV relation displays a break between H[MATH]i- and H[MATH]-dominated regimes.', '1105.4605-1-16-2': 'The increased scatter and weaker correlation at least partially reflects the sensitivity of H[MATH] and FUV emission to absorption by dust.', '1105.4605-1-16-3': 'In the inner ([MATH]), more gas rich parts of galaxies dust reprocesses most H[MATH] and FUV emission into IR emission.', '1105.4605-1-16-4': 'In this regime CO and FUV are to first order uncorrelated ([MATH]).', '1105.4605-1-16-5': 'Outside [MATH] Gyr (including a factor [MATH] to account for heavy elements; see Table [REF] for fit parameters).', '1105.4605-1-16-6': 'This is slightly lower than [MATH] Gyr found by [CITATION] and [CITATION] for a subset of the data analyzed here and [MATH] Gyr recently found by [CITATION] for a sample that is similar to the one analyzed here.', '1105.4605-1-16-7': 'We include (a) more starburst galaxies and (b) more low mass, low metallicity spiral galaxies that [CITATION] and [CITATION] excluded from their CO analysis.', '1105.4605-1-16-8': 'Both dwarfs and starbursts have shorter [MATH] than large spirals .', '1105.4605-1-16-9': 'Moreover, our radial profiles weight these small galaxies more heavily than the pixel sampling used by [CITATION].', '1105.4605-1-17-0': 'As with the scaling relations between CO and tracers of recent star formation (Figure [REF] [REF]), the H[MATH]-SFR relation exhibits significant scatter.', '1105.4605-1-17-1': 'We perform the same procedure to isolate galaxy-to-galaxy variations from scatter within galaxies and again find the scatter in the main relation (upper panels in Figure [REF] [REF]) dominated by galaxy-to-galaxy variations.', '1105.4605-1-17-2': 'We plot the relations after normalization in [MATH] and [MATH] in the middle and bottom panels of Figure [REF] [REF].', '1105.4605-1-17-3': 'Once galaxy-to-galaxy scatter is removed, there is a remarkably tight, uniform linear relation linking molecular gas and star formation across almost three and a half orders of magnitude.', '1105.4605-1-18-0': 'What drives this galaxy-to-galaxy variation?', '1105.4605-1-18-1': 'In both the H[MATH]-SFR and observed intensity relations a large part of the scatter comes from a sub-population of less massive, less metal-rich galaxies that exhibit high SFR-to-CO ratios.', '1105.4605-1-18-2': 'Figure [REF] shows the metallicity dependence of the molecular depletion time, [MATH].', '1105.4605-1-18-3': 'Massive spiral galaxies with high metallicities have considerably longer (median averaged) depletion times: [MATH] Gyr for [MATH], while smaller galaxies with lower metallicities have systematically shorter depletion times: [MATH]24) vs. Gas Phase H[MATH]i & &', '1105.4605-1-19-0': '... all data & [MATH]a &', '1105.4605-1-20-0': '... H[MATH]i-dominated & [MATH] &', '1105.4605-1-21-0': '... H[MATH]-dominated & [MATH] &', '1105.4605-1-22-0': 'H[MATH] & &', '1105.4605-1-23-0': '... all data & [MATH] & [MATH]a', '1105.4605-1-24-0': '... H[MATH]i-dominated & [MATH] & [MATH]', '1105.4605-1-25-0': '... H[MATH]-dominated & [MATH] & [MATH]', '1105.4605-1-26-0': 'H[MATH]i+H[MATH] & &', '1105.4605-1-27-0': '... all data & [MATH] & [MATH]', '1105.4605-1-28-0': '... H[MATH]i-dominated & [MATH] & [MATH]', '1105.4605-1-29-0': '... H[MATH]-dominated & [MATH] & [MATH] aEstimation of uncertainties equal to Table [REF].', '1105.4605-1-30-0': 'The question of which gas component - H[MATH]i , H[MATH] , or total gas - correlates best with recent star formation has received significant attention.', '1105.4605-1-30-1': 'Phrased this way, the question is not particularly well posed: the total gas surface density and the molecular gas fraction are closely related so that the different gas surface densities are not independent quantities.', '1105.4605-1-30-2': 'Therefore we do not necessarily expect a "best" correlation, only different functional forms.', '1105.4605-1-30-3': 'Still, it is illustrative to see how recent star formation relates to each gas tracer.', '1105.4605-1-30-4': 'Table [REF] lists [MATH] and the power law index from an OLS bisector fit between each component and Figure [REF] shows plots for each of the three gas phases and our two SFR prescriptions.', '1105.4605-1-30-5': 'The rank correlation coefficient and the power law fits are determined for regions where we have at least a marginal CO measurement.', '1105.4605-1-30-6': 'If we restrict our analysis to high significance CO measurements, we obtain the same results within the uncertainties.', '1105.4605-1-31-0': 'The table and figure show that recent star formation rate tracers rank-correlate approximately equally well with H[MATH] and H[MATH]i+H[MATH] both across all surface densities and separately in the H[MATH]i- and H[MATH]-dominated regimes.', '1105.4605-1-31-1': 'H[MATH]i does not correlate significantly with star formation in the H[MATH]-dominated inner parts of galaxy disks, though the correlation between H[MATH]i and recent star formation becomes stronger in the H[MATH]i-dominated outer parts of galaxies .', '1105.4605-1-32-0': 'The rank correlation is a non-parametric measure of how well the relative ordering of two data sets align.', '1105.4605-1-32-1': 'A high rank correlation coefficient implies a monotonic relationship but not a fixed functional form law.', '1105.4605-1-32-2': 'For total gas, the power law index relating gas and recent star formation depends fairly strongly on the subset of data used.', '1105.4605-1-32-3': 'If we focus on the regions where [MATH], the best-fit power law relating total gas and recent star formation (from FUV[MATH]24) has an index of [MATH].', '1105.4605-1-32-4': 'Where [MATH], the index is much shallower, [MATH].', '1105.4605-1-32-5': 'By contrast, the power law relating H[MATH] to recent star formation varies less across regimes, from [MATH] where [MATH] to [MATH] where [MATH].', '1105.4605-1-32-6': 'Both of these agree within the uncertainties with the fit to all data, which has slope [MATH].', '1105.4605-1-32-7': 'The slight steepening of the relation in the outer disks ([MATH]) is driven by the interplay of two effects: the rather small dynamic range in gas surface densities and an increased dispersion in SFR-to-H[MATH] ratios driven by the low mass galaxies; those have high SFR-to-H[MATH] ratios and contribute mostly to the H[MATH]i-dominated subset.', '1105.4605-1-32-8': 'The qualitative picture of a break in the total gas-SFR relation but a continuous H[MATH]-SFR relation remains unchanged if SFR(H[MATH][MATH]24) is considered, though there are small changes in the exact numbers (see the left and right panels in Figure [REF]).', '1105.4605-1-33-0': 'Our conclusions thus match those of [CITATION]: a single power law appears to be sufficient to relate [MATH] and [MATH] whereas the relationship between [MATH] and [MATH] varies systematically depending on the subset of data used.', '1105.4605-1-33-1': 'Because we have a dataset that includes significant CO measurements where [MATH] we can extend these findings.', '1105.4605-1-33-2': 'First, total gas and H[MATH] are equally well rank-correlated with recent star formation in all regimes and this correlation is always stronger than the correlation of recent star formation with H[MATH]i .', '1105.4605-1-33-3': 'Second, the H[MATH]-SFR scaling relation extends smoothly into the regime where [MATH] whereas the total gas-SFR relation does not.', '1105.4605-1-33-4': 'Third, the result is independent of our two star formation rate tracers SFR(FUV[MATH]24) and SFR(H[MATH][MATH]24).', '1105.4605-1-34-0': '### Discussion of CO-SFR Scaling Relations', '1105.4605-1-35-0': 'Empirical Results: IR brightness at both 24 and 70 correlates strongly with CO intensity over [MATH]3 orders of magnitude.', '1105.4605-1-35-1': 'Across this range, there is a nearly fixed ratio of CO to IR emission.', '1105.4605-1-35-2': 'FUV and H[MATH] emission show little or no correlation with CO intensity in the inner parts of galaxies, presumably due to extinction, but are found to correlate well with CO in the outer parts of galaxies after galaxy-to-galaxy scatter is removed.', '1105.4605-1-36-0': 'A result of these empirical scaling relations is that the ratio of recent star formation rate, traced either by combining FUV and 24 or H[MATH] and 24 intensities, to molecular gas, traced by CO emission, does not vary strongly between the H[MATH]i-dominated and H[MATH]-dominated ISM.', '1105.4605-1-36-1': 'This result is driven largely by the tight observed correlation between CO and 24 emission.', '1105.4605-1-36-2': 'The tight relation between CO and 70 emission suggests that the CO-IR relation actually holds for a larger range of mid-IR intensities.', '1105.4605-1-36-3': 'The tightening of the CO-FUV and CO-H[MATH] relation in the outer part of galaxies (after removing galaxy-to-galaxy scatter) reinforce the idea of a linear relation extending to large radii.', '1105.4605-1-36-4': 'These tight correlations are consistent with the conclusion of [CITATION] that there is only weak variation in the SFR per unit molecular gas mass with local environment.', '1105.4605-1-37-0': 'Galaxy-to-Galaxy Scatter: Each of the correlations we observe has significant internal scatter.', '1105.4605-1-37-1': 'Breaking this apart into scatter among galaxies and scatter within galaxies, we observe that in every case scatter among galaxies drives the overall scatter in the observed correlation.', '1105.4605-1-37-2': 'This "scatter" among galaxies is not random; less massive, less metal-rich galaxies exhibit a higher ratio of SFR tracer to CO emission and][]Young1996.', '1105.4605-1-37-3': 'There are two straightforward physical interpretations for this.', '1105.4605-1-37-4': 'The efficiency of star formation from H[MATH] gas may be genuinely higher in these systems, a view advocated by [CITATION] and [CITATION].', '1105.4605-1-37-5': 'Alternatively, CO emission may be depressed relative to the true amount of H[MATH] mass due to changes in the dust abundance.', '1105.4605-1-37-6': 'Low mass systems often have lower metallicities and correspondingly less dust, which is required to shield CO .', '1105.4605-1-37-7': 'A precise calibration of the CO-to-H[MATH] conversion factor as a function of metallicity is still lacking, so it is not possible at present to robustly distinguish between these two scenarios.', '1105.4605-1-38-0': 'After removing these galaxy-to-galaxy variations we find a series of extraordinarily tight relationships between CO and tracers of recent star formation.', '1105.4605-1-38-1': 'The most striking - and puzzling - example of this is the emergence of a tight correlation between CO and FUV emission in the outer parts of galaxies ([MATH]).', '1105.4605-1-38-2': 'This is puzzling because one would expect the galaxy to be mostly causally disconnected over the timescales predominantly traced by FUV emission - [MATH] Myr compared to a dynamical (orbital) time of a few [MATH] Myr.', '1105.4605-1-38-3': 'Yet, somehow the differences between galaxies affect the CO-to-FUV ratio much more than the differences between the widely separated rings represented by our data points.', '1105.4605-1-38-4': 'Galaxy-wide variations in metallicity and dust abundance probably offer the best explanation for this.', '1105.4605-1-38-5': 'These may propagate into variations in the CO-to-H[MATH] conversion factor and the average dust extinction.', '1105.4605-1-38-6': 'The latter may lead to scatter in estimates of SFR and both will affect [MATH].', '1105.4605-1-38-7': 'An alternative explanation is that external processes, which affect the whole galaxy, play a large role in setting the star formation rate on timescales traced by FUV emission.', '1105.4605-1-39-0': 'These strong galaxy-to-galaxy variations partially explain the unexpected lack of correlation between CO emission and recent SFR observed by [CITATION].', '1105.4605-1-39-1': 'They averaged across whole galaxies and in doing so conceivably lost the strong internal relations that we observe but preserved the large galaxy-to-galaxy variations that offset internal relations.', '1105.4605-1-39-2': 'The result will be an apparent lack of correlation in galaxy-averaged data that obscures the strong internal relationship.', '1105.4605-1-39-3': 'Whether there is in fact a weaker relationship between H[MATH] and SFR in galaxy-integrated measurements than inside galaxies depends on whether the suggested variations in star formation efficiency are real or a product of a varying CO-to-H[MATH] conversion factor.', '1105.4605-1-40-0': '(The Lack of) A Molecular Star Formation Law: With improved sensitivity to CO emission we now clearly see nearly linear relations between CO and tracers of recent star formation rate spanning from the H[MATH]-dominated to H[MATH]i-dominated parts of galaxies.', '1105.4605-1-40-1': 'Note that the relations will likely depart from these scaling relations if regions of high surface densities ([MATH] M[MATH] pc[MATH]) or starburst galaxies are considered .', '1105.4605-1-40-2': 'The lack of strong variations in the scaling between these two quantities in the "non-starburst regime" reinforces that molecular gas is the key prerequisite for star formation.', '1105.4605-1-40-3': 'A nearly linear correlation over this whole range can also be restated as the absence of a strong relationship between the ratio [MATH] and [MATH].', '1105.4605-1-40-4': 'This implies that [MATH] averaged over a large area is not a key environmental quantity for star formation because it does not affect the rate of star formation per unit molecular gas.', '1105.4605-1-40-5': 'Apparently the global amount of H[MATH] directly sets the global amount of star formation but the surface density of H[MATH] does not affect how quickly molecular gas is converted to stars.', '1105.4605-1-41-0': 'By contrast, the host galaxy does appear to affect the ratio of star formation rate to at least CO intensity.', '1105.4605-1-41-1': 'This indicates important environmental variations but they are not closely linked to surface density.', '1105.4605-1-41-2': 'In this sense, Figure [REF] [REF] offer a counterargument against the idea of a star formation "law" in which gas surface density alone sets the star formation rate.', '1105.4605-1-41-3': 'Instead, over the disks of normal galaxies, we see star formation governed by two processes: (a) the formation of stars in molecular gas, which varies mildly from galaxy to galaxy but appears largely fixed inside a galaxy, and (b) the conversion of H[MATH]i to H[MATH], which does exhibit a strong dependence on environment inside a galaxy, including a strong dependence on surface density (Section [REF]).', '1105.4605-1-41-4': 'In the second part of this paper we will look at this second process by measuring variations in the H[MATH] -H[MATH]i balance as a function of gas surface density and radius.', '1105.4605-1-42-0': 'Systematic Effects: We have interpreted the observed scaling relations in terms of a relationship between molecular gas and recent star formation rate.', '1105.4605-1-42-1': '[CITATION] demonstrated that azimuthally averaged profiles of the FUV[MATH]24 combination that we use here match those of several commonly used star formation rate tracers with [MATH]50% scatter.', '1105.4605-1-42-2': 'Here we have shown that using a combination of H[MATH][MATH]24 to determine the SFR leads to indistinguishable results (see Leroy et al. 2011, in prep.', '1105.4605-1-42-3': 'for more discussion).', '1105.4605-1-42-4': 'We now discuss several systematic effects that may affect the translation from observables to inferred quantities.', '1105.4605-1-43-0': 'The most serious worry is that the IR intensity, which drives the correlations, is acting as a tracer of dust abundance and not recent star formation.', '1105.4605-1-43-1': 'Gas and dust are observed to be well mixed in the ISM, so in the extreme, this would result in plotting gas against gas times some scaling factor (the dust-to-gas ratio).', '1105.4605-1-43-2': 'A more subtle version of the same concern is that CO emission is primarily a function of dust shielding against dissociating UV radiation.', '1105.4605-1-43-3': 'If there are large variations in the abundance of dust in the ISM then it may be likely that dust emission and CO emission emerge from the same regions because that is where CO can form and evade dissociation.', '1105.4605-1-44-0': 'A few considerations suggest that the 24 and the 70 emission are not primarily tracing dust abundance.', '1105.4605-1-44-1': 'First, over whole galaxies, monochromatic IR emission at 24 and 70 does track the SFR .', '1105.4605-1-44-2': 'Second, we observe a linear correlation with CO emission and not with overall gas column, which one might expect for a dust tracer.', '1105.4605-1-44-3': 'Third, both the 24 and 70 bands are well towards the blue side of the peak of the IR SED for dust mixed with non star-forming gas and so are not likely to be direct tracers of the dust optical depth (mass).', '1105.4605-1-44-4': 'Still, a thorough investigation of the interplay between dust abundance, IR emission, and star formation is needed to place SFR tracers involving IR emission on firmer physical footing.', '1105.4605-1-45-0': 'A less severe worry is that using dust and FUV emission makes us sensitive to an old stellar component that might not have formed locally.', '1105.4605-1-45-1': 'Our targets are all actively star-forming systems, so old here means mainly old relative to H[MATH] emission .', '1105.4605-1-45-2': 'The appropriate timescale to use when relating star formation and gas is ambiguous.', '1105.4605-1-45-3': 'When studying an individual region, it may be desirable to use a tracer with the shortest possible time sensitivity.', '1105.4605-1-45-4': 'Averaging over large parts of galaxies, one is implicitly trying to get at the equilibrium relation.', '1105.4605-1-45-5': 'Therefore a tracer with a somewhat longer timescale sensitivity may actually be desirable.', '1105.4605-1-45-6': 'The typical [MATH] Myr timescale over which most UV emission (and dust heating from B stars) occurs is well matched to current estimates for the lifetimes of giant molecular clouds .', '1105.4605-1-45-7': 'This makes for a fairly symmetric measurement - with the spatial and time scales of the two axes matched - though one is comparing recent star formation with the material of future star-forming regions.', '1105.4605-1-46-0': 'The fact that SFRs derived from FUV[MATH]24 and H[MATH][MATH]24 are essentially indistinguishable indicates that the distinction between the time scale probed by H[MATH]4 Myr) and FUV ([MATH] Myr) is not important to this study, probably because of the large spatial scales considered by our azimuthal averages.', '1105.4605-1-47-0': 'There may also be systematic biases in our inferred [MATH].', '1105.4605-1-47-1': 'We have already discussed the dependence of [MATH] on metallicity as a possible explanation for the high SFR-to-CO ratio observed in lower-mass galaxies.', '1105.4605-1-47-2': '[MATH] certainly depends on metallicity.', '1105.4605-1-47-3': 'Current best estimates imply a non-linear relationship, with [MATH] sharply increasing below [MATH] .', '1105.4605-1-47-4': 'For our range of metallicities ([MATH], though the uncertainties on that ratio are large.', '1105.4605-1-47-5': 'More important, [CITATION] suggest variations in the CO line ratios between the inner and outer Milky Way and there are well-established differences between normal disk and starburst galaxies.', '1105.4605-1-47-6': 'Although not immediately apparent from a comparison of HERACLES to literature CO([MATH]) data , such variations could affect our derived [MATH] by as much as [MATH]50%.', '1105.4605-1-47-7': 'Rosolowsky et al. (2011, in prep.)', '1105.4605-1-47-8': 'will present a thorough investigation of how the line ratio varies with environment in HERACLES.', '1105.4605-1-48-0': '## Distribution of Molecular Gas', '1105.4605-1-49-0': 'The tight correlation between SFR tracers and CO emission across all regimes strongly reinforces the primary importance of molecular gas to star formation.', '1105.4605-1-49-1': 'In this section, we therefore examine the distribution of molecular gas in galaxies.', '1105.4605-1-49-2': 'In the outer parts of spiral galaxies where [MATH], the formation of molecular gas from atomic gas appears to represent the bottleneck to star formation and the relative abundance of H[MATH] and H[MATH]i is key to setting the star formation rate .', '1105.4605-1-49-3': 'We can apply the large dynamic range in H[MATH]-to-H[MATH]i ratios achieved by stacking to make improved measurements of how this key quantity varies across galaxies.', '1105.4605-1-50-0': '### Radial Distribution of CO Intensity', '1105.4605-1-51-0': 'Many previous studies have shown that azimuthally averaged CO emission decreases with increasing galactocentric radius .', '1105.4605-1-51-1': 'Whereas galaxy centers often exhibit deviations from the large scale trend, CO emission outside the centers declines approximately uniformly with radius .', '1105.4605-1-51-2': 'A first analysis of the HERACLES data revealed a characteristic exponential decline of CO emission in the inner parts of galaxy disks, with the scale length of CO emission similar to that of old stars and tracers of recent star formation .', '1105.4605-1-51-3': 'With the increased sensitivity from stacking and a larger sample, we can revisit this question and ask if this radial decline in CO intensity continues smoothly out to [MATH].', '1105.4605-1-52-0': 'From exponential fits to the high-significance CO data of each galaxy (solid-dashed lines in Figure [REF] and Appendix), excluding the galaxy centers (inner 30) we find a median exponential scale length, [MATH], of [MATH] with [MATH]% of all [MATH] between [MATH].', '1105.4605-1-52-1': 'This value agrees well with typical scale lengths found in previous studies and the individual galaxy scale lengths agree well with previous work on the HERACLES sample .', '1105.4605-1-53-0': 'The normalizations of these fits reflect galaxy-to-galaxy variations in the total molecular gas content.', '1105.4605-1-53-1': 'In the left panel of Figure [REF] we show all profiles aligned to a common normalization.', '1105.4605-1-53-2': 'We plot the radius in units of [MATH], the 25[MATH] magnitude [MATH]-band isophote, and normalize each profile so that the exponential fits have intensity [MATH] at [MATH].', '1105.4605-1-53-3': 'The figure thus shows the radial variation of CO intensity across our sample, controlled for the overall CO luminosity and absolute size of each galaxy.', '1105.4605-1-53-4': 'Thick crosses mark the median CO intensity and the [MATH]th percentile range in bins [MATH] wide.', '1105.4605-1-53-5': 'The same exponential decline seen in individual profiles is even more evident here, with [MATH] for the average of the sample.', '1105.4605-1-54-0': 'The left panel in Figure [REF] shows that the radial decline of the CO profiles observed previously for the inner part of galaxies extends without significant changes out to our last measured data points.', '1105.4605-1-54-1': 'In most galaxies there is no clear evidence for a sharp cutoff or a change in slope.', '1105.4605-1-54-2': 'As long as the normalized profiles are above the sensitivity limit the decline appears to continue (without significant deviation) with [MATH] on average.', '1105.4605-1-54-3': 'This smooth exponential decline in CO intensity with increasing radius suggests that the observed decline in star formation rate from the inner to outer parts of galaxy disks is driven by a continuous decrease in the supply of molecular gas, rather than a sharp threshold of some kind.', '1105.4605-1-55-0': 'There are several galaxies which deviate from this median exponential trend.', '1105.4605-1-55-1': 'We already noted above that we do not fit exponential profiles to NGC 2798, 2976, 4725 because their gas (H[MATH]i and CO) radial profiles are insufficiently parametrized by exponentials.', '1105.4605-1-55-2': 'There are three galaxies (NGC 2146, 2903, 4569) that we do fit and determine small exponential scale lengths, [MATH].', '1105.4605-1-55-3': 'NGC 2146 hosts an ongoing starburst and both NGC 2903 and NGC 4569 have prominent bars that may be funneling molecular gas to their centers.', '1105.4605-1-55-4': 'For two other galaxies (NGC 2841 and 4579) we determine large exponential scale lengths, [MATH].', '1105.4605-1-55-5': 'These galaxies are better described by a flat distribution (or even a central depression) and a cutoff at larger radii.', '1105.4605-1-56-0': 'The tight correspondence of the CO scale length, [MATH] to [MATH] has been noted before , while other studies have found a close correspondence between CO and near-infrared light .', '1105.4605-1-56-1': 'In our sample there is a fairly good correspondence between [MATH] and the near-infrared scale length, [MATH], measured at [MATH] with [MATH] .', '1105.4605-1-56-2': 'The near-infrared light should approximately trace the distribution of stellar mass, so that our measured scale length is very similar to that of the stellar mass.', '1105.4605-1-56-3': 'This tight coupling has been interpreted to indicate the importance of the stellar potential well to collecting star-forming material .', '1105.4605-1-56-4': 'Here we see this correspondence to continue into the regime where the molecular gas is not the dominant gas component, confirming that molecular gas formation is the bottleneck to star formation.', '1105.4605-1-57-0': 'The right panel in Figure [REF] shows the distribution of enclosed luminosity, mean intensity, and flux at each radius for an exponential disk with scale length of [MATH].', '1105.4605-1-57-1': 'The brightest individual ring for such a disk lies at [MATH] and half the flux is enclosed within [MATH].', '1105.4605-1-57-2': 'This value, [MATH], is fairly close to the radius at which [MATH] in a typical disk galaxy and][]Leroy2008, so that CO emission is about evenly split between the H[MATH]-dominated and H[MATH]i -dominated parts of such a galaxy.', '1105.4605-1-57-3': 'Meanwhile, 90 of the flux lies within [MATH] a value that is very similar to the threshold radius identified by [CITATION].', '1105.4605-1-57-4': 'We do not find evidence to support a true break at this radius, but as an "edge" to the star-forming disk, a 90 contour may have utility.', '1105.4605-1-58-0': '### The H[MATH]-to-HI Ratio', '1105.4605-1-59-0': 'In the outer parts of galaxy disks - and thus over most of the area in galaxy disks - we have [MATH], implying that star-forming H[MATH] gas does not make up most of the interstellar medium.', '1105.4605-1-59-1': 'In this regime the relative abundance of H[MATH] and H[MATH]i is a key quantity to regulate the star formation rate.', '1105.4605-1-59-2': 'Observations over the last decade have revealed strong variations of the fraction of gas in the molecular phase as a function of galactocentric radius, total gas surface density, stellar surface density, disk orbital time, and interstellar pressure .', '1105.4605-1-59-3': 'In Figure [REF] we show the two most basic of these trends, the H[MATH]-to-H[MATH]i ratio, [MATH], as a function of normalized galactocentric radius (left panel) and total gas surface density (right panel).', '1105.4605-1-59-4': 'We focus on [MATH] because it is more easily separated in discrete observables than the fraction of gas that is molecular, [MATH].', '1105.4605-1-59-5': 'This makes it easier to interpret uncertainties and systematic effects like changes in the CO-to-H[MATH] conversion factor.', '1105.4605-1-60-0': 'The left panel of Figure [REF] shows [MATH] as function of galactocentric radius in units of [MATH] for all data detected with high or marginal significance.', '1105.4605-1-60-1': 'For clarity, we do not plot [MATH] for regions where we determined only upper limits in CO intensity; for the inner parts ([MATH]) these upper limits are bounded by [MATH], whereas for outer parts ([MATH]) the upper limits in [MATH] are typically of comparable magnitude as measurements of [MATH] and upper limits are bounded by [MATH].', '1105.4605-1-60-2': 'In agreement with [CITATION], [CITATION], [CITATION], and [CITATION] we find [MATH] to decline with increasing galactocentric radius, a variation that reflects the distinct radial profiles of atomic and molecular gas.', '1105.4605-1-60-3': 'However, radial variations alone do not explain the full range of observed molecular fractions because [MATH] can vary by up to two orders of magnitude at any given galactocentric radius.', '1105.4605-1-61-0': 'A significant part of the variations in [MATH] at a given galactocentric radius corresponds to systematic variations between galaxies.', '1105.4605-1-61-1': 'These are mainly caused by variations in the absolute molecular gas content of a galaxy and can be removed by a normalization procedure similar to one applied in the left panel of Figure [REF].', '1105.4605-1-61-2': 'The result is similar to that seen for the radial profiles of CO: the relationship tightens and we can see that most of the decline of [MATH] inside a galaxy occurs radially.', '1105.4605-1-61-3': 'However, there is significantly more scatter remaining in [MATH] versus radius than we observed for the normalized CO radial profiles, highlighting the importance of parameters other than a combination of radius and host galaxy to set [MATH].', '1105.4605-1-62-0': 'In addition to declining with increasing galactocentric radius, [MATH] increases as the total gas surface density, [MATH], increases.', '1105.4605-1-62-1': 'The more gas that is present along a line of sight, the larger the fraction of gas that is molecular.', '1105.4605-1-62-2': 'The right panel of Figure [REF] shows this result, plotting [MATH] as function of the total gas surface density, [MATH].', '1105.4605-1-62-3': '[MATH] increases with increasing [MATH], with regions of high surface density, [MATH] M[MATH] pc[MATH], being predominately molecular, [MATH].', '1105.4605-1-63-0': 'At high [MATH], the right panel of Figure [REF] is largely a way of visualizing the "saturation" of [MATH] on large scales in galaxies.', '1105.4605-1-63-1': 'A number of authors have found that averaged over hundreds of parsecs to kpc scales, [MATH] rarely exceeds [MATH]10 M[MATH] pc[MATH] .', '1105.4605-1-63-2': 'This limit is clearly violated at high spatial resolution and may vary among classes of galaxies.', '1105.4605-1-63-3': 'Several recent theoretical works have aimed at reproducing this behavior.', '1105.4605-1-63-4': '[CITATION] focused on shielding of H[MATH] inside individual atomic-molecular complexes, whereas [CITATION] examined the interplay between large-scale thermal and dynamical equilibria.', '1105.4605-1-64-0': 'The right panel in Figure [REF] includes also the predicted solar-metallicity [MATH]-[MATH] relation from [CITATION].', '1105.4605-1-64-1': 'They model individual atomic-molecular complexes, however these complexes have a filling factor substantially less than [MATH] inside our beam.', '1105.4605-1-64-2': 'The appropriate [MATH] to input into their model is therefore the average surface densities of the complexes, [MATH], within our beam, which will be related to our observed surface density, [MATH], by the filling factor, [MATH], namely: [MATH].', '1105.4605-1-64-3': 'The model curve (blue short-dashed line) with a filling factor [MATH] is offset towards higher [MATH] (shifted right) or lower [MATH] (shifted down) compared to our data.', '1105.4605-1-64-4': 'To have the model curve intersect our data (red long-dashed line) requires a filling factor of [MATH], so that atomic-molecular complexes fill about half the areas in our beam.', '1105.4605-1-64-5': 'This is before any accounting for the presence of diffuse H[MATH]i not in star-forming atomic-molecular complexes and is assuming a fixed filling factor, both of which are likely oversimplifications.', '1105.4605-1-64-6': 'Nonetheless, the [CITATION] curve does show an overall good correspondence to our data.', '1105.4605-1-65-0': 'As with the radius, the total gas surface density predicts some of the broad behavior of [MATH] but knowing [MATH] does not uniquely specify the amount of molecular gas, particularly at low [MATH].', '1105.4605-1-65-1': 'This are visible as the large scatter in [MATH] at low surface densities in the right panel of Figure [REF].', '1105.4605-1-65-2': 'The scatter reflects a dependence of [MATH] on environmental factors other than gas surface density.', '1105.4605-1-65-3': 'Figure [REF] shows [MATH] over the small range [MATH] M[MATH] pc[MATH] , i.e., the data from the gray highlighted region in Figure [REF].', '1105.4605-1-65-4': 'We plot histograms for several radial bins which are clearly offset, indicating an additional radial dependence of [MATH].', '1105.4605-1-65-5': 'At small radii ([MATH]) we observe a large scatter in [MATH] for [MATH] M[MATH] pc[MATH] whereas at large radii ([MATH]) gas with this surface density is always predominantly H[MATH]i .', '1105.4605-1-66-0': 'As was the case in SFR-H[MATH] space, distinct populations of galaxies are responsible for some of the variations in the right panel of Figure [REF].', '1105.4605-1-66-1': 'Early type (Sab-Sb) spirals (e.g., NGC 2841, 3351, 3627, 4736) often show large molecular fractions, [MATH], but typically have low H[MATH]i and H[MATH] surface densities, [MATH] and [MATH] M[MATH] pc[MATH] .', '1105.4605-1-66-2': 'By contrast, massive Sc galaxies (e.g., NGC 4254, 4321, 5194, 6946) can have comparably high molecular fractions, [MATH], but have higher surface density H[MATH]i disks, [MATH] M[MATH] pc[MATH] , so that these fractions occur at higher [MATH].', '1105.4605-1-66-3': 'A trend with metallicity is not immediately obvious in the data but these differences may reflect the more substantial stellar surface densities found in the earlier-type galaxies.', '1105.4605-1-66-4': 'This increased stellar surface density results in a stronger gravitational field, which could lead to a higher midplane gas pressure and a low fraction of diffuse H[MATH]i gas .', '1105.4605-1-67-0': '### Discussion of [MATH]', '1105.4605-1-68-0': 'Following several recent studies we observe strong systematic variations in the H[MATH]-H[MATH]i balance across galaxies.', '1105.4605-1-68-1': 'Two of the strongest behaviors are an approximately exponential decrease in [MATH] with increasing galactocentric radius and a steady increase in [MATH] with increasing gas surface density.', '1105.4605-1-68-2': 'Our improved sensitivity shows these trends extending to low surface densities and our expanded sample makes clear that neither of these basic parametrizations adequately captures the entire range of [MATH] variations.', '1105.4605-1-68-3': 'The likely physical drivers for the scatter in [MATH] that we observe are metallicity and dust-to-gas ratio , the dissociating radiation field , variations in interstellar gas pressure and density , and external perturbations that drive gas to higher densities .', '1105.4605-1-68-4': 'Each of these quantities are observationally accessible in our sample and estimates of the relative roles of each process will be presented in Leroy et al. (2011, in prep.)', '1105.4605-1-69-0': 'The overall relationship of total gas and star formation rate (middle panels of Figure [REF]) can be reproduced by a roughly fixed ratio of SFR-to-H[MATH] within galaxies (Section [REF]) and the observed scaling of [MATH] with [MATH] (Section [REF]).', '1105.4605-1-69-1': 'This is shown in Figure [REF], where we plot the SFR per unit total gas, [MATH] (the inverse of the total gas depletion time) as function of the molecular-to-atomic ratio, [MATH].', '1105.4605-1-69-2': 'At low surface densities, the [MATH]-[MATH] relation regulates [MATH], at high surface densities where almost all of the gas is molecular the SFR-H[MATH] scaling determines the observed ratio.', '1105.4605-1-69-3': 'This trend is well parametrized by a fixed SFR-to-H[MATH] ratio with a molecular gas depletion time of [MATH] Gyr (blue dotted line) for all of our targets and [MATH] Gyr (red dashed line) for big spirals .', '1105.4605-1-69-4': 'The transition between these two regimes creates the curved shape seen in Figure [REF] [REF].', '1105.4605-1-69-5': 'This offers more support for a modified version of the classical picture of a star formation threshold , in which dense, mostly molecular gas forms stars but the efficiency with which such (molecular) gas forms is a strong function of environment, decreasing steadily with decreasing gas surface density and increasing galactocentric radius.', '1105.4605-1-70-0': '# Summary', '1105.4605-1-71-0': 'We combine HERACLES CO([MATH]) data with H[MATH]i velocity fields, mostly from THINGS, to make sensitive measurements of CO intensity across the disks of 33 nearby star-forming galaxies.', '1105.4605-1-71-1': 'We stack CO spectra across many lines of sight by assuming that the mean H[MATH]i and CO velocities are similar, an assumption that we verify in the inner parts of galaxies.', '1105.4605-1-71-2': 'This approach allows us to detect CO out to galactocentric radii [MATH].', '1105.4605-1-71-3': 'Because we measure integrated CO intensities as low as [MATH] K km s[MATH]1 M[MATH] pc[MATH] , before any correction for inclination) with high significance, we are able to robustly measure CO intensities in parts of galaxies where most of the ISM is atomic.', '1105.4605-1-72-0': 'Using this approach we compare the radially averaged intensities of FUV, H[MATH], IR, CO, and H[MATH]i emission across galaxy disks.', '1105.4605-1-72-1': 'We find an approximately linear relation between CO intensity and monochromatic IR intensity at both 24 and 70 .', '1105.4605-1-72-2': 'For the first time, we show that these scaling relations continue smoothly from the H[MATH]-dominated to H[MATH]i-dominated ISM.', '1105.4605-1-72-3': 'Extinction causes FUV and H[MATH] emission to display a more complex relationship with CO, especially in the inner parts of galaxies.', '1105.4605-1-72-4': 'In the outer parts of galaxy disks FUV and H[MATH] emission do correlate tightly with CO emission after galaxy-to-galaxy variations are removed.', '1105.4605-1-73-0': 'We use two calibration to estimate the recent star formation rate, FUV[MATH]24 and H[MATH][MATH]24, which we compare to H[MATH] derived from CO.', '1105.4605-1-73-1': 'We find an approximately linear relation between [MATH] and [MATH] in the range of [MATH] M[MATH] pc[MATH] with no notable variation between the two SFR estimates.', '1105.4605-1-73-2': 'A number of recent studies have also seen a roughly linear relationship between [MATH] and [MATH] and have argued that it implies that the surface density of H[MATH] averaged over large scales does not strongly affect the efficiency with which molecular gas forms stars.', '1105.4605-1-74-0': 'We do find evidence for variations in the SFR-to-CO ratio among galaxies.', '1105.4605-1-74-1': 'Indeed, most of the scatter in the relations between CO and SFR tracers is driven by galaxy-to-galaxy variations.', '1105.4605-1-74-2': 'These variations are not random, but show the trend observed by [CITATION] that lower mass, lower metallicity galaxies have higher ratios of SFR-to-H[MATH] than massive disk galaxies.', '1105.4605-1-74-3': 'It will take further study to determine whether these are real variations in the efficiency of star formation or reflect changes in the CO-to-H[MATH] conversion factor due to lower metallicities in these systems.', '1105.4605-1-74-4': 'After removing these galaxy-to-galaxy variations the composite H[MATH]-SFR relation is remarkably tight, reinforcing a close link between H[MATH] and star formation inside galaxies.', '1105.4605-1-75-0': 'We compare the scaling between the surface densities of SFR and H[MATH]i , H[MATH] , and total gas (H[MATH]i+H[MATH]).', '1105.4605-1-75-1': 'The relationship between SFR and total gas has roughly the same rank correlation coefficient as that between SFR and H[MATH], but does not obey a single functional form.', '1105.4605-1-75-2': 'Where [MATH] the relationship between [MATH] and [MATH] is steep whereas where [MATH] the relationship is much flatter.', '1105.4605-1-75-3': 'Meanwhile, we observe a linear relationship between [MATH] and [MATH] for the full range of [MATH] M[MATH] pc[MATH] .', '1105.4605-1-75-4': '[MATH] and [MATH] are weakly correlated and exhibit a strongly nonlinear relation, except at very large radii.', '1105.4605-1-76-0': 'The unbroken extension of the [MATH]-[MATH] relation into the H[MATH]i -dominated regime suggests a modified version of the classical picture of a star formation threshold , in which stars form at fixed efficiency out of molecular gas, to first order independent of environment within a galaxy.', '1105.4605-1-76-1': 'The observed turn-over in the relation between SFR and total gas relates to the H[MATH]-to-H[MATH]i ratio which is a strong function of environment.', '1105.4605-1-77-0': 'We therefore investigate the distribution of H[MATH] traced by CO using our stacked data and compare it to the H[MATH]i .', '1105.4605-1-77-1': 'On large scales we observe CO to decrease exponentially with a remarkably uniform scale length of [MATH], again extending previous studies to lower surface densities.', '1105.4605-1-77-2': 'We find the normalization of this exponential decline to vary significantly among galaxies.', '1105.4605-1-77-3': 'The H[MATH]-to-H[MATH]i ratio, traced by the ratio of CO-to-H[MATH]i intensities, also varies systematically across galaxies.', '1105.4605-1-77-4': 'It exhibits significant correlations with both galactocentric radius and total gas surface density and we present high-sensitivity measurements of both of these relationships.', '1105.4605-1-77-5': 'However, neither quantity is sufficient to uniquely predict the H[MATH]-to-H[MATH]i ratio on its own.', '1105.4605-1-78-0': 'We thank the teams of SINGS, LVL, and GALEX NGS for making their outstanding data sets available.', '1105.4605-1-78-1': 'We thank Daniela Calzetti for helpful comments.', '1105.4605-1-78-2': 'We thank the anonymous referee for thoughtful comments that improved the paper.', '1105.4605-1-78-3': 'The work of AS was supported by the Deutsche Forschungsmeinschaft (DFG) Priority Program 1177.', '1105.4605-1-78-4': 'Support for AKL was provided by NASA through Hubble Fellowship grant HST-HF-51258.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.', '1105.4605-1-78-5': 'The work of WJGbB is based upon research supported by the South African Research Chairs Initiative of the Department of Science and Technology and the National Research Foundation.', '1105.4605-1-78-6': 'This research made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the JPL/Caltech, under contract with NASA, NASAs Astrophysical Data System (ADS), and the HyperLeda catalog, located on the WorldWide Web at http://www.obs.univ-lyon1.fr/hypercat/intro.html.', '1105.4605-1-79-0': 'Here we present the radial profiles of CO, H[MATH]i , FUV, H[MATH] , and IR at 24 and 70 used to generate the plots in this paper.', '1105.4605-1-79-1': 'We average the data in 15 wide tilted rings.', '1105.4605-1-79-2': 'For the CO data, we stack the shifted spectra over this area and determine the integrated CO intensity from fitting line profiles to the stacked spectrum.', '1105.4605-1-79-3': 'For the H[MATH]i , FUV, H[MATH] , and IR data we use two-dimensional maps of intensity (Section [REF]).', '1105.4605-1-79-4': 'We determine the [MATH] scatter from the 68[MATH]-percentile from the data inside each ring.', '1105.4605-1-79-5': 'We plot these as error bars but note the distinction from the uncertainty in the mean.', '1105.4605-1-80-0': 'For each galaxy we present two plots: The left panel shows CO and H[MATH]i both in units of observed intensities (K km s[MATH] , left-hand [MATH]-axis) and converted to mass surface densities (M[MATH] pc[MATH] , right-hand [MATH]-axis) of H[MATH] and H[MATH]i .', '1105.4605-1-80-1': 'The color of the CO points indicates the significance with which we could determine the integrated CO intensities: green for high significance measurements, orange for measurements of marginal significance and red for [MATH] upper limits.', '1105.4605-1-80-2': 'To have H[MATH] and H[MATH]i on the same mass surface density scale, we multiplied the observed 21-cm line intensities by a factor of [MATH] (the ratio of Equations [REF] and [REF]).', '1105.4605-1-80-3': 'We also plot the star formation rate (SFR) surface density (M[MATH] yr[MATH] kpc[MATH] ) determined from H[MATH][MATH]24 and FUV[MATH]24.', '1105.4605-1-80-4': 'Black solid-dashed lines show our exponential fit to the radial CO profile.', '1105.4605-1-80-5': 'We fit all high significance data excluding galaxy centers, defined as the the inner 30, which often exhibit breaks from the overall profile .', '1105.4605-1-80-6': 'The derived exponential scale lengths (in units of [MATH], the radius of the 25[MATH] magnitude [MATH]-Band isophote), appear in the lower left corner.', '1105.4605-1-81-0': 'The right panel shows observed intensities (in MJy sr[MATH]) of our SFR tracers - H[MATH] , FUV, 24 and 70 emission.'}

{'1105.4605-2-0-0': 'We use the IRAM HERACLES survey to study CO emission from 33 nearby spiral galaxies down to very low intensities.', '1105.4605-2-0-1': 'Using 21-cm line atomic hydrogen (H[MATH]i) data, mostly from THINGS, we predict the local mean CO velocity based on the mean H[MATH]i velocity.', '1105.4605-2-0-2': 'By re-normalizing the CO velocity axis so that zero corresponds to the local mean H[MATH]i velocity we are able to stack spectra coherently over large regions.', '1105.4605-2-0-3': 'This enables us to measure CO intensities with high significance as low as [MATH] K km s[MATH] M[MATH] pc[MATH]), an improvement of about one order of magnitude over previous studies.', '1105.4605-2-0-4': 'We detect CO out to galactocentric radii [MATH] and find the CO radial profile to follow a remarkably uniform exponential decline with scale length of [MATH] km s[MATH] (Figure [REF] upper right panel).', '1105.4605-2-0-5': 'However, in the central regions of some galaxies the line can be very broad with a flattened or double-horned peak (Figure [REF] upper left panel).', '1105.4605-2-0-6': 'These profiles often coincide with central enhancements like bars or molecular rings and are poorly parametrized by a single Gaussian.', '1105.4605-2-0-7': 'Instead we fit a double-horn profile, a Gaussian scaled by a symmetric second-order polynomial .', '1105.4605-2-0-8': 'Based on "by eye" inspection, the asymmetry in these profiles is small enough that a symmetric function is sufficient.', '1105.4605-2-1-0': 'We derive the best fit profile via a non-linear least squares fit.', '1105.4605-2-1-1': 'We constrain the fit parameters so that the center of the profile lies within [MATH]50 km s[MATH] of zero velocity after shifting, the FWHM is larger than 15 km s[MATH] (to avoid the fit latching onto individual channels), and the amplitude is positive.', '1105.4605-2-1-2': 'We always carry out a Gaussian fit first and in those cases where the FWHM exceeds 60 km s[MATH] we switch to a double-horn profile.', '1105.4605-2-1-3': 'We verify by eye that this yields sensible results.', '1105.4605-2-2-0': 'The integral of the fitted profile gives us the integrated CO line intensity.', '1105.4605-2-2-1': 'We derive the uncertainty in this quantity from the noise, estimated from the signal-free part of the spectrum, and the width of the profile.', '1105.4605-2-2-2': 'It proved useful to define a quality scale for the fit.', '1105.4605-2-2-3': 'The quality is "high" where the peak intensity is larger than [MATH] and its integrated intensity is larger than [MATH] times its uncertainty.', '1105.4605-2-2-4': 'In cases where the peak intensity is less than [MATH] or the integrated intensity is less than [MATH] times its uncertainty, we do not trust the fit and instead determine an upper limit.', '1105.4605-2-2-5': 'We label cases that fall between these regimes as "marginal".', '1105.4605-2-2-6': 'Figure [REF] shows examples of our quality measures and line profile fits.', '1105.4605-2-3-0': 'We derive our upper limits integrating over a Gaussian line profile with FWHM set to 18 km s[MATH] , the typical FWHM found for high SNR spectra at [MATH], and fixed amplitude of 3[MATH].', '1105.4605-2-4-0': '## Stacking as a Function of Radius', '1105.4605-2-5-0': 'We present our stacking technique applied to radial bins.', '1105.4605-2-5-1': 'In principle this method allows us to stack spectra across any region.', '1105.4605-2-5-2': 'For example, we could define regions by total gas column, infrared intensity, or features such as spiral arms and bars.', '1105.4605-2-5-3': 'In practice, radius makes an excellent ordinate.', '1105.4605-2-5-4': 'We wish to study the underlying relationship between CO, H[MATH]i, IR, FUV, and H[MATH] intensity.', '1105.4605-2-5-5': 'Stacking with one of these quantities as the ordinate would require carefully modeling the biases involved to measure the underlying relationships.', '1105.4605-2-5-6': 'Galactocentric radius is a well-determined, independent quantity that is also highly covariant with these other intensities.', '1105.4605-2-5-7': 'This yields a dataset with large dynamic range that is easy to interpret.', '1105.4605-2-6-0': 'Therefore we focus our analysis on data stacked in bins of galactocentric radius.', '1105.4605-2-6-1': 'We average over tilted rings 15 wide, comparable to the angular resolution of our data.', '1105.4605-2-6-2': 'This width corresponds to [MATH] pc for our most distant targets (25 Mpc).', '1105.4605-2-6-3': 'We construct the rings assuming that each galaxy is a thin disk with the inclination and position angle given in Table [REF].', '1105.4605-2-6-4': 'To measure CO with highest sensitivity we construct stacked CO spectra using the procedure described above and fit those to determine the integrated CO line intensities.', '1105.4605-2-6-5': 'For the other observables - H[MATH]i, IR, FUV, and H[MATH] intensities - we use two-dimensional maps of intensity and determine the mean intensity for each tilted ring.', '1105.4605-2-7-0': 'Error bars on the H[MATH]i, IR, FUV, and H[MATH] intensities show the [MATH] scatter within that tilted ring, capturing both statistical noise and deviations from axial symmetry.', '1105.4605-2-7-1': 'We estimate the [MATH] scatter for our CO measurements by integrating the CO cube over a velocity window that is adjusted for each line of sight such that it includes all channels of significant CO emission but at least all channels with velocities within 25 km s[MATH] of the local mean H[MATH]i velocity.', '1105.4605-2-7-2': 'Note that our [MATH] values reflect the scatter in (integrated) intensities of individual lines of sight inside a ring.', '1105.4605-2-7-3': 'They should not be confused with the uncertainty in the determination of the mean intensity inside a ring which is typically much smaller.', '1105.4605-2-8-0': '# Results', '1105.4605-2-9-0': 'In Figure [REF] for NGC 628 and in the Appendix for the rest of the sample, we present stacked radial profiles of integrated CO intensity along with profiles of H[MATH]i , infrared intensity at 24 and 70 , FUV, and H[MATH] intensity.', '1105.4605-2-9-1': 'Following Equations [REF] we combine H[MATH] and 24 intensities to estimate the star formation rate surface density SFR(H[MATH][MATH]24) and compare those to SFR(FUV[MATH]24) derived from FUV and 24 intensities using Equation [REF].', '1105.4605-2-10-0': 'For each galaxy there are two plots: The left panel shows H[MATH]i , H[MATH] (from CO), and SFR for both FUV[MATH]24 and H[MATH][MATH]24.', '1105.4605-2-10-1': 'The right panel shows our SFR tracers - H[MATH] , FUV, 24 and 70 emission.', '1105.4605-2-10-2': 'We present the profiles in both observed intensity (left hand [MATH]-axis) and units of surface density (right hand [MATH]-axis of the left panel) - [MATH], [MATH], and [MATH].', '1105.4605-2-10-3': 'Note that we have projected (only) the surface densities of H[MATH]i , H[MATH] , and SFR to face-on values (i.e., we corrected for inclination).', '1105.4605-2-10-4': 'The observed surface brightnesses are not corrected for the effect of inclination.', '1105.4605-2-10-5': 'Such a correction will just move each galaxy up and down in lockstep and we find it more useful to report the observed values.', '1105.4605-2-10-6': 'The color of a point in the CO profile indicates the significance of the fit to the stacked spectrum: green for high significance, orange for marginal significance, and red for upper limits, these correspond to [MATH] upper limits on the fitted intensity (see Section [REF]).', '1105.4605-2-11-0': '## CO and Star Formation', '1105.4605-2-12-0': 'With these azimuthally averaged data we are able to compare CO to tracers of recent star formation across a large range of H[MATH]-to-H[MATH]i ratios.', '1105.4605-2-12-1': 'In this subsection, we make empirical comparisons between measured intensities, examine the relative roles of H[MATH] and H[MATH]i in the "star formation law," and investigate the origin of the scatter in these relations.', '1105.4605-2-13-0': '### Scaling Relations between CO and IR, FUV, and H[MATH]', '1105.4605-2-14-0': 'Figures [REF] [REF] show scaling relations between observed intensities of CO ([MATH]-axis) and different tracers of recent star formation ([MATH]-axis).', '1105.4605-2-14-1': 'Figure [REF] shows infrared intensities at 24 (top panels) and 70 (bottom panels) and Figure [REF] shows intensities of FUV (top panels) and H[MATH] (bottom panels).', '1105.4605-2-14-2': 'The left hand panels show the relations for all galaxies and all radii.', '1105.4605-2-14-3': 'The panels on the right hand side show only radii [MATH].', '1105.4605-2-14-4': 'H[MATH] and H[MATH]i make up roughly equal parts of the ISM near this radius (see the left panel of Figure [REF]), so most of the points in the right hand panels are H[MATH]i -dominated.', '1105.4605-2-14-5': 'A dotted vertical line shows an integrated CO intensity of [MATH] K km s[MATH] , which corresponds to [MATH] M[MATH] pc[MATH] (assuming [MATH]), which is about the surface density at which H[MATH] and H[MATH]i make up equal parts of the ISM.', '1105.4605-2-14-6': 'A dashed vertical line at [MATH] K km s[MATH] M[MATH] pc[MATH] for [MATH]) shows a conservative sensitivity limit for the whole sample.', '1105.4605-2-14-7': 'Typically our upper limits, which are not displayed in these plots, lie to the left of this line.', '1105.4605-2-14-8': 'They will be systematically higher at large radii when radial rings partially exceed the coverage of our CO maps.', '1105.4605-2-15-0': 'CO emission correlates tightly with IR emission at 24 and 70 (Figure [REF]) (rank correlation coefficient [MATH]).', '1105.4605-2-15-1': 'The correlation extends over three orders of magnitude in CO and IR intensities and crosses the H[MATH]i -to-H[MATH] transition without substantial change in slope or normalization.', '1105.4605-2-15-2': 'Comparing the left panels (all radii) and the right panels ([MATH]) does not reveal any significant radial dependence.', '1105.4605-2-16-0': 'CO emission exhibits a weaker correlation with FUV and H[MATH] (Figure [REF]) ([MATH]) than with IR emission, i.e., both the CO-FUV and CO-H[MATH] relation show much larger scatter than the CO-IR relations.', '1105.4605-2-16-1': 'The CO-FUV relation displays a break between H[MATH]i- and H[MATH]-dominated regimes.', '1105.4605-2-16-2': 'The increased scatter and weaker correlation at least partially reflects the sensitivity of H[MATH] and FUV emission to absorption by dust.', '1105.4605-2-16-3': 'In the inner ([MATH]), more gas rich parts of galaxies dust reprocesses most H[MATH] and FUV emission into IR emission.', '1105.4605-2-16-4': 'In this regime CO and FUV are to first order uncorrelated ([MATH]).', '1105.4605-2-16-5': 'Outside [MATH] Gyr (including a factor [MATH] to account for heavy elements; see Table [REF] for fit parameters).', '1105.4605-2-16-6': 'This is slightly lower than [MATH] Gyr found by [CITATION] and [CITATION] for a subset of the data analyzed here and [MATH] Gyr recently found by [CITATION] for a sample that is similar to the one analyzed here.', '1105.4605-2-16-7': 'We include (a) more starburst galaxies and (b) more low mass, low metallicity spiral galaxies that [CITATION] and [CITATION] excluded from their CO analysis.', '1105.4605-2-16-8': 'Both dwarfs and starbursts have shorter [MATH] than large spirals .', '1105.4605-2-16-9': 'Moreover, our radial profiles weight these small galaxies more heavily than the pixel sampling used by [CITATION].', '1105.4605-2-17-0': 'As with the scaling relations between CO and tracers of recent star formation (Figure [REF] [REF]), the H[MATH]-SFR relation exhibits significant scatter.', '1105.4605-2-17-1': 'We perform the same procedure to isolate galaxy-to-galaxy variations from scatter within galaxies and again find the scatter in the main relation (upper panels in Figure [REF] [REF]) dominated by galaxy-to-galaxy variations.', '1105.4605-2-17-2': 'We plot the relations after normalization in [MATH] and [MATH] in the middle and bottom panels of Figure [REF] [REF].', '1105.4605-2-17-3': 'Once galaxy-to-galaxy scatter is removed, there is a remarkably tight, uniform linear relation linking molecular gas and star formation across almost three and a half orders of magnitude.', '1105.4605-2-18-0': 'What drives this galaxy-to-galaxy variation?', '1105.4605-2-18-1': 'In both the H[MATH]-SFR and observed intensity relations a large part of the scatter comes from a sub-population of less massive, less metal-rich galaxies that exhibit high SFR-to-CO ratios.', '1105.4605-2-18-2': 'Figure [REF] shows the metallicity dependence of the molecular depletion time, [MATH].', '1105.4605-2-18-3': 'Massive spiral galaxies with high metallicities have considerably longer (median averaged) depletion times: [MATH] Gyr for [MATH], while smaller galaxies with lower metallicities have systematically shorter depletion times: [MATH]24) vs. Gas Phase H[MATH]i & &', '1105.4605-2-19-0': '... all data & [MATH]a &', '1105.4605-2-20-0': '... H[MATH]i-dominated & [MATH] &', '1105.4605-2-21-0': '... H[MATH]-dominated & [MATH] &', '1105.4605-2-22-0': 'H[MATH] & &', '1105.4605-2-23-0': '... all data & [MATH] & [MATH]a', '1105.4605-2-24-0': '... H[MATH]i-dominated & [MATH] & [MATH]', '1105.4605-2-25-0': '... H[MATH]-dominated & [MATH] & [MATH]', '1105.4605-2-26-0': 'H[MATH]i+H[MATH] & &', '1105.4605-2-27-0': '... all data & [MATH] & [MATH]', '1105.4605-2-28-0': '... H[MATH]i-dominated & [MATH] & [MATH]', '1105.4605-2-29-0': '... H[MATH]-dominated & [MATH] & [MATH] aEstimation of uncertainties equal to Table [REF].', '1105.4605-2-30-0': 'The question of which gas component - H[MATH]i , H[MATH] , or total gas - correlates best with recent star formation has received significant attention.', '1105.4605-2-30-1': 'Phrased this way, the question is not particularly well posed: the total gas surface density and the molecular gas fraction are closely related so that the different gas surface densities are not independent quantities.', '1105.4605-2-30-2': 'Therefore we do not necessarily expect a "best" correlation, only different functional forms.', '1105.4605-2-30-3': 'Still, it is illustrative to see how recent star formation relates to each gas tracer.', '1105.4605-2-30-4': 'Table [REF] lists [MATH] and the power law index from an OLS bisector fit between each component and Figure [REF] shows plots for each of the three gas phases and our two SFR prescriptions.', '1105.4605-2-30-5': 'The rank correlation coefficient and the power law fits are determined for regions where we have at least a marginal CO measurement.', '1105.4605-2-30-6': 'If we restrict our analysis to high significance CO measurements, we obtain the same results within the uncertainties.', '1105.4605-2-31-0': 'The table and figure show that recent star formation rate tracers rank-correlate approximately equally well with H[MATH] and H[MATH]i+H[MATH] both across all surface densities and separately in the H[MATH]i- and H[MATH]-dominated regimes.', '1105.4605-2-31-1': 'H[MATH]i does not correlate significantly with star formation in the H[MATH]-dominated inner parts of galaxy disks, though the correlation between H[MATH]i and recent star formation becomes stronger in the H[MATH]i-dominated outer parts of galaxies .', '1105.4605-2-32-0': 'The rank correlation is a non-parametric measure of how well the relative ordering of two data sets align.', '1105.4605-2-32-1': 'A high rank correlation coefficient implies a monotonic relationship but not a fixed functional form law.', '1105.4605-2-32-2': 'For total gas, the power law index relating gas and recent star formation depends fairly strongly on the subset of data used.', '1105.4605-2-32-3': 'If we focus on the regions where [MATH], the best-fit power law relating total gas and recent star formation (from FUV[MATH]24) has an index of [MATH].', '1105.4605-2-32-4': 'Where [MATH], the index is much shallower, [MATH].', '1105.4605-2-32-5': 'By contrast, the power law relating H[MATH] to recent star formation varies less across regimes, from [MATH] where [MATH] to [MATH] where [MATH].', '1105.4605-2-32-6': 'Both of these agree within the uncertainties with the fit to all data, which has slope [MATH].', '1105.4605-2-32-7': 'The slight steepening of the relation in the outer disks ([MATH]) is driven by the interplay of two effects: the rather small dynamic range in gas surface densities and an increased dispersion in SFR-to-H[MATH] ratios driven by the low mass galaxies; those have high SFR-to-H[MATH] ratios and contribute mostly to the H[MATH]i-dominated subset.', '1105.4605-2-32-8': 'The qualitative picture of a break in the total gas-SFR relation but a continuous H[MATH]-SFR relation remains unchanged if SFR(H[MATH][MATH]24) is considered, though there are small changes in the exact numbers (see the left and right panels in Figure [REF]).', '1105.4605-2-33-0': 'Our conclusions thus match those of [CITATION]: a single power law appears to be sufficient to relate [MATH] and [MATH] whereas the relationship between [MATH] and [MATH] varies systematically depending on the subset of data used.', '1105.4605-2-33-1': 'Because we have a dataset that includes significant CO measurements where [MATH] we can extend these findings.', '1105.4605-2-33-2': 'First, total gas and H[MATH] are equally well rank-correlated with recent star formation in all regimes and this correlation is always stronger than the correlation of recent star formation with H[MATH]i .', '1105.4605-2-33-3': 'Second, the H[MATH]-SFR scaling relation extends smoothly into the regime where [MATH] whereas the total gas-SFR relation does not.', '1105.4605-2-33-4': 'Third, the result is independent of our two star formation rate tracers SFR(FUV[MATH]24) and SFR(H[MATH][MATH]24).', '1105.4605-2-34-0': '### Discussion of CO-SFR Scaling Relations', '1105.4605-2-35-0': 'Empirical Results: IR brightness at both 24 and 70 correlates strongly with CO intensity over [MATH]3 orders of magnitude.', '1105.4605-2-35-1': 'Across this range, there is a nearly fixed ratio of CO to IR emission.', '1105.4605-2-35-2': 'FUV and H[MATH] emission show little or no correlation with CO intensity in the inner parts of galaxies, presumably due to extinction, but are found to correlate well with CO in the outer parts of galaxies after galaxy-to-galaxy scatter is removed.', '1105.4605-2-36-0': 'A result of these empirical scaling relations is that the ratio of recent star formation rate, traced either by combining FUV and 24 or H[MATH] and 24 intensities, to molecular gas, traced by CO emission, does not vary strongly between the H[MATH]i-dominated and H[MATH]-dominated ISM.', '1105.4605-2-36-1': 'This result is driven largely by the tight observed correlation between CO and 24 emission.', '1105.4605-2-36-2': 'The tight relation between CO and 70 emission suggests that the CO-IR relation actually holds for a larger range of mid-IR intensities.', '1105.4605-2-36-3': 'The tightening of the CO-FUV and CO-H[MATH] relation in the outer part of galaxies (after removing galaxy-to-galaxy scatter) reinforce the idea of a linear relation extending to large radii.', '1105.4605-2-36-4': 'These tight correlations are consistent with the conclusion of [CITATION] that there is only weak variation in the SFR per unit molecular gas mass with local environment.', '1105.4605-2-37-0': 'Galaxy-to-Galaxy Scatter: Each of the correlations we observe has significant internal scatter.', '1105.4605-2-37-1': 'Breaking this apart into scatter among galaxies and scatter within galaxies, we observe that in every case scatter among galaxies drives the overall scatter in the observed correlation.', '1105.4605-2-37-2': 'This "scatter" among galaxies is not random; less massive, less metal-rich galaxies exhibit a higher ratio of SFR tracer to CO emission and][]Young1996.', '1105.4605-2-37-3': 'There are two straightforward physical interpretations for this.', '1105.4605-2-37-4': 'The efficiency of star formation from H[MATH] gas may be genuinely higher in these systems, a view advocated by [CITATION] and [CITATION].', '1105.4605-2-37-5': 'Alternatively, CO emission may be depressed relative to the true amount of H[MATH] mass due to changes in the dust abundance.', '1105.4605-2-37-6': 'Low mass systems often have lower metallicities and correspondingly less dust, which is required to shield CO .', '1105.4605-2-37-7': 'A precise calibration of the CO-to-H[MATH] conversion factor as a function of metallicity is still lacking, so it is not possible at present to robustly distinguish between these two scenarios.', '1105.4605-2-38-0': 'After removing these galaxy-to-galaxy variations we find a series of extraordinarily tight relationships between CO and tracers of recent star formation.', '1105.4605-2-38-1': 'The most striking - and puzzling - example of this is the emergence of a tight correlation between CO and FUV emission in the outer parts of galaxies ([MATH]).', '1105.4605-2-38-2': 'This is puzzling because one would expect the galaxy to be mostly causally disconnected over the timescales predominantly traced by FUV emission - [MATH] Myr compared to a dynamical (orbital) time of a few [MATH] Myr.', '1105.4605-2-38-3': 'Yet, somehow the differences between galaxies affect the CO-to-FUV ratio much more than the differences between the widely separated rings represented by our data points.', '1105.4605-2-38-4': 'Galaxy-wide variations in metallicity and dust abundance probably offer the best explanation for this.', '1105.4605-2-38-5': 'These may propagate into variations in the CO-to-H[MATH] conversion factor and the average dust extinction.', '1105.4605-2-38-6': 'The latter may lead to scatter in estimates of SFR and both will affect [MATH].', '1105.4605-2-38-7': 'An alternative explanation is that external processes, which affect the whole galaxy, play a large role in setting the star formation rate on timescales traced by FUV emission.', '1105.4605-2-39-0': 'These strong galaxy-to-galaxy variations partially explain the unexpected lack of correlation between CO emission and recent SFR observed by [CITATION].', '1105.4605-2-39-1': 'They averaged across whole galaxies and in doing so conceivably lost the strong internal relations that we observe but preserved the large galaxy-to-galaxy variations that offset internal relations.', '1105.4605-2-39-2': 'The result will be an apparent lack of correlation in galaxy-averaged data that obscures the strong internal relationship.', '1105.4605-2-39-3': 'Whether there is in fact a weaker relationship between H[MATH] and SFR in galaxy-integrated measurements than inside galaxies depends on whether the suggested variations in star formation efficiency are real or a product of a varying CO-to-H[MATH] conversion factor.', '1105.4605-2-40-0': '(The Lack of) A Molecular Star Formation Law: With improved sensitivity to CO emission we now clearly see nearly linear relations between CO and tracers of recent star formation rate spanning from the H[MATH]-dominated to H[MATH]i-dominated parts of galaxies.', '1105.4605-2-40-1': 'Note that the relations will likely depart from these scaling relations if regions of high surface densities ([MATH] M[MATH] pc[MATH]) or starburst galaxies are considered .', '1105.4605-2-40-2': 'The lack of strong variations in the scaling between these two quantities in the "non-starburst regime" reinforces that molecular gas is the key prerequisite for star formation.', '1105.4605-2-40-3': 'A nearly linear correlation over this whole range can also be restated as the absence of a strong relationship between the ratio [MATH] and [MATH].', '1105.4605-2-40-4': 'This implies that [MATH] averaged over a large area is not a key environmental quantity for star formation because it does not affect the rate of star formation per unit molecular gas.', '1105.4605-2-40-5': 'Apparently the global amount of H[MATH] directly sets the global amount of star formation but the surface density of H[MATH] does not affect how quickly molecular gas is converted to stars.', '1105.4605-2-41-0': 'By contrast, the host galaxy does appear to affect the ratio of star formation rate to at least CO intensity.', '1105.4605-2-41-1': 'This indicates important environmental variations but they are not closely linked to surface density.', '1105.4605-2-41-2': 'In this sense, Figure [REF] [REF] offer a counterargument against the idea of a star formation "law" in which gas surface density alone sets the star formation rate.', '1105.4605-2-41-3': 'Instead, over the disks of normal galaxies, we see star formation governed by two processes: (a) the formation of stars in molecular gas, which varies mildly from galaxy to galaxy but appears largely fixed inside a galaxy, and (b) the conversion of H[MATH]i to H[MATH], which does exhibit a strong dependence on environment inside a galaxy, including a strong dependence on surface density (Section [REF]).', '1105.4605-2-41-4': 'In the second part of this paper we will look at this second process by measuring variations in the H[MATH] -H[MATH]i balance as a function of gas surface density and radius.', '1105.4605-2-42-0': 'Systematic Effects: We have interpreted the observed scaling relations in terms of a relationship between molecular gas and recent star formation rate.', '1105.4605-2-42-1': '[CITATION] demonstrated that azimuthally averaged profiles of the FUV[MATH]24 combination that we use here match those of several commonly used star formation rate tracers with [MATH]50% scatter.', '1105.4605-2-42-2': 'Here we have shown that using a combination of H[MATH][MATH]24 to determine the SFR leads to indistinguishable results (see Leroy et al. 2011, in prep.', '1105.4605-2-42-3': 'for more discussion).', '1105.4605-2-42-4': 'We now discuss several systematic effects that may affect the translation from observables to inferred quantities.', '1105.4605-2-43-0': 'The most serious worry is that the IR intensity, which drives the correlations, is acting as a tracer of dust abundance and not recent star formation.', '1105.4605-2-43-1': 'Gas and dust are observed to be well mixed in the ISM, so in the extreme, this would result in plotting gas against gas times some scaling factor (the dust-to-gas ratio).', '1105.4605-2-43-2': 'A more subtle version of the same concern is that CO emission is primarily a function of dust shielding against dissociating UV radiation.', '1105.4605-2-43-3': 'If there are large variations in the abundance of dust in the ISM then it may be likely that dust emission and CO emission emerge from the same regions because that is where CO can form and evade dissociation.', '1105.4605-2-44-0': 'A few considerations suggest that the 24 and the 70 emission are not primarily tracing dust abundance.', '1105.4605-2-44-1': 'First, over whole galaxies, monochromatic IR emission at 24 and 70 does track the SFR .', '1105.4605-2-44-2': 'Second, we observe a linear correlation with CO emission and not with overall gas column, which one might expect for a dust tracer.', '1105.4605-2-44-3': 'Third, both the 24 and 70 bands are well towards the blue side of the peak of the IR SED for dust mixed with non star-forming gas and so are not likely to be direct tracers of the dust optical depth (mass).', '1105.4605-2-44-4': 'Still, a thorough investigation of the interplay between dust abundance, IR emission, and star formation is needed to place SFR tracers involving IR emission on firmer physical footing.', '1105.4605-2-45-0': 'A less severe worry is that using dust and FUV emission makes us sensitive to an old stellar component that might not have formed locally.', '1105.4605-2-45-1': 'Our targets are all actively star-forming systems, so old here means mainly old relative to H[MATH] emission .', '1105.4605-2-45-2': 'The appropriate timescale to use when relating star formation and gas is ambiguous.', '1105.4605-2-45-3': 'When studying an individual region, it may be desirable to use a tracer with the shortest possible time sensitivity.', '1105.4605-2-45-4': 'Averaging over large parts of galaxies, one is implicitly trying to get at the equilibrium relation.', '1105.4605-2-45-5': 'Therefore a tracer with a somewhat longer timescale sensitivity may actually be desirable.', '1105.4605-2-45-6': 'The typical [MATH] Myr timescale over which most UV emission (and dust heating from B stars) occurs is well matched to current estimates for the lifetimes of giant molecular clouds .', '1105.4605-2-45-7': 'This makes for a fairly symmetric measurement - with the spatial and time scales of the two axes matched - though one is comparing recent star formation with the material of future star-forming regions.', '1105.4605-2-46-0': 'The fact that SFRs derived from FUV[MATH]24 and H[MATH][MATH]24 are essentially indistinguishable indicates that the distinction between the time scale probed by H[MATH]4 Myr) and FUV ([MATH] Myr) is not important to this study, probably because of the large spatial scales considered by our azimuthal averages.', '1105.4605-2-47-0': 'There may also be systematic biases in our inferred [MATH].', '1105.4605-2-47-1': 'We have already discussed the dependence of [MATH] on metallicity as a possible explanation for the high SFR-to-CO ratio observed in lower-mass galaxies.', '1105.4605-2-47-2': '[MATH] certainly depends on metallicity.', '1105.4605-2-47-3': 'Current best estimates imply a non-linear relationship, with [MATH] sharply increasing below [MATH] .', '1105.4605-2-47-4': 'For our range of metallicities ([MATH], though the uncertainties on that ratio are large.', '1105.4605-2-47-5': 'More important, [CITATION] suggest variations in the CO line ratios between the inner and outer Milky Way and there are well-established differences between normal disk and starburst galaxies.', '1105.4605-2-47-6': 'Although not immediately apparent from a comparison of HERACLES to literature CO([MATH]) data , such variations could affect our derived [MATH] by as much as [MATH]50%.', '1105.4605-2-47-7': 'Rosolowsky et al. (2011, in prep.)', '1105.4605-2-47-8': 'will present a thorough investigation of how the line ratio varies with environment in HERACLES.', '1105.4605-2-48-0': '## Distribution of Molecular Gas', '1105.4605-2-49-0': 'The tight correlation between SFR tracers and CO emission across all regimes strongly reinforces the primary importance of molecular gas to star formation.', '1105.4605-2-49-1': 'In this section, we therefore examine the distribution of molecular gas in galaxies.', '1105.4605-2-49-2': 'In the outer parts of spiral galaxies where [MATH], the formation of molecular gas from atomic gas appears to represent the bottleneck to star formation and the relative abundance of H[MATH] and H[MATH]i is key to setting the star formation rate .', '1105.4605-2-49-3': 'We can apply the large dynamic range in H[MATH]-to-H[MATH]i ratios achieved by stacking to make improved measurements of how this key quantity varies across galaxies.', '1105.4605-2-50-0': '### Radial Distribution of CO Intensity', '1105.4605-2-51-0': 'Many previous studies have shown that azimuthally averaged CO emission decreases with increasing galactocentric radius .', '1105.4605-2-51-1': 'Whereas galaxy centers often exhibit deviations from the large scale trend, CO emission outside the centers declines approximately uniformly with radius .', '1105.4605-2-51-2': 'A first analysis of the HERACLES data revealed a characteristic exponential decline of CO emission in the inner parts of galaxy disks, with the scale length of CO emission similar to that of old stars and tracers of recent star formation .', '1105.4605-2-51-3': 'With the increased sensitivity from stacking and a larger sample, we can revisit this question and ask if this radial decline in CO intensity continues smoothly out to [MATH].', '1105.4605-2-52-0': 'From exponential fits to the high-significance CO data of each galaxy (solid-dashed lines in Figure [REF] and Appendix), excluding the galaxy centers (inner 30) we find a median exponential scale length, [MATH], of [MATH] with [MATH]% of all [MATH] between [MATH].', '1105.4605-2-52-1': 'This value agrees well with typical scale lengths found in previous studies and the individual galaxy scale lengths agree well with previous work on the HERACLES sample .', '1105.4605-2-53-0': 'The normalizations of these fits reflect galaxy-to-galaxy variations in the total molecular gas content.', '1105.4605-2-53-1': 'In the left panel of Figure [REF] we show all profiles aligned to a common normalization.', '1105.4605-2-53-2': 'We plot the radius in units of [MATH], the 25[MATH] magnitude [MATH]-band isophote, and normalize each profile so that the exponential fits have intensity [MATH] at [MATH].', '1105.4605-2-53-3': 'The figure thus shows the radial variation of CO intensity across our sample, controlled for the overall CO luminosity and absolute size of each galaxy.', '1105.4605-2-53-4': 'Thick crosses mark the median CO intensity and the [MATH]th percentile range in bins [MATH] wide.', '1105.4605-2-53-5': 'The same exponential decline seen in individual profiles is even more evident here, with [MATH] for the average of the sample.', '1105.4605-2-54-0': 'The left panel in Figure [REF] shows that the radial decline of the CO profiles observed previously for the inner part of galaxies extends without significant changes out to our last measured data points.', '1105.4605-2-54-1': 'In most galaxies there is no clear evidence for a sharp cutoff or a change in slope.', '1105.4605-2-54-2': 'As long as the normalized profiles are above the sensitivity limit the decline appears to continue (without significant deviation) with [MATH] on average.', '1105.4605-2-54-3': 'This smooth exponential decline in CO intensity with increasing radius suggests that the observed decline in star formation rate from the inner to outer parts of galaxy disks is driven by a continuous decrease in the supply of molecular gas, rather than a sharp threshold of some kind.', '1105.4605-2-55-0': 'There are several galaxies which deviate from this median exponential trend.', '1105.4605-2-55-1': 'We already noted above that we do not fit exponential profiles to NGC 2798, 2976, 4725 because their gas (H[MATH]i and CO) radial profiles are insufficiently parametrized by exponentials.', '1105.4605-2-55-2': 'There are three galaxies (NGC 2146, 2903, 4569) that we do fit and determine small exponential scale lengths, [MATH].', '1105.4605-2-55-3': 'NGC 2146 hosts an ongoing starburst and both NGC 2903 and NGC 4569 have prominent bars that may be funneling molecular gas to their centers.', '1105.4605-2-55-4': 'For two other galaxies (NGC 2841 and 4579) we determine large exponential scale lengths, [MATH].', '1105.4605-2-55-5': 'These galaxies are better described by a flat distribution (or even a central depression) and a cutoff at larger radii.', '1105.4605-2-56-0': 'The tight correspondence of the CO scale length, [MATH] to [MATH] has been noted before , while other studies have found a close correspondence between CO and near-infrared light .', '1105.4605-2-56-1': 'In our sample there is a fairly good correspondence between [MATH] and the near-infrared scale length, [MATH], measured at [MATH] with [MATH] .', '1105.4605-2-56-2': 'The near-infrared light should approximately trace the distribution of stellar mass, so that our measured scale length is very similar to that of the stellar mass.', '1105.4605-2-56-3': 'This tight coupling has been interpreted to indicate the importance of the stellar potential well to collecting star-forming material .', '1105.4605-2-56-4': 'Here we see this correspondence to continue into the regime where the molecular gas is not the dominant gas component, confirming that molecular gas formation is the bottleneck to star formation.', '1105.4605-2-57-0': 'The right panel in Figure [REF] shows the distribution of enclosed luminosity, mean intensity, and flux at each radius for an exponential disk with scale length of [MATH].', '1105.4605-2-57-1': 'The brightest individual ring for such a disk lies at [MATH] and half the flux is enclosed within [MATH].', '1105.4605-2-57-2': 'This value, [MATH], is fairly close to the radius at which [MATH] in a typical disk galaxy and][]Leroy2008, so that CO emission is about evenly split between the H[MATH]-dominated and H[MATH]i -dominated parts of such a galaxy.', '1105.4605-2-57-3': 'Meanwhile, 90 of the flux lies within [MATH] a value that is very similar to the threshold radius identified by [CITATION].', '1105.4605-2-57-4': 'We do not find evidence to support a true break at this radius, but as an "edge" to the star-forming disk, a 90 contour may have utility.', '1105.4605-2-58-0': '### The H[MATH]-to-HI Ratio', '1105.4605-2-59-0': 'In the outer parts of galaxy disks - and thus over most of the area in galaxy disks - we have [MATH], implying that star-forming H[MATH] gas does not make up most of the interstellar medium.', '1105.4605-2-59-1': 'In this regime the relative abundance of H[MATH] and H[MATH]i is a key quantity to regulate the star formation rate.', '1105.4605-2-59-2': 'Observations over the last decade have revealed strong variations of the fraction of gas in the molecular phase as a function of galactocentric radius, total gas surface density, stellar surface density, disk orbital time, and interstellar pressure .', '1105.4605-2-59-3': 'In Figure [REF] we show the two most basic of these trends, the H[MATH]-to-H[MATH]i ratio, [MATH], as a function of normalized galactocentric radius (left panel) and total gas surface density (right panel).', '1105.4605-2-59-4': 'We focus on [MATH] because it is more easily separated in discrete observables than the fraction of gas that is molecular, [MATH].', '1105.4605-2-59-5': 'This makes it easier to interpret uncertainties and systematic effects like changes in the CO-to-H[MATH] conversion factor.', '1105.4605-2-60-0': 'The left panel of Figure [REF] shows [MATH] as function of galactocentric radius in units of [MATH] for all data detected with high or marginal significance.', '1105.4605-2-60-1': 'For clarity, we do not plot [MATH] for regions where we determined only upper limits in CO intensity; for the inner parts ([MATH]) these upper limits are bounded by [MATH], whereas for outer parts ([MATH]) the upper limits in [MATH] are typically of comparable magnitude as measurements of [MATH] and upper limits are bounded by [MATH].', '1105.4605-2-60-2': 'In agreement with [CITATION], [CITATION], [CITATION], and [CITATION] we find [MATH] to decline with increasing galactocentric radius, a variation that reflects the distinct radial profiles of atomic and molecular gas.', '1105.4605-2-60-3': 'However, radial variations alone do not explain the full range of observed molecular fractions because [MATH] can vary by up to two orders of magnitude at any given galactocentric radius.', '1105.4605-2-61-0': 'A significant part of the variations in [MATH] at a given galactocentric radius corresponds to systematic variations between galaxies.', '1105.4605-2-61-1': 'These are mainly caused by variations in the absolute molecular gas content of a galaxy and can be removed by a normalization procedure similar to one applied in the left panel of Figure [REF].', '1105.4605-2-61-2': 'The result is similar to that seen for the radial profiles of CO: the relationship tightens and we can see that most of the decline of [MATH] inside a galaxy occurs radially.', '1105.4605-2-61-3': 'However, there is significantly more scatter remaining in [MATH] versus radius than we observed for the normalized CO radial profiles, highlighting the importance of parameters other than a combination of radius and host galaxy to set [MATH].', '1105.4605-2-62-0': 'In addition to declining with increasing galactocentric radius, [MATH] increases as the total gas surface density, [MATH], increases.', '1105.4605-2-62-1': 'The more gas that is present along a line of sight, the larger the fraction of gas that is molecular.', '1105.4605-2-62-2': 'The right panel of Figure [REF] shows this result, plotting [MATH] as function of the total gas surface density, [MATH].', '1105.4605-2-62-3': '[MATH] increases with increasing [MATH], with regions of high surface density, [MATH] M[MATH] pc[MATH], being predominately molecular, [MATH].', '1105.4605-2-63-0': 'At high [MATH], the right panel of Figure [REF] is largely a way of visualizing the "saturation" of [MATH] on large scales in galaxies.', '1105.4605-2-63-1': 'A number of authors have found that averaged over hundreds of parsecs to kpc scales, [MATH] rarely exceeds [MATH]10 M[MATH] pc[MATH] .', '1105.4605-2-63-2': 'This limit is clearly violated at high spatial resolution and may vary among classes of galaxies.', '1105.4605-2-63-3': 'Several recent theoretical works have aimed at reproducing this behavior.', '1105.4605-2-63-4': '[CITATION] focused on shielding of H[MATH] inside individual atomic-molecular complexes, whereas [CITATION] examined the interplay between large-scale thermal and dynamical equilibria.', '1105.4605-2-64-0': 'The right panel in Figure [REF] includes also the predicted solar-metallicity [MATH]-[MATH] relation from [CITATION].', '1105.4605-2-64-1': 'They model individual atomic-molecular complexes, however these complexes have a filling factor substantially less than [MATH] inside our beam.', '1105.4605-2-64-2': 'The appropriate [MATH] to input into their model is therefore the average surface densities of the complexes, [MATH], within our beam, which will be related to our observed surface density, [MATH], by the filling factor, [MATH], namely: [MATH].', '1105.4605-2-64-3': 'The model curve (blue short-dashed line) with a filling factor [MATH] is offset towards higher [MATH] (shifted right) or lower [MATH] (shifted down) compared to our data.', '1105.4605-2-64-4': 'To have the model curve intersect our data (red long-dashed line) requires a filling factor of [MATH], so that atomic-molecular complexes fill about half the areas in our beam.', '1105.4605-2-64-5': 'This is before any accounting for the presence of diffuse H[MATH]i not in star-forming atomic-molecular complexes and is assuming a fixed filling factor, both of which are likely oversimplifications.', '1105.4605-2-64-6': 'Nonetheless, the [CITATION] curve does show an overall good correspondence to our data.', '1105.4605-2-65-0': 'As with the radius, the total gas surface density predicts some of the broad behavior of [MATH] but knowing [MATH] does not uniquely specify the amount of molecular gas, particularly at low [MATH].', '1105.4605-2-65-1': 'This are visible as the large scatter in [MATH] at low surface densities in the right panel of Figure [REF].', '1105.4605-2-65-2': 'The scatter reflects a dependence of [MATH] on environmental factors other than gas surface density.', '1105.4605-2-65-3': 'Figure [REF] shows [MATH] over the small range [MATH] M[MATH] pc[MATH] , i.e., the data from the gray highlighted region in Figure [REF].', '1105.4605-2-65-4': 'We plot histograms for several radial bins which are clearly offset, indicating an additional radial dependence of [MATH].', '1105.4605-2-65-5': 'At small radii ([MATH]) we observe a large scatter in [MATH] for [MATH] M[MATH] pc[MATH] whereas at large radii ([MATH]) gas with this surface density is always predominantly H[MATH]i .', '1105.4605-2-66-0': 'As was the case in SFR-H[MATH] space, distinct populations of galaxies are responsible for some of the variations in the right panel of Figure [REF].', '1105.4605-2-66-1': 'Early type (Sab-Sb) spirals (e.g., NGC 2841, 3351, 3627, 4736) often show large molecular fractions, [MATH], but typically have low H[MATH]i and H[MATH] surface densities, [MATH] and [MATH] M[MATH] pc[MATH] .', '1105.4605-2-66-2': 'By contrast, massive Sc galaxies (e.g., NGC 4254, 4321, 5194, 6946) can have comparably high molecular fractions, [MATH], but have higher surface density H[MATH]i disks, [MATH] M[MATH] pc[MATH] , so that these fractions occur at higher [MATH].', '1105.4605-2-66-3': 'A trend with metallicity is not immediately obvious in the data but these differences may reflect the more substantial stellar surface densities found in the earlier-type galaxies.', '1105.4605-2-66-4': 'This increased stellar surface density results in a stronger gravitational field, which could lead to a higher midplane gas pressure and a low fraction of diffuse H[MATH]i gas .', '1105.4605-2-67-0': '### Discussion of [MATH]', '1105.4605-2-68-0': 'Following several recent studies we observe strong systematic variations in the H[MATH]-H[MATH]i balance across galaxies.', '1105.4605-2-68-1': 'Two of the strongest behaviors are an approximately exponential decrease in [MATH] with increasing galactocentric radius and a steady increase in [MATH] with increasing gas surface density.', '1105.4605-2-68-2': 'Our improved sensitivity shows these trends extending to low surface densities and our expanded sample makes clear that neither of these basic parametrizations adequately captures the entire range of [MATH] variations.', '1105.4605-2-68-3': 'The likely physical drivers for the scatter in [MATH] that we observe are metallicity and dust-to-gas ratio , the dissociating radiation field , variations in interstellar gas pressure and density , and external perturbations that drive gas to higher densities .', '1105.4605-2-68-4': 'Each of these quantities are observationally accessible in our sample and estimates of the relative roles of each process will be presented in Leroy et al. (2011, in prep.)', '1105.4605-2-69-0': 'The overall relationship of total gas and star formation rate (middle panels of Figure [REF]) can be reproduced by a roughly fixed ratio of SFR-to-H[MATH] within galaxies (Section [REF]) and the observed scaling of [MATH] with [MATH] (Section [REF]).', '1105.4605-2-69-1': 'This is shown in Figure [REF], where we plot the SFR per unit total gas, [MATH] (the inverse of the total gas depletion time) as function of the molecular-to-atomic ratio, [MATH].', '1105.4605-2-69-2': 'At low surface densities, the [MATH]-[MATH] relation regulates [MATH], at high surface densities where almost all of the gas is molecular the SFR-H[MATH] scaling determines the observed ratio.', '1105.4605-2-69-3': 'This trend is well parametrized by a fixed SFR-to-H[MATH] ratio with a molecular gas depletion time of [MATH] Gyr (blue dotted line) for all of our targets and [MATH] Gyr (red dashed line) for big spirals .', '1105.4605-2-69-4': 'The transition between these two regimes creates the curved shape seen in Figure [REF] [REF].', '1105.4605-2-69-5': 'This offers more support for a modified version of the classical picture of a star formation threshold , in which dense, mostly molecular gas forms stars but the efficiency with which such (molecular) gas forms is a strong function of environment, decreasing steadily with decreasing gas surface density and increasing galactocentric radius.', '1105.4605-2-70-0': '# Summary', '1105.4605-2-71-0': 'We combine HERACLES CO([MATH]) data with H[MATH]i velocity fields, mostly from THINGS, to make sensitive measurements of CO intensity across the disks of 33 nearby star-forming galaxies.', '1105.4605-2-71-1': 'We stack CO spectra across many lines of sight by assuming that the mean H[MATH]i and CO velocities are similar, an assumption that we verify in the inner parts of galaxies.', '1105.4605-2-71-2': 'This approach allows us to detect CO out to galactocentric radii [MATH].', '1105.4605-2-71-3': 'Because we measure integrated CO intensities as low as [MATH] K km s[MATH]1 M[MATH] pc[MATH] , before any correction for inclination) with high significance, we are able to robustly measure CO intensities in parts of galaxies where most of the ISM is atomic.', '1105.4605-2-72-0': 'Using this approach we compare the radially averaged intensities of FUV, H[MATH], IR, CO, and H[MATH]i emission across galaxy disks.', '1105.4605-2-72-1': 'We find an approximately linear relation between CO intensity and monochromatic IR intensity at both 24 and 70 .', '1105.4605-2-72-2': 'For the first time, we show that these scaling relations continue smoothly from the H[MATH]-dominated to H[MATH]i-dominated ISM.', '1105.4605-2-72-3': 'Extinction causes FUV and H[MATH] emission to display a more complex relationship with CO, especially in the inner parts of galaxies.', '1105.4605-2-72-4': 'In the outer parts of galaxy disks FUV and H[MATH] emission do correlate tightly with CO emission after galaxy-to-galaxy variations are removed.', '1105.4605-2-73-0': 'We use two calibration to estimate the recent star formation rate, FUV[MATH]24 and H[MATH][MATH]24, which we compare to H[MATH] derived from CO.', '1105.4605-2-73-1': 'We find an approximately linear relation between [MATH] and [MATH] in the range of [MATH] M[MATH] pc[MATH] with no notable variation between the two SFR estimates.', '1105.4605-2-73-2': 'A number of recent studies have also seen a roughly linear relationship between [MATH] and [MATH] and have argued that it implies that the surface density of H[MATH] averaged over large scales does not strongly affect the efficiency with which molecular gas forms stars.', '1105.4605-2-74-0': 'We do find evidence for variations in the SFR-to-CO ratio among galaxies.', '1105.4605-2-74-1': 'Indeed, most of the scatter in the relations between CO and SFR tracers is driven by galaxy-to-galaxy variations.', '1105.4605-2-74-2': 'These variations are not random, but show the trend observed by [CITATION] that lower mass, lower metallicity galaxies have higher ratios of SFR-to-H[MATH] than massive disk galaxies.', '1105.4605-2-74-3': 'It will take further study to determine whether these are real variations in the efficiency of star formation or reflect changes in the CO-to-H[MATH] conversion factor due to lower metallicities in these systems.', '1105.4605-2-74-4': 'After removing these galaxy-to-galaxy variations the composite H[MATH]-SFR relation is remarkably tight, reinforcing a close link between H[MATH] and star formation inside galaxies.', '1105.4605-2-75-0': 'We compare the scaling between the surface densities of SFR and H[MATH]i , H[MATH] , and total gas (H[MATH]i+H[MATH]).', '1105.4605-2-75-1': 'The relationship between SFR and total gas has roughly the same rank correlation coefficient as that between SFR and H[MATH], but does not obey a single functional form.', '1105.4605-2-75-2': 'Where [MATH] the relationship between [MATH] and [MATH] is steep whereas where [MATH] the relationship is much flatter.', '1105.4605-2-75-3': 'Meanwhile, we observe a linear relationship between [MATH] and [MATH] for the full range of [MATH] M[MATH] pc[MATH] .', '1105.4605-2-75-4': '[MATH] and [MATH] are weakly correlated and exhibit a strongly nonlinear relation, except at very large radii.', '1105.4605-2-76-0': 'The unbroken extension of the [MATH]-[MATH] relation into the H[MATH]i -dominated regime suggests a modified version of the classical picture of a star formation threshold , in which stars form at fixed efficiency out of molecular gas, to first order independent of environment within a galaxy.', '1105.4605-2-76-1': 'The observed turn-over in the relation between SFR and total gas relates to the H[MATH]-to-H[MATH]i ratio which is a strong function of environment.', '1105.4605-2-77-0': 'We therefore investigate the distribution of H[MATH] traced by CO using our stacked data and compare it to the H[MATH]i .', '1105.4605-2-77-1': 'On large scales we observe CO to decrease exponentially with a remarkably uniform scale length of [MATH], again extending previous studies to lower surface densities.', '1105.4605-2-77-2': 'We find the normalization of this exponential decline to vary significantly among galaxies.', '1105.4605-2-77-3': 'The H[MATH]-to-H[MATH]i ratio, traced by the ratio of CO-to-H[MATH]i intensities, also varies systematically across galaxies.', '1105.4605-2-77-4': 'It exhibits significant correlations with both galactocentric radius and total gas surface density and we present high-sensitivity measurements of both of these relationships.', '1105.4605-2-77-5': 'However, neither quantity is sufficient to uniquely predict the H[MATH]-to-H[MATH]i ratio on its own.', '1105.4605-2-78-0': 'We thank the teams of SINGS, LVL, and GALEX NGS for making their outstanding data sets available.', '1105.4605-2-78-1': 'We thank Daniela Calzetti for helpful comments.', '1105.4605-2-78-2': 'We thank the anonymous referee for thoughtful comments that improved the paper.', '1105.4605-2-78-3': 'The work of AS was supported by the Deutsche Forschungsmeinschaft (DFG) Priority Program 1177.', '1105.4605-2-78-4': 'Support for AKL was provided by NASA through Hubble Fellowship grant HST-HF-51258.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.', '1105.4605-2-78-5': 'The work of WJGbB is based upon research supported by the South African Research Chairs Initiative of the Department of Science and Technology and the National Research Foundation.', '1105.4605-2-78-6': 'This research made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the JPL/Caltech, under contract with NASA, NASAs Astrophysical Data System (ADS), and the HyperLeda catalog, located on the WorldWide Web at http://www.obs.univ-lyon1.fr/hypercat/intro.html.', '1105.4605-2-79-0': 'Here we present the radial profiles of CO, H[MATH]i , FUV, H[MATH] , and IR at 24 and 70 used to generate the plots in this paper.', '1105.4605-2-79-1': 'We average the data in 15 wide tilted rings.', '1105.4605-2-79-2': 'For the CO data, we stack the shifted spectra over this area and determine the integrated CO intensity from fitting line profiles to the stacked spectrum.', '1105.4605-2-79-3': 'For the H[MATH]i , FUV, H[MATH] , and IR data we use two-dimensional maps of intensity (Section [REF]).', '1105.4605-2-79-4': 'We determine the [MATH] scatter from the 68[MATH]-percentile from the data inside each ring.', '1105.4605-2-79-5': 'We plot these as error bars but note the distinction from the uncertainty in the mean.', '1105.4605-2-80-0': 'For each galaxy we present two plots: The left panel shows CO and H[MATH]i both in units of observed intensities (K km s[MATH] , left-hand [MATH]-axis) and converted to mass surface densities (M[MATH] pc[MATH] , right-hand [MATH]-axis) of H[MATH] and H[MATH]i .', '1105.4605-2-80-1': 'The color of the CO points indicates the significance with which we could determine the integrated CO intensities: green for high significance measurements, orange for measurements of marginal significance and red for [MATH] upper limits.', '1105.4605-2-80-2': 'To have H[MATH] and H[MATH]i on the same mass surface density scale, we multiplied the observed 21-cm line intensities by a factor of [MATH] (the ratio of Equations [REF] and [REF]).', '1105.4605-2-80-3': 'We also plot the star formation rate (SFR) surface density (M[MATH] yr[MATH] kpc[MATH] ) determined from H[MATH][MATH]24 and FUV[MATH]24.', '1105.4605-2-80-4': 'Black solid-dashed lines show our exponential fit to the radial CO profile.', '1105.4605-2-80-5': 'We fit all high significance data excluding galaxy centers, defined as the the inner 30, which often exhibit breaks from the overall profile .', '1105.4605-2-80-6': 'The derived exponential scale lengths (in units of [MATH], the radius of the 25[MATH] magnitude [MATH]-Band isophote), appear in the lower left corner.', '1105.4605-2-81-0': 'The right panel shows observed intensities (in MJy sr[MATH]) of our SFR tracers - H[MATH] , FUV, 24 and 70 emission.'}

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['1105.4605-1-37-4', '1105.4605-2-37-4'], ['1105.4605-1-37-5', '1105.4605-2-37-5'], ['1105.4605-1-37-6', '1105.4605-2-37-6'], ['1105.4605-1-37-7', '1105.4605-2-37-7'], ['1105.4605-1-71-0', '1105.4605-2-71-0'], ['1105.4605-1-71-1', '1105.4605-2-71-1'], ['1105.4605-1-71-2', '1105.4605-2-71-2'], ['1105.4605-1-71-3', '1105.4605-2-71-3'], ['1105.4605-1-30-0', '1105.4605-2-30-0'], ['1105.4605-1-30-1', '1105.4605-2-30-1'], ['1105.4605-1-30-2', '1105.4605-2-30-2'], ['1105.4605-1-30-3', '1105.4605-2-30-3'], ['1105.4605-1-30-4', '1105.4605-2-30-4'], ['1105.4605-1-30-5', '1105.4605-2-30-5'], ['1105.4605-1-30-6', '1105.4605-2-30-6'], ['1105.4605-1-1-0', '1105.4605-2-1-0'], ['1105.4605-1-1-1', '1105.4605-2-1-1'], ['1105.4605-1-1-2', '1105.4605-2-1-2'], ['1105.4605-1-1-3', '1105.4605-2-1-3'], ['1105.4605-1-47-0', '1105.4605-2-47-0'], ['1105.4605-1-47-1', '1105.4605-2-47-1'], ['1105.4605-1-47-2', '1105.4605-2-47-2'], ['1105.4605-1-47-3', '1105.4605-2-47-3'], ['1105.4605-1-47-4', '1105.4605-2-47-4'], ['1105.4605-1-47-5', '1105.4605-2-47-5'], ['1105.4605-1-47-6', '1105.4605-2-47-6'], ['1105.4605-1-47-8', '1105.4605-2-47-8']]

[]

[]

[]

[]

['1105.4605-1-19-0', '1105.4605-1-20-0', '1105.4605-1-21-0', '1105.4605-1-22-0', '1105.4605-1-23-0', '1105.4605-1-24-0', '1105.4605-1-25-0', '1105.4605-1-26-0', '1105.4605-1-27-0', '1105.4605-1-28-0', '1105.4605-1-29-0', '1105.4605-1-42-3', '1105.4605-1-47-7', '1105.4605-2-19-0', '1105.4605-2-20-0', '1105.4605-2-21-0', '1105.4605-2-22-0', '1105.4605-2-23-0', '1105.4605-2-24-0', '1105.4605-2-25-0', '1105.4605-2-26-0', '1105.4605-2-27-0', '1105.4605-2-28-0', '1105.4605-2-29-0', '1105.4605-2-42-3', '1105.4605-2-47-7']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1105.4605

1811.08307

{'1811.08307-1-0-0': 'We derive characteristic functions to determine the number and stability of relaxation oscillations for a class of planar systems.', '1811.08307-1-0-1': 'Applying our criterion, we give conditions under which the chemostat predator-prey system has exactly one or two relaxation oscillations.', '1811.08307-1-0-2': 'By using numerical simulation on our criterion, we verify that, for an epidemic model, the two family of periodic cycles discovered by Li et al. [SIAM J. Appl.', '1811.08307-1-0-3': 'Math, 2016] are a pair of relaxation oscillations.', '1811.08307-1-1-0': '# Introduction', '1811.08307-1-2-0': 'Periodic cycles in ecological models because are important they can be used to explain oscillatory phenomena observed in real-world data.', '1811.08307-1-2-1': 'Relaxation oscillations are periodic cycles formed of sections of both slow- and fast-motion parts of phase trajectories.', '1811.08307-1-2-2': 'In this paper, we extend the criterion for the existence of relaxation oscillations given by Hsu [CITATION] to a class of planar systems.', '1811.08307-1-2-3': 'We apply our criterion on two ecological models: a predator-prey system in a chemostat, and an epidemic model.', '1811.08307-1-2-4': 'Both systems are three-dimensional, and have two-dimensional invariant manifolds.', '1811.08307-1-3-0': 'Predator-prey interaction in a well-stirred chemostat can be modeled by the following ordinary differential equations: [EQUATION]', '1811.08307-1-3-1': 'Taking the limit as [MATH] in [REF] and [REF], we obtain [EQUATION]', '1811.08307-1-3-2': 'Similarly, [EQUATION]'}

{'1811.08307-2-0-0': 'We derive characteristic functions to determine the number and stability of relaxation oscillations for a class of planar systems.', '1811.08307-2-0-1': 'Applying our criterion, we give conditions under which the chemostat predator-prey system has a globally orbitally asymptotically stable limit cycle.', '1811.08307-2-0-2': 'Also we demonstrate that a prescribed number of relaxation oscillations can be constructed by varying the perturbation for an epidemic model studied by Li et al. [SIAM J. Appl.', '1811.08307-2-0-3': 'Math, 2016].', '1811.08307-2-1-0': '# Introduction', '1811.08307-2-2-0': 'Periodic orbits in ecological models are important because they can be used to explain oscillatory phenomena observed in real-world data.', '1811.08307-2-2-1': 'Relaxation oscillations are periodic orbits formed from slow and fast sections.', '1811.08307-2-2-2': 'In this paper, we extend the criterion for the existence of relaxation oscillations given by Hsu [CITATION] to a class of planar systems.', '1811.08307-2-2-3': 'We apply our criterion to two ecological models: a predator-prey system in a chemostat, and an epidemic model.', '1811.08307-2-2-4': 'Both systems are three-dimensional, and have two-dimensional invariant manifolds.', '1811.08307-2-3-0': 'Predator-prey interaction in a well-stirred chemostat (see e.g. [CITATION]) can be modeled by the following ordinary differential equations: [EQUATION] where [MATH] denotes [MATH], [MATH] is the concentration of the nutrient in the growth chamber at time [MATH], [MATH] and [MATH] are the density of prey (which feeds off this nutrient) and predator populations, respectively; [MATH] denotes the concentration of the input nutrient, and [MATH] and [MATH] are constants related to the consumption of the nutrient by the prey population and the consumption of the prey by the predators, respectively.', '1811.08307-2-3-1': 'To ensure that the volume of this vessel remains constant, [MATH] denotes both the rate of inflow from the nutrient reservoir to the growth chamber, as well as the rate of outflow from the growth chamber.', '1811.08307-2-3-2': 'The functional response [MATH], which describes the change in the density of the prey attacked per unit time per predator, is continuously differentiable and satisfies [EQUATION]', '1811.08307-2-3-3': 'It can be verified that system [REF] under assumption [REF] has a unique positive equilibrium for all small [MATH].', '1811.08307-2-3-4': 'From the equations in [REF], [EQUATION] so system [REF] has an invariant simplex [EQUATION] that attracts all points in [MATH].', '1811.08307-2-3-5': 'For system [REF] with [MATH], there is a continuous family of heteroclinic orbits on [MATH] as illustrated in Figure [REF] (see equation [REF] in Section [REF] and its succeeding paragraph for the limiting system on [MATH]).', '1811.08307-2-3-6': 'Each heteroclinic orbit connects two points on the boundary of [MATH], where [MATH].', '1811.08307-2-3-7': 'On the other hand, the restriction of system [REF] on the plane [MATH] is [EQUATION]', '1811.08307-2-3-8': 'Hence the segment [MATH] is a trajectory of system [REF] approaching the positive [MATH]-axis.', '1811.08307-2-3-9': 'The heteroclinic orbits for the limiting system and the trajectory for [REF], forms a continuous family of closed loops.', '1811.08307-2-3-10': 'In Section [REF], we use these loops to construct periodic orbits for the full system (see Theorem [REF]).', '1811.08307-2-3-11': 'Under certain conditions, we show that [REF] has a globally asymptotically periodic orbit in [MATH] for all sufficiently small [MATH].', '1811.08307-2-3-12': 'Moreover, the minimal period of the periodic orbit is of order [MATH] as [MATH], and the trajectory converges to one of the closed loops described above.', '1811.08307-2-3-13': 'That is, this family of periodic orbits forms a relaxation oscillation.', '1811.08307-2-4-0': 'We also study the epidemic model [EQUATION] which was proposed and investigated by Graef et al. [CITATION] and Li et al. [CITATION].', '1811.08307-2-4-1': 'Here [MATH] and [MATH], with [EQUATION] and [EQUATION]', '1811.08307-2-4-2': 'Let [MATH].', '1811.08307-2-4-3': 'System [REF] is equivalent to [EQUATION]', '1811.08307-2-4-4': 'Setting [MATH] in [REF], we obtain the limiting system [EQUATION]', '1811.08307-2-4-5': 'Note that the line [MATH] is a set of equilibria of [REF] in the invariant plane [MATH].', '1811.08307-2-4-6': 'It is known [CITATION] that [MATH] consists of the endpoints of a family of heteroclinic orbits (see Figure [REF]).', '1811.08307-2-4-7': 'The existence of periodic orbits of [REF] was proved by Li et.', '1811.08307-2-4-8': 'al [CITATION].', '1811.08307-2-4-9': 'In Section [REF], we demonstrate that a prescribed number of relaxation oscillations for system [REF] can be obtained by varying the perturbation term [MATH].', '1811.08307-2-5-0': 'Singular perturbations in predator-prey systems were studied in various contexts.', '1811.08307-2-5-1': 'For a model of two predators competing for the same prey, when the prey population grows much faster than the predator populations, the existence of a relaxation oscillation was proved by Liu, Xiao and Yi [CITATION].', '1811.08307-2-5-2': 'For predator-prey systems with Holling type III or IV, when the death and the yield rates of the predator are small and proportional to each other, the canard phenomenon and the cyclicity of limit cycles were investigated by Li and Zhu [CITATION].', '1811.08307-2-5-3': 'For a class of the Holling-Tanner model, when the intrinsic growth rate of the predator is sufficiently small, the existence of a relaxation oscillation was proved by Ghazaryan, Manukian and Schecter [CITATION].', '1811.08307-2-5-4': 'For predator-prey models with eco-evolutionary dynamics in which ecological and evolutionary interactions occur on different time scales, relaxation oscillations were investigated by Piltz et al. [CITATION] and Shen et al. [CITATION].', '1811.08307-2-5-5': 'For the classical predator-prey model with Monod functional response and small predator death rates, the unique periodic orbit, which was proved to exist by Liou and Cheng [CITATION] (who corrected a flaw in the original proof of Cheng [CITATION]) and Kuang and Freedman [CITATION], was proved to form a relaxation oscillation by Hsu and Shi [CITATION], Wang et al. [CITATION], and Lundstrom and Soderbacka [CITATION], and the cyclicity of the limit cycle was investigated by Huzak [CITATION].', '1811.08307-2-5-6': 'For general functional responses, the number of relaxation oscillations depending on the number of local extrema of the prey-isocline was studied by Hsu [CITATION].', '1811.08307-2-6-0': 'This paper is organized as follows.', '1811.08307-2-6-1': 'In Section [REF], we state and prove criteria for the location and stability of relaxation oscillations in a class of planar systems.', '1811.08307-2-6-2': 'In Section [REF] we investigate our criterion for the chemostat predator-prey system [REF], and give conditions under which the system has exactly one or two periodic orbits.', '1811.08307-2-6-3': 'The epidemic model [REF] is studied in Section [REF], in which we compute the characteristic functions in terms of a parametrization of the center manifold, and we use numerical simulation to find the number of relaxation oscillations.', '1811.08307-2-7-0': '# A Criterion for Relaxation Oscillations in Planar Systems', '1811.08307-2-8-0': 'To determine the location and stability of relaxation oscillations for predator-prey systems with small predator death, a criterion was given in [CITATION].', '1811.08307-2-8-1': 'In this section we extend that criteria by considering planar systems of the form [EQUATION] where [MATH], [MATH] and [MATH] are smooth functions satisfying, for some numbers [MATH], [EQUATION]', '1811.08307-2-8-2': 'Setting [MATH] in [REF], we obtain the limiting system [EQUATION]', '1811.08307-2-8-3': 'We assume the following condition (see Figure [REF]):', '1811.08307-2-9-0': '[(H)]', '1811.08307-2-10-0': 'There exists a nonempty open interval [MATH] and smooth functions [MATH] and [MATH], such that, for each [MATH], the points [MATH] and [MATH] are the alpha- and omega-limit points, respectively, of a trajectory of [REF].', '1811.08307-2-11-0': 'Our approach is to use geometric singular perturbation theory [CITATION] to construct periodic orbits.', '1811.08307-2-11-1': 'The idea is that solutions of the full system can potentially be obtained by joining trajectories of the limiting systems.', '1811.08307-2-11-2': 'Under assumption [MATH] for each [MATH], we denote [MATH] the trajectory of [REF] connecting [MATH] and [MATH], and denote [MATH] the segment in the [MATH]-axis that starts at [MATH] and ends [MATH].', '1811.08307-2-11-3': 'Then [MATH] forms a closed loop.', '1811.08307-2-11-4': 'Hence each [MATH] is potentially the limiting configuration of a relaxation oscillation.', '1811.08307-2-11-5': 'Using a variation of bifurcation delay (see [CITATION] and the references therein), in the following theorem we are able to show that only certain [MATH] correspond to relaxation oscillations.', '1811.08307-2-12-0': 'Assume [REF]-[REF] and [MATH].', '1811.08307-2-12-1': 'Set [EQUATION] and [EQUATION]', '1811.08307-2-12-2': 'If [MATH] satisfies [MATH] and [MATH], then for all sufficiently small [MATH], there is a periodic orbit [MATH] of [REF] in a [MATH]-neighborhood of [MATH].', '1811.08307-2-12-3': 'The minimal period of [MATH], denoted by [MATH], satisfies [EQUATION]', '1811.08307-2-12-4': 'Moreover, [MATH] is locally orbitally asymptotically stable if [MATH], and is orbitally unstable if [MATH].', '1811.08307-2-12-5': 'Conversely, if [MATH], then for any point [MATH] in the interior of the trajectory [MATH], there is a neighborhood [MATH] of [MATH] such that no periodic orbit of [REF] intersects [MATH] for any sufficiently small [MATH].', '1811.08307-2-13-0': 'Condition [REF] is not essential and was only provided for convenience.', '1811.08307-2-13-1': 'Without assuming the positivity of [MATH], the results in Theorem [REF] still hold with [MATH] defined using integrals in terms of the time variable [MATH].', '1811.08307-2-13-2': 'Also note that all integrals in [REF] and [REF] exist and are finite under the assumption that [MATH] and [MATH] remain nonzero.', '1811.08307-2-13-3': 'To see that the last integral of [REF], one way is to use the relation [MATH] from [REF] to obtain [EQUATION]', '1811.08307-2-13-4': 'This integral is finite since the integrand in the last expression is bounded.', '1811.08307-2-14-0': 'The function [MATH] defined by [REF] is related to the slow divergence integral studied by De Maesschalck and Dumortier [CITATION].', '1811.08307-2-14-1': 'In their work, the cyclicity of a relaxation oscillation near [MATH] is bounded by an algebraic expression of the the multiplicity of [MATH] as a zero of [MATH].', '1811.08307-2-14-2': 'The function [MATH] in the present work is not necessarily equivalent to [MATH] (in the sense of multiplication by a positive function).', '1811.08307-2-14-3': 'When [MATH], it can be shown that [MATH] and [MATH] have the same sign.', '1811.08307-2-14-4': 'Therefore, the function [MATH] provides an essentially different approach than that in [CITATION] for determining the sign of [MATH].', '1811.08307-2-15-0': 'The function [MATH] defined by [REF] can be expressed in terms of integrals on [MATH]: For any fixed [MATH], let [MATH] be a solution of [REF] with trajectory [MATH].', '1811.08307-2-15-1': 'Then the relations [MATH] and [MATH] yield [EQUATION] and therefore [EQUATION]', '1811.08307-2-15-2': 'These last two identities sometimes can be advantageous for determining the sign of [MATH] (see Theorem [REF]).', '1811.08307-2-16-0': 'For the perspective of modeling, if a planar system of the form [REF] possesses a line of equilibria and a family of heteroclinic orbits connecting points on this line, then by a suitable choice of [MATH] to control the signs of [MATH] and [MATH] in [REF] and [REF], the perturbed system [REF] can have a relaxation oscillation at a prescribed location with a prescribed local stability (see Example [REF]).', '1811.08307-2-17-0': 'In the case that [MATH] and [MATH], Theorem [REF] does not guarantee the existence of a periodic orbit near [MATH].', '1811.08307-2-17-1': 'However, in the region filled by the family of heteroclinic orbits given in [MATH], by Theorem [REF] all possible periodic orbits must lie in an arbitrarily small neighborhood of [MATH] regardless of the sign of [MATH] for all sufficiently small [MATH].', '1811.08307-2-18-0': 'Assume the functions [MATH] and [MATH] in [REF] are separable functions of the form [EQUATION]', '1811.08307-2-18-1': 'Then [MATH] defined in [REF] equals [EQUATION]', '1811.08307-2-18-2': 'Under condition [REF], [EQUATION] and [EQUATION]', '1811.08307-2-18-3': 'Substitute [REF] and [REF] into [REF], we obtain [REF].', '1811.08307-2-19-0': 'Proposition [REF] is the case considered in [CITATION].', '1811.08307-2-20-0': 'Assume that the function [MATH] in [REF] is independent of [MATH], that is, [EQUATION]', '1811.08307-2-20-1': 'Then [MATH] defined in [REF] equals [EQUATION]', '1811.08307-2-20-2': 'The partial derivative [MATH] is identically zero under assumption [REF].', '1811.08307-2-20-3': 'Hence the second integral in [REF] is zero.', '1811.08307-2-21-0': 'If [MATH] for some smooth function [MATH] and [MATH] with [MATH], then [MATH] defined in [REF] is equal to [EQUATION]', '1811.08307-2-21-1': 'From the condition [MATH], [EQUATION]', '1811.08307-2-21-2': 'Note that the [MATH]-coordinates of the endpoints of [MATH] are [MATH].', '1811.08307-2-21-3': 'Since [MATH], [MATH] and consequently [EQUATION]', '1811.08307-2-21-4': 'Therefore [REF] yields [REF].', '1811.08307-2-22-0': 'To prove Theorem [REF], we recall the following two theorems from [CITATION].', '1811.08307-2-23-0': '[Theorem 5.1 in [CITATION]] Consider system [REF], where [MATH], [MATH] and [MATH] are [MATH] functions, [MATH], that satisfy [REF]-[REF].', '1811.08307-2-23-1': 'Assume that [MATH] satisfies relation [EQUATION] and that there exist trajectories [MATH] and [MATH] of the limiting system [EQUATION] such that [MATH] is the omega-limit point of [MATH] and [MATH] is the alpha-limit point of [MATH].', '1811.08307-2-23-2': 'Then for all sufficiently small [MATH] and [MATH], there exists [MATH] such that the following holds.', '1811.08307-2-23-3': 'Let [EQUATION]', '1811.08307-2-23-4': 'Let [EQUATION]', '1811.08307-2-23-5': 'Then the transition mapping from [MATH] to [MATH] of [REF] is well-defined for [MATH], and is [MATH] up to [MATH].', '1811.08307-2-23-6': 'That is, there exists a [MATH] function [EQUATION] such that, for each [MATH] and [MATH], [MATH] and [MATH] are connected by a trajectory of [REF], and [EQUATION]', '1811.08307-2-23-7': 'The time span [MATH] of the trajectory [MATH] connecting [MATH] and [MATH] satisfies [EQUATION]', '1811.08307-2-23-8': 'Moreover, there exists [MATH] such that for each [MATH], if we parameterize [MATH] by [MATH], [MATH], then there exists [MATH] satisfying [EQUATION]', '1811.08307-2-23-9': 'The next theorem is the variation of Floquet Theory.', '1811.08307-2-24-0': '[Theorem 5.2 in [CITATION]]', '1811.08307-2-25-0': 'Consider systems in [MATH], [MATH], of the form [EQUATION] where [MATH] is a [MATH] function of [MATH].', '1811.08307-2-25-1': 'Let [MATH] with [MATH], and let [MATH] be a cross section of [MATH] transversal to [MATH].', '1811.08307-2-25-2': 'Assume that there exist [MATH] and a neighborhood [MATH] of [MATH] such that the return map from [MATH] to [MATH] is well-defined for [MATH] and is [MATH] up to [MATH].', '1811.08307-2-25-3': 'That is, there is a [MATH] function [EQUATION] such that for any [MATH] and [MATH] there is a trajectory of [REF] that starts at [MATH] and returns to [MATH] at [MATH].', '1811.08307-2-26-0': 'Let [MATH], [MATH], be a trajectory of [REF] that starts at [MATH] and ends at [MATH].', '1811.08307-2-26-1': 'Assume that [EQUATION] for some [MATH].', '1811.08307-2-26-2': 'Then [EQUATION] where [MATH] is regarded as a linear transform on the tangent space [MATH].', '1811.08307-2-27-0': 'Now we use Theorems [REF] and [REF] to prove Theorem [REF].', '1811.08307-2-28-0': '[Proof of Theorem [REF]] The proof for the case with [MATH] is similar to that in [CITATION], so we omit it here.', '1811.08307-2-28-1': 'In this case, there is a neighborhood [MATH] of [MATH] such that no periodic orbit of [REF] intersects [MATH] for any sufficiently small [MATH],', '1811.08307-2-29-0': 'Assume [MATH] and [MATH].', '1811.08307-2-29-1': 'Let [MATH] and [MATH].', '1811.08307-2-29-2': 'The condition [MATH] means that [REF] holds.', '1811.08307-2-29-3': 'Let [MATH], [MATH], [MATH] and [MATH] be the points and segments defined in [REF] and [REF].', '1811.08307-2-29-4': 'By Theorem [REF], The transition map [MATH], [MATH] is well defined and is [MATH] up to [MATH].', '1811.08307-2-29-5': 'Let [EQUATION]', '1811.08307-2-29-6': 'Since [REF] is a regular perturbation of [REF] in the region [MATH], the transition map [MATH] for system [REF] is well-defined and is smooth for [MATH].', '1811.08307-2-29-7': 'Since [MATH], we have [MATH].', '1811.08307-2-30-0': 'Let [MATH].', '1811.08307-2-30-1': 'Then [MATH].', '1811.08307-2-30-2': 'That is, [MATH] is a fixed point of [MATH].', '1811.08307-2-30-3': 'To show that this fixed point persists for small [MATH], by the implicit function theorem it suffices to show that the Jacobian matrix of the return map [MATH] evaluated at [MATH] is non-singular.', '1811.08307-2-30-4': 'Let [EQUATION] where [MATH] is the trajectory of [REF] that starts at [MATH] and ends at [MATH].', '1811.08307-2-30-5': 'By Theorem [REF], it suffices to show that [MATH] approaches a nonzero value as [MATH].', '1811.08307-2-31-0': 'Using the equation for [MATH] in [REF], [EQUATION]', '1811.08307-2-31-1': 'For any fixed [MATH], since [MATH] is bounded away from zero on [MATH], by regular perturbation theory we have [EQUATION]', '1811.08307-2-31-2': 'Hence [EQUATION]', '1811.08307-2-31-3': 'Here and in the rest of the proof we use [MATH] to denote constants independent of [MATH] and [MATH].', '1811.08307-2-31-4': 'Next we claim that [EQUATION] and that [EQUATION]', '1811.08307-2-31-5': 'We write [MATH] by [EQUATION]', '1811.08307-2-31-6': 'It can be shown (from the proof of [CITATION]) that, for some [MATH] independent of [MATH], [EQUATION]', '1811.08307-2-31-7': 'Since [MATH] is bounded away from zero, [REF] gives [EQUATION]', '1811.08307-2-31-8': 'Hence [EQUATION]', '1811.08307-2-31-9': 'On the other hand, since [MATH] and [MATH] lies in a neighborhood of [MATH], for any fixed positive number [MATH] we have [MATH] on [MATH] if [MATH] is the entry point of [MATH].', '1811.08307-2-31-10': 'Since [MATH] is bounded away from zero, the equation [MATH] and the estimate [MATH] imply that the time span [MATH] of [MATH] satisfies [MATH].', '1811.08307-2-31-11': 'Therefore [EQUATION]', '1811.08307-2-31-12': 'Similarly, [EQUATION]', '1811.08307-2-31-13': 'Combining estimates [REF], [REF] and [REF], we obtain [REF].', '1811.08307-2-32-0': 'Since [MATH] is bounded away from zero, [REF] gives [EQUATION] as [MATH].', '1811.08307-2-32-1': 'On the other hand, by the equations for [MATH] and [MATH] in [REF], [EQUATION]', '1811.08307-2-32-2': 'Since [MATH] and [MATH], it follows that [EQUATION]', '1811.08307-2-32-3': 'Similarly, from [MATH], [EQUATION]', '1811.08307-2-32-4': 'Combining [REF], [REF] and [REF], we obtain [REF].', '1811.08307-2-33-0': 'By [REF], [REF] and [REF], [EQUATION] where [MATH] is evaluated at [MATH].', '1811.08307-2-33-1': 'Since [MATH] can be arbitrarily small, we obtain [EQUATION]', '1811.08307-2-33-2': 'A similar calculation gives [EQUATION]', '1811.08307-2-33-3': 'By [REF] and [REF] we conclude that the number [MATH] defined by [REF] satisfies [EQUATION]', '1811.08307-2-33-4': 'Using the relation [MATH] from [REF], it follows that [MATH] with [MATH] defined by [REF].', '1811.08307-2-34-0': 'By assumption, [MATH], by Theorem [REF] the return map [MATH] has a unique fixed point near [MATH] for all small [MATH], and [MATH] is a contraction if [MATH], and an expansion if [MATH].', '1811.08307-2-34-1': 'Hence there is a unique periodic orbit [MATH] of [REF] near [MATH] for every small [MATH].', '1811.08307-2-34-2': 'This periodic orbit is locally orbitally asymptotically stable if [MATH], and is unstable if [MATH].', '1811.08307-2-34-3': 'The estimate [REF] follows from [REF].', '1811.08307-2-35-0': '# The Chemostat Predator-Prey System', '1811.08307-2-36-0': 'The restriction of system [REF] on the invariant plane [MATH] is governed by [MATH] and [EQUATION] where [EQUATION]', '1811.08307-2-36-1': 'Define [MATH], that is [EQUATION] and define [EQUATION]', '1811.08307-2-36-2': 'The last inequality follows from [MATH] in condition [REF].', '1811.08307-2-36-3': 'Note that [MATH] is continuous at [MATH] by setting [MATH].', '1811.08307-2-36-4': 'Also note that [MATH] for all [MATH], and that [MATH].', '1811.08307-2-37-0': 'When [MATH], system [REF] reduces to [EQUATION]', '1811.08307-2-37-1': 'Since [MATH] and [MATH], system [REF] has a line of equilibria [MATH] and an isolated saddle equilibrium [MATH].', '1811.08307-2-37-2': 'The unstable manifold of [MATH] is the portion of the line [MATH].', '1811.08307-2-37-3': 'Given any [MATH], since [MATH] for all [MATH], both the forward and backward trajectories starting at [MATH] approach points on the boundary of [MATH] (see Figure [REF]).', '1811.08307-2-37-4': 'This gives a continuous family of heteroclinic orbits parameterized by [MATH] that fills the region in the positive quadrant bounded below by the unstable manifold of [MATH].', '1811.08307-2-38-0': 'Denote the trajectory passing through [MATH] by [MATH], and the [MATH]-values at the omega- and alpha-limit points of [MATH] by [MATH] and [MATH], respectively.', '1811.08307-2-38-1': 'Denote the segment connecting the two endpoints of [MATH] by [MATH].', '1811.08307-2-38-2': 'Then [MATH] is a singular closed orbit for each [MATH]', '1811.08307-2-39-0': 'Applying Theorem [REF] to system [REF] with [MATH] playing the role of [MATH], the formula [REF] gives, up to multiplication by positive constants, [EQUATION]', '1811.08307-2-39-1': 'By Propositions [REF] and [REF], the formula [REF] gives, up to multiplication by positive constants, [EQUATION]', '1811.08307-2-39-2': 'Assume that [MATH] satisfies [MATH] and [MATH], where [MATH] and [MATH] are defined in [REF] and [REF].', '1811.08307-2-39-3': 'Then for any sufficiently small [MATH], there is a periodic orbit [MATH] of [REF] in a [MATH]-neighborhood of [MATH].', '1811.08307-2-39-4': 'The minimal period of [MATH], denoted by [MATH], satisfies [EQUATION]', '1811.08307-2-39-5': 'Moreover, [MATH] is locally orbitally asymptotically stable if [MATH], and is orbitally unstable if [MATH].', '1811.08307-2-39-6': 'Conversely, if [MATH], then for any point [MATH] in the interior of the trajectory [MATH], there is a neighborhood [MATH] of [MATH] such that no periodic orbit of [REF] intersects [MATH] for any sufficiently small [MATH].', '1811.08307-2-40-0': 'If [MATH] and [MATH], then by Theorem [REF], for every small [MATH], system [REF] has unique periodic orbit [MATH] near [MATH].', '1811.08307-2-40-1': 'Since [MATH] is invariant under [REF], [MATH] is also a periodic orbit for [REF].', '1811.08307-2-40-2': 'If [MATH], then [MATH] is orbitally unstable for [REF], and therefore [MATH] is orbitally unstable for [REF].', '1811.08307-2-40-3': 'If [MATH], then [MATH] is locally orbitally asymptotically stable for [REF].', '1811.08307-2-40-4': 'Since [MATH] is a hyperbolic attractor, it follows that [MATH] is locally orbitally asymptotically stable for [REF].', '1811.08307-2-41-0': 'On the other hand, If [MATH] and [MATH], then by Theorem [REF], for every small [MATH], there is no periodic orbit of [REF] near [MATH].', '1811.08307-2-41-1': 'Since [MATH] is a global attractor, it follows that there is no periodic orbit of [REF] near [MATH].', '1811.08307-2-42-0': 'The function [MATH] defined by [REF] can be expressed as a line integral along the trajectory [MATH] as follows.', '1811.08307-2-43-0': 'The the function [MATH] defined by [REF] satisfies [EQUATION]', '1811.08307-2-43-1': 'Fix any [MATH].', '1811.08307-2-43-2': 'Let [MATH] be the solution of [REF] with trajectory [MATH].', '1811.08307-2-43-3': 'Recall that [MATH] and [MATH] are the alpha- and omega-limit point, respectively, of the trajectory [MATH].', '1811.08307-2-43-4': 'From the equation for [MATH] in [REF], equation [REF] can be written as [EQUATION] where [MATH].', '1811.08307-2-43-5': 'On the other hand, from the equation for [MATH] in [REF], [EQUATION]', '1811.08307-2-43-6': 'From [REF] and [REF], it follows that [EQUATION]', '1811.08307-2-43-7': 'Note that [EQUATION] where [MATH], which is continuous at [MATH] because [MATH] and [MATH].', '1811.08307-2-43-8': 'Since [MATH], it follows that [EQUATION]', '1811.08307-2-43-9': 'Equations [REF] and [REF] give [EQUATION]', '1811.08307-2-43-10': 'Therefore, from the equation for [MATH] in [REF], equation [REF] follows.', '1811.08307-2-44-0': 'Next we assume that the function [MATH] in [REF] satisfies the one-hump condition: For some [MATH], [EQUATION]', '1811.08307-2-44-1': 'The following result is similar to [CITATION].', '1811.08307-2-45-0': 'Assume [REF] holds for [MATH] defined by [REF].', '1811.08307-2-45-1': 'Then system [REF] has a globally orbitally asymptotically stable (with respect to all positive non-stationary solutions) periodic orbit for all small [MATH].', '1811.08307-2-46-0': 'We will use the following two lemmas.', '1811.08307-2-47-0': 'Assume [REF].', '1811.08307-2-47-1': 'Then the function [MATH] defined by [REF] has a unique root [MATH] in [MATH], and it satisfies [MATH].', '1811.08307-2-48-0': 'Note that condition [REF] implies that there exists a unique value [MATH] such that [MATH].', '1811.08307-2-49-0': 'First we claim that [EQUATION]', '1811.08307-2-49-1': 'Fix any [MATH].', '1811.08307-2-49-2': 'We parameterize [MATH] by [EQUATION] with [EQUATION]', '1811.08307-2-49-3': 'Then equation [REF] in Proposition [REF] yields [EQUATION]', '1811.08307-2-49-4': 'Since [MATH] for [MATH], by [REF] it follows that [MATH].', '1811.08307-2-50-0': 'Next we claim that [EQUATION]', '1811.08307-2-50-1': 'Fix any [MATH].', '1811.08307-2-50-2': 'From the definition of [MATH] in [REF], using [REF] we have [EQUATION]', '1811.08307-2-50-3': 'Since [MATH] for [MATH], by [REF] we obtain [EQUATION]', '1811.08307-2-50-4': 'Since [MATH] is decreasing and [MATH] is increasing, condition [REF] yields [EQUATION]', '1811.08307-2-50-5': 'The last equality follows from the condition [MATH].', '1811.08307-2-50-6': 'Hence [MATH].', '1811.08307-2-51-0': 'Finally, we claim that [EQUATION]', '1811.08307-2-51-1': 'Note that the expression [REF] of [MATH] can be written as [EQUATION] where [EQUATION]', '1811.08307-2-51-2': 'Note also that the functions [MATH] and [MATH] satisfy [EQUATION]', '1811.08307-2-51-3': 'Since [MATH], [REF] follows from [REF] and[REF].', '1811.08307-2-52-0': 'Since [MATH] for [MATH] and [MATH], the continuous function [MATH] has at least one root in [MATH].', '1811.08307-2-52-1': 'Suppose for contradiction that [MATH] has two distinct roots, say [MATH].', '1811.08307-2-52-2': 'By [REF] and [REF], we have [MATH] and [MATH].', '1811.08307-2-52-3': 'By Theorem [REF] there are locally orbitally asymptotically stable periodic orbits [MATH] and [MATH] near [MATH] and [MATH], respectively.', '1811.08307-2-52-4': 'Note that [MATH] is enclosed by [MATH].', '1811.08307-2-52-5': 'By [REF] and Theorem [REF] there is no unstable periodic orbit between [MATH] and [MATH].', '1811.08307-2-52-6': 'Also note that no equilibrium lies between [MATH] and [MATH].', '1811.08307-2-52-7': 'This contradicts the Poincare-Bendixson Theorem.', '1811.08307-2-52-8': 'Therefore [MATH] has exactly one root [MATH] in [MATH].', '1811.08307-2-52-9': 'By [REF] and [REF], this root satisfies [MATH], and therefore [MATH].', '1811.08307-2-53-0': 'The next lemma was derived by Wolkowicz [CITATION], and we omit its proof here.', '1811.08307-2-54-0': 'If [MATH] on [MATH] or [MATH] on [MATH] for some [MATH], then no periodic orbit of [REF] lies entirely in the strip [MATH] for any sufficiently small [MATH].', '1811.08307-2-55-0': '[Proof of Theorem [REF]] By Lemma [REF], the function [MATH] has a unique root [MATH] in the interval [MATH], and [MATH].', '1811.08307-2-55-1': 'From Theorem [REF] and Lemma [REF], it follows that system [REF] has a unique periodic orbit [MATH] in [MATH] for all small [MATH].', '1811.08307-2-55-2': 'It can be shown by the Butler-McGehee Lemma [CITATION] that the flow [REF] is persistent in the sense that the omega-limit set of any point in [MATH] does not intersect the boundary of [MATH].', '1811.08307-2-55-3': 'Therefore, by the Poincare-Bendixon Theorem, the periodic orbit [MATH] attracts all non-stationary points in [MATH].', '1811.08307-2-56-0': 'Since [MATH] is locally orbitally asymptotically stable in [MATH] for [REF], by [REF] it follows that [MATH] is locally orbitally asymptotically stable in [MATH] for [REF].', '1811.08307-2-56-1': 'Given any solution [MATH] of [REF] with a positive initial value, it can be shown by the Butler-McGehee Lemma that the flow [REF] is persistent in the sense that the omega-limit set of any point in [MATH] does not intersect the boundary of [MATH].', '1811.08307-2-56-2': 'Let [MATH] be the omega-limit set of [MATH].', '1811.08307-2-56-3': 'Then [MATH] since [MATH] is a global attractor and solutions of [REF] are persistent.', '1811.08307-2-56-4': 'By the positive invariance of omega-limit sets, [MATH].', '1811.08307-2-56-5': 'Hence [MATH].', '1811.08307-2-56-6': 'This implies that the trajectory converges to [MATH].', '1811.08307-2-57-0': 'Consider the Holling type II functional response, [EQUATION]', '1811.08307-2-57-1': 'It was shown by Bolger et al. [CITATION] that the function [MATH] defined by [REF] is concave-down.', '1811.08307-2-57-2': 'Hence condition [REF] is satisfied, and the results in Theorem [REF] hold.', '1811.08307-2-58-0': 'Numerical simulations are shown in Figures [REF] and [REF].', '1811.08307-2-58-1': 'In the simulations, the parameters are [MATH] for [REF], and [MATH] and [MATH] in [REF].', '1811.08307-2-58-2': 'Figure [REF] shows that the function [MATH] has a root [MATH], and it satisfies [MATH].', '1811.08307-2-58-3': 'Hence [MATH] corresponds to a stable relaxation oscillation formed by the globally asymptotically stable periodic orbit of [REF] as [MATH].', '1811.08307-2-58-4': 'Figure [REF](A) shows the periodic orbit [MATH] for [REF] with [MATH], and Figure [REF](B) illustrates the trajectory [MATH] for [REF].', '1811.08307-2-58-5': 'The simulation confirms that the location of [MATH] is close to [MATH].', '1811.08307-2-59-0': 'The analysis of [REF] in this section can more generally be applied to systems of the form [EQUATION] that satisfy condition [REF], [MATH] for [MATH], [MATH] is continuous at [MATH], and [EQUATION] for some positive constant [MATH].', '1811.08307-2-59-1': 'Similar to Theorem [REF], a unique locally orbitally asymptotic stable relaxation oscillation for system [REF] in the region filled by a family of heteroclinic orbits can be obtained for all small [MATH] under assumption [REF] with [MATH] and [MATH] replaced by [MATH].', '1811.08307-2-60-0': '# The Epidemic Model', '1811.08307-2-61-0': 'When [MATH], system [REF] reduces to system [REF].', '1811.08307-2-61-1': 'The line [MATH] is a set of equilibria of [REF] in the invariant plane [MATH].', '1811.08307-2-61-2': 'Let [MATH] be the unique value that satisfies [MATH].', '1811.08307-2-61-3': 'It is known [CITATION] that each point on the segment [MATH] is connected to a unique point on the segment [MATH] by a heteroclinic orbit of [REF] (see Figure [REF]).', '1811.08307-2-61-4': 'We define [EQUATION] such that the point [MATH] is the omega-limit point of the heteroclinic orbit, denoted by [MATH], of [REF] starting from [MATH].', '1811.08307-2-62-0': 'It is also known [CITATION] that the invariant manifold [MATH] and the center manifold [MATH] of system [REF] still exist for the perturbed system [REF], and that the perturbed center manifold is a global attractor for [REF] for each small [MATH].', '1811.08307-2-62-1': 'Denote the perturbed manifolds by [MATH] and [MATH].', '1811.08307-2-62-2': 'We parametrize the center manifold [MATH] by [MATH], and define [MATH].', '1811.08307-2-62-3': 'Then the restriction of system [REF] on [MATH] can be written as [EQUATION]', '1811.08307-2-62-4': 'With [MATH] in [REF] playing the role of [MATH] in [REF], and [MATH] playing the role of [MATH], the function [MATH] defined by [REF] is equal to [EQUATION]', '1811.08307-2-62-5': 'By [REF] in Proposition [REF] and [REF] in Remark [REF], the function [MATH] defined by [REF] is equal to [EQUATION]', '1811.08307-2-62-6': 'That is, [EQUATION]', '1811.08307-2-62-7': 'The function [MATH] in [REF] is equivalent to the function [MATH] in [CITATION] in the sense that [MATH] if and only of [MATH].', '1811.08307-2-63-0': 'Assume that [MATH] satisfies [MATH] and [MATH], where [MATH] and [MATH] are defined in [REF] and [REF], respectively.', '1811.08307-2-63-1': 'Then for any sufficiently small [MATH], there is a unique periodic orbit [MATH] of [REF] in a [MATH]-neighborhood of [MATH].', '1811.08307-2-63-2': 'The minimal period of [MATH], denoted by [MATH], satisfies [EQUATION]', '1811.08307-2-63-3': 'Moreover, [MATH] is locally orbitally asymptotically stable if [MATH], and is orbitally unstable if [MATH].', '1811.08307-2-63-4': 'Conversely, if [MATH], then for any point [MATH] in the interior of the trajectory [MATH], there is a neighborhood [MATH] of [MATH] such that no periodic orbit of [REF] intersects [MATH] for any sufficiently small [MATH].', '1811.08307-2-64-0': 'If [MATH], then by Theorem [REF], system [REF] has no periodic orbit near [MATH] in [MATH] for any small [MATH].', '1811.08307-2-64-1': 'Since [MATH] is a global attractor, it follows that system [REF] has no periodic orbit near [MATH] in [MATH].', '1811.08307-2-65-0': 'Next assume that [MATH].', '1811.08307-2-65-1': 'Then by Theorem [REF], system [REF] has a unique periodic orbit [MATH] in a [MATH]-neighborhood of [MATH] in [MATH] for every small [MATH].', '1811.08307-2-65-2': 'If [MATH], then [MATH] is unstable for [REF], and therefore is unstable for [REF].', '1811.08307-2-65-3': 'If [MATH], then [MATH] is locally orbitally asymptotically stable for [REF].', '1811.08307-2-65-4': 'Since [MATH] is a hyperbolic attractor, [MATH] is locally orbitally asymptotically stable for [REF].', '1811.08307-2-66-0': 'The period [MATH] of the limit cycle [MATH] is referred to as interepidemic period (IEP) in [CITATION], where it was shown numerically that [MATH] is proportional to [MATH].', '1811.08307-2-66-1': 'Their observation is consistence with the asymptotic formula [REF].', '1811.08307-2-67-0': 'We are not able to determine the signs of [MATH] and [MATH] in [REF] and [REF] analytically.', '1811.08307-2-67-1': 'Nonetheless, we are able to compute [MATH] and [MATH] numerically.', '1811.08307-2-67-2': 'A numerical difficulty in the computation is to approximate [MATH], since there is no explicit formula for [MATH].', '1811.08307-2-67-3': 'We implement the following algorithm to compute [MATH]: Fix a small number [MATH] and large integers [MATH] and [MATH].', '1811.08307-2-67-4': 'For [MATH], denote by [MATH] the solution of [REF] with initial value [MATH], where [MATH] is an eigenvector corresponding to the unstable eigenvalue of the linearization of [REF] at [MATH], so that the forward trajectory of this solution is near the heteroclinic orbit with the alpha-limit point [MATH].', '1811.08307-2-67-5': 'Let [MATH], [MATH], be a grid of the interval [MATH], and let [MATH], [MATH], be a grid of the interval [MATH].', '1811.08307-2-67-6': 'Define [EQUATION]', '1811.08307-2-67-7': 'Define [MATH] and [MATH].', '1811.08307-2-67-8': 'Then the numerical approximations of [MATH] and [MATH] are [EQUATION]', '1811.08307-2-67-9': 'We use this approximation for [MATH] to evaluate formula [REF] for [MATH].', '1811.08307-2-68-0': 'Following Li et al. [CITATION], we consider [MATH] and parameters [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1811.08307-2-68-1': 'It was proved in [CITATION] that, form small [MATH], system [REF] has a stable periodic orbit and the positive equilibrium is unstable.', '1811.08307-2-68-2': 'Our numerical simulation, illustrated in Figure [REF](A), shows that [MATH] defined by [REF] has a root [MATH] with [MATH].', '1811.08307-2-68-3': 'Hence [MATH] corresponds to a stable relaxation oscillation.', '1811.08307-2-68-4': 'A trajectory with initial data [MATH] and [MATH] is shown in Figure [REF](B).', '1811.08307-2-68-5': 'The trajectory first is attracted by the slow manifold [MATH], and then follows the dynamics on [MATH] to approach the periodic orbit near [MATH].', '1811.08307-2-69-0': 'In the next example we demonstrate that by varying the perturbation term [MATH], system [REF] can obtain two relaxation oscillations.', '1811.08307-2-69-1': 'The idea is that the positive function [MATH] effects the magnitude of the integrand in formula [REF] of [MATH], and hence the function [MATH] can gain extra roots by deforming [MATH].', '1811.08307-2-69-2': 'This method conceptually can be used to construct an arbitrary number of relaxation oscillations.', '1811.08307-2-70-0': 'We replace [MATH] defined by [REF] with [EQUATION] with [MATH].', '1811.08307-2-70-1': 'The function [MATH] differs from [MATH] essentially only in a small interval to the right of [MATH] (see Figure [REF]).', '1811.08307-2-70-2': 'From our numerical simulation, as shown in Figure [REF](A), the function [MATH] defined by [REF] with [MATH] replaced by [MATH] has two roots, [MATH] and [MATH], with [MATH] and [MATH].', '1811.08307-2-70-3': 'Hence [MATH] corresponds to an unstable relaxation oscillation for system [REF] with [MATH] replaced by [MATH], and [MATH] corresponds to a stable relaxation oscillation.', '1811.08307-2-70-4': 'Some trajectories for the system with [MATH] are shown in Figure [REF](B).', '1811.08307-2-71-0': 'Example [REF] is consistent with Li et al. [CITATION], in which two periodic orbits were observed.', '1811.08307-2-71-1': 'It was proved in [CITATION] that, for some parameters, system [REF] exhibits a stable periodic orbit while the positive equilibrium is asymptotically stable, which implies that there must be at least one unstable periodic orbit.', '1811.08307-2-71-2': 'It was commented in [CITATION] that, in general, the unstable periodic orbit is not necessarily a relaxation oscillation but a small periodic orbit through a subcritical Hopf bifurcation.', '1811.08307-2-71-3': 'Using Theorem [REF], we are able to confirm that there is an isolated unstable periodic orbit for small [MATH] that forms a relaxation oscillation.', '1811.08307-2-72-0': '# Discussion', '1811.08307-2-73-0': 'In this paper we derived a criterion to determine the location and stability of relaxation oscillations for the planar system [REF] under assumption [MATH].', '1811.08307-2-73-1': 'In Theorem [REF], characteristic functions [MATH] and [MATH] were obtained from the trajectory [MATH] of the limiting system with [MATH] and satisfy that', '1811.08307-2-74-0': 'In the latter case, the sign of [MATH] determines the stability of the limit cycles.', '1811.08307-2-74-1': 'The proof of the theorem involves geometric singular perturbation theory and a variation of Floquet Theory.', '1811.08307-2-74-2': 'In Propositions [REF], [REF] and [REF], simplified expressions of [MATH] and [MATH] were provided for the case that some terms in the system satisfy certain forms.', '1811.08307-2-75-0': 'We applied this criterion to two systems using two different techniques.', '1811.08307-2-75-1': 'By relating [MATH] with a line integral on the heteroclinic orbit, in Theorem [REF] we derived a condition under which the chemostat predator-prey system [REF] has a unique limit cycle.', '1811.08307-2-75-2': 'By varying the perturbation term, we demonstrate in Examples [REF] and [REF] that the the epidemic model [REF] can have a prescribed number of relaxation oscillations.', '1811.08307-2-75-3': 'In general, it is a numerical challenge to compute unstable periodic orbits in three-dimensional systems.', '1811.08307-2-75-4': 'Our characteristic functions provide a way to locate both stable and unstable periodic orbits.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1811.08307

1601.01160

{'1601.01160-1-0-0': '# Introduction', '1601.01160-1-1-0': 'Investigation on phase transition of an AdS space time has attracted attention of many theoretical physicists recently.', '1601.01160-1-1-1': 'The main motivation maybe steps from the existence of the AdS/CFT correspondence [CITATION], which relates a black hole in the AdS space time to a thermal system without gravity.', '1601.01160-1-1-2': 'For in this case, some interesting but intractable phenomena in strongly coupled system become to be tractable easily in the bulk.', '1601.01160-1-1-3': 'To probe these fascinating phenomena in field theory, one should employ some non-local observables such as two point correlation function, Wilson loop, and holographic entanglement entropy, which are dual to the geodesic length, minimal area surface, and minimal surface area in the bulk individually.', '1601.01160-1-1-4': 'It has been shown that these observables can probe the non-equilibrium thermalization behavior [CITATION], superconducting phase transition [CITATION], and cosmological singularity [CITATION].', '1601.01160-1-2-0': 'In this paper, we intend to use the non-local observables to probe the phase structure of the Born-Infeld-anti-de Sitter black holes.', '1601.01160-1-2-1': 'Usually, phase structure of a black hole is understood from the viewpoint of thermodynamics.', '1601.01160-1-2-2': 'For an uncharged AdS black hole, it has been found that there is a phase transition between the thermal gas in AdS space and Schwarzschild AdS black holes [CITATION], which was interpreted as the confinement/deconfinement phase transition in the dual gauge field theory later [CITATION].', '1601.01160-1-2-3': 'As the charge is endowed with, the AdS black hole will undergo a Van der Waals-like phase transition before it reaches the stable state in the entropy-temperature plane [CITATION].', '1601.01160-1-2-4': 'Specifically speaking, there exists a critical charge, and for the case that the charge of the black hole is smaller than the critical charge, the black hole undertakes a first order phase transition.', '1601.01160-1-2-5': 'As the charge increases to the critical charge, the phase transition is second order, while the charge exceeds the critical charge, there is not phase transition and the black hole is always stable.', '1601.01160-1-2-6': 'The Van der Waals-like phase transition can be observed in many circumstances.', '1601.01160-1-2-7': 'In [CITATION], it was found that a 5-dimensional neutral Gauss-Bonnet black hole demonstrates the Van der Waals-like phase transition in the [MATH] plane, where [MATH] is the Hawking temperature and [MATH] is the Gauss-Bonnet coupling parameter.', '1601.01160-1-2-8': 'In [CITATION], the Van der Waals-like phase transition was also observed in the [MATH] plane, where [MATH] is electric charge and [MATH] is the chemical potential.', '1601.01160-1-2-9': 'Treating the negative cosmological constant as the pressure [MATH] and its conjugate as the thermodynamical volume [MATH], the Van der Waals-like phase transition has also been observed in the [MATH] plane recently[CITATION].', '1601.01160-1-3-0': 'Very recently, with the entanglement entropy as a probe, [CITATION] investigated the phase structure of the Reissner-Nordstrom-AdS black hole and found that there was also a Van der Waals-like phase transition in the entanglement entropy-temperature plane .', '1601.01160-1-3-1': 'They also obtained the critical exponent of the heat capacity for the second order phase transition in the neighborhood of the critical points.', '1601.01160-1-3-2': 'To further confirm the similarity of the phase structure between the thermal entropy and entanglement entropy, [CITATION] checked the equal area law later and found that it held for the first order phase transition in the entanglement entropy-temperature plane.', '1601.01160-1-3-3': 'Now [CITATION] has been generalized to the extended space time [CITATION], massive gravity [CITATION] , as well as Weyl gravity [CITATION], and all the results showed that the entanglement entropy exhibited the same phase structure as that of the thermal entropy.', '1601.01160-1-4-0': 'In this paper, besides the entanglement entropy, we will employ the two point correlation function to probe the phase structure of the black holes.', '1601.01160-1-4-1': 'We choose the Born-Infeld-anti-de Sitter black holes as the gravity model, which is a solution of the Einstein-Born-Infeld action.', '1601.01160-1-4-2': 'There have been many works to study phase structure of the Born-Infeld-anti-de Sitter black holes [CITATION].', '1601.01160-1-4-3': 'The results showed that both the Born-Infeld parameter [MATH] and charge [MATH] affect the phase structure, and to assure the existence of a non-extremal black hole, one should impose the condition [MATH].', '1601.01160-1-4-4': 'In this paper, we find for the case [MATH], the phase structure of the black hole is similar to that of the Reissner-Nordstrom-AdS black hole in the entropy-temperature plane.', '1601.01160-1-4-5': 'While for the case [MATH], there is a novel phase structure, which has not been observed previously.', '1601.01160-1-4-6': 'Specially speaking, a new branch emerges compared with that of the Reissner-Nordstrom-AdS black hole so that there are two unstable regions and two phase transition temperature correspondingly.', '1601.01160-1-4-7': 'All these phase structure are probed by the nonlocal observables such as two point correlation function as well as holographic entanglement entropy, and phase structure of the nonlocal observables are found to be the same as that of the thermal entropy.', '1601.01160-1-5-0': 'Our paper is outlined as follows.', '1601.01160-1-5-1': 'In the next section, we will review the thermodynamic properties of the Born-Infeld-anti-de Sitter black hole firstly, and then study its phase structure in [MATH] plane in a fixed charge ensemble.', '1601.01160-1-5-2': 'In Section [REF], we will employ the two point correlation function and holographic entanglement entropy to probe the phase structure of the Born-Infeld-anti-de Sitter black hole.', '1601.01160-1-5-3': 'In each subsection, the equal area law is checked and the critical exponent of the heat capacity is obtained.', '1601.01160-1-5-4': 'The last section is devoted to discussions and conclusions.', '1601.01160-1-6-0': '# Thermodynamic phase transition of the Born-Infeld-anti-de Sitter black hole', '1601.01160-1-7-0': '## Review of the Born-Infeld-anti-de Sitter black hole', '1601.01160-1-8-0': 'The 4-dimensional Born-Infeld AdS black hole is a solution of the following action [CITATION] [EQUATION] where [EQUATION] in which [MATH] is the hypergeometric function.', '1601.01160-1-8-1': 'From Eq. ([REF]), we know that in the limit [MATH], [MATH], the solution reduces to the the Reissner-Nordstrom-AdS black hole, and in the limit [MATH], it reduces to the Schwarzschild AdS black hole.', '1601.01160-1-9-0': 'The ADM mass of the black hole, defined by [MATH], is given by [EQUATION] in which [MATH] is the event horizon of the black hole.', '1601.01160-1-9-1': 'The Hawking temperature defined by [MATH] can be written as [EQUATION] where we have used Eq. ([REF]).', '1601.01160-1-9-2': 'In addition, according to the Bekenstein-Hawking entropy area relation, we can get the black hole entropy [EQUATION]', '1601.01160-1-9-3': 'Inserting Eq. ([REF]) into Eq. ([REF]), we can get the entropy temperature relation, namely [EQUATION]', '1601.01160-1-9-4': 'It is obvious that besides the charge [MATH], the Born-Infeld parameter [MATH] also affects the phase structure of this space time in the [MATH] plane.', '1601.01160-1-10-0': '## Phase transition of thermal entropy', '1601.01160-1-11-0': 'Based on Eq. ([REF]), we will discuss the phase structure of the Born-Infeld-anti-de Sitter black hole.', '1601.01160-1-11-1': 'We are interested in how [MATH] or [MATH] affects it.', '1601.01160-1-11-2': 'For a fixed charge [MATH], the effect of [MATH] on the phase structure is shown in Figure [REF].', '1601.01160-1-11-3': 'It is obvious that for the small [MATH], the phase structure resembles as that of the Schwarzschild AdS black hole, and for large [MATH], it resembles as that of the Reissner-Nordstrom-AdS black hole as expected.', '1601.01160-1-11-4': 'From (b) of Figure [REF], we can see that as [MATH] increases, the minimum temperature of the space time decreases, and the entropy, also the event horizon, becomes smaller.', '1601.01160-1-11-5': 'That is, the Born-Infeld parameter [MATH] promotes the formation of an AdS space time.', '1601.01160-1-12-0': 'More interestingly, for the smaller charge, we find there is a novel phase transition, which is labeled by the red solid line in (a) of Figure [REF].', '1601.01160-1-12-1': 'From this curve, we know that there are two unstable regions.', '1601.01160-1-12-2': 'Namely the black hole will undergo the following transition: unstable-stable-unstable-stable.', '1601.01160-1-12-3': 'Compared with that of the Reissner-Nordstrom-AdS black hole, a new branch emerges at the onset of the phase transition.', '1601.01160-1-12-4': 'The influence of the charge on the phase structure for a fixed Born-Infeld parameter is plotted in Figure [REF].', '1601.01160-1-13-0': 'Apparently, the phase structure is similar to that of the Schwarzschild AdS black hole for the small charges, and Reissner-Nordstrom-AdS black hole for large charges.', '1601.01160-1-13-1': 'From (a) of Figure [REF], we observe that the larger the charge is, the lower the minimum temperature will be.', '1601.01160-1-13-2': 'In other words, the charge will promote the formation of an AdS space time, which has the same effect as that of the Born-Infeld parameter on the phase structure of the black hole.', '1601.01160-1-14-0': 'For the case [MATH] in (b) of Figure [REF], we see that for some intermediate values of charge [MATH], there is also a similar phase structure as that in (b) in Figure [REF], which is labeled by the red solid line in (b) of Figure [REF].', '1601.01160-1-14-1': 'Note that however, for the large [MATH], the novel phase structure disappears, which is shown in Figure [REF].', '1601.01160-1-14-2': 'For in this case, the Born-Infeld parameter is large enough, so that the phase structure resembles completely as that of the Reissner-Nordstrom-AdS black hole.', '1601.01160-1-15-0': 'Next, we will study detailedly the phase structure of the Born-Infeld AdS black hole for the case [MATH] and [MATH] respectively.', '1601.01160-1-15-1': 'To finish it, we should first find the critical charge for a fixed Born-Infeld parameter by the condition [EQUATION]', '1601.01160-1-15-2': 'However, from Eq. ([REF]), we find it is hard to get the analytical result directly.', '1601.01160-1-15-3': 'Taking the case [MATH] as an example, we will show how to get it numerically next.', '1601.01160-1-15-4': 'We first plot a series of curves for different charges in the [MATH] plane, which is shown in (a) of Figure [REF], and read off the charge which satisfies likely the condition [MATH].', '1601.01160-1-15-5': 'With that rough value, we plot a bunch of curves in the [MATH] plane once with smaller step so that we can get the most likely value of charge.', '1601.01160-1-15-6': 'From (b) of Figure [REF], we find the critical charge should be [MATH], which are labeled by the red dashed lines in (b) of Figure [REF].', '1601.01160-1-15-7': 'Lastly, we adjust the value of [MATH] by hand to find the only solution [MATH] which satisfies Eq.([REF]), which produces [MATH], [MATH].', '1601.01160-1-15-8': 'Substituting these critical values into Eq.([REF]), we further get the critical temperature [MATH].', '1601.01160-1-16-0': 'Having obtained the critical charge, we can plot the isocharge curves for the case [MATH] in the [MATH] plane, which is shown in Figure [REF].', '1601.01160-1-16-1': 'We observe that the black hole undergoes a Hawking-Page phase transition and a Van der Waals-like phase transition.', '1601.01160-1-16-2': 'Specifically speaking, for the case [MATH], there is a minimum temperature [MATH] [CITATION], which is indicated by the red dashed line in (a) of Figure [REF].', '1601.01160-1-16-3': 'When the temperature is higher than [MATH], there are two additional black hole branches.', '1601.01160-1-16-4': 'The small branch is unstable while the large branch is stable.', '1601.01160-1-16-5': 'The Hawking-Page phase transition occurs at the temperature given by [MATH] [CITATION], which is indicated by the red dotted line.', '1601.01160-1-16-6': 'The Hawking-Page phase transition also can be observed in the [MATH] plane, in which [MATH] is the Helmholtz free energy defined by [EQUATION]', '1601.01160-1-16-7': 'From (a) of Figure [REF], we know that there is a minimum temperature [MATH], and above this temperature, there are two branches.', '1601.01160-1-16-8': 'The lower branch is stable always.', '1601.01160-1-16-9': 'The Hawking-Page phase transition occurs at [MATH] for in this case the free energy vanishes.', '1601.01160-1-17-0': 'For the case [MATH], the phase structure is similar to that of the Van der Waals phase transition, which is shown in (b) of Figure [REF].', '1601.01160-1-17-1': 'The solid blue lines from top to down correspond to the isocharges for the case [MATH].', '1601.01160-1-17-2': 'We can see that black holes endowed with different charges have different phase structures.', '1601.01160-1-17-3': 'For the small charge, there is an unstable black hole interpolating between the stable small hole and stable large hole.', '1601.01160-1-18-0': 'The small stable hole will jump to the large stable hole at the critical temperature [MATH].', '1601.01160-1-18-1': 'As the charge increases to the critical charge, the small hole and the large hole merge into one and squeeze out the unstable phase so that an inflection point emerges.', '1601.01160-1-18-2': 'The divergence of the heat capacity in this case implies that the phase transition is second order.', '1601.01160-1-18-3': 'As the charge exceeds the critical charge, we simply have one stable black hole at each temperature.', '1601.01160-1-18-4': 'The Van der Waals-like phase transition can also be observed from the [MATH] relation.', '1601.01160-1-18-5': 'From (b) of Figure [REF], we observe that there is a swallowtail structure, which corresponds to the unstable phase in the top curve in (b) Figure [REF].', '1601.01160-1-18-6': 'The critical temperature [MATH] is apparently the value of the horizontal coordinate of the junction between the small black hole and the large black hole.', '1601.01160-1-18-7': 'As the temperature is lower than the critical temperature [MATH], the free energy of the small black hole is lowest, so the small hole is stable.', '1601.01160-1-18-8': 'As the temperature is higher than [MATH], the free energy of the large black hole is lowest, so that the large hole dominates thereafter.', '1601.01160-1-18-9': 'The non-smoothness of the junction indicates that the phase transition is first order.', '1601.01160-1-18-10': 'From (c) of Figure [REF], we know that there is an inflection point, which corresponds to the inflection point in the middle curve in (b) of Figure [REF].', '1601.01160-1-18-11': 'The horizontal coordinate of the inflection point corresponds to the second order phase transition temperature [MATH].', '1601.01160-1-19-0': 'Similarly, we also can study the phase structure for the case [MATH] in the [MATH] plane.', '1601.01160-1-19-1': 'To finish it, we should first find the critical charge.', '1601.01160-1-19-2': 'Adopting the same strategy as that of the case [MATH], we find [MATH], [MATH], [MATH].', '1601.01160-1-19-3': 'For the case [MATH], the phase structure is the same as that in (a) of Figure [REF].', '1601.01160-1-19-4': 'We will not repeat it here.', '1601.01160-1-19-5': 'For the case [MATH], we find besides the Van der Waals-like phase transition, there is a novel phase transition for [MATH] as observed in Figure [REF] and Figure [REF], which is plotted in (a) of Figure [REF].', '1601.01160-1-19-6': 'That is, at the beginning of the phase transition, an infinitesimally small black hole branch emerges.', '1601.01160-1-19-7': 'It is unstable and its life is very short so that it becomes to a small stable black hole quickly.', '1601.01160-1-19-8': 'In addition, we find there are two unstable region, and correspondingly in the [MATH] plane, we observe two swallowtail structures, which is shown in (a) of Figure [REF].', '1601.01160-1-19-9': 'The horizontal coordinate of the junction of the swallowtail corresponds to the phase transition temperature, thus besides the first order phase transition temperature between the small black hole and large black hole, labeled by [MATH], there is a new phase transition temperature, labeled by [MATH].', '1601.01160-1-19-10': 'It should be stressed that the new phase transition can not take place actually, for above the critical temperature [MATH], the free energy of the large black hole is lowest so that the space time is dominated by the large black hole, which can be seen in (a) of Figure [REF].', '1601.01160-1-19-11': 'Thus, we call thus phase transition pseudo phase transition.', '1601.01160-1-20-0': 'We observe that there is also a Van der Waals-like phase transition, which is plotted in (b) of Figure [REF].', '1601.01160-1-20-1': 'The solid blue lines from top to down correspond to the isocharges for the case [MATH].', '1601.01160-1-20-2': 'The first order phase transition temperature [MATH] can be read off from the horizontal coordinate of the junction of the swallowtail structure in (b) of Figure [REF], and the second order phase transition temperature [MATH] can be read off from the horizontal coordinate of the inflection point in (c) of Figure [REF].', '1601.01160-1-21-0': "As done in [CITATION], we will also check numerically whether Maxwell's equal area law holds for the first order phase transition and pseudo phase transition, which states [EQUATION] in which [MATH] is defined in Eq. ([REF]), [MATH] is the phase transition temperature, and [MATH], [MATH] are the smallest and largest values of unstable region which satisfies the equation [MATH].", '1601.01160-1-21-1': 'Usually for this equation, there are three roots [MATH], so [MATH], [MATH], [MATH].', '1601.01160-1-21-2': 'But for the case [MATH], there are four roots.', '1601.01160-1-21-3': 'One can see from (a) of Figure [REF] that [MATH], [MATH] for the pseudo phase transition with phase transition temperature [MATH], and [MATH], [MATH] for the first order phase transition with phase transition temperature [MATH].', '1601.01160-1-21-4': 'The calculated results for the first order phase transition and pseudo phase transition are listed in Table [REF].', '1601.01160-1-22-0': 'From this table, we can see that [MATH] equals [MATH] for different [MATH] in our numeric accuracy.', '1601.01160-1-22-1': 'The equal area law therefore holds.', '1601.01160-1-23-0': 'For the second order phase transition, we know that the heat capacity is divergent near the critical point and the critical exponent is [MATH].', '1601.01160-1-23-1': 'As stated in [CITATION], near the critical point, there is always a linear relation [EQUATION] with 3 the slope.', '1601.01160-1-23-2': 'It is not difficult to show that for the case [MATH] in our gravity model, the temperature and entropy also satisfy this linear relation near the critical point.', '1601.01160-1-23-3': 'Next, we will take Eq. ([REF]) as a reference to check whether there is a similar relation for the second order phase transition in the two point correlation function-temperature plane as well as entanglement entropy-temperature plane.', '1601.01160-1-24-0': '# Phase transition in the framework of holography', '1601.01160-1-25-0': 'Having obtained the phase structure of thermal entropy of the Born-Infeld AdS black hole in the [MATH] plane, we will study the phase structure of two point correlation function and entanglement entropy in the fled theory to see whether they have the similar phase structure and critical behavior.', '1601.01160-1-26-0': '## Phase structure probed by two point correlation function', '1601.01160-1-27-0': 'According to the AdS/CFT correspondence, in the large [MATH] limit, the equal time two point correlation function can be written as [CITATION] [EQUATION] where [MATH] is the conformal dimension of scalar operator [MATH] in the dual field theory, [MATH] is the length of the bulk geodesic between the points [MATH] and [MATH] on the AdS boundary.', '1601.01160-1-27-1': 'In our gravity model, we can simply choose [MATH] and [MATH] as the two boundary points.', '1601.01160-1-27-2': 'Then with [MATH] to parameterize the trajectory, the proper length is given by [EQUATION] in which [MATH].', '1601.01160-1-27-3': 'Imagining [MATH] as time, and treating [MATH] as the Lagrangian, one can get the equation of motion for [MATH] by making use of the Euler-Lagrange equation.', '1601.01160-1-27-4': 'Then with the following boundary conditions [EQUATION] we can get the numeric result of [MATH].', '1601.01160-1-27-5': 'To explore whether the size of the boundary region affects the phase structure, we will choose [MATH] as two examples.', '1601.01160-1-27-6': 'Note that for a fixed [MATH], the geodesic length is divergent, so it should be regularized by subtracting off the geodesic length in pure AdS with the same boundary region, denoted by [MATH].', '1601.01160-1-27-7': 'To achieve this, we are required to set a UV cutoff for each case, which is chosen to be [MATH] and [MATH], respectively for our two examples.', '1601.01160-1-27-8': 'The regularized geodesic length is labeled as [MATH].', '1601.01160-1-27-9': 'In addition, during the numerics, we will set the AdS radius [MATH] to be 1.', '1601.01160-1-28-0': 'We plot the relation between [MATH] and [MATH] for the case [MATH] in Figure [REF] and Figure [REF] .', '1601.01160-1-28-1': 'The solid blue line in (a) corresponds to the isocharge for the case [MATH], and the dashed blue lines from top to down in (b) correspond to that [MATH].', '1601.01160-1-28-2': 'From these figures, one can see that [MATH] demonstrates a similar phase structure as that of the thermal entropy in Figure [REF].', '1601.01160-1-28-3': 'Namely the two point correlation function can probe both the Hawking-Page phase transition and Van der Waals-like phase transition.', '1601.01160-1-28-4': 'Precisely speaking, we find that the minimum temperature [MATH] as well as Hawking-Page phase transition temperature [MATH] in [MATH], the first order phase transition temperature [MATH], and second order phase transition temperature [MATH] in [MATH] are exactly the same as those in [MATH] plane.', '1601.01160-1-28-5': 'This conclusion will not be affected as [MATH] varies in a reasonable region, which can be seen from Figure [REF] and Figure [REF].', '1601.01160-1-29-0': 'With the two point correlation function, we also can probe the phase structure of the case [MATH], which is shown in Figure [REF] and Figure [REF].', '1601.01160-1-29-1': 'The solid blue line in (a) corresponds to the isocharge for the case [MATH], and the solid blue lines from top to down in (b) correspond to that [MATH].', '1601.01160-1-29-2': 'As the case as the thermal entropy in the [MATH] plane, we find there ia also a novel phase structure besides the Van der Waals-like phase transition in the [MATH] plane.', '1601.01160-1-29-3': 'There are also two unstable region, correspondingly two phase transition temperature, labeled by [MATH], [MATH].', '1601.01160-1-29-4': 'For the phase transition temperature mentioned above, it is easy to check [MATH] by locating the position of local minimum.', '1601.01160-1-29-5': 'But in order to locate [MATH], [MATH], and [MATH] precisely, we are required to examine the equal area law for the first order phase transition as well as pseudo phase transition.', '1601.01160-1-29-6': 'And to locate [MATH], we should obtain the critical exponent [MATH] for the second order phase transition, which are documented as follows.', '1601.01160-1-30-0': 'In the [MATH] plane, we define the equal area law as [EQUATION] in which [MATH] is an Interpolating Function obtained from the numeric result, [MATH] is the phase transition temperature, and [MATH], [MATH] are the smallest and largest values of the unstable region which satisfy the equation [MATH].', '1601.01160-1-30-1': 'Similar to the equal area law in the [MATH] plane, we know that for [MATH] and [MATH], [MATH], while for [MATH], [MATH] for the first order phase transition and [MATH] for the pseudo phase transition.', '1601.01160-1-30-2': 'For different [MATH] and [MATH], the calculated results are listed in Table [REF].', '1601.01160-1-31-0': 'It is obvious that [MATH] equals [MATH] for different [MATH] and [MATH] in a reasonable accuracy.', '1601.01160-1-31-1': 'That is, in the [MATH] plane, the equal area law holds, and it is independent of the Born-Infeld parameter as well as the size of the boundary region.', '1601.01160-1-32-0': 'In order to get the critical exponent for the second order phase transition in the [MATH] plane, we should first define an analogous heat capacity [EQUATION]', '1601.01160-1-32-1': 'Provided a similar relation as showed in Eq. ([REF]) is satisfied, one can get the critical exponent immediately.', '1601.01160-1-32-2': 'So next, we are interested in the logarithm of the quantities [MATH], [MATH], in which [MATH] is the critical temperature mentioned previously, and [MATH] is obtained numerically by the equation [MATH].', '1601.01160-1-32-3': 'We plot the relation between [MATH] and [MATH] for different [MATH] and [MATH] in Figure [REF], where these straight lines can be fitted as [EQUATION]', '1601.01160-1-32-4': 'It is obvious that the slope is about 3, which indicates that the critical exponent is [MATH] nearly for the analogous heat capacity, and the phase transition is also second order with the phase transition temperature [MATH].', '1601.01160-1-33-0': '## Phase structure probed by holographic entanglement entropy', '1601.01160-1-34-0': "According to the formula in [CITATION], holographic entanglement entropy can be given by the area [MATH] of a minimal surface [MATH] anchored on the boundary entangling surface [MATH], namely [EQUATION] where [MATH] is the Newton's constant.", '1601.01160-1-34-1': 'We will take the region [MATH] to be a spherical cap on the boundary delimited by [MATH].', '1601.01160-1-34-2': 'Then based on the definition of area and ([REF]), ([REF]) can be rewritten as [EQUATION] in which [MATH].', '1601.01160-1-34-3': 'Making use of the Euler-Lagrange equation, one can get the equation of motion of [MATH].', '1601.01160-1-34-4': 'With the boundary conditions in Eq. ([REF]), we can obtain the numeric result of [MATH] immediately.', '1601.01160-1-34-5': 'Note that the entanglement entropy is divergent at the boundary, so it should be regularized by subtracting off the entanglement entropy in pure AdS with the same entangling surface and boundary values.', '1601.01160-1-34-6': 'We label the regularized entanglement entropy as [MATH].', '1601.01160-1-34-7': 'We choose the size of the boundary region as [MATH] and set the UV cutoff in the dual field theory to be [MATH].', '1601.01160-1-35-0': 'The isocharges for the cases [MATH] and [MATH] on the entanglement entropy-temperature plane are plotted in Figure [REF] and Figure [REF].', '1601.01160-1-35-1': 'As the same as that of the two point correlation function, for [MATH], we are interested in that the black holes have charges [MATH], and for [MATH], we choose [MATH].', '1601.01160-1-35-2': 'From Figure [REF], we know that the entanglement entropy exhibits a Hawking-Page phase transition, and a Van der Waals-like phase transition as the charge increases form 0 to 0.21.', '1601.01160-1-35-3': 'The entanglement entropy also exhibits the novel phase structure as that of the thermal entropy as well as two point correlation function, which is shown in (a) of Figure [REF].', '1601.01160-1-36-0': "We will also employ Maxwell's equal area law to locate the first order phase transition temperature, namely [MATH] in (b) of Figure [REF] and Figure [REF], [MATH] in (a) of Figure [REF], and pseudo phase transition temperature [MATH].", '1601.01160-1-36-1': 'In the [MATH] plane, we define the equal area law as [EQUATION] in which [MATH] is an Interpolating Function obtained from the numeric result, and [MATH], [MATH] are the smallest and largest roots of the unstable regions which satisfy the equation [MATH].', '1601.01160-1-36-2': 'Surely the values of the phase transition temperature [MATH] depends on the choice of [MATH] and [MATH].', '1601.01160-1-36-3': 'For different [MATH] and [MATH], the results of [MATH], [MATH] and [MATH], [MATH] are listed in Table ([REF]).', '1601.01160-1-37-0': 'It is obvious that [MATH] equals nearly [MATH] for a fixed [MATH] and [MATH].', '1601.01160-1-37-1': 'That is, the equal area law is also valid for the first order phase transition and pseudo phase transition in the entanglement entropy-temperature plane.', '1601.01160-1-37-2': 'This result is the same as that of the thermal entropy as well as two point correlation function.', '1601.01160-1-38-0': 'To confirm that [MATH] is the second order phase transition temperature in the [MATH] plane, we should explore whether it satisfies a similar relation as Eq. ([REF]).', '1601.01160-1-38-1': 'For different [MATH] and [MATH], the relation between [MATH] and [MATH] are plotted in Figure [REF], in which [MATH] is the critical entropy obtained numerically by the equation [MATH].', '1601.01160-1-38-2': 'The analytical result of these curves can be fitted as [EQUATION]', '1601.01160-1-38-3': 'We find for a fixed [MATH] and [MATH], the slope is always about 3, which is consistent with that of the thermal entropy.', '1601.01160-1-38-4': 'That is, the entanglement entropy also exhibits a second order phase transition, with phase transition temperature [MATH], as that of the thermal entropy and two point correlation function.', '1601.01160-1-39-0': '# Concluding remarks', '1601.01160-1-40-0': 'Investigation on the phase structure of a back hole is of great importance for it reveals whether and how a black hole emerges.', '1601.01160-1-40-1': 'Usually, it is realized by studying the relation between some thermodynamic quantities in a fixed ensemble.', '1601.01160-1-40-2': 'In this paper, we found that the phase structure of a back hole also can be probed by the two point correlation function and holographic entanglement entropy, which provides a new strategy to understand the phase structure of the black holes from the viewpoint of holography.', '1601.01160-1-41-0': 'Specially speaking, we first investigated the phase structure of the Born-Infeld-anti-de Sitter black holes in the [MATH] plane in the fixed charge ensemble.', '1601.01160-1-41-1': 'We found that the phase structure of the black hole depends on not only the value of [MATH] or [MATH], but also the combination of [MATH].', '1601.01160-1-41-2': 'For the case [MATH], the black hole resembles as the Reissner-Nordstrom-AdS black hole, and it undergoes the Van der Waals-like phase transition as its charge satisfies the non-extremal condition [MATH].', '1601.01160-1-41-3': 'For the case [MATH], besides the Van der Waals-like phase transition, we also observed a novel phase structure for the case [MATH].', '1601.01160-1-41-4': 'With the two point correlation function and holographic entanglement entropy, we further probed the phase structure of the Born-Infeld-anti-de Sitter black holes and found that both the probes exhibited the same phase structure as that of the thermal entropy for cases [MATH] and [MATH] regardless of the size of the boundary region.', '1601.01160-1-41-5': 'This conclusion was reinforced by checking the equal area law for the first order phase transition as well as pseudo phase transition, and calculating the critical exponent of the analogous heat capacity for the second order phase transition.', '1601.01160-1-42-0': 'In previous investigation on the phase structure of the Born-Infeld-anti-de Sitter black holes, authors concentrated mainly on that of [MATH] [CITATION].', '1601.01160-1-42-1': 'In this paper we found that the phase structure for the case [MATH] is also interesting.', '1601.01160-1-42-2': 'As [MATH], the black hole resembles as the Schwarzschild AdS black hole as expected, and there is a minimum temperature.', '1601.01160-1-42-3': 'We found the larger the charge or the Born-Infeld parameter is, the smaller the minimum temperature will be.', '1601.01160-1-42-4': 'That is, both the charge and the Born-Infeld parameter promote the formation of an AdS black hole.', '1601.01160-1-42-5': 'As [MATH] approaches to 0.5, a new extremal small black hole branch emerges compared with that of the Reissner-Nordstrom-AdS black hole, so that there are two unstable regions and correspondingly two phase transition temperature.', '1601.01160-1-42-6': 'The high temperature phase transition was found to be pseudo for in this case the space time was dominated by the large black hole, which was observed from the [MATH] relation.'}

{'1601.01160-2-0-0': '# Introduction', '1601.01160-2-1-0': 'Investigation on phase transition of an AdS space time has attracted attention of many theoretical physicists recently.', '1601.01160-2-1-1': 'The main motivation maybe steps from the existence of the AdS/CFT correspondence [CITATION], which relates a black hole in the AdS space time to a thermal system without gravity.', '1601.01160-2-1-2': 'For in this case, some interesting but intractable phenomena in strongly coupled system become to be tractable easily in the bulk.', '1601.01160-2-1-3': 'To probe these fascinating phenomena in field theory, one should employ some non-local observables such as two point correlation function, Wilson loop, and holographic entanglement entropy, which are dual to the geodesic length, minimal area surface, and minimal surface area in the bulk individually.', '1601.01160-2-1-4': 'It has been shown that these observables can probe the non-equilibrium thermalization behavior [CITATION], superconducting phase transition [CITATION], and cosmological singularity [CITATION].', '1601.01160-2-2-0': 'In this paper, we intend to use the non-local observables to probe the phase structure of the Born-Infeld-anti-de Sitter black holes.', '1601.01160-2-2-1': 'Usually, phase structure of a black hole is understood from the viewpoint of thermodynamics.', '1601.01160-2-2-2': 'For an uncharged AdS black hole, it has been found that there is a phase transition between the thermal gas in AdS space and Schwarzschild AdS black holes [CITATION], which was interpreted as the confinement/deconfinement phase transition in the dual gauge field theory later [CITATION].', '1601.01160-2-2-3': 'As the charge is endowed with, the AdS black hole will undergo a Van der Waals-like phase transition before it reaches the stable state in the entropy-temperature plane [CITATION].', '1601.01160-2-2-4': 'Specifically speaking, there exists a critical charge, and for the case that the charge of the black hole is smaller than the critical charge, the black hole undertakes a first order phase transition.', '1601.01160-2-2-5': 'As the charge increases to the critical charge, the phase transition is second order, while the charge exceeds the critical charge, there is not phase transition and the black hole is always stable.', '1601.01160-2-2-6': 'The Van der Waals-like phase transition can be observed in many circumstances.', '1601.01160-2-2-7': 'In [CITATION], it was found that a 5-dimensional neutral Gauss-Bonnet black hole demonstrates the Van der Waals-like phase transition in the [MATH] plane, where [MATH] is the Hawking temperature and [MATH] is the Gauss-Bonnet coupling parameter.', '1601.01160-2-2-8': 'In [CITATION], the Van der Waals-like phase transition was also observed in the [MATH] plane, where [MATH] is electric charge and [MATH] is the chemical potential.', '1601.01160-2-2-9': 'Treating the negative cosmological constant as the pressure [MATH] and its conjugate as the thermodynamical volume [MATH], the Van der Waals-like phase transition has also been observed in the [MATH] plane recently[CITATION].', '1601.01160-2-3-0': 'Very recently, with the entanglement entropy as a probe, [CITATION] investigated the phase structure of the Reissner-Nordstrom-AdS black hole and found that there was also a Van der Waals-like phase transition in the entanglement entropy-temperature plane .', '1601.01160-2-3-1': 'They also obtained the critical exponent of the heat capacity for the second order phase transition in the neighborhood of the critical points.', '1601.01160-2-3-2': 'To further confirm the similarity of the phase structure between the thermal entropy and entanglement entropy, [CITATION] checked the equal area law later and found that it held for the first order phase transition in the entanglement entropy-temperature plane.', '1601.01160-2-3-3': 'Now [CITATION] has been generalized to the extended space time [CITATION], massive gravity [CITATION] , as well as Weyl gravity [CITATION], and all the results showed that the entanglement entropy exhibited the same phase structure as that of the thermal entropy.', '1601.01160-2-4-0': 'In this paper, besides the entanglement entropy, we will employ the two point correlation function to probe the phase structure of the black holes.', '1601.01160-2-4-1': 'We choose the Born-Infeld-anti-de Sitter black holes as the gravity model, which is a solution of the Einstein-Born-Infeld action.', '1601.01160-2-4-2': 'There have been many works to study phase structure of the Born-Infeld-anti-de Sitter black holes [CITATION].', '1601.01160-2-4-3': 'The results showed that both the Born-Infeld parameter [MATH] and charge [MATH] affect the phase structure, and to assure the existence of a non-extremal black hole, one should impose the condition [MATH].', '1601.01160-2-4-4': 'In this paper, we find for the case [MATH], the phase structure of the black hole is similar to that of the Reissner-Nordstrom-AdS black hole in the entropy-temperature plane.', '1601.01160-2-4-5': 'While for the case [MATH], there is a novel phase structure, which has not been observed previously.', '1601.01160-2-4-6': 'Specially speaking, a new branch emerges compared with that of the Reissner-Nordstrom-AdS black hole so that there are two unstable regions and two phase transition temperature correspondingly.', '1601.01160-2-4-7': 'All these phase structure are probed by the nonlocal observables such as two point correlation function as well as holographic entanglement entropy, and phase structure of the nonlocal observables are found to be the same as that of the thermal entropy.', '1601.01160-2-5-0': 'Our paper is outlined as follows.', '1601.01160-2-5-1': 'In the next section, we will review the thermodynamic properties of the Born-Infeld-anti-de Sitter black hole firstly, and then study its phase structure in [MATH] plane in a fixed charge ensemble.', '1601.01160-2-5-2': 'In Section [REF], we will employ the two point correlation function and holographic entanglement entropy to probe the phase structure of the Born-Infeld-anti-de Sitter black hole.', '1601.01160-2-5-3': 'In each subsection, the equal area law is checked and the critical exponent of the heat capacity is obtained.', '1601.01160-2-5-4': 'The last section is devoted to discussions and conclusions.', '1601.01160-2-6-0': '# Thermodynamic phase transition of the Born-Infeld-anti-de Sitter black hole', '1601.01160-2-7-0': '## Review of the Born-Infeld-anti-de Sitter black hole', '1601.01160-2-8-0': 'The 4-dimensional Born-Infeld AdS black hole is a solution of the following action [CITATION] [EQUATION] where [EQUATION] in which [MATH] is the hypergeometric function.', '1601.01160-2-8-1': 'From Eq. ([REF]), we know that in the limit [MATH], [MATH], the solution reduces to the the Reissner-Nordstrom-AdS black hole, and in the limit [MATH], it reduces to the Schwarzschild AdS black hole.', '1601.01160-2-9-0': 'The ADM mass of the black hole, defined by [MATH], is given by [EQUATION] in which [MATH] is the event horizon of the black hole.', '1601.01160-2-9-1': 'The Hawking temperature defined by [MATH] can be written as [EQUATION] where we have used Eq. ([REF]).', '1601.01160-2-9-2': 'In addition, according to the Bekenstein-Hawking entropy area relation, we can get the black hole entropy [EQUATION]', '1601.01160-2-9-3': 'Inserting Eq. ([REF]) into Eq. ([REF]), we can get the entropy temperature relation, namely [EQUATION]', '1601.01160-2-9-4': 'It is obvious that besides the charge [MATH], the Born-Infeld parameter [MATH] also affects the phase structure of this space time in the [MATH] plane.', '1601.01160-2-10-0': '## Phase transition of thermal entropy', '1601.01160-2-11-0': 'Based on Eq. ([REF]), we will discuss the phase structure of the Born-Infeld-anti-de Sitter black hole.', '1601.01160-2-11-1': 'We are interested in how [MATH] or [MATH] affects it.', '1601.01160-2-11-2': 'For a fixed charge [MATH], the effect of [MATH] on the phase structure is shown in Figure [REF].', '1601.01160-2-11-3': 'It is obvious that for the small [MATH], the phase structure resembles as that of the Schwarzschild AdS black hole, and for large [MATH], it resembles as that of the Reissner-Nordstrom-AdS black hole as expected.', '1601.01160-2-11-4': 'From (b) of Figure [REF], we can see that as [MATH] increases, the minimum temperature of the space time decreases, and the entropy, also the event horizon, becomes smaller.', '1601.01160-2-11-5': 'That is, the Born-Infeld parameter [MATH] promotes the formation of an AdS space time.', '1601.01160-2-12-0': 'More interestingly, for the smaller charge, we find there is a novel phase transition, which is labeled by the red solid line in (a) of Figure [REF].', '1601.01160-2-12-1': 'From this curve, we know that there are two unstable regions.', '1601.01160-2-12-2': 'Namely the black hole will undergo the following transition: unstable-stable-unstable-stable.', '1601.01160-2-12-3': 'Compared with that of the Reissner-Nordstrom-AdS black hole, a new branch emerges at the onset of the phase transition.', '1601.01160-2-12-4': 'The influence of the charge on the phase structure for a fixed Born-Infeld parameter is plotted in Figure [REF].', '1601.01160-2-13-0': 'Apparently, the phase structure is similar to that of the Schwarzschild AdS black hole for the small charges, and Reissner-Nordstrom-AdS black hole for large charges.', '1601.01160-2-13-1': 'From (a) of Figure [REF], we observe that the larger the charge is, the lower the minimum temperature will be.', '1601.01160-2-13-2': 'In other words, the charge will promote the formation of an AdS space time, which has the same effect as that of the Born-Infeld parameter on the phase structure of the black hole.', '1601.01160-2-14-0': 'For the case [MATH] in (b) of Figure [REF], we see that for some intermediate values of charge [MATH], there is also a similar phase structure as that in (b) in Figure [REF], which is labeled by the red solid line in (b) of Figure [REF].', '1601.01160-2-14-1': 'Note that however, for the large [MATH], the novel phase structure disappears, which is shown in Figure [REF].', '1601.01160-2-14-2': 'For in this case, the Born-Infeld parameter is large enough, so that the phase structure resembles completely as that of the Reissner-Nordstrom-AdS black hole.', '1601.01160-2-15-0': 'Next, we will study detailedly the phase structure of the Born-Infeld AdS black hole for the case [MATH] and [MATH] respectively.', '1601.01160-2-15-1': 'To finish it, we should first find the critical charge for a fixed Born-Infeld parameter by the condition [EQUATION]', '1601.01160-2-15-2': 'However, from Eq. ([REF]), we find it is hard to get the analytical result directly.', '1601.01160-2-15-3': 'Taking the case [MATH] as an example, we will show how to get it numerically next.', '1601.01160-2-15-4': 'We first plot a series of curves for different charges in the [MATH] plane, which is shown in (a) of Figure [REF], and read off the charge which satisfies likely the condition [MATH].', '1601.01160-2-15-5': 'With that rough value, we plot a bunch of curves in the [MATH] plane once with smaller step so that we can get the most likely value of charge.', '1601.01160-2-15-6': 'From (b) of Figure [REF], we find the critical charge should be [MATH], which are labeled by the red dashed lines in (b) of Figure [REF].', '1601.01160-2-15-7': 'Lastly, we adjust the value of [MATH] by hand to find the only solution [MATH] which satisfies Eq.([REF]), which produces [MATH], [MATH].', '1601.01160-2-15-8': 'Substituting these critical values into Eq.([REF]), we further get the critical temperature [MATH].', '1601.01160-2-16-0': 'Having obtained the critical charge, we can plot the isocharge curves for the case [MATH] in the [MATH] plane, which is shown in Figure [REF].', '1601.01160-2-16-1': 'We observe that the black hole undergoes a Hawking-Page phase transition and a Van der Waals-like phase transition.', '1601.01160-2-16-2': 'Specifically speaking, for the case [MATH], there is a minimum temperature [MATH] [CITATION], which is indicated by the red dashed line in (a) of Figure [REF].', '1601.01160-2-16-3': 'When the temperature is higher than [MATH], there are two additional black hole branches.', '1601.01160-2-16-4': 'The small branch is unstable while the large branch is stable.', '1601.01160-2-16-5': 'The Hawking-Page phase transition occurs at the temperature given by [MATH] [CITATION], which is indicated by the red dotted line.', '1601.01160-2-16-6': 'The Hawking-Page phase transition also can be observed in the [MATH] plane, in which [MATH] is the Helmholtz free energy defined by [EQUATION]', '1601.01160-2-16-7': 'From (a) of Figure [REF], we know that there is a minimum temperature [MATH], and above this temperature, there are two branches.', '1601.01160-2-16-8': 'The lower branch is stable always.', '1601.01160-2-16-9': 'The Hawking-Page phase transition occurs at [MATH] for in this case the free energy vanishes.', '1601.01160-2-17-0': 'For the case [MATH], the phase structure is similar to that of the Van der Waals phase transition, which is shown in (b) of Figure [REF].', '1601.01160-2-17-1': 'The solid blue lines from top to down correspond to the isocharges for the case [MATH].', '1601.01160-2-17-2': 'We can see that black holes endowed with different charges have different phase structures.', '1601.01160-2-17-3': 'For the small charge, there is an unstable black hole interpolating between the stable small hole and stable large hole.', '1601.01160-2-18-0': 'The small stable hole will jump to the large stable hole at the critical temperature [MATH].', '1601.01160-2-18-1': 'As the charge increases to the critical charge, the small hole and the large hole merge into one and squeeze out the unstable phase so that an inflection point emerges.', '1601.01160-2-18-2': 'The divergence of the heat capacity in this case implies that the phase transition is second order.', '1601.01160-2-18-3': 'As the charge exceeds the critical charge, we simply have one stable black hole at each temperature.', '1601.01160-2-18-4': 'The Van der Waals-like phase transition can also be observed from the [MATH] relation.', '1601.01160-2-18-5': 'From (b) of Figure [REF], we observe that there is a swallowtail structure, which corresponds to the unstable phase in the top curve in (b) Figure [REF].', '1601.01160-2-18-6': 'The critical temperature [MATH] is apparently the value of the horizontal coordinate of the junction between the small black hole and the large black hole.', '1601.01160-2-18-7': 'As the temperature is lower than the critical temperature [MATH], the free energy of the small black hole is lowest, so the small hole is stable.', '1601.01160-2-18-8': 'As the temperature is higher than [MATH], the free energy of the large black hole is lowest, so that the large hole dominates thereafter.', '1601.01160-2-18-9': 'The non-smoothness of the junction indicates that the phase transition is first order.', '1601.01160-2-18-10': 'From (c) of Figure [REF], we know that there is an inflection point, which corresponds to the inflection point in the middle curve in (b) of Figure [REF].', '1601.01160-2-18-11': 'The horizontal coordinate of the inflection point corresponds to the second order phase transition temperature [MATH].', '1601.01160-2-19-0': 'Similarly, we also can study the phase structure for the case [MATH] in the [MATH] plane.', '1601.01160-2-19-1': 'To finish it, we should first find the critical charge.', '1601.01160-2-19-2': 'Adopting the same strategy as that of the case [MATH], we find [MATH], [MATH], [MATH].', '1601.01160-2-19-3': 'For the case [MATH], the phase structure is the same as that in (a) of Figure [REF].', '1601.01160-2-19-4': 'We will not repeat it here.', '1601.01160-2-19-5': 'For the case [MATH], we find besides the Van der Waals-like phase transition, there is a novel phase transition for [MATH] as observed in Figure [REF] and Figure [REF], which is plotted in (a) of Figure [REF].', '1601.01160-2-19-6': 'That is, at the beginning of the phase transition, an infinitesimally small black hole branch emerges.', '1601.01160-2-19-7': 'It is unstable and its life is very short so that it becomes to a small stable black hole quickly.', '1601.01160-2-19-8': 'In addition, we find there are two unstable region, and correspondingly in the [MATH] plane, we observe two swallowtail structures, which is shown in (a) of Figure [REF].', '1601.01160-2-19-9': 'The horizontal coordinate of the junction of the swallowtail corresponds to the phase transition temperature, thus besides the first order phase transition temperature between the small black hole and large black hole, labeled by [MATH], there is a new phase transition temperature, labeled by [MATH].', '1601.01160-2-19-10': 'It should be stressed that the new phase transition can not take place actually, for above the critical temperature [MATH], the free energy of the large black hole is lowest so that the space time is dominated by the large black hole, which can be seen in (a) of Figure [REF].', '1601.01160-2-19-11': 'Thus, we call thus phase transition pseudo phase transition.', '1601.01160-2-20-0': 'We observe that there is also a Van der Waals-like phase transition, which is plotted in (b) of Figure [REF].', '1601.01160-2-20-1': 'The solid blue lines from top to down correspond to the isocharges for the case [MATH].', '1601.01160-2-20-2': 'The first order phase transition temperature [MATH] can be read off from the horizontal coordinate of the junction of the swallowtail structure in (b) of Figure [REF], and the second order phase transition temperature [MATH] can be read off from the horizontal coordinate of the inflection point in (c) of Figure [REF].', '1601.01160-2-21-0': "As done in [CITATION], we will also check numerically whether Maxwell's equal area law holds for the first order phase transition and pseudo phase transition, which states [EQUATION] in which [MATH] is defined in Eq. ([REF]), [MATH] is the phase transition temperature, and [MATH], [MATH] are the smallest and largest values of unstable region which satisfies the equation [MATH].", '1601.01160-2-21-1': 'Usually for this equation, there are three roots [MATH], so [MATH], [MATH], [MATH].', '1601.01160-2-21-2': 'But for the case [MATH], there are four roots.', '1601.01160-2-21-3': 'One can see from (a) of Figure [REF] that [MATH], [MATH] for the pseudo phase transition with phase transition temperature [MATH], and [MATH], [MATH] for the first order phase transition with phase transition temperature [MATH].', '1601.01160-2-21-4': 'The calculated results for the first order phase transition and pseudo phase transition are listed in Table [REF].', '1601.01160-2-22-0': 'From this table, we can see that [MATH] equals [MATH] for different [MATH] in our numeric accuracy.', '1601.01160-2-22-1': 'The equal area law therefore holds.', '1601.01160-2-23-0': 'For the second order phase transition, we know that the heat capacity is divergent near the critical point and the critical exponent is [MATH].', '1601.01160-2-23-1': 'As stated in [CITATION], near the critical point, there is always a linear relation [EQUATION] with 3 the slope.', '1601.01160-2-23-2': 'It is not difficult to show that for the case [MATH] in our gravity model, the temperature and entropy also satisfy this linear relation near the critical point.', '1601.01160-2-23-3': 'Next, we will take Eq. ([REF]) as a reference to check whether there is a similar relation for the second order phase transition in the two point correlation function-temperature plane as well as entanglement entropy-temperature plane.', '1601.01160-2-24-0': '# Phase transition in the framework of holography', '1601.01160-2-25-0': 'Having obtained the phase structure of thermal entropy of the Born-Infeld AdS black hole in the [MATH] plane, we will study the phase structure of two point correlation function and entanglement entropy in the fled theory to see whether they have the similar phase structure and critical behavior.', '1601.01160-2-26-0': '## Phase structure probed by two point correlation function', '1601.01160-2-27-0': 'According to the AdS/CFT correspondence, in the large [MATH] limit, the equal time two point correlation function can be written as [CITATION] [EQUATION] where [MATH] is the conformal dimension of scalar operator [MATH] in the dual field theory, [MATH] is the length of the bulk geodesic between the points [MATH] and [MATH] on the AdS boundary.', '1601.01160-2-27-1': 'In our gravity model, we can simply choose [MATH] and [MATH] as the two boundary points.', '1601.01160-2-27-2': 'Then with [MATH] to parameterize the trajectory, the proper length is given by [EQUATION] in which [MATH].', '1601.01160-2-27-3': 'Imagining [MATH] as time, and treating [MATH] as the Lagrangian, one can get the equation of motion for [MATH] by making use of the Euler-Lagrange equation.', '1601.01160-2-27-4': 'Then with the following boundary conditions [EQUATION] we can get the numeric result of [MATH].', '1601.01160-2-27-5': 'To explore whether the size of the boundary region affects the phase structure, we will choose [MATH] as two examples.', '1601.01160-2-27-6': 'Note that for a fixed [MATH], the geodesic length is divergent, so it should be regularized by subtracting off the geodesic length in pure AdS with the same boundary region, denoted by [MATH].', '1601.01160-2-27-7': 'To achieve this, we are required to set a UV cutoff for each case, which is chosen to be [MATH] and [MATH], respectively for our two examples.', '1601.01160-2-27-8': 'The regularized geodesic length is labeled as [MATH].', '1601.01160-2-27-9': 'In addition, during the numerics, we will set the AdS radius [MATH] to be 1.', '1601.01160-2-28-0': 'We plot the relation between [MATH] and [MATH] for the case [MATH] in Figure [REF] and Figure [REF] .', '1601.01160-2-28-1': 'The solid blue line in (a) corresponds to the isocharge for the case [MATH], and the dashed blue lines from top to down in (b) correspond to that [MATH].', '1601.01160-2-28-2': 'From these figures, one can see that [MATH] demonstrates a similar phase structure as that of the thermal entropy in Figure [REF].', '1601.01160-2-28-3': 'Namely the two point correlation function can probe both the Hawking-Page phase transition and Van der Waals-like phase transition.', '1601.01160-2-28-4': 'Precisely speaking, we find that the minimum temperature [MATH] as well as Hawking-Page phase transition temperature [MATH] in [MATH], the first order phase transition temperature [MATH], and second order phase transition temperature [MATH] in [MATH] are exactly the same as those in [MATH] plane.', '1601.01160-2-28-5': 'This conclusion will not be affected as [MATH] varies in a reasonable region, which can be seen from Figure [REF] and Figure [REF].', '1601.01160-2-29-0': 'With the two point correlation function, we also can probe the phase structure of the case [MATH], which is shown in Figure [REF] and Figure [REF].', '1601.01160-2-29-1': 'The solid blue line in (a) corresponds to the isocharge for the case [MATH], and the solid blue lines from top to down in (b) correspond to that [MATH].', '1601.01160-2-29-2': 'As the case as the thermal entropy in the [MATH] plane, we find there ia also a novel phase structure besides the Van der Waals-like phase transition in the [MATH] plane.', '1601.01160-2-29-3': 'There are also two unstable region, correspondingly two phase transition temperature, labeled by [MATH], [MATH].', '1601.01160-2-29-4': 'For the phase transition temperature mentioned above, it is easy to check [MATH] by locating the position of local minimum.', '1601.01160-2-29-5': 'But in order to locate [MATH], [MATH], and [MATH] precisely, we are required to examine the equal area law for the first order phase transition as well as pseudo phase transition.', '1601.01160-2-29-6': 'And to locate [MATH], we should obtain the critical exponent [MATH] for the second order phase transition, which are documented as follows.', '1601.01160-2-30-0': 'In the [MATH] plane, we define the equal area law as [EQUATION] in which [MATH] is an Interpolating Function obtained from the numeric result, [MATH] is the phase transition temperature, and [MATH], [MATH] are the smallest and largest values of the unstable region which satisfy the equation [MATH].', '1601.01160-2-30-1': 'Similar to the equal area law in the [MATH] plane, we know that for [MATH] and [MATH], [MATH], while for [MATH], [MATH] for the first order phase transition and [MATH] for the pseudo phase transition.', '1601.01160-2-30-2': 'For different [MATH] and [MATH], the calculated results are listed in Table [REF].', '1601.01160-2-31-0': 'It is obvious that [MATH] equals [MATH] for different [MATH] and [MATH] in a reasonable accuracy.', '1601.01160-2-31-1': 'That is, in the [MATH] plane, the equal area law holds, and it is independent of the Born-Infeld parameter as well as the size of the boundary region.', '1601.01160-2-32-0': 'In order to get the critical exponent for the second order phase transition in the [MATH] plane, we should first define an analogous heat capacity [EQUATION]', '1601.01160-2-32-1': 'Provided a similar relation as showed in Eq. ([REF]) is satisfied, one can get the critical exponent immediately.', '1601.01160-2-32-2': 'So next, we are interested in the logarithm of the quantities [MATH], [MATH], in which [MATH] is the critical temperature mentioned previously, and [MATH] is obtained numerically by the equation [MATH].', '1601.01160-2-32-3': 'We plot the relation between [MATH] and [MATH] for different [MATH] and [MATH] in Figure [REF], where these straight lines can be fitted as [EQUATION]', '1601.01160-2-32-4': 'It is obvious that the slope is about 3, which indicates that the critical exponent is [MATH] nearly for the analogous heat capacity, and the phase transition is also second order with the phase transition temperature [MATH].', '1601.01160-2-33-0': '## Phase structure probed by holographic entanglement entropy', '1601.01160-2-34-0': "According to the formula in [CITATION], holographic entanglement entropy can be given by the area [MATH] of a minimal surface [MATH] anchored on the boundary entangling surface [MATH], namely [EQUATION] where [MATH] is the Newton's constant.", '1601.01160-2-34-1': 'We will take the region [MATH] to be a spherical cap on the boundary delimited by [MATH].', '1601.01160-2-34-2': 'Then based on the definition of area and ([REF]), ([REF]) can be rewritten as [EQUATION] in which [MATH].', '1601.01160-2-34-3': 'Making use of the Euler-Lagrange equation, one can get the equation of motion of [MATH].', '1601.01160-2-34-4': 'With the boundary conditions in Eq. ([REF]), we can obtain the numeric result of [MATH] immediately.', '1601.01160-2-34-5': 'Note that the entanglement entropy is divergent at the boundary, so it should be regularized by subtracting off the entanglement entropy in pure AdS with the same entangling surface and boundary values.', '1601.01160-2-34-6': 'We label the regularized entanglement entropy as [MATH].', '1601.01160-2-34-7': 'We choose the size of the boundary region as [MATH] and set the UV cutoff in the dual field theory to be [MATH].', '1601.01160-2-35-0': 'The isocharges for the cases [MATH] and [MATH] on the entanglement entropy-temperature plane are plotted in Figure [REF] and Figure [REF].', '1601.01160-2-35-1': 'As the same as that of the two point correlation function, for [MATH], we are interested in that the black holes have charges [MATH], and for [MATH], we choose [MATH].', '1601.01160-2-35-2': 'From Figure [REF], we know that the entanglement entropy exhibits a Hawking-Page phase transition, and a Van der Waals-like phase transition as the charge increases form 0 to 0.21.', '1601.01160-2-35-3': 'The entanglement entropy also exhibits the novel phase structure as that of the thermal entropy as well as two point correlation function, which is shown in (a) of Figure [REF].', '1601.01160-2-36-0': "We will also employ Maxwell's equal area law to locate the first order phase transition temperature, namely [MATH] in (b) of Figure [REF] and Figure [REF], [MATH] in (a) of Figure [REF], and pseudo phase transition temperature [MATH].", '1601.01160-2-36-1': 'In the [MATH] plane, we define the equal area law as [EQUATION] in which [MATH] is an Interpolating Function obtained from the numeric result, and [MATH], [MATH] are the smallest and largest roots of the unstable regions which satisfy the equation [MATH].', '1601.01160-2-36-2': 'Surely the values of the phase transition temperature [MATH] depends on the choice of [MATH] and [MATH].', '1601.01160-2-36-3': 'For different [MATH] and [MATH], the results of [MATH], [MATH] and [MATH], [MATH] are listed in Table ([REF]).', '1601.01160-2-37-0': 'It is obvious that [MATH] equals nearly [MATH] for a fixed [MATH] and [MATH].', '1601.01160-2-37-1': 'That is, the equal area law is also valid for the first order phase transition and pseudo phase transition in the entanglement entropy-temperature plane.', '1601.01160-2-37-2': 'This result is the same as that of the thermal entropy as well as two point correlation function.', '1601.01160-2-38-0': 'To confirm that [MATH] is the second order phase transition temperature in the [MATH] plane, we should explore whether it satisfies a similar relation as Eq. ([REF]).', '1601.01160-2-38-1': 'For different [MATH] and [MATH], the relation between [MATH] and [MATH] are plotted in Figure [REF], in which [MATH] is the critical entropy obtained numerically by the equation [MATH].', '1601.01160-2-38-2': 'The analytical result of these curves can be fitted as [EQUATION]', '1601.01160-2-38-3': 'We find for a fixed [MATH] and [MATH], the slope is always about 3, which is consistent with that of the thermal entropy.', '1601.01160-2-38-4': 'That is, the entanglement entropy also exhibits a second order phase transition, with phase transition temperature [MATH], as that of the thermal entropy and two point correlation function.', '1601.01160-2-39-0': '# Concluding remarks', '1601.01160-2-40-0': 'Investigation on the phase structure of a back hole is of great importance for it reveals whether and how a black hole emerges.', '1601.01160-2-40-1': 'Usually, it is realized by studying the relation between some thermodynamic quantities in a fixed ensemble.', '1601.01160-2-40-2': 'In this paper, we found that the phase structure of a back hole also can be probed by the two point correlation function and holographic entanglement entropy, which provides a new strategy to understand the phase structure of the black holes from the viewpoint of holography.', '1601.01160-2-41-0': 'Specially speaking, we first investigated the phase structure of the Born-Infeld-anti-de Sitter black holes in the [MATH] plane in the fixed charge ensemble.', '1601.01160-2-41-1': 'We found that the phase structure of the black hole depends on not only the value of [MATH] or [MATH], but also the combination of [MATH].', '1601.01160-2-41-2': 'For the case [MATH], the black hole resembles as the Reissner-Nordstrom-AdS black hole, and it undergoes the Van der Waals-like phase transition as its charge satisfies the non-extremal condition [MATH].', '1601.01160-2-41-3': 'For the case [MATH], besides the Van der Waals-like phase transition, we also observed a novel phase structure for the case [MATH].', '1601.01160-2-41-4': 'With the two point correlation function and holographic entanglement entropy, we further probed the phase structure of the Born-Infeld-anti-de Sitter black holes and found that both the probes exhibited the same phase structure as that of the thermal entropy for cases [MATH] and [MATH] regardless of the size of the boundary region.', '1601.01160-2-41-5': 'This conclusion was reinforced by checking the equal area law for the first order phase transition as well as pseudo phase transition, and calculating the critical exponent of the analogous heat capacity for the second order phase transition.', '1601.01160-2-42-0': 'In previous investigation on the phase structure of the Born-Infeld-anti-de Sitter black holes, authors concentrated mainly on that of [MATH] [CITATION].', '1601.01160-2-42-1': 'In this paper we found that the phase structure for the case [MATH] is also interesting.', '1601.01160-2-42-2': 'As [MATH], the black hole resembles as the Schwarzschild AdS black hole as expected, and there is a minimum temperature.', '1601.01160-2-42-3': 'We found the larger the charge or the Born-Infeld parameter is, the smaller the minimum temperature will be.', '1601.01160-2-42-4': 'That is, both the charge and the Born-Infeld parameter promote the formation of an AdS black hole.', '1601.01160-2-42-5': 'As [MATH] approaches to 0.5, a new extremal small black hole branch emerges compared with that of the Reissner-Nordstrom-AdS black hole, so that there are two unstable regions and correspondingly two phase transition temperature.', '1601.01160-2-42-6': 'The high temperature phase transition was found to be pseudo for in this case the space time was dominated by the large black hole, which was observed from the [MATH] relation.'}

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'1601.01160-2-41-1'], ['1601.01160-1-41-2', '1601.01160-2-41-2'], ['1601.01160-1-41-3', '1601.01160-2-41-3'], ['1601.01160-1-41-4', '1601.01160-2-41-4'], ['1601.01160-1-41-5', '1601.01160-2-41-5'], ['1601.01160-1-42-0', '1601.01160-2-42-0'], ['1601.01160-1-42-1', '1601.01160-2-42-1'], ['1601.01160-1-42-2', '1601.01160-2-42-2'], ['1601.01160-1-42-3', '1601.01160-2-42-3'], ['1601.01160-1-42-4', '1601.01160-2-42-4'], ['1601.01160-1-42-5', '1601.01160-2-42-5'], ['1601.01160-1-42-6', '1601.01160-2-42-6'], ['1601.01160-1-8-0', '1601.01160-2-8-0'], ['1601.01160-1-8-1', '1601.01160-2-8-1'], ['1601.01160-1-25-0', '1601.01160-2-25-0'], ['1601.01160-1-38-0', '1601.01160-2-38-0'], ['1601.01160-1-38-1', '1601.01160-2-38-1'], ['1601.01160-1-38-2', '1601.01160-2-38-2'], ['1601.01160-1-38-3', '1601.01160-2-38-3'], ['1601.01160-1-38-4', '1601.01160-2-38-4'], ['1601.01160-1-15-0', '1601.01160-2-15-0'], ['1601.01160-1-15-1', '1601.01160-2-15-1'], ['1601.01160-1-15-2', '1601.01160-2-15-2'], ['1601.01160-1-15-3', '1601.01160-2-15-3'], ['1601.01160-1-15-4', '1601.01160-2-15-4'], ['1601.01160-1-15-5', '1601.01160-2-15-5'], ['1601.01160-1-15-6', '1601.01160-2-15-6'], ['1601.01160-1-15-7', '1601.01160-2-15-7'], ['1601.01160-1-15-8', '1601.01160-2-15-8'], ['1601.01160-1-22-0', '1601.01160-2-22-0'], ['1601.01160-1-22-1', '1601.01160-2-22-1'], ['1601.01160-1-3-0', '1601.01160-2-3-0'], ['1601.01160-1-3-1', '1601.01160-2-3-1'], ['1601.01160-1-3-2', '1601.01160-2-3-2'], ['1601.01160-1-3-3', '1601.01160-2-3-3'], ['1601.01160-1-13-0', '1601.01160-2-13-0'], ['1601.01160-1-13-1', '1601.01160-2-13-1'], ['1601.01160-1-13-2', '1601.01160-2-13-2'], ['1601.01160-1-16-0', '1601.01160-2-16-0'], ['1601.01160-1-16-1', '1601.01160-2-16-1'], ['1601.01160-1-16-2', '1601.01160-2-16-2'], ['1601.01160-1-16-3', '1601.01160-2-16-3'], ['1601.01160-1-16-4', '1601.01160-2-16-4'], ['1601.01160-1-16-5', '1601.01160-2-16-5'], ['1601.01160-1-16-6', '1601.01160-2-16-6'], ['1601.01160-1-16-7', '1601.01160-2-16-7'], ['1601.01160-1-16-8', '1601.01160-2-16-8'], ['1601.01160-1-16-9', '1601.01160-2-16-9'], ['1601.01160-1-27-0', '1601.01160-2-27-0'], ['1601.01160-1-27-1', '1601.01160-2-27-1'], ['1601.01160-1-27-2', '1601.01160-2-27-2'], ['1601.01160-1-27-3', '1601.01160-2-27-3'], ['1601.01160-1-27-4', '1601.01160-2-27-4'], ['1601.01160-1-27-5', '1601.01160-2-27-5'], ['1601.01160-1-27-6', '1601.01160-2-27-6'], ['1601.01160-1-27-7', '1601.01160-2-27-7'], ['1601.01160-1-27-8', '1601.01160-2-27-8'], ['1601.01160-1-27-9', '1601.01160-2-27-9'], ['1601.01160-1-9-0', '1601.01160-2-9-0'], ['1601.01160-1-9-1', '1601.01160-2-9-1'], ['1601.01160-1-9-2', '1601.01160-2-9-2'], ['1601.01160-1-9-3', '1601.01160-2-9-3'], ['1601.01160-1-9-4', '1601.01160-2-9-4'], ['1601.01160-1-31-0', '1601.01160-2-31-0'], ['1601.01160-1-31-1', '1601.01160-2-31-1'], ['1601.01160-1-5-0', '1601.01160-2-5-0'], ['1601.01160-1-5-1', '1601.01160-2-5-1'], ['1601.01160-1-5-2', '1601.01160-2-5-2'], ['1601.01160-1-5-3', '1601.01160-2-5-3'], ['1601.01160-1-5-4', '1601.01160-2-5-4'], ['1601.01160-1-40-0', '1601.01160-2-40-0'], ['1601.01160-1-40-1', '1601.01160-2-40-1'], ['1601.01160-1-40-2', '1601.01160-2-40-2'], ['1601.01160-1-11-0', '1601.01160-2-11-0'], ['1601.01160-1-11-1', '1601.01160-2-11-1'], ['1601.01160-1-11-2', '1601.01160-2-11-2'], ['1601.01160-1-11-3', '1601.01160-2-11-3'], ['1601.01160-1-11-4', '1601.01160-2-11-4'], ['1601.01160-1-11-5', '1601.01160-2-11-5'], ['1601.01160-1-20-0', '1601.01160-2-20-0'], ['1601.01160-1-20-1', '1601.01160-2-20-1'], ['1601.01160-1-20-2', '1601.01160-2-20-2'], ['1601.01160-1-36-0', '1601.01160-2-36-0'], ['1601.01160-1-36-1', '1601.01160-2-36-1'], ['1601.01160-1-36-2', '1601.01160-2-36-2'], ['1601.01160-1-36-3', '1601.01160-2-36-3'], ['1601.01160-1-2-0', '1601.01160-2-2-0'], ['1601.01160-1-2-1', '1601.01160-2-2-1'], ['1601.01160-1-2-2', '1601.01160-2-2-2'], ['1601.01160-1-2-3', '1601.01160-2-2-3'], ['1601.01160-1-2-4', '1601.01160-2-2-4'], ['1601.01160-1-2-5', '1601.01160-2-2-5'], ['1601.01160-1-2-6', '1601.01160-2-2-6'], ['1601.01160-1-2-7', '1601.01160-2-2-7'], ['1601.01160-1-2-8', '1601.01160-2-2-8'], ['1601.01160-1-2-9', '1601.01160-2-2-9'], ['1601.01160-1-17-0', '1601.01160-2-17-0'], ['1601.01160-1-17-1', '1601.01160-2-17-1'], ['1601.01160-1-17-2', '1601.01160-2-17-2'], ['1601.01160-1-17-3', '1601.01160-2-17-3'], ['1601.01160-1-23-0', '1601.01160-2-23-0'], ['1601.01160-1-23-1', '1601.01160-2-23-1'], ['1601.01160-1-23-2', '1601.01160-2-23-2'], ['1601.01160-1-23-3', '1601.01160-2-23-3'], ['1601.01160-1-29-0', '1601.01160-2-29-0'], ['1601.01160-1-29-1', '1601.01160-2-29-1'], ['1601.01160-1-29-2', '1601.01160-2-29-2'], ['1601.01160-1-29-3', '1601.01160-2-29-3'], ['1601.01160-1-29-4', '1601.01160-2-29-4'], ['1601.01160-1-29-5', '1601.01160-2-29-5'], ['1601.01160-1-29-6', '1601.01160-2-29-6'], ['1601.01160-1-19-0', '1601.01160-2-19-0'], ['1601.01160-1-19-1', '1601.01160-2-19-1'], ['1601.01160-1-19-2', '1601.01160-2-19-2'], ['1601.01160-1-19-3', '1601.01160-2-19-3'], ['1601.01160-1-19-4', '1601.01160-2-19-4'], ['1601.01160-1-19-5', '1601.01160-2-19-5'], ['1601.01160-1-19-6', '1601.01160-2-19-6'], ['1601.01160-1-19-7', '1601.01160-2-19-7'], ['1601.01160-1-19-8', '1601.01160-2-19-8'], ['1601.01160-1-19-9', '1601.01160-2-19-9'], ['1601.01160-1-19-10', '1601.01160-2-19-10'], ['1601.01160-1-19-11', '1601.01160-2-19-11'], ['1601.01160-1-4-0', '1601.01160-2-4-0'], ['1601.01160-1-4-1', '1601.01160-2-4-1'], ['1601.01160-1-4-2', '1601.01160-2-4-2'], ['1601.01160-1-4-3', '1601.01160-2-4-3'], ['1601.01160-1-4-4', '1601.01160-2-4-4'], ['1601.01160-1-4-5', '1601.01160-2-4-5'], ['1601.01160-1-4-6', '1601.01160-2-4-6'], ['1601.01160-1-4-7', '1601.01160-2-4-7'], ['1601.01160-1-18-0', '1601.01160-2-18-0'], ['1601.01160-1-18-1', '1601.01160-2-18-1'], ['1601.01160-1-18-2', '1601.01160-2-18-2'], ['1601.01160-1-18-3', '1601.01160-2-18-3'], ['1601.01160-1-18-4', '1601.01160-2-18-4'], ['1601.01160-1-18-5', '1601.01160-2-18-5'], ['1601.01160-1-18-6', '1601.01160-2-18-6'], ['1601.01160-1-18-7', '1601.01160-2-18-7'], ['1601.01160-1-18-8', '1601.01160-2-18-8'], ['1601.01160-1-18-9', '1601.01160-2-18-9'], ['1601.01160-1-18-10', '1601.01160-2-18-10'], ['1601.01160-1-18-11', '1601.01160-2-18-11'], ['1601.01160-1-30-0', '1601.01160-2-30-0'], ['1601.01160-1-30-1', '1601.01160-2-30-1'], ['1601.01160-1-30-2', '1601.01160-2-30-2'], ['1601.01160-1-35-0', '1601.01160-2-35-0'], ['1601.01160-1-35-1', '1601.01160-2-35-1'], ['1601.01160-1-35-2', '1601.01160-2-35-2'], ['1601.01160-1-35-3', '1601.01160-2-35-3'], ['1601.01160-1-34-0', '1601.01160-2-34-0'], ['1601.01160-1-34-1', '1601.01160-2-34-1'], ['1601.01160-1-34-2', '1601.01160-2-34-2'], ['1601.01160-1-34-3', '1601.01160-2-34-3'], ['1601.01160-1-34-4', '1601.01160-2-34-4'], ['1601.01160-1-34-5', '1601.01160-2-34-5'], ['1601.01160-1-34-6', '1601.01160-2-34-6'], ['1601.01160-1-34-7', '1601.01160-2-34-7'], ['1601.01160-1-21-0', '1601.01160-2-21-0'], ['1601.01160-1-21-1', '1601.01160-2-21-1'], ['1601.01160-1-21-2', '1601.01160-2-21-2'], ['1601.01160-1-21-3', '1601.01160-2-21-3'], ['1601.01160-1-21-4', '1601.01160-2-21-4'], ['1601.01160-1-28-0', '1601.01160-2-28-0'], ['1601.01160-1-28-1', '1601.01160-2-28-1'], ['1601.01160-1-28-2', '1601.01160-2-28-2'], ['1601.01160-1-28-3', '1601.01160-2-28-3'], ['1601.01160-1-28-4', '1601.01160-2-28-4'], ['1601.01160-1-28-5', '1601.01160-2-28-5']]

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'1601.01160-2-41-1'], ['1601.01160-1-41-2', '1601.01160-2-41-2'], ['1601.01160-1-41-3', '1601.01160-2-41-3'], ['1601.01160-1-41-4', '1601.01160-2-41-4'], ['1601.01160-1-41-5', '1601.01160-2-41-5'], ['1601.01160-1-42-0', '1601.01160-2-42-0'], ['1601.01160-1-42-1', '1601.01160-2-42-1'], ['1601.01160-1-42-2', '1601.01160-2-42-2'], ['1601.01160-1-42-3', '1601.01160-2-42-3'], ['1601.01160-1-42-4', '1601.01160-2-42-4'], ['1601.01160-1-42-5', '1601.01160-2-42-5'], ['1601.01160-1-42-6', '1601.01160-2-42-6'], ['1601.01160-1-8-0', '1601.01160-2-8-0'], ['1601.01160-1-8-1', '1601.01160-2-8-1'], ['1601.01160-1-25-0', '1601.01160-2-25-0'], ['1601.01160-1-38-0', '1601.01160-2-38-0'], ['1601.01160-1-38-1', '1601.01160-2-38-1'], ['1601.01160-1-38-2', '1601.01160-2-38-2'], ['1601.01160-1-38-3', '1601.01160-2-38-3'], ['1601.01160-1-38-4', '1601.01160-2-38-4'], ['1601.01160-1-15-0', '1601.01160-2-15-0'], ['1601.01160-1-15-1', '1601.01160-2-15-1'], ['1601.01160-1-15-2', '1601.01160-2-15-2'], ['1601.01160-1-15-3', '1601.01160-2-15-3'], ['1601.01160-1-15-4', '1601.01160-2-15-4'], ['1601.01160-1-15-5', '1601.01160-2-15-5'], ['1601.01160-1-15-6', '1601.01160-2-15-6'], ['1601.01160-1-15-7', '1601.01160-2-15-7'], ['1601.01160-1-15-8', '1601.01160-2-15-8'], ['1601.01160-1-22-0', '1601.01160-2-22-0'], ['1601.01160-1-22-1', '1601.01160-2-22-1'], ['1601.01160-1-3-0', '1601.01160-2-3-0'], ['1601.01160-1-3-1', '1601.01160-2-3-1'], ['1601.01160-1-3-2', '1601.01160-2-3-2'], ['1601.01160-1-3-3', '1601.01160-2-3-3'], ['1601.01160-1-13-0', '1601.01160-2-13-0'], ['1601.01160-1-13-1', '1601.01160-2-13-1'], ['1601.01160-1-13-2', '1601.01160-2-13-2'], ['1601.01160-1-16-0', '1601.01160-2-16-0'], ['1601.01160-1-16-1', '1601.01160-2-16-1'], ['1601.01160-1-16-2', '1601.01160-2-16-2'], ['1601.01160-1-16-3', '1601.01160-2-16-3'], ['1601.01160-1-16-4', '1601.01160-2-16-4'], ['1601.01160-1-16-5', '1601.01160-2-16-5'], ['1601.01160-1-16-6', '1601.01160-2-16-6'], ['1601.01160-1-16-7', '1601.01160-2-16-7'], ['1601.01160-1-16-8', '1601.01160-2-16-8'], ['1601.01160-1-16-9', '1601.01160-2-16-9'], ['1601.01160-1-27-0', '1601.01160-2-27-0'], ['1601.01160-1-27-1', '1601.01160-2-27-1'], ['1601.01160-1-27-2', '1601.01160-2-27-2'], ['1601.01160-1-27-3', '1601.01160-2-27-3'], ['1601.01160-1-27-4', '1601.01160-2-27-4'], ['1601.01160-1-27-5', '1601.01160-2-27-5'], ['1601.01160-1-27-6', '1601.01160-2-27-6'], ['1601.01160-1-27-7', '1601.01160-2-27-7'], ['1601.01160-1-27-8', '1601.01160-2-27-8'], ['1601.01160-1-27-9', '1601.01160-2-27-9'], ['1601.01160-1-9-0', '1601.01160-2-9-0'], ['1601.01160-1-9-1', '1601.01160-2-9-1'], ['1601.01160-1-9-2', '1601.01160-2-9-2'], ['1601.01160-1-9-3', '1601.01160-2-9-3'], ['1601.01160-1-9-4', '1601.01160-2-9-4'], ['1601.01160-1-31-0', '1601.01160-2-31-0'], ['1601.01160-1-31-1', '1601.01160-2-31-1'], ['1601.01160-1-5-0', '1601.01160-2-5-0'], ['1601.01160-1-5-1', '1601.01160-2-5-1'], ['1601.01160-1-5-2', '1601.01160-2-5-2'], ['1601.01160-1-5-3', '1601.01160-2-5-3'], ['1601.01160-1-5-4', '1601.01160-2-5-4'], ['1601.01160-1-40-0', '1601.01160-2-40-0'], ['1601.01160-1-40-1', '1601.01160-2-40-1'], ['1601.01160-1-40-2', '1601.01160-2-40-2'], ['1601.01160-1-11-0', '1601.01160-2-11-0'], ['1601.01160-1-11-1', '1601.01160-2-11-1'], ['1601.01160-1-11-2', '1601.01160-2-11-2'], ['1601.01160-1-11-3', '1601.01160-2-11-3'], ['1601.01160-1-11-4', '1601.01160-2-11-4'], ['1601.01160-1-11-5', '1601.01160-2-11-5'], ['1601.01160-1-20-0', '1601.01160-2-20-0'], ['1601.01160-1-20-1', '1601.01160-2-20-1'], ['1601.01160-1-20-2', '1601.01160-2-20-2'], ['1601.01160-1-36-0', '1601.01160-2-36-0'], ['1601.01160-1-36-1', '1601.01160-2-36-1'], ['1601.01160-1-36-2', '1601.01160-2-36-2'], ['1601.01160-1-36-3', '1601.01160-2-36-3'], ['1601.01160-1-2-0', '1601.01160-2-2-0'], ['1601.01160-1-2-1', '1601.01160-2-2-1'], ['1601.01160-1-2-2', '1601.01160-2-2-2'], ['1601.01160-1-2-3', '1601.01160-2-2-3'], ['1601.01160-1-2-4', '1601.01160-2-2-4'], ['1601.01160-1-2-5', '1601.01160-2-2-5'], ['1601.01160-1-2-6', '1601.01160-2-2-6'], ['1601.01160-1-2-7', '1601.01160-2-2-7'], ['1601.01160-1-2-8', '1601.01160-2-2-8'], ['1601.01160-1-2-9', '1601.01160-2-2-9'], ['1601.01160-1-17-0', '1601.01160-2-17-0'], ['1601.01160-1-17-1', '1601.01160-2-17-1'], ['1601.01160-1-17-2', '1601.01160-2-17-2'], ['1601.01160-1-17-3', '1601.01160-2-17-3'], ['1601.01160-1-23-0', '1601.01160-2-23-0'], ['1601.01160-1-23-1', '1601.01160-2-23-1'], ['1601.01160-1-23-2', '1601.01160-2-23-2'], ['1601.01160-1-23-3', '1601.01160-2-23-3'], ['1601.01160-1-29-0', '1601.01160-2-29-0'], ['1601.01160-1-29-1', '1601.01160-2-29-1'], ['1601.01160-1-29-2', '1601.01160-2-29-2'], ['1601.01160-1-29-3', '1601.01160-2-29-3'], ['1601.01160-1-29-4', '1601.01160-2-29-4'], ['1601.01160-1-29-5', '1601.01160-2-29-5'], ['1601.01160-1-29-6', '1601.01160-2-29-6'], ['1601.01160-1-19-0', '1601.01160-2-19-0'], ['1601.01160-1-19-1', '1601.01160-2-19-1'], ['1601.01160-1-19-2', '1601.01160-2-19-2'], ['1601.01160-1-19-3', '1601.01160-2-19-3'], ['1601.01160-1-19-4', '1601.01160-2-19-4'], ['1601.01160-1-19-5', '1601.01160-2-19-5'], ['1601.01160-1-19-6', '1601.01160-2-19-6'], ['1601.01160-1-19-7', '1601.01160-2-19-7'], ['1601.01160-1-19-8', '1601.01160-2-19-8'], ['1601.01160-1-19-9', '1601.01160-2-19-9'], ['1601.01160-1-19-10', '1601.01160-2-19-10'], ['1601.01160-1-19-11', '1601.01160-2-19-11'], ['1601.01160-1-4-0', '1601.01160-2-4-0'], ['1601.01160-1-4-1', '1601.01160-2-4-1'], ['1601.01160-1-4-2', '1601.01160-2-4-2'], ['1601.01160-1-4-3', '1601.01160-2-4-3'], ['1601.01160-1-4-4', '1601.01160-2-4-4'], ['1601.01160-1-4-5', '1601.01160-2-4-5'], ['1601.01160-1-4-6', '1601.01160-2-4-6'], ['1601.01160-1-4-7', '1601.01160-2-4-7'], ['1601.01160-1-18-0', '1601.01160-2-18-0'], ['1601.01160-1-18-1', '1601.01160-2-18-1'], ['1601.01160-1-18-2', '1601.01160-2-18-2'], ['1601.01160-1-18-3', '1601.01160-2-18-3'], ['1601.01160-1-18-4', '1601.01160-2-18-4'], ['1601.01160-1-18-5', '1601.01160-2-18-5'], ['1601.01160-1-18-6', '1601.01160-2-18-6'], ['1601.01160-1-18-7', '1601.01160-2-18-7'], ['1601.01160-1-18-8', '1601.01160-2-18-8'], ['1601.01160-1-18-9', '1601.01160-2-18-9'], ['1601.01160-1-18-10', '1601.01160-2-18-10'], ['1601.01160-1-18-11', '1601.01160-2-18-11'], ['1601.01160-1-30-0', '1601.01160-2-30-0'], ['1601.01160-1-30-1', '1601.01160-2-30-1'], ['1601.01160-1-30-2', '1601.01160-2-30-2'], ['1601.01160-1-35-0', '1601.01160-2-35-0'], ['1601.01160-1-35-1', '1601.01160-2-35-1'], ['1601.01160-1-35-2', '1601.01160-2-35-2'], ['1601.01160-1-35-3', '1601.01160-2-35-3'], ['1601.01160-1-34-0', '1601.01160-2-34-0'], ['1601.01160-1-34-1', '1601.01160-2-34-1'], ['1601.01160-1-34-2', '1601.01160-2-34-2'], ['1601.01160-1-34-3', '1601.01160-2-34-3'], ['1601.01160-1-34-4', '1601.01160-2-34-4'], ['1601.01160-1-34-5', '1601.01160-2-34-5'], ['1601.01160-1-34-6', '1601.01160-2-34-6'], ['1601.01160-1-34-7', '1601.01160-2-34-7'], ['1601.01160-1-21-0', '1601.01160-2-21-0'], ['1601.01160-1-21-1', '1601.01160-2-21-1'], ['1601.01160-1-21-2', '1601.01160-2-21-2'], ['1601.01160-1-21-3', '1601.01160-2-21-3'], ['1601.01160-1-21-4', '1601.01160-2-21-4'], ['1601.01160-1-28-0', '1601.01160-2-28-0'], ['1601.01160-1-28-1', '1601.01160-2-28-1'], ['1601.01160-1-28-2', '1601.01160-2-28-2'], ['1601.01160-1-28-3', '1601.01160-2-28-3'], ['1601.01160-1-28-4', '1601.01160-2-28-4'], ['1601.01160-1-28-5', '1601.01160-2-28-5']]

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{'1': 'http://creativecommons.org/publicdomain/zero/1.0/', '2': 'http://creativecommons.org/licenses/by-sa/4.0/'}

https://arxiv.org/abs/1601.01160

1604.06055

{'1604.06055-1-0-0': 'Topological states of matter are peculiar quantum phases showing different edge and bulk transport properties connected by the bulk-boundary correspondence.', '1604.06055-1-0-1': 'While non-interacting fermionic topological insulators are well established by now and have been classified according to a ten-fold scheme, the possible realisation of topological states for bosons has not been much explored yet.', '1604.06055-1-0-2': 'Furthermore, the role of interactions is far from being understood.', '1604.06055-1-0-3': 'Here, we show that a topological state of matter exclusively driven by interactions may occur in the p-band of a Lieb optical lattice filled with ultracold bosons.', '1604.06055-1-0-4': 'The single-particle spectrum of the system displays a remarkable parabolic band-touching point, with both bands exhibiting non-negative curvature.', '1604.06055-1-0-5': 'Although the system is neither topological at the single-particle level, nor for the interacting ground state, on-site interactions induce an anomalous Hall effect for the excitations, carrying a non-zero Chern number.', '1604.06055-1-0-6': 'Our work introduces an experimentally realistic strategy for the formation of interaction-driven topological states of bosons.', '1604.06055-1-1-0': '# Introduction', '1604.06055-1-2-0': 'The seminal work of Kane and Mele for graphene in the presence of a strong spin-orbit coupling [CITATION] has opened the field of topological insulators and brought us a deeper comprehension of phenomena like the quantum Hall effect, which had been known for decades [CITATION].', '1604.06055-1-2-1': 'A universal classification scheme based on symmetries and dimensionality has enabled a solid basis for our understanding of non-interacting fermionic topological systems [CITATION].', '1604.06055-1-2-2': 'In contrast, interaction-driven topological states of matter have remained elusive.', '1604.06055-1-2-3': 'Topological phases were originally predicted to occur in lattice models with dominating next-nearest-neighbor interaction [CITATION], a scenario which appears difficult to envisage in real experiments.', '1604.06055-1-2-4': 'More precise exact diagonalization [CITATION] and DMRG [CITATION] calculations, however, contradicted these mean-field results.', '1604.06055-1-3-0': 'Predictions of topological phases in Dirac-like materials exhibiting a linear dispersion suffer from the additional drawback that theses systems are genuinely quantum critical: due to the zero density of states at the neutrality point, only perturbations exceeding a certain critical value are able to induce a topological phase.', '1604.06055-1-3-1': 'On the other hand, systems with a parabolic band touching point may become topological for infinitesimal values of the interaction [CITATION].', '1604.06055-1-3-2': 'A paradigmatic example is Bernal-stacked bilayer graphene [CITATION], but other cases have also been identified, and this field has attracted much interest recently [CITATION].', '1604.06055-1-4-0': 'The search for interaction-driven topological systems has mostly concentrated on electronic condensed matter [CITATION] or on the equivalent cold-atom fermion system [CITATION].', '1604.06055-1-4-1': 'Only recently, the corresponding bosonic analogs became the subject of theoretical investigations [CITATION].', '1604.06055-1-4-2': 'As for experiments, considerable progress has been achieved in realizing topological states with cold atoms (fermions or bosons) in the non-interacting regime.', '1604.06055-1-4-3': 'Examples range from the Su-Schrieffer-Heeger model [CITATION], to the Harper-Hofstadter model [CITATION] or the long-sought Haldane model [CITATION].', '1604.06055-1-4-4': 'In these systems, topological features of the bands, like a non-trivial Zak phase, Berry curvature, or Chern number, characterize the non-interacting model and interactions play only a marginal role [CITATION].', '1604.06055-1-4-5': 'In this context the intriguing question arises, how to engineer a parabolic band touching with cold atoms, because in this case topological features might emerge exclusively due to interactions.', '1604.06055-1-4-6': 'The main difficulty is that both parabolic bands must have a non-negative curvature in order to provide a band minimum, where the bosons can be condensed.', '1604.06055-1-5-0': 'Here, we show that this scenario can be achieved by employing interacting ultracold bosons in a two-dimensional decorated square lattice potential, known as a model of the copper oxide planes of cuprate superconductors, and sometimes dubbed Lieb lattice in the literature.', '1604.06055-1-5-1': 'The desired band touching arises between the second and third bands at the [MATH]-point of the Brillouin zone.', '1604.06055-1-5-2': 'We show that with proven experimental techniques a stable Bose-Einstein condensate (BEC) can be formed at this high-symmetry point, which exhibits macroscopic angular momentum and superfluid plaquette currents, such that time-reversal symmetry is broken, but translational symmetry is preserved.', '1604.06055-1-5-3': 'This phase, which turns out to be topologically trivial, is the bosonic analog of a phase first proposed by Varma to describe the pseudogap regime of high-[MATH] cuprates [CITATION].', '1604.06055-1-5-4': 'Surprisingly, when we calculate the band structure of the Bogolyubov excitations, we find that they exhibit topological properties that were absent in the non-interacting system and also in the interacting ground state.', '1604.06055-1-6-0': '# Optical lattice', '1604.06055-1-7-0': 'In contrast to previous realizations of a Lieb lattice with light-shift potentials [CITATION], our implementation includes the necessary machinery to prepare atoms in the second Bloch band, and hence to include orbital degrees of freedom.', '1604.06055-1-7-1': 'The Lieb-lattice geometry, depicted in Fig. [REF](a), arises if a unit cell (light blue rectangle in the left detail) with two classes of sites denoted [MATH] and [MATH] is translated via a square Bravais lattice.', '1604.06055-1-7-2': 'In order to excite atoms into the second band of this lattice using the technique demonstrated in Ref. [CITATION], additional functionality (henceforth referred to as band swapping) is required, which allows one to rapidly switch the well depths of the corner sites [MATH] and the bond sites [MATH].', '1604.06055-1-7-3': 'Experimentally, this may be achieved by superimposing the two potentials [EQUATION] thus obtaining [MATH].', '1604.06055-1-7-4': 'Here, [MATH] with [MATH] denoting the wavelengths of the light fields forming [MATH] and [MATH].', '1604.06055-1-7-5': 'As illustrated in the central and right details of Fig. [REF](a), both potentials [MATH] and [MATH] represent simple square lattices, however, rotated with respect to each other by [MATH], and with lattice constants differing by a factor [MATH].', '1604.06055-1-7-6': 'Indicating their potential minima by red and blue disks and their potential maxima by gray disks or grey lines, respectively, Fig. [REF](a) immediately clarifies how their superposition yields the desired Lieb-lattice geometry.', '1604.06055-1-7-7': 'By tuning the ratio [MATH] around unity, the [MATH] wells can be tuned to be deeper than the [MATH] wells or vice versa.', '1604.06055-1-7-8': 'This is shown in Fig. [REF](b), where the potential is plotted for [MATH] and [MATH] on the left- and right-hand sides, respectively.', '1604.06055-1-8-0': 'Experimentally, the required lattice is produced by the simple setup shown in Fig. [REF](c).', '1604.06055-1-8-1': 'Two laser beams with wavelengths [MATH] and [MATH], both detuned to the negative side of an atomic transition with [MATH] corresponding to a few ten MHz, are retro-reflected from the same mirror.', '1604.06055-1-8-2': 'The polarization of the beam at wavelength [MATH] is adjusted to be linear within the lattice plane.', '1604.06055-1-8-3': 'Hence, no interference occurs at the crossing point, where an atomic BEC is prepared, such that the potential [MATH] arises.', '1604.06055-1-8-4': 'Similarly, the beam at wavelength [MATH] exhibits linear polarization perpendicular to the lattice plane, such that maximal interference at the crossing point yields the potential [MATH].', '1604.06055-1-8-5': 'The two potentials can be precisely positioned relative to each other (to better than a few nm) by adjusting [MATH] and the ratio between the distances [MATH] and [MATH] indicated in the Fig. [REF](c) [CITATION].', '1604.06055-1-9-0': '# Theoretical model', '1604.06055-1-10-0': 'We consider the case when the corner sites ([MATH]) are significantly deeper than the bond sites ([MATH]) such that the former host [MATH] and [MATH] orbitals, while the latter host [MATH] orbitals.', '1604.06055-1-10-1': 'The low-lying [MATH] orbitals of the corner sites are neglected.', '1604.06055-1-10-2': 'The corresponding minimal tight-binding Hamiltonian reads [EQUATION] where [MATH] is the nearest-neighbor hopping between [MATH] and [MATH] orbitals, [MATH] is the next-nearest-neighbor hopping between [MATH] orbitals, and [MATH] and [MATH] are, respectively, the on-site energies for [MATH] and [MATH] orbitals (see Fig. [REF](a)).', '1604.06055-1-10-3': 'We will consider [MATH] in the remainder of this work.', '1604.06055-1-10-4': 'Let us define [MATH] and introduce the vector [MATH].', '1604.06055-1-10-5': 'In momentum space, the Hamiltonian can be written as [MATH], where [EQUATION]', '1604.06055-1-10-6': 'Here, we have defined [MATH], [MATH], and introduced units for which the lattice constant [MATH].', '1604.06055-1-10-7': 'At the high-symmetry points, the lowest-energy eigenvectors have the form (up to a normalisation constant): [MATH], [MATH], [MATH], [MATH], corresponding to the energy [MATH].', '1604.06055-1-10-8': 'The [MATH] point is two-fold degenerate.', '1604.06055-1-11-0': 'A comparison between the band structure derived from the Hamiltonian in Eq. ([REF]) and the one obtained from the experimental potential given in Eq. ([REF]) is shown in Fig. [REF](b).', '1604.06055-1-11-1': 'The desired quadratic-band touching at the high-symmetry point [MATH] of the Brillouin zone, arising from two bands with non-negative curvature, may be promptly observed upon inspection of the figure.', '1604.06055-1-12-0': 'In the presence of contact interactions [MATH] and [MATH] for the [MATH]- and [MATH]-orbitals, respectively, and within the harmonic approximation, the interaction Hamiltonian can be cast in the form [CITATION] [EQUATION] where [MATH], [MATH] and the angular-momentum operator is given by [MATH].', '1604.06055-1-12-1': 'Inspection of the Hamiltonian shows that repulsive interactions favour a ground state with finite angular momentum [MATH], thus breaking time-reversal symmetry [MATH].', '1604.06055-1-13-0': '# Ground-state', '1604.06055-1-14-0': 'Lattice systems exhibiting a degenerate set of minima favour condensation at the high-symmetry points, as for instance found for the Kagome lattice [CITATION].', '1604.06055-1-14-1': 'Therefore, we distinguish two types of condensates, depending on whether the condensation occurs at the points [MATH] or at the two-fold degenerate point [MATH].', '1604.06055-1-14-2': 'This scenario is confirmed by solving the Gross-Pitaevskii equation in imaginary time, which in addition allows us to exclude simultaneous condensation at all the three points.', '1604.06055-1-14-3': 'At the momenta corresponding to the high-symmetry points [MATH], [MATH], and [MATH], only some orbitals have a non-vanishing amplitude for the Hamiltonian in Eq. ([REF]).', '1604.06055-1-14-4': 'These considerations lead us to propose the following ansatz for the condensate wave-function [EQUATION]', '1604.06055-1-14-5': 'For the [MATH] condensate, one has [MATH] and [MATH].', '1604.06055-1-14-6': 'For the [MATH] condensate, [MATH].', '1604.06055-1-14-7': 'We make a gauge choice: [MATH].', '1604.06055-1-14-8': 'Minimisation of the mean-field free energy determines the phases [MATH], [MATH], [MATH], [MATH] and the mean-field free energy reads [EQUATION].', '1604.06055-1-14-9': 'Since the condensates break time-reversal symmetry [MATH], each solution is two-fold degenerate.', '1604.06055-1-14-10': 'The time-reversal conjugate solution can be obtained by substituting [MATH] and [MATH].', '1604.06055-1-15-0': 'The angular momentum expectation value is [MATH] in the [MATH] phase and [MATH] in the [MATH] phase, where [MATH] and [MATH] are integers corresponding to [MATH].', '1604.06055-1-15-1': 'Hence, the overall angular momentum vanishes for the [MATH] phase but not for the [MATH] phase.', '1604.06055-1-15-2': 'The [MATH] condensate breaks the translational symmetry of the Bravais lattice, whereas the [MATH] condensate does not (see Fig. [REF]).', '1604.06055-1-15-3': 'Phase gradients induce superfluid currents.', '1604.06055-1-15-4': 'The bond-current operator is defined as [EQUATION] where [MATH] is the creation operator for a boson at site [MATH] and flavor/orbital [MATH] and [MATH] is the hopping amplitude between sites [MATH] and [MATH].', '1604.06055-1-15-5': 'There are currents only along the [MATH] bonds with amplitude [MATH].', '1604.06055-1-15-6': 'We calculated the many-body Chern number associated with each of the ground states [MATH] and [MATH] and we found zero.', '1604.06055-1-15-7': 'However, as we shall discuss below, topological properties arise for the excitations of the [MATH] condensate.', '1604.06055-1-16-0': 'Corrections to the ground-state mean-field energy given by the zero-point fluctuations of the Bogolyubov modes have been also investigated following the procedure outlined in Ref. [CITATION] and we found that the [MATH] condensate has a larger energy compared to the [MATH] condensate, of the order of 1 pK (see Supplemental Material for further details).', '1604.06055-1-16-1': 'This small value makes us confident that a metastable [MATH] condensate could be experimentally realizable.', '1604.06055-1-16-2': 'Nevertheless, the precise estimation of the condensate lifetime is a highly non-trivial problem, which we defer to future investigations.', '1604.06055-1-17-0': '# Excitations and topology', '1604.06055-1-18-0': 'The quadratic band-touching appearing at the [MATH]-point in the non-interacting spectrum is protected by inversion and time-reversal symmetry, and it exhibits a [MATH] Berry flux [CITATION] (see Supplemental Material).', '1604.06055-1-18-1': "Upon condensation, the band degeneracy is lifted because the U(1) symmetry breaking requires the existence of only one gapless mode, according to Goldstone's theorem.", '1604.06055-1-18-2': 'Therefore, the band splitting appearing for the excitations and the further breaking of time-reversal symmetry can allow for a non-vanishing Berry curvature and Chern number.', '1604.06055-1-19-0': 'Within the Bogolyubov approximation, the Hamiltonian can be written as [MATH], with [MATH], where we have introduced the Nambu spinor [MATH].', '1604.06055-1-20-0': 'The Bogolyubov Hamiltonian for the [MATH] condensate reads [EQUATION] where [MATH] was defined in Eq. ([REF]), [EQUATION] and we have introduced the chemical potential [MATH] and defined [MATH] within the harmonic approximation (see Ref. [CITATION]).', '1604.06055-1-20-1': 'We calculate the chemical potential assuming that [MATH], where [MATH] is the fixed total density of condensed particles per unit cell and [MATH].', '1604.06055-1-20-2': 'The non-linearity of the mean-field energy functional [MATH] requires the chemical potential to be found numerically.', '1604.06055-1-21-0': 'The excitation spectrum is obtained by solving the eigenvalue problem [MATH], where [MATH].', '1604.06055-1-21-1': 'The normalization of the eigenvectors is such that [MATH] and [MATH], with [MATH] being the matrix having the eigenvectors [MATH] in its columns.', '1604.06055-1-22-0': 'The excitation spectrum for the [MATH]-condensate is shown in Fig. [REF](a)-(b).', '1604.06055-1-22-1': 'Notice that the lowest branch of the spectrum became non-degenerate, as a consequence of the U(1) symmetry breaking.', '1604.06055-1-22-2': 'This allows us to calculate the associated Chern number [CITATION], [EQUATION] where the Berry curvature [MATH] reads [EQUATION]', '1604.06055-1-22-3': 'The scalar product used in the definition of the Berry potential [MATH] has a signature given by the metric [MATH] [CITATION].', '1604.06055-1-23-0': 'We numerically calculated the Chern number using the procedure described in Ref. [CITATION] and we found [MATH] for the lowest branch of the spectrum.', '1604.06055-1-23-1': 'The Berry curvature for this branch is shown in Fig. [REF](c)-(d).', '1604.06055-1-23-2': 'The non-vanishing [MATH]-space profile of the Berry curvature is analogous to the one of a free-particle with a quadratic band-touching point with an explicit time-reversal symmetry breaking (see Supplemental Material).', '1604.06055-1-23-3': 'The fundamental difference is that the time-reversal symmetry breaking here is emergent and driven by interactions.', '1604.06055-1-24-0': '# Conclusions', '1604.06055-1-25-0': 'Our results provide a novel paradigm to realize topological phases of weakly-interacting bosons.', '1604.06055-1-25-1': 'Essentially, by combining the condensation of atoms at a quadratic band-touching point with the spontaneous time-reversal symmetry breaking induced by the interactions, one can generate a superfluid with topological excitations.', '1604.06055-1-25-2': 'The topological features of the excitations emerge solely because of interactions and are otherwise absent in the non-interacting regime.', '1604.06055-1-26-0': 'According to the topological classification [CITATION], this model belongs to class D, since both time-reversal symmetry and chiral symmetry are broken, while particle-hole symmetry is intrinsically present in the Bogolyubov description.', '1604.06055-1-26-1': 'A realisation of a similar class in bosonic systems has been recently proposed for a Kagome lattice of photonic crystals, where a pairing term breaking U(1) and time-reversal symmetry is introduced by light-squeezing [CITATION].', '1604.06055-1-27-0': 'In our model, instead, interactions induce the symmetry-breaking mechanism that generates the topological features of the excitations.', '1604.06055-1-27-1': 'In particular, the non-vanishing Berry curvature characterizes an anomalous Hall effect for the excitations and affects the collective modes of the gas [CITATION].', '1604.06055-1-27-2': 'It may be directly observed with the interferometric techniques demonstrated in Refs. [CITATION], while the excitation spectra could be measured by momentum resolved Bragg spectroscopy [CITATION].', '1604.06055-1-28-0': 'By diagonalizing the spectrum in a cylindrical geometry, we found the existence of excitations localised at the two edges of the cylinder, as expected from the bulk-boundary correspondence (see red line in Fig. [REF](a)-(b)).', '1604.06055-1-28-1': 'The observation of these finite-size effects could be implemented by using an optical box trap cutting the system along the symmetry axes [CITATION].', '1604.06055-1-28-2': 'The absence of an overall true gap would compromise the detection of the edge states in the presence of disorder, which, however, should not be a problem in the light-shift potential of an optical lattice, since these systems are inherently clean.', '1604.06055-1-29-0': 'Finally, we want to recall that the ground-state of the model investigated here bears strong similarities with the Varma phase, firstly proposed for describing the pseudogap regime of high-[MATH] superconductors [CITATION].', '1604.06055-1-29-1': 'Our findings also open the possibilities to the new exotic states of bosons in the strongly-interacting regime [CITATION], where an insulating Mott phase would be characterised by a gap and may also possess topological features.'}

{'1604.06055-2-0-0': 'Topological states of matter are peculiar quantum phases showing different edge and bulk transport properties connected by the bulk-boundary correspondence.', '1604.06055-2-0-1': 'While non-interacting fermionic topological insulators are well established by now and have been classified according to a ten-fold scheme, the possible realisation of topological states for bosons has not been much explored yet.', '1604.06055-2-0-2': 'Furthermore, the role of interactions is far from being understood.', '1604.06055-2-0-3': 'Here, we show that a topological state of matter exclusively driven by interactions may occur in the p-band of a Lieb optical lattice filled with ultracold bosons.', '1604.06055-2-0-4': 'The single-particle spectrum of the system displays a remarkable parabolic band-touching point, with both bands exhibiting non-negative curvature.', '1604.06055-2-0-5': 'Although the system is neither topological at the single-particle level, nor for the interacting ground state, on-site interactions induce an anomalous Hall effect for the excitations, carrying a non-zero Chern number.', '1604.06055-2-0-6': 'Our work introduces an experimentally realistic strategy for the formation of interaction-driven topological states of bosons.', '1604.06055-2-1-0': 'Introduction.', '1604.06055-2-1-1': 'The seminal work of Kane and Mele for graphene in the presence of a strong spin-orbit coupling [CITATION] has opened the field of topological insulators and brought us a deeper comprehension of phenomena like the quantum Hall effect, which had been known for decades [CITATION].', '1604.06055-2-1-2': 'A universal classification scheme based on symmetries and dimensionality has enabled a solid basis for our understanding of non-interacting fermionic topological systems [CITATION].', '1604.06055-2-1-3': 'In contrast, interaction-driven topological states of matter have remained elusive.', '1604.06055-2-1-4': 'Topological phases were originally predicted to occur in lattice models with dominating next-nearest-neighbor interaction [CITATION], a scenario which appears difficult to envisage in real experiments.', '1604.06055-2-1-5': 'More precise exact diagonalization [CITATION] and DMRG [CITATION] calculations, however, contradicted these mean-field results.', '1604.06055-2-2-0': 'Predictions of topological phases in Dirac-like materials exhibiting a linear dispersion suffer from the additional drawback that theses systems are genuinely quantum critical: due to the zero density of states at the neutrality point, only perturbations exceeding a certain critical value are able to induce a topological phase.', '1604.06055-2-2-1': 'On the other hand, systems with a parabolic band touching point may become topological for infinitesimal values of the interaction [CITATION].', '1604.06055-2-2-2': 'A paradigmatic example is Bernal-stacked bilayer graphene [CITATION], but other cases have also been identified, and this field has attracted much interest recently [CITATION].', '1604.06055-2-3-0': 'The search for interaction-driven topological systems has mostly concentrated on electronic condensed matter [CITATION] or on the equivalent cold-atom fermion system [CITATION].', '1604.06055-2-3-1': 'Only recently, the corresponding bosonic analogs became the subject of theoretical investigations [CITATION].', '1604.06055-2-3-2': 'As for experiments, considerable progress has been achieved in realizing topological states with cold atoms (fermions or bosons) in the non-interacting regime.', '1604.06055-2-3-3': 'Examples range from the Su-Schrieffer-Heeger model [CITATION], to the Harper-Hofstadter model [CITATION] or the long-sought Haldane model [CITATION].', '1604.06055-2-3-4': 'In these systems, topological features of the bands, like a non-trivial Zak phase, Berry curvature, or Chern number, characterize the non-interacting model and interactions play only a marginal role [CITATION].', '1604.06055-2-3-5': 'In this context the intriguing question arises, how to engineer a parabolic band touching with cold atoms, because in this case topological features might emerge exclusively due to interactions.', '1604.06055-2-3-6': 'The main difficulty is that both parabolic bands must have a non-negative curvature in order to provide a band minimum, where the bosons can be condensed.', '1604.06055-2-4-0': 'Here, we show that this scenario can be achieved by employing interacting ultracold bosons in a two-dimensional Lieb lattice, known as a model of the copper oxide planes of cuprate superconductors.', '1604.06055-2-4-1': 'The desired band touching arises between the second and third bands at the [MATH]-point of the Brillouin zone.', '1604.06055-2-4-2': 'We show that with proven experimental techniques a stable Bose-Einstein condensate (BEC) can be formed at this high-symmetry point, which exhibits macroscopic angular momentum and superfluid plaquette currents, such that time-reversal symmetry is broken, but translational symmetry is preserved.', '1604.06055-2-4-3': 'This phase, which turns out to be topologically trivial, is the bosonic analog of a phase first proposed by Varma to describe the pseudogap regime of high-[MATH] cuprates [CITATION].', '1604.06055-2-4-4': 'Surprisingly, when we calculate the band structure of the Bogolyubov excitations, we find that they exhibit emergent topological properties that were absent in the non-interacting system.', '1604.06055-2-5-0': 'Optical lattice.', '1604.06055-2-5-1': 'In contrast to previous realizations of a Lieb lattice with light-shift potentials [CITATION], our implementation includes the necessary machinery to prepare atoms in the second Bloch band, and hence to include orbital degrees of freedom.', '1604.06055-2-5-2': 'The Lieb-lattice geometry, depicted in Fig. [REF](a), arises if a unit cell (light blue rectangle in the left detail) with two classes of sites denoted [MATH] and [MATH] is translated via a square Bravais lattice.', '1604.06055-2-5-3': 'In order to excite atoms into the second band of this lattice using the technique demonstrated in Refs. [CITATION], additional functionality (henceforth referred to as band swapping) is required, which allows one to rapidly switch the well depths of the corner sites [MATH] and the bond sites [MATH].', '1604.06055-2-5-4': 'Experimentally, this may be achieved by superimposing the two potentials [EQUATION] thus obtaining [MATH].', '1604.06055-2-5-5': 'Here, [MATH] with [MATH] denoting the wavelengths of the light fields forming [MATH] and [MATH].', '1604.06055-2-5-6': 'As illustrated in the central and right details of Fig. [REF](a), both potentials [MATH] and [MATH] represent simple square lattices, however, rotated with respect to each other by [MATH], and with lattice constants differing by a factor [MATH].', '1604.06055-2-5-7': 'Indicating their potential minima by red and blue disks and their potential maxima by gray disks or grey lines, respectively, Fig. [REF](a) immediately clarifies how their superposition yields the desired Lieb-lattice geometry.', '1604.06055-2-5-8': 'By tuning the ratio [MATH] around unity, the [MATH] wells can be tuned to be deeper than the [MATH] wells or vice versa.', '1604.06055-2-5-9': 'This is shown in Fig. [REF](b), where the potential is plotted for [MATH] and [MATH] on the left- and right-hand sides, respectively.', '1604.06055-2-6-0': 'Experimentally, the required lattice is produced by the setup shown in Fig. [REF](c).', '1604.06055-2-6-1': 'Two laser beams with wavelengths [MATH] and [MATH], both detuned to the negative side of an atomic transition with [MATH] corresponding to a few ten MHz, are retro-reflected from the same mirror.', '1604.06055-2-6-2': 'The polarization of the beam at wavelength [MATH] is adjusted to be linear within the lattice plane.', '1604.06055-2-6-3': 'Hence, no interference occurs at the crossing point, where an atomic BEC is prepared, such that the potential [MATH] arises.', '1604.06055-2-6-4': 'Similarly, the beam at wavelength [MATH] exhibits linear polarization perpendicular to the lattice plane, such that maximal interference at the crossing point yields the potential [MATH].', '1604.06055-2-6-5': 'The two potentials can be precisely positioned relative to each other (to better than a few nm) by adjusting [MATH] and the ratio between the distances [MATH] and [MATH] indicated in the Fig. [REF](c) [CITATION].', '1604.06055-2-7-0': 'Theoretical model We consider the case when the corner sites ([MATH]) are significantly deeper than the bond sites ([MATH]) such that the former host [MATH] and [MATH] orbitals, while the latter host [MATH] orbitals.', '1604.06055-2-7-1': 'The low-lying [MATH] orbitals of the corner sites are neglected.', '1604.06055-2-7-2': 'The corresponding minimal tight-binding Hamiltonian reads [EQUATION] where [MATH] is the nearest-neighbor hopping between [MATH] and [MATH] orbitals, [MATH] is the next-nearest-neighbor hopping between [MATH] orbitals, and [MATH] and [MATH] are, respectively, the on-site energies for [MATH] and [MATH] orbitals (see Fig. [REF](a)).', '1604.06055-2-7-3': 'We will consider [MATH] in the remainder of this work.', '1604.06055-2-7-4': 'Let us define [MATH] and introduce the vector [MATH].', '1604.06055-2-7-5': 'In momentum space, the Hamiltonian can be written as [MATH], where [EQUATION]', '1604.06055-2-7-6': 'Here, we have defined [MATH], [MATH], and introduced units for which the lattice constant [MATH].', '1604.06055-2-7-7': 'At the high-symmetry points, the lowest-energy eigenvectors have the form (up to a normalisation constant): [MATH], [MATH], [MATH], [MATH], corresponding to the energy [MATH].', '1604.06055-2-7-8': 'The [MATH] point is two-fold degenerate.', '1604.06055-2-8-0': 'A comparison between the band structure derived from the Hamiltonian in Eq. ([REF]) and from the experimental potential given in Eq. ([REF]) is shown in Fig. [REF](b).', '1604.06055-2-8-1': 'The desired quadratic-band touching at the high-symmetry point [MATH] of the Brillouin zone, arising from two bands with non-negative curvature, may be promptly observed upon inspection of the figure.', '1604.06055-2-9-0': 'In the presence of contact interactions [MATH] and [MATH] for the [MATH]- and [MATH]-orbitals, respectively, and within the harmonic approximation, the interaction Hamiltonian can be cast in the form [CITATION] [EQUATION] where [MATH], [MATH] and the angular-momentum operator is given by [MATH].', '1604.06055-2-9-1': 'Inspection of the Hamiltonian shows that repulsive interactions favour a ground state with finite angular momentum [MATH], thus breaking time-reversal symmetry [MATH].', '1604.06055-2-10-0': 'Ground-state.', '1604.06055-2-10-1': 'Lattice systems exhibiting a degenerate set of minima favour condensation at the high-symmetry points, as for instance found for the Kagome lattice [CITATION].', '1604.06055-2-10-2': 'Therefore, we distinguish two types of condensates, depending on whether the condensation occurs at the points [MATH] or at the two-fold degenerate point [MATH].', '1604.06055-2-10-3': 'This scenario is confirmed by solving the Gross-Pitaevskii equation in imaginary time, which in addition allows us to exclude simultaneous condensation at all the three points.', '1604.06055-2-10-4': 'At the momenta corresponding to the high-symmetry points [MATH], [MATH], and [MATH], only some orbitals have a non-vanishing amplitude for the Hamiltonian in Eq. ([REF]).', '1604.06055-2-10-5': 'These considerations lead us to propose the following ansatz for the condensate wave-function [EQUATION]', '1604.06055-2-10-6': 'For the [MATH] condensate, one has [MATH] and [MATH].', '1604.06055-2-10-7': 'For the [MATH] condensate, [MATH].', '1604.06055-2-10-8': 'We make a gauge choice: [MATH].', '1604.06055-2-10-9': 'Minimisation of the mean-field free energy determines the phases [MATH], [MATH], [MATH], [MATH] and the mean-field free energy reads [EQUATION]', '1604.06055-2-10-10': 'Since the condensates break time-reversal symmetry [MATH], each solution is two-fold degenerate.', '1604.06055-2-10-11': 'The time-reversal conjugate solution can be obtained by substituting [MATH] and [MATH].', '1604.06055-2-11-0': 'The angular momentum expectation value is [MATH] in the [MATH] phase and [MATH] in the [MATH] phase, where [MATH] and [MATH] are integers corresponding to [MATH].', '1604.06055-2-11-1': 'Hence, the overall angular momentum vanishes for the [MATH] phase but not for the [MATH] phase.', '1604.06055-2-11-2': 'The [MATH] condensate breaks the translational symmetry of the Bravais lattice, whereas the [MATH] condensate does not (see Fig. [REF]).', '1604.06055-2-11-3': 'Phase gradients induce superfluid currents.', '1604.06055-2-11-4': 'The bond-current operator is defined as [MATH] where [MATH] is the creation operator for a boson at site [MATH] and flavor/orbital [MATH] and [MATH] is the hopping amplitude between sites [MATH] and [MATH].', '1604.06055-2-11-5': 'There are currents only along the [MATH] bonds with amplitude [MATH].', '1604.06055-2-12-0': 'The ground states may possess non-trivial topological properties, such as a Hall response.', '1604.06055-2-12-1': 'The calculation of the transverse (Hall) conductivity [MATH] revealed that this is not the case (see Refs. [CITATION] for more details).', '1604.06055-2-12-2': 'However, as we shall discuss below, topological properties arise for the excitations of the [MATH] condensate.', '1604.06055-2-13-0': 'Corrections to the ground-state mean-field energy given by the zero-point fluctuations of the Bogolyubov modes have been also investigated following the procedure outlined in Ref. [CITATION] and we found that the [MATH] condensate has a larger energy compared to the [MATH] condensate, of the order of 1 pK [CITATION].', '1604.06055-2-13-1': 'This small value makes us confident that a metastable [MATH] condensate could be experimentally realizable.', '1604.06055-2-13-2': 'Nevertheless, the precise estimation of the condensate lifetime is a highly non-trivial problem, which we defer to future investigations.', '1604.06055-2-14-0': 'Excitations and topology.', '1604.06055-2-14-1': 'The quadratic band-touching appearing at the [MATH]-point in the non-interacting spectrum is protected by inversion and time-reversal symmetry, and it exhibits a [MATH] Berry flux [CITATION].', '1604.06055-2-14-2': "Upon condensation, the band degeneracy is lifted because the U(1) symmetry breaking requires the existence of only one gapless mode, according to Goldstone's theorem.", '1604.06055-2-14-3': 'Therefore, the band splitting appearing for the excitations and the further breaking of time-reversal symmetry can allow for a non-vanishing Berry curvature and Chern number.', '1604.06055-2-15-0': 'Within the Bogolyubov approximation, the Hamiltonian can be written as [MATH], with [MATH], where we have introduced the Nambu spinor [MATH].', '1604.06055-2-16-0': 'The Bogolyubov Hamiltonian for the [MATH] condensate reads [EQUATION] where [MATH] was defined in Eq. ([REF]), [MATH], [EQUATION] and we have introduced the chemical potential [MATH] and defined [MATH] within the harmonic approximation (see Ref. [CITATION]).', '1604.06055-2-17-0': 'The excitation spectrum is obtained by solving the eigenvalue problem [MATH], where [MATH].', '1604.06055-2-17-1': 'The normalization of the eigenvectors is such that [MATH] and [MATH], with [MATH] being the matrix having the eigenvectors [MATH] in its columns.', '1604.06055-2-18-0': 'The excitation spectrum for the [MATH]-condensate is shown in Fig. [REF](a).', '1604.06055-2-18-1': 'Notice that the lowest branch of the spectrum became non-degenerate, as a consequence of the U(1) symmetry breaking.', '1604.06055-2-18-2': 'This allows us to calculate the associated Chern number generalised for Bogolyubov Hamiltonians [CITATION], [MATH] where the Berry curvature [MATH] reads [EQUATION].', '1604.06055-2-18-3': 'We numerically calculated the Chern number using the procedure described in Ref. [CITATION] and we found [MATH] for the lowest branch of the spectrum.', '1604.06055-2-18-4': 'The Berry curvature for this branch is shown in Fig. [REF](b).', '1604.06055-2-18-5': 'The non-vanishing [MATH]-space profile of the Berry curvature is analogous to the one of a free-particle with a quadratic band-touching point with an explicit time-reversal symmetry breaking [CITATION].', '1604.06055-2-18-6': 'The fundamental difference is that the time-reversal symmetry breaking here is emergent and driven by interactions.', '1604.06055-2-19-0': 'Conclusions.', '1604.06055-2-19-1': 'Our results provide a novel paradigm to realize topological phases of weakly-interacting bosons.', '1604.06055-2-19-2': 'By combining the condensation of atoms at a quadratic band-touching point with the spontaneous time-reversal symmetry breaking induced by the interactions, one can generate a superfluid with topological excitations.', '1604.06055-2-19-3': 'These topological excitations are non-trivial bogolons, but they are still described in terms of a single-particle Fock space [CITATION].', '1604.06055-2-19-4': 'This is in contrast to purely many-body wave functions with topological properties that lie outside the Altland-Zirnbauer classification, as discussed in Ref. [CITATION].', '1604.06055-2-19-5': 'The effect of interactions on the low-lying excitations is nevertheless crucial for spontaneously generating the topological properties, which are otherwise absent in the non-interacting regime.', '1604.06055-2-20-0': 'According to the topological classification [CITATION], this model belongs to class D: time-reversal symmetry is broken and particle-hole symmetry is intrinsically present in the Bogolyubov description.', '1604.06055-2-20-1': 'Indeed, particle-hole symmetry is represented by the operator [MATH], where [MATH] and [MATH] is the complex conjugation operator, acting as [MATH] with [MATH].', '1604.06055-2-20-2': 'A realisation of a similar class in bosonic systems has been recently proposed for a Kagome lattice of photonic crystals, where a pairing term that breaks U(1) and time-reversal symmetry is introduced by light-squeezing [CITATION].', '1604.06055-2-21-0': 'In our model, instead, interactions induce the symmetry-breaking mechanism that generates the topological features of the excitations.', '1604.06055-2-21-1': 'In particular, the non-vanishing Berry curvature characterizes an anomalous Hall effect for the excitations and affects the collective modes of the gas [CITATION].', '1604.06055-2-21-2': 'It may be directly observed with the interferometric techniques demonstrated in Refs. [CITATION], while the excitation spectra could be measured by momentum resolved Bragg spectroscopy [CITATION].', '1604.06055-2-22-0': 'By diagonalizing the spectrum in a cylindrical geometry, we found the existence of excitations localised at the two edges of the cylinder, as expected from the bulk-boundary correspondence (see red line in Fig. [REF](a)-(b)).', '1604.06055-2-22-1': 'The observation of these finite-size effects could be implemented by using an optical box trap cutting the system along the symmetry axes [CITATION].', '1604.06055-2-22-2': 'The absence of an overall true gap would compromise the detection of the edge states in the presence of disorder, which, however, should not be a problem in the light-shift potential of an optical lattice, since these systems are inherently clean.', '1604.06055-2-23-0': 'Finally, we want to recall that the ground-state of the model investigated here bears strong similarities with the Varma phase, firstly proposed for describing the pseudogap regime of high-[MATH] superconductors [CITATION].', '1604.06055-2-23-1': 'Our findings also open possibilities for new exotic states of bosons in the strongly-interacting regime [CITATION], where an insulating Mott phase would be characterised by a gap and may also possess topological features [CITATION].'}

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[]

[]

[]

['1604.06055-2-1-0', '1604.06055-2-5-0', '1604.06055-2-10-0', '1604.06055-2-14-0', '1604.06055-2-19-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1604.06055

0905.3887

{'0905.3887-1-0-0': 'One of the biggest challenges in biology is to understand how activity at the cellular level of neurons, as a result of their mutual interactions, leads to the observed behavior of an organism responding to a variety of environmental stimuli.', '0905.3887-1-0-1': 'Investigating the intermediate or mesoscopic level of organization in the nervous system is a vital step towards understanding how the integration of micro-level dynamics results in macro-level functioning.', '0905.3887-1-0-2': 'The coordination of many different co-occurring processes at this level underlies the command and control of overall network activity.', '0905.3887-1-0-3': 'In this paper, we have considered the somatic nervous system of the nematode Caenorhabditis elegans, for which the entire neuronal connectivity diagram is known.', '0905.3887-1-0-4': 'We focus on the organization of the system into modules, i.e., neuronal groups having relatively higher connection density compared to that of the overall network.', '0905.3887-1-0-5': 'We show that this mesoscopic feature cannot be explained exclusively in terms of considerations, such as optimizing for resource constraints (viz., total wiring cost) and communication efficiency (i.e., network path length).', '0905.3887-1-0-6': 'Even including information about the genetic relatedness of the cells cannot account for the observed modular structure.', '0905.3887-1-0-7': 'Comparison with other complex networks designed for efficient transport (of signals or resources) implies that neuronal networks form a distinct class.', '0905.3887-1-0-8': 'This suggests that the principal function of the network, viz., processing of sensory information resulting in appropriate motor response, may be playing a vital role in determining the connection topology.', '0905.3887-1-0-9': 'Using modular spectral analysis, we make explicit the intimate relation between function and structure in the nervous system.', '0905.3887-1-0-10': 'This is further brought out by identifying functionally critical neurons purely on the basis of patterns of intra- and inter-modular connections.', '0905.3887-1-0-11': 'Our study reveals how the design of the nervous system reflects several constraints, including its key functional role as a processor of information.', '0905.3887-1-1-0': '# Author Summary', '0905.3887-1-2-0': 'Why are brains wired in the manner that is observed in nature?', '0905.3887-1-2-1': 'To address this, we analyze the structure of the C. elegans neuronal network at an intermediate level of its organization into groups of densely connected neurons.', '0905.3887-1-2-2': 'These structural units, called modules, are composed of multiple neuronal types which rules out the possibility that this is a straightforward partition of the network into sensory and motor regions.', '0905.3887-1-2-3': 'Nor do they have a simple anatomical interpretation (e.g., in terms of ganglia).', '0905.3887-1-2-4': 'We investigate the possibility that this modular structure developed through an evolutionary drive to optimize a nervous system subject to different constraints.', '0905.3887-1-2-5': 'We show that neither wiring cost minimization nor maximization of communication efficiency are adequate in explaining the observed patterns in the network topology.', '0905.3887-1-2-6': 'The distinction with other complex networks designed for efficient signal propagation (such as the Internet) suggests that the neuronal networks key functional requirement of information processing may play a crucial role in determining its overall structure.', '0905.3887-1-2-7': 'Analyzing the network organization at the mesoscopic level also allows us to identify neurons belonging to known functional circuits and predict those that may play a vital part in some as yet undetermined function.', '0905.3887-1-3-0': '# Introduction', '0905.3887-1-4-0': 'The relatively simple nervous systems of invertebrate organisms provide vital insights into how nerve cells integrate sensory information from the environment, resulting in a coordinated response.', '0905.3887-1-4-1': 'Analysing the intermediate or mesoscopic level of organization in such systems is a crucial step in understanding how micro-level activity of single neurons and their interactions eventually result in macro-level behavior of the organism [CITATION].', '0905.3887-1-4-2': 'The nematode Caenorhabditis elegans is a model organism on which such an analysis can be performed, as its entire neuronal wiring layout has been completely mapped [CITATION].', '0905.3887-1-4-3': 'This information enables one to trace in full the course of activity along the neuronal network, from sensory stimulation to motor response [CITATION].', '0905.3887-1-4-4': 'We study its somatic nervous system, comprising 282 neurons that control all activity except the pharyngeal movements.', '0905.3887-1-4-5': 'This can lead to an understanding of the command and control processes occurring at the mesoscopic level that produce specific functional responses, including avoidance behavior and movement along a chemical gradient.', '0905.3887-1-4-6': 'The neuron locations as well as their connections being completely determined by the genetic program, are invariant across individual organisms.', '0905.3887-1-4-7': 'Further, unlike in higher organisms, the connections do not change with time in the adult nematode.', '0905.3887-1-4-8': 'In combination with the possibility of experimenting on the role of single neurons in different functional modalities, these invariances allow one to uniquely identify the important neurons in the system having specific behavioral tasks.', '0905.3887-1-5-0': 'The recent developments in the theory of complex graphs has made available many analytical tools for studying biological networks [CITATION].', '0905.3887-1-5-1': 'The initial emphasis was on developing gross macroscopic descriptions of such systems using measures such as average path length between nodes of the network, the clustering among nodes and the degree sequences (the number of links per node).', '0905.3887-1-5-2': 'However, such global characterizations of systems ignore significant local variations in the connection topology that are often functionally important.', '0905.3887-1-5-3': 'Therefore, investigating the network at a mesoscopic level which consider the broad patterns in the inhomogeneous distribution of connections, may reveal vital clues about the working of an organism that could be hidden in a global analysis.', '0905.3887-1-5-4': 'Further, these large-scale features help in understanding how coordination and integration occurs across different parts of the system, in contrast to a study of microscopic patterns comprising only a few neurons, e.g., motifs [CITATION].', '0905.3887-1-6-0': 'The existence of modules, marked by the occurrence of groups of densely connected nodes with relatively fewer connections between these groups [CITATION], provides a natural meso-level description of many complex systems [CITATION].', '0905.3887-1-6-1': 'In biological networks, such modular organization has been observed across many length scales, from the intra-cellular protein-protein interaction system [CITATION] to food webs comprising various species populations [CITATION].', '0905.3887-1-6-2': 'The relation between certain modules and specific functions, e.g., in metabolic networks [CITATION], helps us to understand how different functions are coordinated in the integral performance of complex biological networks.', '0905.3887-1-7-0': 'Modular organization in the brains of different species have been observed, both in functional networks derived from EEG/MEG and fMRI experiments and in structural networks obtained from tracing anatomical connections [CITATION].', '0905.3887-1-7-1': 'Functionally defined networks are constructed by considering brain areas, comprising a large number of localized neuronal groups, which are linked if they are simultaneously active.', '0905.3887-1-7-2': 'Such system have been shown to be modular for both human [CITATION] and non-human [CITATION] subjects.', '0905.3887-1-7-3': 'Tract-tracing studies in the brains of cat [CITATION] and macaque [CITATION] have also revealed a modular layout in the structural inter-connections between different cortical areas.', '0905.3887-1-7-4': 'However, as neurons are the essential building blocks of the nervous system, ideally one would like to explore the network of interconnections between these most basic elements.', '0905.3887-1-7-5': 'In the extremely complicated mammalian brains, it is so far only possible to analyze such networks for extremely limited regions that do not give a picture of how the system behaves as a whole [CITATION].', '0905.3887-1-7-6': 'The relative simplicity of the nervous system of C. elegans allows a detailed analysis of the network, defined in terms of both electrical (gap junctional) and chemical (synaptic) connections between the neurons (Fig. [REF]).', '0905.3887-1-8-0': 'The ubiquity of modularity in brain networks leads to the obvious question about how to explain the evolution of such a structural organization [CITATION].', '0905.3887-1-8-1': 'One possible reason for the existence of modular architecture is that they may result in low average path length (which is associated with high efficiency of signal communication) and high clustering (that allows local segregation of information processing) in networks [CITATION].', '0905.3887-1-8-2': 'An alternative possibility is that segregation of neurons into spatially localized communities minimizes the total cost associated with the wiring length (the physical distance spanned by connections between neurons).', '0905.3887-1-8-3': 'This cost arises from resources associated with factors such as wiring volume as well as metabolism required for maintenance and propagation of signals across long distances [CITATION].', '0905.3887-1-8-4': 'Developmental constraints, such as the lineage relations between different neurons may also play an important role in determining the connection topology of the neuronal network [CITATION].', '0905.3887-1-8-5': 'In addition, the existence of empirically determined circuits responsible for specific functions (such as, movement associated with exploratory behavior, egg laying, etc.) in the C. elegans nervous system, raises the intriguing possibility that structurally defined modules are associated with definite functional roles [CITATION].', '0905.3887-1-8-6': 'The invariant neuronal connectivity profile of C. elegans allows us to explore the contributions of the above mentioned structural, developmental and functional constraints in governing the mesoscopic organization of the nervous system.', '0905.3887-1-9-0': 'In this paper, we begin our analysis of the organization of the C. elegans nervous system by identifying structurally defined modules in the network of neurons linked by synapses and gap-junctions.', '0905.3887-1-9-1': 'Next, we investigate whether the observed modular structure can be explained by using arguments based on universal principles.', '0905.3887-1-9-2': 'Such criteria, which include minimizing the cost associated with neuronal connections [CITATION] and their genetic encoding [CITATION], or, decreasing the signal propagation path [CITATION], have recently been proposed to explain observed patterns of neuronal position and connectivity.', '0905.3887-1-9-3': 'We determine the role of physical proximity between a pair of neurons in deciding the connection structure, by investigating the correlation between their spatial positions and their modular membership.', '0905.3887-1-9-4': 'We also compare these modules with the existing classification of the nematode nervous system into several ganglia, as the latter have been differentiated in terms of anatomical localization of their constituent neurons.', '0905.3887-1-9-5': 'Results from the above analysis suggests that resource constraints such as wiring cost cannot be the sole deciding factor governing the observed meso-level organization.', '0905.3887-1-9-6': 'We also show that the modules cannot be only a result of the common lineage of their member nodes.', '0905.3887-1-10-0': 'It is natural to expect that the structure of the nervous system is optimized to rapidly process signals from the environment so that the organism can take appropriate action for its survival [CITATION].', '0905.3887-1-10-1': 'By looking at the deviation between the actual network and a system optimized for maximal communication efficiency in conjunction with minimum wiring cost, we infer the existence of additional functional constraints related to processing of information (i.e., other than rapid signal transmission).', '0905.3887-1-10-2': 'This is further supported by the observation of relatively high clustering in C. elegans neuronal network as compared to other information networks (e.g., electronic logic circuits [CITATION]).', '0905.3887-1-10-3': 'Previous investigation of the role of clustering in the performance of neural network models in associative recall of stored patterns, has shown that lower clustering exhibits much better performance [CITATION].', '0905.3887-1-10-4': 'As clustering also increases the wiring cost, while not assisting efficient communication, its presence in a system that has evolved under intense competition for survival may thus imply a key role for clustering in processing information.', '0905.3887-1-11-0': 'This brings us to the possibility that the observed distribution of neurons among modules is closely related to the behavioral requirements of the organism.', '0905.3887-1-11-1': 'For this purpose, we investigate the relation between the modules and the different functional circuits governing specific behavioral aspects of C. elegans.', '0905.3887-1-11-2': 'We identify the functional importance of certain key neurons by observing their relative connectivity within their module as compared to that with neurons belonging to other modules.', '0905.3887-1-11-3': 'By looking at the correlation between local and global connectivity profiles of individual neurons, we observe that the nematode nervous system is significantly different from systems designed for rapid signal transfer, including other information networks occurring in the technological domain, such as the internet.', '0905.3887-1-11-4': 'Further, in contrast to previous observations on the similarity between biological signalling networks having different origins [CITATION], we find that the C. elegans neuronal network has properties distinct from at least one other biological network that is involved in signalling tasks, namely the protein interaction network [CITATION].', '0905.3887-1-12-0': 'Thus, the analysis of the network at the mesoscopic level provides an appropriate framework for identifying the roles that different classes of constraints (developmental, structural and functional) play in determining the organization of a nervous system.', '0905.3887-1-12-1': 'It allows us to infer the existence of criteria related to processing of information governing the observed modular architecture in C. elegans neuronal inter-connections.', '0905.3887-1-12-2': 'It also provides the means for identifying neurons having key roles in the behavioral performance of the organism exclusively from anatomical information about their structural connectivity.', '0905.3887-1-12-3': 'Our results can help experimentalists in focusing their attention to a select group of neurons which may play vital part in as yet undetermined functions.', '0905.3887-1-13-0': '# Results', '0905.3887-1-14-0': '## Modular structure of the C.elegans somatic nervous system', '0905.3887-1-15-0': 'We begin our study of the mesoscopic organization of the network by focusing on identifying its modular arrangement.', '0905.3887-1-15-1': 'In order to determine the community structure of the C. elegans neuronal network, we perform an optimal partitioning of the system into modules, that corresponds to the maximum value of modularity parameter, [MATH] (see Methods for details on modularity measure [MATH] and the algorithm used for modularity determination).', '0905.3887-1-15-2': 'We have considered different cases corresponding to the different types of neuronal connections (viz., gap junctions and synapses) and the nature of such connections (i.e., unweighted or weighted by the number of connections between a given pair).', '0905.3887-1-15-3': 'While the gap junctional network is undirected, the directional nature of signal propagation through a synapse implies that the synaptic network is directed.', '0905.3887-1-15-4': 'For each network, we have obtained the maximum modularity [MATH] and the corresponding number of partitions (Table [REF]).', '0905.3887-1-15-5': 'Note that, the number of modules and their composition is dependent on the type of connections we consider.', '0905.3887-1-16-0': 'As we want to consider all connections in our study, we have also worked with an aggregate network that includes synapses as well as gap junctions.', '0905.3887-1-16-1': 'Throughout this paper, we have reported results for this weighted combined network, unless stated otherwise (see Text S1 for the analysis of the network of synaptic connections).', '0905.3887-1-16-2': 'The link weights in the combined network correspond to the total number of synaptic and gap junctional connections from one neuron to another.', '0905.3887-1-16-3': 'The high value of [MATH] and dense inter-connectivity within modules (Fig. [REF], left) suggest that the network has a modular organization (Table S1).', '0905.3887-1-16-4': 'We further validate our results by calculating the modularity of randomized versions of the network where the degree of each node is kept fixed (see Methods for the network randomization procedure).', '0905.3887-1-16-5': 'The average modularity of these randomized networks is considerably lower than that of the empirical network.', '0905.3887-1-17-0': 'The modules do not have a simple relation with the anatomical layout of the worm.', '0905.3887-1-17-1': 'In particular, they are not a result of a simple division of the nervous system into groups responsible for receiving sensory input, and other groups involved in motor output.', '0905.3887-1-17-2': 'In Fig. [REF] (right), we have analyzed the composition of the different modules in terms of distinct neuron types (viz. sensory, motor and inter-neurons).', '0905.3887-1-17-3': 'None of the modules are exclusively composed of a single type of neuron, although motor neurons do tend to dominate one module.', '0905.3887-1-18-0': '## Modules and spatial localization', '0905.3887-1-19-0': 'To understand why modular structures occur in the neuronal network, we first consider the relation between the optimal partitions and the spatial localization of neurons in each module.', '0905.3887-1-19-1': 'This can tell us whether constraints related to the physical nearness between neurons, such as minimization of the wiring length, dictate the topological organization of the network.', '0905.3887-1-19-2': 'Wiring cost has already been shown to be the decisive factor governing neuron positions in the body of C. elegans [CITATION].', '0905.3887-1-19-3': 'Thus, a plausible hypothesis is that, if most neuronal connections occur within a group of neurons which are physically adjacent to each other, then the wiring cost will be significantly decreased.', '0905.3887-1-19-4': 'In terms of connectivity, this will be manifested as a modular organization of the network, where each module will mostly comprise neurons in close physical proximity.', '0905.3887-1-20-0': 'Fig. [REF] indicates the spatial location of the cell body for each neuron on the nematode body (along the longitudinal axis), segregated according to their membership in different modules.', '0905.3887-1-20-1': 'We see that a large fraction of the neurons belonging to the same module do indeed have their cell bodies close to each other.', '0905.3887-1-20-2': 'However, the large standard deviations for the distribution of positions of the module components, reveal that none of the modules are spatially localized at any specific region on the nematode body axis.', '0905.3887-1-20-3': 'The lack of significant distinction between the modules in terms of their physical location weighs against the hypothesis of wiring length minimization being the dominant factor governing the connectivity.', '0905.3887-1-21-0': 'The above conclusion is further supported by analysing the connectivity pattern of the different ganglia of the nematode nervous system.', '0905.3887-1-21-1': 'The nine anatomically defined ganglia (G1: Anterior, G2: Dorsal, G3: Lateral, G4: Ventral, G5: Retrovesicular, G6: Posterolateral, G7: Preanal, G8: Dorsorectal and G9: Lumbar), in addition to the ventral cord (G10), are defined in terms of physical proximity of their component neurons.', '0905.3887-1-21-2': 'Thus, a lower total connection length between neurons would result in the ganglia having a significantly high density of connections between their constituent neurons.', '0905.3887-1-21-3': 'We, therefore, verify whether the connectivity inside each ganglion is high compared to the corresponding randomized networks.', '0905.3887-1-21-4': 'This is done by measuring the modularity value [MATH], with the partitions of the network corresponding to the ganglionic divisions.', '0905.3887-1-21-5': 'Although the non-zero value of [MATH] indicates that the connection density between the neurons in a ganglia is higher than that for the overall network, it is not as high as the maximum possible value of [MATH], obtained for the optimal partitioning) as seen from Table [REF].', '0905.3887-1-21-6': 'To measure the overlap between the modules obtained by optimal partitioning of the network and the ganglia, we calculate the normalized mutual information index, [MATH] (see Methods).', '0905.3887-1-21-7': 'In the case of perfect match between the two, [MATH], while, if there is no overlap, [MATH].', '0905.3887-1-21-8': 'The low values for [MATH] given in Table [REF] suggest that the composition of the different ganglia is quite distinct from that of the modules.', '0905.3887-1-21-9': 'The overlap between the modules and the ganglia is shown explicitly in Fig. [REF] (top), indicating that most ganglia are composed of neurons belonging to many different modules.', '0905.3887-1-22-0': 'This distribution of the neurons of each ganglion into the [MATH] different modules of the optimal partition allows us to define a modular decomposition spectrum for the different ganglia.', '0905.3887-1-22-1': 'It gives us a metric for inter-ganglionic distance in a [MATH]-dimensional "modular" space.', '0905.3887-1-22-2': 'Thus, in this abstract space, two ganglia are close to each other if they have similar spectral profiles.', '0905.3887-1-22-3': 'Their distance in this "modular" space (Fig. [REF], center) are then compared to their physical distance, as measured in terms of the average separation between the cell bodies of all pairs of neurons [MATH] and [MATH], belonging to different ganglia (Fig. [REF], bottom).', '0905.3887-1-22-4': 'The comparison of the two distance matrices shows that there are indeed certain similarities between these two different concepts of closeness between the ganglia.', '0905.3887-1-22-5': 'For example, the five ganglia located in the head (G1-G5) cluster together, as do the three located towards the tail (G7-G9).', '0905.3887-1-22-6': 'This observation is in accord with previous reports which use the notion of wiring cost for explaining (to a certain extent) the observed relative positions of the ganglia [CITATION].', '0905.3887-1-22-7': 'However, when we consider the corresponding dendrograms that indicate the relative proximity of the different ganglia in physical space and in "modular" space, we observe significant differences between the two.', '0905.3887-1-22-8': 'Ganglia which are close to each other in physical space may not be neighbors in terms of their modular spectra.', '0905.3887-1-22-9': 'For instance, G5 which is located in the head, is closer in "modular" space to the ganglion G8 located in the tail.', '0905.3887-1-22-10': 'This reiterates our previous conclusion that wiring cost minimization, which is related to the physical distance between neurons, is not a dominant factor governing the organization of C. elegans somatic nervous system.', '0905.3887-1-23-0': '## Modules and cell lineage', '0905.3887-1-24-0': 'As developmental processes are believed to play a critical role in determining the structure of the nervous system, we also consider the alternative hypothesis that the structural modules reflect a clustering of neurons that are related in terms of their lineage.', '0905.3887-1-24-1': 'Lineage of a cell is the pattern of successive cellular divisions that occur during its development.', '0905.3887-1-24-2': 'This is invariant in C. elegans, allowing one to trace the individual developmental history of each cell in order to identify the cell-autonomous mechanisms and inter-cellular interactions that define its fate.', '0905.3887-1-24-3': 'We investigate whether a relation exists between the modular structure and the sequence of cell divisions that occur during development, by measuring the average relatedness between neurons occurring in the same module and comparing with that for neurons occurring in different modules.', '0905.3887-1-25-0': 'Fig. [REF] indicates that there is no significant segregation of the modules in terms of the lineage of the neurons comprising them.', '0905.3887-1-25-1': 'Indeed, even coarse distinctions such as AB and non-AB lineage neurons are not apparent from the modular division.', '0905.3887-1-25-2': 'The detailed view of the relatedness between each pair of neurons shown in Fig. S1 indicates that, while in each module there are subgroups of closely related neurons, different subgroups within the same module may be very far from each other in the lineage tree.', '0905.3887-1-25-3': 'Conversely, neurons occurring in different modules can have small distance between each other in terms of lineage.', '0905.3887-1-25-4': 'The fact that C. elegans neurons are largely non-clonally derived from many different parent cells [CITATION] may partly explain this lack of correlation between lineage and modules.', '0905.3887-1-25-5': 'The above results indicate that developmental constraints arising from common ancestry are not exclusively responsible for the observed connection structure of the C. elegans neuronal network.', '0905.3887-1-26-0': '## Optimizing between wiring cost and communication efficiency', '0905.3887-1-27-0': 'In the previous section, we have shown that neither wiring cost minimization nor lineage considerations can by themselves determine the connection topology of the network.', '0905.3887-1-27-1': 'In order to ascertain the possible nature of the additional constraints that gives rise to the observed mesoscopic structure, we now investigate global properties of the neuronal network.', '0905.3887-1-27-2': 'A possible governing factor for network organization is that, rather than decreasing the total wiring length or the average physical distance between connected neurons, the network minimizes the path length for information transfer.', '0905.3887-1-27-3': 'This can be measured by the number of links that must be traversed to go from one neuron to another using the shortest route [CITATION].', '0905.3887-1-27-4': 'We consider this possibility by measuring the communication efficiency of the network, using the harmonic average path length between all pairs of neurons (see Methods).', '0905.3887-1-28-0': 'It is evident that increasing the efficiency requires long-range connections, which however increases the wiring cost of the network (Fig. [REF] top).', '0905.3887-1-28-1': 'Therefore, it is natural to expect that the system would try to optimize between these two constraints.', '0905.3887-1-28-2': 'Thus, we compare the performance of the network as a rapid signal propagation system against the resource cost for the required number of connections.', '0905.3887-1-28-3': 'This cost is measured as the Euclidean length between the cell bodies of all connected pairs of neurons, corresponding to the "dedicated-wire" model of Ref. [CITATION].', '0905.3887-1-28-4': 'It has been shown that the positions of the subset of sensory and motor neurons directly connected to sensory organs and muscles, respectively, can be determined quite accurately by minimizing their total wiring cost [CITATION].', '0905.3887-1-28-5': 'As our focus is on the connection structure of the neuronal network, we keep the neuron positions invariant.', '0905.3887-1-28-6': 'By randomizing the network, keeping the degree of each node unchanged, we can construct a system with a specific wiring cost.', '0905.3887-1-28-7': 'We then measure its communication efficiency.', '0905.3887-1-28-8': 'Fig. [REF] reproduces the expected result that, decreasing the wiring cost of the network causes a decline in its performance in terms of its ability to propagate signals rapidly.', '0905.3887-1-28-9': 'However, it is surprising that the empirical network has a wiring cost much higher than that of the corresponding randomized network having the same communication efficiency.', '0905.3887-1-28-10': 'To see whether this could be an artifact of the measure used to calculate wiring cost, we have considered an alternative method for quantifying it.', '0905.3887-1-29-0': 'Most of the neurons in C. elegans have at most one or two extended processes, on which all the synapses and gap junctions with other neurons are made.', '0905.3887-1-29-1': 'Thus the "dedicated-wire" definition of wiring cost that sums distances between every connected neuronal pair may be a gross over-estimate of the actual usage of resources used in wiring.', '0905.3887-1-29-2': 'Instead, we can use a "common-wire" model to define the wiring cost for connecting to a specific neuron.', '0905.3887-1-29-3': "This is measured by the Euclidean distance between the neuron's cell body and that of the farthest neuron (along the longitudinal axis) it is connected to.", '0905.3887-1-29-4': 'The simple one-dimensional simplification of the C. elegans body that we have assumed here ignores distance along the transverse plane.', '0905.3887-1-29-5': 'Thus, this measure is actually an under-estimate of the actual wiring cost, and should provide an insightful comparison with the above measure obtained from the "dedicated-wire" model.', '0905.3887-1-29-6': 'Fig. [REF] (inset) shows that wiring cost increases with communication efficiency for the randomized networks, which is qualitatively similar to the relation obtained using the preceding definition for wiring cost.', '0905.3887-1-29-7': 'In this case also, we find that the empirical C. elegans network has a much lower efficiency in comparison with an equivalent randomized network having the same wiring cost.', '0905.3887-1-29-8': 'Thus, as this observation is independent of the way wiring cost is defined, it suggests the presence of other constraints that force the neuronal network to have a higher wiring cost or lower efficiency than we would have expected.', '0905.3887-1-29-9': 'These constraints are possibly related to information processing, which is the principal function of the nervous system.', '0905.3887-1-30-0': '## Functional requirement of information processing distinguishes the C. elegans neuronal network', '0905.3887-1-31-0': 'To explore further the possibility that the additional constraints governing the topological structure of C. elegans nervous system may be related to information processing, we investigate how this functional requirement could be responsible for differentiating the system from other complex networks for which communication efficiency is of paramount importance.', '0905.3887-1-31-1': 'Rapid communication of information between different neurons is certainly an important performance criterion.', '0905.3887-1-31-2': 'However, the neuronal network has properties quite distinct from that of (say) the internet or the airline transportation network, which are systems designed for maximum transportation efficiency of signals or physical resources.', '0905.3887-1-32-0': 'For this purpose, we look at the overall network design by decomposing the system into (i) a strongly connected component (SCC), within which it is possible to visit any node from any other node using directed links, (ii) an inward component (IN) and (iii) an outward component (OUT), consisting of nodes from which the SCC can be visited or which can be visited from the SCC, respectively, but not vice versa.', '0905.3887-1-32-1': 'In addition, there can be disconnected components, i.e., nodes which cannot be visited from SCC nor can any visits be made to SCC from there (Fig. [REF]).', '0905.3887-1-32-2': 'A comparison of the C. elegans neuronal network with a similar decomposition of the WWW [CITATION] reveals that, while in the latter the different components are approximately of equal size, the SCC of the nervous system comprises almost the entire network.', '0905.3887-1-32-3': 'Thus, any node can, in principle, affect any other node in the nervous system, suggesting the importance of feedback control for information processing.', '0905.3887-1-33-0': 'Next, we consider the relation between two fundamental properties of the network: the degree of nodes and their betweenness centrality (BC), which characterizes the importance of a node in information propagation over the network (see Methods).', '0905.3887-1-33-1': 'We observe that for both the C. elegans neuronal network and its randomized versions, the degree of a node and its BC are strongly correlated, i.e., highly connected nodes are also the most central (Fig. [REF], left).', '0905.3887-1-33-2': 'This is similar to what has been observed in the internet [CITATION], where the highest degree nodes are also those with the highest betweenness [CITATION], but in sharp contrast to the airport transportation network, where non-hub nodes (low degree) may have very large BC [CITATION].', '0905.3887-1-34-0': 'However, the C. elegans neuronal network differs from both the internet and the protein interaction network (PIN), whose primary function is to allow signal propagation between nodes, in terms of the variation of the degree of a node with the average degree of its neighboring nodes, [MATH].', '0905.3887-1-34-1': 'While in the internet and PIN, [MATH] decays as a power law with node degree, in the neuronal network, this dependence is very weak (Fig [REF], right).', '0905.3887-1-34-2': 'This implies that unlike the internet and PIN, the C. elegans nervous system does not have multiple star-like subnetworks.', '0905.3887-1-34-3': 'Further, it is significantly different from the airline transportation network, where the high degree nodes are closely connected among themselves showing an assortative behavior [CITATION].', '0905.3887-1-34-4': 'Another important distinguishing characteristic of the C. elegans network is that the distributions of link weight and degree do not appear to be scale-free, or even having a long tail (Fig. S2), unlike systems such as the internet and the airline transportation network [CITATION].', '0905.3887-1-34-5': 'Thus, our study shows that there are additional constraints governing the nervous system connection topology in C. elegans, which are unrelated to wiring cost, lineage or communication efficiency.', '0905.3887-1-34-6': 'As the principal function of the system is to process information, the above results suggest it is this functional requirement that provides the additional constraints leading to the observed organization of the nematode neuronal network.', '0905.3887-1-35-0': '## Modules and functional circuits', '0905.3887-1-36-0': 'In order to understand the nature of functional considerations that may govern the network organization, we focus on the overlap of several previously identified functional circuits of C. elegans with the structurally identified modules.', '0905.3887-1-36-1': 'Functional circuits are a subset of neurons which are believed to play a vital role in performing a function, and are distinguished by observing abnormal behavior of the organism when they are individually removed (e.g., by laser ablation).', '0905.3887-1-36-2': 'In biological systems, it has been observed that members of structurally defined communities are often functionally related (e.g., in the intra-cellular protein interaction network [CITATION] and the network of cortical areas in the brain [CITATION]).', '0905.3887-1-36-3': 'Here, we investigate the possibility of a similar correlation between the anatomical modules of C. elegans and its functional circuits.', '0905.3887-1-36-4': 'We consider the functional circuits for (F1) mechanosensation [CITATION], (F2) egg laying [CITATION], (F3) thermotaxis [CITATION], (F4) chemosensation [CITATION], (F5) feeding [CITATION], (F6) exploration [CITATION] and (F7) tap withdrawal [CITATION] (Table S2).', '0905.3887-1-37-0': 'Fig. [REF] shows the modular decomposition of each functional circuit, indicating their overlap with the modules.', '0905.3887-1-37-1': 'The corresponding dendrogram clusters the circuits in terms of the similarity in their modular spectra.', '0905.3887-1-37-2': 'We note that the circuits for chemosensation, feeding and exploration are clustered together.', '0905.3887-1-37-3': 'This is consistent with the fact that most of the neurons belonging to the feeding and exploration circuits are involved in chemosensation.', '0905.3887-1-37-4': 'A surprising observation is that although F6 is a subset of F4, it is actually closer to F5 in "modular" space (distance = 0.18) than to F4 (distance = 0.26), despite the feeding and exploration circuits not having any neuron in common.', '0905.3887-1-37-5': 'This indicates that F5 and F6 are closely related circuits, having a strong connection between their corresponding functions.', '0905.3887-1-37-6': 'Indeed, the feeding behavior of C.elegans is known to be regulated in a context-dependent manner by its chemical milieu.', '0905.3887-1-37-7': 'It integrates external signals [CITATION], such as the availability of food, and nutritional status of the animal, to direct an appropriate response [CITATION].', '0905.3887-1-37-8': 'An example is the avoidance of high CO[MATH] concentrations by satiated animals [CITATION].', '0905.3887-1-37-9': 'Further, the mode of locomotion of the organism is also determined by the quality of food [CITATION].', '0905.3887-1-37-10': 'Another important observation made from the modular decomposition is the proximity of the functional circuit F2 to the group (F4,F5,F6).', '0905.3887-1-37-11': 'This is significant in light of experimental observations that presence of food detected through chemosensory neurons modulates the egg-laying rate in C. elegans [CITATION].', '0905.3887-1-37-12': 'The above results indicate that the relation between the functional circuits, which are essential for the survival of the organism, are reflected in the modular organization of the nematode nervous system.', '0905.3887-1-38-0': '## Functional roles of different neurons', '0905.3887-1-39-0': 'Having looked at the functional circuits and the relations between them in the previous section, we now investigate the importance of individual neurons in terms of their connectivity.', '0905.3887-1-39-1': 'This is revealed by a comparison between the localization of their connections within their own community and their global connectivity profile over the entire network.', '0905.3887-1-39-2': 'In order to do this, we focus on (i) the degree of a node within its module, [MATH], that indicates the number of connections a node has to other members of its module, and (ii) its participation coefficient, [MATH], which measures how dispersed the connections of a node are among the different modules [CITATION].', '0905.3887-1-40-0': 'A node having low within-module degree is called a non-hub ([MATH]) which can be further classified according to their fraction of connections with other modules.', '0905.3887-1-40-1': 'Following Ref. [CITATION], these are classified as (R1) ultra-peripherial nodes ([MATH]), having connections only within their module, (R2) peripherial nodes ([MATH]), which have a majority of their links within their module, (R3) satellite connectors ([MATH]), with many links connecting nodes outside their modules, and (R4) kinless nodes ([MATH]), which form links uniformly across the network.', '0905.3887-1-40-2': 'Hubs, i.e., nodes having relatively large number of connections within their module ([MATH]), are also divided according to their participation coefficient into (R5) provincial hubs ([MATH]), with most connections within their module, (R6) connector hubs ([MATH]), with a significant fraction of links distributed among many modules, and (R7) global hubs ([MATH]), which connect homogeneously to all modules.', '0905.3887-1-40-3': 'This classification allows us to distinguish nodes according to their different roles as brought out by their intra-modular and inter-modular connectivity patterns (Table S3).', '0905.3887-1-41-0': 'We will now use the above methodology on the C. elegans network in order to identify neurons that play a vital role in coordinating activity through sharing information (either locally within their community or globally over the entire network).', '0905.3887-1-41-1': 'Fig. [REF] shows the comparison between the empirical network and a corresponding randomized network (obtained by keeping the degree of each node fixed).', '0905.3887-1-41-2': 'We immediately notice that the randomized network does not have any nodes having the roles R1 and R5, indicating that the modular nature of the original network has been lost.', '0905.3887-1-41-3': 'In fact, in the randomized system, most nodes have higher participation coefficient, with a large majority being satellite connectors (R3).', '0905.3887-1-41-4': 'More interesting is the fact that, the empirical neural network does not posses any neuron having the global roles played by R4 and R7, whereas these regions may be populated in randomized networks.', '0905.3887-1-41-5': 'This implies that modular identity in the C. elegans neuronal network is very pronounced.', '0905.3887-1-42-0': 'It is possible that, neurons having the role of provincial hubs may be involved in local coordination of neural activity, while, the connector hubs may be responsible for integration of local excitations to produce a coherent response of the entire system.', '0905.3887-1-42-1': 'This hypothesis is supported by noting that all command interneurons (of the class AVA, AVB, AVD, AVE, PVC), which control forward and backward locomotion of the worm by regulating motor output, play the role of connector hubs.', '0905.3887-1-42-2': 'In fact, out of the 23 neurons in the class R6, 20 are known to belong to different functional circuits.', '0905.3887-1-42-3': 'Among the rest, although DVA does not belong to any of the known circuits, it has recently been identified as being involved in mechanosensory response and in its absence, the frequency and magnitude of the tap-induced reversal, as well as, the acceleration magnitude is diminished [CITATION].', '0905.3887-1-42-4': 'The two remaining neurons, AVKL and SMBVL, have not been implicated so far in any known functional circuit.', '0905.3887-1-42-5': 'However, their occurrence in this class suggests that they may be important for some, as yet unknown, function.', '0905.3887-1-42-6': 'This is a potentially interesting prediction that may be verified in the laboratory.', '0905.3887-1-43-0': 'The significance of these results is underlined by a comparison with the randomized network.', '0905.3887-1-43-1': 'For instance, in the random realization shown in Fig. [REF] (b), of the 49 neurons playing the role of connector or global hubs, less than half (viz., 23) actually belong to any of the known functional circuits.', '0905.3887-1-43-2': 'The appearance of most of the command interneurons in the high-[MATH] region of both the empirical and randomized networks indicates that their high overall degree is responsible for their observed role of "connecting hubs".', '0905.3887-1-44-0': 'We now turn to the 28 neurons which play the role of provincial hubs.', '0905.3887-1-44-1': 'Half of all the inhibitory D-class motorneurons (viz., DD1-DD3 and VD1-VD6) are found to belong to this class.', '0905.3887-1-44-2': 'This is significant as these neurons have already been implicated in the ability of the worm to initiate backward motion.', '0905.3887-1-44-3': 'While they also contribute to forward locomotion, previous experiments have shown that they are not essential [CITATION].', '0905.3887-1-44-4': 'This fits with our hypothesis that, R5 neurons are important for local coordination but may not be crucial for the global integration of activity.', '0905.3887-1-44-5': 'A pair of excitatory B-class motorneurons that sustain coordinated forward locomotion in the worm also appear as provincial hubs.', '0905.3887-1-44-6': 'Of the remaining R5 neurons, 9 have been previously identified as belonging to various functional circuits.', '0905.3887-1-44-7': 'It will be interesting to verify the functional relevance of the remaining 8 neurons (OLLL/R, RMDVL/R, SMDVR, RIH, RMDDL/R) in the laboratory.', '0905.3887-1-44-8': 'Thus, overall, we find a very good correlation between the connectivity pattern and the functional importance of different neurons.', '0905.3887-1-45-0': 'Analysis of neurons having different roles in terms of their membership in the different ganglia (Fig. S3) indicates that the lateral ganglion provides the majority of neurons acting as connector hubs (R6).', '0905.3887-1-45-1': 'This is consistent with an earlier study where this ganglion was found to be the principal highway for information flowing between neurons responsible for receiving sensory stimuli and those involved in motor response [CITATION].', '0905.3887-1-45-2': 'To check the significance of the above result, we observe the membership of the set of connector hubs in the corresponding randomized network (Fig. S3).', '0905.3887-1-45-3': 'While this also shows many neurons from the lateral ganglion, unlike in the empirical network it has significant representation from other ganglia too (e.g., the retrovesicular ganglion).', '0905.3887-1-46-0': 'We have also carried out an analysis of the probability of connections between neurons having different roles, relative to the randomized network (figure not shown).', '0905.3887-1-46-1': 'This allows us to compare the C. elegans neuronal network with other networks involved in (a) transportation and (b) information propagation.', '0905.3887-1-46-2': 'We do not find any significant overlap of the nematode nervous system with networks in either of these two classes, suggesting that the neuronal network does not belong exclusively to either class of networks.', '0905.3887-1-46-3': 'This assumes significance in light of recent work distinguishing information (or signalling) networks, such as the Internet and protein interactome, on the one hand, and transportation networks, such as metabolic and airport networks, on the other, into two classes [CITATION].', '0905.3887-1-46-4': 'Our results suggest that neuronal networks which have to process information, in addition to transferring signals, may constitute a different category from either of the above classes.', '0905.3887-1-47-0': '# Discussion', '0905.3887-1-48-0': 'In this paper, we have carried out a detailed analysis of the mesoscopic structure in the connection topology of the C. elegans neuronal network.', '0905.3887-1-48-1': 'Inferring the organizing principles underlying the network may give us an understanding of the way in which an organism makes sense of the external world.', '0905.3887-1-48-2': 'We have focused primarily on the existence of modules, i.e., groups of neurons having higher connection density among themselves than with neurons in other groups.', '0905.3887-1-48-3': 'Presence of such mesoscopic organization naturally prompts us to ask the reasons behind the evolution of these features in the network.', '0905.3887-1-49-0': 'In lower invertebrates like nematodes, the genome is the dominant factor which governs the development of the organism, including its nervous system.', '0905.3887-1-49-1': 'The neuronal network structure is formed early in the life-cycle of the organism, when most of the cells and their connections are configured permanently.', '0905.3887-1-49-2': 'Although external cues may play a role, the relative absence of individual variations in the network organization makes C. elegans an ideal system for studying how the system has evolved to optimize for various constraints, such as minimizing resource use and maximizing performance.', '0905.3887-1-50-0': 'There have been recent attempts at explaining neuronal position and structural layout of the network by using static constraints, such as wiring economy and communication path minimization.', '0905.3887-1-50-1': 'Although we find that membership of neurons in specific modules are correlated with their physical nearness, the empirical network is sub-optimal in terms of both the above-mentioned constraints.', '0905.3887-1-50-2': 'By comparing the system with other complex networks that have been either designed or have evolved for rapid transportation while being subject to wiring economy, we find that the C. elegans nervous system stands apart as a distinct class.', '0905.3887-1-50-3': 'This suggests that the principal function of neuronal networks, viz., the processing of information, distinguishes it from the other networks considered, and plays a vital role in governing its arrangement.', '0905.3887-1-50-4': 'Considering the importance of this constraint in ensuring the survival of an organism, it is natural that this should be key to the organizing principles underlying the design of the network.', '0905.3887-1-50-5': 'The intimate relation between function and structure of the nervous system is further brought out by our use of structural analysis to distinguish neurons that are critical for the survival of the organism.', '0905.3887-1-50-6': 'In addition to identifying neurons that have been already empirically implicated in different functions (which serve as a verification of our method), we also predict several neurons which can be potentially crucial for certain, as yet unidentified, functions.', '0905.3887-1-51-0': 'Biological systems are distinguished by the occurrence of discrete entities with a distinct function, which are often termed as functional modules [CITATION].', '0905.3887-1-51-1': 'For several networks that occur in the biological context (such as that of protein interactions), the components of structural modules are seen to be functionally related.', '0905.3887-1-51-2': 'This suggests that modules provide a framework for relating mesoscopic patterns in the connection topology to subsystems responsible for specific functions.', '0905.3887-1-51-3': 'Although there is no unique correspondence between the structural modules of C. elegans and the known functional circuits [CITATION], we use the overlap of the circuits with the modular membership of their constituent neurons to discover correlations between them.', '0905.3887-1-51-4': 'Our results reveal non-trivial association between circuits whose corresponding functions are closely connected, even when they do not share any common neurons.', '0905.3887-1-51-5': 'As such relations could not have been revealed by a micro-scale study which focuses on individual neurons and their connections, this result highlights one of the significant advantages of investigating the network at the mesoscopic level.', '0905.3887-1-52-0': 'When we compare the nervous system of C. elegans with the brains of higher organisms, we observe the modular organization of the latter to be more prominent [CITATION].', '0905.3887-1-52-1': 'For example, the network of cortical areas in the cat and macaque brains exhibit distinct modules [CITATION], with each module being identified with specific functions [CITATION].', '0905.3887-1-52-2': 'A possible reason for the relatively weak modular structure in the nematode could be due to the existence of extended processes for the neurons of C. elegans.', '0905.3887-1-52-3': 'Many of these span almost the entire body length, an effect that is enhanced by the approximately linear nature of the nematode body plan.', '0905.3887-1-52-4': 'As a result, connections are not constrained by the physical distance between soma of the neurons, as would be the case in mammalian brains.', '0905.3887-1-52-5': 'It is apparent that such constraints on the geographical distance spanned by links between nodes (viz., cost of wiring length) can give rise to clustering of connections among physically adjacent elements.', '0905.3887-1-52-6': 'In addition, the small nervous system of C. elegans, comprising only 302 neurons, lacks redundancy.', '0905.3887-1-52-7': 'Therefore, individual neurons may often have to perform a set of tasks which in higher organisms are performed by several different neurons.', '0905.3887-1-52-8': 'Thus, functional modularity is less prominent in the nematode, as some neurons belong to multiple behavioral circuits.', '0905.3887-1-53-0': 'Another principal distinction between the C. elegans nervous system and the brains of higher organisms such as human beings, is the relative high connectivity in the former (the connection density being [MATH]).', '0905.3887-1-53-1': 'By contrast, the connectance for human brain is around [MATH] [CITATION], which leads us to the question of how communication efficiency can remain high in such a sparsely connected network.', '0905.3887-1-53-2': 'It is possible that the more intricate hierarchical and modular structures seen in the brains of higher organisms is a response to the above problem.', '0905.3887-1-53-3': 'The fact that the rate at which the number of neurons [MATH] increase across species, is not matched by a corresponding increase in the number of links (which increases slower than [MATH]) implies the existence of constraints on the latter, which is a resource cost in addition to the earlier mentioned cost of wiring length.', '0905.3887-1-54-0': 'Networks provide the scaffolding for the computational architectures that mediate cognitive functions.', '0905.3887-1-54-1': 'The pattern of connections of a neuron (or a neuronal cluster) defines its functionality not just locally but also as an integrated part of the nervous system.', '0905.3887-1-54-2': 'This is because neurocognitive networks across the evolutionary tree consist of interconnected clusters of neurons that are simultaneously activated for generating a single or a series of cognitive outputs.', '0905.3887-1-54-3': 'However, while some of the neurons (or clusters of neurons) are essential for the relevant outcome, others are ancillary.', '0905.3887-1-54-4': 'Thus, they work in a collaborative mode but are not interchangeable, each displaying relative specializations for separate behavioral components.', '0905.3887-1-54-5': 'Since the most prominent neuronal pathways are those that interconnect components of functional circuit, our analysis validates the intrinsic connection between network structure and the functions of the nervous system.', '0905.3887-1-54-6': 'The concept that behavioral consequences of damaging a region of the brain will reflect the disruption of the underlying network architecture may be intuitive but is particularly important when one considers brain dysfunction through neurodegeneration [CITATION], where atrophy is seen to propagate preferentially through networks of functionally related neurons [CITATION].', '0905.3887-1-54-7': 'While the idea that damage in one part of the brain can affect other areas connected to that region is not new, our work on the mesoscopic network organization maybe extended to look at disease progression through the network, beyond the current focus on the pathology within individual cells.', '0905.3887-1-55-0': '# Materials and Methods', '0905.3887-1-56-0': 'Connectivity Data: We have used information about the network connectivity, positions of neurons and the lineage distance between cells for the C. elegans nervous system, from the database published in Ref. [CITATION] and available from the online Atlas of C. elegans anatomy (www.wormatlas.org).', '0905.3887-1-56-1': 'This is an updated and revised version of the wiring data originally published in Ref. [CITATION].', '0905.3887-1-56-2': 'The connectivity between neurons and the positions of the neuron cell bodies along the longitudinal axis of the worm is reconstructed based on serial section electromicrography.', '0905.3887-1-56-3': 'Note that, the new database adds or updates about 3000 connections to the previous version.', '0905.3887-1-56-4': 'The lineage data indicates the relatedness between every pair of neurons in terms of distances in the embryonic and post-embryonic lineage trees.', '0905.3887-1-56-5': 'This is measured by identifying the last common progenitor cells of the two neurons, and then counting the number of cell divisions from this common ancestor.', '0905.3887-1-56-6': 'Each cell division adds a single unit to the total distance, with the initial division from the common progenitor counted only once.', '0905.3887-1-57-0': 'Modularity (Q): To decompose a given network into modules (where a module or community is defined as a subnetwork having a higher density of connections relative to the entire network) we use a method introduced in Ref. [CITATION].', '0905.3887-1-57-1': 'We compute a quantitative measure of modularity, [MATH], for a given partitioning of the network into several communities, [EQUATION]', '0905.3887-1-57-2': 'Here, [MATH] is the adjacency matrix ([MATH] is 1, if neurons [MATH], [MATH] are connected, and [MATH], otherwise).', '0905.3887-1-57-3': 'The degree of each node [MATH] is given by [MATH].', '0905.3887-1-57-4': '[MATH] is the total number of links in the network, [MATH] is the Kronecker delta ([MATH], if [MATH], and [MATH], otherwise), and [MATH] is the label of the community to which vertex [MATH] is assigned.', '0905.3887-1-57-5': 'In the case of directed and weighted network, the above measure can be generalized as [EQUATION] where, [MATH] is the sum of weights of all links in the network ([MATH] is the weight of the link from neuron [MATH] to neuron [MATH]), and the weighted in-degree and out-degree of node [MATH] are given by [MATH] and [MATH], respectively.', '0905.3887-1-58-0': 'The optimal partitioning of the network is the one which maximizes the modularity measure [MATH] (or [MATH]).', '0905.3887-1-58-1': 'We obtain this using a generalization of the spectral method [CITATION].', '0905.3887-1-58-2': 'We first define a modularity matrix [MATH], [EQUATION]', '0905.3887-1-58-3': 'To split the network into modules, the eigenvectors corresponding to the largest positive eigenvalue of the symmetric matrix ([MATH]) is calculated and the communities are assigned based on the sign of the elements of the eigenvector.', '0905.3887-1-58-4': 'This divides the network into two parts, which is refined further by exchanging the module membership of each node in turn if it results in an increase in the modularity.', '0905.3887-1-58-5': 'The process is then repeated by splitting each of the two divisions into further subdivisions.', '0905.3887-1-58-6': 'This recursive bisection of the network is carried out until no further increase of [MATH] is possible.', '0905.3887-1-59-0': 'Modular spectra: We analyze different neuronal groups, defined in terms of functions, anatomy (e.g., ganglia), etc., by proposing a decomposition in terms of the overlap of their constituent neurons with the different modules.', '0905.3887-1-59-1': 'Let the set of all neurons be optimally partitioned into [MATH] modules.', '0905.3887-1-59-2': 'We then define a overlap matrix, [MATH], where the rows correspond to the different neuronal groups, and the columns correspond to the different modules.', '0905.3887-1-59-3': 'An element of this overlap matix, [MATH], is the number of neurons in the group [MATH] that appear in the module [MATH].', '0905.3887-1-59-4': 'Then, the decomposition of the [MATH]-th group in the abstract [MATH]-dimensional basis space formed by the modules is [MATH].', '0905.3887-1-59-5': 'We use the distance between two groups in this "modular" space as a metric to measure their closeness to each other.', '0905.3887-1-60-0': 'Betweenness centrality (BC): To measure the importance of a node in facilitating communication across a network, we consider how frequently a node is used to convey information from any part of the network to any other part using the shortest available path.', '0905.3887-1-60-1': 'The betweenness centrality of a node [MATH] is defined as the fraction of shortest paths between the pairs of all other nodes in the network that pass through [MATH] [CITATION].', '0905.3887-1-60-2': 'If the total number of shortest paths between nodes [MATH] and [MATH] is [MATH], of which [MATH] paths pass through node [MATH], then the betweenness centrality of node [MATH] is [EQUATION]', '0905.3887-1-60-3': 'Communication Efficiency (E): To measure the speed of information transfer over the network, one can define the efficiency [MATH] of communication between vertices [MATH] and [MATH] to be inversely proportional to the shortest distance [MATH].', '0905.3887-1-60-4': 'Therefore, the efficiency of communication across the whole network is [EQUATION]', '0905.3887-1-60-5': 'This is the harmonic mean geodesic distance between all pairs, which does not diverge even when the network is disconnected [CITATION].', '0905.3887-1-61-0': 'Network Randomization: An ensemble of randomized versions of the empirical network is constructed keeping the in-degree and out-degree of each node unchanged.', '0905.3887-1-61-1': 'Each such network is created by rewiring randomly selected pairs of directed edges, [MATH] and [MATH], such that, in the randomized network, the corresponding directed edges are [MATH] and [MATH].', '0905.3887-1-61-2': 'However, if these new links already exist in the empirical network, this step is aborted and a new pair of edges are chosen in order to prevent the occurrence of multiple edges [CITATION].', '0905.3887-1-61-3': 'The above procedure is repeated [MATH] times for a single realization of the randomized network.', '0905.3887-1-61-4': 'In order to compare the properties of the empirical network with its randomized version, an ensemble of [MATH] realizations is considered.', '0905.3887-1-62-0': 'Determining the intra- and inter-modular role of a neuron: The role played by each neuron in terms of its connectivity within its own module and in the entire network is determined according to two properties: (i) the relative within module degree, [MATH], and (ii) the participation coefficient, [MATH].', '0905.3887-1-63-0': 'The [MATH]-score of the within module degree distinguishes nodes that are hubs of their communities from those that are non-hubs.', '0905.3887-1-63-1': 'It is defined as [EQUATION] where [MATH] is the number of links of node [MATH] to other nodes in its community [MATH] and [MATH] are taken over all nodes in module [MATH].', '0905.3887-1-63-2': 'The within-community degree [MATH]-score measures how well-connected node [MATH] is to other nodes in the community.', '0905.3887-1-64-0': 'The nodes are also distinguished based on their connectivity profile over the entire network, in particular, their connections to nodes in other communities.', '0905.3887-1-64-1': 'Two nodes with same within module degree [MATH]-score can play different roles, if one of them has significantly higher inter-modular connections compared to the other.', '0905.3887-1-64-2': 'This is measured by the participation coefficient [MATH] of node [MATH], defined as [EQUATION] where [MATH] is the number of links from node [MATH] to other nodes in its community [MATH] and [MATH] is the total degree of node [MATH].', '0905.3887-1-64-3': 'Therefore, the participation coefficient of a node is close to 1, if its links are uniformly distributed among all the communities, and is 0, if all its links are within its own community.', '0905.3887-1-65-0': 'Normalized mutual information ([MATH]): To measure the overlap of the membership of nodes in a ganglion with their membership in a specific module, we use the normalized mutual information measure [CITATION].', '0905.3887-1-65-1': 'First, we define a overlap matrix, [MATH], where the rows correspond to the different ganglia, and the columns correspond to the modules obtained by optimal partitioning of the network.', '0905.3887-1-65-2': 'An element of this overlap matix, [MATH], is the number of neurons in the ganglion [MATH] that appear in the module [MATH].', '0905.3887-1-65-3': 'An information-theoretic measure of similarity between the partitions can then be defined as [EQUATION] where [MATH] and [MATH] are the numbers of ganglia and modules respectively.', '0905.3887-1-65-4': 'The sum over row [MATH] of matrix [MATH] is denoted by [MATH] and the sum over column [MATH] is denoted by [MATH].', '0905.3887-1-65-5': 'If the modules are identical to the ganglia, then [MATH] takes its maximum value of 1.', '0905.3887-1-65-6': 'On the other hand, if the modular partitioning is independent of the division of the network into ganglia, [MATH].'}

{'0905.3887-2-0-0': 'One of the biggest challenges in biology is to understand how activity at the cellular level of neurons, as a result of their mutual interactions, leads to the observed behavior of an organism responding to a variety of environmental stimuli.', '0905.3887-2-0-1': 'Investigating the intermediate or mesoscopic level of organization in the nervous system is a vital step towards understanding how the integration of micro-level dynamics results in macro-level functioning.', '0905.3887-2-0-2': 'The coordination of many different co-occurring processes at this level underlies the command and control of overall network activity.', '0905.3887-2-0-3': 'In this paper, we have considered the somatic nervous system of the nematode Caenorhabditis elegans, for which the entire neuronal connectivity diagram is known.', '0905.3887-2-0-4': 'We focus on the organization of the system into modules, i.e., neuronal groups having relatively higher connection density compared to that of the overall network.', '0905.3887-2-0-5': 'We show that this mesoscopic feature cannot be explained exclusively in terms of considerations such as, optimizing for resource constraints (viz., total wiring cost) and communication efficiency (i.e., network path length).', '0905.3887-2-0-6': 'Even including information about the genetic relatedness of the cells cannot account for the observed modular structure.', '0905.3887-2-0-7': 'Comparison with other complex networks designed for efficient transport (of signals or resources) implies that neuronal networks form a distinct class.', '0905.3887-2-0-8': 'This suggests that the principal function of the network, viz., processing of sensory information resulting in appropriate motor response, may be playing a vital role in determining the connection topology.', '0905.3887-2-0-9': 'Using modular spectral analysis we make explicit the intimate relation between function and structure in the nervous system.', '0905.3887-2-0-10': 'This is further brought out by identifying functionally critical neurons purely on the basis of patterns of intra- and inter-modular connections.', '0905.3887-2-0-11': 'Our study reveals how the design of the nervous system reflects several constraints, including its key functional role as a processor of information.', '0905.3887-2-1-0': '# Author Summary', '0905.3887-2-2-0': 'Why are brains wired in the manner that is observed in nature?', '0905.3887-2-2-1': 'To address this, we analyze the structure of the Caenorhabditis elegans neuronal network at an intermediate level of its organization into groups of densely connected neurons.', '0905.3887-2-2-2': 'These structural units, called modules, are composed of multiple neuronal types which rules out the possibility that this is a straightforward partition of the network into sensory and motor regions.', '0905.3887-2-2-3': 'Nor do they have a simple anatomical interpretation (e.g., in terms of ganglia).', '0905.3887-2-2-4': 'We investigate the possibility that this modular structure developed through an evolutionary drive to optimize a nervous system subject to different constraints.', '0905.3887-2-2-5': 'We show that neither wiring cost minimization nor maximization of communication efficiency are adequate in explaining the observed patterns in the network topology.', '0905.3887-2-2-6': "The distinction with other complex networks designed for efficient signal propagation (such as the Internet) suggests that the neuronal network's key functional requirement of information processing may play a crucial role in determining its overall structure.", '0905.3887-2-2-7': 'Analyzing the network organization at the mesoscopic level also allows us to identify neurons belonging to known functional circuits and predict those that may play a vital part in some as yet undetermined function.', '0905.3887-2-3-0': '# Introduction', '0905.3887-2-4-0': 'The relatively simple nervous systems of invertebrate organisms provide vital insights into how nerve cells integrate sensory information from the environment, resulting in a coordinated response.', '0905.3887-2-4-1': 'Analysing the intermediate or mesoscopic level of organization in such systems is a crucial step in understanding how micro-level activity of single neurons and their interactions eventually result in macro-level behavior of the organism [CITATION].', '0905.3887-2-4-2': 'The nematode Caenorhabditis elegans is a model organism on which such an analysis can be performed, as its entire neuronal wiring layout has been completely mapped [CITATION].', '0905.3887-2-4-3': 'This information enables one to trace in full the course of activity along the neuronal network, from sensory stimulation to motor response [CITATION].', '0905.3887-2-4-4': 'We study its somatic nervous system, comprising 282 neurons that control all activity except the pharyngeal movements.', '0905.3887-2-4-5': 'This can lead to an understanding of the command and control processes occurring at the mesoscopic level that produce specific functional responses, including avoidance behavior and movement along a chemical gradient.', '0905.3887-2-4-6': 'The neuron locations as well as their connections being completely determined by the genetic program, are invariant across individual organisms [CITATION].', '0905.3887-2-4-7': 'Further, unlike in higher organisms, the connections do not change with time in the adult nematode [CITATION].', '0905.3887-2-4-8': 'In combination with the possibility of experimenting on the role of single neurons in different functional modalities, these invariances allow one to uniquely identify the important neurons in the system having specific behavioral tasks.', '0905.3887-2-5-0': 'The recent developments in the theory of complex graphs has made available many analytical tools for studying biological networks [CITATION].', '0905.3887-2-5-1': 'The initial emphasis was on developing gross macroscopic descriptions of such systems using measures such as average path length between nodes of the network, the clustering among nodes and the degree sequences (the number of links per node).', '0905.3887-2-5-2': 'However, such global characterizations of systems ignore significant local variations in the connection topology that are often functionally important.', '0905.3887-2-5-3': 'Therefore, investigating the network at a mesoscopic level which consider the broad patterns in the inhomogeneous distribution of connections, may reveal vital clues about the working of an organism that could be hidden in a global analysis.', '0905.3887-2-5-4': 'Further, these large-scale features help in understanding how coordination and integration occurs across different parts of the system, in contrast to a study of microscopic patterns comprising only a few neurons, e.g., motifs [CITATION].', '0905.3887-2-6-0': 'The existence of modules, marked by the occurrence of groups of densely connected nodes with relatively fewer connections between these groups [CITATION], provides a natural meso-level description of many complex systems [CITATION].', '0905.3887-2-6-1': 'In biological networks, such modular organization has been observed across many length scales, from the intra-cellular protein-protein interaction system [CITATION] to food webs comprising various species populations [CITATION].', '0905.3887-2-6-2': 'The relation between certain modules and specific functions, e.g., in metabolic networks [CITATION], helps us to understand how different functions are coordinated in the integral performance of complex biological networks.', '0905.3887-2-7-0': 'Modular organization in the brains of different species have been observed, both in functional networks derived from EEG/MEG and fMRI experiments and in structural networks obtained from tracing anatomical connections [CITATION].', '0905.3887-2-7-1': 'Functionally defined networks are constructed by considering brain areas, comprising a large number of localized neuronal groups, which are linked if they are simultaneously active.', '0905.3887-2-7-2': 'Such systems have been shown to be modular for both human [CITATION] and non-human [CITATION] subjects.', '0905.3887-2-7-3': 'Tract-tracing studies in the brains of cat [CITATION] and macaque [CITATION] have also revealed a modular layout in the structural inter-connections between different cortical areas.', '0905.3887-2-7-4': 'However, as neurons are the essential building blocks of the nervous system, ideally one would like to explore the network of interconnections between these most basic elements.', '0905.3887-2-7-5': 'In the extremely complicated mammalian brains, it is so far only possible to analyze such networks for extremely limited regions that do not give a picture of how the system behaves as a whole [CITATION].', '0905.3887-2-7-6': 'The relative simplicity of the nervous system of C. elegans allows a detailed analysis of the network, defined in terms of both electrical (gap junctional) and chemical (synaptic) connections between the neurons (Fig. [REF]).', '0905.3887-2-8-0': 'The ubiquity of modularity in brain networks leads to the obvious question about how to explain the evolution of such a structural organization [CITATION].', '0905.3887-2-8-1': 'One possible reason for the existence of modular architecture is that it may result in low average path length (which is associated with high efficiency of signal communication) and high clustering (that allows local segregation of information processing) in networks [CITATION].', '0905.3887-2-8-2': 'An alternative possibility is that segregation of neurons into spatially localized communities minimizes the total cost associated with the wiring length (the physical distance spanned by connections between neurons).', '0905.3887-2-8-3': 'This cost arises from resources associated with factors such as wiring volume as well as metabolism required for maintenance and propagation of signals across long distances [CITATION].', '0905.3887-2-8-4': 'Developmental constraints, such as the lineage relations between different neurons may also play an important role in determining the connection topology of the neuronal network [CITATION].', '0905.3887-2-8-5': 'In addition, the existence of empirically determined circuits responsible for specific functions (such as, movement associated with exploratory behavior, egg laying, etc.) in the C. elegans nervous system, raises the intriguing possibility that structurally defined modules are associated with definite functional roles [CITATION].', '0905.3887-2-8-6': 'The invariant neuronal connectivity profile of C. elegans allows us to explore the contributions of the above mentioned structural, developmental and functional constraints in governing the mesoscopic organization of the nervous system.', '0905.3887-2-9-0': 'In this paper, we begin our analysis of the organization of the C. elegans nervous system by identifying structurally defined modules in the network of neurons linked by synapses and gap-junctions.', '0905.3887-2-9-1': 'Next, we investigate whether the observed modular structure can be explained by using arguments based on universal principles.', '0905.3887-2-9-2': 'Such criteria, which include minimizing the cost associated with neuronal connections [CITATION] and their genetic encoding [CITATION], or, decreasing the signal propagation path [CITATION], have recently been proposed to explain observed patterns of neuronal position and connectivity.', '0905.3887-2-9-3': 'Complementing the above studies, we seek to understand the factors determining the topological arrangement of the nervous system, given the physical locations of the neurons.', '0905.3887-2-9-4': 'We determine the role of physical proximity between a pair of neurons in deciding the connection structure, by investigating the correlation between their spatial positions and their modular membership.', '0905.3887-2-9-5': 'We also compare these modules with the existing classification of the nematode nervous system into several ganglia, as the latter have been differentiated in terms of anatomical localization of their constituent neurons.', '0905.3887-2-9-6': 'Results from the above analysis suggests that resource constraints such as wiring cost cannot be the sole deciding factor governing the observed meso-level organization.', '0905.3887-2-9-7': 'We also show that the modules cannot be only a result of the common lineage of their member nodes.', '0905.3887-2-10-0': 'It is natural to expect that the structure of the nervous system is optimized to rapidly process signals from the environment so that the organism can take appropriate action for its survival [CITATION].', '0905.3887-2-10-1': 'By looking at the deviation between the actual network and a system optimized for maximal communication efficiency in conjunction with minimum wiring cost, we infer the existence of additional functional constraints possibly related to processing of information (i.e., other than rapid signal transmission).', '0905.3887-2-10-2': 'Information processing refers to the transformation of signals [CITATION], such as selective suppression of activation in specific pathways, which allows different sensory stimuli to initiate response in distinct sets of motor neurons.', '0905.3887-2-10-3': 'Absence of such transformations would result in non-specific arousal of a large number of connected neurons, and will not achieve the high level of control and coordination necessary for performing a variety of specific functional tasks.', '0905.3887-2-10-4': 'This is also related to the observation of relatively high clustering in C. elegans neuronal network as compared to other information networks (e.g., electronic logic circuits [CITATION]).', '0905.3887-2-10-5': 'Previous investigation of the role of clustering in the performance of neural network models in associative recall of stored patterns, has shown that lower clustering exhibits much better performance [CITATION].', '0905.3887-2-10-6': 'It has also been shown that the clustering in C. elegans neuronal network is higher than that in degree-conserved randomized networks having the same wiring cost [CITATION].', '0905.3887-2-10-7': 'Therefore, the presence of enhanced clustering in a system that has evolved under intense competition for survival may imply it plays a key role in processing information.', '0905.3887-2-11-0': 'This brings us to the possibility that the observed distribution of neurons among modules is closely related to the behavioral requirements of the organism.', '0905.3887-2-11-1': 'For this purpose, we investigate the relation between the modules and the different functional circuits governing specific behavioral aspects of C. elegans.', '0905.3887-2-11-2': 'We identify the functional importance of certain key neurons by observing their relative connectivity within their module as compared to that with neurons belonging to other modules.', '0905.3887-2-11-3': 'By looking at the correlation between local and global connectivity profiles of individual neurons, we observe that the nematode nervous system is different from systems designed for rapid signal transfer, including other information networks occurring in the technological domain, such as the Internet.', '0905.3887-2-11-4': 'Further, in contrast to previous observations on the similarity between biological signalling networks having different origins [CITATION], we find that the C. elegans neuronal network has properties distinct from at least one other biological network involved in signalling, the protein interaction network (viz., in terms of the assortativity and the role-to-role connectivity profile) [CITATION].', '0905.3887-2-12-0': 'Thus, the analysis of the network at the mesoscopic level provides an appropriate framework for identifying the roles that different classes of constraints (developmental, structural and functional) play in determining the organization of a nervous system.', '0905.3887-2-12-1': 'It allows us to infer the existence of criteria related to processing of information governing the observed modular architecture in C. elegans neuronal inter-connections.', '0905.3887-2-12-2': 'It also provides the means for identifying neurons having key roles in the behavioral performance of the organism exclusively from anatomical information about their structural connectivity.', '0905.3887-2-12-3': 'Our results can help experimentalists in focusing their attention to a select group of neurons which may play a vital part in as yet undetermined functions.', '0905.3887-2-13-0': '# Results', '0905.3887-2-14-0': '## Modular structure of the C.elegans somatic nervous system', '0905.3887-2-15-0': 'We begin our study of the mesoscopic organization of the network by focusing on identifying its modular arrangement.', '0905.3887-2-15-1': 'In order to determine the community structure of the C. elegans neuronal network, we perform an optimal partitioning of the system into modules, that corresponds to the maximum value of modularity parameter, [MATH] (see Methods for details on modularity measure [MATH] and the algorithm used for modularity determination).', '0905.3887-2-15-2': 'We have considered different cases corresponding to the different types of neuronal connections (viz., gap junctions and synapses) and the nature of such connections (i.e., unweighted or weighted by the number of connections between a given pair).', '0905.3887-2-15-3': 'While the gap junctional network is undirected, the directional nature of signal propagation through a synapse implies that the synaptic network is directed.', '0905.3887-2-15-4': 'For each network, we have obtained using the spectral method, the maximum modularity [MATH] and the corresponding number of partitions (Table [REF]).', '0905.3887-2-15-5': 'Note that, the number of modules and their composition is dependent on the type of connections we consider.', '0905.3887-2-16-0': 'As we want to consider all connections in our study, we have also worked with an aggregate network that includes synapses as well as gap junctions.', '0905.3887-2-16-1': 'Throughout this paper, we have reported results for this weighted combined network, unless stated otherwise (see Text S1 for the analysis of the network of synaptic connections).', '0905.3887-2-16-2': 'The link weights in the combined network correspond to the total number of synaptic and gap junctional connections from one neuron to another.', '0905.3887-2-16-3': 'The high value of [MATH] and dense inter-connectivity within modules (Fig. [REF], a) suggest that the network has a modular organization (Table S1).', '0905.3887-2-16-4': 'We further validate our results by calculating the modularity of randomized versions of the network where the degree of each node is kept fixed (see Methods for the network randomization procedure).', '0905.3887-2-16-5': 'The average modularity of these randomized networks is considerably lower than that of the empirical network.', '0905.3887-2-16-6': 'We examine the robustness of the modular partitioning with respect to the method used for detecting communities by using another module determination algorithm [CITATION].', '0905.3887-2-16-7': 'However, this produces a lower value for [MATH] compared to the spectral method.', '0905.3887-2-16-8': 'The three modules identified using this alternative method have a substantial overlap (normalized mutual information [MATH] = 0.496) with the six modules obtained using the spectral method.', '0905.3887-2-16-9': 'It suggests that the alternative algorithm yields a coarser-grained picture of the network communities that are considered in the rest of the paper.', '0905.3887-2-17-0': 'The modules do not have a simple relation with the anatomical layout of the worm.', '0905.3887-2-17-1': 'In particular, they are not a result of a simple division of the nervous system into groups responsible for receiving sensory input, and other groups involved in motor output.', '0905.3887-2-17-2': 'In Fig. [REF] (b), we have analyzed the composition of the different modules in terms of distinct neuron types (viz. sensory, motor and inter-neurons).', '0905.3887-2-17-3': 'None of the modules are exclusively composed of a single type of neuron, although motor neurons do tend to dominate one module.', '0905.3887-2-18-0': '## Modules and spatial localization', '0905.3887-2-19-0': 'To understand why modular structures occur in the neuronal network, we first consider the relation between the optimal partitions and the spatial localization of neurons in each module.', '0905.3887-2-19-1': 'This can tell us whether constraints related to the physical nearness between neurons, such as minimization of the wiring length, dictate the topological organization of the network.', '0905.3887-2-19-2': 'Wiring cost has already been shown to be the decisive factor governing neuron positions in the body of C. elegans [CITATION].', '0905.3887-2-19-3': 'Thus, a plausible hypothesis is that, if most neuronal connections occur within a group of neurons which are physically adjacent to each other, then the wiring cost will be significantly decreased.', '0905.3887-2-19-4': 'In terms of connectivity, this will be manifested as a modular organization of the network, where each module will mostly comprise neurons in close physical proximity.', '0905.3887-2-20-0': 'Fig. [REF] indicates the spatial location of the cell body for each neuron on the nematode body (along the longitudinal axis), segregated according to their membership in different modules.', '0905.3887-2-20-1': 'We see that a large fraction of the neurons belonging to the same module do indeed have their cell bodies close to each other.', '0905.3887-2-20-2': 'A one-way ANOVA test, comparing the positions of the neurons in the different modules with the null hypothesis that they are drawn from the same population, shows that it can be rejected at confidence level of [MATH].', '0905.3887-2-20-3': 'However, the large standard deviations for the distribution of positions of the module components reveal that none of the modules are spatially localized at any specific region on the nematode body axis.', '0905.3887-2-20-4': 'A more detailed multiple comparison procedure carried out for every pair of modules shows the absence of statistically significant spatial separation for the module pairs (I, III), (I, V), (II, VI), (III, IV), (III, V), (III, VI) and (IV, VI).', '0905.3887-2-20-5': 'The lack of significant distinction between the modules in terms of their physical location weighs against the hypothesis of wiring length minimization being the dominant factor governing the connectivity.', '0905.3887-2-21-0': 'The above conclusion is further supported by analysing the connectivity pattern of the different ganglia of the nematode nervous system.', '0905.3887-2-21-1': 'The nine anatomically defined ganglia (G1: Anterior, G2: Dorsal, G3: Lateral, G4: Ventral, G5: Retrovesicular, G6: Posterolateral, G7: Preanal, G8: Dorsorectal and G9: Lumbar), in addition to the ventral cord (G10), are defined in terms of physical proximity of their component neurons.', '0905.3887-2-21-2': 'Thus, a lower total connection length between neurons would result in the ganglia having a relatively higher density of connections between their constituent neurons.', '0905.3887-2-21-3': 'This would imply that the existence of ganglia imposes a modular structure in the connection topology.', '0905.3887-2-21-4': 'To examine how well the ganglionic arrangement explains the observed modularity of the neuronal network, we have measured the modularity value [MATH] where the network communities correspond to the different ganglia.', '0905.3887-2-21-5': 'Although the non-zero value of [MATH] indicates that the connection density between the neurons in a ganglion is higher than that for the overall network, it is not as high as the maximum possible value of [MATH], obtained for the optimal partitioning) as seen from Table [REF].', '0905.3887-2-21-6': 'To measure the overlap between the modules obtained by optimal partitioning of the network and the ganglia, we calculate the normalized mutual information index, [MATH] (see Methods).', '0905.3887-2-21-7': 'In the case of perfect match between the two, [MATH], while, if there is no overlap, [MATH].', '0905.3887-2-21-8': 'The low values for [MATH] given in Table [REF] suggest that the composition of the different ganglia is quite distinct from that of the modules.', '0905.3887-2-21-9': 'The overlap between the modules and the ganglia is shown explicitly in Fig. [REF] (a), indicating that most ganglia are composed of neurons belonging to many different modules.', '0905.3887-2-22-0': 'This distribution of the neurons of each ganglion into the [MATH] different modules of the optimal partition allows us to define a modular decomposition spectrum for the different ganglia.', '0905.3887-2-22-1': 'It gives us a metric for inter-ganglionic distance in a [MATH]-dimensional "modular" space.', '0905.3887-2-22-2': 'Thus, in this abstract space, two ganglia are close to each other if they have similar spectral profiles.', '0905.3887-2-22-3': 'Their distance in this "modular" space (Fig. [REF], b) are then compared to their physical distance, as measured in terms of the average separation between the cell bodies of all pairs of neurons [MATH] and [MATH], belonging to different ganglia (Fig. [REF], c).', '0905.3887-2-22-4': 'The comparison of the two distance matrices shows that there are indeed certain similarities between these two different concepts of closeness between the ganglia.', '0905.3887-2-22-5': 'For example, the five ganglia located in the head (G1-G5) cluster together, as do the three located towards the tail (G7-G9).', '0905.3887-2-22-6': 'This similarity can be quantified by computing the correlation between these two distance measures (i.e., Euclidean and modular), [MATH]).', '0905.3887-2-22-7': 'Our observation is in accord with previous reports which use the notion of wiring cost for explaining (to a certain extent) the observed relative positions of the ganglia [CITATION].', '0905.3887-2-22-8': 'However, when we consider the corresponding dendrograms that indicate the relative proximity of the different ganglia in physical space and in "modular" space, we observe significant differences between the two.', '0905.3887-2-22-9': 'Ganglia which are close to each other in physical space may not be neighbors in terms of their modular spectra.', '0905.3887-2-22-10': 'For instance, G5 which is located in the head, is closer in "modular" space to the ganglion G8 located in the tail.', '0905.3887-2-22-11': 'On computing the correlation coefficient between the two trees by considering the distances between every pair of ganglia (measured as the path length between the pair in the dendrogram), we obtain [MATH]).', '0905.3887-2-22-12': 'This value is substantially lower than 1, the value expected had the two dendrograms been identical.', '0905.3887-2-22-13': 'It reiterates our previous conclusion that wiring cost minimization, which is related to the physical distance between neurons, is not a dominant factor governing the organization of C. elegans somatic nervous system.', '0905.3887-2-23-0': '## Modules and cell lineage', '0905.3887-2-24-0': 'As developmental processes are believed to play a critical role in determining the structure of the nervous system, we also consider the alternative hypothesis that the structural modules reflect a clustering of neurons that are related in terms of their lineage.', '0905.3887-2-24-1': 'Lineage of a cell is the pattern of successive cellular divisions that occur during its development.', '0905.3887-2-24-2': 'This is invariant in C. elegans, allowing one to trace the individual developmental history of each cell in order to identify the cell-autonomous mechanisms and inter-cellular interactions that define its fate [CITATION].', '0905.3887-2-24-3': 'We investigate whether a relation exists between the modular structure and the sequence of cell divisions that occur during development, by measuring the average relatedness between neurons occurring in the same module and comparing with that for neurons occurring in different modules.', '0905.3887-2-25-0': 'Fig. [REF] indicates that it is difficult to distinguish the modules in terms of the lineage of the neurons comprising them.', '0905.3887-2-25-1': 'Indeed, even coarse distinctions such as AB and non-AB lineage neurons are not apparent from the modular division.', '0905.3887-2-25-2': 'A one-way ANOVA test, with the null hypothesis that the average lineage distance between neurons within the same module and that between neurons belonging to different modules are obtained from the same distribution, shows that it cannot be rejected at confidence level of [MATH].', '0905.3887-2-25-3': 'We next analyse each pair of modules using a multiple comparison procedure for the intra-modular and inter-modular lineage distances of their constituent neurons.', '0905.3887-2-25-4': 'This reveals that the neurons belonging to modules III and VI have within-module lineage distance distribution that is statistically indistinguishable from the distribution of their lineage distance with neurons belonging to any of the other modules.', '0905.3887-2-25-5': 'Thus, for at least two of the modules, one cannot segregate them in terms of cell lineage.', '0905.3887-2-25-6': 'The detailed view of the relatedness between each pair of neurons shown in Fig. S1 indicates that, while in each module there are subgroups of closely related neurons, different subgroups within the same module may be very far from each other in the lineage tree.', '0905.3887-2-25-7': 'Conversely, neurons occurring in different modules can have small distance between each other in terms of lineage.', '0905.3887-2-25-8': 'This observation is supported by the low correlation [MATH]) between the physical and lineage distances of neurons.', '0905.3887-2-25-9': 'The fact that C. elegans neurons are largely non-clonally derived from many different parent cells [CITATION] may partly explain this lack of correlation between lineage and modules.', '0905.3887-2-25-10': 'The above results indicate that developmental constraints arising from common ancestry are not exclusively responsible for the observed connection structure of the C. elegans neuronal network.', '0905.3887-2-26-0': '## Optimizing between wiring cost and communication efficiency', '0905.3887-2-27-0': 'In the previous section, we have shown that neither wiring cost minimization nor lineage considerations can by themselves determine the connection topology of the network.', '0905.3887-2-27-1': 'In order to ascertain the possible nature of the additional constraints that gives rise to the observed mesoscopic structure, we now investigate global properties of the neuronal network.', '0905.3887-2-27-2': 'A possible governing factor for network organization is that, rather than decreasing the total wiring length or the average physical distance between connected neurons, the network minimizes the path length for information transfer.', '0905.3887-2-27-3': 'This can be measured by the number of links that must be traversed to go from one neuron to another using the shortest route [CITATION].', '0905.3887-2-27-4': 'We consider this possibility by measuring the communication efficiency of the network, using the harmonic average path length between all pairs of neurons (see Methods).', '0905.3887-2-28-0': 'It is evident that increasing the efficiency requires topological long-range connections, which however increases the wiring cost of the network (Fig. [REF] top).', '0905.3887-2-28-1': 'Therefore, it is natural to expect that the system would try to optimize between these two constraints.', '0905.3887-2-28-2': 'Thus, we compare the performance of the network as a rapid signal propagation system against the resource cost for the required number of connections.', '0905.3887-2-28-3': 'This cost is measured as the Euclidean length between the cell bodies of all connected pairs of neurons, corresponding to the "dedicated-wire" model of Ref. [CITATION].', '0905.3887-2-28-4': 'It has been shown that the positions of the subset of sensory and motor neurons directly connected to sensory organs and muscles, respectively, can be determined quite accurately by minimizing their total wiring cost [CITATION].', '0905.3887-2-28-5': 'As our focus is on the connection structure of the neuronal network, we keep the neuron positions invariant.', '0905.3887-2-28-6': 'By randomizing the network, keeping the degree of each node unchanged, we can construct a system with a specific wiring cost.', '0905.3887-2-28-7': 'We then measure its communication efficiency.', '0905.3887-2-28-8': 'Fig. [REF] reproduces the expected result that, decreasing the wiring cost of the network causes a decline in its performance in terms of its ability to propagate signals rapidly.', '0905.3887-2-28-9': 'However, it is surprising that the empirical network has a wiring cost much higher than that of the corresponding randomized network having the same communication efficiency.', '0905.3887-2-28-10': 'To see whether this could be an artifact of the measure used to calculate wiring cost, we have considered an alternative method for quantifying it.', '0905.3887-2-29-0': 'Most of the neurons in C. elegans have at most one or two extended processes, on which all the synapses and gap junctions with other neurons are made.', '0905.3887-2-29-1': 'Thus the "dedicated-wire" definition of wiring cost that sums Euclidean distances between every connected neuronal pair may be a gross over-estimate of the actual usage of resources used in wiring.', '0905.3887-2-29-2': 'Instead, we can use a "common-wire" model to define the wiring cost for connecting to a specific neuron.', '0905.3887-2-29-3': "This is measured by the Euclidean distance between the neuron's cell body and that of the farthest neuron (along the longitudinal axis) it is connected to.", '0905.3887-2-29-4': 'The simple one-dimensional simplification of the C. elegans body that we have assumed here ignores distance along the transverse plane.', '0905.3887-2-29-5': 'Thus, this measure is actually an under-estimate of the actual wiring cost, and should provide an insightful comparison with the above measure obtained from the "dedicated-wire" model.', '0905.3887-2-29-6': 'Fig. [REF] (inset) shows that wiring cost increases with communication efficiency for the randomized networks, which is qualitatively similar to the relation obtained using the preceding definition for wiring cost.', '0905.3887-2-29-7': 'In this case also, we find that the empirical C. elegans network has a much lower efficiency in comparison with an equivalent randomized network having the same wiring cost.', '0905.3887-2-29-8': 'Thus, as this observation is independent of these two definitions of wiring cost, it suggests the presence of other constraints that force the neuronal network to have a higher wiring cost or lower efficiency than we would have expected.', '0905.3887-2-29-9': 'These constraints are possibly related to information processing, which is the principal function of the nervous system.', '0905.3887-2-30-0': '## Information processing is a distinctive feature of the C. elegans neuronal network', '0905.3887-2-31-0': 'To explore further the possibility that the additional constraints governing the topological structure of C. elegans nervous system may be related to information processing, we investigate how this functional requirement could be responsible for differentiating the system from other complex networks for which communication efficiency is of paramount importance.', '0905.3887-2-31-1': 'Rapid communication of information between different neurons is certainly an important performance criterion.', '0905.3887-2-31-2': 'However, the neuronal network has properties quite distinct from that of (say) the Internet or the airline transportation network, which are systems designed for maximum transportation efficiency of signals or physical resources.', '0905.3887-2-32-0': 'For this purpose, we look at the overall network design by decomposing the system into (i) a strongly connected component (SCC), within which it is possible to visit any node from any other node using directed links, (ii) an inward component (IN) and (iii) an outward component (OUT), consisting of nodes from which the SCC can be visited or which can be visited from the SCC, respectively, but not vice versa.', '0905.3887-2-32-1': 'In addition, there can be disconnected components, i.e., nodes which cannot be visited from SCC nor can any visits be made to SCC from there (Fig. [REF]).', '0905.3887-2-32-2': 'A comparison of the C. elegans neuronal network with a similar decomposition of the WWW [CITATION] reveals that, while in the latter the different components are approximately of equal size (WWW: SCC [MATH] 56 million pages, while IN, OUT consist of [MATH] 43 million pages each), the SCC of the nervous system comprises almost the entire network (SCC has 274 neurons, IN has 4 neurons and OUT has 1 neuron).', '0905.3887-2-32-3': 'Thus, any node can, in principle, affect any other node in the nervous system, suggesting the importance of feedback control for information processing.', '0905.3887-2-33-0': 'Next, we consider the relation between two fundamental properties of the network: the degree of nodes and their betweenness centrality (BC), which characterizes the importance of a node in information propagation over the network (see Methods).', '0905.3887-2-33-1': 'We observe that for both the C. elegans neuronal network and its randomized versions, the degree of a node and its BC are strongly correlated, i.e., highly connected nodes are also the most central (Fig. [REF], left).', '0905.3887-2-33-2': 'This is similar to what has been observed in the Internet [CITATION], where the highest degree nodes are also those with the highest betweenness [CITATION], but in sharp contrast to the airport transportation network, where non-hub nodes (low degree) may have very large BC [CITATION].', '0905.3887-2-34-0': 'However, the C. elegans neuronal network differs from other networks whose primary function is to allow signal propagation between nodes (viz., the Internet and the protein interaction network (PIN)), in terms of the variation of the degree of a node with the average degree of its neighboring nodes, [MATH].', '0905.3887-2-34-1': 'While in the Internet and PIN, [MATH] decays as a power law with node degree, in the neuronal network, this dependence is very weak (Fig [REF], right), especially when contrasted with the randomized ensemble.', '0905.3887-2-34-2': 'This implies that the C. elegans nervous system does not have multiple star-like subnetworks as seen in the Internet and PIN [CITATION].', '0905.3887-2-34-3': 'Further, it is different from the airline transportation network, where the high degree nodes are closely connected among themselves showing an assortative behavior [CITATION].', '0905.3887-2-34-4': 'In fact, computation of the assortativity coefficient [MATH] indicates that the network is disassortative (as previously stated in Refs. [CITATION]), although comparison with that of the degree-conserved randomized ensemble ([MATH]) indicates that this is predominantly a result of the degree sequence.', '0905.3887-2-34-5': 'Another important distinguishing characteristic of the C. elegans network is that the distributions of link weight and degree do not appear to be scale-free, or even having a long tail (Fig. S2), unlike systems such as the Internet and the airline transportation network [CITATION].', '0905.3887-2-34-6': 'Thus, our study shows that there are additional constraints governing the nervous system connection topology in C. elegans, which are unrelated to wiring cost, lineage or communication efficiency.', '0905.3887-2-34-7': 'As the principal function of the system is to process information, the above results suggest it is this functional requirement that provides the additional constraints leading to the observed organization of the nematode neuronal network.', '0905.3887-2-35-0': '## Modules and functional circuits', '0905.3887-2-36-0': 'In order to understand the nature of functional considerations that may govern the network organization, we focus on the overlap of several previously identified functional circuits of C. elegans with the structurally identified modules.', '0905.3887-2-36-1': 'Functional circuits are a subset of neurons which are believed to play a vital role in performing a function, and are distinguished by observing abnormal behavior of the organism when they are individually removed (e.g., by laser ablation).', '0905.3887-2-36-2': 'In biological systems, it has been observed that members of structurally defined communities are often functionally related (e.g., in the intra-cellular protein interaction network [CITATION] and the network of cortical areas in the brain [CITATION]).', '0905.3887-2-36-3': 'Here, we investigate the possibility of a similar correlation between the anatomical modules of C. elegans and its functional circuits.', '0905.3887-2-36-4': 'We consider the functional circuits for (F1) mechanosensation [CITATION], (F2) egg laying [CITATION], (F3) thermotaxis [CITATION], (F4) chemosensation [CITATION], (F5) feeding [CITATION], (F6) exploration [CITATION] and (F7) tap withdrawal [CITATION] (Table S2).', '0905.3887-2-37-0': 'Fig. [REF] shows the modular decomposition of each functional circuit, indicating their overlap with the modules.', '0905.3887-2-37-1': 'The corresponding dendrogram clusters the circuits in terms of the similarity in their modular spectra.', '0905.3887-2-37-2': 'We note that the circuits for chemosensation, feeding and exploration are clustered together.', '0905.3887-2-37-3': 'This is consistent with the fact that most of the neurons belonging to the feeding and exploration circuits are involved in chemosensation.', '0905.3887-2-37-4': 'A surprising observation is that although F6 is a subset of F4, it is actually closer to F5 in "modular" space (distance = 0.18) than to F4 (distance = 0.26), despite the feeding and exploration circuits not having any neuron in common.', '0905.3887-2-37-5': 'This close relation between F5 and F6 in modular space is suggestive of a relation between modularity and functionality, as it is known from experiments that there is a strong connection between their corresponding functions.', '0905.3887-2-37-6': 'The feeding behavior of C.elegans is known to be regulated in a context-dependent manner by its chemical milieu.', '0905.3887-2-37-7': 'It integrates external signals [CITATION], such as the availability of food, and nutritional status of the animal, to direct an appropriate response [CITATION].', '0905.3887-2-37-8': 'An example is the avoidance of high CO[MATH] concentrations by satiated animals [CITATION].', '0905.3887-2-37-9': 'Further, the mode of locomotion of the organism is also determined by the quality of food [CITATION].', '0905.3887-2-37-10': 'Another important observation made from the modular decomposition is the proximity of the functional circuit F2 to the group (F4,F5,F6).', '0905.3887-2-37-11': 'This is significant in light of experimental observations that presence of food detected through chemosensory neurons modulates the egg-laying rate in C. elegans [CITATION].', '0905.3887-2-37-12': 'The above results indicate that the relation between the functional circuits, which are essential for the survival of the organism, are reflected in the modular organization of the nematode nervous system.', '0905.3887-2-38-0': '## Functional roles of different neurons', '0905.3887-2-39-0': 'Having looked at the functional circuits and the relations between them in the previous section, we now investigate the importance of individual neurons in terms of their connectivity.', '0905.3887-2-39-1': 'This is revealed by a comparison between the localization of their connections within their own community and their global connectivity profile over the entire network.', '0905.3887-2-39-2': 'In order to do this, we focus on (i) the degree of a node within its module, [MATH], that indicates the number of connections a node has to other members of its module, and (ii) its participation coefficient, [MATH], which measures how dispersed the connections of a node are among the different modules [CITATION].', '0905.3887-2-40-0': 'A node having low within-module degree is called a non-hub ([MATH]) which can be further classified according to their fraction of connections with other modules.', '0905.3887-2-40-1': 'Following Ref. [CITATION], these are classified as (R1) ultra-peripheral nodes ([MATH]), having connections only within their module, (R2) peripheral nodes ([MATH]), which have a majority of their links within their module, (R3) satellite connectors ([MATH]), with many links connecting nodes outside their modules, and (R4) kinless nodes ([MATH]), which form links uniformly across the network.', '0905.3887-2-40-2': 'Hubs, i.e., nodes having relatively large number of connections within their module ([MATH]), are also divided according to their participation coefficient into (R5) provincial hubs ([MATH]), with most connections within their module, (R6) connector hubs ([MATH]), with a significant fraction of links distributed among many modules, and (R7) global hubs ([MATH]), which connect homogeneously to all modules.', '0905.3887-2-40-3': 'This classification allows us to distinguish nodes according to their different roles as brought out by their intra-modular and inter-modular connectivity patterns (Table S3).', '0905.3887-2-41-0': 'We will now use the above methodology on the C. elegans network in order to identify neurons that play a vital role in coordinating activity through sharing information (either locally within their community or globally over the entire network).', '0905.3887-2-41-1': 'Fig. [REF] shows the comparison between the empirical network and a corresponding randomized network (obtained by keeping the degree of each node fixed).', '0905.3887-2-41-2': 'Results for a randomized ensemble, comparing the number of neurons in each role against that for the empirical network, are given in Table S4.', '0905.3887-2-41-3': 'We immediately notice that the randomized networks have relatively very few nodes having the roles R1 and R5, indicating that the modular nature of the original network has been lost.', '0905.3887-2-41-4': 'In fact, in the randomized system, most nodes have higher participation coefficient, with a large majority being satellite connectors (R3).', '0905.3887-2-41-5': 'More interesting is the fact that, the empirical neural network does not possess any neuron having the global roles played by R4 and R7, whereas these regions may be populated in randomized networks.', '0905.3887-2-41-6': 'This implies that modular identity in the C. elegans neuronal network is very pronounced.', '0905.3887-2-42-0': 'It is possible that, neurons having the role of provincial hubs may be involved in local coordination of neural activity, while, the connector hubs may be responsible for integration of local excitations to produce a coherent response of the entire system.', '0905.3887-2-42-1': 'This hypothesis is supported by noting that all command interneurons (of the class AVA, AVB, AVD, AVE, PVC), which control forward and backward locomotion of the worm by regulating motor output, play the role of connector hubs.', '0905.3887-2-42-2': 'In fact, out of the 23 neurons in the class R6, 20 are known to belong to different functional circuits.', '0905.3887-2-42-3': 'Among the rest, although DVA does not belong to any of the known circuits, it has recently been identified as being involved in mechanosensory response and in its absence, the frequency and magnitude of the tap-induced reversal, as well as the acceleration magnitude, is diminished [CITATION].', '0905.3887-2-42-4': 'The two remaining neurons, AVKL and SMBVL, have not been implicated so far in any known functional circuit.', '0905.3887-2-42-5': 'However, their occurrence in this class suggests that they may be important for some, as yet unknown, function.', '0905.3887-2-42-6': 'This is a potentially interesting prediction that may be verified in the laboratory.', '0905.3887-2-43-0': 'The significance of these results is underlined by a comparison with the randomized network.', '0905.3887-2-43-1': 'For instance, in the random realization shown in Fig. [REF] (b), of the 49 neurons playing the role of connector or global hubs, less than half (viz., 23) actually belong to any of the known functional circuits.', '0905.3887-2-43-2': 'The appearance of most of the command interneurons in the high-[MATH] region of both the empirical and randomized networks indicates that their high overall degree is responsible for their observed role of "connecting hubs".', '0905.3887-2-44-0': 'We now turn to the 28 neurons which play the role of provincial hubs.', '0905.3887-2-44-1': 'Half of all the inhibitory D-class motorneurons (viz., DD1-DD3 and VD1-VD6) are found to belong to this class.', '0905.3887-2-44-2': 'This is significant as these neurons have already been implicated in the ability of the worm to initiate backward motion.', '0905.3887-2-44-3': 'While they also contribute to forward locomotion, previous experiments have shown that they are not essential [CITATION].', '0905.3887-2-44-4': 'This fits with our hypothesis that, R5 neurons are important for local coordination but may not be crucial for the global integration of activity.', '0905.3887-2-44-5': 'A pair of excitatory B-class motorneurons that sustain coordinated forward locomotion in the worm also appear as provincial hubs.', '0905.3887-2-44-6': 'Of the remaining R5 neurons, 9 have been previously identified as belonging to various functional circuits.', '0905.3887-2-44-7': 'It will be interesting to verify the functional relevance of the remaining 8 neurons (OLLL/R, RMDVL/R, SMDVR, RIH, RMDDL/R) in the laboratory.', '0905.3887-2-44-8': 'Thus, overall, we find a very good correlation between the connectivity pattern and the functional importance of different neurons.', '0905.3887-2-45-0': 'Analysis of neurons having different roles in terms of their membership in the different ganglia (Fig. S3) indicates that the lateral ganglion provides the majority of neurons acting as connector hubs (R6).', '0905.3887-2-45-1': 'This is consistent with an earlier study where this ganglion was found to be the principal highway for information flowing between neurons responsible for receiving sensory stimuli and those involved in motor response [CITATION].', '0905.3887-2-45-2': 'To check the significance of the above result, we observe the membership of the set of connector hubs in the corresponding randomized network (Fig. S3).', '0905.3887-2-45-3': 'While this also shows many neurons from the lateral ganglion, unlike in the empirical network it has significant representation from other ganglia too (e.g., the retrovesicular ganglion).', '0905.3887-2-46-0': 'We have also carried out an analysis of the frequency of links between neurons having different roles, relative to the randomized network (Fig. S4).', '0905.3887-2-46-1': 'This allows us to compare the C. elegans neuronal network with other networks involved in (a) transportation and (b) information propagation.', '0905.3887-2-46-2': 'It has been shown that networks of class (b) shows significant under-representation of links between R1-R1, R5-R6 and R6-R6, whereas networks of class (a) exhibit over-representation of all three [CITATION].', '0905.3887-2-46-3': 'These patterns have been related to the occurrence of stringy periphery in class (a) and multi-star structures in class (b) [CITATION].', '0905.3887-2-46-4': 'In the case of C. elegans, R1-R1 does seem to be over-represented.', '0905.3887-2-46-5': 'However, R6-R6 shows very little over-representation, while both R5-R6 and R6-R5 show slight under-representation.', '0905.3887-2-46-6': 'This difference in the role-to-role connectivity pattern for the nematode nervous system with the networks in the above-mentioned two classes suggests that its structure is not exclusively characterized by either a stringy periphery or multiple stars.', '0905.3887-2-46-7': 'This assumes significance in light of recent work distinguishing information (or signalling) networks, such as the Internet and protein interactome, on the one hand, and transportation networks, such as metabolic and airport networks, on the other, into two classes [CITATION].', '0905.3887-2-46-8': 'Our results suggest that neuronal networks which have to process information, in addition to transferring signals, may constitute a different category from either of the above classes.', '0905.3887-2-47-0': '# Discussion', '0905.3887-2-48-0': 'In this paper, we have carried out a detailed analysis of the mesoscopic structure in the connection topology of the C. elegans neuronal network.', '0905.3887-2-48-1': 'Inferring the organizing principles underlying the network may give us an understanding of the way in which an organism makes sense of the external world.', '0905.3887-2-48-2': 'We have focused primarily on the existence of modules, i.e., groups of neurons having higher connection density among themselves than with neurons in other groups.', '0905.3887-2-48-3': 'The presence of such mesoscopic organization naturally prompts us to ask the reasons behind the evolution of these features in the network.', '0905.3887-2-49-0': 'In lower invertebrates like nematodes, the genome is the dominant factor which governs the development of the organism, including its nervous system.', '0905.3887-2-49-1': 'The neuronal network structure is formed early in the life-cycle of the organism, when most of the cells and their connections are configured permanently.', '0905.3887-2-49-2': 'Although external cues may play a role, the relative absence of individual variations in the network organization makes C. elegans an ideal system for studying how the system has evolved to optimize for various constraints, such as minimizing resource use and maximizing performance.', '0905.3887-2-50-0': 'There have been recent attempts at explaining neuronal position and structural layout of the network by using static constraints, such as wiring economy and communication path minimization.', '0905.3887-2-50-1': 'Although we find that membership of neurons in specific modules are correlated with their physical nearness, the empirical network is sub-optimal in terms of both the above-mentioned constraints.', '0905.3887-2-50-2': 'By comparing the system with other complex networks that have been either designed or have evolved for rapid transportation while being subject to wiring economy, we find that the C. elegans nervous system stands apart as a distinct class.', '0905.3887-2-50-3': 'This suggests that the principal function of neuronal networks, viz., the processing of information, distinguishes it from the other networks considered, and plays a vital role in governing its arrangement.', '0905.3887-2-50-4': 'Considering the importance of this constraint in ensuring the survival of an organism, it is natural that this should be key to the organizing principles underlying the design of the network.', '0905.3887-2-50-5': 'The intimate relation between function and structure of the nervous system is further brought out by our use of structural analysis to distinguish neurons that are critical for the survival of the organism.', '0905.3887-2-50-6': 'In addition to identifying neurons that have been already empirically implicated in different functions (which serve as a verification of our method), we also predict several neurons which can be potentially crucial for certain, as yet unidentified, functions.', '0905.3887-2-51-0': 'Biological systems are distinguished by the occurrence of discrete entities with a distinct function, which are often termed as functional modules [CITATION].', '0905.3887-2-51-1': 'For several networks that occur in the biological context (such as that of protein interactions), the components of structural modules are seen to be functionally related.', '0905.3887-2-51-2': 'This suggests that modules provide a framework for relating mesoscopic patterns in the connection topology to subsystems responsible for specific functions.', '0905.3887-2-51-3': 'Although there is no unique correspondence between the structural modules of C. elegans and the known functional circuits [CITATION], we use the overlap of the circuits with the modular membership of their constituent neurons to discover correlations between them.', '0905.3887-2-51-4': 'Our results reveal non-trivial association between circuits whose corresponding functions are closely connected, even when they do not share any common neurons.', '0905.3887-2-51-5': 'As such relations could not have been revealed by a micro-scale study which focuses on individual neurons and their connections, this result highlights one of the significant advantages of investigating the network at the mesoscopic level.', '0905.3887-2-52-0': 'When we compare the nervous system of C. elegans with the brains of higher organisms, we observe the modular organization of the latter to be more prominent [CITATION].', '0905.3887-2-52-1': 'For example, the network of cortical areas in the cat and macaque brains exhibit distinct modules [CITATION], with each module being identified with specific functions [CITATION].', '0905.3887-2-52-2': 'A possible reason for the relatively weak modular structure in the nematode could be due to the existence of extended processes for the neurons of C. elegans.', '0905.3887-2-52-3': 'Many of these span almost the entire body length, an effect that is enhanced by the approximately linear nature of the nematode body plan.', '0905.3887-2-52-4': 'As a result, connections are not constrained by the physical distance between soma of the neurons, as is mostly the case in mammalian brains.', '0905.3887-2-52-5': 'It is apparent that such constraints on the geographical distance spanned by links between nodes (viz., cost of wiring length) can give rise to clustering of connections among physically adjacent elements.', '0905.3887-2-52-6': 'In addition, the small nervous system of C. elegans, comprising only 302 neurons, lacks redundancy.', '0905.3887-2-52-7': 'Therefore, individual neurons may often have to perform a set of tasks which in higher organisms are performed by several different neurons.', '0905.3887-2-52-8': 'Thus, functional modularity is less prominent in the nematode, as some neurons belong to multiple behavioral circuits.', '0905.3887-2-53-0': 'Another principal distinction between the C. elegans nervous system and the brains of higher organisms such as human beings, is the relative high connectivity in the former (the connection density being [MATH]).', '0905.3887-2-53-1': 'By contrast, the connectance for human brain is around [MATH] [CITATION], which leads us to the question of how communication efficiency can remain high in such a sparsely connected network.', '0905.3887-2-53-2': 'It is possible that the more intricate hierarchical and modular structures seen in the brains of higher organisms is a response to the above problem.', '0905.3887-2-53-3': 'The fact that the rate at which the number of neurons [MATH] increase across species, is not matched by a corresponding increase in the number of links (which increases slower than [MATH]) implies the existence of constraints on the latter, which is a resource cost in addition to the earlier mentioned cost of wiring length.', '0905.3887-2-53-4': 'Note that, in the present work we have focused on a single level of modular decomposition of the nematode neuronal network.', '0905.3887-2-53-5': 'It is possible that the system may have multiple levels of hierarchically arranged inter-nested modules [CITATION].', '0905.3887-2-53-6': 'Investigating the existence of such organization in the C. elegans nervous system is a potential topic for future exploration.', '0905.3887-2-54-0': 'Networks provide the scaffolding for the computational architectures that mediate cognitive functions.', '0905.3887-2-54-1': 'The pattern of connections of a neuron (or a neuronal cluster) defines its functionality not just locally but also as an integrated part of the nervous system.', '0905.3887-2-54-2': 'This is because neurocognitive networks across the evolutionary tree consist of interconnected clusters of neurons that are simultaneously activated for generating a single or a series of cognitive outputs.', '0905.3887-2-54-3': 'However, while some of the neurons (or clusters of neurons) are essential for the relevant outcome, others are ancillary.', '0905.3887-2-54-4': 'Thus, they work in a collaborative mode but are not interchangeable, each displaying relative specializations for separate behavioral components.', '0905.3887-2-54-5': 'Since the most prominent neuronal pathways are those that interconnect components of functional circuits, our analysis validates the intrinsic connection between network structure and the functions of the nervous system.', '0905.3887-2-54-6': 'The concept that behavioral consequences of damaging a region of the brain will reflect the disruption of the underlying network architecture may be intuitive but is particularly important when one considers brain dysfunction through neurodegeneration [CITATION], where atrophy is seen to propagate preferentially through networks of functionally related neurons [CITATION].', '0905.3887-2-54-7': 'While the idea that damage in one part of the brain can affect other areas connected to that region is not new, our work on the mesoscopic network organization may be extended to look at disease progression through the network, beyond the current focus on the pathology within individual cells.', '0905.3887-2-55-0': '# Materials and Methods', '0905.3887-2-56-0': 'Connectivity Data: We have used information about the network connectivity, positions of neurons and the lineage distance between cells for the C. elegans nervous system, from the database published in Ref. [CITATION] and available from the online Atlas of C. elegans anatomy (www.wormatlas.org).', '0905.3887-2-56-1': 'This is an updated and revised version of the wiring data originally published in Ref. [CITATION].', '0905.3887-2-56-2': 'The connectivity between neurons and the positions of the neuron cell bodies along the longitudinal axis of the worm is reconstructed based on serial section electromicrography.', '0905.3887-2-56-3': 'Note that, the new database adds or updates about 3000 connections to the previous version.', '0905.3887-2-56-4': 'The lineage data indicates the relatedness between every pair of neurons in terms of distances in the embryonic and post-embryonic lineage trees.', '0905.3887-2-56-5': 'This is measured by identifying the last common progenitor cells of the two neurons, and then counting the number of cell divisions from this common ancestor.', '0905.3887-2-56-6': 'Each cell division adds a single unit to the total lineage distance, with the initial division from the common progenitor counted only once.', '0905.3887-2-57-0': 'Modularity (Q): To decompose a given network into modules (where a module or community is defined as a subnetwork having a higher density of connections relative to the entire network) we use a method introduced in Ref. [CITATION].', '0905.3887-2-57-1': 'We compute a quantitative measure of modularity, [MATH], for a given partitioning of the network into several communities, [EQUATION]', '0905.3887-2-57-2': 'Here, [MATH] is the adjacency matrix ([MATH] is 1, if neurons [MATH], [MATH] are connected, and [MATH], otherwise).', '0905.3887-2-57-3': 'The degree of each node [MATH] is given by [MATH].', '0905.3887-2-57-4': '[MATH] is the total number of links in the network, [MATH] is the Kronecker delta ([MATH], if [MATH], and [MATH], otherwise), and [MATH] is the label of the community to which vertex [MATH] is assigned.', '0905.3887-2-57-5': 'In the case of directed and weighted network, the above measure can be generalized as [EQUATION] where, [MATH] is the sum of weights of all links in the network ([MATH] is the weight of the link from neuron [MATH] to neuron [MATH]), and the weighted in-degree and out-degree of node [MATH] are given by [MATH] and [MATH], respectively.', '0905.3887-2-58-0': 'The optimal partitioning of the network is the one which maximizes the modularity measure [MATH] (or [MATH]).', '0905.3887-2-58-1': 'We obtain this using a generalization of the spectral method [CITATION].', '0905.3887-2-58-2': 'We first define a modularity matrix [MATH], [EQUATION]', '0905.3887-2-58-3': 'To split the network into modules, the eigenvectors corresponding to the largest positive eigenvalue of the symmetric matrix ([MATH]) is calculated and the communities are assigned based on the sign of the elements of the eigenvector.', '0905.3887-2-58-4': 'This divides the network into two parts, which is refined further by exchanging the module membership of each node in turn if it results in an increase in the modularity.', '0905.3887-2-58-5': 'The process is then repeated by splitting each of the two divisions into further subdivisions.', '0905.3887-2-58-6': 'This recursive bisection of the network is carried out until no further increase of [MATH] is possible.', '0905.3887-2-59-0': 'Modular spectra: We analyze different neuronal groups, defined in terms of functions, anatomy (e.g., ganglia), etc., by proposing a decomposition in terms of the overlap of their constituent neurons with the different modules.', '0905.3887-2-59-1': 'Let the set of all neurons be optimally partitioned into [MATH] modules.', '0905.3887-2-59-2': 'We then define an overlap matrix, [MATH], where the rows correspond to the different neuronal groups, and the columns correspond to the different modules.', '0905.3887-2-59-3': 'An element of this overlap matrix, [MATH], is the number of neurons in group [MATH] that are from the module [MATH].', '0905.3887-2-59-4': 'Then, the decomposition of the [MATH]-th group in the abstract [MATH]-dimensional basis space formed by the modules is [MATH], where [MATH] is the total number of neurons in the [MATH]-th group.', '0905.3887-2-59-5': 'The distance between two groups [MATH] and [MATH] in this "modular" space is defined as [EQUATION]', '0905.3887-2-59-6': 'Thus, this measure can be used as a metric for closeness between different neuronal groups.', '0905.3887-2-60-0': 'Decomposition of the network into SCC, IN and OUT components: In order to determine the Strongly Connected Component (SCC) of the network, we first calculate the graph distance matrix containing the shortest directed path between every pair of nodes in the network.', '0905.3887-2-60-1': 'A finite path length from neuron [MATH] to neuron [MATH] indicates the existence of a connected path from one neuron to the other.', '0905.3887-2-60-2': 'By grouping together all nodes which have finite path length with all other members of the group, we determine the SCC.', '0905.3887-2-60-3': "In general, one can use Tarjan's algorithm for SCC determination [CITATION].", '0905.3887-2-60-4': 'Next, we identify all neurons not belonging to SCC but which can be reached via a directed path starting from a node in SCC.', '0905.3887-2-60-5': 'This constitutes the OUT component of the network.', '0905.3887-2-60-6': 'Similarly, the group of neurons which do not belong to SCC but which have finite directed path length to a member of SCC, constitutes the set of IN neurons.', '0905.3887-2-61-0': 'Betweenness centrality (BC): To measure the importance of a node in facilitating communication across a network, we consider how frequently a node is used to convey information from any part of the network to any other part using the shortest available path.', '0905.3887-2-61-1': 'The betweenness centrality of a node [MATH] is defined as the fraction of shortest paths between the pairs of all other nodes in the network that pass through [MATH] [CITATION].', '0905.3887-2-61-2': 'If the total number of shortest paths between nodes [MATH] and [MATH] is [MATH], of which [MATH] paths pass through node [MATH], then the betweenness centrality of node [MATH] is [EQUATION]', '0905.3887-2-61-3': 'Communication Efficiency (E): To measure the speed of information transfer over the network, one can define the efficiency [MATH] of communication between vertices [MATH] and [MATH] to be inversely proportional to the shortest graph distance [MATH].', '0905.3887-2-61-4': 'Therefore, the efficiency of communication across the whole network is [EQUATION]', '0905.3887-2-61-5': 'This is the harmonic mean of graph distances between all pairs, which does not diverge even when the network is disconnected [CITATION].', '0905.3887-2-62-0': 'Network Randomization: An ensemble of randomized versions of the empirical network is constructed keeping the in-degree and out-degree of each node unchanged.', '0905.3887-2-62-1': 'Each such network is created by rewiring randomly selected pairs of directed edges, [MATH] and [MATH], such that, in the randomized network, the corresponding directed edges are [MATH] and [MATH].', '0905.3887-2-62-2': 'However, if these new links already exist in the empirical network, this step is aborted and a new pair of edges are chosen in order to prevent the occurrence of multiple edges [CITATION].', '0905.3887-2-62-3': 'The above procedure is repeated [MATH] times for a single realization of the randomized network.', '0905.3887-2-62-4': 'In order to compare the properties of the empirical network with its randomized version, an ensemble of [MATH] realizations is considered.', '0905.3887-2-63-0': 'Determining the intra- and inter-modular role of a neuron: The role played by each neuron in terms of its connectivity within its own module and in the entire network is determined according to two properties [CITATION]: (i) the relative within module degree, [MATH], and (ii) the participation coefficient, [MATH].', '0905.3887-2-64-0': 'The [MATH]-score of the within module degree distinguishes nodes that are hubs of their communities from those that are non-hubs.', '0905.3887-2-64-1': 'It is defined as [EQUATION] where [MATH] is the number of links of node [MATH] to other nodes in its community [MATH] and [MATH] are taken over all nodes in module [MATH].', '0905.3887-2-64-2': 'The within-community degree [MATH]-score measures how well-connected node [MATH] is to other nodes in the community.', '0905.3887-2-65-0': 'The nodes are also distinguished based on their connectivity profile over the entire network, in particular, their connections to nodes in other communities.', '0905.3887-2-65-1': 'Two nodes with same within module degree [MATH]-score can play different roles, if one of them has significantly higher inter-modular connections compared to the other.', '0905.3887-2-65-2': 'This is measured by the participation coefficient [MATH] of node [MATH], defined as [EQUATION] where [MATH] is the number of links from node [MATH] to other nodes in its community [MATH] and [MATH] is the total degree of node [MATH].', '0905.3887-2-65-3': 'Therefore, the participation coefficient of a node is close to 1, if its links are uniformly distributed among all the communities, and is 0, if all its links are within its own community.', '0905.3887-2-66-0': 'Normalized mutual information ([MATH]): To measure the overlap of the membership of nodes in a ganglion with their membership in a specific module, we use the normalized mutual information measure [CITATION].', '0905.3887-2-66-1': 'First, we define a overlap matrix, [MATH], where the rows correspond to the different ganglia, and the columns correspond to the modules obtained by optimal partitioning of the network.', '0905.3887-2-66-2': 'An element of this overlap matrix, [MATH], is the number of neurons in the ganglion [MATH] that appear in the module [MATH].', '0905.3887-2-66-3': 'An information-theoretic measure of similarity between the partitions can then be defined as [EQUATION] where [MATH] and [MATH] are the numbers of ganglia and modules respectively.', '0905.3887-2-66-4': 'The sum over row [MATH] of matrix [MATH] is denoted by [MATH] and the sum over column [MATH] is denoted by [MATH].', '0905.3887-2-66-5': 'If the modules are identical to the ganglia, then [MATH] takes its maximum value of 1.', '0905.3887-2-66-6': 'On the other hand, if the modular partitioning is independent of the division of the network into ganglia, [MATH].'}

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['0905.3887-1-40-3', '0905.3887-2-40-3']]

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0905.3887

1305.1953

{'1305.1953-1-0-0': 'Localized Majorana fermions emerge in many topologically ordered systems and exhibit exchange statistics of Ising anyons.', '1305.1953-1-0-1': 'This enables noise-resistant implementation of a limited set of operations by braiding and fusing Majorana fermions.', '1305.1953-1-0-2': 'Unfortunately, these operations are incapable of implementing universal quantum computation.', '1305.1953-1-0-3': 'We show that, regardless of these limitations, Majorana fermions could be used to demonstrate non-locality (correlations incompatible with a local hidden variable theory) in experiments using only topologically protected operations.', '1305.1953-1-0-4': 'We also demonstrate that our proposal is optimal in terms of resources, with 10 Majorana fermions shown to be both necessary and sufficient for demonstrating bipartite non-locality.', '1305.1953-1-0-5': 'Furthermore, we identify severe restrictions on the possibility of tripartite non-locality.', '1305.1953-1-0-6': 'We comment on the potential of such entangled systems to be used in quantum information protocols.', '1305.1953-1-1-0': '# Introduction', '1305.1953-1-2-0': 'Fermions that are their own anti-particle are known as Majorana, as opposed to Dirac, fermions.', '1305.1953-1-2-1': 'While no fundamental particles in the Standard Model are Majorana fermions, they frequently emerge as localised quasi-particles in models of condensed matter systems [CITATION].', '1305.1953-1-2-2': 'Recent years have seen a race to experimentally confirm their existence, with some evidence already found [CITATION].', '1305.1953-1-2-3': 'Research into these systems is driven, at least partially, by their potential applications in topological quantum computing.', '1305.1953-1-2-4': 'For localised and well-separated Majorana fermions with zero energy, the system is protected from noise effects that could otherwise prove devastating in quantum computers.', '1305.1953-1-3-0': 'Suitable Majorana fermions emerge in many two-dimensional (2D) systems including the Kitaev honeycomb lattice [CITATION], fractional quantum hall systems [CITATION], topological insulators [CITATION], and a variety of other systems [CITATION].', '1305.1953-1-3-1': 'Adiabatically exchanging these fermions, and so braiding their world-lines, gives rise to the non-abelian exchange statistics of Ising anyons.', '1305.1953-1-3-2': 'Majorana fermions can also emerge as edge modes in one-dimensional (1D) systems [CITATION], such as the Kiteav wire [CITATION].', '1305.1953-1-3-3': 'While braiding is not a meaningful concept in strict 1D systems, networks of wires also enable braiding with Ising statistics [CITATION].', '1305.1953-1-3-4': 'The unitary evolution from braiding is geometric in origin, and so robust against small experimental imperfections.', '1305.1953-1-3-5': 'However, these topologically protected braiding operations are not computationally powerful enough for universal quantum computing.', '1305.1953-1-3-6': 'Indeed, any free fermionic system can be efficiently classically simulated [CITATION].', '1305.1953-1-3-7': 'Access to some non-topological operations - which may be noisy, but not too noisy [CITATION] - can be used to promote the system to full universality [CITATION].', '1305.1953-1-3-8': 'However, here we are interested in understanding the purely topological protected capabilities of Majorana fermions, and in particular their capacity for demonstrating non-locality [CITATION].', '1305.1953-1-4-0': 'Non-locality is the inability for a local hidden variable (LHV) theory to reproduce the correlations of space-like separated measurements.', '1305.1953-1-4-1': 'Until now, non-locality has only been investigated in more exotic topological systems by Brennen et al. [CITATION].', '1305.1953-1-4-2': 'Having only considered systems capable of universal quantum computation, Brennen et al. concluded their work saying, "it is intriguing to ask whether one could find intermediate anyonic theories which have the power to generate Bell violating states by topologically protected gates, but are not universal for topological quantum computation".', '1305.1953-1-4-3': 'We resolve this mystery by showing that Majorana fermions, and equivalently Ising anyons, could be used in an experiment demonstrating the non-locality of quantum mechanics.', '1305.1953-1-4-4': 'It appears that the standard and ubiquitous CHSH inequality cannot be violated with only topological operations, and the non-topological resources sufficient for a CHSH violation have been investigated [CITATION].', '1305.1953-1-4-5': 'We turn instead to a non-local experiment proposed by Cabello [CITATION] where 2 parties each select from 3 possible measurements, with each measurement producing 3 bits of classical information as outcomes.', '1305.1953-1-4-6': 'This non-locality proposal was built on the idea of the Mermin-Peres "magic square", which was originally used to show the contextuality of quantum mechanics [CITATION].', '1305.1953-1-4-7': 'We find that these experiments could be implemented with each of two parties holding 5 Majorana fermions, and present a Bell inequality based on the magic square.', '1305.1953-1-4-8': 'We also present no-go results showing for two parties each holding 4 Majoranas, all experimental statistics can be produced by a local hidden variable theory.', '1305.1953-1-4-9': 'If two parties share an unequal number of Majoranas, say [MATH] and [MATH] with [MATH], we find that the resource is locally equivalent to both parties holding just [MATH] Majorana fermions.', '1305.1953-1-4-10': 'Hence, 10 Majorana fermions are both necessary and sufficient for the phenomenon of bipartite non-locality to be demonstrated.', '1305.1953-1-4-11': 'Furthermore, we find that the correlations required to demonstrate the three-party Greenberger-Horne-Zeilinger (GHZ) paradox [CITATION] cannot be implemented with any number of Majoranas.', '1305.1953-1-4-12': 'This indicates that our proposal is the simplest possible non-locality experiment with Majorana fermions.', '1305.1953-1-5-0': 'Bell proposed experiments initially to falsify alternative theories that claim the world to be local in Nature.', '1305.1953-1-5-1': 'Now these Bell-type non-locality experiments are also known to have practical applications.', '1305.1953-1-5-2': 'Such experiments can form the basis of quantum cryptography when the devices used are faulty or even untrusted [CITATION].', '1305.1953-1-5-3': 'Typically, such proposals are envisaged for photonic systems, since they provide an easier means of accomplishing space-like separation of measurement events (a requirement for a non-local experiment).', '1305.1953-1-5-4': "Polarisation-entangled photons have, for many years, been used to test Bell's proposal, proving successful if one ignores the detector loophole [CITATION].", '1305.1953-1-5-5': 'Experiments with trapped ions have efficient enough measurements to avoid the detector loophole, but do not satisfy space-like measurement separation [CITATION].', '1305.1953-1-5-6': 'Designing better experiments, hopefully closing all loopholes, is an active area of research [CITATION].', '1305.1953-1-5-7': 'We shall not argue that our topological proposal is more promising than the aforementioned approaches.', '1305.1953-1-5-8': 'Rather, we take the first step by showing theoretical feasibility under ideal conditions.', '1305.1953-1-5-9': 'Implicit in our approach is the assumption that the physical system can be shared between two parties over sufficient distances that measurements are space-like separated; this is a similar technical difficulty faced by ion trap designs [CITATION].', '1305.1953-1-6-0': '# Majorana fermions and braiding', '1305.1953-1-7-0': '## Majorana fermions', '1305.1953-1-8-0': 'We begin with a review of Majorana fermions and their available dynamics, largely following Refs. [CITATION].', '1305.1953-1-8-1': 'Generally, Majorana fermions are described by a set of Hermitian operators [MATH] satisfying [EQUATION] and [MATH] for all [MATH], acting on the physical Hilbert space [MATH] constituting a direct sum between the even and odd parity sectors.', '1305.1953-1-8-2': 'The algebra of physical operators [MATH] is spanned by products of an even number of Majorana fermion operators.', '1305.1953-1-8-3': 'Taking two such fermions and clockwise braiding their world lines results in a unitary [MATH]', '1305.1953-1-9-0': 'that maps the operators as [EQUATION] for [MATH].', '1305.1953-1-9-1': 'Composing these braid operations results in a permutation [MATH], with possible phase change [MATH], so that [MATH].', '1305.1953-1-9-2': 'The phases are constrained so that the global parity is preserved, leaving [MATH] unchanged.', '1305.1953-1-10-0': 'Such braidings are a special case of unitary transformations [MATH] acting on [MATH] that reflect linear mode transformation [EQUATION] for [MATH].', '1305.1953-1-10-1': 'Such unitary transformations are the ones commonly considered in the context of fermionic linear optics.', '1305.1953-1-10-2': 'All states encountered in this work are Gaussian fermionic states [CITATION].', '1305.1953-1-10-3': 'They are entirely described by their anti-symmetric covariance matrix [MATH], [MATH], which has entries [MATH].', '1305.1953-1-11-0': '## Stabilizer language', '1305.1953-1-12-0': 'We primarily describe quantum states in the Heisenberg picture by specifying a sufficient number of eigenvalue equations.', '1305.1953-1-12-1': 'We say an operator [MATH] stabilises a state vector [MATH] if [EQUATION]', '1305.1953-1-12-2': 'We assume that initialisation of the system prepares a state vector [MATH] stabilised by [MATH] for all [MATH].', '1305.1953-1-13-0': 'Therefore, any state prepared from the initialisation [MATH] by braiding will be stabilised by [MATH] for some permutation [MATH].', '1305.1953-1-13-1': 'Note that, since permutations are one-to-one mappings, if [MATH], then [MATH].', '1305.1953-1-13-2': 'Products of stabilizers are again stabilizers, and so [MATH] generate a group [MATH] associated with the state.', '1305.1953-1-13-3': 'Similarly, topologically protected measurements, also called charge measurements, are those of the form [MATH].', '1305.1953-1-13-4': 'Throughout, we say a state is accessible if and only if it can be prepared with the above described topological operations.', '1305.1953-1-13-5': 'Again, states obtained by performing such measurements on Gaussian states are Gaussian.', '1305.1953-1-14-0': 'Collective charge measurements, such as of [MATH], cannot be measured non-destructively and in a topological manner.', '1305.1953-1-14-1': 'The collective charge observable can, however, be inferred by measuring [MATH] and [MATH] and multiplying the outcomes.', '1305.1953-1-14-2': 'However, such a process is destructive in that it will always disentangle these Majoranas from all other systems.', '1305.1953-1-14-3': 'We labour this point because the capacity to make non-destructive charge measurements increases the computational power of the system [CITATION] beyond that assumed in our later no-go theorems.', '1305.1953-1-15-0': '## Anyonic formalism', '1305.1953-1-16-0': 'We shall restate some of the above in the anyonic formalism, which may be of benefit to some readers.', '1305.1953-1-16-1': 'Using, [MATH] to denote an Ising anyon, [MATH] for another fermion, and [MATH] for the vacuum, we have the fusion channel [MATH].', '1305.1953-1-16-2': 'Fusing Ising anyons [MATH] and [MATH] is equivalent to measuring [MATH], where producing a [MATH] particle is equivalent to the [MATH] measurement outcome (eigenvalue) and producing the vacuum [MATH] outcome denotes the [MATH] eigenvalue.', '1305.1953-1-16-3': 'Conversely, if we begin with a vacuum and create [MATH] pairs of Ising anyons, we have the initialisation state described above if anyons from the [MATH] pair are labeled as [MATH] and [MATH].', '1305.1953-1-17-0': '## Entanglement properties', '1305.1953-1-18-0': 'We proceed by identifying the equivalence classes of entangled states that are accessible under the operations described above.', '1305.1953-1-18-1': 'Consider two parties, Alice and Bob, holding respective sets of Majorana fermions [MATH] and [MATH].', '1305.1953-1-18-2': 'Clearly, any state stabilized by [MATH] can be locally prepared by Alice for [MATH] labeling a pair of Majorana fermions.', '1305.1953-1-18-3': 'A similar statement holds for Bob, and so the interesting stabilizers have [MATH] and [MATH] with each party holding one half of a pair of Majorana fermions (henceforth referred to as Majorana pairs).', '1305.1953-1-18-4': 'Assuming they hold [MATH] such pairs, they can always locally braid such that the state is stabilized by [MATH] for all [MATH].', '1305.1953-1-18-5': 'We see that the entanglement is entirely captured by the number of such Majorana pairs shared by Alice and Bob, and throughout we assume this canonical form for the stabilizers.', '1305.1953-1-18-6': 'For notational simplicity, it is beneficial to use [MATH] and [MATH], so Majorana pairs are stabilized by [MATH].', '1305.1953-1-19-0': 'Crucially, the number of Majorana pairs is distinct from the number of Bell pairs that give rise to useful entanglement.', '1305.1953-1-19-1': 'To investigate the useful correlations between Alice and Bob, we must consider how such pairs respond to measurements.', '1305.1953-1-19-2': 'Consider a pair of stabilizers [MATH] and [MATH] for [MATH], the state is also an eigenstate of their product [MATH] which using anti-commutation of fermions equals [MATH].', '1305.1953-1-19-3': 'The factors [MATH] and [MATH] are locally measurable, and so their outcomes must be anti-correlated.', '1305.1953-1-19-4': 'Hence, the state has the flavor of a singlet state.', '1305.1953-1-20-0': 'To make measurements Alice and Bob must share at least two Majorana pairs, but two alone is trivial since there is only one possible local measurement and with only one measurement we cannot construct a test of non-locality.', '1305.1953-1-20-1': 'With three Majorana pairs, Alice has three possible measurement observables that mutually anti-commute and are isomorphic to the Pauli operators.', '1305.1953-1-20-2': 'An encoding is a collection of maps [MATH], identifying the set of fermionic modes with qubit or spin systems.', '1305.1953-1-20-3': 'The identification of products of Majorana fermions with Pauli operators of an associated qubit system can be taken as [EQUATION]', '1305.1953-1-20-4': 'From the anti-commutation relations of the Majorana fermions, one can readily verify that these operators generate the Pauli group of a single qubit.', '1305.1953-1-20-5': 'Bob similarly has measurement options isomorphic to the Pauli operators.', '1305.1953-1-20-6': 'The correlations between Alice and Bob resulting from the above measurements on three Majorana pairs match those of Pauli measurements on the two-qubit singlet state (for Alice and Bob each having a single qubit).', '1305.1953-1-20-7': 'Hence, three Majorana pairs can be said to reproduce the entanglement of a Bell pair.', '1305.1953-1-20-8': 'However, it is well-known that the measurement statistics under the operations allowed here can be reproduced by a local hidden variable (LHV) theory.', '1305.1953-1-20-9': 'Later we present a strengthened proof that even four Majorana pairs are incapable of demonstrating non-locality.', '1305.1953-1-21-0': '# Non-locality', '1305.1953-1-22-0': 'Before proceeding we refine the concepts of non-locality.', '1305.1953-1-22-1': 'Assume Alice and Bob hold some quantum state [MATH] and are able to freely choose from a set of measurements [MATH] and [MATH] respectively, where [MATH] denotes the set of different kinds of measurement settings.', '1305.1953-1-22-2': 'The measurements will have possible outcomes that we denote [MATH] for a suitable [MATH] that occur with probability [EQUATION] where [MATH]) is the positive-operator valued measure for setting [MATH]) and outcome [MATH]).', '1305.1953-1-22-3': 'There are many different experiments that can achieve the same probability distributions, and each of these is a realisation of [MATH].', '1305.1953-1-22-4': 'Conversely, we say a probability distribution [MATH] is quantum if there exists a choice of measurements and a quantum state [MATH] that realises [MATH].', '1305.1953-1-23-0': 'When are these observations a proof of non-locality?', '1305.1953-1-23-1': 'Imagine that Alice and Bob are space-like separated and are allowed to make their choice of measurement freely, they now want to know whether any classical model can produce these observations.', '1305.1953-1-23-2': "Since they cannot communicate, this classical model has some pre-determined instructions that given Alice and Bob's choice of measurement will output some value.", '1305.1953-1-23-3': 'Such an instruction set is called a LHV theory.', '1305.1953-1-23-4': 'An arbitrary hidden variable [MATH] takes values in some space [MATH] equipped with a probability measure, and which determines local probability functions [EQUATION]', '1305.1953-1-23-5': 'The probability of obtaining the outcome pair [MATH] given that [MATH] have been chosen by Alice and Bob [EQUATION]', '1305.1953-1-23-6': 'A probability distribution is local (also called LHV correlations) if and only if there exists a decomposition of the above form, and we denote this as [MATH].', '1305.1953-1-23-7': 'If no such local model exists, [MATH], we say the probability distribution is non-local.', '1305.1953-1-23-8': 'Furthermore, any experiment realizing a non-local probability distribution is a non-locality experiment.', '1305.1953-1-24-0': 'Typically, we confirm non-locality by checking whether a probability distribution violates a Bell inequality.', '1305.1953-1-24-1': 'Let us fix the number of the measurement settings and outcomes, and denote the entire set of possible probability distributions by [MATH].', '1305.1953-1-24-2': 'The Bell inequalities follow by first defining a real-valued linear function, [MATH], which is sometimes called a non-local game [CITATION], the respective parties are referred to as players and the action taken are strategies.', '1305.1953-1-24-3': 'The non-local game as the term is used here is described by a real-valued function, [MATH] such that [EQUATION]', '1305.1953-1-24-4': 'For any such non-local game, there exists a Bell inequality that holds for all local [MATH] [EQUATION]', '1305.1953-1-24-5': 'The above is true simply by definition.', '1305.1953-1-24-6': 'However, for any [MATH] and [MATH] where [MATH] we can conclude [MATH] is non-local.', '1305.1953-1-25-0': 'Let us consider a particular non-local game [MATH], and its classical limit [MATH].', '1305.1953-1-25-1': 'Since non-local games are linear functions and [MATH] is a convex set, the classical limit can always be achieved by an extremal point in [MATH].', '1305.1953-1-25-2': 'That is, the value of [MATH] can always be achieved by a probability distribution of the form [EQUATION] where Alice and Bob deterministically assign measurement outcomes.', '1305.1953-1-25-3': 'There are only finitely many such distributions, so this greatly simplifies the evaluation of [MATH].', '1305.1953-1-26-0': '# Magic squares with Majoranas', '1305.1953-1-27-0': 'To demonstrate that these Majorana fermions exhibit non-locality we adapt a specific proof of "non-locality without inequalities" first devised by Cabello [CITATION].', '1305.1953-1-27-1': 'There exist other proofs of non-locality for qubits without the use of a Bell inequality, such as those by Hardy [CITATION] and the GHZ paradox [CITATION].', '1305.1953-1-27-2': "Crucially, Hardy's argument does not work for maximally entangled states and so it cannot be applied.", '1305.1953-1-27-3': 'Furthermore we show later that the GHZ argument does not apply here either.', '1305.1953-1-27-4': "Cabello's proof works for two Bell states, a four-qubit state, shared by Alice and Bob each of which can make a measurement on two qubits of this state.", '1305.1953-1-27-5': 'The analysis shares much of its character with the proof that quantum mechanics is non-contextual derived by Peres and Mermin, often referred to as the "Magic Square Game".', '1305.1953-1-27-6': "We will follow the simplification of Cabello's approach as presented by Aravind [CITATION] so that the result may be of more general appeal, but optimally tailored to systems of Majorana fermions.", '1305.1953-1-28-0': '## The set-up', '1305.1953-1-29-0': 'We now describe the set-up.', '1305.1953-1-29-1': 'We assume that Alice and Bob share five Majorana pairs, prepared in the canonical form outlined earlier.', '1305.1953-1-30-0': 'Each party then makes a choice of three different measurement settings.', '1305.1953-1-30-1': 'Each measurement setting relates to three measurements involving a subset of the Majorana modes held by the respective party.', '1305.1953-1-30-2': 'Hence, for each measurement setting, a string of three values of [MATH] is obtained, so [MATH].', '1305.1953-1-30-3': 'For Alice we label the choice of measurement as [MATH] for [MATH] and for Bob the choice of measurement is written as [MATH]) for [MATH].', '1305.1953-1-30-4': 'Alice and Bob then obtain the output strings [MATH], [MATH], respectively, with [MATH] labeling the choice of measurements [MATH] and [MATH].', '1305.1953-1-30-5': 'The measurements are fixed by the columns (for Alice) and rows (for Bob), of square tables (so-called "magic squares"), as shown in Table [REF].', '1305.1953-1-30-6': "In these magic squares, Alice and Bob's respective square tables contain, up-to a phase, the same measurements performed by the respective parties.", '1305.1953-1-30-7': "It can easily be verified that an element of Alice's table multiplied with the same element of Bob's table (i.e. the same [MATH] row and [MATH] column in each table) gives a stabilizer of their shared quantum state.", '1305.1953-1-30-8': 'Formally, this enforces that [MATH] for all [MATH].', '1305.1953-1-30-9': 'Furthermore, the product of observables in any row of both tables gives [MATH], and so for any measurement setting [MATH].', '1305.1953-1-30-10': 'Whereas the product of observables in any column of both tables gives [MATH], and so [MATH].', '1305.1953-1-31-0': 'Aside from these constraints, the measurement outcomes are entirely random and so the experiment realises [EQUATION] and in the next section we confirm this to be non-local and robust against some experimental noise.', '1305.1953-1-32-0': '## The magic square game without inequalities', '1305.1953-1-33-0': 'Earlier, we saw that for any non-local game the best local strategy can be achieved by Alice and Bob deterministically assigning measurement outcomes.', '1305.1953-1-33-1': 'In the problem at hand, it is convenient to describe these local probability distributions by a [MATH] matrix with entries in [MATH].', '1305.1953-1-33-2': "We use [MATH] to denote Alice's table, which has entries such that [EQUATION]", '1305.1953-1-33-3': 'Whereas for Bob, we take [EQUATION]', '1305.1953-1-33-4': 'Clearly, there is a one-to-one correspondence between these tables and deterministic probability distributions.', '1305.1953-1-33-5': 'Note also the slight difference in definitions for Alice and Bob.', '1305.1953-1-33-6': 'For Alice, a single measurement setting will output a column of her table, whereas for Bob, a single measurement setting will output a row of his table.', '1305.1953-1-33-7': 'This convention is justified because the quantum correlations satisfy [MATH] for all [MATH], and so if the LHV theory replicates this correlation we have that [EQUATION]', '1305.1953-1-33-8': 'Therefore, Alice and Bob must share identical tables to reproduce this feature of the quantum correlations.', '1305.1953-1-33-9': 'However, the quantum correlations have an additional feature, they satisfy parity constraints.', '1305.1953-1-33-10': "Alice's parity constraint entails that for all [MATH], we have [MATH], and hence the parity of the entire table is [MATH].", '1305.1953-1-33-11': "In contrast, Bob's parity constraint entails that for all [MATH], we have [MATH], and so [MATH].", '1305.1953-1-33-12': 'We conclude that Alice and Bob cannot hold identical classical tables and also satisfy all the parity constraints, and so they can never perfectly reproduce the quantum correlations.', '1305.1953-1-33-13': 'Since quantum observables do not commute, finding the parity of the table of operators can change depending on the order we multiply the entries.', '1305.1953-1-34-0': '## The magic square game with Bell inequalities', '1305.1953-1-35-0': 'We have seen that classical experiments can never reproduce the quantum correlations demonstrated in the previous section.', '1305.1953-1-35-1': 'However, we are interested in what level of imperfection can be tolerated by any quantum experiment while still violating locality.', '1305.1953-1-35-2': 'To quantify this we must specify a particular non-local game, see Eq. ([REF]), by specifying a function [MATH].', '1305.1953-1-35-3': 'The goal is to have a larger value when the correlations match those of the quantum predications, and smaller when they fail.', '1305.1953-1-35-4': 'The simplest choice is a function that takes two values, and it is conventional to take these as [MATH], and so we have [EQUATION]', '1305.1953-1-35-5': 'This non-local game we call the magic square game.', '1305.1953-1-35-6': 'It is easy to verify that using majorana fermions, which realise the probability distribution [MATH] (see Eq. [REF]), we find [MATH].', '1305.1953-1-36-0': 'We are now in a position to identify [MATH], the maximum classical value.', '1305.1953-1-36-1': 'Alice and Bob could use identical tables, but they then contravene some parity constraints, and we find this results in at most [MATH].', '1305.1953-1-36-2': 'However, the maximum is achieved if Alice and Bob use classical tables that differ only in a single entry and satisfy all the parity constraints, such as in Table [REF].', '1305.1953-1-36-3': 'This yields [MATH] since for 8 of the measurement settings we acquire a contribution of [MATH], but for one setting we have [MATH].', '1305.1953-1-36-4': 'Hence, any imperfect experiment with Majorana fermions that attains [MATH] is sufficient to demonstrate non-locality.', '1305.1953-1-37-0': 'We also comment on how the imperfect quantum setting can, sometimes, economically be described.', '1305.1953-1-37-1': 'For perfect operations of the type considered here, the covariance matrix [MATH] will only contain entries contained in [MATH] and satisfies [MATH].', '1305.1953-1-37-2': 'If errors and imperfections are present, the entries of [MATH] will also attain values different from those, while it is still true that [EQUATION]', '1305.1953-1-37-3': 'Under braiding transformations of the form ([REF]), covariance matrices transform as congruences [MATH] with orthogonal matrices [MATH], which can easily be kept track of.', '1305.1953-1-37-4': 'The statistics of measurements can still be determined based on the covariance matrix only, even if errors are taken into account in the preparation step.', '1305.1953-1-38-0': '# Too few Majorana pairs', '1305.1953-1-39-0': '## Setting', '1305.1953-1-40-0': 'Earlier we saw that if Alice and Bob share three Majorana pairs, the available measurements are isomorphic to Pauli measurements on a single Bell pair.', '1305.1953-1-40-1': 'It is well-known folklore that such a system can be modeled by a local-hidden variable theory.', '1305.1953-1-40-2': 'Here we present a stronger argument covering up to 4 Majorana pairs.', '1305.1953-1-40-3': 'This shows that five or more Majorana pairs are necessary, as well as sufficient, to demonstrate non-locality.', '1305.1953-1-40-4': 'We begin by characterizing the full set of possible measurements.', '1305.1953-1-40-5': 'For Alice, there are three pairs of commuting measurements she can perform [EQUATION]', '1305.1953-1-40-6': 'If Alice measures a pair of observables [MATH], she get two random bits, which we denote [MATH].', '1305.1953-1-40-7': 'Similar measurement sets, labeled [MATH], are available to Bob, with [MATH] replacing [MATH].', '1305.1953-1-40-8': 'If Bob measures the analogous set, so [MATH], he gets perfectly anticorrelated outcomes, such that [MATH] and [MATH].', '1305.1953-1-40-9': 'Next we consider when Bob measures a different set of operators, so [MATH].', '1305.1953-1-40-10': 'We need to determine which products of measurements correspond to stabilizers of the 4 Majorana pairs.', '1305.1953-1-40-11': 'We observe that for Alice and any measurement set, the product of the observables is [MATH], and similarly for Bob the product is [MATH].', '1305.1953-1-40-12': 'Since the observables are matching and contain an even number of fermionic operators, these observables will have correlated outcomes.', '1305.1953-1-40-13': 'Formally, these correlations entail that [MATH] for all measurement settings [MATH] and [MATH].', '1305.1953-1-40-14': 'However, when [MATH], this is the only correlation and otherwise the measurement outcomes are entirely random.', '1305.1953-1-40-15': 'From these observations we can deduce that the outcome probabilities, as a function of measurement settings [MATH] and [MATH], are [EQUATION]', '1305.1953-1-40-16': 'The structure is richer than for Pauli measurements on a Bell pair.', '1305.1953-1-40-17': 'However, it can still be explained by an LHV theory, which we now turn to.', '1305.1953-1-41-0': '## Local hidden variable model', '1305.1953-1-42-0': 'For our LHV theory we use a set of four hidden variables, which we label as [MATH], each of which takes values [MATH].', '1305.1953-1-42-1': 'Hence, we can write [EQUATION]', '1305.1953-1-42-2': 'We distinguish [MATH] from the other variables as it plays a unique role.', '1305.1953-1-42-3': "Next we fix Alice's probabilities to depend deterministically on the hidden variables as follows [EQUATION] and for Bob we take [EQUATION]", '1305.1953-1-42-4': 'Notice that, for all measurement settings, the measurement outcomes obey [EQUATION]', '1305.1953-1-42-5': 'Hence, for all choices of the hidden variables [MATH], the measurements outcomes satisfy [MATH].', '1305.1953-1-42-6': 'Furthermore, when [MATH] we have the added constraint that measurements satisfy [MATH] and [MATH].', '1305.1953-1-42-7': 'This tells us that for all choices of hidden variables, the distributions satisfy the second and fourth lines of Eqs. ([REF]).', '1305.1953-1-42-8': 'To achieve the correct weighting of the non-zero probabilities we simply take a uniform distribution over the hidden variables so that [MATH] for all [MATH] and [MATH].', '1305.1953-1-42-9': 'This completes our account of a LHV theory for all possible measurements on 4 Majorana pairs.', '1305.1953-1-43-0': '# Impossibility of GHZ state preparation', '1305.1953-1-44-0': 'Here we show that the correlations of 3-party GHZ states cannot be prepared using topologically protected operations and Majorana fermions, blocking attempts to violate locality using the Mermin-GHZ paradox.', '1305.1953-1-44-1': 'The proof technique can be easily extended to larger GHZ states, and the associated generalizations of the of the GHZ paradox [CITATION].', '1305.1953-1-44-2': 'We begin with some definitions and general observations.', '1305.1953-1-44-3': 'We consider products of fermionic operators, of the form [EQUATION] where [MATH] and [MATH].', '1305.1953-1-44-4': 'For two such operators, [MATH] and [MATH], denote with [MATH] the overlap in fermionic operators they hold in common, so that [MATH].', '1305.1953-1-44-5': 'Furthermore, let [MATH] denote the overlap shared by all three operators, which is [MATH].', '1305.1953-1-44-6': 'We will show that, for any accessible state with stabilizer [MATH], both the following hold', '1305.1953-1-45-0': '[(i)] for all [MATH], we have that [MATH] is even;', '1305.1953-1-46-0': '[(ii)] for all [MATH], we have that [MATH] is even.', '1305.1953-1-47-0': 'Accessible states possess a stabilizer [MATH] generated by [MATH], where [MATH].', '1305.1953-1-47-1': 'The structure of the generators imposes a structure on the whole group, which we use to prove the above.', '1305.1953-1-47-2': 'First, note that distinct generators [MATH] share no fermionic operators in common, and so [MATH].', '1305.1953-1-47-3': 'Expressing the stabilizers of an accessible state in terms of the generators, we have [EQUATION] for some [MATH].', '1305.1953-1-47-4': 'For two such operators, each generator they share contributes a pair of fermionic operators in common, and so [EQUATION]', '1305.1953-1-47-5': 'Clearly, this is always even.', '1305.1953-1-47-6': 'Furthermore, [EQUATION] which is again even.', '1305.1953-1-47-7': 'Overlaps between larger collections of operators must, for accessible states, again be even.', '1305.1953-1-47-8': 'Note also that, property (i) also follows from the necessary commutativity of the group [MATH], and is true for all quantum states.', '1305.1953-1-47-9': 'Whereas property (ii) is a genuine constraint on the whole set of quantum states.', '1305.1953-1-48-0': 'The correlations required for the Mermin paradox can be achieved by measuring Pauli operators on a GHZ state stabilized by [EQUATION] using some encoding [MATH].', '1305.1953-1-48-1': 'From the anti-commutation of the Pauli operators, that is [MATH], we see that the last stabilizer follows from the first three, such that [MATH].', '1305.1953-1-48-2': 'We consider all possibilities where the Pauli operators are replaced by appropriate products of Majorana operators.', '1305.1953-1-48-3': 'To be measurable, every [MATH] and [MATH] must consist of a product of an even number of Majorana operators.', '1305.1953-1-48-4': 'As remarked earlier, it is essential [MATH] anti-commutes with [MATH] and so [MATH] where [MATH] is odd.', '1305.1953-1-48-5': 'Finally, they must be local operators, so operators associated with different parties must be supported on distinct subsets of Majorana modes.', '1305.1953-1-48-6': 'From locality we deduce that [EQUATION]', '1305.1953-1-48-7': 'From anti-commutivity, all of the [MATH] are odd, and the sum of 3 odd numbers is again odd.', '1305.1953-1-48-8': 'We conclude that such a set of stabilizers contradicts property (ii) of accessible states.', '1305.1953-1-48-9': 'Under very general assumptions we have shown that no accessible states are isomorphic to the 3-qubit GHZ state.', '1305.1953-1-48-10': 'In particular, the most natural encoding would be to take [MATH] and [MATH], for which [MATH] and [MATH].', '1305.1953-1-48-11': 'However, more exotic encoding beyond this more canonical choice are also covered by the above argument.', '1305.1953-1-49-0': '# Quantum information protocols with Majorana fermions', '1305.1953-1-50-0': 'We finally briefly comment on the perspective of using Majorana fermions in the way described here in basic protocols of quantum information processing involving entanglement.', '1305.1953-1-50-1': 'Surely, all of the consequences of braiding operations can be classically efficiently simulated, by virtue of the observation of them constituting a subset of those operations in fermionic linear optics.', '1305.1953-1-50-2': 'Still, a number of interesting quantum information protocols involving entanglement can readily be conceived which are sketched here.', '1305.1953-1-50-3': 'This analysis complements the findings of Ref. [CITATION], in which measurement-only topological quantum computation has been considered.', '1305.1953-1-51-0': '## Teleportation', '1305.1953-1-52-0': 'Notably, instances of teleportation [CITATION] are possible.', '1305.1953-1-52-1': 'Meaningful variants of teleportation involving Majorana fermions should share the features that (i) an unknown state is considered the input, taken from a set of at least two non-orthogonal quantum states, and (ii) the output should be statistically indistinguishable from the input to the protocol.', '1305.1953-1-53-0': 'Here, Alice and Bob not only share [MATH] Majorana pairs, so as usual, the initial state is stabilized by [MATH] for [MATH].', '1305.1953-1-53-1': 'But furthermore, Alice holds further [MATH] Majorana modes that are not entangled with Bob.', '1305.1953-1-53-2': 'These local ancilla are stabilized by [EQUATION] and represent the modes that will be teleported, herein called the input modes.', '1305.1953-1-53-3': 'Such a scheme can indeed be devised by making use of 4 Majorana pairs, so [MATH].', '1305.1953-1-53-4': 'The input register consists of 4 Majorana modes, initially stabilized by [MATH] and [MATH].', '1305.1953-1-53-5': 'Many different inputs could be prepared by braiding the input modes.', '1305.1953-1-53-6': 'For instance, the input can be set to one of two non-orthogonal inputs by (I) either braiding [MATH] and [MATH] or (II) not.', '1305.1953-1-53-7': 'It is not difficult to see that these two situations (I) and (II) cannot be perfectly distinguished with unit probability.', '1305.1953-1-54-0': 'The remainder of the protocol does not depend on this input, and neither is knowledge of it required.', '1305.1953-1-54-1': 'In the next step, [MATH], [MATH], [MATH], [MATH] are measured and the results, elements of [MATH], classically communicated to Bob.', '1305.1953-1-54-2': 'One can then show that the output statistics of measurements performed by Bob, [MATH] and [MATH], appropriately interpreted using the received classical bits, is indistinguishable from the respective measurements on [MATH] and [MATH].', '1305.1953-1-54-3': 'For example, in scenario (II), if the outcomes [MATH] are communicated, the outcomes of measurements [MATH] and [MATH] are completely determined to be [MATH].', '1305.1953-1-55-0': '## Dense coding', '1305.1953-1-56-0': 'In a similar fashion, dense coding [CITATION] can be performed with Majorana fermions and the above specified operations.', '1305.1953-1-56-1': 'Again, let us be specific what a fair analogue of such a scheme would be.', '1305.1953-1-56-2': 'A valid dense coding scheme is one in which the entanglement-assisted single shot classical capacity of a quantum channel is higher compared to the corresponding capacity in the absence of entanglement.', '1305.1953-1-56-3': 'We hence would like to introduce a protocol where starting from entangled shared resources, with local braiding operations and a subsequent transmission of some modes, one can encode more bits of classical information than what is possible without having shared entangled resources available, but transmitting the same number of modes.', '1305.1953-1-56-4': 'For qudits, it is known that the entanglement-assisted capacity is exactly double the one reachable without assistance [CITATION].', '1305.1953-1-56-5': 'For Majorana fermions, it turns out, the same holds true.', '1305.1953-1-57-0': 'This is readily possible with 3 Majorana pairs, again partially shared by Alice and Bob and partially held by Alice.', '1305.1953-1-57-1': 'Both parties share 2 Majorana pairs, stabilized by [MATH], [MATH].', '1305.1953-1-57-2': 'In addition, Alice holds 2 Majorana modes in a state stabilized by [MATH].', '1305.1953-1-57-3': 'By local braiding, Alice can achieve a state stabilized by [EQUATION] in a scheme that maps [EQUATION] [MATH] for [MATH], [MATH] and [MATH].', '1305.1953-1-57-4': 'It is easy to see that with a suitable choice of [MATH], 2 bits of information can be encoded.', '1305.1953-1-57-5': 'Furthermore, after Alice transmits Majorana modes [MATH] and [MATH] to Bob, he can reliably retrieve [MATH] and [MATH] by locally measuring [MATH] and [MATH].', '1305.1953-1-57-6': 'At the same time, with two Majorana modes and no shared entanglement, a single bit of information can be encoded only.', '1305.1953-1-57-7': 'This constitutes a valid dense coding scheme based on Majorana fermions.', '1305.1953-1-57-8': 'It also resembles the situation of transmitting two bits of classical information with a single transmitted qubit, if entangled resources are initially available.', '1305.1953-1-58-0': '# Closing remarks', '1305.1953-1-59-0': 'We have seen that locality can be violated using only the topologically protected operations of Majorana fermions or Ising anyons.', '1305.1953-1-59-1': 'These operations are a subclass of fermionic linear optics [CITATION], and so we conclude that these systems can also violate locality.', '1305.1953-1-59-2': 'The model of fermionic linear optics was proposed as an analog of bosonic linear optics, where a well-known LHV theory prevents any non-locality experiment [CITATION].', '1305.1953-1-59-3': 'Seen in this light, our result is quite surprising.', '1305.1953-1-59-4': 'Furthermore, we remark that that fractionalized analogs of Majorana fermions, known as parafermions [CITATION], may face similar obstacles from the existence of certain hidden variable theories [CITATION].', '1305.1953-1-59-5': 'It is natural to ask more generally what other anyonic systems are non-local, again without the requirement of universal quantum computing capabilities.', '1305.1953-1-59-6': 'Finally, we speculate that these experiments could prove useful as a probe to certify the presence of Majorana fermions and potentially help dispel some of the present ambiguity surrounding experiments.'}

{'1305.1953-2-0-0': 'EARL T. CAMPBELL *0.015truein Dahlem Center for Complex Quantum Systems, Freie Universitat Berlin, Berlin, Germany', '1305.1953-2-1-0': 'MATTY J. HOBAN *0.015truein ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, Castelldefels (Barcelona), Spain *10PT JENS EISERT *0.015truein Dahlem Center for Complex Quantum Systems, Freie Universitat Berlin, Berlin, Germany *0.225truein', '1305.1953-2-2-0': '*0.21truein', '1305.1953-2-3-0': '4.5in Localized Majorana fermions emerge in many topologically ordered systems and exhibit exchange statistics of Ising anyons.', '1305.1953-2-3-1': 'This enables noise-resistant implementation of a limited set of operations by braiding and fusing Majorana fermions.', '1305.1953-2-3-2': 'Unfortunately, these operations are incapable of implementing universal quantum computation.', '1305.1953-2-3-3': 'We show that, regardless of these limitations, Majorana fermions could be used to demonstrate non-locality (correlations incompatible with a local hidden variable theory) in experiments using only topologically protected operations.', '1305.1953-2-3-4': 'We also demonstrate that our proposal is optimal in terms of resources, with 10 Majorana fermions shown to be both necessary and sufficient for demonstrating bipartite non-locality.', '1305.1953-2-3-5': 'Furthermore, we identify severe restrictions on the possibility of tripartite non-locality.', '1305.1953-2-3-6': 'We comment on the potential of such entangled systems to be used in quantum information protocols.', '1305.1953-2-4-0': '# Introduction', '1305.1953-2-5-0': 'Fermions that are their own anti-particle are known as Majorana, as opposed to Dirac, fermions.', '1305.1953-2-5-1': 'While presently there is no evidence that any fundamental particles are Majorana fermions, they frequently emerge as localised quasi-particles in models of condensed matter systems [CITATION].', '1305.1953-2-5-2': 'Recent years have seen a race to experimentally confirm their existence, with some evidence already found [CITATION].', '1305.1953-2-5-3': 'Research into these systems is driven, at least partially, by their potential applications in topological quantum computing.', '1305.1953-2-5-4': 'For localised and well-separated Majorana fermions with zero energy, the system is protected from noise effects that could otherwise prove devastating in quantum computers.', '1305.1953-2-6-0': 'Suitable Majorana fermions emerge in many two-dimensional (2D) systems including the Kitaev honeycomb lattice [CITATION], fractional quantum hall systems [CITATION], topological insulators [CITATION], and a variety of other systems [CITATION].', '1305.1953-2-6-1': 'Adiabatically exchanging these fermions, and so braiding their world-lines, gives rise to the non-abelian exchange statistics of Ising anyons.', '1305.1953-2-6-2': 'Majorana fermions can also emerge as edge modes in one-dimensional (1D) systems [CITATION], such as the Kiteav wire [CITATION].', '1305.1953-2-6-3': 'While braiding is not a meaningful concept in strict 1D systems, networks of wires also enable braiding with Ising statistics [CITATION].', '1305.1953-2-6-4': 'The unitary evolution from braiding is geometric in origin, and so robust against small experimental imperfections.', '1305.1953-2-6-5': 'However, these topologically protected braiding operations are not computationally powerful enough for universal quantum computing.', '1305.1953-2-6-6': 'Indeed, any free fermionic system can be efficiently classically simulated [CITATION].', '1305.1953-2-6-7': 'Access to some non-topological operations - which may be noisy, but not too noisy [CITATION] - can be used to promote the system to full universality [CITATION].', '1305.1953-2-6-8': 'However, here we are interested in understanding the purely topologically protected capabilities of Majorana fermions, and in particular their capacity for demonstrating non-locality [CITATION].', '1305.1953-2-7-0': 'Non-locality is the inability for a local hidden variable (LHV) theory to reproduce the correlations of space-like separated measurements.', '1305.1953-2-7-1': 'Until now, non-locality has only been investigated in more exotic topological systems by Brennen et al. [CITATION].', '1305.1953-2-7-2': 'Having only considered systems capable of universal quantum computation, Brennen et al. concluded their work saying, "it is intriguing to ask whether one could find intermediate anyonic theories which have the power to generate Bell violating states by topologically protected gates, but are not universal for topological quantum computation".', '1305.1953-2-7-3': 'We resolve this mystery by showing that Majorana fermions, and equivalently Ising anyons, could be used in an experiment demonstrating the non-locality of quantum mechanics.', '1305.1953-2-7-4': 'It appears that the standard and ubiquitous Clauser-Horne-Shimony-Holt (CHSH) inequality cannot be violated with only topological operations, and the non-topological resources sufficient for a CHSH violation have been investigated [CITATION].', '1305.1953-2-7-5': 'We turn instead to a non-local experiment proposed by Cabello [CITATION] where 2 parties each select from 3 possible measurements, with each measurement producing 3 bits of classical information as outcomes.', '1305.1953-2-7-6': 'This non-locality proposal was built on the idea of the Mermin-Peres "magic square", which was originally used to show the contextuality of quantum mechanics [CITATION].', '1305.1953-2-7-7': 'We find that a variant of these experiments could be implemented with each of two parties holding 5 Majorana fermions, and present a Bell inequality based on the magic square.', '1305.1953-2-7-8': 'We also present no-go results showing for two parties holding 4 Majoranas each, all experimental statistics can be produced by a local hidden variable theory.', '1305.1953-2-7-9': 'If two parties unequally share Majoranas, say Alice holds [MATH] and Bob holds [MATH] with [MATH], we find that the resource is locally equivalent to both parties holding just [MATH] Majorana fermions.', '1305.1953-2-7-10': 'Hence, 10 Majorana fermions are both necessary and sufficient for the phenomenon of bipartite non-locality to be topologically demonstrated.', '1305.1953-2-7-11': 'Furthermore, we find that the correlations required to demonstrate the three-party Greenberger-Horne-Zeilinger (GHZ) paradox [CITATION] cannot be implemented with any number of Majoranas.', '1305.1953-2-7-12': 'This indicates that our proposal is the simplest possible non-locality experiment with Majorana fermions.', '1305.1953-2-8-0': 'Bell experiments were intended, initially, to falsify alternative theories that claim the world to be local in Nature.', '1305.1953-2-8-1': 'Now these Bell-type non-locality experiments are also known to have practical applications.', '1305.1953-2-8-2': 'Such experiments can form the basis of quantum cryptography when the devices used are faulty or even untrusted [CITATION].', '1305.1953-2-8-3': 'Typically, such proposals are envisaged for photonic systems, since they provide an easier means of accomplishing space-like separation of measurement events (a requirement for a non-local experiment).', '1305.1953-2-8-4': "Polarisation-entangled photons have, for many years, been used to test Bell's proposal, proving successful if one ignores the detector loophole [CITATION].", '1305.1953-2-8-5': 'Experiments with trapped ions have efficient enough measurements to avoid the detector loophole, but do not satisfy space-like measurement separation [CITATION].', '1305.1953-2-8-6': 'Designing better experiments, hopefully closing all loopholes, is an active area of research [CITATION].', '1305.1953-2-8-7': 'We shall not argue that our topological proposal is more promising than the aforementioned approaches.', '1305.1953-2-8-8': 'Rather, we take the first step by showing theoretical feasibility under ideal conditions.', '1305.1953-2-8-9': 'Implicit in our approach is the assumption that the physical system can be shared between two parties over sufficient distances that measurements are space-like separated; this is a similar technical difficulty faced by ion trap designs [CITATION].', '1305.1953-2-9-0': '# Majorana fermions and braiding', '1305.1953-2-10-0': '## Majorana fermions', '1305.1953-2-11-0': 'We begin with a review of Majorana fermions and their available dynamics, largely following Refs. [CITATION].', '1305.1953-2-11-1': 'Generally, Majorana fermions are described by a set of Hermitian operators [MATH] satisfying [EQUATION] and [MATH] for all [MATH], acting on the physical Hilbert space [MATH] constituting a direct sum between the even and odd parity sectors.', '1305.1953-2-11-2': 'The algebra of physical operators [MATH] is spanned by products of an even number of Majorana fermion operators.', '1305.1953-2-11-3': 'Taking two such fermions and clockwise braiding their world lines results in a unitary [MATH]', '1305.1953-2-12-0': 'that maps the operators as [EQUATION] for [MATH].', '1305.1953-2-12-1': 'Composing these braid operations results in a permutation [MATH], with possible phase change [MATH], so that [MATH].', '1305.1953-2-12-2': 'The phases are constrained so that the global parity is preserved, leaving [MATH] unchanged.', '1305.1953-2-13-0': 'Such braidings are a special case of unitary transformations [MATH] acting on [MATH] that reflect linear mode transformation [EQUATION] for [MATH].', '1305.1953-2-13-1': 'Such unitary transformations are the ones commonly considered in the context of fermionic linear optics.', '1305.1953-2-13-2': 'All states encountered in this work are Gaussian fermionic states [CITATION].', '1305.1953-2-13-3': 'They are entirely described by their anti-symmetric covariance matrix [MATH], [MATH], which has entries [MATH].', '1305.1953-2-14-0': '## Stabiliser language', '1305.1953-2-15-0': 'We primarily describe quantum states in the Heisenberg picture by specifying a sufficient number of eigenvalue equations.', '1305.1953-2-15-1': 'We say an operator [MATH] stabilises a state vector [MATH] if [EQUATION]', '1305.1953-2-15-2': 'We assume that initialisation of the system prepares a state vector [MATH] stabilised by [MATH] for all [MATH].', '1305.1953-2-16-0': 'Therefore, any state prepared from the initialisation [MATH] by braiding will be stabilised by [MATH] for some permutation [MATH].', '1305.1953-2-16-1': 'Note that, since permutations are one-to-one mappings, if [MATH], then [MATH].', '1305.1953-2-16-2': 'Products of stabilisers are again stabilisers, and so [MATH] generate a group [MATH] associated with the state.', '1305.1953-2-16-3': 'Similarly, topologically protected measurements, also called charge measurements, are those of the form [MATH].', '1305.1953-2-16-4': 'Throughout, we say a state is accessible if and only if it can be prepared with the above described topological operations.', '1305.1953-2-16-5': 'Again, states obtained by performing such measurements on Gaussian states are Gaussian.', '1305.1953-2-17-0': 'Collective charge measurements, such as of [MATH], cannot be measured non-destructively and in a topological manner.', '1305.1953-2-17-1': 'The collective charge observable can, however, be inferred by measuring [MATH] and [MATH] and multiplying the outcomes.', '1305.1953-2-17-2': 'However, such a process is destructive in that it will always disentangle these Majoranas from all other systems.', '1305.1953-2-17-3': 'We labour this point because the capacity to make non-destructive charge measurements increases the computational power of the system [CITATION] beyond that assumed in our later no-go theorems.', '1305.1953-2-18-0': '## Anyonic formalism', '1305.1953-2-19-0': 'We shall restate some of the above in the anyonic formalism, which may be of benefit to some readers.', '1305.1953-2-19-1': 'Using, [MATH] to denote an Ising anyon, [MATH] for another fermion, and [MATH] for the vacuum, we have the fusion channel [MATH].', '1305.1953-2-19-2': 'Fusing Ising anyons [MATH] and [MATH] is equivalent to measuring [MATH], where producing a [MATH] particle is equivalent to the [MATH] measurement outcome (eigenvalue) and producing the vacuum [MATH] outcome denotes the [MATH] eigenvalue.', '1305.1953-2-19-3': 'Conversely, if we begin with a vacuum and create [MATH] pairs of Ising anyons, we have the initialisation state described above if anyons from the [MATH] pair are labeled as [MATH] and [MATH].', '1305.1953-2-20-0': '## Entanglement properties', '1305.1953-2-21-0': 'We proceed by identifying the equivalence classes of entangled states that are accessible under the operations described above.', '1305.1953-2-21-1': 'Consider two parties, Alice and Bob, holding respective sets of Majorana fermions [MATH] and [MATH].', '1305.1953-2-21-2': 'Clearly, any state stabilised by [MATH] can be locally prepared by Alice for [MATH] labeling a pair of Majorana fermions.', '1305.1953-2-21-3': 'A similar statement holds for Bob, and so the interesting stabilisers have [MATH] and [MATH] with each party holding one half of a pair of Majorana fermions (henceforth referred to as Majorana pairs).', '1305.1953-2-21-4': 'Assuming they hold [MATH] such pairs, they can always locally braid such that the state is stabilised by [MATH] for all [MATH].', '1305.1953-2-21-5': 'We see that the entanglement is entirely captured by the number of such Majorana pairs shared by Alice and Bob, and throughout we assume this canonical form for the stabilisers.', '1305.1953-2-21-6': 'For notational simplicity, it is beneficial to use [MATH] and [MATH], so Majorana pairs are stabilised by [MATH].', '1305.1953-2-22-0': 'Crucially, the number of Majorana pairs is distinct from the number of Bell pairs that give rise to useful entanglement.', '1305.1953-2-22-1': 'To investigate the useful correlations between Alice and Bob, we must consider how such pairs respond to measurements.', '1305.1953-2-22-2': 'Consider a pair of stabilisers [MATH] and [MATH] for [MATH], the state is also an eigenstate of their product [MATH] which using anti-commutation of fermions equals [MATH].', '1305.1953-2-22-3': 'The factors [MATH] and [MATH] are locally measurable, and so their outcomes must be anti-correlated.', '1305.1953-2-22-4': 'Hence, the state has the flavor of a singlet state.', '1305.1953-2-23-0': 'To make measurements Alice and Bob must share at least two Majorana pairs, but two alone is trivial since there is only one possible local measurement and with only one measurement we cannot construct a test of non-locality.', '1305.1953-2-23-1': 'With three Majorana pairs, Alice has three possible measurement observables that mutually anti-commute and are isomorphic to the Pauli operators.', '1305.1953-2-23-2': 'An encoding is a collection of maps [MATH], identifying the set of fermionic modes with qubit or spin systems.', '1305.1953-2-23-3': 'The identification of products of Majorana fermions with Pauli operators of an associated qubit system can be taken as [EQUATION]', '1305.1953-2-23-4': 'From the anti-commutation relations of the Majorana fermions, one can readily verify that these operators generate the Pauli group of a single qubit.', '1305.1953-2-23-5': 'Bob similarly has measurement options isomorphic to the Pauli operators.', '1305.1953-2-23-6': 'The correlations between Alice and Bob resulting from the above measurements on three Majorana pairs match those of Pauli measurements on the two-qubit singlet state (for Alice and Bob each having a single qubit).', '1305.1953-2-23-7': 'Hence, three Majorana pairs can be said to reproduce the entanglement of a Bell pair.', '1305.1953-2-23-8': 'However, it is well-known that the measurement statistics under the operations allowed here can be reproduced by a local hidden variable (LHV) theory.', '1305.1953-2-23-9': 'Later we present a strengthened proof that even four Majorana pairs are incapable of demonstrating non-locality.', '1305.1953-2-24-0': '# Non-locality', '1305.1953-2-25-0': 'Before proceeding we refine the concepts of non-locality.', '1305.1953-2-25-1': 'Assume Alice and Bob hold some quantum state [MATH] and are able to freely choose from a set of measurements [MATH] and [MATH] respectively, where [MATH] denotes the set of different kinds of measurement settings.', '1305.1953-2-25-2': 'The measurements will have possible outcomes that we denote [MATH] for a suitable [MATH] that occur with probability [EQUATION] where [MATH]) is the positive-operator valued measure for setting [MATH]) and outcome [MATH]).', '1305.1953-2-25-3': 'There are many different experiments that can achieve the same probability distributions, and each of these is a realisation of [MATH].', '1305.1953-2-25-4': 'Conversely, we say a probability distribution [MATH] is quantum if there exists a choice of measurements and a quantum state [MATH] that realises [MATH].', '1305.1953-2-26-0': 'When are these observations a proof of non-locality?', '1305.1953-2-26-1': 'Imagine that Alice and Bob are space-like separated and are allowed to make their choice of measurement freely, they now want to know whether any classical model can produce these observations.', '1305.1953-2-26-2': "Since they cannot communicate, this classical model has some pre-determined instructions that, given Alice and Bob's choice of measurements, will output some value.", '1305.1953-2-26-3': 'Such an instruction set is called a LHV theory.', '1305.1953-2-26-4': 'An arbitrary hidden variable [MATH] takes values in some space [MATH] equipped with a probability measure, and which determines local probability functions [EQUATION]', '1305.1953-2-26-5': 'The probability of obtaining the outcome pair [MATH], given that [MATH] have been chosen by Alice and Bob, is then [EQUATION]', '1305.1953-2-26-6': 'A probability distribution is local (also called a LHV) if and only if there exists a decomposition of the above form, and we denote this as [MATH].', '1305.1953-2-26-7': 'If no such local model exists, [MATH], we say the probability distribution is non-local.', '1305.1953-2-26-8': 'Furthermore, any experiment realizing a non-local probability distribution is a non-locality experiment.', '1305.1953-2-27-0': 'Typically, we confirm non-locality by checking whether a probability distribution violates a Bell inequality.', '1305.1953-2-27-1': 'Let us fix the number of measurement settings and outcomes, and denote the entire set of possible probability distributions by [MATH].', '1305.1953-2-27-2': 'The Bell inequalities follow by first defining a real-valued linear function, [MATH], which is sometimes called a non-local game [CITATION], the respective parties are referred to as players and the action taken are strategies.', '1305.1953-2-27-3': 'The non-local game, as the term is used here, is described by a real-valued function, [MATH] such that [EQUATION]', '1305.1953-2-27-4': 'For any such non-local game, there exists a Bell inequality that holds for all local [MATH] [EQUATION]', '1305.1953-2-27-5': 'The above is true simply by definition.', '1305.1953-2-27-6': 'However, for any [MATH] and [MATH] where [MATH] we can conclude [MATH] is non-local.', '1305.1953-2-28-0': 'Let us consider a particular non-local game [MATH], and its classical limit [MATH].', '1305.1953-2-28-1': 'Since non-local games are linear functions and [MATH] is a convex set, the classical limit can always be achieved by an extremal point in [MATH].', '1305.1953-2-28-2': 'That is, the value of [MATH] can always be achieved by a probability distribution of the form [EQUATION] where Alice and Bob deterministically assign measurement outcomes.', '1305.1953-2-28-3': 'There are only finitely many such distributions, so this greatly simplifies the evaluation of [MATH].', '1305.1953-2-29-0': '# Magic squares with Majoranas', '1305.1953-2-30-0': 'To demonstrate that these Majorana fermions exhibit non-locality we adapt a specific proof of "non-locality without inequalities" first devised by Cabello [CITATION].', '1305.1953-2-30-1': 'There exist other proofs of non-locality for qubits without the use of a Bell inequality, such as those by Hardy [CITATION] and the GHZ paradox [CITATION].', '1305.1953-2-30-2': "Crucially, Hardy's argument does not work for maximally entangled states and so it cannot be applied.", '1305.1953-2-30-3': 'Furthermore we show later that the GHZ argument does not apply here either.', '1305.1953-2-30-4': "Cabello's proof works for two Bell states, a four-qubit state, shared by Alice and Bob each of which can make a measurement on two qubits of this state.", '1305.1953-2-30-5': 'The analysis shares much of its character with the proof that quantum mechanics is non-contextual derived by Peres and Mermin, often referred to as the "Magic Square Game".', '1305.1953-2-30-6': "We will follow the simplification of Cabello's approach as presented by Aravind [CITATION] so that the result may be of more general appeal, but optimally tailored to systems of Majorana fermions.", '1305.1953-2-31-0': '## The set-up', '1305.1953-2-32-0': 'We now describe the set-up.', '1305.1953-2-32-1': 'We assume that Alice and Bob share five Majorana pairs, prepared in the canonical form outlined earlier.', '1305.1953-2-33-0': 'Each party then makes a choice of three different measurement settings.', '1305.1953-2-33-1': 'Each measurement setting relates to three measurements involving a subset of the Majorana modes held by the respective party.', '1305.1953-2-33-2': 'Hence, for each measurement setting, a string of three values of [MATH] is obtained, so [MATH].', '1305.1953-2-33-3': 'For Alice we label the choice of measurement as [MATH] for [MATH] and for Bob the choice of measurement is written as [MATH]) for [MATH].', '1305.1953-2-33-4': 'Alice and Bob then obtain the output strings [MATH], [MATH], respectively, with [MATH] labeling the choice of measurements [MATH] and [MATH].', '1305.1953-2-33-5': 'The measurements are fixed by the columns (for Alice) and rows (for Bob), of square tables (so-called "magic squares"), as shown in Table [REF].', '1305.1953-2-33-6': "In these magic squares, Alice and Bob's respective square tables contain, up-to a phase, the same measurements performed by the respective parties.", '1305.1953-2-33-7': "It can easily be verified that an element of Alice's table multiplied with the same element of Bob's table (i.e. the same [MATH] row and [MATH] column in each table) gives a stabiliser of their shared quantum state.", '1305.1953-2-33-8': 'Formally, this enforces that [MATH] for all [MATH].', '1305.1953-2-33-9': 'Furthermore, the product of observables in any column of both tables gives [MATH], and so for any measurement setting [MATH].', '1305.1953-2-33-10': 'Whereas the product of observables in any row of both tables gives [MATH], and so [MATH].', '1305.1953-2-33-11': 'Notice that any triple of observables, for either Alice or Bob, always contains a single measurement observable acting on 4 Majorana modes.', '1305.1953-2-33-12': 'As remarked earlier, such measurements cannot be directly measured but can be inferred from other measurement outcomes.', '1305.1953-2-33-13': 'Here this poses no problem as the required information is provided by the remaining pair of observables.', '1305.1953-2-33-14': 'For instance, the first column for Alice corresponds to measuring [MATH], and while [MATH] cannot be directly measured, we see that [MATH] and so we can simply infer the outcome from [MATH].', '1305.1953-2-33-15': 'In any actual experiment, Alice and Bob only measure a pair of observables, but for clarity of exposition it is convenient to speak of each party measuring a whole triple of observables.', '1305.1953-2-34-0': 'Aside from these constraints, the measurement outcomes are entirely random and so the experiment realises [EQUATION] and in the next section we confirm this to be non-local and robust against some experimental noise.', '1305.1953-2-35-0': '## The magic square game without inequalities', '1305.1953-2-36-0': 'Earlier, we saw that for any non-local game the best local strategy can be achieved by Alice and Bob deterministically assigning measurement outcomes.', '1305.1953-2-36-1': 'In the problem at hand, it is convenient to describe these local probability distributions by a [MATH] table with entries in [MATH].', '1305.1953-2-36-2': "We use [MATH] to denote Alice's table, which has entries such that [EQUATION]", '1305.1953-2-36-3': 'Whereas for Bob, we take [EQUATION]', '1305.1953-2-36-4': 'Clearly, there is a one-to-one correspondence between these tables and deterministic probability distributions.', '1305.1953-2-36-5': 'Note also the slight difference in definitions for Alice and Bob.', '1305.1953-2-36-6': 'For Alice, a single measurement setting will output a column of her table, whereas for Bob, a single measurement setting will output a row of his table.', '1305.1953-2-36-7': 'This convention is justified because the quantum correlations satisfy [MATH] for all [MATH], and so if the LHV theory replicates this correlation we have that [EQUATION]', '1305.1953-2-36-8': 'Therefore, Alice and Bob must share identical tables to reproduce this feature of the quantum correlations.', '1305.1953-2-36-9': 'However, the quantum correlations have an additional feature, they satisfy parity constraints.', '1305.1953-2-36-10': "Alice's parity constraint entails that for all [MATH], we have [MATH], and hence the parity of the entire table is [MATH].", '1305.1953-2-36-11': "In contrast, Bob's parity constraint entails that for all [MATH], we have [MATH], and so [MATH].", '1305.1953-2-36-12': 'We conclude that Alice and Bob cannot hold identical classical tables and also satisfy all the parity constraints, and so they can never perfectly reproduce the quantum correlations.', '1305.1953-2-36-13': 'Since quantum observables do not commute, finding the parity of the table of operators can change depending on the order we multiply the entries.', '1305.1953-2-37-0': '## The magic square game with Bell inequalities', '1305.1953-2-38-0': 'We have seen that classical experiments can never reproduce the quantum correlations demonstrated in the previous section.', '1305.1953-2-38-1': 'However, we are interested in what level of imperfection can be tolerated by any quantum experiment while still violating locality.', '1305.1953-2-38-2': 'To quantify this we must specify a particular non-local game, see Eq. ([REF]), by specifying a function [MATH].', '1305.1953-2-38-3': 'The goal is to have a larger value when the correlations match those of the quantum predications, and smaller when they fail.', '1305.1953-2-38-4': 'The simplest choice is a function that takes two values, and it is conventional to take these as [MATH], and so we have [EQUATION]', '1305.1953-2-38-5': 'This non-local game we call the magic square game.', '1305.1953-2-38-6': 'It is easy to verify that using Majorana fermions, which realise the probability distribution [MATH] (see Eq. [REF]), we find [MATH].', '1305.1953-2-39-0': 'We are now in a position to identify [MATH], the maximum classical value.', '1305.1953-2-39-1': 'Alice and Bob could use identical tables, but they then contravene many parity constraints, and we find this results in at most [MATH].', '1305.1953-2-39-2': 'However, the maximum is achieved if Alice and Bob use classical tables that differ only in a single entry and satisfy all the parity constraints, such as in Table [REF].', '1305.1953-2-39-3': 'This yields [MATH] since for 8 of the measurement settings we acquire a contribution of [MATH], but for one setting we have [MATH].', '1305.1953-2-39-4': 'Hence, any imperfect experiment with Majorana fermions that attains [MATH] is sufficient to demonstrate non-locality.', '1305.1953-2-40-0': 'We also comment on how the imperfect quantum setting can, sometimes, economically be described.', '1305.1953-2-40-1': 'For perfect operations of the type considered here, the covariance matrix [MATH] will only contain entries contained in [MATH] and satisfies [MATH].', '1305.1953-2-40-2': 'If errors and imperfections are present, the entries of [MATH] will also attain values different from those, while it is still true that [EQUATION]', '1305.1953-2-40-3': 'Under braiding transformations of the form ([REF]), covariance matrices transform as congruences [MATH] with orthogonal matrices [MATH], which can easily be kept track of.', '1305.1953-2-40-4': 'The statistics of measurements can still be determined based on the covariance matrix only, even if errors are taken into account in the preparation step.', '1305.1953-2-41-0': '# Too few Majorana pairs', '1305.1953-2-42-0': '## Setting', '1305.1953-2-43-0': 'Earlier we saw that if Alice and Bob share three Majorana pairs, the available measurements are isomorphic to Pauli measurements on a single Bell pair.', '1305.1953-2-43-1': 'It is well-known folklore that such a system can be modeled by a local-hidden variable theory.', '1305.1953-2-43-2': 'Here we present a stronger argument covering up to 4 Majorana pairs.', '1305.1953-2-43-3': 'This shows that five or more Majorana pairs are necessary, as well as sufficient, to demonstrate non-locality.', '1305.1953-2-43-4': 'We begin by characterizing the full set of possible measurements.', '1305.1953-2-43-5': 'For Alice, there are three pairs of commuting measurements she can perform [EQUATION]', '1305.1953-2-43-6': 'If Alice measures a pair of observables [MATH], she get two random bits, which we denote [MATH].', '1305.1953-2-43-7': 'Similar measurement sets, labeled [MATH], are available to Bob, with [MATH] replacing [MATH].', '1305.1953-2-43-8': 'If Bob measures the analogous set, so [MATH], he gets perfectly anticorrelated outcomes, such that [MATH] and [MATH].', '1305.1953-2-43-9': 'Next we consider when Bob measures a different set of operators, so [MATH].', '1305.1953-2-43-10': 'We need to determine which products of measurements correspond to stabilisers of the 4 Majorana pairs.', '1305.1953-2-43-11': 'We observe that for Alice and any measurement set, the product of the observables is [MATH], and similarly for Bob the product is [MATH].', '1305.1953-2-43-12': 'Since the observables are matching and contain an even number of fermionic operators, these observables will have correlated outcomes.', '1305.1953-2-43-13': 'Formally, these correlations entail that [MATH] for all measurement settings [MATH] and [MATH].', '1305.1953-2-43-14': 'However, when [MATH], this is the only correlation and otherwise the measurement outcomes are entirely random.', '1305.1953-2-43-15': 'From these observations we can deduce that the outcome probabilities, as a function of measurement settings [MATH] and [MATH], are [EQUATION]', '1305.1953-2-43-16': 'The structure is richer than for Pauli measurements on a Bell pair.', '1305.1953-2-43-17': 'However, it can still be explained by an LHV theory, which we now turn to.', '1305.1953-2-44-0': '## Local hidden variable model', '1305.1953-2-45-0': 'For our LHV theory we use a set of four hidden variables, which we label as [MATH], each of which takes values [MATH].', '1305.1953-2-45-1': 'Hence, we can write [EQUATION]', '1305.1953-2-45-2': 'We distinguish [MATH] from the other variables as it plays a unique role.', '1305.1953-2-45-3': "Next we fix Alice's probabilities to depend deterministically on the hidden variables as follows [EQUATION] and for Bob we take [EQUATION]", '1305.1953-2-45-4': 'Notice that, for all measurement settings, the measurement outcomes obey [EQUATION]', '1305.1953-2-45-5': 'Hence, for all choices of the hidden variables [MATH], the measurements outcomes satisfy [MATH].', '1305.1953-2-45-6': 'Furthermore, when [MATH] we have the added constraint that measurements satisfy [MATH] and [MATH].', '1305.1953-2-45-7': 'This tells us that for all choices of hidden variables, the distributions satisfy the second and fourth lines of Eqs. ([REF]).', '1305.1953-2-45-8': 'To achieve the correct weighting of the non-zero probabilities we simply take a uniform distribution over the hidden variables so that [MATH] for all [MATH] and [MATH].', '1305.1953-2-45-9': 'This completes our account of a LHV theory for all possible measurements on 4 Majorana pairs.', '1305.1953-2-46-0': '# Impossibility of GHZ state preparation', '1305.1953-2-47-0': 'Here we show that the correlations of 3-party GHZ states cannot be prepared using topologically protected operations and Majorana fermions, blocking attempts to violate locality using the Mermin-GHZ paradox.', '1305.1953-2-47-1': 'The proof technique can be easily extended to larger GHZ states, and the associated generalizations of the GHZ paradox [CITATION].', '1305.1953-2-47-2': 'We begin with some definitions and general observations.', '1305.1953-2-47-3': 'We consider products of fermionic operators, of the form [EQUATION] where [MATH] and [MATH].', '1305.1953-2-47-4': 'For two such operators, [MATH] and [MATH], denote with [MATH] the overlap in fermionic operators they hold in common, so that [MATH].', '1305.1953-2-47-5': 'Furthermore, let [MATH] denote the overlap shared by all three operators, which is [MATH].', '1305.1953-2-47-6': 'We will show that, for any accessible state with stabiliser [MATH], both the following hold', '1305.1953-2-48-0': '[(i)] for all [MATH], we have that [MATH] is even; [(ii)] for all [MATH], we have that [MATH] is even.', '1305.1953-2-49-0': 'Accessible states possess a stabiliser [MATH] generated by [MATH], where [MATH].', '1305.1953-2-49-1': 'The structure of the generators imposes a structure on the whole group, which we use to prove the above.', '1305.1953-2-49-2': 'First, note that distinct generators [MATH] share no fermionic operators in common, and so [MATH].', '1305.1953-2-49-3': 'Expressing the stabilisers of an accessible state in terms of the generators, we have [EQUATION] for some [MATH].', '1305.1953-2-49-4': 'For two such operators, each generator they share contributes a pair of fermionic operators in common, and so [EQUATION]', '1305.1953-2-49-5': 'Clearly, this is always even.', '1305.1953-2-49-6': 'Furthermore, [EQUATION] which is again even.', '1305.1953-2-49-7': 'More generally, overlaps between larger collections of operators must, for accessible states, again be even.', '1305.1953-2-49-8': 'Note also that, property (i) also follows from the necessary commutativity of the group [MATH], and is true for all quantum states.', '1305.1953-2-49-9': 'Whereas property (ii) is a genuine constraint on the whole set of quantum states.', '1305.1953-2-50-0': 'The correlations required for the Mermin paradox can be achieved by measuring Pauli operators on a GHZ state stabilised by [EQUATION] using some encoding [MATH].', '1305.1953-2-50-1': 'From the anti-commutation of the Pauli operators, that is [MATH], we see that the last stabiliser follows from the first three, such that [MATH].', '1305.1953-2-50-2': 'We consider all possibilities where the Pauli operators are replaced by appropriate products of Majorana operators.', '1305.1953-2-50-3': 'To be measurable, every [MATH] and [MATH] must consist of a product of an even number of Majorana operators.', '1305.1953-2-50-4': 'As remarked earlier, it is essential [MATH] anti-commutes with [MATH] and so [MATH] where [MATH] is odd.', '1305.1953-2-50-5': 'Finally, they must be local operators, so operators associated with different parties must be supported on distinct subsets of Majorana modes.', '1305.1953-2-50-6': 'From locality we deduce that [EQUATION]', '1305.1953-2-50-7': 'From anti-commutivity, all of the [MATH] are odd, and the sum of 3 odd numbers is again odd.', '1305.1953-2-50-8': 'We conclude that such a set of stabilisers contradicts property (ii) of accessible states.', '1305.1953-2-50-9': 'Under very general assumptions we have shown that no accessible states are isomorphic to the 3-qubit GHZ state.', '1305.1953-2-50-10': 'In particular, the most natural encoding would be to take [MATH] and [MATH], for which [MATH] and [MATH].', '1305.1953-2-50-11': 'However, more exotic encoding beyond this more canonical choice are also covered by the above argument.', '1305.1953-2-50-12': "Alternative proofs based on the fermionic Gaussian nature of the states generated by topological operations are conceivable, and could potentially make use of Wick's theorem.", '1305.1953-2-51-0': '# Quantum information protocols with Majorana fermions', '1305.1953-2-52-0': 'We finally comment on the perspective of using Majorana fermions in basic protocols of quantum information processing involving entanglement.', '1305.1953-2-52-1': 'Surely, all of the consequences of braiding operations can be classically efficiently simulated, by virtue of the observation that they constitute a subset of those operations in fermionic linear optics.', '1305.1953-2-52-2': 'Still, a number of interesting quantum information protocols involving entanglement can readily be conceived which are sketched here.', '1305.1953-2-52-3': 'This analysis complements the findings of Ref. [CITATION], in which measurement-only topological quantum computation has been considered.', '1305.1953-2-53-0': '## Teleportation', '1305.1953-2-54-0': 'Notably, instances of teleportation [CITATION] are possible.', '1305.1953-2-54-1': 'Meaningful variants of teleportation involving Majorana fermions should share the features that (i) an unknown state is considered the input, taken from a set of at least two non-orthogonal quantum states, and (ii) the output should be statistically indistinguishable from the input to the protocol.', '1305.1953-2-55-0': 'Here, Alice and Bob not only share [MATH] Majorana pairs, so as usual, the initial state is stabilised by [MATH] for [MATH].', '1305.1953-2-55-1': 'But furthermore, Alice holds further [MATH] Majorana modes that are not entangled with Bob.', '1305.1953-2-55-2': 'These local ancilla are stabilised by [EQUATION] and represent the modes that will be teleported, herein called the input modes.', '1305.1953-2-55-3': 'Such a scheme can indeed be devised by making use of 4 Majorana pairs, so [MATH].', '1305.1953-2-55-4': 'The input register consists of 4 Majorana modes, initially stabilised by [MATH] and [MATH].', '1305.1953-2-55-5': 'Many different inputs could be prepared by braiding the input modes.', '1305.1953-2-55-6': 'For instance, the input can be set to one of two non-orthogonal inputs by either (I) braiding [MATH] and [MATH] or (II) not.', '1305.1953-2-55-7': 'It is not difficult to see that these two situations (I) and (II) cannot be perfectly distinguished with unit probability.', '1305.1953-2-56-0': 'The remainder of the protocol does not depend on this input, and neither is knowledge of it required.', '1305.1953-2-56-1': 'In the next step, [MATH], [MATH], [MATH], [MATH] are measured and the results, elements of [MATH], classically communicated to Bob.', '1305.1953-2-56-2': 'One can then show that the output statistics of measurements performed by Bob, [MATH] and [MATH], appropriately interpreted using the received classical bits, is indistinguishable from the respective measurements on [MATH] and [MATH].', '1305.1953-2-56-3': 'For example, in scenario (II), if the outcomes [MATH] are communicated, the outcomes of measurements [MATH] and [MATH] are completely determined to be [MATH].', '1305.1953-2-57-0': '## Dense coding', '1305.1953-2-58-0': 'In a similar fashion, dense coding [CITATION] can be performed with Majorana fermions and the above specified operations.', '1305.1953-2-58-1': 'Again, let us be specific what a fair analogue of such a scheme would be.', '1305.1953-2-58-2': 'A valid dense coding scheme is one in which the entanglement-assisted single shot classical capacity of a quantum channel is higher compared to the corresponding capacity in the absence of entanglement.', '1305.1953-2-58-3': 'We hence would like to introduce a protocol where starting from entangled shared resources, with local braiding operations and a subsequent transmission of some modes, one can encode more bits of classical information than is possible without having shared entangled resources available, but transmitting the same number of modes.', '1305.1953-2-58-4': 'For qudits, it is known that the entanglement-assisted capacity is exactly double the one reachable without assistance [CITATION].', '1305.1953-2-58-5': 'For Majorana fermions, it turns out, the same holds true.', '1305.1953-2-59-0': 'This is readily possible with 3 Majorana pairs, again partially shared by Alice and Bob and partially held by Alice.', '1305.1953-2-59-1': 'Both parties share 2 Majorana pairs, stabilised by [MATH], [MATH].', '1305.1953-2-59-2': 'In addition, Alice holds 2 Majorana modes in a state stabilised by [MATH].', '1305.1953-2-59-3': 'By local braiding, Alice can achieve a state stabilised by [EQUATION] in a scheme that maps [EQUATION] [MATH] for [MATH], and fixing [MATH] and [MATH].', '1305.1953-2-59-4': 'It is easy to see that with a suitable choice of [MATH], 2 bits of information can be encoded.', '1305.1953-2-59-5': 'Furthermore, after Alice transmits Majorana modes [MATH] and [MATH] to Bob, he can reliably retrieve [MATH] and [MATH] by locally measuring [MATH] and [MATH].', '1305.1953-2-59-6': 'At the same time, with two Majorana modes and no shared entanglement, a single bit of information can be encoded only.', '1305.1953-2-59-7': 'This constitutes a valid dense coding scheme based on Majorana fermions.', '1305.1953-2-59-8': 'It also resembles the situation of transmitting two bits of classical information with a single transmitted qubit, if entangled resources are initially available.', '1305.1953-2-60-0': '# Closing remarks', '1305.1953-2-61-0': 'We have seen that locality can be violated using only the topologically protected operations of Majorana fermions or Ising anyons.', '1305.1953-2-61-1': 'These operations are a subclass of fermionic linear optics [CITATION], and so we conclude that these systems can also violate locality.', '1305.1953-2-61-2': 'The model of fermionic linear optics was proposed as an analog of bosonic linear optics, where a well-known LHV theory prevents any non-locality experiment [CITATION].', '1305.1953-2-61-3': 'Seen in this light, our result is quite surprising.', '1305.1953-2-61-4': 'Furthermore, we remark that that fractionalized analogs of Majorana fermions, known as parafermions [CITATION], may face similar obstacles from the existence of certain hidden variable theories [CITATION].', '1305.1953-2-61-5': 'It is natural to ask more generally what other anyonic systems are non-local, again without the requirement of universal quantum computing capabilities.', '1305.1953-2-61-6': 'We know of only one related study [CITATION], where a six-level classical spin model was used to simulate the charge submodel of [MATH] (the quantum double model based on the symmetric group of 3 objects).', '1305.1953-2-61-7': 'Although non-locality was not explicitly discussed, it is clear that the classical spin model amounts to a LHV theory.', '1305.1953-2-61-8': 'Finally, we speculate that these experiments could prove useful as a probe to certify the presence of Majorana fermions and potentially help dispel some of the present ambiguity surrounding experiments.', '1305.1953-2-62-0': 'In the final stages of completing this research we became aware of Ref. [CITATION], which addresses to some extent similar questions and concerns the non-locality of Majorana fermions when nondestructive collective-charge measurements are available.', '1305.1953-2-62-1': 'This represents an additional experimental capability, posing additional challenges, beyond braiding and standard measurements.', '1305.1953-2-62-2': 'However, without this capability, the GHZ correlations required by Ref. [CITATION] cannot be implemented by virtue of our no-go result of Sec. [REF].'}

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'1305.1953-2-43-12'], ['1305.1953-1-40-13', '1305.1953-2-43-13'], ['1305.1953-1-40-14', '1305.1953-2-43-14'], ['1305.1953-1-40-15', '1305.1953-2-43-15'], ['1305.1953-1-40-16', '1305.1953-2-43-16'], ['1305.1953-1-40-17', '1305.1953-2-43-17'], ['1305.1953-1-56-0', '1305.1953-2-58-0'], ['1305.1953-1-56-1', '1305.1953-2-58-1'], ['1305.1953-1-56-2', '1305.1953-2-58-2'], ['1305.1953-1-56-4', '1305.1953-2-58-4'], ['1305.1953-1-56-5', '1305.1953-2-58-5'], ['1305.1953-1-18-0', '1305.1953-2-21-0'], ['1305.1953-1-18-1', '1305.1953-2-21-1'], ['1305.1953-1-0-1', '1305.1953-2-3-1'], ['1305.1953-1-0-2', '1305.1953-2-3-2'], ['1305.1953-1-0-3', '1305.1953-2-3-3'], ['1305.1953-1-0-4', '1305.1953-2-3-4'], ['1305.1953-1-0-5', '1305.1953-2-3-5'], ['1305.1953-1-0-6', '1305.1953-2-3-6'], ['1305.1953-1-4-0', '1305.1953-2-7-0'], ['1305.1953-1-4-1', '1305.1953-2-7-1'], ['1305.1953-1-4-2', '1305.1953-2-7-2'], ['1305.1953-1-4-3', '1305.1953-2-7-3'], ['1305.1953-1-4-5', '1305.1953-2-7-5'], ['1305.1953-1-4-6', '1305.1953-2-7-6'], ['1305.1953-1-4-11', '1305.1953-2-7-11'], ['1305.1953-1-4-12', '1305.1953-2-7-12'], ['1305.1953-1-10-0', '1305.1953-2-13-0'], ['1305.1953-1-10-1', '1305.1953-2-13-1'], ['1305.1953-1-10-2', '1305.1953-2-13-2'], ['1305.1953-1-10-3', '1305.1953-2-13-3'], ['1305.1953-1-59-0', '1305.1953-2-61-0'], ['1305.1953-1-59-1', '1305.1953-2-61-1'], ['1305.1953-1-59-2', '1305.1953-2-61-2'], ['1305.1953-1-59-3', '1305.1953-2-61-3'], ['1305.1953-1-59-4', '1305.1953-2-61-4'], ['1305.1953-1-59-5', '1305.1953-2-61-5'], ['1305.1953-1-59-6', '1305.1953-2-61-8'], ['1305.1953-1-3-0', '1305.1953-2-6-0'], ['1305.1953-1-3-1', '1305.1953-2-6-1'], ['1305.1953-1-3-2', '1305.1953-2-6-2'], ['1305.1953-1-3-3', '1305.1953-2-6-3'], ['1305.1953-1-3-4', '1305.1953-2-6-4'], ['1305.1953-1-3-5', '1305.1953-2-6-5'], ['1305.1953-1-3-6', '1305.1953-2-6-6'], ['1305.1953-1-3-7', '1305.1953-2-6-7'], ['1305.1953-1-33-0', '1305.1953-2-36-0'], ['1305.1953-1-33-2', '1305.1953-2-36-2'], ['1305.1953-1-33-3', 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'1305.1953-2-49-9'], ['1305.1953-1-13-0', '1305.1953-2-16-0'], ['1305.1953-1-13-1', '1305.1953-2-16-1'], ['1305.1953-1-13-3', '1305.1953-2-16-3'], ['1305.1953-1-13-4', '1305.1953-2-16-4'], ['1305.1953-1-13-5', '1305.1953-2-16-5'], ['1305.1953-1-52-0', '1305.1953-2-54-0'], ['1305.1953-1-52-1', '1305.1953-2-54-1'], ['1305.1953-1-14-0', '1305.1953-2-17-0'], ['1305.1953-1-14-1', '1305.1953-2-17-1'], ['1305.1953-1-14-2', '1305.1953-2-17-2'], ['1305.1953-1-14-3', '1305.1953-2-17-3'], ['1305.1953-1-54-0', '1305.1953-2-56-0'], ['1305.1953-1-54-1', '1305.1953-2-56-1'], ['1305.1953-1-54-2', '1305.1953-2-56-2'], ['1305.1953-1-54-3', '1305.1953-2-56-3'], ['1305.1953-1-22-0', '1305.1953-2-25-0'], ['1305.1953-1-22-1', '1305.1953-2-25-1'], ['1305.1953-1-22-2', '1305.1953-2-25-2'], ['1305.1953-1-22-3', '1305.1953-2-25-3'], ['1305.1953-1-22-4', '1305.1953-2-25-4'], ['1305.1953-1-57-0', '1305.1953-2-59-0'], ['1305.1953-1-57-4', '1305.1953-2-59-4'], ['1305.1953-1-57-5', '1305.1953-2-59-5'], ['1305.1953-1-57-6', '1305.1953-2-59-6'], ['1305.1953-1-57-7', '1305.1953-2-59-7'], ['1305.1953-1-57-8', '1305.1953-2-59-8'], ['1305.1953-1-53-1', '1305.1953-2-55-1'], ['1305.1953-1-53-3', '1305.1953-2-55-3'], ['1305.1953-1-53-5', '1305.1953-2-55-5'], ['1305.1953-1-53-7', '1305.1953-2-55-7'], ['1305.1953-1-48-2', '1305.1953-2-50-2'], ['1305.1953-1-48-3', '1305.1953-2-50-3'], ['1305.1953-1-48-4', '1305.1953-2-50-4'], ['1305.1953-1-48-5', '1305.1953-2-50-5'], ['1305.1953-1-48-6', '1305.1953-2-50-6'], ['1305.1953-1-48-7', '1305.1953-2-50-7'], ['1305.1953-1-48-9', '1305.1953-2-50-9'], ['1305.1953-1-48-10', '1305.1953-2-50-10'], ['1305.1953-1-48-11', '1305.1953-2-50-11'], ['1305.1953-1-35-0', '1305.1953-2-38-0'], ['1305.1953-1-35-1', '1305.1953-2-38-1'], ['1305.1953-1-35-2', '1305.1953-2-38-2'], ['1305.1953-1-35-3', '1305.1953-2-38-3'], ['1305.1953-1-35-4', '1305.1953-2-38-4'], ['1305.1953-1-35-5', '1305.1953-2-38-5'], ['1305.1953-1-5-1', '1305.1953-2-8-1'], ['1305.1953-1-5-2', '1305.1953-2-8-2'], ['1305.1953-1-5-3', '1305.1953-2-8-3'], ['1305.1953-1-5-4', '1305.1953-2-8-4'], ['1305.1953-1-5-5', '1305.1953-2-8-5'], ['1305.1953-1-5-6', '1305.1953-2-8-6'], ['1305.1953-1-5-7', '1305.1953-2-8-7'], ['1305.1953-1-5-8', '1305.1953-2-8-8'], ['1305.1953-1-5-9', '1305.1953-2-8-9'], ['1305.1953-1-27-0', '1305.1953-2-30-0'], ['1305.1953-1-27-1', '1305.1953-2-30-1'], ['1305.1953-1-27-2', '1305.1953-2-30-2'], ['1305.1953-1-27-3', '1305.1953-2-30-3'], ['1305.1953-1-27-4', '1305.1953-2-30-4'], ['1305.1953-1-27-5', '1305.1953-2-30-5'], ['1305.1953-1-27-6', '1305.1953-2-30-6'], ['1305.1953-1-37-0', '1305.1953-2-40-0'], ['1305.1953-1-37-1', '1305.1953-2-40-1'], ['1305.1953-1-37-2', '1305.1953-2-40-2'], ['1305.1953-1-37-3', '1305.1953-2-40-3'], ['1305.1953-1-37-4', '1305.1953-2-40-4'], ['1305.1953-1-23-0', '1305.1953-2-26-0'], ['1305.1953-1-23-1', '1305.1953-2-26-1'], ['1305.1953-1-23-3', '1305.1953-2-26-3'], ['1305.1953-1-23-4', '1305.1953-2-26-4'], ['1305.1953-1-23-7', '1305.1953-2-26-7'], ['1305.1953-1-23-8', '1305.1953-2-26-8'], ['1305.1953-1-2-0', '1305.1953-2-5-0'], ['1305.1953-1-2-2', '1305.1953-2-5-2'], ['1305.1953-1-2-3', '1305.1953-2-5-3'], ['1305.1953-1-2-4', '1305.1953-2-5-4'], ['1305.1953-1-24-0', '1305.1953-2-27-0'], ['1305.1953-1-24-2', '1305.1953-2-27-2'], ['1305.1953-1-24-4', '1305.1953-2-27-4'], ['1305.1953-1-24-5', '1305.1953-2-27-5'], ['1305.1953-1-24-6', '1305.1953-2-27-6'], ['1305.1953-1-50-2', '1305.1953-2-52-2'], ['1305.1953-1-50-3', '1305.1953-2-52-3'], ['1305.1953-1-42-0', '1305.1953-2-45-0'], ['1305.1953-1-42-1', '1305.1953-2-45-1'], ['1305.1953-1-42-2', '1305.1953-2-45-2'], ['1305.1953-1-42-3', '1305.1953-2-45-3'], ['1305.1953-1-42-4', '1305.1953-2-45-4'], ['1305.1953-1-42-5', '1305.1953-2-45-5'], ['1305.1953-1-42-6', '1305.1953-2-45-6'], ['1305.1953-1-42-7', '1305.1953-2-45-7'], ['1305.1953-1-42-8', '1305.1953-2-45-8'], ['1305.1953-1-42-9', '1305.1953-2-45-9'], ['1305.1953-1-36-0', '1305.1953-2-39-0'], ['1305.1953-1-36-2', '1305.1953-2-39-2'], ['1305.1953-1-36-3', '1305.1953-2-39-3'], ['1305.1953-1-36-4', '1305.1953-2-39-4'], ['1305.1953-1-29-0', '1305.1953-2-32-0'], ['1305.1953-1-29-1', '1305.1953-2-32-1'], ['1305.1953-1-25-0', '1305.1953-2-28-0'], ['1305.1953-1-25-1', '1305.1953-2-28-1'], ['1305.1953-1-25-2', '1305.1953-2-28-2'], ['1305.1953-1-25-3', '1305.1953-2-28-3']]

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[]

[]

[]

['1305.1953-1-45-0', '1305.1953-1-46-0', '1305.1953-2-2-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1305.1953

1806.00361

{'1806.00361-1-0-0': 'In this paper, we discuss the charged anti-de Sitter(AdS) black holes where metric depends on the energy of a probe.', '1806.00361-1-0-1': 'By the Heisenberg uncertainty principle and the modified dispersion relation, we achieve deformed temperature.', '1806.00361-1-0-2': 'Moreover, in rainbow gravity we calculate the heat capacity in a fixed charge and discuss the thermal stability.', '1806.00361-1-0-3': 'We also achieve similar behaviour with this system of the liquid-gas system in extending phase space (including [MATH] and [MATH] ) and study its critical behavior with the pressure given by the cosmological constant and with a fixed black hole charge [MATH] .', '1806.00361-1-0-4': 'Furthermore, we study the Gibbs function and find characteristic swallow tail behavior, which indicates there is the phase transition.', '1806.00361-1-0-5': 'We also find there is a special value about the the mass of test particle which would make the black hole zero temperature and diverging heat capacity with fixed charge.', '1806.00361-1-1-0': '# Introduction', '1806.00361-1-2-0': 'It is known Lorentz symmetry is one of most important symmetries in nature, however, some researches indicate the Lorentz symmetry might be violated in the ultraviolet limit [CITATION].', '1806.00361-1-2-1': 'Since the standard energy-momentum dispersion relation relates to the Lorentz symmetry, the deformation of Lorentz symmetry would impel the modification of energy-momentum dispersion relation.', '1806.00361-1-2-2': 'In fact, some calculations in loop quantum gravity have showed the dispersion relations may be deformed.', '1806.00361-1-2-3': 'Meanwhile, based on the deformed energy-momentum dispersion relation the double special relativity has arisen [CITATION].', '1806.00361-1-2-4': 'In this theory, in addition to the velocity of light being the maximum velocity attainable there is another constant for maximum energy scale in nature which is the Planck energy [MATH] .', '1806.00361-1-2-5': 'It gives different picture for the special relativity in microcosmic physics.', '1806.00361-1-2-6': "The theory has been generalized to curved spacetime by Joao Magueijo and Lee Smolin [CITATION] and is called gravity's rainbow.", '1806.00361-1-2-7': 'In this theory, the geometry of spacetime depends on the energy of the test particle and observers of different energy would see different geometry of spacetime.', '1806.00361-1-2-8': 'Hence, a family of energy-dependent metrics named as rainbow metrics will describe the geometry of spacetime, which is different from general gravity theory.', '1806.00361-1-2-9': 'Based on the non-linear of Lorentz transformation, the energy-momentum dispersion relation rewrites as [EQUATION] where [MATH] is the Planck energy.', '1806.00361-1-2-10': 'The rainbow functions [MATH] and [MATH] are required to satisfy [EQUATION]', '1806.00361-1-2-11': 'In this case, the deformed energy moment relationship Eq. ([REF]) will go back to classical one when the energy of the test particle is much lower than [MATH] .', '1806.00361-1-2-12': "Due to this energy-dependent modification to the dispersion relation, the metric [MATH] in gravity's rainbow will be rewritten as [CITATION] [EQUATION] here the energy dependence of the frame fields can be written as [EQUATION] where the tilde quantities refer to the energy-independent frame fields.", '1806.00361-1-2-13': 'This then leads to a one-parameter Einstein equation [EQUATION] where [MATH] and [MATH] is energy-dependent Einstein tensor and energy-momentum tensor, [MATH] and [MATH] is an energy-dependent Newton constant and cosmological constant.', '1806.00361-1-2-14': 'Generally, lots of forms of rainbow function have been discussed in literatures, in this paper we will mainly employ following rainbow function [EQUATION] which has been widely used in Refs [CITATION].', '1806.00361-1-3-0': 'Recently, some literatures have also studied the Schwarzschild black holes, Schwarzschild AdS black holes and Reissner-Nordstrom black holes in rainbow gravity [CITATION].', '1806.00361-1-3-1': 'In this paper, we will extend the discussion of the Ref.[CITATION] for charged AdS black holes in rainbow gravity.', '1806.00361-1-3-2': 'Compared with Ref.[CITATION], we will further study the charged AdS black holes in rainbow gravity and discuss the critical behavior and phase transition.', '1806.00361-1-4-0': 'The paper is organized as follows.', '1806.00361-1-4-1': 'In the next section, by using the Heisenberg Uncertainty Principle(HUP) and the modified dispersion relation, we obtain deformed temperature, we also calculate heat capacity with a fixed charge and discuss the thermal stability.', '1806.00361-1-4-2': 'In Sec.3, we find the charged AdS black holes have similar behaviour with the liquid-gas system with the pressure given by the cosmological constant while we treat the black holes charge [MATH] as a fixed external parameter, not a thermodynamic variable.', '1806.00361-1-4-3': 'We also calculate the Gibbs free energy and find characteristic swallow tail behavior.', '1806.00361-1-4-4': 'Finally, the conclusion and discussion will be offered in Sec.4.', '1806.00361-1-5-0': '# The thermal stability', '1806.00361-1-6-0': 'In rainbow gravity the line element of the modified charged AdS black holes can be described as [CITATION] [EQUATION] where [EQUATION] here, [MATH] which [MATH] is cosmological constant.', '1806.00361-1-6-1': 'Because all energy dependence in the energy-independent coordinates must be in the rainbow functions [MATH] and [MATH] , [MATH] is independent on the energy of test particle [CITATION].', '1806.00361-1-6-2': "In gravity's rainbow, the modified temperature related to the standard temperature [MATH] was [CITATION] [EQUATION] here, [MATH] is event horizon.", '1806.00361-1-7-0': "In gravity's rainbow, although the metric depends on the energy of test particle, the usual HUP still holds [CITATION].", '1806.00361-1-7-1': 'The uncertainty of momentum near the horizon can be described as [CITATION] [MATH] .', '1806.00361-1-7-2': 'So when we choose [MATH] in rainbow function, through Eq. ([REF]) and Eq. ([REF]) we can get [EQUATION] here, [MATH] , [MATH] is the mass of test particle and [MATH] is a constant parameter.', '1806.00361-1-7-3': 'When the test particle is photon, [MATH] .', '1806.00361-1-8-0': 'When [MATH] , the standard temperature was given as [CITATION] [EQUATION]', '1806.00361-1-8-1': 'So when using the Eq([REF]) and [MATH] , we can get the temperature of charged AdS black holes in rainbow gravity [EQUATION] here [MATH] .', '1806.00361-1-8-2': 'It is easy to find [MATH] when [MATH] .', '1806.00361-1-8-3': 'The Eq. ([REF]) shows there are two solution when [MATH] , one corresponds to extreme black hole, another corresponds to [MATH] .', '1806.00361-1-8-4': 'In terms of the second solution, we will give some discussions.', '1806.00361-1-8-5': 'The result indicates the temperature of black holes completely depends on the mass of the test particle when the black holes keep with fixed mass, charge and anti-de Sitter radius.', '1806.00361-1-8-6': 'The bigger the mass of test particle is, the smeller the temperature of black holes is.', '1806.00361-1-8-7': 'When [MATH] , the temperature keeps zero.', '1806.00361-1-8-8': 'Generally, due to gravitys rainbow a minimum radius is given with respect to black holes when the temperature tends to zero, however, the paper shows all black holes can keep zero temperature when the test particle mass approaches a value, such as, [MATH] .', '1806.00361-1-8-9': 'But due to [MATH] in general condition, so it may be difficult to test the phenomenon with zero temperature about black holes.', '1806.00361-1-9-0': 'In generally, the thermal stability can be determined by the heat capacity, the thought also be used to the systems of black holes [CITATION].', '1806.00361-1-9-1': 'In other words, the positive heat capacity for black holes corresponds to a stable state and the negative heat capacity for black holes corresponds to unstable state.', '1806.00361-1-9-2': 'In following discussion, we will focus on the heat capacity to discuss the stability of black holes.', '1806.00361-1-9-3': 'We can calculate the mass of charged AdS black holes through [MATH] as [EQUATION]', '1806.00361-1-9-4': 'The modified entropy can also be computed from the first law [MATH] with the modified temperature as[CITATION] [EQUATION]', '1806.00361-1-9-5': 'Note that the next leading order is logarithmic as [MATH] , which is similar to the quantum correction in Refs[CITATION].', '1806.00361-1-9-6': 'With [MATH] we can achieve [MATH] .', '1806.00361-1-9-7': 'We can find the result is in agreement with standard entropy [MATH] when [MATH] which is standard condition.', '1806.00361-1-10-0': 'The heat capacity with a fixed charge can be calculated as [EQUATION] which shows that [MATH] reduces to standard condition [CITATION] with [MATH] .', '1806.00361-1-10-1': 'It is easy to find the heat capacity is diverging when [MATH] .', '1806.00361-1-10-2': 'Generally, when the temperature tends to zero, the heat capacity also tends to zero.', '1806.00361-1-10-3': 'However, our paper show a different and anomalous phenomenon.', '1806.00361-1-10-4': 'It may be explained as the black holes could not be observed when test particle mass encounters [MATH] .', '1806.00361-1-10-5': 'It is lucky the phenomenon is just a observation effect, but the result gives us a way to test the theory of rainbow gravity.', '1806.00361-1-11-0': 'The numerical methods indicate there is special point corresponding to one divergent point of heat capacity with [MATH] , there are two divergent points when [MATH] and no divergent point with [MATH] which have been described in Fig.[REF], Fig.[REF], Fig.[REF].', '1806.00361-1-11-1': 'Through Eq.([REF]), we get the different expression compared with Ref.[CITATION].', '1806.00361-1-11-2': "The Fig.[REF] shows there is a continuous phase and does't appear phase transition with [MATH] .", '1806.00361-1-11-3': 'In the Fig.[REF], there is one divergent point with [MATH] , it shows two stable phases for [MATH] ,phase 1 and phase 2, which individually represents a phase of large black hole(LBH) and a small black hole(SBH).', '1806.00361-1-11-4': 'In Fig.[REF], we can find there are three phases and two divergent points with [MATH] .', '1806.00361-1-11-5': 'Phase 1 experiences a continuous process from a unstable phase for [MATH] to a stable phase for [MATH] ; phase 2 is a pure unstable phase for [MATH] ; and phase 3 is a stable phase for [MATH] .', '1806.00361-1-11-6': 'It is easy to find phase 1 represents the phase of SBH and phase 3 represents the phase of LBH.', '1806.00361-1-11-7': 'However, there is a special unstable phase 2 between phase 1 and phase 3.', '1806.00361-1-11-8': 'It indicates if the system experiences from phase 3 to phase 1, the system must experience a medium unstable state.', '1806.00361-1-11-9': 'The state may be explained as an exotic quark-gluon plasma with negative heat capacity [CITATION].', '1806.00361-1-12-0': '# The phase transition of Charged AdS black holes in extending phase space', '1806.00361-1-13-0': "Surprisingly, although rainbow functions modify the [MATH] term, they don't affect thermodynamical pressure related to the cosmological constant [CITATION].", '1806.00361-1-13-1': 'So we can take the following relation [EQUATION]', '1806.00361-1-13-2': 'Since David Kubiznak and Robert B. Mann have showed the critical behaviour of charged-AdS black holes and completed the analogy of this system with the liquid-gas system [CITATION], in what follows we will study whether the critical behavior of the charged AdS black holes system in rainbow gravity was kept.', '1806.00361-1-13-3': 'Using Eq.([REF]) and Eq.([REF]) in extended phase space, we can get [EQUATION]', '1806.00361-1-13-4': 'We will follow the method discussed by Ref.[CITATION] to study the critical behaviour.', '1806.00361-1-13-5': 'The critical point is obtained from [EQUATION] which leads to [EQUATION] here [MATH] , [MATH] .', '1806.00361-1-13-6': 'We can achieve [EQUATION] which shows the critical ratio was deformed due to the existence of rainbow gravity.', '1806.00361-1-13-7': 'It is notable that Eq. ([REF]) will back to the usual universal ratio with [MATH] .', '1806.00361-1-13-8': 'Generally, in charged AdS black holes the pressure and temperature were demanded as positive real value.', '1806.00361-1-13-9': 'From Eq. ([REF]), we can find with [MATH] and [MATH] [EQUATION]', '1806.00361-1-13-10': 'It indicates there is a restriction between [MATH] and [MATH] when there are the positive real critical values.', '1806.00361-1-14-0': 'The [MATH] diagram has been described in Fig.[REF].', '1806.00361-1-14-1': 'From Fig.[REF], we can find that charged AdS black holes in rainbow gravity have an analogy with the Van der Waals system and have a first-order phase transition with [MATH] .', '1806.00361-1-14-2': 'Namely, when considering rainbow gravity with the form of Eq.([REF]), the behavior like Van der Waals system also reserve.', '1806.00361-1-15-0': 'We also can discuss the phase transition from the Gibbs free energy, through Ref.[CITATION] the black hole mass is then identified with the enthalpy, rather than the internal energy, so the Gibbs free energy for fixed charge in the rainbow gravity will be [EQUATION]', '1806.00361-1-15-1': 'It has been showed in the Fig.[REF].', '1806.00361-1-15-2': 'Since the picture of [MATH] demonstrates the characteristic swallow tail behaviour, there is a first order transition in the system.', '1806.00361-1-16-0': '# Conclusion', '1806.00361-1-17-0': 'In this paper, we have discussed the charged AdS black holes with the test particle with mass [MATH] .', '1806.00361-1-17-1': 'By the modified dispersion relation and HUP, we got deformed temperature in charged AdS black holes.', '1806.00361-1-17-2': 'We have found that the temperature is just a rescaling for different mass of test particle.', '1806.00361-1-17-3': 'We have discussed the divergence about the heat capacity with a fixed charge.', '1806.00361-1-17-4': 'It indicates the numbers of phase will be changing as the [MATH] takes different values.', '1806.00361-1-17-5': 'When ( [MATH] ), there is only one divergent point about heat capacity; with [MATH] , we have found there are two divergent points and three phases including two stable phases and one unstable phase.', '1806.00361-1-17-6': 'In particular, we have finished an analogy between the charged AdS black holes in the rainbow gravity in extending phase space and the liquid-gas system.', '1806.00361-1-17-7': 'We have also studied [MATH] critical behavior about the charged AdS black holes in the rainbow gravity.', '1806.00361-1-17-8': 'The consequence shows there is the Van der Waals like behavior in the rainbow gravity when [MATH] and [MATH] coincide with Eq. ([REF]).', '1806.00361-1-17-9': 'But the rainbow function deform the consequence of critical pressure,temperature and radius.', '1806.00361-1-17-10': "At last, we have discussed the Gibbs free energy and have obtained characteristic 'swallow tail' behaviour that can be explained first-order phase transition.", '1806.00361-1-18-0': 'We also find the temperature tends to zero and heat capacity with a fixed charge tends to diverging with [MATH] .', '1806.00361-1-18-1': 'Those results indicate there is a special value of mass about test particle encountered [MATH] which would test zero temperature and diverging heat capacity.', '1806.00361-1-18-2': 'Those results also indicate whether there is a black hole remnant depends on how to choose rainbow function [MATH] and [MATH] and HUP.', '1806.00361-1-18-3': 'In generally, when we choose [MATH] , the black hole remnant may exit.', '1806.00361-1-18-4': 'But when we choose [MATH] , the result may be different.', '1806.00361-1-19-0': '# Conflicts of Interest', '1806.00361-1-20-0': 'The authors declare that there are no conflicts of interest regarding the publication of this paper.'}

{'1806.00361-2-0-0': 'In this paper, the thermodynamic property of charged AdS black holes is studied in rainbow gravity.', '1806.00361-2-0-1': 'By the Heisenberg Uncertainty Principle and the modified dispersion relation, we obtain deformed temperature.', '1806.00361-2-0-2': 'Moreover, in rainbow gravity we calculate the heat capacity in a fixed charge and discuss the thermal stability.', '1806.00361-2-0-3': 'We also obtain a similar behaviour with the liquid-gas system in extending phase space (including [MATH] and [MATH] ) and study its critical behavior with the pressure given by the cosmological constant and with a fixed black hole charge [MATH] .', '1806.00361-2-0-4': 'Furthermore, we study the Gibbs function and find its characteristic swallow tail behavior, which indicates the phase transition.', '1806.00361-2-0-5': 'We also find there is a special value about the mass of test particle which would lead the black hole to zero temperature and a diverging heat capacity with a fixed charge.', '1806.00361-2-1-0': '# Introduction', '1806.00361-2-2-0': 'It is known Lorentz symmetry is one of most important symmetries in nature, however, some researches indicate the Lorentz symmetry might be violated in the ultraviolet limit [CITATION].', '1806.00361-2-2-1': 'Since the standard energy-momentum dispersion relation relates to the Lorentz symmetry, the deformation of Lorentz symmetry would lead to the modification of energy-momentum dispersion relation.', '1806.00361-2-2-2': 'In fact, some calculations in loop quantum gravity have showed the dispersion relations may be deformed.', '1806.00361-2-2-3': 'Meanwhile, based on the deformed energy-momentum dispersion relation the double special relativity has arisen [CITATION].', '1806.00361-2-2-4': 'In this theory, in addition to the velocity of light being the maximum velocity attainable there is another constant for maximum energy scale in nature which is the Planck energy [MATH] .', '1806.00361-2-2-5': 'It gives different picture for the special relativity in microcosmic physics.', '1806.00361-2-2-6': "The theory has been generalized to curved spacetime by Joao Magueijo and Lee Smolin, called gravity's rainbow [CITATION].", '1806.00361-2-2-7': 'In their theory, the geometry of spacetime depends on the energy of the test particle and observers of different energy would see different geometry of spacetime.', '1806.00361-2-2-8': 'Hence, a family of energy-dependent metrics named as rainbow metrics will describe the geometry of spacetime, which is different from general gravity theory.', '1806.00361-2-2-9': 'Based on the non-linear of Lorentz transformation, the energy-momentum dispersion relation can be rewritten as [EQUATION] where [MATH] is the Planck energy.', '1806.00361-2-2-10': 'The rainbow functions [MATH] and [MATH] are required to satisfy [EQUATION]', '1806.00361-2-2-11': 'In this case, the deformed energy-momentum dispersion relation Eq. ([REF]) will go back to classical one when the energy of the test particle is much lower than [MATH] .', '1806.00361-2-2-12': "Due to this energy-dependent modification to the dispersion relation, the metric [MATH] in gravity's rainbow could be rewritten as [CITATION] [EQUATION] where the energy dependence of the frame fields are [EQUATION] here the tilde quantities refer to the energy-independent frame fields.", '1806.00361-2-2-13': 'This leads to a one-parameter Einstein equation [EQUATION] where [MATH] and [MATH] are energy-dependent Einstein tensor and energy-momentum tensor, [MATH] and [MATH] are energy-dependent cosmological constant and Newton constant.', '1806.00361-2-2-14': 'Generally, many forms of rainbow functions have been discussed in literatures, in this paper we will mainly employ the following rainbow functions [EQUATION] which has been widely used in Refs [CITATION].', '1806.00361-2-3-0': 'Recently, Schwarzschild black holes, Schwarzschild AdS black holes and Reissner-Nordstrom black holes in rainbow gravity [CITATION] have been studied.', '1806.00361-2-3-1': 'Ahmed Farag Alia, Mir Faizald and Mohammed M.Khalile [CITATION] studied the deformed temperature about charged AdS black holes in rainbow gravity based on Heisenberg Uncertainty Principle(HUP), [MATH].', '1806.00361-2-3-2': 'In this paper, we study the thermodynamical property about the charged AdS black holes in rainbow gravity based on the usual HUP, [MATH] .', '1806.00361-2-3-3': 'Moreover, we study how the mass of test particle influences thermodynamical property for charged AdS black holes.', '1806.00361-2-4-0': 'The paper is organized as follows.', '1806.00361-2-4-1': 'In the next section, by using the HUP and the modified dispersion relation, we obtain deformed temperature, we also calculate heat capacity with a fixed charge and discuss the thermal stability.', '1806.00361-2-4-2': 'In Sec.3, we find the charged AdS black holes have similar behaviour with the liquid-gas system with the pressure given by the cosmological constant while we treat the black holes charge [MATH] as a fixed external parameter, not a thermodynamic variable.', '1806.00361-2-4-3': 'We also calculate the Gibbs free energy and find characteristic swallow tail behavior.', '1806.00361-2-4-4': 'Finally, the conclusion and discussion will be offered in Sec.4.', '1806.00361-2-5-0': '# The thermal stability', '1806.00361-2-6-0': 'In rainbow gravity the line element of the modified charged AdS black holes can be described as [CITATION] [EQUATION] where [EQUATION]', '1806.00361-2-6-1': 'Generally, [MATH] which is cosmological constant.', '1806.00361-2-6-2': 'Because all energy dependence in the energy-independent coordinates must be in the rainbow functions [MATH] and [MATH] , [MATH] is independent on the energy of test particle [CITATION].', '1806.00361-2-6-3': "In gravity's rainbow, the deformed temperature related to the standard temperature [MATH] was [CITATION] [EQUATION] where [MATH] is horizon radius.", '1806.00361-2-7-0': "In gravity's rainbow, although the metric depends on the energy of test particle, the usual HUP can be still used [CITATION].", '1806.00361-2-7-1': 'For simplicity we take [MATH] in the following discussion, by combining Eq. ([REF]) with Eq. ([REF]) we can get [EQUATION] where [MATH] , [MATH] is the mass of test particle and [MATH] is a constant parameter.', '1806.00361-2-8-0': 'Generally, the standard temperature was given by [CITATION] [EQUATION]', '1806.00361-2-8-1': 'When using Eqs. ([REF]) and ([REF]), we can get the temperature of charged AdS black holes in rainbow gravity [EQUATION] where [MATH] .', '1806.00361-2-8-2': 'It is easy to find [MATH] when [MATH] .', '1806.00361-2-8-3': 'Eq. ([REF]) shows there are two solutions when [MATH] , one corresponds to extreme black hole, the other to [MATH] .', '1806.00361-2-8-4': 'The second solution indicates the temperature of black holes completely depends on the mass of test particle when the black holes keep with fixed mass, charge and anti-de Sitter radius.', '1806.00361-2-8-5': 'The bigger the mass of test particle is, the smaller the temperature of black holes is.', '1806.00361-2-8-6': 'When [MATH] , the temperature keeps zero.', '1806.00361-2-8-7': "Generally, due to gravity's rainbow, a minimum radius with respect to the black hole is given and is related to a radius of black hole remnant when the temperature tends to zero [CITATION].", '1806.00361-2-8-8': 'However, our paper shows all black holes can keep zero temperature when the test particle mass approaches a value, such as, [MATH] .', '1806.00361-2-8-9': 'But due to [MATH] in general condition, it may be difficult to test the phenomenon with zero temperature about black holes.', '1806.00361-2-9-0': 'In general, the thermal stability can be determined by the heat capacity, which is also used to the systems of black holes [CITATION].', '1806.00361-2-9-1': 'In other words, the positive heat capacity corresponds to a stable state and the negative heat capacity corresponds to unstable state.', '1806.00361-2-9-2': 'In following discussions, we will focus on the heat capacity to discuss the stability of black holes.', '1806.00361-2-9-3': 'When [MATH] , the mass of charged AdS black holes can be calculated as [EQUATION]', '1806.00361-2-9-4': 'Based on the first law [MATH] with the deformed temperature[CITATION], the modified entropy can be computed [EQUATION]', '1806.00361-2-9-5': 'Note that the next leading order is logarithmic as [MATH] , which is similar to the quantum correction in Refs. [CITATION].', '1806.00361-2-9-6': 'With [MATH] we can get [MATH] .', '1806.00361-2-9-7': 'We can find the result is in agreement with the standard entropy [MATH] when [MATH] , which is standard condition.', '1806.00361-2-10-0': 'The heat capacity with a fixed charge can be calculated as [EQUATION] which shows that [MATH] reduces to standard condition [CITATION] with [MATH] .', '1806.00361-2-10-1': 'Obviously, the heat capacity is diverging when [MATH] .', '1806.00361-2-10-2': 'Generally, when the temperature vanishes, the heat capacity also tends to zero.', '1806.00361-2-10-3': 'However, our paper shows a different and anomalous phenomenon.', '1806.00361-2-10-4': 'Fortunately, the phenomenon is just an observation effect, the result gives us a way to test the theory of rainbow gravity.', '1806.00361-2-10-5': 'Some of the conditions above indicate that the mass of test particle does not influence the forms of temperature, entropy and heat capacity but only changes their amplitudes.', '1806.00361-2-11-0': 'The numerical methods indicate there are three situations corresponding to zero, one and two diverging points of heat capacity respectively, which have been described in Fig.[REF], Fig.[REF], Fig.[REF].', '1806.00361-2-11-1': 'Fig.[REF] shows a continuous phase and does not appear phase transition with [MATH] .', '1806.00361-2-11-2': 'In Fig.[REF], there is one diverging point and two stable phases for [MATH] with [MATH] , phase 1 and phase 2, which individually represents a phase of large black hole (LBH) and a small black hole (SBH).', '1806.00361-2-11-3': 'In Fig.[REF], one can find there are three phases and two diverging points with [MATH] .', '1806.00361-2-11-4': 'Phase 1 experiences a continuous process from a unstable phase [MATH] to a stable phase [MATH] ; phase 2 is a pure unstable phase with [MATH] ; phase 3 is a stable phase with [MATH] .', '1806.00361-2-11-5': 'It is easy to see phase 1 represents the phase of SBH and phase 3 represents the phase of LBH.', '1806.00361-2-11-6': 'However, there is a special unstable phase 2 between phase 1 and phase 3.', '1806.00361-2-11-7': 'This indicates when the system evolutes from phase 3 to phase 1, the system must experience a medium unstable state which could be explained as an exotic quark-gluon plasma with negative heat capacity [CITATION].', '1806.00361-2-12-0': '# The phase transition of Charged AdS black holes in extending phase space', '1806.00361-2-13-0': 'Surprisingly, although rainbow functions modify the [MATH] term, they do not affect thermodynamical pressure related to the cosmological constant [CITATION].', '1806.00361-2-13-1': 'So we can take the following relation [EQUATION]', '1806.00361-2-13-2': 'Since David Kubiznak and Robert B. Mann have showed the critical behaviour of charged AdS black holes and completed the analogy of this system with the liquid-gas system [CITATION], in what follows we will study whether the critical behavior of the charged AdS black holes system in rainbow gravity is kept.', '1806.00361-2-13-3': 'Using Eq.([REF]) and Eq.([REF]) in extended phase space, we can get [EQUATION]', '1806.00361-2-13-4': 'Similarly with Ref.[CITATION], the critical point is obtained from [EQUATION] which leads to [EQUATION] where [MATH] , [MATH] .', '1806.00361-2-13-5': 'We can obtain [EQUATION] which shows the critical ratio is deformed due to the existence of rainbow gravity.', '1806.00361-2-13-6': 'It is notable that Eq. ([REF]) will back to the usual ratio with [MATH] .', '1806.00361-2-13-7': 'Generally, for charged AdS black holes, the pressure and temperature are demanded as positive real value.', '1806.00361-2-13-8': 'From Eq.([REF]), when [MATH] and [MATH] , we have [EQUATION] which indicates a restriction between [MATH] and [MATH] .', '1806.00361-2-13-9': 'The [MATH] diagram has been described in Fig.[REF].', '1806.00361-2-13-10': 'From Fig.[REF], we can find that charged AdS black holes in rainbow gravity have an analogy with the Van-der-Waals system and have a first-order phase transition with [MATH] .', '1806.00361-2-13-11': 'Namely, when considering rainbow gravity with the form of Eq.([REF]), the behavior like Van-der-Waals system can also be obtained.', '1806.00361-2-14-0': 'Based on Ref.[CITATION] the black hole mass is identified with the enthalpy, rather than the internal energy, so the Gibbs free energy for fixed charge in the rainbow gravity will be [EQUATION] which has been showed in the Fig.[REF].', '1806.00361-2-14-1': 'Because the picture of [MATH] demonstrates the characteristic swallow tail behaviour, there is a first order transition in the system.', '1806.00361-2-15-0': '# Conclusion', '1806.00361-2-16-0': 'In this paper, we have studied the thermodynamic behavior of charged AdS black holes in rainbow gravity.', '1806.00361-2-16-1': 'By the modified dispersion relation and HUP, we got deformed temperature in charged AdS black holes using no-zero mass of test particle.', '1806.00361-2-16-2': 'We have discussed the divergence about the heat capacity with a fixed charge.', '1806.00361-2-16-3': 'Our result shows that the phase structure has a relationship with AdS radius [MATH].', '1806.00361-2-16-4': 'When [MATH], there is only one diverging point about heat capacity; when [MATH], we have found there are two diverging points and three phases including two stable phases and one unstable phase.', '1806.00361-2-16-5': 'In particular, an analogy between the charged AdS black holes in the rainbow gravity and the liquid-gas system is discussed.', '1806.00361-2-16-6': 'We have also showed [MATH] critical behavior about the charged AdS black holes in the rainbow gravity.', '1806.00361-2-16-7': 'The consequence shows there is the Van-der-Waals like behavior in the rainbow gravity when [MATH] and [MATH] coincide with Eq. ([REF]).', '1806.00361-2-16-8': 'The rainbow functions deform the forms of critical pressure,temperature and radius.', '1806.00361-2-16-9': "At last, we have discussed the Gibbs free energy and have obtained characteristic 'swallow tail' behaviour which can be the explanation of first-order phase transition.", '1806.00361-2-17-0': 'We find the mass of test particle does not influence the forms of temperature, entropy and heat capacity but only changes their amplitudes.', '1806.00361-2-17-1': 'Moreover, there is a special value about the mass of test particle encountered [MATH] , which would lead to zero temperature and diverging heat capacity for charged AdS black holes in rainbow gravity.', '1806.00361-2-18-0': '# Conflicts of Interest', '1806.00361-2-19-0': 'The authors declare that there are no conflicts of interest regarding the publication of this paper.'}

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[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1806.00361

0805.3481

{'0805.3481-1-0-0': 'The possibility to have an extension of the cosmological standard model with a Hubble expansion rate [MATH] constrained to a finite interval is considered.', '0805.3481-1-0-1': 'Two epochs of accelerated expansion arise naturally when the Hubble expansion rate approaches to the two limiting values.', '0805.3481-1-0-2': 'The new description of the history of the universe is confronted with cosmological data and with several theoretical ideas going beyond the standard cosmological model.', '0805.3481-1-1-0': '# Introduction', '0805.3481-1-2-0': 'According to General Relativity, if the universe is filled with the particles of the Standard Model of particle physics, gravity should lead to a slowing down of the expansion of the universe.', '0805.3481-1-2-1': 'However, in 1998 two independent evidences of present accelerated expansion were presented [CITATION] and later confirmed by different observations [CITATION].', '0805.3481-1-2-2': 'On the other hand measurements of large scale structure [CITATION] and CMB anisotropy [CITATION] also indicate that the Universe evolved through a period of accelerated expansion early on (inflation).', '0805.3481-1-3-0': 'There is no compelling explanation for any of these cosmic accelerations, but many intriguing ideas are being explored.', '0805.3481-1-3-1': 'In the case of inflation the origin of the accelerated expansion can be a modification of gravity at small scales [CITATION] or a coupling of the expansion of the universe to the progress of phase transitions [CITATION].', '0805.3481-1-3-2': 'In the case of the present accelerated expansion these ideas can be classified in three main groups: new exotic sources of the gravitational field with large negative pressure [CITATION] (Dark Energy), modifications of gravity at large scales [CITATION] and rejection of the spatial homogeneity as a good approximation in the description of the present universe [CITATION].', '0805.3481-1-4-0': 'Different models (none of them compelling) for the source responsible of each of the two periods of acceleration have been considered.', '0805.3481-1-4-1': 'Einstein equations admit a cosmological constant [MATH], which can be realised as the stress-energy tensor of empty space.', '0805.3481-1-4-2': 'This [MATH] together with Cold Dark Matter, Standard Model particles and General Relativiy form the current cosmological model, [MATH]CDM.', '0805.3481-1-4-3': 'However, quantum field theory predicts a value of [MATH] which is 120 orders of magnitude higher than observed.', '0805.3481-1-4-4': 'Supersymmetry can lower this value 60 orders of magnitude, which is still ridiculous [CITATION].', '0805.3481-1-4-5': 'In order to solve this paradox, dynamical Dark Energy models have been proposed.', '0805.3481-1-5-0': 'This has also lead to explore the possibility that cosmic acceleration arises fron new gravitational physics.', '0805.3481-1-5-1': 'Again here several alternatives for a modification of the Einstein-Hilbert action at large and small curvatures [CITATION], or even higher dimensional models [CITATION], producing an accelerated expansion have been identified.', '0805.3481-1-5-2': 'All these analysis include an adhoc restriction to actions involving simple functions of the scalar curvature and or the Gauss-Bonnet tensor.', '0805.3481-1-5-3': 'This discussion is sufficient to stablish the point that cosmic acceleration can be made compatible with a standard source for the gravitational field but it is convenient to consider a more general framework in order to make a systematic analysis of the cosmological effects of a modification of general relativity.', '0805.3481-1-6-0': 'In this paper we parametrise the evolution of the universe (considered isotropic and homogeneous) with the Hubble parameter [MATH].', '0805.3481-1-6-1': 'One finds that it is possible to restrict the domain of [MATH] to a bounded interval.', '0805.3481-1-6-2': 'This restriction naturally produces an accelarated expansion when the Hubble expansion rate approaches any of the two edges of the interval.', '0805.3481-1-6-3': 'Therefore we find a new way to incorporate two periods of cosmic acceleration produced by a modification of general relativity.', '0805.3481-1-6-4': 'This simple phenomenological approach to the problem of accelerated expansion in cosmology proves to be equivalent at the homogeneous level to other descriptions based on modifications of the Einstein-Hilbert action or the introduction of exotic components in the matter lagrangian.', '0805.3481-1-7-0': 'In the next section we will review how a general modification of the gravitational action lead to a generalised first Friedman equation at the homogeneous level.', '0805.3481-1-7-1': 'In the third section we will present a specific model (ACM) based on the simplest way to implement a bounded interval of [MATH].', '0805.3481-1-7-2': 'In the fourth section we will contrast the predictions of ACM for the present acceleration with astrophysical observations.', '0805.3481-1-7-3': 'In the fifth section we will show that it is always possible to find modified gravitational actions which lead to a given generalised first Friedman equation, and present some simple examples.', '0805.3481-1-7-4': 'In the sixth section we will show that it is also always possible to find Dark Energy models which are equivalent to a given generalised first Friedman equation, and present some simple examples.', '0805.3481-1-7-5': 'The last section is devoted to summary and conclusions.', '0805.3481-1-8-0': '# Action of the cosmological standard model extension', '0805.3481-1-9-0': 'The spatial homogeneity and isotropy allow to reduce the gravitational system to a mechanical sytem with two variables [MATH], [MATH] which parametrize the Robertson-Walker geometry [EQUATION]', '0805.3481-1-9-1': 'Invariance under reparametrisations of the time variable imply that the invariant time diferential [MATH] must be used.', '0805.3481-1-9-2': 'Also a rescaling of the space variables [MATH] together with [MATH] and [MATH] is a symmetry that must be kept in the lagrangian.', '0805.3481-1-9-3': 'The action of the reduced homogeneous gravitational system can be then written as [EQUATION] with [MATH].', '0805.3481-1-10-0': 'If we keep the standard definition of the gravitational coupling and the density ([MATH]) and pressure ([MATH]) of a cosmological homogeneous and isotropic fluid as a source of the gravitational field, we have the equations of the reduced system [EQUATION]', '0805.3481-1-10-1': 'It is possible to choose a new time coordinate [MATH] such that [EQUATION]', '0805.3481-1-10-2': 'This amounts to make [MATH] in the action ([REF]) of the gravitational system and the evolution equations of the cosmological model reduce to a set of equations for the scale factor [MATH].', '0805.3481-1-10-3': 'If we introduce the notation [EQUATION] with [MATH] denoting the partial derivatives of the Lagrangian as a function of the variables [MATH] we have [EQUATION]', '0805.3481-1-10-4': 'In order to determine the evolution of the Universe one has to use the general covariance of the action of the system which acts as a source of the gravitational field.', '0805.3481-1-10-5': 'In the homogeneous and isotropic approximation the vanishing of the covariant divergence of the energy-momentum tensor leads to the continuity equation [EQUATION]', '0805.3481-1-10-6': 'In the radiation dominated era one has [EQUATION] where [MATH] is a constant parametrising the general solution of the continuity equation.', '0805.3481-1-10-7': 'In a period dominated by matter one has a presureless fluid and then [EQUATION] with [MATH] a constant.', '0805.3481-1-10-8': 'When these expressions for the energy density and pressure are plugged in ([REF]-[REF]) one ends up with two compatible differential equations for the scale factor [MATH] which describes the evolution of the Universe.', '0805.3481-1-10-9': 'From now on we will use the more common notation [MATH],...', '0805.3481-1-11-0': '# The Asymptotic Cosmological Model', '0805.3481-1-12-0': 'Let us assume that the Lagrangian L of the gravitational system is such that the evolution equations ([REF]-[REF]) admit a solution such that [MATH] (absence of singularities), [MATH] (perpetual expansion) and [MATH].', '0805.3481-1-12-1': 'In that case one has a different value for the scale factor [MATH] and the Hubble rate [MATH] at each time and then one has a one to one correspondence between the scale factor and the Hubble rate.', '0805.3481-1-12-2': 'Since the continuity equation (together with the equation of state) gives a relation between the scale factor and the density then one can describe a solution of the evolution equations of the generalised cosmological model through a relation between the energy density and the Hubble rate, i.e. through a generalised first Friedman equation [EQUATION] with [MATH] a smooth function which parametrises the different algebraic relations corresponding to different solutions of different cosmological models.', '0805.3481-1-12-3': 'Each choice for the function [MATH] defines a phenomenological description of a cosmological model.', '0805.3481-1-12-4': 'Then one can take it as an starting point trying to translate any observation into a partial information on the function [MATH] which parametrises the cosmological model.', '0805.3481-1-13-0': 'Eq. ([REF]) is all one needs in order to reconstruct the evolution of the Universe at the homogeneous level.', '0805.3481-1-13-1': 'The generalised second Friedman equation is obtained by using the continuity equation ([REF]) and the expression for the density as a function of [MATH].', '0805.3481-1-13-2': 'One has [EQUATION]', '0805.3481-1-13-3': 'In the matter dominated era one has [EQUATION] and then the assumed properties of the solution of the evolution equations require the consistency conditions [EQUATION]', '0805.3481-1-13-4': 'In the period dominated by radiation one has [EQUATION] instead of ([REF]) and the same consistency conditions ([REF]) for the function [MATH] which defines the generalised first Friedman equation.', '0805.3481-1-14-0': 'We introduce now a class of phenomenological cosmological models defined by the condition that the Hubble rate has an upper bound [MATH] and a lower bound [MATH].', '0805.3481-1-14-1': 'This can be implemented through a function [MATH] going to infinity when [MATH] approaches [MATH] and going to zero when [MATH] approaches [MATH].', '0805.3481-1-14-2': 'We will also assume that there is an interval of [MATH] in which the behaviour of the energy density with the Hubble parameter is to a good approximation scale free, i.e. [MATH].', '0805.3481-1-14-3': 'The source of the gravitational field will be an homogeneous and isotropic fluid composed of relativistic and nonrelativistic particles.', '0805.3481-1-14-4': 'The moment of evolution history when the energy density of relativistic and nonrelativistic particles coincide falls in the scale free evolution interval, therefore recovering the Cosmological Standard Model without curvature in the limit [MATH] and [MATH].', '0805.3481-1-15-0': 'Notice that this interpretation is independent of the underlying theory of gravitation.', '0805.3481-1-15-1': 'The Hubble parameter can be used to parametrise the history of universe as long as [MATH].', '0805.3481-1-15-2': 'The total density can be thus expressed as a function of [MATH].', '0805.3481-1-15-3': 'If [MATH] is bounded, then [MATH] will have a pole at [MATH] and a zero at [MATH].', '0805.3481-1-15-4': 'Far from these scales the behaviour of [MATH] can be assumed to be approximately scale free.', '0805.3481-1-16-0': 'Under these conditions we can parametrize the dependence of the cosmological model on the lower bound [MATH] when the Hubble rate approaches to it by [EQUATION] and similarly for the dependence on the upper bound [MATH] [EQUATION] where the two functions [MATH] satisfy the conditions [EQUATION] [MATH] is a constant allowed in principle by dimensional arguments.', '0805.3481-1-16-1': 'If [MATH] then it can be moved to the lhs of the Friedman equation, turning [MATH], and can be interpreted as the ratio between an effective cosmological value of the gravitational coupling and the value measured with local tests.', '0805.3481-1-16-2': 'But [MATH] would be in conflict with Nucleosynthesis, through the relic abundances of [MATH] and other heavy elements (for 3 neutrino species) [CITATION], so we set [MATH] .', '0805.3481-1-16-3': 'Therefore [EQUATION]', '0805.3481-1-16-4': 'The consistency conditions ([REF]) translate into [EQUATION] for the two functions [MATH] defined in the interval [MATH].', '0805.3481-1-16-5': 'Thus we can divide the cosmic evolution history in three periods.', '0805.3481-1-16-6': 'In the earliest, relativistic particles dominate the energy density of the universe and the generalised first Friedman equation shows a dependence on the upper bound [MATH].', '0805.3481-1-16-7': 'There is also a transition period in which the effect of the bounds can be neglected and the rhs of the first Friedman equation is scale free; this period includes the transition from a radiation dominated universe to a matter dominated universe.', '0805.3481-1-16-8': 'In the third present period, nonrelativistic particles dominate de energy density of the universe but the dependence on the lower bound [MATH] must be accounted for in the generalised first Friedman equation.', '0805.3481-1-17-0': 'From the definition of the Hubble parameter one has [EQUATION]', '0805.3481-1-17-1': 'Then in order to see if there is an accelerated or deccelerated expansion one has to see whether [MATH] is greater or smaller than [MATH].', '0805.3481-1-18-0': 'In the period dominated by radiation one has [EQUATION] with [MATH], where we have used ([REF]) assuming that only the dependence on the upper bound ([MATH]) of the Hubble parameter is relevant.', '0805.3481-1-18-1': 'A very simple choice for this dependence is given by [EQUATION] with [MATH] a (positive) exponent which parametrizes the departure from the standard cosmological model when the Hubble rate approaches its upper bound.', '0805.3481-1-18-2': 'With this simple choice one has a transition from an accelerated expansion for [MATH] into a deccelerated expansion when [MATH] which includes the domain of validity of the standar cosmological model ([MATH]).', '0805.3481-1-19-0': 'In the period dominated by matter (which corresponds to lower values of the Hubble rate) we assume that only the dependence on the lower bound ([MATH]) of the Hubble rate is relevant.', '0805.3481-1-19-1': 'Then one has [EQUATION] with [MATH].', '0805.3481-1-19-2': 'We can also consider a dependence on [MATH] parametrised simply by an exponent [MATH] [EQUATION]', '0805.3481-1-19-3': 'With this choice one has a transition from a deccelerated expansion for [MATH] leaving the domain of validity of the standard cosmological model and entering into an accelerated expansion when the Hubble rate approaches its lower bound for [MATH].', '0805.3481-1-20-0': 'The possibility of describing [MATH] as a function of [MATH] is independent of the existence of spatial curvature [MATH].', '0805.3481-1-20-1': 'However, in the kinematics of observables in the expanding universe we do need to specify the value of [MATH].', '0805.3481-1-20-2': 'In the rest of the paper we will assume that the universe is flat [MATH], although the same analisis could be done for arbitrary [MATH].', '0805.3481-1-21-0': 'The properties of the expansion obtained in this simple example (a period of deccelerated expansion separating two periods of accelerated expansion) are general to the class of phenomenological models with a generalized first Friedman equation ([REF]) with [MATH] satisfying the consistency conditions ([REF]) and a Hubble rate constrained to a finite interval.', '0805.3481-1-21-1': 'What is specific of the example defined by ([REF],[REF]) is the simple dependence on the Hubble rate bounds and the values of the Hubble rate at the transitions between the three periods of expansion. >', '0805.3481-1-21-2': 'From now on we will name this description the Asymptotic Cosmological Model (ACM).', '0805.3481-1-21-3': 'With respect to the late accelerated expansion, ACM can be seen as a generalisation of [MATH]CDM, which can be recovered by setting [MATH].', '0805.3481-1-21-4': 'It also includes an early epoch of exponential expansion which can be seen as a phenomenological description of the evolution of the universe at inflation in the homogeneous level.', '0805.3481-1-22-0': '## Horizon problem', '0805.3481-1-23-0': 'One can see that the horizon of a radiation-dominated universe can be made arbitrarely large as a consequence of an upper bound on the Hubble parameter and in this way one can understand the observed isotropy of the cosmic microwave background at large angular scales.', '0805.3481-1-24-0': 'Let us consider the effect of the modification of the cosmological model on the calculation of the distance [MATH] of a source of a light signal emitted at time [MATH] and observed at time [MATH] [EQUATION]', '0805.3481-1-24-1': 'We have [EQUATION] where in the first step we have used the definition of the Hubble expansion rate [MATH] and in the second step we have used the relation between the scale factor [MATH] and [MATH] as given by ([REF]-[REF]).', '0805.3481-1-24-2': 'We are considering both times [MATH] and [MATH] in the radiation dominated period.', '0805.3481-1-25-0': 'The distance [MATH] is then given by [EQUATION]', '0805.3481-1-25-1': 'If [MATH] is very close to [MATH] (i.e. if we choose the time [MATH] when the light signal is emitted well inside the period of accelerated expansion) then the integral is dominated by the region around [MATH] which is very close to [MATH].', '0805.3481-1-25-2': 'Then one can approximate in the integrand [EQUATION]', '0805.3481-1-25-3': 'On the other hand if the observation is made at a time [MATH] within the domain of validity of the cosmological standard model ([MATH]) then one can aproximate the factor [MATH] in front of the integral by [MATH] and then one has [EQUATION] where [MATH] is the incomplete Beta function, and it can be made arbitrarely large by choosing [MATH] sufficiently close to [MATH].', '0805.3481-1-25-4': 'In this way we see that a cosmological model with a finite interval of variation for [MATH] solves the horizon problem.', '0805.3481-1-26-0': '# Constraints of ACM by Observations', '0805.3481-1-27-0': 'In this section we will carry out a more technical analysis about how the astrophysical observations constrain the parameter space of ACM in the matter dominance period.', '0805.3481-1-27-1': 'This analysis is based on the use of (assumed) standard candles, basicly Type Ia Supernovae [CITATION] and CMB [CITATION].', '0805.3481-1-27-2': 'These observations constrain the parameter space to confidence regions in which the combination [MATH] is constrained to be around one quarter.', '0805.3481-1-27-3': 'The consideration of both Type Ia SNe and CMB together favour [MATH].', '0805.3481-1-27-4': 'The results of this analysis can be seen in figures ([REF], [REF], [REF]).', '0805.3481-1-27-5': 'A reader not interested in technical details might well skip this section.', '0805.3481-1-28-0': 'We center our discussion of the experimental tests of the Asymptotic Cosmological Model in the late accelerated expansion produced when [MATH] approaches its lower bound [MATH].', '0805.3481-1-28-1': 'The vast amount of supernovae data colected by [CITATION] and [CITATION], the data from ths SDSS Baryon Acoustic Oscillation [CITATION], the mismatch between total energy density and total matter energy density seen at CMB anisotropies [CITATION] and the meassurements of present local mass density by 2dF and SDSS [CITATION] compared with the measurements of [MATH] from the HST Cepheids [CITATION] show that the universe undergoes a surprising accelerated expansion at the present time.', '0805.3481-1-29-0': 'We will firstly confront the model with the Supernovae Ia data from Riess et al. and SNLS collaboration.', '0805.3481-1-29-1': 'The usefulness of the Supernovae data as a test of Dark Energy models relies on the assumption that Type Ia SNe behave as standard candles, i.e., they have a well defined environment-independent luminosity [MATH] and spectrum.', '0805.3481-1-29-2': 'Therefore, we can use measured bolometric flux [MATH] and frequency to determine luminosity distance [MATH] and redshift [MATH].', '0805.3481-1-29-3': 'The luminosity distance is given now by [EQUATION]', '0805.3481-1-29-4': 'The computed value must be compared with the one obtained experimentally from the measured extinction-corrected distance moduli ([MATH]) for each SN.', '0805.3481-1-29-5': 'The SNe data have been compiled in references [CITATION], and we have limited the lowest redshift at [MATH] in order to avoid a possible "Hubble Bubble" [CITATION].', '0805.3481-1-29-6': 'Therefore our sample consists on 182 SNe.', '0805.3481-1-29-7': 'We will determine the likelihood of the parameters from a [MATH] statistic, [EQUATION] where [MATH] is the dispersion in supernova redshift due to peculiar velocities (we adopt [MATH] in units of distance moduli), [MATH] is the uncertainity in the individual measured distance moduli [MATH], and [MATH] is the value of [MATH] at [MATH] computed with a certain value of the set of parameters [MATH].', '0805.3481-1-29-8': 'This [MATH] has been marginalised over the nuisance parameter [MATH] using the adaptative method in reference [CITATION].', '0805.3481-1-29-9': 'The resulting likelihood distribution function [MATH] has been explored using Monte Carlo Markov Chains.', '0805.3481-1-29-10': 'We get a best fit of ACM at [MATH] and [MATH], for which [MATH].', '0805.3481-1-29-11': 'In contrast, fixing [MATH], we get [MATH] and [MATH] for the best fit [MATH]CDM.', '0805.3481-1-29-12': 'The confidence regions are shown in Fig. 1 (top).', '0805.3481-1-30-0': 'We can add new constraints for the model coming from measurements of the present local matter energy density from the combination of 2dF and SDSS with HST Cepheids, rendering [MATH]; and the distance to the last scattering surface from WMAP, which leads to [MATH] [CITATION].', '0805.3481-1-30-1': 'Including this priors we get a best fit of ACM at [MATH] and [MATH] (our simulation explored the region with [MATH]), for which [MATH].', '0805.3481-1-30-2': 'In contrast we get [MATH] and [MATH] for the best fit [MATH]CDM, which is outside the [MATH] confidence region shown in Fig. 1 (bottom).', '0805.3481-1-30-3': 'The fits of the best fit [MATH]CDM and ACM taking into account the priors to the SNe data are compared in Fig. 2.', '0805.3481-1-31-0': 'It can be realised that the information which can be stracted from the data is limited.', '0805.3481-1-31-1': 'This can be seen in the Fig. 3, in which it is explicit that the data constrain mainly the value of the present matter energy density, [MATH].', '0805.3481-1-32-0': '# Generalized First Friedman Equation and [MATH] gravity', '0805.3481-1-33-0': 'The extension of the cosmological model considered here could be compared with recent works on a modification of gravity at large or very short distances [CITATION].', '0805.3481-1-33-1': 'It has been shown that by considering a correction to the Eintein Hilbert action including positive and negative powers of the scalar curvature it is possible to reproduce an accelerated expansion at large and small values of the curvature in the cosmological model.', '0805.3481-1-33-2': 'Some difficulties to make these modifications of the gravitational action compatible with the solar system tests of general relativity have lead to consider a more general gravitational action including the three posible scalars that one can construct with the Riemann curvature tensor [CITATION], although the Gauss-Bonnet scalar is the only combination which is free from ghosts and other pathologies.', '0805.3481-1-33-3': 'In fact there is no clear reason why to restrict the extension of general relativity in this way.', '0805.3481-1-33-4': 'Once one goes beyond the derivative expansion there is no reason to forget the scalars that can be constructed with more than two derivatives of the metric and then one does not have a good justification to restrict in this way the modification of the gravitational theory.', '0805.3481-1-33-5': 'It does not seem difficult to find an appropriate function of the scalar curvature or the Gauss-Bonnet scalar, which leads to a cosmology with a bounded Hubble expansion rate.', '0805.3481-1-34-0': 'One may ask if a set of metric f(R) theories which include ACM as an homogeneous and isotropic solution exists.', '0805.3481-1-34-1': 'The answer is that a biparametric family of f(R) actions which lead to an ACM solution exists.', '0805.3481-1-34-2': 'In the following section we will derive them and we will discuss some examples.', '0805.3481-1-34-3': 'The derivation follows the same steps of modified f(R)-gravity reconstruction from any FRW cosmology [CITATION].', '0805.3481-1-35-0': 'We start from the action [EQUATION] which leads to the "generalized first Friedman equation" [EQUATION]', '0805.3481-1-35-1': 'If the energy density of the universe is mainly due to matter (as in the late accelerated expansion), we can use ([REF]) and ([REF]) to express [MATH], [MATH], [MATH] and [MATH] as functions of [MATH].', '0805.3481-1-35-2': 'Thus we get [EQUATION] [MATH] can be obtained from ([REF]) and set into ([REF]); then we get an inhomogeneous second order linear differential equation with nonconstant coefficients.', '0805.3481-1-35-3': 'Therefore, there will always be a biparametric family of [MATH] actions which present ACM as their homogeneous and isotropic solution.', '0805.3481-1-35-4': 'The difference between these actions will appear in the behaviour of perturbations, which is not fixed by ([REF]).', '0805.3481-1-35-5': 'Some of these actions are particularly easy to solve.', '0805.3481-1-35-6': 'If [MATH] then [MATH] and the differential equation becomes [EQUATION]', '0805.3481-1-35-7': 'Its general solution is [EQUATION] which will give ([REF]) as First Friedman equation as long as the radiation energy density can be neglected.', '0805.3481-1-35-8': 'In general the differential equation ([REF]) will not be able to be solved analytically, and only approximate solutions can be found as power series around a certain singular point [MATH] (a value of [MATH] such that [MATH] cancels out the coefficient of [MATH] in ([REF])).', '0805.3481-1-35-9': 'These solutions will be of the form [MATH] with [EQUATION] where [MATH], [MATH] and the series will converge inside a certain radius of convergence.', '0805.3481-1-35-10': 'An interesting choice of [MATH] is [MATH], which is the value of [MATH] at [MATH].', '0805.3481-1-35-11': 'One can also find the approximate solution of the differential equation for [MATH], which will be of the form [EQUATION]', '0805.3481-1-35-12': 'One can in principle assume that the action ([REF]) could be considered as valid also in the region in which radiation begins to dominate, but this action reproduces ([REF]) only if matter dominates.', '0805.3481-1-35-13': 'However, some of the new terms appearing in ([REF]) will be negligible against [MATH] when radiation begins to dominate.', '0805.3481-1-35-14': 'The others can be canceled out by zeroing the appropriate integration constant and therefore the Cosmological Standard Model will be recovered as a good approximation when radiation begins to dominate.', '0805.3481-1-36-0': '[MATH]-theories are not the only modified gravity theories studied in the literature; [MATH]-theories [CITATION] are also a popular field of research.', '0805.3481-1-36-1': 'In these theories the Einstein-Hilbert action is suplemented by a function of the Gauss-Bonnet scalar [MATH] (not to be confused with the gravitational coupling [MATH]).', '0805.3481-1-36-2': 'The same approach can be used to answer what [MATH] actions are able to reproduce ACM homogeneous evolution once more in analogy with the modified f(G)-gravity reconstruction of a FRW cosmology [CITATION].', '0805.3481-1-36-3': 'The form of the action will be [EQUATION] from which we derive the modified Friedman equation [EQUATION]', '0805.3481-1-36-4': 'Following the same proccedure as in the case of [MATH]-theories we arrive to the differential equation [EQUATION]', '0805.3481-1-36-5': 'For a given [MATH] one can use ([REF]) to get [MATH], then set it into ([REF]) and solve the second order linear differential equation.', '0805.3481-1-36-6': 'As in the previous case there will be a biparametric family of solutions for [MATH] which will have ([REF]) as their homogeneous isotropic solution as long as matter dominates.', '0805.3481-1-36-7': 'The parameters will need to be fixed in order to make the contribution of undesired terms in the Friedman equation to be negligible when radiation dominates.', '0805.3481-1-36-8': 'Again [MATH] is an example which can be solved analytically.', '0805.3481-1-36-9': 'The differential equation turns to be [EQUATION] which has the solution [EQUATION]', '0805.3481-1-36-10': 'In most cases this proccedure will not admit an analytical solution and an approximate solution will need to be found numerically.', '0805.3481-1-37-0': 'A similar discussion can be made to explain the early accelerated expansion (inflation) as the result of an [MATH] or [MATH] action, taking into account that radiation and not matter dominates the energy density at early epoch.', '0805.3481-1-37-1': 'The analogue to ([REF]) will be now a solution of the form [EQUATION] for [MATH] with [MATH].', '0805.3481-1-37-2': 'Terms in the action which dominate over [MATH] when [MATH] becomes small enough should be eliminated in order to recover the Standard Cosmological model before matter starts to dominate.', '0805.3481-1-38-0': 'Both accelerated expansions admit to be described together in the homogeneous limit by an [MATH] action with terms [MATH] with [MATH] coming from ([REF]) and terms [MATH] with [MATH] coming from ([REF]).', '0805.3481-1-38-1': 'The action will contain a term, the Einstein-Hilbert action [MATH], which will be dominant for [MATH] including the period when matter and radiation have comparable energy densities.', '0805.3481-1-39-0': '# Alternative Descriptions of the ACM', '0805.3481-1-40-0': 'In general, by virtue of the gravitational field equations, it is always possible to convert a modification in the gravitational term of the action to a modification in the matter content of the universe.', '0805.3481-1-40-1': 'In particular, at the homogeneous level, it is possible to convert a generalized first Friedman equation of the type ([REF]) to an equation in which appart of the usual matter term there is a dark energy component with an unusual equation of state [MATH] and in which General Relativity is not modified.', '0805.3481-1-41-0': 'The trivial procedure is the following.', '0805.3481-1-41-1': 'Assume that the source of the gravitational field in the modified gravity theory behaves as [MATH] (matter or radiation).', '0805.3481-1-41-2': 'In both epochs in which the modification of gravity is relevant, this is the case.', '0805.3481-1-41-3': 'We can define a dark energy or alternative gravity energy density as [EQUATION]', '0805.3481-1-41-4': 'The continuity equation ([REF]) allows to define a pressure for this dark energy component as [MATH].', '0805.3481-1-41-5': 'The equation of state of the dominant content of the universe leads to [EQUATION] which can be used to express [MATH] as a function of [MATH], and one gets the final expression for [MATH], [EQUATION]', '0805.3481-1-41-6': 'Then, for any given [MATH] we can use ([REF]) to get [MATH] and substitute in ([REF]) to get an expression of [MATH] which can be interpreted as the equation of state of a dark energy component.', '0805.3481-1-41-7': 'In the simple case of a matter dominated universe with ACM, [MATH] gives obviously a dark energy component verifying [MATH].', '0805.3481-1-41-8': 'Another simple example is [MATH] for which [EQUATION]', '0805.3481-1-41-9': 'The inverse proccedure is also straightforward.', '0805.3481-1-41-10': 'Suppose we have a universe filled with a standard component [MATH] and a dark energy fluid [MATH].', '0805.3481-1-41-11': 'Using the continuity equation of both fluids one can express their energy densities as a function of the scale factor [MATH] and then use this relation to express [MATH] as a function of [MATH].', '0805.3481-1-41-12': 'Then the Friedman equation reads [EQUATION] which, solving for [MATH], is trivially equivalent to ([REF]).', '0805.3481-1-41-13': 'This argument could be applied to reformulate any cosmological model based on a modification of the equation of state of the dark energy component [CITATION] as a generalised first Friedman equation ([REF]).', '0805.3481-1-42-0': 'Until now we have considered descriptions in which there is an exotic constituent of the universe besides a standard component (pressureless matter or radiation).', '0805.3481-1-42-1': 'In these cases pressureless matter includes both baryons and Dark Matter.', '0805.3481-1-42-2': 'However, there are also descriptions in which Dark Matter is unified with Dark Energy in a single constituent of the universe.', '0805.3481-1-42-3': 'One of these examples is the Chaplygin gas [MATH] [CITATION].', '0805.3481-1-42-4': 'The use of the continuity equation leads to [EQUATION] where [MATH] and [MATH] are integration constants.', '0805.3481-1-42-5': 'The previous method can be used to find the generalized first Friedman equation for the baryon density in this model, [EQUATION] where [MATH], [MATH], and [MATH] is the present value of [MATH].', '0805.3481-1-42-6': 'In the [MATH] limit this model can be interpreted as a universe filled with baryons and dark matter or as a universe filled with baryons and with a higher effective value of [MATH].', '0805.3481-1-42-7': "Equation ([REF]) doesn't fulfill ([REF]) because it describes the behaviour of just baryon density.", '0805.3481-1-42-8': 'In the [MATH] limit, the model can be interpreted as a universe filled with baryons and a cosmological constant or as an ACM model with [MATH] and filled only with baryons.', '0805.3481-1-43-0': 'An analogous example in the early epoch would be a universe with standard Einsteinian gravity but filled with a fluid which behaves as a fluid of ultrarelativistic particles if the energy density is low enough but whose density has an upper bound [EQUATION]', '0805.3481-1-43-1': 'This dependence for the energy density in the scale factor follows from the equation of state [EQUATION]', '0805.3481-1-43-2': 'This model turns out to be equivalent to a universe filled with radiation following eq. ([REF]) with [MATH] and [MATH].', '0805.3481-1-44-0': 'Another equivalent description would be to consider that the universe is also filled with some self interacting scalar field [MATH] which acconunts for the discrepancy between the standard energy-momentum and GR Einstein tensors.', '0805.3481-1-44-1': 'Given an arbitrary modified Friedman equation ([REF]) a potential [MATH] can be found such that the cosmologies described by both models are the same.', '0805.3481-1-44-2': 'The procedure is similar to the one used to reconstruct a potential from a given cosmology [CITATION].', '0805.3481-1-44-3': 'From the point of view of the scalar field, the cosmology is defined by a set of three coupled differential equations [EQUATION]', '0805.3481-1-44-4': 'The solution of these equations for a certain [MATH] will give [MATH] and [MATH], and therefore [MATH] which is [MATH] up to a factor [MATH].', '0805.3481-1-44-5': 'In this way one finds the generalized first Friedman equation associated with the introduction of a self interacting scalar field.', '0805.3481-1-44-6': 'Alternatively, given a function [MATH] in a generalized first Friedman equation ([REF]) for the density [MATH], one can find a scalar field theory leading to the same cosmology in the homogeneous limit.', '0805.3481-1-44-7': 'By considering the time derivative of ([REF]) and using ([REF]) and ([REF]) one gets [EQUATION] where we can use ([REF]) and ([REF]) to get [MATH] as a function of [MATH].', '0805.3481-1-44-8': 'Setting this on ([REF]) we get [MATH] as a function of [MATH].', '0805.3481-1-44-9': 'On the other hand, [MATH], so [EQUATION] and [EQUATION]', '0805.3481-1-44-10': 'If [MATH] is such that the rhs of ([REF]) is positive definite, it can be solved and the solution [MATH] inverted and substituted into ([REF]) in order to get [MATH].', '0805.3481-1-44-11': 'This is the case of a function of the type ([REF]) with [MATH].', '0805.3481-1-44-12': 'For the particular case of an ACM expansion with [MATH], the solution is a flat potential [MATH] and a constant value of [MATH].', '0805.3481-1-45-0': 'If [MATH] is such that the rhs of ([REF]) is negative definite, as it happens in ([REF]) with [MATH] or in ([REF]), the problem can be cured by changing the sign of the kinetic term in ([REF]).', '0805.3481-1-45-1': 'The result is a phantom quintesence in the case of ([REF]) with [MATH].', '0805.3481-1-45-2': 'The case ([REF]) is more complicated because the energy density of the associated inflaton turns out to be negative.', '0805.3481-1-45-3': 'Moreover, it is of the same order as the energy density of ultrarelativistic particles during the whole accelerated expansion epoch.', '0805.3481-1-45-4': 'Therefore it seems that this model is inequivalent to other inflation scenarios previously studied.', '0805.3481-1-46-0': "In summary, there are many equivalent ways to describe the discrepancy between observed matter content of the universe and Eintein's General Relativity.", '0805.3481-1-46-1': 'At the homogeneous level, it is trivial to find relations among them.', '0805.3481-1-46-2': 'The possibility to stablish the equivalence of different descriptions is not a peculiarity of a description of the discrepancy in terms of a generalised first Friedman equation ([REF]).', '0805.3481-1-46-3': 'The same relations can be found among generalised equations of state, scalar-tensor theories and f(R) modifed gravity [CITATION].', '0805.3481-1-47-0': '# Summary and discussion', '0805.3481-1-48-0': 'It may be interesting to go beyond [MATH]CDM in the description of the history of the universe in order to identify the origin of the two epochs of accelerated expansion.', '0805.3481-1-48-1': 'We have proposed to use the expression of the energy density as a function of the Hubble parameter as the best candidate to describe the history of the universe.', '0805.3481-1-48-2': 'In this context the late time accelerated epoch and the early time inflation can be easily parametrised.', '0805.3481-1-49-0': 'We have considered a simple modification of the cosmological equations characterised by the appearance of an upper and a lower bound on the Hubble expansion rate.', '0805.3481-1-49-1': 'A better fit of the experimental data can be obtained with this extended cosmological model as compared with the [MATH]CDM fit.', '0805.3481-1-49-2': 'Once more precise data are available, it will be possible to identify the behaviour of the energy density as a function of the Hubble parameter and then look for a theoretical derivation of such behaviour.', '0805.3481-1-50-0': 'We plan to continue with a systematic analysis of different alternatives incorporating the main features of the example considered in this work.', '0805.3481-1-50-1': 'Although the details of the departures from the standard cosmology can change we expect a general patern of the effects induced by the presence of the two bounds on [MATH].', '0805.3481-1-50-2': 'We also plan to go further considering the evolution of inhomogeneities looking for new consequences of the bounds on [MATH].', '0805.3481-1-51-0': 'The discussion presented in this work, which is based on a new description of the accelerated expansion epochs of the universe, can open a new way to explore either modifications of the theory of gravity or new components in the universe homogeneous fluid.', '0805.3481-1-51-1': 'Lacking theoretical criteria to select among the possible ways to go beyond [MATH]CDM, we think that the phenomenological approach proposed in this work is justified.', '0805.3481-1-52-0': 'We are grateful to Paola Arias and Justo Lopez-Sarrion for discussions in the first stages of this work.', '0805.3481-1-52-1': 'We also acknowledge discussions with Julio Fabris, Roberto Emparan, Sergei Odintsov, Aurelio Grillo and Fernando Mendez.', '0805.3481-1-52-2': 'This work has been partially supported by CICYT (grant FPA2006-02315) and DGIID-DGA (grant2008-E24/2).', '0805.3481-1-52-3': 'J.I. acknowledges a FPU grant from MEC.'}

{'0805.3481-2-0-0': 'The possibility of having an extension of the cosmological standard model with a Hubble expansion rate [MATH] constrained to a finite interval is considered.', '0805.3481-2-0-1': 'Two periods of accelerated expansion arise naturally when the Hubble expansion rate approaches to the two limiting values.', '0805.3481-2-0-2': 'The new description of the history of the universe is confronted with cosmological data and with several theoretical ideas going beyond the standard cosmological model.', '0805.3481-2-1-0': '# Introduction', '0805.3481-2-2-0': 'According to General Relativity, if the universe is filled with the particles of the Standard Model of particle physics, gravity should lead to a deceleration of the expansion of the universe.', '0805.3481-2-2-1': 'However, in 1998 two independent evidences of present accelerated expansion were presented [CITATION] and later confirmed by different observations [CITATION].', '0805.3481-2-2-2': 'On the other hand, measurements of large scale structure [CITATION] and CMB anisotropy [CITATION] also indicate that the universe evolved through a period of early accelerated expansion (inflation).', '0805.3481-2-3-0': 'There is no compelling explanation for any of these cosmic accelerations, but many intriguing ideas are being explored.', '0805.3481-2-3-1': 'In the case of inflation, the origin of the accelerated expansion can be either a modification of gravity at small scales [CITATION] or a coupling of the expansion of the universe to the progress of phase transitions [CITATION].', '0805.3481-2-3-2': 'In the case of the present accelerated expansion these ideas can be classified into three main groups: new exotic sources of the gravitational field with large negative pressure [CITATION] (Dark Energy), modifications of gravity at large scales [CITATION] and rejection of the spatial homogeneity as a good approximation in the description of the present universe [CITATION].', '0805.3481-2-4-0': 'Different models (none of them compelling) of the source responsible for each of the two periods of accelerated expansion have been considered.', '0805.3481-2-4-1': 'Einstein equations admit a cosmological constant [MATH], which can be realized as the stress-energy tensor of empty space.', '0805.3481-2-4-2': 'This [MATH], together with Cold Dark Matter, Standard Model particles and General Relativity, form the current cosmological model ([MATH]CDM).', '0805.3481-2-4-3': 'However, quantum field theory predicts a value of [MATH] which is 120 orders of magnitude higher than observed.', '0805.3481-2-4-4': 'Supersymmetry can lower this value 60 orders of magnitude, which is still ridiculous [CITATION].', '0805.3481-2-4-5': 'In order to solve this paradox, dynamical Dark Energy models have been proposed.', '0805.3481-2-5-0': 'This has also lead to explore the possibility that cosmic acceleration arises from new gravitational physics.', '0805.3481-2-5-1': 'Also here several alternative modifications of the Einstein-Hilbert action at large and small curvatures [CITATION], or even higher dimensional models [CITATION], producing an accelerated expansion have been identified.', '0805.3481-2-5-2': 'All these analysis include an ad hoc restriction to actions involving simple functions of the scalar curvature and/or the Gauss-Bonnet tensor.', '0805.3481-2-5-3': 'This discussion is sufficient to establish the point that cosmic acceleration can be made compatible with a standard source for the gravitational field, but it is convenient to consider a more general framework in order to make a systematic analysis of the cosmological effects of a modification of general relativity.', '0805.3481-2-6-0': 'In this paper we parametrize the evolution of the universe (considered isotropic and homogeneous) with the Hubble parameter [MATH].', '0805.3481-2-6-1': 'One finds that it is possible to restrict the domain of [MATH] to a bounded interval.', '0805.3481-2-6-2': 'This restriction naturally produces an accelerated expansion when the Hubble expansion rate approaches any of the two edges of the interval.', '0805.3481-2-6-3': 'Therefore, we find a new way to incorporate two periods of cosmic acceleration produced by a modification of general relativity.', '0805.3481-2-6-4': 'But the dependence on the two limiting values of [MATH] can be chosen independently.', '0805.3481-2-6-5': 'One could even consider a Hubble expansion rate constrained to a semi-infinite interval with a unique period of accelerated expansion.', '0805.3481-2-6-6': 'In this sense the aim to have a unified explanation of both periods of accelerated expansion is only partially achieved.', '0805.3481-2-6-7': 'This simple phenomenological approach to the problem of accelerated expansion in cosmology proves to be equivalent at the homogeneous level to other descriptions based on modifications of the Einstein-Hilbert action or the introduction of exotic components in the matter Lagrangian.', '0805.3481-2-7-0': 'In the next section we will review how a general modification of the gravitational action leads to a generalized first Friedman equation at the homogeneous level.', '0805.3481-2-7-1': 'In the third section we will present a specific model (ACM) based on the simplest way to implement a bounded interval of [MATH].', '0805.3481-2-7-2': 'In the fourth section we will contrast the predictions of ACM for the present acceleration with astrophysical observations.', '0805.3481-2-7-3': 'In the fifth section we will show that it is always possible to find modified gravitational actions which lead to a given generalized first Friedman equation, and present some simple examples.', '0805.3481-2-7-4': 'In the sixth section we will show that it is also always possible to find Dark Energy models which are equivalent to a given generalized first Friedman equation, and present some simple examples.', '0805.3481-2-7-5': 'The last section is devoted to summary and conclusions.', '0805.3481-2-8-0': '# Action of the cosmological standard model extension', '0805.3481-2-9-0': 'The spatial homogeneity and isotropy allow to reduce the gravitational system to a mechanical system with two variables [MATH], [MATH] which parametrize the Robertson-Walker geometry [EQUATION]', '0805.3481-2-9-1': 'Invariance under parameterizations of the time variable imply that the invariant time differential [MATH] must be used.', '0805.3481-2-9-2': 'Also a rescaling of the spatial variables [MATH] together with [MATH] and [MATH] is a symmetry that must be kept in the Lagrangian.', '0805.3481-2-9-3': 'The action of the reduced homogeneous gravitational system can be then written as [EQUATION] with [MATH].', '0805.3481-2-10-0': 'If we keep the standard definition of the gravitational coupling and the density ([MATH]) and pressure ([MATH]) of a cosmological homogeneous and isotropic fluid as a source of the gravitational field, we have the equations of the reduced system [EQUATION]', '0805.3481-2-10-1': 'It is possible to choose a new time coordinate [MATH] such that [EQUATION]', '0805.3481-2-10-2': 'This is equivalent to set [MATH] in the action ([REF]) of the gravitational system and the evolution equations of the cosmological model reduce to a set of equations for the scale factor [MATH].', '0805.3481-2-10-3': 'If we introduce the notation [EQUATION] with [MATH] denoting the partial derivatives of the Lagrangian as a function of the variables [MATH] we have [EQUATION]', '0805.3481-2-10-4': 'In the homogeneous and isotropic approximation, the vanishing of the covariant divergence of the energy-momentum tensor leads to the continuity equation [EQUATION]', '0805.3481-2-10-5': 'In the radiation dominated era one has [EQUATION] where [MATH] is a constant parameterizing the general solution of the continuity equation.', '0805.3481-2-10-6': 'In a period dominated by matter one has a pressureless fluid and then [EQUATION] with constant [MATH].', '0805.3481-2-10-7': 'When these expressions for the energy density and pressure are plugged in ([REF]-[REF]), one ends up with two compatible differential equations for the scale factor [MATH] which describe the evolution of the universe.', '0805.3481-2-10-8': 'From now on we will use the more common notation [MATH],...', '0805.3481-2-11-0': '# The Asymptotic Cosmological Model', '0805.3481-2-12-0': 'Let us assume that the Lagrangian L of the gravitational system is such that the evolution equations ([REF]-[REF]) admit a solution such that [MATH] (absence of singularities), [MATH] (perpetual expansion) and [MATH].', '0805.3481-2-12-1': 'In that case, one has a different value of the scale factor [MATH] and the Hubble rate [MATH] at each time and then one has a one to one correspondence between the scale factor and the Hubble rate.', '0805.3481-2-12-2': 'Since the continuity equation (together with the equation of state) gives a relation between the scale factor and the density, one can describe a solution of the evolution equations of the generalized cosmological model through a relation between the energy density and the Hubble rate, i.e. through a generalized first Friedman equation [EQUATION] with [MATH] a smooth function which parametrizes the different algebraic relations corresponding to different solutions of different cosmological models.', '0805.3481-2-12-3': 'Each choice for the function [MATH] defines a phenomenological description of a cosmological model.', '0805.3481-2-12-4': 'Then one can take it as a starting point trying to translate any observation into a partial information on the function [MATH] which parametrizes the cosmological model.', '0805.3481-2-13-0': 'Eq. ([REF]) is all one needs in order to reconstruct the evolution of the universe at the homogeneous level.', '0805.3481-2-13-1': 'The generalized second Friedman equation is obtained by using the continuity equation ([REF]) and the expression for [MATH] as a function of [MATH].', '0805.3481-2-13-2': 'One has [EQUATION]', '0805.3481-2-13-3': 'In the matter dominated era one has [EQUATION] and then the assumed properties of the solution for the evolution equations require the consistency conditions [EQUATION]', '0805.3481-2-13-4': 'In the period dominated by radiation one has [EQUATION] instead of ([REF]) and the same consistency conditions ([REF]) for the function [MATH] which defines the generalized first Friedman equation.', '0805.3481-2-14-0': 'We introduce now a phenomenological cosmological model defined by the condition that the Hubble rate has an upper bound [MATH] and a lower bound [MATH].', '0805.3481-2-14-1': 'This can be implemented through a function [MATH] going to infinity when [MATH] approaches [MATH] and going to zero when [MATH] approaches [MATH].', '0805.3481-2-14-2': 'We will also assume that there is an interval of [MATH] in which the behavior of the energy density with the Hubble parameter is, to a good approximation, scale-free i.e. [MATH].', '0805.3481-2-14-3': 'The source of the gravitational field will be a homogeneous and isotropic fluid composed of relativistic and non-relativistic particles.', '0805.3481-2-14-4': 'The Cosmological Standard Model without curvature is recovered in the limit [MATH] and [MATH] which is a good approximation for the period of decelerated expansion.', '0805.3481-2-15-0': 'Notice that this interpretation is independent of the underlying theory of gravitation.', '0805.3481-2-15-1': 'The Hubble parameter can be used to parametrize the history of universe as long as [MATH].', '0805.3481-2-15-2': 'The total density can be thus expressed as a function of [MATH].', '0805.3481-2-15-3': 'If [MATH] is bounded, then [MATH] will have a pole at [MATH] and a zero at [MATH].', '0805.3481-2-15-4': 'Far from these scales, the behavior of [MATH] can be assumed to be approximately scale-free.', '0805.3481-2-16-0': 'Under these conditions, we can parametrize the dependence of the cosmological model on the lower bound [MATH] by [EQUATION] and similarly for the dependence on the upper bound [MATH] [EQUATION] where the two functions [MATH] satisfy the conditions [EQUATION] [MATH] is a constant allowed in principle by dimensional arguments.', '0805.3481-2-16-1': 'If [MATH] then it can be moved to the lhs of the Friedman equation, turning [MATH], and can be interpreted as the ratio between an effective cosmological value of the gravitational coupling and the value measured with local tests.', '0805.3481-2-16-2': 'But [MATH] would be in conflict with Nucleosynthesis, through the relic abundances of [MATH] and other heavy elements (for 3 neutrino species) [CITATION], so we set [MATH] .', '0805.3481-2-16-3': 'Therefore [EQUATION]', '0805.3481-2-16-4': 'The consistency conditions ([REF]) result in [EQUATION] for the two functions [MATH] defined in the interval [MATH].', '0805.3481-2-16-5': 'Thus, we can divide the cosmic evolution history into three periods.', '0805.3481-2-16-6': 'In the earliest, relativistic particles dominate the energy density of the universe and the generalized first Friedman equation shows a dependence on the upper bound [MATH].', '0805.3481-2-16-7': 'There is also a transition period in which the effect of the bounds can be neglected and the rhs of the first Friedman equation is scale-free; this period includes the transition from a radiation dominated universe to a matter dominated universe.', '0805.3481-2-16-8': 'In the third present period, non-relativistic particles dominate the energy density of the universe but the dependence on the lower bound [MATH] must be accounted for in the generalized first Friedman equation.', '0805.3481-2-17-0': 'From the definition of the Hubble parameter, one has [EQUATION]', '0805.3481-2-17-1': 'Then, in order to see if there is an accelerated or decelerated expansion, one has to determine whether [MATH] is greater or smaller than [MATH].', '0805.3481-2-18-0': 'In the period dominated by radiation one has [EQUATION] with [MATH], where we have used ([REF]) assuming that only the dependence on the upper bound ([MATH]) of the Hubble parameter is relevant.', '0805.3481-2-18-1': 'A very simple choice for this dependence is given by [EQUATION] with [MATH] a (positive) exponent which parametrizes the departure from the standard cosmological model when the Hubble rate approaches its upper bound.', '0805.3481-2-18-2': 'With this simple choice one has a transition from an accelerated expansion for [MATH] into a decelerated expansion when [MATH], which includes the domain of validity of the standard cosmological model ([MATH]).', '0805.3481-2-19-0': 'In the period dominated by matter (which corresponds to lower values of the Hubble rate) we assume that only the dependence on the lower bound ([MATH]) of the Hubble rate is relevant.', '0805.3481-2-19-1': 'Then one has [EQUATION] with [MATH].', '0805.3481-2-19-2': 'We can also consider a dependence on [MATH] parametrized simply by an exponent [MATH] [EQUATION]', '0805.3481-2-19-3': 'With this choice one has a transition from a decelerated expansion for [MATH] leaving the domain of validity of the standard cosmological model and entering into an accelerated expansion when the Hubble rate approaches its lower bound for [MATH].', '0805.3481-2-20-0': 'The possibility of describing [MATH] as a function of [MATH] is independent of the existence of spatial curvature [MATH].', '0805.3481-2-20-1': 'However, in the kinematics of observables in the expanding universe we do need to specify the value of [MATH].', '0805.3481-2-20-2': 'In the rest of the paper we will assume that the universe is flat ([MATH]), although the same analysis could be done for arbitrary [MATH].', '0805.3481-2-21-0': 'The properties of the expansion obtained in this simple example (a period of decelerated expansion separating two periods of accelerated expansion) are general to the class of phenomenological models with a generalized first Friedman equation ([REF]) with [MATH] satisfying the consistency conditions ([REF]) and a Hubble rate constrained to a finite interval.', '0805.3481-2-21-1': 'The specific part of the example defined by ([REF],[REF]) is the simple dependence on the Hubble rate bounds and the values of the Hubble rate at the transitions between the three periods of expansion.', '0805.3481-2-21-2': 'From now on we will name this description the Asymptotic Cosmological Model (ACM).', '0805.3481-2-21-3': 'With respect to the late accelerated expansion, ACM can be seen as a generalization of [MATH]CDM, which can be recovered by setting [MATH].', '0805.3481-2-21-4': 'It also includes an early period of exponential expansion which can be seen as a phenomenological description of the evolution of the universe at inflation in the homogeneous approximation.', '0805.3481-2-22-0': '## Horizon problem', '0805.3481-2-23-0': 'One can see that the horizon of a radiation-dominated universe can be made arbitrarily large as a consequence of an upper bound on the Hubble parameter and in this way one can understand the observed isotropy of the cosmic microwave background at large angular scales.', '0805.3481-2-24-0': 'Let us consider the effect of the modification of the cosmological model on the calculation of the distance [MATH] of a source of a light signal emitted at time [MATH] and observed at time [MATH] [EQUATION]', '0805.3481-2-24-1': 'We have [EQUATION] where in the first step we have used the definition of the Hubble expansion rate [MATH] and in the second step we have used the relation between the scale factor [MATH] and [MATH] as given by ([REF]-[REF]).', '0805.3481-2-24-2': 'We are considering both times [MATH] and [MATH] in the radiation dominated period.', '0805.3481-2-25-0': 'The distance [MATH] is then given by [EQUATION]', '0805.3481-2-25-1': 'If [MATH] is very close to [MATH] (i.e. if we choose the time [MATH] when the light signal is emitted well inside the period of accelerated expansion) then the integral is dominated by the region around [MATH] which is very close to [MATH].', '0805.3481-2-25-2': 'Then one can approximate in the integrand [EQUATION]', '0805.3481-2-25-3': 'On the other hand if the observation is made at a time [MATH] within the domain of validity of the cosmological standard model ([MATH]) then the factor [MATH] in front of the integral can be approximated by [MATH] and then one has [EQUATION] where [MATH] is the incomplete Beta function, and it can be made arbitrarily large by choosing [MATH] sufficiently close to [MATH].', '0805.3481-2-25-4': 'In this way we see that a cosmological model with a finite interval of variation for [MATH] solves the horizon problem.', '0805.3481-2-26-0': '# Constraints of ACM by Observations', '0805.3481-2-27-0': 'In this section we will carry out a more technical analysis about how the astrophysical observations constrain the parameter space of ACM in the matter dominance period.', '0805.3481-2-27-1': 'This analysis is based on the use of (assumed) standard candles, basically Type Ia Supernovae [CITATION] and CMB [CITATION].', '0805.3481-2-27-2': 'These observations constrain the parameter space to confidence regions in which the combination [MATH] is constrained to be around one quarter.', '0805.3481-2-27-3': 'The consideration of both Type Ia SNe and CMB together favor [MATH].', '0805.3481-2-27-4': 'The results of this analysis can be seen in figures (FIG. 1-3).', '0805.3481-2-27-5': 'A reader not interested in technical details might well skip this section.', '0805.3481-2-28-0': 'We center our discussion of the experimental tests of the Asymptotic Cosmological Model in the late accelerated expansion produced when [MATH] approaches its lower bound [MATH].', '0805.3481-2-28-1': 'The vast amount of supernovae data collected by [CITATION] and [CITATION], the data from the SDSS Baryon Acoustic Oscillation [CITATION], the mismatch between total energy density and total matter energy density seen at CMB anisotropies [CITATION] and the measurements of present local mass density by 2dF and SDSS [CITATION] compared with the measurements of [MATH] from the HST Cepheids [CITATION] show that the universe undergoes a surprising accelerated expansion at the present time.', '0805.3481-2-29-0': 'We will firstly confront the model with the Supernovae Ia data from Riess et al. and SNLS collaboration.', '0805.3481-2-29-1': 'The usefulness of the Supernovae data as a test of Dark Energy models relies on the assumption that Type Ia SNe behave as standard candles, i.e., they have a well defined environment-independent luminosity [MATH] and spectrum.', '0805.3481-2-29-2': 'Therefore, we can use measured bolometric flux [MATH] and frequency to determine luminosity distance [MATH] and redshift [MATH].', '0805.3481-2-29-3': 'The luminosity distance is given now by [EQUATION]', '0805.3481-2-29-4': 'The computed value must be compared with the one obtained experimentally from the measured extinction-corrected distance moduli ([MATH]) for each SN.', '0805.3481-2-29-5': 'The SNe data have been compiled in references [CITATION], and we have limited the lowest redshift at [MATH] in order to avoid a possible "Hubble Bubble" [CITATION].', '0805.3481-2-29-6': 'Therefore our sample consists on 182 SNe.', '0805.3481-2-29-7': 'We will determine the likelihood of the parameters from a [MATH] statistic, [EQUATION] where [MATH] is the dispersion in supernova redshift due to peculiar velocities (we adopt [MATH] in units of distance moduli), [MATH] is the uncertainty in the individual measured distance moduli [MATH], and [MATH] is the value of [MATH] at [MATH] computed with a certain value of the set of parameters [MATH].', '0805.3481-2-29-8': 'This [MATH] has been marginalized over the nuisance parameter [MATH] using the adaptive method in reference [CITATION].', '0805.3481-2-29-9': 'The resulting likelihood distribution function [MATH] has been explored using Monte Carlo Markov Chains.', '0805.3481-2-29-10': 'We get a best fit of ACM at [MATH] and [MATH], for which [MATH].', '0805.3481-2-29-11': 'In contrast, fixing [MATH], we get [MATH] and [MATH] for the best fit [MATH]CDM.', '0805.3481-2-29-12': 'The confidence regions are shown in Fig. 1 (top).', '0805.3481-2-30-0': 'We can add new constraints for the model coming from measurements of the present local matter energy density from the combination of 2dF and SDSS with HST Cepheids, rendering [MATH]; and the distance to the last scattering surface from WMAP, which leads to [MATH] [CITATION].', '0805.3481-2-30-1': 'Including these priors we get a best fit of ACM at [MATH] and [MATH] (our simulation explored the region with [MATH]), for which [MATH].', '0805.3481-2-30-2': 'In contrast, we get [MATH] and [MATH] for the best fit [MATH]CDM, which is outside the [MATH] confidence region shown in Fig. 1 (bottom).', '0805.3481-2-30-3': 'The fits of the best fit [MATH]CDM and ACM taking into account the priors to the SNe data are compared in Fig. 2.', '0805.3481-2-30-4': 'The information which can be extracted from the data is limited.', '0805.3481-2-30-5': 'This can be seen in Fig. 3, in which it is explicit that the data constrain mainly the value of the present matter energy density, [MATH].', '0805.3481-2-31-0': '# Generalized First Friedman Equation and [MATH] gravity', '0805.3481-2-32-0': 'The extension of the cosmological model considered here could be compared with recent works on a modification of gravity at large or very short distances [CITATION].', '0805.3481-2-32-1': 'It has been shown that, by considering a correction to the Einstein Hilbert action including positive and negative powers of the scalar curvature, it is possible to reproduce an accelerated expansion at large and small values of the curvature in the cosmological model.', '0805.3481-2-32-2': 'Some difficulties to make these modifications of the gravitational action compatible with the solar system tests of general relativity have lead to consider a more general gravitational action, including the possible scalars that one can construct with the Riemann curvature tensor [CITATION], although the Gauss-Bonnet scalar is the only combination which is free from ghosts and other pathologies.', '0805.3481-2-32-3': 'In fact, there is no clear reason to restrict the extension of general relativity in this way.', '0805.3481-2-32-4': 'Once one goes beyond the derivative expansion, one should consider scalars that can be constructed with more than two derivatives of the metric and then one does not have a good justification to restrict in this way the modification of the gravitational theory.', '0805.3481-2-32-5': 'It does not seem difficult to find an appropriate function of the scalar curvature or the Gauss-Bonnet scalar, which leads to a cosmology with a bounded Hubble expansion rate.', '0805.3481-2-33-0': 'One may ask if a set of metric f(R) theories which include ACM as an homogeneous and isotropic solution exists.', '0805.3481-2-33-1': 'The answer is that a bi-parametric family of f(R) actions which lead to an ACM solution exists.', '0805.3481-2-33-2': 'In the following section we will derive them and we will discuss some examples.', '0805.3481-2-33-3': 'The derivation follows the same steps of modified f(R)-gravity reconstruction from any FRW cosmology [CITATION].', '0805.3481-2-34-0': 'We start from the action [EQUATION] which leads to the "generalized first Friedman equation" [EQUATION]', '0805.3481-2-34-1': 'If the energy density of the universe is mainly due to matter (as in the late accelerated expansion), we can use ([REF]) and ([REF]) to express [MATH], [MATH], [MATH] and [MATH] as functions of [MATH].', '0805.3481-2-34-2': 'Thus we get [EQUATION] [MATH] can be obtained from ([REF]) and set into ([REF]); then we get an inhomogeneous second order linear differential equation with non-constant coefficients.', '0805.3481-2-34-3': 'Therefore, there will always be a bi-parametric family of [MATH] actions which present ACM as their homogeneous and isotropic solution.', '0805.3481-2-34-4': 'The difference between these actions will appear in the behavior of perturbations, which is not fixed by ([REF]).', '0805.3481-2-34-5': 'Some of these actions are particularly easy to solve.', '0805.3481-2-34-6': 'If [MATH] then [MATH] and the differential equation becomes [EQUATION]', '0805.3481-2-34-7': 'Its general solution is [EQUATION] which will give ([REF]) as First Friedman equation as long as the radiation energy density can be neglected.', '0805.3481-2-34-8': 'In general the differential equation ([REF]) will not be solvable analytically, and only approximate solutions can be found as power series around a certain singular point [MATH] (a value of [MATH] such that [MATH] cancels out the coefficient of [MATH] in ([REF])).', '0805.3481-2-34-9': 'These solutions will be of the form [MATH] with [EQUATION] where [MATH], [MATH] and the series will converge inside a certain radius of convergence.', '0805.3481-2-34-10': 'An interesting choice of [MATH] is [MATH], which is the value of [MATH] at [MATH].', '0805.3481-2-34-11': 'One can also find the approximate solution of the differential equation for [MATH], which will be of the form [EQUATION]', '0805.3481-2-34-12': 'One can in principle assume that the action ([REF]) could be considered as valid also in the region in which radiation begins to dominate, but this action reproduces ([REF]) only if matter dominates.', '0805.3481-2-34-13': 'However, some of the new terms appearing in ([REF]) will be negligible against [MATH] when radiation begins to dominate.', '0805.3481-2-34-14': 'The others can be canceled out by setting to zero the appropriate integration constant and therefore the Cosmological Standard Model will be recovered as a good approximation when radiation begins to dominate.', '0805.3481-2-35-0': '[MATH]-theories are not the only modified gravity theories studied in the literature; [MATH]-theories [CITATION] are also a popular field of research.', '0805.3481-2-35-1': 'In these theories, the Einstein-Hilbert action is supplemented by a function of the Gauss-Bonnet scalar [MATH] (not to be confused with the gravitational coupling [MATH]).', '0805.3481-2-35-2': 'The same approach can be used to answer what [MATH] actions are able to reproduce ACM homogeneous evolution, once more in analogy with the modified f(G)-gravity reconstruction of a FRW cosmology [CITATION].', '0805.3481-2-35-3': 'The form of the action will be [EQUATION] from which we derive the modified Friedman equation [EQUATION]', '0805.3481-2-35-4': 'Following the same procedure as in the case of [MATH]-theories we arrive to the differential equation [EQUATION]', '0805.3481-2-35-5': 'For a given [MATH], one can use ([REF]) to get [MATH], then set it into ([REF]) and solve the second order linear differential equation.', '0805.3481-2-35-6': 'As in the previous case, there will be a bi-parametric family of solutions for [MATH] which will have ([REF]) as their homogeneous isotropic solution as long as matter dominates.', '0805.3481-2-35-7': 'The parameters will need to be fixed in order to make the contribution of undesired terms in the Friedman equation to be negligible when radiation dominates.', '0805.3481-2-35-8': 'Again [MATH] is an example which can be solved analytically.', '0805.3481-2-35-9': 'The differential equation turns to be [EQUATION] which has the solution [EQUATION]', '0805.3481-2-35-10': 'In most cases this procedure will not admit an analytical solution and an approximate solution will need to be found numerically.', '0805.3481-2-36-0': 'A similar discussion can be made to explain the early accelerated expansion (inflation) as a result of an [MATH] or [MATH] action, taking into account that in this case the dominant contribution to the energy density is radiation instead of matter.', '0805.3481-2-36-1': 'The analogue to ([REF]) will be now a solution of the form [EQUATION] for [MATH] with [MATH].', '0805.3481-2-36-2': 'Terms in the action which dominate over [MATH] when [MATH] becomes small enough should be eliminated in order to recover the Standard Cosmological model before matter starts to dominate.', '0805.3481-2-37-0': 'Both accelerated expansions can be described together in the homogeneous limit by an [MATH] action with terms [MATH] with [MATH] coming from ([REF]) and terms [MATH] with [MATH] coming from ([REF]).', '0805.3481-2-37-1': 'The action will contain a term, the Einstein-Hilbert action [MATH], which will be dominant for [MATH] including the period when matter and radiation have comparable energy densities.', '0805.3481-2-38-0': '# Alternative Descriptions of the ACM', '0805.3481-2-39-0': 'In general, by virtue of the gravitational field equations, it is always possible to convert a modification in the gravitational term of the action to a modification in the matter content of the universe.', '0805.3481-2-39-1': 'In particular, at the homogeneous level, it is possible to convert a generalized first Friedman equation of the type ([REF]) to an equation in which, apart from the usual matter term, there is a dark energy component with an unusual equation of state [MATH] and in which General Relativity is not modified.', '0805.3481-2-40-0': 'The trivial procedure is the following.', '0805.3481-2-40-1': 'Assume that the source of the gravitational field in the modified gravity theory behaves as [MATH] (matter or radiation).', '0805.3481-2-40-2': 'This is the case when the modification of gravity is relevant.', '0805.3481-2-40-3': 'We can then define a dark energy or effective gravitational energy density as [EQUATION]', '0805.3481-2-40-4': 'The continuity equation ([REF]) allows to define a pressure for this dark energy component as [MATH].', '0805.3481-2-40-5': 'The equation of state of the dominant content of the universe leads to [EQUATION] which can be used to express [MATH] as a function of [MATH], and one gets the final expression for [MATH], [EQUATION]', '0805.3481-2-40-6': 'Then, for any given [MATH] we can use ([REF]) to get [MATH] and substitute in ([REF]) to get an expression of [MATH] which can be interpreted as the equation of state of a dark energy component.', '0805.3481-2-40-7': 'In the simple case of a matter dominated universe with ACM, [MATH] gives obviously a dark energy component verifying [MATH].', '0805.3481-2-40-8': 'Another simple example is [MATH], for which [EQUATION]', '0805.3481-2-40-9': 'The inverse procedure is also straightforward.', '0805.3481-2-40-10': 'Suppose we have a universe filled with a standard component [MATH] and a dark energy fluid [MATH].', '0805.3481-2-40-11': 'Using the continuity equation of both fluids one can express their energy densities as a function of the scale factor [MATH] and then use this relation to express [MATH] as a function of [MATH].', '0805.3481-2-40-12': 'Then the Friedman equation reads [EQUATION] which, solving for [MATH], is trivially equivalent to ([REF]).', '0805.3481-2-40-13': 'This argument could be applied to reformulate any cosmological model based on a modification of the equation of state of the dark energy component [CITATION] as a generalized first Friedman equation ([REF]).', '0805.3481-2-41-0': 'Until now we have considered descriptions in which there is an exotic constituent of the universe besides a standard component (pressureless matter or radiation).', '0805.3481-2-41-1': 'In these cases, pressureless matter includes both baryons and Dark Matter.', '0805.3481-2-41-2': 'However, there are also descriptions in which Dark Matter is unified with Dark Energy in a single constituent of the universe.', '0805.3481-2-41-3': 'One of these examples is the Chaplygin gas [MATH] [CITATION].', '0805.3481-2-41-4': 'The use of the continuity equation leads to [EQUATION] where [MATH] and [MATH] are integration constants.', '0805.3481-2-41-5': 'The previous method can be used to find the generalized first Friedman equation for the baryon density in this model, [EQUATION] where [MATH], [MATH], and [MATH] is the present value of [MATH].', '0805.3481-2-41-6': 'In the [MATH] limit this model can be interpreted as a universe filled with baryons and dark matter or as a universe filled with baryons and with a higher effective value of [MATH].', '0805.3481-2-41-7': 'Equation ([REF]) does not fulfill ([REF]) because it describes the behavior of just baryon density.', '0805.3481-2-41-8': 'In the [MATH] limit, the model can be interpreted as a universe filled with baryons and a cosmological constant or as an ACM model with [MATH] and filled only with baryons.', '0805.3481-2-42-0': 'A similar example in the early period of accelerated expansion would be a universe filled with a fluid which behaves as a fluid of ultra-relativistic particles if the energy density is low enough but whose density has an upper bound [EQUATION]', '0805.3481-2-42-1': 'This dependence for the energy density in the scale factor follows from the equation of state [EQUATION]', '0805.3481-2-42-2': 'This model turns out to be equivalent to a universe filled with radiation following eq. ([REF]) with [MATH] and [MATH].', '0805.3481-2-43-0': 'Another equivalent description would be to consider that the universe is also filled with some self interacting scalar field [MATH] which accounts for the discrepancy between the standard energy-momentum and GR Einstein tensors.', '0805.3481-2-43-1': 'Given an arbitrary modified Friedman equation ([REF]) a potential [MATH] can be found such that the cosmologies described by both models are the same.', '0805.3481-2-43-2': 'The procedure is similar to the one used to reconstruct a potential from a given cosmology [CITATION].', '0805.3481-2-43-3': 'From the point of view of the scalar field, the cosmology is defined by a set of three coupled differential equations [EQUATION]', '0805.3481-2-43-4': 'The solution of these equations for a certain [MATH] will give [MATH] and [MATH], and therefore [MATH] which is [MATH] up to a factor [MATH].', '0805.3481-2-43-5': 'In this way one finds the generalized first Friedman equation associated with the introduction of a self interacting scalar field.', '0805.3481-2-43-6': 'Alternatively, given a function [MATH] in a generalized first Friedman equation ([REF]) for the density [MATH], one can find a scalar field theory leading to the same cosmology in the homogeneous limit.', '0805.3481-2-43-7': 'By considering the time derivative of ([REF]) and using ([REF]) and ([REF]) one gets [EQUATION] where we can use ([REF]) and ([REF]) to get [MATH] as a function of [MATH].', '0805.3481-2-43-8': 'Setting this on ([REF]) we get [MATH] as a function of [MATH].', '0805.3481-2-43-9': 'On the other hand, [MATH], so [EQUATION] and [EQUATION]', '0805.3481-2-43-10': 'If [MATH] is such that the rhs of ([REF]) is positive definite, it can be solved and the solution [MATH] inverted and substituted into ([REF]) in order to get [MATH].', '0805.3481-2-43-11': 'This is the case of a function of the type ([REF]) with [MATH].', '0805.3481-2-43-12': 'For the particular case of an ACM expansion with [MATH], the solution is a flat potential [MATH] and a constant value of [MATH].', '0805.3481-2-44-0': 'If [MATH] is such that the rhs of ([REF]) is negative definite, as it happens in ([REF]) with [MATH] or in ([REF]), the problem can be solved by changing the sign of the kinetic term in ([REF]).', '0805.3481-2-44-1': 'The result is a phantom quintessence in the case of ([REF]) with [MATH].', '0805.3481-2-44-2': 'The case ([REF]) is more complicated because the energy density of the associated inflaton turns out to be negative.', '0805.3481-2-44-3': 'Moreover, it is of the same order as the energy density of ultra-relativistic particles during the whole period of accelerated expansion.', '0805.3481-2-44-4': 'Therefore, it seems that this model is inequivalent to other inflation scenarios previously studied.', '0805.3481-2-45-0': "In summary, there are many equivalent ways to describe the discrepancy between observed matter content of the universe and Einstein's General Relativity.", '0805.3481-2-45-1': 'At the homogeneous level, it is trivial to find relations among them.', '0805.3481-2-45-2': 'The possibility to establish the equivalence of different descriptions is not a peculiarity of the description of this discrepancy in terms of a generalized first Friedman equation ([REF]).', '0805.3481-2-45-3': 'The same relations can be found among generalized equations of state, scalar-tensor theories and f(R) modified gravity [CITATION].', '0805.3481-2-46-0': '# Summary and discussion', '0805.3481-2-47-0': 'It may be interesting to go beyond [MATH]CDM in the description of the history of the universe in order to identify the origin of the two periods of accelerated expansion.', '0805.3481-2-47-1': 'We have proposed to use the expression of the energy density as a function of the Hubble parameter as the best candidate to describe the history of the universe.', '0805.3481-2-47-2': 'In this context the late time period of accelerated expansion and the early time inflation period can be easily parametrized.', '0805.3481-2-48-0': 'We have considered a simple modification of the cosmological equations characterized by the appearance of an upper and a lower bound on the Hubble expansion rate.', '0805.3481-2-48-1': 'A better fit of the experimental data can be obtained with this extended cosmological model as compared with the [MATH]CDM fit.', '0805.3481-2-48-2': 'Once more precise data are available, it will be possible to identify the behavior of the energy density as a function of the Hubble parameter and then look for a theoretical derivation of such behavior.', '0805.3481-2-49-0': 'We plan to continue with a systematic analysis of different alternatives incorporating the main features of the example considered in this work.', '0805.3481-2-49-1': 'Although the details of the departures from the standard cosmology can change, we expect a general pattern of the effects induced by the presence of the two bounds on [MATH].', '0805.3481-2-49-2': 'We also plan to go further, considering the evolution of inhomogeneities looking for new consequences of the bounds on [MATH].', '0805.3481-2-50-0': 'The discussion presented in this work, which is based on a new description of the periods of accelerated expansion of the universe, can open a new way to explore either modifications of the theory of gravity or new components in the universe homogeneous fluid.', '0805.3481-2-50-1': 'Lacking theoretical criteria to select among the possible ways to go beyond [MATH]CDM, we think that the phenomenological approach proposed in this work is justified.', '0805.3481-2-51-0': 'We are grateful to Paola Arias and Justo Lopez-Sarrion for discussions in the first stages of this work.', '0805.3481-2-51-1': 'We also acknowledge discussions with Julio Fabris, Antonio Segui, Roberto Emparan, Sergei Odintsov, Aurelio Grillo and Fernando Mendez.', '0805.3481-2-51-2': 'This work has been partially supported by CICYT (grant FPA2006-02315) and DGIID-DGA (grant2008-E24/2).', '0805.3481-2-51-3': 'J.I. acknowledges a FPU grant from MEC.'}

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'0805.3481-2-34-9'], ['0805.3481-1-35-10', '0805.3481-2-34-10'], ['0805.3481-1-35-11', '0805.3481-2-34-11'], ['0805.3481-1-35-12', '0805.3481-2-34-12'], ['0805.3481-1-35-13', '0805.3481-2-34-13'], ['0805.3481-1-4-3', '0805.3481-2-4-3'], ['0805.3481-1-4-4', '0805.3481-2-4-4'], ['0805.3481-1-4-5', '0805.3481-2-4-5'], ['0805.3481-1-49-1', '0805.3481-2-48-1'], ['0805.3481-1-25-0', '0805.3481-2-25-0'], ['0805.3481-1-25-1', '0805.3481-2-25-1'], ['0805.3481-1-25-2', '0805.3481-2-25-2'], ['0805.3481-1-25-4', '0805.3481-2-25-4'], ['0805.3481-1-44-1', '0805.3481-2-43-1'], ['0805.3481-1-44-2', '0805.3481-2-43-2'], ['0805.3481-1-44-3', '0805.3481-2-43-3'], ['0805.3481-1-44-4', '0805.3481-2-43-4'], ['0805.3481-1-44-5', '0805.3481-2-43-5'], ['0805.3481-1-44-6', '0805.3481-2-43-6'], ['0805.3481-1-44-7', '0805.3481-2-43-7'], ['0805.3481-1-44-8', '0805.3481-2-43-8'], ['0805.3481-1-44-9', '0805.3481-2-43-9'], ['0805.3481-1-44-10', '0805.3481-2-43-10'], ['0805.3481-1-44-11', '0805.3481-2-43-11'], 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[]

[['0805.3481-1-16-5', '0805.3481-2-16-5'], ['0805.3481-1-41-2', '0805.3481-2-40-2'], ['0805.3481-1-41-3', '0805.3481-2-40-3'], ['0805.3481-1-4-2', '0805.3481-2-4-2'], ['0805.3481-1-45-3', '0805.3481-2-44-3'], ['0805.3481-1-48-2', '0805.3481-2-47-2'], ['0805.3481-1-27-4', '0805.3481-2-27-4'], ['0805.3481-1-37-0', '0805.3481-2-36-0'], ['0805.3481-1-43-0', '0805.3481-2-42-0'], ['0805.3481-1-10-7', '0805.3481-2-10-6'], ['0805.3481-1-42-7', '0805.3481-2-41-7'], ['0805.3481-1-31-0', '0805.3481-2-30-4']]

[]

['0805.3481-1-13-2', '0805.3481-1-16-3', '0805.3481-2-13-2', '0805.3481-2-16-3']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0805.3481

1803.10436

{'1803.10436-1-0-0': 'Let [MATH] be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form [MATH].', '1803.10436-1-0-1': 'We assume that [MATH] is nil-invariant.', '1803.10436-1-0-2': 'This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of [MATH] is an infinitesimal isometry for [MATH].', '1803.10436-1-0-3': 'Among these Lie algebras are the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume.', '1803.10436-1-0-4': 'We prove a strong invariance property for nil-invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non-compact type of [MATH] act by infinitesimal isometries for [MATH].', '1803.10436-1-0-5': 'Moreover, we study properties of the kernel of [MATH] and the totally isotropic ideals in [MATH] in relation to the index of [MATH].', '1803.10436-1-0-6': 'Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most two.', '1803.10436-1-0-7': 'Examples show that the theory becomes significantly more complicated for index greater than two.', '1803.10436-1-1-0': '# Introduction and main results', '1803.10436-1-2-0': 'Let [MATH] be a finite-dimensional Lie algebra equipped with a symmetric bilinear form [MATH].', '1803.10436-1-2-1': 'The pair is called a metric Lie algebra.', '1803.10436-1-2-2': 'Traditionally, the bilinear form [MATH] is called invariant if the adjoint representation of [MATH] acts by skew linear maps.', '1803.10436-1-2-3': 'We will call [MATH] nil-invariant, if every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of [MATH] is a skew linear map.', '1803.10436-1-2-4': 'This nil-invariance condition appears to be significantly weaker than the requirement that [MATH] is invariant.', '1803.10436-1-3-0': 'Recall that the dimension of a maximal totally isotropic subspace is called the index of a symmetric bilinear form, and that the form is called definite if its index is zero.', '1803.10436-1-3-1': 'Since definite bilinear forms do not admit nilpotent skew maps, the condition of nil-invariance is less restrictive and therefore more interesting for metric Lie algebras with bilinear forms of higher index.', '1803.10436-1-4-0': 'In this paper, we mainly study finite-dimensional real Lie algebras [MATH] with a nil-invariant symmetric bilinear form.', '1803.10436-1-4-1': 'We will discuss the general properties of these metric Lie algebras, compare them with Lie algebras with invariant symmetric bilinear form, and derive elements of a classification theory, which give a complete description for low index, in particular, in the situation of index less than three.', '1803.10436-1-5-0': 'Nil-invariant bilinear forms and isometry Lie algebras The motivation for this article mainly stems from the theory of geometric transformation groups and automorphism groups of geometric structures.', '1803.10436-1-6-0': 'Namely, consider a Lie group [MATH] acting by isometries on a pseudo-Riemannian manifold [MATH] of finite volume.', '1803.10436-1-6-1': 'Then at each point [MATH], the scalar product [MATH] naturally induces a symmetric bilinear form [MATH] on the Lie algebra [MATH] of [MATH].', '1803.10436-1-6-2': 'As we show in Section [REF] of this paper, the bilinear form [MATH] is nil-invariant on [MATH].', '1803.10436-1-6-3': 'Note that, in general, [MATH] will be degenerate, since the subalgebra [MATH] of [MATH] tangent to the stabilizer [MATH] of [MATH] is contained in its kernel.', '1803.10436-1-7-0': 'Isometry groups of Lorentzian metrics (where the scalar products [MATH] are of index one) have been studied intensely.', '1803.10436-1-7-1': 'Results obtained by Adams and Stuck [CITATION] in the compact situation, and by Zeghib [CITATION] amount to a classification of the isometry Lie algebras of Lorentzian manifolds of finite volume.', '1803.10436-1-8-0': 'In these works it is used prominently that, for Lorentzian finite volume manifolds, the scalar products [MATH] are invariant by the elements of the nilpotent radical of [MATH], cf[MATH] [CITATION].', '1803.10436-1-8-1': 'The latter condition is closely related to nil-invariance, but it is also significantly less restrictive.', '1803.10436-1-8-2': 'The role played by the stronger nil-invariance condition seems to have gone unnoticed so far.', '1803.10436-1-9-0': 'Aside from Lorentzian manifolds, the classification problem for isometry Lie algebras of finite volume geometric manifolds with metric [MATH] of arbitrary index appears to be much more difficult.', '1803.10436-1-10-0': 'Some more specific results have been obtained in the context of homogeneous pseudo-Riemannian manifolds.', '1803.10436-1-10-1': 'Here, [MATH] can be described as a coset space [MATH], and any associated metric Lie algebra [MATH] locally determines [MATH] and [MATH], as well as the geometry of [MATH].', '1803.10436-1-10-2': 'These pseudo-Riemannian manifolds are model spaces of particular interest.', '1803.10436-1-11-0': 'Based on [CITATION], a structure theory for Lorentzian homogeneous spaces of finite volume is given by Zeghib [CITATION].', '1803.10436-1-12-0': 'Pseudo-Riemannian homogeneous spaces of arbitrary index were studied by Baues and Globke [CITATION] for solvable Lie groups [MATH].', '1803.10436-1-12-1': 'They found that, for solvable [MATH], the finite volume condition implies that the stabilizer [MATH] is a lattice in [MATH] and that the metric on [MATH] is induced by a bi-invariant metric on [MATH].', '1803.10436-1-12-2': 'Also it was observed in [CITATION] that the nil-invariance condition holds for the isometry Lie algebras of finite volume homogeneous spaces, where it appears as a direct consequence of the Borel density theorem.', '1803.10436-1-12-3': 'The main result in [CITATION] amounts to showing the surprising fact that any nil-invariant symmetric bilinear form on a solvable Lie algebra [MATH] is, in fact, an invariant form.', '1803.10436-1-13-0': 'By studying metric Lie algebras with nil-invariant symmetric bilinear form, the present work aims to further understand the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume.', '1803.10436-1-13-1': 'We will derive a structure theory which allows to completely describe such algebras in index less than three.', '1803.10436-1-13-2': 'In particular, this classification contains all local models for pseudo-Riemannian homogeneous spaces of finite volume of index less than three.', '1803.10436-1-14-0': '## Main results and structure of the paper', '1803.10436-1-15-0': 'In Section [REF], we prove that the orbit maps of isometric actions of Lie groups on pseudo-Riemannian manifolds of finite volume give rise to nil-invariant scalar products on their tangent Lie algebras.', '1803.10436-1-16-0': 'Some basic definitions and properties of metric Lie algebras are reviewed in Section [REF].', '1803.10436-1-17-0': 'In favourable cases, nil-invariance of [MATH] already implies invariance.', '1803.10436-1-17-1': 'For solvable Lie algebras [MATH], this is always the case, as was shown in [CITATION].', '1803.10436-1-17-2': 'These results are briefly summarized in Section [REF].', '1803.10436-1-17-3': 'In this section, we will also review the classification of solvable Lie algebras with invariant scalar products of index one and two.', '1803.10436-1-17-4': 'Their properties will be needed further on.', '1803.10436-1-18-0': '## Strong invariance properties', '1803.10436-1-19-0': 'In Section [REF] we begin our investigation of nil-invariant symmetric bilinear forms [MATH] on arbitrary Lie algebras.', '1803.10436-1-19-1': 'For any Lie algebra [MATH], we let [EQUATION] denote a Levi decomposition of [MATH], where [MATH] is semisimple of compact type, [MATH] is semisimple of non-compact type and [MATH] is the solvable radical of [MATH].', '1803.10436-1-19-2': 'For this, recall that [MATH] is called of compact type if the Killing form of [MATH] is definite and that [MATH] is of non-compact type if it has no ideal of compact type.', '1803.10436-1-19-3': 'We also write [EQUATION].', '1803.10436-1-20-0': 'Our first main result is a strong invariance property for nil-invariant symmetric bilinear forms:', '1803.10436-1-21-0': 'Let [MATH] be a real finite-dimensional Lie algebra, let [MATH] be a nil-invariant symmetric bilinear form on [MATH] and [MATH] the restriction of [MATH] to [MATH].', '1803.10436-1-21-1': 'Then:', '1803.10436-1-22-0': '[MATH] is invariant by the adjoint action of [MATH] on [MATH].', '1803.10436-1-22-1': '[MATH] is invariant by [MATH].', '1803.10436-1-23-0': 'This result is as good as one can hope for.', '1803.10436-1-23-1': 'For example, any scalar product on a semisimple Lie algebra [MATH] of compact type is already nil-invariant, without any further invariance property required.', '1803.10436-1-23-2': 'Note that Theorem [REF] does not require any assumption on the index of [MATH].', '1803.10436-1-24-0': 'We would like to point out that the proof of Theorem [REF] works for Lie algebras over any field of characteristic zero if the notion of subalgebra of compact type [MATH] is replaced by the appropriate notion of maximal anisotropic semisimple subalgebra of [MATH].', '1803.10436-1-24-1': 'The latter condition is equivalent to the requirement that the Cartan subalgebras of [MATH] do not contain any elements split over the ground field.', '1803.10436-1-25-0': 'We obtain the following striking corollary to Theorem [REF], or rather to its proof:', '1803.10436-1-26-0': 'Let [MATH] be a finite-dimensional Lie algebra over the field of complex numbers and [MATH] a nil-invariant symmetric bilinear form on [MATH].', '1803.10436-1-26-1': 'Then [MATH] is invariant.', '1803.10436-1-27-0': 'For any nil-invariant symmetric bilinear form [MATH], it is important to consider its kernel [EQUATION] also called the metric radical of [MATH].', '1803.10436-1-27-1': 'If [MATH] is invariant, then [MATH] is an ideal of [MATH].', '1803.10436-1-27-2': 'If [MATH] is nil-invariant, then, in general, [MATH] is not even a subalgebra of [MATH].', '1803.10436-1-27-3': 'Nevertheless, a considerable simplification of the exposition may be obtained by restricting results to metric Lie algebras whose radical [MATH] does not contain any non-trivial ideals of [MATH].', '1803.10436-1-27-4': 'Such metric Lie algebras will be called effective.', '1803.10436-1-27-5': 'This condition is, of course, natural from the geometric motivation.', '1803.10436-1-27-6': 'Moreover, it is not a genuine restriction, since by dividing out the maximal ideal of [MATH] contained in [MATH], one may pass from any metric Lie algebra to a quotient metric Lie algebra that is effective.', '1803.10436-1-28-0': 'Theorem [REF] determines the properties of [MATH] significantly as is shown in the following:', '1803.10436-1-29-0': 'Let [MATH] be a finite-dimensional real Lie algebra with a nil-invariant symmetric bilinear form [MATH].', '1803.10436-1-29-1': 'Assume that the metric radical [MATH] does not contain any non-trivial ideal of [MATH].', '1803.10436-1-29-2': 'Let [MATH] denote the center of [MATH].', '1803.10436-1-29-3': 'Then [EQUATION].', '1803.10436-1-30-0': 'The proof of Corollary [REF] can be found in Section [REF], which is at the technical heart of our paper.', '1803.10436-1-30-1': 'We start out by studying the totally isotropic ideals in [MATH], and in particular properties of the metric radical [MATH].', '1803.10436-1-30-2': 'Our first main result in Section [REF] is the proof of Corollary [REF] in Subsection [REF].', '1803.10436-1-31-0': 'As the form [MATH] may be degenerate, it is useful to introduce its relative index.', '1803.10436-1-31-1': 'By definition, this is the index of the induced scalar product on the vector space [MATH].', '1803.10436-1-31-2': 'The relative index mostly determines the geometric and algebraic type of the bilinear form [MATH].', '1803.10436-1-32-0': 'For effective metric Lie algebras with relative index [MATH], we further strengthen Corollary [REF] by showing that, with this additional requirement, [MATH] does not intersect [MATH].', '1803.10436-1-32-1': 'This is formulated in Corollary [REF].', '1803.10436-1-33-0': '## Classifications for small index', '1803.10436-1-34-0': 'Section [REF] culminates in Subsection [REF], where we give an analysis of the action of semisimple subalgebras on the solvable radical of [MATH].', '1803.10436-1-34-1': 'This imposes strong restrictions on the structure of [MATH] for small relative index.', '1803.10436-1-35-0': 'The combined results are summarized in Section [REF], leading to the following general structure theorem for the case [MATH]:', '1803.10436-1-36-0': 'Let [MATH] be a real finite-dimensional Lie algebra with nil-invariant symmetric bilinear form [MATH] of relative index [MATH], and assume that [MATH] does not contain a non-trivial ideal of [MATH].', '1803.10436-1-36-1': 'Then:', '1803.10436-1-37-0': 'The Levi decomposition [REF] of [MATH] is a direct sum of ideals: [MATH].', '1803.10436-1-38-0': '[MATH] is contained in [MATH] and [MATH].', '1803.10436-1-38-1': '[MATH] and [MATH].', '1803.10436-1-39-0': 'Examples in Section [REF] illustrate that the statements in Theorem [REF] may fail for relative index [MATH].', '1803.10436-1-40-0': 'We specialize Theorem [REF] to obtain classifications of the Lie algebras [MATH] in the cases [MATH] (Theorem [REF]) and [MATH] (Theorem [REF]).', '1803.10436-1-40-1': 'As follows from the discussion at the beginning, these theorems also describe the structure of isometry Lie algebras of pseudo-Riemannian homogeneous spaces of finite volume with index one or two (real signatures of type [MATH] or [MATH], respectively).', '1803.10436-1-40-2': 'Our first result concerns the Lorentzian case:', '1803.10436-1-41-0': 'Let [MATH] be a Lie algebra with nil-invariant symmetric bilinear form [MATH] of relative index [MATH], and assume that [MATH] does not contain a non-trivial ideal of [MATH].', '1803.10436-1-41-1': 'Then one of the following cases occurs:', '1803.10436-1-42-0': '[(I)] [MATH], where [MATH] is abelian and either semidefinite or Lorentzian.', '1803.10436-1-42-1': '[(II)] [MATH], where [MATH] is Lorentzian of oscillator type.', '1803.10436-1-42-2': '[(III)] [MATH], where [MATH] is abelian and definite, [MATH] is Lorentzian.', '1803.10436-1-43-0': "This classification of isometry Lie algebras for finite volume homogeneous Lorentzian manifolds is contained in Zeghib's [CITATION], which uses a somewhat different approach in its proof.", '1803.10436-1-43-1': 'Moreover, the list in [CITATION] contains two additional cases of metric Lie algebras (Heisenberg algebra and tangent algebra of the affine group, compare Example [REF] of the present paper) that cannot appear as Lie algebras of transitive Lorentzian isometry groups, since they do not satisfy the effectivity condition.', '1803.10436-1-43-2': 'According to [CITATION], models of all three types (I)-(III) actually occur as isometry Lie algebras of homogeneous spaces [MATH], in which case [MATH] is a subalgebra tangent to a closed subgroup [MATH] of [MATH].', '1803.10436-1-44-0': 'The algebraic methods developed here also lead to a complete understanding in the case of signature [MATH]:', '1803.10436-1-45-0': 'Let [MATH] be a Lie algebra with nil-invariant symmetric bilinear form [MATH] of relative index [MATH], and assume that [MATH] does not contain a non-trivial ideal of [MATH].', '1803.10436-1-45-1': 'Then one of the following cases occurs:', '1803.10436-1-46-0': '[(I)] [MATH], where [MATH] is one of the following:', '1803.10436-1-47-0': '[MATH] is abelian.', '1803.10436-1-47-1': '[MATH] is Lorentzian of oscillator type.', '1803.10436-1-47-2': '[MATH] is solvable but non-abelian with invariant scalar product of index [MATH].', '1803.10436-1-48-0': '[(II)] [MATH].', '1803.10436-1-48-1': 'Here, [MATH] is abelian, [MATH] with a non-degenerate invariant scalar product of index [MATH].', '1803.10436-1-48-2': 'Moreover, [MATH] is definite.', '1803.10436-1-48-3': '[(III)] [MATH], where [MATH] is Lorentzian, and [MATH] is one of the following:', '1803.10436-1-49-0': '[MATH] is abelian and either semidefinite or Lorentzian.', '1803.10436-1-49-1': '[MATH] is Lorentzian of oscillator type.', '1803.10436-1-50-0': 'For the definition of an oscillator algebra, see Example [REF].', '1803.10436-1-50-1': 'The possibilities for [MATH] in case (I-c) of Theorem [REF] above are discussed in Section [REF].', '1803.10436-1-50-2': 'Note further that the orthogonality relations of Theorem [REF] part (3) are always satisfied.', '1803.10436-1-51-0': 'Theorem [REF] contains no information which of the possible algebraic models actually do occur as isometry Lie algebras of homogeneous spaces of index two.', '1803.10436-1-51-1': 'This question needs to be considered on another occasion.', '1803.10436-1-52-0': '## Notations and conventions', '1803.10436-1-53-0': 'The identity element of a group [MATH] is denoted by [MATH].', '1803.10436-1-54-0': 'Let [MATH] be a subgroup of a Lie group [MATH].', '1803.10436-1-54-1': 'We write [MATH] for the adjoint representation of [MATH] on the Lie algebra [MATH] of [MATH], to distinguish it from the adjoint representation [MATH] on its own Lie algebra [MATH].', '1803.10436-1-55-0': 'The centralizer and the normalizer of [MATH] in [MATH] are denoted by [MATH] and [MATH], respectively.', '1803.10436-1-55-1': 'The center of [MATH] is denoted by [MATH].', '1803.10436-1-56-0': 'If [MATH] and [MATH] are two Lie algebras, the notation [MATH] denotes the direct product of Lie algebras.', '1803.10436-1-56-1': 'The notations [MATH] and [MATH] are used to indicate sums and direct sums of vector spaces.', '1803.10436-1-57-0': 'The solvable radical [MATH] of [MATH] is the maximal solvable ideal of [MATH].', '1803.10436-1-57-1': 'The semisimple Lie algebra [MATH] is a direct product [MATH] of Lie algebras, where [MATH] is a semisimple Lie algebra of compact type, meaning that the Killing form of [MATH] is definite, and [MATH] is semisimple without factors of compact type.', '1803.10436-1-58-0': 'For any linear operator [MATH], [MATH] denotes its Jordan decomposition, where [MATH] is semisimple, and [MATH] is nilpotent.', '1803.10436-1-58-1': 'Further notation will be introduced in Section [REF].'}

{'1803.10436-2-0-0': 'Let [MATH] be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form [MATH].', '1803.10436-2-0-1': 'We assume that [MATH] is nil-invariant.', '1803.10436-2-0-2': 'This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of [MATH] is an infinitesimal isometry for [MATH].', '1803.10436-2-0-3': 'Among these Lie algebras are the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume.', '1803.10436-2-0-4': 'We prove a strong invariance property for nil-invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non-compact type of [MATH] act by infinitesimal isometries for [MATH].', '1803.10436-2-0-5': 'Moreover, we study properties of the kernel of [MATH] and the totally isotropic ideals in [MATH] in relation to the index of [MATH].', '1803.10436-2-0-6': 'Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most two.', '1803.10436-2-0-7': 'Examples show that the theory becomes significantly more complicated for index greater than two.', '1803.10436-2-0-8': 'We apply our results to study simply connected pseudo-Riemannian homogeneous spaces of finite volume.', '1803.10436-2-1-0': '# Introduction and main results', '1803.10436-2-2-0': 'Let [MATH] be a finite-dimensional Lie algebra equipped with a symmetric bilinear form [MATH].', '1803.10436-2-2-1': 'The pair is called a metric Lie algebra.', '1803.10436-2-2-2': 'Traditionally, the bilinear form [MATH] is called invariant if the adjoint representation of [MATH] acts by skew linear maps.', '1803.10436-2-2-3': 'We will call [MATH] nil-invariant, if every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of [MATH] is a skew linear map.', '1803.10436-2-2-4': 'This nil-invariance condition appears to be significantly weaker than the requirement that [MATH] is invariant.', '1803.10436-2-3-0': 'Recall that the dimension of a maximal totally isotropic subspace is called the index of a symmetric bilinear form, and that the form is called definite if its index is zero.', '1803.10436-2-3-1': 'Since definite bilinear forms do not admit nilpotent skew maps, the condition of nil-invariance is less restrictive and therefore more interesting for metric Lie algebras with bilinear forms of higher index.', '1803.10436-2-4-0': 'In this paper, we mainly study finite-dimensional real Lie algebras [MATH] with a nil-invariant symmetric bilinear form.', '1803.10436-2-4-1': 'We will discuss the general properties of these metric Lie algebras, compare them with Lie algebras with invariant symmetric bilinear form, and derive elements of a classification theory, which give a complete description for low index, in particular, in the situation of index less than three.', '1803.10436-2-5-0': 'Nil-invariant bilinear forms and isometry Lie algebras The motivation for this article mainly stems from the theory of geometric transformation groups and automorphism groups of geometric structures.', '1803.10436-2-6-0': 'Namely, consider a Lie group [MATH] acting by isometries on a pseudo-Riemannian manifold [MATH] of finite volume.', '1803.10436-2-6-1': 'Then at each point [MATH], the scalar product [MATH] naturally induces a symmetric bilinear form [MATH] on the Lie algebra [MATH] of [MATH].', '1803.10436-2-6-2': 'As we show in Section [REF] of this paper, the bilinear form [MATH] is nil-invariant on [MATH].', '1803.10436-2-6-3': 'Note that, in general, [MATH] will be degenerate, since the subalgebra [MATH] of [MATH] tangent to the stabilizer [MATH] of [MATH] is contained in its kernel.', '1803.10436-2-7-0': 'Isometry groups of Lorentzian metrics (where the scalar products [MATH] are of index one) have been studied intensely.', '1803.10436-2-7-1': 'Results obtained by Adams and Stuck [CITATION] in the compact situation, and by Zeghib [CITATION] amount to a classification of the isometry Lie algebras of Lorentzian manifolds of finite volume.', '1803.10436-2-8-0': 'In these works it is used prominently that, for Lorentzian finite volume manifolds, the scalar products [MATH] are invariant by the elements of the nilpotent radical of [MATH], cf[MATH] [CITATION].', '1803.10436-2-8-1': 'The latter condition is closely related to nil-invariance, but it is also significantly less restrictive.', '1803.10436-2-8-2': 'The role played by the stronger nil-invariance condition seems to have gone unnoticed so far.', '1803.10436-2-9-0': 'Aside from Lorentzian manifolds, the classification problem for isometry Lie algebras of finite volume geometric manifolds with metric [MATH] of arbitrary index appears to be much more difficult.', '1803.10436-2-10-0': 'Some more specific results have been obtained in the context of homogeneous pseudo-Riemannian manifolds.', '1803.10436-2-10-1': 'Here, [MATH] can be described as a coset space [MATH], and any associated metric Lie algebra [MATH] locally determines [MATH] and [MATH], as well as the geometry of [MATH].', '1803.10436-2-10-2': 'These pseudo-Riemannian manifolds are model spaces of particular interest.', '1803.10436-2-11-0': 'Based on [CITATION], a structure theory for Lorentzian homogeneous spaces of finite volume is given by Zeghib [CITATION].', '1803.10436-2-12-0': 'Pseudo-Riemannian homogeneous spaces of arbitrary index were studied by Baues and Globke [CITATION] for solvable Lie groups [MATH].', '1803.10436-2-12-1': 'They found that, for solvable [MATH], the finite volume condition implies that the stabilizer [MATH] is a lattice in [MATH] and that the metric on [MATH] is induced by a bi-invariant metric on [MATH].', '1803.10436-2-12-2': 'Also it was observed in [CITATION] that the nil-invariance condition holds for the isometry Lie algebras of finite volume homogeneous spaces, where it appears as a direct consequence of the Borel density theorem.', '1803.10436-2-12-3': 'The main result in [CITATION] amounts to showing the surprising fact that any nil-invariant symmetric bilinear form on a solvable Lie algebra [MATH] is, in fact, an invariant form (a concise proof is also provided in Appendix [REF]).', '1803.10436-2-13-0': 'By studying metric Lie algebras with nil-invariant symmetric bilinear form, the present work aims to further understand the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume.', '1803.10436-2-13-1': 'We will derive a structure theory which allows to completely describe such algebras in index less than three.', '1803.10436-2-13-2': 'In particular, this classification contains all local models for pseudo-Riemannian homogeneous spaces of finite volume of index less than three.', '1803.10436-2-14-0': '## Main results and structure of the paper', '1803.10436-2-15-0': 'In Section [REF], we prove that the orbit maps of isometric actions of Lie groups on pseudo-Riemannian manifolds of finite volume give rise to nil-invariant scalar products on their tangent Lie algebras.', '1803.10436-2-16-0': 'Some basic definitions and properties of metric Lie algebras are reviewed in Section [REF].', '1803.10436-2-17-0': 'In favourable cases, nil-invariance of [MATH] already implies invariance.', '1803.10436-2-17-1': 'For solvable Lie algebras [MATH], this is always the case, as was first shown in [CITATION].', '1803.10436-2-17-2': 'These results are briefly summarized in Section [REF].', '1803.10436-2-17-3': 'In this section, we will also review the classification of solvable Lie algebras with invariant scalar products of index one and two.', '1803.10436-2-17-4': 'Their properties will be needed further on.', '1803.10436-2-18-0': '## Strong invariance properties', '1803.10436-2-19-0': 'In Section [REF] we begin our investigation of nil-invariant symmetric bilinear forms [MATH] on arbitrary Lie algebras.', '1803.10436-2-19-1': 'For any Lie algebra [MATH], we let [EQUATION] denote a Levi decomposition of [MATH], where [MATH] is semisimple of compact type, [MATH] is semisimple of non-compact type and [MATH] is the solvable radical of [MATH].', '1803.10436-2-19-2': 'For this, recall that [MATH] is called of compact type if the Killing form of [MATH] is definite and that [MATH] is of non-compact type if it has no ideal of compact type.', '1803.10436-2-19-3': 'We also write [EQUATION].', '1803.10436-2-20-0': 'Our first main result is a strong invariance property for nil-invariant symmetric bilinear forms:', '1803.10436-2-21-0': 'Let [MATH] be a real finite-dimensional Lie algebra, let [MATH] be a nil-invariant symmetric bilinear form on [MATH] and [MATH] the restriction of [MATH] to [MATH].', '1803.10436-2-21-1': 'Then:', '1803.10436-2-22-0': '[MATH] is invariant by the adjoint action of [MATH] on [MATH].', '1803.10436-2-22-1': '[MATH] is invariant by [MATH].', '1803.10436-2-23-0': 'Note that any scalar product on a semisimple Lie algebra [MATH] of compact type is already nil-invariant, without any further invariance property required.', '1803.10436-2-23-1': 'Therefore, Theorem [REF] is as strong as one can hope for.', '1803.10436-2-24-0': 'We would like to point out that the proof of Theorem [REF] works for Lie algebras over any field of characteristic zero if the notion of subalgebra of compact type [MATH] is replaced by the appropriate notion of maximal anisotropic semisimple subalgebra of [MATH].', '1803.10436-2-24-1': 'The latter condition is equivalent to the requirement that the Cartan subalgebras of [MATH] do not contain any elements split over the ground field.', '1803.10436-2-25-0': 'We obtain the following striking corollary to Theorem [REF], or rather to its proof:', '1803.10436-2-26-0': 'Let [MATH] be a finite-dimensional Lie algebra over the field of complex numbers and [MATH] a nil-invariant symmetric bilinear form on [MATH].', '1803.10436-2-26-1': 'Then [MATH] is invariant.', '1803.10436-2-27-0': 'For any nil-invariant symmetric bilinear form [MATH], it is important to consider its kernel [EQUATION] also called the metric radical of [MATH].', '1803.10436-2-27-1': 'If [MATH] is invariant, then [MATH] is an ideal of [MATH].', '1803.10436-2-27-2': 'If [MATH] is nil-invariant, then, in general, [MATH] is not even a subalgebra of [MATH].', '1803.10436-2-27-3': 'Nevertheless, a considerable simplification of the exposition may be obtained by restricting results to metric Lie algebras whose radical [MATH] does not contain any non-trivial ideals of [MATH].', '1803.10436-2-27-4': 'Such metric Lie algebras will be called effective.', '1803.10436-2-27-5': 'This condition is, of course, natural from the geometric motivation.', '1803.10436-2-27-6': 'Moreover, it is not a genuine restriction, since by dividing out the maximal ideal of [MATH] contained in [MATH], one may pass from any metric Lie algebra to a quotient metric Lie algebra that is effective.', '1803.10436-2-28-0': 'Theorem [REF] determines the properties of [MATH] significantly as is shown in the following:', '1803.10436-2-29-0': 'Let [MATH] be a finite-dimensional real Lie algebra with a nil-invariant symmetric bilinear form [MATH].', '1803.10436-2-29-1': 'Assume that the metric radical [MATH] does not contain any non-trivial ideal of [MATH].', '1803.10436-2-29-2': 'Let [MATH] denote the center of [MATH].', '1803.10436-2-29-3': 'Then [EQUATION].', '1803.10436-2-30-0': 'The proof of Corollary [REF] can be found in Section [REF], which is at the technical heart of our paper.', '1803.10436-2-30-1': 'In Subsection [REF] we start out by studying the totally isotropic ideals in [MATH], and in particular properties of the metric radical [MATH].', '1803.10436-2-30-2': 'The main part of the proof of Corollary [REF] is given in Subsection [REF].', '1803.10436-2-30-3': 'We then also prove that if in addition [MATH] is [MATH]-invariant, then [MATH].', '1803.10436-2-31-0': 'As the form [MATH] may be degenerate, it is useful to introduce its relative index.', '1803.10436-2-31-1': 'By definition, this is the index of the induced scalar product on the vector space [MATH].', '1803.10436-2-31-2': 'The relative index mostly determines the geometric and algebraic type of the bilinear form [MATH].', '1803.10436-2-32-0': 'For effective metric Lie algebras with relative index [MATH], we further strengthen Corollary [REF] by showing that, with this additional requirement, [MATH] does not intersect [MATH].', '1803.10436-2-32-1': 'This is formulated in Corollary [REF].', '1803.10436-2-33-0': '## Classifications for small index', '1803.10436-2-34-0': 'Section [REF] culminates in Subsection [REF], where we give an analysis of the action of semisimple subalgebras on the solvable radical of [MATH].', '1803.10436-2-34-1': 'This imposes strong restrictions on the structure of [MATH] for small relative index.', '1803.10436-2-35-0': 'The combined results are summarized in Section [REF], leading to the following general structure theorem for the case [MATH]:', '1803.10436-2-36-0': 'Let [MATH] be a real finite-dimensional Lie algebra with nil-invariant symmetric bilinear form [MATH] of relative index [MATH], and assume that [MATH] does not contain a non-trivial ideal of [MATH].', '1803.10436-2-36-1': 'Then:', '1803.10436-2-37-0': 'The Levi decomposition [REF] of [MATH] is a direct sum of ideals: [MATH].', '1803.10436-2-38-0': '[MATH] is contained in [MATH] and [MATH].', '1803.10436-2-38-1': '[MATH] and [MATH].', '1803.10436-2-39-0': 'Examples in Section [REF] illustrate that the statements in Theorem [REF] may fail for relative index [MATH].', '1803.10436-2-40-0': 'We specialize Theorem [REF] to obtain classifications of the Lie algebras [MATH] in the cases [MATH] and [MATH].', '1803.10436-2-40-1': 'As follows from the discussion at the beginning, these theorems also describe the structure of isometry Lie algebras of pseudo-Riemannian homogeneous spaces of finite volume with index one or two (real signatures of type [MATH] or [MATH], respectively).', '1803.10436-2-40-2': 'Our first result concerns the Lorentzian case:', '1803.10436-2-41-0': 'Let [MATH] be a Lie algebra with nil-invariant symmetric bilinear form [MATH] of relative index [MATH], and assume that [MATH] does not contain a non-trivial ideal of [MATH].', '1803.10436-2-41-1': 'Then one of the following cases occurs:', '1803.10436-2-42-0': '[(I)] [MATH], where [MATH] is abelian and either semidefinite or Lorentzian.', '1803.10436-2-42-1': '[(II)] [MATH], where [MATH] is Lorentzian of oscillator type.', '1803.10436-2-42-2': '[(III)] [MATH], where [MATH] is abelian and definite, [MATH] is Lorentzian.', '1803.10436-2-43-0': "This classification of isometry Lie algebras for finite volume homogeneous Lorentzian manifolds is contained in Zeghib's [CITATION], which uses a somewhat different approach in its proof.", '1803.10436-2-43-1': 'Moreover, the list in [CITATION] contains two additional cases of metric Lie algebras (Heisenberg algebra and tangent algebra of the affine group, compare Example [REF] of the present paper) that cannot appear as Lie algebras of transitive Lorentzian isometry groups, since they do not satisfy the effectivity condition.', '1803.10436-2-43-2': 'According to [CITATION], models of all three types (I)-(III) actually occur as isometry Lie algebras of homogeneous spaces [MATH], in which case [MATH] is a subalgebra tangent to a closed subgroup [MATH] of [MATH].', '1803.10436-2-44-0': 'The algebraic methods developed here also lead to a complete understanding in the case of signature [MATH]:', '1803.10436-2-45-0': 'Let [MATH] be a Lie algebra with nil-invariant symmetric bilinear form [MATH] of relative index [MATH], and assume that [MATH] does not contain a non-trivial ideal of [MATH].', '1803.10436-2-45-1': 'Then one of the following cases occurs:', '1803.10436-2-46-0': '[(I)] [MATH], where [MATH] is one of the following:', '1803.10436-2-47-0': '[MATH] is abelian.', '1803.10436-2-47-1': '[MATH] is Lorentzian of oscillator type.', '1803.10436-2-47-2': '[MATH] is solvable but non-abelian with invariant scalar product of index [MATH].', '1803.10436-2-48-0': '[(II)] [MATH].', '1803.10436-2-48-1': 'Here, [MATH] is abelian, [MATH] with a non-degenerate invariant scalar product of index [MATH].', '1803.10436-2-48-2': 'Moreover, [MATH] is definite.', '1803.10436-2-48-3': '[(III)] [MATH], where [MATH] is Lorentzian, and [MATH] is one of the following:', '1803.10436-2-49-0': '[MATH] is abelian and either semidefinite or Lorentzian.', '1803.10436-2-49-1': '[MATH] is Lorentzian of oscillator type.', '1803.10436-2-50-0': 'For the definition of an oscillator algebra, see Example [REF].', '1803.10436-2-50-1': 'The possibilities for [MATH] in case (I-c) of Theorem [REF] above are discussed in Section [REF].', '1803.10436-2-50-2': 'Note further that the orthogonality relations of Theorem [REF] part (3) are always satisfied.', '1803.10436-2-51-0': 'Theorem [REF] contains no information which of the possible algebraic models actually do occur as isometry Lie algebras of homogeneous spaces of index two.', '1803.10436-2-51-1': 'This question needs to be considered on another occasion.', '1803.10436-2-52-0': 'We apply our results to study the isometry groups of simply connected homogeneous pseudo-Riemannian manifolds of finite volume.', '1803.10436-2-52-1': "D'Ambra [CITATION] showed that a simply connected compact analytic Lorentzian manifold (not necessarily homogeneous) has compact isometry group, and she also gave an example of a simply connected compact analytic manifold of metric signature [MATH] that has a non-compact isometry group.", '1803.10436-2-53-0': 'Here, we study homogeneous spaces for arbitrary metric signature.', '1803.10436-2-53-1': 'The main result is the following theorem:', '1803.10436-2-54-0': 'Let [MATH] be a connected and simply connected pseudo-Riemannian homogeneous space of finite volume, [MATH], and let [MATH] be the stabilizer subgroup in [MATH] of a point in [MATH].', '1803.10436-2-54-1': 'Let [MATH] be a Levi decomposition, where [MATH] is the solvable radical of [MATH].', '1803.10436-2-54-2': 'Then:', '1803.10436-2-55-0': '[MATH] is compact.', '1803.10436-2-55-1': '[MATH] is compact and acts transitively on [MATH].', '1803.10436-2-55-2': '[MATH] is abelian.', '1803.10436-2-55-3': 'Let [MATH] be the maximal compact subgroup of [MATH].', '1803.10436-2-55-4': 'Then [MATH].', '1803.10436-2-55-5': 'More explicitely, [MATH] where [MATH] and [MATH].', '1803.10436-2-55-6': '[MATH] is connected.', '1803.10436-2-55-7': 'If [MATH], then [MATH], where [MATH] and [MATH] are normal subgroups in [MATH], [MATH] is finite, and [MATH] is the graph of a non-trivial homomorphism [MATH], where the restriction [MATH] is injective.', '1803.10436-2-56-0': 'In Section [REF] we give examples of isometry groups of compact simply connected homogeneous [MATH] with non-compact radical.', '1803.10436-2-56-1': 'However, for metric index [MATH] or [MATH] the isometry group of a simply connected [MATH] is always compact:', '1803.10436-2-57-0': 'The isometry group of any simply connected pseudo-Riemannian homogeneous manifold of finite volume with metric index [MATH] is compact.', '1803.10436-2-58-0': 'As follows from Theorem [REF], the isometry Lie algebra of a simply connected pseudo-Riemannian homogeneous space of finite volume has abelian radical.', '1803.10436-2-58-1': 'This motivates a closer investigation of Lie algebras with abelian radical that admit nil-invariant symmetric bilinear forms in Section [REF].', '1803.10436-2-58-2': 'In this direction, we prove:', '1803.10436-2-59-0': 'Let [MATH] be a Lie algebra whose solvable radical [MATH] is abelian.', '1803.10436-2-59-1': 'Suppose [MATH] is equipped with a nil-invariant symmetric bilinear form [MATH] such that the metric radical [MATH] of [MATH] does not contain a non-trivial ideal of [MATH].', '1803.10436-2-59-2': 'Let [MATH] be a Levi subalgebra of [MATH], where [MATH] is of compact type and [MATH] has no simple factors of compact type.', '1803.10436-2-59-3': 'Then [MATH] is an orthogonal direct product of ideals [EQUATION] with [EQUATION] where [MATH] and [MATH] are orthogonal direct products, and [MATH] is a metric cotangent algebra.', '1803.10436-2-59-4': 'The restrictions of [MATH] to [MATH] and [MATH] are invariant and non-degenerate.', '1803.10436-2-59-5': 'In particular, [MATH].', '1803.10436-2-60-0': 'For the definition of metric cotangent algebra, see Example [REF].', '1803.10436-2-60-1': 'We call an algebra [MATH] with [MATH] semisimple of compact type and [MATH] abelian a Lie algebra of Euclidean type.', '1803.10436-2-60-2': 'By Theorem [REF], isometry Lie algebras of compact simply connected pseudo-Riemannian homogeneous spaces are of Euclidean type.', '1803.10436-2-60-3': 'However, not every Lie algebra of Euclidean type appears as the isometry Lie algebra of a compact pseudo-Riemannian homogeneous space.', '1803.10436-2-60-4': 'In fact, this is the case for the Euclidean Lie algebras [MATH] with [MATH].', '1803.10436-2-61-0': 'The Euclidean group [MATH], [MATH], does not have compact quotients with a pseudo-Riemannian metric such that [MATH] acts isometrically and almost effectively.', '1803.10436-2-62-0': 'Note that [MATH] acts transitively and effectively on compact manifolds with finite fundamental group, as we remark at the end of Section [REF].', '1803.10436-2-63-0': '## Notations and conventions', '1803.10436-2-64-0': 'The identity element of a group [MATH] is denoted by [MATH].', '1803.10436-2-64-1': 'We let [MATH] denote the connected component of the identity of [MATH].', '1803.10436-2-65-0': 'Let [MATH] be a subgroup of a Lie group [MATH].', '1803.10436-2-65-1': 'We write [MATH] for the adjoint representation of [MATH] on the Lie algebra [MATH] of [MATH], to distinguish it from the adjoint representation [MATH] on its own Lie algebra [MATH].', '1803.10436-2-66-0': 'If [MATH] is a [MATH]-module, then we write [MATH] for the module of [MATH]-invariants.', '1803.10436-2-66-1': 'Similary, [MATH] for a [MATH]-module.', '1803.10436-2-67-0': 'The centralizer and the normalizer of [MATH] in [MATH] are denoted by [MATH] and [MATH], respectively.', '1803.10436-2-67-1': 'The center of [MATH] is denoted by [MATH].', '1803.10436-2-67-2': 'We use similar notation for Lie groups.', '1803.10436-2-68-0': 'If [MATH] and [MATH] are two Lie algebras, the notation [MATH] denotes the direct product of Lie algebras.', '1803.10436-2-68-1': 'The notations [MATH] and [MATH] are used to indicate sums and direct sums of vector spaces.', '1803.10436-2-69-0': 'The solvable radical [MATH] of [MATH] is the maximal solvable ideal of [MATH].', '1803.10436-2-69-1': 'The semisimple Lie algebra [MATH] is a direct product [MATH] of Lie algebras, where [MATH] is a semisimple Lie algebra of compact type, meaning that the Killing form of [MATH] is definite, and [MATH] is semisimple without factors of compact type.', '1803.10436-2-70-0': 'For any linear operator [MATH], [MATH] denotes its Jordan decomposition, where [MATH] is semisimple, and [MATH] is nilpotent.', '1803.10436-2-70-1': 'Further notation will be introduced in Section [REF].'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1803.10436

0907.3814

{'0907.3814-1-0-0': 'We report on the observation of a quenched moment of inertia as resulting from superfluidity in a strongly interacting Fermi gas.', '0907.3814-1-0-1': 'Our method is based on setting the hydrodynamic gas in slow rotation and determining its moment of inertia by detecting the precession of a radial quadrupole excitation.', '0907.3814-1-0-2': 'The measurements distinguish between the superfluid or collisional origin of hydrodynamic behavior, and show the phase transition.', '0907.3814-1-1-0': 'Superfluidity is a striking property of quantum fluids at very low temperatures.', '0907.3814-1-1-1': 'For bosonic systems, important examples are liquids and clusters of [MATH]He and atomic Bose-Einstein condensates.', '0907.3814-1-1-2': 'In fermionic systems, superfluidity is a more intricate phenomenon as it requires pairing of particles.', '0907.3814-1-1-3': 'Fermionic superfluidity is known to occur in atomic nuclei and [MATH]He liquids and it is also at the heart of superconductivity, thus being of great technological importance.', '0907.3814-1-1-4': 'Recent advances with ultracold Fermi gases have opened up unprecedented possibilities to study the properties of strongly interacting fermionic superfluids [CITATION].', '0907.3814-1-1-5': 'Early experiments on ultracold Fermi gases with resonant interparticle interactions compiled increasing evidence for superfluidity [CITATION] until the phenomenon was firmly established by the observation of vortex lattices [CITATION].', '0907.3814-1-2-0': 'In this Letter, we report on the manifestation of superfluidity in a quenched moment of inertia (MOI) in a strongly interacting Fermi gas undergoing slow rotation.', '0907.3814-1-2-1': 'The basic idea of a quenched MOI as a signature of superfluidity goes back more than 50 years in nuclear physics, where MOIs below the classical, rigid-body value were attributed to superfluidity [CITATION].', '0907.3814-1-2-2': 'The quenching of the MOI was also shown in liquid [MATH]He [CITATION] and has, more recently, served for the discovery of a supersolid phase [CITATION].', '0907.3814-1-2-3': 'Here we introduce the observation of the quenched MOI as a new method to study superfluidity in ultracold Fermi gases.', '0907.3814-1-3-0': 'Our basic experimental situation is illustrated in Fig. [REF].', '0907.3814-1-3-1': 'At a finite temperature below the critical temperature [MATH], the harmonically trapped cloud consists of a superfluid core centered in a collisionally hydrodynamic cloud.', '0907.3814-1-3-2': 'We assume that the trapping potential is close to cylindrical symmetry, but with a slight, controllable deformation that rotates around the corresponding axis with an angular velocity [MATH].', '0907.3814-1-3-3': 'The nonsuperfluid part of the cloud is then subject to friction with the trap and follows its rotation with an angular velocity [MATH], which in a steady state ideally reaches [MATH].', '0907.3814-1-3-4': 'The corresponding angular momentum can be expressed as [MATH], where [MATH] denotes the MOI.', '0907.3814-1-3-5': 'The superfluid core cannot carry angular momentum, assuming that vortex nucleation is avoided, and does therefore not contribute to the MOI of the system.', '0907.3814-1-3-6': 'Thus [MATH] represents the effective MOI of the whole system.', '0907.3814-1-4-0': 'The case of a rotating system in a steady state, where the normal part carries the maximum possible angular momentum, allows us to distinguish the superfluid quenching of the MOI from another effect studied in Ref. [CITATION].', '0907.3814-1-4-1': 'The latter originates from irrotational flow and may also be discussed in terms of a MOI below the rigid-body value, but it can occur for both the superfluid and the collisionally hydrodynamic normal phase.', '0907.3814-1-5-0': 'Our measurements rely on the possibility to determine the total angular momentum [MATH] of a rotating hydrodynamic cloud by detecting the precession of a radial quadrupole excitation.', '0907.3814-1-5-1': 'This method is well established and has been extensively used in the context of atomic Bose-Einstein condensates [CITATION] and recently also with strongly interacting Fermi gases [CITATION].', '0907.3814-1-5-2': 'The precession frequency can be written as [MATH] [CITATION], where [MATH] corresponds to a moment of inertia as calculated from the density distribution under the assumption that the whole cloud, including the superfluid part, would perform a rigid rotation.', '0907.3814-1-5-3': 'Substituting [MATH] for [MATH], we obtain [MATH], with [MATH] for the full MOI in a normal system, and [MATH] for a MOI that is quenched because of the superfluid core.', '0907.3814-1-6-0': 'The starting point of our experiments is an optically trapped, strongly interacting Fermi gas consisting of an equal mixture of [MATH]Li atoms in the lowest two atomic states [CITATION].', '0907.3814-1-6-1': 'The broad 834-G Feshbach resonance [CITATION] allows us to control the [MATH]-wave interaction.', '0907.3814-1-6-2': 'If not otherwise stated, the measurements presented here refer to the resonance center, where a unitarity-limited Fermi gas is realized [CITATION].', '0907.3814-1-6-3': 'The cigar-shaped quantum gas is confined in a far red-detuned, single-beam optical dipole trap with additional axial magnetic confinement.', '0907.3814-1-6-4': 'The trap can be well approximated by a harmonic potential with radial oscillation frequencies [MATH]Hz and an axial frequency of [MATH]Hz.', '0907.3814-1-6-5': 'The Fermi energy of the noninteracting gas is given by [MATH], where [MATH] is the total atom number.', '0907.3814-1-6-6': 'The Fermi temperature is [MATH]K, with [MATH] denoting the Boltzmann constant.', '0907.3814-1-7-0': 'Our scheme to study the rotational properties is described in detail in Ref. [CITATION].', '0907.3814-1-7-1': 'It is based on a rotating elliptical deformation of the trap, characterized by a small ellipticity parameter [CITATION] [MATH].', '0907.3814-1-7-2': 'In contrast to our previous work, we use a lower rotation frequency of [MATH]Hz [MATH].', '0907.3814-1-7-3': 'This low value allows us to avoid a resonant quadrupole mode excitation, which is known as an efficient mechanism for vortex nucleation [CITATION].', '0907.3814-1-7-4': 'The excitation and detection of the precessing radial quadrupole mode also follows the procedures described in Ref. [CITATION].', '0907.3814-1-8-0': 'At this point it is important to discuss the consequences of residual trap imperfections, still present when we attempt to realize a cylindrically symmetric optical potential.', '0907.3814-1-8-1': 'As we showed in previous work [CITATION], we can control the ellipticity down to a level of [MATH]1%.', '0907.3814-1-8-2': 'Moreover, deviations from perfect cylindrical symmetry may occur because of other residual effects, such as corrugations of the optical trapping potential.', '0907.3814-1-8-3': 'As a consequence, a certain rotational damping is unavoidable, but damping times can reach typically one second [CITATION].', '0907.3814-1-8-4': 'This has two main effects for our observations.', '0907.3814-1-8-5': 'First, our measurements yield precession frequencies slightly below [MATH].', '0907.3814-1-8-6': 'This is because of a delay time of [MATH]ms between turning off the rotating trap ellipticity and applying the surface mode excitation.', '0907.3814-1-8-7': 'It is introduced to make sure that any possible collective excitation resulting from the rotating trap has damped out when the mode precession is measured.', '0907.3814-1-8-8': 'Because of rotational damping during this delay time, the measured precession frequencies [MATH] are somewhat below [MATH].', '0907.3814-1-8-9': 'To compensate for this effect, we determine the corresponding damping parameter [MATH] for each set of measurements, finding day-to-day variations with typical values between [MATH] and [MATH].', '0907.3814-1-8-10': 'The second effect is induced by friction with static (nonrotating) trap imperfections when the rotating ellipticity is applied.', '0907.3814-1-8-11': 'This leads to equilibrium values for [MATH] typically a few percent below [MATH].', '0907.3814-1-8-12': 'For this second effect there is no straightforward compensation, and it needs to be explicitly discussed when interpreting the experimental results.', '0907.3814-1-9-0': 'Thermometry is performed after the whole experimental sequence.', '0907.3814-1-9-1': 'We damp out the rotation by stopping the trap rotation and keeping the ellipticity [CITATION].', '0907.3814-1-9-2': 'We convert the gas into a weakly interacting one by a slow magnetic field ramp to [MATH]G, and we finally measure the temperature [MATH] [CITATION].', '0907.3814-1-9-3': 'Note that the isentropic conversion tends to decrease the temperature such that [MATH] is always somewhat below the temperature [MATH] at unitarity [CITATION].', '0907.3814-1-9-4': 'The relative statistical uncertainty of the temperature measurement is about [MATH] in the relevant temperature range.', '0907.3814-1-10-0': 'For our further discussion, we introduce a dimensionless precession parameter [MATH] by normalizing our observable [MATH] to its maximum possible value of [MATH], [EQUATION]', '0907.3814-1-10-1': 'Values [MATH] show the presence of at least one of the two effects, namely the incomplete rotation of the normal part ([MATH]) or the superfluid quenching of the MOI ([MATH]).', '0907.3814-1-10-2': 'Our basic idea to distinguish between these two effects is based on the fact that [MATH] represents a temperature-dependent equilibrium property, whereas [MATH] reflects the spin-up dynamics.', '0907.3814-1-10-3': 'Therefore an increase of [MATH] observed at a fixed temperature can be attributed to the latter effect.', '0907.3814-1-10-4': 'This, however, is experimentally not straightforward because of the presence of residual heating.', '0907.3814-1-10-5': 'In the rotating trap we always observe some heating, which under all our experimental conditions can be well described by a constant rate [MATH]nK/s = [MATH]/s.', '0907.3814-1-11-0': 'In a first set of experiments, to observe the spin-up dynamics at a fixed temperature, we apply a special procedure based on the timing scheme illustrated on top of Fig. [REF].', '0907.3814-1-11-1': 'Our procedure takes advantage of the constant heating rate [MATH] to control the final temperature of the gas.', '0907.3814-1-11-2': 'We apply the trap rotation in two separate stages of duration [MATH] and [MATH].', '0907.3814-1-11-3': 'In an intermediate time interval of [MATH]ms we damp out the rotation that is induced by the first stage [CITATION].', '0907.3814-1-11-4': 'The angular momentum disappears, but the heating effect remains [CITATION].', '0907.3814-1-11-5': 'The second stage spins the cloud up again and induces further heating.', '0907.3814-1-11-6': 'When [MATH] is kept constant, we find that the total heating by the two rotation stages is [MATH].', '0907.3814-1-11-7': 'As only the second stage leads to a final angular momentum, [MATH] reveals the spin-up dynamics.', '0907.3814-1-11-8': 'A variation of the parameter [MATH] controls [MATH] and thus allows for a variation of the final temperature [MATH]; the temperature offset [MATH] is set by the initial cooling and some unavoidable heating during the experimental sequence without trap rotation.', '0907.3814-1-12-0': 'Our experimental results for [MATH] are shown in Fig. [REF] for four different values of the heating parameter [MATH] in a range between [MATH] to [MATH].', '0907.3814-1-12-1': 'This corresponds to a temperature range between about [MATH] and [MATH], where we expect the superfluid phase transition to occur.', '0907.3814-1-12-2': 'All four curves show qualitatively the same behavior.', '0907.3814-1-12-3': 'Within a few 100ms, [MATH] rises before it saturates.', '0907.3814-1-12-4': 'It is straightforward to interpret this time-dependent increase of [MATH] in terms of the spin-up dynamics.', '0907.3814-1-12-5': 'The observed behavior can be well fit by simple exponential curves (solid lines) approaching a steady-state value.', '0907.3814-1-12-6': 'This fit yields a time constant of [MATH]50ms for all four curves and allows us to extract the different saturation values for [MATH].', '0907.3814-1-12-7': 'Note that a characteristic spin-up time of about 50ms is consistent with theoretical predictions for a collisionally hydrodynamic gas [CITATION].', '0907.3814-1-13-0': 'The four curves in Fig. [REF] clearly show a general temperature dependence of the saturation value of [MATH].', '0907.3814-1-13-1': 'For the two highest temperatures, however, both curves approach nearly the same saturation value of [MATH]0.95.', '0907.3814-1-13-2': 'This indicates that a superfluid core is very small or absent and that in both cases the hydrodynamic cloud rotates with an angular velocity [MATH] near the maximum possible value of [MATH].', '0907.3814-1-13-3': 'For the lower two temperatures, however, the saturation value shows a clear temperature dependence.', '0907.3814-1-13-4': 'At [MATH] and [MATH], we observe [MATH] to saturate at [MATH] and [MATH].', '0907.3814-1-13-5': 'Normalizing these values to the maximum observed [MATH] at higher temperatures, we derive quenching factors for the MOI of [MATH] and [MATH], respectively.', '0907.3814-1-14-0': 'Let us comment on the possible influence of vortices.', '0907.3814-1-14-1': 'We cannot exclude their presence, as their nucleation can proceed not only via a resonant quadrupole mode excitation [CITATION], but also via a coupling to the thermal cloud [CITATION].', '0907.3814-1-14-2': 'Vortices would result in additional angular momentum in the rotating cloud and its collective behavior would be closer to the normal case.', '0907.3814-1-14-3': 'This would tend to increase [MATH] at lower temperatures, and would counteract the behavior that we observe.', '0907.3814-1-15-0': 'In a second set of experiments, we study the superfluid phase transition in a way which is experimentally simpler, but which requires the knowledge of the spin-up dynamics as obtained from the measurements presented before.', '0907.3814-1-15-1': 'Here the trap rotation is applied without any interruption and we observe the increase of [MATH] with the rotation time [MATH]; all other parameters and procedures are essentially the same as in the measurements before.', '0907.3814-1-15-2': 'In this procedure the temperature is not constant, but rises according to [MATH], where the heating rate [MATH]nK/s is the same as before and [MATH] is somewhat lower because of the less complex timing sequence.', '0907.3814-1-16-0': 'Fig.[REF] shows how [MATH] increases with the rotation time [MATH]; the upper scale shows the corresponding temperature [MATH].', '0907.3814-1-16-1': 'The observed increase in [MATH] generally results from both factors in Eq.([REF]), the rising [MATH] and the rising [MATH].', '0907.3814-1-16-2': 'However, both effects contribute on different time scales, which allows us to distinguish between three different stages (I-III), as indicated in Fig.[REF].', '0907.3814-1-16-3': 'For [MATH]ms (stage I) the spin-up dynamics dominates and leads to a fast increase in [MATH].', '0907.3814-1-16-4': 'For intermediate times, [MATH]ms (stage II), [MATH] still increases, but at a much slower rate.', '0907.3814-1-16-5': 'Here, for a fully rotating normal part, the superfluid quenching of the MOI ([MATH]) is the dominant effect.', '0907.3814-1-16-6': 'For [MATH]ms (stage III), [MATH] saturates and acquires a value that is very close to the maximum, indicating the full rotation of a gas without any superfluid component.', '0907.3814-1-17-0': 'For the superfluid phase transition, which occurs between stages II and III, we obtain a critical temperature of [MATH].', '0907.3814-1-17-1': 'The conversion of this value, measured in the weakly interacting regime, to the actual temperature in the unitarity limit yields values for [MATH] between [MATH] [CITATION] and [MATH] [CITATION].', '0907.3814-1-17-2': 'This range for [MATH] is consistent with previous experimental results [CITATION] and theoretical predictions [CITATION].', '0907.3814-1-18-0': 'The superfluid quenching effect in stage II can be quantitatively discussed in terms of the slope [MATH] near the phase transition.', '0907.3814-1-18-1': 'Our data in Fig. [REF] show an increase of the MOI according to [MATH].', '0907.3814-1-18-2': 'This is fully consistent with the observed temperature dependence of the MOI in Fig. [REF], from which we obtain [MATH].', '0907.3814-1-18-3': 'In terms of the critical temperature [MATH] this slope corresponds to [MATH].', '0907.3814-1-18-4': 'We can compare this behavior with available predictions for the limits of BEC and BCS theory.', '0907.3814-1-18-5': 'For a BEC, the superfluid quenching effect is very small [CITATION].', '0907.3814-1-18-6': 'For a BCS-type system, Refs. [CITATION] predict a slope of [MATH].', '0907.3814-1-18-7': 'Our observed behavior thus lies in between the two extremes of BEC and BCS.', '0907.3814-1-19-0': 'In conclusion, we have demonstrated the quenching of the moment of inertia that occurs as a consequence of superfluidity in a strongly interacting Fermi gas.', '0907.3814-1-19-1': 'This effect provides us with a novel probe for the system as, in contrast to other common methods such as expansion measurements and studies of collective modes, it allows us to distinguish between the two possible origins of hydrodynamic behavior, namely collisions in a normal phase and superfluidity.', '0907.3814-1-20-0': 'We thank L. Sidorenkov for discussions.', '0907.3814-1-20-1': 'We acknowledge support by the Austrian Science Fund (FWF) within SFB 15 (project part 21) and SFB 40 (project part 4).'}

{'0907.3814-2-0-0': 'We report on the observation of a quenched moment of inertia as resulting from superfluidity in a strongly interacting Fermi gas.', '0907.3814-2-0-1': 'Our method is based on setting the hydrodynamic gas in slow rotation and determining its angular momentum by detecting the precession of a radial quadrupole excitation.', '0907.3814-2-0-2': 'The measurements distinguish between the superfluid or collisional origin of hydrodynamic behavior, and show the phase transition.', '0907.3814-2-1-0': '# Introduction', '0907.3814-2-2-0': 'Superfluidity is a striking property of quantum fluids at very low temperatures.', '0907.3814-2-2-1': 'For bosonic systems, important examples are liquids and clusters of [MATH]He and atomic Bose-Einstein condensates.', '0907.3814-2-2-2': 'In fermionic systems, superfluidity is a more intricate phenomenon as it requires pairing of particles.', '0907.3814-2-2-3': 'Fermionic superfluidity is known to occur in atomic nuclei and [MATH]He liquids and it is also at the heart of superconductivity, thus being of great technological importance.', '0907.3814-2-2-4': 'Recent advances with ultracold Fermi gases have opened up unprecedented possibilities to study the properties of strongly interacting fermionic superfluids [CITATION].', '0907.3814-2-2-5': 'Early experiments on ultracold Fermi gases with resonant interparticle interactions compiled increasing evidence for superfluidity [CITATION] until the phenomenon was firmly established by the observation of vortex lattices [CITATION].', '0907.3814-2-3-0': 'Here we report on the manifestation of superfluidity in a quenched moment of inertia (MOI) in a strongly interacting Fermi gas that undergoes slow rotation.', '0907.3814-2-3-1': 'The basic idea of a quenched MOI as a signature of superfluidity goes back more than 50 years in nuclear physics, where MOIs below the classical, rigid-body value were attributed to superfluidity [CITATION].', '0907.3814-2-3-2': 'The quenching of the MOI was also shown in liquid [MATH]He [CITATION] and has, more recently, served for the discovery of a supersolid phase [CITATION].', '0907.3814-2-3-3': 'Here we introduce the observation of the quenched MOI as a new method to study superfluidity in ultracold Fermi gases.', '0907.3814-2-4-0': '# Basic idea of the measurement', '0907.3814-2-5-0': 'The basic situation that underlies our experiments is illustrated in Fig. [REF].', '0907.3814-2-5-1': 'At a finite temperature below the critical temperature [MATH], the harmonically trapped cloud consists of a superfluid core centered in a collisionally hydrodynamic cloud.', '0907.3814-2-5-2': 'We assume that the trapping potential is close to cylindrical symmetry, but with a slight, controllable deformation that rotates around the corresponding axis with an angular velocity [MATH].', '0907.3814-2-5-3': 'The nonsuperfluid part of the cloud is then subject to friction with the trap and follows its rotation with an angular velocity [MATH] [CITATION], which in a steady state ideally reaches [MATH].', '0907.3814-2-5-4': 'The corresponding angular momentum can be expressed as [MATH], where [MATH] denotes the MOI.', '0907.3814-2-5-5': 'The superfluid core cannot carry angular momentum, assuming that vortex nucleation is avoided, and therefore does not contribute to the MOI of the system.', '0907.3814-2-5-6': 'Thus [MATH] represents the MOI of the whole system.', '0907.3814-2-6-0': 'The case of a rotating system in a steady state, where the normal part carries the maximum possible angular momentum, allows us to distinguish the superfluid quenching of the MOI from another effect studied in Ref. [CITATION].', '0907.3814-2-6-1': 'The latter originates from irrotational flow and may also be discussed in terms of a MOI below the rigid-body value, but it can occur for both the superfluid and the collisionally hydrodynamic normal phase.', '0907.3814-2-7-0': 'Our measurements rely on the possibility to determine the total angular momentum [MATH] of a rotating hydrodynamic cloud by detecting the precession of a radial quadrupole excitation.', '0907.3814-2-7-1': 'This method is well established and has been extensively used in the context of atomic Bose-Einstein condensates [CITATION].', '0907.3814-2-7-2': 'We have recently applied it to a rotating, strongly interacting Fermi gas to investigate the slow decay of angular momentum [CITATION].', '0907.3814-2-7-3': 'The method works under the general condition that the gas behaves hydrodynamically.', '0907.3814-2-7-4': 'Then the precession frequency can be written as [MATH] [CITATION], where [MATH] corresponds to a moment of inertia as calculated from the density distribution under the assumption that the whole cloud, including the superfluid part, would perform a rigid rotation.', '0907.3814-2-7-5': 'Substituting [MATH] for [MATH], we obtain [MATH], with [MATH] for the full MOI in a normal system, and [MATH] for a MOI that is quenched because of the superfluid core.', '0907.3814-2-8-0': '# Experimental setup and procedures', '0907.3814-2-9-0': 'The starting point of our experiments is an optically trapped, strongly interacting Fermi gas consisting of an equal mixture of [MATH]Li atoms in the lowest two atomic states [CITATION].', '0907.3814-2-9-1': 'The broad 834-G Feshbach resonance [CITATION] allows us to control the [MATH]-wave interaction.', '0907.3814-2-9-2': 'If not otherwise stated, the measurements presented here refer to the resonance center.', '0907.3814-2-9-3': 'Here a unitarity-limited Fermi gas [CITATION] is realized, which is known to exhibit deep hydrodynamic behavior even well above the critical temperature for superfluidity, see e.g. [CITATION].', '0907.3814-2-9-4': 'The cigar-shaped quantum gas is confined in a far red-detuned, single-beam optical dipole trap with additional axial magnetic confinement.', '0907.3814-2-9-5': 'The trap can be well approximated by a harmonic potential with radial oscillation frequencies [MATH]Hz and an axial frequency of [MATH]Hz.', '0907.3814-2-9-6': 'The Fermi energy of the noninteracting gas is given by [MATH], where [MATH] is the total atom number.', '0907.3814-2-9-7': 'The Fermi temperature is [MATH]K, with [MATH] denoting the Boltzmann constant.', '0907.3814-2-10-0': 'Our scheme to study the rotational properties is described in detail in Ref. [CITATION].', '0907.3814-2-10-1': 'It is based on a rotating elliptical deformation of the trap, characterized by a small ellipticity parameter [CITATION] [MATH].', '0907.3814-2-10-2': 'In contrast to our previous work, we use a lower rotation frequency of [MATH]Hz [MATH].', '0907.3814-2-10-3': 'This low value allows us to avoid a resonant quadrupole mode excitation, which is known as an efficient mechanism for vortex nucleation [CITATION].', '0907.3814-2-10-4': 'The excitation and detection of the precessing radial quadrupole mode also follows the procedures described in Ref. [CITATION].', '0907.3814-2-11-0': 'At this point it is important to discuss the consequences of residual trap imperfections, still present when we attempt to realize a cylindrically symmetric optical potential.', '0907.3814-2-11-1': 'As we showed in previous work [CITATION], we can control the ellipticity down to a level of [MATH]1%.', '0907.3814-2-11-2': 'Moreover, deviations from perfect cylindrical symmetry may occur because of other residual effects, such as corrugations of the optical trapping potential.', '0907.3814-2-11-3': 'As a consequence, a certain rotational damping is unavoidable, but damping times can reach typically one second [CITATION].', '0907.3814-2-11-4': 'This has two main effects for our observations.', '0907.3814-2-11-5': 'First, our measurements yield precession frequencies slightly below [MATH].', '0907.3814-2-11-6': 'This is because of a delay time of [MATH]ms between turning off the rotating trap ellipticity and applying the quadrupole mode excitation.', '0907.3814-2-11-7': 'It is introduced to make sure that any possible collective excitation resulting from the rotating trap has damped out when the mode precession is measured.', '0907.3814-2-11-8': 'Because of rotational damping during this delay time, the measured precession frequencies [MATH] are somewhat below [MATH].', '0907.3814-2-11-9': 'To compensate for this effect, we determine the corresponding damping parameter [MATH] for each set of measurements, finding day-to-day variations with typical values between [MATH] and [MATH].', '0907.3814-2-11-10': 'The second effect is induced by friction with static (nonrotating) trap imperfections when the rotating ellipticity is applied.', '0907.3814-2-11-11': 'This leads to equilibrium values for [MATH] typically a few percent below [MATH].', '0907.3814-2-11-12': 'For this second effect there is no straightforward compensation, and it needs to be explicitly discussed when interpreting the experimental results.', '0907.3814-2-12-0': 'Thermometry is performed after the whole experimental sequence.', '0907.3814-2-12-1': 'We damp out the rotation by stopping the trap rotation and keeping the ellipticity [CITATION].', '0907.3814-2-12-2': 'We convert the gas into a weakly interacting one by a slow magnetic field ramp to [MATH]G, and we finally measure the temperature [MATH] [CITATION].', '0907.3814-2-12-3': 'Note that the isentropic conversion tends to decrease the temperature such that [MATH] is always somewhat below the temperature [MATH] at unitarity [CITATION].', '0907.3814-2-12-4': 'The relative statistical uncertainty of the temperature measurement is about [MATH] in the relevant temperature range.', '0907.3814-2-13-0': '# Experimental results', '0907.3814-2-14-0': 'To discuss our experimental results we introduce a dimensionless precession parameter [MATH] by normalizing our observable [MATH] to its maximum possible value of [MATH], [EQUATION]', '0907.3814-2-14-1': 'The maximum possible value of [MATH] corresponds to a fully rotating, classically hydrodynamic cloud.', '0907.3814-2-14-2': 'Values [MATH] show the presence of at least one of the two effects, namely the incomplete rotation of the normal part ([MATH]) or the superfluid quenching of the MOI ([MATH]).', '0907.3814-2-14-3': 'It is crucial for the interpretation of our experimental results to distinguish between these two effects.', '0907.3814-2-14-4': 'Our basic idea to achieve this relies on the fact that [MATH] represents a temperature-dependent equilibrium property, whereas [MATH] depends on the dynamics of the spin-up before the system has reached an equilibrium.', '0907.3814-2-14-5': 'Experimentally, however, measurements of equilibrium properties at a fixed temperature are not straightforward because of the presence of residual heating leading to a slow, steady temperature increase.', '0907.3814-2-14-6': 'In the rotating trap we always observe some heating, which under all our experimental conditions can be well described by a constant rate [MATH]nK/s = [MATH]/s.', '0907.3814-2-15-0': '## Equilibrium state of rotation', '0907.3814-2-16-0': 'To identify the conditions under which our cloud reaches its equilibrium state of rotation, we have developed a special procedure based on the timing scheme illustrated on top of Fig. [REF].', '0907.3814-2-16-1': 'Our procedure takes advantage of the constant heating rate [MATH] to control the final temperature of the gas when [MATH] is measured.', '0907.3814-2-16-2': 'We apply the trap rotation in two separate stages of duration [MATH] and [MATH].', '0907.3814-2-16-3': 'In an intermediate time interval of [MATH]ms [CITATION] we damp out the rotation that is induced by the first stage.', '0907.3814-2-16-4': 'The angular momentum disappears, but the heating effect remains [CITATION].', '0907.3814-2-16-5': 'The second stage spins up the cloud again and induces further heating.', '0907.3814-2-16-6': 'When [MATH] is kept constant, we find that the total heating by the two rotation stages is [MATH].', '0907.3814-2-16-7': 'As only the second stage leads to a final angular momentum, the equilibrium state reached at a constant temperature can be identified when [MATH] reaches a constant value for increasing [MATH] and fixed [MATH].', '0907.3814-2-16-8': 'The temperature can be controlled by a variation of the parameter [MATH] and is obtained as [MATH].', '0907.3814-2-16-9': 'The temperature offset [MATH] is set by the initial cooling and some unavoidable heating during the experimental sequence without trap rotation.', '0907.3814-2-16-10': 'Under our conditions [MATH].', '0907.3814-2-17-0': 'Our experimental results for [MATH] are shown in Fig. [REF] for four different values of the heating parameter [MATH] in a range between [MATH] to [MATH], which corresponds to a range of [MATH] between about [MATH] and [MATH].', '0907.3814-2-17-1': 'All four curves show qualitatively the same behavior.', '0907.3814-2-17-2': 'Within a few 100ms, [MATH] rises before reaching a final equilibrium value.', '0907.3814-2-17-3': 'This time-dependent increase of [MATH] is related to the spin-up dynamics [CITATION].', '0907.3814-2-17-4': 'We find that the observed increase and saturation of [MATH] can be well fit by simple exponential curves (solid lines), and we use these fits to extract the different equilibrium values [MATH].', '0907.3814-2-18-0': 'The equilibrium values [MATH] exhibit an interesting temperature dependence.', '0907.3814-2-18-1': 'The lower three values show a pronounced increase with temperature, [MATH], [MATH], and [MATH] for [MATH], [MATH], and [MATH], respectively.', '0907.3814-2-18-2': 'We interpret this increase as a consequence of the decreasing superfluid core and thus the decreasing MOI quenching effect.', '0907.3814-2-18-3': 'For our highest temperature ([MATH]) we only observe a marginal further increase to [MATH].', '0907.3814-2-18-4': 'This indicates that the superfluid core is very small or absent leading to a disappearance of the quenching effect.', '0907.3814-2-18-5': 'The fact that the maximum [MATH] stays a few percent below [MATH] can be explained by trap imperfections as discussed in Sec. [REF].', '0907.3814-2-19-0': 'Let us comment on the possible influence of vortices [CITATION].', '0907.3814-2-19-1': 'We cannot exclude their presence, as their nucleation can proceed not only via a resonant quadrupole mode excitation [CITATION], but also via a coupling to the thermal cloud [CITATION].', '0907.3814-2-19-2': 'Vortices would result in additional angular momentum in the rotating cloud and its collective behavior would be closer to the normal case.', '0907.3814-2-19-3': 'This would tend to increase [MATH] at lower temperatures, counteracting the behavior that we observe.', '0907.3814-2-20-0': '## Superfluid phase transition', '0907.3814-2-21-0': 'In a second set of experiments, we study the superfluid phase transition in a way which is experimentally simpler, but which requires information on the equilibrium state as obtained from the measurements presented before.', '0907.3814-2-21-1': 'The trap rotation is applied continuously, and we observe the increase of [MATH] with the rotation time [MATH].', '0907.3814-2-21-2': 'All other parameters and procedures are essentially the same as in the measurements before.', '0907.3814-2-21-3': 'Here the temperature is not constant, but rises according to [MATH], where the heating rate [MATH]nK/s is the same as before and [MATH] is somewhat lower because of the less complex timing sequence.', '0907.3814-2-22-0': 'Figure [REF] shows how [MATH] increases with the rotation time [MATH] (filled symbols); the upper scale shows the corresponding temperature [MATH].', '0907.3814-2-22-1': 'The observed increase of [MATH] generally results from both factors in Eq. ([REF]), corresponding to the rising [MATH] (spin-up dynamics) and the rising [MATH] (decrease of the superfluid MOI quenching).', '0907.3814-2-22-2': 'Figure [REF] also shows the values [MATH] as determined from Fig. [REF] (crosses), for which we know that the spin-up of the normal component has established an equilibrium with [MATH] being close to one.', '0907.3814-2-22-3': 'The comparison shows that already for [MATH]s the data set obtained with the simpler procedure follows essentially the same behavior.', '0907.3814-2-22-4': 'The small quantitative difference that the crosses are slightly below the open symbols can be explained by a somewhat stronger influence of trap imperfections in the earlier measurements of Sec. [REF] [CITATION] or by the uncertainty in the initial temperature [MATH].', '0907.3814-2-22-5': 'For [MATH]s, we can assume that the system is in an equilibrium state, which follows the slowly increasing temperature, and we can fully attribute the further increase of [MATH] to the quenching of the MOI.', '0907.3814-2-23-0': 'The superfluid phase transition corresponds to the point where the precession parameter [MATH] reaches its saturation value.', '0907.3814-2-23-1': 'This is observed for a time [MATH]s, when [MATH].', '0907.3814-2-23-2': 'The conversion of this temperature parameter (measured in the weakly interacting regime after an isentropic change) to the actual temperature in the unitarity-limit regime [CITATION] yields a value for the critical temperature [MATH] of about [MATH].', '0907.3814-2-23-3': 'This result is consistent with previous experimental results [CITATION], the range of which is indicated by the shaded region in Fig. [REF].', '0907.3814-2-23-4': 'The result is also consistent with theoretical predictions [CITATION].', '0907.3814-2-24-0': 'For a more precise extraction of [MATH] from experimental MOI quenching data, a theoretical model would be required that describes the saturation behavior of [MATH] as [MATH] is approached.', '0907.3814-2-24-1': 'Theoretical predictions are available for the BEC limit [CITATION] and the BCS limit [CITATION].', '0907.3814-2-24-2': 'In the unitarity limit it should, in principle, be possible to extract the MOI from spatial profiles of the normal and the superfluid fraction [CITATION].', '0907.3814-2-24-3': 'Clearly, more work is necessary to quantitatively understand the quenching effect in the strongly interacting regime.', '0907.3814-2-25-0': '# Conclusion', '0907.3814-2-26-0': 'We have demonstrated the quenching of the moment of inertia that occurs in a slowly rotating, strongly interacting Fermi gas as a consequence of superfluidity.', '0907.3814-2-26-1': 'This effect provides us with a novel probe for the system as, in contrast to other common methods such as expansion measurements and studies of collective modes, it allows us to distinguish between the two possible origins of hydrodynamic behavior, namely collisions in a normal phase and superfluidity.'}

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'0907.3814-2-16-4'], ['0907.3814-1-11-6', '0907.3814-2-16-6'], ['0907.3814-1-1-0', '0907.3814-2-2-0'], ['0907.3814-1-1-1', '0907.3814-2-2-1'], ['0907.3814-1-1-2', '0907.3814-2-2-2'], ['0907.3814-1-1-3', '0907.3814-2-2-3'], ['0907.3814-1-1-4', '0907.3814-2-2-4'], ['0907.3814-1-1-5', '0907.3814-2-2-5'], ['0907.3814-1-12-2', '0907.3814-2-17-1'], ['0907.3814-1-7-0', '0907.3814-2-10-0'], ['0907.3814-1-7-1', '0907.3814-2-10-1'], ['0907.3814-1-7-2', '0907.3814-2-10-2'], ['0907.3814-1-7-3', '0907.3814-2-10-3'], ['0907.3814-1-7-4', '0907.3814-2-10-4'], ['0907.3814-1-10-1', '0907.3814-2-14-2'], ['0907.3814-1-10-5', '0907.3814-2-14-6'], ['0907.3814-1-3-1', '0907.3814-2-5-1'], ['0907.3814-1-3-2', '0907.3814-2-5-2'], ['0907.3814-1-3-4', '0907.3814-2-5-4'], ['0907.3814-1-8-0', '0907.3814-2-11-0'], ['0907.3814-1-8-1', '0907.3814-2-11-1'], ['0907.3814-1-8-2', '0907.3814-2-11-2'], ['0907.3814-1-8-3', '0907.3814-2-11-3'], ['0907.3814-1-8-4', '0907.3814-2-11-4'], ['0907.3814-1-8-5', '0907.3814-2-11-5'], ['0907.3814-1-8-7', '0907.3814-2-11-7'], ['0907.3814-1-8-8', '0907.3814-2-11-8'], ['0907.3814-1-8-9', '0907.3814-2-11-9'], ['0907.3814-1-8-10', '0907.3814-2-11-10'], ['0907.3814-1-8-11', '0907.3814-2-11-11'], ['0907.3814-1-8-12', '0907.3814-2-11-12'], ['0907.3814-2-7-0', '0907.3814-3-7-0'], ['0907.3814-2-7-1', '0907.3814-3-7-1'], ['0907.3814-2-7-2', '0907.3814-3-7-2'], ['0907.3814-2-7-3', '0907.3814-3-7-3'], ['0907.3814-2-7-4', '0907.3814-3-7-4'], ['0907.3814-2-7-5', '0907.3814-3-7-5'], ['0907.3814-2-19-0', '0907.3814-3-19-0'], ['0907.3814-2-19-2', '0907.3814-3-19-2'], ['0907.3814-2-19-3', '0907.3814-3-19-3'], ['0907.3814-2-10-0', '0907.3814-3-10-0'], ['0907.3814-2-10-1', '0907.3814-3-10-1'], ['0907.3814-2-10-2', '0907.3814-3-10-2'], ['0907.3814-2-10-3', '0907.3814-3-10-3'], ['0907.3814-2-5-0', '0907.3814-3-5-0'], ['0907.3814-2-5-1', '0907.3814-3-5-1'], ['0907.3814-2-5-2', '0907.3814-3-5-2'], ['0907.3814-2-5-3', '0907.3814-3-5-3'], ['0907.3814-2-5-4', '0907.3814-3-5-4'], ['0907.3814-2-5-5', '0907.3814-3-5-5'], ['0907.3814-2-5-6', '0907.3814-3-5-6'], ['0907.3814-2-14-0', '0907.3814-3-14-0'], ['0907.3814-2-14-1', '0907.3814-3-14-1'], ['0907.3814-2-14-2', '0907.3814-3-14-2'], ['0907.3814-2-14-3', '0907.3814-3-14-3'], ['0907.3814-2-14-4', '0907.3814-3-14-4'], ['0907.3814-2-14-5', '0907.3814-3-14-5'], ['0907.3814-2-11-0', '0907.3814-3-11-0'], ['0907.3814-2-11-1', '0907.3814-3-11-1'], ['0907.3814-2-11-2', '0907.3814-3-11-2'], ['0907.3814-2-11-3', '0907.3814-3-11-3'], ['0907.3814-2-11-4', '0907.3814-3-11-4'], ['0907.3814-2-11-5', '0907.3814-3-11-5'], ['0907.3814-2-11-6', '0907.3814-3-11-6'], ['0907.3814-2-11-7', '0907.3814-3-11-7'], ['0907.3814-2-11-8', '0907.3814-3-11-8'], ['0907.3814-2-11-10', '0907.3814-3-11-10'], ['0907.3814-2-11-12', '0907.3814-3-11-12'], ['0907.3814-2-9-0', '0907.3814-3-9-0'], ['0907.3814-2-9-1', '0907.3814-3-9-1'], ['0907.3814-2-9-2', '0907.3814-3-9-2'], ['0907.3814-2-9-3', '0907.3814-3-9-3'], ['0907.3814-2-9-4', '0907.3814-3-9-4'], ['0907.3814-2-9-5', '0907.3814-3-9-5'], ['0907.3814-2-9-6', '0907.3814-3-9-6'], ['0907.3814-2-9-7', '0907.3814-3-9-7'], ['0907.3814-2-23-0', '0907.3814-3-23-0'], ['0907.3814-2-23-1', '0907.3814-3-23-1'], ['0907.3814-2-23-2', '0907.3814-3-23-2'], ['0907.3814-2-23-3', '0907.3814-3-23-3'], ['0907.3814-2-23-4', '0907.3814-3-23-4'], ['0907.3814-2-2-0', '0907.3814-3-2-0'], ['0907.3814-2-2-1', '0907.3814-3-2-1'], ['0907.3814-2-2-2', '0907.3814-3-2-2'], ['0907.3814-2-2-3', '0907.3814-3-2-3'], ['0907.3814-2-2-4', '0907.3814-3-2-4'], ['0907.3814-2-2-5', '0907.3814-3-2-5'], ['0907.3814-2-12-0', '0907.3814-3-12-0'], ['0907.3814-2-12-1', '0907.3814-3-12-1'], ['0907.3814-2-12-2', '0907.3814-3-12-2'], ['0907.3814-2-12-3', '0907.3814-3-12-3'], ['0907.3814-2-12-4', '0907.3814-3-12-4'], ['0907.3814-2-3-0', '0907.3814-3-3-0'], ['0907.3814-2-3-3', '0907.3814-3-3-3'], ['0907.3814-2-18-0', '0907.3814-3-18-0'], ['0907.3814-2-18-1', '0907.3814-3-18-1'], ['0907.3814-2-18-2', '0907.3814-3-18-2'], ['0907.3814-2-18-3', '0907.3814-3-18-3'], ['0907.3814-2-18-4', '0907.3814-3-18-4'], ['0907.3814-2-18-5', '0907.3814-3-18-5'], ['0907.3814-2-21-0', '0907.3814-3-21-0'], ['0907.3814-2-21-1', '0907.3814-3-21-1'], ['0907.3814-2-21-2', '0907.3814-3-21-2'], ['0907.3814-2-21-3', '0907.3814-3-21-3'], ['0907.3814-2-17-0', '0907.3814-3-17-0'], ['0907.3814-2-17-1', '0907.3814-3-17-1'], ['0907.3814-2-17-2', '0907.3814-3-17-2'], ['0907.3814-2-17-3', '0907.3814-3-17-3'], ['0907.3814-2-17-4', '0907.3814-3-17-4'], ['0907.3814-2-24-0', '0907.3814-3-24-0'], ['0907.3814-2-24-1', '0907.3814-3-24-1'], ['0907.3814-2-24-2', '0907.3814-3-24-2'], ['0907.3814-2-24-3', '0907.3814-3-24-3'], ['0907.3814-2-22-0', '0907.3814-3-22-0'], ['0907.3814-2-22-1', '0907.3814-3-22-1'], ['0907.3814-2-22-2', '0907.3814-3-22-2'], ['0907.3814-2-22-3', '0907.3814-3-22-3'], ['0907.3814-2-22-4', '0907.3814-3-22-4'], ['0907.3814-2-22-5', '0907.3814-3-22-5'], ['0907.3814-2-26-0', '0907.3814-3-26-0'], ['0907.3814-2-26-1', '0907.3814-3-26-1'], ['0907.3814-2-0-0', '0907.3814-3-0-0'], ['0907.3814-2-0-1', '0907.3814-3-0-1'], ['0907.3814-2-16-0', '0907.3814-3-16-0'], ['0907.3814-2-16-1', '0907.3814-3-16-1'], ['0907.3814-2-16-2', '0907.3814-3-16-2'], ['0907.3814-2-16-3', '0907.3814-3-16-3'], ['0907.3814-2-16-4', '0907.3814-3-16-4'], ['0907.3814-2-16-5', '0907.3814-3-16-5'], ['0907.3814-2-16-6', '0907.3814-3-16-6'], ['0907.3814-2-16-7', '0907.3814-3-16-7'], ['0907.3814-2-16-8', '0907.3814-3-16-8'], ['0907.3814-2-16-9', '0907.3814-3-16-9'], ['0907.3814-2-16-10', '0907.3814-3-16-10'], ['0907.3814-1-0-1', '0907.3814-2-0-1'], ['0907.3814-1-14-0', '0907.3814-2-19-0'], ['0907.3814-1-14-3', '0907.3814-2-19-3'], ['0907.3814-1-5-2', '0907.3814-2-7-4'], ['0907.3814-1-2-0', '0907.3814-2-3-0'], ['0907.3814-1-11-1', '0907.3814-2-16-1'], ['0907.3814-1-11-3', '0907.3814-2-16-3'], ['0907.3814-1-11-5', '0907.3814-2-16-5'], ['0907.3814-1-12-0', '0907.3814-2-17-0'], ['0907.3814-1-15-0', '0907.3814-2-21-0'], ['0907.3814-1-15-2', '0907.3814-2-21-3'], ['0907.3814-1-10-0', '0907.3814-2-14-0'], ['0907.3814-1-3-3', '0907.3814-2-5-3'], ['0907.3814-1-3-5', '0907.3814-2-5-5'], ['0907.3814-1-3-6', '0907.3814-2-5-6'], ['0907.3814-1-8-6', '0907.3814-2-11-6'], ['0907.3814-2-19-1', '0907.3814-3-19-1'], ['0907.3814-2-14-6', '0907.3814-3-14-6'], ['0907.3814-2-6-0', '0907.3814-3-6-0'], ['0907.3814-2-11-9', '0907.3814-3-11-9'], ['0907.3814-2-3-1', '0907.3814-3-3-1'], ['0907.3814-2-3-2', '0907.3814-3-3-2'], ['0907.3814-2-0-2', '0907.3814-3-0-2'], ['0907.3814-1-6-2', '0907.3814-2-9-2'], ['0907.3814-1-6-2', '0907.3814-2-9-3'], ['0907.3814-1-17-1', '0907.3814-2-23-2'], ['0907.3814-1-17-2', '0907.3814-2-23-3'], ['0907.3814-1-17-2', '0907.3814-2-23-4'], ['0907.3814-1-5-1', '0907.3814-2-7-1'], ['0907.3814-1-19-0', '0907.3814-2-26-0'], ['0907.3814-1-11-0', '0907.3814-2-16-0'], ['0907.3814-1-11-7', '0907.3814-2-16-7'], ['0907.3814-1-11-8', '0907.3814-2-16-9'], ['0907.3814-1-12-3', '0907.3814-2-17-2'], ['0907.3814-1-12-4', '0907.3814-2-17-3'], ['0907.3814-1-12-5', '0907.3814-2-17-4'], ['0907.3814-1-15-1', '0907.3814-2-21-1'], ['0907.3814-1-15-1', '0907.3814-2-21-2'], ['0907.3814-1-10-2', '0907.3814-2-14-4'], ['0907.3814-1-10-4', '0907.3814-2-14-5'], ['0907.3814-1-16-0', '0907.3814-2-22-0'], ['0907.3814-1-16-1', '0907.3814-2-22-1'], ['0907.3814-2-6-1', '0907.3814-3-6-3'], ['0907.3814-2-11-11', '0907.3814-3-11-11']]

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0907.3814

{'0907.3814-3-0-0': 'We report on the observation of a quenched moment of inertia as resulting from superfluidity in a strongly interacting Fermi gas.', '0907.3814-3-0-1': 'Our method is based on setting the hydrodynamic gas in slow rotation and determining its angular momentum by detecting the precession of a radial quadrupole excitation.', '0907.3814-3-0-2': 'The measurements distinguish between the superfluid and collisional origins of hydrodynamic behavior, and show the phase transition.', '0907.3814-3-1-0': '# Introduction', '0907.3814-3-2-0': 'Superfluidity is a striking property of quantum fluids at very low temperatures.', '0907.3814-3-2-1': 'For bosonic systems, important examples are liquids and clusters of [MATH]He and atomic Bose-Einstein condensates.', '0907.3814-3-2-2': 'In fermionic systems, superfluidity is a more intricate phenomenon as it requires pairing of particles.', '0907.3814-3-2-3': 'Fermionic superfluidity is known to occur in atomic nuclei and [MATH]He liquids and it is also at the heart of superconductivity, thus being of great technological importance.', '0907.3814-3-2-4': 'Recent advances with ultracold Fermi gases have opened up unprecedented possibilities to study the properties of strongly interacting fermionic superfluids [CITATION].', '0907.3814-3-2-5': 'Early experiments on ultracold Fermi gases with resonant interparticle interactions compiled increasing evidence for superfluidity [CITATION] until the phenomenon was firmly established by the observation of vortex lattices [CITATION].', '0907.3814-3-3-0': 'Here we report on the manifestation of superfluidity in a quenched moment of inertia (MOI) in a strongly interacting Fermi gas that undergoes slow rotation.', '0907.3814-3-3-1': 'The basic idea of a quenched MOI as a signature of superfluidity dates back to more than 50 years ago in nuclear physics, where MOIs below the classical, rigid-body value were attributed to superfluidity [CITATION].', '0907.3814-3-3-2': 'The quenching of the MOI was also shown in liquid [MATH]He [CITATION] and has, more recently, served for the discovery of a possible supersolid phase [CITATION].', '0907.3814-3-3-3': 'Here we introduce the observation of the quenched MOI as a new method to study superfluidity in ultracold Fermi gases.', '0907.3814-3-4-0': '# Basic idea of the measurement', '0907.3814-3-5-0': 'The basic situation that underlies our experiments is illustrated in Fig. [REF].', '0907.3814-3-5-1': 'At a finite temperature below the critical temperature [MATH], the harmonically trapped cloud consists of a superfluid core centered in a collisionally hydrodynamic cloud.', '0907.3814-3-5-2': 'We assume that the trapping potential is close to cylindrical symmetry, but with a slight, controllable deformation that rotates around the corresponding axis with an angular velocity [MATH].', '0907.3814-3-5-3': 'The nonsuperfluid part of the cloud is then subject to friction with the trap and follows its rotation with an angular velocity [MATH] [CITATION], which in a steady state ideally reaches [MATH].', '0907.3814-3-5-4': 'The corresponding angular momentum can be expressed as [MATH], where [MATH] denotes the MOI.', '0907.3814-3-5-5': 'The superfluid core cannot carry angular momentum, assuming that vortex nucleation is avoided, and therefore does not contribute to the MOI of the system.', '0907.3814-3-5-6': 'Thus [MATH] represents the MOI of the whole system.', '0907.3814-3-6-0': 'The case of a rotating system in a steady state, where the normal part carries the maximum possible angular momentum, allows us to distinguish the superfluid quenching of the MOI from a non-equilibrium quenching effect as studied in Ref. [CITATION].', '0907.3814-3-6-1': 'There the authors investigated the hydrodynamic expansion of a gas with a known angular momentum.', '0907.3814-3-6-2': 'This situation, where the velocity fields of the normal and superfluid components are not in a steady state, can also be discussed in terms of a MOI below the rigid-body value.', '0907.3814-3-6-3': 'In contrast to the phenomenon investigated in our present work, the effect of Ref. [CITATION] is related to irrotational flow and can occur for both the superfluid and the collisionally hydrodynamic normal phase.', '0907.3814-3-7-0': 'Our measurements rely on the possibility to determine the total angular momentum [MATH] of a rotating hydrodynamic cloud by detecting the precession of a radial quadrupole excitation.', '0907.3814-3-7-1': 'This method is well established and has been extensively used in the context of atomic Bose-Einstein condensates [CITATION].', '0907.3814-3-7-2': 'We have recently applied it to a rotating, strongly interacting Fermi gas to investigate the slow decay of angular momentum [CITATION].', '0907.3814-3-7-3': 'The method works under the general condition that the gas behaves hydrodynamically.', '0907.3814-3-7-4': 'Then the precession frequency can be written as [MATH] [CITATION], where [MATH] corresponds to a moment of inertia as calculated from the density distribution under the assumption that the whole cloud, including the superfluid part, would perform a rigid rotation.', '0907.3814-3-7-5': 'Substituting [MATH] for [MATH], we obtain [MATH], with [MATH] for the full MOI in a normal system, and [MATH] for a MOI that is quenched because of the superfluid core.', '0907.3814-3-8-0': '# Experimental setup and procedures', '0907.3814-3-9-0': 'The starting point of our experiments is an optically trapped, strongly interacting Fermi gas consisting of an equal mixture of [MATH]Li atoms in the lowest two atomic states [CITATION].', '0907.3814-3-9-1': 'The broad 834-G Feshbach resonance [CITATION] allows us to control the [MATH]-wave interaction.', '0907.3814-3-9-2': 'If not otherwise stated, the measurements presented here refer to the resonance center.', '0907.3814-3-9-3': 'Here a unitarity-limited Fermi gas [CITATION] is realized, which is known to exhibit deep hydrodynamic behavior even well above the critical temperature for superfluidity, see e.g. [CITATION].', '0907.3814-3-9-4': 'The cigar-shaped quantum gas is confined in a far red-detuned, single-beam optical dipole trap with additional axial magnetic confinement.', '0907.3814-3-9-5': 'The trap can be well approximated by a harmonic potential with radial oscillation frequencies [MATH]Hz and an axial frequency of [MATH]Hz.', '0907.3814-3-9-6': 'The Fermi energy of the noninteracting gas is given by [MATH], where [MATH] is the total atom number.', '0907.3814-3-9-7': 'The Fermi temperature is [MATH]K, with [MATH] denoting the Boltzmann constant.', '0907.3814-3-10-0': 'Our scheme to study the rotational properties is described in detail in Ref. [CITATION].', '0907.3814-3-10-1': 'It is based on a rotating elliptical deformation of the trap, characterized by a small ellipticity parameter [CITATION] [MATH].', '0907.3814-3-10-2': 'In contrast to our previous work, we use a lower rotation frequency of [MATH]Hz [MATH].', '0907.3814-3-10-3': 'This low value allows us to avoid a resonant quadrupole mode excitation, which is known as an efficient mechanism for vortex nucleation [CITATION].', '0907.3814-3-10-4': 'To excite the quadrupole mode [CITATION] we switch on an elliptic trap deformation for 50[MATH]s.', '0907.3814-3-10-5': 'We detect the resulting oscillation by taking absorption images of the cloud after a variable hold time in the trap and a short free expansion time after release from the trap.', '0907.3814-3-10-6': 'More details on this excitation and detection scheme are given in Ref. [CITATION].', '0907.3814-3-11-0': 'At this point it is important to discuss the consequences of residual trap imperfections, still present when we attempt to realize a cylindrically symmetric optical potential.', '0907.3814-3-11-1': 'As we showed in previous work [CITATION], we can control the ellipticity down to a level of [MATH]1%.', '0907.3814-3-11-2': 'Moreover, deviations from perfect cylindrical symmetry may occur because of other residual effects, such as corrugations of the optical trapping potential.', '0907.3814-3-11-3': 'As a consequence, a certain rotational damping is unavoidable, but damping times can reach typically one second [CITATION].', '0907.3814-3-11-4': 'This has two main effects for our observations.', '0907.3814-3-11-5': 'First, our measurements yield precession frequencies slightly below [MATH].', '0907.3814-3-11-6': 'This is because of a delay time of [MATH]ms between turning off the rotating trap ellipticity and applying the quadrupole mode excitation.', '0907.3814-3-11-7': 'It is introduced to make sure that any possible collective excitation resulting from the rotating trap has damped out when the mode precession is measured.', '0907.3814-3-11-8': 'Because of rotational damping during this delay time, the measured precession frequencies [MATH] are somewhat below [MATH].', '0907.3814-3-11-9': 'To compensate for this effect, we directly measure the reduction of [MATH] that occurs during a 20ms hold time to determine the corresponding damping parameter [MATH] for each set of measurements, finding day-to-day variations with typical values between [MATH] and [MATH].', '0907.3814-3-11-10': 'The second effect is induced by friction with static (nonrotating) trap imperfections when the rotating ellipticity is applied.', '0907.3814-3-11-11': 'This leads to equilibrium values for [MATH] typically a few percent below [MATH], depending on the ratio between the time constants for spin up and damping [CITATION].', '0907.3814-3-11-12': 'For this second effect there is no straightforward compensation, and it needs to be explicitly discussed when interpreting the experimental results.', '0907.3814-3-12-0': 'Thermometry is performed after the whole experimental sequence.', '0907.3814-3-12-1': 'We damp out the rotation by stopping the trap rotation and keeping the ellipticity [CITATION].', '0907.3814-3-12-2': 'We convert the gas into a weakly interacting one by a slow magnetic field ramp to [MATH]G, and we finally measure the temperature [MATH] [CITATION].', '0907.3814-3-12-3': 'Note that the isentropic conversion tends to decrease the temperature such that [MATH] is always somewhat below the temperature [MATH] at unitarity [CITATION].', '0907.3814-3-12-4': 'The relative statistical uncertainty of the temperature measurement is about [MATH] in the relevant temperature range.', '0907.3814-3-13-0': '# Experimental results', '0907.3814-3-14-0': 'To discuss our experimental results we introduce a dimensionless precession parameter [MATH] by normalizing our observable [MATH] to its maximum possible value of [MATH], [EQUATION]', '0907.3814-3-14-1': 'The maximum possible value of [MATH] corresponds to a fully rotating, classically hydrodynamic cloud.', '0907.3814-3-14-2': 'Values [MATH] show the presence of at least one of the two effects, namely the incomplete rotation of the normal part ([MATH]) or the superfluid quenching of the MOI ([MATH]).', '0907.3814-3-14-3': 'It is crucial for the interpretation of our experimental results to distinguish between these two effects.', '0907.3814-3-14-4': 'Our basic idea to achieve this relies on the fact that [MATH] represents a temperature-dependent equilibrium property, whereas [MATH] depends on the dynamics of the spin-up before the system has reached an equilibrium.', '0907.3814-3-14-5': 'Experimentally, however, measurements of equilibrium properties at a fixed temperature are not straightforward because of the presence of residual heating leading to a slow, steady temperature increase.', '0907.3814-3-14-6': 'In the rotating trap we always observe some heating, which under all our experimental conditions can be well described by a constant rate [MATH]nK/s = [MATH]/s [CITATION].', '0907.3814-3-15-0': '## Equilibrium state of rotation', '0907.3814-3-16-0': 'To identify the conditions under which our cloud reaches its equilibrium state of rotation, we have developed a special procedure based on the timing scheme illustrated on top of Fig. [REF].', '0907.3814-3-16-1': 'Our procedure takes advantage of the constant heating rate [MATH] to control the final temperature of the gas when [MATH] is measured.', '0907.3814-3-16-2': 'We apply the trap rotation in two separate stages of duration [MATH] and [MATH].', '0907.3814-3-16-3': 'In an intermediate time interval of [MATH]ms [CITATION] we damp out the rotation that is induced by the first stage.', '0907.3814-3-16-4': 'The angular momentum disappears, but the heating effect remains [CITATION].', '0907.3814-3-16-5': 'The second stage spins up the cloud again and induces further heating.', '0907.3814-3-16-6': 'When [MATH] is kept constant, we find that the total heating by the two rotation stages is [MATH].', '0907.3814-3-16-7': 'As only the second stage leads to a final angular momentum, the equilibrium state reached at a constant temperature can be identified when [MATH] reaches a constant value for increasing [MATH] and fixed [MATH].', '0907.3814-3-16-8': 'The temperature can be controlled by a variation of the parameter [MATH] and is obtained as [MATH].', '0907.3814-3-16-9': 'The temperature offset [MATH] is set by the initial cooling and some unavoidable heating during the experimental sequence without trap rotation.', '0907.3814-3-16-10': 'Under our conditions [MATH].', '0907.3814-3-17-0': 'Our experimental results for [MATH] are shown in Fig. [REF] for four different values of the heating parameter [MATH] in a range between [MATH] to [MATH], which corresponds to a range of [MATH] between about [MATH] and [MATH].', '0907.3814-3-17-1': 'All four curves show qualitatively the same behavior.', '0907.3814-3-17-2': 'Within a few 100ms, [MATH] rises before reaching a final equilibrium value.', '0907.3814-3-17-3': 'This time-dependent increase of [MATH] is related to the spin-up dynamics [CITATION].', '0907.3814-3-17-4': 'We find that the observed increase and saturation of [MATH] can be well fit by simple exponential curves (solid lines), and we use these fits to extract the different equilibrium values [MATH].', '0907.3814-3-18-0': 'The equilibrium values [MATH] exhibit an interesting temperature dependence.', '0907.3814-3-18-1': 'The lower three values show a pronounced increase with temperature, [MATH], [MATH], and [MATH] for [MATH], [MATH], and [MATH], respectively.', '0907.3814-3-18-2': 'We interpret this increase as a consequence of the decreasing superfluid core and thus the decreasing MOI quenching effect.', '0907.3814-3-18-3': 'For our highest temperature ([MATH]) we only observe a marginal further increase to [MATH].', '0907.3814-3-18-4': 'This indicates that the superfluid core is very small or absent leading to a disappearance of the quenching effect.', '0907.3814-3-18-5': 'The fact that the maximum [MATH] stays a few percent below [MATH] can be explained by trap imperfections as discussed in Sec. [REF].', '0907.3814-3-19-0': 'Let us comment on the possible influence of vortices [CITATION].', '0907.3814-3-19-1': 'We cannot exclude their presence [CITATION], as their nucleation can proceed not only via a resonant quadrupole mode excitation [CITATION], but also via a coupling to the thermal cloud [CITATION].', '0907.3814-3-19-2': 'Vortices would result in additional angular momentum in the rotating cloud and its collective behavior would be closer to the normal case.', '0907.3814-3-19-3': 'This would tend to increase [MATH] at lower temperatures, counteracting the behavior that we observe.', '0907.3814-3-20-0': '## Superfluid phase transition', '0907.3814-3-21-0': 'In a second set of experiments, we study the superfluid phase transition in a way which is experimentally simpler, but which requires information on the equilibrium state as obtained from the measurements presented before.', '0907.3814-3-21-1': 'The trap rotation is applied continuously, and we observe the increase of [MATH] with the rotation time [MATH].', '0907.3814-3-21-2': 'All other parameters and procedures are essentially the same as in the measurements before.', '0907.3814-3-21-3': 'Here the temperature is not constant, but rises according to [MATH], where the heating rate [MATH]nK/s is the same as before and [MATH] is somewhat lower because of the less complex timing sequence.', '0907.3814-3-22-0': 'Figure [REF] shows how [MATH] increases with the rotation time [MATH] (filled symbols); the upper scale shows the corresponding temperature [MATH].', '0907.3814-3-22-1': 'The observed increase of [MATH] generally results from both factors in Eq. ([REF]), corresponding to the rising [MATH] (spin-up dynamics) and the rising [MATH] (decrease of the superfluid MOI quenching).', '0907.3814-3-22-2': 'Figure [REF] also shows the values [MATH] as determined from Fig. [REF] (crosses), for which we know that the spin-up of the normal component has established an equilibrium with [MATH] being close to one.', '0907.3814-3-22-3': 'The comparison shows that already for [MATH]s the data set obtained with the simpler procedure follows essentially the same behavior.', '0907.3814-3-22-4': 'The small quantitative difference that the crosses are slightly below the open symbols can be explained by a somewhat stronger influence of trap imperfections in the earlier measurements of Sec. [REF] [CITATION] or by the uncertainty in the initial temperature [MATH].', '0907.3814-3-22-5': 'For [MATH]s, we can assume that the system is in an equilibrium state, which follows the slowly increasing temperature, and we can fully attribute the further increase of [MATH] to the quenching of the MOI.', '0907.3814-3-23-0': 'The superfluid phase transition corresponds to the point where the precession parameter [MATH] reaches its saturation value.', '0907.3814-3-23-1': 'This is observed for a time [MATH]s, when [MATH].', '0907.3814-3-23-2': 'The conversion of this temperature parameter (measured in the weakly interacting regime after an isentropic change) to the actual temperature in the unitarity-limit regime [CITATION] yields a value for the critical temperature [MATH] of about [MATH].', '0907.3814-3-23-3': 'This result is consistent with previous experimental results [CITATION], the range of which is indicated by the shaded region in Fig. [REF].', '0907.3814-3-23-4': 'The result is also consistent with theoretical predictions [CITATION].', '0907.3814-3-24-0': 'For a more precise extraction of [MATH] from experimental MOI quenching data, a theoretical model would be required that describes the saturation behavior of [MATH] as [MATH] is approached.', '0907.3814-3-24-1': 'Theoretical predictions are available for the BEC limit [CITATION] and the BCS limit [CITATION].', '0907.3814-3-24-2': 'In the unitarity limit it should, in principle, be possible to extract the MOI from spatial profiles of the normal and the superfluid fraction [CITATION].', '0907.3814-3-24-3': 'Clearly, more work is necessary to quantitatively understand the quenching effect in the strongly interacting regime.', '0907.3814-3-25-0': '# Conclusion', '0907.3814-3-26-0': 'We have demonstrated the quenching of the moment of inertia that occurs in a slowly rotating, strongly interacting Fermi gas as a consequence of superfluidity.', '0907.3814-3-26-1': 'This effect provides us with a novel probe for the system as, in contrast to other common methods such as expansion measurements and studies of collective modes, it allows us to distinguish between the two possible origins of hydrodynamic behavior, namely collisions in a normal phase and superfluidity.'}

1012.2247

{'1012.2247-1-0-0': 'The cross-Kerr effect is studied for two pulses propagating in an atomic medium in a tripod configuration, dressed by a strong standing wave coupling beam.', '1012.2247-1-0-1': 'Nonlinear phase shifts for both transmitted and reflected beams are calculated taking into account the redistribution of the population among the atomic levels which allows one to study the phase shifts in a wide range of pulse detunings.', '1012.2247-1-0-2': 'The influence of other parameters, e.g., the intensity of the control field and the relaxation rates, on the cross-Kerr effect is also examined in detail.', '1012.2247-1-1-0': '# Introduction', '1012.2247-1-2-0': 'A large family of papers devoted to the properties of optically dressed atomic media and pulse propagation through them focus on the regime of the electromagnetically induced transparency (EIT) [CITATION] in which a strong control field, coupling two unpopulated levels of a [MATH] system, creates a transparency window for a weak probe beam.', '1012.2247-1-2-1': 'This generic model was extended by including additional atomic levels coupled by laser or microwave fields in the double-[MATH] [CITATION], tripod [CITATION], [MATH] [CITATION] or inverted-[MATH] [CITATION] configurations, by admitting different initial conditions or controlling the processes in time; the latter could lead, e.g., to light slowdown, storage and release [CITATION].', '1012.2247-1-3-0': 'Using the control field in the standing wave configuration opens new flexible possibilities of creating metamaterials and photonic crystals in cold atom samples.', '1012.2247-1-3-1': 'An incoming probe light encounters a periodic optical structure and is subject to Bragg scattering.', '1012.2247-1-3-2': 'For particular frequency ranges there appear transmission and stop bands, the properties of which can easily be steered by a proper choice of the control field.', '1012.2247-1-3-3': 'The field-induced dispersion of the medium and the transmission and reflection spectra have been analyzed, e.g., in Refs. [CITATION].', '1012.2247-1-3-4': 'It was also possible to stop a pulse inside the medium in the form of atomic excitations and then to turn it back into photons which, however, could not leave the medium due to the standing character of the releasing control beam [CITATION].', '1012.2247-1-4-0': 'Another important development was taking advantage of the fact that in EIT-like conditions a usually weak interaction between two photons (or weak classical pulses) may become enhanced by many orders of magnitude.', '1012.2247-1-4-1': 'In particular it was possible to predict cross nonlinear effects and thus to create new types of a polarization phase gate in a properly prepared optically dressed medium in the [MATH] [CITATION], tripod [CITATION] or inverted-[MATH] [CITATION] configuration.', '1012.2247-1-4-2': 'The authors of those papers have shown that it is possible to significantly change the phase of one of the propagating pulses due to a quadratic susceptibility dependence on the other pulse.', '1012.2247-1-4-3': 'Their theoretical prediction was experimentally confirmed in Ref. [CITATION].', '1012.2247-1-4-4': 'Similar results, both theoretical and experimental, have recently been obtained for an inverted-[MATH] system [CITATION], which closely resembles the tripod system but allows for some simplifications concerning the level populations.', '1012.2247-1-4-5': 'The cross-phase modulation has also been studied for a four-level [MATH] system [CITATION], a five-level system [CITATION].', '1012.2247-1-4-6': 'It has also been shown that cross-phase modulation can be performed for stored light pulses [CITATION].', '1012.2247-1-4-7': 'Those results can pave the way towards constructing all-optical logical devices.', '1012.2247-1-5-0': 'We have recently pointed out [CITATION] that the cross-Kerr nonlinear phase shifts appear also for a beam reflected due to the periodic structure induced by a strong standing control field in a medium in the tripod configuration.', '1012.2247-1-5-1': 'Our previous results depend on the assumption of a symmetric population distribution on the two atomic levels coupled with the upper level by the two propagating pulses.', '1012.2247-1-5-2': 'Here, this restrictive condition is waived and we are thus able to examine the nonlinear phase shifts in a wide range of the probe pulse detuning.', '1012.2247-1-6-0': 'In Section II we present the theoretical approach, some details of which are shown in Appendix.', '1012.2247-1-6-1': 'In Section III we discuss the numerical results for the nonlinear phase shifts as functions of the detuning for various combinations of control field intensities, relaxation rates and sample length.', '1012.2247-1-7-0': '# Theory', '1012.2247-1-8-0': 'We consider a tripod system driven by a strong control field [MATH], coupling the ground level [MATH] and the upper level [MATH], and by two weaker laser fields called probe and trigger (denoted by [MATH] and [MATH]), coupling [MATH] with [MATH] and [MATH], respectively (see Fig. [REF]).', '1012.2247-1-8-1': 'We express the fields through their complex amplitudes [MATH] slowly varying in time: [EQUATION].', '1012.2247-1-8-2': 'Each of the couplings is detuned by [MATH], [MATH].', '1012.2247-1-8-3': 'Here [MATH] stands for the energy of the state [MATH] and [MATH] denotes the frequency of the [MATH]-th field.', '1012.2247-1-9-0': 'All the fields propagate along the [MATH] axis.', '1012.2247-1-9-1': 'In general the control wave is allowed to have two counter-propagating components, of arbitrary ratio.', '1012.2247-1-9-2': 'In the case of a running control wave, the component antiparallel to the incoming probe and trigger fields is zero.', '1012.2247-1-9-3': 'In the case of a perfect standing wave, both components are equal.', '1012.2247-1-9-4': "Thus, the control field provides a spatial lattice of period [MATH], [MATH] being the field's wave vector.", '1012.2247-1-9-5': "In the configuration proposed below [MATH] may be slightly different from the probe and trigger fields' wave vectors [MATH], but that can be corrected by tilting both components of the control field by an angle [MATH] with respect to the [MATH]-axis so that [MATH] [CITATION].", '1012.2247-1-10-0': 'Such a model can be realized in a cold gas of [MATH]Rb atoms.', '1012.2247-1-10-1': 'The states [MATH] and [MATH] may correspond, e.g., to Zeeman sublevels [MATH], state [MATH] to [MATH], and the upper state [MATH] to the level [MATH].', '1012.2247-1-10-2': 'Such a system could be used as a polarization phase gate, as was first proposed in Ref. [CITATION].', '1012.2247-1-10-3': 'Each of the two pulses interacts with the medium when it has the proper circular polarization: right for the probe and left for the trigger.', '1012.2247-1-10-4': 'When both pulses are properly polarized the nonlinear interaction between them gives rise to a cross-phase modulation.', '1012.2247-1-11-0': '## Bloch-Maxwell equations', '1012.2247-1-12-0': 'The Bloch equations for the atom + field system in the rotating wave approximation read: [EQUATION] where [MATH], [MATH], [MATH], and [MATH] is the density matrix of the atoms in the Schrodinger picture, [MATH], [MATH], [MATH], [MATH], [MATH].', '1012.2247-1-12-1': 'The Rabi frequencies are defined by [MATH], where [MATH] is the electric dipole moment matrix element.', '1012.2247-1-13-0': 'The simplified model of relaxations adopted in the above equations takes into account spontaneous emission from the upper state [MATH] to a lower state [MATH], described by the relaxation rates [MATH] and [MATH] with [MATH], as well as deexcitation and decoherence between the lower levels [MATH], [MATH], due to atomic collisions.', '1012.2247-1-13-1': 'The relaxation rates for the coherences [MATH], [MATH], and those for the collision-induced population relaxations were taken equal for simplicity (cf. Ref. [CITATION]).', '1012.2247-1-14-0': 'To find the propagation equation for both probe and trigger, we start with the Maxwell wave equation, written below for the probe pulse: [EQUATION] where [MATH] denotes the medium polarization component connected with the probe.', '1012.2247-1-14-1': 'For the positive frequency term [MATH] we find for the pulse envelope slowly varying in time, i.e. when [MATH]: [EQUATION]', '1012.2247-1-14-2': 'To obtain the polarization we introduce the medium electric susceptibility for the probe by: [EQUATION] where [MATH] is the vacuum permittivity.', '1012.2247-1-14-3': 'Thus the medium susceptibility for the probe can be written as: [EQUATION] where [MATH] is the number of atoms per unit volume.', '1012.2247-1-14-4': 'The time limit means that we need to find the steady state solutions to the Bloch equations.', '1012.2247-1-15-0': 'If Eq. ([REF]) is used in the Fourier transform of Eq. ([REF]), with the Fourier variable identified with the probe detuning, we obtain: [EQUATION]', '1012.2247-1-15-1': 'The assumption that the probe and trigger are weak with respect to the control field [MATH] allows us to expand the susceptibility into the Taylor series: [EQUATION]', '1012.2247-1-15-2': 'The explicit form of the Taylor expansion coefficients can be found in Appendix.', '1012.2247-1-15-3': 'The first term above corresponds to the linear susceptibility, while the third-order terms [MATH] and [MATH] represent the self- and cross-Kerr effect, respectively.', '1012.2247-1-15-4': 'The latter is of particular importance, as we are interested in the cross-phase modulation.', '1012.2247-1-16-0': 'We now write the control field as: [EQUATION] where [MATH] are its forward and backward propagating parts.', '1012.2247-1-16-1': 'As the field is periodic in space, so are the medium optical properties.', '1012.2247-1-16-2': 'Therefore we expand the susceptibilities for both probe and trigger into the Fourier series: [EQUATION]', '1012.2247-1-16-3': 'The pulses propagating in a periodic medium acquire their reflected components.', '1012.2247-1-16-4': 'In the two-mode approximation (for details see [CITATION]) we write: [EQUATION] where [MATH] are slowly varying.', '1012.2247-1-17-0': 'The Fourier expansion of the susceptibilities given by Eq. ([REF]) can be then rewritten in the form: [EQUATION] where [MATH].', '1012.2247-1-18-0': 'We now insert Eqs. ([REF],[REF]) into Eq. ([REF]) and drop terms rapidly oscillating in space.', '1012.2247-1-18-1': 'To write the propagation equations in the final form we make use of the relation [MATH] and set [MATH].', '1012.2247-1-18-2': 'For the probe we find: [EQUATION] where: [EQUATION]', '1012.2247-1-18-3': 'The trigger equations are obtained by interchanging the indices [MATH].', '1012.2247-1-18-4': 'Explicit formulae for the susceptibilities Fourier components as well as their detailed deriviation can be found in Appendix.', '1012.2247-1-19-0': '## Population distribution', '1012.2247-1-20-0': 'As we will change the probe detuning [MATH] in a wide range while keeping the trigger detuning [MATH] constant, we introduce an asymmetry in the populations [MATH] and [MATH] even for [MATH].', '1012.2247-1-20-1': 'Additionally, the symmetry may be spoiled by the fact that [MATH] (the energy of the state [MATH]) is slightly smaller than [MATH], and in a cold medium transitions from [MATH] to [MATH] are possible while those from [MATH] to [MATH] are not.', '1012.2247-1-20-2': 'In previous works (see Refs. [CITATION]) the probe and trigger fields were only allowed detunings such that [MATH].', '1012.2247-1-20-3': 'In that case the stationary population distribution was assumed to be [MATH].', '1012.2247-1-20-4': 'Such an approach is simple, but it is well justified only in the resonance region of frequencies where [MATH].', '1012.2247-1-20-5': 'As we show in the section [REF], for a wider range of pulse detunings it is necessary to make a better estimation of the population distribution.', '1012.2247-1-21-0': 'To evaluate the populations we proceed as follows.', '1012.2247-1-21-1': 'We first use equations 2-4 of the stationary version of Eqs. (1) to express [MATH] through [MATH] and [MATH]: [EQUATION] where [MATH], [MATH].', '1012.2247-1-21-2': 'Note that here we do not assume any of the populations negligible.', '1012.2247-1-21-3': 'Next we express [MATH] through the diagonal elements of [MATH] in the lowest-order approximation with respect to the probe and trigger: [EQUATION] where [MATH].', '1012.2247-1-21-4': 'Combining the two latter steps we obtain a set of linear equations for the diagonal elements: [EQUATION] where: [EQUATION]', '1012.2247-1-21-5': 'The solutions of Eqs. ([REF]) are [EQUATION] while the value of [MATH] follows from the probability conservation.', '1012.2247-1-21-6': 'The denominator [MATH] reads: [EQUATION]', '1012.2247-1-21-7': 'Using the above equations with constant initial values of probe and trigger fields and a given value of the control field we are able to estimate the population distribution.', '1012.2247-1-21-8': 'Note however that, strictly speaking, the populations in the standing wave case are also modulated in space.', '1012.2247-1-21-9': 'However, as discussed below, the latter effect should not play a very important role in the case of an imperfect standing wave.', '1012.2247-1-22-0': '# Numerical results', '1012.2247-1-23-0': 'In this section we present a variety of numerical results illustrating the cross-Kerr effect in our system and its dependence on the system parameters.', '1012.2247-1-23-1': 'In particular, we discuss how the cross-phase modulation is affected by the population redistribution.', '1012.2247-1-24-0': 'We consider the propagation equations ([REF],[REF]) with the boundary conditions corresponding to both probe and trigger originally propagating towards positive [MATH]: [EQUATION] where [MATH] is the sample length.', '1012.2247-1-24-1': 'The equations are solved iteratively until self-consistency is achieved, alternately for the probe and trigger, with the susceptibilities calculated using the fields obtained in the preceding step.', '1012.2247-1-24-2': 'In the first iteration for the probe the initial constant value of the trigger has been used.', '1012.2247-1-24-3': 'The numerical task in each step is thus solving a linear equation.', '1012.2247-1-24-4': 'We find two independent solutions with one-point boundary conditions and combine them to obtain the solution with the proper two-point boundary conditions given by Eqs. ([REF]).', '1012.2247-1-25-0': 'The transmission and reflection coefficients for the probe and trigger beams are defined by: [EQUATION] while the phases of the transmitted and reflected fields are given by: [EQUATION]', '1012.2247-1-25-1': 'First we present the results obtained for a running control field.', '1012.2247-1-25-2': 'We compare them for two cases: when [MATH] and when all the populations are given by Eqs. ([REF]).', '1012.2247-1-25-3': 'In this way we can check in which regime the approximation of equal populations is reliable and for which detunings of the probe field from the resonance conditions we still remain in this regime.', '1012.2247-1-25-4': 'We choose the set of parameters which are of order of those typical of real atomic systems: [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] mm, [MATH] C[MATH]m, [MATH] cm[MATH].', '1012.2247-1-26-0': 'The results are shown in Fig. [REF]a. Note that shapes of the obtained [MATH] functions are different for the two approximations.', '1012.2247-1-26-1': 'The results are in agreement at three points: when the calculated population distribution is almost exactly: [MATH] and within the resonance region.', '1012.2247-1-26-2': 'The trigger-induced phase shift obtained in the case when the populations have been calculated according to Eqs. ([REF]) is larger for some frequencies.', '1012.2247-1-27-0': 'In the imperfect standing wave case we take [MATH] MHz, [MATH] MHz.', '1012.2247-1-27-1': 'All the other parameters remain unchanged.', '1012.2247-1-27-2': "The shape of the transmitted field's phase dependence on the probe detuning is similar to that in the running wave case, so we do not show it here.", '1012.2247-1-27-3': "The reflected field's phase shift is shown in Fig. [REF]b.", '1012.2247-1-28-0': 'Again we find some significant differences in the shape of the function and the values of the calculated phase shift.', '1012.2247-1-28-1': 'The results are in agreement at the same three points as before.', '1012.2247-1-28-2': 'In Fig. [REF] it is shown that at two of those points the calculated population is equally distributed between the levels [MATH] and [MATH] while the third one occurs close to the resonance, i.e. for [MATH].', '1012.2247-1-28-3': 'At the latter point, however, due to a numerical instability in the algorithm used to find the solutions of Eq. ([REF]), the calculated density matrix is no longer positive definite and our results in this very narrow spectral region are unphysical (see the sharp spikes appearing in Fig. [REF]).', '1012.2247-1-28-4': 'The present results show that taking into account the redistribution of the population among the atomic levels is crucial and that our setting [MATH] is reliable.', '1012.2247-1-28-5': 'If the redistribution were ignored the phase shifts could only be estimated within rather limited ranges of frequencies.', '1012.2247-1-28-6': 'On the contrary, we present in the following several numerical results obtained with this method over a large detuning range.', '1012.2247-1-29-0': "We check how the phase shifts of both transmitted field and the reflected one depend on 'how much standing' the control field is.", '1012.2247-1-29-1': 'The case of a perfect standing wave, however, is tricky, as in the node regions the steady state population is trapped in the state [MATH] due to relaxation effects and the medium becomes transparent to both probe and trigger.', '1012.2247-1-29-2': 'Therefore it is reasonable to only consider the cases when the control field at the quasinodes is still strong enough to pump the population from the level [MATH].', '1012.2247-1-29-3': 'We change the right-propagating part of the control field [MATH] while keeping the left-moving part constant: [MATH] MHz.', '1012.2247-1-29-4': 'All the other parameters remain unchanged.', '1012.2247-1-29-5': 'The results are shown in Fig. [REF]a. All the plots cross in the resonance region where the phase shift is small.', '1012.2247-1-29-6': 'Note that the crossing point is slightly moved to the left from the exact resonance.', '1012.2247-1-29-7': 'The phase shift [MATH] reaches a maximal value for some [MATH] which depends on [MATH].', '1012.2247-1-29-8': "The more intense is the control field the flatter is the plot and the larger is the peak's shift towards higher frequencies, which provides a way of controlling the phases.", '1012.2247-1-29-9': 'We find the same effect of the peak moving to the right when decreasing the left-propagating part of the control field [MATH] while keeping [MATH] constant (not shown).', '1012.2247-1-29-10': 'The more intense is the control field the smaller is the trigger-induced phase shift.', '1012.2247-1-30-0': "In Fig. [REF]b we plot the reflected field's phase shifts for a few (the same as in Fig. [REF]a) values of [MATH].", '1012.2247-1-30-1': 'Again all the curves overlap in the resonance region where the phase shift is small.', '1012.2247-1-30-2': 'Around the resonance the reflected phase shift shows an oscillatory dependence on the probe detuning.', '1012.2247-1-30-3': "The width of the frequency range where the oscillations are present grows with [MATH] but the number of peaks remains constant - again the phase shift's plot becomes flatter when the control field is increased.", '1012.2247-1-30-4': 'The curves are complicated now but roughly we can say that the trigger-induced phase shift decreases when the control field becomes stronger.', '1012.2247-1-30-5': 'In general a too strong control field is not advantegous for generating considerable trigger-induced phase shifts.', '1012.2247-1-30-6': 'The results obtained then gradually turn into those typical of the usual EIT case.', '1012.2247-1-31-0': 'As expected, the transmission coefficient (see Fig. [REF]a) grows with [MATH] and so does the width of the transparency window.', '1012.2247-1-31-1': 'Again the plots corresponding to the reflection are more complex, as shown in Fig. [REF]b.', '1012.2247-1-31-2': 'In general the reflection coefficient decreases when [MATH] is increased, but there are some frequency ranges where the dependence is more complicated.', '1012.2247-1-31-3': 'Yet if both parts of the control field are increased simultaneously, so that [MATH] is constant, then not only the transmission but also the reflection coefficient increases for most of the probe frequencies (as shown in Fig. [REF]c).', '1012.2247-1-31-4': 'This is due to the absorption cancellation by a strong control field.', '1012.2247-1-32-0': 'It is interesting to check how our results depend on the relaxation rates [MATH], [MATH], due to interatomic collisions.', '1012.2247-1-32-1': 'We take [MATH] MHz, [MATH] MHz.', '1012.2247-1-32-2': 'All the other parameters are chosen as before, except for [MATH] which take several values.', '1012.2247-1-32-3': 'In Fig. [REF] we compare the trigger influence on the phase shift of the transmitted and reflected probe parts, for a few values of the relaxation constants.', '1012.2247-1-32-4': 'The trigger-induced phase shift decreases in general for growing [MATH].', '1012.2247-1-32-5': "On the other hand, as the relaxation rates grow, the frequency range where the reflected field's phase dependence is oscillatory becomes wider, the number of oscillations increases and both [MATH] and [MATH] plots become steeper.", '1012.2247-1-33-0': 'As expected, the transmission and reflection coefficients decrease in general when the relaxation rates grow, but there are some frequency ranges where the reflection coefficient is a nonmonotonic function of [MATH].', '1012.2247-1-34-0': 'In Fig. [REF] we observe the behaviour of the population of the level [MATH], depending on the value of [MATH].', '1012.2247-1-34-1': 'We do not show the complementary plot for [MATH].', '1012.2247-1-34-2': 'The populations of [MATH] and [MATH] are negligible, except for very large values of the relaxations ([MATH] MHz) when they are of the order of a few percent.', '1012.2247-1-34-3': 'Note that the populations [MATH] (and [MATH]) may vary rapidly within the transparency window.', '1012.2247-1-34-4': 'For increasing [MATH] the curves become less steep and an approximation consisting in adopting constant values of the two populations may be better justified.', '1012.2247-1-34-5': 'The maximum values of [MATH] may be close to unity which apparently occurs when the frequency of the probe suits the energy interval between [MATH] and the energy one of the lower states dressed by [MATH] and [MATH].', '1012.2247-1-34-6': 'Between the maxima there is a deep minimum of [MATH] due to the population trapping in the state [MATH].', '1012.2247-1-35-0': 'If there are no relaxations in the system, then the steady state solution for the no-trigger case is trivial: [MATH], [MATH] due to spontaneous emission from the upper level to all the lower levels.', '1012.2247-1-35-1': 'There is no mechanism yet pumping the population out from [MATH].', '1012.2247-1-35-2': 'The nonnegligible relaxations provide such a mechanism and the plot illustrating the [MATH] dependence on the probe detuning is of a similar shape as in the trigger-present case.', '1012.2247-1-36-0': 'We have checked the dependence of our results on [MATH] relaxation rates, [MATH], responsible for the spontaneous emission.', '1012.2247-1-36-1': 'There is hardly any influence of [MATH] on the trigger-induced phase shift of the transmitted field.', '1012.2247-1-36-2': 'This is also true for the reflected field for most of the probe detunings.', '1012.2247-1-36-3': 'There are, however, some frequency ranges when there is a nonmonotonic dependence on [MATH] (not shown).', '1012.2247-1-37-0': 'As shown in Fig. [REF]a, increasing the length of the sample leads to increasing the trigger-induced phase shift of the transmitted field.', '1012.2247-1-37-1': 'For the considered lengths of the sample the nonlinear phase shift increases for growing [MATH].', '1012.2247-1-37-2': 'However, for an extended sample both pulses become partially absorbed and their mutual impact is smaller.', '1012.2247-1-38-0': "Elongating the sample leads to no significant change in the amplitude of oscillations in the reflected field's phase but it does influence the reflection coefficient in a nonmonotonic way (see Fig. [REF]b,c), which provides another way of controlling the cross-Kerr effect.", '1012.2247-1-39-0': '# Conclusions', '1012.2247-1-40-0': 'We have developed the study of the cross Kerr effect in the propagation of two pulses, named trigger and probe, in a medium of four-level atoms in the tripod configuration, dressed by a third strong field in the running- or imperfect standing-wave configuration.', '1012.2247-1-40-1': 'The first- and third-order susceptibilities have been obtained by solving the Bloch equations in the stationary case and expanded into the Fourier series.', '1012.2247-1-40-2': 'We have numerically solved the propagation equations for the right- and left-running components of the two pulses in the two-mode approximation and we have obtained the phase shifts of all the fields.', '1012.2247-1-40-3': 'We have estimated the populations of the atomic levels, waiving the approximation of earlier papers.', '1012.2247-1-40-4': 'We have shown their significant influence on the phase shifts of the transmitted and reflected beams in a considerable range of the frequencies.', '1012.2247-1-41-0': 'We have extended the possibility of constructing phase gates for both dressing-field configurations.', '1012.2247-1-41-1': 'We have shown that the values of the nonlinear phase shifts are significantly modified when the correction of the populations has been taken into account.', '1012.2247-1-41-2': 'The phase gate can operate for the probe detuning in the whole transparency window.', '1012.2247-1-41-3': 'The shifts for the reflected probe wave exhibit oscillations as functions of the probe detuning, allowing for its optimum choice, i.e. such that the nonlinear phase shift is largest; in our examples it could be in some cases of the order of one radian.', '1012.2247-1-42-0': "We have also checked the Kerr effect's dependence on the strength of the coupling field, the relaxation rates and the sample's length.", '1012.2247-1-42-1': 'To have a considerable phase shift the coupling field cannot be too strong and the relaxation rates between the lower tripod states should not be too large (though for their zero value the medium would become transparent in the zero-trigger case).', '1012.2247-1-42-2': 'For not too long samples increasing their lengths may increase the nonlinear phase shifts but not proportionally.', '1012.2247-1-43-0': 'We have proven that in the imperfect standing wave configuration the amplitudes and nonlinear phase shifts of the transmitted and reflected parts of the probe beam can be controlled and optimized by choosing various parameters, in particular the probe detuning.', '1012.2247-1-44-0': 'K. Slowik is grateful to G. C. La Rocca and Scuola Normale Superiore in Pisa for their kind hospitality.', '1012.2247-1-44-1': 'The work of K. Slowik was sponsored by the scholarship for doctoral students ZPORR 2008/2009 of the Marshal of the Kuyavian-Pomeranian Voivodeship.', '1012.2247-1-45-0': 'APPENDIX', '1012.2247-1-46-0': 'The medium susceptibilities can be written as: [EQUATION] and can be calculated with the use of the steady state solutions of the Bloch equations.', '1012.2247-1-47-0': '>From the last three of Eqs. ([REF]) we find the steady state spin coherences: [EQUATION]', '1012.2247-1-47-1': 'We substitute these expressions into the stationary form of the equations [MATH] in Eqs. ([REF]) and thus obtain a set of three coupled equations for [MATH], [MATH], [MATH].', '1012.2247-1-47-2': 'As the control field is much stronger than any other field present in the system, it prevents the states [MATH] and [MATH] from being populated.', '1012.2247-1-47-3': 'Therefore we set [MATH], as was done in Ref. [CITATION].', '1012.2247-1-47-4': 'However, contrary to Ref. [CITATION], here we do not assume the probe and trigger detunings to be almost equal.', '1012.2247-1-47-5': 'Hence the population distribution between the states [MATH] and [MATH] may be asymmetric and in general [MATH].', '1012.2247-1-47-6': 'This crucial issue has been discussed in Section [REF].', '1012.2247-1-47-7': 'Next we eliminate [MATH] and arrive at: [EQUATION] where [MATH].', '1012.2247-1-48-0': 'We combine the above expressions to obtain [MATH] and [MATH].', '1012.2247-1-48-1': 'The solutions of the above set of equations expanded into the Taylor series have the form of Eq. ([REF]), with: [EQUATION]', '1012.2247-1-48-2': 'The trigger susceptibilities are obtained by interchanging the indices [MATH].', '1012.2247-1-49-0': 'For the control field being in the form of a quasi-standing wave [MATH] the medium susceptibilities can be then expanded into the Fourier series: [EQUATION] and similarly for [MATH], etc.', '1012.2247-1-49-1': 'To find the explicit form of the Fourier coefficients we rewrite the expressions ([REF] - [REF]): [EQUATION] where: [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH].', '1012.2247-1-50-0': 'Let us denote: [EQUATION]', '1012.2247-1-50-1': 'The above integrals have been calculated with the use of the residues method.', '1012.2247-1-50-2': 'The Fourier coefficients of the susceptibilities read then: [EQUATION]', '1012.2247-1-50-3': 'As shown above, only these coefficients are present in the pulse propagation equations.', '1012.2247-1-51-0': '# References'}

{'1012.2247-2-0-0': 'The cross-Kerr effect is studied for two weak beams, probe and trigger, propagating in an atomic medium in a tripod configuration, dressed by a strong standing wave coupling beam in a regime of electromagnetically induced transparency.', '1012.2247-2-0-1': "The nonlinear phase shifts for both transmitted and reflected probe beams induced by the trigger's presence are found to depend on the probe detuning, the control beams intensity, the relaxation rates and, in particular, on the redistribution of the population among the atomic levels.", '1012.2247-2-0-2': 'Such a quantitative analysis indicates that the transmitted and reflected probe beam components and their respective phase shifts can be easily controlled and optimized.', '1012.2247-2-1-0': '# Introduction', '1012.2247-2-2-0': 'All-optical nonlinearities at low light levels are instrumental to the optical implementation of quantum information processing systems.', '1012.2247-2-2-1': 'Large nonlinearities can be achieved when the light-matter interaction is resonant, in which case however absorption typically plays a detrimental role.', '1012.2247-2-2-2': 'A great deal of attention has thus been devoted to a variety of schemes based on the regime of electromagnetically induced transparency (EIT) [CITATION] in which this counterbalance can be overcome.', '1012.2247-2-2-3': 'In a typical EIT configuration, a strong control field, coupling two unpopulated levels of a [MATH] system, creates a transparency window for a weak probe beam.', '1012.2247-2-2-4': 'This basic scheme was extended by including additional atomic levels coupled by laser or microwave fields in the double-[MATH] [CITATION], tripod [CITATION], [MATH] [CITATION] or inverted-[MATH] [CITATION] configurations, just to name a few.', '1012.2247-2-2-5': 'Under EIT-like conditions the weak interaction between two photons (or weak classical pulses) may become largely enhanced.', '1012.2247-2-2-6': 'In particular, very large cross nonlinear effects have been predicted leading to new types of polarization phase gates in an optically dressed medium in the [MATH] [CITATION], tripod [CITATION] or inverted-[MATH] [CITATION] configuration.', '1012.2247-2-2-7': 'Relevant experimental work has very recently been reported in Ref. [CITATION].', '1012.2247-2-2-8': 'In these instances, a significant change of the phase of one of the propagating pulses is achieved due to the cross-Kerr effect induced by the other pulse.', '1012.2247-2-2-9': 'Analogous effects have recently been studied, both theoretically and experimentally, for an inverted-[MATH] system [CITATION], as well as for a four-level [MATH]-type [CITATION] and five-level system [CITATION].', '1012.2247-2-2-10': 'Important developments concern also the dynamic control of the process leading, e.g., to light slowdown, storage and release [CITATION].', '1012.2247-2-2-11': 'These results are expected to pave the way towards constructing all-optical logical devices.', '1012.2247-2-3-0': 'Unlike most typical EIT configurations employing a running wave coupling field, using a control standing wave coupling beam opens the possibility of creating an all-optically tunable Bragg mirror.', '1012.2247-2-3-1': 'In such a novel kind of dynamically tunable metamaterial [CITATION] an incoming probe light encounters a spatially periodic optical structure and is subject to Bragg scattering.', '1012.2247-2-3-2': 'For specific frequency ranges transmission and stop bands appear, the properties of which can be steered by a proper choice of the control field.', '1012.2247-2-3-3': 'The field-induced dispersion of the medium and the transmission and reflection spectra have been analyzed, e.g., in Refs. [CITATION], including the case of a quasi-standing wave coupling field [CITATION].', '1012.2247-2-3-4': 'It was also possible to stop and store a pulse inside the medium, and then retrieve it in the form of a stationary light pulse which could not leave the medium due to the standing wave character of the releasing control beam [CITATION].', '1012.2247-2-3-5': 'It has also been shown that high nonlinearities can be achieved for stored light pulses [CITATION].', '1012.2247-2-3-6': 'An all-optical dynamic cavity for the confinement of light pulses based on standing wave EIT Bragg mirrors has recently been proposed [CITATION].', '1012.2247-2-4-0': 'Very recently, we have pointed out that in the presence of a trigger beam in a tripod configuration, a probe partially reflected from the periodic structure induced by a strong standing wave EIT coupling field undergoes large cross-Kerr nonlinear phase shifts.', '1012.2247-2-4-1': 'Such a novel configuration is specifically apt for developing a phase-tunable beam splitter in which both the amplitudes and the phases of the trasmitted and reflected beam can be controlled.', '1012.2247-2-4-2': 'We have given some illustrative results valid in a narrow range of frequencies owing to the restrictive assumption that the atomic population is symmetrically distributed between the two lower levels that are coupled with the upper level respectively by the probe and trigger pulses [CITATION].', '1012.2247-2-4-3': 'Here, we provide a comprehensive theoretical study of the problem, including a proper treatment of the population redistribution.', '1012.2247-2-4-4': 'The latter point turns out to be crucial to extend the range of possible probe detunings and, thus, optimize the control possibilities.', '1012.2247-2-4-5': "Our approach combines analytical methodologies used to describe propagation effects in both nonlinear and spatially periodic media, and allow us to numerically examine the phase shifts of the reflected and transmitted probe field induced by the trigger's presence over a wide range of probe detunings as well as their dependence on various parameters characterizing both the atomic medium and the three laser fields.", '1012.2247-2-4-6': 'Our results indicate that the tripod standing wave EIT configuration makes control over the cross-Kerr effect more versatile than for a running wave EIT configuration [CITATION] in which no reflected beam appears.', '1012.2247-2-5-0': 'The paper is organized as follows.', '1012.2247-2-5-1': 'In Section II, we present the basic theoretical approach to calculate the cross-Kerr effect in a medium additionally dressed by a quasi-standing wave EIT coupling field, based on expanding the medium susceptibility into a power series with respect to the probe and trigger fields and into a Fourier series appropriate to the spatial periodicity imposed by the coupling field.', '1012.2247-2-5-2': "In Appendix A we derive in detail the formulae for the susceptibility's expansion coefficients, while in Appendix B we present the way to evaluate the population redistribution.", '1012.2247-2-5-3': 'In Section III, we present and discuss the numerical results for the trigger induced nonlinear phase shifts of the reflected and transmited probe fields as functions of the probe detuning for various combinations of control field intensities, relaxation rates and sample length.', '1012.2247-2-5-4': 'Finally, we draw our conclusions.', '1012.2247-2-6-0': '# Theory', '1012.2247-2-7-0': 'We consider a tripod system driven by a strong control field ([MATH]) coupling the ground level [MATH] and the upper level [MATH], and by two weaker laser fields called probe ([MATH]) and trigger ([MATH]), coupling [MATH] with [MATH] and [MATH], respectively, as shown in Fig. [REF].', '1012.2247-2-7-1': 'We express the fields through their complex amplitudes [MATH] slowly varying in time: [EQUATION].', '1012.2247-2-7-2': 'Each of the couplings is detuned by [MATH], [MATH].', '1012.2247-2-7-3': 'Here [MATH] stands for the energy of the state [MATH] and [MATH] denotes the frequency of the [MATH]-th field.', '1012.2247-2-8-0': 'All the fields propagate along the [MATH] axis.', '1012.2247-2-8-1': 'In general the control wave is allowed to have two counter-propagating components of arbitrary ratio.', '1012.2247-2-8-2': 'In the case of a running control wave, the component antiparallel to the incoming probe and trigger fields is zero.', '1012.2247-2-8-3': 'In the case of a perfect standing wave, both components are equal.', '1012.2247-2-8-4': "Thus, the control field provides a spatial lattice of period [MATH], [MATH] being the field's wave number.", '1012.2247-2-8-5': "In the configuration proposed below [MATH] may be slightly different from the probe and trigger fields' wave vectors [MATH], but that can be corrected by tilting both components of the control field by an angle [MATH] with respect to the [MATH]-axis so that [MATH] [CITATION].", '1012.2247-2-9-0': 'Such a configuration can be realized in a cold gas of [MATH]Rb atoms.', '1012.2247-2-9-1': 'The states [MATH] and [MATH] may correspond, e.g., to Zeeman sublevels [MATH], state [MATH] to [MATH], and the upper state [MATH] to the level [MATH].', '1012.2247-2-9-2': 'The scheme could be used as a polarization phase gate, as was first proposed in Ref. [CITATION].', '1012.2247-2-9-3': 'Each of the two pulses interacts with the medium when it has the proper circular polarization: right for the probe and left for the trigger.', '1012.2247-2-9-4': 'When both pulses are properly polarized the nonlinear interaction between them gives rise to a cross-phase modulation.', '1012.2247-2-10-0': 'The Bloch equations for the atom + field system in the rotating wave approximation read: [EQUATION] where [MATH], [MATH], [MATH], and [MATH] is the density matrix of the atoms in the Schrodinger picture, [MATH], [MATH], [MATH], [MATH], [MATH].', '1012.2247-2-10-1': 'The Rabi frequencies are defined by [MATH], where [MATH] is the electric dipole moment matrix element.', '1012.2247-2-11-0': 'The simplified model of relaxations adopted in the above equations takes into account spontaneous emission from the upper state [MATH] to a lower state [MATH], described by the relaxation rates [MATH] and [MATH] with [MATH], as well as deexcitation and decoherence between the lower levels [MATH], [MATH], due to atomic collisions.', '1012.2247-2-11-1': 'The relaxation rates for the coherences [MATH], [MATH], and those for the collision-induced population relaxations were taken equal for simplicity (cf. Ref. [CITATION]).', '1012.2247-2-12-0': 'The propagation equation for the positive frequency part of the probe field, slowly varying in time, i.e. when [MATH], reads in the Fourier picture, with the Fourier frequency variable identified with the probe detuning [EQUATION] with the medium susceptibility for the probe given by [EQUATION] where [MATH] is the number of atoms per unit volume.', '1012.2247-2-12-1': 'The time limit means that we need to find the steady state solutions to the Bloch equations.', '1012.2247-2-13-0': 'The assumption that the probe and trigger are weak with respect to the control field [MATH] allows us to expand the susceptibility into the Taylor series: [EQUATION]', '1012.2247-2-13-1': 'The explicit form of the Taylor expansion coefficients can be found in Appendix A.', '1012.2247-2-13-2': 'The term [MATH] corresponds to the linear susceptibility, while the third-order terms [MATH] and [MATH] represent the self- and cross-Kerr effect, respectively.', '1012.2247-2-13-3': 'The latter is of particular importance, as we are interested in the cross-phase modulation.', '1012.2247-2-14-0': 'We now write the control field as: [EQUATION] where [MATH] of constant values are its forward and backward propagating parts.', '1012.2247-2-14-1': 'As the field is periodic in space, so are the medium optical properties.', '1012.2247-2-14-2': 'Therefore we expand the susceptibilities for both probe and trigger into the Fourier series: [EQUATION]', '1012.2247-2-14-3': 'The pulses propagating in a periodic medium acquire their reflected components.', '1012.2247-2-14-4': 'In the two-mode approximation (for details see [CITATION]) we write: [EQUATION] where [MATH] are slowly varying in space.', '1012.2247-2-15-0': 'We now insert Eqs. ([REF],[REF]) into Eq. ([REF]) and drop terms rapidly oscillating in space.', '1012.2247-2-15-1': 'To write the propagation equations in the final form we make use of the relation [MATH] and set [MATH].', '1012.2247-2-15-2': 'For the probe we find: [EQUATION] where [EQUATION] with [MATH].', '1012.2247-2-15-3': 'The trigger equations are obtained by interchanging the indices [MATH].', '1012.2247-2-15-4': 'Explicit formulae for the susceptibilities Fourier components as well as their detailed deriviation can be found in Appendix A.', '1012.2247-2-16-0': 'As we will change the probe detuning [MATH] in a wide range while keeping the trigger detuning [MATH] constant, we introduce an asymmetry in the populations [MATH] and [MATH] even for [MATH].', '1012.2247-2-16-1': 'Additionally, the symmetry may be spoiled by the fact that [MATH] (the energy of the state [MATH]) is slightly smaller than [MATH], and in a cold medium transitions from [MATH] to [MATH] are possible while those from [MATH] to [MATH] are not.', '1012.2247-2-16-2': 'In previous works (see Refs. [CITATION]) the probe and trigger fields were only allowed detunings such that [MATH].', '1012.2247-2-16-3': 'In that case the stationary population distribution was assumed to be [MATH].', '1012.2247-2-16-4': 'Such an approach is simple, but it is well justified only in the resonance region of frequencies.', '1012.2247-2-16-5': 'As we show in the section [REF], for a wider range of pulse detunings it is necessary to make a better estimation of the population distribution.', '1012.2247-2-16-6': 'In Appendix B we present the way to evaluate the populations in the case of a quasi-standing coupling beam ([MATH], as in Ref. [CITATION]).', '1012.2247-2-17-0': '# Results and discussion', '1012.2247-2-18-0': 'In this section we present numerical results illustrating the cross-Kerr effect in our system and its dependence on the system parameters.', '1012.2247-2-18-1': 'In particular, we discuss how the cross-phase modulation is affected by the population redistribution.', '1012.2247-2-19-0': 'We have solved the propagation equations ([REF],[REF]) with the boundary conditions corresponding to both probe and trigger originally propagating towards positive [MATH]: [EQUATION] where [MATH] describes the amplitude of the incoming probe/trigger pulse and [MATH] is the sample length.', '1012.2247-2-19-1': "We have used the susceptibilities' Fourier components as presented in Appendix A.", '1012.2247-2-19-2': 'The level populations have been evaluated as described in Appendix B, using the initial amplitudes of the probe and trigger fields as input data.', '1012.2247-2-19-3': 'Note that, strictly speaking, the populations in the standing wave case are also modulated in space.', '1012.2247-2-19-4': 'However, as discussed below, the latter effect should not play a very important role in the case of an quasi-standing wave.', '1012.2247-2-20-0': 'The equations are solved iteratively until self-consistency is achieved, alternately for the probe and trigger, with the susceptibilities calculated using the fields obtained in the preceding step.', '1012.2247-2-20-1': 'In the first iteration for the probe the initial constant value of the trigger has been used.', '1012.2247-2-20-2': 'The numerical task in each step is thus solving a linear equation.', '1012.2247-2-20-3': 'We find two independent solutions with one-point boundary conditions and combine them to obtain the solution with the proper two-point boundary conditions given by Eqs. ([REF]).', '1012.2247-2-21-0': 'The transmission and reflection coefficients for the probe and trigger beams are defined by: [EQUATION] while the phases of the transmitted and reflected fields are given by: [EQUATION]', '1012.2247-2-21-1': 'Unless otherwise specified, we use the following set of input data which are of order of those of typical atomic systems: [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] MHz, [MATH] mm, [MATH] C[MATH]m, [MATH] cm[MATH].', '1012.2247-2-21-2': 'Our choosing a quasi-standing wave is connected with the fact that for a perfect one the steady state population in the node regions is trapped in the state [MATH] due to relaxation effects and the medium becomes transparent to both probe and trigger.', '1012.2247-2-21-3': 'Therefore, it is reasonable to only consider the cases in which the control field at the quasinodes is still strong enough to pump the population from the level [MATH].', '1012.2247-2-22-0': 'We first calculate the corrected values of the populations which are shown in Fig. [REF] in a narrow (plot (a)) and wide (plot (b)) ranges of the probe detunings.', '1012.2247-2-22-1': 'Note that the populations [MATH] and [MATH] may significantly differ from [MATH].', '1012.2247-2-22-2': 'They may vary rapidly within the transparency window and the maximum value of [MATH] may be close to unity which apparently occurs when the frequency of the probe suits the energy interval between [MATH] and the energy of one of the lower states dressed by [MATH] and [MATH].', '1012.2247-2-22-3': 'Between the maxima there is a deep minimum of [MATH] due to the population trapping in the state [MATH].', '1012.2247-2-22-4': 'The populations of [MATH] and [MATH] are negligible, which is consistent with our assumption.', '1012.2247-2-23-0': 'If there are no relaxations in the system, then the steady state solution for the no-trigger case is trivial: [MATH], [MATH] due to spontaneous emission from the upper level to all the lower levels.', '1012.2247-2-23-1': 'There is no mechanism yet pumping the population out from [MATH].', '1012.2247-2-23-2': 'The nonnegligible relaxations provide such a mechanism and the plot illustrating the [MATH] dependence on the probe detuning is of a similar shape as in the trigger-present case (not shown).', '1012.2247-2-24-0': 'The phase shifts for a running control field ([MATH] MHz, [MATH], the other parameters unchanged) are shown in Fig. [REF]a.', '1012.2247-2-24-1': 'We compare the results for the transmitted probe beam in the cases of equal or corrected values of the populations [MATH] and [MATH] and for the trigger pulse present or absent.', '1012.2247-2-24-2': 'The values of the obtained phase shifts [MATH] differ significantly for the two approximations concerning the populations, except for three points: when the calculated population distribution is almost exact: [MATH] and within the resonance region [MATH].', '1012.2247-2-24-3': 'The trigger-induced (Kerr) phase shifts (the differences between the values given by pairs of curves) in the two cases are also different and there are frequency intervals in which the Kerr shifts are significantly larger in the case in which the populations have been calculated according to Eqs. ([REF]).', '1012.2247-2-24-4': 'In this way the results of Ref. [CITATION] can be generalized for a wider range of the probe detunings.', '1012.2247-2-25-0': 'In Fig. [REF]b we make a similar comparison for a reflected beam in the case of a quasi - standing control field ([MATH] MHz, [MATH] MHz, the other parameters unchanged).', '1012.2247-2-25-1': 'Again taking into account the population redistribution (lower plot) caused a qualitative change of the behaviour of the phase shifts as well as of their differences (the Kerr shifts) compared with the case of equal populations.', '1012.2247-2-25-2': 'The results are in agreement at the same three points as before.', '1012.2247-2-25-3': 'However, at [MATH], due to a numerical instability in the algorithm used to find the solutions of Eq. ([REF]), the calculated density matrix is no longer positive definite and our results in this very narrow spectral region are unphysical (see the sharp spikes appearing in Fig. [REF]).', '1012.2247-2-26-0': "In Fig. [REF] we show the cross-Kerr phase shifts [MATH] due to the trigger's presence for both the transmitted and reflected probe beams defined as [MATH](trigger on)[MATH](trigger off).", '1012.2247-2-26-1': 'One can see that the shifts may be of order of one radian and there is a frequency range in which they do not change rapidly.', '1012.2247-2-26-2': 'The narrow minima or maxima in Fig. [REF] correspond to the situation in which the phases [MATH] vary rapidly.', '1012.2247-2-26-3': 'The present results show how our system behaves as an all-optically controlled Kerr medium.', '1012.2247-2-26-4': 'In particular, it may serve as a tunable beam splitter for the probe beam in which the phases of the reflected and/or trasmitted beam are significantly affected by the trigger beam.', '1012.2247-2-26-5': 'For example, for detunings of [MATH] MHz or of [MATH] MHz at which both the transmission and reflection coefficients are large (see Fig. [REF]), the trigger induced phase shift of the trasmitted and reflected probe beams are both appreciable.', '1012.2247-2-26-6': 'On the contrary, for a detuning of [MATH] MHz at which both the transmission and reflection coefficients are considerable (see Fig. [REF]), while the trigger induced phase shift for the transmitted probe beam is significant, that of the reflected beam is negligible.', '1012.2247-2-26-7': 'This beam splitting action accompanied by a trigger induced phase shift of the reflected and/or transmitted probe components is a unique feature of the tripod standing wave EIT configuration with respect to others previously considered.', '1012.2247-2-26-8': 'Taking into account the redistribution of the population among the atomic levels is however crucial.', '1012.2247-2-26-9': 'If the redistribution were ignored the phase shifts could only be estimated within rather limited ranges of frequencies.', '1012.2247-2-26-10': 'In particular, while for [MATH] MHz the approximation [MATH] is still reasonable as shown in Fig. [REF], for [MATH] MHz or [MATH] MHz it fails.', '1012.2247-2-26-11': 'In the following, in order to illustrate the dependence of the Kerr effect on various parameters, we discuss several numerical results obtained over a large detuning range using the method here developed to calculate the population redistribution.', '1012.2247-2-27-0': "We now check how the phase shifts of both the transmitted field and reflected one depend on 'how much standing' the control field is.", '1012.2247-2-27-1': 'We change the right-propagating part of the control field [MATH] while keeping the left-moving part constant: [MATH] MHz.', '1012.2247-2-27-2': 'All the other parameters remain unchanged.', '1012.2247-2-27-3': 'The results are shown in Fig. [REF]a for the transmitted beam and Fig. [REF]b for the reflected one.', '1012.2247-2-27-4': 'All the plots cross in the resonance region where the phase shift is small.', '1012.2247-2-27-5': 'The phase shift [MATH] reaches a maximal value for some [MATH] which depends on [MATH].', '1012.2247-2-27-6': "The more intense is the control field the flatter is the plot and the larger is the peak's shift towards higher frequencies, which provides a way of controlling the phases.", '1012.2247-2-27-7': 'We find the same effect of the peak moving to the right when decreasing the left-propagating part of the control field [MATH] while keeping [MATH] constant (not shown).', '1012.2247-2-27-8': 'The more intense is the control field the smaller is the trigger-induced phase shift.', '1012.2247-2-27-9': 'Around the resonance the reflected phase shift [MATH] shows an oscillatory dependence on the probe detuning.', '1012.2247-2-27-10': "The width of the frequency range where the oscillations are present grows with [MATH] but the number of peaks remains constant - again the phase shift's plot becomes flatter when the control field is increased.", '1012.2247-2-27-11': 'The curves are complicated now but roughly we can say that the trigger-induced phase shift decreases when the control field becomes stronger.', '1012.2247-2-27-12': 'The extrema of the Kerr phase shifts [MATH] in the intervals in which [MATH] vary rapidly become less prominent and the distance between them increases for growing control fields.', '1012.2247-2-27-13': 'In general a too strong control field is not advantegous for generating considerable trigger-induced phase shifts.', '1012.2247-2-27-14': 'Then the results gradually turn into those typical of the usual EIT case.', '1012.2247-2-28-0': 'As expected, the transmission coefficient (see Fig. [REF]a) grows with [MATH] and so does the width of the transparency window.', '1012.2247-2-28-1': 'Again the plots corresponding to the reflection are more complex, as shown in Fig. [REF]b.', '1012.2247-2-28-2': 'In general the reflection coefficient decreases when [MATH] is increased, but there are some frequency ranges where the dependence is more complicated.', '1012.2247-2-28-3': 'Yet if both parts of the control field are increased simultaneously, so that [MATH] is constant, then not only the transmission but also the reflection coefficient increases for most of the probe frequencies.', '1012.2247-2-28-4': 'This is due to the absorption cancellation by a strong control field.', '1012.2247-2-29-0': 'We have also checked how our results depend on the relaxation rates [MATH], [MATH], due to interatomic collisions.', '1012.2247-2-29-1': 'The main observation is that the trigger-induced phase shift decreases for growing [MATH].', '1012.2247-2-29-2': "On the other hand, as the relaxation rates grow, the frequency range where the reflected field's phase dependence is oscillatory becomes wider and the number of oscillations increases.", '1012.2247-2-29-3': 'As expected, the transmission and reflection coefficients decrease in general when the relaxation rates grow, but there are some frequency ranges where the reflection coefficient is a nonmonotonic function of [MATH].', '1012.2247-2-29-4': 'Populations [MATH] and [MATH] are negligible except for very large values of the relaxations ([MATH] MHz) when they are of the order of a few percent.', '1012.2247-2-29-5': 'For increasing [MATH] the phase shifts become less steep and an approximation consisting in adopting constant values of the populations [MATH] and [MATH] may be better justified.', '1012.2247-2-30-0': 'Increasing the length of the sample leads to increasing the trigger-induced phase shift of the transmitted field.', '1012.2247-2-30-1': 'For the considered lengths of the sample the nonlinear phase shift increases for growing [MATH].', '1012.2247-2-30-2': 'However, for an extended sample both pulses become partially absorbed and their mutual impact is smaller.', '1012.2247-2-30-3': "Elongating the sample leads to no significant change in the amplitude of oscillations in the reflected field's phase but it does influence the reflection coefficient in a nonmonotonic way, which provides another way of controlling the cross-Kerr effect.", '1012.2247-2-30-4': 'Thus, in order to have a considerable phase shift the coupling field cannot be too strong and the relaxation rates between the lower tripod states should not be too large; also the length of the sample should be appropriately optimized.', '1012.2247-2-31-0': '# Conclusions', '1012.2247-2-32-0': 'We have presented a comprehensive study of the cross Kerr effect in the propagation of two weak laser fields, a trigger and a probe, in a medium of four-level atoms in the tripod configuration, dressed by a third strong coupling field in the quasi-standing wave configuration.', '1012.2247-2-32-1': 'This has been done by taking into the proper account the population redistribution.', '1012.2247-2-32-2': 'Using the relevant terms of the Fourier expansion of the linear and nonlinear susceptibilities, we have numerically solved the propagation equations for the right- (incoming) and left- (reflected) running components of the two weak fields.', '1012.2247-2-32-3': 'In particular, the phase shifts of the transmitted and reflected probe induced by the trigger turn out to be as large as one radian.', '1012.2247-2-32-4': 'We have further shown that the population redistribution significantly affects the phase shifts of the transmitted and reflected beams unless very specific values of probe detuning are used.', '1012.2247-2-32-5': 'Over a wide range of probe detunings, on the contrary, we find that the trigger induced phase shifts for the reflected and transmitted probe can be flexibly and independently changed.', '1012.2247-2-32-6': "We have finally characterized the Kerr effect's dependence on the strength of the coupling field, the relaxation rates and the sample's length.", '1012.2247-2-33-0': 'In summary, by using a quasi-standing wave configuration the amplitudes of the transmitted and reflected parts of the probe beam and their respective trigger-induced phase shifts can be controlled and optimized by acting on various parameters, and especially on the probe detuning.', '1012.2247-2-33-1': 'While previous work has shown the usefulness of the tripod configuration in a running wave EIT regime to achieve a large cross-Kerr effect on the transmitted probe beam, our results extend such findings to the standing wave EIT regime in which both the reflected and transmitted probe beams can be separately controlled.', '1012.2247-2-34-0': 'K. Slowik is grateful to G. C. La Rocca and Scuola Normale Superiore in Pisa for their kind hospitality.', '1012.2247-2-34-1': 'The work of K. Slowik was sponsored by the scholarship for doctoral students ZPORR 2008/2009 of the Marshal of the Kuyavian-Pomeranian Voivodeship.', '1012.2247-2-34-2': 'Financial support from grant Azione integrata IT09L244H5 of MIUR is also acknowledged.', '1012.2247-2-35-0': 'APPENDIX A', '1012.2247-2-36-0': 'The medium susceptibilities can be written as: [EQUATION] and can be calculated with the use of the steady state solutions of the Bloch equations.', '1012.2247-2-37-0': 'From the last three of Eqs. ([REF]) we find the steady state spin coherences: [EQUATION]', '1012.2247-2-37-1': 'We substitute these expressions into the stationary form of the equations [MATH] in Eqs. ([REF]) and thus obtain a set of three coupled equations for [MATH], [MATH], [MATH].', '1012.2247-2-37-2': 'As the control field is much stronger than any other field present in the system, it prevents the states [MATH] and [MATH] from being populated.', '1012.2247-2-37-3': 'Therefore we set [MATH], as was done in Ref. [CITATION].', '1012.2247-2-37-4': 'However, contrary to Ref. [CITATION], here we do not assume the probe and trigger detunings to be almost equal.', '1012.2247-2-37-5': 'Hence the population distribution between the states [MATH] and [MATH] may be asymmetric and in general [MATH].', '1012.2247-2-37-6': 'This crucial issue has been discussed in Section [REF].', '1012.2247-2-37-7': 'Next we eliminate [MATH] and arrive at: [EQUATION] where [MATH].', '1012.2247-2-38-0': 'We combine the above expressions to obtain [MATH] and [MATH].', '1012.2247-2-38-1': 'The solutions of the above set of equations expanded into the Taylor series have the form of Eq. ([REF]), with: [EQUATION]', '1012.2247-2-38-2': 'The trigger susceptibilities are obtained by interchanging the indices [MATH].', '1012.2247-2-39-0': 'For the control field being in the form of a quasi-standing wave [MATH] the medium susceptibilities can be then expanded into the Fourier series: [EQUATION] and similarly for [MATH], etc.', '1012.2247-2-39-1': 'To find the explicit form of the Fourier coefficients we rewrite the expressions ([REF] - [REF]): [EQUATION] where: [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH].', '1012.2247-2-40-0': 'Let us denote: [EQUATION]', '1012.2247-2-40-1': 'The above integrals have been calculated with the use of the residues method.', '1012.2247-2-40-2': 'The Fourier coefficients of the susceptibilities read then: [EQUATION]', '1012.2247-2-40-3': 'As shown above, only these coefficients are present in the pulse propagation equations.', '1012.2247-2-40-4': 'Note that the corresponding formulae given in Ref. [CITATION] included some misprints.', '1012.2247-2-41-0': 'APPENDIX B', '1012.2247-2-42-0': 'To evaluate the population redistribution we proceed as follows.', '1012.2247-2-42-1': 'We first use equations 2-4 of the stationary version of Eqs. (1) to express [MATH] through [MATH] and [MATH]: [EQUATION] where [MATH], [MATH].', '1012.2247-2-42-2': 'Note that here we do not assume any of the populations negligible.', '1012.2247-2-42-3': 'Next we express [MATH] through the diagonal elements of [MATH] in the lowest-order approximation with respect to the probe and trigger: [EQUATION] where [MATH].', '1012.2247-2-42-4': 'Combining the two latter steps we obtain a set of linear equations for the diagonal elements: [EQUATION] where: [EQUATION]', '1012.2247-2-42-5': 'The solutions of Eqs. ([REF]) are [EQUATION] while the value of [MATH] follows from the probability conservation.', '1012.2247-2-42-6': 'The denominator [MATH] reads: [EQUATION]', '1012.2247-2-42-7': 'Using the above equations with constant initial values of probe and trigger fields and a given value of the control field we are able to estimate the population distribution.', '1012.2247-2-43-0': '# References'}

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'1012.2247-2-7-1'], ['1012.2247-1-8-2', '1012.2247-2-7-2'], ['1012.2247-1-8-3', '1012.2247-2-7-3'], ['1012.2247-1-9-0', '1012.2247-2-8-0'], ['1012.2247-1-9-2', '1012.2247-2-8-2'], ['1012.2247-1-9-3', '1012.2247-2-8-3'], ['1012.2247-1-9-5', '1012.2247-2-8-5'], ['1012.2247-1-31-0', '1012.2247-2-28-0'], ['1012.2247-1-31-1', '1012.2247-2-28-1'], ['1012.2247-1-31-2', '1012.2247-2-28-2'], ['1012.2247-1-31-4', '1012.2247-2-28-4'], ['1012.2247-1-47-1', '1012.2247-2-37-1'], ['1012.2247-1-47-2', '1012.2247-2-37-2'], ['1012.2247-1-47-3', '1012.2247-2-37-3'], ['1012.2247-1-47-4', '1012.2247-2-37-4'], ['1012.2247-1-47-5', '1012.2247-2-37-5'], ['1012.2247-1-47-6', '1012.2247-2-37-6'], ['1012.2247-1-47-7', '1012.2247-2-37-7'], ['1012.2247-1-25-0', '1012.2247-2-21-0'], ['1012.2247-1-13-0', '1012.2247-2-11-0'], ['1012.2247-1-13-1', '1012.2247-2-11-1'], ['1012.2247-1-46-0', '1012.2247-2-36-0'], ['1012.2247-1-28-1', '1012.2247-2-25-2'], ['1012.2247-1-48-0', '1012.2247-2-38-0'], ['1012.2247-1-48-1', '1012.2247-2-38-1'], ['1012.2247-1-48-2', '1012.2247-2-38-2'], ['1012.2247-1-34-6', '1012.2247-2-22-3'], ['1012.2247-1-16-1', '1012.2247-2-14-1'], ['1012.2247-1-16-2', '1012.2247-2-14-2'], ['1012.2247-1-16-3', '1012.2247-2-14-3'], ['1012.2247-1-12-0', '1012.2247-2-10-0'], ['1012.2247-1-12-1', '1012.2247-2-10-1'], ['1012.2247-1-24-1', '1012.2247-2-20-0'], ['1012.2247-1-24-2', '1012.2247-2-20-1'], ['1012.2247-1-24-3', '1012.2247-2-20-2'], ['1012.2247-1-24-4', '1012.2247-2-20-3'], ['1012.2247-1-23-1', '1012.2247-2-18-1'], ['1012.2247-1-14-4', '1012.2247-2-12-1'], ['1012.2247-1-21-1', '1012.2247-2-42-1'], ['1012.2247-1-21-2', '1012.2247-2-42-2'], ['1012.2247-1-21-3', '1012.2247-2-42-3'], ['1012.2247-1-21-4', '1012.2247-2-42-4'], ['1012.2247-1-21-5', '1012.2247-2-42-5'], ['1012.2247-1-21-6', '1012.2247-2-42-6'], ['1012.2247-1-21-7', '1012.2247-2-42-7'], ['1012.2247-1-27-1', '1012.2247-2-27-2'], ['1012.2247-1-29-3', '1012.2247-2-27-1'], ['1012.2247-1-29-7', '1012.2247-2-27-5'], ['1012.2247-1-29-8', '1012.2247-2-27-6'], ['1012.2247-1-29-9', '1012.2247-2-27-7'], ['1012.2247-1-29-10', '1012.2247-2-27-8'], ['1012.2247-1-30-3', '1012.2247-2-27-10'], ['1012.2247-1-30-4', '1012.2247-2-27-11'], ['1012.2247-1-30-5', '1012.2247-2-27-13'], ['1012.2247-1-33-0', '1012.2247-2-29-3'], ['1012.2247-1-37-1', '1012.2247-2-30-1'], ['1012.2247-1-37-2', '1012.2247-2-30-2'], ['1012.2247-1-20-4', '1012.2247-2-16-4'], ['1012.2247-1-35-2', '1012.2247-2-23-2'], ['1012.2247-1-15-2', '1012.2247-2-13-1'], ['1012.2247-1-15-3', '1012.2247-2-13-2'], ['1012.2247-1-10-0', '1012.2247-2-9-0'], ['1012.2247-1-10-2', '1012.2247-2-9-2'], ['1012.2247-1-18-4', '1012.2247-2-15-4'], ['1012.2247-1-8-0', '1012.2247-2-7-0'], ['1012.2247-1-9-1', '1012.2247-2-8-1'], ['1012.2247-1-9-4', '1012.2247-2-8-4'], ['1012.2247-1-31-3', '1012.2247-2-28-3'], ['1012.2247-1-47-0', '1012.2247-2-37-0'], ['1012.2247-1-28-3', '1012.2247-2-25-3'], ['1012.2247-1-6-1', '1012.2247-2-5-3'], ['1012.2247-1-34-5', '1012.2247-2-22-2'], ['1012.2247-1-16-0', '1012.2247-2-14-0'], ['1012.2247-1-16-4', '1012.2247-2-14-4'], ['1012.2247-1-23-0', '1012.2247-2-18-0'], ['1012.2247-1-3-2', '1012.2247-2-3-2'], ['1012.2247-1-21-0', '1012.2247-2-42-0'], ['1012.2247-1-21-8', '1012.2247-2-19-3'], ['1012.2247-1-21-9', '1012.2247-2-19-4'], ['1012.2247-1-29-0', '1012.2247-2-27-0'], ['1012.2247-1-29-5', '1012.2247-2-27-4'], ['1012.2247-1-30-2', '1012.2247-2-27-9'], ['1012.2247-1-30-6', '1012.2247-2-27-14'], ['1012.2247-1-32-5', '1012.2247-2-29-2'], ['1012.2247-1-37-0', '1012.2247-2-30-0'], ['1012.2247-1-38-0', '1012.2247-2-30-3'], ['1012.2247-1-40-0', '1012.2247-2-32-0'], ['1012.2247-1-40-4', '1012.2247-2-32-4'], ['1012.2247-1-0-0', '1012.2247-2-0-0'], ['1012.2247-1-0-1', '1012.2247-2-0-1'], ['1012.2247-1-43-0', '1012.2247-2-33-0'], ['1012.2247-1-18-2', '1012.2247-2-15-2'], ['1012.2247-1-25-4', '1012.2247-2-21-1'], ['1012.2247-1-34-2', '1012.2247-2-22-4'], ['1012.2247-1-34-3', '1012.2247-2-22-1'], ['1012.2247-1-26-1', '1012.2247-2-24-2'], ['1012.2247-1-5-0', '1012.2247-2-4-0'], ['1012.2247-1-5-1', '1012.2247-2-4-2'], ['1012.2247-1-3-1', '1012.2247-2-3-1'], ['1012.2247-1-3-3', '1012.2247-2-3-3'], ['1012.2247-1-3-4', '1012.2247-2-3-4'], ['1012.2247-1-14-3', '1012.2247-2-12-0'], ['1012.2247-1-27-3', '1012.2247-2-27-3'], ['1012.2247-1-32-0', '1012.2247-2-29-0'], ['1012.2247-1-32-4', '1012.2247-2-29-1'], ['1012.2247-1-42-1', '1012.2247-2-30-4'], ['1012.2247-1-2-0', '1012.2247-2-2-2'], ['1012.2247-1-2-0', '1012.2247-2-2-3'], ['1012.2247-1-2-1', '1012.2247-2-2-4'], ['1012.2247-1-2-1', '1012.2247-2-2-10'], ['1012.2247-1-4-0', '1012.2247-2-2-5'], ['1012.2247-1-4-1', '1012.2247-2-2-6'], ['1012.2247-1-4-2', '1012.2247-2-2-8'], ['1012.2247-1-4-4', '1012.2247-2-2-9'], ['1012.2247-1-4-5', '1012.2247-2-2-9'], ['1012.2247-1-4-7', '1012.2247-2-2-11']]

[['1012.2247-1-20-0', '1012.2247-2-16-0'], ['1012.2247-1-20-1', '1012.2247-2-16-1'], ['1012.2247-1-20-2', '1012.2247-2-16-2'], ['1012.2247-1-20-3', '1012.2247-2-16-3'], ['1012.2247-1-20-5', '1012.2247-2-16-5'], ['1012.2247-1-35-0', '1012.2247-2-23-0'], ['1012.2247-1-35-1', '1012.2247-2-23-1'], ['1012.2247-1-49-0', '1012.2247-2-39-0'], ['1012.2247-1-49-1', '1012.2247-2-39-1'], ['1012.2247-1-44-0', '1012.2247-2-34-0'], ['1012.2247-1-44-1', '1012.2247-2-34-1'], ['1012.2247-1-50-0', '1012.2247-2-40-0'], ['1012.2247-1-50-1', '1012.2247-2-40-1'], ['1012.2247-1-50-2', '1012.2247-2-40-2'], ['1012.2247-1-50-3', '1012.2247-2-40-3'], ['1012.2247-1-15-1', '1012.2247-2-13-0'], ['1012.2247-1-15-4', '1012.2247-2-13-3'], ['1012.2247-1-10-1', '1012.2247-2-9-1'], ['1012.2247-1-10-3', '1012.2247-2-9-3'], ['1012.2247-1-10-4', '1012.2247-2-9-4'], ['1012.2247-1-18-0', '1012.2247-2-15-0'], ['1012.2247-1-18-1', '1012.2247-2-15-1'], ['1012.2247-1-18-3', '1012.2247-2-15-3'], ['1012.2247-1-8-1', '1012.2247-2-7-1'], ['1012.2247-1-8-2', '1012.2247-2-7-2'], ['1012.2247-1-8-3', '1012.2247-2-7-3'], ['1012.2247-1-9-0', '1012.2247-2-8-0'], ['1012.2247-1-9-2', '1012.2247-2-8-2'], ['1012.2247-1-9-3', '1012.2247-2-8-3'], ['1012.2247-1-9-5', '1012.2247-2-8-5'], ['1012.2247-1-31-0', '1012.2247-2-28-0'], ['1012.2247-1-31-1', '1012.2247-2-28-1'], ['1012.2247-1-31-2', '1012.2247-2-28-2'], ['1012.2247-1-31-4', '1012.2247-2-28-4'], ['1012.2247-1-47-1', '1012.2247-2-37-1'], ['1012.2247-1-47-2', '1012.2247-2-37-2'], ['1012.2247-1-47-3', '1012.2247-2-37-3'], ['1012.2247-1-47-4', '1012.2247-2-37-4'], ['1012.2247-1-47-5', '1012.2247-2-37-5'], ['1012.2247-1-47-6', '1012.2247-2-37-6'], ['1012.2247-1-47-7', '1012.2247-2-37-7'], ['1012.2247-1-25-0', '1012.2247-2-21-0'], ['1012.2247-1-13-0', '1012.2247-2-11-0'], ['1012.2247-1-13-1', '1012.2247-2-11-1'], ['1012.2247-1-46-0', '1012.2247-2-36-0'], ['1012.2247-1-28-1', '1012.2247-2-25-2'], ['1012.2247-1-48-0', '1012.2247-2-38-0'], ['1012.2247-1-48-1', '1012.2247-2-38-1'], ['1012.2247-1-48-2', '1012.2247-2-38-2'], ['1012.2247-1-34-6', '1012.2247-2-22-3'], ['1012.2247-1-16-1', '1012.2247-2-14-1'], ['1012.2247-1-16-2', '1012.2247-2-14-2'], ['1012.2247-1-16-3', '1012.2247-2-14-3'], ['1012.2247-1-12-0', '1012.2247-2-10-0'], ['1012.2247-1-12-1', '1012.2247-2-10-1'], ['1012.2247-1-24-1', '1012.2247-2-20-0'], ['1012.2247-1-24-2', '1012.2247-2-20-1'], ['1012.2247-1-24-3', '1012.2247-2-20-2'], ['1012.2247-1-24-4', '1012.2247-2-20-3'], ['1012.2247-1-23-1', '1012.2247-2-18-1'], ['1012.2247-1-14-4', '1012.2247-2-12-1'], ['1012.2247-1-21-1', '1012.2247-2-42-1'], ['1012.2247-1-21-2', '1012.2247-2-42-2'], ['1012.2247-1-21-3', '1012.2247-2-42-3'], ['1012.2247-1-21-4', '1012.2247-2-42-4'], ['1012.2247-1-21-5', '1012.2247-2-42-5'], ['1012.2247-1-21-6', '1012.2247-2-42-6'], ['1012.2247-1-21-7', '1012.2247-2-42-7'], ['1012.2247-1-27-1', '1012.2247-2-27-2'], ['1012.2247-1-29-3', '1012.2247-2-27-1'], ['1012.2247-1-29-7', '1012.2247-2-27-5'], ['1012.2247-1-29-8', '1012.2247-2-27-6'], ['1012.2247-1-29-9', '1012.2247-2-27-7'], ['1012.2247-1-29-10', '1012.2247-2-27-8'], ['1012.2247-1-30-3', '1012.2247-2-27-10'], ['1012.2247-1-30-4', '1012.2247-2-27-11'], ['1012.2247-1-30-5', '1012.2247-2-27-13'], ['1012.2247-1-33-0', '1012.2247-2-29-3'], ['1012.2247-1-37-1', '1012.2247-2-30-1'], ['1012.2247-1-37-2', '1012.2247-2-30-2']]

[['1012.2247-1-20-4', '1012.2247-2-16-4'], ['1012.2247-1-35-2', '1012.2247-2-23-2'], ['1012.2247-1-15-2', '1012.2247-2-13-1'], ['1012.2247-1-15-3', '1012.2247-2-13-2'], ['1012.2247-1-10-0', '1012.2247-2-9-0'], ['1012.2247-1-10-2', '1012.2247-2-9-2'], ['1012.2247-1-18-4', '1012.2247-2-15-4'], ['1012.2247-1-8-0', '1012.2247-2-7-0'], ['1012.2247-1-9-1', '1012.2247-2-8-1'], ['1012.2247-1-9-4', '1012.2247-2-8-4'], ['1012.2247-1-31-3', '1012.2247-2-28-3'], ['1012.2247-1-47-0', '1012.2247-2-37-0'], ['1012.2247-1-28-3', '1012.2247-2-25-3'], ['1012.2247-1-6-1', '1012.2247-2-5-3'], ['1012.2247-1-34-5', '1012.2247-2-22-2'], ['1012.2247-1-16-0', '1012.2247-2-14-0'], ['1012.2247-1-16-4', '1012.2247-2-14-4'], ['1012.2247-1-23-0', '1012.2247-2-18-0'], ['1012.2247-1-3-2', '1012.2247-2-3-2'], ['1012.2247-1-21-0', '1012.2247-2-42-0'], ['1012.2247-1-21-8', '1012.2247-2-19-3'], ['1012.2247-1-21-9', '1012.2247-2-19-4'], ['1012.2247-1-29-0', '1012.2247-2-27-0'], ['1012.2247-1-29-5', '1012.2247-2-27-4'], ['1012.2247-1-30-2', '1012.2247-2-27-9'], ['1012.2247-1-30-6', '1012.2247-2-27-14'], ['1012.2247-1-32-5', '1012.2247-2-29-2'], ['1012.2247-1-37-0', '1012.2247-2-30-0'], ['1012.2247-1-38-0', '1012.2247-2-30-3']]

[]

[['1012.2247-1-40-0', '1012.2247-2-32-0'], ['1012.2247-1-40-4', '1012.2247-2-32-4'], ['1012.2247-1-0-0', '1012.2247-2-0-0'], ['1012.2247-1-0-1', '1012.2247-2-0-1'], ['1012.2247-1-43-0', '1012.2247-2-33-0'], ['1012.2247-1-18-2', '1012.2247-2-15-2'], ['1012.2247-1-25-4', '1012.2247-2-21-1'], ['1012.2247-1-34-2', '1012.2247-2-22-4'], ['1012.2247-1-34-3', '1012.2247-2-22-1'], ['1012.2247-1-26-1', '1012.2247-2-24-2'], ['1012.2247-1-5-0', '1012.2247-2-4-0'], ['1012.2247-1-5-1', '1012.2247-2-4-2'], ['1012.2247-1-3-1', '1012.2247-2-3-1'], ['1012.2247-1-3-3', '1012.2247-2-3-3'], ['1012.2247-1-3-4', '1012.2247-2-3-4'], ['1012.2247-1-14-3', '1012.2247-2-12-0'], ['1012.2247-1-27-3', '1012.2247-2-27-3'], ['1012.2247-1-32-0', '1012.2247-2-29-0'], ['1012.2247-1-32-4', '1012.2247-2-29-1'], ['1012.2247-1-42-1', '1012.2247-2-30-4'], ['1012.2247-1-2-0', '1012.2247-2-2-2'], ['1012.2247-1-2-0', '1012.2247-2-2-3'], ['1012.2247-1-2-1', '1012.2247-2-2-4'], ['1012.2247-1-2-1', '1012.2247-2-2-10'], ['1012.2247-1-4-0', '1012.2247-2-2-5'], ['1012.2247-1-4-1', '1012.2247-2-2-6'], ['1012.2247-1-4-2', '1012.2247-2-2-8'], ['1012.2247-1-4-4', '1012.2247-2-2-9'], ['1012.2247-1-4-5', '1012.2247-2-2-9'], ['1012.2247-1-4-7', '1012.2247-2-2-11']]

[]

['1012.2247-1-45-0', '1012.2247-2-35-0', '1012.2247-2-41-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1012.2247

1507.02705

{'1507.02705-1-0-0': 'We have used a combination of numerical modeling and experiments to study carbon etching in the presence of a hydrogen plasma.', '1507.02705-1-0-1': 'We model the evolution of a low density EUV-induced plasma during and after the EUV pulse to obtain the energy resolved ion fluxes from the plasma to the surface.', '1507.02705-1-0-2': 'By relating the computed ion fluxes to the experimentally observed etching rate at various pressures and ion energies, we show that at low pressure and energy, carbon etching is due to chemical sputtering, while at high pressure and energy a reactive ion etching process is likely to dominate.', '1507.02705-1-1-0': '# Introduction', '1507.02705-1-2-0': 'Many vacuum mirror optical tools suffer from the build-up of carbon contamination due to cracking of hydrocarbons under powerful vacuum ultraviolet radiation [CITATION].', '1507.02705-1-2-1': 'The large absorption of EUV radiation by carbon becomes significant in the case of multi-element optical systems, where throughput can be greatly reduced by even a very thin layer of carbon contamination on the top of each mirror.', '1507.02705-1-3-0': 'The problem of EUV induced carbon contamination has been addressed in a series of publications [CITATION].', '1507.02705-1-3-1': 'The reduction of carbon films in a hydrogen atmosphere or hydrogen plasma has also been extensively studied [CITATION].', '1507.02705-1-3-2': 'Despite numerous studies, however, it is still difficult to predict the carbon removal rate, because there are many contributing factors.', '1507.02705-1-3-3': 'Several aspects that significantly affect the carbon removal rate include: many different allotropes and compounds of the carbon (e.g. soft black or hard graphite), many different contributing reaction paths (e.g. physical sputtering, chemical sputtering, reactive ion etching etc.), and, last but not least, small admixtures to the background gas, which, while residing on the carbon surface, can produce reactive species once irradiated by EUV.', '1507.02705-1-3-4': 'Despite this complexity, experiments have shown that carbon etching can be achieved under certain EUV-induced plasma conditions.', '1507.02705-1-3-5': 'Nevertheless, it has proven difficult to fully understand the etch process, because the characteristics of the EUV-induced plasma are poorly known, and the plasma-surface interaction has many contributing factors [CITATION].', '1507.02705-1-4-0': 'In this paper, we use a model of the EUV-induced plasma to numerically analyse the fluxes from the plasma to the sample surface.', '1507.02705-1-4-1': 'Our model is a self-consistent 2D Particle-in-Cell model of the weakly ionized low pressure hydrogen plasma that is formed during the EUV pulse due to ionization by EUV photons and secondary electrons from the surface.', '1507.02705-1-4-2': 'As described in Section [REF], our model provides an accurate estimate the ion flux composition and energy distribution.', '1507.02705-1-4-3': 'However, a considerable number of parameters required for accurate simulations are not well known, therefore, the modeling results were combined with experimental observations.', '1507.02705-1-4-4': 'With the aid of simulations, we show that the shape of the energy distribution function of the ion fluxes in the considered experimental setup are mainly defined by the setup geometry, background pressure, and externally applied bias voltage.', '1507.02705-1-4-5': 'The ion dose, on the other hand is sensitive to the variations of many other parameters (e.g. EUV dose, secondary electron yield, etc.).', '1507.02705-1-5-0': 'By combining the computed energy distribution function of the ion flux with the experimentally measured ion dose, insight into the mechanism for carbon removal was gained.', '1507.02705-1-5-1': 'By analysing the differences in yield between EUV-induced plasma and surface wave discharge plasma experiments in combination with numerical simulations, we show that chemical sputtering dominates for low pressures and energies.', '1507.02705-1-5-2': 'It was found that the carbon removal yield for both the surface wave discharge and EUV-induced plasmas was similar in the overlapping energy range.', '1507.02705-1-5-3': 'Hence, the effect of the EUV radiation on carbon removal should be significantly smaller than was estimated previously [CITATION].', '1507.02705-1-6-0': '# Experimental setup', '1507.02705-1-7-0': 'The ISAN EUV experimental setup is based on a tin EUV radiation source, which is a Z-pinch discharge plasma with [MATH]Hz repetition rate, which has been described in detail elsewhere [CITATION].', '1507.02705-1-7-1': 'In brief, EUV radiation is introduced into a so-called "clean" chamber (see Fig. [REF]), separated from the source and collector optics by a Si:Zr spectral purity filter (SPF).', '1507.02705-1-7-2': 'The clean chamber is differentially pumped to pressure of [MATH] torr.', '1507.02705-1-7-3': 'Under vacuum conditions, the background hydrocarbon carbon growth rate was measured to be 0.4[MATH]0.2 nm/10 Mshot.', '1507.02705-1-8-0': 'The diameter of the EUV beam at the sample was 5 mm.', '1507.02705-1-8-1': 'In addition to the direct beam, some scattered EUV radiation was also incident on the sample.', '1507.02705-1-8-2': 'The EUV pulse duration is about 100 ns (FWHM), with a tail, as described in [CITATION].', '1507.02705-1-9-0': 'The incident EUV power was measured using a sensitive thermo-couple attached to a thin copper disk.', '1507.02705-1-9-1': 'It was found that the radiation intensity was [MATH] W/cm[MATH] after the SPF and approximately [MATH] W/cm[MATH] without the SPF.', '1507.02705-1-9-2': 'The ratio of EUV intensities with and without the SPF corresponds to the calculated transmission of the 100 nm Si:Zr SPF filter.', '1507.02705-1-10-0': 'The EUV intensity on the sample decreases with time because of carbon growth on the SPF filter and focusing optics.', '1507.02705-1-10-1': 'In later experiments, the EUV intensity was measured to be 0.1 W/cm[MATH] after the SPF.', '1507.02705-1-11-0': 'During the experiments, the hydrogen pressure was set in the range of 2.8 Pa - 86.5 Pa.', '1507.02705-1-11-1': 'To ensure that each radiation pulse excited a plasma in an atmosphere dominated by hydrogen, hydrogen flowed through a liquid nitrogen trap (to remove water) and into the chamber at 100 liters[MATH]torr/minute .', '1507.02705-1-12-0': 'To control the energy of the ion flux, the sample holder assembly was biased in the range of -200 - 0 V, while all other metallic electrodes were grounded.', '1507.02705-1-12-1': 'The samples consisted of carbon, deposited by magnetron sputtering on a silicon wafer to a thickness of [MATH]30 nm.', '1507.02705-1-12-2': 'Each sample was exposed to the 10[MATH] EUV pulses at a different combination of bias and hydrogen pressure.', '1507.02705-1-12-3': 'After exposure, the amount of carbon removed was measured by X-Ray florescence (XRF) (EDS) analysis and spectroscopic ellipsometry.', '1507.02705-1-13-0': '# Model', '1507.02705-1-14-0': 'A two dimensional particle in cell (PIC) model with [MATH] geometry was used to model the experiment.', '1507.02705-1-14-1': 'Our model follows the general PIC scheme, described elsewhere [CITATION].', '1507.02705-1-15-0': 'Ionization is initiated by an EUV pulse, which directly ionizes the background gas, and produces electrons by photo emission from the SPF and the sample.', '1507.02705-1-15-1': 'This process leads to the formation of an EUV induced plasma in the chamber.', '1507.02705-1-15-2': 'The ionization and photo-emission process and their inclusion in the model is described below.', '1507.02705-1-16-0': 'Although the plasma is continually re-ignited by the pulsed EUV radiation, we consider each EUV pulse to induce a plasma in a cold neutral gas that is at equilibrium, because the characteristic time of plasma decay in the given geometry is about [MATH], which is much shorter than the 660 [MATH]s between pulses.', '1507.02705-1-16-1': 'The restoration of thermal equilibrium in the background gas is also much faster than pulse repetition rate.', '1507.02705-1-17-0': '## chamber configuration', '1507.02705-1-18-0': 'The configuration of vacuum chamber is presented in Fig. [REF].', '1507.02705-1-18-1': 'Because the chamber is axially symmetric, it is possible to include the structure of the internal chamber in the simulations.', '1507.02705-1-18-2': 'To accurately model the plasma dynamics when the sample is biased, the space between the SPF and the metal grid is included in simulations.', '1507.02705-1-18-3': 'The grid is included in the model as a number of metallic rings.', '1507.02705-1-18-4': 'When a bias is applied to the sample, ions from the plasma, which is formed between SPF and metal grid, are pulled towards the sample.', '1507.02705-1-19-0': '## Deielectric model', '1507.02705-1-20-0': 'In the experiment, a dielectric mica diaphragm was used to decrease electron emission from the sample holder structure.', '1507.02705-1-20-1': 'In the model, this feature is included with a simple dielectric model: a dielectric is not conductive, it can accumulate charge and the secondary electron yield under EUV irradiation is order of magnitude lower than for the sample.', '1507.02705-1-21-0': 'The electron reflection and secondary electron emission under electron impact (SEE) are included in the model.', '1507.02705-1-21-1': 'These two processes are combined into SEE with the probability ([MATH]) defined as follows.', '1507.02705-1-21-2': '[EQUATION]', '1507.02705-1-21-3': 'Here E - is the energy of incoming electron.', '1507.02705-1-21-4': 'The parameters of the used mica are unknown, therefore, we choses [MATH] eV and [MATH], these parameters corresponds to linear fit for SEE yield for quarts in [CITATION].', '1507.02705-1-22-0': 'This approximation blends between significant slow electron backscattering for low energy electrons and secondary electron emission for higher impact energies.', '1507.02705-1-22-1': 'We do not use higher order approximations, since both effects are known to depend strongly on the surface conditions [CITATION].', '1507.02705-1-23-0': 'This model for the dielectric is over simplified, but sufficient to allow us to to take the out of focus EUV radiation, which is incident on the diaphragm, into account in a consistent manner.', '1507.02705-1-24-0': '## Length scales and grid resolution', '1507.02705-1-25-0': 'The time evolution of one EUV pulse has two distinct stages: negative space-charge dominated during the beginning of the EUV pulse, and decay of the positively charged plasma after the pulse.', '1507.02705-1-26-0': 'In order to model the negative space-charge dominated part of the plasma evolution, the potential well near the surface, which is formed by the electrons that have escaped to the volume, must be resolved.', '1507.02705-1-26-1': 'The length scale of the space-charge potential well depends on the energy distribution of photoelectrons from the surface and on the current which passes through the system.', '1507.02705-1-27-0': 'For the purpose of estimation, it is possible to simplify the formulas from [CITATION] and obtain [EQUATION]', '1507.02705-1-27-1': 'Here, [MATH] is the initial temperature of the emitted photoelectrons, [MATH] is the current density near the cathode, and [MATH] is the distance from the cathode to the bottom of the space charge potential well.', '1507.02705-1-27-2': 'For [MATH]1 eV and [MATH]20 mA/cm[MATH], one obtains [MATH]0.02 cm.', '1507.02705-1-28-0': 'For the plasma dominated part of the plasma evolution we need to resolve the Debye length in the volume, and also have a finer grid near the surface to resolve the plasma sheath.', '1507.02705-1-29-0': 'From the EUV intensity ([MATH]0.02 mJ per pulse) and background hydrogen pressure (2.8 Pa, 11.2 Pa), the plasma density can be estimated to be about 10[MATH] cm[MATH].', '1507.02705-1-29-1': 'For [MATH]0.5 eV and [MATH] 1/cm[MATH] one obtains [MATH]0.015 cm, which is comparable to the space charge length scale estimated above.', '1507.02705-1-29-2': 'To achieve this, the grid is refined near the sample surface and SPF.', '1507.02705-1-29-3': 'The minimum grid cell size is [MATH], and is gradually increased in steps of 5% until the bulk cell size of [MATH] is reached.', '1507.02705-1-29-4': 'Several tests were performed to ensure that the chosen grid resolution does not affect the plasma dynamics.', '1507.02705-1-30-0': '## photo-electron emission', '1507.02705-1-31-0': 'The energy spectrum of photoelectrons emitted from the SPF and sample surface are calculated from [CITATION]: [EQUATION]', '1507.02705-1-31-1': 'Here [MATH] is the probability of emitting an electron with energy, [MATH], from a surface with a work function, [MATH].', '1507.02705-1-31-2': 'For carbon [MATH] eV [CITATION].', '1507.02705-1-31-3': 'For electrons emitted from the surface, we assume an angular dependence given by a cosine emission law [CITATION].', '1507.02705-1-31-4': 'For purpose of estimation, we used the same approximation for the spectrum of photoelectrons emitted from the dielectric.', '1507.02705-1-32-0': '## EUV spectrum and photoionization', '1507.02705-1-33-0': 'To calculate the distribution of ion species, the spectrum of the EUV radiation must be included in the model, along with the energy-dependent cross sections for each photoionization process.', '1507.02705-1-33-1': 'Direct photoionisation of hydrogen by EUV photons is included as two processes: [EQUATION]', '1507.02705-1-33-2': 'The photoionisation cross-sections were taken from [CITATION].', '1507.02705-1-34-0': 'In the experiment, the EUV spectrum was measured and found to be in agreement with data from [CITATION].', '1507.02705-1-34-1': 'The EUV spectrum, however, was measured before reflection from the Zr collector mirrors, and transmission through the SPF.', '1507.02705-1-35-0': 'The calculated transparency curve that was provided with the Si:Zr SPF was used to calculate the EUV spectrum after transmission through the SPF (See Fig. [REF]).', '1507.02705-1-36-0': 'The SPF transmission curve has a transmission of about 1% in the range of 20 - 30 eV, which is important because the photoabsorption cross-section is very large in this energy range (see Fig. [REF]).', '1507.02705-1-37-0': 'The contribution due to radiation in the 20 - 30 eV range is difficult to quantify for two reasons: the intensity of the radiation varies significantly due to carbon growth on the SPF, and the accuracy of the SPF transmission curve in this energy range is unknown.', '1507.02705-1-37-1': 'Nevertheless, we include it because, even a small transmission will result in additional photoinization, which leads to a reduced space-charge potential barrier at low pressures (e.g. 2.8 Pa, 11.2 Pa).', '1507.02705-1-38-0': '## Cross-sections set', '1507.02705-1-39-0': 'The cross-sections set used in the model consists of electron collisions with hydrogen and ion collisions with hydrogen.', '1507.02705-1-39-1': 'The collisions between plasma species and three body processes are neglected due to their low probability under the conditions considered here.', '1507.02705-1-39-2': 'To accurately model electron collision related processes in hydrogen discharges with a Monte Carlo (MC) model, one needs to take into account the differential cross-sections for ionisation and excitations processes.', '1507.02705-1-39-3': 'As described in detail in [CITATION], the particular choice of the angular dependence of cross-sections significantly influences the simulation results.', '1507.02705-1-39-4': 'We adopt a set of cross sections found in Ref. [CITATION].', '1507.02705-1-40-0': 'We use an approach similar to that described in [CITATION].', '1507.02705-1-40-1': 'The reaction probability is sampled using the integrated reaction cross-section.', '1507.02705-1-40-2': 'The differential cross-section data is used to determine the collision kinematics and energy redistribution between products.', '1507.02705-1-40-3': 'Electron elastic scattering and hydrogen electronic excitations, and angular scattering data is taken from [CITATION].', '1507.02705-1-40-4': 'For electron impact ionization of hydrogen, we use an experimentally determined doubly differential cross-section [CITATION].', '1507.02705-1-40-5': 'The set of cross-sections for collisions between ions and hydrogen is based on [CITATION] because this set provides good agreement with swarm data for ions in hydrogen.', '1507.02705-1-40-6': 'We neglect the formation of H[MATH], because the cross-section of dissociative electron attachment is very low, and the density of vibrationally exited hydrogen molecules too low to make a significant contribution to the production of H[MATH].', '1507.02705-1-41-0': 'We make use of the procedure described in [CITATION] to perform Monte Carlo collisions with the background gas.', '1507.02705-1-41-1': 'We tested the consistency of our implementation by modeling swarm experiments and found good agreement with experimental values [CITATION] for the first Townsend electron ionization coefficient, the electron mobility, and for H[MATH] and H[MATH] mobility in hydrogen [CITATION].', '1507.02705-1-42-0': '# Analysis of the charge - bias characteristic', '1507.02705-1-43-0': 'Let us begin with the analysis of the charge - bias voltage characteristics (Q - V), which were measured during the experiments.', '1507.02705-1-43-1': 'The ion dose on the sample per pulse for the range of pressures and bias voltages considered here was estimated from the experimental results as a difference between the collected charge for given pressure and bias voltage and the collected charge for vacuum conditions [CITATION].', '1507.02705-1-43-2': 'The measured curves are presented in Fig. [REF].', '1507.02705-1-43-3': 'It is worth noting that these [MATH] characteristics have some peculiarities.', '1507.02705-1-44-0': 'Firstly, all the characteristics reach saturation, and the turning point for the almost all curves is approximately -75 V.', '1507.02705-1-44-1': 'If there is a significant contribution from ionization by the accelerated secondary electrons from the sample in the drift regime, one would expect a rapid increase of the collected charge for biases in the range of -200 .', '1507.02705-1-44-2': '-50 V.', '1507.02705-1-45-0': 'Secondly, the increase of the collected charge for -200 V bias does not depend linearly on pressure in the range of 2.8 Pa to 22.4 Pa, i.e. the collected charge increases less than two-fold while the pressure increases eight times.', '1507.02705-1-45-1': 'But, one would expect at least linear growth of the collected charge with pressure if the volume ionization provides a significant contribution to the total collected charge.', '1507.02705-1-45-2': 'Thus, the overall contribution of photoionization in the volume is small compared with the other factors.', '1507.02705-1-46-0': 'Thirdly, for -200 V bias and 22.4 Pa background pressure, due to the photoionization process, all the external bias would be applied over a very small layer near the sample, thus, all the secondary electrons will gain 200 eV energy.', '1507.02705-1-46-1': 'The H[MATH] ionization cross-section for this energy is approximately [MATH] cm[MATH], thus, in the space between the sample and the grid ([MATH] cm), these electrons will, on average, have 0.8 ionization inducing collisions.', '1507.02705-1-46-2': 'If we assume that, under vacuum conditions and -200 V bias, the SE current from the sample saturates, and we obtain that, just due to the direct electron induced ionization, without cascade process and photoionization, the collected charge should be greater than 1.8 nC.', '1507.02705-1-46-3': 'in the experiment, however, the collected charge is approximately 1.9 nC.', '1507.02705-1-47-0': '## Average secondary electron yield', '1507.02705-1-48-0': 'The above analysis suggests that the combination of EUV power per pulse, and effective secondary electron yield (SEY) produces approximately 1 nC of electrons from the sample.', '1507.02705-1-49-0': 'It is instructive to estimate the total SE charge from the experimental parameters.', '1507.02705-1-49-1': 'From the EUV intensity, repetition rate and spot size, the average dose per pulse was approximately [MATH]0.017 mJ.', '1507.02705-1-49-2': 'The secondary electron yield for carbon under EUV radiation is estimated to be approximately [MATH] for the photon energy range of 60 eV - 100 eV [CITATION], which leads to a saturation charge of 2 nC, which would lead to a significant disagreement between the simulated and measured charge bias characteristics.', '1507.02705-1-49-3': 'It is also worth noting that effective SEY from mica was small, since, in the experiment, the charge collected from a sample made from mica was measured to be an order of magnitude smaller than for the carbon sample.', '1507.02705-1-50-0': 'However, the measured charge bias characteristics are the only experimental data that provides a reference point for the simulations of the experiment dynamics.', '1507.02705-1-50-1': 'Since the spatial distribution of EUV intensity is subject to an unknown systematic measurement error, and the SEY is known to vary widely, depending on the surface conditions, we chose to keep the product of incident EUV and SEY a constant, chosen to provide 1 nC total SE charge from the sample.', '1507.02705-1-51-0': '## Role of dielectric ring', '1507.02705-1-52-0': 'During the simulations we found that the mica accumulated charge over many EUV pulses, significantly effecting the local field distribution, and, hence changing the flux incident on the sample.', '1507.02705-1-52-1': 'If charging is neglected, the Q - V response under vacuum and 2.8 Pa conditions cannot be reproduced for any reasonable parameter values.', '1507.02705-1-52-2': 'The discrepancy is caused by the potential barrier near the sample surface, which is created by the negative space charge generated during the EUV pulse.', '1507.02705-1-52-3': 'Although the potential is low, it produces a significant effect because the SE energy spectrum ([REF]) is strongly peaked at rather low energies: approximately [MATH] eV.', '1507.02705-1-53-0': 'The charge accumulation on the mica significantly increases the local field strength near the sample surface (see Fig. [REF]).', '1507.02705-1-53-1': 'This removes the space charge potential barrier for biases of -100V bias and higher under the conditions that we consider.', '1507.02705-1-54-0': 'The charging of the mica should saturate, leading to an unchanging charge density distribution on the mica.', '1507.02705-1-54-1': 'To estimate the charge density, we simulated one hundred pulses under vacuum conditions for all experimentally applied bias values.', '1507.02705-1-54-2': 'In these calculations, the effective SEY from the mica is assumed to be about 0.001, which corresponds to the experimentally measured value.', '1507.02705-1-55-0': 'This approach corresponds to the experimental procedure, since the Q - V characteristics were measured via averaging a large number of pulses for every combination of bias and pressure.', '1507.02705-1-55-1': 'No special means were used in the experiment to remove the accumulated charge from the mica between pulses.', '1507.02705-1-56-0': 'For negative biases, simulations show that the mica potential rapidly reaches approximately 0 V potential.', '1507.02705-1-56-1': 'For these conditions, further charging is very slow, because (a) the electric field strength near the mica is very small, therefore, even a small amount of emitted charge creates a space charge potential barrier, and, (b), because the mica potential is close to zero, there is a small flux of electrons from the sample to the mica.', '1507.02705-1-57-0': 'The continued slow variation in charge distribution is impractical to simulate, however.The final charge distribution requires a very long time to calculate and is sensitive to the combination of the SEY from both the sample and the mica, and to the corresponding SE energy distribution functions.', '1507.02705-1-57-1': 'Nevertheless, the additional errors due to these limitations are expected to be small.', '1507.02705-1-57-2': "In practice the maximum potential of the mica under vacuum conditions is bounded, since, as the mica's potential increases to [MATH]10 V, the electron current from the sample to mica becomes significant.", '1507.02705-1-58-0': 'It is worth noting that the EUV plasma itself can contribute to charging the mica, since, for stationary discharges, the dielectric charges to the plasma potential.', '1507.02705-1-58-1': 'Despite the the fact that the potential of the EUV-induced plasma can be high (e.g. 20 - 30 V or more after EUV pulse), it rapidly decreases to several volts due to electron cooling due to collisions with the background gas.', '1507.02705-1-58-2': 'Most probably, the mica potential would converge to the time averaged potential of the plasma, e.g. several volts.', '1507.02705-1-58-3': 'Therefore, in the following results, the mica was pre-charged to the value that was obtained from simulations under vacuum conditions for the appropriate bias.', '1507.02705-1-59-0': 'The optimum parameter combination that reproduces the experimentally measured Q - V characteristics were chosen as follows: the mica was pre-charged as described above, the EUV intensity was kept constant at the measured value, but the amount of the scattered EUV radiation incident on the sample holder was chosen so that 40[MATH] of EUV radiation was incident on the dielectric mica, and the effective SEY from the sample was kept at to 0.01 for the photon energy range of 60 eV - 100 eV, while the SE contribution from the VUV part of the spectrum was neglected.', '1507.02705-1-59-1': 'The simulated Q - V characteristic, after parameter optimization, and comparison with the measurements are presented in Fig. [REF].', '1507.02705-1-60-0': '# Ion fluxes to the sample surface', '1507.02705-1-61-0': 'The removal of carbon from the surface material should be directly dependent on the energy distribution function (EDF) of the ion flux incident on the surface.', '1507.02705-1-61-1': 'The conditions of the experimental study of carbon cleaning in the ISAN EUV experiment are summarized in Table [REF].', '1507.02705-1-61-2': 'We computed the plasma conditions, ion fluxes, and ion EDF for these conditions.', '1507.02705-1-61-3': 'The simulated time for all cases was 10 [MATH]s. For 2.8 Pa - 11.2 Pa and biases -200 V - -100 V, the plasma had completely decayed and all ions were collected during the simulation time.', '1507.02705-1-61-4': 'But, for the 60 Pa and 86.5 Pa cases some plasma was still left in the simulation domain.', '1507.02705-1-62-0': 'The EDFs, integrated over the simulation window, and averaged over the sample surface, are presented in Fig. [REF].', '1507.02705-1-62-1': 'The composition of the computed ion flux depends significantly on the pressure.', '1507.02705-1-62-2': 'For the 3 Pa case, the main ion is H[MATH], because the characteristic time for H[MATH] to H[MATH] conversion in 3 Pa H[MATH] is approximately 0.5 [MATH]s. With increasing pressure, H[MATH] becomes the main ion, as expected.', '1507.02705-1-62-3': 'There is also a non-negligible contribution from fast H[MATH] which is produced due to a resonant charge exchange reaction.', '1507.02705-1-63-0': 'The maximum ion energy is larger than the applied bias voltage due to the build-up of the plasma potential.', '1507.02705-1-63-1': 'But the number of such fast ions is small, due to the fast plasma decay.', '1507.02705-1-63-2': 'For the same reason, the most energetic ion is H[MATH] because its small mass allows it to accelerate during the decay of the plasma potential.', '1507.02705-1-64-0': 'It was observed, that contrary to the ion dose, the shape of the ion flux EDF was mainly defined by bias, pressure and the accumulated charge on the dielectric.', '1507.02705-1-64-1': 'However, as discussed in previous section, we expect that the equilibrium mica potential does not significantly differ from 0 V. For small variations of the mica potential (if all other parameters are the same), the variations of the EDF shape are small for all simulations where a sample bias was applied.', '1507.02705-1-64-2': 'Moreover, the EDF was barely sensitive to reasonable variations of other parameters.', '1507.02705-1-65-0': 'This insensitivity is a consequence of the experimental procedure.', '1507.02705-1-65-1': 'The applied bias allows ions to be collected from the entire chamber volume.', '1507.02705-1-65-2': 'Hence, the characteristics of the plasma are important only for the short time when the sheath between plasma and sample is small.', '1507.02705-1-65-3': 'As ions are collected at the sample, the sheath size becomes larger and larger.', '1507.02705-1-65-4': 'Therefore, as time progresses, the instantantaneous ion flux EDF starts to depend only on the bias potential, mica potential, and the distance that the ions travel between the plasma and the sample.', '1507.02705-1-65-5': 'This is because these parameters determine the maximum possible energy of the ions, while the background pressure determines the number of collisions (e.g energy loss), and, hence, the average energy of the ion at the sample.', '1507.02705-1-65-6': 'Therefore, for the considered conditions, the shape of the ion flux EDF is mostly determined by the bias, mica potential, pressure, and the chamber geometry.', '1507.02705-1-66-0': 'It is notable, that the mica charge, due to exposure to EUV radiation and plasma, leads to the ion flux being focused on the sample (see Fig. [REF]).', '1507.02705-1-66-1': 'The ion flux to the mica itself becomes negligible compared to the simulation with zero charge on the mica.', '1507.02705-1-66-2': 'Therefore, the measured ion doses should represent accurately the ion doses on the sample.', '1507.02705-1-67-0': 'Hence, for the analysis of the experiments, it is reasonable to take the computed EDF shape, normed to the experimentally measured ion dose (if it is available).', '1507.02705-1-67-1': 'This approach allows us to compare the experimental results at different pressures.', '1507.02705-1-68-0': '# Discussion', '1507.02705-1-69-0': 'The maximum etched depths and total etched volumes per exposure of 10[MATH] pulses for different pressures and biases are presented in Table. [REF].', '1507.02705-1-69-1': 'The etched volumes are the integrals over etch profiles, which were determined by XRF measurement (see Fig. [REF]).', '1507.02705-1-69-2': 'In the cases of 60 Pa and 86.5 Pa pressure, the etch profile was only measured at the center of EUV spot.', '1507.02705-1-69-3': 'We assumed, for these two cases, a uniform etching profile to estimate the carbon removal yield.', '1507.02705-1-69-4': 'Hence, for these cases, the amount of removed carbon is most probably overestimated.', '1507.02705-1-70-0': 'As was discussed in the previous section, we used the experimentally measured ion doses for yield estimation.', '1507.02705-1-70-1': 'However, for some experiments, the ion dose was not measured.', '1507.02705-1-70-2': 'In these cases, we used the ion doses obtained from simulation results.', '1507.02705-1-70-3': 'This approach is valid at high pressures because the uncertainty in the simulations is on the order of 30% (see Fig. [REF]).', '1507.02705-1-71-0': 'For all samples, the average carbon removal yield was larger than 0.2 carbon atoms per ion.', '1507.02705-1-71-1': 'Thus, physical sputtering is not the main process, since for energies below 200 eV, the expected sputtering yield is lower than [MATH] carbon atoms per hydrogen ion [CITATION].', '1507.02705-1-71-2': 'Therefore, carbon is removed via a chemical sputtering process, or a reactive ion etching process, because the effective yield for these processes is known to be large under certain conditions [CITATION].', '1507.02705-1-72-0': 'However, the results for -200 V bias, especially at low pressure (2.8 - 11.2 Pa), cannot be described by a chemical sputtering process, since the average yield is larger than one carbon atom per ion.', '1507.02705-1-72-1': 'Hence, reactive ion etching may be responsible for these very high etch rates.', '1507.02705-1-72-2': 'However, for reactive ion etching there should be weakly bound radicals on the top of the carbon, which are desorbed from surface due to ion impact, which, in turn, lead to a very high yield per ion.', '1507.02705-1-72-3': 'Although, we have no data about these radicals, it is likely to be some form of methane radical (e.g., CH[MATH]).', '1507.02705-1-72-4': 'The alternative: oxidation (e.g., CO and CO[MATH]) [CITATION] is unlikely because water, which is the main source of oxygen, is removed from the hydrogen flow, and, during experiments, the chamber walls are kept at liquid nitrogen temperatures, trapping the majority of the residual water.', '1507.02705-1-72-5': 'Hence, the contribution of water to the carbon removal was limited by the initial coverage which is typically sub mono layer for amorphous carbon for the considered temperature range [CITATION].', '1507.02705-1-73-0': 'In the case of experiments with -50 V and -100 V biases, the results can be compared with carbon etching in a surface wave discharge plasma [CITATION], where similar magnetron deposited carbon on silicon substrate samples were used.', '1507.02705-1-73-1': 'The carbon removal yield for experiments with biases of -50 V and -100 V was estimated by convolving the simulated EDF with the energy dependent yield, as measured in the SWD experiments (see Fig. [REF] and column "SWD recomputed" in table [REF]).', '1507.02705-1-73-2': 'In spite of H[MATH] being the main ion in SWD plasma, we used the same yield for estimation, since we do not expect dramatic difference between H[MATH] and H[MATH] for carbon removal.', '1507.02705-1-73-3': 'These estimates show that the yields found in EUV experiments with low biases agree with the recomputed yield from the SWD experiment.', '1507.02705-1-73-4': 'The margin of error is, however, large enough that we cannot completely exclude direct influence from EUV radiation.', '1507.02705-1-74-0': '# Conclusion', '1507.02705-1-75-0': 'The evolution of an EUV induced hydrogen plasma was simulated.', '1507.02705-1-75-1': 'The simulations, due to their close coupling to experimental conditions, allowed the magnitude, composition, and energy spectrum of the flux from the plasma to the surface to be estimated.', '1507.02705-1-75-2': 'This was used to quantitatively compare carbon etch rates between different EUV plasma conditions, as well as compare carbon etching under EUV-induced plasma to etching under SWD plasmas.', '1507.02705-1-76-0': 'It was observed that the carbon etching mechanism at low bias and pressure was different than that at high bias and pressure.', '1507.02705-1-76-1': 'For the higher energy range, the carbon removal yield in EUV-induced plasma was larger than one carbon atom per ion.', '1507.02705-1-76-2': 'Most probably, etching is dominated by a reactive ion etching process, which may be due to the production of methane radicals (e.g. CH[MATH]) that can desorb under energetic ion flux.', '1507.02705-1-77-0': 'However, at low bias energies, the etch rates found in EUV-induced plasma agree with those found in SWD plasma to within model and measurement uncertainties.', '1507.02705-1-77-1': 'Therefore, the role of the EUV radiation in a previous study [CITATION] was overestimated at low bias energies and pressures.', '1507.02705-1-78-0': 'However, our study does not exclude EUV enhancing etch rates at low biases or having a small ([MATH] C/ion) enhancing effect over the range of biases and pressures that we studied.', '1507.02705-1-78-1': 'Therefore, experiments to determine the effect of EUV in the limit of low-to-no bias are required.', '1507.02705-1-79-0': 'The authors would like to thank Tatiana Rakhimova for helpful discussion.', '1507.02705-1-80-0': 'This work is part of the research program “Controlling photon and plasma induced processes at EUV optical surfaces (CP3E)” of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)” which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).', '1507.02705-1-80-1': 'The CP3E programme is cofinanced by ASML (Veldhoven) and the AgentschapNL through the Catrene EXEPT program.'}

{'1507.02705-2-0-0': 'We have used a combination of numerical modeling and experiments to study carbon etching in the presence of a hydrogen plasma.', '1507.02705-2-0-1': 'We model the evolution of a low density EUV-induced plasma during and after the EUV pulse to obtain the energy resolved ion fluxes from the plasma to the surface.', '1507.02705-2-0-2': 'By relating the computed ion fluxes to the experimentally observed etching rate at various pressures and ion energies, we show that at low pressure and energy, carbon etching is due to chemical sputtering, while at high pressure and energy a reactive ion etching process is likely to dominate.', '1507.02705-2-1-0': '# Introduction', '1507.02705-2-2-0': 'Many vacuum mirror optical tools suffer from the build-up of carbon contamination due to cracking of hydrocarbons under powerful vacuum ultraviolet radiation [CITATION].', '1507.02705-2-2-1': 'The large absorption of EUV radiation by carbon becomes significant in the case of multi-element optical systems, where throughput can be greatly reduced by even a very thin layer of carbon contamination on the top of each mirror.', '1507.02705-2-3-0': 'The problem of EUV induced carbon contamination has been addressed in a series of publications [CITATION].', '1507.02705-2-3-1': 'The reduction of carbon films in a hydrogen atmosphere or hydrogen plasma has also been extensively studied [CITATION].', '1507.02705-2-3-2': 'Despite numerous studies, however, it is still difficult to predict the carbon removal rate, because there are many contributing factors.', '1507.02705-2-3-3': 'Several aspects that significantly affect the carbon removal rate include: many different allotropes and compounds of the carbon (e.g. soft black or hard graphite), many different contributing reaction paths (e.g. physical sputtering, chemical sputtering, reactive ion etching etc.), and, last but not least, small admixtures to the background gas, which, while residing on the carbon surface, can produce reactive species once irradiated by EUV.', '1507.02705-2-3-4': 'Despite this complexity, experiments have shown that carbon etching can be achieved under certain EUV-induced plasma conditions.', '1507.02705-2-3-5': 'Nevertheless, it has proven difficult to fully understand the etch process, because the characteristics of the EUV-induced plasma are poorly known, and the plasma-surface interaction has many contributing factors [CITATION].', '1507.02705-2-4-0': 'In this paper, we use a model of the EUV-induced plasma to numerically analyse the fluxes from the plasma to the sample surface.', '1507.02705-2-4-1': 'Our model is a self-consistent 2D Particle-in-Cell model of the weakly ionized low pressure hydrogen plasma that is formed during the EUV pulse due to ionization by EUV photons and secondary electrons from the surface.', '1507.02705-2-4-2': 'As described in Section [REF], our model provides an accurate estimate the ion flux composition and energy distribution.', '1507.02705-2-4-3': 'However, a considerable number of parameters required for accurate simulations are not well known, therefore, the modeling results were combined with experimental observations.', '1507.02705-2-4-4': 'With the aid of simulations, we show that the shape of the energy distribution function of the ion fluxes in the considered experimental setup are mainly defined by the setup geometry, background pressure, and externally applied bias voltage.', '1507.02705-2-4-5': 'The ion dose, on the other hand is sensitive to the variations of many other parameters (e.g. EUV dose, secondary electron yield, etc.).', '1507.02705-2-5-0': 'By combining the computed energy distribution function of the ion flux with the experimentally measured ion dose, insight into the mechanism for carbon removal was gained.', '1507.02705-2-5-1': 'By analysing the differences in yield between EUV-induced plasma and surface wave discharge plasma experiments in combination with numerical simulations, we show that chemical sputtering dominates for low pressures and energies.', '1507.02705-2-5-2': 'It was found that the carbon removal yield for both the surface wave discharge and EUV-induced plasmas was similar in the overlapping energy range.', '1507.02705-2-5-3': 'Hence, the effect of the EUV radiation on carbon removal is found to be significantly smaller than was estimated previously [CITATION].', '1507.02705-2-6-0': '# Experimental setup', '1507.02705-2-7-0': 'The ISAN EUV experimental setup is based on a tin EUV radiation source, which is a Z-pinch discharge plasma with [MATH] Hz repetition rate, which has been described in detail elsewhere [CITATION].', '1507.02705-2-7-1': 'This source is a good tool for exploring EUV-induced surface processes over a large number of pulses ([MATH] 1 MShot).', '1507.02705-2-7-2': 'In brief, EUV radiation is introduced into a so-called "clean" chamber (see Fig. [REF]), separated from the source and collector optics by a Si:Zr spectral purity filter (SPF).', '1507.02705-2-7-3': 'The clean chamber is differentially pumped to pressure of [MATH] torr.', '1507.02705-2-7-4': 'Under vacuum conditions, the background hydrocarbon carbon growth rate was measured to be 0.4[MATH]0.2 nm/10 Mshot.', '1507.02705-2-8-0': 'The diameter of the EUV beam at the sample was 5 mm.', '1507.02705-2-8-1': 'In addition to the direct beam, some scattered EUV radiation was also incident on the sample.', '1507.02705-2-8-2': 'The EUV pulse duration is about 100 ns (FWHM), with a tail, as described in [CITATION].', '1507.02705-2-9-0': 'The incident EUV power was measured using a sensitive thermo-couple attached to a thin copper disk.', '1507.02705-2-9-1': 'It was found that the radiation intensity was [MATH] W/cm[MATH] after the SPF and approximately [MATH] W/cm[MATH] without the SPF.', '1507.02705-2-9-2': 'The ratio of EUV intensities with and without the SPF corresponds to the calculated transmission of the 100 nm Si:Zr SPF filter over an energy range of 60 to 100 eV (see Fig [REF]).', '1507.02705-2-10-0': 'The EUV intensity on the sample decreases with time because of carbon growth on the SPF filter and focusing optics.', '1507.02705-2-10-1': 'In later experiments, the EUV intensity was measured to be 0.1 W/cm[MATH] after the SPF.', '1507.02705-2-11-0': 'During the experiments, the hydrogen pressure was set in the range of 2.8 Pa - 86.5 Pa.', '1507.02705-2-11-1': 'To ensure that each radiation pulse excited a plasma in an atmosphere dominated by hydrogen, hydrogen flowed through a liquid nitrogen trap (to remove water) and into the chamber at 100 liters[MATH]torr/minute .', '1507.02705-2-12-0': 'To control the energy of the ion flux, the sample holder assembly was biased in the range of -200 - 0 V, while all other metallic electrodes were grounded.', '1507.02705-2-12-1': 'The samples consisted of carbon, deposited by magnetron sputtering on a silicon wafer to a thickness of [MATH]30 nm.', '1507.02705-2-12-2': 'Each sample was exposed to the 10[MATH] EUV pulses at a different combination of bias and hydrogen pressure.', '1507.02705-2-12-3': 'After exposure, the amount of carbon removed was measured by X-Ray fluorescence (XRF) (EDS) analysis and spectroscopic ellipsometry.', '1507.02705-2-13-0': '# Model', '1507.02705-2-14-0': 'A two dimensional particle in cell (PIC) model with [MATH] geometry was used to model the experiment.', '1507.02705-2-14-1': 'Our model follows the general PIC scheme, described elsewhere [CITATION].', '1507.02705-2-15-0': 'Ionization is initiated by an EUV pulse, which directly ionizes the background gas, and produces electrons by photo emission from the SPF and the sample.', '1507.02705-2-15-1': 'This process leads to the formation of an EUV induced plasma in the chamber.', '1507.02705-2-15-2': 'The ionization and photo-emission process and their inclusion in the model is described below.', '1507.02705-2-16-0': 'Although the plasma is continually re-ignited by the pulsed EUV radiation, we consider each EUV pulse to induce a plasma in a cold neutral gas that is at equilibrium, because the characteristic time of plasma decay in the given geometry is about [MATH], which is much shorter than the 660 [MATH]s between pulses.', '1507.02705-2-16-1': 'The restoration of thermal equilibrium in the background gas is also much shorter then the time between pulses.', '1507.02705-2-17-0': '## Chamber configuration', '1507.02705-2-18-0': 'The configuration of vacuum chamber is presented in Fig. [REF].', '1507.02705-2-18-1': 'Because the chamber is axially symmetric, it is possible to include the structure of the internal chamber in the simulations.', '1507.02705-2-18-2': 'To accurately model the plasma dynamics when the sample is biased, the space between the SPF and the metal grid is included in our simulations.', '1507.02705-2-18-3': 'The grid is included in the model as a number of metallic rings.', '1507.02705-2-18-4': 'When a bias is applied to the sample, ions from the plasma, which is formed between SPF and metal grid, are pulled towards the sample.', '1507.02705-2-19-0': '## Deielectric model', '1507.02705-2-20-0': 'In the experiment, a dielectric mica diaphragm was used to prevent EUV radiation being incident on the sample holder structure, thereby decreasing electron emission from the sample holder structure.', '1507.02705-2-20-1': 'Therefore, allowing plasma parameters to be estimated from the discharge characteristics of the sample.', '1507.02705-2-20-2': 'In the model, this feature is included with a simple dielectric model: a dielectric is not conductive, it can accumulate charge and the secondary electron yield under EUV irradiation is order of magnitude lower than for the sample.', '1507.02705-2-21-0': 'The electron reflection and secondary electron emission under electron impact (SEE) are included in the model.', '1507.02705-2-21-1': 'These two processes are combined into SEE with the probability ([MATH]) defined as follows.', '1507.02705-2-21-2': '[EQUATION]', '1507.02705-2-21-3': 'Here E - is the energy of incoming electron.', '1507.02705-2-21-4': 'The parameters of the used mica are unknown, therefore, we choses [MATH] eV and [MATH], these parameters corresponds to linear fit for SEE yield for quarts in [CITATION].', '1507.02705-2-22-0': 'This approximation blends between significant slow electron backscattering for low energy electrons and secondary electron emission for higher impact energies.', '1507.02705-2-22-1': 'We do not use higher order approximations, since both effects are known to depend strongly on the surface conditions [CITATION].', '1507.02705-2-23-0': 'This model for the dielectric is over simplified, but sufficient to allow us to to take the out of focus EUV radiation, which is incident on the diaphragm, into account in a consistent manner.', '1507.02705-2-24-0': '## Length scales and grid resolution', '1507.02705-2-25-0': 'The time evolution of one EUV pulse has two distinct stages: negative space-charge dominated during the beginning of the EUV pulse, and decay of the positively charged plasma after the pulse [CITATION].', '1507.02705-2-26-0': 'In order to model the negative space-charge dominated part of the plasma evolution, the potential well near the surface, which is formed by the electrons that have escaped to the volume, must be resolved.', '1507.02705-2-26-1': 'The length scale of the space-charge potential well depends on the energy distribution of photoelectrons from the surface and on the current which passes through the system.', '1507.02705-2-27-0': 'For the purpose of estimation, it is possible to simplify the formulas from [CITATION] and obtain [EQUATION]', '1507.02705-2-27-1': 'Here, [MATH] is the initial temperature of the emitted photoelectrons, [MATH] is the current density near the cathode, and [MATH] is the distance from the cathode to the bottom of the space charge potential well.', '1507.02705-2-27-2': 'For [MATH]1 eV and [MATH]20 mA/cm[MATH], one obtains [MATH]0.02 cm.', '1507.02705-2-28-0': 'For the plasma dominated part of the plasma evolution we need to resolve the Debye length in the volume, and also have a finer grid near the surface to resolve the plasma sheath.', '1507.02705-2-29-0': 'From the EUV intensity ([MATH]0.02 mJ per pulse) and background hydrogen pressure (2.8 Pa, 11.2 Pa), the plasma density can be estimated to be about 10[MATH] cm[MATH].', '1507.02705-2-29-1': 'For [MATH]0.5 eV and [MATH] 1/cm[MATH] one obtains [MATH]0.015 cm, which is comparable to the space charge length scale estimated above.', '1507.02705-2-29-2': 'To achieve this, the grid is refined near the sample surface and SPF.', '1507.02705-2-29-3': 'The minimum grid cell size is [MATH], and is gradually increased in steps of 5% until the bulk cell size of [MATH] is reached.', '1507.02705-2-29-4': 'Several tests were performed to ensure that the chosen grid resolution does not affect the plasma dynamics.', '1507.02705-2-30-0': '## photo-electron emission', '1507.02705-2-31-0': 'The energy spectrum of photoelectrons emitted from the SPF and sample surface are calculated from [CITATION]: [EQUATION]', '1507.02705-2-31-1': 'Here [MATH] is the probability of emitting an electron with energy, [MATH], from a surface with a work function, [MATH].', '1507.02705-2-31-2': 'For carbon [MATH] eV [CITATION].', '1507.02705-2-31-3': 'For electrons emitted from the surface, we assume an angular dependence given by a cosine emission law [CITATION].', '1507.02705-2-31-4': 'For purpose of estimation, we used the same approximation for the spectrum of photoelectrons emitted from the dielectric.', '1507.02705-2-32-0': '## EUV spectrum and photoionization', '1507.02705-2-33-0': 'To calculate the distribution of ion species, the spectrum of the EUV radiation must be included in the model, along with the energy-dependent cross sections for each photoionization process.', '1507.02705-2-33-1': 'Direct photoionisation of hydrogen by EUV photons is included as two processes: [EQUATION]', '1507.02705-2-33-2': 'The photoionisation cross-sections were taken from [CITATION].', '1507.02705-2-34-0': 'In the experiment, the EUV spectrum was measured (green curve Fig. [REF]) and found to be in agreement with data from [CITATION].', '1507.02705-2-34-1': 'The EUV spectrum, however, was measured before reflection from the Zr collector mirrors, and transmission through the SPF.', '1507.02705-2-35-0': 'The calculated transparency curve that was provided with the Si:Zr SPF was used to calculate the EUV spectrum after transmission through the SPF (See Fig. [REF] blue curve).', '1507.02705-2-36-0': 'The SPF transmission curve has a transmission of about 1% in the range of 20 - 30 eV, which is important because the photoabsorption cross-section is very large in this energy range (see Fig. [REF] red curve).', '1507.02705-2-37-0': 'The contribution due to radiation in the 20 - 30 eV range is difficult to quantify for two reasons: the intensity of the radiation varies significantly due to carbon growth on the SPF, and the accuracy of the SPF transmission curve in this energy range is unknown.', '1507.02705-2-37-1': 'Nevertheless, we include it because, even a small transmission will result in additional photoinization, which leads to a reduced space-charge potential barrier at low pressures (e.g. 2.8 Pa, 11.2 Pa).', '1507.02705-2-38-0': 'Although the photoionization due to the 20 - 30 eV radiation range is significant compared to the 13 nm band, it has significance only for cases where no bias is applied to the sample.', '1507.02705-2-38-1': 'For cases with bias, the presence of this radiation leads to a decrease of the space-charge potential.', '1507.02705-2-39-0': '## Cross-sections set', '1507.02705-2-40-0': 'The cross-sections set used in the model consists of electron collisions with hydrogen and ion collisions with hydrogen.', '1507.02705-2-40-1': 'The collisions between plasma species and three body processes are neglected due to their low probability under the conditions considered here.', '1507.02705-2-40-2': 'To accurately model electron collision related processes in hydrogen discharges with a Monte Carlo (MC) model, one needs to take into account the differential cross-sections for ionisation and excitations processes.', '1507.02705-2-40-3': 'As described in detail in [CITATION], the particular choice of the angular dependence of cross-sections significantly influences the simulation results.', '1507.02705-2-40-4': 'We adopt a set of cross sections found in Ref. [CITATION].', '1507.02705-2-41-0': 'We use an approach similar to that described in [CITATION].', '1507.02705-2-41-1': 'The reaction probability is sampled using the integrated reaction cross-section.', '1507.02705-2-41-2': 'The differential cross-section data is used to determine the collision kinematics and energy redistribution between products.', '1507.02705-2-41-3': 'Electron elastic scattering and hydrogen electronic excitations, and angular scattering data is taken from [CITATION].', '1507.02705-2-41-4': 'For electron impact ionization of hydrogen, we use an experimentally determined doubly differential cross-section [CITATION].', '1507.02705-2-41-5': 'The set of cross-sections for collisions between ions and hydrogen is based on [CITATION] because this set provides good agreement with swarm data for ions in hydrogen.', '1507.02705-2-41-6': 'We neglect the formation of H[MATH], because the cross-section of dissociative electron attachment is very low, and the density of vibrationally exited hydrogen molecules too low to make a significant contribution to the production of H[MATH].', '1507.02705-2-42-0': 'We make use of the procedure described in [CITATION] to perform Monte Carlo collisions with the background gas.', '1507.02705-2-42-1': 'We tested the consistency of our implementation by modeling swarm experiments and found good agreement with experimental values [CITATION] for the first Townsend electron ionization coefficient, the electron mobility, and for H[MATH] and H[MATH] mobility in hydrogen [CITATION].', '1507.02705-2-43-0': '# Analysis of the charge - bias characteristic', '1507.02705-2-44-0': 'Let us begin with the analysis of the charge - bias voltage characteristics (Q - V), which were measured during the experiments.', '1507.02705-2-44-1': 'The ion dose on the sample per pulse for the range of pressures and bias voltages considered here was estimated from the experimental results as a difference between the collected charge for given pressure and bias voltage and the collected charge for vacuum conditions [CITATION].', '1507.02705-2-44-2': 'The measured curves are presented in Fig. [REF].', '1507.02705-2-44-3': 'It is worth noting that these [MATH] characteristics have some peculiarities.', '1507.02705-2-45-0': 'Firstly, all the characteristics reach saturation, and the turning point for the almost all curves is approximately -75 V.', '1507.02705-2-45-1': 'If there is a significant contribution from ionization by the accelerated secondary electrons from the sample in the drift regime, one would expect a rapid increase of the collected charge for biases in the range of -200 .', '1507.02705-2-45-2': '-50 V.', '1507.02705-2-46-0': 'Secondly, the increase of the collected charge for -200 V bias does not depend linearly on pressure in the range of 2.8 Pa to 22.4 Pa, i.e. the collected charge increases less than two-fold while the pressure increases eight times.', '1507.02705-2-46-1': 'But, one would expect at least linear growth of the collected charge with pressure if the volume ionization provides a significant contribution to the total collected charge.', '1507.02705-2-46-2': 'Thus, the overall contribution of photoionization in the volume is small compared with the other factors.', '1507.02705-2-47-0': 'Thirdly, for -200 V bias and 22.4 Pa background pressure, due to the photoionization process, all the external bias would be applied over a very small layer near the sample, thus, all the secondary electrons will gain 200 eV energy.', '1507.02705-2-47-1': 'The H[MATH] ionization cross-section for this energy is approximately [MATH] cm[MATH], thus, in the space between the sample and the grid ([MATH] cm), these electrons will, on average, have 0.8 ionization inducing collisions.', '1507.02705-2-47-2': 'If we assume that, under vacuum conditions and -200 V bias, the SE current from the sample saturates, and we obtain that, just due to the direct electron induced ionization, without cascade process and photoionization, the collected charge should be greater than 1.8 nC.', '1507.02705-2-47-3': 'in the experiment, however, the collected charge is approximately 1.9 nC.', '1507.02705-2-48-0': '## Average secondary electron yield', '1507.02705-2-49-0': 'The above analysis suggests that the combination of EUV power per pulse, and effective secondary electron yield (SEY) produces approximately 1 nC of electrons from the sample.', '1507.02705-2-50-0': 'It is instructive to estimate the total SE charge from the experimental parameters.', '1507.02705-2-50-1': 'From the EUV intensity, repetition rate and spot size, the average dose per pulse was approximately [MATH]0.017 mJ.', '1507.02705-2-50-2': 'The secondary electron yield for carbon under EUV radiation is estimated to be approximately [MATH] for the photon energy range of 60 eV - 100 eV [CITATION], which leads to a saturation charge of 2 nC, which would lead to a significant disagreement between the simulated and measured charge bias characteristics.', '1507.02705-2-50-3': 'It is also worth noting that effective SEY from mica was small, since, in the experiment, the charge collected from a sample made from mica was measured to be an order of magnitude smaller than for the carbon sample.', '1507.02705-2-51-0': 'However, the measured charge bias characteristics are the only experimental data that provides a reference point for the simulations of the experiment dynamics.', '1507.02705-2-51-1': 'Since the spatial distribution of EUV intensity is subject to an unknown systematic measurement error, and the SEY is known to vary widely, depending on the surface conditions, we chose to keep the product of incident EUV and SEY a constant, chosen to provide 1 nC total SE charge from the sample.', '1507.02705-2-52-0': '## Role of dielectric ring', '1507.02705-2-53-0': 'During the simulations we found that the mica accumulated charge over many EUV pulses, significantly effecting the local field distribution, and, hence changing the flux incident on the sample.', '1507.02705-2-53-1': 'If charging is neglected, the Q - V response under vacuum and 2.8 Pa conditions cannot be reproduced for any reasonable parameter values.', '1507.02705-2-53-2': 'The discrepancy is caused by the potential barrier near the sample surface, which is created by the negative space charge generated during the EUV pulse.', '1507.02705-2-53-3': 'Although the potential is low, it produces a significant effect because the SE energy spectrum ([REF]) is strongly peaked at rather low energies: approximately [MATH] eV.', '1507.02705-2-54-0': 'The charge accumulation on the mica significantly increases the local field strength near the sample surface (see Fig. [REF]).', '1507.02705-2-54-1': 'This removes the space charge potential barrier for biases of -100V bias and higher under the conditions that we consider.', '1507.02705-2-55-0': 'The charging of the mica should saturate, leading to an unchanging charge density distribution on the mica.', '1507.02705-2-55-1': 'To estimate the charge density, we simulated one hundred pulses under vacuum conditions for all experimentally applied bias values.', '1507.02705-2-55-2': 'In these calculations, the effective SEY from the mica is assumed to be about 0.001, which corresponds to the experimentally measured value.', '1507.02705-2-56-0': 'This approach corresponds to the experimental procedure, since the Q - V characteristics were measured via averaging a large number of pulses for every combination of bias and pressure.', '1507.02705-2-56-1': 'No special means were used in the experiment to remove the accumulated charge from the mica between pulses.', '1507.02705-2-57-0': 'For negative biases, simulations show that the mica potential rapidly reaches approximately 0 V potential.', '1507.02705-2-57-1': 'For these conditions, further charging is very slow, because (a) the electric field strength near the mica is very small, therefore, even a small amount of emitted charge creates a space charge potential barrier, and, (b), because the mica potential is close to zero, there is a small flux of electrons from the sample to the mica.', '1507.02705-2-58-0': 'The continued slow variation in charge distribution is impractical to simulate, however.The final charge distribution requires a very long time to calculate and is sensitive to the combination of the SEY from both the sample and the mica, and to the corresponding SE energy distribution functions.', '1507.02705-2-58-1': 'Nevertheless, the additional errors due to these limitations are expected to be small.', '1507.02705-2-58-2': "In practice the maximum potential of the mica under vacuum conditions is bounded, since, as the mica's potential increases to [MATH]10 V, the electron current from the sample to mica becomes significant.", '1507.02705-2-59-0': 'It is worth noting that the EUV plasma itself can contribute to charging the mica, since, for stationary discharges, the dielectric charges to the plasma potential.', '1507.02705-2-59-1': 'Despite the the fact that the potential of the EUV-induced plasma can be high (e.g. 20 - 30 V or more after EUV pulse), it rapidly decreases to several volts due to electron cooling due to collisions with the background gas.', '1507.02705-2-59-2': 'Most probably, the mica potential would converge to the time averaged potential of the plasma, e.g. several volts.', '1507.02705-2-59-3': 'Therefore, in the following results, the mica was pre-charged to the value that was obtained from simulations under vacuum conditions for the appropriate bias.', '1507.02705-2-60-0': 'The optimum parameter combination that reproduces the experimentally measured Q - V characteristics were chosen as follows: the mica was pre-charged as described above, the EUV intensity was kept constant at the measured value, but the amount of the scattered EUV radiation incident on the sample holder was chosen so that 40[MATH] of EUV radiation was incident on the dielectric mica, and the effective SEY from the sample was kept at to 0.01 for the photon energy range of 60 eV - 100 eV, while the SE contribution from the VUV part of the spectrum was neglected.', '1507.02705-2-60-1': 'The simulated Q - V characteristic, after parameter optimization, and comparison with the measurements are presented in Fig. [REF].', '1507.02705-2-61-0': '# Ion fluxes to the sample surface', '1507.02705-2-62-0': 'The removal of carbon from the surface material should be directly dependent on the energy distribution function (EDF) of the ion flux incident on the surface.', '1507.02705-2-62-1': 'The conditions of the experimental study of carbon cleaning in the ISAN EUV experiment are summarized in Table [REF].', '1507.02705-2-62-2': 'We computed the plasma conditions, ion fluxes, and ion EDF for these conditions.', '1507.02705-2-62-3': 'The simulated time for all cases was 10 [MATH]s. For 2.8 Pa - 11.2 Pa and biases -200 V - -100 V, the plasma had completely decayed and all ions were collected during the simulation time.', '1507.02705-2-62-4': 'But, for the 60 Pa and 86.5 Pa cases some plasma was still left in the simulation domain.', '1507.02705-2-63-0': 'The EDFs, integrated over the simulation window, and averaged over the sample surface, are presented in Fig. [REF].', '1507.02705-2-63-1': 'The composition of the computed ion flux depends significantly on the pressure.', '1507.02705-2-63-2': 'For the 3 Pa case, the main ion is H[MATH], because the characteristic time for H[MATH] to H[MATH] conversion in 3 Pa H[MATH] is approximately 0.5 [MATH]s. With increasing pressure, H[MATH] becomes the main ion, as expected.', '1507.02705-2-63-3': 'There is also a non-negligible contribution from fast H[MATH] which is produced due to a resonant charge exchange reaction.', '1507.02705-2-64-0': 'The maximum ion energy is larger than the applied bias voltage due to the build-up of the plasma potential.', '1507.02705-2-64-1': 'But the number of such fast ions is small, due to the fast plasma decay.', '1507.02705-2-64-2': 'For the same reason, the most energetic ion is H[MATH] because its small mass allows it to accelerate during the decay of the plasma potential.', '1507.02705-2-65-0': 'It was observed, that contrary to the ion dose, the shape of the ion flux EDF was mainly defined by bias, pressure and the accumulated charge on the dielectric.', '1507.02705-2-65-1': 'However, as discussed in previous section, we expect that the equilibrium mica potential does not significantly differ from 0 V. For small variations of the mica potential (if all other parameters are the same), the variations of the EDF shape are small for all simulations where a sample bias was applied.', '1507.02705-2-65-2': 'Moreover, the EDF was barely sensitive to reasonable variations of other parameters.', '1507.02705-2-66-0': 'This insensitivity is a consequence of the experimental procedure.', '1507.02705-2-66-1': 'The applied bias allows ions to be collected from the entire chamber volume.', '1507.02705-2-66-2': 'Hence, the characteristics of the plasma are important only for the short time when the sheath between plasma and sample is small.', '1507.02705-2-66-3': 'As ions are collected at the sample, the sheath size becomes larger and larger.', '1507.02705-2-66-4': 'Therefore, as time progresses, the instantantaneous ion flux EDF starts to depend only on the bias potential, mica potential, and the distance that the ions travel between the plasma and the sample.', '1507.02705-2-66-5': 'This is because these parameters determine the maximum possible energy of the ions, while the background pressure determines the number of collisions (e.g energy loss), and, hence, the average energy of the ion at the sample.', '1507.02705-2-66-6': 'Therefore, for the considered conditions, the shape of the ion flux EDF is mostly determined by the bias, mica potential, pressure, and the chamber geometry.', '1507.02705-2-67-0': 'It is notable, that the mica charge, due to exposure to EUV radiation and plasma, leads to the ion flux being focused on the sample (see Fig. [REF]).', '1507.02705-2-67-1': 'The ion flux to the mica itself becomes negligible compared to the simulation with zero charge on the mica.', '1507.02705-2-67-2': 'Therefore, the measured ion doses should represent accurately the ion doses on the sample.', '1507.02705-2-68-0': 'Hence, for the analysis of the experiments, it is reasonable to take the computed EDF shape, normed to the experimentally measured ion dose (if it is available).', '1507.02705-2-68-1': 'This approach allows us to compare the experimental results at different pressures.', '1507.02705-2-69-0': '# Discussion', '1507.02705-2-70-0': 'The maximum etched depths and total etched volumes per exposure of 10[MATH] pulses for different pressures and biases are presented in Table. [REF].', '1507.02705-2-70-1': 'The etched volumes are the integrals over etch profiles, which were determined by XRF measurement (see Fig. [REF]).', '1507.02705-2-70-2': 'In the cases of 60 Pa and 86.5 Pa pressure, the etch profile was only measured at the center of the EUV spot.', '1507.02705-2-70-3': 'We assumed, for these two cases, a uniform etching profile to estimate the carbon removal yield.', '1507.02705-2-70-4': 'Hence, for these cases, the amount of removed carbon is most probably overestimated.', '1507.02705-2-71-0': 'As was discussed in the previous section, we used the experimentally measured ion doses for yield estimation.', '1507.02705-2-71-1': 'However, for some experiments, the ion dose was not measured.', '1507.02705-2-71-2': 'In these cases, we used the ion doses obtained from simulation results.', '1507.02705-2-71-3': 'This approach is valid at high pressures because the uncertainty in the simulations is on the order of 30% (see Fig. [REF]).', '1507.02705-2-72-0': 'For all samples, the average carbon removal yield was larger than 0.2 carbon atoms per ion.', '1507.02705-2-72-1': 'Thus, physical sputtering is not the main process, since for energies below 200 eV, the expected sputtering yield is lower than [MATH] carbon atoms per hydrogen ion [CITATION].', '1507.02705-2-72-2': 'Therefore, carbon is removed via a chemical sputtering process, or a reactive ion etching process, because the effective yield for these processes is known to be large under certain conditions [CITATION].', '1507.02705-2-73-0': 'However, the results for -200 V bias, especially at low pressure (2.8 - 11.2 Pa), cannot be described by a chemical sputtering process, since the average yield is larger than one carbon atom per ion.', '1507.02705-2-73-1': 'Hence, reactive ion etching may be responsible for these very high etch rates.', '1507.02705-2-73-2': 'However, for reactive ion etching there should be weakly bound radicals on the top of the carbon, which are desorbed from surface due to ion impact, which, in turn, lead to a very high yield per ion.', '1507.02705-2-73-3': 'Although, we have no data about these radicals, it is likely to be some form of methane radical (e.g., CH[MATH]).', '1507.02705-2-73-4': 'The alternative: oxidation (e.g., CO and CO[MATH]) [CITATION] is unlikely because water, which is the main source of oxygen, is removed from the hydrogen flow, and, during experiments, the chamber walls are kept at liquid nitrogen temperatures, trapping the majority of the residual water.', '1507.02705-2-73-5': 'Hence, the contribution of water to the carbon removal was limited by the initial coverage which is typically sub mono layer for amorphous carbon for the considered temperature range [CITATION].', '1507.02705-2-74-0': 'In the case of experiments with -50 V and -100 V biases, the results can be compared with carbon etching in a surface wave discharge plasma [CITATION], where similar magnetron deposited carbon on silicon substrate samples were used.', '1507.02705-2-74-1': 'The carbon removal yield for experiments with biases of -50 V and -100 V was estimated by convolving the simulated EDF with the energy dependent yield, as measured in the SWD experiments (see Fig. [REF] and column "SWD recomputed" in table [REF]).', '1507.02705-2-74-2': 'In spite of H[MATH] being the main ion in SWD plasma, we used the same yield for estimation, since we do not expect dramatic difference between H[MATH] and H[MATH] for carbon removal.', '1507.02705-2-74-3': 'These estimates show that the yields found in EUV experiments with low biases agree with the recomputed yield from the SWD experiment.', '1507.02705-2-74-4': 'The margin of error is, however, large enough that we cannot completely exclude direct influence from EUV radiation.', '1507.02705-2-75-0': '# Conclusion', '1507.02705-2-76-0': 'The evolution of an EUV induced hydrogen plasma was simulated.', '1507.02705-2-76-1': 'The simulations, due to their close coupling to experimental conditions, allowed the magnitude, composition, and energy spectrum of the flux from the plasma to the surface to be estimated.', '1507.02705-2-76-2': 'Our model successfully computes the charge bias characteristics of the sample to an accuracy of a factor of two.', '1507.02705-2-76-3': 'This was used to quantitatively compare carbon etch rates between different EUV plasma conditions, as well as compare carbon etching under EUV-induced plasma to etching under SWD plasmas.', '1507.02705-2-76-4': 'In addition, the model describes the focusing effect of the dielectric surround, which allows the measured carbon etch profiles to be better understood in a quantitative manner.', '1507.02705-2-77-0': 'It was observed that the carbon etching mechanism at low bias and pressure was different than that at high bias and pressure.', '1507.02705-2-77-1': 'For the higher energy range, the carbon removal yield in EUV-induced plasma was larger than one carbon atom per ion.', '1507.02705-2-77-2': 'Most probably, etching is dominated by a reactive ion etching process, which may be due to the production of methane radicals (e.g. CH[MATH]) that can desorb under energetic ion flux.', '1507.02705-2-78-0': 'However, at low bias energies, the etch rates found in EUV-induced plasma agree with those found in SWD plasma to within model and measurement uncertainties.', '1507.02705-2-78-1': 'Therefore, the role of the EUV radiation in a previous study [CITATION] was overestimated at low bias energies and pressures.', '1507.02705-2-79-0': 'However, our study does not exclude EUV enhancing etch rates at low biases or having a small ([MATH] C/ion) enhancing effect over the range of biases and pressures that we studied.', '1507.02705-2-79-1': 'Therefore, experiments to determine the effect of EUV in the limit of low-to-no bias are required.', '1507.02705-2-80-0': 'The authors would like to thank Tatiana Rakhimova (Skobeltsyn Institute of Nuclear Physics) for helpful discussion.', '1507.02705-2-81-0': 'This work is part of the research program "Controlling photon and plasma induced processes at EUV optical surfaces (CP3E)" of the "Stichting voor Fundamenteel Onderzoek der Materie (FOM)" which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).', '1507.02705-2-81-1': 'The CP3E programme is cofinanced by industrial partners, including ASML (Veldhoven), and the AgentschapNL through the Catrene EXEPT program.'}

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'1507.02705-2-40-4'], ['1507.02705-1-61-0', '1507.02705-2-62-0'], ['1507.02705-1-61-1', '1507.02705-2-62-1'], ['1507.02705-1-61-2', '1507.02705-2-62-2'], ['1507.02705-1-61-3', '1507.02705-2-62-3'], ['1507.02705-1-61-4', '1507.02705-2-62-4'], ['1507.02705-1-27-0', '1507.02705-2-27-0'], ['1507.02705-1-27-1', '1507.02705-2-27-1'], ['1507.02705-1-65-0', '1507.02705-2-66-0'], ['1507.02705-1-65-1', '1507.02705-2-66-1'], ['1507.02705-1-65-2', '1507.02705-2-66-2'], ['1507.02705-1-65-3', '1507.02705-2-66-3'], ['1507.02705-1-65-4', '1507.02705-2-66-4'], ['1507.02705-1-65-5', '1507.02705-2-66-5'], ['1507.02705-1-65-6', '1507.02705-2-66-6'], ['1507.02705-1-44-0', '1507.02705-2-45-0'], ['1507.02705-1-44-1', '1507.02705-2-45-1'], ['1507.02705-1-9-0', '1507.02705-2-9-0'], ['1507.02705-1-9-1', '1507.02705-2-9-1'], ['1507.02705-1-43-0', '1507.02705-2-44-0'], ['1507.02705-1-43-1', '1507.02705-2-44-1'], ['1507.02705-1-43-2', '1507.02705-2-44-2'], ['1507.02705-1-43-3', '1507.02705-2-44-3'], ['1507.02705-1-62-0', '1507.02705-2-63-0'], ['1507.02705-1-62-1', '1507.02705-2-63-1'], ['1507.02705-1-62-2', '1507.02705-2-63-2'], ['1507.02705-1-62-3', '1507.02705-2-63-3'], ['1507.02705-1-28-0', '1507.02705-2-28-0'], ['1507.02705-1-8-0', '1507.02705-2-8-0'], ['1507.02705-1-8-1', '1507.02705-2-8-1'], ['1507.02705-1-8-2', '1507.02705-2-8-2'], ['1507.02705-1-15-0', '1507.02705-2-15-0'], ['1507.02705-1-15-1', '1507.02705-2-15-1'], ['1507.02705-1-15-2', '1507.02705-2-15-2'], ['1507.02705-1-4-0', '1507.02705-2-4-0'], ['1507.02705-1-4-1', '1507.02705-2-4-1'], ['1507.02705-1-4-2', '1507.02705-2-4-2'], ['1507.02705-1-4-3', '1507.02705-2-4-3'], ['1507.02705-1-4-4', '1507.02705-2-4-4'], ['1507.02705-1-4-5', '1507.02705-2-4-5'], ['1507.02705-1-41-0', '1507.02705-2-42-0'], ['1507.02705-1-41-1', '1507.02705-2-42-1'], ['1507.02705-1-14-0', '1507.02705-2-14-0'], ['1507.02705-1-14-1', '1507.02705-2-14-1'], ['1507.02705-1-57-0', '1507.02705-2-58-0'], ['1507.02705-1-57-1', '1507.02705-2-58-1'], ['1507.02705-1-57-2', '1507.02705-2-58-2'], ['1507.02705-1-77-0', '1507.02705-2-78-0'], ['1507.02705-1-77-1', '1507.02705-2-78-1'], ['1507.02705-1-50-0', '1507.02705-2-51-0'], ['1507.02705-1-50-1', '1507.02705-2-51-1'], ['1507.02705-1-64-0', '1507.02705-2-65-0'], ['1507.02705-1-64-1', '1507.02705-2-65-1'], ['1507.02705-1-64-2', '1507.02705-2-65-2']]

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[]

[['1507.02705-1-79-0', '1507.02705-2-80-0'], ['1507.02705-1-20-0', '1507.02705-2-20-0'], ['1507.02705-1-16-1', '1507.02705-2-16-1'], ['1507.02705-1-9-2', '1507.02705-2-9-2']]

[]

['1507.02705-1-21-2', '1507.02705-1-27-2', '1507.02705-1-44-2', '1507.02705-2-21-2', '1507.02705-2-27-2', '1507.02705-2-45-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1507.02705

1602.01333

{'1602.01333-1-0-0': '# Introduction', '1602.01333-1-1-0': 'The one-loop UV/IR mixing structure of noncommutative (NC) [MATH]=1 super Yang-Mills theory defined in terms of the noncommutative fields was studied some years ago in a number of papers [CITATION].', '1602.01333-1-1-1': 'The outcome was the famous quadratic noncommutative IR divergences which occur in the one-loop gauge field propagator of the non-supersymmetric version of the theory cancel here due to Supersymmetry.', '1602.01333-1-1-2': 'The one-loop gauge field propagator still carries a logarithmic UV divergence -a simple pole in Dimensional Regularization- and the dual logarithmic noncommutative IR divergence [MATH] as a result of the UV/IR mixing being at work.', '1602.01333-1-1-3': 'By increasing the number of supersymmetries of the noncommutative Yang-Mills theory one makes the UV behaviour of the theory softer and eventually finite for [MATH] [CITATION], at which point noncommutative IR divergences cease to exist [CITATION].', '1602.01333-1-1-4': 'In the [MATH] super Yang-Mills case, there still remain logarithmic UV divergences at one-loop in the two-point function which give rise via UV/IR mixing to the corresponding IR divergences [CITATION].', '1602.01333-1-1-5': 'That noncommutative [MATH] super Yang-Mills has a smooth commutative limit has been shown in Ref. [CITATION].', '1602.01333-1-2-0': 'It is known that classically noncommutative gauge field theories admit a dual formulation in terms of ordinary fields, a formulation that is obtained by using the celebrated Seiberg-Witten map [CITATION].', '1602.01333-1-2-1': 'However we still do not know whether this duality holds at the quantum level, ie, whether the quantum theory defined in terms of noncommutative fields is the same as the ordinary quantum theory -called the dual ordinary theory- whose classical action is obtained from the noncommutative action by using the Seiberg-Witten map.', '1602.01333-1-2-2': 'The existence of UV/IR mixing effects in noncommutative field theory defined terms of noncommutative fields is thought to be a characteristic feature of those field theories.', '1602.01333-1-2-3': 'It is thus sensible to think such effects should also occur in the ordinary dual theory obtained, as previously explained, by using the Seiberg-Witten map.', '1602.01333-1-2-4': 'That these UV/IR mixing effects actually occur in the propagator of the gauge field of the dual ordinary theory was first shown in Ref. [CITATION] by using the [MATH]-exact Seiberg-Witten map expansion [CITATION].', '1602.01333-1-2-5': 'The analysis of the properties and phenomenological implications of the UV/IR mixing effects that occur in noncommutative gauge theories defined by means of the [MATH]-exact Seiberg-Witten map has been pursued in Refs. [CITATION].', '1602.01333-1-3-0': 'Up to best of our knowledge no analysis of the UV and the noncommutative IR structure of noncommutative super Yang-Mills theory defined by means of the [MATH]-exact Seiberg-Witten map has been carried out as yet.', '1602.01333-1-3-1': 'In particular, it is not known whether the cancellation of the quadratic noncommutative IR divergences of the gauge-field propagator that occurs, as we mentioned above, in noncommutative super Yang-Mills theory defined in terms of noncommutative fields also works in its dual ordinary theory.', '1602.01333-1-3-2': 'The answer to this question is far from obvious since Supersymmetry is linearly realized in terms of the noncommutative fields -and thus there exist a superfield formalism- but is non-linearly realized -see Ref. [CITATION]- in terms of the ordinary fields of the dual ordinary theory defined by means of the Seiberg-Witten map.', '1602.01333-1-3-3': 'It has long been known that the proper definition of theories with non-linearly realized symmetries is a highly non-trivial process.', '1602.01333-1-4-0': 'The purpose of this paper is to work out all the one-loop 1PI two-point functions, and analyze the UV and noncommutative IR structure of those functions, in noncommutative U(1) [MATH]=1,2 and 4 super Yang-Mills theories in the Wess-Zumino gauge, when those theories are defined in terms of ordinary fields by means of the [MATH]-exact Seiberg-Witten maps.', '1602.01333-1-4-1': 'To analyze the gauge dependence of the UV and noncommutative IR of the gauge field two-point functions we shall consider to types of gauge-fixing terms for the ordinary gauge field: the standard ordinary Feynman gauge-fixing term and the noncommutative Feynman gauge-fixing term.', '1602.01333-1-5-0': 'The layout of this paper is as follows.', '1602.01333-1-5-1': 'Section 2 is devoted to the computation of the one-loop contributions to the photon and photino propagators in [MATH] super Yang-Mills U(1) theory in the ordinary Feynman gauge.', '1602.01333-1-5-2': 'In Section 3 we discuss, for later use, the construction by using the [MATH]-exact Seiberg-Witten map of a noncommutative U(1) theory with a noncommutative field transforming under the adjoint.', '1602.01333-1-5-3': 'The one-loop propagators of the the ordinary fields of noncommutative [MATH]=2 and 4 super Yang-Mills U(1) theories defined by using the [MATH]-exact Seiberg-Witten map are worked out in Sections 4 and 5 in the ordinary Feynman gauge.', '1602.01333-1-5-4': 'In sections 6 and 7 we use a noncommutative Feynman gauge to quantize [MATH] super Yang-Mills U(1) theory and compute the one-loop photon propagator.', '1602.01333-1-5-5': 'Sections 6 and 7 are introduced to analyze the dependence on the gauge-fixing term of the UV and noncommutative IR contributions found in previous sections.', '1602.01333-1-5-6': 'The overall discussion of our results is carried out in Section 8.', '1602.01333-1-5-7': 'We also include several appendices which are needed to follow properly the analysis and computations carried out in the body of the paper.', '1602.01333-1-6-0': '# Noncommutative [MATH]=1 SYM U(1) theory and the [MATH]-exact SW map', '1602.01333-1-7-0': 'The noncommutative field content of the noncommutative U(1) super Yang-Mills theory in the Wess-Zumino gauge is the noncommutative gauge field [MATH], its supersymmetric fermion partner [MATH] and the noncommutative SUSY-auxiliary field [MATH].', '1602.01333-1-7-1': 'The ordinary/commutative counterparts of [MATH], [MATH] and [MATH] will be denoted by [MATH] (photon), [MATH] (photino) and [MATH], respectively.', '1602.01333-1-7-2': 'Regarding dotted and undotted fermions and [MATH] traces, we shall follow the conventions of [CITATION].', '1602.01333-1-8-0': 'In terms of the noncommutative fields and in the Wess-Zumino gauge the action of U(1) super Yang-Mills theory reads [EQUATION] where [MATH] and [MATH].', '1602.01333-1-9-0': 'The above action [MATH] ([REF]), is invariant under the following noncommutative supersymmetry transformations [EQUATION]', '1602.01333-1-9-1': 'These supersymmetry transformations close on translations modulo noncommutative gauge transformations and tell us that supersymmetry is linearly realized on the noncommutative fields-see [CITATION] for further discussion.', '1602.01333-1-9-2': 'The action in [REF], is also invariant under noncommutative U(1) transformations, which in the noncommutative BRST form read [EQUATION] with [MATH] being the noncommutative U(1) ghost field.', '1602.01333-1-9-3': 'The above action [MATH] can be expressed in terms of ordinary fields, [MATH], [MATH] and [MATH], by means of the SW map.', '1602.01333-1-9-4': 'The resulting functional is invariant under ordinary U(1) BRST transformations: [EQUATION] where [MATH] is the ordinary U(1) ghost field.', '1602.01333-1-9-5': 'Indeed, the SW map maps ordinary BRST orbits into the noncommutative BRST orbits.', '1602.01333-1-10-0': 'The [MATH]-exact SW map for [MATH] has been worked out in [CITATION] up to three ordinary U(1) gauge fields [MATH].', '1602.01333-1-10-1': 'It reads [EQUATION] where to [MATH] order the gauge field strength SW map [MATH] expansion is fairly universal [CITATION] [EQUATION]', '1602.01333-1-10-2': 'The [MATH] order SW map for the gauge field strength from [CITATION] in that case reads [EQUATION]', '1602.01333-1-10-3': 'The generalized star products relevant for this work are defined as follows [CITATION]', '1602.01333-1-11-0': '(f_2 g)(x)&=e^-i(p+q)xf(p)g(q)f__2(p,q),', '1602.01333-1-12-0': "[f g h]__3'(x)&=e^-i(p+q+k)xf(p)g(q)h(k)f__3(p,q,k),", '1602.01333-1-13-0': '[f g h]_M_(I)(x)&=e^-i(p+q+k)xf(p)g(q)h(k)f_(I)(p,q,k),', '1602.01333-1-14-0': 'with', '1602.01333-1-15-0': 'f__2(p,q)&=pq2pq2,', '1602.01333-1-16-0': "f__3'(p,q,k)&=(pq2+pk2-qk2)-1(pq2+pk2-qk2)qk2-(pq2+pk2+qk2)-1(pq2+pk2+qk2)qk2,", '1602.01333-1-17-0': "f_(I)(p,q,k)&=1pq(f__3'[p,q,-(p+q+k)]-f__3'[p,q,k]).", '1602.01333-1-18-0': 'The [MATH]-exact SW map for [MATH] up to two ordinary fields [MATH] can be retrieved from the expression for [MATH] in [REF], [REF] and [REF] as explained in the Appendix A: [EQUATION] where [EQUATION] and [EQUATION]', '1602.01333-1-18-1': 'To compute the full one-loop photon two-point function, one also needs the SW maps for the [MATH] and [MATH] fields.', '1602.01333-1-18-2': 'They read', '1602.01333-1-19-0': "_=&_-e^ija_i_2_j _+e^24^ij^kl([a_i_j(a_k_l _)]__3'", '1602.01333-1-20-0': "&-[a_i(f_jk_l_-a_k_l _j_)]__3' +[_j _a_k(_l a_i+f_li)]__3')+O(e^3),", '1602.01333-1-21-0': "D^(nc)=&D-e^ija_i_2_j D+e^24^ij^kl([a_i_j(a_k_l D)]__3'", '1602.01333-1-22-0': "&-[a_i(f_jk_lD-a_k_l _jD)]__3' +[_j Da_k(_l a_i+f_li)]__3')+O(e^3).", '1602.01333-1-23-0': 'Let us stress that the way we have constructed-by appropriate restriction of the SW map for the gauge-field-the SW map for [MATH] and [MATH] is very much in harmony with the idea that if supersymmetry and gauge symmetry are not to clash, the superpartners must have similar behavior with regard to the gauge group.', '1602.01333-1-24-0': 'As discussed in [CITATION] the noncommutative supersymmetric transformations in [REF] can be understood as the push-forward under the SW map of the appropriate [MATH]-dependent supersymmetric transformations of the ordinary fields.', '1602.01333-1-24-1': 'Here we have worked out the [MATH]-exact expression for such deformed transformations up to order [MATH]: [EQUATION] where [EQUATION]', '1602.01333-1-24-2': 'The reader shall find in the Appendix A the values of objects in the previous equations that have not been given yet.', '1602.01333-1-25-0': 'The [MATH]-exact deformed supersymmetry transformations given in [REF] and [REF] can be rightly called supersymmetry transformations since, as shown in [CITATION], they close on translations modulo gauge transformations and, hence, they carry a representation of the supersymmetry algebra.', '1602.01333-1-25-1': 'However notice that these deformed supersymmetry transformations of the ordinary field do not realize the supersymmetry algebra linearly.', '1602.01333-1-25-2': 'Finally, since these supersymmetry transformations generate the noncommutative supersymmetry transformations of [REF], we conclude that the total [MATH]-exact action (given explicitly in the next subsection) has to be invariant at second order in [MATH], under the deformed supersymmetry transformations in [REF].', '1602.01333-1-26-0': '## The action', '1602.01333-1-27-0': 'Now, substituting in [REF], the Seiberg-Witten maps in [REF], [REF] and [REF] and dropping any contribution of order [MATH], one obtains the SYM U(1) action in terms of commutative fields: [EQUATION] where [EQUATION] and [EQUATION]', '1602.01333-1-27-1': 'First we note that, since the 3-photon and the 4-photon self-couplings described by the action [MATH] ([REF]), Feynman rules are already given in previous papers [CITATION] and [CITATION], respectively, so we shall not repeat them here.', '1602.01333-1-28-0': '## The photon one-loop contributions to the photon polarization tensor', '1602.01333-1-29-0': 'Most generally speaking, the total photon one-loop 1PI two-point function [MATH] in the [MATH] SYM theory is the sum of following contributions [EQUATION] where [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] refer to the contributions from photon bubble, photon tadpole, photino bubble, photino tadpole, adjoint scalar bubble and adjoint scalar tadpole diagrams, respectively.', '1602.01333-1-29-1': 'The last two diagrams appear only in the extended SUSY, of course.', '1602.01333-1-29-2': 'We use [MATH] for the number of photinos (Weyl fermions) and [MATH] for the number of real adjoint scalar bosons (one complex scalar is counted as two real scalars), which are uniquely determined by [MATH] SUSY.', '1602.01333-1-30-0': 'Explicit computation revolves that each of these diagrams can be expressed as a linear combinations of five transverse tensor structures [EQUATION]', '1602.01333-1-30-1': 'The sum [REF] can be equivalent, in the language of the five tensor decomposition to [EQUATION]', '1602.01333-1-30-2': 'In the subsequent sections we are going to compute and give the coefficients [MATH], [MATH], [MATH], [MATH] and [MATH] in detail via equations [REF], [REF], [REF], [REF] and[REF], respectively.', '1602.01333-1-31-0': 'For the [MATH] theory [MATH], [MATH], thus we have [EQUATION]', '1602.01333-1-31-1': "In this section we are going to show that all quadratic IR divergences cancel in each of the [MATH]'s, then we extend our results to the [MATH] theories as well.", '1602.01333-1-32-0': 'We choose one specific set of four (five in the sections 6 and 7) nonplanar/special function integrals [MATH], [MATH], [MATH] and [MATH] alongside the usual planar/commutative UV divergent integrals to express all loop diagrams/coefficients in this article.', '1602.01333-1-32-1': 'This decomposition enjoys the advantage that each nonplanar integral bear distinctive asymptotic behavior in the IR regime: [MATH] carries all the quadratic IR divergence [MATH], with a pre-factor [MATH], while [MATH] and [MATH] contain the dual logarithmic noncommutative IR divergence [MATH], with pre-factors [MATH] and [MATH], respectively.', '1602.01333-1-32-2': 'The last integral [MATH] is finite at the IR limit.', '1602.01333-1-32-3': 'Further details of these integrals are given in the Appendix B.', '1602.01333-1-33-0': '### The photon-bubble diagram', '1602.01333-1-34-0': 'From the photon bubble diagram Fig. [REF] we obtain the following loop-coefficients [CITATION] [EQUATION]', '1602.01333-1-34-1': "Extracting the divergent parts from each of the [MATH]'s", '1602.01333-1-35-0': 'B_1(p)&=+(43 +3 p^2(trtheta)(p)^2 +4 p^2(p)^2(p)^4) (2 + (^2(p)^2)) +finite terms,', '1602.01333-1-36-0': 'B_2(p)&=+2p^2(p)^2(2-p^2(trtheta)(p)^2 )(2 + (^2(p)^2))+3231(p)^4+finite terms,', '1602.01333-1-37-0': 'B_3(p)&=+2p^2(p)^2 (2 + (^2(p)^2))-3231(p)^4+finite terms,', '1602.01333-1-38-0': 'B_4(p)&=-4p^4(p)^4(2 + (^2(p)^2))-323p^2(p)^6+finite terms,', '1602.01333-1-39-0': 'B_5(p)&=+4p^4(p)^4 (2 + (^2(p)^2))+finite terms,', '1602.01333-1-40-0': 'we observe the presence of the UV plus logarithmic IR divergences in all of them.', '1602.01333-1-40-1': 'The logarithmic IR divergences from both planar and nonplanar sources in the bubble diagram appear to have identical coefficient and combine into a single [MATH] term, confirming the UV/IR mixing expected.', '1602.01333-1-40-2': 'The quadratic IR divergence, on the other hand, exists only in the [MATH] terms.', '1602.01333-1-41-0': '### The photon-tadpole diagram', '1602.01333-1-42-0': 'From tadpole Fig. [REF] we obtain the same tensor structure as from the photon bubble diagram (Fig. [REF]) with the following coefficients [MATH]: [EQUATION]', '1602.01333-1-42-1': 'We notice immediately the absence of UV plus logarithmic divergent terms contrary to the photon-bubble diagram results ([REF]).', '1602.01333-1-42-2': 'In addition, the tadpole diagram produces no finite terms either, and the quadratic IR are again present in the second, third and fourth term!', '1602.01333-1-43-0': '## The photino one-loop contributions to the photon polarization tensor', '1602.01333-1-44-0': 'The photino sector contains two diagrams: photino tadpole Fig. [REF] and photino bubble Fig. [REF].', '1602.01333-1-44-1': 'We are going to see below that only the latter contributes to the photon polarization tensor.', '1602.01333-1-45-0': '### The photino-tadpole diagram', '1602.01333-1-46-0': 'The photino-tadpole contribution is computed using vertex ([REF]).', '1602.01333-1-47-0': 'It produces only the quadratic IR divergent terms which cancel each other internally, thus giving vanishing contribution to the photon polarization tensor [EQUATION]', '1602.01333-1-48-0': '### The photino-bubble diagram', '1602.01333-1-49-0': 'Taking the photino-photon Feynman rules from Appendix C we obtain', '1602.01333-1-50-0': 'the following photino-bubble contribution to the photon polarization tensor, [MATH], from Fig. [REF]: [EQUATION]', '1602.01333-1-50-1': 'Taking into account the trace [EQUATION] and that [MATH] cannot have, at the end of the day, contributions depending on [MATH], one arrives at [EQUATION]', '1602.01333-1-50-2': 'After some amount of computations we find that only first two of the general five-terms structure ([REF]) survive in the [MATH]-dimensions: [EQUATION]', '1602.01333-1-50-3': 'After inspecting the divergences in these two terms we also found that the first of the two, [MATH], contain the logarithmic UV/IR mixing terms, while [MATH] possesses only quadratic IR divergence and finite terms, as the dual NC logarithmic divergences from integrals [MATH] and [MATH] cancel each other.', '1602.01333-1-51-0': "Summing over [REF], [REF] and [REF] one can see that the total quadratic IR divergences in all [MATH]'s are zero.", '1602.01333-1-51-1': 'The total UV divergences for the [MATH] theory are as follows', '1602.01333-1-52-0': '_1(p)_UV=p^2(p)^4(3(trtheta)(p)^2+4(p)^2)(2+(^2(p)^2)),', '1602.01333-1-53-0': '_2(p)_UV=p^2(p)^2(2-p^2(trtheta)(p)^2 )(2+(^2(p)^2)),', '1602.01333-1-54-0': '_3(p)_UV=p^2(p)^2(2+(^2(p)^2)),', '1602.01333-1-55-0': '_4(p)_UV=-4p^4(p)^4(2+(^2(p)^2)),', '1602.01333-1-56-0': '_5(p)_UV=4p^4(p)^4 (2 + (^2(p)^2)).', '1602.01333-1-57-0': '## The one-loop SUSY-auxiliary field contributions to the photon propagator', '1602.01333-1-58-0': 'The free two-point function of the SUSY-auxiliary field [MATH] reads [EQUATION] hence, the integrals to be computed in dimensional regularization are of the type [EQUATION]', '1602.01333-1-58-1': 'The integrals in [REF] vanish in dimensional regularization and hence the SUSY-auxiliary field [MATH] does not contribute to the one-loop photon propagator.', '1602.01333-1-58-2': 'Indeed, following [CITATION], we split first the [MATH]-dimensional [MATH] into [EQUATION] where [MATH] belongs to [MATH] dimensional space orthogonal to [MATH], and [MATH] is the modulus of [MATH] -recall that [MATH] is a space-like vector, since [MATH].', '1602.01333-1-58-3': 'Then introduce the following definition of the dimensionally regularized integral in [REF]: [EQUATION]', '1602.01333-1-58-4': 'However, in dimensional regularization -see [CITATION]- [EQUATION] which in turn leads to the conclusion that the integral [REF] vanishes under the dimensional regularization procedure.', '1602.01333-1-59-0': 'It is not difficult to see that the argument above can be generalized to integrals with positive [MATH] powers too, i.e. [EQUATION]', '1602.01333-1-59-1': 'One can explicitly verify two special cases of the identity above [EQUATION] using a generalization of the n-nested zero regulator method [CITATION].', '1602.01333-1-60-0': '## The one-loop photino 1-PI two point function', '1602.01333-1-61-0': 'The photino self-energy consists two diagrams, a tadpole Fig. [REF] and a bubble Fig. [REF].', '1602.01333-1-61-1': 'Explicit computation shows that the tadpole diagram Fig. [REF] vanishes: [EQUATION]', '1602.01333-1-61-2': 'The bubble diagram was computed in [CITATION], which boils down to following expressions [EQUATION] with [EQUATION] and [EQUATION]', '1602.01333-1-61-3': 'One can easily notice the absence of quadratic IR divergent integral [MATH].', '1602.01333-1-61-4': 'The UV divergence can be expressed as follows [EQUATION]', '1602.01333-1-62-0': '# Minimal action of the noncommutative adjoint scalar field', '1602.01333-1-63-0': 'It is commonly known that extended, [MATH], super YM contain not only fermion (photino) but also scalar bosons in the adjoint representation.', '1602.01333-1-63-1': 'These scalar bosons couple minimally to the gauge field, their action is then [EQUATION] for real scalar, or [EQUATION] for complex scalar.', '1602.01333-1-63-2': "We study the minimal interacting scalar boson's contribution to 1-PI photon two point function as well as the scalar's own 1-PI two point function in this section.", '1602.01333-1-63-3': 'These results will be used for our discussion on [MATH] SYM in the subsequent sections.', '1602.01333-1-64-0': 'It is straightforward to derive the SW map expansion of either [MATH] or [MATH] using the method described in the Appendix A[EQUATION] with the products [MATH] being defined in [CITATION].', '1602.01333-1-64-1': 'One can show that [EQUATION] if we express one complex scalar in terms of two real scalars.', '1602.01333-1-64-2': 'For this reason one complex scalar contribution to the photon 1-PI two point function is twice as one real scalar while the photon contribution to the complex scalar two point function is the same as real scalar.', '1602.01333-1-64-3': 'We will then only compute those for the real scalar boson.', '1602.01333-1-64-4': 'The scalar-photon Feynman rules are given in Appendix D.', '1602.01333-1-65-0': '## Scalar one-loop contributions to the photon polarization tensor', '1602.01333-1-66-0': 'Like the photino sector, the adjoint scalar sector contains also two diagrams that contributes to the photon polarization tensor, the scalar-bubble diagram Fig. [REF] and scalar-tadpole diagram Fig. [REF].', '1602.01333-1-66-1': 'They both follow the five tensor structure decomposition [REF] and stay nonzero at the [MATH] limit.', '1602.01333-1-67-0': '### The scalar-bubble diagram', '1602.01333-1-68-0': 'Using the Feynman rule ([REF]) and employing the dimensional regularization techniques we obtain the following loop-coefficielnts from the photon bubble diagram Fig. [REF]: [EQUATION]', '1602.01333-1-69-0': '### The scalar-tadpole diagram', '1602.01333-1-70-0': 'Using the Feynman rule ([REF]) we compute the photon tadpole diagram in Fig. [REF],', '1602.01333-1-71-0': "S^(p)_tad=_k'=1^4S^_k'(p)_tad& =12d^D (2)^D i ^2_k'=1^4S^_k'(,p,-p,-).", '1602.01333-1-72-0': 'Starting with the first integral under dimensional regularization:', '1602.01333-1-73-0': "&S^_1'(p)_tad=-e^2d^D (2)^D f__2(,p) ^2 ((p)((p) g^-p^()^-()^p^)+p^2()^()^)_D4", '1602.01333-1-74-0': '&=e^2(4)^283(p)^4[3(g^(p)^2-()^p^2+ p^(p)^)+4p^2(p)^2(()^(p)^2+(p)^(p)^)],', '1602.01333-1-75-0': 'we obtained the IR result.', '1602.01333-1-75-1': 'Next to compute [MATH], [MATH] and [MATH] we first need to establish the following identities:', '1602.01333-1-76-0': "&f__3'(,p,-p)=f__3'(,-p,p)=f__3'(-,p,-p)=f__3'(-,-p,p)= 1,", '1602.01333-1-77-0': "& f__3'(p,-p,-)=f__3'(p,-p,)=f__3'(-p,p,-)=f__3'(-p,p,)= f^2__2(, p),", '1602.01333-1-78-0': "& f__3'(-,p,-p)+f__3'(p,-p,-)-2f^2__2(, p) -f^2__2(, p),", '1602.01333-1-79-0': "& f__3'(p,-p,-)+f__3'(-p,p,-)-2f^2__2(, p)=0,", '1602.01333-1-80-0': "& f__3'(,p,-p)+f__3'(p,-p,)-2f^2__2(, p) -f^2__2(, p),", '1602.01333-1-81-0': "& f__3'(p,-p,)+f__3'(-p,p,)-2f^2__2(, p)=0.", '1602.01333-1-82-0': 'We then find the following pure IR divergent terms:', '1602.01333-1-83-0': "&S^_4'(p)_tad=-2S^_2'(p)_tad=-2S^_3'(p)_tad =-S^_2'(p)_tad-S^_3'(p)_tad", '1602.01333-1-84-0': '& =e^2d^D (2)^D f__2(,p) ^2 ((p )((p)^()^+()^(p)^) -(p)(^(p)^+(p)^^))_D4', '1602.01333-1-85-0': '& =e^2(4)^2323(p)^(p)^(p)^4, [EQUATION]', '1602.01333-1-85-1': 'Using ([REF]) and ([REF]) and by comparing with general tensor structure ([REF]) we have found that from scalar-photon tadpole diagram only two terms survives: [EQUATION]', '1602.01333-1-85-2': 'Finally summing up the IR parts of bubble ([REF]) and tadpole ([REF]) contributions we get: [EQUATION] where all IR terms from both diagrams, except the one arising from the bubble, cancels out.', '1602.01333-1-85-3': 'Interesting enough is that within this noncommutaive scalar-photon action in the adjont ([REF]) we are facing the exact cancelations of all divergences of the higher order terms of noncommutative tensor-parameter [MATH], showing thus the consistency of our computations.', '1602.01333-1-86-0': '## The photon one-loop contribution to scalar 1-PI two point function', '1602.01333-1-87-0': 'The one-loop adjoint scalar 1-PI two point function in the minimal coupled model consists the tadpole diagram Fig. [REF] and the bubble diagram Fig. [REF].', '1602.01333-1-88-0': 'The evaluation is straightforward.', '1602.01333-1-88-1': 'We obtain in the end [EQUATION] and [EQUATION]', '1602.01333-1-88-2': 'The total quadratic IR divergence reads [EQUATION]', '1602.01333-1-88-3': 'We will soon see in the next section that this divergence is exactly canceled by the contributions from scalar-photino and scalar self-interaction diagrams.', '1602.01333-1-89-0': '# Noncommutative [MATH]=2 SYM U(1) theory and the [MATH]-exact SW map', '1602.01333-1-90-0': 'The noncommutative U(1) [MATH]=2 super Yang-Mills theory has the following action [EQUATION] in the Wess-Zumino gauge.', '1602.01333-1-90-1': 'The noncommutative fields in the previous action constitute the noncommutative U(1) supermultiplet [MATH].', '1602.01333-1-90-2': '[MATH] and [MATH] are Weyl fermion fields and [MATH] is a complex scalar field.', '1602.01333-1-90-3': 'Each field [MATH], [MATH] and [MATH] transforms under the adjoint action of the NC U(1), so that the NC covariant derivative is [MATH].', '1602.01333-1-91-0': 'By replacing the noncommutative fields of the action in ([REF]) with the [MATH]-exact Seiberg-Witten maps -see Appendix A- [MATH], [MATH], [MATH], [MATH], the action [MATH] is turned into the action of a theory which is an interacting deformation of the free ordinary U(1) supersymmetric theory for the U(1) supermultiplet [MATH].', '1602.01333-1-91-1': 'This deformation is supersymmetric although supersymmetry -[MATH]- is nonlinearly realized on the ordinary multiplet [MATH]; a feature we have already seen in the [MATH] superYangMills case.', '1602.01333-1-92-0': 'The contributions to the action in ([REF]) that are needed to compute one-loop 1PI two-point function of each field in [MATH] can be readily obtained by using ([REF]), ([REF]), ([REF]) and [EQUATION]', '1602.01333-1-92-1': 'The terms in ([REF]) yields the scalar-fermion [MATH]=2 Feynman rules ([REF]) given in Appendix E.', '1602.01333-1-92-2': 'Now we are ready to display the value of each one-loop Feynman diagram contributing to the two-point functions of the ordinary fields of the theory.', '1602.01333-1-93-0': '## The one-loop 1PI two-point function for photon field [MATH]', '1602.01333-1-94-0': 'For [MATH] theory one has [MATH].', '1602.01333-1-94-1': 'Substituting these numbers as well as the scalar bubble and tadpole results to [REF], and then restricting to quadratic IR divergence only, gives [EQUATION] i.e. clean quadratic IR divergence cancellation.', '1602.01333-1-94-2': 'Remaining UV divergences can be expressed using the five-term notation in [REF] as follows', '1602.01333-1-95-0': '_1(p)_UV=(43-43n_f-13n_s+p^2(p)^4(3trtheta(p)^2+4(p)^2))(2+(^2(p)^2))_n_f=n_s=2,', '1602.01333-1-96-0': '_2(p)_UV=p^2(p)^2(2-p^2(trtheta)(p)^2 )(2+(^2(p)^2)),', '1602.01333-1-97-0': '_3(p)_UV=p^2(p)^2(2+(^2(p)^2)),', '1602.01333-1-98-0': '_4(p)_UV=-4p^4(p)^4(2+(^2(p)^2)),', '1602.01333-1-99-0': '_5(p)_UV=4p^4(p)^4 (2 + (^2(p)^2)).', '1602.01333-1-100-0': '## The one-loop 1PI two-point function for the scalar [MATH]', '1602.01333-1-101-0': 'The one-loop 1PI two-point function, [MATH], of field [MATH] is the sum of the four diagrams (Fig. [REF], Fig. [REF], Fig. [REF] and Fig. [REF]).', '1602.01333-1-101-1': 'The first two are already given as equations [REF] and [REF] in the last section.', '1602.01333-1-101-2': 'The values of the third and fourth diagrams read [EQUATION]', '1602.01333-1-101-3': 'Hence, one gets the following full scalar two-point function: [EQUATION] which is again quadratic IR divergence free, as [EQUATION]', '1602.01333-1-101-4': 'The UV divergence reads [EQUATION]', '1602.01333-1-102-0': '## The one-loop 1PI two-point function for photinos [MATH] and [MATH]', '1602.01333-1-103-0': 'In the [MATH]=2 theory there is scalar-photino loop (Fig. [REF]) alongside the photon-photino loop contribution which is identical to the [MATH] theory value [REF] for each of the two photinos.', '1602.01333-1-104-0': 'The scalar-photino loop integral gives following contribution [EQUATION]', '1602.01333-1-104-1': 'Total photino two-point function is finally given as a sum: [EQUATION]', '1602.01333-1-104-2': 'It is quadratic IR divergence free and has following UV divergence [EQUATION]', '1602.01333-1-105-0': '# Noncommutative [MATH]=4 SYM U(1) theory and the [MATH]-exact SW map', '1602.01333-1-106-0': 'Let [MATH], [MATH], [MATH] the noncommutative U(1) [MATH]=4 supermultiplet; then, the action of the noncommutative U(1) [MATH]=4 super Yang-Mills theory reads [EQUATION]', '1602.01333-1-106-1': 'The matrices [MATH], [MATH] and [MATH], give rise to the IRREP of the Dirac matrices in 8 Euclidean dimensions; further details can be found in [CITATION].', '1602.01333-1-106-2': 'Let us recall that [MATH] is the noncomutative gauge field, that [MATH] is a noncommutative Weyl field and that [MATH] is a noncommutative real scalar field.', '1602.01333-1-106-3': 'The noncommutative U(1) acts by the adjoint action on [MATH] and [MATH], and hence [MATH].', '1602.01333-1-107-0': 'By replacing, in [MATH] above, the fields [MATH], [MATH] and [MATH] with the corresponding [MATH]-exact Seiberg-Witten maps -namely, [MATH], [MATH] and [MATH], respectively, we obtain an action which defines an interacting deformation of the ordinary [MATH] super Yang-Mills theory in the Wess-Zumino gauge.', '1602.01333-1-107-1': 'This deformed action is expressed in terms of the fields of the ordinary [MATH] Yang-Mills supermultiplet [MATH], [MATH], [MATH] and it is invariant (on-shell) under the deformed supersymmetric transformations of the ordinary supermultiplet [MATH] which gives rise to the [MATH] supersymmetric transformations of the fields in [MATH].', '1602.01333-1-107-2': 'As in the [MATH] and [MATH] cases, the supersymmetry transformations of the ordinary fields that leave [MATH] in ([REF]) invariance gives rise to a on-shell nonlinear realization of [MATH] supersymmetry algebra.', '1602.01333-1-108-0': 'The contributions to the action in ([REF]) that are needed to compute one-loop 1PI two-point function of each field in [MATH] can easily obtained by using ([REF]), ([REF]), ([REF]) and [EQUATION]', '1602.01333-1-108-1': 'The terms in ([REF]) yields the Feynman rules Appendix D.', '1602.01333-1-109-0': 'Below we shall display the value of each one-loop Feynman diagram contributing to the two-point functions of the ordinary fields of the theory.', '1602.01333-1-110-0': '## The one-loop 1PI two-point function for massless vector field [MATH]', '1602.01333-1-111-0': 'The [MATH] 1PI two-point function follows the general formula [REF], only with [MATH] and [MATH].', '1602.01333-1-111-1': 'One can immediately recognize clean cancelation of the quadratic IR divergences after substituting [MATH] into [REF].', '1602.01333-1-112-0': '## The one-loop 1PI two-point function for the scalar [MATH]', '1602.01333-1-113-0': 'The one-loop 1PI two-point function, [MATH], of field [MATH] is the sum of five diagrams whose values read [EQUATION] hence [EQUATION] is again IR divergence free, only this time we have [EQUATION]', '1602.01333-1-113-1': 'The UV part reads [EQUATION]', '1602.01333-1-114-0': '## The one-loop 1PI two-point function for [MATH]', '1602.01333-1-115-0': 'The one-loop 1PI two-point function, [MATH], of the field [MATH] is the sum of the three diagrams Fig. [REF], Fig. [REF] and Fig. [REF] whose values read [EQUATION] with [MATH] being given in ([REF]) and ([REF]), respectively, and [EQUATION]', '1602.01333-1-115-1': 'Hence, [EQUATION] is quadratic IR divergence free, and the total UV divergences is presented below [EQUATION]', '1602.01333-1-116-0': '# Effect of gauge fixing on photon two point function', '1602.01333-1-117-0': 'In the prior sections we have shown that the quadratic IR divergent contribution to the photon two point function can be canceled by introducing supersymmetry.', '1602.01333-1-117-1': 'Yet we have still two unanswered question: First, all our computations above are carried out in the commutative Feynman gauge, which, albeit convenient, is just one specific choice.', '1602.01333-1-117-2': 'We do not know whether the cancelation we found would be changed by a change of gauge fixing.', '1602.01333-1-117-3': 'Second, we have quite complicated UV divergence in the Feynman gauge in general, which may be modified by changing gauge fixing, as in the commutative gauge theories.', '1602.01333-1-117-4': 'To study these two issues we introduce in this section a new, non-local and nonlinear gauge fixing based on the Seiberg-Witten map then evaluate its effect to the photon two point function.', '1602.01333-1-118-0': '## The noncommutative Feynman gauge fixing action', '1602.01333-1-119-0': 'We introduce a new gauge fixing for non-local U(1) gauge theory via the [MATH]-exact Seiberg-Witten map.', '1602.01333-1-119-1': 'In terms of BRST language, this gauge fixing contains BRST-auxiliary field [MATH], and it is given by', '1602.01333-1-120-0': 'S&=S_U(1)+S_gf=S_U(1) +s(_A^(a_,^ij)+B2),', '1602.01333-1-121-0': 'S_gf&=B(_A^+B2)- s(_A^) =12(B+_A^)^2 -12(_A^)^2-_(sA^),', '1602.01333-1-122-0': 'with [MATH] being regular U(1) BRST transformations [MATH], where [MATH] is the U(1) ghost.', '1602.01333-1-122-1': 'Next we use consistency condition for SW map to get [MATH], where [MATH] is [MATH] ghost and [MATH] the [MATH] covariant derivative in the adjoint representation [MATH].', '1602.01333-1-123-0': 'Since the SW map for [MATH] is actually the same as for the NC gauge parameter [MATH] from [CITATION] we can derive the photon-photon and photon-ghost coupling in this gauge.', '1602.01333-1-123-1': 'So by using the following BRST transformations [EQUATION] from ([REF]) is produced the gauge fixing action [EQUATION] which after the application of the SW map resulting Feynman rules for the gauge fixing and ghost induced diagrams given in Appendix F.', '1602.01333-1-123-2': 'We name this gauge as "noncommutative (NC) Feynman gauge" as it is formally identical to the Feynman gauge in a [MATH] gauge theory.', '1602.01333-1-124-0': '## One-loop contributions from the new NC gauge fixing action', '1602.01333-1-125-0': 'The new gauge fixing action [REF] introduces additional terms to the three and four photon self-couplings as well as photon-ghost couplings, as summarized in [REF].', '1602.01333-1-125-1': 'Unlike the three and four photon couplings in the commutative Feynman gauge [CITATION], these new interaction terms are no longer transverse.', '1602.01333-1-125-2': 'It then becomes intriguing how the sum of the resulting loop integrals behave.', '1602.01333-1-126-0': 'From Feynman rules [REF] we find following diagrams Fig.[REF]-[REF] contributing to the one loop photon two point function.', '1602.01333-1-127-0': 'Denoting the total sum of Fig.[REF]-[REF] as [MATH], it turns out to be convenient to split it into two partial sums [EQUATION]', '1602.01333-1-127-1': 'Here [MATH] presents the sum over sums Figures [REF] and [REF], which contain one 3-photon vertex from the classical action and the other from gauge fixing action, while [MATH] sums over the rest of them which are solely from the gauge fixing.', '1602.01333-1-128-0': 'Evaluation of the diagrams Fig. [REF]-[REF] follows substantially the standard procedure used in the prior section, except the rising of two new types of tadpole integrals.', '1602.01333-1-128-1': 'The first one takes the form of the second term in [REF] so it can be removed by our regularization prescription.', '1602.01333-1-128-2': 'The second one is a tadpole integral without any loop momenta in the numerator i.e. [MATH].', '1602.01333-1-128-3': 'This integral contains a total effective loop momenta power [MATH] because of the additional denominator [MATH] from the nonlocal factor [MATH], which is below the minimal power for the commutative tadpole integral to vanish [CITATION].', '1602.01333-1-128-4': 'Consequently we observe unregularized UV divergence when computing this integral by transforming it to bubble or applying the n-nested zero regulator method [CITATION].', '1602.01333-1-128-5': 'We develop an alternative prescription [REF] based on the parametrization [REF] which is capable of dimensionally-regularizing this integral into a [MATH] divergence plus the logarithmic UV/IR mixing term [MATH] at the [MATH] limit [REF].', '1602.01333-1-129-0': 'We are capable of express both [MATH] and [MATH] appropriately once [MATH] is added to the prior basis integral set [MATH], [MATH], [MATH] and [MATH].', '1602.01333-1-129-1': 'The outcome is listed as below [EQUATION]', '1602.01333-1-129-2': 'One can immediately notice that [MATH] contains only two tensor structures [MATH] and [MATH] which can not be combined into a transverse sum.', '1602.01333-1-129-3': 'The loss of transversality appears to be, of course, surprising.', '1602.01333-1-129-4': 'However we are going to develop the understanding to this seeming odd fact in the next section and show that it is actually reasonable.', '1602.01333-1-130-0': '# Gauge fixing contribution without integrating out BRST-auxiliary field', '1602.01333-1-131-0': 'In order to achieve simple transversality we conclude that one have to keep BRST auxiliary field [MATH] from being integrated out.', '1602.01333-1-131-1': 'Arguments for that are as follows.', '1602.01333-1-131-2': 'Starting with the action ([REF]) we write a generating functional [EQUATION] from where we have effective action in terms of "currents": [EQUATION]', '1602.01333-1-131-3': 'Regular BRST transformation [MATH] acting on [MATH] vanish, thus we have: [EQUATION]', '1602.01333-1-131-4': 'Since the transfersality condition means [MATH], which is satisfied in equation ([REF]) only for [MATH].', '1602.01333-1-131-5': 'However this is not allowed if we integrate out the [MATH] field.', '1602.01333-1-131-6': 'Thus we do not do that, instead we construct propagator from the following doublet combination [MATH].', '1602.01333-1-132-0': '## Formal analysis', '1602.01333-1-133-0': 'In order to compute the two point function(s) within the presence of the B-field, we must define the propagator(s) for the "kind of strange" vector-scalar field [MATH] doublet.', '1602.01333-1-133-1': 'Starting with [EQUATION] we get a quadratic part of [MATH] [EQUATION] whose Fourier transform is as follows [EQUATION]', '1602.01333-1-133-2': 'The Hermitian matrix [EQUATION] is then the inverse of the propagators in the momentum space.', '1602.01333-1-133-3': 'Next we inverse [MATH] to get the propagators [EQUATION]', '1602.01333-1-133-4': 'From ([REF]) and ([REF]) we have four equations:', '1602.01333-1-134-0': '& _^=-k^2 G_+k^G_ k^+iA_k^,0=-ik^G_+A_ & 0=-k^2 B^+k^B_k^+iG k^,1=-iB_k^+ 1,', '1602.01333-1-135-0': 'which we solve by using simple Ansatz: [MATH], [MATH].', '1602.01333-1-135-1': 'Taking into account a text book convention [MATH] for the phase factor we add overall factor [MATH] and obtained correct photon propagator, as is illustrated in Fig [REF].', '1602.01333-1-136-0': 'One can write down the usual field redefinition [MATH] in [REF] in the fourier transformed context as [EQUATION]', '1602.01333-1-136-1': 'The [MATH] then diagonalizes the bilinear form [REF] [EQUATION]', '1602.01333-1-136-2': 'It is easy to see that inverting [MATH] gives the expected Feynman propagator.', '1602.01333-1-136-3': 'From this viewpoint the B-integration can be achieved by diagonalization.', '1602.01333-1-137-0': 'Next, one can formally generalize the tree level diagonalization procedure to one loop.', '1602.01333-1-137-1': 'Consider in general the one loop corrections as another Hermitian matrix [MATH] adding to the [MATH] matrix, we can write the quadratic part of the 1-loop corrected effective action in the momentum space as follows [EQUATION]', '1602.01333-1-137-2': 'Next we express [MATH] in terms of its components [EQUATION]', '1602.01333-1-137-3': 'The Slavnov-Taylor identity [REF] then requires that [MATH] must be transverse, while others not.', '1602.01333-1-137-4': 'One can now replace [MATH] by a new transformation [EQUATION] with [MATH] diagonalizing the 1-loop corrected bilinear form [MATH] [EQUATION] where [EQUATION]', '1602.01333-1-137-5': 'We then conjecture that the formal leading order expansion of [MATH] with respect to coupling constant [MATH] corresponds to the gauge fixing corrections to the 1-loop 1-PI photon two point function, i.e. [EQUATION]', '1602.01333-1-137-6': 'And, as we will see below, this relation/conjecture indeed holds.', '1602.01333-1-138-0': '## The action of the gauge and BRST-auxiliary fields and Feynman rules', '1602.01333-1-139-0': 'We define the noncommutative photon-auxiliary field action by using the first and second order SW maps for the NC gauge field, [MATH] and [MATH], respectivly: [EQUATION] from where we obtain the corresponding photon-auxiliary field interaction vertices.', '1602.01333-1-139-1': 'Corresponding Feynman rules from the above action generates one-loop correction to the quadratic effective action and are given in Appendix G.', '1602.01333-1-140-0': '## One-loop contributions to photon propagator up to the quadratic order, from the auxiliary BRST field [MATH]', '1602.01333-1-141-0': 'Based on our vertex read-out convention, we the following correspondence rule between the matrix elements of one loop correction [MATH] and the 1-PI loop diagrams [EQUATION]', '1602.01333-1-141-1': 'Note that [MATH] denotes all contributions from the classical action, which is the same as summing over all contributions to photon two point function computed in sections 2-5 and thus transverse.', '1602.01333-1-141-2': 'Explicit computation first revolves that [EQUATION] thus the Slavnov-Taylor identity is actually trivially fulfilled.', '1602.01333-1-142-0': 'The rest of the matrix elements listed in [REF] are nonzero and boils down to the following expressions [EQUATION] where', '1602.01333-1-143-0': '_I=(4^2)^2-D2(p^2)^D2-2(D-2)(3-D2)B(D2-1, D2-1)_D4--(4I_K^0+6 I_H),', '1602.01333-1-144-0': '_II=p^2(p)^2(&(4^2)^2-D2(p^2)^D2-2(D-1)(3-D2)B(D2-1, D2-1)_D4-', '1602.01333-1-145-0': '&-18(p)^2 T_-2-(6I_K^0+6I_H)),', '1602.01333-1-146-0': 'and', '1602.01333-1-147-0': '_mix^e^2(4)^2(p^+(p)^p^2(p)^2)(4+4 I_H).', '1602.01333-1-148-0': 'Finally [EQUATION]', '1602.01333-1-148-1': 'Next we start to verify our conjecture [REF].', '1602.01333-1-148-2': 'First we derive the following relations from it [EQUATION]', '1602.01333-1-148-3': 'One then immediately observes that the second and third relations do fulfill.', '1602.01333-1-148-4': 'As for the first one we can compute its right hand side [EQUATION] which is in agreement with ([REF]).', '1602.01333-1-148-5': 'Thus the conjectured relation [REF] is proven.', '1602.01333-1-149-0': 'Retrospectively, we see that the gauge fixing contribution to the 1-loop correction is a shift of the on-shell value of the gauge fixing functional.', '1602.01333-1-149-1': 'Therefore it does not need to be transverse.', '1602.01333-1-149-2': 'The fact that the pure gauge fixing and mixing contributions satisfies [REF] independently is because the former can be considered as gauge fixing to a free [MATH] gauge theory.', '1602.01333-1-150-0': "Finally let's briefly discuss the divergences in the gauge fixing configuration(s).", '1602.01333-1-150-1': 'Using the results in Appendix B we can see that [MATH] contains no quadratic IR divergent term, therefore the quadratic IR divergence cancelation we found in the prior sections are also preserved under this gauge fixing choice.', '1602.01333-1-150-2': 'We can also extract the UVlogarithmic divergence at the [MATH] limit', '1602.01333-1-151-0': '_A_UV=-(2-p^2(p)^2 (3trtheta(p)^2+4(p)^2))(2+(^2(p)^2)),', '1602.01333-1-152-0': '_B_UV=-72p^2(p)^2(2+(^2(p)^2)),', '1602.01333-1-153-0': 'while the [MATH] is finite at this limit.', '1602.01333-1-153-1': 'We conclude our analysis by listing of all UV plus log divergences in [MATH], which can be decomposed into seven symmetric tensor structures [EQUATION]', '1602.01333-1-153-2': 'Explicit computation then yields', '1602.01333-1-154-0': '_1_UV= p^2(43-43n_f-13n_s+p^2(p)^4(3trtheta(p)^2+4(p)^2))(2+(^2(p)^2)),', '1602.01333-1-155-0': '_2_UV=(-2-43+43n_f+13n_s)(2+(^2(p)^2)),', '1602.01333-1-156-0': '_3_UV=B_2_UV=2p^2(p)^2(2-p^2(trtheta)(p)^2 )(2+(^2(p)^2)),', '1602.01333-1-157-0': '_4_UV=B_3_UV=2p^2(p)^2(2+(^2(p)^2)),', '1602.01333-1-158-0': '_5_UV=B_4_UV=-4p^4(p)^4(2+(^2(p)^2)),', '1602.01333-1-159-0': '_6_UV=B_5_UV=4p^4(p)^4 (2 + (^2(p)^2)),', '1602.01333-1-160-0': '_7_UV=_B_UV=-72p^2(p)^2(2 +(^2(p)^2)).', '1602.01333-1-161-0': '# Summary and discussion', '1602.01333-1-162-0': 'In this paper we have computed the one-loop contributions to all the propagators of the noncommutative super Yang-Mills U(1) theory with [MATH]=1, 2 and 4 supersymmetry and defined by means of the [MATH]-exact Seiberg-Witten map.', '1602.01333-1-162-1': 'We have shown that -for [MATH]=1, 2 and 4- the quadratic noncommutative IR divergence, [EQUATION] -a trade-mark of noncommutative gauge theories- which occur in the bosonic, fermionic and scalar contributions to photon propagator cancel each other, rendering a photon propagator free of them as befits Supersymmetry.', '1602.01333-1-162-2': 'Indeed, from [REF] for [MATH], one gets: [EQUATION]', '1602.01333-1-162-3': 'That this cancellation occurs in the case at hand is nontrivial since Supersymmetry acts nonlinearly -see ([REF])- on the ordinary fields.', '1602.01333-1-162-4': 'Let us recall that the cancellation of quadratic noncommutative IR divergences is also a feature of noncommutative super Yang-Mills theories when formulated in terms of noncommutative field [CITATION].', '1602.01333-1-162-5': 'Hence, our result concerning the cancellation of the quadratic noncommutative IR divergences really points in the direction that the [MATH]-exact Seiberg-Witten map really provides quantum duals of the same underlying theory.', '1602.01333-1-163-0': 'We have also shown -see ([REF]) and ([REF])- that the characteristic quadratic noncommutative IR divergences, [EQUATION] which arise in the individual contributions to the one-loop propagators of the scalar fields in the [MATH]=2 and 4 Supersymmetry, also cancel each other at the end of the day.', '1602.01333-1-164-0': 'We have seen that the previous cancellations occur both in the ordinary Feynman gauge and in the noncommutative Feynman gauge -see subsection 7.4-, so our computations further indicate that the cancellation is robust against changing the gauge fixing and may have real physical, and therefore gauge invariant, content.', '1602.01333-1-164-1': 'Let us recall that independence of gauge-fixing parameter of the cancellation of noncommutative IR divergences in the dual theory, i.e., in [MATH]=1 U(1) super Yang-Mills theory formulated in terms of the noncommutative fields, has been shown to hold -see Ref. [CITATION].', '1602.01333-1-165-0': 'In this paper we have also worked out explicitly the one-loop UV divergent contributions -which show as poles at [MATH]- to all propagator of the theory: see ([REF]), ([REF])-([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF])-([REF]).', '1602.01333-1-165-1': 'It is noticeable that the pole parts displayed in the equations we have just quoted contain non-polynomial, ie, non-local, terms whose denominator is a power of [MATH].', '1602.01333-1-165-2': "In this regard we would like to point out that, in keeping with Weinberg's power counting theorem [CITATION], Feynman integrals whose degree of UV divergence is not the same along all directions are liable to give rise to pole contributions which are non-polynomial.", '1602.01333-1-165-3': 'This is the situation in our case, for our integrands contain factors of the type [EQUATION] and these factors approach zero as [MATH] along the direction parallel to [MATH] and as [MATH] along any direction orthogonal to [MATH].', '1602.01333-1-165-4': 'Hence UV divergences with a non-polynomial dependence on the momenta may occur and our computations show that indeed they do occur.', '1602.01333-1-165-5': 'We would like to recall that a similar situation, -i.e. the non-polynomial UV divergences- happens in ordinary Yang-Mills theories in the light-cone gauge [CITATION].', '1602.01333-1-166-0': 'Now, the UV divergences of two-point functions are in general gauge dependent quantities.', '1602.01333-1-166-1': 'We have verified that this is so in our case by computing the one-loop propagator of the gauge field both in the ordinary Feynman gauge and in the noncommutative Feynman gauge -see Section 6.', '1602.01333-1-166-2': 'The result for the first type of gauge is in ([REF])-([REF]) and in ([REF])-([REF]) for the second type of gauge fixing term: their differences stand out.', '1602.01333-1-166-3': 'Hence, extracting gauge invariant information from the UV divergences is the next challenge along this line of research and it will require the computation of three and higher point functions.', '1602.01333-1-167-0': 'Let us finally remark that UV/IR mixing effects also work for the non-polynomial UV divergent contributions we have obtained.', '1602.01333-1-167-1': 'Indeed, as seen in ([REF]), ([REF])-([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF])-([REF]) every pole in [MATH] comes hand in hand with the logarithmic noncommutative IR divergence [MATH].', '1602.01333-1-167-2': 'The reader is referred to the final part of Appendix B for further information regarding this issue.'}

{'1602.01333-2-0-0': '# Introduction', '1602.01333-2-1-0': 'The one-loop UV/IR mixing structure of noncommutative (NC) [MATH]=1 super Yang-Mills theory defined in terms of the noncommutative fields was studied some years ago in a number of papers [CITATION].', '1602.01333-2-1-1': 'The outcome was the famous quadratic noncommutative IR divergences which occur in the one-loop gauge field propagator of the non-supersymmetric version of the theory cancel here due to Supersymmetry.', '1602.01333-2-1-2': 'The one-loop gauge field propagator still carries a logarithmic UV divergence -a simple pole in Dimensional Regularization- and the dual logarithmic noncommutative IR divergence [MATH] as a result of the UV/IR mixing being at work.', '1602.01333-2-1-3': 'By increasing the number of supersymmetries of the noncommutative Yang-Mills theory one makes the UV behaviour of the theory softer and eventually finite for [MATH] [CITATION], at which point noncommutative IR divergences cease to exist [CITATION].', '1602.01333-2-1-4': 'In the [MATH] super Yang-Mills case, there still remain logarithmic UV divergences at one-loop in the two-point function which give rise via UV/IR mixing to the corresponding IR divergences [CITATION].', '1602.01333-2-1-5': 'That noncommutative [MATH] super Yang-Mills has a smooth commutative limit has been shown in Ref. [CITATION].', '1602.01333-2-2-0': 'It is known that classically noncommutative gauge field theories admit a dual formulation in terms of ordinary fields, a formulation that is obtained by using the celebrated Seiberg-Witten map [CITATION].', '1602.01333-2-2-1': 'However we still do not know whether this duality holds at the quantum level, i.e., whether the quantum theory defined in terms of noncommutative fields is the same as the ordinary quantum theory -called the dual ordinary theory- whose classical action is obtained from the noncommutative action by using the Seiberg-Witten map.', '1602.01333-2-2-2': 'The existence of the UV/IR mixing effects in noncommutative field theory defined in terms of the noncommutative fields is thought to be a characteristic feature of those field theories.', '1602.01333-2-2-3': 'It is thus sensible to think such effects should also occur in the ordinary dual theory obtained, as previously explained, by using the Seiberg-Witten map.', '1602.01333-2-2-4': 'That these UV/IR mixing effects actually occur in the propagator of the gauge field of the dual ordinary theory was first shown in Ref. [CITATION] by using the [MATH]-exact Seiberg-Witten map expansion [CITATION].', '1602.01333-2-2-5': 'The analysis of the properties and phenomenological implications of the UV/IR mixing effects that occur in noncommutative gauge theories defined by means of the [MATH]-exact Seiberg-Witten map has been pursued in Refs. [CITATION].', '1602.01333-2-3-0': 'Up to the best of our knowledge no analysis of the UV and the noncommutative IR structures of the noncommutative super Yang-Mills theory defined by means of the [MATH]-exact Seiberg-Witten map has been carried out as yet.', '1602.01333-2-3-1': 'In particular, it is not known whether the cancellation of the quadratic noncommutative IR divergences of the gauge-field propagator that occurs, as we mentioned above, in noncommutative super Yang-Mills theory defined in terms of noncommutative fields also works in its dual ordinary theory.', '1602.01333-2-3-2': 'Answer to this question is far from obvious since Supersymmetry is linearly realized in terms of the noncommutative fields -and thus there exists a superfield formalism- but is non-linearly realized -see Ref. [CITATION]- in terms of the ordinary fields of the dual ordinary theory defined by means of the Seiberg-Witten map.', '1602.01333-2-3-3': 'It has long been known that the proper definition of theories with non-linearly realized symmetries is a highly non-trivial process.', '1602.01333-2-4-0': 'The purpose of this paper is to work out all the one-loop 1PI two-point functions, and analyze the UV and noncommutative IR structures of those functions, in noncommutative U(1) [MATH]=1,2 and 4 super Yang-Mills theories in the Wess-Zumino gauge, when those theories are defined in terms of ordinary fields by means of the [MATH]-exact Seiberg-Witten maps.', '1602.01333-2-4-1': 'To analyze the gauge dependence of the UV and noncommutative IR of the gauge field two-point functions we shall consider two types of gauge-fixing terms for the ordinary gauge field: the standard ordinary Feynman gauge-fixing term and the noncommutative Feynman gauge-fixing term.', '1602.01333-2-5-0': 'The layout of this paper is as follows.', '1602.01333-2-5-1': 'Section 2 is devoted to the computation of the one-loop contributions to the photon and photino propagators in [MATH] super Yang-Mills U(1) theory in the ordinary Feynman gauge.', '1602.01333-2-5-2': 'In section 3 we discuss, for later use, the construction by using the [MATH]-exact Seiberg-Witten map of a noncommutative U(1) theory with a noncommutative scalar field transforming under the adjoint representation.', '1602.01333-2-5-3': 'The one-loop propagators of the ordinary fields of noncommutative [MATH]=2 and 4 super Yang-Mills U(1) theories defined by using the [MATH]-exact Seiberg-Witten map are worked out in sections 4 and 5 in the ordinary Feynman gauge.', '1602.01333-2-5-4': 'In sections 6 and 7 we use a noncommutative Feynman gauge to quantize [MATH] super Yang-Mills U(1) theory and compute the one-loop photon propagator.', '1602.01333-2-5-5': 'Sections 6 and 7 are introduced to analyze the dependence on the gauge-fixing term of the UV and noncommutative IR contributions found in previous sections.', '1602.01333-2-5-6': 'The overall discussion of our results is carried out in Section 8.', '1602.01333-2-5-7': 'We also include several appendices which are needed to complete properly the analysis and computations carried out in the body of the paper.', '1602.01333-2-6-0': '# Noncommutative [MATH]=1 SYM U(1) theory and the [MATH]-exact SW map', '1602.01333-2-7-0': 'The noncommutative field content of the noncommutative U(1) super Yang-Mills theory in the Wess-Zumino gauge is the noncommutative gauge field [MATH], its supersymmetric fermion partner [MATH] and the noncommutative SUSY-auxiliary field [MATH].', '1602.01333-2-7-1': 'The ordinary/commutative counterparts of [MATH], [MATH] and [MATH] will be denoted by [MATH] (photon), [MATH] (photino) and [MATH], respectively.', '1602.01333-2-7-2': 'Regarding dotted and undotted fermions and [MATH] traces, we shall follow the conventions of [CITATION].', '1602.01333-2-8-0': 'In terms of the noncommutative fields and in the Wess-Zumino gauge the action of U(1) super Yang-Mills theory reads [EQUATION] where [MATH] and [MATH].', '1602.01333-2-9-0': 'The above action [MATH] ([REF]), is invariant under the following noncommutative supersymmetry transformations [EQUATION]', '1602.01333-2-9-1': 'These supersymmetry transformations close on translations modulo noncommutative gauge transformations and tell us that supersymmetry is linearly realized on the noncommutative fields-see [CITATION] for further discussion.', '1602.01333-2-9-2': 'The action in [REF], is also invariant under noncommutative U(1) transformations, which in the noncommutative BRST form read [EQUATION] with [MATH] being the noncommutative U(1) ghost field.', '1602.01333-2-9-3': 'The above action [MATH] can be expressed in terms of ordinary fields, [MATH], [MATH] and [MATH], by means of the SW map.', '1602.01333-2-9-4': 'The resulting functional is invariant under ordinary U(1) BRST transformations: [EQUATION] where [MATH] is the ordinary U(1) ghost field.', '1602.01333-2-9-5': 'Indeed, the SW map maps ordinary BRST orbits into the noncommutative BRST orbits.', '1602.01333-2-10-0': 'The [MATH]-exact SW map for [MATH] has been worked out in [CITATION] up to the three ordinary U(1) gauge fields [MATH].', '1602.01333-2-10-1': 'It reads [EQUATION] where, up to the [MATH] order, the gauge field strength SW map [MATH] expansion is fairly universal [CITATION] [EQUATION]', '1602.01333-2-10-2': 'The [MATH] order SW map for the gauge field strength from [CITATION] in that case reads [EQUATION]', '1602.01333-2-10-3': 'The generalized star products relevant for this work are defined as follows [CITATION]', '1602.01333-2-11-0': '(f_2 g)(x)&=e^-i(p+q)xf(p)g(q)f__2(p,q),', '1602.01333-2-12-0': "[f g h]__3'(x)&=e^-i(p+q+k)xf(p)g(q)h(k)f__3(p,q,k),", '1602.01333-2-13-0': '[f g h]_M_(I)(x)&=e^-i(p+q+k)xf(p)g(q)h(k)f_(I)(p,q,k),', '1602.01333-2-14-0': 'with', '1602.01333-2-15-0': 'f__2(p,q)&=pq2pq2,', '1602.01333-2-16-0': "f__3'(p,q,k)&=(pq2+pk2-qk2)-1(pq2+pk2-qk2)qk2-(pq2+pk2+qk2)-1(pq2+pk2+qk2)qk2,", '1602.01333-2-17-0': "f_(I)(p,q,k)&=1pq(f__3'[p,q,-(p+q+k)]-f__3'[p,q,k]).", '1602.01333-2-18-0': 'The [MATH]-exact SW map for [MATH] up to the two ordinary fields [MATH] can be retrieved from the expression for [MATH] in [REF], [REF] and [REF] as explained in the appendix A: [EQUATION] where [EQUATION] and [EQUATION]', '1602.01333-2-18-1': 'To compute the full one-loop photon two-point function, one also needs the SW maps for the [MATH] and [MATH] fields.', '1602.01333-2-18-2': 'They read', '1602.01333-2-19-0': "_=&_-e^ija_i_2_j _+e^24^ij^kl([a_i_j(a_k_l _)]__3'", '1602.01333-2-20-0': "&-[a_i(f_jk_l_-a_k_l _j_)]__3' +[_j _a_k(_l a_i+f_li)]__3')+O(e^3),", '1602.01333-2-21-0': "D^(nc)=&D-e^ija_i_2_j D+e^24^ij^kl([a_i_j(a_k_l D)]__3'", '1602.01333-2-22-0': "&-[a_i(f_jk_lD-a_k_l _jD)]__3' +[_j Da_k(_l a_i+f_li)]__3')+O(e^3).", '1602.01333-2-23-0': 'Let us stress that the way we have constructed-by appropriate restriction of the SW map for the gauge-field-the SW map for [MATH] and [MATH] is very much in harmony with the idea that if supersymmetry and gauge symmetry are not to clash, the superpartners must have similar behavior with respect to the gauge group.', '1602.01333-2-24-0': 'As discussed in [CITATION] the noncommutative supersymmetric transformations in [REF] can be understood as the push-forward under the SW map of the appropriate [MATH]-dependent supersymmetric transformations of the ordinary fields.', '1602.01333-2-24-1': 'Here we have worked out the [MATH]-exact expression for such deformed transformations up to the order [MATH], [EQUATION] where [EQUATION]', '1602.01333-2-24-2': 'The reader shall find in the appendix A the values of objects in the previous equations that have not been given yet.', '1602.01333-2-25-0': 'The [MATH]-exact deformed supersymmetry transformations given in [REF] and [REF] can be rightly called supersymmetry transformations since, as shown in [CITATION], they close on translations modulo gauge transformations and, hence, they carry a representation of the supersymmetry algebra.', '1602.01333-2-25-1': 'However notice that these deformed supersymmetry transformations of the ordinary field do not realize the supersymmetry algebra linearly.', '1602.01333-2-25-2': 'Finally, since these supersymmetry transformations generate the noncommutative supersymmetry transformations of [REF], we conclude that the total [MATH]-exact action (given explicitly in the next subsection) has to be invariant up to the second order in [MATH], under the deformed supersymmetry transformations in [REF].', '1602.01333-2-26-0': '## The action', '1602.01333-2-27-0': 'Now, substituting into [REF], the Seiberg-Witten maps from [REF], [REF] and [REF] and dropping any contribution of order [MATH], one obtains the SYM U(1) action in terms of commutative fields: [EQUATION] where [EQUATION] and [EQUATION]', '1602.01333-2-27-1': 'First we note that, since the Feynman rules of the 3- and 4-photon self-couplings ([REF]), are already given in previous papers [CITATION] and [CITATION], respectively, we shall not repeat them here.', '1602.01333-2-27-2': 'Photino-photon Feynman rules, obtained from ([REF]), are given explicitly in the appendix C.', '1602.01333-2-28-0': '## The photon one-loop contributions to the photon polarization tensor', '1602.01333-2-29-0': 'Most generally speaking, the total photon one-loop 1PI two-point function [MATH] in the [MATH] SYM theory is the sum of the following contributions [EQUATION] where [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] refer to the contributions from the photon bubble and tadpole, the photino bubble and tadpole, and the adjoint scalar bubble and tadpole diagrams, respectively.', '1602.01333-2-29-1': 'The last two diagrams appear only in the extended SUSY, of course.', '1602.01333-2-29-2': 'We use [MATH] for the number of photinos (Weyl fermions) and [MATH] for the number of real adjoint scalar bosons (one complex scalar is counted as two real scalars), which are uniquely determined by [MATH] supersymmetry.', '1602.01333-2-30-0': 'Explicit computation revolves that each of these diagrams can be expressed as a linear combinations of five transverse tensor structures [EQUATION]', '1602.01333-2-30-1': 'The sum [REF] can be expressed, in the language of the five tensor decomposition ([REF]), as [EQUATION]', '1602.01333-2-30-2': 'In the subsequent sections we are going to compute and give the coefficients [MATH], [MATH], [MATH] and [MATH], [MATH] and [MATH] in detail via equations [REF], [REF], [REF], [REF] and [REF], respectively.', '1602.01333-2-31-0': 'For the [MATH] theory [MATH], [MATH], thus from ([REF]) we have [EQUATION]', '1602.01333-2-31-1': "In this section we are going to show that all quadratic IR divergences cancel in each of the [MATH]'s, then we extend our results to the [MATH] theories as well.", '1602.01333-2-32-0': 'We choose one specific set of four (five in the sections 6 and 7) nonplanar/special function integrals [MATH], [MATH], [MATH] and [MATH] alongside the usual planar/commutative UV divergent integrals to express all loop diagrams/coefficients in this article.', '1602.01333-2-32-1': 'This decomposition enjoys the advantage that each nonplanar integral bear distinctive asymptotic behavior in the IR regime: [MATH] carries all the quadratic IR divergence [MATH], with a pre-factor [MATH], while [MATH] and [MATH] contain the dual logarithmic noncommutative IR divergence [MATH], with pre-factors [MATH] and [MATH], respectively.', '1602.01333-2-32-2': 'The last integral [MATH] is finite at the IR limit.', '1602.01333-2-32-3': 'Further details of these integrals are given in the appendix B.', '1602.01333-2-33-0': '### The photon-bubble diagram', '1602.01333-2-34-0': 'From the photon bubble diagram Fig. [REF] we obtain the following loop-coefficients [CITATION] [EQUATION]', '1602.01333-2-34-1': "Extracting the divergent parts from each of the [MATH]'s", '1602.01333-2-35-0': 'B_1(p)&=+(43 +3 p^2(trtheta)(p)^2 +4 p^2(p)^2(p)^4) (2 + (^2(p)^2)) +finite terms,', '1602.01333-2-36-0': 'B_2(p)&=+2p^2(p)^2(2-p^2(trtheta)(p)^2 )(2 + (^2(p)^2))+3231(p)^4+finite terms,', '1602.01333-2-37-0': 'B_3(p)&=+2p^2(p)^2 (2 + (^2(p)^2))-3231(p)^4+finite terms,', '1602.01333-2-38-0': 'B_4(p)&=-4p^4(p)^4(2 + (^2(p)^2))-323p^2(p)^6+finite terms,', '1602.01333-2-39-0': 'B_5(p)&=+4p^4(p)^4 (2 + (^2(p)^2))+finite terms,', '1602.01333-2-40-0': 'we observe the presence of the UV plus logarithmic IR divergences in all of them.', '1602.01333-2-40-1': 'The logarithmic IR divergences from both planar and nonplanar sources in the bubble diagram appear to have identical coefficient and combine into a single [MATH] term, confirming the expected UV/IR mixing.', '1602.01333-2-40-2': 'The quadratic IR divergence, on the other hand, exists only in the [MATH] terms.', '1602.01333-2-41-0': '### The photon-tadpole diagram', '1602.01333-2-42-0': 'From tadpole Fig. [REF] we obtain the same tensor structure as from the photon bubble diagram (Fig. [REF]) with the following loop-coefficients [MATH]: [EQUATION]', '1602.01333-2-42-1': 'We notice immediately the absence of UV plus logarithmic divergent terms contrary to the photon-bubble diagram results ([REF]).', '1602.01333-2-42-2': 'In addition, the tadpole diagram produces no finite terms either, and the quadratic IR are again present in the second, third and fourth term!', '1602.01333-2-43-0': '## The photino one-loop contributions to the photon polarization tensor', '1602.01333-2-44-0': 'The photino sector contains two diagrams: photino tadpole Fig. [REF] and photino bubble Fig. [REF].', '1602.01333-2-44-1': 'We are going to see below that only the latter contributes to the photon polarization tensor.', '1602.01333-2-45-0': '### The photino-tadpole diagram', '1602.01333-2-46-0': 'The photino-tadpole contribution is computed using vertex ([REF]).', '1602.01333-2-47-0': 'It produces only the quadratic IR divergent terms which cancel each other internally, thus giving vanishing contribution to the photon polarization tensor [EQUATION]', '1602.01333-2-48-0': '### The photino-bubble diagram', '1602.01333-2-49-0': 'Taking the photino-photon Feynman rules from appendix C we obtain', '1602.01333-2-50-0': 'the following photino-bubble contribution to the photon polarization tensor, [MATH], from Fig. [REF]: [EQUATION]', '1602.01333-2-50-1': 'Taking into account the trace [EQUATION] and that [MATH] cannot have, at the end of the day, contributions depending on [MATH], one arrives at [EQUATION]', '1602.01333-2-50-2': 'After some amount of computations we find that only first two of the general five-terms structure ([REF]) survive in [MATH] dimensions: [EQUATION]', '1602.01333-2-50-3': 'After inspecting the divergences in these two terms we also find that the first of the two, [MATH], contain the logarithmic UV/IR mixing terms, while [MATH] possesses only quadratic IR divergence and finite terms, as the dual NC logarithmic divergences from integrals [MATH] and [MATH] cancel each other.', '1602.01333-2-51-0': "Summing over [REF], [REF] and [REF] one can see that the total quadratic IR divergences in all [MATH]'s are zero.", '1602.01333-2-51-1': 'The total UV divergences for the [MATH] theory are as follows', '1602.01333-2-52-0': '_1(p)_UV=p^2(p)^4(3(trtheta)(p)^2+4(p)^2)(2+(^2(p)^2)),', '1602.01333-2-53-0': '_2(p)_UV=p^2(p)^2(2-p^2(trtheta)(p)^2 )(2+(^2(p)^2)),', '1602.01333-2-54-0': '_3(p)_UV=p^2(p)^2(2+(^2(p)^2)),', '1602.01333-2-55-0': '_4(p)_UV=-4p^4(p)^4(2+(^2(p)^2)),', '1602.01333-2-56-0': '_5(p)_UV=4p^4(p)^4 (2 + (^2(p)^2)).', '1602.01333-2-57-0': '## The one-loop SUSY-auxiliary field contributions to the photon propagator', '1602.01333-2-58-0': 'The free two-point function of the SUSY-auxiliary field [MATH] reads [EQUATION] hence, the integrals to be computed in dimensional regularization are of the type [EQUATION]', '1602.01333-2-58-1': 'The integrals in [REF] vanish in dimensional regularization and hence the SUSY-auxiliary field [MATH] does not contribute to the one-loop photon propagator.', '1602.01333-2-58-2': 'Indeed, following [CITATION], we first split the [MATH]-dimensional [MATH] into [EQUATION] where [MATH] belongs to [MATH] dimensional space orthogonal to [MATH], and [MATH] is the modulus of [MATH] -recall that [MATH] is a space-like vector, since [MATH].', '1602.01333-2-58-3': 'Then introduce the following definition of the dimensionally regularized integral in [REF]: [EQUATION]', '1602.01333-2-58-4': 'However, in dimensional regularization -see [CITATION]- [EQUATION] which in turn leads to the conclusion that the integral [REF] vanishes under the dimensional regularization procedure.', '1602.01333-2-59-0': 'It is not difficult to see that the argument above can be generalized to integrals with positive [MATH] powers too, i.e. [EQUATION]', '1602.01333-2-59-1': 'One can explicitly verify two special cases of the identity above [EQUATION] using a generalization of the n-nested zero regulator method [CITATION].', '1602.01333-2-60-0': '## The one-loop photino 1-PI two point function', '1602.01333-2-61-0': 'The photino self-energy consists two diagrams, a tadpole Fig. [REF] and a bubble Fig. [REF].', '1602.01333-2-61-1': 'Explicit computation shows that the tadpole diagram Fig. [REF] vanishes: [EQUATION]', '1602.01333-2-61-2': 'The bubble diagram was computed in [CITATION], which boils down to the following expressions [EQUATION] with [EQUATION] and [EQUATION]', '1602.01333-2-61-3': 'One can easily notice the absence of the quadratic IR divergent integral [MATH].', '1602.01333-2-61-4': 'The UV divergence can be expressed as follows [EQUATION]', '1602.01333-2-62-0': '# Minimal action of the noncommutative adjoint scalar field', '1602.01333-2-63-0': 'It is commonly known that extended, [MATH], super YM theories contain not only fermion (photino) but also scalar bosons in the adjoint representation.', '1602.01333-2-63-1': 'These scalar bosons couple minimally to the gauge field, and their action for the real scalar is [EQUATION] or [EQUATION] for the complex scalar.', '1602.01333-2-63-2': "We study the minimal interacting scalar boson's contribution to 1-PI photon two point function as well as the scalar's own 1-PI two point function in this section.", '1602.01333-2-63-3': 'These results will be used for our discussion on [MATH] SYM in the subsequent sections.', '1602.01333-2-64-0': 'It is straightforward to derive the SW map expansion of either [MATH] or [MATH] using the method described in the appendix A[EQUATION] with the product [MATH] being defined in [CITATION].', '1602.01333-2-64-1': 'One can show that [EQUATION] if we express one complex scalar in terms of two real scalars.', '1602.01333-2-64-2': 'For this reason one complex scalar contribution to the photon 1-PI two point function is twice as one real scalar, while the photon contribution to the complex scalar two point function is the same as for the real scalar two point function.', '1602.01333-2-64-3': 'Thus, we shall compute only those for the real scalar field.', '1602.01333-2-64-4': 'The scalar-photon Feynman rules are given in the appendix D.', '1602.01333-2-65-0': '## Scalar one-loop contributions to the photon polarization tensor', '1602.01333-2-66-0': 'Like the photino sector, the adjoint scalar sector contains also two diagrams that contribute to the photon polarization tensor, the scalar-bubble diagram Fig. [REF] and scalar-tadpole diagram Fig. [REF].', '1602.01333-2-66-1': 'They both follow the five tensor structure decomposition [REF] and stay nonzero at the [MATH] limit.', '1602.01333-2-67-0': '### The scalar-bubble diagram', '1602.01333-2-68-0': 'Using Feynman rule ([REF]) and employing the dimensional regularization techniques we obtain the following loop-coefficients from the photon-scalar bubble diagram Fig. [REF]: [EQUATION]', '1602.01333-2-69-0': '### The scalar-tadpole diagram', '1602.01333-2-70-0': 'Next, with Feynman rule ([REF]) we compute the photon-scalar tadpole diagram in Fig. [REF],', '1602.01333-2-71-0': "S^(p)_tad=_k'=1^4S^_k'(p)_tad& =12d^D (2)^D i ^2_k'=1^4S^_k'(,p,-p,-).", '1602.01333-2-72-0': 'Starting with the first integral under dimensional regularization:', '1602.01333-2-73-0': "&S^_1'(p)_tad=-e^2d^D (2)^D f__2(,p) ^2 ((p)((p) g^-p^()^-()^p^)+p^2()^()^)_D4", '1602.01333-2-74-0': '&=e^2(4)^283(p)^4[3(g^(p)^2-()^p^2+ p^(p)^)+4p^2(p)^2(()^(p)^2+(p)^(p)^)],', '1602.01333-2-75-0': 'we obtained the IR result.', '1602.01333-2-75-1': 'To evaluate [MATH], [MATH] and [MATH] we first need to establish the following identities:', '1602.01333-2-76-0': "&f__3'(,p,-p)=f__3'(,-p,p)=f__3'(-,p,-p)=f__3'(-,-p,p)= 1,", '1602.01333-2-77-0': "& f__3'(p,-p,-)=f__3'(p,-p,)=f__3'(-p,p,-)=f__3'(-p,p,)= f^2__2(, p),", '1602.01333-2-78-0': "& f__3'(-,p,-p)+f__3'(p,-p,-)-2f^2__2(, p) -f^2__2(, p),", '1602.01333-2-79-0': "& f__3'(p,-p,-)+f__3'(-p,p,-)-2f^2__2(, p)=0,", '1602.01333-2-80-0': "& f__3'(,p,-p)+f__3'(p,-p,)-2f^2__2(, p) -f^2__2(, p),", '1602.01333-2-81-0': "& f__3'(p,-p,)+f__3'(-p,p,)-2f^2__2(, p)=0.", '1602.01333-2-82-0': 'We then find the following pure IR divergent terms:', '1602.01333-2-83-0': "&S^_4'(p)_tad=-2S^_2'(p)_tad=-2S^_3'(p)_tad =-S^_2'(p)_tad-S^_3'(p)_tad", '1602.01333-2-84-0': '& =e^2d^D (2)^D f__2(,p) ^2 ((p )((p)^()^+()^(p)^) -(p)(^(p)^+(p)^^))_D4', '1602.01333-2-85-0': '& =e^2(4)^2323(p)^(p)^(p)^4, [EQUATION]', '1602.01333-2-85-1': 'Using ([REF]) and ([REF]) and by comparing with general tensor structure ([REF]) we have found that from scalar-photon tadpole diagram only two terms survive: [EQUATION]', '1602.01333-2-85-2': 'Finally summing up the IR parts of bubble ([REF]) and tadpole ([REF]) contributions we get: [EQUATION] where all IR terms from both diagrams, except the one arising from the bubble, cancels out.', '1602.01333-2-85-3': 'Interesting enough is that within this noncommutaive scalar-photon action in the adjont ([REF]) we are facing the exact cancelations of all divergences of the higher order terms of noncommutative tensor-parameter [MATH], showing thus the consistency of our computations.', '1602.01333-2-86-0': '## The photon one-loop contribution to scalar 1-PI two point function', '1602.01333-2-87-0': 'The one-loop adjoint scalar 1-PI two point function in the minimal coupled model consists the tadpole diagram Fig. [REF] and the bubble diagram Fig. [REF].', '1602.01333-2-88-0': 'The evaluation is straightforward.', '1602.01333-2-88-1': 'We obtain in the end [EQUATION] and [EQUATION]', '1602.01333-2-88-2': 'The total quadratic IR divergence reads [EQUATION]', '1602.01333-2-88-3': 'We will soon see in the next section that this divergence is exactly canceled by the contributions from scalar-photino and scalar self-interaction diagrams.', '1602.01333-2-89-0': '# Noncommutative [MATH]=2 SYM U(1) theory and the [MATH]-exact SW map', '1602.01333-2-90-0': 'The noncommutative U(1) [MATH]=2 super Yang-Mills theory has the following action [EQUATION] in the Wess-Zumino gauge.', '1602.01333-2-90-1': 'The noncommutative fields in the previous action constitute the noncommutative U(1) supermultiplet [MATH].', '1602.01333-2-90-2': '[MATH] and [MATH] are Weyl fermion fields and [MATH] is a complex scalar field.', '1602.01333-2-90-3': 'Each field [MATH], [MATH] and [MATH] transforms under the adjoint action of the NC U(1), so that the NC covariant derivative is [MATH].', '1602.01333-2-91-0': 'By replacing the noncommutative fields of the action in ([REF]) with the [MATH]-exact Seiberg-Witten maps -see appendix A- [MATH], [MATH], [MATH], [MATH], the action [MATH] is turned into the action of a theory which is an interacting deformation of the free ordinary U(1) supersymmetric theory for the U(1) supermultiplet [MATH].', '1602.01333-2-91-1': 'This deformation is supersymmetric although supersymmetry -[MATH]- is nonlinearly realized on the ordinary multiplet [MATH]; a feature we have already seen in the [MATH] SYM case.', '1602.01333-2-92-0': 'The contributions to the action in ([REF]) that are needed to compute one-loop 1PI two-point function of each field in [MATH] can be readily obtained by using ([REF]), ([REF]), ([REF]) and [EQUATION]', '1602.01333-2-92-1': 'The terms in ([REF]) yields the scalar-fermion [MATH]=2 Feynman rules ([REF]) given in appendix E.', '1602.01333-2-92-2': 'Now we are ready to display the value of each one-loop Feynman diagram contributing to the two-point functions of the ordinary fields of the theory.', '1602.01333-2-93-0': '## The one-loop 1PI two-point function for photon field [MATH]', '1602.01333-2-94-0': 'For [MATH] theory one has [MATH].', '1602.01333-2-94-1': 'Substituting these numbers as well as the scalar bubble and tadpole results to [REF], and then restricting to quadratic IR divergence only, gives [EQUATION] i.e. clean quadratic IR divergence cancellation.', '1602.01333-2-94-2': 'Remaining UV divergences can be expressed using the five-term notation in [REF] as follows', '1602.01333-2-95-0': '_1(p)_UV=(43-43n_f-13n_s+p^2(p)^4(3trtheta(p)^2+4(p)^2))(2+(^2(p)^2))_n_f=n_s=2,', '1602.01333-2-96-0': '_2(p)_UV=p^2(p)^2(2-p^2(trtheta)(p)^2 )(2+(^2(p)^2)),', '1602.01333-2-97-0': '_3(p)_UV=p^2(p)^2(2+(^2(p)^2)),', '1602.01333-2-98-0': '_4(p)_UV=-4p^4(p)^4(2+(^2(p)^2)),', '1602.01333-2-99-0': '_5(p)_UV=4p^4(p)^4 (2 + (^2(p)^2)).', '1602.01333-2-100-0': '## The one-loop 1PI two-point function for the scalar [MATH]', '1602.01333-2-101-0': 'The one-loop 1PI two-point function, [MATH], of field [MATH] is the sum of the four diagrams (Fig. [REF], Fig. [REF], Fig. [REF] and Fig. [REF]).', '1602.01333-2-101-1': 'The first two are already given as equations [REF] and [REF] in the last section.', '1602.01333-2-101-2': 'The values of the third and fourth diagrams read [EQUATION]', '1602.01333-2-101-3': 'Hence, one gets the following full scalar two-point function: [EQUATION] which is again quadratic IR divergence free, as [EQUATION]', '1602.01333-2-101-4': 'The UV divergence reads [EQUATION]', '1602.01333-2-102-0': '## The one-loop 1PI two-point function for photinos [MATH] and [MATH]', '1602.01333-2-103-0': 'In the [MATH]=2 theory there is a photino-scalar loop (Fig. [REF]) alongside the photino-photon loop contribution which is identical to the [MATH] theory value [REF] for each of the two photinos.', '1602.01333-2-104-0': 'The photino-scalar loop-integral gives the following contribution [EQUATION]', '1602.01333-2-104-1': 'Total photino two-point function is finally given as a sum: [EQUATION]', '1602.01333-2-104-2': 'It is quadratic IR divergence free and it has the following UV divergence [EQUATION]', '1602.01333-2-105-0': '# Noncommutative [MATH]=4 SYM U(1) theory and the [MATH]-exact SW map', '1602.01333-2-106-0': 'Let [MATH], [MATH], [MATH] define the noncommutative U(1) [MATH]=4 supermultiplet; then, the action of the noncommutative U(1) [MATH]=4 super Yang-Mills theory reads [EQUATION]', '1602.01333-2-106-1': 'The matrices [MATH], [MATH] and [MATH] give rise to the IRREP of the Dirac matrices in 8 Euclidean dimensions; further details can be found in [CITATION].', '1602.01333-2-106-2': 'Let us recall that [MATH] is the noncomutative gauge field, that [MATH] is a noncommutative Weyl field and that [MATH] is a noncommutative real scalar field.', '1602.01333-2-106-3': 'The noncommutative U(1) acts by the adjoint action on [MATH] and [MATH], and hence [MATH].', '1602.01333-2-107-0': 'By replacing, in [MATH] above, the fields [MATH], [MATH] and [MATH] with the corresponding [MATH]-exact Seiberg-Witten maps -namely, [MATH], [MATH] and [MATH], respectively, we obtain an action which defines an interacting deformation of the ordinary [MATH] SYM theory in the Wess-Zumino gauge.', '1602.01333-2-107-1': 'This deformed action is expressed in terms of the fields of the ordinary [MATH] Yang-Mills supermultiplet [MATH], [MATH], [MATH] and it is invariant (on-shell) under the deformed supersymmetric transformations of the ordinary supermultiplet [MATH] which give rise to the [MATH] supersymmetric transformations of the fields in [MATH].', '1602.01333-2-107-2': 'As in the [MATH] and [MATH] cases, the supersymmetry transformations of the ordinary fields that leave [MATH] in ([REF]) invariant gives rise to on-shell nonlinear realization of [MATH] supersymmetry algebra.', '1602.01333-2-108-0': 'The contributions to the action in ([REF]) that are needed to compute one-loop 1PI two-point function of each field in [MATH] can easily be obtained by using ([REF]), ([REF]), ([REF]) and [EQUATION]', '1602.01333-2-108-1': 'The terms in ([REF]) yields the Feynman rules given in appendix D.', '1602.01333-2-109-0': 'Below we shall display the value of each one-loop Feynman diagram contributing to the two-point functions of the ordinary fields of the theory.', '1602.01333-2-110-0': '## The one-loop 1PI two-point function for massless vector field [MATH]', '1602.01333-2-111-0': 'The [MATH] 1PI two-point function follows the general formula [REF], only with [MATH] and [MATH].', '1602.01333-2-111-1': 'One can immediately recognize clean cancelation of the quadratic IR divergences after substituting [MATH] into [REF].', '1602.01333-2-112-0': '## The one-loop 1PI two-point function for the scalar [MATH]', '1602.01333-2-113-0': 'The one-loop 1PI two-point function, [MATH], of the field [MATH] is the sum of five diagrams Fig. [REF], Fig. [REF], Fig. [REF] and Fig. [REF], whose values read [EQUATION] hence [EQUATION] is again IR divergence free, only this time we have [EQUATION]', '1602.01333-2-113-1': 'The UV part reads [EQUATION]', '1602.01333-2-114-0': '## The one-loop 1PI two-point function for [MATH]', '1602.01333-2-115-0': 'The one-loop 1PI two-point function, [MATH], of the field [MATH] is the sum of the three diagrams Fig. [REF], Fig. [REF] and Fig. [REF] whose values read [EQUATION] with [MATH] being given in ([REF]) and ([REF]), respectively, and [EQUATION]', '1602.01333-2-115-1': 'Hence, [EQUATION] is quadratic IR divergence free, and the total UV divergences is presented below [EQUATION]', '1602.01333-2-116-0': '# Effect of gauge fixing on photon two point function', '1602.01333-2-117-0': 'In the prior sections we have shown that the quadratic IR divergent contribution to the photon two point function can be canceled by introducing supersymmetry.', '1602.01333-2-117-1': 'Yet we have still two unanswered question: First, all of our computations above are carried out in the commutative Feynman gauge, which, albeit convenient, is just one specific choice.', '1602.01333-2-117-2': 'We do not know whether the cancelation we found would be changed by a change of gauge fixing.', '1602.01333-2-117-3': 'Second, we have quite complicated UV divergence in the Feynman gauge in general, which may be modified by changing gauge fixing, as in the commutative gauge theories.', '1602.01333-2-117-4': 'To study these two issues we introduce in this section a new, non-local and nonlinear gauge fixing based on the Seiberg-Witten map then evaluate its effect to the photon two point function.', '1602.01333-2-118-0': '## The noncommutative Feynman gauge fixing action', '1602.01333-2-119-0': 'We introduce a new gauge fixing for non-local U(1) gauge theory via the [MATH]-exact Seiberg-Witten map.', '1602.01333-2-119-1': 'In terms of BRST language, this gauge fixing contains BRST-auxiliary field [MATH], and it is given by', '1602.01333-2-120-0': 'S&=S_U(1)+S_gf=S_U(1) +s(_A^(a_,^ij)+B2),', '1602.01333-2-121-0': 'S_gf&=B(_A^+B2)- s(_A^) =12(B+_A^)^2 -12(_A^)^2-_(sA^),', '1602.01333-2-122-0': 'with [MATH] being regular U(1) BRST transformation [MATH], where [MATH] is the U(1) ghost.', '1602.01333-2-122-1': 'Next we use consistency condition for SW map to get [MATH], where [MATH] is [MATH] ghost and [MATH] the [MATH] covariant derivative in the adjoint representation [MATH].', '1602.01333-2-123-0': 'Since the SW map for [MATH] is actually the same as for the NC gauge parameter [MATH] from [CITATION] we can derive the photon-photon and photon-ghost coupling in this gauge.', '1602.01333-2-123-1': 'So by using the BRST transformations [EQUATION] the following gauge fixing action is produced from ([REF]) [EQUATION] which after the application of the SW map resulting Feynman rules for the gauge fixing and ghost induced diagrams given in the appendix F.', '1602.01333-2-123-2': 'We name this gauge as "the noncommutative Feynman gauge" as it is formally identical to the Feynman gauge in the [MATH] gauge theory.', '1602.01333-2-124-0': '## One-loop contributions from the new NC gauge fixing action', '1602.01333-2-125-0': 'The new gauge fixing action [REF] introduces additional terms to the three and four photon self-couplings as well as photon-ghost couplings, as summarized in [REF].', '1602.01333-2-125-1': 'Unlike the three and four photon couplings in the commutative Feynman gauge [CITATION], these new interaction terms are no longer transverse.', '1602.01333-2-125-2': 'It then becomes intriguing how the sum of the resulting loop integrals behave.', '1602.01333-2-126-0': 'From Feynman rules [REF] we find the following diagrams Fig. [REF]-[REF] contributions to the one loop photon two point function.', '1602.01333-2-127-0': "Denoting the total sum of Fig's.", '1602.01333-2-127-1': '[REF] to [REF] as [MATH], it turns out to be convenient to split it into two partial sums, [EQUATION]', '1602.01333-2-127-2': 'Here [MATH] presents the sum over Figures [REF] and [REF], which contain one 3-photon vertex from the classical action and the other from gauge fixing action, while [MATH] sums over the rest of them which are solely from the gauge fixing.', '1602.01333-2-128-0': 'Evaluation of diagrams in Fig. [REF]-[REF] follows substantially the standard procedure used in the prior section, except the rising of the two new types of tadpole integrals.', '1602.01333-2-128-1': 'The first one takes the form of the second term in [REF] so it can be removed by our regularization prescription.', '1602.01333-2-128-2': 'The second one is a tadpole integral without any loop momenta in the numerator i.e. [MATH].', '1602.01333-2-128-3': 'This integral contains total effective loop momenta power [MATH] because of the additional denominator [MATH] from the nonlocal factor [MATH], which is below the minimal power for the commutative tadpole integral to vanish [CITATION].', '1602.01333-2-128-4': 'Consequently we observe unregularized UV divergence when computing this integral by transforming it into the bubble or applying the n-nested zero regulator method [CITATION].', '1602.01333-2-128-5': 'We develop an alternative prescription [REF] based on the parametrization [REF] which is capable of dimensionally-regularizing this integral into a [MATH] divergence plus the logarithmic UV/IR mixing term [MATH] at the [MATH] limit [REF].', '1602.01333-2-129-0': 'We are able to express both [MATH] and [MATH] appropriately once [MATH] is added to the prior basis integral set [MATH], [MATH], [MATH] and [MATH].', '1602.01333-2-129-1': 'The outcome is listed as below [EQUATION]', '1602.01333-2-129-2': 'One can immediately notice that [MATH] contains only two tensor structures [MATH] and [MATH] which can not be combined into a transverse sum.', '1602.01333-2-129-3': 'The loss of transversality appears to be, of course, surprising.', '1602.01333-2-129-4': 'However we are going to develop reasoning/arguments for this seeming odd behavior in the next section and show that it is in fact understandable.', '1602.01333-2-130-0': '# Gauge fixing contribution without integrating out BRST-auxiliary field', '1602.01333-2-131-0': 'In order to achieve simple transversality we conclude that one has to keep BRST auxiliary field [MATH] from being integrated out.', '1602.01333-2-131-1': 'Arguments for that are as follows.', '1602.01333-2-132-0': 'Starting with the action ([REF]) we write a generating functional [EQUATION] from where we have effective action in terms of "currents": [EQUATION]', '1602.01333-2-132-1': 'Regular BRST transformation [MATH] acting on [MATH] vanishes, thus we have: [EQUATION]', '1602.01333-2-132-2': 'Since the transversality condition means [MATH], which is satisfied in equation ([REF]) only for [MATH].', '1602.01333-2-132-3': 'This however is not allowed if we do integrate out the [MATH] field.', '1602.01333-2-132-4': 'Thus we do not perform that, instead we construct a propagator from the following doublet combination [MATH].', '1602.01333-2-133-0': '## Formal analysis', '1602.01333-2-134-0': 'In order to compute the two point function(s) within the presence of the B-field, we must define the propagator(s) for the "kind of strange" vector-scalar field [MATH] doublet.', '1602.01333-2-134-1': 'Starting with [EQUATION] we get a quadratic part of [MATH] [EQUATION] whose Fourier transform is as follows: [EQUATION]', '1602.01333-2-134-2': 'The Hermitian matrix [EQUATION] is then the inverse of the propagator in the momentum space.', '1602.01333-2-134-3': 'Next we inverse [MATH] to obtain [EQUATION]', '1602.01333-2-134-4': 'From ([REF]) and ([REF]) we have four equations:', '1602.01333-2-135-0': '& _^=-k^2 G_+k^G_ k^+iA_k^,0=-ik^G_+A_ & 0=-k^2 B^+k^B_k^+iG k^,1=-iB_k^+ 1,', '1602.01333-2-136-0': 'which we solve by using simple Ansatz: [MATH], [MATH].', '1602.01333-2-136-1': 'Taking into account a text book convention [MATH] for the phase factor we add overall factor [MATH] and obtained correct photon propagator, as illustrated in Fig [REF].', '1602.01333-2-137-0': 'One can write down the usual field redefinition [MATH] in [REF] in the Fourier transformed context as [EQUATION]', '1602.01333-2-137-1': 'The [MATH] then diagonalizes the bilinear form [REF] into [EQUATION]', '1602.01333-2-137-2': 'It is easy to see that inverting this [MATH] gives the expected Feynman propagator.', '1602.01333-2-137-3': 'From that viewpoint the B-integration can be achieved by diagonalization.', '1602.01333-2-137-4': 'One can formally generalize the tree level diagonalization procedure to the one loop.', '1602.01333-2-137-5': 'Consider in general the one loop corrections as another Hermitian matrix [MATH] adding to the [MATH] matrix, we write the quadratic part of the 1-loop corrected effective action in the momentum space as [EQUATION]', '1602.01333-2-137-6': 'Next we express [MATH] in terms of its components [EQUATION]', '1602.01333-2-137-7': 'The Slavnov-Taylor identity [REF] then requires that [MATH] has to be transverse, while others not.', '1602.01333-2-137-8': 'One can now replace [MATH] by a new transformation [EQUATION] with [MATH] diagonalizing the 1-loop corrected bilinear form [MATH] [EQUATION] where [EQUATION]', '1602.01333-2-137-9': 'We then conjecture that the formal leading order expansion of [MATH] with respect to the coupling constant [MATH] corresponds to the gauge fixing corrections to the 1-loop 1-PI photon two point function, i.e. [EQUATION]', '1602.01333-2-137-10': 'And, as we shall see below, this relation/conjecture indeed holds.', '1602.01333-2-138-0': '## The action of the gauge and BRST-auxiliary fields and Feynman rules', '1602.01333-2-139-0': 'We define the noncommutative photon-auxiliary field action by using the first and the second order SW maps for the NC gauge field, [MATH] and [MATH], respectively: [EQUATION] from where we obtain the corresponding photon-auxiliary field interaction vertices.', '1602.01333-2-139-1': 'Corresponding Feynman rules from the above action generate one-loop correction to the quadratic effective action and are given in appendix G.', '1602.01333-2-140-0': '## One-loop contributions to the photon effective action up to the quadratic order, from the BRST auxiliary field [MATH]', '1602.01333-2-141-0': 'Based on our vertex read-out convention, [MATH], we obtain the following correspondence rule between the matrix elements of one loop correction [MATH] and the 1-PI loop diagrams [EQUATION]', '1602.01333-2-141-1': 'Note that [MATH] denotes all contributions from the classical action, which is the same as summing over all contributions to the photon two point function computed in sections 2-5 and thus transverse.', '1602.01333-2-141-2': 'Explicit computation first revolves that [EQUATION] thus the Slavnov-Taylor identity is actually trivially fulfilled.', '1602.01333-2-142-0': 'The rest of the matrix elements listed in [REF] are nonzero and boil down to the following expressions [EQUATION] where', '1602.01333-2-143-0': '_I=(4^2)^2-D2(p^2)^D2-2(D-2)(3-D2)B(D2-1, D2-1)_D4--(4I_K^0+6 I_H),', '1602.01333-2-144-0': '_II=p^2(p)^2(&(4^2)^2-D2(p^2)^D2-2(D-1)(3-D2)B(D2-1, D2-1)_D4-', '1602.01333-2-145-0': '&-18(p)^2 T_-2-(6I_K^0+6I_H)),', '1602.01333-2-146-0': 'and', '1602.01333-2-147-0': '_mix^e^2(4)^2(p^+(p)^p^2(p)^2)(4+4 I_H).', '1602.01333-2-148-0': 'Finally we have [EQUATION]', '1602.01333-2-148-1': 'Next we start to verify our conjecture [REF].', '1602.01333-2-148-2': 'First we derive the following relations from it, [EQUATION]', '1602.01333-2-148-3': 'One then immediately observes that the second and third relations do fulfill.', '1602.01333-2-148-4': 'As for the first one we can compute its right hand side [EQUATION] which is in agreement with ([REF]).', '1602.01333-2-148-5': 'Thus the conjectured relation [REF] is proven.', '1602.01333-2-149-0': 'In fact the gauge fixing contribution to the 1-loop correction is a shift out of the on-shell point of the gauge fixing functional.', '1602.01333-2-149-1': 'Therefore it does not need to be transverse.', '1602.01333-2-149-2': 'The fact that the pure gauge fixing and mixing contributions satisfies [REF] independently is because the former can be considered as a gauge fixing to the free [MATH] gauge theory.', '1602.01333-2-150-0': "Finally let's briefly discuss the divergences in the gauge fixing configuration(s).", '1602.01333-2-150-1': 'Using the results from appendix B we can see that [MATH] contains no quadratic IR divergent term, therefore the quadratic IR divergence cancelation we found in the prior sections are also preserved under this gauge fixing choice point.', '1602.01333-2-150-2': 'We can also extract the UV logarithmic divergence at the [MATH] limit', '1602.01333-2-151-0': '_A_UV=-(2-p^2(p)^2 (3trtheta(p)^2+4(p)^2))(2+(^2(p)^2)),', '1602.01333-2-152-0': '_B_UV=-72p^2(p)^2(2+(^2(p)^2)),', '1602.01333-2-153-0': 'while the [MATH] is finite at this limit.', '1602.01333-2-153-1': 'We conclude our analysis by listing of all UV plus logarithmic divergences in [MATH], which can be decomposed into seven symmetric tensor structures [EQUATION]', '1602.01333-2-153-2': 'Explicit computation then yields', '1602.01333-2-154-0': '_1_UV= p^2(43-43n_f-13n_s+p^2(p)^4(3trtheta(p)^2+4(p)^2))(2+(^2(p)^2)),', '1602.01333-2-155-0': '_2_UV=(-2-43+43n_f+13n_s)(2+(^2(p)^2)),', '1602.01333-2-156-0': '_3_UV=B_2_UV=2p^2(p)^2(2-p^2(trtheta)(p)^2 )(2+(^2(p)^2)),', '1602.01333-2-157-0': '_4_UV=B_3_UV=2p^2(p)^2(2+(^2(p)^2)),', '1602.01333-2-158-0': '_5_UV=B_4_UV=-4p^4(p)^4(2+(^2(p)^2)),', '1602.01333-2-159-0': '_6_UV=B_5_UV=4p^4(p)^4 (2 + (^2(p)^2)),', '1602.01333-2-160-0': '_7_UV=_B_UV=-72p^2(p)^2(2 +(^2(p)^2)).', '1602.01333-2-161-0': '# Summary and discussion', '1602.01333-2-162-0': 'In this paper we have computed the one-loop contributions to all propagators of the noncommutative super Yang-Mills U(1) theory with [MATH]=1, 2 and 4 supersymmetry and defined by the means of the [MATH]-exact Seiberg-Witten map.', '1602.01333-2-162-1': 'We have shown that for [MATH]=1, 2 and 4 the quadratic noncommutative IR divergence, [EQUATION] -a trade-mark of the noncommutative gauge theories- which occur in the bosonic, fermionic and scalar loop-contributions to the photon propagator cancel each other, rendering photon propagator free of them as befits of the Supersymmetry.', '1602.01333-2-162-2': 'Indeed, from [REF] for [MATH], one gets: [EQUATION]', '1602.01333-2-162-3': 'This cancellation, occuring in the case at hand, is nontrivial since Supersymmetry acts nonlinearly -see ([REF])- on the ordinary fields.', '1602.01333-2-162-4': 'Let us recall that the cancellation of quadratic noncommutative IR divergences is also a feature of noncommutative super Yang-Mills theories when formulated in terms of noncommutative fields [CITATION].', '1602.01333-2-162-5': 'Hence, our result concerning the cancellation of the quadratic noncommutative IR divergences really points into direction that the [MATH]-exact Seiberg-Witten map really provides quantum duals of the same underlying theory.', '1602.01333-2-163-0': 'We have shown -see ([REF]) and ([REF])- that the characteristic quadratic noncommutative IR divergences, [EQUATION] which arise in the individual contributions to the one-loop propagators of the scalar fields in the [MATH]=2 and 4 Supersymmetry, also cancel each other at the end of the day.', '1602.01333-2-163-1': 'The same holds for the photino field as well.', '1602.01333-2-164-0': 'Since the previous cancellations occur both in the ordinary Feynman gauge and in the noncommutative Feynman gauge -see section 7-, our computations further indicate that the cancellation is robust against changing the gauge fixing and may have real physical, and therefore gauge invariant, content.', '1602.01333-2-164-1': 'Let us recall that independence of gauge-fixing parameter of the cancellation of noncommutative IR divergences in the dual theory, i.e., in [MATH]=1 U(1) super Yang-Mills theory formulated in terms of the noncommutative fields, has been shown to hold -see Ref. [CITATION].', '1602.01333-2-165-0': 'In this paper we have also worked out explicitly the one-loop UV divergent contributions -which show as poles at [MATH]- to all propagators of the theory: see ([REF]), ([REF])-([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF])-([REF]).', '1602.01333-2-165-1': 'It is noticeable that the pole parts displayed in the equations we have just quoted contain non-polynomial, i.e., non-local, terms whose denominator is a power of [MATH].', '1602.01333-2-165-2': "With respect to this we would like to point out that, in keeping with Weinberg's power counting theorem [CITATION], Feynman integrals whose degree of UV divergence are not the same along all directions are liable to give rise to the pole contributions which are non-polynomial.", '1602.01333-2-165-3': 'This is exactly our situation since our integrands contain factors of the type [EQUATION] and these factors approach to zero as [MATH] along the direction parallel to [MATH], and as [MATH] along any direction orthogonal to [MATH].', '1602.01333-2-165-4': 'Hence UV divergences with a non-polynomial dependence on the momenta may occur and our computations show that indeed they do occur.', '1602.01333-2-165-5': 'We would like to recall that a similar situation, -i.e. the non-polynomial UV divergences- happens in ordinary Yang-Mills theories in the light-cone gauge [CITATION].', '1602.01333-2-166-0': 'Now, the UV divergences of two-point functions are in general gauge dependent quantities.', '1602.01333-2-166-1': 'We have verified that this is so in our case by computing the one-loop propagator of the gauge field both in the ordinary Feynman gauge and in the noncommutative Feynman gauge -see Section 6.', '1602.01333-2-166-2': 'The result for the first type of gauge is in ([REF])-([REF]) and in ([REF])-([REF]) for the second type of gauge fixing term: their differences stand out.', '1602.01333-2-166-3': 'Hence, extracting gauge invariant information from the UV divergences is our next challenge along this line of research and it will require the computation of three and higher point functions.', '1602.01333-2-167-0': 'Let us finally remark that UV/IR mixing effects also work for the non-polynomial UV divergent contributions we have obtained.', '1602.01333-2-167-1': 'Indeed, as seen in ([REF]), ([REF])-([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF])-([REF]) every pole in [MATH] comes hand in hand with the logarithmic noncommutative IR divergence [MATH].', '1602.01333-2-167-2': 'The reader is referred to the final part of Appendix B for further information regarding this issue.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1602.01333

hep-th-0302160

{'hep-th-0302160-1-0-0': 'We discuss a new mechanism of obtaining a period of cosmological inflation in the context of string theory.', 'hep-th-0302160-1-0-1': 'This mechanism is based on embedded defects which form dynamically on higher dimensional D-branes.', 'hep-th-0302160-1-0-2': 'Such defects generate topological inflation, but unlike topological inflation from stable defects, here there is a natural graceful exit from inflation: the decay of the embedded defect.', 'hep-th-0302160-1-0-3': 'We demonstrate the idea in the context of a brane-antibrane annihilation process.', 'hep-th-0302160-1-0-4': 'The graceful exit mechanism suggested here applies generically to all realizations of inflation on D-branes.', 'hep-th-0302160-1-1-0': ']', 'hep-th-0302160-1-2-0': '# Introduction', 'hep-th-0302160-1-3-0': 'There has been a lot of interest recently in exploring ways of obtaining a period of cosmological inflation from string theory.', 'hep-th-0302160-1-3-1': 'A realization of inflation in string theory should address some of the conceptual problems of quantum field theory realizations of inflationary cosmology [CITATION].', 'hep-th-0302160-1-3-2': 'Since string theory contains many moduli fields which are massless at the perturbative level, stringy inflation may yield a natural mechanism to produce fluctuations of the required small amplitude [CITATION].', 'hep-th-0302160-1-3-3': 'A description of inflation in the context of string theory would also automatically resolve the trans-Planckian problem [CITATION] of inflationary cosmology, since it would yield a complete description of the dynamics of fluctuations during the entire inflationary period.', 'hep-th-0302160-1-3-4': 'Furthermore, since one of the goals of string theory is to provide a nonsingular cosmology, a stringy inflation model should also allow one to resolve the singularity problem [CITATION] of inflationary cosmology.', 'hep-th-0302160-1-4-0': 'Recent developments in particle phenomenology have explored the possibility that the our matter fields are confined to a four-dimensional space-time hypersurface in a higher-dimensional bulk space-time [CITATION].', 'hep-th-0302160-1-4-1': 'In the context of string theory this hypersurface arises naturally as a D-brane, one of possibly many branes that reside in the bulk spacetime [CITATION].', 'hep-th-0302160-1-4-2': 'However, for our purposes here, a sufficient starting point is some higher dimensional field theory (coupled weakly to gravity).', 'hep-th-0302160-1-4-3': 'This higher dimensional theory can be completed in the ultraviolet by embedding in critical string theory, but also by using the [MATH] fixed point [CITATION], little string theory [CITATION], or by deconstruction [CITATION].', 'hep-th-0302160-1-4-4': 'In order to have a definite mental picture we assume the higher dimensional theory is embedded in string theory, and is realized on D-branes.', 'hep-th-0302160-1-5-0': 'This "brane-world" scenario has generated new realizations of cosmic inflation.', 'hep-th-0302160-1-5-1': 'In one setup [CITATION], the inflaton is the separation between a brane-antibrane pair.', 'hep-th-0302160-1-5-2': 'However, it has been shown [CITATION] that in this context it is not easy to obtain initial conditions which lead to a sufficient period of inflation.', 'hep-th-0302160-1-5-3': 'Another possibility is to have inflation generated by branes at an angle [CITATION], or by a configuration of defects of different dimensionalities [CITATION].', 'hep-th-0302160-1-5-4': 'This has the advantage of possibly providing a mechanism for localization of chiral matter on the inflating locus.', 'hep-th-0302160-1-6-0': 'An alternative way to obtain inflation from brane collisions was suggested in [CITATION], based on the realization [CITATION] that topological defects of co-dimension greater or equal to one are formed on the brane worldvolume during the collision of a brane-antibrane pair.', 'hep-th-0302160-1-6-1': 'In the simplest example, the order parameter of the phase transition which occurs when the brane-antibrane pair meets is given by a complex tachyon field [CITATION] which condenses at a nonvanishing expectation value.', 'hep-th-0302160-1-6-2': 'The vacuum manifold of the tachyon field ground state values has the topology of [MATH].', 'hep-th-0302160-1-6-3': 'By causality [CITATION], the values of the condensate in the vacuum manifold are uncorrelated on large length scales, and hence a network of co-dimension 2 defects inevitably will form.', 'hep-th-0302160-1-6-4': 'In the case of a [MATH]-[MATH] brane pair, the resulting topological defect is a [MATH] brane which could be our world.', 'hep-th-0302160-1-6-5': 'Provided that the thickness of the topological defect is larger than the Hubble radius, the defect core can undergo inflation (topological inflation [CITATION]).', 'hep-th-0302160-1-7-0': 'A problem with the mechanism of [CITATION] (and of models of 4 dimensional topological inflation [CITATION]) is the graceful exit problem: how does inflation end, and a transition to the usual late time cosmological evolution occur?', 'hep-th-0302160-1-8-0': 'In this paper, we provide a simple solution to this problem, which generalizes to all above mentioned "brane-world" models of inflation.', 'hep-th-0302160-1-8-1': 'We point out that the defects which form in brane collisions will often be embedded defects (which are non-BPS branes in the string context) (see [CITATION] for a discussion of embedded defects in field theory).', 'hep-th-0302160-1-8-2': 'These embedded defects can be stabilized at high temperatures by plasma effects [CITATION], but decay at a sufficiently low temperature, thus providing a natural graceful exit mechanism from inflation.', 'hep-th-0302160-1-8-3': 'In the context of field theory inflation, such a scenario was suggested in [CITATION].', 'hep-th-0302160-1-9-0': 'We may call the proposed inflationary model "Higher Dimensional (Non-) Topological Inflation" since inflation is taking place in the higher-dimensional brane worldvolume, not only in the directions of the defect (which are the dimensions of our four-dimensional physical space-time).', 'hep-th-0302160-1-9-1': 'Assuming that there is a mechanism to stabilize the radion in the two (compact) distinguished extra dimensions (the directions of the original brane orthogoal to the defect), we would have found a way of making two of the internal dimensions larger than the others, thus making contact to the scenario of [CITATION] .', 'hep-th-0302160-1-9-2': 'Alternatively, the two extra dimensions may be taken to be infinite; as one moves away from the core of the defect, inflation proceeds more slowly, and ends earlier.', 'hep-th-0302160-1-9-3': 'Thus the resulting spacetime is expected to be warped, making contact with the work of [CITATION].', 'hep-th-0302160-1-9-4': 'The question of which alternative is realized in specific examples needs a detailed, model dependent analysis, which we hope to return to in the future.', 'hep-th-0302160-1-10-0': 'The causality argument (Kibble mechanism [CITATION]) which ensures that in a phase transition leading to the possibility of topological defect formation such defects inevitably will form, with a separation smaller or comparable to the Hubble length, also ensures the formation of stabilized embedded defects in models which admit them, as will be explained later.', 'hep-th-0302160-1-10-1': 'Thus, there is no initial condition problem for our model of inflation.', 'hep-th-0302160-1-11-0': 'In field theory models with a symmetry breaking scale [MATH] much smaller than the Planck scale, the width of a defect is typically much smaller than the Hubble length.', 'hep-th-0302160-1-11-1': 'Thus, topological inflation will not occur.', 'hep-th-0302160-1-11-2': 'An advantage of inflation in the context of higher dimensions is that there is a new scale, the brane tension, which is naturally higher than the Planck mass scale .', 'hep-th-0302160-1-11-3': 'As will be shown in Section III, in this context the defect width can easily be larger than the Hubble length, thus making defect-driven inflation possible.', 'hep-th-0302160-1-12-0': 'Our proposed mechanism of topological inflation from embedded defects can apply in situations more general than that of a brane-antibrane annihilation process.', 'hep-th-0302160-1-12-1': 'For example, in the context of brane intersections (the intersection region undergoing topological inflation), the branes may partially unwind around each other, leaving behind some non-inflating components.', 'hep-th-0302160-1-12-2': 'This process excites the off-diagonal open strings, which have no geometrical interpretation.', 'hep-th-0302160-1-13-0': 'The outline of this paper is as follows.', 'hep-th-0302160-1-13-1': 'In the next section, we discuss a particular example in which one can realize topological inflation from an embedded defect, namely embedded defects forming in a brane-antibrane annihilation process.', 'hep-th-0302160-1-13-2': 'Next, we discuss how topological inflation from embedded defects has a natural graceful exit mechanism.', 'hep-th-0302160-1-13-3': 'Finally.', 'hep-th-0302160-1-13-4': 'we summarize our results, and discuss further applications of the basic idea of topological inflation from embedded defects in the context of brane physics.', 'hep-th-0302160-1-14-0': '# Embedded Defects from Higher Dimensional D-branes', 'hep-th-0302160-1-15-0': 'We will start with a brief review of the original scenario of [CITATION].', 'hep-th-0302160-1-15-1': 'The scenario of [CITATION] takes as a starting point a pair of parallel [MATH] and [MATH] branes approaching each other.', 'hep-th-0302160-1-15-2': 'As studied in [CITATION], when the pair gets sufficiently close, a tachyon instability sets in.', 'hep-th-0302160-1-15-3': 'For minimal gauge content of the branes, the tachyon can be described by a complex scalar field [MATH] with a standard symmetry breaking (Mexican hat) potential [EQUATION] where the point [MATH] corresponds to branes at zero separation, and [MATH] is the symmetry breaking scale.', 'hep-th-0302160-1-15-4': 'The vacuum at [MATH] corresponds to the closed string vacuum, where the original branes have annihilated.', 'hep-th-0302160-1-16-0': 'The vacuum manifold of the tachyon condensates is [MATH], and hence according to the usual Kibble argument [CITATION], the fact that the condensate values must be uncorrelated on scales larger than the Hubble radius inevitably leads to the formation of stable co-dimension 2 defects, in this case [MATH] branes, on the worldvolume of the original [MATH] branes.', 'hep-th-0302160-1-16-1': 'Unlike most of the discussion of defects on annihilating D-branes, we are interested in the case where the resulting D3 brane is not BPS saturated, so that inflation is possible.', 'hep-th-0302160-1-17-0': 'In terms of gauge theory on the original brane worldvolume, the above process corresponds to the symmetry breaking [EQUATION] where one of the [MATH] factors on the left hand side of the equation lives on either of the branes, and the unbroken subgroup corresponds to the diagonal [MATH] factor.', 'hep-th-0302160-1-17-1': 'The standard homotopy arguments (see e.g. [CITATION] for a discussion of such arguments in the context of topological defects) yield [EQUATION] where [MATH] is the vacuum manifold, and [MATH] and [MATH] are the full gauge group and the residual gauge group after symmetry breaking, respectively .', 'hep-th-0302160-1-18-0': 'This set-up is not restricted to annihilating D-branes.', 'hep-th-0302160-1-18-1': 'Suppose we start with any higher dimensional D-brane configuration, and that at initial times the set-up is removed from its vacuum.', 'hep-th-0302160-1-18-2': 'This vacuum may be the closed string vacuum, but also an open string vacuum, where the branes align in a supersymmetric fashion.', 'hep-th-0302160-1-18-3': 'In the process of relaxation to the vacuum, the Kibble mechanism will ensure the creation of local defects, for example brane intersections.', 'hep-th-0302160-1-18-4': 'It is those defects which seed inflation in the higher-dimensional topological context, and which are the subject of our discussion.', 'hep-th-0302160-1-18-5': 'In particular, the energy scale associated with this initial phase transition can be much smaller than the string scale, so the above (higher dimensional) field theory considerations apply.', 'hep-th-0302160-1-19-0': 'Our idea, in this context, is to concentrate on the case of embedded, non-topological, defects.', 'hep-th-0302160-1-19-1': 'For this end we enhance the gauge symmetry on each of the branes, e.g. from [MATH] to [MATH].', 'hep-th-0302160-1-19-2': 'Indeed, this arises naturally in the context of fivebranes in type I theory.', 'hep-th-0302160-1-19-3': 'In this case the worldvolume gauge group is [MATH].', 'hep-th-0302160-1-19-4': 'The tachyonic scalar fields transform as [MATH] under [MATH].', 'hep-th-0302160-1-20-0': 'Now, the symmetry breaking pattern during tachyon condensation becomes [EQUATION]', 'hep-th-0302160-1-20-1': 'In this case, no stable topological defects (in particular no co-dimension 2 defects) form during this phase transition since the vacuum manifold is [MATH] and hence [MATH].', 'hep-th-0302160-1-20-2': 'Note that in the context of studies of branes in Type I string theory it is already known that the [MATH] brane is unstable [CITATION].', 'hep-th-0302160-1-21-0': 'The situation is similar to what happens in the electroweak theory where (for vanishing Weinberg angle), the symmetry breaking is [MATH], as effectively in the above case ([REF]).', 'hep-th-0302160-1-21-1': 'However, as is well known from analysis of the electroweak theory [CITATION], it is possible to construct embedded defects, solutions of the equations of motion which correspond to unstable defects.', 'hep-th-0302160-1-21-2': 'In the case of the electroweak theory in four space-time dimensions, the defects are electroweak strings, an example of embedded strings .', 'hep-th-0302160-1-22-0': 'As was recently realized [CITATION], such embedded defects can be stabilized at finite density by plasma effects.', 'hep-th-0302160-1-22-1': 'Consider again first the example of the standard electroweak theory.', 'hep-th-0302160-1-22-2': 'The order parameter of the symmetry breaking phase transition has four real components, two of which are electrically charged, two are electrically neutral.', 'hep-th-0302160-1-22-3': 'In the presence of a thermal bath of photons, the effective potential will then be lifted in the charged scalar field directions by more than it is lifted in the neutral scalar field directions.', 'hep-th-0302160-1-22-4': 'Thus, the vacuum manifold corresponding to the theory in a photon plasma is [MATH] and not [MATH].', 'hep-th-0302160-1-22-5': 'This can stabilize a subset of embedded strings, namely the string configurations constructed from the neutral scalar fields (these are the electroweak Z-strings in the case of the standard electroweak theory).', 'hep-th-0302160-1-23-0': 'Note that if the embedded defects are thermally stabilized at the temperature just below the phase transition temperature, then the causality argument (Kibble mechanism [CITATION]) applies and states that at least of the order one such defect will form per correlation volume (which in turn is smaller than the Hubble volume).', 'hep-th-0302160-1-23-1': 'The argument is as follows: during the phase transition, the order parameter relaxes to the vacuum manifold [MATH].', 'hep-th-0302160-1-23-2': 'However, in the presence of thermal stabilizing effects it will relax to the reduced vacuum manifold [MATH].', 'hep-th-0302160-1-23-3': 'However, which value in [MATH] is taken on is random on scales larger than the correlation length.', 'hep-th-0302160-1-23-4': 'Hence, defects will form.', 'hep-th-0302160-1-23-5': 'This process was studied in detail in the case of semilocal strings (related to embedded strings) in [CITATION].', 'hep-th-0302160-1-24-0': 'Our idea is to apply this mechanism to brane cosmology.', 'hep-th-0302160-1-24-1': 'We need to assume that the branes contain a thermal bath of gauge fields which couple in the same way to the [MATH] symmetry breaking field of our brane setup as the photon does to the [MATH] symmetry breaking order parameter in the electroweak theory.', 'hep-th-0302160-1-24-2': 'Ideally, the usual photon field could play this role.', 'hep-th-0302160-1-24-3': 'In this case, it would be natural to have only this field in thermal equilibrium at late times, since it would be the only massless gauge field.', 'hep-th-0302160-1-24-4': 'The situation would then be identical to the electroweak theory discussed above.', 'hep-th-0302160-1-24-5': 'Given the above assumptions about field content and coupling to the [MATH] symmetry breaking order parameter, it is completely natural from the point of view of cosmology (hot initial state in the very early Universe) to have the field excited in the required way.', 'hep-th-0302160-1-25-0': 'In this case, the phase transition corresponding to brane collision will, by the Kibble mechanism [CITATION], inevitably produce a network of co-dimension 2 branes.', 'hep-th-0302160-1-25-1': 'These defects will remain stable until the matter density on the branes has sufficiently diluted.', 'hep-th-0302160-1-26-0': 'Let us now be a bit more specific.', 'hep-th-0302160-1-26-1': 'Given an [MATH] gauge group on each of the [MATH] branes, the world volume theory of a coincident [MATH]-[MATH] brane pair is [MATH], and any gauge field configuration can be written as a [MATH] matrix of complex entries, the upper diagonal [MATH] entries corresponding to a [MATH] matrix describing open strings which start and end on the first brane, the lower diagonal [MATH] entries describing a [MATH] matrix corresponding to the open strings starting and ending on the other brane.', 'hep-th-0302160-1-26-2': 'The two off-diagonal [MATH] sub-matrices are [MATH] matrices corresponding to open strings beginning and ending on different branes.', 'hep-th-0302160-1-26-3': 'It is these latter sectors which contain the tachyon [CITATION].', 'hep-th-0302160-1-26-4': 'In the case of [MATH] gauge fields on each of the branes, the tachyon is a complex scalar field (one real component from each of the sectors), in the case of [MATH] gauge fields, there are two real components in each sector.', 'hep-th-0302160-1-27-0': 'Let us now assume that on each of the branes there is a thermal bath of a [MATH] subgroup of [MATH], and the other gauge fields assumed to be vanishing.', 'hep-th-0302160-1-27-1': 'In this case, the gauge fields excited on the branes will only couple to one of the two real tachyon fields in each sector.', 'hep-th-0302160-1-27-2': 'Let us combine these two fields into a complex field [MATH] (charged with respect to the excited gauge field), the other two into a complex field [MATH] (neutral with respect to the excited gauge field).', 'hep-th-0302160-1-27-3': 'The effective world volume Lagrangian then becomes [EQUATION] where the [MATH] stands for the gauge covariant derivative, the gauge field being the one excited on the brane.', 'hep-th-0302160-1-27-4': 'Hence, the gauge field directly couples only to the charged scalar field [MATH] and not to the neutral field [MATH].', 'hep-th-0302160-1-27-5': 'The field strength tensor of the gauge field [MATH] excited on the brane is denoted by [MATH], but the [MATH] term in the Lagrangian will not be important in the following.', 'hep-th-0302160-1-27-6': 'The tachyon potential [MATH] has the typical symmetry breaking form [EQUATION] where [MATH] is a coupling constant, and [MATH] is the ground state expectation value of the tachyon field magnitude.', 'hep-th-0302160-1-27-7': 'The corresponding vacuum manifold is [MATH].', 'hep-th-0302160-1-28-0': 'The above effective Lagrangian is the same which describes the sigma model of low energy QCD in the chiral limit, coupled to an external bath of photons [CITATION].', 'hep-th-0302160-1-28-1': 'It also describes the standard electroweak theory with only the photon field excited [CITATION].', 'hep-th-0302160-1-28-2': 'In the presence of a thermal bath of [MATH] gauge fields, we can take the thermal average of the Lagrangian.', 'hep-th-0302160-1-28-3': 'Terms linear in [MATH] vanish, the quadratic term [MATH] becomes [MATH], where [MATH] is the temperature of the bath.', 'hep-th-0302160-1-28-4': 'Thus, an effective mass term for the [MATH] field is generated.', 'hep-th-0302160-1-28-5': 'The effective potential becomes [EQUATION]', 'hep-th-0302160-1-28-6': 'Hence, the vacuum manifold is lifted in the charged scalar field directions.', 'hep-th-0302160-1-28-7': 'The vacuum manifold of the effective potential becomes [MATH].', 'hep-th-0302160-1-28-8': 'Thus, co-dimension 2 defects in which the neutral scalar field configuration takes on the Nielsen-Olesen [CITATION] form become important as metastable defects.', 'hep-th-0302160-1-29-0': '# Embedded Defects, Inflation, and Graceful Exit', 'hep-th-0302160-1-30-0': 'The idea of topological inflation is quite simple [CITATION]: provided that the core of the defect is comparable or larger than the Hubble radius, then the potential energy density in the defect core will be larger than the tension energy, and thus via standard minimal coupling to gravity, the core will commence inflationary expansion.', 'hep-th-0302160-1-30-1': 'For defects associates with an energy scale much smaller than the Planck scale, the width is much too small to obtain topological inflation.', 'hep-th-0302160-1-30-2': 'However, if the tension of the defect is given by an energy scale [MATH] which is comparable of larger than [MATH], then the condition for topological inflation is satisfied.', 'hep-th-0302160-1-30-3': 'A quantitative analysis [CITATION] yields as condition for topological inflation [EQUATION]', 'hep-th-0302160-1-30-4': 'Let us give a brief derivation of this result.', 'hep-th-0302160-1-30-5': 'For a typical symmetry breaking potential of the form ([REF]), the defect width [MATH] is of the order [CITATION] [EQUATION] which can be seen by balancing potential and tension energies.', 'hep-th-0302160-1-30-6': 'In the core, the potential energy density dominates, in the outside regions of the defect the tension energy density is larger.', 'hep-th-0302160-1-30-7': 'The condition for defect inflation is that the core width is greater than the Hubble length [MATH].', 'hep-th-0302160-1-30-8': 'Applying the Friedmann equation [EQUATION] (where [MATH] is the energy density) to the defect core (where [MATH]), the condition becomes [EQUATION]', 'hep-th-0302160-1-30-9': 'Thus, for values of [MATH] comparable or larger than [MATH] but coupling constant [MATH], both the condition ([REF]) for topological inflation and the condition [MATH] for applicability of the Friedmann equations for the evolution of the background space-time are satisfied.', 'hep-th-0302160-1-31-0': 'The above conditions can easily be satisfied for defect arising on D-branes in superstring theory.', 'hep-th-0302160-1-31-1': 'In this context, it has been shown that the tachyon potential, computed in superstring field theory, has the form [CITATION] (see also [CITATION]) [EQUATION] where [MATH] is proportional to the brane tension and [MATH] is the string tension.', 'hep-th-0302160-1-31-2': 'Thus, if [MATH], then the conditions for topological inflation are satisfied.', 'hep-th-0302160-1-31-3': 'This condition requires the (six dimensional) string coupling to be small.', 'hep-th-0302160-1-32-0': 'Topological inflation from stable defects suffers from a graceful exit problem.', 'hep-th-0302160-1-32-1': 'Inflation in the defect core continues forever, and in order to make a successful transition to late time cosmology one usually has to postulate that our region of space originated from a region close to the original defect boundary.', 'hep-th-0302160-1-32-2': 'This is already problematic in the case of field theory inflation in four dimensional space-time.', 'hep-th-0302160-1-32-3': 'Here the problem becomes even more acute, since in the context of brane world models, our matter fields have to be localized on the locus that is to become our physical spacetime.', 'hep-th-0302160-1-33-0': 'However, if the inflationary expansion takes place in the core of an embedded defect, the graceful exit problem is naturally resolved: after some finite time, the defect decays, the potential energy density disappears, and inflation stops.', 'hep-th-0302160-1-34-0': 'As was shown in [CITATION], embedded defects typically acquire superconducting current.', 'hep-th-0302160-1-34-1': 'These currents help stabilize the embedded defects down to very low temperatures.', 'hep-th-0302160-1-34-2': 'At the temperature [MATH] below which the thermal barrier becomes too low to stabilize the embedded defect, a core phase transition [CITATION] occurs (the charged scalar fields acquire nonvanishing expectation values), but the defect persists [CITATION].', 'hep-th-0302160-1-34-3': 'The defect decays only when the gauge field responsible for the stabilization of the defect ceases to be in thermal equilibrium.', 'hep-th-0302160-1-34-4': 'Let us assume that it is the photon which is responsible for the stabilization of our embedded [MATH] brane.', 'hep-th-0302160-1-34-5': 'In this case, the decay temperature [MATH] could be as low as [EQUATION] the temperature of recombination .', 'hep-th-0302160-1-34-6': 'If the temperature at the onset of inflation is about [MATH], in light of the formula ([REF]) for the tachyon potential in string field theory quite a reasonable value (it is also the upper bound on the scale of inflation in order not to overproduce gravitational waves), then we (optimistically) obtain about 62 e-foldings of inflation, only slightly larger than the minimum number of e-foldings required [CITATION] for inflation to solve the horizon and flatness problems of standard cosmology.', 'hep-th-0302160-1-34-7': 'Note that for this small number of e-foldings, the physical wavelength of comoving scales which correspond to present day cosmological scales was larger than the Planck length at the beginning of inflation.', 'hep-th-0302160-1-34-8': 'Thus, there is no trans-Planckian problem [CITATION] in this model.', 'hep-th-0302160-1-35-0': 'As mentioned in the Introduction, in this mechanism of topological inflation from embedded defects produced during [MATH] and [MATH] brane annihilations, two of the internal dimensions (namely those parallel to the worldvolume of the initial branes) also inflate.', 'hep-th-0302160-1-35-1': 'Given the low number of e-foldings of inflation which result in this scenario, the size of the two large extra dimensions is comfortably below the observational bounds (assuming that there is a radion stabilization mechanism which sets after inflation in for these directions).', 'hep-th-0302160-1-36-0': '# Summary and Discussion', 'hep-th-0302160-1-37-0': 'In this paper we have proposed a new way of obtaining inflation in the context of brane physics.', 'hep-th-0302160-1-37-1': 'The mechanism is based on inflation taking place in the core of an embedded defect.', 'hep-th-0302160-1-37-2': 'The embedded defect is stabilized at high temperatures by plasma effects, as studied in [CITATION], but becomes unstable and decays at lower temperatures, thus providing a natural graceful exit mechanism from inflation.', 'hep-th-0302160-1-38-0': 'We have suggested a specific realization of this idea in the context of a brane-antibrane setup, where our world volume corresponds to that of an embedded 3 brane which is generated in the process of tachyon condensation of an annihilating 5 brane - antibrane pair.', 'hep-th-0302160-1-38-1': 'To obtain an embedded defect in this setup, the gauge groups on the 5 branes must be enhanced from the minimal gauge group [MATH].', 'hep-th-0302160-1-39-0': 'Clearly this idea is more general than the brane-antibrane setup.', 'hep-th-0302160-1-39-1': 'For example, in the context of brane intersections [CITATION] an embedded defect can be realized if the gauge content on the branes is enlarged.', 'hep-th-0302160-1-39-2': 'Some intersections can unwind as the temperature cools, a process described in the field theory as the decay of the embedded defect, and requires one to excite off-diagonal, non-geometrical open strings (see for example [CITATION] for a detailed discussion).', 'hep-th-0302160-1-40-0': 'More generally, it is well-known, at least in the BPS sector, that stability of defects depends on parameters of the theory (see e.g. [CITATION]).', 'hep-th-0302160-1-40-1': 'This is well controlled for supersymmetric states, by using the BPS formula, but is expected to be the case generally.', 'hep-th-0302160-1-40-2': 'In this case, one can use the topological inflation idea, and provide a natural graceful exit mechanism along the lines of this paper.', 'hep-th-0302160-1-40-3': 'In this scenario the initial stable defect causes inflation.', 'hep-th-0302160-1-40-4': 'As inflation proceeds, and effective parameters of the theory change in response to cooling down, causing the initial defect to become unstable.', 'hep-th-0302160-1-40-5': 'This paper is a concrete example of this more general scenario.', 'hep-th-0302160-1-41-0': 'Obviously, our scenario of "topological inflation" from an embedded defect can also be realized in ordinary four dimensional quantum field theory.', 'hep-th-0302160-1-41-1': 'However, in that context it appears unnatural to obtain defects with the width required for topological inflation, unless field values larger than the Planck scale are invoked.', 'hep-th-0302160-1-41-2': 'As we have seen, in the context of the brane setup, the existence of new physical scales (e.g. the brane tensions) makes it easy to obtain defects of sufficient width to support topological inflation.', 'hep-th-0302160-1-42-0': 'We have also seen that in our realization of embedded defect inflation in the context of an annihilating [MATH] and [MATH] brane pair, inflation can produce a hierarchy in the sizes of the internal dimensions (or a warped geometry), since the two extra spatial dimensions parallel to the worldvolume of the [MATH] branes also expand exponentially.', 'hep-th-0302160-1-43-0': 'Recently, it was argued that de Sitter space vacua in string theory will suffer from conceptual problems [CITATION], if the lifetime of de Sitter space lasts longer than the Poincare recurrence time [MATH], where [MATH] is the entropy of the space.', 'hep-th-0302160-1-43-1': 'For example, Kachru et.', 'hep-th-0302160-1-43-2': 'al provided a stringy realization of a meta-stable de Sitter space which tunnels to flat space-time in a timescale shorter than [MATH][CITATION].', 'hep-th-0302160-1-43-3': 'Our scenario is more reminiscent of the new inflationary models, in that no tunneling is required to stop inflation.', 'hep-th-0302160-1-43-4': 'Consequently our inflationary spacetime turns into a hot FRW space-time after [MATH] e-foldings, a time scale exponentially shorter than [MATH].'}

{'hep-th-0302160-2-0-0': 'We discuss a new mechanism of obtaining a period of cosmological inflation in the context of string theory.', 'hep-th-0302160-2-0-1': 'This mechanism is based on embedded defects which form dynamically on higher dimensional D-branes.', 'hep-th-0302160-2-0-2': 'Such defects generate topological inflation, but unlike topological inflation from stable defects, here there is a natural graceful exit from inflation: the decay of the embedded defect.', 'hep-th-0302160-2-0-3': 'We demonstrate the idea in the context of a brane-antibrane annihilation process.', 'hep-th-0302160-2-0-4': 'The graceful exit mechanism suggested here applies generically to all realizations of inflation on D-branes.', 'hep-th-0302160-2-1-0': ']', 'hep-th-0302160-2-2-0': '# Introduction', 'hep-th-0302160-2-3-0': 'There has been a lot of interest recently in exploring ways of obtaining a period of cosmological inflation from string theory.', 'hep-th-0302160-2-3-1': 'A realization of inflation in string theory should address some of the conceptual problems of quantum field theory realizations of inflationary cosmology [CITATION].', 'hep-th-0302160-2-3-2': 'Since string theory contains many moduli fields which are massless at the perturbative level, stringy inflation may yield a natural mechanism to produce fluctuations of the required small amplitude [CITATION].', 'hep-th-0302160-2-3-3': 'A description of inflation in the context of string theory would also automatically resolve the trans-Planckian problem [CITATION] of inflationary cosmology, since it would yield a complete description of the dynamics of fluctuations during the entire inflationary period.', 'hep-th-0302160-2-3-4': 'Furthermore, since one of the goals of string theory is to provide a nonsingular cosmology, a stringy inflation model should also allow one to resolve the singularity problem [CITATION] of inflationary cosmology.', 'hep-th-0302160-2-4-0': 'Recent developments in particle phenomenology have explored the possibility that the our matter fields are confined to a four-dimensional space-time hypersurface in a higher-dimensional bulk space-time [CITATION].', 'hep-th-0302160-2-4-1': 'In the context of string theory this hypersurface arises naturally as a D-brane, one of possibly many branes that reside in the bulk spacetime [CITATION].', 'hep-th-0302160-2-4-2': 'However, for our purposes here, a sufficient starting point is some higher dimensional field theory (coupled weakly to gravity).', 'hep-th-0302160-2-4-3': 'This higher dimensional theory can be completed in the ultraviolet by embedding in critical string theory, but also by using the [MATH] fixed point [CITATION], little string theory [CITATION], or by deconstruction [CITATION].', 'hep-th-0302160-2-4-4': 'In order to have a definite mental picture we assume the higher dimensional theory is embedded in string theory, and is realized on D-branes.', 'hep-th-0302160-2-5-0': 'This "brane-world" scenario has generated new realizations of cosmic inflation.', 'hep-th-0302160-2-5-1': 'In one setup [CITATION], the inflaton is the separation between a brane-antibrane pair.', 'hep-th-0302160-2-5-2': 'However, it has been shown [CITATION] that in this context it is not easy to obtain initial conditions which lead to a sufficient period of inflation.', 'hep-th-0302160-2-5-3': 'Another possibility is to have inflation generated by branes at an angle [CITATION], or by a configuration of defects of different dimensionalities [CITATION].', 'hep-th-0302160-2-5-4': 'This has the advantage of possibly providing a mechanism for localization of chiral matter on the inflating locus.', 'hep-th-0302160-2-6-0': 'An alternative way to obtain inflation from brane collisions was suggested in [CITATION], based on the realization [CITATION] that topological defects of co-dimension greater or equal to one are formed on the brane worldvolume during the collision of a brane-antibrane pair.', 'hep-th-0302160-2-6-1': 'In the simplest example, the order parameter of the phase transition which occurs when the brane-antibrane pair meets is given by a complex tachyon field [CITATION] which condenses at a nonvanishing expectation value.', 'hep-th-0302160-2-6-2': 'The vacuum manifold of the tachyon field ground state values has the topology of [MATH].', 'hep-th-0302160-2-6-3': 'By causality [CITATION], the values of the condensate in the vacuum manifold are uncorrelated on large length scales, and hence a network of co-dimension 2 defects inevitably will form.', 'hep-th-0302160-2-6-4': 'In the case of a [MATH]-[MATH] brane pair, the resulting topological defect is a [MATH] brane which could be our world.', 'hep-th-0302160-2-6-5': 'Provided that the thickness of the topological defect is larger than the Hubble radius, the defect core can undergo inflation (topological inflation [CITATION]).', 'hep-th-0302160-2-7-0': 'A problem with the mechanism of [CITATION] (and of models of 4 dimensional topological inflation [CITATION]) is the graceful exit problem: how does inflation end, and a transition to the usual late time cosmological evolution occur?', 'hep-th-0302160-2-8-0': 'In this paper, we provide a simple solution to this problem, which generalizes to all above mentioned "brane-world" models of inflation.', 'hep-th-0302160-2-8-1': 'We point out that the defects which form in brane collisions will often be embedded defects (which are non-BPS branes in the string context) (see [CITATION] for a discussion of embedded defects in field theory).', 'hep-th-0302160-2-8-2': 'These embedded defects can be stabilized at high temperatures by plasma effects [CITATION], but decay at a sufficiently low temperature, thus providing a natural graceful exit mechanism from inflation.', 'hep-th-0302160-2-8-3': 'In the context of field theory inflation, such a scenario was suggested in [CITATION].', 'hep-th-0302160-2-9-0': 'We may call the proposed inflationary model "Higher Dimensional (Non-) Topological Inflation" since inflation is taking place in the higher-dimensional brane worldvolume, not only in the directions of the defect (which are the dimensions of our four-dimensional physical space-time).', 'hep-th-0302160-2-9-1': 'Assuming that there is a mechanism to stabilize the radion in the two (compact) distinguished extra dimensions (the directions of the original brane orthogoal to the defect), we would have found a way of making two of the internal dimensions larger than the others, thus making contact to the scenario of [CITATION] .', 'hep-th-0302160-2-9-2': 'Alternatively, the two extra dimensions may be taken to be infinite; as one moves away from the core of the defect, inflation proceeds more slowly, and ends earlier.', 'hep-th-0302160-2-9-3': 'Thus the resulting spacetime is expected to be warped, making contact with the work of [CITATION].', 'hep-th-0302160-2-9-4': 'The question of which alternative is realized in specific examples needs a detailed, model dependent analysis, which we hope to return to in the future.', 'hep-th-0302160-2-10-0': 'The causality argument (Kibble mechanism [CITATION]) which ensures that in a phase transition leading to the possibility of topological defect formation such defects inevitably will form, with a separation smaller or comparable to the Hubble length, also ensures the formation of stabilized embedded defects in models which admit them, as will be explained later.', 'hep-th-0302160-2-10-1': 'Thus, there is no initial condition problem for our model of inflation.', 'hep-th-0302160-2-11-0': 'In field theory models with a symmetry breaking scale [MATH] much smaller than the Planck scale, the width of a defect is typically much smaller than the Hubble length.', 'hep-th-0302160-2-11-1': 'Thus, topological inflation will not occur.', 'hep-th-0302160-2-11-2': 'An advantage of inflation in the context of higher dimensions is that there is a new scale, the brane tension, which is naturally higher than the Planck mass scale .', 'hep-th-0302160-2-11-3': 'As will be shown in Section III, in this context the defect width can easily be larger than the Hubble length, thus making defect-driven inflation possible.', 'hep-th-0302160-2-12-0': 'Our proposed mechanism of topological inflation from embedded defects can apply in situations more general than that of a brane-antibrane annihilation process.', 'hep-th-0302160-2-12-1': 'For example, in the context of brane intersections (the intersection region undergoing topological inflation), the branes may partially unwind around each other, leaving behind some non-inflating components.', 'hep-th-0302160-2-12-2': 'This process excites the off-diagonal open strings, which have no geometrical interpretation.', 'hep-th-0302160-2-13-0': 'The outline of this paper is as follows.', 'hep-th-0302160-2-13-1': 'In the next section, we discuss a particular example in which one can realize topological inflation from an embedded defect, namely embedded defects forming in a brane-antibrane annihilation process.', 'hep-th-0302160-2-13-2': 'Next, we discuss how topological inflation from embedded defects has a natural graceful exit mechanism.', 'hep-th-0302160-2-13-3': 'Finally.', 'hep-th-0302160-2-13-4': 'we summarize our results, and discuss further applications of the basic idea of topological inflation from embedded defects in the context of brane physics.', 'hep-th-0302160-2-14-0': '# Embedded Defects from Higher Dimensional D-branes', 'hep-th-0302160-2-15-0': 'We will start with a brief review of the original scenario of [CITATION].', 'hep-th-0302160-2-15-1': 'The scenario of [CITATION] takes as a starting point a pair of parallel [MATH] and [MATH] branes approaching each other.', 'hep-th-0302160-2-15-2': 'As studied in [CITATION], when the pair gets sufficiently close, a tachyon instability sets in.', 'hep-th-0302160-2-15-3': 'For minimal gauge content of the branes, the tachyon can be described by a complex scalar field [MATH] with a standard symmetry breaking (Mexican hat) potential [EQUATION] where the point [MATH] corresponds to branes at zero separation, and [MATH] is the symmetry breaking scale.', 'hep-th-0302160-2-15-4': 'The vacuum at [MATH] corresponds to the closed string vacuum, where the original branes have annihilated.', 'hep-th-0302160-2-16-0': 'The vacuum manifold of the tachyon condensates is [MATH], and hence according to the usual Kibble argument [CITATION], the fact that the condensate values must be uncorrelated on scales larger than the Hubble radius inevitably leads to the formation of stable co-dimension 2 defects, in this case [MATH] branes, on the worldvolume of the original [MATH] branes.', 'hep-th-0302160-2-16-1': 'Unlike most of the discussion of defects on annihilating D-branes, we are interested in the case where the resulting D3 brane is not BPS saturated, so that inflation is possible.', 'hep-th-0302160-2-17-0': 'In terms of gauge theory on the original brane worldvolume, the above process corresponds to the symmetry breaking [EQUATION] where one of the [MATH] factors on the left hand side of the equation lives on either of the branes, and the unbroken subgroup corresponds to the diagonal [MATH] factor.', 'hep-th-0302160-2-17-1': 'The standard homotopy arguments (see e.g. [CITATION] for a discussion of such arguments in the context of topological defects) yield [EQUATION] where [MATH] is the vacuum manifold, and [MATH] and [MATH] are the full gauge group and the residual gauge group after symmetry breaking, respectively .', 'hep-th-0302160-2-18-0': 'This set-up is not restricted to annihilating D-branes.', 'hep-th-0302160-2-18-1': 'Suppose we start with any higher dimensional D-brane configuration, and that at initial times the set-up is removed from its vacuum.', 'hep-th-0302160-2-18-2': 'This vacuum may be the closed string vacuum, but also an open string vacuum, where the branes align in a supersymmetric fashion.', 'hep-th-0302160-2-18-3': 'In the process of relaxation to the vacuum, the Kibble mechanism will ensure the creation of local defects, for example brane intersections.', 'hep-th-0302160-2-18-4': 'It is those defects which seed inflation in the higher-dimensional topological context, and which are the subject of our discussion.', 'hep-th-0302160-2-18-5': 'In particular, the energy scale associated with this initial phase transition can be much smaller than the string scale, so the above (higher dimensional) field theory considerations apply.', 'hep-th-0302160-2-19-0': 'Our idea, in this context, is to concentrate on the case of embedded, non-topological, defects.', 'hep-th-0302160-2-19-1': 'For this end we enhance the gauge symmetry on each of the branes, e.g. from [MATH] to [MATH].', 'hep-th-0302160-2-19-2': 'Indeed, this arises naturally in the context of fivebranes in type I theory.', 'hep-th-0302160-2-19-3': 'In this case the worldvolume gauge group is [MATH].', 'hep-th-0302160-2-19-4': 'The tachyonic scalar fields transform as [MATH] under [MATH].', 'hep-th-0302160-2-20-0': 'Now, the symmetry breaking pattern during tachyon condensation becomes [EQUATION]', 'hep-th-0302160-2-20-1': 'In this case, no stable topological defects (in particular no co-dimension 2 defects) form during this phase transition since the vacuum manifold is [MATH] and hence [MATH].', 'hep-th-0302160-2-20-2': 'Note that in the context of studies of branes in Type I string theory it is already known that the [MATH] brane is unstable [CITATION].', 'hep-th-0302160-2-21-0': 'The situation is similar to what happens in the electroweak theory where (for vanishing Weinberg angle), the symmetry breaking is [MATH], as effectively in the above case ([REF]).', 'hep-th-0302160-2-21-1': 'However, as is well known from analysis of the electroweak theory [CITATION], it is possible to construct embedded defects, solutions of the equations of motion which correspond to unstable defects.', 'hep-th-0302160-2-21-2': 'In the case of the electroweak theory in four space-time dimensions, the defects are electroweak strings, an example of embedded strings .', 'hep-th-0302160-2-22-0': 'As was recently realized [CITATION], such embedded defects can be stabilized at finite density by plasma effects.', 'hep-th-0302160-2-22-1': 'Consider again first the example of the standard electroweak theory.', 'hep-th-0302160-2-22-2': 'The order parameter of the symmetry breaking phase transition has four real components, two of which are electrically charged, two are electrically neutral.', 'hep-th-0302160-2-22-3': 'In the presence of a thermal bath of photons, the effective potential will then be lifted in the charged scalar field directions by more than it is lifted in the neutral scalar field directions.', 'hep-th-0302160-2-22-4': 'Thus, the vacuum manifold corresponding to the theory in a photon plasma is [MATH] and not [MATH].', 'hep-th-0302160-2-22-5': 'This can stabilize a subset of embedded strings, namely the string configurations constructed from the neutral scalar fields (these are the electroweak Z-strings in the case of the standard electroweak theory).', 'hep-th-0302160-2-23-0': 'Note that if the embedded defects are thermally stabilized at the temperature just below the phase transition temperature, then the causality argument (Kibble mechanism [CITATION]) applies and states that at least of the order one such defect will form per correlation volume (which in turn is smaller than the Hubble volume).', 'hep-th-0302160-2-23-1': 'The argument is as follows: during the phase transition, the order parameter relaxes to the vacuum manifold [MATH].', 'hep-th-0302160-2-23-2': 'However, in the presence of thermal stabilizing effects it will relax to the reduced vacuum manifold [MATH].', 'hep-th-0302160-2-23-3': 'However, which value in [MATH] is taken on is random on scales larger than the correlation length.', 'hep-th-0302160-2-23-4': 'Hence, defects will form.', 'hep-th-0302160-2-23-5': 'This process was studied in detail in the case of semilocal strings (related to embedded strings) in [CITATION].', 'hep-th-0302160-2-24-0': 'Our idea is to apply this mechanism to brane cosmology.', 'hep-th-0302160-2-24-1': 'We need to assume that the branes contain a thermal bath of gauge fields which couple in the same way to the [MATH] symmetry breaking field of our brane setup as the photon does to the [MATH] symmetry breaking order parameter in the electroweak theory.', 'hep-th-0302160-2-24-2': 'Ideally, the usual photon field could play this role.', 'hep-th-0302160-2-24-3': 'In this case, it would be natural to have only this field in thermal equilibrium at late times, since it would be the only massless gauge field.', 'hep-th-0302160-2-24-4': 'The situation would then be identical to the electroweak theory discussed above.', 'hep-th-0302160-2-24-5': 'Given the above assumptions about field content and coupling to the [MATH] symmetry breaking order parameter, it is completely natural from the point of view of cosmology (hot initial state in the very early Universe) to have the field excited in the required way.', 'hep-th-0302160-2-25-0': 'In this case, the phase transition corresponding to brane collision will, by the Kibble mechanism [CITATION], inevitably produce a network of co-dimension 2 branes.', 'hep-th-0302160-2-25-1': 'These defects will remain stable until the matter density on the branes has sufficiently diluted.', 'hep-th-0302160-2-26-0': 'Let us now be a bit more specific.', 'hep-th-0302160-2-26-1': 'Given an [MATH] gauge group on each of the [MATH] branes, the world volume theory of a coincident [MATH]-[MATH] brane pair is [MATH], and any gauge field configuration can be written as a [MATH] matrix of complex entries, the upper diagonal [MATH] entries corresponding to a [MATH] matrix describing open strings which start and end on the first brane, the lower diagonal [MATH] entries describing a [MATH] matrix corresponding to the open strings starting and ending on the other brane.', 'hep-th-0302160-2-26-2': 'The two off-diagonal [MATH] sub-matrices are [MATH] matrices corresponding to open strings beginning and ending on different branes.', 'hep-th-0302160-2-26-3': 'It is these latter sectors which contain the tachyon [CITATION].', 'hep-th-0302160-2-26-4': 'In the case of [MATH] gauge fields on each of the branes, the tachyon is a complex scalar field (one real component from each of the sectors), in the case of [MATH] gauge fields, there are two real components in each sector.', 'hep-th-0302160-2-27-0': 'Let us now assume that on each of the branes there is a thermal bath of a [MATH] subgroup of [MATH], and the other gauge fields assumed to be vanishing.', 'hep-th-0302160-2-27-1': 'In this case, the gauge fields excited on the branes will only couple to one of the two real tachyon fields in each sector.', 'hep-th-0302160-2-27-2': 'Let us combine these two fields into a complex field [MATH] (charged with respect to the excited gauge field), the other two into a complex field [MATH] (neutral with respect to the excited gauge field).', 'hep-th-0302160-2-27-3': 'The effective world volume Lagrangian then becomes [EQUATION] where the [MATH] stands for the gauge covariant derivative, the gauge field being the one excited on the brane.', 'hep-th-0302160-2-27-4': 'Hence, the gauge field directly couples only to the charged scalar field [MATH] and not to the neutral field [MATH].', 'hep-th-0302160-2-27-5': 'The field strength tensor of the gauge field [MATH] excited on the brane is denoted by [MATH], but the [MATH] term in the Lagrangian will not be important in the following.', 'hep-th-0302160-2-27-6': 'The tachyon potential [MATH] has the typical symmetry breaking form [EQUATION] where [MATH] is a coupling constant, and [MATH] is the ground state expectation value of the tachyon field magnitude.', 'hep-th-0302160-2-27-7': 'The corresponding vacuum manifold is [MATH].', 'hep-th-0302160-2-28-0': 'The above effective Lagrangian is the same which describes the sigma model of low energy QCD in the chiral limit, coupled to an external bath of photons [CITATION].', 'hep-th-0302160-2-28-1': 'It also describes the standard electroweak theory with only the photon field excited [CITATION].', 'hep-th-0302160-2-28-2': 'In the presence of a thermal bath of [MATH] gauge fields, we can take the thermal average of the Lagrangian.', 'hep-th-0302160-2-28-3': 'Terms linear in [MATH] vanish, the quadratic term [MATH] becomes [MATH], where [MATH] is the temperature of the bath.', 'hep-th-0302160-2-28-4': 'Thus, an effective mass term for the [MATH] field is generated.', 'hep-th-0302160-2-28-5': 'The effective potential becomes [EQUATION]', 'hep-th-0302160-2-28-6': 'Hence, the vacuum manifold is lifted in the charged scalar field directions.', 'hep-th-0302160-2-28-7': 'The vacuum manifold of the effective potential becomes [MATH].', 'hep-th-0302160-2-28-8': 'Thus, co-dimension 2 defects in which the neutral scalar field configuration takes on the Nielsen-Olesen [CITATION] form become important as metastable defects.', 'hep-th-0302160-2-29-0': '# Embedded Defects, Inflation, and Graceful Exit', 'hep-th-0302160-2-30-0': 'The idea of topological inflation is quite simple [CITATION]: provided that the core of the defect is comparable or larger than the Hubble radius, then the potential energy density in the defect core will be larger than the tension energy, and thus via standard minimal coupling to gravity, the core will commence inflationary expansion.', 'hep-th-0302160-2-30-1': 'For defects associates with an energy scale much smaller than the Planck scale, the width is much too small to obtain topological inflation.', 'hep-th-0302160-2-30-2': 'However, if the tension of the defect is given by an energy scale [MATH] which is comparable of larger than [MATH], then the condition for topological inflation is satisfied.', 'hep-th-0302160-2-30-3': 'A quantitative analysis [CITATION] yields as condition for topological inflation [EQUATION]', 'hep-th-0302160-2-30-4': 'Let us give a brief derivation of this result.', 'hep-th-0302160-2-30-5': 'For a typical symmetry breaking potential of the form ([REF]), the defect width [MATH] is of the order [CITATION] [EQUATION] which can be seen by balancing potential and tension energies.', 'hep-th-0302160-2-30-6': 'In the core, the potential energy density dominates, in the outside regions of the defect the tension energy density is larger.', 'hep-th-0302160-2-30-7': 'The condition for defect inflation is that the core width is greater than the Hubble length [MATH].', 'hep-th-0302160-2-30-8': 'Applying the Friedmann equation [EQUATION] (where [MATH] is the energy density) to the defect core (where [MATH]), the condition becomes [EQUATION]', 'hep-th-0302160-2-30-9': 'Thus, for values of [MATH] comparable or larger than [MATH] but coupling constant [MATH], both the condition ([REF]) for topological inflation and the condition [MATH] for applicability of the Friedmann equations for the evolution of the background space-time are satisfied.', 'hep-th-0302160-2-31-0': 'The above conditions can easily be satisfied for defect arising on D-branes in superstring theory.', 'hep-th-0302160-2-31-1': 'In this context, it has been shown that the tachyon potential, computed in superstring field theory, has the form [CITATION] (see also [CITATION]) [EQUATION] where [MATH] is proportional to the brane tension and [MATH] is the string tension.', 'hep-th-0302160-2-31-2': 'Thus, if [MATH], then the conditions for topological inflation are satisfied.', 'hep-th-0302160-2-31-3': 'This condition requires the (six dimensional) string coupling to be small.', 'hep-th-0302160-2-32-0': 'Topological inflation from stable defects suffers from a graceful exit problem.', 'hep-th-0302160-2-32-1': 'Inflation in the defect core continues forever, and in order to make a successful transition to late time cosmology one usually has to postulate that our region of space originated from a region close to the original defect boundary.', 'hep-th-0302160-2-32-2': 'This is already problematic in the case of field theory inflation in four dimensional space-time.', 'hep-th-0302160-2-32-3': 'Here the problem becomes even more acute, since in the context of brane world models, our matter fields have to be localized on the locus that is to become our physical spacetime.', 'hep-th-0302160-2-33-0': 'However, if the inflationary expansion takes place in the core of an embedded defect, the graceful exit problem is naturally resolved: after some finite time, the defect decays, the potential energy density disappears, and inflation stops.', 'hep-th-0302160-2-34-0': 'As was shown in [CITATION], embedded defects typically acquire superconducting current.', 'hep-th-0302160-2-34-1': 'These currents help stabilize the embedded defects down to very low temperatures.', 'hep-th-0302160-2-34-2': 'At the temperature [MATH] below which the thermal barrier becomes too low to stabilize the embedded defect, a core phase transition [CITATION] (see also [CITATION]) occurs (the charged scalar fields acquire nonvanishing expectation values), but the defect persists [CITATION].', 'hep-th-0302160-2-34-3': 'The defect decays only when the gauge field responsible for the stabilization of the defect ceases to be in thermal equilibrium.', 'hep-th-0302160-2-34-4': 'Let us assume that it is the photon which is responsible for the stabilization of our embedded [MATH] brane.', 'hep-th-0302160-2-34-5': 'In this case, the decay temperature [MATH] could be as low as [EQUATION] the temperature of recombination .', 'hep-th-0302160-2-34-6': 'If the temperature at the onset of inflation is about [MATH], in light of the formula ([REF]) for the tachyon potential in string field theory quite a reasonable value (it is also the upper bound on the scale of inflation in order not to overproduce gravitational waves), then we (optimistically) obtain about 62 e-foldings of inflation, only slightly larger than the minimum number of e-foldings required [CITATION] for inflation to solve the horizon and flatness problems of standard cosmology.', 'hep-th-0302160-2-34-7': 'Note that for this small number of e-foldings, the physical wavelength of comoving scales which correspond to present day cosmological scales was larger than the Planck length at the beginning of inflation.', 'hep-th-0302160-2-34-8': 'Thus, there is no trans-Planckian problem [CITATION] in this model.', 'hep-th-0302160-2-35-0': 'As mentioned in the Introduction, in this mechanism of topological inflation from embedded defects produced during [MATH] and [MATH] brane annihilations, two of the internal dimensions (namely those parallel to the worldvolume of the initial branes) also inflate.', 'hep-th-0302160-2-35-1': 'Given the low number of e-foldings of inflation which result in this scenario, the size of the two large extra dimensions is comfortably below the observational bounds (assuming that there is a radion stabilization mechanism which sets after inflation in for these directions).', 'hep-th-0302160-2-36-0': '# Summary and Discussion', 'hep-th-0302160-2-37-0': 'In this paper we have proposed a new way of obtaining inflation in the context of brane physics.', 'hep-th-0302160-2-37-1': 'The mechanism is based on inflation taking place in the core of an embedded defect.', 'hep-th-0302160-2-37-2': 'The embedded defect is stabilized at high temperatures by plasma effects, as studied in [CITATION], but becomes unstable and decays at lower temperatures, thus providing a natural graceful exit mechanism from inflation.', 'hep-th-0302160-2-38-0': 'We have suggested a specific realization of this idea in the context of a brane-antibrane setup, where our world volume corresponds to that of an embedded 3 brane which is generated in the process of tachyon condensation of an annihilating 5 brane - antibrane pair.', 'hep-th-0302160-2-38-1': 'To obtain an embedded defect in this setup, the gauge groups on the 5 branes must be enhanced from the minimal gauge group [MATH].', 'hep-th-0302160-2-39-0': 'Clearly this idea is more general than the brane-antibrane setup.', 'hep-th-0302160-2-39-1': 'For example, in the context of brane intersections [CITATION] an embedded defect can be realized if the gauge content on the branes is enlarged.', 'hep-th-0302160-2-39-2': 'Some intersections can unwind as the temperature cools, a process described in the field theory as the decay of the embedded defect, and requires one to excite off-diagonal, non-geometrical open strings (see for example [CITATION] for a detailed discussion).', 'hep-th-0302160-2-40-0': 'More generally, it is well-known, at least in the BPS sector, that stability of defects depends on parameters of the theory (see e.g. [CITATION]).', 'hep-th-0302160-2-40-1': 'This is well controlled for supersymmetric states, by using the BPS formula, but is expected to be the case generally.', 'hep-th-0302160-2-40-2': 'In this case, one can use the topological inflation idea, and provide a natural graceful exit mechanism along the lines of this paper.', 'hep-th-0302160-2-40-3': 'In this scenario the initial stable defect causes inflation.', 'hep-th-0302160-2-40-4': 'As inflation proceeds, and effective parameters of the theory change in response to cooling down, causing the initial defect to become unstable.', 'hep-th-0302160-2-40-5': 'This paper is a concrete example of this more general scenario.', 'hep-th-0302160-2-41-0': 'Obviously, our scenario of "topological inflation" from an embedded defect can also be realized in ordinary four dimensional quantum field theory.', 'hep-th-0302160-2-41-1': 'However, in that context it appears unnatural to obtain defects with the width required for topological inflation, unless field values larger than the Planck scale are invoked.', 'hep-th-0302160-2-41-2': 'As we have seen, in the context of the brane setup, the existence of new physical scales (e.g. the brane tensions) makes it easy to obtain defects of sufficient width to support topological inflation.', 'hep-th-0302160-2-42-0': 'We have also seen that in our realization of embedded defect inflation in the context of an annihilating [MATH] and [MATH] brane pair, inflation can produce a hierarchy in the sizes of the internal dimensions (or a warped geometry), since the two extra spatial dimensions parallel to the worldvolume of the [MATH] branes also expand exponentially.', 'hep-th-0302160-2-43-0': 'Recently, it was argued that de Sitter space vacua in string theory will suffer from conceptual problems [CITATION], if the lifetime of de Sitter space lasts longer than the Poincare recurrence time [MATH], where [MATH] is the entropy of the space.', 'hep-th-0302160-2-43-1': 'For example, Kachru et.', 'hep-th-0302160-2-43-2': 'al provided a stringy realization of a meta-stable de Sitter space which tunnels to flat space-time in a timescale shorter than [MATH][CITATION].', 'hep-th-0302160-2-43-3': 'Our scenario is more reminiscent of the new inflationary models, in that no tunneling is required to stop inflation.', 'hep-th-0302160-2-43-4': 'Consequently our inflationary spacetime turns into a hot FRW space-time after [MATH] e-foldings, a time scale exponentially shorter than [MATH].'}

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'hep-th-0302160-2-27-1'], ['hep-th-0302160-1-27-2', 'hep-th-0302160-2-27-2'], ['hep-th-0302160-1-27-3', 'hep-th-0302160-2-27-3'], ['hep-th-0302160-1-27-4', 'hep-th-0302160-2-27-4'], ['hep-th-0302160-1-27-5', 'hep-th-0302160-2-27-5'], ['hep-th-0302160-1-27-6', 'hep-th-0302160-2-27-6'], ['hep-th-0302160-1-27-7', 'hep-th-0302160-2-27-7'], ['hep-th-0302160-1-24-0', 'hep-th-0302160-2-24-0'], ['hep-th-0302160-1-24-1', 'hep-th-0302160-2-24-1'], ['hep-th-0302160-1-24-2', 'hep-th-0302160-2-24-2'], ['hep-th-0302160-1-24-3', 'hep-th-0302160-2-24-3'], ['hep-th-0302160-1-24-4', 'hep-th-0302160-2-24-4'], ['hep-th-0302160-1-24-5', 'hep-th-0302160-2-24-5'], ['hep-th-0302160-1-33-0', 'hep-th-0302160-2-33-0'], ['hep-th-0302160-1-18-0', 'hep-th-0302160-2-18-0'], ['hep-th-0302160-1-18-1', 'hep-th-0302160-2-18-1'], ['hep-th-0302160-1-18-2', 'hep-th-0302160-2-18-2'], ['hep-th-0302160-1-18-3', 'hep-th-0302160-2-18-3'], ['hep-th-0302160-1-18-4', 'hep-th-0302160-2-18-4'], ['hep-th-0302160-1-18-5', 'hep-th-0302160-2-18-5'], ['hep-th-0302160-1-30-0', 'hep-th-0302160-2-30-0'], ['hep-th-0302160-1-30-1', 'hep-th-0302160-2-30-1'], ['hep-th-0302160-1-30-2', 'hep-th-0302160-2-30-2'], ['hep-th-0302160-1-30-3', 'hep-th-0302160-2-30-3'], ['hep-th-0302160-1-30-4', 'hep-th-0302160-2-30-4'], ['hep-th-0302160-1-30-5', 'hep-th-0302160-2-30-5'], ['hep-th-0302160-1-30-6', 'hep-th-0302160-2-30-6'], ['hep-th-0302160-1-30-7', 'hep-th-0302160-2-30-7'], ['hep-th-0302160-1-30-8', 'hep-th-0302160-2-30-8'], ['hep-th-0302160-1-30-9', 'hep-th-0302160-2-30-9'], ['hep-th-0302160-1-38-0', 'hep-th-0302160-2-38-0'], ['hep-th-0302160-1-38-1', 'hep-th-0302160-2-38-1'], ['hep-th-0302160-1-28-0', 'hep-th-0302160-2-28-0'], ['hep-th-0302160-1-28-1', 'hep-th-0302160-2-28-1'], ['hep-th-0302160-1-28-2', 'hep-th-0302160-2-28-2'], ['hep-th-0302160-1-28-3', 'hep-th-0302160-2-28-3'], ['hep-th-0302160-1-28-4', 'hep-th-0302160-2-28-4'], ['hep-th-0302160-1-28-5', 'hep-th-0302160-2-28-5'], ['hep-th-0302160-1-28-6', 'hep-th-0302160-2-28-6'], ['hep-th-0302160-1-28-7', 'hep-th-0302160-2-28-7'], ['hep-th-0302160-1-28-8', 'hep-th-0302160-2-28-8'], ['hep-th-0302160-1-34-0', 'hep-th-0302160-2-34-0'], ['hep-th-0302160-1-34-1', 'hep-th-0302160-2-34-1'], ['hep-th-0302160-1-34-3', 'hep-th-0302160-2-34-3'], ['hep-th-0302160-1-34-4', 'hep-th-0302160-2-34-4'], ['hep-th-0302160-1-34-5', 'hep-th-0302160-2-34-5'], ['hep-th-0302160-1-34-6', 'hep-th-0302160-2-34-6'], ['hep-th-0302160-1-34-7', 'hep-th-0302160-2-34-7'], ['hep-th-0302160-1-34-8', 'hep-th-0302160-2-34-8'], ['hep-th-0302160-1-8-0', 'hep-th-0302160-2-8-0'], ['hep-th-0302160-1-8-1', 'hep-th-0302160-2-8-1'], ['hep-th-0302160-1-8-2', 'hep-th-0302160-2-8-2'], ['hep-th-0302160-1-8-3', 'hep-th-0302160-2-8-3'], ['hep-th-0302160-1-39-0', 'hep-th-0302160-2-39-0'], ['hep-th-0302160-1-39-1', 'hep-th-0302160-2-39-1'], ['hep-th-0302160-1-39-2', 'hep-th-0302160-2-39-2'], ['hep-th-0302160-1-11-0', 'hep-th-0302160-2-11-0'], ['hep-th-0302160-1-11-1', 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'hep-th-0302160-2-40-3'], ['hep-th-0302160-1-40-4', 'hep-th-0302160-2-40-4'], ['hep-th-0302160-1-40-5', 'hep-th-0302160-2-40-5'], ['hep-th-0302160-1-41-0', 'hep-th-0302160-2-41-0'], ['hep-th-0302160-1-41-1', 'hep-th-0302160-2-41-1'], ['hep-th-0302160-1-41-2', 'hep-th-0302160-2-41-2'], ['hep-th-0302160-1-35-0', 'hep-th-0302160-2-35-0'], ['hep-th-0302160-1-35-1', 'hep-th-0302160-2-35-1'], ['hep-th-0302160-1-32-0', 'hep-th-0302160-2-32-0'], ['hep-th-0302160-1-32-1', 'hep-th-0302160-2-32-1'], ['hep-th-0302160-1-32-2', 'hep-th-0302160-2-32-2'], ['hep-th-0302160-1-32-3', 'hep-th-0302160-2-32-3'], ['hep-th-0302160-1-5-0', 'hep-th-0302160-2-5-0'], ['hep-th-0302160-1-5-1', 'hep-th-0302160-2-5-1'], ['hep-th-0302160-1-5-2', 'hep-th-0302160-2-5-2'], ['hep-th-0302160-1-5-3', 'hep-th-0302160-2-5-3'], ['hep-th-0302160-1-5-4', 'hep-th-0302160-2-5-4'], ['hep-th-0302160-1-17-0', 'hep-th-0302160-2-17-0'], ['hep-th-0302160-1-17-1', 'hep-th-0302160-2-17-1'], ['hep-th-0302160-1-23-0', 'hep-th-0302160-2-23-0'], ['hep-th-0302160-1-23-1', 'hep-th-0302160-2-23-1'], ['hep-th-0302160-1-23-2', 'hep-th-0302160-2-23-2'], ['hep-th-0302160-1-23-3', 'hep-th-0302160-2-23-3'], ['hep-th-0302160-1-23-4', 'hep-th-0302160-2-23-4'], ['hep-th-0302160-1-23-5', 'hep-th-0302160-2-23-5'], ['hep-th-0302160-1-6-0', 'hep-th-0302160-2-6-0'], ['hep-th-0302160-1-6-1', 'hep-th-0302160-2-6-1'], ['hep-th-0302160-1-6-2', 'hep-th-0302160-2-6-2'], ['hep-th-0302160-1-6-3', 'hep-th-0302160-2-6-3'], ['hep-th-0302160-1-6-4', 'hep-th-0302160-2-6-4'], ['hep-th-0302160-1-6-5', 'hep-th-0302160-2-6-5'], ['hep-th-0302160-1-12-0', 'hep-th-0302160-2-12-0'], ['hep-th-0302160-1-12-1', 'hep-th-0302160-2-12-1'], ['hep-th-0302160-1-12-2', 'hep-th-0302160-2-12-2'], ['hep-th-0302160-1-37-0', 'hep-th-0302160-2-37-0'], ['hep-th-0302160-1-37-1', 'hep-th-0302160-2-37-1'], ['hep-th-0302160-1-37-2', 'hep-th-0302160-2-37-2'], ['hep-th-0302160-1-31-0', 'hep-th-0302160-2-31-0'], ['hep-th-0302160-1-31-1', 'hep-th-0302160-2-31-1'], ['hep-th-0302160-1-31-2', 'hep-th-0302160-2-31-2'], ['hep-th-0302160-1-31-3', 'hep-th-0302160-2-31-3'], ['hep-th-0302160-1-19-0', 'hep-th-0302160-2-19-0'], ['hep-th-0302160-1-19-1', 'hep-th-0302160-2-19-1'], ['hep-th-0302160-1-19-2', 'hep-th-0302160-2-19-2'], ['hep-th-0302160-1-19-3', 'hep-th-0302160-2-19-3'], ['hep-th-0302160-1-19-4', 'hep-th-0302160-2-19-4'], ['hep-th-0302160-1-13-0', 'hep-th-0302160-2-13-0'], ['hep-th-0302160-1-13-1', 'hep-th-0302160-2-13-1'], ['hep-th-0302160-1-13-2', 'hep-th-0302160-2-13-2'], ['hep-th-0302160-1-13-4', 'hep-th-0302160-2-13-4'], ['hep-th-0302160-1-9-0', 'hep-th-0302160-2-9-0'], ['hep-th-0302160-1-9-1', 'hep-th-0302160-2-9-1'], ['hep-th-0302160-1-9-2', 'hep-th-0302160-2-9-2'], ['hep-th-0302160-1-9-3', 'hep-th-0302160-2-9-3'], ['hep-th-0302160-1-9-4', 'hep-th-0302160-2-9-4'], ['hep-th-0302160-1-34-2', 'hep-th-0302160-2-34-2']]

[['hep-th-0302160-1-7-0', 'hep-th-0302160-2-7-0'], ['hep-th-0302160-1-4-0', 'hep-th-0302160-2-4-0'], ['hep-th-0302160-1-4-1', 'hep-th-0302160-2-4-1'], ['hep-th-0302160-1-4-2', 'hep-th-0302160-2-4-2'], ['hep-th-0302160-1-4-3', 'hep-th-0302160-2-4-3'], ['hep-th-0302160-1-4-4', 'hep-th-0302160-2-4-4'], ['hep-th-0302160-1-25-0', 'hep-th-0302160-2-25-0'], ['hep-th-0302160-1-25-1', 'hep-th-0302160-2-25-1'], ['hep-th-0302160-1-26-0', 'hep-th-0302160-2-26-0'], ['hep-th-0302160-1-26-1', 'hep-th-0302160-2-26-1'], ['hep-th-0302160-1-26-2', 'hep-th-0302160-2-26-2'], ['hep-th-0302160-1-26-3', 'hep-th-0302160-2-26-3'], ['hep-th-0302160-1-26-4', 'hep-th-0302160-2-26-4'], ['hep-th-0302160-1-3-0', 'hep-th-0302160-2-3-0'], ['hep-th-0302160-1-3-1', 'hep-th-0302160-2-3-1'], ['hep-th-0302160-1-3-2', 'hep-th-0302160-2-3-2'], ['hep-th-0302160-1-3-3', 'hep-th-0302160-2-3-3'], ['hep-th-0302160-1-3-4', 'hep-th-0302160-2-3-4'], ['hep-th-0302160-1-27-0', 'hep-th-0302160-2-27-0'], ['hep-th-0302160-1-27-1', 'hep-th-0302160-2-27-1'], ['hep-th-0302160-1-27-2', 'hep-th-0302160-2-27-2'], ['hep-th-0302160-1-27-3', 'hep-th-0302160-2-27-3'], ['hep-th-0302160-1-27-4', 'hep-th-0302160-2-27-4'], ['hep-th-0302160-1-27-5', 'hep-th-0302160-2-27-5'], ['hep-th-0302160-1-27-6', 'hep-th-0302160-2-27-6'], ['hep-th-0302160-1-27-7', 'hep-th-0302160-2-27-7'], ['hep-th-0302160-1-24-0', 'hep-th-0302160-2-24-0'], ['hep-th-0302160-1-24-1', 'hep-th-0302160-2-24-1'], ['hep-th-0302160-1-24-2', 'hep-th-0302160-2-24-2'], ['hep-th-0302160-1-24-3', 'hep-th-0302160-2-24-3'], ['hep-th-0302160-1-24-4', 'hep-th-0302160-2-24-4'], ['hep-th-0302160-1-24-5', 'hep-th-0302160-2-24-5'], ['hep-th-0302160-1-33-0', 'hep-th-0302160-2-33-0'], ['hep-th-0302160-1-18-0', 'hep-th-0302160-2-18-0'], ['hep-th-0302160-1-18-1', 'hep-th-0302160-2-18-1'], ['hep-th-0302160-1-18-2', 'hep-th-0302160-2-18-2'], ['hep-th-0302160-1-18-3', 'hep-th-0302160-2-18-3'], ['hep-th-0302160-1-18-4', 'hep-th-0302160-2-18-4'], ['hep-th-0302160-1-18-5', 'hep-th-0302160-2-18-5'], ['hep-th-0302160-1-30-0', 'hep-th-0302160-2-30-0'], ['hep-th-0302160-1-30-1', 'hep-th-0302160-2-30-1'], ['hep-th-0302160-1-30-2', 'hep-th-0302160-2-30-2'], ['hep-th-0302160-1-30-3', 'hep-th-0302160-2-30-3'], ['hep-th-0302160-1-30-4', 'hep-th-0302160-2-30-4'], ['hep-th-0302160-1-30-5', 'hep-th-0302160-2-30-5'], ['hep-th-0302160-1-30-6', 'hep-th-0302160-2-30-6'], ['hep-th-0302160-1-30-7', 'hep-th-0302160-2-30-7'], ['hep-th-0302160-1-30-8', 'hep-th-0302160-2-30-8'], ['hep-th-0302160-1-30-9', 'hep-th-0302160-2-30-9'], ['hep-th-0302160-1-38-0', 'hep-th-0302160-2-38-0'], ['hep-th-0302160-1-38-1', 'hep-th-0302160-2-38-1'], ['hep-th-0302160-1-28-0', 'hep-th-0302160-2-28-0'], ['hep-th-0302160-1-28-1', 'hep-th-0302160-2-28-1'], ['hep-th-0302160-1-28-2', 'hep-th-0302160-2-28-2'], ['hep-th-0302160-1-28-3', 'hep-th-0302160-2-28-3'], ['hep-th-0302160-1-28-4', 'hep-th-0302160-2-28-4'], ['hep-th-0302160-1-28-5', 'hep-th-0302160-2-28-5'], ['hep-th-0302160-1-28-6', 'hep-th-0302160-2-28-6'], ['hep-th-0302160-1-28-7', 'hep-th-0302160-2-28-7'], ['hep-th-0302160-1-28-8', 'hep-th-0302160-2-28-8'], ['hep-th-0302160-1-34-0', 'hep-th-0302160-2-34-0'], ['hep-th-0302160-1-34-1', 'hep-th-0302160-2-34-1'], ['hep-th-0302160-1-34-3', 'hep-th-0302160-2-34-3'], ['hep-th-0302160-1-34-4', 'hep-th-0302160-2-34-4'], ['hep-th-0302160-1-34-5', 'hep-th-0302160-2-34-5'], ['hep-th-0302160-1-34-6', 'hep-th-0302160-2-34-6'], ['hep-th-0302160-1-34-7', 'hep-th-0302160-2-34-7'], ['hep-th-0302160-1-34-8', 'hep-th-0302160-2-34-8'], ['hep-th-0302160-1-8-0', 'hep-th-0302160-2-8-0'], ['hep-th-0302160-1-8-1', 'hep-th-0302160-2-8-1'], ['hep-th-0302160-1-8-2', 'hep-th-0302160-2-8-2'], ['hep-th-0302160-1-8-3', 'hep-th-0302160-2-8-3'], ['hep-th-0302160-1-39-0', 'hep-th-0302160-2-39-0'], ['hep-th-0302160-1-39-1', 'hep-th-0302160-2-39-1'], ['hep-th-0302160-1-39-2', 'hep-th-0302160-2-39-2'], ['hep-th-0302160-1-11-0', 'hep-th-0302160-2-11-0'], ['hep-th-0302160-1-11-1', 'hep-th-0302160-2-11-1'], ['hep-th-0302160-1-11-2', 'hep-th-0302160-2-11-2'], ['hep-th-0302160-1-11-3', 'hep-th-0302160-2-11-3'], ['hep-th-0302160-1-43-0', 'hep-th-0302160-2-43-0'], ['hep-th-0302160-1-43-1', 'hep-th-0302160-2-43-1'], ['hep-th-0302160-1-43-2', 'hep-th-0302160-2-43-2'], ['hep-th-0302160-1-43-3', 'hep-th-0302160-2-43-3'], ['hep-th-0302160-1-43-4', 'hep-th-0302160-2-43-4'], ['hep-th-0302160-1-15-0', 'hep-th-0302160-2-15-0'], ['hep-th-0302160-1-15-1', 'hep-th-0302160-2-15-1'], ['hep-th-0302160-1-15-2', 'hep-th-0302160-2-15-2'], ['hep-th-0302160-1-15-3', 'hep-th-0302160-2-15-3'], ['hep-th-0302160-1-15-4', 'hep-th-0302160-2-15-4'], ['hep-th-0302160-1-0-0', 'hep-th-0302160-2-0-0'], ['hep-th-0302160-1-0-1', 'hep-th-0302160-2-0-1'], ['hep-th-0302160-1-0-2', 'hep-th-0302160-2-0-2'], ['hep-th-0302160-1-0-3', 'hep-th-0302160-2-0-3'], ['hep-th-0302160-1-0-4', 'hep-th-0302160-2-0-4'], ['hep-th-0302160-1-22-0', 'hep-th-0302160-2-22-0'], ['hep-th-0302160-1-22-1', 'hep-th-0302160-2-22-1'], ['hep-th-0302160-1-22-2', 'hep-th-0302160-2-22-2'], ['hep-th-0302160-1-22-3', 'hep-th-0302160-2-22-3'], ['hep-th-0302160-1-22-4', 'hep-th-0302160-2-22-4'], ['hep-th-0302160-1-22-5', 'hep-th-0302160-2-22-5'], ['hep-th-0302160-1-16-0', 'hep-th-0302160-2-16-0'], ['hep-th-0302160-1-16-1', 'hep-th-0302160-2-16-1'], ['hep-th-0302160-1-42-0', 'hep-th-0302160-2-42-0'], ['hep-th-0302160-1-10-0', 'hep-th-0302160-2-10-0'], ['hep-th-0302160-1-10-1', 'hep-th-0302160-2-10-1'], ['hep-th-0302160-1-21-0', 'hep-th-0302160-2-21-0'], ['hep-th-0302160-1-21-1', 'hep-th-0302160-2-21-1'], ['hep-th-0302160-1-21-2', 'hep-th-0302160-2-21-2'], ['hep-th-0302160-1-20-0', 'hep-th-0302160-2-20-0'], ['hep-th-0302160-1-20-1', 'hep-th-0302160-2-20-1'], ['hep-th-0302160-1-20-2', 'hep-th-0302160-2-20-2'], ['hep-th-0302160-1-40-0', 'hep-th-0302160-2-40-0'], ['hep-th-0302160-1-40-1', 'hep-th-0302160-2-40-1'], ['hep-th-0302160-1-40-2', 'hep-th-0302160-2-40-2'], ['hep-th-0302160-1-40-3', 'hep-th-0302160-2-40-3'], ['hep-th-0302160-1-40-4', 'hep-th-0302160-2-40-4'], ['hep-th-0302160-1-40-5', 'hep-th-0302160-2-40-5'], ['hep-th-0302160-1-41-0', 'hep-th-0302160-2-41-0'], ['hep-th-0302160-1-41-1', 'hep-th-0302160-2-41-1'], ['hep-th-0302160-1-41-2', 'hep-th-0302160-2-41-2'], ['hep-th-0302160-1-35-0', 'hep-th-0302160-2-35-0'], ['hep-th-0302160-1-35-1', 'hep-th-0302160-2-35-1'], ['hep-th-0302160-1-32-0', 'hep-th-0302160-2-32-0'], ['hep-th-0302160-1-32-1', 'hep-th-0302160-2-32-1'], ['hep-th-0302160-1-32-2', 'hep-th-0302160-2-32-2'], ['hep-th-0302160-1-32-3', 'hep-th-0302160-2-32-3'], ['hep-th-0302160-1-5-0', 'hep-th-0302160-2-5-0'], ['hep-th-0302160-1-5-1', 'hep-th-0302160-2-5-1'], ['hep-th-0302160-1-5-2', 'hep-th-0302160-2-5-2'], ['hep-th-0302160-1-5-3', 'hep-th-0302160-2-5-3'], ['hep-th-0302160-1-5-4', 'hep-th-0302160-2-5-4'], ['hep-th-0302160-1-17-0', 'hep-th-0302160-2-17-0'], ['hep-th-0302160-1-17-1', 'hep-th-0302160-2-17-1'], ['hep-th-0302160-1-23-0', 'hep-th-0302160-2-23-0'], ['hep-th-0302160-1-23-1', 'hep-th-0302160-2-23-1'], ['hep-th-0302160-1-23-2', 'hep-th-0302160-2-23-2'], ['hep-th-0302160-1-23-3', 'hep-th-0302160-2-23-3'], ['hep-th-0302160-1-23-4', 'hep-th-0302160-2-23-4'], ['hep-th-0302160-1-23-5', 'hep-th-0302160-2-23-5'], ['hep-th-0302160-1-6-0', 'hep-th-0302160-2-6-0'], ['hep-th-0302160-1-6-1', 'hep-th-0302160-2-6-1'], ['hep-th-0302160-1-6-2', 'hep-th-0302160-2-6-2'], ['hep-th-0302160-1-6-3', 'hep-th-0302160-2-6-3'], ['hep-th-0302160-1-6-4', 'hep-th-0302160-2-6-4'], ['hep-th-0302160-1-6-5', 'hep-th-0302160-2-6-5'], ['hep-th-0302160-1-12-0', 'hep-th-0302160-2-12-0'], ['hep-th-0302160-1-12-1', 'hep-th-0302160-2-12-1'], ['hep-th-0302160-1-12-2', 'hep-th-0302160-2-12-2'], ['hep-th-0302160-1-37-0', 'hep-th-0302160-2-37-0'], ['hep-th-0302160-1-37-1', 'hep-th-0302160-2-37-1'], ['hep-th-0302160-1-37-2', 'hep-th-0302160-2-37-2'], ['hep-th-0302160-1-31-0', 'hep-th-0302160-2-31-0'], ['hep-th-0302160-1-31-1', 'hep-th-0302160-2-31-1'], ['hep-th-0302160-1-31-2', 'hep-th-0302160-2-31-2'], ['hep-th-0302160-1-31-3', 'hep-th-0302160-2-31-3'], ['hep-th-0302160-1-19-0', 'hep-th-0302160-2-19-0'], ['hep-th-0302160-1-19-1', 'hep-th-0302160-2-19-1'], ['hep-th-0302160-1-19-2', 'hep-th-0302160-2-19-2'], ['hep-th-0302160-1-19-3', 'hep-th-0302160-2-19-3'], ['hep-th-0302160-1-19-4', 'hep-th-0302160-2-19-4'], ['hep-th-0302160-1-13-0', 'hep-th-0302160-2-13-0'], ['hep-th-0302160-1-13-1', 'hep-th-0302160-2-13-1'], ['hep-th-0302160-1-13-2', 'hep-th-0302160-2-13-2'], ['hep-th-0302160-1-13-4', 'hep-th-0302160-2-13-4'], ['hep-th-0302160-1-9-0', 'hep-th-0302160-2-9-0'], ['hep-th-0302160-1-9-1', 'hep-th-0302160-2-9-1'], ['hep-th-0302160-1-9-2', 'hep-th-0302160-2-9-2'], ['hep-th-0302160-1-9-3', 'hep-th-0302160-2-9-3'], ['hep-th-0302160-1-9-4', 'hep-th-0302160-2-9-4']]

[['hep-th-0302160-1-34-2', 'hep-th-0302160-2-34-2']]

[]

[]

[]

['hep-th-0302160-1-1-0', 'hep-th-0302160-1-13-3', 'hep-th-0302160-2-1-0', 'hep-th-0302160-2-13-3']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/0302160

1103.3516

{'1103.3516-1-0-0': 'We present a spectral weight conserving formalism for Fermionic thermal Green\'s functions that are discretized in imaginary time [MATH] and thus periodic in imaginary ("Matsubara") frequency [MATH].', '1103.3516-1-0-1': 'The formalism requires a generalization of the Dyson equation [MATH] and the Baym-Kadanoff-Luttinger-Ward functional for the free energy [MATH].', '1103.3516-1-0-2': "A conformal transformation is used to analytically continue the periodized Matsubara Green's function to the continuous real axis in a way that conserves the discontinuity at [MATH] of the corresponding real-time Green's function.", '1103.3516-1-0-3': "For given discretization the method allows numerical Green's function calculations of very high precision and it appears to give a well controlled convergent approximation as we decrease the discretization interval.", '1103.3516-1-0-4': 'The ideas are tested on Dynamical Mean Field Theory calculations of the paramagnetic Hubbard model.', '1103.3516-1-1-0': "The analytic properties of finite temperature Green's functions is a cornerstone of the theory of quantum many body theory.", '1103.3516-1-1-1': "[CITATION] The Green's functions can be represented either in continuous complex time or by discrete values on an infinite set of Matsubara frequencies.", '1103.3516-1-1-2': "The physical retarded Green's function is given by the analytic continuation of this discrete function of Matsubara frequencies to a continuous function on the real frequency axis.", '1103.3516-1-1-3': 'Although it is extremely elegant the method poses a numerical challenge due to the slow convergence of the Matsubara functions for large frequency.', '1103.3516-1-2-0': "Here we develop a method to work with a Fermion Green's function defined at [MATH] equally spaced points [MATH] in imaginary time.", '1103.3516-1-2-1': 'The discretization results in the "Matsubara frequency" representation becoming [MATH] dimensional and the Green\'s function periodic in this basis.', '1103.3516-1-2-2': 'We are able to construct a scheme that satisfies the following properties.', '1103.3516-1-2-3': "First that (a) the limit [MATH] yields the standard theory of thermal Green's function.", '1103.3516-1-2-4': "In addition, for all values of [MATH] the following three constraints hold: (b) the Green's functions are consistent with a Luttinger-Ward variational principle (c) the free energy is exact for noninteracting particles (d) an analytic continuation to the continuous real axis with the proper discontinuity at [MATH] exists.", '1103.3516-1-3-0': 'Let us consider Fermions described by a Hamiltonian [MATH] with [MATH], and where [MATH] represents Fermion creation operators of a state [MATH] and where [MATH] may be momentum and spin.', '1103.3516-1-3-1': 'The chemical potential [MATH] is absorbed in [MATH].', '1103.3516-1-3-2': 'The interaction is [MATH].', '1103.3516-1-3-3': "We consider the one-particle Green's function [MATH] where [MATH] is the sum over a complete set of states, [MATH] is time ordering, [MATH] is inverse temperature, and [MATH] is the partition function [MATH].", '1103.3516-1-3-4': 'We assume [MATH] is diagonal in [MATH] and there are no anomalous "superconducting" terms.', '1103.3516-1-3-5': 'The function [MATH] is defined on [MATH] and obeys [MATH].', '1103.3516-1-3-6': 'This antiperiodicity condition allows [MATH] to be in written in terms of the Matsubara frequencies [MATH] for integer [MATH].', '1103.3516-1-4-0': "The expansion of [MATH] in powers of [MATH] can be expressed as sums of connected diagrams consisting of the vertex [MATH] at a time [MATH] and the non-interacting Green's function [MATH], where internal times are integrated over as [MATH] and where an [MATH]'th order diagram has a prefactor [MATH] and [MATH] the number of Fermion loops.", '1103.3516-1-4-1': "[CITATION] In this work we discretize [MATH], effectively replacing integrals over continuous time [MATH] with sums over [MATH], by carefully defining a discretized Green's function and self energy.", '1103.3516-1-5-0': "Start by defining [MATH] and [MATH] and discretize the non-interacting Green's function [MATH], with [MATH], to times [MATH] .", '1103.3516-1-5-1': 'The value at [MATH] is defined as the average of the two limits so that [MATH].', '1103.3516-1-5-2': 'We can expand [MATH] and the "Matsubara transform" [MATH] is [EQUATION]', '1103.3516-1-5-3': 'For compactness we will usually drop the index [MATH] and frequency [MATH] when it is clear from the context.', '1103.3516-1-5-4': 'We see that [MATH] is periodic under [MATH] and in the limit [MATH]) the continuum expression [MATH] is appropriately recovered.', '1103.3516-1-5-5': "We also define the additional Green's functions [MATH] corresponding to the [MATH] limit of [MATH].", '1103.3516-1-6-0': "Let us now define the periodized full Green's function [MATH] and the self-energy [MATH] through the two expressions [EQUATION] and [EQUATION]", '1103.3516-1-6-1': 'The object [MATH] is the functional[CITATION] [EQUATION] defined in terms of skeleton diagrams of [MATH] except for the 1st order direct and exchange diagrams where we use [MATH].', '1103.3516-1-6-2': 'The self energy is thus given by the amputated skeleton diagrams such as shown in Figure [REF], and the resulting set of equations given by inserting Eq. [REF].', '1103.3516-1-7-0': 'The expression [REF] is a generalization of the Dyson equation for the standard formalism, [MATH], and reduces to the latter for [MATH].', '1103.3516-1-7-1': 'It also preserves the property that a constant, [MATH] and [MATH] independent, self energy acts a chemical potential.', '1103.3516-1-8-0': 'Defining a "periodized" self energy [MATH] by [MATH] which reduces to [MATH] in the limit [MATH].', '1103.3516-1-8-1': 'Eq. [REF] can also be rewritten in the more suggestive form [EQUATION] which again reduces to the ordinary Dyson equation for [MATH].', '1103.3516-1-9-0': 'Consider now the free energy [MATH].', '1103.3516-1-9-1': 'As shown by Luttinger and Ward [CITATION], for the standard formalism, the free energy can be expressed as [MATH], where [MATH] and [MATH] as in Eq. [REF] (limit [MATH]).', '1103.3516-1-9-2': 'The expression [MATH] also provides a variational formulation where [MATH] corresponds to a stationary point [MATH] that yields the Dyson equation [MATH].', '1103.3516-1-9-3': 'We now describe how to generalize this variational formulation to be consistent with Eq. [REF] and demands (a)-(c) stated in the introduction.', '1103.3516-1-10-0': 'We define the following generalization [EQUATION] where [MATH].', '1103.3516-1-10-1': 'The expression [MATH] reduces to the standard expression in the limit [MATH] and as readily shown using Eq. [REF] the stationarity condition [MATH] gives the generalized Dyson equation Eq. [REF].', '1103.3516-1-10-2': 'In addition, it can be shown that in the noninteracting limit when [MATH] the free energy in Eq. [REF] is exact for all values of [MATH] through the expression [EQUATION]', '1103.3516-1-10-3': 'The functional [MATH] can also be shown to give a value for the total particle number which is consistent with the formalism i.e. [MATH].', '1103.3516-1-10-4': 'Baym and Kadanoff[CITATION] noted that conserving approximations can be found by including only a finite number of diagrams in the Luttinger-Ward function.', '1103.3516-1-10-5': "This construction can now be used for the discretized Green's functions to include only a subset of diagrams in [MATH].", '1103.3516-1-11-0': "Let us now explore the analytic structure of the periodized Green's function.", '1103.3516-1-11-1': 'Starting with the spectral representation in terms of a complete set of eigenstates [MATH], for [MATH], [MATH].', '1103.3516-1-11-2': 'Using Eq. [REF] we can compute [MATH] as [EQUATION] where the spectral function is given by the conventional expression [MATH].', '1103.3516-1-11-3': 'Eq. [REF] gives the analytic continuation to [MATH] by letting [MATH] and shows that [MATH] is analytic except where [MATH] is an integer multiple of [MATH].', '1103.3516-1-11-4': "Using [MATH] and defining [MATH] and [MATH], in analogy with the standard retarded and advanced Green's functions, we find [EQUATION] where [MATH].", '1103.3516-1-11-5': 'It follows that [MATH] has a branch cut on the real axis, and repeated with period [MATH], where [MATH] changes sign.', '1103.3516-1-11-6': 'In addition the imaginary and real part are related by a generalized Kramers-Kronig relation [EQUATION]', '1103.3516-1-11-7': "We will now show how to use a conformal transformation to make an analytic continuation of the periodized Green's function to a rational function with simple poles.", '1103.3516-1-11-8': 'Consider a spectral function of the form [EQUATION] which reduces to the Lorentzian [MATH] in the limit [MATH].', '1103.3516-1-11-9': 'Because of the antiperiodicity and to ensure positive spectral weight we can assume [MATH].', '1103.3516-1-11-10': 'We can evaluate the integral in Eq. [REF] by a closed contour containing three poles, as shown in Figure [REF].', '1103.3516-1-11-11': 'The result, for [MATH], is [EQUATION] and using the same countour integral it can also shown be that [MATH] is properly normalized: [MATH].', '1103.3516-1-11-12': 'The function [MATH] is thus completely free from singularities in the strip [MATH] and has poles outside the strip at [MATH] and [MATH] and obeys [MATH] as shown in Figure [REF]a.', '1103.3516-1-11-13': "In the limit [MATH] the function reduces to [MATH] which is the usual retarded Green's function from a Lorentzian spectral weight.", '1103.3516-1-11-14': "Evaluating [MATH] in the strip [MATH] gives an expression which is analytic in that strip and which analogously corresponds to the advanced Green's function.", '1103.3516-1-11-15': "The full analytic Green's function with periodically repeated branch cuts corresponds to gluing together the two branches as indicated in Figure [REF]b.", '1103.3516-1-11-16': "In the limit [MATH], [MATH] thus reproducing the standard structure of the analytic Green's function with a branch cut on the real axis.", '1103.3516-1-12-0': 'Let us now consider a spectral function defined by a set of such "periodized Lorentzians" [MATH].', '1103.3516-1-12-1': 'For [MATH] we find [EQUATION] where we have allowed for a complex prefactor [MATH] that must satisfy [MATH] to conserve total spectral weight.', '1103.3516-1-12-2': 'Given a set of values of [MATH] we would like to extract the values of [MATH] and [MATH] that solves Eq. [REF].', '1103.3516-1-12-3': 'The latter can be cast into the form of a rational function containing only simple poles (as opposed to periodically repeated) by means of the conformal mapping [EQUATION] which gives [EQUATION] with [MATH].', '1103.3516-1-12-4': 'The transformation maps the strip [MATH] to the entire complex plane.', '1103.3516-1-12-5': 'The strip [MATH] that contains the singularities is mapped to the interior of the unit circle and the strip [MATH] that is free from singularities is outside, as exemplified in Figure [REF]c and d.', '1103.3516-1-12-6': 'The points [MATH] on the real axis map to [MATH] and the point [MATH] maps to [MATH].', '1103.3516-1-12-7': 'The location and residues of poles of [MATH] can be identified with [MATH] and [MATH].', '1103.3516-1-13-0': 'The function [MATH] should obey a number of properties.', '1103.3516-1-13-1': 'Since Eq. [REF] can be written in Pade form as [MATH] where [MATH] is an [MATH] degree and [MATH] is an [MATH] degree polynomial of [MATH] and [MATH], we can identify [MATH] with the roots of [MATH] and [MATH] as the residues of [MATH].', '1103.3516-1-13-2': 'It is important to pick [MATH] to be odd, since this gives a precise boundary condition [MATH].', '1103.3516-1-13-3': 'We also require [MATH] to obtain the proper normalization.', '1103.3516-1-13-4': 'A crucial observation is that Eq. [REF] preserves this condition independent of [MATH] and demonstrates that total spectral weight is preserved by the periodized Dyson equation.', '1103.3516-1-13-5': 'We thus find [MATH] and [MATH] by fitting [MATH] to [MATH] Pade form, yielding [MATH].', '1103.3516-1-13-6': 'In addition we have the symmetry [MATH] resulting in [MATH] independent poles.', '1103.3516-1-13-7': 'Thus [MATH] in Eq. [REF].', '1103.3516-1-13-8': 'Having identified the parameters [MATH] and [MATH] in the fit, the spectral function can be evaluated through [MATH] using Eq [REF].', '1103.3516-1-13-9': "For every value of [MATH] we can also obtain a properly normalized spectral weight using [MATH] resulting in Green's function with the proper discontinuity [MATH].", '1103.3516-1-14-0': 'We tested our method on the Dynamical mean field theory (DMFT) method using the (IPT) approximation with the half filled paramagnetic Hubbard model and used [MATH].', '1103.3516-1-14-1': 'The IPT method has been described elsewhere.', '1103.3516-1-14-2': '[CITATION] A self-consistent momentum independent self energy is found for an effective quantum impurity model in second order perturbation theory with an onsite interaction [MATH] (units of half-bandwidth [MATH]).', '1103.3516-1-14-3': 'We found that the convergence was significantly enhanced by evaluating the dynamic part of the self energy as [MATH].', '1103.3516-1-14-4': 'where Eq. [REF] has been used.', '1103.3516-1-14-5': '(The constant [MATH] contribution is absorbed in the chemical potential.)', '1103.3516-1-14-6': 'This expression ensures that [MATH] lies within the appropriate bounds and by expanding [MATH] in powers of [MATH] it is equivalent to the standard expression to order [MATH].', '1103.3516-1-14-7': 'The detailed implications of this procedure should be explored further.', '1103.3516-1-15-0': 'Figure [REF] shows the spectral function for varying bare energy [MATH] at [MATH]), in the metallic, [MATH], and insulating, [MATH] phases using the standard semicircular bare density of states.', '1103.3516-1-15-1': 'We performed the calculation to 40 significant digits which allows for a very accurate Pade fit to all [MATH] Matsubara values [MATH] as well as the two spectral weight constraints at [MATH].', '1103.3516-1-15-2': 'The location of poles are indicated in the figure for [MATH].', '1103.3516-1-15-3': 'Since [MATH] is [MATH]-independent we can extract it with a single Pade fit and use this to generate the spectral function for any [MATH].', '1103.3516-1-15-4': 'An alternative use of the discretization scheme is detailed elsewhere.', '1103.3516-1-15-5': '[CITATION]', '1103.3516-1-16-0': "We have presented a method to work with discrete-time Matsubara Green's functions.", '1103.3516-1-16-1': 'In spite of working in a finite dimensional space the method yields an analytic continuation that obeys the boundary condition at [MATH], a problem that has plagued other previous discretization schemes.', '1103.3516-1-16-2': 'Within this finite dimensional space, numerical calculations can be done to the extremely high accuracy that is necessary to numerically obtain a meaningful analytic continuations from imaginary to the real time axis.', '1103.3516-1-16-3': "The method should have wide applicability to all problems requiring numerical evaluation of Matsubara Green's function in condensed matter physics and quantum field theory."}

{'1103.3516-2-0-0': 'We present a spectral weight conserving formalism for Fermionic thermal Green\'s functions that are discretized in imaginary time [MATH] and thus periodic in imaginary ("Matsubara") frequency [MATH].', '1103.3516-2-0-1': 'The formalism requires a generalization of the Dyson equation [MATH] and the Baym-Kadanoff-Luttinger-Ward functional for the free energy [MATH].', '1103.3516-2-0-2': "A conformal transformation is used to analytically continue the periodized Matsubara Green's function to real frequencies in a way that conserves the discontinuity at [MATH] of the corresponding real-time Green's function.", '1103.3516-2-0-3': "This allows numerical Green's function calculations of very high precision and it appears to give a well controlled convergent approximation in the [MATH] discretization.", '1103.3516-2-0-4': 'The formalism is tested on dynamical mean field theory calculations of the paramagnetic Hubbard model.', '1103.3516-2-1-0': "The analytic properties of finite temperature Green's functions is a cornerstone of the theory of quantum many body theory.", '1103.3516-2-1-1': "[CITATION] The Green's functions can be represented either in continuous complex time or by discrete values on an infinite set of Matsubara frequencies.", '1103.3516-2-1-2': "The physical retarded Green's function and the corresponding spectral function is given by the analytic continuation of this discrete function of Matsubara frequencies to a continuous function on the real frequency axis.", '1103.3516-2-1-3': 'Although it is extremely elegant the method poses a numerical challenge.', '1103.3516-2-1-4': 'It was early recognized that the Pade series could be used fit data from a finite number of Matsubara frequencies.', '1103.3516-2-1-5': '[CITATION] This scheme is numerically quite ill conditioned and much effort has been devoted to resolving this difficulty.', '1103.3516-2-1-6': 'For quantum Monte Carlo methods the Maximum Entropy method [CITATION] that directly computes the spectral function as a probabilility distribution is instead widely used.', '1103.3516-2-2-0': "We develop a method to work with a Fermion Green's function defined exactly only at [MATH] equally spaced points [MATH] in imaginary time.", '1103.3516-2-2-1': 'In this space of dimension [MATH] a discrete solution is computed numerically exactly.', '1103.3516-2-2-2': 'A conformal transformation is used to construct an exact analytic continuation using a rational function.', '1103.3516-2-2-3': 'Since [MATH] does not have to be large to yield stable results, calculations can be done using very high precision.', '1103.3516-2-2-4': 'This resolves many of the difficulties others have had with an ill conditioned Pade series.', '1103.3516-2-2-5': '[CITATION]', '1103.3516-2-3-0': 'Our approach assures that several basic conditions are satisfied.', '1103.3516-2-3-1': 'In addition to the obvious demand that the limit [MATH] yields a proper continuum limit, we obtain three additional properties valid for all [MATH] that are not present in previous approaches and which give a well controlled and rapidly convergent result as [MATH] is increased.', '1103.3516-2-3-2': "For all values of [MATH] we find that (a) the Green's function obeys exactly a Luttinger-Ward variational principle (b) the free energy is exact for noninteracting particles (c) a numerically exact analytic continuation exists that reproduces the data on the imaginary frequency axis and has the proper discontinuity and analytic structure at [MATH].", '1103.3516-2-4-0': 'Let us consider Fermions described by a Hamiltonian [MATH] with [MATH], and where [MATH] represents Fermion creation operators of a state [MATH] and where [MATH] may be momentum and spin.', '1103.3516-2-4-1': 'The chemical potential [MATH] is absorbed in [MATH].', '1103.3516-2-4-2': 'The interaction is [MATH].', '1103.3516-2-4-3': "We consider the one-particle Green's function [MATH] where [MATH] is the sum over a complete set of states, [MATH] is time ordering, [MATH] is inverse temperature, and [MATH] is the partition function [MATH].", '1103.3516-2-4-4': 'We assume [MATH] is diagonal in [MATH] and there are no anomalous "superconducting" terms.', '1103.3516-2-4-5': 'The function [MATH] is defined on [MATH] and obeys [MATH].', '1103.3516-2-4-6': 'This antiperiodicity condition allows [MATH] to be transformed to Matsubara frequencies [MATH] for integer [MATH].', '1103.3516-2-5-0': "The expansion of [MATH] in powers of [MATH] can be expressed as sums of connected diagrams consisting of the vertex [MATH] at a time [MATH] and the non-interacting Green's function [MATH], where internal times are integrated over as [MATH] and an [MATH]'th order diagram has a prefactor [MATH] with [MATH] the number of Fermion loops.", '1103.3516-2-5-1': "[CITATION] In this work we discretize [MATH] and replace integrals over continuous time [MATH] with sums over [MATH], by carefully defining a discretized Green's function and self energy.", '1103.3516-2-6-0': "We start by defining [MATH] and [MATH] and discretize the non-interacting Green's function [MATH], with [MATH], to times [MATH] .", '1103.3516-2-6-1': 'The value at [MATH] is defined as the average of the two limits so that [MATH].', '1103.3516-2-6-2': 'We can expand [MATH] and the "Matsubara transform" [MATH] is [EQUATION]', '1103.3516-2-6-3': 'For compactness we will usually drop the index [MATH] and frequency [MATH] when it is clear from the context.', '1103.3516-2-6-4': 'We see that [MATH] is periodic[CITATION] under [MATH] and in the limit [MATH]) the continuum expression [MATH] is appropriately recovered.', '1103.3516-2-6-5': "We also define the additional Green's functions [MATH] in analogy to the [MATH] limit of [MATH].", '1103.3516-2-7-0': "Let us now define the periodized full Green's function [MATH] and the self-energy [MATH] through the two expressions [EQUATION] and [MATH].", '1103.3516-2-7-1': 'The object [MATH] is the functional[CITATION] defined as the sum of linked closed skeleton diagrams of [MATH], except for the 1st order diagrams where we use [MATH].', '1103.3516-2-7-2': 'The self energy is thus given by the amputated skeleton diagrams and the resulting set of equations using Eq. [REF].', '1103.3516-2-8-0': 'Equation [REF] is a generalization of the standard Dyson equation [MATH], and reduces to the latter as [MATH].', '1103.3516-2-8-1': 'It also preserves the property that a constant, [MATH] and [MATH] independent, self energy acts a chemical potential.', '1103.3516-2-9-0': 'Defining [MATH] which reduces to [MATH] in the limit [MATH].', '1103.3516-2-9-1': 'Eq. [REF] can also be rewritten in the more suggestive form [EQUATION] which again reduces to the ordinary Dyson equation for [MATH].', '1103.3516-2-10-0': 'Consider now the free energy [MATH].', '1103.3516-2-10-1': 'As shown by Luttinger and Ward [CITATION], for the standard formalism, the free energy can be expressed as [MATH], where [MATH].', '1103.3516-2-10-2': 'The expression [MATH] also provides a variational formulation where [MATH] corresponds to a stationary point [MATH] that yields the Dyson equation [MATH].', '1103.3516-2-10-3': 'We now describe how to generalize this variational formulation to be consistent with Eq. [REF] and demands (a)-(b) stated in the introduction.', '1103.3516-2-11-0': 'We define the following generalization [EQUATION] where [MATH].', '1103.3516-2-11-1': 'The expression [MATH] reduces to the standard expression in the limit [MATH] and the stationarity condition [MATH] gives the generalized Dyson equation Eq. [REF].', '1103.3516-2-11-2': 'In addition, it can be shown that in the noninteracting limit when [MATH] the free energy in Eq. [REF] is exact for all values of [MATH] through the expression [MATH].', '1103.3516-2-11-3': 'The functional [MATH] can also be shown to give a value for the total particle number which is consistent with the formalism i.e. [MATH].', '1103.3516-2-11-4': 'Baym and Kadanoff [CITATION] noted that conserving approximations can be found by including only a finite number of diagrams in the Luttinger-Ward function.', '1103.3516-2-11-5': "This construction can now be used for the discretized Green's functions to include only a subset of diagrams in [MATH].", '1103.3516-2-12-0': "Let us now explore the analytic structure of the periodized Green's function.", '1103.3516-2-12-1': 'Starting with the spectral representation in terms of a complete set of eigenstates [MATH], for [MATH], [MATH] and using Eq. [REF], we can compute [MATH] as [EQUATION] where the spectral function is given by the conventional expression [MATH].', '1103.3516-2-12-2': 'Eq. [REF] gives the analytic continuation to [MATH] by letting [MATH] and shows that [MATH] is analytic except where [MATH] is an integer multiple of [MATH].', '1103.3516-2-12-3': 'Using [MATH] and defining [MATH] and [MATH], we formally recover the standard expression [MATH].', '1103.3516-2-12-4': 'Inserting into Eq. [REF] gives the generalized Kramers-Kronig relation [EQUATION]', '1103.3516-2-12-5': "We will now show how to use a conformal transformation to make an analytic continuation of the periodized Green's function to a rational function with simple poles.", '1103.3516-2-12-6': 'Consider a spectral function of the form [EQUATION] which reduces to the Lorentzian [MATH] in the limit [MATH].', '1103.3516-2-12-7': 'Because of the antiperiodicity and to ensure positive spectral weight we can assume [MATH].', '1103.3516-2-12-8': 'We can evaluate the integral in Eq. [REF] by a closed contour containing three poles, as shown in Figure [REF].', '1103.3516-2-12-9': 'The result, for [MATH], is [EQUATION] and using the same countour integral it can also shown be that [MATH] is properly normalized: [MATH].', '1103.3516-2-12-10': 'The function [MATH] is thus completely free from singularities in the strip [MATH] and has poles outside the strip at [MATH] and [MATH] and obeys [MATH] as shown in Figure [REF]a.', '1103.3516-2-12-11': "In the limit [MATH] the function reduces to [MATH] which is the usual retarded Green's function from a Lorentzian spectral weight.", '1103.3516-2-12-12': "Evaluating [MATH] in the strip [MATH] gives an expression which is analytic in that strip and which analogously corresponds to the advanced Green's function.", '1103.3516-2-12-13': "The full analytic Green's function with periodically repeated branch cuts corresponds to gluing together the two branches as indicated in Figure [REF]b.", '1103.3516-2-12-14': "In the limit [MATH], [MATH] thus reproducing the standard structure of the analytic Green's function with a branch cut on the real axis.", '1103.3516-2-13-0': 'Let us now consider a spectral function defined by a set of such "periodized Lorentzians" [MATH].', '1103.3516-2-13-1': 'For [MATH] we find [EQUATION] where we have allowed for a complex prefactor [MATH] that must satisfy [MATH] to conserve total spectral weight.', '1103.3516-2-13-2': 'Given a set of values of [MATH] we would like to extract the values of [MATH], [MATH] and [MATH] that solves Eq. [REF].', '1103.3516-2-13-3': 'The latter can be cast into the form of a rational function containing only simple poles (as opposed to periodically repeated) by means of the conformal mapping [EQUATION] which gives [EQUATION] with [MATH].', '1103.3516-2-13-4': 'The transformation maps the strip [MATH] to the entire complex plane.', '1103.3516-2-13-5': 'The strip [MATH] that contains the singularities is mapped to the interior of the unit circle and the strip [MATH] that is free from singularities is outside, as exemplified in Figure [REF]c and d.', '1103.3516-2-13-6': 'The points [MATH] on the real axis map to [MATH] and the point [MATH] maps to [MATH].', '1103.3516-2-14-0': 'The function [MATH] should obey a number of properties.', '1103.3516-2-14-1': 'Since Eq. [REF] can be written as a rational function [MATH] where [MATH] is an [MATH] degree and [MATH] is an [MATH] degree polynomial of [MATH] and [MATH], we can identify [MATH] with the roots of [MATH] and [MATH] as the residues of [MATH].', '1103.3516-2-14-2': 'Taking [MATH] odd gives a precise boundary condition [MATH].', '1103.3516-2-14-3': 'We also require [MATH] to obtain the proper normalization.', '1103.3516-2-14-4': 'A crucial observation is that Eq. [REF] preserves this condition independent of [MATH] and demonstrates that total spectral weight is preserved by the periodized Dyson equation.', '1103.3516-2-14-5': 'By straightforward calculation from Eq. [REF] we also conclude that [MATH] which further constrains the analytic continuation.', '1103.3516-2-14-6': 'We thus find [MATH] and [MATH] by fitting [MATH] to [MATH] Pade form, yielding [MATH].', '1103.3516-2-14-7': 'In addition we have the symmetry [MATH] resulting in [MATH] independent poles.', '1103.3516-2-14-8': 'Thus [MATH] in Eq. [REF].', '1103.3516-2-14-9': 'Having identified the parameters [MATH] and [MATH] in the fit, the spectral function can be evaluated through [MATH] using Eq [REF].', '1103.3516-2-14-10': "For every value of [MATH] we can also obtain a properly normalized spectral weight using [MATH] resulting in Green's function with the proper discontinuity [MATH].", '1103.3516-2-15-0': 'We tested our method on the Dynamical mean field theory (DMFT) method using the IPT approximation with the half filled paramagnetic Hubbard model.', '1103.3516-2-15-1': '[CITATION] We found that the convergence was significantly enhanced by evaluating the dynamic part of the self energy as [MATH] where Eq. [REF] has been used.', '1103.3516-2-15-2': 'This expression ensures that [MATH] lies within the appropriate bounds and by expanding [MATH] in powers of [MATH] it is equivalent to the standard expression to order [MATH].', '1103.3516-2-15-3': 'The detailed implications of this procedure should be explored further.', '1103.3516-2-16-0': 'Figure [REF] shows the spectral function for varying bare energy [MATH] at [MATH]), in the metallic, [MATH] (with [MATH]), and insulating, [MATH]) phases using the standard semicircular bare density of states[CITATION] and 40 significant digits to compute the rational function.', '1103.3516-2-16-1': 'This gives 7 and 12 independent complex poles respectively.', '1103.3516-2-16-2': 'The location of poles are indicated in Fig. [REF] for [MATH].', '1103.3516-2-16-3': 'Since [MATH] is [MATH]-independent we can extract it with a single Pade fit and use this to generate the spectral function for any [MATH].', '1103.3516-2-16-4': 'We have found that the number of frequencies needed is roughly [MATH] for convergence of the IPT recursion algorithm and as further test we have used the method at low temperature [MATH] and a number of significant digits roughly also equal to [MATH].', '1103.3516-2-17-0': 'As a second test of the analytic continuation method, we considered a noninteracting single impurity Anderson model whose spectral function consists of a sharp resonance as well as a continuum.', '1103.3516-2-17-1': 'These features have been found to be difficult to reproduce in detail using ordinary Pade methods.', '1103.3516-2-17-2': '[CITATION] We used Eq. [REF] to compute [MATH] from the exact [MATH] using [MATH], [MATH], and 60 significant figures.', '1103.3516-2-17-3': 'Using this we find 16 independent complex poles that to the eye reproduces the exact spectral function.', '1103.3516-2-17-4': 'It has an integrated rms deviation over total weight (excluding the resonance) of [MATH].', '1103.3516-2-17-5': 'The resonance has a width of [MATH] and a normalization which is within [MATH] of the exact value.', '1103.3516-2-17-6': 'A major reason for our success is the high precision for the Pade fit rather than a large number of poles.', '1103.3516-2-17-7': 'Fitting a greater number of less accurate data points does not yield comparable accuracy.', '1103.3516-2-17-8': "This motivates using our method to compute a Green's function to extremely high precision in a relatively small dimensional space and to use the combination of conformal transformation and Pade fit to infer the analytic continuation to the real axis.", '1103.3516-2-18-0': "We have presented a method to work with discrete-time Matsubara Green's functions.", '1103.3516-2-18-1': 'In spite of working in a finite dimensional space the method yields an analytic continuation that obeys the boundary condition at [MATH], a problem that has plagued other previous discretization schemes.', '1103.3516-2-18-2': 'Within this small space, numerical calculations can be done to the extremely high accuracy that is necessary to numerically obtain a meaningful analytic continuations from imaginary to the real time axis.', '1103.3516-2-18-3': "The method should have wide applicability to all problems requiring numerical evaluation of Matsubara Green's function in condensed matter physics nuclear physics and quantum field theory."}

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[]

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[]

['1103.3516-1-13-7', '1103.3516-1-15-5', '1103.3516-2-2-5', '1103.3516-2-14-8']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1103.3516

1604.04698

{'1604.04698-1-0-0': 'A sudden quantum quench of a Bloch band from one topological phase toward another has been shown to exhibit an intimate connection with the notion of a dynamical quantum phase transition (DQPT), where the returning probability of the quenched state to the initial state-i.e. the Loschmidt echo-vanishes at critical times [MATH].', '1604.04698-1-0-1': 'Analytical results so far are limited to two-band models, leaving the exact relation between topology and DQPT unclear.', '1604.04698-1-0-2': 'In this work, we show that for a general multi-band system, a robust DQPT relies on the existence of nodes (i.e. zeros) in the wavefunction overlap between the initial band and the post-quench energy eigenstates.', '1604.04698-1-0-3': 'These nodes are topologically protected if the two participating wavefunctions have distinctive topological indices.', '1604.04698-1-0-4': 'We demonstrate these ideas in detail for both one and two spatial dimensions using a three-band generalized Hofstadter model.', '1604.04698-1-0-5': 'We also discuss possible experimental observations.', '1604.04698-1-1-0': 'Introduction Advances in experimental techniques, in particular in cold atom systems [CITATION], have reinvigorated recent interest in quantum dynamics [CITATION].', '1604.04698-1-1-1': 'A paradigmatic setup in this context is a quantum quench [CITATION], wherein a system is prepared as an eigenstate [MATH] of an initial Hamiltonian [MATH], but evolved under a different Hamiltonian [MATH].', '1604.04698-1-1-2': 'In a slow ramp [CITATION], one has in addition the control over how fast the switching between [MATH] and [MATH] can be, as well as what path to take in the space of Hamiltonians.', '1604.04698-1-1-3': 'Since [MATH] typically consists of many excited states of [MATH] with a non-thermal distribution, its time evolution provides a unique venue for investigating issues in nonequilibrium quantum statistical mechanics such as thermalization, equilibration, or the lack thereof [CITATION].', '1604.04698-1-1-4': 'A particularly fruitful approach to understanding dynamics after a quantum quench is by exploiting the formal similarity between the time evolution operator [MATH], and the thermal density operator [MATH].', '1604.04698-1-1-5': 'This enables one to leverage and extend notions in equilibrium statistical mechanics to the realm of quantum dynamics.', '1604.04698-1-1-6': 'In this spirit, the return amplitude', '1604.04698-1-2-0': 'G(t) = e^-iH_F t = _n ^(n) ^2 e^-iE_n t', '1604.04698-1-3-0': 'can be thought of as a partition function along imaginary temperature [MATH], with the prepared state [MATH] as a fixed boundary [CITATION].', '1604.04698-1-3-1': 'Here [MATH] and [MATH] are eigenstates and eigenvalues of the post-quench [MATH], respectively.', '1604.04698-1-3-2': 'Heyl et al showed [CITATION] that analogous to the thermal free energy, a dynamical free energy density [CITATION] can be defined, [MATH], where [MATH] is system size.', '1604.04698-1-3-3': 'Singularities in [MATH] then signifies the onset of what was proposed as a dynamical quantum phase transition (DQPT).', '1604.04698-1-3-4': 'In statistical mechanics, phase transitions are closely related to the zeros of the partition function-known as Fisher zeros-in the complex temperature plane [CITATION].', '1604.04698-1-3-5': 'Historically, Yang and Lee were the first to connect phase transitions with zeros of the partition function in complexified parameter space [CITATION].', '1604.04698-1-3-6': 'While Fisher zeros are always complex for finite systems, they may coalesce into a continuum (line in one parameter dimension, area in two parameter dimensions, etc) that cuts through the real temperature axis in the thermodynamic limit, giving rise to an equilibrium phase transition.', '1604.04698-1-3-7': 'Investigations on DQPT have followed a similar route by first solving the Fisher zeros in the complex temperature plane, and then identifying conditions for them to cross the axis of imaginary temperature (real time).', '1604.04698-1-3-8': 'DQPT is thus mathematically identified as [MATH] at critical time(s) [MATH].', '1604.04698-1-3-9': 'DQPTs occur in both integrable [CITATION] and non-integrable [CITATION] spin systems for quenches across quantum critical points.', '1604.04698-1-3-10': 'They can further be classified by discontinuities in different orders of time derivatives of [MATH] [CITATION] vis-a-vis their thermal counterparts.', '1604.04698-1-3-11': 'Very recently DQPTs have also been shown to constitute unstable fixed points in the renormalization group flow, and are therefore subject to the notion of universality class and scaling [CITATION].', '1604.04698-1-4-0': 'Physically, the return amplitude [MATH] is related to the power spectrum of work performed during a quench, [MATH], which is the Fourier transform of [MATH], and [MATH] is the energy of the initial state [CITATION].', '1604.04698-1-4-1': 'This in principle makes [MATH], and hence DQPT, a measurable phenomenon.', '1604.04698-1-4-2': 'A pratically more viable route to experimental verification is through measuring time evolution of thermodynamic quantities, which may exhibit post-quench oscillations at a time scale commensurate with the DQPT critical time [MATH], and universal scaling near [MATH] [CITATION].', '1604.04698-1-4-3': 'In band systems, as we will show, they may also be identified by a complete depletion at [MATH] of sublattice or spin-polarized particle density at certain crystal momenta, see Eq. [REF].', '1604.04698-1-5-0': 'Parallel to the development of DQPT as the dynamical analogue of equilibrium phase transitions is the investigation on its relation with topology [CITATION].', '1604.04698-1-5-1': 'This issue arises naturally because in the transverse field Ising model, in which DQPT was first discovered, the quantum critical point can be mapped to a topological phase transition at which the quantized Berry phase of the fermionized Hamiltonian jumps between [MATH] and [MATH].', '1604.04698-1-5-2': 'DQPT in this two-band fermion model was attributed to the occurrence of "population inversion" [CITATION] where it becomes equally probable to find the initial state in either of the two post-quench bands, a consequence of the Berry phase jump [CITATION].', '1604.04698-1-5-3': 'The same analysis has been extended to various two-band models in one- and two-spatial dimensions (1D/2D) [CITATION], where definitive connection was found between DQPT and quench across topological transitions, although some complications exist .', '1604.04698-1-5-4': 'DQPTs have also been demonstrated to occur for quenches within the same topological phase [CITATION], although from the point of view of topological protection, these are not robust as they require fine-tuning of the Hamiltonians.', '1604.04698-1-6-0': 'The purpose of this work is to develop a general theory beyond two band models to clarify the relation between robust DQPT and topology.', '1604.04698-1-6-1': 'We will show that a robust DQPT-one which is insensitive to the details of the pre- and post-quench Hamiltonians other than the phases to which they belong-relies on the existence of zeros (or nodes) in the wavefunction overlap between the initial band and all eigenstates of the post-quench Hamiltonian.', '1604.04698-1-6-2': 'These nodes are topologically protected if the two participating wavefunctions have distinctive topological indices: for example, the Chern number difference [MATH] provides a lower bound to the number of [MATH]-space nodes in the overlap [MATH], see Theorem [REF].', '1604.04698-1-6-3': 'These considerations lead to the notion of topological and symmetry-protected DQPTs which we will demonstrate in detail using a [MATH]-band generalized Hofstadter model.', '1604.04698-1-7-0': 'Amplitude and phase conditions of DQPT The DQPT condition [MATH] can be interpreted geometrically as the complex numbers [MATH] forming a closed polygon in the complex plane at [MATH], see Fig. [REF].', '1604.04698-1-7-1': 'The time-independent content of this observation is that the amplitudes [MATH] satisfy a generalized triangle inequality, [MATH].', '1604.04698-1-7-2': 'Invoking [MATH], one has the amplitude condition,', '1604.04698-1-8-0': 'z_n = ^(n) ^2 !', '1604.04698-1-8-1': '12 n .', '1604.04698-1-9-0': 'For [MATH] that satisfy Eq. [REF], solutions to [MATH] exist and form a subspace [MATH] on the [MATH]-torus,', '1604.04698-1-10-0': 'M_z_nT^N: e^-i_n _n=1^N z_ne^-i_n = 0 .', '1604.04698-1-11-0': 'To set off DQPT, the dynamical phases must be able to evolve into [MATH].', '1604.04698-1-11-1': 'This constitutes the phase condition,', '1604.04698-1-12-0': 't^* : e^-iE_n t^* M_z_n .', '1604.04698-1-13-0': 'DQPT requires both conditions to hold simultaneously.', '1604.04698-1-14-0': 'Phase ergodicity in few-level systems At first glance, the phase condition may seem to be the more stringent one.', '1604.04698-1-14-1': 'After a quench across a quantum phase transition, a many-body initial state [MATH] typically has overlap with an extensive amount of eigenstates of the post-quench Hamiltonian [MATH] and therefore the amplitudes [MATH] are generically exponentially small in system size, rendering Eq. [REF] satisfied in general.', '1604.04698-1-14-2': 'Existence of DQPT then relies entirely on the phase condition.', '1604.04698-1-14-3': 'Integrable systems, however, point to the possibility that the amplitude and phase conditions may be intricately related and traded for one another.', '1604.04698-1-14-4': 'Such systems can effectively be broken down into few-level subsystems labeled by quantum numbers [MATH], say [MATH] levels [MATH] for [MATH] in the [MATH] sector.', '1604.04698-1-14-5': 'Correspondingly [MATH].', '1604.04698-1-14-6': "For the transverse field Ising model, Kitaev's honeycomb model [CITATION], and band insulator models, [MATH] is the Bloch momentum.", '1604.04698-1-14-7': 'It is known that as long as the [MATH] gaps, [MATH], are not rationally related, the dynamical phases [MATH] are ergodic on the [MATH]-torus up to an overall phase [CITATION], and will therefore evolve into its subspace [MATH] (Eq. [REF]).', '1604.04698-1-14-8': 'Phase ergodicity thus guarantees the phase condition Eq. [REF], and DQPT in each [MATH] sector depends entirely on the amplitude condition.', '1604.04698-1-15-0': 'Robust DQPT protected by nodes in wavefunction overlap Hereafter, we focus on quenches in multi-band Bloch systems with [MATH] bands.', '1604.04698-1-15-1': 'For simplicity we use a single filled band [MATH] as the pre-quench state.', '1604.04698-1-15-2': 'Generalization to multiple filled bands is straightforward.', '1604.04698-1-15-3': 'The post-quench return amplitude is [MATH],', '1604.04698-1-16-0': 'G(k, t) = _n=1^N_B ^(n)(k) (k)^2 e^-i_n(k) t ,', '1604.04698-1-17-0': 'where [MATH] and [MATH] are respectively the post-quench energy eigenstates and eigenvalues.', '1604.04698-1-17-1': 'Assume phase ergodicity holds at all [MATH] points-this is a very relaxed requirement provided there is no degeneracy at any [MATH] point.', '1604.04698-1-17-2': 'Then DQPT amounts to the existence of at least one [MATH] at which Eq. [REF] is satisfied, namely', '1604.04698-1-18-0': 'k Brillouin Zone: ^(n)(k) (k)^2 12 n .', '1604.04698-1-19-0': 'We now discuss how Eq. [REF] and hence DQPT can arise from nodes in wavefunction overlaps.', '1604.04698-1-19-1': 'Note that this is not the only way to get DQPT.', '1604.04698-1-19-2': 'Its virtue lies in its robustness against perturbations to the Hamiltonians.', '1604.04698-1-19-3': 'In Supplemental Materials (SM), we show examples where DQPTs with no overlap node can be easily avoided simply by Hamiltonian parameter tuning without crossing a phase boundary.', '1604.04698-1-19-4': 'The overlap nodes are, on the other hand, typically topologically protected, a point we will return to later.', '1604.04698-1-19-5': 'Now consider the following quench.', '1604.04698-1-19-6': 'Let [MATH] label "sublattices", which in general may also include other degrees of freedom, e.g., orbitals, spins, etc.', '1604.04698-1-19-7': 'Prepare the pre-quench state by filling [MATH],', '1604.04698-1-20-0': '= _r _r,1^ = _k _k,1^ ,', '1604.04698-1-21-0': 'where [MATH] creates an electron on sublattice [MATH] in unit cell [MATH], [MATH] is the vacuum, [MATH], and [MATH] is the total number of unit cells.', '1604.04698-1-21-1': 'The system is then time-evolved under an integer quantum Hall Hamiltonian [MATH] where [MATH], and we assume the Chern number of all bands of [MATH] are non-zero, [MATH].', '1604.04698-1-21-2': 'The overlap in Eq. [REF] is [MATH], where [MATH] is the [MATH] component of [MATH], an eigenvector of the post-quench Hamiltonian matrix [MATH].', '1604.04698-1-21-3': 'It is known that any component [MATH] must have at least [MATH] zeros in the Brillouin zone [CITATION], see also Thm.', '1604.04698-1-21-4': '[REF].', '1604.04698-1-21-5': 'Now assume at an arbitrary Bloch momentum [MATH], [MATH] has the highest weight: [MATH].', '1604.04698-1-21-6': 'The existence of node means [MATH] cannot remain as the highest weight element over the entire Brillouin zone, and hence must switch rank with the second highest weight element, say [MATH], at some point [MATH]: [MATH] .', '1604.04698-1-21-7': 'Together with the normalization [MATH], one concludes that at [MATH], Eq. [REF] is satisfied.', '1604.04698-1-22-0': 'Note that in this case, the return amplitude [MATH] is related to the [MATH]-space sublattice particle density,', '1604.04698-1-23-0': '_k,a(t) (t) _k, a^ _ k,a^ (t)= G(k,t)^2 .', '1604.04698-1-24-0': 'A DQPT can thus be identified by [MATH], i.e., a complete depletion of particles with momentum [MATH] on sublattice [MATH] (or orbital, spin, etc.), which may be experimentally measurable.', '1604.04698-1-25-0': 'The argument above for node-protected DQPT applies to any pre-/post-quench combinations.', '1604.04698-1-25-1': 'In general, if the overlap of the pre-quench band [MATH] with every eigenstate [MATH] of [MATH] has nodes in the Brillouin zone, then the triangle inequality Eq. [REF] is guaranteed, and a robust DQPT would occur.', '1604.04698-1-25-2': 'This criterion can be written in a form more amenable to numerical test,', '1604.04698-1-26-0': '_MaxMin _n[_k ^(n)(k) (k)] ,', '1604.04698-1-27-0': '_MaxMin = 0 Robust DQPT .', '1604.04698-1-28-0': 'Topological protection of nodes in wavefunction overlaps There is a curious connection between wavefunction zeros and quantization.', '1604.04698-1-28-1': 'In elementary quantum mechanics, nodes in the radial wavefunction is related to the principal quantum number [CITATION].', '1604.04698-1-28-2': 'In continuum integer quantum Hall systems, the number of nodes in the wavefunction [MATH] for [MATH] in a magnetic unit cell is given by its Chern number magnitude [MATH] [CITATION].', '1604.04698-1-28-3': 'These nodes persist even in the presence of weak disorder [CITATION].', '1604.04698-1-28-4': 'On a lattice, [MATH] gives the number of [MATH]-space nodes in all wavefunction components [MATH] [CITATION], a phenomenon closely related to the energetic spectral flow of the edge states [CITATION].', '1604.04698-1-28-5': 'Note that the relation between [MATH] and wavefunction nodes relies on one participant of the overlap, namely the basis states [MATH] and [MATH], to be topologically trivial.', '1604.04698-1-28-6': 'If both participants can be nontrivial, the number of nodes in their overlap should depend on both topological indices on an equal footing.', '1604.04698-1-28-7': 'Indeed we have the following theorems,', '1604.04698-1-29-0': 'In 2D, the overlap of Bloch bands [MATH] and [MATH], with Chern numbers [MATH] and [MATH] respectively, must have at least [MATH] nodes in the Brillouin zone.', '1604.04698-1-30-0': 'In 1D, the Berry phase [MATH] of a real Bloch band, [MATH], is quantized to [MATH] or [MATH].', '1604.04698-1-30-1': 'The overlap of two real bands [MATH] and [MATH], with Berry phases [MATH] and [MATH] respectively, must have at least one node if [MATH].', '1604.04698-1-31-0': 'See SM for proof.', '1604.04698-1-31-1': 'Note that symmetry protection may enforce a Hamiltonian to be real [CITATION], leading to the real bands in Thm.', '1604.04698-1-31-2': '[REF].', '1604.04698-1-31-3': 'This prompts the notion of symmetry-protected DQPT, reminiscent of symmetry-protected topological phases that may be classified by topological numbers at high-symmetry hyper-surfaces [CITATION].', '1604.04698-1-31-4': 'An example will be given later.', '1604.04698-1-32-0': 'Generalized Hofstadter model We demonstrate ideas discussed above using a generalized Hofstadter model,', '1604.04698-1-33-0': 'H(k, t, m) =', '1604.04698-1-34-0': 'd_1 & v_1 & v_3e^ik_y', '1604.04698-1-35-0': 'v_1 & d_2 & v_2', '1604.04698-1-36-0': 'v_3 e^-ik_y & v_2 & d_3 ,', '1604.04698-1-37-0': 'd_a = 2(k_x + a ) + am ,', '1604.04698-1-38-0': 'v_a = 1+2t[k_x + (a +12)] ,', '1604.04698-1-39-0': 'a = 1,2,3 , 23 .', '1604.04698-1-40-0': 'The nearest neighbor hopping is set as [MATH].', '1604.04698-1-40-1': 'At [MATH], we recover the Hofstadter model [CITATION] on a square lattice with magnetic flux [MATH] per structural unit cell, and its magnetic unit cell consists of [MATH] structural unit cells along the [MATH] direction.', '1604.04698-1-40-2': '[MATH] allows for second neighbor (i.e. diagonal) hopping, and [MATH] describes a flux-commensurate onsite sawtooth potential.', '1604.04698-1-40-3': 'See SM for phase diagram.', '1604.04698-1-40-4': 'At [MATH] and [MATH], [MATH] is invariant under the combined transformation of time-reversal, [MATH], and inversion, [MATH], and is hence real.', '1604.04698-1-40-5': 'Eigenstates there are subject to Thm.', '1604.04698-1-40-6': '[REF].', '1604.04698-1-41-0': 'Now consider quenches in which the initial state is prepared by filling one of the three bands of a pre-quench Hamiltonian parameterized by [MATH], and evolved using a post-quench Hamiltonian with [MATH] .', '1604.04698-1-41-1': 'In Fig. [REF], we keep [MATH] fixed, and plot the [MATH] (Eq. [REF]) of the three pre-quench bands as functions of the post-quench [MATH].', '1604.04698-1-41-2': 'By varying [MATH], the post-quench [MATH] is swept through six different topological phases as labeled in Fig. [REF].', '1604.04698-1-42-0': 'Let us illustrate topological and symmetry-protected DQPTs with two examples, using [MATH] as the pre-quench state (blue circled line in Fig. [REF]): [(i)] Topological DQPT protected by 2D Chern number.', '1604.04698-1-42-1': 'Consider the quench from [MATH] to phase 5.', '1604.04698-1-42-2': 'In this case, the Chern number of the pre-quench state ([MATH]) differs from all three Chern numbers of the post-quench Hamiltonian ([MATH]), thus from Thm.', '1604.04698-1-42-3': '[REF], all three overlaps have nodes, and Eq. [REF] is satisfied.', '1604.04698-1-42-4': 'Symmetry-protected DQPT.', '1604.04698-1-42-5': 'Consider the quench from [MATH] to phase 2.', '1604.04698-1-42-6': 'In this case, the pre-quench Chern number ([MATH]) is identical to at least one of the post-quench Chern numbers ([MATH]), hence not all overlaps have nodes originating from Thm.', '1604.04698-1-42-7': '[REF].', '1604.04698-1-42-8': 'Nevertheless, at [MATH] and [MATH] where the Hamiltonian is real, its eigenstates can be classified by their Berry phases.', '1604.04698-1-42-9': 'One can find numerically that at [MATH], the Berry phase for [MATH] is [MATH], whereas that of the post-quench [MATH] (the one with [MATH]) is [MATH].', '1604.04698-1-42-10': 'According to Thm.', '1604.04698-1-42-11': '[REF], therefore, [MATH] has node along [MATH].', '1604.04698-1-42-12': 'Nodes in overlaps of [MATH] with [MATH] and [MATH] are still protected by Thm.', '1604.04698-1-42-13': '[REF].', '1604.04698-1-42-14': 'Thus all three overlaps have nodes and DQPT is protected.', '1604.04698-1-43-0': 'Details of all [MATH] quench types ([MATH] pre-quench states [MATH] post-quench phases) can be found in SM.', '1604.04698-1-43-1': 'We should note here that out of all 18 types, 2 robust DQPTs ([MATH] to phases 2 and 5) exhibit an even number of overlap nodes at [MATH] and/or [MATH] not accounted for by Thms.', '1604.04698-1-43-2': '[REF] and [REF].', '1604.04698-1-43-3': 'By tuning [MATH] and [MATH], we were able to shift the nodes along [MATH] as well as to change the total number of nodes by an even number, but could not entirely eliminate them.', '1604.04698-1-43-4': 'We suspect however that they could eventually be eliminated in an enlarged parameter space.', '1604.04698-1-44-0': 'Conclusion and discussion In this work, we showed that for quantum quenches between gapped phases in a generic multi-band system, a robust dynamical quantum phase transition (DQPT) is a consequence of momentum-space nodes (or zeros) in the wavefunction overlap between the pre-quench state and all post-quench energy eigenstates.', '1604.04698-1-44-1': 'Nodes in wavefunction overlaps are topologically protected if the topological indices of the two participating wavefunctions-such as Chern number in 2D and Berry phase in 1D-are different.', '1604.04698-1-45-0': 'Our main tenets here are the triangle inequality Eq. [REF], and phase ergodicity.', '1604.04698-1-45-1': 'It is interesting to note that collapsing a band gap would affect both conditions: right at the gap collapsing point [MATH], the two phases become mutually locked; as the gap re-opens, the system has gone through a topological transition, which changes the node structure in wavefunction overlaps.', '1604.04698-1-45-2': 'We also note that while the existence of topological and symmetry-protected DQPT is insensitive to details of the energy band structure, the exact times at which it would occur will inevitably depend on the latter.', '1604.04698-1-45-3': 'The shortest critical time will be bounded by the recurrence time of the phases, which, for few-level systems such as band insulators, should remain physically relevant.', '1604.04698-1-46-0': 'DQPT in band systems is in principle experimentally measurable.', '1604.04698-1-46-1': 'As shown in Eq. [REF], DQPT can be identified as the depletion of "sublattice" particle density [MATH] where sublattice [MATH] can also refer to spin, orbital, etc.', '1604.04698-1-46-2': 'Particle density [MATH] can already be measured in cold atom systems by time-of-flight experiments [CITATION].', '1604.04698-1-46-3': 'It is not hard to envisage an additional procedure of "sublattice" isolation in such measurements, e.g., by using a magnetic field for spin filtering, or by releasing other sublattices [MATH] slightly earlier than [MATH].'}

{'1604.04698-2-0-0': 'A sudden quantum quench of a Bloch band from one topological phase toward another has been shown to exhibit an intimate connection with the notion of a dynamical quantum phase transition (DQPT), where the returning probability of the quenched state to the initial state-i.e. the Loschmidt echo-vanishes at critical times [MATH].', '1604.04698-2-0-1': 'Analytical results so far are limited to two-band models, leaving the exact relation between topology and DQPT unclear.', '1604.04698-2-0-2': 'In this work, we show that for a general multi-band system, a robust DQPT relies on the existence of nodes (i.e. zeros) in the wavefunction overlap between the initial band and the post-quench energy eigenstates.', '1604.04698-2-0-3': 'These nodes are topologically protected if the two participating wavefunctions have distinctive topological indices.', '1604.04698-2-0-4': 'We demonstrate these ideas in detail for both one and two spatial dimensions using a three-band generalized Hofstadter model.', '1604.04698-2-0-5': 'We also discuss possible experimental observations.', '1604.04698-2-1-0': 'Introduction Advances in experimental techniques, in particular in cold atom systems [CITATION], have reinvigorated recent interest in quantum dynamics [CITATION].', '1604.04698-2-1-1': 'A paradigmatic setup in this context is a quantum quench [CITATION], wherein a system is prepared as an eigenstate [MATH] of an initial Hamiltonian [MATH], but evolved under a different Hamiltonian [MATH].', '1604.04698-2-1-2': 'In a slow ramp [CITATION], one has in addition the control over how fast the switching between [MATH] and [MATH] can be, as well as what path to take in the space of Hamiltonians.', '1604.04698-2-1-3': 'Since [MATH] typically consists of many excited states of [MATH] with a non-thermal distribution, its time evolution provides a unique venue for investigating issues in nonequilibrium quantum statistical mechanics such as thermalization, equilibration, or the lack thereof [CITATION].', '1604.04698-2-1-4': 'A particularly fruitful approach to understanding dynamics after a quantum quench is by exploiting the formal similarity between the time evolution operator [MATH], and the thermal density operator [MATH].', '1604.04698-2-1-5': 'This enables one to leverage and extend notions in equilibrium statistical mechanics to the realm of quantum dynamics.', '1604.04698-2-1-6': 'In this spirit, the return amplitude', '1604.04698-2-2-0': 'G(t) = e^-iH_F t = _n ^(n) ^2 e^-iE_n t', '1604.04698-2-3-0': 'can be thought of as a partition function along imaginary temperature [MATH], with the prepared state [MATH] as a fixed boundary [CITATION].', '1604.04698-2-3-1': 'Here [MATH] and [MATH] are eigenstates and eigenvalues of the post-quench [MATH], respectively.', '1604.04698-2-3-2': 'Heyl et al showed [CITATION] that analogous to the thermal free energy, a dynamical free energy density [CITATION] can be defined, [MATH], where [MATH] is system size.', '1604.04698-2-3-3': 'Singularities in [MATH] then signifies the onset of what was proposed as a dynamical quantum phase transition (DQPT).', '1604.04698-2-3-4': 'In statistical mechanics, phase transitions are closely related to the zeros of the partition function-known as Fisher zeros-in the complex temperature plane [CITATION].', '1604.04698-2-3-5': 'Historically, Yang and Lee were the first to connect phase transitions with zeros of the partition function in complexified parameter space [CITATION].', '1604.04698-2-3-6': 'While Fisher zeros are always complex for finite systems, they may coalesce into a continuum (line in one parameter dimension, area in two parameter dimensions, etc) that cuts through the real temperature axis in the thermodynamic limit, giving rise to an equilibrium phase transition.', '1604.04698-2-3-7': 'Investigations on DQPT have followed a similar route by first solving the Fisher zeros in the complex temperature plane, and then identifying conditions for them to cross the axis of imaginary temperature (real time).', '1604.04698-2-3-8': 'DQPT is thus mathematically identified as [MATH] at critical time(s) [MATH] .', '1604.04698-2-3-9': 'DQPTs occur in both integrable [CITATION] and non-integrable [CITATION] spin systems for quenches across quantum critical points.', '1604.04698-2-3-10': 'They can further be classified by discontinuities in different orders of time derivatives of [MATH] [CITATION] vis-a-vis their thermal counterparts.', '1604.04698-2-3-11': 'Very recently DQPTs have also been shown to constitute unstable fixed points in the renormalization group flow, and are therefore subject to the notion of universality class and scaling [CITATION].', '1604.04698-2-4-0': 'Physically, the return amplitude [MATH] is related to the power spectrum of work performed during a quench, [MATH], which is the Fourier transform of [MATH], and [MATH] is the energy of the initial state [CITATION].', '1604.04698-2-4-1': 'This in principle makes [MATH], and hence DQPT, a measurable phenomenon.', '1604.04698-2-4-2': 'A pratically more viable route to experimental verification is through measuring time evolution of thermodynamic quantities, which may exhibit post-quench oscillations at a time scale commensurate with the DQPT critical time [MATH], and universal scaling near [MATH] [CITATION].', '1604.04698-2-4-3': 'In band systems, as we will show, they may also be identified by a complete depletion at [MATH] of sublattice or spin-polarized particle density at certain crystal momenta, see Eq. [REF].', '1604.04698-2-5-0': 'Parallel to the development of DQPT as the dynamical analogue of equilibrium phase transitions is the investigation on its relation with topology [CITATION].', '1604.04698-2-5-1': 'This issue arises naturally because in the transverse field Ising model, in which DQPT was first discovered, the quantum critical point can be mapped to a topological phase transition at which the quantized Berry phase of the fermionized Hamiltonian jumps between [MATH] and [MATH].', '1604.04698-2-5-2': 'DQPT in this two-band fermion model was attributed to the occurrence of "population inversion" [CITATION] where it becomes equally probable to find the initial state in either of the two post-quench bands, a consequence of the Berry phase jump [CITATION].', '1604.04698-2-5-3': 'The same analysis has been extended to various two-band models in one- and two-spatial dimensions (1D/2D) [CITATION], where definitive connection was found between DQPT and quench across topological transitions, although some complications exist .', '1604.04698-2-5-4': 'DQPTs have also been demonstrated to occur for quenches within the same topological phase [CITATION], although from the point of view of topological protection, these are not robust as they require fine-tuning of the Hamiltonians.', '1604.04698-2-6-0': 'The purpose of this work is to develop a general theory beyond two band models to clarify the relation between robust DQPT and topology.', '1604.04698-2-6-1': 'We will show that a robust DQPT-one which is insensitive to the details of the pre- and post-quench Hamiltonians other than the phases to which they belong-relies on the existence of zeros (or nodes) in the wavefunction overlap between the initial band and all eigenstates of the post-quench Hamiltonian.', '1604.04698-2-6-2': 'These nodes are topologically protected if the two participating wavefunctions have distinctive topological indices: for example, the Chern number difference [MATH] provides a lower bound to the number of [MATH]-space nodes in the overlap [MATH], see Theorem [REF].', '1604.04698-2-6-3': 'These considerations lead to the notion of topological and symmetry-protected DQPTs which we will demonstrate in detail using a [MATH]-band generalized Hofstadter model.', '1604.04698-2-6-4': 'Analysis of a 1D 3-band model exhibiting symmetry-protected DQPT can be found in Supplemental Materials (SM).', '1604.04698-2-6-5': 'Amplitude and phase conditions of DQPT The DQPT condition [MATH] can be interpreted geometrically as the complex numbers [MATH] forming a closed polygon in the complex plane at [MATH], see Fig. [REF].', '1604.04698-2-6-6': 'The time-independent content of this observation is that the amplitudes [MATH] satisfy a generalized triangle inequality, [MATH].', '1604.04698-2-6-7': 'Invoking [MATH], one has the amplitude condition,', '1604.04698-2-7-0': 'z_n = ^(n) ^2 !', '1604.04698-2-7-1': '12 n .', '1604.04698-2-8-0': 'For [MATH] that satisfy Eq. [REF], solutions to [MATH] exist and form a subspace [MATH] on the [MATH]-torus,', '1604.04698-2-9-0': 'M_z_nT^N: e^-i_n _n=1^N z_ne^-i_n = 0 .', '1604.04698-2-10-0': 'To set off DQPT, the dynamical phases must be able to evolve into [MATH].', '1604.04698-2-10-1': 'This constitutes the phase condition,', '1604.04698-2-11-0': 't^* : e^-iE_n t^* M_z_n .', '1604.04698-2-12-0': 'DQPT requires both conditions to hold simultaneously.', '1604.04698-2-13-0': 'Phase ergodicity in few-level systems At first glance, the phase condition may seem to be the more stringent one.', '1604.04698-2-13-1': 'After a quench across a quantum phase transition, a many-body initial state [MATH] typically has overlap with an extensive amount of eigenstates of the post-quench Hamiltonian [MATH] and therefore the amplitudes [MATH] are generically exponentially small in system size, rendering Eq. [REF] satisfied in general.', '1604.04698-2-13-2': 'Existence of DQPT then relies entirely on the phase condition.', '1604.04698-2-13-3': 'Integrable systems, however, point to the possibility that the amplitude and phase conditions may be intricately related and traded for one another.', '1604.04698-2-13-4': 'Such systems can effectively be broken down into few-level subsystems labeled by quantum numbers [MATH], say [MATH] levels [MATH] for [MATH] in the [MATH] sector.', '1604.04698-2-13-5': 'Correspondingly [MATH].', '1604.04698-2-13-6': "For the transverse field Ising model, Kitaev's honeycomb model [CITATION], and band insulator models, [MATH] is the Bloch momentum.", '1604.04698-2-13-7': 'It is known that as long as the [MATH] gaps, [MATH], are not rationally related, the dynamical phases [MATH] are ergodic on the [MATH]-torus up to an overall phase [CITATION], and will therefore evolve into its subspace [MATH] (Eq. [REF]).', '1604.04698-2-13-8': 'Phase ergodicity thus guarantees the phase condition Eq. [REF], and DQPT in each [MATH] sector depends entirely on the amplitude condition.', '1604.04698-2-14-0': 'Robust DQPT protected by nodes in wavefunction overlap Hereafter, we focus on quenches in multi-band Bloch systems with [MATH] bands.', '1604.04698-2-14-1': 'For simplicity we use a single filled band [MATH] as the pre-quench state.', '1604.04698-2-14-2': 'Generalization to multiple filled bands is straightforward.', '1604.04698-2-14-3': 'The post-quench return amplitude is [MATH],', '1604.04698-2-15-0': 'G(k, t) = _n=1^N_B ^(n)(k) (k)^2 e^-i_n(k) t ,', '1604.04698-2-16-0': 'where [MATH] and [MATH] are respectively the post-quench energy eigenstates and eigenvalues.', '1604.04698-2-16-1': 'Assume phase ergodicity holds at all [MATH] points-this is a very relaxed requirement provided there is no degeneracy at any [MATH] point.', '1604.04698-2-16-2': 'Then DQPT amounts to the existence of at least one [MATH] at which Eq. [REF] is satisfied, namely', '1604.04698-2-17-0': 'k Brillouin Zone: ^(n)(k) (k)^2 12 n .', '1604.04698-2-18-0': 'We now discuss how Eq. [REF] and hence DQPT can arise from nodes in wavefunction overlaps.', '1604.04698-2-18-1': 'Note that this is not the only way to get DQPT.', '1604.04698-2-18-2': 'Its virtue lies in its robustness against perturbations to the Hamiltonians.', '1604.04698-2-18-3': 'In SM, we provide examples where DQPTs with no overlap node can be easily avoided simply by Hamiltonian parameter tuning without crossing a phase boundary.', '1604.04698-2-18-4': 'The overlap nodes are, on the other hand, typically topologically protected, a point we will return to later.', '1604.04698-2-18-5': 'Now consider the following quench.', '1604.04698-2-18-6': 'Let [MATH] label "sublattices", which in general may also include other degrees of freedom, e.g., orbitals, spins, etc.', '1604.04698-2-18-7': 'Prepare the pre-quench state by filling [MATH],', '1604.04698-2-19-0': '= _r _r,1^ = _k _k,1^ ,', '1604.04698-2-20-0': 'where [MATH] creates an electron on sublattice [MATH] in unit cell [MATH], [MATH] is the vacuum, [MATH], and [MATH] is the total number of unit cells.', '1604.04698-2-20-1': 'The system is then time-evolved under an integer quantum Hall Hamiltonian [MATH] where [MATH], and we assume the Chern number of all bands of [MATH] are non-zero, [MATH].', '1604.04698-2-20-2': 'The overlap in Eq. [REF] is [MATH], where [MATH] is the [MATH] component of [MATH], an eigenvector of the post-quench Hamiltonian matrix [MATH].', '1604.04698-2-20-3': 'It is known that any component [MATH] must have at least [MATH] zeros in the Brillouin zone [CITATION], see also Thm.', '1604.04698-2-20-4': '[REF].', '1604.04698-2-20-5': 'Now assume at an arbitrary Bloch momentum [MATH], [MATH] has the highest weight: [MATH].', '1604.04698-2-20-6': 'The existence of node means [MATH] cannot remain as the highest weight element over the entire Brillouin zone, and hence must switch rank with the second highest weight element, say [MATH], at some point [MATH]: [MATH] .', '1604.04698-2-20-7': 'Together with the normalization [MATH], one concludes that at [MATH], Eq. [REF] is satisfied.', '1604.04698-2-21-0': 'Note that in this case, the return amplitude [MATH] is related to the [MATH]-space sublattice particle density,', '1604.04698-2-22-0': '_k,a(t) (t) _k, a^ _ k,a^ (t)= G(k,t)^2 .', '1604.04698-2-23-0': 'A DQPT can thus be identified by [MATH], i.e., a complete depletion of particles with momentum [MATH] on sublattice [MATH] (or orbital, spin, etc.), which may be experimentally measurable.', '1604.04698-2-24-0': 'The argument above for node-protected DQPT applies to any pre-/post-quench combinations.', '1604.04698-2-24-1': 'In general, if the overlap of the pre-quench band [MATH] with every eigenstate [MATH] of [MATH] has nodes in the Brillouin zone, then the triangle inequality Eq. [REF] is guaranteed, and a robust DQPT would occur.', '1604.04698-2-24-2': 'This criterion can be written in a form more amenable to numerical test,', '1604.04698-2-25-0': '_MaxMin _n[_k ^(n)(k) (k)] ,', '1604.04698-2-26-0': '_MaxMin = 0 Robust DQPT .', '1604.04698-2-27-0': 'Topological protection of nodes in wavefunction overlaps There is a curious connection between wavefunction zeros and quantization.', '1604.04698-2-27-1': 'In elementary quantum mechanics, nodes in the radial wavefunction is related to the principal quantum number [CITATION].', '1604.04698-2-27-2': 'In continuum integer quantum Hall systems, the number of nodes in the wavefunction [MATH] for [MATH] in a magnetic unit cell is given by its Chern number magnitude [MATH] [CITATION].', '1604.04698-2-27-3': 'These nodes persist even in the presence of weak disorder [CITATION].', '1604.04698-2-27-4': 'On a lattice, [MATH] gives the number of [MATH]-space nodes in all wavefunction components [MATH] [CITATION], a phenomenon closely related to the energetic spectral flow of the edge states [CITATION].', '1604.04698-2-27-5': 'Note that the relation between [MATH] and wavefunction nodes relies on one participant of the overlap, namely the basis states [MATH] and [MATH], to be topologically trivial.', '1604.04698-2-27-6': 'If both participants can be nontrivial, the number of nodes in their overlap should depend on both topological indices on an equal footing.', '1604.04698-2-27-7': 'Indeed we have the following theorems,', '1604.04698-2-28-0': 'In 2D, the overlap of Bloch bands [MATH] and [MATH], with Chern numbers [MATH] and [MATH] respectively, must have at least [MATH] nodes in the Brillouin zone.', '1604.04698-2-29-0': 'In 1D, the Berry phase [MATH] of a real Bloch band, [MATH], is quantized to [MATH] or [MATH].', '1604.04698-2-29-1': 'The overlap of two real bands [MATH] and [MATH], with Berry phases [MATH] and [MATH] respectively, must have at least one node if [MATH].', '1604.04698-2-30-0': 'See SM for proof.', '1604.04698-2-30-1': 'Note that symmetry protection may enforce a Hamiltonian to be real [CITATION], leading to the real bands in Thm.', '1604.04698-2-30-2': '[REF].', '1604.04698-2-30-3': 'This prompts the notion of symmetry-protected DQPT, reminiscent of symmetry-protected topological phases that may be classified by topological numbers at high-symmetry hyper-surfaces [CITATION].', '1604.04698-2-30-4': 'An example will be given later, see also SM.', '1604.04698-2-31-0': 'Generalized Hofstadter model We demonstrate ideas discussed above using a generalized Hofstadter model,', '1604.04698-2-32-0': 'H(k, t, m) =', '1604.04698-2-33-0': 'd_1 & v_1 & v_3e^ik_y', '1604.04698-2-34-0': 'v_1 & d_2 & v_2', '1604.04698-2-35-0': 'v_3 e^-ik_y & v_2 & d_3 ,', '1604.04698-2-36-0': 'd_a = 2(k_x + a ) + am ,', '1604.04698-2-37-0': 'v_a = 1+2t[k_x + (a +12)] ,', '1604.04698-2-38-0': 'a = 1,2,3 , 23 .', '1604.04698-2-39-0': 'The nearest neighbor hopping is set as [MATH].', '1604.04698-2-39-1': 'At [MATH], we recover the Hofstadter model [CITATION] on a square lattice with magnetic flux [MATH] per structural unit cell, and its magnetic unit cell consists of [MATH] structural unit cells along the [MATH] direction.', '1604.04698-2-39-2': '[MATH] allows for second neighbor (i.e. diagonal) hopping, and [MATH] describes a flux-commensurate onsite sawtooth potential.', '1604.04698-2-39-3': 'See SM for phase diagram.', '1604.04698-2-39-4': 'At [MATH] and [MATH], [MATH] is invariant under the combined transformation of time-reversal, [MATH], and inversion, [MATH], and is hence real.', '1604.04698-2-39-5': 'Eigenstates there are subject to Thm.', '1604.04698-2-39-6': '[REF].', '1604.04698-2-40-0': 'Now consider quenches in which the initial state is prepared by filling one of the three bands of a pre-quench Hamiltonian parameterized by [MATH], and evolved using a post-quench Hamiltonian with [MATH] .', '1604.04698-2-40-1': 'In Fig. [REF], we keep [MATH] fixed, and plot the [MATH] (Eq. [REF]) of the three pre-quench bands as functions of the post-quench [MATH].', '1604.04698-2-40-2': 'By varying [MATH], the post-quench [MATH] is swept through six different topological phases as labeled in Fig. [REF].', '1604.04698-2-41-0': 'Let us illustrate topological and symmetry-protected DQPTs with two examples, using [MATH] as the pre-quench state (blue circled line in Fig. [REF]): [(i)] Topological DQPT protected by 2D Chern number.', '1604.04698-2-41-1': 'Consider the quench from [MATH] to phase 5.', '1604.04698-2-41-2': 'In this case, the Chern number of the pre-quench state ([MATH]) differs from all three Chern numbers of the post-quench Hamiltonian ([MATH]), thus from Thm.', '1604.04698-2-41-3': '[REF], all three overlaps have nodes, and Eq. [REF] is satisfied.', '1604.04698-2-41-4': 'Symmetry-protected DQPT.', '1604.04698-2-41-5': 'Consider the quench from [MATH] to phase 2.', '1604.04698-2-41-6': 'In this case, the pre-quench Chern number ([MATH]) is identical to at least one of the post-quench Chern numbers ([MATH]), hence not all overlaps have nodes originating from Thm.', '1604.04698-2-41-7': '[REF].', '1604.04698-2-41-8': 'Nevertheless, at [MATH] and [MATH] where the Hamiltonian is real, its eigenstates can be classified by their Berry phases.', '1604.04698-2-41-9': 'One can find numerically that at [MATH], the Berry phase for [MATH] is [MATH], whereas that of the post-quench [MATH] (the one with [MATH]) is [MATH].', '1604.04698-2-41-10': 'According to Thm.', '1604.04698-2-41-11': '[REF], therefore, [MATH] has node along [MATH].', '1604.04698-2-41-12': 'Nodes in overlaps of [MATH] with [MATH] and [MATH] are still protected by Thm.', '1604.04698-2-41-13': '[REF].', '1604.04698-2-41-14': 'Thus all three overlaps have nodes and DQPT is protected.', '1604.04698-2-42-0': 'Details of all [MATH] quench types ([MATH] pre-quench states [MATH] post-quench phases) can be found in SM.', '1604.04698-2-42-1': 'We should note here that out of all 18 types, 2 robust DQPTs ([MATH] to phases 2 and 5) exhibit an even number of overlap nodes at [MATH] and/or [MATH] not accounted for by Thms.', '1604.04698-2-42-2': '[REF] and [REF].', '1604.04698-2-42-3': 'By tuning [MATH] and [MATH], we were able to shift the nodes along [MATH] as well as to change the total number of nodes by an even number, but could not entirely eliminate them.', '1604.04698-2-42-4': 'We suspect however that they could eventually be eliminated in an enlarged parameter space.', '1604.04698-2-43-0': 'Conclusion and discussion In this work, we showed that for quantum quenches between gapped phases in a generic multi-band system, a robust dynamical quantum phase transition (DQPT) is a consequence of momentum-space nodes (or zeros) in the wavefunction overlap between the pre-quench state and all post-quench energy eigenstates.', '1604.04698-2-43-1': 'Nodes in wavefunction overlaps are topologically protected if the topological indices of the two participating wavefunctions-such as Chern number in 2D and Berry phase in 1D-are different.', '1604.04698-2-44-0': 'Our main tenets here are the triangle inequality Eq. [REF], and phase ergodicity.', '1604.04698-2-44-1': 'It is interesting to note that collapsing a band gap would affect both conditions: right at the gap collapsing point [MATH], the two phases become mutually locked; as the gap re-opens, the system has gone through a topological transition, which changes the node structure in wavefunction overlaps.', '1604.04698-2-44-2': 'We also note that while the existence of topological and symmetry-protected DQPT is insensitive to details of the energy band structure, the exact times at which it would occur will inevitably depend on the latter.', '1604.04698-2-44-3': 'The shortest critical time will be upper-bounded by the recurrence time of the phases, which, for few-level systems such as band insulators, should remain physically relevant .', '1604.04698-2-45-0': 'DQPT in band systems is in principle experimentally measurable.', '1604.04698-2-45-1': 'As shown in Eq. [REF], DQPT can be identified as the depletion of "sublattice" particle density [MATH] where sublattice [MATH] can also refer to spin, orbital, etc.', '1604.04698-2-45-2': 'Particle density [MATH] can already be measured in cold atom systems by time-of-flight experiments [CITATION].', '1604.04698-2-45-3': 'It is not hard to envisage an additional procedure of "sublattice" isolation in such measurements, e.g., by using a magnetic field for spin filtering, or by releasing other sublattices [MATH] slightly earlier than [MATH].'}

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[]

[['1604.04698-1-31-4', '1604.04698-2-30-4']]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1604.04698

1105.6010

{'1105.6010-1-0-0': '# Introduction', '1105.6010-1-1-0': 'In the context of component-based embedded systems, the management of reconfiguration in adaptive systems is an increasingly important feature.', '1105.6010-1-1-1': 'The Fractal component-based framework, and its industrial instantiation MIND, provide for support for control operations in the lifecycle of components.', '1105.6010-1-1-2': 'Nevertheless, the use of complex and integrated architectures make the management of this reconfiguration operations difficult to handle by programmers.', '1105.6010-1-1-3': 'To address this issue, we propose to use synchronous languages, which are a complete approach to the design of reactive systems, based on behavior models in the form of transition systems.', '1105.6010-1-1-4': 'Furthermore, the design of closed-loop reactive controllers of reconfigurations can benefit from formal tools like Discrete Controller Synthesis (DCS).', '1105.6010-1-2-0': 'Using DCS, integrated in a programming language, provides designers for support in the correct design of controllers.', '1105.6010-1-2-1': 'This method is different from the usual method of first programming and then verifying.', '1105.6010-1-2-2': 'It involves the automated generation of part of the control logic of a system.', '1105.6010-1-2-3': 'Discrete control has until now been applied to computing systems only very rarely [CITATION].', '1105.6010-1-2-4': 'An important challenge is to integrate this formal reactive systems design in actual, practical operating systems.', '1105.6010-1-2-5': 'An open issue is the identification and correct use of the practical sensors and monitors providing for reliable and significant information; the control points and actuators available in the API of the OS, enabling enforcement of a management policy; the firing conditions for the transitions of the automata.', '1105.6010-1-3-0': 'In this paper we describe an approach to concretely integrate synchronous reconfiguration controllers in Fractal component-based systems.', '1105.6010-1-3-1': 'Our contribution is: (i) a synchronous model of the behavior of the reconfigurable components, in the form of the state space of the control problem, and the specification of the control objectives (Section [REF]); (ii) a component-based architecture for the implementation of the resulting controller with the Fractal/Cecilia programming environment, taking advantage of the Comete middleware.', '1105.6010-1-3-2': '(Sections [REF] and [REF]).', '1105.6010-1-3-3': 'We validate it by the case study of the Comanche HTTP server deployed on a multi-core execution platform.', '1105.6010-1-4-0': '# Background', '1105.6010-1-5-0': '## The Fractal Component Model', '1105.6010-1-6-0': 'We introduce Fractal[CITATION][CITATION], a hierarchical and reflective component model and Cecilia[CITATION], a component-base software engineering framework providing a C implementation of this model.', '1105.6010-1-6-1': 'Fractaldefines components as entities encompassing behaviours and data.', '1105.6010-1-6-2': 'A component can be dismentled in two parts: a membrane and a content.', '1105.6010-1-6-3': 'The content is either a set of operations or a finite number of sub-components, which are under the control of the enclosing membrane.', '1105.6010-1-6-4': 'Components can be nested at an arbitrary level in a recursive fashion.', '1105.6010-1-7-0': 'Components interact with their environment through interfaces.', '1105.6010-1-7-1': 'Interfaces are typed collections of operations.', '1105.6010-1-7-2': 'They can be of two sorts: client interfaces emit operation invocations, server interfaces receive operation invocations.', '1105.6010-1-7-3': 'One-way operation invocations may carry arguments, two-way operations consist of an invocation followed by the return of a result.', '1105.6010-1-7-4': 'Components can expose several interfaces.', '1105.6010-1-7-5': 'Client and server interfaces are connected through explicit bindings.', '1105.6010-1-7-6': 'Functional interfaces are access points to content operations, while Controller interfaces define membrane operations.', '1105.6010-1-7-7': 'The membrane embodies the control behaviour associated with a particular component.', '1105.6010-1-8-0': 'The Fractalmodel defines a set of optional controller interfaces to adress minimal requirements in terms of introspection, composition and life-cycle.', '1105.6010-1-8-1': 'Among others are:', '1105.6010-1-9-0': "Life-Cycle Controller: controls component's behavioural phases such as starting and stopping.", '1105.6010-1-10-0': "Binding Controller: establish/break bindings between component's interfaces and its environment.", '1105.6010-1-11-0': 'The Ceciliaframework is a coherent toolchain to design, compile and deploy component-based applications.', '1105.6010-1-11-1': 'It allows the description of hierarchical Fractalarchitectures using an xml ADL, and the implementation of the primitive components using the C langage.', '1105.6010-1-11-2': "By automaticaly generating the controllers related glue-code, the toolchain allow the developer to focus on the implementation of the primitives' content operations i.e. the functional interfaces' methods in figure [REF].", '1105.6010-1-11-3': 'From an application definition and its implementation, the toolchain generate a standalone executable or a set of independent component binaries.', '1105.6010-1-11-4': 'The Cometemiddleware described in the next section relies on these independent bricks to dynamically load and bind instances of components to build arbitrary architectures.', '1105.6010-1-12-0': '## Comete', '1105.6010-1-13-0': 'Comete [CITATION] is a minimal middleware and run-time layer engineered by STMicroelectronics to dynamicaly deploy, run and reconfigure Cecilia components over distributed platforms.', '1105.6010-1-13-1': 'Cometeis providing a distributed event-driven architecture.', '1105.6010-1-13-2': 'Applications deployed using Cometecan take advantage of the high-level abstraction of this platform to communicate asynchronous messages between components.', '1105.6010-1-13-3': "The middleware models a distributed platform as a set of processing elements and handles communication between them so that application developpers don't have to know about the underlying communication channels and protocols.", '1105.6010-1-14-0': 'The first step of any application deployment using Cometelies in the instantiation of a loader component (Figure [REF](a)) bound to the CometeAPI interface (this interface is further detailed in Listing [REF]).', '1105.6010-1-14-1': 'This mandatory component is in charge of deploying the application by invoking Cometeoperations such as instantiations and bindings to build the initial configuration (Figure [REF](b)).', '1105.6010-1-14-2': 'Once the deployment process done, the loader can remotely manipulate LifeCycleController (lcc) interface of any instantiated component through the start/stop methods to initiate the execution.', '1105.6010-1-14-3': 'Then, the application is free to diverge from its initial architecture by successive reconfigurations.', '1105.6010-1-14-4': 'The initial deployment is like a reconfiguration from a single primitive component to a potentially more complex architecture.', '1105.6010-1-15-0': 'Each processing element executes message handlers related to components it embodies.', '1105.6010-1-15-1': 'The runtime layer consists of a task queue associated with a FIFO scheduler (Figure [REF]).', '1105.6010-1-15-2': 'The scheduler is non preemptive.', '1105.6010-1-15-3': 'Every message directed to a given component will be handled as task on the respective processing element.', '1105.6010-1-15-4': 'The execution model guaranties that: a) At any given time only one method is executing on a processing element.', '1105.6010-1-15-5': 'b) Components deployed on different processing element may execute methods concurrently.', '1105.6010-1-16-0': 'At its lowest level, the runtime layer is dealing with platform heterogeneity in terms of operating systems, hardware platforms and communication protocols.', '1105.6010-1-16-1': 'This heterogeneity is then abstracted by the middleware layer.', '1105.6010-1-16-2': "Internally, a remote binding is handled by a couple of stub/skeleton components loaded respectively on the client's and server's processing element.", '1105.6010-1-16-3': 'On the client side, the stub transparently intercepts and serializes emitted messages.', '1105.6010-1-16-4': 'Data are then transmitted through a platform channel to the server side.', '1105.6010-1-16-5': 'The skeleton reads serialized message data, operates the inverse process and pushes a handler task in the local runtime queue.', '1105.6010-1-16-6': 'The message arguments are stored until the associated task is scheduled.', '1105.6010-1-16-7': 'Thanks to a generic, scalable, component-based architecture, Cometeis targeting a wide range of platforms, from embedded System-On-Chip to distributed computers.', '1105.6010-1-16-8': 'The middleware is built over an extensible library of components providing support for various processing elements and communication channels.', '1105.6010-1-16-9': 'Noticeable specializations of Cometeare: the STm8010(Traviata) board including three ST200 cores, the xStream many-cores streaming prototype, posix-compliant operating systems easing deployment over multi-threaded hardware, computers distributed over a TCP/IP network.', '1105.6010-1-17-0': '## Heptagon and BZR', '1105.6010-1-18-0': '### Heptagon language', '1105.6010-1-19-0': 'In this work, we use the language Heptagon/BZR [CITATION] .', '1105.6010-1-19-1': 'The Heptagon language allows to describe reactive systems by means of generalized Moore machines, i.e., mixed synchronous dataflow equations and automata [CITATION], with parallel and hierarchical composition.', '1105.6010-1-19-2': 'The basic behavior is that at each reaction step, values in the input flows are used in order to compute the values in the output flows for that step.', '1105.6010-1-19-3': "Inside the nodes, this is expressed as a set of declarations, which takes the form of equations defining, for each output, the values that the flow takes, in terms of an expression on other flows' instantaneous values, possibly using values computed in preceding steps (also known as state values).", '1105.6010-1-20-0': 'Figure [REF] shows a small program in this language.', '1105.6010-1-20-1': 'It describes the control of a task, which can either be idle or active.', '1105.6010-1-20-2': 'When it is idle, i.e., in the initial Idle state, then the occurrence of the input r requests the launch of the task.', '1105.6010-1-20-3': 'Another input c (which will be controlled further by the synthesized controller) can either allow the activation, or temporary block the request and make the automaton go to a waiting state.', '1105.6010-1-20-4': 'When active, the task can be ended with the input e.', '1105.6010-1-20-5': 'This delayable node has two outputs, a featuring the instantaneous activity of the task, and s being emitted on the instant when it becomes active: [EQUATION]', '1105.6010-1-21-0': '### BZR and controller synthesis', '1105.6010-1-22-0': 'BZR is an extension of Heptagon, allowing its compilation to involve discrete controller synthesis (DCS) using the DCS tool Sigali [CITATION].', '1105.6010-1-22-1': 'DCS allows to compute automatically a controller, i.e., a function which will act on the initial program so as to enforce a given temporal property.', '1105.6010-1-22-2': 'Concretely, the BZR language allows the declaration of controllable variables, which are not defined by the programmer.', '1105.6010-1-22-3': 'These free variables can be used in the program so as to let some choices undecided (e.g., choice between several transitions).', '1105.6010-1-22-4': 'The controller, computed by DCS, is then able to avoid undesired states of the application by setting and updating appropriate values for these variables at runtime.', '1105.6010-1-23-0': 'Figure [REF] shows an example of use of these controllable variables.', '1105.6010-1-23-1': 'It consists in two instances of the delayable node, as in Figure [REF].', '1105.6010-1-23-2': 'They run in parallel, defined by synchronous composition: one global step corresponds to one local step for every equation, i.e., here, for every instance of the automaton in the delayable node.', '1105.6010-1-23-3': 'Then, the twotasks node so defined is given a contract composed of two parts: the part allowing the declaration of controllable variables ([MATH] and [MATH]), and the part allowing the programmer to assert the property to be enforced by DCS, using the controllable variables.', '1105.6010-1-23-4': "Here, we want to ensure that the two tasks running in parallel won't be both active at the same time.", '1105.6010-1-23-5': 'Thus, [MATH] and [MATH] will be used by the computed controller to block some requests, leading automata of tasks to the waiting state whenever the other task is active.', '1105.6010-1-24-0': '### Heptagon/BZR compilation', '1105.6010-1-25-0': 'The compilation of a Heptagon/BZR program produces sequential code in a target general programming language (C, Java or Caml).', '1105.6010-1-25-1': 'This code takes the form of two functions (or methods), named reset and step.', '1105.6010-1-25-2': 'reset initializes the internal state of the program.', '1105.6010-1-25-3': 'The step function is evaluated at each logical instant to compute output values from an input vector and the current state, possibly updated.', '1105.6010-1-25-4': 'A typical way of using these functions is to enclose this step call in an infinite loop:', '1105.6010-1-26-0': 'current_state [MATH] reset() // state initialization for each step do inputs [MATH] gather current events outputs, next_state [MATH] step(inputs, current_state) handle outputs current_state [MATH] next_state', '1105.6010-1-27-0': 'Eventually, such infinite loop will not be as clearly stated, but hidden within, e.g., events managers, threads, interrupts, depending on the application context.', '1105.6010-1-27-1': 'In our context, we will use the C generated code.', '1105.6010-1-27-2': 'The API, for a node f with inputs [MATH] (typed [MATH]) and outputs [MATH] (typed [MATH]) is given below.', '1105.6010-1-27-3': 'The additional input mem is a pointer towards the internal state, to be used and updated.', '1105.6010-1-27-4': 'The result of the fstep call is a structure in which are placed the outputs values.', '1105.6010-1-28-0': 'void f_reset(f_mem* mem); f_res f_step( [MATH] , ..., [MATH] , f_mem* mem);', '1105.6010-1-29-0': '## System/manager Interaction', '1105.6010-1-30-0': 'Our approach applies to systems supporting dynamic reconfiguration and providing some events describing their observable states.', '1105.6010-1-30-1': 'It relies on the modeling of the system (and possibly the environment) by means of Heptagon automata and describing the control objectives.', '1105.6010-1-30-2': 'The BZR compilation tool-chain compiles the automata and the control objectives into a synchronous program which will be referred to as managerin the sequel.', '1105.6010-1-30-3': 'The managerencapsulates a model of the system and a controllerto enforce the objectives.', '1105.6010-1-31-0': 'Figure [REF] illustrates the use of a manager (i.e., synchronous program) to manage the reconfiguration of a system.', '1105.6010-1-31-1': 'The managerreceives some input events from the system and the environment.', '1105.6010-1-31-2': 'According to these inputs and the current state of the model, the controllermay update the state of the model.', '1105.6010-1-31-3': 'Internally, this reduces to fill the controllable variables with the appropriate computed values.', '1105.6010-1-31-4': 'The computation of the new state of the model corresponds to a step of the managerand may fire some commands.', '1105.6010-1-31-5': 'The commands are meant to reconfigure the managed system in order for this latter to be coherent with its model.', '1105.6010-1-32-0': '# The Comanche Http Server', '1105.6010-1-33-0': 'This section describes an example of a reconfigurable component-based software application written in Fractal together with the execution platform on which the application will be run.', '1105.6010-1-33-1': 'The complete system will be used as a case-study in order to introduce our approach for the management of reconfigurable systems in Section [REF].', '1105.6010-1-34-0': '## Components architecture', '1105.6010-1-35-0': 'Figure [REF] describes the architecture of a HTTP server written in Fractal.', '1105.6010-1-35-1': 'It is a variant of the Comanche server used as an example in tutorials.', '1105.6010-1-35-2': 'Incoming requests are received by the frontendcomponent, which transmits them to the analyzercomponent.', '1105.6010-1-35-3': 'The latter forward well-formed requests to the dispatcherwhich queries fileserver[MATH]or fileserver[MATH]to solve the requests.', '1105.6010-1-35-4': 'The analyzercomponent can also send requests to the logger to keep track of HTTP requests.', '1105.6010-1-36-0': 'The required components for this application to work are frontend, analyzer, dispatcherand fileserver[MATH].', '1105.6010-1-36-1': 'A first available degree of dynamical reconfiguration lies in fileserver[MATH]and logger.', '1105.6010-1-36-2': 'As illustrated by the dashed lines, fileserver[MATH]and loggermay be activated (resp., deactivated) and connected to (resp., disconnected from) the rest of the components.', '1105.6010-1-37-0': '## Execution architecture', '1105.6010-1-38-0': 'A second degree of reconfiguration concerns component mapping over processing elements.', '1105.6010-1-38-1': 'For our experiments, we used a server equipped with two Intel Xeon dual-core processors, each one running at 1.86GHz clock frequency.', '1105.6010-1-38-2': 'The four available cores enable the execution of four tasks in parallel.', '1105.6010-1-38-3': 'In the sequel, we will refer to each core as a processing element (pe[MATH], ..., pe[MATH]).', '1105.6010-1-39-0': 'Figure [REF] describes the initial deployment of Comanche components using Comete middleware (Section [REF]) on the execution platform.', '1105.6010-1-39-1': 'Each component is associated with a processing element, on which it will execute in order to handle messages in its associated FIFO.', '1105.6010-1-39-2': 'Component bindings are implemented by the Comete middleware as asynchronous communication channels.', '1105.6010-1-39-3': 'For the sake of clarity of the figure, the bindings are made implicit.', '1105.6010-1-39-4': 'Notice that the managerand Comete loader also execute on the same platform.', '1105.6010-1-40-0': '## Renconfiguration policy', '1105.6010-1-41-0': 'The control objectives we want to enforce are related to distinct aspects of the Comanche application and the execution platform.', '1105.6010-1-41-1': 'We are able to design a manager to fire reconfiguration commands in order to enforce the objectives.', '1105.6010-1-41-2': 'Comete together with Fractal API provide means to reflect manager commands on the managed system.', '1105.6010-1-41-3': 'Those means we are interested in are component migration provided by Comete API and lifecycle (resp., binding) control interface of the Fractal API in order to start/stop (resp., bind/unbind) components.', '1105.6010-1-41-4': 'In the following, we list the control objectives:', '1105.6010-1-42-0': '[Processing element availability] A processing element is a shared resource that may be unavailable for some reasons (e.g., energy saving, higher priority task, fault, etc.).', '1105.6010-1-42-1': 'We want that no component is running on an unavailable processing element (component migration).', '1105.6010-1-43-0': '[Workload balancing] There is a maximum workload, a processing element should not exceed.', '1105.6010-1-43-1': 'Migrate components in order to decrease the workload.', '1105.6010-1-44-0': '[Quality-of-Service] When the fileserver[MATH]is overloaded, start the fileserver[MATH]in order to keep the average response time low (lifecycle and binding control).', '1105.6010-1-45-0': '[Exclusiveness] The loggershould be started upon the request of the user, and then fileserver[MATH]should not be running (lifecycle and binding control).', '1105.6010-1-46-0': 'In order to ensure the objectives we described above, a manager is implemented and put in a closed-loop with the software application as described in section [REF].', '1105.6010-1-46-1': 'A simple scenario is as follows.', '1105.6010-1-46-2': 'The manager receives an event stating that the fileserver[MATH]is overloaded.', '1105.6010-1-46-3': 'It performs some computation and emits the command to start the fileserver[MATH]and bind it to the rest of the components.', '1105.6010-1-46-4': 'We measure the workload of fileserver[MATH]by comparing the size of the FIFO associated to it with a specific threshold.', '1105.6010-1-46-5': 'Next, the controller receives from the environment that the processing element pe[MATH]is no longer available.', '1105.6010-1-46-6': 'The controller reacts by emitting the command to migrate the components running on pe[MATH]to other processing elements, balancing the workload as described by the related property.', '1105.6010-1-47-0': '# Designing the manager', '1105.6010-1-48-0': 'We follow a modeling method for the design of a manager to enforce the properties we described above; we will elaborate on it by improving the treament of reconfiguration actions in Section [REF].', '1105.6010-1-48-1': 'The use of DCS ensures that the manager is correct with respect to the properties given as objectives.', '1105.6010-1-48-2': 'It mainly consists of the following steps: (1) Providing a synchronous model of the behavior of the reconfigurable components and the processing elements, giving the state space of the control problem; (2) Specifying the control objectives and identifying the controllable variables; (3) Compiling: controller synthesis and code generation.', '1105.6010-1-49-0': '## Modeling Components With Heptagon', '1105.6010-1-50-0': 'We adopt a modular modeling approach to enable reusing models.', '1105.6010-1-50-1': 'Moreover, we distinguish between the application models (i.e., software) and the execution platform (e.g., hardware) models.', '1105.6010-1-50-2': 'Instances of these models will be associated in the main synchronous program (see Section [REF]) to obtain the global model.', '1105.6010-1-50-3': 'Hence, the approach facilitates replacing hardware models without modifying software ones in case the application is ported to other execution platforms.', '1105.6010-1-51-0': '### Modeling Software Components', '1105.6010-1-52-0': 'Component Lifecycle', '1105.6010-1-53-0': 'Figure [REF]-(a) shows the automaton modeling a Fractal component lifecycle.', '1105.6010-1-53-1': 'A component may be in one of three possible states.', '1105.6010-1-53-2': 'A Running (R) component is connected to the other components and may handle incoming messages in its associated FIFO.', '1105.6010-1-53-3': 'A Stopped (S) component is disconnected from the others and can not execute to handle messages in its input FIFO.', '1105.6010-1-53-4': 'Finally, a component may be put in a Safe Stopping (SS) in order to handle the remaining messages in its FIFO before completely stopping.', '1105.6010-1-54-0': 'The automaton has three inputs (ch, fe and s for change, FIFOs empty and stop respectively), and two outputs run and disc.', '1105.6010-1-54-1': 'Initially, the component is stopped (i.e., state S).', '1105.6010-1-54-2': 'It goes to R(i.e., running) upon receiving ch.', '1105.6010-1-54-3': 'From this state, it goes to state SSupon receiving ch.', '1105.6010-1-54-4': 'At state SS, it goes to state Supon receiving fe which means that the input FIFOs are empty.', '1105.6010-1-54-5': 'At any state, receiving s forces the automaton to go to the state S.', '1105.6010-1-54-6': 'The two outputs of the automaton run and disc (for running and disconnect) take their values as described by the figure.', '1105.6010-1-54-7': 'These outputs tell whether a component is running (resp., disconnected) or not.', '1105.6010-1-55-0': 'As described by the transitions of the automaton, some reconfiguration commands (start, stop, connect, disconnect), represented as additional outputs, are fired upon changing state.', '1105.6010-1-55-1': 'These commands are meant to reconfigure the related component in order to be coherent with its lifecycle model.', '1105.6010-1-55-2': 'For instance the transition from Sto R, outputs the commands start and connect in order to start the component and connect it to the rest of the components.', '1105.6010-1-56-0': 'FIFO State', '1105.6010-1-57-0': 'With each server interface of a component is associated a FIFO.', '1105.6010-1-57-1': 'The FIFO stores the input events before they are handled by the component.', '1105.6010-1-57-2': 'Figure [REF]-(b) models the state of such FIFOs.', '1105.6010-1-57-3': 'It is a two state automaton with one input and one output (f and full respectively).', '1105.6010-1-57-4': 'Initially, the FIFO is empty.', '1105.6010-1-57-5': 'It goes to the state FF (resp., FE) upon receiving f (resp., [MATH]f).', '1105.6010-1-57-6': 'The value output full indicates the state of the FIFO: true (resp., false) means that the FIFO size is above (resp., below) a given threshold.', '1105.6010-1-58-0': '### Modeling Hardware Components', '1105.6010-1-59-0': 'Component mapping on Processing Elements', '1105.6010-1-60-0': 'Figure [REF] models the mapping of a component on the available processing elements.', '1105.6010-1-60-1': 'It is a four state automaton.', '1105.6010-1-60-2': 'Each state represents mapping on one processing element, on which a component may run.', '1105.6010-1-60-3': 'It may change from one state to another depending on the Boolean inputs a and b.', '1105.6010-1-60-4': 'That is, each state has three outgoing transitions (one to each remaining state).', '1105.6010-1-60-5': 'The transition to take depends on the value of a and b (two Boolean inputs to encode four possibilities).', '1105.6010-1-60-6': 'For the sake of clarity, not all of the transitions are present in the figure.', '1105.6010-1-60-7': 'The output of the automaton is of enumerated type; it takes its values in pe[MATH], pe[MATH], pe[MATH], pe[MATH] depending on the state of the automaton.', '1105.6010-1-61-0': 'The transitions of the automaton are associated with outputs (mig[MATH], mig[MATH], mig[MATH], mig[MATH]).', '1105.6010-1-61-1': 'They are associated to migration commands, in order to migrate the corresponding component to its new processing element.', '1105.6010-1-62-0': 'Processing Element Availability', '1105.6010-1-63-0': 'We need a model of the availability of a processing element in order to know whether a component may run on it or not.', '1105.6010-1-63-1': 'Figure [REF] describes such a model.', '1105.6010-1-63-2': 'It is a two states automaton with one input dis (for disable) and one output on which tells on the availability of the processing element.', '1105.6010-1-63-3': 'Initially, a processing element is available (i.e., at state ON).', '1105.6010-1-63-4': 'It goes to the state OFF(resp., ON) upon receiving dis (resp., [MATH]dis).', '1105.6010-1-63-5': 'The output on takes its value depending on the state of the automaton.', '1105.6010-1-64-0': 'Processing Element Workload', '1105.6010-1-65-0': 'The workload of a pe[MATH]depends on the components running on it and their bindings to the other components.', '1105.6010-1-65-1': 'Two components [MATH] and [MATH] (client and server), bound through their respective interfaces [MATH] and [MATH] induce some workload on their associated processing elements pe[MATH]and pe[MATH].', '1105.6010-1-65-2': 'Based on some benchmarks, we estimate the workload related to pe[MATH]and pe[MATH]as described by the following equations:', '1105.6010-1-66-0': 'a) [MATH].', '1105.6010-1-66-1': 'b) [MATH].', '1105.6010-1-67-0': 'The workload of the processing element running the server component [MATH] depends on the cost [MATH] of the executed function, and the function call frequency [MATH].', '1105.6010-1-67-1': 'The processing element running the client component is charged with the communication effort.', '1105.6010-1-67-2': 'That is, the processing element workload depends on the call frequency and a parameter [MATH] which depends on the position of the two communicating processing elements.', '1105.6010-1-67-3': 'In our case-study, the processing elements are processor cores.', '1105.6010-1-67-4': 'The communication uses cache memories of distinct levels.', '1105.6010-1-67-5': 'The values we associate to [MATH] represent the fact that the communication cost differs depending on the type of cache used between the processing elements.', '1105.6010-1-67-6': 'These values are: [MATH] in case pe[MATH]and pe[MATH]refer to the same core, [MATH] in case the processing elements are distinct cores of the same processor, [MATH] in case the two processing elements refer to distinct cores of distinct processors.', '1105.6010-1-68-0': 'In Listing [REF], we give the interface of the node cost implementing workload equations.', '1105.6010-1-68-1': 'For a given binding between two components, the node takes as input the position of the client and the server component, the frequency [MATH] of the calls between them and the cost of the function executed by the server component.', '1105.6010-1-68-2': 'The outputs are integer values associating workloads induced by the two components with the available processing elements.', '1105.6010-1-68-3': 'The involved processing elements will be associated with positive integer values, the others will take the value zero.', '1105.6010-1-69-0': '## Complete system model', '1105.6010-1-70-0': 'The main program (partially described in listing of Listing [REF]) consists of the synchronous composition of the automata modeling each component (hardware and software).', '1105.6010-1-70-1': 'The inputs of the main node consist of the events stating on: (pes1) the availability of the processing element pe[MATH]; (addL) the request of the logger; (fS1, fS2, fL) the load of the FIFO associated with fileserver[MATH], fileserver[MATH]and loggerrespectively.', '1105.6010-1-71-0': 'The outputs of the main program correspond to the reconfiguration/migration commands fired by the automata.', '1105.6010-1-71-1': 'For instance migLp0 is fired by the automaton modeling the position of the component logger(line 22).', '1105.6010-1-71-2': 'When migLp0 takes the value true, the component loggershould be migrated to pe[MATH].', '1105.6010-1-71-3': 'The output startS2 is fired by the automaton modeling fileserver[MATH]lifecycle (line 17).', '1105.6010-1-71-4': 'When it is true, it states that the component should be started.', '1105.6010-1-72-0': 'The main program is composed of: An instance of node fifo for each server interface of the components (lines 11-13).', '1105.6010-1-72-1': 'An instance of node lifecycle for each component (lines 15-17).', '1105.6010-1-72-2': 'An instance of processing element availability for each processing element (lines 19-20).', '1105.6010-1-72-3': 'An instance of node position for each component (lines 22-24).', '1105.6010-1-72-4': 'An instance of node cost for each binding between components (lines 26-28).', '1105.6010-1-72-5': 'Finally, the cost imposed on each processing element corresponds to the sum of all the cost induced by the bindings (lines 29-30).', '1105.6010-1-73-0': 'numbers=left,numberstyle=,stepnumber=2,numbersep=5pt', '1105.6010-1-74-0': '## Describing Control Objectives with BZR', '1105.6010-1-75-0': 'We now have a complete model of the possible behaviors of the system, in the absence of control.', '1105.6010-1-75-1': 'We want to obtain a controller that will enforce the policy given informally in Section [REF].', '1105.6010-1-75-2': 'We do this in the form of a contract for each of the points of the policy as follows.', '1105.6010-1-76-0': 'Processing element availability We want no component running on an unavailable processing element: this can be achieved by component migration.', '1105.6010-1-77-0': 'The corresponding expression in the contract to be enforced i.e., to be controlled for invariance, is: [EQUATION]', '1105.6010-1-77-1': 'The involved controllable variables defined in the part will be, for each component:', '1105.6010-1-78-0': 'a[MATH],', '1105.6010-1-79-0': 'b[MATH], [MATH].', '1105.6010-1-79-1': 'The effect of the control will be that, when some processing element becomes unavailable, appropriate migrations will be fired.', '1105.6010-1-79-2': 'More precisely: upon reception of input pes[MATH] at value , the availability model makes a transition as shown in Figure [REF].', '1105.6010-1-79-3': 'In the global model, a transition will be taken to a next state where the above expression is .', '1105.6010-1-79-4': 'For components on the unavailable PE, the automata shown in Figure [REF] can not stay in the current state without violating the property, therefore controllables will take a value such that a local transition occurs, hence firing a migration action towards another available PE.', '1105.6010-1-80-0': 'Workload balancing Workload on each PE[MATH] is bounded by [MATH]: this is achieved by component migration.The expression is:', '1105.6010-1-81-0': '[MATH] .', '1105.6010-1-82-0': 'The involved controllables are:', '1105.6010-1-83-0': 'a[MATH],', '1105.6010-1-84-0': 'b[MATH], [MATH].', '1105.6010-1-84-1': 'The effect of the control will be that, if some migration or starting of a component happens on a PE, the choice encoded by controllables will be between PEs for which this addition would not violate the bound.', '1105.6010-1-85-0': 'Quality-of-Service When the fileserver[MATH]is overloaded, start the fileserver[MATH]: this is achieved by lifecycle and binding control.', '1105.6010-1-85-1': 'The expressions are:', '1105.6010-1-86-0': 'qos= [MATH] .', '1105.6010-1-87-0': 'The involved controllables are: sS2, chS2.', '1105.6010-1-87-1': 'The effect of the control will be that, if the FIFO of fileserver[MATH]reaches its threshold, the fileserver[MATH]is started unless the Logger runs; and when the FIFO goes back under the threshold, it is disconnected, and eventually stopped.', '1105.6010-1-88-0': "Exclusiveness When the loggerruns, fileserver[MATH]doesn't: this is achieved by lifecycle and binding control.", '1105.6010-1-88-1': 'The corresponding expression is: [EQUATION]', '1105.6010-1-88-2': 'The involved controllable is: sS2.', '1105.6010-1-88-3': 'The effect of the control is that, when the Logger is started, the fileserver[MATH]must be stopping or idle.', '1105.6010-1-89-0': '# Manager Integration', '1105.6010-1-90-0': 'The C code generated by the BZR compilation tool-chain consists of two functions step(...) and reset(...).', '1105.6010-1-90-1': 'Listing [REF] is a sketch of a program using these functions in order to manage the reconfiguration of a system.', '1105.6010-1-90-2': 'At line 2, reset() is called once to initialize the memory of the program.', '1105.6010-1-90-3': 'The piece of code ranging from line 5 to line 10 is made sensitive to incoming events; i.e., this part of the code is executed each time new events are received.', '1105.6010-1-90-4': 'The function prepareevents() prepares the events and provides the inputs for the function step().', '1105.6010-1-90-5': 'The signature of the function step() corresponds to the one of the main program of Listing [REF].', '1105.6010-1-90-6': 'A call to this function corresponds to a step of the synchronous program.', '1105.6010-1-90-7': 'The outputs of step() are given to the function generatecommands() in order to translate the outputs into the consequent reconfiguration commands.', '1105.6010-1-91-0': 'numbers=left,numberstyle=,stepnumber=1,numbersep=5pt, basicstyle=', '1105.6010-1-92-0': '## Wrapping the manager into a component', '1105.6010-1-93-0': 'The functions listed in Listing [REF] are wrapped into a Fractal component in order to be connected to the rest of application and middleware components (see Figure [REF]).', '1105.6010-1-93-1': 'This component provides one server interface Events and one client interface Commands.', '1105.6010-1-93-2': 'The signature of the interface Events is described in Listing [REF].', '1105.6010-1-93-3': 'The method setEvent provided by this interface enables components to register event values.', '1105.6010-1-93-4': 'Indeed, the manager hides a buffer containing the last value registered of each event.', '1105.6010-1-93-5': 'The function prepareevents() uses this buffer to provide the inputs for the function step().', '1105.6010-1-94-0': 'The function generatecommands() translates the outputs of the function step() into method calls through the Commands interface.', '1105.6010-1-94-1': 'Indeed, the signature of the required interface Commands is that of ComponentManager provided by Comete API (see Listing [REF]).', '1105.6010-1-94-2': 'It provides methods for component migration, reconfiguration and lifecycle management.', '1105.6010-1-95-0': 'numbers=none', '1105.6010-1-96-0': '## Concrete integration of the manager', '1105.6010-1-97-0': 'Figure [REF] describes the concrete integration of the manager together with the application and Comete middleware components.', '1105.6010-1-97-1': 'The component user is connected to the Events interface of the manager.', '1105.6010-1-97-2': 'It intercepts user requests (add logger for instance) and registers them as events.', '1105.6010-1-97-3': 'The FIFO components (part of Comete) are made implicit in the figure.', '1105.6010-1-97-4': 'As illustrated by the dotted lines, the FIFOs associated with fileserver[MATH], fileserver[MATH]and loggerare connected to the Events interface of the manager in order to register events related to their size.', '1105.6010-1-98-0': 'Upon receiving events, the manager performs some computation and fires some reconfiguration commands in the form of method calls.', '1105.6010-1-98-1': 'These method calls are handled by the loader of Comete.', '1105.6010-1-98-2': 'Indeed, as illustrated by the dashed lines, the loader is connected to the control interface of the application components in order to start/stop bind/unbind them.', '1105.6010-1-98-3': 'Moreover, the loader knows how migrating components should be achieved regarding the execution platform.', '1105.6010-1-99-0': 'Comete middleware hides complex mechanisms and provides a simple API to manage component lifecycle, migration, and architecture transparently, whatever the execution platform is.', '1105.6010-1-99-1': 'For that purpose, we adopted a centralized version of the manageremitting reconfiguration commands to Comete instead of application components.', '1105.6010-1-99-2': 'We could make the managercommunicate directly with the components (modulo re-engineering Comete) to enable a distributed implementation of the manager.', '1105.6010-1-99-3': 'This means that parts of the managerwould be integrated into Fractal component membranes.', '1105.6010-1-100-0': 'In this section we presented a simple version of the use of a synchronous manager for the reconfiguration of a parallel and asynchronous application.', '1105.6010-1-100-1': 'The reconfiguration commands were considered to be short enough to complete before the next reaction of the manager.', '1105.6010-1-100-2': 'However, this is not always the case.', '1105.6010-1-100-3': 'Indeed, some reconfiguration commands may take non negligible time to complete.', '1105.6010-1-100-4': 'During this time interval, the system model inside the manager does not reflect the actual state of the system.', '1105.6010-1-100-5': 'For instance, when the command to start a component C[MATH] is fired, the model of C[MATH] is at state Running (R) but the actual component is not yet started.', '1105.6010-1-100-6': 'The manager reaction to other incoming events during reconfiguration progress may lead the managed system to undesirable states.', '1105.6010-1-100-7': 'Next section proposes an extension to our modeling approach in order to solve issues related to asynchronous commands.', '1105.6010-1-101-0': '# Asynchronous commands', '1105.6010-1-102-0': 'We propose a controller architecture together with some guidelines for writing synchronous models in order to overcome the issues related to the transient incoherence of the model with respect to the state of the system.', '1105.6010-1-102-1': 'Our approach relies on the explicit representation of the states where some commands are being processed but not completed, together with some synchronization mechanisms.', '1105.6010-1-103-0': 'The purpose of identifying the states where some reconfiguration commands are in progress is to make it possible for the programmer to decide what should happen during reconfigurations.', '1105.6010-1-103-1': 'The programmer should rewrite objectives properties taking into account these situations.', '1105.6010-1-104-0': '## Modeling command execution', '1105.6010-1-105-0': '### Behavioral models', '1105.6010-1-106-0': 'Some synchronous languages (e.g., Esterel) provide built-in constructs in order to perform asynchronous calls within a synchronous program [CITATION].', '1105.6010-1-106-1': 'We follow the same principle of such constructs and model the asynchronous aspect of reconfiguration commands by means of an automaton.', '1105.6010-1-107-0': 'Such an automaton is illustrated by Figure [REF].', '1105.6010-1-107-1': 'At state P(for pending), the command is emitted but not yet completed.', '1105.6010-1-107-2': 'At state D, the command has finished.', '1105.6010-1-107-3': 'The automaton has two inputs do and done.', '1105.6010-1-107-4': 'The output pending tells whether the command is pending or not.', '1105.6010-1-107-5': 'From the initial state, upon receiving do which corresponds to the firing of the command, the automaton goes to state P.', '1105.6010-1-107-6': 'The automaton stay at state Puntil receiving the input done.', '1105.6010-1-107-7': 'The input done signals that the command has been completed.', '1105.6010-1-108-0': 'We associate an instance of such an automaton with each asynchcronous command.', '1105.6010-1-108-1': 'Hence the modification w.r.t. model of Section [REF] is systematic and modular.', '1105.6010-1-109-0': '### Objectives and contracts', '1105.6010-1-110-0': 'Being aware of the state where a command is in progress but not finished, one can add some properties to be enforced and modify the previous ones in order to decide what should happen during reconfigurations.', '1105.6010-1-110-1': 'For illustration purpose, we consider the same program as in Section [REF] with the command startS2 as the only command taking non-negligible time to complete.', '1105.6010-1-110-2': 'In the following fragment of program, we associate an instance of the node command to startS2 (line 5) and modify the contract:', '1105.6010-1-111-0': 'numbers=left,numberstyle=,stepnumber=1,numbersep=5pt', '1105.6010-1-112-0': 'The contract changes as follows:', '1105.6010-1-113-0': 'The properties qos and exc are now dependent to the pending of the command.', '1105.6010-1-113-1': 'pensS2[MATH](qos exc) tells that qos and exc may not be enforced when the command startS2 is in progress.', '1105.6010-1-113-2': 'But, they must be enforced once the command completes.', '1105.6010-1-113-3': 'pend (defined at line 6) is a new property.', '1105.6010-1-113-4': 'It forbids the controller to modify the state of the fileserver[MATH]when the command to start it is in progress.', '1105.6010-1-113-5': 'This property is explained below.', '1105.6010-1-114-0': 'In order to understand the property pend, consider the automata modeling fileserver[MATH]lifecycle and startS2 command execution in Figure [REF].', '1105.6010-1-114-1': 'Adding the property pend = (pensS2 discS2) forces the controller to avoid the states where the command is pending (i.e., at state p) and any state where discS2 is true (i.e., at states s and ss).', '1105.6010-1-114-2': 'That is, to avoid the states (ss,p) and (s,p) in the product of the two automata.', '1105.6010-1-115-0': 'At the beginning the automata are at state (s,d).', '1105.6010-1-115-1': 'Upon receiving ch, automaton (a) changes states and fires the command startS2 which makes automaton (b) change state at the same instant.', '1105.6010-1-115-2': 'That is, the global state is (r,p).', '1105.6010-1-115-3': 'Unless receiving done, which is uncontrollable, the controller will not make automaton (a) change state in order not to violate the property pend.', '1105.6010-1-116-0': '## Controller Architecture', '1105.6010-1-117-0': 'The actual implementation of the manager is component-based.', '1105.6010-1-117-1': 'Figure [REF] describes the internals of such a component.', '1105.6010-1-117-2': 'It consists of three subcomponents: evt, ctrl, and cmd.', '1105.6010-1-117-3': 'The functions prepareevents(), reset(), step(), and generatecommands() are spread over the subcomponents as follows: evtencapsulates prepareevents().', '1105.6010-1-117-4': 'It prepares the inputs to ctrlthat encapsulates reset() and step() generated by the BZR compilation.', '1105.6010-1-117-5': 'The component cmdtranslates the outputs of ctrlinto reconfiguration methods calls.', '1105.6010-1-118-0': 'The component evtis in charge of implementing the method provided by the interface Events from which components and the environment may register events.', '1105.6010-1-118-1': 'The component cmdis in charge of calling reconfiguration methods through Commands interface.', '1105.6010-1-118-2': 'Moreover, each time a reconfiguration command is completed, cmdnotices done events related to this command.', '1105.6010-1-118-3': 'That is why cmdis bound to evt.', '1105.6010-1-119-0': '# Related work', '1105.6010-1-120-0': 'A lot of work has been devoted to dynamic reconfiguration of component-based software systems.', '1105.6010-1-120-1': 'In the case of Fractal, one can refer to the tools for component introspection and the languages for specifying reconfigurations [CITATION] or integration in parallel frameworks [CITATION].', '1105.6010-1-121-0': 'Our work fits in the context of applying formal methods for dynamic reconfiguration [CITATION] in order to ensure properties related to components and reconfigurations.', '1105.6010-1-121-1': 'In [CITATION] the authors specify reconfiguration properties by means of temporal logic in order to apply model checking or runtime monitoring of system reconfigurations.', '1105.6010-1-121-2': 'In comparison, our approach benefits from the discrete controller synthesis which provides a correct by construction manager ensuring properties on components and reconfigurations.', '1105.6010-1-121-3': 'In [CITATION], the authors give a general software engineering framework, with some indications of integration, but no complete implementation.', '1105.6010-1-121-4': 'In comparison, we perform a concrete integration of synchronous managers with Fractal, in the form of its Cecilia programming environment, and the Comete middleware.', '1105.6010-1-121-5': 'In addition, we consider and treat the case of asynchronous reconfiguration actions.', '1105.6010-1-122-0': 'Other related work concern the concrete integration of synchronous programs for the management of asynchronous systems.', '1105.6010-1-122-1': 'In [CITATION], the authors provides a synchronous controller for configuring device drivers aiming at global power management of embedded systems.', '1105.6010-1-122-2': 'Apart from their not using DCS techniques, the difference with our approach lies in the call of reconfiguration functions.', '1105.6010-1-122-3': 'Indeed, in [CITATION], function calls are performed inside the reaction of the controller.', '1105.6010-1-122-4': 'This has the benefit of always keeping a model reflecting the exact state of the system.', '1105.6010-1-122-5': 'In our case we use to follow the principle of asynchronous function calls as in [CITATION].', '1105.6010-1-123-0': '# Conclusion', '1105.6010-1-124-0': 'Our contribution is on the one hand a synchronous model of the behavior of the reconfigurable components, in the form of the state space of the control problem, and the specification of the control objectives (Section [REF]).', '1105.6010-1-124-1': 'On the other hand, we contribute a component-based architecture for the implementation of the resulting controller with the Fractal/Cecilia programming environment, taking advantage of the Comete middleware.', '1105.6010-1-124-2': '(Sections [REF] and [REF])', '1105.6010-1-125-0': 'In this work, we apply formal techniques issued from academic research to an open, stable and available framework supported by an independent industry consortium [CITATION].', '1105.6010-1-125-1': 'The Fractal model is currently being implemented in the MINDproject [CITATION], a sequel to the Cecilia framework.', '1105.6010-1-125-2': 'The MINDframework is a collaborative initiative to spread component-based software engineering to an even larger community of academic institutions and industries.', '1105.6010-1-125-3': 'The present contribution keeps its legitimacy in the MINDcontext as most of the concepts and semantics introduced with Cecilia and Comete remain.', '1105.6010-1-125-4': 'Therefore, applying our contribution to MINDshould be limited to a straightforward adaptation .', '1105.6010-1-126-0': 'We have ongoing work on another interesting case study: an H264 video processing application, implemented on a multicore architecture using Cecilia and Comete.', '1105.6010-1-126-1': 'It could be handled following the very same methodology we propose, because it follows essentially the same structure: degrees of reconfiguration concern migrations, and adding and removal of components performing video effects on the stream.', '1105.6010-1-127-0': 'Perspectives are in the line of generalizing our proposal at the language-level, by extending the Fractal ADL with a way to incorporate automata notation for reconfiguration description, as well as the behavioral contracts, and integrating the application of BZR in a global Fractal compilation flow.', '1105.6010-1-127-1': 'Also, the integration of synchronous controllers in a Fractal components architecture should be facilitated by identifying general programming guidelines, providing end-users with informal rules such that components are controllable, and the synchronous instant and step (and the states in the automata) fits well with the event granularity.', '1105.6010-1-128-0': '# Complete Example and Integration', '1105.6010-1-129-0': 'In what follows, we describe the complete integration process of the code produced by the bzr compilation for the control of the comanche application software.', '1105.6010-1-129-1': 'Writing the synchronous model of the manager has been discussed above.', '1105.6010-1-129-2': 'Here we give more details providing simulation results of the manager and the process of integration in the Fractal application.', '1105.6010-1-130-0': '# Simulation of the Synchronous Program', '1105.6010-1-131-0': 'The interface of the synchronous program is described in Listing [REF].', '1105.6010-1-131-1': 'The input disable states on the availability of the processing element PE[MATH].', '1105.6010-1-131-2': 'This input takes the value false as long as PE[MATH] is available (false otherwise).', '1105.6010-1-131-3': 'fifoH1F, fifoH2F, fifoL2F are Booleans stating on the size of the FIFOs associated with FileServer1, FileServer2 and Logger respectively.', '1105.6010-1-131-4': 'They take the value false as long as the size of the associated FIFOs is under their specific threshold.', '1105.6010-1-131-5': 'The input addlog tells whether the logger is required or not.', '1105.6010-1-131-6': 'The input cstartH2done notifies that the command for starting FileServer2 has been actually completed.', '1105.6010-1-132-0': 'The outputs of the main program are associated with the possible commands provided by the reconfigurable system.', '1105.6010-1-132-1': 'An output associated with a command takes the value false until the command has to be fired.', '1105.6010-1-132-2': 'Then, it takes the value true during one step.', '1105.6010-1-133-0': 'Figure [REF] illustrates part of the simulation results.', '1105.6010-1-133-1': 'The inputs (in blue color) are given (by the user) in order to simulate the events received from the application.', '1105.6010-1-133-2': 'The outputs (in red color) corresponding to reconfiguration commands take the value true each time a command has to be fired.', '1105.6010-1-134-0': 'The interaction scenario as illustrated by the figure is as follows:', '1105.6010-1-135-0': '## BZR Program Compilation into C Code', '1105.6010-1-136-0': 'The BZR compilation tool chain compiles the synchronous program into C code.', '1105.6010-1-136-1': 'The generated functions of our interest are the reset() and step() functions of the main program.', '1105.6010-1-136-2': 'reset() initializes the memory associated with the synchronous program.', '1105.6010-1-136-3': 'The function step() computes the outputs of the program depending on the inputs.', '1105.6010-1-136-4': 'It also updates the internal memory.', '1105.6010-1-137-0': 'The prototype of the function step() is presented in Listing [REF].', '1105.6010-1-137-1': 'It has the same input parameters as the main function of the synchronous program in Listing [REF].', '1105.6010-1-137-2': 'The parameter self refer to the memory of the complete program.', '1105.6010-1-137-3': 'The returned value of the function step() is of type mainres it is a structure.', '1105.6010-1-137-4': 'Each field of this structure corresponds to an output of the main program.', '1105.6010-1-138-0': '# Integration of the step() function', '1105.6010-1-139-0': 'The integration process of a synchronous manager for the control of reconfiguration lies in the integration of the step() function generated by the synchronous compilation.', '1105.6010-1-139-1': 'In order to achieve this, we have to make some choices.', '1105.6010-1-139-2': 'In particular, we need to decide when the function should be called, and how to apply reconfiguration commands:', '1105.6010-1-140-0': 'Our choice for this case study is to consider event-triggered calls of the step together with asynchronous firing of commands.', '1105.6010-1-140-1': 'The choice may be discussed.', '1105.6010-1-140-2': 'But this is out of the scope of the paper.', '1105.6010-1-141-0': '## Wrapping The Generated Code Into a Fractal Component', '1105.6010-1-142-0': 'The C generated code is wrapped into a Fractal component in order to receive application events and fire reconfiguration commands.', '1105.6010-1-142-1': 'Figure [REF] describes the internals of such a component and the type of the provided (resp. required) interface of the components.', '1105.6010-1-142-2': 'Notice that two interfaces may be bound if an only if they are of the same type.', '1105.6010-1-143-0': '### The Component EventReceiver', '1105.6010-1-144-0': 'Listing [REF] illustrates part of the implementation of the method setValue of the EventReceiver component.', '1105.6010-1-144-1': 'This method is called by other components in order to notify events.', '1105.6010-1-145-0': 'The component manages an array of Boolean values (DATA.stepinputs).', '1105.6010-1-145-1': 'Each cell of the array corresponds to one input event.', '1105.6010-1-145-2': 'Each time an event is notified through this method, its value is updated, then the method step is called.', '1105.6010-1-145-3': 'This method is implemented by the component SynchronousProgram.', '1105.6010-1-145-4': 'The parameter of the call is the array of event values.', '1105.6010-1-146-0': '### The Component SynchronousProgram', '1105.6010-1-147-0': 'Listing [REF] illustrates the code associated with the method step of the component SynchronousProgram.', '1105.6010-1-147-1': 'This method contains a call to the step method actually generated by the BZR compilation (at line 7).', '1105.6010-1-147-2': 'The result of the synchronous step (i.e., DATA.res) are forwarded to the CommandsGenerator in order to translate them into the consequent reconfiguration commands.', '1105.6010-1-148-0': '### The Component CommandsGenerator', '1105.6010-1-149-0': 'Listing [REF] illustrates the implementation of the method doAction of the component CommandsGenerator.', '1105.6010-1-149-1': 'It receives a structure containing the Boolean values associated with the possible commands.', '1105.6010-1-149-2': 'This method parses the received values and fire a command each time the associated output variable has the value true.', '1105.6010-1-150-0': '# Retrieving Application/Environment Events', '1105.6010-1-151-0': 'As explained above, the component EventReceiver of the manager offers a method for event notification.', '1105.6010-1-151-1': 'The components in charge of notifying these events are connected to this interface.', '1105.6010-1-151-2': 'Among the application components, only the FIFOs associated with fileserver[MATH], fileserver[MATH]and loggerare supposed to send events to the manager.', '1105.6010-1-151-3': 'Moreover, there is a component modeling the user and the environment.', '1105.6010-1-151-4': 'The component is in charge of notifying the events related to adding the logger and the availability of a processing element.', '1105.6010-1-151-5': 'These components call the method setValue() of the component EventReceiver whenever a particular condition holds.', '1105.6010-1-151-6': 'Listing [REF] is a sample piece of program for notifying events.', '1105.6010-1-151-7': 'The condition depends on the nature of the event to be notified.', '1105.6010-1-151-8': 'In case of the FIFOs, the condition concerns the size of the FIFO regarding to its associated threshold.'}

{'1105.6010-2-0-0': '# Introduction', '1105.6010-2-1-0': 'In the context of component-based embedded systems, the management of reconfiguration in adaptive systems is an increasingly important feature.', '1105.6010-2-1-1': 'The Fractal component-based framework, and its industrial instantiation MIND, provide for support for control operations in the lifecycle of components.', '1105.6010-2-1-2': 'Nevertheless, the use of complex and integrated architectures make the management of this reconfiguration operations difficult to handle by programmers.', '1105.6010-2-1-3': 'To address this issue, we propose to use synchronous languages, which are a complete approach to the design of reactive systems, based on behavior models in the form of transition systems.', '1105.6010-2-1-4': 'Furthermore, the design of closed-loop reactive controllers of reconfigurations can benefit from formal tools like Discrete Controller Synthesis (DCS).', '1105.6010-2-2-0': 'Using DCS, integrated in a programming language, provides designers for support in the correct design of controllers.', '1105.6010-2-2-1': 'This method is different from the usual method of first programming and then verifying.', '1105.6010-2-2-2': 'It involves the automated generation of part of the control logic of a system.', '1105.6010-2-2-3': 'Discrete control has until now been applied to computing systems only very rarely [CITATION].', '1105.6010-2-2-4': 'An important challenge is to integrate this formal reactive systems design in actual, practical operating systems.', '1105.6010-2-2-5': 'An open issue is the identification and correct use of the practical sensors and monitors providing for reliable and significant information; the control points and actuators available in the API of the OS, enabling enforcement of a management policy; the firing conditions for the transitions of the automata.', '1105.6010-2-3-0': 'In this paper we describe an approach to concretely integrate synchronous reconfiguration controllers in Fractal component-based systems.', '1105.6010-2-3-1': 'Our contribution is: (i) a synchronous model of the behavior of the reconfigurable components, in the form of the state space of the control problem, and the specification of the control objectives (Section [REF]); (ii) a component-based architecture for the implementation of the resulting controller with the Fractal/Cecilia programming environment, taking advantage of the Comete middleware.', '1105.6010-2-3-2': '(Sections [REF] and [REF]).', '1105.6010-2-3-3': 'We validate it by the case study of the Comanche HTTP server deployed on a multi-core execution platform.', '1105.6010-2-4-0': '# Background', '1105.6010-2-5-0': '## The Fractal Component Model', '1105.6010-2-6-0': 'We introduce Fractal[CITATION][CITATION], a hierarchical and reflective component model and Cecilia[CITATION], a component-base software engineering framework providing a C implementation of this model.', '1105.6010-2-6-1': 'Fractaldefines components as entities encompassing behaviours and data.', '1105.6010-2-6-2': 'A component can be dismentled in two parts: a membrane and a content.', '1105.6010-2-6-3': 'The content is either a set of operations or a finite number of sub-components, which are under the control of the enclosing membrane.', '1105.6010-2-6-4': 'Components can be nested at an arbitrary level in a recursive fashion.', '1105.6010-2-7-0': 'Components interact with their environment through interfaces.', '1105.6010-2-7-1': 'Interfaces are typed collections of operations.', '1105.6010-2-7-2': 'They can be of two sorts: client interfaces emit operation invocations, server interfaces receive operation invocations.', '1105.6010-2-7-3': 'One-way operation invocations may carry arguments, two-way operations consist of an invocation followed by the return of a result.', '1105.6010-2-7-4': 'Components can expose several interfaces.', '1105.6010-2-7-5': 'Client and server interfaces are connected through explicit bindings.', '1105.6010-2-7-6': 'Functional interfaces are access points to content operations, while Controller interfaces define membrane operations.', '1105.6010-2-7-7': 'The membrane embodies the control behaviour associated with a particular component.', '1105.6010-2-8-0': 'The Fractalmodel defines a set of optional controller interfaces to adress minimal requirements in terms of introspection, composition and life-cycle.', '1105.6010-2-8-1': 'Among others are:', '1105.6010-2-9-0': "Life-Cycle Controller: controls component's behavioural phases such as starting and stopping.", '1105.6010-2-10-0': "Binding Controller: establish/break bindings between component's interfaces and its environment.", '1105.6010-2-11-0': 'The Ceciliaframework is a coherent toolchain to design, compile and deploy component-based applications.', '1105.6010-2-11-1': 'It allows the description of hierarchical Fractalarchitectures using an xml ADL, and the implementation of the primitive components using the C langage.', '1105.6010-2-11-2': "By automaticaly generating the controllers related glue-code, the toolchain allow the developer to focus on the implementation of the primitives' content operations i.e. the functional interfaces' methods in figure [REF].", '1105.6010-2-11-3': 'From an application definition and its implementation, the toolchain generate a standalone executable or a set of independent component binaries.', '1105.6010-2-11-4': 'The Cometemiddleware described in the next section relies on these independent bricks to dynamically load and bind instances of components to build arbitrary architectures.', '1105.6010-2-12-0': '## Comete', '1105.6010-2-13-0': 'Comete [CITATION] is a minimal middleware and run-time layer engineered by STMicroelectronics to dynamicaly deploy, run and reconfigure Cecilia components over distributed platforms.', '1105.6010-2-13-1': 'Cometeis providing a distributed event-driven architecture.', '1105.6010-2-13-2': 'Applications deployed using Cometecan take advantage of the high-level abstraction of this platform to communicate asynchronous messages between components.', '1105.6010-2-13-3': "The middleware models a distributed platform as a set of processing elements and handles communication between them so that application developpers don't have to know about the underlying communication channels and protocols.", '1105.6010-2-14-0': 'The first step of any application deployment using Cometelies in the instantiation of a loader component (Figure [REF](a)) bound to the CometeAPI interface (this interface is further detailed in Listing [REF]).', '1105.6010-2-14-1': 'This mandatory component is in charge of deploying the application by invoking Cometeoperations such as instantiations and bindings to build the initial configuration (Figure [REF](b)).', '1105.6010-2-14-2': 'Once the deployment process done, the loader can remotely manipulate LifeCycleController (lcc) interface of any instantiated component through the start/stop methods to initiate the execution.', '1105.6010-2-14-3': 'Then, the application is free to diverge from its initial architecture by successive reconfigurations.', '1105.6010-2-14-4': 'The initial deployment is like a reconfiguration from a single primitive component to a potentially more complex architecture.', '1105.6010-2-15-0': 'Each processing element executes message handlers related to components it embodies.', '1105.6010-2-15-1': 'The runtime layer consists of a task queue associated with a FIFO scheduler (Figure [REF]).', '1105.6010-2-15-2': 'The scheduler is non preemptive.', '1105.6010-2-15-3': 'Every message directed to a given component will be handled as task on the respective processing element.', '1105.6010-2-15-4': 'The execution model guaranties that: a) At any given time only one method is executing on a processing element.', '1105.6010-2-15-5': 'b) Components deployed on different processing element may execute methods concurrently.', '1105.6010-2-16-0': 'At its lowest level, the runtime layer is dealing with platform heterogeneity in terms of operating systems, hardware platforms and communication protocols.', '1105.6010-2-16-1': 'This heterogeneity is then abstracted by the middleware layer.', '1105.6010-2-16-2': "Internally, a remote binding is handled by a couple of stub/skeleton components loaded respectively on the client's and server's processing element.", '1105.6010-2-16-3': 'On the client side, the stub transparently intercepts and serializes emitted messages.', '1105.6010-2-16-4': 'Data are then transmitted through a platform channel to the server side.', '1105.6010-2-16-5': 'The skeleton reads serialized message data, operates the inverse process and pushes a handler task in the local runtime queue.', '1105.6010-2-16-6': 'The message arguments are stored until the associated task is scheduled.', '1105.6010-2-16-7': 'Thanks to a generic, scalable, component-based architecture, Cometeis targeting a wide range of platforms, from embedded System-On-Chip to distributed computers.', '1105.6010-2-16-8': 'The middleware is built over an extensible library of components providing support for various processing elements and communication channels.', '1105.6010-2-16-9': 'Noticeable specializations of Cometeare: the STm8010(Traviata) board including three ST200 cores, the xStream many-cores streaming prototype, posix-compliant operating systems easing deployment over multi-threaded hardware, computers distributed over a TCP/IP network.', '1105.6010-2-17-0': '## Heptagon and BZR', '1105.6010-2-18-0': '### Heptagon language', '1105.6010-2-19-0': 'In this work, we use the language Heptagon/BZR [CITATION] .', '1105.6010-2-19-1': 'The Heptagon language allows to describe reactive systems by means of generalized Moore machines, i.e., mixed synchronous dataflow equations and automata [CITATION], with parallel and hierarchical composition.', '1105.6010-2-19-2': 'The basic behavior is that at each reaction step, values in the input flows are used in order to compute the values in the output flows for that step.', '1105.6010-2-19-3': "Inside the nodes, this is expressed as a set of declarations, which takes the form of equations defining, for each output, the values that the flow takes, in terms of an expression on other flows' instantaneous values, possibly using values computed in preceding steps (also known as state values).", '1105.6010-2-20-0': 'Figure [REF] shows a small program in this language.', '1105.6010-2-20-1': 'It describes the control of a task, which can either be idle or active.', '1105.6010-2-20-2': 'When it is idle, i.e., in the initial Idle state, then the occurrence of the input r requests the launch of the task.', '1105.6010-2-20-3': 'Another input c (which will be controlled further by the synthesized controller) can either allow the activation, or temporary block the request and make the automaton go to a waiting state.', '1105.6010-2-20-4': 'When active, the task can be ended with the input e.', '1105.6010-2-20-5': 'This delayable node has two outputs, a featuring the instantaneous activity of the task, and s being emitted on the instant when it becomes active: [EQUATION]', '1105.6010-2-21-0': '### BZR and controller synthesis', '1105.6010-2-22-0': 'BZR is an extension of Heptagon, allowing its compilation to involve discrete controller synthesis (DCS) using the DCS tool Sigali [CITATION].', '1105.6010-2-22-1': 'DCS allows to compute automatically a controller, i.e., a function which will act on the initial program so as to enforce a given temporal property.', '1105.6010-2-22-2': 'Concretely, the BZR language allows the declaration of controllable variables, which are not defined by the programmer.', '1105.6010-2-22-3': 'These free variables can be used in the program so as to let some choices undecided (e.g., choice between several transitions).', '1105.6010-2-22-4': 'The controller, computed by DCS, is then able to avoid undesired states of the application by setting and updating appropriate values for these variables at runtime.', '1105.6010-2-23-0': 'Figure [REF] shows an example of use of these controllable variables.', '1105.6010-2-23-1': 'It consists in two instances of the delayable node, as in Figure [REF].', '1105.6010-2-23-2': 'They run in parallel, defined by synchronous composition: one global step corresponds to one local step for every equation, i.e., here, for every instance of the automaton in the delayable node.', '1105.6010-2-23-3': 'Then, the twotasks node so defined is given a contract composed of two parts: the part allowing the declaration of controllable variables ([MATH] and [MATH]), and the part allowing the programmer to assert the property to be enforced by DCS, using the controllable variables.', '1105.6010-2-23-4': "Here, we want to ensure that the two tasks running in parallel won't be both active at the same time.", '1105.6010-2-23-5': 'Thus, [MATH] and [MATH] will be used by the computed controller to block some requests, leading automata of tasks to the waiting state whenever the other task is active.', '1105.6010-2-24-0': '### Heptagon/BZR compilation', '1105.6010-2-25-0': 'The compilation of a Heptagon/BZR program produces sequential code in a target general programming language (C, Java or Caml).', '1105.6010-2-25-1': 'This code takes the form of two functions (or methods), named reset and step.', '1105.6010-2-25-2': 'reset initializes the internal state of the program.', '1105.6010-2-25-3': 'The step function is evaluated at each logical instant to compute output values from an input vector and the current state, possibly updated.', '1105.6010-2-25-4': 'A typical way of using these functions is to enclose this step call in an infinite loop:', '1105.6010-2-26-0': 'current_state [MATH] reset() // state initialization for each step do inputs [MATH] gather current events outputs, next_state [MATH] step(inputs, current_state) handle outputs current_state [MATH] next_state', '1105.6010-2-27-0': 'Eventually, such infinite loop will not be as clearly stated, but hidden within, e.g., events managers, threads, interrupts, depending on the application context.', '1105.6010-2-27-1': 'In our context, we will use the C generated code.', '1105.6010-2-27-2': 'The API, for a node f with inputs [MATH] (typed [MATH]) and outputs [MATH] (typed [MATH]) is given below.', '1105.6010-2-27-3': 'The additional input mem is a pointer towards the internal state, to be used and updated.', '1105.6010-2-27-4': 'The result of the fstep call is a structure in which are placed the outputs values.', '1105.6010-2-28-0': 'void f_reset(f_mem* mem); f_res f_step( [MATH] , ..., [MATH] , f_mem* mem);', '1105.6010-2-29-0': '## System/manager Interaction', '1105.6010-2-30-0': 'Our approach applies to systems supporting dynamic reconfiguration and providing some events describing their observable states.', '1105.6010-2-30-1': 'It relies on the modeling of the system (and possibly the environment) by means of Heptagon automata and describing the control objectives.', '1105.6010-2-30-2': 'The BZR compilation tool-chain compiles the automata and the control objectives into a synchronous program which will be referred to as managerin the sequel.', '1105.6010-2-30-3': 'The managerencapsulates a model of the system and a controllerto enforce the objectives.', '1105.6010-2-31-0': 'Figure [REF] illustrates the use of a manager (i.e., synchronous program) to manage the reconfiguration of a system.', '1105.6010-2-31-1': 'The managerreceives some input events from the system and the environment.', '1105.6010-2-31-2': 'According to these inputs and the current state of the model, the controllermay update the state of the model.', '1105.6010-2-31-3': 'Internally, this reduces to fill the controllable variables with the appropriate computed values.', '1105.6010-2-31-4': 'The computation of the new state of the model corresponds to a step of the managerand may fire some commands.', '1105.6010-2-31-5': 'The commands are meant to reconfigure the managed system in order for this latter to be coherent with its model.', '1105.6010-2-32-0': '# The Comanche Http Server', '1105.6010-2-33-0': 'This section describes an example of a reconfigurable component-based software application written in Fractal together with the execution platform on which the application will be run.', '1105.6010-2-33-1': 'The complete system will be used as a case-study in order to introduce our approach for the management of reconfigurable systems in Section [REF].', '1105.6010-2-34-0': '## Components architecture', '1105.6010-2-35-0': 'Figure [REF] describes the architecture of a HTTP server written in Fractal.', '1105.6010-2-35-1': 'It is a variant of the Comanche server used as an example in tutorials.', '1105.6010-2-35-2': 'Incoming requests are received by the frontendcomponent, which transmits them to the analyzercomponent.', '1105.6010-2-35-3': 'The latter forward well-formed requests to the dispatcherwhich queries fileserver[MATH]or fileserver[MATH]to solve the requests.', '1105.6010-2-35-4': 'The analyzercomponent can also send requests to the logger to keep track of HTTP requests.', '1105.6010-2-36-0': 'The required components for this application to work are frontend, analyzer, dispatcherand fileserver[MATH].', '1105.6010-2-36-1': 'A first available degree of dynamical reconfiguration lies in fileserver[MATH]and logger.', '1105.6010-2-36-2': 'As illustrated by the dashed lines, fileserver[MATH]and loggermay be activated (resp., deactivated) and connected to (resp., disconnected from) the rest of the components.', '1105.6010-2-37-0': '## Execution architecture', '1105.6010-2-38-0': 'A second degree of reconfiguration concerns component mapping over processing elements.', '1105.6010-2-38-1': 'For our experiments, we used a server equipped with two Intel Xeon dual-core processors, each one running at 1.86GHz clock frequency.', '1105.6010-2-38-2': 'The four available cores enable the execution of four tasks in parallel.', '1105.6010-2-38-3': 'In the sequel, we will refer to each core as a processing element (pe[MATH], ..., pe[MATH]).', '1105.6010-2-39-0': 'Figure [REF] describes the initial deployment of Comanche components using Comete middleware (Section [REF]) on the execution platform.', '1105.6010-2-39-1': 'Each component is associated with a processing element, on which it will execute in order to handle messages in its associated FIFO.', '1105.6010-2-39-2': 'Component bindings are implemented by the Comete middleware as asynchronous communication channels.', '1105.6010-2-39-3': 'For the sake of clarity of the figure, the bindings are made implicit.', '1105.6010-2-39-4': 'Notice that the managerand Comete loader also execute on the same platform.', '1105.6010-2-40-0': '## Renconfiguration policy', '1105.6010-2-41-0': 'The control objectives we want to enforce are related to distinct aspects of the Comanche application and the execution platform.', '1105.6010-2-41-1': 'We are able to design a manager to fire reconfiguration commands in order to enforce the objectives.', '1105.6010-2-41-2': 'Comete together with Fractal API provide means to reflect manager commands on the managed system.', '1105.6010-2-41-3': 'Those means we are interested in are component migration provided by Comete API and lifecycle (resp., binding) control interface of the Fractal API in order to start/stop (resp., bind/unbind) components.', '1105.6010-2-41-4': 'In the following, we list the control objectives:', '1105.6010-2-42-0': '[Processing element availability] A processing element is a shared resource that may be unavailable for some reasons (e.g., energy saving, higher priority task, fault, etc.).', '1105.6010-2-42-1': 'We want that no component is running on an unavailable processing element (component migration).', '1105.6010-2-43-0': '[Workload balancing] There is a maximum workload, a processing element should not exceed.', '1105.6010-2-43-1': 'Migrate components in order to decrease the workload.', '1105.6010-2-44-0': '[Quality-of-Service] When the fileserver[MATH]is overloaded, start the fileserver[MATH]in order to keep the average response time low (lifecycle and binding control).', '1105.6010-2-45-0': '[Exclusiveness] The loggershould be started upon the request of the user, and then fileserver[MATH]should not be running (lifecycle and binding control).', '1105.6010-2-46-0': 'In order to ensure the objectives we described above, a manager is implemented and put in a closed-loop with the software application as described in section [REF].', '1105.6010-2-46-1': 'A simple scenario is as follows.', '1105.6010-2-46-2': 'The manager receives an event stating that the fileserver[MATH]is overloaded.', '1105.6010-2-46-3': 'It performs some computation and emits the command to start the fileserver[MATH]and bind it to the rest of the components.', '1105.6010-2-46-4': 'We measure the workload of fileserver[MATH]by comparing the size of the FIFO associated to it with a specific threshold.', '1105.6010-2-46-5': 'Next, the controller receives from the environment that the processing element pe[MATH]is no longer available.', '1105.6010-2-46-6': 'The controller reacts by emitting the command to migrate the components running on pe[MATH]to other processing elements, balancing the workload as described by the related property.', '1105.6010-2-47-0': '# Designing the manager', '1105.6010-2-48-0': 'We follow a modeling method for the design of a manager to enforce the properties we described above; we will elaborate on it by improving the treament of reconfiguration actions in Section [REF].', '1105.6010-2-48-1': 'The use of DCS ensures that the manager is correct with respect to the properties given as objectives.', '1105.6010-2-48-2': 'It mainly consists of the following steps: (1) Providing a synchronous model of the behavior of the reconfigurable components and the processing elements, giving the state space of the control problem; (2) Specifying the control objectives and identifying the controllable variables; (3) Compiling: controller synthesis and code generation.', '1105.6010-2-49-0': '## Modeling Components With Heptagon', '1105.6010-2-50-0': 'We adopt a modular modeling approach to enable reusing models.', '1105.6010-2-50-1': 'Moreover, we distinguish between the application models (i.e., software) and the execution platform (e.g., hardware) models.', '1105.6010-2-50-2': 'Instances of these models will be associated in the main synchronous program (see Section [REF]) to obtain the global model.', '1105.6010-2-50-3': 'Hence, the approach facilitates replacing hardware models without modifying software ones in case the application is ported to other execution platforms.', '1105.6010-2-51-0': '### Modeling Software Components', '1105.6010-2-52-0': 'Component Lifecycle', '1105.6010-2-53-0': 'Figure [REF]-(a) shows the automaton modeling a Fractal component lifecycle.', '1105.6010-2-53-1': 'A component may be in one of three possible states.', '1105.6010-2-53-2': 'A Running (R) component is connected to the other components and may handle incoming messages in its associated FIFO.', '1105.6010-2-53-3': 'A Stopped (S) component is disconnected from the others and can not execute to handle messages in its input FIFO.', '1105.6010-2-53-4': 'Finally, a component may be put in a Safe Stopping (SS) in order to handle the remaining messages in its FIFO before completely stopping.', '1105.6010-2-54-0': 'The automaton has three inputs (ch, fe and s for change, FIFOs empty and stop respectively), and two outputs run and disc.', '1105.6010-2-54-1': 'Initially, the component is stopped (i.e., state S).', '1105.6010-2-54-2': 'It goes to R(i.e., running) upon receiving ch.', '1105.6010-2-54-3': 'From this state, it goes to state SSupon receiving ch.', '1105.6010-2-54-4': 'At state SS, it goes to state Supon receiving fe which means that the input FIFOs are empty.', '1105.6010-2-54-5': 'At any state, receiving s forces the automaton to go to the state S.', '1105.6010-2-54-6': 'The two outputs of the automaton run and disc (for running and disconnect) take their values as described by the figure.', '1105.6010-2-54-7': 'These outputs tell whether a component is running (resp., disconnected) or not.', '1105.6010-2-55-0': 'As described by the transitions of the automaton, some reconfiguration commands (start, stop, connect, disconnect), represented as additional outputs, are fired upon changing state.', '1105.6010-2-55-1': 'These commands are meant to reconfigure the related component in order to be coherent with its lifecycle model.', '1105.6010-2-55-2': 'For instance the transition from Sto R, outputs the commands start and connect in order to start the component and connect it to the rest of the components.', '1105.6010-2-56-0': 'FIFO State', '1105.6010-2-57-0': 'With each server interface of a component is associated a FIFO.', '1105.6010-2-57-1': 'The FIFO stores the input events before they are handled by the component.', '1105.6010-2-57-2': 'Figure [REF]-(b) models the state of such FIFOs.', '1105.6010-2-57-3': 'It is a two state automaton with one input and one output (f and full respectively).', '1105.6010-2-57-4': 'Initially, the FIFO is empty.', '1105.6010-2-57-5': 'It goes to the state FF (resp., FE) upon receiving f (resp., [MATH]f).', '1105.6010-2-57-6': 'The value output full indicates the state of the FIFO: true (resp., false) means that the FIFO size is above (resp., below) a given threshold.', '1105.6010-2-58-0': '### Modeling Hardware Components', '1105.6010-2-59-0': 'Component mapping on Processing Elements', '1105.6010-2-60-0': 'Figure [REF] models the mapping of a component on the available processing elements.', '1105.6010-2-60-1': 'It is a four state automaton.', '1105.6010-2-60-2': 'Each state represents mapping on one processing element, on which a component may run.', '1105.6010-2-60-3': 'It may change from one state to another depending on the Boolean inputs a and b.', '1105.6010-2-60-4': 'That is, each state has three outgoing transitions (one to each remaining state).', '1105.6010-2-60-5': 'The transition to take depends on the value of a and b (two Boolean inputs to encode four possibilities).', '1105.6010-2-60-6': 'For the sake of clarity, not all of the transitions are present in the figure.', '1105.6010-2-60-7': 'The output of the automaton is of enumerated type; it takes its values in pe[MATH], pe[MATH], pe[MATH], pe[MATH] depending on the state of the automaton.', '1105.6010-2-61-0': 'The transitions of the automaton are associated with outputs (mig[MATH], mig[MATH], mig[MATH], mig[MATH]).', '1105.6010-2-61-1': 'They are associated to migration commands, in order to migrate the corresponding component to its new processing element.', '1105.6010-2-62-0': 'Processing Element Availability', '1105.6010-2-63-0': 'We need a model of the availability of a processing element in order to know whether a component may run on it or not.', '1105.6010-2-63-1': 'Figure [REF] describes such a model.', '1105.6010-2-63-2': 'It is a two states automaton with one input dis (for disable) and one output on which tells on the availability of the processing element.', '1105.6010-2-63-3': 'Initially, a processing element is available (i.e., at state ON).', '1105.6010-2-63-4': 'It goes to the state OFF(resp., ON) upon receiving dis (resp., [MATH]dis).', '1105.6010-2-63-5': 'The output on takes its value depending on the state of the automaton.', '1105.6010-2-64-0': 'Processing Element Workload', '1105.6010-2-65-0': 'The workload of a pe[MATH]depends on the components running on it and their bindings to the other components.', '1105.6010-2-65-1': 'Two components [MATH] and [MATH] (client and server), bound through their respective interfaces [MATH] and [MATH] induce some workload on their associated processing elements pe[MATH]and pe[MATH].', '1105.6010-2-65-2': 'Based on some benchmarks, we estimate the workload related to pe[MATH]and pe[MATH]as described by the following equations:', '1105.6010-2-66-0': 'a) [MATH].', '1105.6010-2-66-1': 'b) [MATH].', '1105.6010-2-67-0': 'The workload of the processing element running the server component [MATH] depends on the cost [MATH] of the executed function, and the function call frequency [MATH].', '1105.6010-2-67-1': 'The processing element running the client component is charged with the communication effort.', '1105.6010-2-67-2': 'That is, the processing element workload depends on the call frequency and a parameter [MATH] which depends on the position of the two communicating processing elements.', '1105.6010-2-67-3': 'In our case-study, the processing elements are processor cores.', '1105.6010-2-67-4': 'The communication uses cache memories of distinct levels.', '1105.6010-2-67-5': 'The values we associate to [MATH] represent the fact that the communication cost differs depending on the type of cache used between the processing elements.', '1105.6010-2-67-6': 'These values are: [MATH] in case pe[MATH]and pe[MATH]refer to the same core, [MATH] in case the processing elements are distinct cores of the same processor, [MATH] in case the two processing elements refer to distinct cores of distinct processors.', '1105.6010-2-68-0': 'In Listing [REF], we give the interface of the node cost implementing workload equations.', '1105.6010-2-68-1': 'For a given binding between two components, the node takes as input the position of the client and the server component, the frequency [MATH] of the calls between them and the cost of the function executed by the server component.', '1105.6010-2-68-2': 'The outputs are integer values associating workloads induced by the two components with the available processing elements.', '1105.6010-2-68-3': 'The involved processing elements will be associated with positive integer values, the others will take the value zero.', '1105.6010-2-69-0': '## Complete system model', '1105.6010-2-70-0': 'The main program (partially described in listing of Listing [REF]) consists of the synchronous composition of the automata modeling each component (hardware and software).', '1105.6010-2-70-1': 'The inputs of the main node consist of the events stating on: (pes1) the availability of the processing element pe[MATH]; (addL) the request of the logger; (fS1, fS2, fL) the load of the FIFO associated with fileserver[MATH], fileserver[MATH]and loggerrespectively.', '1105.6010-2-71-0': 'The outputs of the main program correspond to the reconfiguration/migration commands fired by the automata.', '1105.6010-2-71-1': 'For instance migLp0 is fired by the automaton modeling the position of the component logger(line 22).', '1105.6010-2-71-2': 'When migLp0 takes the value true, the component loggershould be migrated to pe[MATH].', '1105.6010-2-71-3': 'The output startS2 is fired by the automaton modeling fileserver[MATH]lifecycle (line 17).', '1105.6010-2-71-4': 'When it is true, it states that the component should be started.', '1105.6010-2-72-0': 'The main program is composed of: An instance of node fifo for each server interface of the components (lines 11-13).', '1105.6010-2-72-1': 'An instance of node lifecycle for each component (lines 15-17).', '1105.6010-2-72-2': 'An instance of processing element availability for each processing element (lines 19-20).', '1105.6010-2-72-3': 'An instance of node position for each component (lines 22-24).', '1105.6010-2-72-4': 'An instance of node cost for each binding between components (lines 26-28).', '1105.6010-2-72-5': 'Finally, the cost imposed on each processing element corresponds to the sum of all the cost induced by the bindings (lines 29-30).', '1105.6010-2-73-0': 'numbers=left,numberstyle=,stepnumber=2,numbersep=5pt', '1105.6010-2-74-0': '## Describing Control Objectives with BZR', '1105.6010-2-75-0': 'We now have a complete model of the possible behaviors of the system, in the absence of control.', '1105.6010-2-75-1': 'We want to obtain a controller that will enforce the policy given informally in Section [REF].', '1105.6010-2-75-2': 'We do this in the form of a contract for each of the points of the policy as follows.', '1105.6010-2-76-0': 'Processing element availability We want no component running on an unavailable processing element: this can be achieved by component migration.', '1105.6010-2-77-0': 'The corresponding expression in the contract to be enforced i.e., to be controlled for invariance, is: [EQUATION]', '1105.6010-2-77-1': 'The involved controllable variables defined in the part will be, for each component:', '1105.6010-2-78-0': 'a[MATH],', '1105.6010-2-79-0': 'b[MATH], [MATH].', '1105.6010-2-79-1': 'The effect of the control will be that, when some processing element becomes unavailable, appropriate migrations will be fired.', '1105.6010-2-79-2': 'More precisely: upon reception of input pes[MATH] at value , the availability model makes a transition as shown in Figure [REF].', '1105.6010-2-79-3': 'In the global model, a transition will be taken to a next state where the above expression is .', '1105.6010-2-79-4': 'For components on the unavailable PE, the automata shown in Figure [REF] can not stay in the current state without violating the property, therefore controllables will take a value such that a local transition occurs, hence firing a migration action towards another available PE.', '1105.6010-2-80-0': 'Workload balancing Workload on each PE[MATH] is bounded by [MATH]: this is achieved by component migration.The expression is:', '1105.6010-2-81-0': '[MATH] .', '1105.6010-2-82-0': 'The involved controllables are:', '1105.6010-2-83-0': 'a[MATH],', '1105.6010-2-84-0': 'b[MATH], [MATH].', '1105.6010-2-84-1': 'The effect of the control will be that, if some migration or starting of a component happens on a PE, the choice encoded by controllables will be between PEs for which this addition would not violate the bound.', '1105.6010-2-85-0': 'Quality-of-Service When the fileserver[MATH]is overloaded, start the fileserver[MATH]: this is achieved by lifecycle and binding control.', '1105.6010-2-85-1': 'The expressions are:', '1105.6010-2-86-0': 'qos= [MATH] .', '1105.6010-2-87-0': 'The involved controllables are: sS2, chS2.', '1105.6010-2-87-1': 'The effect of the control will be that, if the FIFO of fileserver[MATH]reaches its threshold, the fileserver[MATH]is started unless the Logger runs; and when the FIFO goes back under the threshold, it is disconnected, and eventually stopped.', '1105.6010-2-88-0': "Exclusiveness When the loggerruns, fileserver[MATH]doesn't: this is achieved by lifecycle and binding control.", '1105.6010-2-88-1': 'The corresponding expression is: [EQUATION]', '1105.6010-2-88-2': 'The involved controllable is: sS2.', '1105.6010-2-88-3': 'The effect of the control is that, when the Logger is started, the fileserver[MATH]must be stopping or idle.', '1105.6010-2-89-0': '# Manager Integration', '1105.6010-2-90-0': 'The C code generated by the BZR compilation tool-chain consists of two functions step(...) and reset(...).', '1105.6010-2-90-1': 'Listing [REF] is a sketch of a program using these functions in order to manage the reconfiguration of a system.', '1105.6010-2-90-2': 'At line 2, reset() is called once to initialize the memory of the program.', '1105.6010-2-90-3': 'The piece of code ranging from line 5 to line 10 is made sensitive to incoming events; i.e., this part of the code is executed each time new events are received.', '1105.6010-2-90-4': 'The function prepareevents() prepares the events and provides the inputs for the function step().', '1105.6010-2-90-5': 'The signature of the function step() corresponds to the one of the main program of Listing [REF].', '1105.6010-2-90-6': 'A call to this function corresponds to a step of the synchronous program.', '1105.6010-2-90-7': 'The outputs of step() are given to the function generatecommands() in order to translate the outputs into the consequent reconfiguration commands.', '1105.6010-2-91-0': 'numbers=left,numberstyle=,stepnumber=1,numbersep=5pt, basicstyle=', '1105.6010-2-92-0': '## Wrapping the manager into a component', '1105.6010-2-93-0': 'The functions listed in Listing [REF] are wrapped into a Fractal component in order to be connected to the rest of application and middleware components (see Figure [REF]).', '1105.6010-2-93-1': 'This component provides one server interface Events and one client interface Commands.', '1105.6010-2-93-2': 'The signature of the interface Events is described in Listing [REF].', '1105.6010-2-93-3': 'The method setEvent provided by this interface enables components to register event values.', '1105.6010-2-93-4': 'Indeed, the manager hides a buffer containing the last value registered of each event.', '1105.6010-2-93-5': 'The function prepareevents() uses this buffer to provide the inputs for the function step().', '1105.6010-2-94-0': 'The function generatecommands() translates the outputs of the function step() into method calls through the Commands interface.', '1105.6010-2-94-1': 'Indeed, the signature of the required interface Commands is that of ComponentManager provided by Comete API (see Listing [REF]).', '1105.6010-2-94-2': 'It provides methods for component migration, reconfiguration and lifecycle management.', '1105.6010-2-95-0': 'numbers=none', '1105.6010-2-96-0': '## Concrete integration of the manager', '1105.6010-2-97-0': 'Figure [REF] describes the concrete integration of the manager together with the application and Comete middleware components.', '1105.6010-2-97-1': 'The component user is connected to the Events interface of the manager.', '1105.6010-2-97-2': 'It intercepts user requests (add logger for instance) and registers them as events.', '1105.6010-2-97-3': 'The FIFO components (part of Comete) are made implicit in the figure.', '1105.6010-2-97-4': 'As illustrated by the dotted lines, the FIFOs associated with fileserver[MATH], fileserver[MATH]and loggerare connected to the Events interface of the manager in order to register events related to their size.', '1105.6010-2-98-0': 'Upon receiving events, the manager performs some computation and fires some reconfiguration commands in the form of method calls.', '1105.6010-2-98-1': 'These method calls are handled by the loader of Comete.', '1105.6010-2-98-2': 'Indeed, as illustrated by the dashed lines, the loader is connected to the control interface of the application components in order to start/stop bind/unbind them.', '1105.6010-2-98-3': 'Moreover, the loader knows how migrating components should be achieved regarding the execution platform.', '1105.6010-2-99-0': 'Comete middleware hides complex mechanisms and provides a simple API to manage component lifecycle, migration, and architecture transparently, whatever the execution platform is.', '1105.6010-2-99-1': 'For that purpose, we adopted a centralized version of the manageremitting reconfiguration commands to Comete instead of application components.', '1105.6010-2-99-2': 'We could make the managercommunicate directly with the components (modulo re-engineering Comete) to enable a distributed implementation of the manager.', '1105.6010-2-99-3': 'This means that parts of the managerwould be integrated into Fractal component membranes.', '1105.6010-2-100-0': 'In this section we presented a simple version of the use of a synchronous manager for the reconfiguration of a parallel and asynchronous application.', '1105.6010-2-100-1': 'The reconfiguration commands were considered to be short enough to complete before the next reaction of the manager.', '1105.6010-2-100-2': 'However, this is not always the case.', '1105.6010-2-100-3': 'Indeed, some reconfiguration commands may take non negligible time to complete.', '1105.6010-2-100-4': 'During this time interval, the system model inside the manager does not reflect the actual state of the system.', '1105.6010-2-100-5': 'For instance, when the command to start a component C[MATH] is fired, the model of C[MATH] is at state Running (R) but the actual component is not yet started.', '1105.6010-2-100-6': 'The manager reaction to other incoming events during reconfiguration progress may lead the managed system to undesirable states.', '1105.6010-2-100-7': 'Next section proposes an extension to our modeling approach in order to solve issues related to asynchronous commands.', '1105.6010-2-101-0': '# Asynchronous commands', '1105.6010-2-102-0': 'We propose a controller architecture together with some guidelines for writing synchronous models in order to overcome the issues related to the transient incoherence of the model with respect to the state of the system.', '1105.6010-2-102-1': 'Our approach relies on the explicit representation of the states where some commands are being processed but not completed, together with some synchronization mechanisms.', '1105.6010-2-103-0': 'The purpose of identifying the states where some reconfiguration commands are in progress is to make it possible for the programmer to decide what should happen during reconfigurations.', '1105.6010-2-103-1': 'The programmer should rewrite objectives properties taking into account these situations.', '1105.6010-2-104-0': '## Modeling command execution', '1105.6010-2-105-0': '### Behavioral models', '1105.6010-2-106-0': 'Some synchronous languages (e.g., Esterel) provide built-in constructs in order to perform asynchronous calls within a synchronous program [CITATION].', '1105.6010-2-106-1': 'We follow the same principle of such constructs and model the asynchronous aspect of reconfiguration commands by means of an automaton.', '1105.6010-2-107-0': 'Such an automaton is illustrated by Figure [REF].', '1105.6010-2-107-1': 'At state P(for pending), the command is emitted but not yet completed.', '1105.6010-2-107-2': 'At state D, the command has finished.', '1105.6010-2-107-3': 'The automaton has two inputs do and done.', '1105.6010-2-107-4': 'The output pending tells whether the command is pending or not.', '1105.6010-2-107-5': 'From the initial state, upon receiving do which corresponds to the firing of the command, the automaton goes to state P.', '1105.6010-2-107-6': 'The automaton stay at state Puntil receiving the input done.', '1105.6010-2-107-7': 'The input done signals that the command has been completed.', '1105.6010-2-108-0': 'We associate an instance of such an automaton with each asynchcronous command.', '1105.6010-2-108-1': 'Hence the modification w.r.t. model of Section [REF] is systematic and modular.', '1105.6010-2-109-0': '### Objectives and contracts', '1105.6010-2-110-0': 'Being aware of the state where a command is in progress but not finished, one can add some properties to be enforced and modify the previous ones in order to decide what should happen during reconfigurations.', '1105.6010-2-110-1': 'For illustration purpose, we consider the same program as in Section [REF] with the command startS2 as the only command taking non-negligible time to complete.', '1105.6010-2-110-2': 'In the following fragment of program, we associate an instance of the node command to startS2 (line 5) and modify the contract:', '1105.6010-2-111-0': 'numbers=left,numberstyle=,stepnumber=1,numbersep=5pt', '1105.6010-2-112-0': 'The contract changes as follows:', '1105.6010-2-113-0': 'The properties qos and exc are now dependent to the pending of the command.', '1105.6010-2-113-1': 'pensS2[MATH](qos exc) tells that qos and exc may not be enforced when the command startS2 is in progress.', '1105.6010-2-113-2': 'But, they must be enforced once the command completes.', '1105.6010-2-113-3': 'pend (defined at line 6) is a new property.', '1105.6010-2-113-4': 'It forbids the controller to modify the state of the fileserver[MATH]when the command to start it is in progress.', '1105.6010-2-113-5': 'This property is explained below.', '1105.6010-2-114-0': 'In order to understand the property pend, consider the automata modeling fileserver[MATH]lifecycle and startS2 command execution in Figure [REF].', '1105.6010-2-114-1': 'Adding the property pend = (pensS2 discS2) forces the controller to avoid the states where the command is pending (i.e., at state p) and any state where discS2 is true (i.e., at states s and ss).', '1105.6010-2-114-2': 'That is, to avoid the states (ss,p) and (s,p) in the product of the two automata.', '1105.6010-2-115-0': 'At the beginning the automata are at state (s,d).', '1105.6010-2-115-1': 'Upon receiving ch, automaton (a) changes states and fires the command startS2 which makes automaton (b) change state at the same instant.', '1105.6010-2-115-2': 'That is, the global state is (r,p).', '1105.6010-2-115-3': 'Unless receiving done, which is uncontrollable, the controller will not make automaton (a) change state in order not to violate the property pend.', '1105.6010-2-116-0': '## Controller Architecture', '1105.6010-2-117-0': 'The actual implementation of the manager is component-based.', '1105.6010-2-117-1': 'Figure [REF] describes the internals of such a component.', '1105.6010-2-117-2': 'It consists of three subcomponents: evt, ctrl, and cmd.', '1105.6010-2-117-3': 'The functions prepareevents(), reset(), step(), and generatecommands() are spread over the subcomponents as follows: evtencapsulates prepareevents().', '1105.6010-2-117-4': 'It prepares the inputs to ctrlthat encapsulates reset() and step() generated by the BZR compilation.', '1105.6010-2-117-5': 'The component cmdtranslates the outputs of ctrlinto reconfiguration methods calls.', '1105.6010-2-118-0': 'The component evtis in charge of implementing the method provided by the interface Events from which components and the environment may register events.', '1105.6010-2-118-1': 'The component cmdis in charge of calling reconfiguration methods through Commands interface.', '1105.6010-2-118-2': 'Moreover, each time a reconfiguration command is completed, cmdnotices done events related to this command.', '1105.6010-2-118-3': 'That is why cmdis bound to evt.', '1105.6010-2-119-0': '# Related work', '1105.6010-2-120-0': 'A lot of work has been devoted to dynamic reconfiguration of component-based software systems.', '1105.6010-2-120-1': 'In the case of Fractal, one can refer to the tools for component introspection and the languages for specifying reconfigurations [CITATION] or integration in parallel frameworks [CITATION].', '1105.6010-2-121-0': 'Our work fits in the context of applying formal methods for dynamic reconfiguration [CITATION] in order to ensure properties related to components and reconfigurations.', '1105.6010-2-121-1': 'In [CITATION] the authors specify reconfiguration properties by means of temporal logic in order to apply model checking or runtime monitoring of system reconfigurations.', '1105.6010-2-121-2': 'In comparison, our approach benefits from the discrete controller synthesis which provides a correct by construction manager ensuring properties on components and reconfigurations.', '1105.6010-2-121-3': 'In [CITATION], the authors give a general software engineering framework, with some indications of integration, but no complete implementation.', '1105.6010-2-121-4': 'In comparison, we perform a concrete integration of synchronous managers with Fractal, in the form of its Cecilia programming environment, and the Comete middleware.', '1105.6010-2-121-5': 'In addition, we consider and treat the case of asynchronous reconfiguration actions.', '1105.6010-2-122-0': 'Other related work concern the concrete integration of synchronous programs for the management of asynchronous systems.', '1105.6010-2-122-1': 'In [CITATION], the authors provides a synchronous controller for configuring device drivers aiming at global power management of embedded systems.', '1105.6010-2-122-2': 'Apart from their not using DCS techniques, the difference with our approach lies in the call of reconfiguration functions.', '1105.6010-2-122-3': 'Indeed, in [CITATION], function calls are performed inside the reaction of the controller.', '1105.6010-2-122-4': 'This has the benefit of always keeping a model reflecting the exact state of the system.', '1105.6010-2-122-5': 'In our case we use to follow the principle of asynchronous function calls as in [CITATION].', '1105.6010-2-123-0': '# Conclusion', '1105.6010-2-124-0': 'Our contribution is on the one hand a synchronous model of the behavior of the reconfigurable components, in the form of the state space of the control problem, and the specification of the control objectives (Section [REF]).', '1105.6010-2-124-1': 'On the other hand, we contribute a component-based architecture for the implementation of the resulting controller with the Fractal/Cecilia programming environment, taking advantage of the Comete middleware.', '1105.6010-2-124-2': '(Sections [REF] and [REF])', '1105.6010-2-125-0': 'In this work, we apply formal techniques issued from academic research to an open, stable and available framework supported by an independent industry consortium [CITATION].', '1105.6010-2-125-1': 'The Fractal model is currently being implemented in the MINDproject [CITATION], a sequel to the Cecilia framework.', '1105.6010-2-125-2': 'The MINDframework is a collaborative initiative to spread component-based software engineering to an even larger community of academic institutions and industries.', '1105.6010-2-125-3': 'The present contribution keeps its legitimacy in the MINDcontext as most of the concepts and semantics introduced with Cecilia and Comete remain.', '1105.6010-2-125-4': 'Therefore, applying our contribution to MINDshould be limited to a straightforward adaptation .', '1105.6010-2-126-0': 'We have ongoing work on another interesting case study: an H264 video processing application, implemented on a multicore architecture using Cecilia and Comete.', '1105.6010-2-126-1': 'It could be handled following the very same methodology we propose, because it follows essentially the same structure: degrees of reconfiguration concern migrations, and adding and removal of components performing video effects on the stream.', '1105.6010-2-127-0': 'Perspectives are in the line of generalizing our proposal at the language-level, by extending the Fractal ADL with a way to incorporate automata notation for reconfiguration description, as well as the behavioral contracts, and integrating the application of BZR in a global Fractal compilation flow.', '1105.6010-2-127-1': 'Also, the integration of synchronous controllers in a Fractal components architecture should be facilitated by identifying general programming guidelines, providing end-users with informal rules such that components are controllable, and the synchronous instant and step (and the states in the automata) fits well with the event granularity.', '1105.6010-2-128-0': '# Complete Example and Integration', '1105.6010-2-129-0': 'In what follows, we describe the complete integration process of the code produced by the bzr compilation for the control of the comanche application software.', '1105.6010-2-129-1': 'Writing the synchronous model of the manager has been discussed above.', '1105.6010-2-129-2': 'Here we give more details providing simulation results of the manager and the process of integration in the Fractal application.', '1105.6010-2-130-0': '# Simulation of the Synchronous Program', '1105.6010-2-131-0': 'The interface of the synchronous program is described in Listing [REF].', '1105.6010-2-131-1': 'The input disable states on the availability of the processing element PE[MATH].', '1105.6010-2-131-2': 'This input takes the value false as long as PE[MATH] is available (false otherwise).', '1105.6010-2-131-3': 'fifoH1F, fifoH2F, fifoL2F are Booleans stating on the size of the FIFOs associated with FileServer1, FileServer2 and Logger respectively.', '1105.6010-2-131-4': 'They take the value false as long as the size of the associated FIFOs is under their specific threshold.', '1105.6010-2-131-5': 'The input addlog tells whether the logger is required or not.', '1105.6010-2-131-6': 'The input cstartH2done notifies that the command for starting FileServer2 has been actually completed.', '1105.6010-2-132-0': 'The outputs of the main program are associated with the possible commands provided by the reconfigurable system.', '1105.6010-2-132-1': 'An output associated with a command takes the value false until the command has to be fired.', '1105.6010-2-132-2': 'Then, it takes the value true during one step.', '1105.6010-2-133-0': 'Figure [REF] illustrates part of the simulation results.', '1105.6010-2-133-1': 'The inputs (in blue color) are given (by the user) in order to simulate the events received from the application.', '1105.6010-2-133-2': 'The outputs (in red color) corresponding to reconfiguration commands take the value true each time a command has to be fired.', '1105.6010-2-134-0': 'The interaction scenario as illustrated by the figure is as follows:', '1105.6010-2-135-0': '## BZR Program Compilation into C Code', '1105.6010-2-136-0': 'The BZR compilation tool chain compiles the synchronous program into C code.', '1105.6010-2-136-1': 'The generated functions of our interest are the reset() and step() functions of the main program.', '1105.6010-2-136-2': 'reset() initializes the memory associated with the synchronous program.', '1105.6010-2-136-3': 'The function step() computes the outputs of the program depending on the inputs.', '1105.6010-2-136-4': 'It also updates the internal memory.', '1105.6010-2-137-0': 'The prototype of the function step() is presented in Listing [REF].', '1105.6010-2-137-1': 'It has the same input parameters as the main function of the synchronous program in Listing [REF].', '1105.6010-2-137-2': 'The parameter self refer to the memory of the complete program.', '1105.6010-2-137-3': 'The returned value of the function step() is of type mainres it is a structure.', '1105.6010-2-137-4': 'Each field of this structure corresponds to an output of the main program.', '1105.6010-2-138-0': '# Integration of the step() function', '1105.6010-2-139-0': 'The integration process of a synchronous manager for the control of reconfiguration lies in the integration of the step() function generated by the synchronous compilation.', '1105.6010-2-139-1': 'In order to achieve this, we have to make some choices.', '1105.6010-2-139-2': 'In particular, we need to decide when the function should be called, and how to apply reconfiguration commands:', '1105.6010-2-140-0': 'Our choice for this case study is to consider event-triggered calls of the step together with asynchronous firing of commands.', '1105.6010-2-140-1': 'The choice may be discussed.', '1105.6010-2-140-2': 'But this is out of the scope of the paper.', '1105.6010-2-141-0': '## Wrapping The Generated Code Into a Fractal Component', '1105.6010-2-142-0': 'The C generated code is wrapped into a Fractal component in order to receive application events and fire reconfiguration commands.', '1105.6010-2-142-1': 'Figure [REF] describes the internals of such a component and the type of the provided (resp. required) interface of the components.', '1105.6010-2-142-2': 'Notice that two interfaces may be bound if an only if they are of the same type.', '1105.6010-2-143-0': '### The Component EventReceiver', '1105.6010-2-144-0': 'Listing [REF] illustrates part of the implementation of the method setValue of the EventReceiver component.', '1105.6010-2-144-1': 'This method is called by other components in order to notify events.', '1105.6010-2-145-0': 'The component manages an array of Boolean values (DATA.stepinputs).', '1105.6010-2-145-1': 'Each cell of the array corresponds to one input event.', '1105.6010-2-145-2': 'Each time an event is notified through this method, its value is updated, then the method step is called.', '1105.6010-2-145-3': 'This method is implemented by the component SynchronousProgram.', '1105.6010-2-145-4': 'The parameter of the call is the array of event values.', '1105.6010-2-146-0': '### The Component SynchronousProgram', '1105.6010-2-147-0': 'Listing [REF] illustrates the code associated with the method step of the component SynchronousProgram.', '1105.6010-2-147-1': 'This method contains a call to the step method actually generated by the BZR compilation (at line 7).', '1105.6010-2-147-2': 'The result of the synchronous step (i.e., DATA.res) are forwarded to the CommandsGenerator in order to translate them into the consequent reconfiguration commands.', '1105.6010-2-148-0': '### The Component CommandsGenerator', '1105.6010-2-149-0': 'Listing [REF] illustrates the implementation of the method doAction of the component CommandsGenerator.', '1105.6010-2-149-1': 'It receives a structure containing the Boolean values associated with the possible commands.', '1105.6010-2-149-2': 'This method parses the received values and fire a command each time the associated output variable has the value true.', '1105.6010-2-150-0': '# Retrieving Application/Environment Events', '1105.6010-2-151-0': 'As explained above, the component EventReceiver of the manager offers a method for event notification.', '1105.6010-2-151-1': 'The components in charge of notifying these events are connected to this interface.', '1105.6010-2-151-2': 'Among the application components, only the FIFOs associated with fileserver[MATH], fileserver[MATH]and loggerare supposed to send events to the manager.', '1105.6010-2-151-3': 'Moreover, there is a component modeling the user and the environment.', '1105.6010-2-151-4': 'The component is in charge of notifying the events related to adding the logger and the availability of a processing element.', '1105.6010-2-151-5': 'These components call the method setValue() of the component EventReceiver whenever a particular condition holds.', '1105.6010-2-151-6': 'Listing [REF] is a sample piece of program for notifying events.', '1105.6010-2-151-7': 'The condition depends on the nature of the event to be notified.', '1105.6010-2-151-8': 'In case of the FIFOs, the condition concerns the size of the FIFO regarding to its associated threshold.'}

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[]

[]

[]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1105.6010

1406.7815

{'1406.7815-1-0-0': 'We introduce a general expression that quantifies the entanglement production rate in continuous variable systems in which entanglement depends on frequency.', '1406.7815-1-0-1': 'To exemplify its meaning and potential, we apply it to a four-mode optomechanical setup that enables the simultaneous up- and down-conversion of photons from a drive laser into entangled photon pairs.', '1406.7815-1-0-2': 'This setup is efficient in that both the drive and the optomechanical up- and down-conversion can be fully resonant.', '1406.7815-1-1-0': 'Introduction.', '1406.7815-1-1-1': 'Entanglement is an essential feature of quantum mechanics and a crucial resource for quantum communication and information processing.', '1406.7815-1-1-2': 'Although many quantum information protocols exploit qubits, with their discrete state space, the original Einstein-Podolsky-Rosen [CITATION] entanglement involves continuous variables (CV), and CV entanglement has many modern applications [CITATION].', '1406.7815-1-2-0': 'To enhance the generation of CV-entangled radiation beams, resonant modes can be exploited (e.g. [CITATION] in the microwave domain, or [CITATION] in the optical domain).', '1406.7815-1-2-1': 'An essential feature of such setups is that the output spectrum is peaked, and strong entanglement may persist only for narrow frequency intervals.', '1406.7815-1-2-2': 'This raises the general question how to characterize the overall rate of CV entanglement generation.', '1406.7815-1-3-0': 'In this work, we introduce an "entanglement rate" to quantify CV entanglement in such settings.', '1406.7815-1-3-1': 'Afterwards, we illustrate this measure by applying it to a four-mode optomechanical setup that allows the fully resonant, and thereby efficient, generation of entanglement.', '1406.7815-1-4-0': 'The rapidly developing field of cavity optomechanics focuses on the dynamics of photons and phonons coupled via radiation forces, see [CITATION] for a recent review.', '1406.7815-1-4-1': 'The optomechanical interaction has been predicted to produce entanglement, e.g. between optics and mechanics [CITATION] or between light modes [CITATION].', '1406.7815-1-4-2': 'Recently, optomechanical entanglement was demonstrated experimentally for the first time, in a microwave circuit [CITATION], making its analysis especially timely.', '1406.7815-1-5-0': 'The entanglement rate.', '1406.7815-1-5-1': 'For any source of CV-entangled beams, we can imagine to apply a frequency filter to each of the beams.', '1406.7815-1-5-2': 'Subsequently, the entanglement between filtered modes of the two beams can be calculated, e.g. using the logarithmic negativity [CITATION] as an entanglement measure.', '1406.7815-1-5-3': 'This filtering analysis has been applied to several settings [CITATION], for a detailed explanation see e.g. [CITATION].', '1406.7815-1-6-0': 'However, if the source emits only in a narrow bandwidth (Fig. [REF]), it can be argued that while the entanglement between individual wave packets may be large, the rate for emitting these is small, producing only a limited total amount of entanglement per unit time.', '1406.7815-1-6-1': 'We thus face the question of how to quantify the entanglement rate for a source with an arbitrary spectrum.', '1406.7815-1-6-2': 'This will allow us to optimize that rate, which, depending on the intended application, may actually be the prime figure of merit.', '1406.7815-1-7-0': 'A simple approach consists in defining the temporal extent [MATH] of the filter function, quantifying the entanglement [MATH], and then calculating [MATH].', '1406.7815-1-7-1': 'This was employed recently to analyze the first experiment on spatially separated CV entanglement in superconducting circuits [CITATION].', '1406.7815-1-7-2': 'However, examples show that this does not always cover the full entanglement (especially when several spectral peaks are involved) and there is some arbitrariness in the choice of filter function and the definition of [MATH].', '1406.7815-1-7-3': 'We now turn to introducing a general definition that remedies those problems.', '1406.7815-1-8-0': 'The natural way to make these notions precise is by using the Wannier-basis of wave packets [MATH] that live on a regular grid both in time, [MATH], and in frequency-space, [MATH], where [MATH].', '1406.7815-1-8-1': 'The Fourier-transform [MATH] of these wave packets is nonzero only in the interval [MATH], where [MATH].', '1406.7815-1-8-2': 'The [MATH], and therefore also the [MATH], form an orthonormal basis.', '1406.7815-1-8-3': 'That makes it possible to decompose uniquely the output field into modes defined by these wave packets: [MATH], where [MATH] annihilates a photon in mode [MATH], and [MATH].', '1406.7815-1-8-4': 'This construction can be applied to each of the entangled beams [CITATION].', '1406.7815-1-9-0': 'We now calculate the logarithmic negativity [MATH] between two wave packets of the two output fields, at the same time slot [MATH] and in suitable frequency slots.', '1406.7815-1-9-1': 'In a typical application with a stationary-state source producing beams via parametric down-conversion or four-wave mixing, energy conservation dictates that pairs of frequencies [MATH] and [MATH] are entangled only when they obey [MATH].', '1406.7815-1-10-0': 'To prepare for our definition of the entanglement rate, we have to discuss the dependence of [MATH] on the filter time [MATH], especially for the limit [MATH], where [MATH] is much longer than the physical correlation time [MATH] of the source ([MATH] is the width of the spectral peaks).', '1406.7815-1-10-1': 'First, this ensures that there will be no correlations between wave packets located at different time slots, such that we capture the full amount of entanglement.', '1406.7815-1-10-2': 'Second, we will now explain in this wave packet picture why [MATH] tends to a well-defined limit for [MATH], which is consistent with direct calculations[CITATION].', '1406.7815-1-11-0': 'Consider enlarging the filter time [MATH] by a factor [MATH], shrinking the filter frequency interval by [MATH].', '1406.7815-1-11-1': 'In effect, the new wave packets of size [MATH] encompass [MATH] of the old wave packets.', '1406.7815-1-11-2': 'That this coarse-graining keeps the correlators, and thus the entanglement [MATH], unchanged can be understood already from a very simple consideration.', '1406.7815-1-11-3': 'Take a suitably normalized sum of [MATH] operators, [MATH] and likewise [MATH].', '1406.7815-1-11-4': 'Then the correlator between these "averaged" operators will equal the original correlator: [MATH].', '1406.7815-1-11-5': 'This holds provided there have been no cross-correlations and [MATH] is independent of [MATH].', '1406.7815-1-11-6': 'The same logic applies to our case, where the new modes are properly normalized averages over the old modes: [MATH], with [MATH] encoding the overlap between the two basis sets.', '1406.7815-1-11-7': 'The detailed structure of [MATH] is not important for our argument, but essentially [MATH] is nonzero only in a range of size [MATH], for [MATH], and it is normalized: [MATH].', '1406.7815-1-11-8': 'As shown in [CITATION], this ensures a well-defined limit [MATH].', '1406.7815-1-12-0': 'The entanglement rate for a given frequency slot [MATH] may now be defined naturally as [MATH], the ratio between the entanglement contained in a pair of wave packets at that frequency and the time [MATH] between those wave packets.', '1406.7815-1-12-1': '(Here [MATH] would denote the index for beam 1 to which corresponds uniquely an index [MATH] for beam 2, as explained above.)', '1406.7815-1-12-2': 'As the logarithmic negativity is additive, it makes sense to add up [MATH] for all the small frequency intervals into which the emission of the source has been decomposed.', '1406.7815-1-13-0': 'We thus arrive at the total entanglement rate, which we define as: [EQUATION]', '1406.7815-1-13-1': 'Here [MATH] has been the entanglement calculated for the frequency slot [MATH], we have exploited that the limit [MATH] is well-defined, where [MATH] is kept fixed, and we have used [MATH] to convert the sum into an integral.', '1406.7815-1-13-2': 'We may call [MATH] the "spectral density of entanglement" [CITATION].', '1406.7815-1-13-3': 'In the limit [MATH], the details of the Wannier basis have become unimportant, and [MATH] can now be calculated using any filter.', '1406.7815-1-14-0': 'This definition has a direct operational meaning, corresponding to splitting each of the two entangled beams into many frequency-filtered output beams, where the frequency resolution has been chosen fine enough, such that [MATH].', '1406.7815-1-14-1': 'The rate [MATH] quantifies the total entanglement per unit time contained in the sum of those streams.', '1406.7815-1-14-2': 'Our definition also shows that [MATH] itself may be interpreted as the entanglement rate per frequency interval.', '1406.7815-1-15-0': 'A specific optomechanical example, and its implementation.', '1406.7815-1-15-1': 'We illustrate the features of the entanglement rate in a model describing the effective interaction between two optical modes ([MATH] and [MATH]) and a mechanical mode ([MATH]): [EQUATION]', '1406.7815-1-15-2': 'Similar Hamiltonians have been studied previously in the context of entanglement generation between light modes [CITATION].', '1406.7815-1-15-3': 'For instance, Wang and Clerk [CITATION] studied intra-cavity entanglement, while Tian [CITATION] investigated also the stationary output entanglement.', '1406.7815-1-16-0': 'Before proceeding, we note how to implement this model using three equidistant optical modes, enhancing the efficiency beyond previous suggestions.', '1406.7815-1-16-1': 'The optical mode spacing [MATH] is nearly resonant with the vibration frequency [MATH], with a frequency mismatch [MATH] (Fig. [REF]a).', '1406.7815-1-16-2': 'A laser drives the center optical mode at [MATH], with a detuning [MATH].', '1406.7815-1-16-3': 'An optomechanical interaction of the kind [MATH], and similarly for [MATH], scatters photons up and down, into modes [MATH] and [MATH], while simultaneously destroying (creating) phonons.', '1406.7815-1-16-4': 'When a phonon is virtually emitted and re-absorbed, an effective four-wave mixing process is induced, generating a pair of [MATH] and [MATH] photons out of a pair of [MATH] photons.', '1406.7815-1-16-5': 'Thus, two-mode vacuum squeezing (EPR entanglement) is produced.', '1406.7815-1-16-6': 'We assume that the drive is strong and we can replace [MATH] by the coherent amplitude [MATH].', '1406.7815-1-16-7': 'This yields the Hamiltonian ([REF]) with [MATH], provided we choose frames rotating at [MATH] for the modes [MATH], and at [MATH] for [MATH].', '1406.7815-1-16-8': 'Moreover, only nearly resonant terms are kept, which is allowed if [MATH], where [MATH] is the optical intensity decay rate.', '1406.7815-1-17-0': 'Possible experimental implementations include a membrane-in-the-middle setup tuned to a point with three equidistant modes [CITATION] (Fig. [REF]b) or coupled optomechanical cells, e.g. in an optomechanical crystal [CITATION] (Fig. [REF]c), see [CITATION].', '1406.7815-1-17-1': 'Such a triply-resonant setup enhances the efficiency: As compared to previous suggestions with only one resonantly driven mode, generating entangled Stokes and anti-Stokes sidebands (similar to [CITATION]), one wins a factor [MATH] in the intensity of the entangled output beams, while compared to setups with two optical ouput modes (e.g. [CITATION]), one wins a factor [MATH], for fixed input laser power and [MATH].', '1406.7815-1-18-0': 'We use standard input-output theory[CITATION] for our analysis: [EQUATION]', '1406.7815-1-18-1': 'Here [MATH] is the mechanical damping rate and [MATH].', '1406.7815-1-18-2': 'As usual, [MATH] and [MATH], with [MATH] the thermal occupation, and likewise for [MATH] (but without thermal noise).', '1406.7815-1-18-3': 'Solving Eqs. ([REF]) and employing [MATH], we find the linear relation between output and input fields in terms of a scattering matrix.', '1406.7815-1-19-0': 'To calculate the logarithmic negativity [MATH], we need the correlators of the quadratures of simultaneously ejected filtered field modes, for a filtering time [MATH].', '1406.7815-1-19-1': 'Entangled photon pairs are emitted at physical frequencies [MATH], corresponding to [MATH] in our rotating frame.', '1406.7815-1-19-2': 'Thus, we evaluate [CITATION] correlators like [MATH].', '1406.7815-1-20-0': 'Results.', '1406.7815-1-20-1': 'We first address some general features.', '1406.7815-1-20-2': 'The system can become unstable (Fig. [REF]a), both towards mechanical and optical oscillations.', '1406.7815-1-20-3': 'The optical stability boundary is approximately given by [MATH].', '1406.7815-1-20-4': 'Eliminating the mechanical mode (for [MATH]), we obtain an effective optical model, which yields the photon pair creation rate [MATH].', '1406.7815-1-20-5': 'This diverges at the optical stability boundary.', '1406.7815-1-21-0': 'We now focus on the output entanglement [MATH] and especially the entanglement rate.', '1406.7815-1-21-1': 'In contrast to the intra-cavity entanglement (discussed in [CITATION]), we find that [MATH] is not bounded.', '1406.7815-1-21-2': 'This is similar to the difference between intra-cavity and output squeezing [CITATION].', '1406.7815-1-21-3': 'Numerical plots of [MATH] for the Hamiltonian ([REF]) have been shown so far [CITATION] only for the special case [MATH] and asymmetric optomechanical couplings.', '1406.7815-1-21-4': 'Entanglement of temporal modes was also discussed for a pulsed scheme [CITATION] in the case [MATH].', '1406.7815-1-22-0': 'The output entanglement grows significantly at the stability boundary (Fig. [REF]b), even diverging in the effective optical model.', '1406.7815-1-22-1': 'This is typical near an instability but it comes at the price of linewidth narrowing, reducing the entanglement rate.', '1406.7815-1-23-0': 'In order to appreciate this, we now discuss the output intensity spectrum, [MATH], in Fig. [REF]a,b,c.', '1406.7815-1-23-1': 'We expect that any mechanical noise contributing to the optical output is deleterious for entanglement, which is confirmed by explicit calculation.', '1406.7815-1-23-2': 'Thus, we plot the spectrum in an instructive way, distinguishing optical and mechanical contributions, as obtained from the linear relation [MATH].', '1406.7815-1-23-3': 'There are typically two peaks, separated by [MATH] and containing primarily optical or mechanical noise, respectively.', '1406.7815-1-23-4': 'Near the optical instability (Fig. [REF]b), the optical peak gets strong and narrow.', '1406.7815-1-24-0': 'When the optical mode spacing matches the mechanical frequency ([MATH]) and the laser is on resonance ([MATH]), the two peaks merge and have a narrow linewidth set by the mechanical damping rate [MATH] ( Fig. [REF]c).', '1406.7815-1-24-1': 'It will turn out that the entanglement rate is maximized near this point.', '1406.7815-1-24-2': 'This is entirely counterintuitive: One might expect that at [MATH] mechanical noise is injected into the output beams, destroying entanglement.', '1406.7815-1-24-3': 'However, we find that the optical noise can completely overwhelm the mechanical noise for strong driving, when the cooperativity is sufficiently large, [MATH].', '1406.7815-1-25-0': 'The output entanglement [MATH] is shown in Fig. [REF]d,e,f. Typically, [MATH] is maximal at the optical peak near [MATH].', '1406.7815-1-25-1': 'For the important case [MATH], we find an analytical expression, [MATH], where [EQUATION] depends on both driving (via [MATH]) and temperature.', '1406.7815-1-25-2': 'We checked that choosing different optomechanical couplings for the two modes [MATH] will not increase [MATH], in contrast to the intra-cavity case [CITATION].', '1406.7815-1-26-0': 'In Fig. [REF]a,b, we compare the maximal output entanglement [MATH] and the entanglement rate [MATH].', '1406.7815-1-26-1': 'While [MATH] becomes large near the optical boundary of stability, the entanglement rate there remains small, due to the narrow bandwidth.', '1406.7815-1-26-2': 'This will be a general feature in many similar systems.', '1406.7815-1-26-3': 'Rather, [MATH] is optimal near [MATH].', '1406.7815-1-26-4': 'We found that, as the mechanical damping rate increases, the optimum shifts away from [MATH].', '1406.7815-1-27-0': 'The entanglement rate depends on the full shape of [MATH], in particular the peak width(s).', '1406.7815-1-27-1': 'Crucially, those widths are distinct from those in the output spectrum, due to the nonlinear (logarithmic) dependence of [MATH] on parameters.', '1406.7815-1-27-2': 'For example, in the case [MATH] of greatest interest, the peak width (see Fig. [REF]f, Fig. [REF]c) is not set by the small mechanical linewidth [MATH], unlike for the output spectrum discussed above.', '1406.7815-1-27-3': 'Thus, the values of the entanglement rate [MATH] for the parameters explored here are much larger, of the order [MATH].', '1406.7815-1-27-4': 'For increasing temperatures, [MATH] decreases, but only slowly, indicating robust entanglement in this setup: [MATH], with the prefactor set by the cooperativity [MATH] (Fig. [REF]d).', '1406.7815-1-28-0': 'Conclusions.', '1406.7815-1-28-1': '- We have introduced an entanglement rate as a quantitative measure for the CV entanglement production per unit time in setups involving resonant modes.', '1406.7815-1-28-2': 'Moreover, we have studied an optomechanical setup with one mechanical and three optical modes that allows fully resonant production of optical entanglement.', '1406.7815-1-28-3': 'The output entanglement and the entanglement rate are maximized for different parameter choices.', '1406.7815-1-28-4': 'The concept introduced here should be useful for analyzing setups in other domains, like cavities with a Kerr medium [CITATION] or microwave resonators with nonlinearities [CITATION].'}

{'1406.7815-2-0-0': 'We derive a general expression that quantifies the total entanglement production rate in continuous variable systems, where a source emits two entangled Gaussian beams with arbitrary correlators.', '1406.7815-2-0-1': 'This expression is especially useful for situations where the source emits an arbitrary frequency spectrum, e.g. when cavities are involved.', '1406.7815-2-0-2': 'To exemplify its meaning and potential, we apply it to a four-mode optomechanical setup that enables the simultaneous up- and down-conversion of photons from a drive laser into entangled photon pairs.', '1406.7815-2-0-3': 'This setup is efficient in that both the drive and the optomechanical up- and down-conversion can be fully resonant.', '1406.7815-2-1-0': '# Introduction', '1406.7815-2-2-0': 'Entanglement is an essential feature of quantum mechanics and a crucial resource for quantum communication and information processing.', '1406.7815-2-2-1': 'The most common situation involves a source that continuously produces entangled beams.', '1406.7815-2-2-2': 'One of the most natural characteristics of such a source is obviously the rate at which it generates entanglement.', '1406.7815-2-2-3': 'If the source sends out pairs of entangled particles, with subsequent pairs completely independent, this rate can simply be defined as the entanglement of each pair, divided by the time between pairs.', '1406.7815-2-2-4': 'However, such a naive approach fails if there are correlations between subsequent pairs, or if we consider entangled beams of radiation that cannot be naturally decomposed into well-defined pairs of particles.', '1406.7815-2-2-5': 'In particular, this is true for the very important case of continuous variable (CV) entangled beams.', '1406.7815-2-2-6': 'Although many quantum information protocols exploit qubits, with their discrete state space, the original Einstein-Podolsky-Rosen [CITATION] entanglement involves continuous variables, and CV entanglement has many modern applications [CITATION].', '1406.7815-2-3-0': 'In this article, we set out to provide a general definition for an entanglement rate in such nontrivial situations.', '1406.7815-2-3-1': 'It will turn out that our general definition, when applied to stationary Gaussian CV beams, gives rise to a frequency integral over what we call a "spectral density of entanglement".', '1406.7815-2-3-2': 'We show how to obtain this from the two-point time correlators of the entangled beams, using a suitable additive entanglement measure (the logarithmic negativity [CITATION]).', '1406.7815-2-3-3': 'For one of the most common situations, we also provide explicit analytical expressions.', '1406.7815-2-3-4': 'Our definition of the entanglement rate is particularly important for setups where the output spectrum is arbitrary (e.g. containing one or several peaks).', '1406.7815-2-3-5': 'This is a widespread case, since the generation of CV-entangled radiation beams is often enhanced by using cavity modes (e.g. [CITATION] in the microwave domain, or [CITATION] in the optical domain).', '1406.7815-2-3-6': 'Recently, the authors of the first experiment on spatially separated CV entanglement in superconducting circuits even quantified their source by quoting the effective number of entangled bits per second [CITATION], estimated from the bandwidth of the circuit and the entanglement between two modes.', '1406.7815-2-3-7': 'Our entanglement rate would provide a precisely defined way to quantitatively characterize such situations.', '1406.7815-2-4-0': 'After we introduce and discuss the general definition, we illustrate our entanglement rate by applying it to a four-mode optomechanical setup that allows the fully resonant, and thereby efficient, generation of entanglement.', '1406.7815-2-4-1': 'The rapidly developing field of cavity optomechanics focuses on the dynamics of photons and phonons coupled via radiation forces, see [CITATION] for a recent review.', '1406.7815-2-4-2': 'The optomechanical interaction has been predicted to produce entanglement, e.g. between optics and mechanics [CITATION] or between light modes [CITATION].', '1406.7815-2-4-3': 'Recently, optomechanical entanglement was demonstrated experimentally for the first time, in a microwave circuit [CITATION], making its analysis especially timely.', '1406.7815-2-5-0': '# The entanglement rate', '1406.7815-2-6-0': 'The situation we have in mind is very general: A source emits two CV-entangled beams, described by bosonic fields [MATH] and [MATH], see Fig. [REF]a.', '1406.7815-2-6-1': 'These could be, for example, two light beams of different polarization or fields propagating along two different waveguides.', '1406.7815-2-6-2': 'Typical sources might be a nonlinear crystal optical resonator driven by a pump beam, a driven optomechanical cavity, or a driven on-chip microwave cavity containing nonlinear elements like Josephson junctions.', '1406.7815-2-7-0': 'We focus on the important regime of generating stationary Gaussian CV-entanglement.', '1406.7815-2-7-1': 'In that regime, one treats the pump as classical and then obtains a quadratic Hamiltonian, leading to Gaussian statistics of the emitted beams.', '1406.7815-2-7-2': 'Because of the pump, that Hamiltonian will be time-dependent, containing terms of the form [MATH], where [MATH] would be a cavity mode (our analysis also applies for several different modes).', '1406.7815-2-7-3': 'Here [MATH] is the pump frequency if the original nonlinearity was of the [MATH]-type, [MATH], whereas [MATH] would be twice the pump frequency for a [MATH]-type nonlinearity [MATH].', '1406.7815-2-7-4': 'As is usual in such situations, it will be most useful to switch to a frame rotating at the frequency [MATH], such that the Hamiltonian becomes time-independent and we are dealing with a stationary problem.', '1406.7815-2-7-5': 'In that new frame, [MATH] relates to the pump frequency.', '1406.7815-2-8-0': 'Our analysis then focusses on the entanglement properties of the fields [MATH] emitted from any such source.', '1406.7815-2-8-1': 'Being Gaussian, these fields are completely characterized by their two-time correlators.', '1406.7815-2-8-2': 'The details of the source do not matter, except that it is assumed to produce Gaussian beams that are stationary, i.e. where the correlators only depend on the time-difference.', '1406.7815-2-9-0': 'At this point we note that in some situations it may also be natural to consider only a single field [MATH], propagating along a single waveguide.', '1406.7815-2-9-1': 'Then, frequency components centered symmetrically around the pump frequency can be entangled, and they may afterwards be directed to two different output ports by frequency-filtering.', '1406.7815-2-9-2': 'For such a situation, we can still apply our approach if we define [MATH] to contain the positive frequency ([MATH]) components of the field [MATH], while likewise [MATH] would contain the negative frequency ([MATH]) components, where [MATH] is already determined with respect to the rotating frame.', '1406.7815-2-10-0': 'Since the situation is stationary, it is natural to try and define an entanglement rate, i.e. the entanglement per unit time emitted from the source.', '1406.7815-2-10-1': 'We propose to do this in the most natural way by calculating the overall amount of entanglement between the two beams in a long time interval [MATH] and then dividing by [MATH], in the end sending [MATH] to infinity: [EQUATION]', '1406.7815-2-10-2': 'This definition is not constrained to CV entangled beams or to Gaussian states.', '1406.7815-2-10-3': 'It only requires (a) stationarity of the source, (b) an entanglement measure that is additive for product states, and (c) a finite correlation time [MATH] for the beams.', '1406.7815-2-10-4': 'Given such a finite correlation time, the fields on two subsequent time-intervals of length [MATH] are not correlated to a very good approximation.', '1406.7815-2-10-5': 'This holds because even though there may be remaining correlations near the boundary between the time-intervals, these are appreciable only up to a distance of order [MATH] from the boundary and can be neglected in the limit [MATH].', '1406.7815-2-10-6': 'Due to the additivity (and stationarity), we then get [MATH], such that the entanglement rate calculated for time intervals of sizes [MATH] or [MATH] is the same (up to the small corrections which we can neglect in the limit of large [MATH]).', '1406.7815-2-11-0': 'From now on, we specialize to stationary Gaussian CV entangled beams.', '1406.7815-2-11-1': 'We will use the logarithmic negativity [CITATION] [MATH] as an entanglement measure, since it is both straightforwardly evaluated for Gaussian multi-mode states and has the important property of additivity.', '1406.7815-2-12-0': 'Let us now consider the fields [MATH] on the interval [MATH], even though they are defined for all times [MATH].', '1406.7815-2-12-1': 'We can define discrete frequency modes ([MATH]): [EQUATION]', '1406.7815-2-12-2': 'The normalization is chosen such that the [MATH] fulfill bosonic commutation relations, [MATH], where [MATH] is discrete, with [MATH] an integer.', '1406.7815-2-13-0': 'We now want to calculate the full logarithmic negativity [MATH] between the two beams on that time interval, which is equivalent to the entanglement between two sets of harmonic oscillators [CITATION].', '1406.7815-2-13-1': 'In our case, we are even considering infinitely many harmonic oscillators [MATH].', '1406.7815-2-13-2': 'We stress that the entanglement [MATH] is (of course) independent of our choice of basis for each of the beams, as a different choice of basis amounts to implementing a local unitary transformation.', '1406.7815-2-13-3': 'The correlations between the two beams can be arbitrary, except that they are supposed to decay beyond some correlation time [MATH] (which is true in any reasonable physical situation).', '1406.7815-2-13-4': 'As already mentioned above, we will assume that [MATH], such that the correlations between subsequent intervals of size [MATH] can be neglected.', '1406.7815-2-13-5': 'In that limit, we can use stationarity to show that only the following types of correlators may be nonzero in the present situation, up to corrections that are small in [MATH]: [MATH] and [MATH] (and their conjugates).', '1406.7815-2-13-6': 'For example, we find [EQUATION] where we have defined the Fourier transform of the correlator: [MATH].', '1406.7815-2-13-7': 'Likewise, we have [MATH].', '1406.7815-2-14-0': 'In summary, for any given [MATH], only the correlators involving the modes [MATH] and [MATH] are nonzero (see Fig. [REF]b).', '1406.7815-2-14-1': 'There is entanglement [MATH] between the two modes [MATH] and [MATH] of beam 1 with the two modes [MATH] and [MATH] of beam 2.', '1406.7815-2-14-2': 'Further below, we will show how [MATH] can be calculated from the correlators.', '1406.7815-2-14-3': 'The overall entanglement [MATH] between the two beams can then be decomposed into a sum, because of the additivity of the logarithmic negativity: [EQUATION]', '1406.7815-2-14-4': 'We note that [MATH] in the sum does not depend on [MATH] (after adopting the approximation of Eq. [REF]), although the number of summation terms does scale with [MATH] due to the discretization of [MATH].', '1406.7815-2-15-0': 'The sum over discrete frequencies can be converted into an integral.', '1406.7815-2-15-1': 'To leading order at large [MATH], we have [MATH].', '1406.7815-2-15-2': 'Therefore, we can identify [MATH] as the "spectral density of entanglement" [CITATION].', '1406.7815-2-15-3': 'In addition, this confirms that the total entanglement indeed grows linearly in [MATH] for sufficiently large times.', '1406.7815-2-15-4': 'As a consequence we finally obtain the entanglement rate: [EQUATION]', '1406.7815-2-15-5': 'This shows that [MATH] itself may be interpreted as the entanglement rate per frequency interval.', '1406.7815-2-15-6': 'It is possible to give an explicit expression for [MATH] in terms of the covariance matrix containing the correlators [CITATION].', '1406.7815-2-15-7': 'The general case, involving the four modes [MATH], [MATH], [MATH], and [MATH], is a bit cumbersome, requiring the symplectic diagonalization of a [MATH] covariance matrix.', '1406.7815-2-15-8': 'However, the expressions simplify considerably if there is purely two-mode squeezing, i.e. if the intra-mode correlators [MATH] vanish.', '1406.7815-2-15-9': 'In that case, we find [EQUATION]', '1406.7815-2-15-10': 'For positive [MATH], [MATH] is the entanglement between [MATH] and [MATH] while [MATH] is the entanglement between [MATH] and [MATH].', '1406.7815-2-15-11': 'In contrast to [MATH], the density [MATH] is double-sided (has contributions both at negative and positive frequencies).', '1406.7815-2-15-12': 'We note that in the following we will also commonly refer to [MATH] as the spectral density of entanglement, since it is closely related to [MATH].', '1406.7815-2-15-13': 'Setting [MATH], [MATH], and [MATH], we have: [EQUATION]', '1406.7815-2-15-14': 'As an aside we note that we choose to work with the natural logarithm in our discussion (some articles use [MATH], which is more natural for discrete qubits [CITATION]).', '1406.7815-2-16-0': 'In this special case, the entanglement rate is therefore: [EQUATION]', '1406.7815-2-17-0': '# Relation to Entanglement between Wave Packets', '1406.7815-2-18-0': 'We now want to connect our general result to previously applied approaches for quantifying the entanglement in such situations.', '1406.7815-2-18-1': 'It is a common procedure to employ normalized mode functions ("filter functions") [MATH] that have the shape of wave packets, in order to define two harmonic oscillator modes, one for each beam: [EQUATION]', '1406.7815-2-18-2': 'Here [MATH].', '1406.7815-2-18-3': 'The entanglement between [MATH] and [MATH] can then be calculated, e.g. again using the logarithmic negativity as an entanglement measure.', '1406.7815-2-18-4': 'This filtering analysis has been applied to several settings [CITATION], for a detailed explanation see e.g. [CITATION].', '1406.7815-2-18-5': 'The catch is that this procedure introduces a filtering time [MATH] (the extent of the wave packets), and the results will be a function of [MATH], which is usually taken to be arbitrary.', '1406.7815-2-19-0': 'How would one loosely define an entanglement rate based on this procedure?', '1406.7815-2-19-1': 'We can imagine that there is a stream of such wave packets, with a spacing of about [MATH] (where care would have to be taken to define them to be orthogonal).', '1406.7815-2-19-2': 'A simple, though phenomenological approach to define an entanglement rate would be to simply calculate the ratio [MATH].', '1406.7815-2-20-0': 'However, it is clear that this approach is not systematic.', '1406.7815-2-20-1': 'In fact, it cannot always cover the full entanglement, since there may be entanglement in components of the beam that are orthogonal to the filter functions which are employed.', '1406.7815-2-20-2': 'In addition, there is some arbitrariness in the choice of filter function (and, thus, even in the definition of [MATH]).', '1406.7815-2-20-3': 'Moreover, one may have situations where there are temporal correlations extending beyond [MATH].', '1406.7815-2-20-4': 'Then, the entanglement present in the beams may be underestimated.', '1406.7815-2-20-5': 'If one tries to remedy this problem by choosing larger [MATH], then the filter bandwidth shrinks and one may miss entanglement present at other frequencies.', '1406.7815-2-21-0': 'It turns out that an approach based on wave packets can be made to work, but only if one constructs a suitable complete basis that has a clear physical meaning.', '1406.7815-2-21-1': 'It is more systematic than the naive filtering approach described so far and it covers all the entanglement.', '1406.7815-2-21-2': 'In addition, it can be related to a direct physical prescription, and we will see that it leads to the same results as our general, basis-independent definition discussed in the previous section.', '1406.7815-2-22-0': 'For simplicity, focus on the situation with only cross-correlations (no intra-beam squeezing) that we discussed at the end of section [REF].', '1406.7815-2-22-1': 'Imagine one sends one beam through a frequency filter [MATH], where [MATH].', '1406.7815-2-22-2': 'Likewise, the other beam will be sent through another filter, at negative frequencies [MATH].', '1406.7815-2-22-3': 'Now construct a complete set of orthogonal wave packet modes ("Wannier basis") with a spacing [MATH] in time, which are able to fully represent the filtered beams (see Fig. [REF]).', '1406.7815-2-22-4': 'As we show in the appendix [REF], the logarithmic negativity [MATH] between two such wave packet modes (one in each beam, at equal time-slots) is related to [MATH] in the limit [MATH]: [EQUATION]', '1406.7815-2-22-5': 'As a consequence, the general definition of the previous section agrees with the entanglement rate calculated from such a wave packet picture.', '1406.7815-2-22-6': 'This wave packet approach can also be viewed as representing the following physical procedure: Split each of the two entangled beams into many frequency-filtered output beams, where the frequency resolution [MATH] has been chosen fine enough, such that [MATH].', '1406.7815-2-22-7': 'The rate [MATH] quantifies the total entanglement per unit time contained in the sum of those streams (since it was defined that way in the previous section).', '1406.7815-2-22-8': 'Each pair of wave packets (of length [MATH]) can in principle be exploited for an application such as CV quantum teleportation.', '1406.7815-2-22-9': 'A concrete physical measurement of the entanglement between any two wave packets could be performed in a standard way, using homodyne measurements.', '1406.7815-2-22-10': 'For example, the local oscillator can provide strong pulses that are shaped in the form of the wave packet modes that we want to consider.', '1406.7815-2-23-0': '# A specific optomechanical example, and its implementation', '1406.7815-2-24-0': '## Model', '1406.7815-2-25-0': 'We illustrate the features of the entanglement rate in a model describing the effective interaction between two localized optical modes ([MATH] and [MATH]) and a mechanical mode ([MATH]): [EQUATION]', '1406.7815-2-25-1': 'Similar Hamiltonians have been studied previously in the context of entanglement generation between light modes [CITATION].', '1406.7815-2-25-2': 'For instance, Wang and Clerk [CITATION] studied intra-cavity entanglement, while Tian [CITATION] investigated also the stationary output entanglement.', '1406.7815-2-25-3': 'We note that, more recently, an analysis of the output entanglement in the setup with [MATH] (but with unequal couplings for up- and down-conversion) was provided in [CITATION].', '1406.7815-2-26-0': 'Before proceeding, we note how to implement this model using three equidistant optical modes, enhancing the efficiency beyond previous suggestions.', '1406.7815-2-26-1': 'The optical mode spacing [MATH] is nearly resonant with the vibration frequency [MATH], with a frequency mismatch [MATH] (Fig. [REF]a).', '1406.7815-2-26-2': 'A laser drives the center optical mode at [MATH], with a detuning [MATH].', '1406.7815-2-26-3': 'An optomechanical interaction of the kind [MATH], and similarly for [MATH], scatters photons up and down, into modes [MATH] and [MATH], while simultaneously destroying (creating) phonons.', '1406.7815-2-26-4': 'When a phonon is virtually emitted and re-absorbed, an effective four-wave mixing process is induced, generating a pair of [MATH] and [MATH] photons out of a pair of [MATH] photons.', '1406.7815-2-26-5': 'Thus, two-mode vacuum squeezing (EPR entanglement) is produced.', '1406.7815-2-26-6': 'We assume that the drive is strong and we can replace [MATH] by the coherent amplitude [MATH].', '1406.7815-2-26-7': 'This yields the Hamiltonian ([REF]) with [MATH], provided we choose frames rotating at [MATH] for the modes [MATH], and at [MATH] for [MATH].', '1406.7815-2-26-8': 'Moreover, only nearly resonant terms are kept, which is allowed if [MATH], where [MATH] is the optical intensity decay rate.', '1406.7815-2-27-0': 'Possible experimental implementations include a membrane-in-the-middle setup tuned to a point with three equidistant modes [CITATION] (Fig. [REF]b) or coupled optomechanical cells, e.g. in an optomechanical crystal [CITATION] (Fig. [REF]c), see the appendix.', '1406.7815-2-27-1': 'Such a triply-resonant setup enhances the efficiency: As compared to previous suggestions with only one resonantly driven mode, generating entangled Stokes and anti-Stokes sidebands (similar to [CITATION]), one wins a factor [MATH] in the intensity of the entangled output beams, while compared to setups with two optical ouput modes (e.g. [CITATION]), one wins a factor [MATH], for fixed input laser power and [MATH].', '1406.7815-2-28-0': 'We use standard input-output theory[CITATION] for our analysis: [EQUATION]', '1406.7815-2-28-1': 'Here [MATH] is the mechanical damping rate and [MATH].', '1406.7815-2-28-2': 'As usual, [MATH] and [MATH], with [MATH] the thermal occupation, and likewise for [MATH] (but without thermal noise).', '1406.7815-2-28-3': 'Solving Eqs. ([REF]) and employing [MATH], we find the linear relation between output and input fields in terms of a scattering matrix.', '1406.7815-2-29-0': 'Coming back to our general definition of the entanglement rate, we would consider [MATH] as the first beam [MATH] and [MATH] as the second beam [MATH].', '1406.7815-2-29-1': 'It is not difficult to show (and can be confirmed by direct calculation) that there is no intra-beam squeezing, i.e. [MATH] and likewise for beam 2.', '1406.7815-2-29-2': 'Thus, we want to employ the formulas Eqs. ([REF]) and ([REF]) in order to find the contributions to the spectral density of entanglement and the entanglement rate.', '1406.7815-2-30-0': 'To calculate [MATH], we need the correlators of the two beams.', '1406.7815-2-30-1': 'Entangled photon pairs are emitted at physical frequencies [MATH], corresponding to [MATH] in our rotating frame.', '1406.7815-2-30-2': 'We have to evaluate correlators like [MATH] as shown in appendix [REF].', '1406.7815-2-31-0': '## Results', '1406.7815-2-32-0': 'We first address some general features.', '1406.7815-2-32-1': 'The system can become unstable (Fig. [REF]a), both towards mechanical and optical oscillations.', '1406.7815-2-32-2': 'The optical stability boundary is approximately given by [MATH].', '1406.7815-2-32-3': 'Eliminating the mechanical mode (for [MATH]), we obtain an effective optical model, which yields the photon pair creation rate [MATH].', '1406.7815-2-32-4': 'This diverges at the optical stability boundary.', '1406.7815-2-33-0': 'We now focus on the spectral density of entanglement [MATH] that characterizes the output beams, and especially the entanglement rate.', '1406.7815-2-33-1': 'In contrast to the intra-cavity entanglement (discussed in [CITATION]), we find that [MATH] is not bounded.', '1406.7815-2-33-2': 'This is similar to the difference between intra-cavity and output squeezing [CITATION].', '1406.7815-2-33-3': 'Numerical plots of [MATH] for the Hamiltonian ([REF]) have been shown so far [CITATION] only for the special case [MATH] and asymmetric optomechanical couplings.', '1406.7815-2-33-4': 'Entanglement of temporal modes was also discussed for a pulsed scheme [CITATION] in the case [MATH].', '1406.7815-2-34-0': 'The output entanglement grows significantly at the stability boundary (Fig. [REF]b), even diverging in the effective optical model.', '1406.7815-2-34-1': 'This is typical near an instability but it comes at the price of linewidth narrowing, reducing the entanglement rate.', '1406.7815-2-35-0': 'In order to appreciate this, we now discuss the output intensity spectrum, [MATH], in Fig. [REF]a,b,c.', '1406.7815-2-35-1': 'We expect that any mechanical noise contributing to the optical output is deleterious for entanglement, which is confirmed by explicit calculation.', '1406.7815-2-35-2': 'Thus, we plot the spectrum in an instructive way, distinguishing optical and mechanical contributions, as obtained from the linear relation [MATH].', '1406.7815-2-35-3': 'There are typically two peaks, separated by [MATH] and containing primarily optical or mechanical noise, respectively.', '1406.7815-2-35-4': 'Near the optical instability (Fig. [REF]b), the optical peak gets strong and narrow.', '1406.7815-2-36-0': 'When the optical mode spacing matches the mechanical frequency ([MATH]) and the laser is on resonance ([MATH]), the two peaks merge and have a narrow linewidth set by the mechanical damping rate [MATH] ( Fig. [REF]c).', '1406.7815-2-36-1': 'It will turn out that the entanglement rate is maximized near this point.', '1406.7815-2-36-2': 'This is entirely counterintuitive: One might expect that at [MATH] mechanical noise is injected into the output beams, destroying entanglement.', '1406.7815-2-36-3': 'However, we find that the optical noise can completely overwhelm the mechanical noise for strong driving, when the cooperativity is sufficiently large, [MATH].', '1406.7815-2-37-0': 'The spectral density [MATH] is shown in Fig. [REF]d,e,f. Typically, [MATH] is maximal at the optical peak near [MATH].', '1406.7815-2-37-1': 'For the important case [MATH], we find an analytical expression, [MATH], where [EQUATION] depends on both driving (via [MATH]) and temperature.', '1406.7815-2-37-2': 'We checked that choosing different optomechanical couplings for the two modes [MATH] will not increase [MATH], in contrast to the intra-cavity case [CITATION].', '1406.7815-2-38-0': 'In Fig. [REF]a,b, we compare the maximum of the spectral density of entanglement, [MATH], and the entanglement rate [MATH].', '1406.7815-2-38-1': 'While [MATH] becomes large near the optical boundary of stability, the entanglement rate there remains small, due to the narrow bandwidth.', '1406.7815-2-38-2': 'This will be a general feature in many similar systems.', '1406.7815-2-38-3': 'Rather, [MATH] is optimal near [MATH].', '1406.7815-2-38-4': 'We found that, as the mechanical damping rate increases, the optimum shifts away from [MATH].', '1406.7815-2-39-0': 'The entanglement rate depends on the full shape of [MATH], in particular the peak width(s).', '1406.7815-2-39-1': 'Crucially, those widths are distinct from those in the output spectrum, due to the nonlinear (logarithmic) dependence of [MATH] on parameters.', '1406.7815-2-39-2': 'For example, in the case [MATH] of greatest interest, the peak width (see Fig. [REF]f, Fig. [REF]c) is not set by the small mechanical linewidth [MATH], unlike for the output spectrum discussed above.', '1406.7815-2-39-3': 'Thus, the values of the entanglement rate [MATH] for the parameters explored here are much larger, of the order [MATH].', '1406.7815-2-39-4': 'For increasing temperatures, [MATH] decreases, but only slowly, indicating robust entanglement in this setup: [MATH], with the prefactor set by the cooperativity [MATH] (Fig. [REF]d).', '1406.7815-2-40-0': '# Conclusions', '1406.7815-2-41-0': 'We have introduced an entanglement rate as a quantitative measure for the CV entanglement production per unit time in setups involving resonant modes.', '1406.7815-2-41-1': 'The definition is natural, in that it simply characterizes the total entanglement between two beams within a time-interval of size [MATH], in the limit [MATH].', '1406.7815-2-41-2': 'In principle, it is also more general than the Gaussian CV case studied in the present manuscript.', '1406.7815-2-42-0': 'Moreover, we have studied an optomechanical setup with one mechanical and three optical modes that allows fully resonant production of optical entanglement.', '1406.7815-2-42-1': 'The spectral density of entanglement and the overall entanglement rate are optimized for different parameter choices.', '1406.7815-2-42-2': 'The concept introduced here should be useful for analyzing setups in other domains, like cavities with a Kerr medium [CITATION] or microwave resonators with nonlinearities [CITATION].'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1406.7815

nucl-ex-0701003

{'nucl-ex-0701003-1-0-0': 'Precision measurements were carried out to test the similarities between the ground states of [MATH]Ge and [MATH]Se.', 'nucl-ex-0701003-1-0-1': 'The extent to which these two nuclei can be characterized as consisting of correlated pairs of neutrons in a BCS-like ground state was studied.', 'nucl-ex-0701003-1-0-2': 'The pair removal (p,t) reaction was measured at the far forward angle of 3[MATH].', 'nucl-ex-0701003-1-0-3': 'The relative cross sections are consistent (at the 5% level) with the description of these nuclei in terms of a correlated pairing state outside the N=28 closed shells with no pairing vibrations.', 'nucl-ex-0701003-1-0-4': 'Data were also obtained for [MATH]Ge and [MATH]Se.', 'nucl-ex-0701003-1-1-0': '# introduction', 'nucl-ex-0701003-1-2-0': 'Interest in the possibility of observing neutrinoless double beta decay (0[MATH]) is considerable.', 'nucl-ex-0701003-1-2-1': 'If this decay were to be definitively observed, it would show that the neutrinos are their own antiparticles.', 'nucl-ex-0701003-1-2-2': 'In addition, the rate of the decay would be a measure of the neutrino rest mass, if the nuclear matrix element were known.', 'nucl-ex-0701003-1-2-3': 'Unfortunately, theoretical calculations of this do not agree well with each other [CITATION].', 'nucl-ex-0701003-1-2-4': 'It seems appropriate to determine additional properties of the ground states of the possible 0[MATH] systems by experiment, and thus help constrain and test theoretical calculations of this exotic decay mode.', 'nucl-ex-0701003-1-2-5': 'One of the likely candidate nuclei is [MATH]Ge decaying to the ground state of [MATH]Se.', 'nucl-ex-0701003-1-2-6': 'We have started with a study of the properties of the ground states of these nuclei, and especially the similarities and differences between them, using transfer reactions.', 'nucl-ex-0701003-1-2-7': 'One part of this study is an accurate measurement of one-nucleon transfer in order to probe the occupation numbers of valence orbits for both neutrons and protons, with particular attention to changes in these occupations.', 'nucl-ex-0701003-1-2-8': 'The other part is to study pair-correlations in these nuclei by nucleon pair transfer.', 'nucl-ex-0701003-1-2-9': 'Here we report on a comparison of neutron pair transfer from the (p,t) reaction.', 'nucl-ex-0701003-1-3-0': 'Pair transfer between 0[MATH] states in the (p,t) or (t,p) reaction proceeds via L=0 transfer and the angular distribution is sharply forward peaked.', 'nucl-ex-0701003-1-3-1': 'This feature was recognized early [CITATION] and was crucial in exploring the importance of such correlations and their excitations, the so-called pairing vibrations [CITATION].', 'nucl-ex-0701003-1-3-2': 'The latter are an indication of deviations from the simplest pairing picture and can occur in regions of changing shapes, or when there is a gap in single-particle states, such as near a shell closure.', 'nucl-ex-0701003-1-4-0': 'We have carried out measurements of the neutron pair-removal (p,t) reaction on targets of [MATH]Ge and [MATH]Se.', 'nucl-ex-0701003-1-4-1': 'The reaction on [MATH]Se was measured because the [MATH]Se(p,t)[MATH]Se leads to [MATH]Se, while [MATH]Se(p,t)[MATH]Se starts from the ground state of [MATH]Se.', 'nucl-ex-0701003-1-4-2': 'Thus both reactions are relevant to the pairing structure of this ground state.', 'nucl-ex-0701003-1-4-3': 'The [MATH]Ge target was included as a check.', 'nucl-ex-0701003-1-5-0': 'There are two relevant aspects to these measurements.', 'nucl-ex-0701003-1-5-1': 'The first is the matter of pairing vibrations.', 'nucl-ex-0701003-1-5-2': 'The even Ge and Se isotopes are well studied and evidence for excited 0[MATH] states has been established [CITATION] .', 'nucl-ex-0701003-1-5-3': 'In some of the lighter isotopes of Ge and Se, two-neutron transfer reactions have shown significant strength populating excited 0[MATH] states.', 'nucl-ex-0701003-1-5-4': 'These pairing vibrations indicate that there are significant BCS-like pair correlations in connecting the target ground state in an even initial nucleus to excited 0[MATH] states in the final.', 'nucl-ex-0701003-1-5-5': 'If there were significant differences in this regard between reactions leading to or from [MATH]Ge and [MATH]Se, this would be an indication that the pair correlations in the initial and final states in double-beta decay differ.', 'nucl-ex-0701003-1-5-6': 'Any reliable calculations of the process would presumably have to reproduce such differences in order to obtain a reasonable matrix element for the 0[MATH] decay.', 'nucl-ex-0701003-1-5-7': 'Secondly, if accurate cross sections were available for pair transfer further checks could be made of the similarity between the initial and final wave functions used in such calculations.', 'nucl-ex-0701003-1-6-0': '# Experimental Considerations', 'nucl-ex-0701003-1-7-0': 'Given that the L=0 (p,t) transitions are the strongest in the spectrum of final states at very forward angles, and this is also where the approximations inherent in the DWBA should be the best, it is desirable to carry out these measurements as close to 0[MATH] as feasible.', 'nucl-ex-0701003-1-7-1': 'The (p,t) reaction had been studied previously on Ge [CITATION] and Se [CITATION] isotopes; relevant (t,p) measurements have also been made [CITATION].', 'nucl-ex-0701003-1-7-2': 'Both the experimental methodology and the data analysis methods in the two (p,t) studies were different (the target thickness was estimated from high-energy elastic scattering where optical model predictions are ambiguous, the angular distributions start at 7.5 degrees, and only angle-integrated cross sections are quoted, etc.), making a reliable systematic comparison of Ge and Se data difficult.', 'nucl-ex-0701003-1-7-3': 'Our purpose here was to measure the cross sections at as far forward angles as feasible and to obtain a consistent set of accurate cross sections with particular care taken to reduce relative systematic uncertainties.', 'nucl-ex-0701003-1-8-0': 'The choice of energy was governed by the desirability of having both protons and tritons well above the Coulomb barrier.', 'nucl-ex-0701003-1-8-1': 'For the present measurement we therefore chose 23-MeV protons from the Yale ESTU tandem Van de Graaf accelerator.', 'nucl-ex-0701003-1-8-2': 'This energy is similar to that used in earlier experiments, both for (p,t) and (t,p) reactions.', 'nucl-ex-0701003-1-8-3': 'Because one of the objectives of the present measurement was to obtain accurate cross sections, we chose to measure the thickness of the evaporated germanium and selenium targets in situ by simply lowering the proton beam energy to 6 MeV, where the elastic scattering cross sections are very close to Rutherford values.', 'nucl-ex-0701003-1-8-4': 'The angle should not be so far forward that small uncertainties in angle would become significant and 30[MATH] was chosen because calculations with several optical potentials showed that the deviation from Rutherford scattering was less than 2%.', 'nucl-ex-0701003-1-8-5': 'The [MATH]Ge and [MATH]Se target thicknesses were found to range between 160 and 400 [MATH]g/cm[MATH].', 'nucl-ex-0701003-1-8-6': 'Since Se can sublimate at a relatively low temperature, this low-energy target thickness measurement was made at the beginning and then at the end of the experiment; no significant differences ([MATH]3%) were observed.', 'nucl-ex-0701003-1-8-7': 'The highest beam currents used were about 35 nA, though considerably lower (2-3 nA) for the 3[MATH] measurements.', 'nucl-ex-0701003-1-9-0': 'The Yale Enge split-pole spectrograph was used for the measurements with a focal-plane detector that cleanly separates tritons from other reaction products.', 'nucl-ex-0701003-1-9-1': 'As monitors, two Si surface barrier detectors at [MATH] were used.', 'nucl-ex-0701003-1-9-2': 'They were calibrated in terms of beam intensity using a current integrator connected to a Faraday cup.', 'nucl-ex-0701003-1-9-3': 'The beam integrator was set to the same scale which was used for the low-energy measurements and the solid-angle setting for the aperture of the spectrograph was also the same.', 'nucl-ex-0701003-1-9-4': 'This established the relationship between the monitor elastic scattering yield and current integrator readings for each target.', 'nucl-ex-0701003-1-10-0': 'The 3[MATH] setting for the spectrograph required that the Faraday cup be retracted so that the beam-current measurement had to rely on the previously calibrated monitor counters.', 'nucl-ex-0701003-1-10-1': 'Removal of the Faraday cup meant that the beam entered the spectrometer.', 'nucl-ex-0701003-1-10-2': 'The magnetic rigidity of the tritons from the reaction is such that, with the magnetic field set for observing tritons, protons from the target cannot directly reach the focal plane and are intercepted inside the spectrometer.', 'nucl-ex-0701003-1-10-3': 'Protons scattered at this point can enter the focal plane and, whilst they can be distinguished from tritons by their ionization density, they do impose a counting rate limit.', 'nucl-ex-0701003-1-10-4': 'As a result, the furthest forward angle where measurements could be made was 3[MATH].', 'nucl-ex-0701003-1-10-5': 'Distorted-wave calculations indicate that the cross section at this angle is lower than that at 0[MATH] by about 8%.', 'nucl-ex-0701003-1-10-6': 'Spectra were also measured at the laboratory angle of 22[MATH] , which is close to the minimum for L=0 angular distributions, though the location and depth of the minimum is very sensitive to the Q-value and the distorting parameters as is seen in Figure [REF].', 'nucl-ex-0701003-1-10-7': 'Nevertheless, the ratio of the 3[MATH] to the 22[MATH] cross sections is huge compared to other L values, and is therefore an excellent identifier of L=0 transitions, though the precise value of this ratio will depend on the exact location and depth of the sharp minimum in the angular distribution.', 'nucl-ex-0701003-1-11-0': 'Representative spectra from the 3[MATH] measurements are shown in Figure [REF] where the ground-state transitions are clearly seen to dominate.', 'nucl-ex-0701003-1-11-1': 'The results of the cross-section measurements are shown in Table [REF] below, with ratios to the ground-state cross sections given only for states with yields at 3[MATH] larger than 1% of that for the ground states.', 'nucl-ex-0701003-1-11-2': 'The transitions where the ratio of cross sections between 3[MATH] and 22[MATH] is consistent with L=0 are shown in bold.', 'nucl-ex-0701003-1-11-3': 'More complete data, though at further back angles and with less attention to accurate relative cross sections had been reported, but the emphasis in these previous measurements was on angular distributions starting at 7.5[MATH], and only angle-integrated cross sections are quoted [CITATION].', 'nucl-ex-0701003-1-11-4': 'The uncertainty in the present experimental cross sections is believed to be [MATH]10, while that in the relative values is estimated at [MATH]5 .', 'nucl-ex-0701003-1-11-5': 'These uncertainties are dominated by estimates of systematic errors, (constancy of the beam spot on target, accuracy of angle determinations in the monitors, possible small drifts in monitor calibration, possible inefficiency in the focal plane detector, uniformity of target thickness etc.), while the statistical contribution is of the order of 1.', 'nucl-ex-0701003-1-12-0': 'The (p,t) differential cross sections at 3[MATH] for populating the 0[MATH] ground states for the four targets are very similar: 6.4, 6.7, 6.0, and 7.1 mb/sr for [MATH]Ge and [MATH]Se respectively.', 'nucl-ex-0701003-1-12-1': 'Excited 0[MATH] states stand out in the ratio between the 3[MATH] and 22[MATH] yields, which is an order of magnitude larger than for any other excited state.', 'nucl-ex-0701003-1-12-2': 'With the exception of the [MATH]Ge target, none of these excited 0[MATH] states is populated with a cross section at 3[MATH] that is more than 2% of that leading to the ground states.', 'nucl-ex-0701003-1-12-3': 'In the [MATH]Ge(p,t)[MATH]Ge reaction the cross section to the first-excited 0[MATH] state is 1.9 mb/sr.', 'nucl-ex-0701003-1-12-4': 'This feature is well known [CITATION] as an example of a pairing vibration.', 'nucl-ex-0701003-1-12-5': 'The case of [MATH]Ge is illustrative of effects that can be problematic; however the context of the current work is related only to the [MATH]Ge/[MATH]Se double-beta-decay system.', 'nucl-ex-0701003-1-13-0': '# DWBA Calculations', 'nucl-ex-0701003-1-14-0': 'The Q-values for the (p,t) reaction on the various targets are different.', 'nucl-ex-0701003-1-14-1': 'Therefore, the dependence of the reaction mechanism on those values has to be taken into account in order to use these cross sections to compare the extent of pair correlations in the ground states of these nuclei.', 'nucl-ex-0701003-1-15-0': 'The calculations were carried out with the DWBA program PTOLEMY [CITATION].', 'nucl-ex-0701003-1-15-1': 'The form factor for the neutron pair was calculated assuming a mass-2, [MATH] di-neutron bound in a Woods-Saxon potential with the appropriate binding energy and having 3 nodes in its wave function.', 'nucl-ex-0701003-1-15-2': 'The proton potentials were those of Ref. [CITATION] and the triton potential that of Ref. [CITATION].', 'nucl-ex-0701003-1-15-3': 'The DWBA calculations, were multiplied by C[MATH]S, where S is the number of pairs in the active orbits outside N=28 (S=8 for [MATH]Ge and [MATH]Se, 7 for [MATH]Ge and [MATH]Se), and C[MATH] is a small correction for isospin.', 'nucl-ex-0701003-1-15-4': 'The values are given in Table [REF] together with the ratio of the experimental cross sections to the calculated values.', 'nucl-ex-0701003-1-15-5': 'The absolute magnitude of the DWBA cross section is very sensitive to the choice of the distorting potentials and so is the location of the first minimum in the angular distribution.', 'nucl-ex-0701003-1-15-6': 'However, all calculations indicate that the relative values between these targets of the calculated cross sections at 3[MATH] remain the same (within 10%) with various reasonable potentials for protons and tritons.', 'nucl-ex-0701003-1-16-0': 'With the proton potential of Ref. [CITATION], the average ratio changes from about 32 to about 20.', 'nucl-ex-0701003-1-16-1': 'The experimental cross sections are remarkably constant to start with, without any correction for reaction dynamics they vary by [MATH]5%.', 'nucl-ex-0701003-1-16-2': 'Dividing the experimental numbers by the spectroscopic factors and the negligibly small variation of calculated reaction cross sections reduces this difference between [MATH]Ge and [MATH]Se even further.', 'nucl-ex-0701003-1-16-3': 'However, this improvement is probably not significant, in view of the estimated 5% relative error.', 'nucl-ex-0701003-1-16-4': 'The peak cross sections are also shown in Figure [REF].', 'nucl-ex-0701003-1-17-0': '# Pair-adding reactions', 'nucl-ex-0701003-1-18-0': 'The pair-adding (t,p) reaction had been studied previously, for both the [MATH]Ge(t,p)[MATH]Ge [CITATION] and the [MATH]Se(t,p)[MATH]Se [CITATION] reactions.', 'nucl-ex-0701003-1-18-1': 'While the ground-state transitions are the same as the ones studied here, transitions to excited states could, in principle, show up and indicate pairing vibrations.', 'nucl-ex-0701003-1-18-2': 'While the angular distributions were not measured at as far forward angles as in the present work, some estimate can be obtained by comparing integral cross sections.', 'nucl-ex-0701003-1-18-3': 'No excited 0[MATH] states were seen in these (t,p) studies with a strength greater than 4% of the ground-state transition, confirming the dominance of pair correlations in the ground states of these nuclei, with no splitting into pairing vibrations.', 'nucl-ex-0701003-1-19-0': '# Conclusions', 'nucl-ex-0701003-1-20-0': 'Our experimental cross sections are remarkably constant for ground-state transitions with the various targets.', 'nucl-ex-0701003-1-20-1': 'The ratio of the ground-state transitions to DWBA calculations is constant to better than the 5% estimated accuracy of the measurements, with the simplest assumptions about spectroscopic factors for pair transfer outside N=28.', 'nucl-ex-0701003-1-20-2': 'Transitions to excited 0[MATH] states from [MATH]Ge and [MATH]Se targets are no more than a few percent of the ground-state transitions indicating no sign of the pairing vibrations that appear in some of the lighter isotopes, and that would complicate a simple picture of the ground states.', 'nucl-ex-0701003-1-20-3': 'The constancy of the ground-state strength in pair correlations seems to be as true for neutron-pair-adding transfers leading to these nuclei as it is for pair removal from them.', 'nucl-ex-0701003-1-20-4': 'The present results suggest that the ground states of [MATH]Ge and [MATH]Se exhibit quantitatively very similar neutron pair-correlations.', 'nucl-ex-0701003-1-20-5': 'Changes in pairing are thus unlikely to be a significant complicating factor in the wave functions of these states for calculations of neutrinoless double [MATH] decay.', 'nucl-ex-0701003-1-21-0': 'We wish to acknowledge helpful discussions with S.C. Pieper and M.H. Macfarlane.', 'nucl-ex-0701003-1-21-1': 'The work was supported by the U.S. Department of Energy, Office of Nuclear Physics, under contracts DE-FG02-91ER-40609 and DE-AC02-06CH11357, the UK Engineering and Physical Sciences Research Council, and IN2P3/CNRS-France.', 'nucl-ex-0701003-1-22-0': 'Ba04 J.N. Bahcall, H. Murayama, and C. Pena-Garay, Phys.', 'nucl-ex-0701003-1-22-1': 'Yo62 S. Yoshida, Nucl.', 'nucl-ex-0701003-1-22-2': 'Be66 D.R. Bes and R.A. Broglia, Nucl.', 'nucl-ex-0701003-1-22-3': 'Gu77 D. Guilbault et al., Phys.', 'nucl-ex-0701003-1-22-4': 'Ar78 D. Ardouin et al., Phys.', 'nucl-ex-0701003-1-22-5': 'Bo77 M. Borsaru et al., Nucl.', 'nucl-ex-0701003-1-22-6': 'Le79 C. Lebrun et al., Phys.', 'nucl-ex-0701003-1-22-7': 'Wa87 D.L. Watson and H.T. Fortune, Phys.', 'nucl-ex-0701003-1-22-8': 'PT M.H. Macfarlane and Steven C. Pieper, ANL-76-11 Rev. 1 Argonne National Laboratory Report (1978) unpublished.', 'nucl-ex-0701003-1-22-9': 'Be69 F.D. Becchetti and G.W. Greenlees, Phys.', 'nucl-ex-0701003-1-22-10': 'Pe81 R. Perry, Phys.'}

{'nucl-ex-0701003-2-0-0': 'Precision measurements were carried out to test the similarities between the ground states of [MATH]Ge and [MATH]Se.', 'nucl-ex-0701003-2-0-1': 'The extent to which these two nuclei can be characterized as consisting of correlated pairs of neutrons in a BCS-like ground state was studied.', 'nucl-ex-0701003-2-0-2': 'The pair removal (p,t) reaction was measured at the far forward angle of 3[MATH].', 'nucl-ex-0701003-2-0-3': 'The relative cross sections are consistent (at the 5% level) with the description of these nuclei in terms of a correlated pairing state outside the N=28 closed shells with no pairing vibrations.', 'nucl-ex-0701003-2-0-4': 'Data were also obtained for [MATH]Ge and [MATH]Se.', 'nucl-ex-0701003-2-1-0': '# introduction', 'nucl-ex-0701003-2-2-0': 'Interest in the possibility of observing neutrinoless double beta decay (0[MATH]) is considerable.', 'nucl-ex-0701003-2-2-1': 'If this decay were to be definitively observed, it would show that the neutrinos are their own antiparticles.', 'nucl-ex-0701003-2-2-2': 'In addition, the rate of the decay would be a measure of the neutrino rest mass, if the nuclear matrix element were known.', 'nucl-ex-0701003-2-2-3': 'Unfortunately, theoretical calculations of this do not agree well with each other [CITATION].', 'nucl-ex-0701003-2-2-4': 'It seems appropriate to determine additional properties of the ground states of the possible 0[MATH] systems by experiment, and thus help constrain and test theoretical calculations of this exotic decay mode.', 'nucl-ex-0701003-2-2-5': 'One of the likely candidate nuclei is [MATH]Ge decaying to the ground state of [MATH]Se.', 'nucl-ex-0701003-2-2-6': 'We have started with a study of the properties of the ground states of these nuclei, and especially the similarities and differences between them, using transfer reactions.', 'nucl-ex-0701003-2-2-7': 'One part of this study is an accurate measurement of one-nucleon transfer in order to probe the occupation numbers of valence orbits for both neutrons and protons, with particular attention to changes in these occupations.', 'nucl-ex-0701003-2-2-8': 'The other part is to study pair-correlations in these nuclei by nucleon pair transfer.', 'nucl-ex-0701003-2-2-9': 'Here we report on a comparison of neutron pair transfer from the (p,t) reaction.', 'nucl-ex-0701003-2-2-10': 'We hope to obtain similar data on proton pair correlations from ([MATH]He,n) reactions in a future experiment.', 'nucl-ex-0701003-2-3-0': 'Pair transfer between 0[MATH] states in the (p,t) or (t,p) reaction proceeds via L=0 transfer and the angular distribution, for energies above the Coulomb barrier, is sharply forward peaked.', 'nucl-ex-0701003-2-3-1': 'This feature was recognized early [CITATION] and was crucial in exploring the importance of such correlations and their excitations, the so-called pairing vibrations [CITATION].', 'nucl-ex-0701003-2-3-2': 'The latter are an indication of deviations from the simplest pairing picture and can occur in regions of changing shapes, or when there is a gap in single-particle states, such as near a shell closure.', 'nucl-ex-0701003-2-4-0': 'We have carried out measurements of the neutron pair-removal (p,t) reaction on targets of [MATH]Ge and [MATH]Se.', 'nucl-ex-0701003-2-4-1': 'The reaction on [MATH]Se was measured because the [MATH]Se(p,t)[MATH]Se leads to [MATH]Se, while [MATH]Se(p,t)[MATH]Se starts from the ground state of [MATH]Se.', 'nucl-ex-0701003-2-4-2': 'Thus both reactions are relevant to the pairing structure of this ground state.', 'nucl-ex-0701003-2-4-3': 'The [MATH]Ge target was included as a check.', 'nucl-ex-0701003-2-5-0': 'There are two relevant aspects to these measurements.', 'nucl-ex-0701003-2-5-1': 'The first is the matter of pairing vibrations.', 'nucl-ex-0701003-2-5-2': 'The even Ge and Se isotopes are well studied and evidence for excited 0[MATH] states has been established [CITATION] .', 'nucl-ex-0701003-2-5-3': 'In some of the lighter isotopes of Ge and Se, two-neutron transfer reactions have shown significant strength populating excited 0[MATH] states.', 'nucl-ex-0701003-2-5-4': 'These pairing vibrations indicate that there are significant BCS-like pair correlations in connecting the target ground state in an even initial nucleus to excited 0[MATH] states in the final.', 'nucl-ex-0701003-2-5-5': 'If there were significant differences in this regard between reactions leading to or from [MATH]Ge and [MATH]Se, this would be an indication that the pair correlations in the initial and final states in double-beta decay differ.', 'nucl-ex-0701003-2-5-6': 'Any reliable calculations of the process would presumably have to reproduce such differences in order to obtain a reasonable matrix element for the 0[MATH] decay.', 'nucl-ex-0701003-2-5-7': 'Secondly, if accurate cross sections were available for pair transfer further checks could be made of the similarity between the initial and final wave functions used in such calculations.', 'nucl-ex-0701003-2-6-0': '# Experimental Considerations', 'nucl-ex-0701003-2-7-0': 'Given that the L=0 (p,t) transitions are the strongest in the spectrum of final states at very forward angles, which is also the region where the approximations inherent in the distorted wave Born approximation (DWBA) are best satisfied, it is desirable to carry out these measurements as close to 0[MATH] as feasible.', 'nucl-ex-0701003-2-7-1': 'The (p,t) reaction had been studied previously on Ge [CITATION] and Se [CITATION] isotopes; relevant (t,p) measurements have also been made [CITATION].', 'nucl-ex-0701003-2-7-2': 'Both the experimental methodology and the data analysis methods in the two (p,t) studies were different (the target thickness was estimated from high-energy elastic scattering where optical model predictions are ambiguous, the angular distributions start at 7.5 degrees, and only angle-integrated cross sections are quoted, etc.), making a reliable systematic comparison of Ge and Se data difficult.', 'nucl-ex-0701003-2-7-3': 'Our purpose here was to measure the cross sections at as far forward angles as feasible and to obtain a consistent set of accurate cross sections with particular care taken to reduce relative systematic uncertainties.', 'nucl-ex-0701003-2-8-0': 'The choice of energy was governed by the desirability of having both protons and tritons well above the Coulomb barrier.', 'nucl-ex-0701003-2-8-1': 'For the present measurement we therefore chose 23-MeV protons from the Yale ESTU tandem Van de Graaf accelerator.', 'nucl-ex-0701003-2-8-2': 'This energy is similar to that used in earlier experiments, both for (p,t) and (t,p) reactions.', 'nucl-ex-0701003-2-8-3': 'Because one of the objectives of the present measurement was to obtain accurate cross sections, we chose to measure the thickness of the evaporated germanium and selenium targets in situ by simply lowering the proton beam energy to 6 MeV, where the elastic scattering cross sections are very close to Rutherford values.', 'nucl-ex-0701003-2-8-4': 'The angle should not be so far forward that small uncertainties in angle would become significant and 30[MATH] was chosen because calculations with several optical potentials showed that the deviation from Rutherford scattering was less than 2%.', 'nucl-ex-0701003-2-8-5': 'The [MATH]Ge and [MATH]Se target thicknesses were found to range between 160 and 400 [MATH]g/cm[MATH].', 'nucl-ex-0701003-2-8-6': 'Since Se can sublimate at a relatively low temperature, this low-energy target thickness measurement was made at the beginning and then at the end of the experiment; no significant differences ([MATH]3%) were observed.', 'nucl-ex-0701003-2-8-7': 'The highest beam currents used were about 35 nA, though considerably lower (2-3 nA) for the 3[MATH] measurements.', 'nucl-ex-0701003-2-9-0': 'The Yale Enge split-pole spectrograph was used for the measurements with a focal-plane detector that cleanly separates tritons from other reaction products.', 'nucl-ex-0701003-2-9-1': 'As monitors, two Si surface barrier detectors at [MATH] were used.', 'nucl-ex-0701003-2-9-2': 'They were calibrated in terms of beam intensity using a current integrator connected to a Faraday cup.', 'nucl-ex-0701003-2-9-3': 'The beam integrator was set to the same scale that was used for the low-energy measurements and the solid-angle setting for the aperture of the spectrograph was also the same, thus establishing a relationship between an absolute cross section scale in terms of the monitor yields, instead of the beam intergrator, for each target.', 'nucl-ex-0701003-2-10-0': 'The 3[MATH] setting for the spectrograph required that the Faraday cup be retracted so that the beam-current measurement had to rely on the previously calibrated monitor counters.', 'nucl-ex-0701003-2-10-1': 'Removal of the Faraday cup meant that the beam entered the spectrometer.', 'nucl-ex-0701003-2-10-2': 'The magnetic rigidity of the tritons from the reaction is such that, with the magnetic field set for observing tritons, protons from the target cannot directly reach the focal plane and are intercepted inside the spectrometer.', 'nucl-ex-0701003-2-10-3': 'Protons scattered at this point can enter the focal plane and, whilst they can be distinguished from tritons by their ionization density, they do impose a counting rate limit.', 'nucl-ex-0701003-2-10-4': 'As a result, the furthest forward angle where measurements could be made was 3[MATH].', 'nucl-ex-0701003-2-10-5': 'Distorted-wave calculations indicate that the cross section at this angle is lower than that at 0[MATH] by about 8%.', 'nucl-ex-0701003-2-10-6': 'Spectra were also measured at the laboratory angle of 22[MATH] , which is close to the minimum for L=0 angular distributions, though the location and depth of the minimum is very sensitive to the Q-value and the distorting parameters as is seen in Figure [REF].', 'nucl-ex-0701003-2-10-7': 'Nevertheless, the ratio of the 3[MATH] to the 22[MATH] cross sections is huge compared to that for the other L values, and is therefore an excellent identifier of L=0 transitions, though the precise value of this ratio will depend on the exact location and depth of the sharp minimum in the angular distribution.', 'nucl-ex-0701003-2-11-0': 'Representative spectra from the 3[MATH] measurements are shown in Figure [REF] where the ground-state transitions are clearly seen to dominate.', 'nucl-ex-0701003-2-11-1': 'The results of the cross-section measurements are shown in Table [REF] below, with ratios to the ground-state cross sections given only for states with yields, at 3[MATH], larger than 1% of the ground state yield.', 'nucl-ex-0701003-2-11-2': 'The transitions where the ratio of cross sections between 3[MATH] and 22[MATH] is consistent with L=0 are shown in bold.', 'nucl-ex-0701003-2-11-3': 'More complete data, though at further back angles and with less attention to accurate relative cross sections had been reported, but the emphasis in these previous measurements was on angular distributions starting at 7.5[MATH], and only angle-integrated cross sections are quoted [CITATION].', 'nucl-ex-0701003-2-11-4': 'The uncertainty in the present experimental cross sections is believed to be [MATH]10, while that in the relative values is estimated at [MATH]5 .', 'nucl-ex-0701003-2-11-5': 'These uncertainties are dominated by estimates of systematic errors (constancy of the beam spot on target, accuracy of angle determinations in the monitors, possible small drifts in monitor calibration, possible inefficiency in the focal plane detector, uniformity of target thickness etc.) while the statistical contribution is of the order of 1.', 'nucl-ex-0701003-2-12-0': 'The (p,t) differential cross sections at 3[MATH] for populating the 0[MATH] ground states for the four targets are very similar: 6.4, 6.7, 6.0, and 7.1 mb/sr for [MATH]Ge and [MATH]Se respectively.', 'nucl-ex-0701003-2-12-1': 'Excited 0[MATH] states stand out in the ratio between the 3[MATH] and 22[MATH] yields, which is an order of magnitude larger than for any other excited state.', 'nucl-ex-0701003-2-12-2': 'With the exception of the [MATH]Ge target, none of these excited 0[MATH] states is populated with a cross section at 3[MATH] that is more than 2% of that leading to the ground states.', 'nucl-ex-0701003-2-12-3': 'In the [MATH]Ge(p,t)[MATH]Ge reaction the cross section to the first-excited 0[MATH] state is 1.9 mb/sr.', 'nucl-ex-0701003-2-12-4': 'This feature is well known [CITATION] as an example of a pairing vibration.', 'nucl-ex-0701003-2-12-5': 'The case of [MATH]Ge is illustrative of effects that can be problematic; however the context of the current work is related only to the [MATH]Ge/[MATH]Se double-beta-decay system.', 'nucl-ex-0701003-2-13-0': '# DWBA Calculations', 'nucl-ex-0701003-2-14-0': 'The calculations were carried out with the DWBA program PTOLEMY [CITATION] to correct the dependence of the reaction on Q-values.', 'nucl-ex-0701003-2-14-1': 'The consideration of the details of nuclear structure is beyond the scope of this study, even though [MATH]Ge and [MATH]Se have 6 neutron vacancies in the N=50 shell, [MATH]Ge and [MATH]Se have 8.', 'nucl-ex-0701003-2-14-2': 'The form factor for the neutron pair was calculated assuming a mass-2, [MATH] di-neutron bound in a Woods-Saxon potential with the appropriate binding energy and having 3 nodes in its wave function.', 'nucl-ex-0701003-2-14-3': 'The proton potentials were those of Ref. [CITATION] and the triton potential that of Ref. [CITATION].', 'nucl-ex-0701003-2-14-4': 'The measured cross sections at 3 degrees are given in Table [REF] together with the ratio of the experimental cross sections to the calculated values.', 'nucl-ex-0701003-2-14-5': 'The absolute magnitude of the DWBA cross section is very sensitive to the choice of the distorting potentials as is the location of the first minimum in the angular distribution.', 'nucl-ex-0701003-2-14-6': 'However, all calculations indicate that the relative values between these targets of the calculated cross sections at 3[MATH] remain the same (within 10%) with various reasonable potentials for protons and tritons.', 'nucl-ex-0701003-2-15-0': 'With the proton potential of Ref. [CITATION], the average ratio changes from 136 to 217.', 'nucl-ex-0701003-2-15-1': 'The experimental cross sections by themselves are remarkably constant; without any correction for reaction dynamics they vary by [MATH]6%.', 'nucl-ex-0701003-2-15-2': 'Dividing the experimental numbers by the calculated reaction cross sections, reduced the difference between [MATH]Ge and [MATH]Se even further.', 'nucl-ex-0701003-2-15-3': 'However, this improvement is probably not significant, in view of the estimated 5% relative error and the neglect of the different numbers of neutrons.', 'nucl-ex-0701003-2-15-4': 'The peak cross sections and DWBA trends are also shown in Figure [REF].', 'nucl-ex-0701003-2-16-0': '# Pair-adding reactions', 'nucl-ex-0701003-2-17-0': 'The pair-adding (t,p) reaction had been studied previously, for both the [MATH]Ge(t,p)[MATH]Ge [CITATION] and the [MATH]Se(t,p)[MATH]Se [CITATION] reactions.', 'nucl-ex-0701003-2-17-1': 'While the ground-state transitions are the same as the ones studied here, transitions to excited states could, in principle, show up and indicate pairing vibrations.', 'nucl-ex-0701003-2-17-2': 'While the angular distributions were not measured at as far forward angles as in the present work, some estimate can be obtained by comparing integral cross sections.', 'nucl-ex-0701003-2-17-3': 'No excited 0[MATH] states were seen in these (t,p) studies with a strength greater than 4% of the ground-state transition, confirming the dominance of pair correlations in the ground states of these nuclei, with no splitting into pairing vibrations.', 'nucl-ex-0701003-2-18-0': '# Conclusions', 'nucl-ex-0701003-2-19-0': 'The experimental cross sections measured in this experiment are remarkably constant for ground-state transitions with the various targets.', 'nucl-ex-0701003-2-19-1': 'The difference between [MATH]Ge and [MATH]Se, in the ratio of the experimental ground-state transition strengths divided by the appropriate DWBA calculations, is less than the 5% estimated accuracy of the measurements.', 'nucl-ex-0701003-2-19-2': 'Transitions to excited 0[MATH] states from [MATH]Ge and [MATH]Se targets are no more than a few percent of the ground-state transitions indicating no sign of the pairing vibrations that appear in some of the lighter isotopes.', 'nucl-ex-0701003-2-19-3': 'Had such admixtures been present, this would have complicated a simple comparison of the ground states.', 'nucl-ex-0701003-2-19-4': 'The constancy of the ground-state strength in pair correlations seems to be as true for neutron-pair-adding transfers leading to these nuclei as it is for pair removal from them.', 'nucl-ex-0701003-2-19-5': 'The present results suggest that the ground states of [MATH]Ge and [MATH]Se exhibit quantitatively very similar neutron pair-correlations.', 'nucl-ex-0701003-2-19-6': 'Changes in pairing are thus unlikely to be a significant complicating factor in the wave functions of these states for calculations of neutrinoless double [MATH] decay.', 'nucl-ex-0701003-2-20-0': 'We wish to acknowledge helpful discussions with S.C. Pieper, Ben Bayman, and M.H. Macfarlane.', 'nucl-ex-0701003-2-20-1': 'The help of John Greene in meticulous work on target preparation is also acknowledged.', 'nucl-ex-0701003-2-20-2': 'The work was supported by the U.S. Department of Energy, Office of Nuclear Physics, under contracts DE-FG02-91ER-40609 and DE-AC02-06CH11357, the UK Engineering and Physical Sciences Research Council, and IN2P3/CNRS-France.', 'nucl-ex-0701003-2-21-0': 'Ba04 J.N. Bahcall, H. Murayama, and C. Pena-Garay, Phys.', 'nucl-ex-0701003-2-21-1': 'Yo62 S. Yoshida, Nucl.', 'nucl-ex-0701003-2-21-2': 'Be66 D.R. Bes and R.A. Broglia, Nucl.', 'nucl-ex-0701003-2-21-3': 'Gu77 D. Guilbault et al., Phys.', 'nucl-ex-0701003-2-21-4': 'Ar78 D. Ardouin et al., Phys.', 'nucl-ex-0701003-2-21-5': 'Bo77 M. Borsaru et al., Nucl.', 'nucl-ex-0701003-2-21-6': 'Le79 C. Lebrun et al., Phys.', 'nucl-ex-0701003-2-21-7': 'Wa87 D.L. Watson and H.T. Fortune, Phys.', 'nucl-ex-0701003-2-21-8': 'PT M.H. Macfarlane and Steven C. Pieper, ANL-76-11 Rev. 1 Argonne National Laboratory Report (1978) unpublished.', 'nucl-ex-0701003-2-21-9': 'Be69 F.D. Becchetti and G.W. Greenlees, Phys.', 'nucl-ex-0701003-2-21-10': 'Pe81 R. Perry, Phys.'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/nucl-ex/0701003

1905.10120

{'1905.10120-1-0-0': 'We consider a transitive action of a finitely generated group [MATH] and the Schreier graph [MATH] it defines.', '1905.10120-1-0-1': 'For a probability measure [MATH] on [MATH] with a finite first moment we show that if the induced random walk is transient, it converges towards the space of ends of [MATH].', '1905.10120-1-0-2': "As a corollary we obtain that for a probability measure on Thompson's group [MATH] with a finite first moment, the support of which generates [MATH] as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary.", '1905.10120-1-0-3': 'Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.', '1905.10120-1-1-0': "Keywords- Random walks on groups, Poisson boundary, Schreier graph, Thompson's group [MATH]", '1905.10120-1-2-0': '# Introduction', '1905.10120-1-3-0': 'Consider a finitely generated group [MATH] acting on a space [MATH] (on the right).', '1905.10120-1-3-1': 'For a point [MATH] and a generating set [MATH], the Schreier graph [MATH] is the graph the vertex set of which is the orbit [MATH] of [MATH], and the edges [MATH] are the couples of the form [MATH] for [MATH] and [MATH].', '1905.10120-1-3-2': 'Throughout this article, we will assume the action to be transitive, that is for every [MATH], [MATH].', '1905.10120-1-3-3': 'We take a measure [MATH] on [MATH] and will study for which [MATH] the induced random walk on [MATH] converges towards an end of the graph.', '1905.10120-1-3-4': 'We recall the definition of the end space.', '1905.10120-1-3-5': 'Consider an exhaustive increasing sequence [MATH] of finite subsets of [MATH].', '1905.10120-1-3-6': 'An end of [MATH] is a sequence [MATH] where [MATH] is an infinite connected component of the subgraph obtained by deleting the vertices in [MATH] and adjacent edges.', '1905.10120-1-3-7': 'For more details, see Definition [REF].', '1905.10120-1-4-0': 'Consider a finitely generated group [MATH] acting transitively on a space [MATH].', '1905.10120-1-4-1': 'Fix a generating set [MATH] and let [MATH] be the associated Schreier graph.', '1905.10120-1-4-2': 'Let [MATH] be a measure on [MATH] with a finite first moment such that the induced random walk on [MATH] is transient.', '1905.10120-1-4-3': 'Then the random walk almost surely converges towards a (random) end of the graph.', '1905.10120-1-5-0': 'Notice that for measures with finite support, the result is straightforward.', '1905.10120-1-5-1': 'It is also already known for the case where the action of [MATH] on [MATH] is non-amenable (see Woess [CITATION]), under the condition of a finite first moment.', '1905.10120-1-5-2': 'Non-amenable action means that there is no invariant mean on [MATH], or equivalently, that for the induced random walk on [MATH], the probability of return at the origin decreases exponentially.', '1905.10120-1-6-0': 'For a non-amenable action and non-degenerate measure, the random walk is always transient (see [CITATION]).', '1905.10120-1-6-1': 'In the general case, transience can sometimes be obtained from the graph geometry using a comparison Lemma [REF] due to Baldi-Lohoué-Peyrière [CITATION].', '1905.10120-1-6-2': 'Combining this lemma and the theorem we obtain:', '1905.10120-1-7-0': 'Consider a finitely generated group [MATH] acting transitively on a space [MATH].', '1905.10120-1-7-1': 'Fix a generating set [MATH] and let [MATH] be the associated Schreier graph.', '1905.10120-1-7-2': 'Assume that [MATH] is a transient graph.', '1905.10120-1-7-3': 'Then for all measures [MATH] on [MATH] with finite first moments such that [MATH] generates [MATH] as a semigroup, the induced random walk almost surely converges towards an end of the graph.', '1905.10120-1-8-0': "We will also explain how this result can be applied to Thompson's group [MATH].", '1905.10120-1-8-1': 'Let us recall the definition of this group.', '1905.10120-1-8-2': 'The set of dyadic rationals [MATH] is the set of numbers of the form [MATH] with [MATH].', '1905.10120-1-8-3': "Thompson's group [MATH] is the group of orientation-preserving piecewise linear self-isomorphisms of the closed unit interval with dyadic slopes, with a finite number of break points, all break points being in [MATH].", '1905.10120-1-8-4': 'It is a finitely generated group with a canonical generating set (with two elements).', '1905.10120-1-8-5': "See Cannon-Floyd-Perry [CITATION] or Meier's book [CITATION] for details and properties.", '1905.10120-1-8-6': 'Its amenability is a celebrated open question.', '1905.10120-1-8-7': 'It is well known that amenability is equivalent to the existence of a non-degenerate measure with trivial Poisson boundary (see Kaimanovich-Vershik [CITATION], Rosenblatt [CITATION]).', '1905.10120-1-8-8': 'The boundary of a random walk induced by an action is a quotient of the boundary on the group.', '1905.10120-1-9-0': 'The Schreier graph on [MATH] (of a conjugate action of [MATH]) has been described by Savchuk [CITATION].', '1905.10120-1-9-1': 'It is a tree that can be understood as a combination of a skeleton quasi-isometric to a binary tree, and rays attached at each point of the skeleton (see Figure [REF]).', '1905.10120-1-9-2': 'Understanding the geometry of the graph directly shows that it is transient.', '1905.10120-1-9-3': 'Kaimanovich [CITATION] also proves this result without using the geometry of the graph.', '1905.10120-1-9-4': 'Hence by Corollary [REF] we obtain', '1905.10120-1-10-0': "Consider a measure on Thompson's group [MATH] with a finite first moment, the support of which generates [MATH] as a semigroup.", '1905.10120-1-10-1': 'Then the induced random walk on [MATH] has non-trivial Poisson boundary.', '1905.10120-1-11-0': 'This extends the following previous results.', '1905.10120-1-11-1': 'Kaimanovich [CITATION] and Mishchenko [CITATION] prove that the simple random walk on the Schreier graph given by that action has non-trivial boundary.', '1905.10120-1-11-2': 'Kaimanovich [CITATION] further shows that it is non-trivial for walks induced by measures with supports that are finite and generate [MATH] as a semigroup.', '1905.10120-1-11-3': 'We have also shown [CITATION] that for any measure with a finite first moment on [MATH], the support of which generates [MATH] as a semigroup, the walk on the group has non-trivial Poisson boundary.', '1905.10120-1-12-0': 'The result of the corollary is false without assuming a finite first moment.', '1905.10120-1-12-1': 'Juschenko and Zheng [CITATION] have proven that there exists a symmetric non-degenerate measure on [MATH] such that the induced random walk has trivial Poisson boundary.', '1905.10120-1-12-2': 'If the trajectories almost surely converge towards points on the end space, the end space endowed the exit measure on it is a quotient of the Poisson boundary.', '1905.10120-1-12-3': 'However, the self-similarity of the graph implies that the exit measure cannot be trivial, as we prove in Lemma [REF].', '1905.10120-1-12-4': 'Combining the result of Juschenko-Zheng with this lemma we obtain:', '1905.10120-1-13-0': 'There exists a finitely generated group [MATH], a space [MATH] and a symmetric non-degenerate measure on [MATH] such that', '1905.10120-1-14-0': "In particular, the measure described by Juschenko and Zheng [CITATION] provides an example for the action of Thompson's group [MATH] on [MATH].", '1905.10120-1-15-0': "Concerning Thompson's group [MATH], studying the Poisson boundary of random walks on it has been highlighted as a possible approach to proving non-amenability in the work of Kaimanovich.", '1905.10120-1-15-1': 'The results by him and Mischenko further suggested that one could consider the boundary of induced random walks [MATH], but that was shown impossible by the result of Juschenko-Zheng.', '1905.10120-1-15-2': 'In more recent results, Juschenko [CITATION] studied walks on the space of [MATH]-element subsets of [MATH] and gave a combinatorial necessary and sufficient condition for the Poisson boundary of induced walks on that space to be non-trivial for all non-degenerate measures.', '1905.10120-1-15-3': 'In that situation, the existence a measure with trivial boundary is due to Juschenko for [MATH] and to Schneider and Thom [CITATION] for a general [MATH].'}

{'1905.10120-2-0-0': 'We consider a transitive action of a finitely generated group [MATH] and the Schreier graph [MATH] it defines.', '1905.10120-2-0-1': 'For a probability measure [MATH] on [MATH] with a finite first moment we show that if the induced random walk is transient, it converges towards the space of ends of [MATH].', '1905.10120-2-0-2': "As a corollary we obtain that for a probability measure on Thompson's group [MATH] with a finite first moment, the support of which generates [MATH] as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary.", '1905.10120-2-0-3': 'Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.', '1905.10120-2-1-0': "Keywords- Random walks on groups, Poisson boundary, Schreier graph, Thompson's group [MATH]", '1905.10120-2-2-0': '# Introduction', '1905.10120-2-3-0': 'Consider a finitely generated group [MATH] acting on a space [MATH] (on the right).', '1905.10120-2-3-1': 'For a point [MATH] and a generating set [MATH], the Schreier graph [MATH] is the graph the vertex set of which is the orbit [MATH] of [MATH], and the edges [MATH] are the couples of the form [MATH] for [MATH] and [MATH].', '1905.10120-2-3-2': 'Throughout this article, we will assume the action to be transitive, that is for every [MATH], [MATH].', '1905.10120-2-3-3': 'We take a measure [MATH] on [MATH] and will study for which [MATH] the induced random walk on [MATH] converges towards an end of the graph.', '1905.10120-2-3-4': 'We recall the definition of the end space.', '1905.10120-2-3-5': 'Consider an exhaustive increasing sequence [MATH] of finite subsets of [MATH].', '1905.10120-2-3-6': 'An end of [MATH] is a sequence [MATH] where [MATH] is an infinite connected component of the subgraph obtained by deleting the vertices in [MATH] and adjacent edges.', '1905.10120-2-3-7': 'For more details, see Definition [REF].', '1905.10120-2-4-0': 'Consider a finitely generated group [MATH] acting transitively on a space [MATH].', '1905.10120-2-4-1': 'Fix a generating set [MATH] and let [MATH] be the associated Schreier graph.', '1905.10120-2-4-2': 'Let [MATH] be a measure on [MATH] with a finite first moment such that the induced random walk on [MATH] is transient.', '1905.10120-2-4-3': 'Then the random walk almost surely converges towards a (random) end of the graph.', '1905.10120-2-5-0': 'Notice that for measures with finite support, the result is straightforward.', '1905.10120-2-5-1': 'It is also already known for the case where the action of [MATH] on [MATH] is non-amenable (see Woess [CITATION]), under the condition of a finite first moment.', '1905.10120-2-5-2': 'Non-amenable action means that there is no invariant mean on [MATH], or equivalently, that for the induced random walk on [MATH], the probability of return at the origin decreases exponentially.', '1905.10120-2-6-0': 'For a non-amenable action and non-degenerate measure, the random walk is always transient (see [CITATION]).', '1905.10120-2-6-1': 'In the general case, transience can sometimes be obtained from the graph geometry using a comparison Lemma [REF] due to Baldi-Lohoué-Peyrière [CITATION].', '1905.10120-2-6-2': 'Combining this lemma and the theorem we obtain:', '1905.10120-2-7-0': 'Consider a finitely generated group [MATH] acting transitively on a space [MATH].', '1905.10120-2-7-1': 'Fix a generating set [MATH] and let [MATH] be the associated Schreier graph.', '1905.10120-2-7-2': 'Assume that [MATH] is a transient graph.', '1905.10120-2-7-3': 'Then for all measures [MATH] on [MATH] with finite first moments such that [MATH] generates [MATH] as a semigroup, the induced random walk almost surely converges towards an end of the graph.', '1905.10120-2-8-0': "We will also explain how this result can be applied to Thompson's group [MATH].", '1905.10120-2-8-1': 'Let us recall the definition of this group.', '1905.10120-2-8-2': 'The set of dyadic rationals [MATH] is the set of numbers of the form [MATH] with [MATH].', '1905.10120-2-8-3': "Thompson's group [MATH] is the group of orientation-preserving piecewise linear self-isomorphisms of the closed unit interval with dyadic slopes, with a finite number of break points, all break points being in [MATH].", '1905.10120-2-8-4': 'It is a finitely generated group with a canonical generating set (with two elements).', '1905.10120-2-8-5': "See Cannon-Floyd-Perry [CITATION] or Meier's book [CITATION] for details and properties.", '1905.10120-2-8-6': 'Its amenability is a celebrated open question.', '1905.10120-2-8-7': 'It is well known that amenability is equivalent to the existence of a non-degenerate measure with trivial Poisson boundary (see Kaimanovich-Vershik [CITATION], Rosenblatt [CITATION]).', '1905.10120-2-8-8': 'The boundary of a random walk induced by an action is a quotient of the boundary on the group.', '1905.10120-2-9-0': 'The Schreier graph on [MATH] (of a conjugate action of [MATH]) has been described by Savchuk [CITATION].', '1905.10120-2-9-1': 'It is a tree that can be understood as a combination of a skeleton quasi-isometric to a binary tree, and rays attached at each point of the skeleton (see Figure [REF]).', '1905.10120-2-9-2': 'Understanding the geometry of the graph directly shows that it is transient.', '1905.10120-2-9-3': 'Kaimanovich [CITATION] also proves this result without using the geometry of the graph.', '1905.10120-2-9-4': 'Hence by Corollary [REF] we obtain', '1905.10120-2-10-0': "Consider a measure on Thompson's group [MATH] with a finite first moment, the support of which generates [MATH] as a semigroup.", '1905.10120-2-10-1': 'Then the induced random walk on [MATH] has non-trivial Poisson boundary.', '1905.10120-2-11-0': 'This extends the following previous results.', '1905.10120-2-11-1': 'Kaimanovich [CITATION] and Mishchenko [CITATION] prove that the simple random walk on the Schreier graph given by that action has non-trivial boundary.', '1905.10120-2-11-2': 'Kaimanovich [CITATION] further shows that it is non-trivial for walks induced by measures with supports that are finite and generate [MATH] as a semigroup.', '1905.10120-2-11-3': 'We have also shown [CITATION] that for any measure with a finite first moment on [MATH], the support of which generates [MATH] as a semigroup, the walk on the group has non-trivial Poisson boundary.', '1905.10120-2-12-0': 'The result of the corollary is false without assuming a finite first moment.', '1905.10120-2-12-1': 'Juschenko and Zheng [CITATION] have proven that there exists a symmetric non-degenerate measure on [MATH] such that the induced random walk has trivial Poisson boundary.', '1905.10120-2-12-2': 'If the trajectories almost surely converge towards points on the end space, the end space endowed the exit measure on it is a quotient of the Poisson boundary.', '1905.10120-2-12-3': 'However, the self-similarity of the graph implies that the exit measure cannot be trivial, as we prove in Lemma [REF].', '1905.10120-2-12-4': 'Combining the result of Juschenko-Zheng with this lemma we obtain:', '1905.10120-2-13-0': 'There exists a finitely generated group [MATH], a space [MATH] and a symmetric non-degenerate measure on [MATH] such that', '1905.10120-2-14-0': "In particular, the measure described by Juschenko and Zheng [CITATION] provides an example for the action of Thompson's group [MATH] on [MATH].", '1905.10120-2-15-0': "Concerning Thompson's group [MATH], studying the Poisson boundary of random walks on it has been highlighted as a possible approach to proving non-amenability in the work of Kaimanovich.", '1905.10120-2-15-1': 'The results by him and Mischenko further suggested that one could consider the boundary of induced random walks [MATH], but that was shown impossible by the result of Juschenko-Zheng.', '1905.10120-2-15-2': 'In more recent results, Juschenko [CITATION] studied walks on the space of [MATH]-element subsets of [MATH] and gave a combinatorial necessary and sufficient condition for the Poisson boundary of induced walks on that space to be non-trivial for all non-degenerate measures.', '1905.10120-2-15-3': 'In that situation, the existence a measure with trivial boundary is due to Juschenko for [MATH] and to Schneider and Thom [CITATION] for a general [MATH].'}

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[]

[]

[]

[]

['1905.10120-1-6-2', '1905.10120-1-12-4', '1905.10120-2-6-2', '1905.10120-2-12-4']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1905.10120

1602.05881

{'1602.05881-1-0-0': 'We study the effective cosmological dynamics, emerging as the hydrodynamics of simple condensate states, of a group field theory model for quantum gravity coupled to a massless scalar field and reduced to its isotropic sector.', '1602.05881-1-0-1': 'The quantum equations of motion for these group field theory condensate states are given in relational terms with respect to the scalar field, from which effective dynamics for spatially flat, homogeneous and isotropic space-times can be extracted.', '1602.05881-1-0-2': 'The result is a generalization of the Friedmann equations, including quantum gravity modifications, in a specific regime of the theory.', '1602.05881-1-0-3': 'The classical Friedmann equations of general relativity are recovered in a suitable semi-classical limit for some range of parameters of the microscopic dynamics.', '1602.05881-1-0-4': 'An important result is that the quantum geometries associated with these GFT condensate states are non-singular: a bounce generically occurs in the Planck regime.', '1602.05881-1-0-5': 'For some choices of condensate states, these modified Friedmann equations are very similar to those of loop quantum cosmology.', '1602.05881-1-1-0': '# Introduction', '1602.05881-1-2-0': 'One of the main challenges for any fundamental theory of quantum gravity that proposes a candidate for the microscopic degrees of freedom for a quantum space-time is to extract its effective macroscopic physics, which should agree with those of general relativity in some suitable approximation.', '1602.05881-1-2-1': "This is often referred to as the 'problem of the continuum' in quantum gravity, since in many approaches the fundamental microscopic degrees of freedom are of a discrete nature (and of no direct spatio-temporal interpretation).", '1602.05881-1-2-2': 'Furthermore, the extraction of effective continuum physics at macroscopic scales is also needed in order to make contact with observations by identifying testable signatures of the proposed microscopic structure of space-time that could be tested at larger scales.', '1602.05881-1-2-3': "The 'problem of the continuum' in quantum gravity is hard, even after one has a promising candidate definition for the microscopic degrees of freedom and their dynamics (which is of course the first hard challenge), because the number of these degrees of freedom in any macroscopic space-time is expected to be extremely large.", '1602.05881-1-2-4': 'This implies that it is necessary to determine the (quantum) dynamics of suitably identified many-body quantum states in the fundamental theory, another difficult task.', '1602.05881-1-3-0': 'One key topic of macroscopic gravitational physics, that on the one hand would benefit from a solid foundation in quantum gravity, and on the other hand is a natural setting where the predictions of quantum gravity effects could realistically be confronted to observations, is cosmology.', '1602.05881-1-3-1': 'Therefore the extraction of an effective cosmological dynamics from fundamental quantum gravity models is a particularly interesting and important problem.', '1602.05881-1-4-0': 'In a cosmological context, for homogeneous space-times a few global observables are enough to characterize the macroscopic configurations and dynamics of the space-time and geometry, despite the large number of fundamental degrees of freedom involved.', '1602.05881-1-4-1': 'Therefore some kind of truncation, in both the description of the relevant quantum states and of their dynamics, is necessary.', '1602.05881-1-4-2': 'Two main types of truncations are typically considered: coarse-graining (roughly, to use ensembles of microscopic degrees of freedom collectively parametrized by a few variables capturing the relevant macroscopic observables) and symmetry reduction (roughly, to reduce the degrees of freedom of the theory in order to include only the kinematical variables corresponding to these macroscopic observables).', '1602.05881-1-4-3': 'Of course, both strategies involve some approximations to the full theory.', '1602.05881-1-5-0': 'In the coarse-graining approach, the effective dynamics of the collective (cosmologically relevant) observables implicitly capture some effects of the other (coarse-grained away) degrees of freedom; how exactly and how well it manages to do so both depend on the precise coarse-graining scheme as well as on the other approximations that are required to make it work.', '1602.05881-1-5-1': 'From this perspective, cosmology (being the theory of the most global, large-scale degrees of freedom) should arise as a sort of quantum gravity hydrodynamics [CITATION].', '1602.05881-1-5-2': 'In the symmetry reduction approach the relevant cosmological symmetries (e.g., homogeneity and isotropy) are imposed exactly, in order to obtain a system with a minimal number of degrees of freedom, while the others are simply projected out of the resulting theory.', '1602.05881-1-5-3': 'If this is done at the microscopic level, it clearly leads to entirely neglecting any effect that the removed degrees of freedom have on the physics of the remaining (cosmological) degrees of freedom.', '1602.05881-1-5-4': 'It may thus be seen, in a way, as a particularly brutal form of coarse-graining.', '1602.05881-1-5-5': "Any form of coarse-graining would then be 'better' than such a drastic reduction, as a matter of principle, and especially so in a quantum context.", '1602.05881-1-6-0': 'However, coarse-graining (especially in the full quantum theory) is cumbersome and technically challenging, and in some cases it is certainly true that a symmetry reduction procedure is enough to capture all of the relevant physics in a (sometimes significantly) more straightforward manner.', '1602.05881-1-6-1': 'Whether symmetry reduction does in fact capture the relevant physics can be judged only on a case by case basis; for example, coarse-graining is usually preferred in condensed matter physics, but the easier symmetry reduction works for simple systems like the hydrogen atom.', '1602.05881-1-7-0': 'In the classical gravitational context, symmetry reduction is the main framework used to study the large-scale dynamics of space-time, but even there it is a hotly debated issue whether symmetry reduction gives the same results as coarse-graining gravitational inhomogeneities [CITATION].', '1602.05881-1-7-1': 'In a quantum gravitational context, of course, the issue is even murkier and under less control.', '1602.05881-1-7-2': 'Further, even under the assumption that symmetry reduction is enough, it is not clear whether the symmetry reduction should be done at the classical level and one should then quantize the resulting reduced system, or directly at the quantum level.', '1602.05881-1-7-3': 'The two procedures cannot be expected to commute, in general, and much depends on the exact details of the microscopic quantum theory and on its interpretation in relation to continuum general relativity (in particular, whether the full quantum theory is understood as a quantization of general relativity or a quantum theory whose effective description only is given by general relativity).', '1602.05881-1-8-0': 'The above considerations apply to any quantum gravity formalism in which the microscopic degrees of freedom cannot be directly interpreted as quantized continuum geometries, and therefore should be related to continuum geometries through a more careful analysis.', '1602.05881-1-9-0': 'In particular, these considerations are relevant for loop quantum gravity (LQG) [CITATION] which, despite its historical roots as a canonical quantization of general relativity in connection variables, ends up with fundamental quantum degrees of freedom which are purely combinatorial and algebraic (e.g., s-knots).', '1602.05881-1-9-1': 'This feature is more apparent in its covariant, or spin foam, formulation, where the same type of combinatorial and algebraic degrees of freedom -directly obtained from a discretization of the continuum theory- are used in order to define a prescription for the quantum gravitational path integral.', '1602.05881-1-10-0': 'This is also the case for group field theories (GFTs) [CITATION], which can be seen on the one hand as a second quantized formulation of (the kinematics and dynamics of) loop quantum gravity degrees of freedom [CITATION], although organized in a different Fock-type Hilbert space, and on the other hand as an enrichment of lattice gravity approaches like tensor models [CITATION] (in turn a higher-dimensional generalization of matrix models for 2D gravity, and closely related to the dynamical triangulations formalism for quantum gravity [CITATION]) by group-theoretic data, i.e., the type of data characterizing quantum geometry in loop quantum gravity.', '1602.05881-1-10-1': 'The GFT framework provides the context for the results presented in this paper.', '1602.05881-1-11-0': "Due to the nature of the fundamental degrees of freedom in both LQG and GFT, it is reasonable to expect cosmological states to correspond to highly excited states with respect to the natural background independent vacuum state of the theory, which is a 'no space' state (e.g., the Ashtekar-Lewandowski vacuum state [CITATION] in LQG, and the simple Fock vacuum in GFT).", '1602.05881-1-11-1': 'Thus, to study cosmological space-times in LQG, it will be necessary to only consider a small number of the degrees of freedom of the relevant highly-excited states, and it is clear that some form of coarse-graining is required (perhaps of the simple symmetry reduction type).', '1602.05881-1-12-0': 'In fact, the problem of the continuum (and the related problem of extracting cosmological dynamics from the full theory) is attracting increasing attention both in GFT and in LQG.', '1602.05881-1-12-1': 'At the more formal level, it is tied to the study of renormalization flows of quantum gravity models, which is now an active area of research for spin foam models both in their lattice gravity interpretation [CITATION] and in their GFT formulation [CITATION], and to the issue of identifying a suitable phase of the theory in which a continuum geometric rewriting of its dynamics is possible (by defining a non-degenerate geometric vacuum), which has seen considerable progress both in canonical LQG [CITATION] and in GFT, where it has been suggested that condensate states are natural candidates for non-degenerate geometric vacua.', '1602.05881-1-13-0': 'In the GFT formalism, the hypothesis that the process responsible for the emergence of a continuum geometric space-time is a condensation of the GFT microscopic quanta of space-time (i.e., spin network nodes or equivalently abstract labelled polyhedra) has suggested a promising strategy for extracting an effective cosmological dynamics from the full theory.', '1602.05881-1-13-1': 'Quantum GFT condensate states have been studied in some detail, including both simple condensates corresponding to a gas of uncorrelated GFT quanta (i.e., spin network nodes with no information on their mutual connectivity) and more complex condensates retaining topological information and defined by (infinite) sums over spin network graphs [CITATION] (see also [CITATION] for a detailed overview of the methods and the results).', '1602.05881-1-13-2': "In all cases, the key feature is that the same 'condensate wave function' is assigned to all of the GFT quanta defining the quantum state and hence acts as a collective variable.", '1602.05881-1-13-3': 'This is understood as the coarse-grained quantum gravity analogue of the classical spatial homogeneity condition that characterizes the simplest cosmological space-times.', '1602.05881-1-13-4': 'The effective condensate dynamics can then be extracted directly from the full theory and are given by a non-linear equation for the condensate wave function.', '1602.05881-1-13-5': 'Since the domain of definition of this condensate wave function is isomorphic to the minisuperspace of homogeneous (and potentially anisotropic) geometries, the GFT condensate hydrodynamics has the form of a non-linear extension of loop quantum cosmology and can be studied with similar methods.', '1602.05881-1-13-6': 'Several other results have also been obtained in the study of GFT condensate states, some of which will be reviewed briefly in the following since in this paper we will build on these earlier results.', '1602.05881-1-13-7': 'See [CITATION] for further details.', '1602.05881-1-14-0': 'Furthermore, the importance of studying many-particle states in quantum gravity for the cosmological sector is also motivated directly from the purely canonical LQG point of view, following from some important insights provided by loop quantum cosmology (LQC) [CITATION].', '1602.05881-1-14-1': 'In order to clearly describe the input that LQC offers concerning the quanta of geometry in a cosmological space-time, it is helpful to briefly review LQC and explain what is known about the relation between LQC and LQG.', '1602.05881-1-15-0': 'In LQC, symmetry-reduced space-times are quantized in a non-perturbative fashion following the procedures developed in loop quantum gravity.', '1602.05881-1-15-1': 'A lot of progress has been achieved in the field of LQC over the past few years, with the main results being that the big-bang and big-crunch singularities of general relativity are resolved due to quantum gravity effects [CITATION] and are replaced by a bounce [CITATION].', '1602.05881-1-15-2': 'To be precise, in LQC the dynamics of a contracting space-time are given by general relativity until the space-time curvature nears the Planck scale, at which point quantum gravity effects become important and cause a bounce which can be seen as a quantum gravity bridge between the contracting and expanding branches of the space-time that can each be treated classically.', '1602.05881-1-16-0': 'Although LQC is based on the same quantization techniques as LQG, LQC is not derived from LQG and the precise relation between LQC and LQG is currently unknown.', '1602.05881-1-16-1': 'This is an important issue to address which is given additional urgency since some predictions of LQC effects have been calculated in a number of cosmological scenarios including inflation [CITATION] and the matter bounce scenario [CITATION] that are of phenomenological interest.', '1602.05881-1-16-2': 'It is therefore important to see whether the results of LQC can be provided further support by a proper (if approximate) derivation of LQC from full LQG.', '1602.05881-1-16-3': 'One of the goals of this paper is to shed some light on this problem by studying the dynamics of cosmological states in the GFT reformulation of LQG.', '1602.05881-1-17-0': 'While the issue of the relation of LQC with the full theory has not yet been fully resolved, there has been some important work in this direction including the result that the basis states of LQC can be embedded in the kinematical Hilbert space of LQG [CITATION] (even though the kinematical Hilbert space of LQC is not the symmetry-reduced kinematical Hilbert space of LQG [CITATION]).', '1602.05881-1-17-1': 'However, the main results regarding the relation between LQC and LQG concern their kinematical Hilbert spaces and their basis states, and much less is known about the dynamical sector.', '1602.05881-1-18-0': 'Still, a heuristic argument concerning the relation between LQG and LQC provides key insights which are in close agreement with the discussion above concerning the importance of using highly excited states (with respect to the no space vacuum) in order to study the cosmological sector.', '1602.05881-1-18-1': 'This heuristic relation is based on an observation concerning the kinematical Hilbert space of LQG and the construction of the field strength operator in LQC.', '1602.05881-1-19-0': 'First, quantum states in the kinematical LQG Hilbert space (and in GFT) can be viewed [CITATION] as a collection of polyhedra that are glued together across faces with equal areas (although the faces glued together need not have the same shape).', '1602.05881-1-19-1': 'In a typical piece-wise flat interpretation of the resulting discrete structures, space-time curvature is understood to lie on the faces of the polyhedra and the amplitude of the curvature lying on any given face can be evaluated via a holonomy of the Ashtekar-Barbero connection along the edges that form the boundary of that face.', '1602.05881-1-20-0': 'Second, one of the key steps in LQC is the definition of the operator corresponding to the space-time curvature.', '1602.05881-1-20-1': 'This is constructed by evaluating the holonomy of the Ashtekar-Barbero connection around a loop of minimal area, as it would be in full LQG.', '1602.05881-1-20-2': 'The choice of the minimal area (rather than some other non-zero', '1602.05881-1-21-0': 'area) is justified by assuming that in the LQG wave function corresponding to the LQC states, there is a large number of polyhedra and that each of their surface areas is given by the minimal area eigenvalue of LQG, [MATH] (where [MATH] is a dimensionless number of order 1) [CITATION].', '1602.05881-1-22-0': 'The combination of these two ingredients leads to the following heuristic relation between LQC and LQG.', '1602.05881-1-22-1': 'In the simplest, homogeneous and isotropic case, LQC states correspond to wave functions in LQG with a large number of quanta of geometry, or polyhedra, whose surface areas are all given by the minimal non-zero area possible in LQG [CITATION].', '1602.05881-1-22-2': 'This key insight from LQC strongly motivates the use of the family of GFT states we will use in this paper, with the polyhedra specialized for simplicity to tetrahedra.', '1602.05881-1-23-0': 'In other words, the LQG wave function for cosmological space-times, as suggested by LQC, would be a state composed of a large number of identical tetrahedra whose four areas are all equal to [MATH].', '1602.05881-1-23-1': 'An important point is that these tetrahedra are not only equiareal but also equilateral, a stronger condition.', '1602.05881-1-23-2': 'LQC neglects the connectivity information of the tetrahedra, and without the connectivity information, the LQG wave function of a large number of identical tetrahedra is exactly a simple condensate state.', '1602.05881-1-23-3': 'This is the family of GFT states that we shall consider here.', '1602.05881-1-24-0': 'Now, ignoring the connectivity information of the tetrahedra forming the space-time, for the sake of simplicity but also because this information is not immediately relevant for purely homogeneous and isotropic configurations (and furthermore is ignored in the heuristic relation between LQC and LQG), this type of wave function has only one degree of freedom given by the number of tetrahedra [MATH].', '1602.05881-1-24-1': 'This is very restrictive and in order to allow more general states we will remove the requirement that the areas of the faces of each tetrahedron be [MATH].', '1602.05881-1-24-2': 'Instead, we will assume that the LQG wave functions (in the GFT reformulation) that correspond to cosmological space-times are condensate states composed of a large number of identical equilateral tetrahedra, with no a priori restriction on their face areas.', '1602.05881-1-24-3': 'On the basis of these insights, and since condensate states with varying number of quanta are most easily handled in a field theory formalism where there exist creation and annihilation operators, the discussion above suggests that group field theory (GFT) -which provides a field theory reformulation of LQG, as reviewed in Sec. [REF]- is the most natural framework to study the cosmological sector of LQG, and that the coarse-grained cosmological dynamics come from the hydrodynamics of GFT condensate states.', '1602.05881-1-25-0': 'Before presenting our results, we point out that there do exist a number of other approaches to the problem of extracting the cosmological sector of LQG, including so-called spin foam cosmology [CITATION] as well as work on symmetry-reduced canonical loop quantum gravity, either with a large number of spin network nodes [CITATION] (see also [CITATION] for related earlier work), or just one node [CITATION].', '1602.05881-1-25-1': 'The approach considered in [CITATION], in particular, despite the clear difference in strategy and of formalism, considers quantum states quite similar to our GFT condensates.', '1602.05881-1-25-2': 'However, all of these alternative approaches assume that the relevant physics is correctly captured by a fixed number of quanta of geometry (which is moreover very small in some cases), an assumption which is seemingly at odds with the lattice-refinement interpretation of LQC which seems to indicate that the number of quanta of geometry increases as the universe expands (or decreases if the space-time is contracting) [CITATION], as the heuristic relation between LQC and LQG suggests.', '1602.05881-1-25-3': 'See also [CITATION] for further arguments concerning the potential importance for allowing for the number of quanta to change in the cosmological sector of LQG.', '1602.05881-1-25-4': 'In the condensate states studied here there is no restriction on the number of quanta, indeed the number of quanta of geometry will be determined dynamically as it necessarily enters the effective cosmological equations [CITATION].', '1602.05881-1-25-5': 'This is one of the important advantages of using GFT condensate states to study the cosmological sector of LQG, in addition to directly addressing the possibility of cosmological space-times being constituted of a large number of quanta of geometry.', '1602.05881-1-26-0': 'In this paper, we will extend previous work on the cosmology of GFT condensates in a number of directions: (i) we work with a GFT that corresponds to (a slight generalization of) the Engle-Pereira-Rovelli-Livine (EPRL) spin foam model of LQG [CITATION], which is the most developed Lorentzian model for 4D quantum gravity and is thus a bona fide candidate for the fundamental definition of the quantum dynamics rather than a simple toy model; (ii) we generalize the GFT formalism in order to include a massless scalar field, (iii) in an isotropic restriction, we extract an effective (relational) cosmological dynamics directly from the fundamental dynamics of the model, showing that it reproduces the Friedmann equations in a classical limit; (iv) we derive the leading order quantum gravity corrections to the Friedmann dynamics of the GFT condensate states, which generically predict that the classical big-bang singularity is resolved and is replaced by a non-singular bounce; (v) interestingly, the modified Friedmann equations are qualitatively similar to the effective dynamics of LQC for a simple type of GFT condensate state.', '1602.05881-1-26-1': 'It is worth stressing that, while in this paper we focus on those contributions to the effective dynamics that are expected to be dominant in our approximation (in particular, we consider the interactions as subdominant compared to the kinetic term in the quantum equations of motion as is required in the Gross-Pitaevskii approximation), all of the terms coming from the microscopic dynamics are known and subdominant contributions can be computed explicitly.', '1602.05881-1-27-0': 'More precisely, the outline of the paper is the following.', '1602.05881-1-27-1': 'We start by giving a brief introduction to GFTs in general and reviewing the results of the earlier papers on GFT condensate states [CITATION] in Sec. [REF], and explain how to include a scalar field in a GFT in Sec. [REF].', '1602.05881-1-27-2': 'Then in Sec. [REF] we define the condensate states we take as an ansatz to represent the homogeneous and isotropic sector of LQG, and explain how the scalar field can be used as a relational clock.', '1602.05881-1-27-3': 'In Sec. [REF] we derive and study the relational Friedmann equations of the GFT condensate states, showing in particular that the big-bang and big-crunch singularities are resolved and are replaced by a bounce, a result analogous to the main qualitative results of LQC.', '1602.05881-1-27-4': 'We end with some comments on the limiting case of the matter energy density vanishing in Sec. [REF] and a more general discussion in Sec. [REF].', '1602.05881-1-28-0': '# Basic Elements of the Group Field Theory Formalism', '1602.05881-1-29-0': 'Group Field Theories (GFTs) [CITATION] are a special class of quantum (or statistical) field theories defined over group manifolds (hence the name).', '1602.05881-1-29-1': 'Motivated by quantum Regge calculus [CITATION], dynamical triangulations [CITATION], tensor models [CITATION], and loop quantum gravity (LQG) [CITATION], the action for GFTs is typically chosen specifically so that the perturbative expansion of the generating functional generates a sum over diagrams that are dual to simplicial complexes and whose vertices, edges and faces are decorated with group theoretic data.', '1602.05881-1-29-2': 'Then, the Feynman amplitude for each simplicial complex in this sum (once the group theoretic data on the vertices, edges and faces are summed over) corresponds precisely to the quantum gravity path integral evaluated on that particular simplicial complex, assuming appropriate choices are made for the GFT action [CITATION].', '1602.05881-1-29-3': 'The continuum limit is obtained through the sum over all allowed simplicial complexes.', '1602.05881-1-30-0': 'Significantly, GFTs are related to both the canonical and covariant forms of LQG.', '1602.05881-1-30-1': 'On the one hand, GFTs can be understood as a second quantization of canonical LQG [CITATION], with a different organization of the quantum states so to obtain a Fock space, while the GFT Feynman amplitudes can be equivalently written as spin foam models [CITATION], the covariant form of LQG (see [CITATION] for an introduction to spin foam models).', '1602.05881-1-31-0': 'Furthermore, as quantum field theories, GFTs admit an operatorial representation in terms of ladder operators acting on conventional Fock spaces.', '1602.05881-1-31-1': 'In this Fock representation, it is clear that GFTs are a second-quantized language for LQG since the GFT quanta are in fact spin network nodes', '1602.05881-1-32-0': 'What will be particularly useful here is that the presence of a Fock space structure renders available some of the powerful methods used in condensed matter physics to construct approximate solutions involving many-body degrees of freedom (which are typically very difficult to access using simple perturbative techniques or attempting to solve directly the many-body Schrodinger equation) and to study the corresponding physics [CITATION].', '1602.05881-1-32-1': 'These methods may become particularly important in quantum gravity if the continuum limit, in background independent theories of quantum gravity like LQG, does in fact correspond to states with many quanta of geometry as is often expected [CITATION].', '1602.05881-1-33-0': 'In this section, we will make the above discussion more precise by considering a specific GFT model.', '1602.05881-1-33-1': 'We will first define the field operators in the GFT and explain their geometric interpretation, then introduce the Fock space and operators on the Fock space, and finally present the quantum equations of motion.', '1602.05881-1-34-0': '## The Field Operators', '1602.05881-1-35-0': 'In what follows we will consider a simple bosonic GFT based on the group [MATH] (the most common choice for quantum gravity GFTs).', '1602.05881-1-35-1': 'The Fock space is built starting from the single particle Hilbert space [MATH] corresponding to four-valent spin network nodes, where the ladder operators are [EQUATION] and these ladder operators are required to be translation invariant under diagonal group multiplication from the right, i.e., [EQUATION]', '1602.05881-1-35-2': 'The identification of this Hilbert space with four-valent spin network nodes is clear: the four [MATH] are the parallel transport of an [MATH]-valued connection along each of the four links emanating from the same node of a spin network, and the translation invariance encodes the gauge-invariance at the node in spin networks.', '1602.05881-1-35-3': 'For this reason, although the right translation invariance is a global translation in [MATH] it is often referred to as gauge invariance', '1602.05881-1-36-0': 'Note that the choice of four copies of [MATH] in the definition of the field operators amounts to a restriction to four-valent spin network nodes.', '1602.05881-1-36-1': 'This is a minor restriction in the formalism for the sake of simplicity; the theory can be extended, in principle, to include spin network nodes of arbitrary valence [CITATION].', '1602.05881-1-37-0': 'Using the notational shorthand [MATH] to denote the [MATH] group elements on which the field operator is evaluated, the commutation relations for the ladder operators are [EQUATION] which include the correct right gauge invariance.', '1602.05881-1-37-1': 'This shows that the creation operator [MATH] can be interpreted to create a single four-valent spin network node with data given by [MATH] up to a gauge transformation on the right.', '1602.05881-1-38-0': 'Equivalently, the same quanta can be interpreted as tetrahedra dual to the four-valent spin network nodes, in which case the four triangular faces of the tetrahedra are dual to the open links leaving the spin network node and are labelled by the same group elements.', '1602.05881-1-38-1': 'To make this dual simplicial interpretation more transparent, it is convenient to rewrite the theory as a field theory on the Lie algebra [MATH] via a noncommutative Fourier transform, [EQUATION] where [MATH] are noncommutative plane waves and [MATH] denotes the Haar measure on [MATH] [CITATION].', '1602.05881-1-38-2': 'The Lie algebra elements should be understood as the normal vectors to each face of the tetrahedron (with their norm corresponding to the area of each face), that in turn are in one-to-one correspondence with the face bivectors.', '1602.05881-1-38-3': 'This rewriting (of both GFT and LQG) depends explicitly on the choice of the quantization map for the flux observables, which will dictate the choice of non-commutative plane waves and their star product [CITATION].', '1602.05881-1-39-0': 'Using the properties of the plane waves [MATH], it can be verified that the Lie algebra counterpart of the right gauge invariance is the closure constraint for the four faces of the tetrahedron [CITATION].', '1602.05881-1-39-1': 'For this reason, the domain of definition of the field operators [MATH] is the subspace of [MATH] obeying the closure relation [EQUATION]', '1602.05881-1-39-2': 'This clearly shows that the quanta created by the field operators can indeed be interpreted as quantum tetrahedra with face bivectors given by the [MATH].', '1602.05881-1-39-3': 'Furthermore, it is clear that (at the price of a more complicated theory) it is possible to define a more general GFT than this one which also includes quanta interpreted as higher-valent spin network nodes, or equivalently as convex polyhedra with a greater number of faces.', '1602.05881-1-40-0': '## The Fock Space', '1602.05881-1-41-0': "In GFTs, the Fock space is constructed in the standard fashion (assuming Bose statistics) from the 'one-particle' Hilbert space described in Sec. [REF] (for details see [CITATION]).", '1602.05881-1-41-1': 'Note that the Fock vacuum represents the state with no spin network nodes, and thus no topological nor geometrical information; the usual spin network states of LQG can be obtained by acting on this vacuum with creation operators convoluted with the desired spin network wave function [CITATION].', '1602.05881-1-42-0': 'In this second-quantized framework, all operators of interest are generated by polynomials of ladder operators convoluted with suitable kernels where these kernels are the matrix elements of the corresponding first-quantized operators (i.e., the operators of standard canonical LQG like area or volume).', '1602.05881-1-42-1': 'For example, the second-quantized volume operator is (expressed in terms of the Fourier-transformed field operators) [EQUATION] where [MATH] is the volume of a tetrahedron where the normal vectors to each of its faces are [MATH] and [MATH].', '1602.05881-1-42-2': 'This operator gives the total volume of all of the quanta of geometry.', '1602.05881-1-43-0': 'It is also possible to consider more general one-body operators of the type [EQUATION] where the field operators do not have the same arguments.', '1602.05881-1-43-1': 'Here [MATH] is the LQG matrix element of the operator [MATH], evaluated between two nodes decorated with the elements [MATH].', '1602.05881-1-44-0': 'There is also a very important new one-body observable that only appears in the second-quantized framework.', '1602.05881-1-44-1': 'This is the number operator [EQUATION] which counts the number of quanta present in a given state: its eigenvalues distinguish the N-body sectors of the GFT Fock space.', '1602.05881-1-44-2': 'This is also an example of a purely combinatorial observable that characterizes the underlying graph of spin network states, regardless of the attached group theoretic data, which is a type of observable not usually considered in the standard LQG framework.', '1602.05881-1-45-0': 'More general N-body operators that act on more than one spin network node (or tetrahedron), or resulting in more than one, can also be constructed and are of obvious interest.', '1602.05881-1-45-1': 'For example, curvature operators that calculate the parallel transport around a loop containing several spin network nodes will generically be of this type.', '1602.05881-1-45-2': 'Of course, the form of this type of operator is more complicated and, since in this paper we are interested in homogeneous degrees of freedom (to describe homogeneous cosmological space-times), we will restrict our attention to one-body operators like those corresponding to the volume; these will be enough to totally characterize the relevant homogeneous degrees of freedom.', '1602.05881-1-46-0': '## The Quantum Equations of Motion', '1602.05881-1-47-0': 'The previous two sections present the key concepts underlining the kinematics of GFTs.', '1602.05881-1-47-1': 'Now, the GFT will be fully specified by giving its dynamics, which are encoded in the GFT action [MATH].', '1602.05881-1-47-2': 'The action typically includes involving a kinetic term quadratic in the field operators and some interaction terms which are higher order in the field operators.', '1602.05881-1-48-0': 'Given the action, the path integral for the theory can be (formally) defined, and from the path integral formalism the Schwinger-Dyson equations for correlation functions can be derived, which represent the complete (although formal) specification of the quantum dynamics.', '1602.05881-1-49-0': 'The same quantum dynamics can also be given in operator form, starting from the operator corresponding (up to operator ordering ambiguities) to the classical equations of motion written in terms of field operators, the first equation of motion being [EQUATION] and the second being the corresponding equation obtained from the variation of the action with respect to [MATH].', '1602.05881-1-50-0': 'As already mentioned above, the GFT action can be defined to reproduce a spin foam model by choosing the action so that the perturbative expansion of the GFT partition function around the Fock vacuum matches the expansion of the spin foam model.', '1602.05881-1-50-1': 'To be precise, the kinetic term and the interaction term in the GFT action are respectively chosen to reproduce the edge and vertex amplitudes of the spin foam model.', '1602.05881-1-51-0': 'For the most common type of spin foam models which are based on simplicial interactions (i.e., where the spin foam vertices are associated to 4-simplices), the only interaction terms are five-valent and therefore the GFT action will have the general form of [EQUATION] where the complex-valued functions [MATH] and [MATH] depend on the [MATH].', '1602.05881-1-51-1': 'Note that here each [MATH] denotes 4 group elements [MATH] at each spin network node [MATH].', '1602.05881-1-51-2': 'The notation is the following: indices [MATH] label the node and run from [MATH] with [MATH] being the total number of nodes (so in the kinetic term [MATH] and in the interaction term [MATH]), while indices [MATH] label the links at a given node and take the values [MATH] since only four-valent spin-network nodes are considered in this family of GFT models.', '1602.05881-1-52-0': 'Also, for the sake of simplicity only two interaction terms are included in the above action, one of which contains five creation operators and no annihilation operators and the other the opposite.', '1602.05881-1-52-1': 'Clearly, there are four other interaction terms that could be included (with some relations between them necessary in order to ensure that the action be real).', '1602.05881-1-52-2': 'For now we will restrict our attention to these two terms as they shall be sufficient to show how to include any relevant interaction term in any further analysis.', '1602.05881-1-53-0': 'The perturbative expansion of this model gives Feynman diagrams dual to four-dimensional simplicial complexes, and the amplitudes are determined by convolutions of the interaction kernels [MATH] associated to the 4-simplices with the inverse kernels of the kinetic term (i.e., the propagator of the theory which connects neighbouring 4-simplices).', '1602.05881-1-54-0': 'Finally, for the action [REF] corresponding to a GFT based on simplicial interactions, the first quantum equation of motion is [EQUATION] with the second quantum equation of motion being the adjoint of this equation.', '1602.05881-1-55-0': 'The specific form of [MATH] and [MATH] will depend on the choices in the construction of the GFT, and in this case we will be interested in the GFT based on the EPRL spin foam model, which is the most studied Lorentzian spin foam model.', '1602.05881-1-56-0': 'One of the main principles used in determining the form of [MATH] and [MATH] is that the imposition of the simplicity constraints -which transform the 4D topological BF field theory into a geometric theory with local degrees of freedom [CITATION]- can be achieved by modifying the kernels of the 4D [MATH] BF Ooguri GFT model with conditions that restrict the four bivectors [MATH] labeling the triangular faces appearing in the simplicial complexes dual to the Feynman diagrams of the theory to be simple bivectors [CITATION].', '1602.05881-1-56-1': 'This is the GFT counterpart of the standard procedure followed in defining spin foam models [CITATION], and following this procedure gives the EPRL GFT model.', '1602.05881-1-57-0': 'However, here we will not go through the details of this lengthy procedure; the interested reader is instead referred to [CITATION].', '1602.05881-1-57-1': 'Instead, it will be sufficient for our purposes to highlight one of the key properties of the GFT based on the EPRL spin foam model.', '1602.05881-1-58-0': 'To do this, it is convenient to work in the spin representation, where the field operators [REF] are rewritten via the Peter-Weyl decomposition [EQUATION] where [MATH].', '1602.05881-1-58-1': 'Note while a generic function of [MATH] would have a more general form, the gauge-invariance of the field operators is translated in the spin representation to the presence of the intertwiners [MATH] labeled by [MATH].', '1602.05881-1-59-0': 'Then, using the shorthand notation [EQUATION] the general GFT action for the case of simplicial interactions in the spin representation has the form [EQUATION] where [MATH], and of course each [MATH] and [MATH] represent the four [MATH] and [MATH] labels colouring the four links leaving the [MATH] spin network node.', '1602.05881-1-60-0': 'The key property of the GFT based on the EPRL spin foam model is that (i) the kinetic term contains a Kronecker delta between the [MATH] and intertwiner labels, and (ii) the interaction term contains a Kronecker delta for the [MATH] labels colouring the links that meet in the interaction.', '1602.05881-1-60-1': 'Thus, in this case the kinetic term has the simple form [EQUATION] and the interaction term is similarly simplified, with the first interaction term having the form [EQUATION] while the second interaction term is simply [MATH].', '1602.05881-1-60-2': 'The input from the EPRL model here is in the combinatoric form of the [MATH] and [MATH] arguments in the field variables which is due to the presence of Kronecker delta functions in the interaction term which have been taken into account in the above expression (in particular, [MATH]).', '1602.05881-1-60-3': 'The presence of these Kronecker deltas will play an important role in what follows, but the remaining details of the functional form of [MATH] and [MATH] will not be necessary.', '1602.05881-1-60-4': 'For their detailed expression we refer to the literature [CITATION], and we only mention here that they encode a specific relation between the [MATH] representation labels that we use as variables here and [MATH] representations, as well as a condition of invariance of the same functions under [MATH].', '1602.05881-1-60-5': 'This specific relation between [MATH] and [MATH] data is the end result of the EPRL prescription for imposing the constraints reducing topological BF theory to gravity, and which are nothing more than the conditions enforcing geometricity of the simplicial structures on which the model is based.', '1602.05881-1-61-0': 'While it is not necessary to give the exact functional form of [MATH] and [MATH] here, it is nonetheless important to be aware of a number of ambiguities that arise in the definition of GFT (and spin foam) models.', '1602.05881-1-61-1': 'First, there are the standard factor-ordering ambiguities that generically arise in quantum mechanics (note also that in this setting even operators corresponding to momentum space variables, i.e., the discretized fluxes of the triad field, will not necessarily commute amongst themselves).', '1602.05881-1-61-2': 'Second, the specific choices that are made for [MATH] and [MATH] are motivated by a discretization of (in this case) the Plebanski-Holst action, and some ambiguities necessarily arise that depend on the specific discretization method.', '1602.05881-1-61-3': 'Third, generically, while the implementation of the simplicity constraints alone (which reduce the topological BF theory to gravity) fixes the type of variables that should appear in the theory so that there is a proper discrete geometric interpretation, and may further suggest the amplitudes associated to vertices, additional requirements (e.g., concerning the composition law of transition amplitudes [CITATION], or the renormalizability of the resulting model) are needed in order to determine the amplitudes associated to edges and faces.', '1602.05881-1-61-4': 'Finally, even once the vertex, edge and face amplitudes are determined, it is possible to redefine these in such a fashion so that the total amplitudes remain unchanged, since these last depend only on convolutions of vertex, edge and face amplitudes in specific combinations.', '1602.05881-1-62-0': 'The existence of construction and quantization ambiguities is of course not surprising as they affect the definition of any quantum theory.', '1602.05881-1-62-1': 'It is hoped that -and there are general reasons to expect this [CITATION]- (assuming the continuum and semi-classical limits to be properly defined and shown to exist) most of the large-scale properties of the effective description will be insensitive to many of the details of the construction of the microscopic model, and in particular that they will be insensitive to many of the ambiguities listed above.', '1602.05881-1-62-2': 'Of course, the extent of this independence from microphysics will depend on the specific class of phenomena of interest.', '1602.05881-1-63-0': 'We will see that, for the family of GFTs and for the particular class of states that will be considered here, on the one hand there will be a certain degree of universality, but on the other hand a number of these subtle choices in the definition of the GFT model will directly affect the effective cosmological dynamics and thus give different predictions.', '1602.05881-1-64-0': '# Coupling to a Scalar Field', '1602.05881-1-65-0': 'So far we have presented the simplest incarnation of GFT models, those which encode only (quantum) geometric degrees of freedom.', '1602.05881-1-65-1': 'However, in order to study most physical applications, and to extract concrete predictions from GFTs, it is necessary to introduce matter fields.', '1602.05881-1-65-2': 'This is especially true in the cosmological context where vacuum homogeneous and isotropic space-times are either Minkowski or (anti-)de Sitter.', '1602.05881-1-65-3': 'Furthermore, when matter fields are present, it is possible to define relational observables (which are diffeomorphism-invariant).', '1602.05881-1-65-4': 'On the other hand, in the vacuum setting it is often very difficult to define Dirac observables.', '1602.05881-1-66-0': 'While the inclusion of matter fields in GFTs (and also in spin foam models) has not been extensively explored (although see [CITATION]), here we will show how it is possible to include a scalar field in a GFT model.', '1602.05881-1-67-0': '## The Field Operators and the Fock Space', '1602.05881-1-68-0': 'The path that we shall follow here is to simply modify the definition of the field operators so that each GFT quanta carries a value of the scalar field.', '1602.05881-1-68-1': 'To be specific, considering the case of a single real scalar field, the field operators become [EQUATION] with the domain of the GFT field operators now [MATH].', '1602.05881-1-69-0': 'Clearly, the Fock space for the GFT with a scalar field can be constructed in exactly the fashion as in the vacuum case, except that in this case the one-particle Hilbert space is [MATH].', '1602.05881-1-70-0': 'An immediate consequence of having a scalar field as an argument in the GFT field is that the presence of the scalar field permits the definition of relational observables, i.e., operators evaluated for a specific value of the scalar field.', '1602.05881-1-70-1': 'Relational observables are particularly useful in cosmology, where matter fields can often act as relational clocks that are well-defined and evolve monotonically globally.', '1602.05881-1-70-2': "Thus it is often possible to avoid the problem of time in quantum gravity by defining physical evolution with respect to the scalar field 'clock'.", '1602.05881-1-71-0': "For example, besides the 'total number' operator [EQUATION] it is also possible to define the number operator at a fixed value of [MATH], [EQUATION]", '1602.05881-1-71-1': 'Clearly, analogous definitions will hold for other relational operators.', '1602.05881-1-72-0': 'The scalar field is also, of course, a true physical degree of freedom.', '1602.05881-1-72-1': 'There are a number of new observables associated to the scalar field, which are the second-quantized counterpart of the standard observables of a scalar field, namely polynomials in the scalar field and its derivatives.', '1602.05881-1-72-2': 'The two fundamental second-quantized operators are the scalar field operator [EQUATION] and the conjugate momentum operator (written in the momentum representation of the scalar field) [EQUATION]', '1602.05881-1-72-3': 'The [MATH] can be rewritten in the [MATH] representation via a Fourier transform, [EQUATION] here it is given in a manifestly self-adjoint form.', '1602.05881-1-72-4': 'From this it is easy to define the self-adjoint relational conjugate momentum operator [EQUATION] which will play an important role in extracting the cosmological dynamics from the GFT condensate states.', '1602.05881-1-73-0': '## The Quantum Equations of Motion', '1602.05881-1-74-0': 'The dynamics for the GFT models with a scalar field will be determined by the GFT action which is to be chosen following the same procedure as in the vacuum case.', '1602.05881-1-74-1': 'We will once again only consider simplicial interactions, and therefore the action will contain a kinetic term together with interaction terms containing five copies of the field variable.', '1602.05881-1-75-0': 'In this section, we will explain what choices are possible in the GFT action in order to appropriately account for the scalar field.', '1602.05881-1-75-1': 'The construction will be such that, for states where the scalar field vanishes the quantum equations of motion will be precisely those obtained in Sec. [REF].', '1602.05881-1-75-2': 'Of course, the same quantization ambiguities present in the vacuum case will also arise when a scalar field is present.', '1602.05881-1-76-0': 'In order to determine how the scalar field should appear in the kinetic and interaction terms in the action for the case of a GFT with simplicial interactions, it is necessary to understand how a scalar field is discretized on a simplicial discretization of a space-time.', '1602.05881-1-77-0': 'Since a scalar field should be discretized on 0-dimensional elements of the chosen lattice, there are two choices: either real numbers representing the value of the scalar field can be associated to vertices of the simplicial lattice or to nodes of its dual complex.', '1602.05881-1-77-1': 'In the first case, the scalar field will propagate along the edges of the simplicial complex, while in the second case the scalar field will propagate along the links of the dual complex.', '1602.05881-1-77-2': 'While it is reasonable to expect that these two choices lead to equivalent results in the continuum limit, one choice may be better than the other.', '1602.05881-1-77-3': "First, which choice is most 'natural' depends to some extent on the interpretation given to the simplicial complex (and its dual), and second, one choice may give a GFT where calculations are much easier than in the other.", '1602.05881-1-78-0': "If the basic 4-simplices constituting the simplicial complex are viewed as the finitary substitute of the notion of points in a continuum manifold [CITATION] -in which case the data at each 4-simplex constitute the 'local' data in a discrete setting- then it is most natural to discretize the scalar field on nodes of the dual complex.", '1602.05881-1-78-1': 'In this case a constant value of the scalar field is assigned to the 4D elements of the simplicial lattice, and the gradient terms are given by the difference of the value of the scalar field in neighbouring 4-simplices.', '1602.05881-1-78-2': 'Note that it follows that these discrete gradients are naturally associated to the common boundaries of the neighbouring 4-simplices (i.e., the tetrahedra), and therefore the scalar field propagates along the links of the dual complex.', '1602.05881-1-79-0': 'Moreover, and perhaps even more importantly, this discretization choice also generically leads to a much simpler extension of the GFT (and spin foam) framework to include scalar matter, as shall become clear.', '1602.05881-1-80-0': 'Locating the discretization on the dual node to the simplicial discretization means that the scalar field has a single value over any 4-simplex.', '1602.05881-1-80-1': 'This directly implies that the five fields in the GFT interaction -which correspond to the five tetrahedra in a single 4-simplex- all share the same value of the scalar field.', '1602.05881-1-80-2': 'For this reason the interaction term must impose that the value of the scalar be the same in all of the five interacting fields, in which case the (first) interaction term has the form [EQUATION] where the Dirac [MATH] distributions are shown explicitly for the sake of clarity.', '1602.05881-1-80-3': 'Clearly, the integrals over [MATH] and [MATH] are easily evaluated, giving [EQUATION]', '1602.05881-1-80-4': 'The second interaction term is simply the adjoint of this one.', '1602.05881-1-80-5': 'It is clear that, for GFT models with this interaction term, the scalar field degrees of freedom enter locally in the GFT action (unlike the quantum geometric degrees of freedom that are non-local), and this will simplify a number of calculations.', '1602.05881-1-81-0': 'Since the scalar field propagates along dual links between neighbouring 4-simplices, the gradients of the scalar field will be encoded in the GFT kinetic term, which in general will have the form [EQUATION]', '1602.05881-1-81-1': 'This is the completely general form of a GFT model for quantum gravity coupled to a real scalar field, whose action is given by [EQUATION] which follows directly from the discretization strategy described above.', '1602.05881-1-82-0': '## A Massless Scalar Field', '1602.05881-1-83-0': 'We will now focus on the case of the simplest scalar field coupled to gravity: a minimally coupled, massless scalar field.', '1602.05881-1-83-1': 'While this is a particularly simple case, it is nonetheless enough in order to extract non-trivial cosmological dynamics directly from the GFT model.', '1602.05881-1-83-2': 'Furthermore, a minimally coupled, massless scalar field in a homogeneous space-time acts like a stiff perfect fluid which is of particular interest in early universe cosmology as it is one of the few matter fields that remains relevant (in the sense that its effect on the dynamics of the geometrical degrees of freedom does not become negligible) in the high curvature regime when anisotropies are present.', '1602.05881-1-83-3': 'Therefore, a minimally coupled massless scalar field is of particular interest in studies of quantum gravity effects in cosmology.', '1602.05881-1-84-0': 'An additional advantage of the minimally coupled massless scalar field is that symmetry considerations will be enough to strongly restrict the form of the GFT action.', '1602.05881-1-84-1': '(Note that for more general cases, whether a non-minimal coupling or the presence of a potential, or both, a more detailed analysis will be necessary in order to determine the appropriate form of the GFT action.)', '1602.05881-1-85-0': 'The matter contribution to the classical (continuum) Lagrangian for a minimally coupled massless scalar field on a curved background is [EQUATION] and this Lagrangian clearly exhibits the shift and sign reversal symmetries [EQUATION]', '1602.05881-1-85-1': 'Note that non-minimal coupling or any non-trivial potential [MATH] in the scalar field action would kill the first symmetry, and could kill the second, depending the specific form of the non-minimal coupling or potential.', '1602.05881-1-86-0': 'The key step here is that we shall assume that these symmetries are also present at the quantum level in the GFT, and this places strong restrictions on the GFT action.', '1602.05881-1-86-1': 'To be specific, the kinetic term can only depend on the square of scalar field differences [MATH], while [MATH] must be independent of [MATH]: [EQUATION]', '1602.05881-1-86-2': 'It is clear that any dependance on the field values themselves in the kinetic term or in the interaction kernel would break the shift symmetry, while a dependence of the kinetic term on odd powers of field differences would break the reflection symmetry.', '1602.05881-1-87-0': 'Assuming analyticity of the field variable with respect to the scalar field, it is possible to express the kinetic term as a derivative expansion.', '1602.05881-1-87-1': 'This derivative expansion will be particularly useful in the case where the difference [MATH] is small (compared to the Planck mass) in which case the first few terms will provide a good approximation to the full kinetic term.', '1602.05881-1-87-2': 'This derivative expansion can be obtained by rewriting [MATH] and [MATH], and the kinetic term becomes [EQUATION] and the [MATH] field variable can be Taylor-expanded in its scalar field argument around [MATH].', '1602.05881-1-87-3': 'Then, after evaluating the integral over [MATH], the kinetic term becomes [EQUATION] where [EQUATION]', '1602.05881-1-87-4': 'Note that all of the odd terms in the Taylor expansion do not contribute to the sum in [REF] since the integral over [MATH] of an odd power of [MATH] multiplying the even function [MATH] gives zero.', '1602.05881-1-88-0': 'The functional form of the [MATH], which is of course determined by the kinetic term in the GFT action, encodes order by order in derivatives of [MATH] how the quantum geometric and matter degrees of freedom propagate.', '1602.05881-1-88-1': 'In particular, their exact form could be determined by ensuring that the GFT Feynman amplitudes match term by term the discrete path integral for gravity coupled to a minimally coupled massless scalar field; we leave this analysis for future work.', '1602.05881-1-89-0': 'It will not be necessary to determine the exact functional form of the [MATH] for our purposes.', '1602.05881-1-89-1': 'Instead, having ensured that the GFT action has the correct symmetries, we will work in the small derivative limit of the scalar field (i.e., where differences in the scalar field are small compared to the Planck mass), and truncate the kinetic term in the action to the lowest two orders, keeping only the terms corresponding to [MATH] and [MATH] in the sum [REF].', '1602.05881-1-89-2': 'Then, it will be possible to obtain some constraints on their form by studying the dynamics of condensate states of this GFT model -which as shall be explained shortly, are expected to capture the degrees of freedom of the spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time- and comparing these dynamics, in the appropriate semi-classical limit, to the Friedmann equations of general relativity.', '1602.05881-1-90-0': 'Finally, here we are interested in the GFT model based on the EPRL spin foam model (whose kinetic and interaction terms in the vacuum case were given respectively in [REF] and [REF]) with a minimally coupled massless scalar field.', '1602.05881-1-90-1': 'It is easy to add repeat the procedure outlined in this section starting from the GFT action for the vacuum EPRL model, this gives (in the spin representation) [EQUATION] with [EQUATION] where the [MATH] and [MATH] field variables having the same arguments in the kinetic terms in the action having imposed the Kronecker deltas, and [EQUATION]', '1602.05881-1-90-2': 'Note that the kinetic term has been truncated since, as explained above, we are considering the small derivative limit in [MATH].', '1602.05881-1-91-0': 'A few comments on this GFT model are in order.', '1602.05881-1-91-1': 'First, it was not necessary to assume the presence of any space-time symmetries in order to derive this action; in particular neither homogeneity nor isotropy were imposed.', '1602.05881-1-91-2': 'The simplicity of the GFT action is a result of not only working with a particularly simple matter field (the minimally coupled massless scalar field), but also the chosen discretization where the scalar field is discretized on each 4-simplex.', '1602.05881-1-91-3': 'The simplicity of the GFT classical equations of motion can be understood as the result of coarse-graining the small-scale complexity and obtaining the hydrodynamical equations of motion for the collective behaviour.', '1602.05881-1-91-4': 'This interpretation will be relevant in the following sections.', '1602.05881-1-91-5': 'Second, somewhat surprisingly from a purely formal GFT viewpoint, the minimally coupled massless scalar field enters the GFT action in exactly the same fashion as the standard time coordinate in ordinary quantum field theory.', '1602.05881-1-91-6': 'The presence of the scalar field allows for the definition of a host of new relational observations, but the above observation is stronger: the minimally coupled massless scalar field can be used to define a global relational clock and thus provides a well-defined notion of global time evolution in a diffeomorphism invariant context.', '1602.05881-1-91-7': 'This will be particularly useful in the cosmological context, but it is likely that this will be a powerful tool in a number of other physical settings as well.', '1602.05881-1-92-0': '# GFT Condensates', '1602.05881-1-93-0': 'As in any interacting quantum field theory, it would be naive to expect to be able to solve the quantum dynamics exactly for realistic GFT models (and note that the situation is potentially worse in the GFT context due to the background independence of GFT models as well as the non-local nature of the quantum geometric interactions).', '1602.05881-1-93-1': 'Instead, the appropriate strategy is to study simplified trial states and look for approximate solutions that may capture the relevant properties of the physical setting of interest.', '1602.05881-1-93-2': 'Then, if these approximate solutions are chosen judiciously, they will provide approximate answers to the physical questions at hand.', '1602.05881-1-94-0': 'In this paper, we will focus on the problem of extracting effective cosmological dynamics from the full quantum gravity formalism, using the GFT model with a minimally coupled massless scalar field based on the EPRL spin foam model described in the previous sections.', '1602.05881-1-94-1': 'In this case, the approximate solutions of interest are GFT condensate states, which are believed to capture the relevant degrees of freedom of the homogeneous FLRW and Bianchi space-times [CITATION].', '1602.05881-1-94-2': 'We will now summarize the motivation for considering GFT condensate states, and specifically their relevance to the cosmological sector of GFT.', '1602.05881-1-94-3': 'Further details and background information can be found in the literature [CITATION].', '1602.05881-1-95-0': "The key idea is that, by definition in a condensate state, all spin network nodes are characterized by the same condensate wave function and in this sense condensate states have a 'wave function homogeneity'.", '1602.05881-1-95-1': 'For this reason, condensate states are naturally adapted to the notion of homogeneity, and the coarse-graining procedure which determines the hydrodynamical equations of motion of homogeneous space-times from the microscopic GFT dynamics will be more straightforward in this case than it is in general.', '1602.05881-1-95-2': 'In addition, in GFT condensate states, the condensate wave function for each spin network node stores the intrinsic geometric data of a quantum tetrahedron in 4D, which is isomorphic to the space of homogeneous (and possibly anisotropic) space-times [CITATION].', '1602.05881-1-95-3': 'This will further simplify the coarse-graining procedure.', '1602.05881-1-96-0': 'In the coarse-graining procedure then, the fact that all of the fundamental GFT quanta are in the same state naturally implements the idea of spatial homogeneity of the continuum (quantum) geometry one can reconstruct from them.', '1602.05881-1-96-1': "The notion of 'continuum' here refers to the fact that an infinite number of fundamental degrees of freedom is captured by the definition of the condensate states; they do not correspond in this sense to any (say, lattice) truncation of the theory but rather to a coarse-graining of it, in which an infinite number of fundamental degrees of freedom is captured by a single collective wave function depending on a small number of collective variables.", '1602.05881-1-96-2': 'This is in full analogy with the use of condensate states in many-body quantum theory, e.g., to extract the relevant macroscopic behaviour of quantum liquids in the condensate phase.', '1602.05881-1-96-3': "In fact, this particular class of quantum states would be physically (and not only mathematically) appropriate for describing our universe at macroscopic scales in a scenario in which the 'geometrogenesis' transition -between the pre-geometric phase of quantum gravity (which would not admit a continuum description in terms of geometric fields and general relativity) and the geometric one (described by continuous semi-classical geometries)- is a Bose-Einstein condensation process for the fundamental quantum gravity building blocks of space-time [CITATION].", '1602.05881-1-97-0': '## The Simplest Condensate States: The Gross-Pitaevskii Approximation', '1602.05881-1-98-0': 'Since the aim here is to describe the simplest cosmological space-times -namely the spatially flat, homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology- the appropriate degrees of freedom can already be captured in the simplest condensate states.', '1602.05881-1-99-0': 'In particular, since on the one hand the only geometrical observable of interest is the spatial volume', '1602.05881-1-100-0': '(as a function of relational time [MATH]) which can easily be calculated without knowing how the spin network nodes are connected amongst themselves, and on the other hand there is no need to encode the spatial curvature in the connectivity of the spin network nodes, it is in fact reasonable to entirely ignore the connectivity of the spin network nodes in the condensate state.', '1602.05881-1-101-0': 'Although this is a drastic approximation, it appears quite reasonable for the spatially flat FLRW space-time.', '1602.05881-1-101-1': 'In addition, this approximation will significantly simplify both the form of the resulting condensate state as well as its quantum equations of motion.', '1602.05881-1-102-0': 'Following the arguments above, we shall neglect the connectivity amongst the spin network nodes, and in this case it is possible to restrict our attention to the relatively simple Gross-Pitaevskii condensate states.', '1602.05881-1-102-1': 'These are coherent states for the GFT field operator, [EQUATION] where [EQUATION]', '1602.05881-1-102-2': 'As can be seen here, the condensate wave function [MATH] is not normalized.', '1602.05881-1-102-3': 'Indeed, its norm captures an important physical quantity, the number of fundamental GFT quanta.', '1602.05881-1-102-4': 'Note that an important property of condensate states is that they are eigenstates of the field operator [MATH], [EQUATION]', '1602.05881-1-102-5': 'In order to restrict the condensate wave function [MATH] to the space of intrinsic geometries of a tetrahedron in 4D (which as already noted above is isomorphic to the space of spatially homogeneous space-times), an additional restriction is imposed upon the condensate wave function.', '1602.05881-1-102-6': 'Besides the right gauge invariance given in [REF], the condensate wave function is also taken to be invariant under diagonal left gauge transformations, [EQUATION]', '1602.05881-1-102-7': 'This choice is rather natural, from a geometric point of view: requiring invariance under diagonal left gauge invariance is equivalent to group-averaging over the relative embedding of the tetrahedron in [MATH], which is not a physically relevant observable.', '1602.05881-1-103-0': 'It is clear that condensate states satisfying the requirement [REF] encode no explicit information about the connectivity of the graphs underlying microscopic states, and therefore also no information about the topology of the space-time itself.', '1602.05881-1-103-1': 'As a result, the appropriate procedure to coarse-grain these states and extract the correct hydrodynamical equations of motion is rather straightforward, but is also less manifest; the coarse-graining is more apparent when working with more general condensate states [CITATION].', '1602.05881-1-103-2': 'While the limitations of the approximation of neglecting the connectivity information should be kept in mind, it is also important to remember that in condensed matter physics the same type of drastic simplification is commonly employed and Gross-Pitaevskii condensate states capture a lot of relevant physics even in their simple form.', '1602.05881-1-104-0': 'Nonetheless, in some settings of physical interest it is to be expected that correlations among GFT quanta may be relevant (in which case quantum fluctuations cannot be neglected), and then the Gross-Pitaevskii approximation will fail.', '1602.05881-1-104-1': 'Indeed, the Gross-Pitaevskii hydrodynamics -as captured by the simple condensate states of the type [REF]- will break down when the interactions between the GFT quanta become sufficiently large.', '1602.05881-1-104-2': 'This can happen either if the coupling constants in front of the interaction terms in the GFT action are big, or when the condensate density (i.e., the modulus of the condensate wave function or equivalently the expectation value of the GFT number operator) grows too large.', '1602.05881-1-104-3': 'We will discuss this point in greater detail in Sec. [REF].', '1602.05881-1-105-0': 'In order to study situations where the interactions become important, it will be necessary to use more complicated condensate states, for example the generalizations defined in [CITATION].', '1602.05881-1-105-1': 'These condensate states contain information about the connectivity of the spin network nodes and hence they are easier to coarse-grain in order to recover the geometry of the resulting space-time; in particular, these generalized condensate states directly provide the topological information of the space-time which is absent in the Gross-Pitaevskii GFT condensate states [REF].', '1602.05881-1-106-0': 'Still, even in the Gross-Pitaevskii approximation the condensate wave function [MATH] encodes both the number of quanta in the state and the distribution over the possible values of the geometric data of homogeneous (and potentially anisotropic) cosmologies.', '1602.05881-1-106-1': 'Therefore, these condensate states provide a natural point of contact with the usual Wheeler-DeWitt picture of quantum cosmology, and have the additional advantages of (i) not requiring any symmetry reduction, and (ii) including the many-body features of the full Hilbert space of the microscopic theory.', '1602.05881-1-107-0': 'Furthermore, for these simple condensate states it is straightforward to extract their effective dynamics.', '1602.05881-1-107-1': 'An exact solution to the quantum equations of motion would satisfy all Schwinger-Dyson equations.', '1602.05881-1-107-2': 'However, here we have made an important approximation by assuming that the relevant state is a condensate state of the simplest type.', '1602.05881-1-107-3': 'Therefore, we can only hope for the Schwinger-Dyson equations to be solved approximately.', '1602.05881-1-107-4': 'For this reason we concentrate on only one Schwinger-Dyson equation - the simplest one which corresponds to the classical equations of motion for the condensate state [MATH], which is [EQUATION]', '1602.05881-1-107-5': 'Note that these equations are non-linear in the condensate wave function and can be non-local on minisuperspace (e.g., if the condensate wave function can be rewritten in terms of minisuperspace variables like the scale factor [MATH], the non-linear terms may couple the condensate wave function at different values of [MATH]).', '1602.05881-1-108-0': 'For the action [REF], the dynamics for the condensate state [REF] corresponding to its classical equations of motion is simply (in the spin representation): [EQUATION] with the interaction term contributing a non-linear term of the order of [MATH], the details of which are unimportant for now.', '1602.05881-1-108-1': 'Also note that there is no sum over [MATH] or [MATH] here.', '1602.05881-1-109-0': 'This equation of motion clearly shows that for the condensate state [MATH], the microscopic dynamics of the GFT quanta can be described hydrodynamically in terms of the collective variable [MATH], the condensate wave function.', '1602.05881-1-109-1': 'The hydrodynamic nature of the resulting effective dynamics is the result of the Schwinger-Dyson equations effectively collapsing, due to the simple form of the state [MATH], to the one-particle correlation function (which is exactly [MATH] in the condensate approximation [REF]).', '1602.05881-1-109-2': 'The continuum nature of the equation arises from the fact that [MATH] is given by an infinite sum over numbers of spin network nodes (and hence implicitly over graphs also), and therefore [MATH] is a non-perturbative state with respect to the Fock vacuum.', '1602.05881-1-110-0': '## The Large Density Case', '1602.05881-1-111-0': 'As has already been mentioned, in the large density regime -i.e., when the state has a large number of quanta- the Gross-Pitaevskii approximation encoded in the simple condensate state [REF] breaks down.', '1602.05881-1-111-1': 'This is because, while solutions to the complete GFT quantum dynamics satisfy the operator equations [REF], the coherent state ansatz [REF] only satisfies the expectation value of that same operator equation and thus necessarily fails to capture enough information about the dynamics in regimes where quantum fluctuations and correlations among the GFT quanta become important.', '1602.05881-1-112-0': 'This will be the case, in particular, if the interaction term is large.', '1602.05881-1-112-1': 'To illustrate this point in a bit more detail, let us consider a simple GFT model whose equation of motion is [EQUATION]', '1602.05881-1-112-2': 'While this is a vacuum GFT model, the results obtained here are rather general and also hold for GFT models with a scalar field.', '1602.05881-1-113-0': 'For the coherent state ansatz [REF], the expectation value of the above equation of motion gives [EQUATION] and it can be seen that the condensate state does not exactly satisfy the quantum equations of motion [REF].', '1602.05881-1-113-1': 'This is not surprising since the condensate ansatz is only expected to provide an approximate solution to the equations of motion.', '1602.05881-1-113-2': 'An estimate of the amplitude of the error can be assessed in a number of ways; in the GFT framework, one possibility is to look at the norm of the quantity [EQUATION]', '1602.05881-1-113-3': 'Using the field equations for [MATH], the state [MATH] can be rewritten as [EQUATION] with [EQUATION]', '1602.05881-1-113-4': 'Then the norm of [MATH] can easily be calculated, giving [EQUATION]', '1602.05881-1-113-5': 'The second term is exactly the result obtained from normal ordering all of the operators in the expectation values.', '1602.05881-1-113-6': 'Therefore, the norm of [MATH] is controlled by the functional of the wave function that is obtained by taking the necessary Wick contractions.', '1602.05881-1-113-7': 'Although the expressions are cumbersome and not particularly illuminating, it is clear that this functional will contain several terms obtained from combining positive powers of the condensate wave function.', '1602.05881-1-113-8': 'When [MATH] becomes large, then the norm of [MATH] could become large as well, in general, and at a much faster rate, given the presence of higher powers.', '1602.05881-1-113-9': 'In this case the Gross-Pitaevskii approximation would clearly break down.', '1602.05881-1-113-10': 'Whether the Gross-Pitaevskii approximation fails for large [MATH] has to be considered on a case by case basis, as it depends on the interplay between the solution and the interaction kernels of the specific GFT model of interest.', '1602.05881-1-114-0': 'This limitation is the analogue of the failure of the dilute gas approximation in the description of the Bose gas whenever the strength of the interactions becomes large with respect to the interatomic distance.', '1602.05881-1-114-1': 'This means that the coherent state approximation can be trusted only in a mesoscopic regime, in which the number of quanta is large enough to admit a continuous geometry approximation but small enough to avoid large deviations from the correct physical state.', '1602.05881-1-114-2': 'In settings where the GFT analogue of the Gross-Pitaevskii ansatz fails, one will have to resort to more involved condensate states (e.g., of the type described in [CITATION]) and/or include the dynamics of fluctuations and their coupling to the background condensate dynamics.', '1602.05881-1-115-0': '## Restriction to Isotropy', '1602.05881-1-116-0': 'In the same spirit of coarse-graining to cosmological observables (rather than symmetry-reduction), we will now impose a further restriction on the form of the condensate wave function so that only isotropic quantities such as the spatial volume or the Hubble rate can be reconstructed from [MATH].', '1602.05881-1-116-1': 'Clearly, this restriction on the type of microscopic states allowed will lead to further simplifications in the effective continuum dynamics.', '1602.05881-1-116-2': 'Note that imposing isotropy on the condensate wave function is a restriction on the macroscopic variable that collectively describes all the microscopic configurations entering the definition of the state - it does not correspond to a symmetry reduction of these microscopic configurations to isotropic ones (this property of the coarse-graining is more apparent for the case of generalized condensate states [CITATION], but also holds in this case.)', '1602.05881-1-117-0': 'Imposing isotropy can be done in a number of different ways.', '1602.05881-1-117-1': 'Since the GFT action for the EPRL spin foam model [REF] is expressed in the spin representation, it is convenient to determine how isotropy is to be imposed in the spin representation.', '1602.05881-1-117-2': "Since spin network nodes can be interpreted as tetrahedra, it is natural to require that the tetrahedra be as 'isotropic' as possible, and therefore we shall define isotropic condensate wave functions to be those that correspond to equilateral tetrahedra (which are the most 'isotropic' configurations).", '1602.05881-1-117-3': 'As a result, an isotropic condensate wave function [MATH] will depend only on the volume of the tetrahedron or, equivalently, the surface area of one of its faces (as well as on the scalar field [MATH])', '1602.05881-1-118-0': 'Since isotropic condensate wave functions are to be those that correspond to equilateral tetrahedra, it is helpful to recall some of their properties.', '1602.05881-1-118-1': 'The classical geometry of an equilateral tetrahedron is entirely described by its four area vectors [MATH], which all have the same norm and are related to each via the discrete subgroup of [MATH] called the tetrahedral group.', '1602.05881-1-118-2': 'Equilateral tetrahedra have two properties that will be relevant for the construction of isotropic condensate wave functions: (i) the areas of the faces are equal, [MATH] and (ii) for given a total surface area [MATH], the volume of the tetrahedron is maximized in the equilateral case.', '1602.05881-1-119-0': 'We can use these two properties to display more clearly what are the true dynamical degrees of freedom that survive in the condensate wave function.', '1602.05881-1-119-1': 'First, using the Peter-Weyl theorem to decompose the wave function into a basis of orthonormal functions given by the Wigner matrices, we see that left and right gauge invariance reduce the wave function to a very simple form: [EQUATION] in terms of linear combinations of pairs of intertwiners, one associated to the left gauge invariance and one to the right closure condition.', '1602.05881-1-119-2': 'The geometric properties are therefore entirely contained in the scalar functions [MATH].', '1602.05881-1-120-0': 'The first geometric condition on the isotropic restriction is encoded in the quantum theory by requiring that [MATH] must have support only on spin network nodes where the four [MATH] on each link are equal, [EQUATION]', '1602.05881-1-120-1': 'The second condition, the requirement that the volume be maximized in equilateral tetrahedra, is imposed by demanding that for a given [MATH] the condensate wave function only has support on the intertwiner that (i) is an eigenvector of the LQG volume operator, and (ii) has the largest eigenvalue.', '1602.05881-1-120-2': 'Denoting this (normalized) intertwiner by [MATH], [EQUATION] where the [MATH] are determined by the above two conditions together with [MATH].', '1602.05881-1-121-0': 'Then, the requirement that the wave function has components only along the equilateral tetrahedra components implies that [EQUATION]', '1602.05881-1-121-1': "Finally, the geometric interpretation of the [MATH] intertwiner (and of the 'left flux operators' also) is not immediately clear.", '1602.05881-1-121-2': 'However, some heuristic arguments do provide some guidance.', '1602.05881-1-121-3': "Since the connectivity in GFTs is imposed (in the spin representation) by requiring that the 'left' angular momentum quantum numbers agree between connected links, it may be possible to interpret the left flux operators as the right flux operators parallel-transported from the node of the spin network to the ends of the spin network links.", '1602.05881-1-121-4': 'Since scalar quantities like the volume should behave trivially under parallel transport, this suggests requiring that the volume eigenstate of [MATH] be equal to that of [MATH], and hence that [MATH].', '1602.05881-1-121-5': 'We leave a more detailed study of this condition for future work.', '1602.05881-1-122-0': 'Then, implementing all of the above conditions, [EQUATION]', '1602.05881-1-122-1': 'With all these restrictions, coming from geometric reasoning and symmetry arguments, only the form of [MATH] remains to be solved for.', '1602.05881-1-122-2': 'As indicated at the beginning of this section, the condensate wave function (at a given [MATH]) is entirely determined by its dependence on just one geometric quantity, the spin [MATH].', '1602.05881-1-122-3': 'For this reason, the only geometric quantities that can be extracted from a condensate wave function of this type are isotropic quantities like the total spatial volume and the Hubble rate, and so [REF] is a natural candidate to extract the homogeneous, isotropic and spatially flat cosmological sector from quantum gravity.', '1602.05881-1-123-0': 'Finally, for the sake of completeness, note that in the group representation an isotropic condensate wave function has the form [EQUATION]', '1602.05881-1-123-1': 'The form of an isotropic condensate wave function in the flux representation can be obtained via a non-commutative Fourier transform.', '1602.05881-1-124-0': '# Effective Cosmological Dynamics of Isotropic Quantum Gravity Condensates', '1602.05881-1-125-0': 'As already outlined in Sec. [REF], in the Gross-Pitaevskii approximation, given a condensate wave function for a specific GFT model, it is a straightforward procedure to obtain the (non-linear) equations of motion for that condensate wave function.', '1602.05881-1-125-1': 'In this section, we will derive the equations of motion for the isotropic condensate wave function [MATH] for a GFT model based on the Lorentzian EPRL spin foam model and with a massless scalar field, whose action is given in [REF].', '1602.05881-1-125-2': 'From these equations of motion it will be possible to extract effective Friedmann equations, expressed in a relational form with respect to the massless scalar field [MATH].', '1602.05881-1-126-0': '## Relational Dynamics', '1602.05881-1-127-0': 'As already explained in Sec. [REF], the equation of motion for a condensate in the Gross-Pitaevskii approximation is given by [REF].', '1602.05881-1-127-1': 'It is worth repeating that the condensate state is expected to be an approximate solution, and therefore not all of the Schwinger-Dyson equations will hold.', '1602.05881-1-127-2': 'Instead, we only impose the first Schwinger-Dyson equation exactly.', '1602.05881-1-127-3': 'For an isotropic condensate wave function of the type [REF], and for the GFT action [REF], based on the EPRL spin foam model and extended in order to include a massless scalar field, this equation has the simple form [EQUATION] which are non-linear in [MATH], and the constants [MATH] and [MATH] are obtained from combinations of the kinetic and interaction terms in the GFT action with the non-[MATH] terms in the condensate wave function [MATH].', '1602.05881-1-127-4': 'To be specific, decomposing the various terms in the action in representations, [EQUATION] using a condensed notation in order to avoid writing explicitly all the tensor indices, and here the [MATH] symbol of [MATH] is that of, e.g., [CITATION].', '1602.05881-1-128-0': 'Note that these equations of motion for the condensate wave function can also be obtained from the action [EQUATION] obtained by replacing the GFT field in the GFT action by the condensate wave function.', '1602.05881-1-128-1': 'Here it is clear that the scalar field [MATH] plays the role of a relational time variable.', '1602.05881-1-129-0': 'The condensate equations of motion depend directly on the details of the GFT action, since these determine in part the coefficients [MATH] and [MATH].', '1602.05881-1-129-1': 'It will be possible to constrain their form by requiring that the Friedmann equation be recovered in an appropriate semi-classical limit.', '1602.05881-1-130-0': 'Crucially, the interaction term does not couple [MATH] with different [MATH].', '1602.05881-1-130-1': 'This is due to the combination of the isotropic restriction and the form of the EPRL vertex amplitude which contain Kronecker deltas [MATH] for all edges that meet in the four-simplex.', '1602.05881-1-130-2': 'Thus, if five equilateral tetrahedra are combined in a four-simplex, and the vertex amplitude is the EPRL one (or one with an analogous property) then it immediately follows that all of the five equilateral tetrahedra must have the same [MATH].', '1602.05881-1-130-3': 'This decoupling does not occur generically, even in the isotropic restriction, for other spin foam models, e.g., those like the Baratin-Oriti model [CITATION] involving more elaborate fusion coefficients.', '1602.05881-1-130-4': "For this reason, the interaction term is 'local' in the spin label since it has the form [MATH] rather than [MATH].", '1602.05881-1-130-5': 'Clearly, this significantly simplifies the equations of motion.', '1602.05881-1-131-0': 'As true in general for GFT condensates, we have thus obtained a quantum cosmology-like equation for a cosmological wave function on the space of (isotropic) homogeneous geometries.', '1602.05881-1-131-1': "This equation is however non-linear, as to be expected in a hydrodynamic context, with the non-linearities effectively encoding the microscopic interactions between the fundamental 'atoms of space', which are also ultimately responsible for the generation of inhomogeneities at both microscopic and macroscopic scales (see also [CITATION] for a similar construction).", '1602.05881-1-132-0': "Before we start analyzing the effective dynamical equations, we point out that, from the symmetries of [MATH], it is obvious that there is a conserved quantity for every [MATH], the 'energy' [MATH] of the condensate wave function [MATH] with respect to the relational time [MATH], [EQUATION]", '1602.05881-1-132-1': 'In addition, in the regime in which the interaction term is small (which is necessary for the Gross-Pitaevskii approximation to hold), the [MATH] charge [MATH] related to the symmetry [MATH] emerges as another conserved quantity [EQUATION]', '1602.05881-1-132-2': 'Note that, following from the definition of the momentum of the massless scalar field, it is easy to check that [MATH] and therefore [MATH] is a conserved quantity also in the limit where the Gross-Pitaevskii approximation holds.', '1602.05881-1-133-0': 'The equation [EQUATION] is exactly the continuity equation in cosmology, for the case of a massless scalar field.', '1602.05881-1-133-1': 'This is a particularly simple example of how the large-scale, coarse-grained effective dynamics can be extracted from the GFT quantum equations of motion for condensate states.', '1602.05881-1-133-2': 'This result is also a first confirmation of the consistency of the identification of the GFT condensate state and an emergent FLRW space-time geometry.', '1602.05881-1-134-0': 'Note that the condition that the interactions be subdominant is required in order to recover the continuity equation for the isotropic condensate state and, as we will show, the Friedmann equations.', '1602.05881-1-134-1': 'While this is to some extent only a technical restriction to a regime where simple condensate states can be trusted, it is not unreasonable from a physical point of view.', '1602.05881-1-134-2': 'One expects that generic interactions would generate correlations between GFT quanta, and there is no reason to expect these to respect any homogeneity condition, but rather to produce inhomogeneities both the microscopic and macroscopic level.', '1602.05881-1-134-3': 'And when inhomogeneities are included in cosmology (even at linear order) the continuity equation is modified.', '1602.05881-1-134-4': 'Note that the heuristic arguments above do not necessarily imply that GFT non-linearities at the level of the hydrodynamic equation encode inhomogeneities (as has been suggested in [CITATION]), but this is an interesting hypothesis to explore, especially considering how similar equations (again inspired by BEC theory) have been obtained as an effective description of inhomogeneities in a non-linear extension of (loop) quantum cosmology [CITATION].', '1602.05881-1-135-0': 'For the remainder of this paper, we will only consider the limit where the interaction term is much smaller than the linear terms.', '1602.05881-1-135-1': 'This is not because the non-linear case is difficult to solve (in fact, for the simple condensate equations of motion considered here, it is relatively straightforward to study the dynamics of the condensate wave function even in the presence of the non-linear term) but rather because in that limit the Gross-Pitaevskii approximation is expected to fail, in the sense that it cannot be justified from a microscopic point of view since the simple condensate state we use here cannot be expected to be a good approximation to a realistic vacuum of the theory, and it is necessary to consider more complex condensate states than [REF].', '1602.05881-1-136-0': 'Therefore, we will study the regime where [MATH] is sufficiently small so that the interaction term is subdominant, but at the same time not so small that the hydrodynamic approximation ceases to make sense: after all, [MATH] corresponds to the average number of GFT quanta at the relational time [MATH], and a large number of quanta is necessary for the hydrodynamic approximation to be valid.', '1602.05881-1-136-1': 'In this regime, the equation of motion for [MATH] reduces to [EQUATION] with [MATH].', '1602.05881-1-137-0': 'At this point, it is convenient to separate [MATH] into its modulus and phase, [EQUATION] with [MATH] and [MATH] both assumed to be real, and [MATH] to be positive.', '1602.05881-1-137-1': 'From now on, we will drop the argument [MATH], and denote derivatives with respect to [MATH] with primes, e.g., [MATH].', '1602.05881-1-137-2': 'Then, in terms of [MATH] and [MATH], the equation of motion [REF] splits into a real and an imaginary part, which are respectively [EQUATION] and [EQUATION]', '1602.05881-1-137-3': 'The last equation, coming from the imaginary part of [REF], can easily be solved and shows that the combination [MATH] is a constant of the motion, and in fact is precisely the conserved [MATH] charge [REF], [EQUATION]', '1602.05881-1-137-4': "Note that the other conserved charge, the 'GFT energy' for each [MATH], also has a simple form, [EQUATION]", '1602.05881-1-137-5': 'Finally, using [REF], the remaining equation of motion [REF] can be rewritten as [EQUATION] and this has the form of the equation of motion of a particle in a central potential.', '1602.05881-1-137-6': 'In particular, note that the effective potential diverges as [MATH]; this implies that [MATH] remains non-zero at all times (for non-zero [MATH]).', '1602.05881-1-137-7': 'This is what will lead to the resolution of the big-bang and big-crunch singularities in the cosmological space-time, as is explained in detail in the next section, so long as the cosmological dynamics are captured by the above equation.', '1602.05881-1-138-0': 'However, before studying the dynamics in more detail and extracting the equations of motion for geometric quantities, it is important to recall the assumptions that were necessary in order to derive [REF].', '1602.05881-1-138-1': 'First, we have assumed that a cosmological state in quantum gravity is well-approximated by a simple condensate that in particular ignores connectivity information, which is in general a very important set of dynamical degrees of freedom.', '1602.05881-1-138-2': 'However, in the case of isotropic cosmology we expect these degrees of freedom to play a less important role since the only relevant geometric observables are the spatial volume and its conjugate.', '1602.05881-1-138-3': 'Second, we further imposed that the quanta of geometry in the condensate be isotropic, and we are working in the limit where the scalar field [MATH] is assumed to evolve slowly.', '1602.05881-1-138-4': 'Finally, we are considering the regime where the interaction term in [REF] is subdominant, and hence where the [MATH] are sufficiently small.', '1602.05881-1-139-0': 'On the other hand, for there to exist a continuum interpretation of the condensate state as a space-time, there must be a large number of quanta of geometry in the condensate state, which requires the [MATH] to be large.', '1602.05881-1-139-1': '(Also, in order for a consistent continuum geometric interpretation to be valid at least for large total spatial volumes of the universe, a few more conditions are needed, namely that there be a small curvature and a small volume associated to each individual GFT quantum.', '1602.05881-1-139-2': 'These last conditions are not necessary for the mathematical consistency of the condensate approximation, but are necessary to have a clear space-time interpretation for the condensate state.)', '1602.05881-1-140-0': 'A delicate interplay between the values of [MATH] and the coupling constants (and kernels) of the theory is required for the condensate approximation to be valid while at the same time neglecting the interactions.', '1602.05881-1-140-1': 'It is only when all of these assumptions hold that a reliable cosmological interpretation of the condensate state exists and that the effective dynamics extracted here from the full theory can be trusted.', '1602.05881-1-141-0': '## Condensate Friedmann Equations', '1602.05881-1-142-0': 'The effective dynamics of the GFT condensates is (part of) the hydrodynamics of the GFT model we are studying, and is encoded in an equation for the mean field [MATH] (and its complex conjugate) or, in more conventional hydrodynamic form, for a density [MATH] and a phase [MATH], which in turn can be decomposed in terms of modes associated to representations [MATH].', '1602.05881-1-142-1': 'This type of equation has the form of a non-linear extension of a quantum cosmology dynamics, even though the physical interpretation is different.', '1602.05881-1-142-2': 'From this type of equation, just as in (loop) quantum cosmology, it is possible to extract the gravitational dynamics in the form of equations for geometric quantities.', '1602.05881-1-142-3': 'In particular, for homogeneous and isotropic configurations, a natural choice is to derive an effective equation that governs the dynamics of the volume of the universe, coupled to the scalar field.', '1602.05881-1-143-0': 'This can be done in a straightforward fashion in this case starting from the equations of motion for [MATH] obtained in the previous section and relating the spatial volume to the [MATH].', '1602.05881-1-143-1': 'By using the massless scalar field [MATH] as a relational clock, the resulting equations of motion for [MATH] can be compared to the Friedmann equations of cosmology, which are presented in the Appendix [REF].', '1602.05881-1-144-0': 'The quantity of interest here is the total volume of the universe in the condensate state, at a given moment of the relational time [MATH], [EQUATION] where [MATH] is the eigenvalue of the volume operator in canonical loop quantum gravity acting on an equilateral (as defined in Sec. [REF]) four-valent spin network node in the representation [MATH].', '1602.05881-1-144-1': '(Clearly, it follows from the definition of equilateral spin network nodes that [MATH] is the largest eigenvalue of the LQG volume operator possible for a node with all [MATH].)', '1602.05881-1-144-2': 'Note that the scaling mentioned here is approximate, and for a detailed analysis it would be necessary to explicitly calculate [MATH] for each [MATH].', '1602.05881-1-144-3': 'However, this will not be necessary here.', '1602.05881-1-145-0': 'A technical comment is also in order here.', '1602.05881-1-145-1': 'The LQG volume operator depends on the Barbero-Immirzi parameter [MATH], which only appears in spin foam models after the simplicity constraints have been imposed.', '1602.05881-1-145-2': 'In the GFT models based on spin foam models, the simplicity constraints are imposed in the interaction term in the GFT action, whose effect in the equations of motion has been assumed to be negligible.', '1602.05881-1-145-3': 'However, an operator in GFT can only be interpreted as a geometric operator after simplicity has been imposed.', '1602.05881-1-145-4': 'This is why it is important to remember that we are not ignoring the effect of the interaction term but instead we are considering the case where the contribution of the interaction term to the equations of motion is negligible compared to that of the kinetic terms.', '1602.05881-1-145-5': 'The interaction term is nonetheless present and imposes simplicity, but its contribution to the equations of motion of the condensate wave function is negligible and can be ignored.', '1602.05881-1-146-0': 'Now, given [REF], and using the notation of Sec. [REF], [EQUATION] and [EQUATION]', '1602.05881-1-146-1': 'Both [MATH] and [MATH] depend also on the [MATH] interaction term in the equations of motion, but the contribution from the interaction term is assumed to be subdominant in the Gross-Pitaevskii approximation and therefore we neglect these terms here.', '1602.05881-1-147-0': 'From the equations above it follows immediately that the generalised Friedmann equations in terms of the relational time [MATH] are given by [EQUATION] and [EQUATION]', '1602.05881-1-147-1': 'These effective Friedmann equations for the GFT condensate include the correct classical limit (i.e., they reproduce the standard Friedmann equations of general relativity, justifying their name), as shall be shown in Sec. [REF], as well as some quantum corrections coming from the microscopic GFT theory.', '1602.05881-1-147-2': 'Interestingly, some of these corrections have a clear geometric meaning, which shall be discussed shortly.', '1602.05881-1-147-3': 'From these equations, it is possible to solve for the dynamics of the total volume, given some initial state [MATH] at an initial time [MATH].', '1602.05881-1-148-0': 'An important point here is that, for the energy density of the massless scalar field, which is defined in terms of the expectation values of scalar field momentum and volume operators as [EQUATION] to be non-zero, at least one of the [MATH] must be non-zero', '1602.05881-1-149-0': 'The condition that at least one of the [MATH] be non-zero is necessary for the relational dynamics to be well-defined, and also to ensure that the homogeneous and isotropic space-time is an FLRW space-time, not the vacuum Minkowski space-time.', '1602.05881-1-150-0': 'This restriction has important consequences.', '1602.05881-1-150-1': 'Obviously, the condition that at least one of the [MATH] be non-zero is a necessary (although not sufficient) condition for the existence of solutions with a good cosmological interpretation, and also for the consistency of the relational description in the first place.', '1602.05881-1-150-2': 'On the other hand, this is not in itself a necessary condition for the mathematical consistency of the condensate dynamics.', '1602.05881-1-150-3': 'This means that there may be solutions which do not satisfy this condition, but are still mathematically well-defined and within the regime of validity of the condensate hydrodynamics we are studying.', '1602.05881-1-150-4': 'Therefore, this is an additional requirement beyond the assumptions for a condensate which is necessary for the condensate state to be interpreted as a cosmological space-time.', '1602.05881-1-151-0': 'An open question is whether setting all [MATH] (but still having large [MATH]) gives Minkowski space, in which case the condensate state would correspond to a large space-time although there would be no relational dynamics.', '1602.05881-1-151-1': 'We comment further on the vacuum limit in Sec. [REF].', '1602.05881-1-152-0': 'Requiring that the energy density of the massless scalar field be non-vanishing has a very important consequence: since at least one [MATH] must be non-zero to have a solution that can be interpreted as a cosmological space-time, it follows from [REF] that at least one [MATH] will always remain greater than zero.', '1602.05881-1-152-1': 'In turn, since [MATH], it follows that [MATH] will always remain non-zero.', '1602.05881-1-152-2': 'Therefore, we find that for all cosmological solutions, the volume will never become zero.', '1602.05881-1-153-0': 'In this way, the big-bang and big-crunch singularities of classical FLRW space-times that occur generically in general relativity are resolved in the GFT condensate states studied here.', '1602.05881-1-153-1': 'The equation of motion for [MATH] [REF] clearly shows that the individual [MATH] will reach a minimal value at which point they will bounce (and it is clear that there is only a single bounce since [MATH] has only one turning point), and thus the cosmological space-time that emerges from the GFT condensate state is that of a bouncing FLRW space-time.', '1602.05881-1-154-0': 'In order to see exactly how the singularity is resolved, and better understand the nature of the quantum effects causing this resolution, it is necessary to solve our modified Friedmann equations for [MATH] for some initial conditions.', '1602.05881-1-154-1': 'Unfortunately, it is difficult to provide an exact solution to these equations of motion for generic initial conditions, but there are two special cases when an explicit solution can be found.', '1602.05881-1-155-0': '## Classical Limit', '1602.05881-1-156-0': 'As already mentioned, the momentum of the scalar field, defined as the expectation value of the operator [REF] in the condensate state, is given by [MATH] and therefore [MATH] is a conserved quantity: this is exactly the continuity equation for a massless scalar field in an FLRW space-time.', '1602.05881-1-156-1': 'Therefore, the only other requirement in order to verify that the correct semi-classical limit is obtained is to ensure that the correct Friedmann equation is recovered.', '1602.05881-1-157-0': 'The classical limit of the generalised Friedmann equations is obtained when the Hubble rate is small compared to the inverse Planck time, and this will occur at sufficiently large volumes, i.e., when [MATH] and [MATH] (note that the semi-classical limit is not the limit of large volume, but of small space-time curvature; nonetheless, the space-time curvature decreases as the space-time expands and therefore the dominant term in the Friedmann equation at large volumes is also the dominant term when the space-time curvature is small).', '1602.05881-1-157-1': 'As shall be seen in the next section, the terms containing [MATH] and [MATH] can be understood as quantum corrections.', '1602.05881-1-158-0': 'In this limit, the generalised Friedmann equations become [EQUATION] and [EQUATION]', '1602.05881-1-158-1': 'We immediately see from these equations that, in order to recover the classical Friedmann equations of general relativity in terms of the relational time [MATH], which are given in Appendix [REF], (in this specific context where the FLRW space-time emerges as a condensate of isotropic GFT quanta) it is necessary to identify [MATH] for all [MATH].', '1602.05881-1-158-2': 'For these values of [MATH], the GFT condensate dynamics reproduce the classical Friedmann equations of general relativity.', '1602.05881-1-158-3': '(As an aside, note that while it may be possible, at a specific relational instant [MATH], to choose a different set of values for [MATH] that also gives the correct limit, this identification will not be preserved by the dynamics and hence the correct classical Friedmann equations would in this case only be recovered in a small neighbourhood of relational time around [MATH].)', '1602.05881-1-159-0': 'The condition that [MATH] is a requirement on the form of the terms [MATH] and [MATH] that are determined by the GFT action: if [MATH] for some [MATH], then it follows that the correct Friedmann equations are not recovered in the classical limit.', '1602.05881-1-159-1': 'Note also that this should be understood as a definition of [MATH] which arises as a hydrodynamic parameter and it is thus a function of the microscopic GFT parameters, and not as an interpretation of the microscopic parameters.', '1602.05881-1-159-2': 'This is an important conceptual point since this identification has no reason to be valid in a generic regime of the dynamics (e.g., for non-condensate GFT states) and may be different in other settings.', '1602.05881-1-160-0': 'So, if all [MATH], then the generalised Friedmann equations of the GFT condensate become, in the classical limit, [EQUATION] which are exactly the Friedmann equations of general relativity for a spatially flat FLRW space-time with a massless scalar field [MATH], used as a relational time (see Appendix [REF] for details).', '1602.05881-1-161-0': 'The solution to these equations of motion is the standard one of classical general relativity, [EQUATION] as expected, with the sign in the exponent depending on whether the universe is expanding or contracting, and [MATH] depending on the initial conditions.', '1602.05881-1-162-0': '## Single Spin Condensates', '1602.05881-1-163-0': 'The other case where the equations of motion for [MATH] can be solved exactly, and for generic initial conditions, is when only one [MATH] is non-zero, which corresponds to a condensate wave function that is very sharply (infinitely) peaked in [MATH], [EQUATION]', '1602.05881-1-163-1': 'Then the sum over [MATH] in all of the expressions trivializes and an exact solution can be found which includes quantum corrections.', '1602.05881-1-164-0': 'This assumption mirrors the situation that is thought to be relevant in LQC, where there is also the extra assumption that the underlying LQG state consists of a graph with a very large number of nodes and links, and that the spins on all of the links are identical (often chosen to be [MATH]).', '1602.05881-1-164-1': 'It follows that in the LQC picture, a cosmological space-time expands or contracts by modifications to the combinatorial structure of the spin network that consist of adding or removing nodes, rather than by changing the spin labels on the spin network; this is analogous to the volume dynamics extracted from the underlying GFT model where changes in [MATH] correspond to changes in the number of GFT quanta, rather than transitions between GFT quanta coloured by different spin representations.', '1602.05881-1-164-2': "In the limiting case [REF] considered here, the volume dynamics is entirely dictated by the number of GFT quanta via [MATH]; this is essentially identical to the heuristic interpretation suggested by the LQC 'improved dynamics' relating LQC to the underlying LQG spin networks.", '1602.05881-1-164-3': 'Finally, note that the missing connectivity information in the simple GFT condensates considered here does not play any role in LQC either.', '1602.05881-1-165-0': 'Of course, if only one mode [MATH] contributes to the effective dynamics, then the correct classical limit requires a milder condition on the microscopic dynamics to reproduce the classical Friedmann equation with respect to the more general case considered in the previous subsection, namely that [MATH] (and there are no requirements on the other [MATH]).', '1602.05881-1-166-0': 'Therefore, we set [MATH] in the following so that the correct classical limit is ensured.', '1602.05881-1-166-1': 'Now, since [MATH] for all [MATH], only [MATH] is non-zero and [EQUATION]', '1602.05881-1-166-2': 'Given [REF], the total volume is simply given by [EQUATION] and the first modified Friedmann equation simplifies to [EQUATION] which can be rewritten, using the relation for the energy density of a massless scalar field [MATH], as [EQUATION] with [MATH].', '1602.05881-1-166-3': 'It is clear that the first term is the classical limit, and that the second term is a quantum gravity correction.', '1602.05881-1-166-4': 'In addition, from scaling arguments (the Friedmann equation must be invariant under [MATH] and [MATH]) and dimensional analysis, it follows that [MATH].', '1602.05881-1-166-5': 'Since [MATH], it follows that [MATH] and the third term is also a quantum gravity correction to the classical Friedmann equation.', '1602.05881-1-167-0': 'Similarly, the second modified Friedmann equation is [EQUATION]', '1602.05881-1-167-1': 'Since the last term here also vanishes in the limit of [MATH], it is also to be understood as coming from quantum gravity corrections.', '1602.05881-1-168-0': 'By comparing these modified Friedmann equations for the isotropic GFT condensate [REF], under the condition [REF], to the effective Friedmann equations of loop quantum cosmology given in Appendix [REF], it is clear that they are identical in the case that [MATH], while there are differences of order [MATH] if [MATH] is not zero.', '1602.05881-1-168-1': 'Therefore, the condensate states considered here reproduce, within the full GFT theory, a similar type of quantum-corrected cosmological dynamics as the ones found in LQC.', '1602.05881-1-168-2': 'In fact, even the form of [MATH] obtained here is closely analogous to that of the critical energy density of LQC, which according to the heuristic relation between LQC and LQG also goes as [MATH].', '1602.05881-1-169-0': 'These results therefore provide strong support for the qualitative results of LQC, obtained through the assumption of a particular heuristic relation between full loop quantum gravity and the minisuperspace models of LQC.', '1602.05881-1-169-1': 'Remarkably, very similar effective dynamics arise for the GFT condensate states that mirror the heuristic relation that lies at the heart of LQC.', '1602.05881-1-170-0': 'A few differences must be stressed as well.', '1602.05881-1-170-1': 'First, GFT condensate states provide a significantly more general framework than the specific LQG states used to motivate various constructions in LQC; for example, it is possible and even straightforward to consider states with contributions coming from many different [MATH], as is clear from the discussion in Sec. [REF].', '1602.05881-1-170-2': "Another important difference is the presence of a new quantum number [MATH] which corresponds to an 'energy' of the GFT condensate in the spin [MATH] (with respect to the relational time [MATH]), but whose fundamental geometric interpretation remains, for now, unclear.", '1602.05881-1-170-3': 'However, the effect of [MATH] on the dynamics is relatively simple: if [MATH] then the bounce occurs at an energy density greater than [MATH], on the other hand, if [MATH] then the bounce occurs at an energy density less than [MATH].', '1602.05881-1-170-4': 'Importantly, no matter the value of [MATH], a bounce similar to that found in LQC will always occur: the resolution of the big-bang and big-crunch singularities in GFT condensate cosmology is robust to changes in [MATH] (of course, so long as the regime [MATH] falls within the regime of approximations used to obtain the above cosmological dynamics from the full microscopic dynamics).', '1602.05881-1-171-0': 'A final comment is in order regarding the space-time interpretation of the GFT condensate.', '1602.05881-1-171-1': 'If there are only a few GFT quanta, it is doubtful that it is possible to associate a continuum space-time to such a quantum state, because the approximation between GFT condensate states and continuum geometries should be expected to fail.', '1602.05881-1-171-2': 'Moreover, there is no reason to expect that the hydrodynamic approximation used here to extract the effective cosmological dynamics would still be viable if only a few fundamental GFT quanta are excited from the Fock vacuum.', '1602.05881-1-171-3': '(Although note that the hydrodynamic equations of motion would formally still apply even in this case.)', '1602.05881-1-171-4': 'Thus, a large number of GFT quanta are necessary for the approximations used here to be valid.', '1602.05881-1-171-5': 'Note however that this is not necessarily problematic in a bouncing cosmological context: as is well known in LQC, the bounce occurs when the space-time curvature nears the Planck curvature scale, not when the volume of the universe is comparable to the Planck volume (another way to put it is that it is the Planck energy density that sets the scale of the bounce, not the Planck volume).', '1602.05881-1-171-6': 'The same is true here: the bounce occurs when the space-time curvature is large, and therefore the space-time can (and typically does) bounce at volumes that are much larger than the Planck volume even though the space-time curvature is Planckian.', '1602.05881-1-171-7': 'When this is the case, there will be a large numbers of fundamental GFT quanta even during the bounce, and therefore, modulo other considerations, there will be room for a space-time interpretation of the GFT condensate at all times, including the bounce point.', '1602.05881-1-172-0': '# The Vacuum Limit', '1602.05881-1-173-0': 'In principle, it is also an interesting problem to study the effective cosmology of GFT condensates, and of quantum gravity more generally, in absence of matter fields.', '1602.05881-1-173-1': 'However, there are several difficulties in doing so (some of which have already been noticed in LQC [CITATION]).', '1602.05881-1-174-0': 'In the present context, the vacuum limit of the FLRW space-time, which should give Minkowksi space, is non-trivial for a number of reasons.', '1602.05881-1-174-1': "The main problem is that, in the absence of a matter field, there is no longer a relational clock available and therefore the issue of defining and using diffeomorphism invariant observables (often subsumed in the label 'problem of time') in quantum gravity must be faced head-on, as one should not expect to have any reliable physics otherwise.", '1602.05881-1-174-2': "In particular, the geometric interpretation of the condensate wave function [MATH] is not clear: if [MATH] is to be interpreted as a three-dimensional spatial slice at the relational 'instant of time' [MATH], then it is not immediately obvious how to extract a four-dimensional space-time, or even a reliable phase space interpretation, from [MATH].", '1602.05881-1-174-3': 'This is because there is no control over which spatial slice it refers to, and any apparent physical feature could be attributed instead to some peculiar slicing of spacetime.', '1602.05881-1-175-0': 'Even starting from the Friedmann equation [REF], it is not obvious how to obtain the vacuum limit.', '1602.05881-1-175-1': 'While the right-hand side of [REF] is easily handled by simply setting [MATH], the left-hand side is much trickier, since the derivative with respect to [MATH] becomes ill-defined.', '1602.05881-1-175-2': 'The fact that this is a singular limit is obvious from the classical theory, where the Friedmann equation is simply [MATH]: there is no straightforward way to take the vacuum limit (at least, not without using the relation [REF] which holds in general relativity but cannot be assumed blindly for a GFT condensate state).', '1602.05881-1-176-0': 'Furthermore, we already know that the Gross-Pitaevskii approximation we use is only the simplest truncation available, which may capture some relevant features of the emergent continuum spacetime, as we have shown, but certainly neglects a lot of important aspects of the theory (e.g., connectivity information).', '1602.05881-1-176-1': 'While one can argue that these simple condensates encode enough information to reproduce the dynamics of homogeneous cosmologies, how much is actually captured can only be checked a posteriori, by studying the emergent cosmological dynamics, as we have done.', '1602.05881-1-176-2': 'Then, recovering the Friedmann equations in an appropriate semi-classical limit provides strong evidence that in fact neglecting some of the microsopic details like the connectivity information of the spin network nodes was a reasonable approximation.', '1602.05881-1-176-3': 'However, in Minkowski space there is no dynamical equation that can be recovered in order to provide this type of check.', '1602.05881-1-177-0': 'An additional difficulty with the Gross-Pitaevskii condensate wave function for a vacuum GFT model of EPRL type -in which case there is no derivative operator on the group manifold in the kinetic term of the GFT action- is that the interactions cannot be negligible.', '1602.05881-1-177-1': 'This is obvious from the equation of motion for the (isotropic) condensate wave function for a vacuum GFT model, which has the general form [EQUATION]', '1602.05881-1-177-2': 'Clearly, for this equation to hold, the amplitude of the interaction term must be the same as that of the linear term.', '1602.05881-1-177-3': 'Therefore, the interactions in this case are necessarily large, so we have to work exactly in the regime where the Gross-Pitaevskii approximation cannot be justified from the microscopic theory.', '1602.05881-1-177-4': 'For the vacuum case, it is then anyway necessary to consider more complicated condensate states, perhaps such as those constructed in [CITATION].', '1602.05881-1-178-0': 'One can still learn some interesting lessons about GFTs and spin foam models from studying these equations.', '1602.05881-1-178-1': 'For example, any solution of these equations is necessarily non-perturbative in the GFT coupling constant (if only one interaction is included), and thus it is not reproducible (nor is its putative physics) via the spin foam expansion.', '1602.05881-1-178-2': 'This is in fact the case for all the classical solutions to GFT models that have been studied to date (see e.g., [CITATION]).', '1602.05881-1-178-3': 'However, for the reasons we have explained above, it is unclear how these formal insights can be turned into physical ones.', '1602.05881-1-179-0': '# Discussion', '1602.05881-1-180-0': 'In this paper, we started from a rather general definition of the EPRL spin foam model for the microscopic dynamics of quantum geometry in the GFT framework (completing the spin foam definition).', '1602.05881-1-180-1': 'We parametrized some of the ambiguities entering the definition of the model, by working with general kinetic kernels, and with the aim of constraining these ambiguities by studying the effective cosmological dynamics they give rise to.', '1602.05881-1-180-2': 'While we focused on one specific model, we expect many of our results to hold more generally, and there are indeed several indications that this is the case.', '1602.05881-1-181-0': 'Then, we explained how to couple quantum geometry with a free massless scalar field and gave the definition of the corresponding extension of the GFT model.', '1602.05881-1-181-1': 'One of the key steps here was enforcing at the GFT level the basic symmetries characterizing the scalar field path integral in the free massless case and then performing a small derivative expansion keeping only the lowest orders.', '1602.05881-1-181-2': 'This gives a relatively simple model that we are confident captures the correct semi-classical continuum physics.', '1602.05881-1-181-3': 'At the same time, this allows us to parametrize some ambiguities by working with more general coefficients encoding the matter-geometry coupling.', '1602.05881-1-181-4': 'This is our first new result.', '1602.05881-1-181-5': 'Beside the intrinsic interest of coupling matter and geometry in a GFT model, it was pivotal for the study of the effective cosmological dynamics since it allows for a formulation of the evolution of the universe in terms of purely diffeomorphism invariant observables, as the theoretical context demands, by using the scalar field as a relational clock; this is also a traditional strategy in quantum cosmology.', '1602.05881-1-181-6': 'Interestingly, the scalar field degrees of freedom also enter the GFT action in the same way in which a local time variable would do in standard quantum field theory.', '1602.05881-1-182-0': 'The next step was to implement a restriction to purely isotropic degrees of freedom for the quantum geometric condensate states to be used for the extraction of cosmological dynamics.', '1602.05881-1-182-1': 'These already implement a notion of homogeneity, as discussed in Sec. [REF], and are thus adapted for the simplest cosmological space-times.', '1602.05881-1-182-2': 'We define what it means to restrict the condensate wave function to isotropic configurations by analysing the simplicial geometry of the GFT states.', '1602.05881-1-182-3': 'Indeed, pure volume information can be most directly extracted by the use of quantum states whose quanta can be described by equilateral tetrahedra.', '1602.05881-1-182-4': 'This is our second, intermediate, result.', '1602.05881-1-183-0': 'Armed with the resulting simple condensate states, their effective cosmological dynamics follow directly from the full quantum dynamics of the model, in the GFT analogue of a Gross-Pitaevskii approximation (amounting to the simplest mean field treatment), and correspond to the hydrodynamics of the GFT model.', '1602.05881-1-183-1': 'Despite the simplicity of the GFT condensate states, the emerging cosmological dynamics are extremely interesting.', '1602.05881-1-183-2': 'The equations of motion are non-linear with respect to the condensate wave function, and they can be interpreted in analogy with a quantum cosmology wave function.', '1602.05881-1-183-3': 'Thus, this equation has the form of an extension of a quantum cosmology dynamics as used, e.g., in loop quantum cosmology, and with analogous geometric variables.', '1602.05881-1-183-4': 'Due to the isotropic restriction on the condensate wave function, the actual form of the equation simplifies greatly and it can be manipulated in a straightforward manner.', '1602.05881-1-184-0': 'By defining appropriate relational volume observables, we obtained two equations governing the dynamics of the volume of the universe as a function of the massless scalar field, and depending on the particular solution of the effective GFT condensate hydrodynamics through the values of [MATH] and [MATH].', '1602.05881-1-184-1': 'These two new quantities correspond to state-dependent conserved charges of the system (in the Gross-Pitaevskii approximation when interactions are subdominant).', '1602.05881-1-184-2': 'The two equations take the form of modified Friedmann equations and are our third main result.', '1602.05881-1-185-0': 'Their interpretation is confirmed by their subsequent analysis.', '1602.05881-1-185-1': 'First of all we have shown that in an appropriate semi-classical limit they reduce to the standard Friedmann equations of general relativity.', '1602.05881-1-185-2': 'This is true for any state chosen (among the solutions of the GFT condensate dynamics) and at all times (i.e., for all values of the scalar field) if and only if precise conditions on the initial kinetic kernels of the microscopic GFT models are satisfied.', '1602.05881-1-185-3': "These conditions also provide a definition of Newton's constant [MATH] as a function of the microscopic parameter of the theory, as expected in such a hydrodynamics context.", '1602.05881-1-185-4': 'This is our fourth main result.', '1602.05881-1-186-0': 'Beyond this classical approximation, the theory provides a number of quantum corrections to the classical Friedmann dynamics.', '1602.05881-1-186-1': 'All of them can be computed from first principles starting from the microscopic GFT theory.', '1602.05881-1-186-2': 'We have shown that they share one common feature: the cosmological singularities predicted by the classical theory are generically avoided and a quantum gravity bounce takes their place.', '1602.05881-1-186-3': 'This is our fifth main result.', '1602.05881-1-187-0': 'The scenario suggested by the GFT condensate ansatz for cosmological dynamics is therefore similar to the one found in loop quantum cosmology, which is therefore corroborated by the results found here.', '1602.05881-1-187-1': 'In fact, further evidence in support of LQC is obtained by considering GFT condensate states chosen to only have support on one spin [MATH] - this is the type of state that LQC suggests is relevant for the cosmological sector of LQG.', '1602.05881-1-187-2': 'For this type of condensate state, the resulting effective cosmological dynamics also simplify greatly and can be analyzed in detail, with the result that the effective dynamics are extremely similar to the LQC effective Friedmann equations.', '1602.05881-1-187-3': 'This is our last main result.', '1602.05881-1-188-0': 'Obviously, several approximations were required in order to obtain the results summarized above from the full theory.', '1602.05881-1-188-1': 'As a consequence, these results can only be trusted so long as the GFT state remains within the domain of validity of these approximations.', '1602.05881-1-188-2': 'Let us point them out, for completeness.', '1602.05881-1-189-0': 'First, aside from the choice of the microscopic GFT model itself, we used a small derivative approximation with respect to the scalar field, which is then assumed to be slowly varying throughout the evolution of the universe.', '1602.05881-1-190-0': 'Second, even assuming that the relevant quantum states to study cosmological dynamics within the full GFT theory are condensate states, we used a very simple type of such states, i.e., coherent states of the GFT field operator.', '1602.05881-1-190-1': 'This is a drastic approximation; generic condensate states are much richer.', '1602.05881-1-190-2': 'In particular, we have neglected to include any information about the connectivity of the fundamental spin network degrees of freedom.', '1602.05881-1-190-3': 'While the connectivity information is not required in order to characterize homogeneous and isotropic space-times, this nevertheless represents an important restriction.', '1602.05881-1-190-4': 'For example, it implies that the only information about the topology of the universe is the one that can be inferred from the effective cosmological dynamics itself and that the topology of the space-time cannot be constrained at level of the quantum states.', '1602.05881-1-190-5': 'Still, as is well known from the study of real Bose condensates, even these drastic simplifications in the choice of the condensate states that are considered can provide a wealth of information about the relevant physics.', '1602.05881-1-191-0': 'The restriction to isotropic degrees of freedom is of course an additional approximation.', '1602.05881-1-191-1': 'While it does restrict the GFT condensate states to only depend on the degrees of freedom of interest, it has the disadvantage of being imposed as a starting point and for this reason it may prevent the understanding of some aspects of cosmological dynamics and in particular it makes it impossible to study how an isotropic configuration may be generated from a more general initial state.', '1602.05881-1-192-0': 'Given these approximations in the definition of the GFT model and the choice of the condensate states, we then extracted the effective cosmological dynamics directly from the full quantum dynamics of the microscopic GFT theory.', '1602.05881-1-192-1': 'However, the equation we used is the first Schwinger-Dyson equations, corresponding to the classical GFT equations of motion, and is the simplest formulation of the hydrodynamics of the theory.', '1602.05881-1-192-2': 'Ignoring the other Schwinger-Dyson equations is a truncation of the full quantum dynamics.', '1602.05881-1-193-0': 'Having obtained the equations of motion for the condensate wave function, we further assumed that the GFT interaction terms (corresponding to the non-linear terms in the condensate hydrodynamics) in this equation are subdominant compared to the GFT kinetic terms.', '1602.05881-1-193-1': 'This is the case for small values of the coupling constants (which are subject to the renormalization group flow) and for state densities that are not too large.', '1602.05881-1-193-2': 'This is the direct GFT counterpart of the dilute gas approximation used for standard Bose liquids in the Gross-Pitaevskii approximation.', '1602.05881-1-193-3': 'In fact, this approximation is required for consistency with the previous approximation that connectivity information is negligible, since stronger interactions would imply stronger correlations between the GFT quanta, in which case it would become necessary to take into account the connectivity between the spin network nodes and to work with more complicated condensate states.', '1602.05881-1-194-0': 'A last type of approximation was not explicitly stated since it does not enter directly any of the calculations and it is not in fact required by the formalism itself.', '1602.05881-1-194-1': 'This is the condition that curvatures associated to individual GFT quanta are not too big and that their individual contribution to the total volume is rather small.', '1602.05881-1-194-2': 'This is required for a straightforward interpretation of the corresponding quantum states as associated to a continuum geometry.', '1602.05881-1-194-3': 'At this point, it is not yet clear how strictly this condition should be enforced and at what point it fails.', '1602.05881-1-194-4': 'A more careful inspection of explicit solutions may provide some insight in this direction.', '1602.05881-1-195-0': 'Within the regime of the full theory in which the above approximations hold, our results hold as well and, we believe, provide interesting insights into the cosmological sector of LQG and GFT.', '1602.05881-1-196-0': 'This being said, there remain many open questions to be addressed in order to further corroborate these results and also to develop the path they open.', '1602.05881-1-196-1': 'We shall conclude by outlining some of the open issues and directions for future research.', '1602.05881-1-197-0': 'It is important to repeat the analysis done here for the GFT based on the EPRL spin foam model for another GFT based on a different spin foam model for 4D quantum gravity.', '1602.05881-1-197-1': 'We do not anticipate very different results, though, due to the fact that the main difference between the different GFT models will be encoded in the GFT interaction terms and we have assumed their contributions to the equations of motion of the condensate wave function to be subdominant.', '1602.05881-1-197-2': 'Therefore, a different interaction term would only give different types of quantum corrections to the general cosmological dynamics we have found.', '1602.05881-1-197-3': 'Moreover, we expressed the kinetic terms of the GFT action in a rather general form in order to minimize the reliance on the details of the microscopic model.', '1602.05881-1-197-4': 'However, any difference would be worth investigating.', '1602.05881-1-197-5': 'For example, it is not obvious that models enforcing a weaker relation between [MATH] and [MATH] data than the EPRL model would give an effective equation as simple as the one we obtained in the isotropic restriction, and this may offer interesting insights on the role of Lorentzian structures at the effective cosmological level.', '1602.05881-1-198-0': 'Also concerning the choice of the fundamental GFT model, a more detailed analysis of the coupling of the scalar field to (quantum) geometry is needed involving a careful analysis of the Feynman amplitudes of the coupled GFT model, and in particular it is necessary to go beyond the approximation of small derivatives.', '1602.05881-1-198-1': 'One would like the GFT model coupled to a free massless scalar field to define a proper simplicial path integral for discrete gravity coupled to a scalar field discretized on the same cellular complex corresponding to the GFT Feynman diagram.', '1602.05881-1-198-2': 'This analysis should also suggest how to go beyond the case of a free massless scalar field.', '1602.05881-1-198-3': 'While a scalar field with a non-trivial potential would not be a good global clock, it would be interesting to understand how generic scalar fields should be treated in GFT models.', '1602.05881-1-198-4': 'Moreover, it would be interesting to see how the new terms encoding its potential at the level of the GFT action modify its role as a time variable for the same field theory.', '1602.05881-1-198-5': 'Going even further, it would also be good to include other types of matter fields like Maxwell and Dirac fields.', '1602.05881-1-199-0': 'Another important next step is to perform a more detailed analysis of the solutions to the cosmological equations we have obtained.', '1602.05881-1-199-1': 'It is only by having at hand explicit solutions of the dynamics that we will be able to study the detailed evolution of geometric observables (and specifically the spatial volume of the space-time as well as the relational Hubble rate) and obtain a detailed picture of the evolution of the universe as dictated by the fundamental GFT.', '1602.05881-1-199-2': 'In addition, this analysis will also allow us to check the exact regimes of validity of the various approximations we employed.', '1602.05881-1-200-0': 'In particular, a careful analysis of this type will provide a better understanding of the quantum corrections to the classical Friedmann equations as predicted by the GFT, and in particular of those coming from the GFT interaction terms (which could perhaps be expressed in terms of geometric observables).', '1602.05881-1-200-1': 'Clearly, the interaction terms will generate quantum gravity corrections to the classical equations of general relativity which will be interesting to study, and in addition, as already explained, the interaction terms encode the differences between different microscopic definitions of the quantum gravity dynamics (i.e., different spin foam models or GFT actions) can be elucidated.', '1602.05881-1-200-2': 'So this is a further reason to study the effect of the interaction term in the GFT action on the emergent cosmological dynamics.', '1602.05881-1-200-3': 'It would also be interesting to derive an effective field theory that captures the same corrections to classical general relativity for homogeneous and isotropic space-times, since this would be a nice way to characterize the type of effective continuum dynamics of geometry that is predicted by the GFT models.', '1602.05881-1-201-0': 'The isotropic restriction on the condensate wave functions could also be lifted.', '1602.05881-1-201-1': 'This would allow for a larger class of operators on the condensate wave function, and could potentially be used in order to study anisotropic (and homogeneous) space-times.', '1602.05881-1-201-2': 'In that case, removing the isotropic restriction could perhaps also give some insight into the transition between isotropic and anisotropic regimes in full quantum gravity.', '1602.05881-1-202-0': 'Finally, there remain two related important open problems concerning extensions of the study of cosmology using GFT condensate states.', '1602.05881-1-202-1': 'The first is to tackle the regime of the dynamics where the GFT interactions cease to be subdominant and have to be included explicitly in the effective cosmological dynamics.', '1602.05881-1-202-2': 'This is not so much a technical issue (since it would be rather straightforward to include the interaction terms in the equations derived in this paper and solve them in the case when the interactions are large), but a conceptual one since the Gross-Pitaevskii approximation fails to be justified by the microscopic theory when interactions become important.', '1602.05881-1-202-3': 'In fact, in the regime where the interaction term is no longer negligible it will likely be necessary to choose a different class of condensate states and, in particular, it may be necessary to use condensate states which take fully into account the connectivity of the underlying fundamental spin network states.', '1602.05881-1-202-4': 'This can be done, but the degree of complexity of the analysis necessary in order to extract the effective cosmological dynamics will be considerably higher than what was needed here (among other complications, a number of graph theoretic issues will need to be addressed).', '1602.05881-1-203-0': 'At a more physical level, the second longer-term direction of research will be to develop a generalization of GFT condensate cosmology in order to include perturbations over the condensate state representing an homogeneous universe, as this would allow for a treatment of cosmological perturbations.', '1602.05881-1-203-1': 'A first step would be to reproduce in the context of the full GFT formalism the basic set-up of the analysis of cosmological perturbations performed in loop quantum cosmology; in particular, the separate universe picture [CITATION] appears rather straightforward to implement.', '1602.05881-1-203-2': 'The real goal, however, is more ambitious: to develop cosmological perturbation theory from first principles in a fundamental theory of quantum gravity, in this case GFTs.', '1602.05881-1-204-0': '# Relational Dynamics in Cosmology', '1602.05881-1-205-0': 'This appendix contains a brief review of the Hamiltonian formalism applied to a spatially flat FLRW space-time, and the resulting Friedmann equations in general relativity and loop quantum cosmology.', '1602.05881-1-205-1': 'As in the main text, we consider the case where the matter content is a minimally coupled massless scalar field [MATH].', '1602.05881-1-206-0': 'In order to avoid divergent integrals in the Hamiltonian formulation, we assume that the topology of the spatial surfaces is [MATH] and that the volume of the 3-torus with respect to the coordinates [MATH] is 1.', '1602.05881-1-207-0': '## General Relativity', '1602.05881-1-208-0': 'The line element for a flat FLRW space-time is [EQUATION] with [MATH] being the lapse and [MATH] the scale factor, and the Hamiltonian constraint -coming from the Einstein-Hilbert action, given the line element [REF]- is given by [EQUATION] where [MATH] is the volume, [MATH] is the variable canonically conjugate to [MATH] with the Poisson bracket [MATH], and [MATH] is the momentum of the scalar field and satisfies [MATH].', '1602.05881-1-209-0': 'The equations of motion are easily derived from the Hamiltonian; the two that are necessary for our purposes are those for [MATH] and [MATH].', '1602.05881-1-209-1': 'For the massless scalar field, [EQUATION] which is clearly monotonic since [MATH] is a constant of motion as [MATH].', '1602.05881-1-209-2': 'Thus, the scalar field can act as a good relational clock.', '1602.05881-1-209-3': 'The dynamics of [MATH] is given by [EQUATION] showing that [MATH] is the Hubble rate in proper time.', '1602.05881-1-209-4': 'Then, using the vanishing of the Hamiltonian constraint [REF] and rewriting the time derivative in terms of the massless scalar field via [REF] gives the first Friedmann equation in terms of the relational clock [MATH], [EQUATION]', '1602.05881-1-209-5': 'It can be checked that the continuity equation, once the energy density and pressure have been expressed as [MATH], is simply [MATH] which is trivially satisfied since [MATH] is a constant of the motion and therefore the remaining relevant equation of motion is the second Friedmann equation obtained by differentiating [REF] with respect to [MATH], giving [EQUATION]', '1602.05881-1-209-6': 'The equations [REF] and [REF] are the two Friedmann equations for a spatially flat FLRW space-time expressed in terms of the relational clock [MATH], which can be directly compared to the Gross-Pitaevski equation of the condensate states studied in this paper.', '1602.05881-1-210-0': '## Loop Quantum Cosmology', '1602.05881-1-211-0': 'Using the non-perturbative quantization techniques of loop quantum gravity, in loop quantum cosmology (LQC) it has been possible to obtain a well-defined quantum theory for FLRW space-times where the big-bang and big-crunch singularities are resolved [CITATION].', '1602.05881-1-211-1': 'An important result in LQC (and in quantum cosmology in general) is that states that are semi-classical (i.e., sharply peaked around a classical solution) in the low curvature regime remain sharply-peaked throughout their evolution, for the reason that observables in the quantum cosmology are global quantities and hence are heavy degrees of freedom [CITATION].', '1602.05881-1-212-0': 'For these semi-classical states, to determine the main features of the quantum dynamics it is sufficient to calculate the evolution of the expectation values of only a few observables, for example the total volume [MATH] and the momentum of the scalar field [MATH].', '1602.05881-1-212-1': 'Interestingly, the dynamics of these observables, for sharply peaked states, are given by a simple modification to the Friedmann equations.', '1602.05881-1-212-2': 'These are called the LQC effective Friedmann equations, and in terms of proper time [MATH] (i.e., for [MATH]) they are [CITATION] [EQUATION] and [EQUATION] where [MATH] is the Hubble rate, a dot denotes a derivative with respect to proper time, and [MATH] is the critical energy density of LQC.', '1602.05881-1-212-3': 'From these equations it is clear that a bounce in the scale factor occurs when the energy density of the matter field equals the critical energy density [MATH], and hence in LQC a contracting universe bounces into an expanding universe, with a quantum gravity bridge linking the two classical branches of the space-time.', '1602.05881-1-213-0': 'Also, from these equations it can easily be checked that the continuity equation is unchanged, [EQUATION]', '1602.05881-1-213-1': 'Recall that for a massless scalar field, [MATH] and, as in the case of general relativity, the continuity equation reduces to [MATH].', '1602.05881-1-214-0': 'In order to make contact with the equations of motion for the GFT condensate, it is necessary to express the LQC effective Friedmann equations in terms of the relational clock [MATH].', '1602.05881-1-214-1': '(Note that the continuity equation [MATH] shows that [MATH] is a good global clock here.)', '1602.05881-1-214-2': 'This can be done via the relation [CITATION] [EQUATION] which is unchanged from the classical relation.', '1602.05881-1-215-0': 'Therefore, the LQC effective Friedmann equations in terms of the relational clock [MATH] are [EQUATION] and [EQUATION]', '1602.05881-1-215-1': 'Interestingly, when expressed in terms of the relational clock [MATH], only the first LQC effective Friedmann equation is modified from its classical form.', '1602.05881-1-215-2': 'This is simply due to what is effectively a choice of the lapse [MATH] when [MATH] is used as a relational clock.', '1602.05881-1-216-0': 'We would like to thank the participants of the "Cosmology from Quantum Gravity" workshop, and in particular Mairi Sakellariadou, Martin Bojowald, Jakub Mielczarek, Steffen Gielen and Andreas Pithis, for helpful discussions.', '1602.05881-1-216-1': 'This work was partially supported by the John Templeton Foundation through the grant PS-GRAV/1401.'}

{'1602.05881-2-0-0': 'We study the effective cosmological dynamics, emerging as the hydrodynamics of simple condensate states, of a group field theory model for quantum gravity coupled to a massless scalar field and reduced to its isotropic sector.', '1602.05881-2-0-1': 'The quantum equations of motion for these group field theory condensate states are given in relational terms with respect to the scalar field, from which effective dynamics for spatially flat, homogeneous and isotropic space-times can be extracted.', '1602.05881-2-0-2': 'The result is a generalization of the Friedmann equations, including quantum gravity modifications, in a specific regime of the theory corresponding to a Gross-Pitaevskii approximation where interactions are subdominant.', '1602.05881-2-0-3': 'The classical Friedmann equations of general relativity are recovered in a suitable semi-classical limit for some range of parameters of the microscopic dynamics.', '1602.05881-2-0-4': 'An important result is that the quantum geometries associated with these GFT condensate states are non-singular: a bounce generically occurs in the Planck regime.', '1602.05881-2-0-5': 'For some choices of condensate states, these modified Friedmann equations are very similar to those of loop quantum cosmology.', '1602.05881-2-1-0': '# Introduction', '1602.05881-2-2-0': 'One of the main challenges for any fundamental theory of quantum gravity that proposes a candidate for the microscopic degrees of freedom for a quantum space-time is to extract its effective macroscopic physics, which should agree with those of general relativity in some suitable approximation.', '1602.05881-2-2-1': "This is often referred to as the 'problem of the continuum' in quantum gravity, since in many approaches the fundamental microscopic degrees of freedom are of a discrete nature (and of no direct spatio-temporal interpretation).", '1602.05881-2-2-2': 'Furthermore, the extraction of effective continuum physics at macroscopic scales is also needed in order to make contact with observations by identifying testable signatures of the proposed microscopic structure of space-time that could be tested at larger scales.', '1602.05881-2-2-3': "The 'problem of the continuum' in quantum gravity is hard, even after one has a promising candidate definition for the microscopic degrees of freedom and their dynamics (which is of course the first hard challenge), because the number of these degrees of freedom in any macroscopic space-time is expected to be extremely large.", '1602.05881-2-2-4': 'This implies that it is necessary to determine the (quantum) dynamics of suitably identified many-body quantum states in the fundamental theory, another difficult task.', '1602.05881-2-3-0': 'One key topic of macroscopic gravitational physics, that on the one hand would benefit from a solid foundation in quantum gravity, and on the other hand is a natural setting where the predictions of quantum gravity effects could realistically be confronted to observations, is cosmology.', '1602.05881-2-3-1': 'Therefore the extraction of an effective cosmological dynamics from fundamental quantum gravity models is a particularly interesting and important problem.', '1602.05881-2-4-0': 'In a cosmological context, for homogeneous space-times a few global observables are enough to characterize the macroscopic configurations and dynamics of the space-time and geometry, despite the large number of fundamental degrees of freedom involved.', '1602.05881-2-4-1': 'Therefore some kind of truncation, in both the description of the relevant quantum states and of their dynamics, is necessary.', '1602.05881-2-4-2': 'Two main types of truncations are typically considered: coarse-graining (roughly, to use ensembles of microscopic degrees of freedom collectively parametrized by a few variables capturing the relevant macroscopic observables) and symmetry reduction (roughly, to reduce the degrees of freedom of the theory in order to include only the kinematical variables corresponding to these macroscopic observables).', '1602.05881-2-4-3': 'Of course, both strategies involve some approximations to the full theory.', '1602.05881-2-5-0': 'In the coarse-graining approach, the effective dynamics of the collective (cosmologically relevant) observables implicitly capture some effects of the other (coarse-grained away) degrees of freedom; how exactly and how well it manages to do so both depend on the precise coarse-graining scheme as well as on the other approximations that are required to make it work.', '1602.05881-2-5-1': 'From this perspective, cosmology (being the theory of the most global, large-scale degrees of freedom) should arise as a sort of quantum gravity hydrodynamics [CITATION].', '1602.05881-2-5-2': 'In the symmetry reduction approach the relevant cosmological symmetries (e.g., homogeneity and isotropy) are imposed exactly, in order to obtain a system with a minimal number of degrees of freedom, while the others are simply projected out of the resulting theory.', '1602.05881-2-5-3': 'If this is done at the microscopic level, it clearly leads to entirely neglecting any effect that the removed degrees of freedom have on the physics of the remaining (cosmological) degrees of freedom.', '1602.05881-2-5-4': 'It may thus be seen, in a way, as a particularly brutal form of coarse-graining.', '1602.05881-2-5-5': "Any form of coarse-graining would then be 'better' than such a drastic reduction, as a matter of principle, and especially so in a quantum context.", '1602.05881-2-6-0': 'However, coarse-graining (especially in the full quantum theory) is cumbersome and technically challenging, and in some cases it is certainly true that a symmetry reduction procedure is enough to capture all of the relevant physics in a (sometimes significantly) more straightforward manner.', '1602.05881-2-6-1': 'Whether symmetry reduction does in fact capture the relevant physics can be judged only on a case by case basis; for example, coarse-graining is usually preferred in condensed matter physics, but the easier symmetry reduction works for simple systems like the hydrogen atom.', '1602.05881-2-7-0': 'In the classical gravitational context, symmetry reduction is the main framework used to study the large-scale dynamics of space-time, but even there it is a hotly debated issue whether symmetry reduction gives the same results as coarse-graining gravitational inhomogeneities [CITATION].', '1602.05881-2-7-1': 'In a quantum gravitational context, of course, the issue is even murkier and under less control.', '1602.05881-2-7-2': 'Further, even under the assumption that symmetry reduction is enough, it is not clear whether the symmetry reduction should be done at the classical level and one should then quantize the resulting reduced system, or directly at the quantum level.', '1602.05881-2-7-3': 'The two procedures cannot be expected to commute, in general, and much depends on the exact details of the microscopic quantum theory and on its interpretation in relation to continuum general relativity (in particular, whether the full quantum theory is understood as a quantization of general relativity or a quantum theory whose effective description only is given by general relativity).', '1602.05881-2-8-0': 'The above considerations apply to any quantum gravity formalism in which the microscopic degrees of freedom cannot be directly interpreted as quantized continuum geometries, and therefore should be related to continuum geometries through a more careful analysis.', '1602.05881-2-9-0': 'In particular, these considerations are relevant for loop quantum gravity (LQG) [CITATION] which, despite its historical roots as a canonical quantization of general relativity in connection variables, ends up with fundamental quantum degrees of freedom which are purely combinatorial and algebraic (e.g., s-knots).', '1602.05881-2-9-1': 'This feature is more apparent in its covariant, or spin foam, formulation, where the same type of combinatorial and algebraic degrees of freedom -directly obtained from a discretization of the continuum theory- are used in order to define a prescription for the quantum gravitational path integral.', '1602.05881-2-10-0': 'This is also the case for group field theories (GFTs) [CITATION], which can be seen on the one hand as a second quantized formulation of (the kinematics and dynamics of) loop quantum gravity degrees of freedom [CITATION], although organized in a different Fock-type Hilbert space, and on the other hand as an enrichment of lattice gravity approaches like tensor models [CITATION] (in turn a higher-dimensional generalization of matrix models for 2D gravity, and closely related to the dynamical triangulations formalism for quantum gravity [CITATION]) by group-theoretic data, i.e., the type of data characterizing quantum geometry in loop quantum gravity.', '1602.05881-2-10-1': 'The GFT framework provides the context for the results presented in this paper.', '1602.05881-2-11-0': "Due to the nature of the fundamental degrees of freedom in both LQG and GFT, it is reasonable to expect cosmological states to correspond to highly excited states with respect to the natural background independent vacuum state of the theory, which is a 'no space' state (e.g., the Ashtekar-Lewandowski vacuum state [CITATION] in LQG, and the simple Fock vacuum in GFT).", '1602.05881-2-11-1': 'Thus, to study cosmological space-times in LQG, it will be necessary to only consider a small number of the degrees of freedom of the relevant highly-excited states, and it is clear that some form of coarse-graining is required (perhaps of the simple symmetry reduction type).', '1602.05881-2-12-0': 'In fact, the problem of the continuum (and the related problem of extracting cosmological dynamics from the full theory) is attracting increasing attention both in GFT and in LQG.', '1602.05881-2-12-1': 'At the more formal level, it is tied to the study of renormalization flows of quantum gravity models, which is now an active area of research for spin foam models both in their lattice gravity interpretation [CITATION] and in their GFT formulation [CITATION], and to the issue of identifying a suitable phase of the theory in which a continuum geometric rewriting of its dynamics is possible (by defining a non-degenerate geometric vacuum), which has seen considerable progress both in canonical LQG [CITATION] and in GFT, where it has been suggested that condensate states are natural candidates for non-degenerate geometric vacua.', '1602.05881-2-13-0': 'In the GFT formalism, the hypothesis that the process responsible for the emergence of a continuum geometric space-time is a condensation of the GFT microscopic quanta of space-time (i.e., spin network nodes or equivalently abstract labelled polyhedra) has suggested a promising strategy for extracting an effective cosmological dynamics from the full theory.', '1602.05881-2-13-1': 'Quantum GFT condensate states have been studied in some detail, including both simple condensates corresponding to a gas of uncorrelated GFT quanta (i.e., spin network nodes with no information on their mutual connectivity) and more complex condensates retaining topological information and defined by (infinite) sums over spin network graphs [CITATION] (see also [CITATION] for a detailed overview of the methods and the results).', '1602.05881-2-13-2': "In all cases, the key feature is that the same 'condensate wave function' is assigned to all of the GFT quanta defining the quantum state and hence acts as a collective variable.", '1602.05881-2-13-3': 'This is understood as the coarse-grained quantum gravity analogue of the classical spatial homogeneity condition that characterizes the simplest cosmological space-times.', '1602.05881-2-13-4': 'The effective condensate dynamics can then be extracted directly from the full theory and are given by a non-linear equation for the condensate wave function.', '1602.05881-2-13-5': 'Since the domain of definition of this condensate wave function is isomorphic to the minisuperspace of homogeneous (and potentially anisotropic) geometries, the GFT condensate hydrodynamics has the form of a non-linear extension of loop quantum cosmology and can be studied with similar methods.', '1602.05881-2-13-6': 'Several other results have also been obtained in the study of GFT condensate states, some of which will be reviewed briefly in the following since in this paper we will build on these earlier results.', '1602.05881-2-13-7': 'See [CITATION] for further details.', '1602.05881-2-14-0': 'Furthermore, the importance of studying many-particle states in quantum gravity for the cosmological sector is also motivated directly from the purely canonical LQG point of view, following from some important insights provided by loop quantum cosmology (LQC) [CITATION].', '1602.05881-2-14-1': 'In order to clearly describe the input that LQC offers concerning the quanta of geometry in a cosmological space-time, it is helpful to briefly review LQC and explain what is known about the relation between LQC and LQG.', '1602.05881-2-15-0': 'In LQC, symmetry-reduced space-times are quantized in a non-perturbative fashion following the procedures developed in loop quantum gravity.', '1602.05881-2-15-1': 'A lot of progress has been achieved in the field of LQC over the past few years, with the main results being that the big-bang and big-crunch singularities of general relativity are resolved due to quantum gravity effects [CITATION] and are replaced by a bounce [CITATION].', '1602.05881-2-15-2': 'To be precise, in LQC the dynamics of a contracting space-time are given by general relativity until the space-time curvature nears the Planck scale, at which point quantum gravity effects become important and cause a bounce which can be seen as a quantum gravity bridge between the contracting and expanding branches of the space-time that can each be treated classically.', '1602.05881-2-16-0': 'Although LQC is based on the same quantization techniques as LQG, LQC is not derived from LQG and the precise relation between LQC and LQG is currently unknown.', '1602.05881-2-16-1': 'This is an important issue to address which is given additional urgency since some predictions of LQC effects have been calculated in a number of cosmological scenarios including inflation [CITATION] and the matter bounce scenario [CITATION] that are of phenomenological interest.', '1602.05881-2-16-2': 'It is therefore important to see whether the results of LQC can be provided further support by a proper (if approximate) derivation of LQC from full LQG.', '1602.05881-2-16-3': 'One of the goals of this paper is to shed some light on this problem by studying the dynamics of cosmological states in the GFT reformulation of LQG.', '1602.05881-2-17-0': 'While the issue of the relation of LQC with the full theory has not yet been fully resolved, there has been some important work in this direction including the result that the basis states of LQC can be embedded in the kinematical Hilbert space of LQG [CITATION] (even though the kinematical Hilbert space of LQC is not the symmetry-reduced kinematical Hilbert space of LQG [CITATION]).', '1602.05881-2-17-1': 'However, the main results regarding the relation between LQC and LQG concern their kinematical Hilbert spaces and their basis states, and much less is known about the dynamical sector.', '1602.05881-2-18-0': 'Still, a heuristic argument concerning the relation between LQG and LQC provides key insights which are in close agreement with the discussion above concerning the importance of using highly excited states (with respect to the no space vacuum) in order to study the cosmological sector.', '1602.05881-2-18-1': 'This heuristic relation is based on an observation concerning the kinematical Hilbert space of LQG and the construction of the field strength operator in LQC.', '1602.05881-2-19-0': 'First, quantum states in the kinematical LQG Hilbert space (and in GFT) can be viewed [CITATION] as a collection of polyhedra that are glued together across faces with equal areas (although the faces glued together need not have the same shape).', '1602.05881-2-19-1': 'In a typical piece-wise flat interpretation of the resulting discrete structures, space-time curvature is understood to lie on the faces of the polyhedra and the amplitude of the curvature lying on any given face can be evaluated via a holonomy of the Ashtekar-Barbero connection along the edges that form the boundary of that face.', '1602.05881-2-20-0': 'Second, one of the key steps in LQC is the definition of the operator corresponding to the space-time curvature.', '1602.05881-2-20-1': 'This is constructed by evaluating the holonomy of the Ashtekar-Barbero connection around a loop of minimal area, as it would be in full LQG.', '1602.05881-2-20-2': 'The choice of the minimal area (rather than some other non-zero', '1602.05881-2-21-0': 'area) is justified by assuming that in the LQG wave function corresponding to the LQC states, there is a large number of polyhedra and that each of their surface areas is given by the minimal area eigenvalue of LQG, [MATH] (where [MATH] is a dimensionless number of order 1) [CITATION].', '1602.05881-2-22-0': 'The combination of these two ingredients leads to the following heuristic relation between LQC and LQG.', '1602.05881-2-22-1': 'In the simplest, homogeneous and isotropic case, LQC states correspond to wave functions in LQG with a large number of quanta of geometry, or polyhedra, whose surface areas are all given by the minimal non-zero area possible in LQG [CITATION].', '1602.05881-2-22-2': 'This key insight from LQC strongly motivates the use of the family of GFT states we will use in this paper, with the polyhedra specialized for simplicity to tetrahedra.', '1602.05881-2-23-0': 'In other words, the LQG wave function for cosmological space-times, as suggested by LQC, would be a state composed of a large number of identical tetrahedra whose four areas are all equal to [MATH].', '1602.05881-2-23-1': 'An important point is that these tetrahedra are not only equiareal but also equilateral, a stronger condition.', '1602.05881-2-23-2': 'LQC neglects the connectivity information of the tetrahedra, and without the connectivity information, the LQG wave function of a large number of identical tetrahedra is exactly a simple condensate state.', '1602.05881-2-23-3': 'This is the family of GFT states that we shall consider here.', '1602.05881-2-24-0': 'Now, ignoring the connectivity information of the tetrahedra forming the space-time, for the sake of simplicity but also because this information is not immediately relevant for purely homogeneous and isotropic configurations (and furthermore is ignored in the heuristic relation between LQC and LQG), this type of wave function has only one degree of freedom given by the number of tetrahedra [MATH].', '1602.05881-2-24-1': 'This is very restrictive and in order to allow more general states we will remove the requirement that the areas of the faces of each tetrahedron be [MATH].', '1602.05881-2-24-2': 'Instead, we will assume that the LQG wave functions (in the GFT reformulation) that correspond to cosmological space-times are condensate states composed of a large number of identical equilateral tetrahedra, with no a priori restriction on their face areas.', '1602.05881-2-24-3': 'On the basis of these insights, and since condensate states with varying number of quanta are most easily handled in a field theory formalism where there exist creation and annihilation operators, the discussion above suggests that group field theory (GFT) -which provides a field theory reformulation of LQG, as reviewed in Sec. [REF]- is the most natural framework to study the cosmological sector of LQG, and that the coarse-grained cosmological dynamics come from the hydrodynamics of GFT condensate states.', '1602.05881-2-25-0': 'Before presenting our results, we point out that there do exist a number of other approaches to the problem of extracting the cosmological sector of LQG, including so-called spin foam cosmology [CITATION] as well as work on symmetry-reduced canonical loop quantum gravity, either with a large number of spin network nodes [CITATION] (see also [CITATION] for related earlier work), or just one node [CITATION].', '1602.05881-2-25-1': 'The approach considered in [CITATION], in particular, despite the clear difference in strategy and of formalism, considers quantum states quite similar to our GFT condensates.', '1602.05881-2-25-2': 'However, all of these alternative approaches assume that the relevant physics is correctly captured by a fixed number of quanta of geometry (which is moreover very small in some cases), an assumption which is seemingly at odds with the lattice-refinement interpretation of LQC which seems to indicate that the number of quanta of geometry increases as the universe expands (or decreases if the space-time is contracting) [CITATION], as the heuristic relation between LQC and LQG suggests.', '1602.05881-2-25-3': 'See also [CITATION] for further arguments concerning the potential importance for allowing for the number of quanta to change in the cosmological sector of LQG.', '1602.05881-2-25-4': 'In the condensate states studied here there is no restriction on the number of quanta, indeed the number of quanta of geometry will be determined dynamically as it necessarily enters the effective cosmological equations [CITATION].', '1602.05881-2-25-5': 'This is one of the important advantages of using GFT condensate states to study the cosmological sector of LQG, in addition to directly addressing the possibility of cosmological space-times being constituted of a large number of quanta of geometry.', '1602.05881-2-26-0': 'In this paper, we will extend previous work on the cosmology of GFT condensates in a number of directions: (i) we work with a GFT that corresponds to (a slight generalization of) the Engle-Pereira-Rovelli-Livine (EPRL) spin foam model of LQG [CITATION], which is the most developed Lorentzian model for 4D quantum gravity and is thus a bona fide candidate for the fundamental definition of the quantum dynamics rather than a simple toy model; (ii) we generalize the GFT formalism in order to include a massless scalar field, (iii) in an isotropic restriction, and working with Gross-Pitaevskii condensate states, we extract an effective (relational) cosmological dynamics directly from the fundamental dynamics of the model, showing that it reproduces the Friedmann equations in a classical limit, at least as long as interactions are subdominant; (iv) in the same regime of approximations we derive the leading order quantum gravity corrections to the Friedmann dynamics of the GFT condensate states, which generically predict that the classical big-bang singularity is resolved and is replaced by a non-singular bounce; (v) interestingly, the modified Friedmann equations are qualitatively similar to the effective dynamics of LQC for a simple type of GFT condensate state.', '1602.05881-2-26-1': 'It is worth stressing that, while in this paper we focus on those contributions to the effective dynamics that are expected to be dominant in our approximation (in particular, we consider the interactions as subdominant compared to the kinetic term in the quantum equations of motion as is required in the Gross-Pitaevskii approximation), all of the terms coming from the microscopic dynamics are known and subdominant contributions can be computed explicitly.', '1602.05881-2-27-0': 'More precisely, the outline of the paper is the following.', '1602.05881-2-27-1': 'We start by giving a brief introduction to GFTs in general and reviewing the results of the earlier papers on GFT condensate states [CITATION] in Sec. [REF], and explain how to include a scalar field in a GFT in Sec. [REF].', '1602.05881-2-27-2': 'Then in Sec. [REF] we define the condensate states we take as an ansatz to represent the homogeneous and isotropic sector of LQG, and explain how the scalar field can be used as a relational clock.', '1602.05881-2-27-3': 'In Sec. [REF] we derive and study the relational Friedmann equations of the GFT condensate states, showing in particular that the big-bang and big-crunch singularities are resolved and are replaced by a bounce, a result analogous to the main qualitative results of LQC.', '1602.05881-2-27-4': 'We end with some comments on the difficulties associated with taking the limiting case of vanishing matter energy density in Sec. [REF] and a more general discussion in Sec. [REF].', '1602.05881-2-28-0': '# Basic Elements of the Group Field Theory Formalism', '1602.05881-2-29-0': 'Group Field Theories (GFTs) [CITATION] are a special class of quantum (or statistical) field theories defined over group manifolds (hence the name).', '1602.05881-2-29-1': 'Motivated by quantum Regge calculus [CITATION], dynamical triangulations [CITATION], tensor models [CITATION], and loop quantum gravity (LQG) [CITATION], the action for GFTs is typically chosen specifically so that the perturbative expansion of the generating functional generates a sum over diagrams that are dual to simplicial complexes and whose vertices, edges and faces are decorated with group theoretic data.', '1602.05881-2-29-2': 'Then, the Feynman amplitude for each simplicial complex in this sum (once the group theoretic data on the vertices, edges and faces are summed over) corresponds precisely to the quantum gravity path integral evaluated on that particular simplicial complex, assuming appropriate choices are made for the GFT action [CITATION].', '1602.05881-2-29-3': 'The continuum limit is obtained through the sum over all allowed simplicial complexes.', '1602.05881-2-30-0': 'Significantly, GFTs are related to both the canonical and covariant forms of LQG.', '1602.05881-2-30-1': 'On the one hand, GFTs can be understood as a second quantization of canonical LQG [CITATION], with a different organization of the quantum states so to obtain a Fock space, while the GFT Feynman amplitudes can be equivalently written as spin foam models [CITATION], the covariant form of LQG (see [CITATION] for an introduction to spin foam models).', '1602.05881-2-31-0': 'Furthermore, as quantum field theories, GFTs admit an operatorial representation in terms of ladder operators acting on conventional Fock spaces.', '1602.05881-2-31-1': 'In this Fock representation, it is clear that GFTs are a second-quantized language for LQG since the GFT quanta are in fact spin network nodes', '1602.05881-2-32-0': 'What will be particularly useful here is that the presence of a Fock space structure renders available some of the powerful methods used in condensed matter physics to construct approximate solutions involving many-body degrees of freedom (which are typically very difficult to access using simple perturbative techniques or attempting to solve directly the many-body Schrodinger equation) and to study the corresponding physics [CITATION].', '1602.05881-2-32-1': 'These methods may become particularly important in quantum gravity if the continuum limit, in background independent theories of quantum gravity like LQG, does in fact correspond to states with many quanta of geometry as is often expected [CITATION].', '1602.05881-2-33-0': 'In this section, we will make the above discussion more precise by considering a specific GFT model.', '1602.05881-2-33-1': 'We will first define the field operators in the GFT and explain their geometric interpretation, then introduce the Fock space and operators on the Fock space, and finally present the quantum equations of motion.', '1602.05881-2-34-0': '## The Field Operators', '1602.05881-2-35-0': 'In what follows we will consider a simple bosonic GFT based on the group [MATH] (the most common choice for quantum gravity GFTs).', '1602.05881-2-35-1': 'The Fock space is built starting from the single particle Hilbert space [MATH] corresponding to four-valent spin network nodes, where the ladder operators are [EQUATION] and these ladder operators are required to be translation invariant under diagonal group multiplication from the right, i.e., [EQUATION]', '1602.05881-2-35-2': 'The identification of this Hilbert space with four-valent spin network nodes is clear: the four [MATH] are the parallel transport of an [MATH]-valued connection along each of the four links emanating from the same node of a spin network, and the translation invariance encodes the gauge-invariance at the node in spin networks.', '1602.05881-2-35-3': 'For this reason, although the right translation invariance is a global translation in [MATH] it is often referred to as gauge invariance', '1602.05881-2-36-0': 'Note that the choice of four copies of [MATH] in the definition of the field operators amounts to a restriction to four-valent spin network nodes.', '1602.05881-2-36-1': 'This is a minor restriction in the formalism for the sake of simplicity; the theory can be extended, in principle, to include spin network nodes of arbitrary valence [CITATION].', '1602.05881-2-37-0': 'Using the notational shorthand [MATH] to denote the [MATH] group elements on which the field operator is evaluated, the commutation relations for the ladder operators are [EQUATION] which include the correct right gauge invariance.', '1602.05881-2-37-1': 'This shows that the creation operator [MATH] can be interpreted to create a single four-valent spin network node with data given by [MATH] up to a gauge transformation on the right.', '1602.05881-2-38-0': 'Equivalently, the same quanta can be interpreted as tetrahedra dual to the four-valent spin network nodes, in which case the four triangular faces of the tetrahedra are dual to the open links leaving the spin network node and are labelled by the same group elements.', '1602.05881-2-38-1': 'To make this dual simplicial interpretation more transparent, it is convenient to rewrite the theory as a field theory on the Lie algebra [MATH] via a noncommutative Fourier transform, [EQUATION] where [MATH] are noncommutative plane waves and [MATH] denotes the Haar measure on [MATH] [CITATION].', '1602.05881-2-38-2': 'The Lie algebra elements should be understood as the normal vectors to each face of the tetrahedron (with their norm corresponding to the area of each face), that in turn are in one-to-one correspondence with the face bivectors.', '1602.05881-2-38-3': 'This rewriting (of both GFT and LQG) depends explicitly on the choice of the quantization map for the flux observables, which will dictate the choice of non-commutative plane waves and their star product [CITATION].', '1602.05881-2-39-0': 'Using the properties of the plane waves [MATH], it can be verified that the Lie algebra counterpart of the right gauge invariance is the closure constraint for the four faces of the tetrahedron [CITATION].', '1602.05881-2-39-1': 'For this reason, the domain of definition of the field operators [MATH] is the subspace of [MATH] obeying the closure relation [EQUATION]', '1602.05881-2-39-2': 'This clearly shows that the quanta created by the field operators can indeed be interpreted as quantum tetrahedra with face bivectors given by the [MATH].', '1602.05881-2-39-3': 'Furthermore, it is clear that (at the price of a more complicated theory) it is possible to define a more general GFT than this one which also includes quanta interpreted as higher-valent spin network nodes, or equivalently as convex polyhedra with a greater number of faces.', '1602.05881-2-40-0': '## The Fock Space', '1602.05881-2-41-0': "In GFTs, the Fock space is constructed in the standard fashion (assuming Bose statistics) from the 'one-particle' Hilbert space described in Sec. [REF] (for details see [CITATION]).", '1602.05881-2-41-1': 'Note that the Fock vacuum represents the state with no spin network nodes, and thus no topological nor geometrical information; the usual spin network states of LQG can be obtained by acting on this vacuum with creation operators convoluted with the desired spin network wave function [CITATION].', '1602.05881-2-42-0': 'In this second-quantized framework, all operators of interest are generated by polynomials of ladder operators convoluted with suitable kernels where these kernels are the matrix elements of the corresponding first-quantized operators (i.e., the operators of standard canonical LQG like area or volume).', '1602.05881-2-42-1': 'For example, the second-quantized volume operator is (expressed in terms of the Fourier-transformed field operators) [EQUATION] where [MATH] is the volume of a tetrahedron where the normal vectors to each of its faces are [MATH] and [MATH].', '1602.05881-2-42-2': 'This operator gives the total volume of all of the quanta of geometry.', '1602.05881-2-43-0': 'It is also possible to consider more general one-body operators of the type [EQUATION] where the field operators do not have the same arguments.', '1602.05881-2-43-1': 'Here [MATH] is the LQG matrix element of the operator [MATH], evaluated between two nodes decorated with the elements [MATH].', '1602.05881-2-44-0': 'There is also a very important new one-body observable that only appears in the second-quantized framework.', '1602.05881-2-44-1': 'This is the number operator [EQUATION] which counts the number of quanta present in a given state: its eigenvalues distinguish the N-body sectors of the GFT Fock space.', '1602.05881-2-44-2': 'This is also an example of a purely combinatorial observable that characterizes the underlying graph of spin network states, regardless of the attached group theoretic data, which is a type of observable not usually considered in the standard LQG framework.', '1602.05881-2-45-0': 'More general N-body operators that act on more than one spin network node (or tetrahedron), or resulting in more than one, can also be constructed and are of obvious interest.', '1602.05881-2-45-1': 'For example, curvature operators that calculate the parallel transport around a loop containing several spin network nodes will generically be of this type.', '1602.05881-2-45-2': 'Of course, the form of this type of operator is more complicated and, since in this paper we are interested in homogeneous degrees of freedom (to describe homogeneous cosmological space-times), we will restrict our attention to one-body operators like those corresponding to the volume; these will be enough to totally characterize the relevant homogeneous degrees of freedom.', '1602.05881-2-46-0': '## The Quantum Equations of Motion', '1602.05881-2-47-0': 'The previous two sections present the key concepts underlining the kinematics of GFTs.', '1602.05881-2-47-1': 'Now, the GFT will be fully specified by giving its dynamics, which are encoded in the GFT action [MATH].', '1602.05881-2-47-2': 'The action typically includes involving a kinetic term quadratic in the field operators and some interaction terms which are higher order in the field operators.', '1602.05881-2-48-0': 'Given the action, the path integral for the theory can be (formally) defined, and from the path integral formalism the Schwinger-Dyson equations for correlation functions can be derived, which represent the complete (although formal) specification of the quantum dynamics.', '1602.05881-2-49-0': 'The same quantum dynamics can also be given in operator form, starting from the operator corresponding (up to operator ordering ambiguities) to the classical equations of motion written in terms of field operators, the first equation of motion being [EQUATION] and the second being the corresponding equation obtained from the variation of the action with respect to [MATH].', '1602.05881-2-50-0': 'As already mentioned above, the GFT action can be defined to reproduce a spin foam model by choosing the action so that the perturbative expansion of the GFT partition function around the Fock vacuum matches the expansion of the spin foam model.', '1602.05881-2-50-1': 'To be precise, the kinetic term and the interaction term in the GFT action are respectively chosen to reproduce the edge and vertex amplitudes of the spin foam model.', '1602.05881-2-51-0': 'For the most common type of spin foam models which are based on simplicial interactions (i.e., where the spin foam vertices are associated to 4-simplices), the only interaction terms are five-valent and therefore the GFT action will have the general form of [EQUATION] where the complex-valued functions [MATH] and [MATH] depend on the [MATH].', '1602.05881-2-51-1': 'Note that here each [MATH] denotes 4 group elements [MATH] at each spin network node [MATH].', '1602.05881-2-51-2': 'The notation is the following: indices [MATH] label the node and run from [MATH] with [MATH] being the total number of nodes (so in the kinetic term [MATH] and in the interaction term [MATH]), while indices [MATH] label the links at a given node and take the values [MATH] since only four-valent spin-network nodes are considered in this family of GFT models.', '1602.05881-2-52-0': 'Also, for the sake of simplicity only two interaction terms are included in the above action, one of which contains five creation operators and no annihilation operators and the other the opposite.', '1602.05881-2-52-1': 'Clearly, there are four other interaction terms that could be included (with some relations between them necessary in order to ensure that the action be real).', '1602.05881-2-52-2': 'For now we will restrict our attention to these two terms as they shall be sufficient to show how to include any relevant interaction term in any further analysis.', '1602.05881-2-53-0': 'The perturbative expansion of this model gives Feynman diagrams dual to four-dimensional simplicial complexes, and the amplitudes are determined by convolutions of the interaction kernels [MATH] associated to the 4-simplices with the inverse kernels of the kinetic term (i.e., the propagator of the theory which connects neighbouring 4-simplices).', '1602.05881-2-54-0': 'Finally, for the action [REF] corresponding to a GFT based on simplicial interactions, the first quantum equation of motion is [EQUATION] with the second quantum equation of motion being the adjoint of this equation.', '1602.05881-2-55-0': 'The specific form of [MATH] and [MATH] will depend on the choices in the construction of the GFT, and in this case we will be interested in the GFT based on the EPRL spin foam model, which is the most studied Lorentzian spin foam model.', '1602.05881-2-56-0': 'One of the main principles used in determining the form of [MATH] and [MATH] is that the imposition of the simplicity constraints -which transform the 4D topological BF field theory into a geometric theory with local degrees of freedom [CITATION]- can be achieved by modifying the kernels of the 4D [MATH] BF Ooguri GFT model with conditions that restrict the four bivectors [MATH] labeling the triangular faces appearing in the simplicial complexes dual to the Feynman diagrams of the theory to be simple bivectors [CITATION].', '1602.05881-2-56-1': 'This is the GFT counterpart of the standard procedure followed in defining spin foam models [CITATION], and following this procedure gives the EPRL GFT model.', '1602.05881-2-57-0': 'However, here we will not go through the details of this lengthy procedure; the interested reader is instead referred to [CITATION].', '1602.05881-2-57-1': 'Instead, it will be sufficient for our purposes to highlight one of the key properties of the GFT based on the EPRL spin foam model.', '1602.05881-2-58-0': 'To do this, it is convenient to work in the spin representation, where the field operators [REF] are rewritten via the Peter-Weyl decomposition [EQUATION] where [MATH].', '1602.05881-2-58-1': 'Note while a generic function of [MATH] would have a more general form, the gauge-invariance of the field operators is translated in the spin representation to the presence of the intertwiners [MATH] labeled by [MATH].', '1602.05881-2-59-0': 'Then, using the shorthand notation [EQUATION] the general GFT action for the case of simplicial interactions in the spin representation has the form [EQUATION] where [MATH], and of course each [MATH] and [MATH] represent the four [MATH] and [MATH] labels colouring the four links leaving the [MATH] spin network node.', '1602.05881-2-60-0': 'The key property of the GFT based on the EPRL spin foam model is that (i) the kinetic term contains a Kronecker delta between the [MATH] and intertwiner labels, and (ii) the interaction term contains a Kronecker delta for the [MATH] labels colouring the links that meet in the interaction.', '1602.05881-2-60-1': 'Thus, in this case the kinetic term has the simple form [EQUATION] and the interaction term is similarly simplified, with the first interaction term having the form [EQUATION] while the second interaction term is simply [MATH].', '1602.05881-2-60-2': 'The input from the EPRL model here is in the combinatoric form of the [MATH] and [MATH] arguments in the field variables which is due to the presence of Kronecker delta functions in the interaction term which have been taken into account in the above expression (in particular, [MATH]).', '1602.05881-2-60-3': 'The presence of these Kronecker deltas will play an important role in what follows, but the remaining details of the functional form of [MATH] and [MATH] will not be necessary.', '1602.05881-2-60-4': 'For their detailed expression we refer to the literature [CITATION], and we only mention here that they encode a specific relation between the [MATH] representation labels that we use as variables here and [MATH] representations, as well as a condition of invariance of the same functions under [MATH].', '1602.05881-2-60-5': 'This specific relation between [MATH] and [MATH] data is the end result of the EPRL prescription for imposing the constraints reducing topological BF theory to gravity, and which are nothing more than the conditions enforcing geometricity of the simplicial structures on which the model is based.', '1602.05881-2-61-0': 'While it is not necessary to give the exact functional form of [MATH] and [MATH] here, it is nonetheless important to be aware of a number of ambiguities that arise in the definition of GFT (and spin foam) models.', '1602.05881-2-61-1': 'First, there are the standard factor-ordering ambiguities that generically arise in quantum mechanics (note also that in this setting even operators corresponding to momentum space variables, i.e., the discretized fluxes of the triad field, will not necessarily commute amongst themselves).', '1602.05881-2-61-2': 'Second, the specific choices that are made for [MATH] and [MATH] are motivated by a discretization of (in this case) the Plebanski-Holst action, and some ambiguities necessarily arise that depend on the specific discretization method.', '1602.05881-2-61-3': 'Third, generically, while the implementation of the simplicity constraints alone (which reduce the topological BF theory to gravity) fixes the type of variables that should appear in the theory so that there is a proper discrete geometric interpretation, and may further suggest the amplitudes associated to vertices, additional requirements (e.g., concerning the composition law of transition amplitudes [CITATION], or the renormalizability of the resulting model) are needed in order to determine the amplitudes associated to edges and faces.', '1602.05881-2-61-4': 'Finally, even once the vertex, edge and face amplitudes are determined, it is possible to redefine these in such a fashion so that the total amplitudes remain unchanged, since these last depend only on convolutions of vertex, edge and face amplitudes in specific combinations.', '1602.05881-2-62-0': 'The existence of construction and quantization ambiguities is of course not surprising as they affect the definition of any quantum theory.', '1602.05881-2-62-1': 'It is hoped that -and there are general reasons to expect this [CITATION]- (assuming the continuum and semi-classical limits to be properly defined and shown to exist) most of the large-scale properties of the effective description will be insensitive to many of the details of the construction of the microscopic model, and in particular that they will be insensitive to many of the ambiguities listed above.', '1602.05881-2-62-2': 'Of course, the extent of this independence from microphysics will depend on the specific class of phenomena of interest.', '1602.05881-2-63-0': 'We will see that, for the family of GFTs and for the particular class of states that will be considered here, on the one hand there will be a certain degree of universality, but on the other hand a number of these subtle choices in the definition of the GFT model will directly affect the effective cosmological dynamics and thus give different predictions.', '1602.05881-2-64-0': '# Coupling to a Scalar Field', '1602.05881-2-65-0': 'So far we have presented the simplest incarnation of GFT models, those which encode only (quantum) geometric degrees of freedom.', '1602.05881-2-65-1': 'However, in order to study most physical applications, and to extract concrete predictions from GFTs, it is necessary to introduce matter fields.', '1602.05881-2-65-2': 'This is especially true in the cosmological context where vacuum homogeneous and isotropic space-times are either Minkowski or (anti-)de Sitter.', '1602.05881-2-65-3': 'Furthermore, when matter fields are present, it is possible to define relational observables (which are diffeomorphism-invariant).', '1602.05881-2-65-4': 'On the other hand, in the vacuum setting it is often very difficult to define Dirac observables.', '1602.05881-2-66-0': 'While the inclusion of matter fields in GFTs (and also in spin foam models) has not been extensively explored (although see [CITATION]), here we will show how it is possible to include a scalar field in a GFT model.', '1602.05881-2-67-0': '## The Field Operators and the Fock Space', '1602.05881-2-68-0': 'The path that we shall follow here is to simply modify the definition of the field operators so that each GFT quanta carries a value of the scalar field.', '1602.05881-2-68-1': 'To be specific, considering the case of a single real scalar field, the field operators become [EQUATION] with the domain of the GFT field operators now [MATH].', '1602.05881-2-69-0': 'Clearly, the Fock space for the GFT with a scalar field can be constructed in exactly the fashion as in the vacuum case, except that in this case the one-particle Hilbert space is [MATH].', '1602.05881-2-70-0': 'An immediate consequence of having a scalar field as an argument in the GFT field is that the presence of the scalar field permits the definition of relational observables, i.e., operators evaluated for a specific value of the scalar field.', '1602.05881-2-70-1': 'Relational observables are particularly useful in cosmology, where matter fields can often act as relational clocks that are well-defined and evolve monotonically globally.', '1602.05881-2-70-2': "Thus it is often possible to avoid the problem of time in quantum gravity by defining physical evolution with respect to the scalar field 'clock'.", '1602.05881-2-71-0': "For example, besides the 'total number' operator [EQUATION] it is also possible to define the number operator at a fixed value of [MATH], [EQUATION]", '1602.05881-2-71-1': 'This is a distributional operator, and it can be regulated by smearing it over a small interval [MATH].', '1602.05881-2-71-2': 'Clearly, analogous definitions will hold for other relational operators.', '1602.05881-2-72-0': 'The scalar field is also, of course, a true physical degree of freedom.', '1602.05881-2-72-1': 'There are a number of new observables associated to the scalar field, which are the second-quantized counterpart of the standard observables of a scalar field, namely polynomials in the scalar field and its derivatives.', '1602.05881-2-72-2': 'The two fundamental second-quantized operators are the scalar field operator [EQUATION] and the conjugate momentum operator (written in the momentum representation of the scalar field) [EQUATION]', '1602.05881-2-72-3': 'The [MATH] can be rewritten in the [MATH] representation via a Fourier transform, [EQUATION] here it is given in a manifestly self-adjoint form.', '1602.05881-2-72-4': 'From this it is easy to define the self-adjoint relational conjugate momentum operator [EQUATION] which will play an important role in extracting the cosmological dynamics from the GFT condensate states.', '1602.05881-2-73-0': '## The Quantum Equations of Motion', '1602.05881-2-74-0': 'The dynamics for the GFT models with a scalar field will be determined by the GFT action which is to be chosen following the same procedure as in the vacuum case.', '1602.05881-2-74-1': 'We will once again only consider simplicial interactions, and therefore the action will contain a kinetic term together with interaction terms containing five copies of the field variable.', '1602.05881-2-75-0': 'In this section, we will explain what choices are possible in the GFT action in order to appropriately account for the scalar field.', '1602.05881-2-75-1': 'The construction will be such that, for states where the scalar field vanishes the quantum equations of motion will be precisely those obtained in Sec. [REF].', '1602.05881-2-75-2': 'Of course, the same quantization ambiguities present in the vacuum case will also arise when a scalar field is present.', '1602.05881-2-76-0': 'In order to determine how the scalar field should appear in the kinetic and interaction terms in the action for the case of a GFT with simplicial interactions, it is necessary to understand how a scalar field is discretized on a simplicial discretization of a space-time.', '1602.05881-2-77-0': 'Since a scalar field should be discretized on 0-dimensional elements of the chosen lattice, there are two choices: either real numbers representing the value of the scalar field can be associated to vertices of the simplicial lattice or to nodes of its dual complex.', '1602.05881-2-77-1': 'In the first case, the scalar field will propagate along the edges of the simplicial complex, while in the second case the scalar field will propagate along the links of the dual complex.', '1602.05881-2-77-2': 'While it is reasonable to expect that these two choices lead to equivalent results in the continuum limit, one choice may be better than the other.', '1602.05881-2-77-3': "First, which choice is most 'natural' depends to some extent on the interpretation given to the simplicial complex (and its dual), and second, one choice may give a GFT where calculations are much easier than in the other.", '1602.05881-2-78-0': "If the basic 4-simplices constituting the simplicial complex are viewed as the finitary substitute of the notion of points in a continuum manifold [CITATION] -in which case the data at each 4-simplex constitute the 'local' data in a discrete setting- then it is most natural to discretize the scalar field on nodes of the dual complex.", '1602.05881-2-78-1': 'In this case a constant value of the scalar field is assigned to the 4D elements of the simplicial lattice, and the gradient terms are given by the difference of the value of the scalar field in neighbouring 4-simplices.', '1602.05881-2-78-2': 'Note that it follows that these discrete gradients are naturally associated to the common boundaries of the neighbouring 4-simplices (i.e., the tetrahedra), and therefore the scalar field propagates along the links of the dual complex.', '1602.05881-2-79-0': 'Moreover, and perhaps even more importantly, this discretization choice also generically leads to a much simpler extension of the GFT (and spin foam) framework to include scalar matter, as shall become clear.', '1602.05881-2-80-0': 'Locating the discretization on the dual node to the simplicial discretization means that the scalar field has a single value over any 4-simplex.', '1602.05881-2-80-1': 'This directly implies that the five fields in the GFT interaction -which correspond to the five tetrahedra in a single 4-simplex- all share the same value of the scalar field.', '1602.05881-2-80-2': 'For this reason the interaction term must impose that the value of the scalar be the same in all of the five interacting fields, in which case the (first) interaction term has the form [EQUATION] where the Dirac [MATH] distributions are shown explicitly for the sake of clarity.', '1602.05881-2-80-3': 'Clearly, the integrals over [MATH] and [MATH] are easily evaluated, giving [EQUATION]', '1602.05881-2-80-4': 'The second interaction term is simply the adjoint of this one.', '1602.05881-2-80-5': 'It is clear that, for GFT models with this interaction term, the scalar field degrees of freedom enter locally in the GFT action (unlike the quantum geometric degrees of freedom that are non-local), and this will simplify a number of calculations.', '1602.05881-2-81-0': 'Since the scalar field propagates along dual links between neighbouring 4-simplices, the gradients of the scalar field will be encoded in the GFT kinetic term, which in general will have the form [EQUATION]', '1602.05881-2-81-1': 'This is the completely general form of a GFT model for quantum gravity coupled to a real scalar field, whose action is given by [EQUATION] which follows directly from the discretization strategy described above.', '1602.05881-2-82-0': '## A Massless Scalar Field', '1602.05881-2-83-0': 'We will now focus on the case of the simplest scalar field coupled to gravity: a minimally coupled, massless scalar field.', '1602.05881-2-83-1': 'While this is a particularly simple case, it is nonetheless enough in order to extract non-trivial cosmological dynamics directly from the GFT model.', '1602.05881-2-83-2': 'Furthermore, a minimally coupled, massless scalar field in a homogeneous space-time acts like a stiff perfect fluid which is of particular interest in early universe cosmology as it is one of the few matter fields that remains relevant (in the sense that its effect on the dynamics of the geometrical degrees of freedom does not become negligible) in the high curvature regime when anisotropies are present.', '1602.05881-2-83-3': 'Therefore, a minimally coupled massless scalar field is of particular interest in studies of quantum gravity effects in cosmology.', '1602.05881-2-84-0': 'An additional advantage of the minimally coupled massless scalar field is that symmetry considerations will be enough to strongly restrict the form of the GFT action.', '1602.05881-2-84-1': '(Note that for more general cases, whether a non-minimal coupling or the presence of a potential, or both, a more detailed analysis will be necessary in order to determine the appropriate form of the GFT action.)', '1602.05881-2-85-0': 'The matter contribution to the classical (continuum) Lagrangian for a minimally coupled massless scalar field on a curved background is [EQUATION] and this Lagrangian clearly exhibits the shift and sign reversal symmetries [EQUATION]', '1602.05881-2-85-1': 'Note that non-minimal coupling or any non-trivial potential [MATH] in the scalar field action would kill the first symmetry, and could kill the second, depending the specific form of the non-minimal coupling or potential.', '1602.05881-2-86-0': 'The key step here is that we shall assume that these symmetries are also present at the quantum level in the GFT, and this places strong restrictions on the GFT action.', '1602.05881-2-86-1': 'To be specific, the kinetic term can only depend on the square of scalar field differences [MATH], while [MATH] must be independent of [MATH]: [EQUATION]', '1602.05881-2-86-2': 'It is clear that any dependance on the field values themselves in the kinetic term or in the interaction kernel would break the shift symmetry, while a dependence of the kinetic term on odd powers of field differences would break the reflection symmetry.', '1602.05881-2-87-0': 'Assuming analyticity of the field variable with respect to the scalar field, it is possible to express the kinetic term as a derivative expansion.', '1602.05881-2-87-1': 'This derivative expansion will be particularly useful in the case where the difference [MATH] is small (compared to the Planck mass) in which case the first few terms will provide a good approximation to the full kinetic term.', '1602.05881-2-87-2': 'This derivative expansion can be obtained by rewriting [MATH] and [MATH], and the kinetic term becomes [EQUATION] and the [MATH] field variable can be Taylor-expanded in its scalar field argument around [MATH].', '1602.05881-2-87-3': 'Then, after evaluating the integral over [MATH], the kinetic term becomes [EQUATION] where [EQUATION]', '1602.05881-2-87-4': 'Note that all of the odd terms in the Taylor expansion do not contribute to the sum in [REF] since the integral over [MATH] of an odd power of [MATH] multiplying the even function [MATH] gives zero.', '1602.05881-2-88-0': 'The functional form of the [MATH], which is of course determined by the kinetic term in the GFT action, encodes order by order in derivatives of [MATH] how the quantum geometric and matter degrees of freedom propagate.', '1602.05881-2-88-1': 'In particular, their exact form could be determined by ensuring that the GFT Feynman amplitudes match term by term the discrete path integral for gravity coupled to a minimally coupled massless scalar field; we leave this analysis for future work.', '1602.05881-2-89-0': 'It will not be necessary to determine the exact functional form of the [MATH] for our purposes.', '1602.05881-2-89-1': 'Instead, having ensured that the GFT action has the correct symmetries, we will work in the small derivative limit of the scalar field (i.e., where differences in the scalar field are small compared to the Planck mass), and truncate the kinetic term in the action to the lowest two orders, keeping only the terms corresponding to [MATH] and [MATH] in the sum [REF].', '1602.05881-2-89-2': 'Then, it will be possible to obtain some constraints on their form by studying the dynamics of condensate states of this GFT model -which as shall be explained shortly, are expected to capture the degrees of freedom of the spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time- and comparing these dynamics, in the appropriate semi-classical limit, to the Friedmann equations of general relativity.', '1602.05881-2-90-0': 'Finally, here we are interested in the GFT model based on the EPRL spin foam model (whose kinetic and interaction terms in the vacuum case were given respectively in [REF] and [REF]) with a minimally coupled massless scalar field.', '1602.05881-2-90-1': 'It is easy to add repeat the procedure outlined in this section starting from the GFT action for the vacuum EPRL model, this gives (in the spin representation) [EQUATION] with [EQUATION] where the [MATH] and [MATH] field variables having the same arguments in the kinetic terms in the action having imposed the Kronecker deltas, and [EQUATION]', '1602.05881-2-90-2': 'Note that the kinetic term has been truncated since, as explained above, we are considering the small derivative limit in [MATH].', '1602.05881-2-91-0': 'A few comments on this GFT model are in order.', '1602.05881-2-91-1': 'First, it was not necessary to assume the presence of any space-time symmetries in order to derive this action; in particular neither homogeneity nor isotropy were imposed.', '1602.05881-2-91-2': 'The simplicity of the GFT action is a result of not only working with a particularly simple matter field (the minimally coupled massless scalar field), but also the chosen discretization where the scalar field is discretized on each 4-simplex.', '1602.05881-2-91-3': 'The simplicity of the GFT classical equations of motion can be understood as the result of coarse-graining the small-scale complexity and obtaining the hydrodynamical equations of motion for the collective behaviour.', '1602.05881-2-91-4': 'This interpretation will be relevant in the following sections.', '1602.05881-2-91-5': 'Second, somewhat surprisingly from a purely formal GFT viewpoint, the minimally coupled massless scalar field enters the GFT action in exactly the same fashion as the standard time coordinate in ordinary quantum field theory.', '1602.05881-2-91-6': 'The presence of the scalar field allows for the definition of a host of new relational observations, but the above observation is stronger: the minimally coupled massless scalar field can be used to define a global relational clock and thus provides a well-defined notion of global time evolution in a diffeomorphism invariant context.', '1602.05881-2-91-7': 'This will be particularly useful in the cosmological context, but it is likely that this will be a powerful tool in a number of other physical settings as well.', '1602.05881-2-92-0': '# GFT Condensates', '1602.05881-2-93-0': 'As in any interacting quantum field theory, it would be naive to expect to be able to solve the quantum dynamics exactly for realistic GFT models (and note that the situation is potentially worse in the GFT context due to the background independence of GFT models as well as the non-local nature of the quantum geometric interactions).', '1602.05881-2-93-1': 'Instead, the appropriate strategy is to study simplified trial states and look for approximate solutions that may capture the relevant properties of the physical setting of interest.', '1602.05881-2-93-2': 'Then, if these approximate solutions are chosen judiciously, they will provide approximate answers to the physical questions at hand.', '1602.05881-2-94-0': 'In this paper, we will focus on the problem of extracting effective cosmological dynamics from the full quantum gravity formalism, using the GFT model with a minimally coupled massless scalar field based on the EPRL spin foam model described in the previous sections.', '1602.05881-2-94-1': 'In this case, the approximate solutions of interest are GFT condensate states, which are believed to capture the relevant degrees of freedom of the homogeneous FLRW and Bianchi space-times [CITATION].', '1602.05881-2-94-2': 'We will now summarize the motivation for considering GFT condensate states, and specifically their relevance to the cosmological sector of GFT.', '1602.05881-2-94-3': 'Further details and background information can be found in the literature [CITATION].', '1602.05881-2-95-0': "The key idea is that, by definition in a condensate state, all spin network nodes are characterized by the same condensate wave function and in this sense condensate states have a 'wave function homogeneity'.", '1602.05881-2-95-1': 'For this reason, condensate states are naturally adapted to the notion of homogeneity, and the coarse-graining procedure which determines the hydrodynamical equations of motion of homogeneous space-times from the microscopic GFT dynamics will be more straightforward in this case than it is in general.', '1602.05881-2-95-2': 'In addition, in GFT condensate states, the condensate wave function for each spin network node stores the intrinsic geometric data of a quantum tetrahedron in 4D, which is isomorphic to the space of homogeneous (and possibly anisotropic) space-times [CITATION].', '1602.05881-2-95-3': 'This will further simplify the coarse-graining procedure.', '1602.05881-2-96-0': 'In the coarse-graining procedure then, the fact that all of the fundamental GFT quanta are in the same state naturally implements the idea of spatial homogeneity of the continuum (quantum) geometry one can reconstruct from them.', '1602.05881-2-96-1': "The notion of 'continuum' here refers to the fact that an infinite number of fundamental degrees of freedom is captured by the definition of the condensate states; they do not correspond in this sense to any (say, lattice) truncation of the theory but rather to a coarse-graining of it, in which an infinite number of fundamental degrees of freedom is captured by a single collective wave function depending on a small number of collective variables.", '1602.05881-2-96-2': 'This is in full analogy with the use of condensate states in many-body quantum theory, e.g., to extract the relevant macroscopic behaviour of quantum liquids in the condensate phase.', '1602.05881-2-96-3': "In fact, this particular class of quantum states would be physically (and not only mathematically) appropriate for describing our universe at macroscopic scales in a scenario in which the 'geometrogenesis' transition -between the pre-geometric phase of quantum gravity (which would not admit a continuum description in terms of geometric fields and general relativity) and the geometric one (described by continuous semi-classical geometries)- is a Bose-Einstein condensation process for the fundamental quantum gravity building blocks of space-time [CITATION].", '1602.05881-2-97-0': '## The Simplest Condensate States: The Gross-Pitaevskii Approximation', '1602.05881-2-98-0': 'Since the aim here is to describe the simplest cosmological space-times -namely the spatially flat, homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology- the appropriate degrees of freedom can already be captured in the simplest condensate states.', '1602.05881-2-99-0': 'In particular, since on the one hand the only geometrical observable of interest is the spatial volume', '1602.05881-2-100-0': '(as a function of relational time [MATH]) which can easily be calculated without knowing how the spin network nodes are connected amongst themselves, and on the other hand there is no need to encode the spatial curvature in the connectivity of the spin network nodes, it is in fact reasonable to entirely ignore the connectivity of the spin network nodes in the condensate state.', '1602.05881-2-101-0': 'Although this is a drastic approximation, it appears quite reasonable for the spatially flat FLRW space-time.', '1602.05881-2-101-1': 'In addition, this approximation will significantly simplify both the form of the resulting condensate state as well as its quantum equations of motion.', '1602.05881-2-102-0': 'Following the arguments above, we shall neglect the connectivity amongst the spin network nodes, and in this case it is possible to restrict our attention to the relatively simple Gross-Pitaevskii condensate states.', '1602.05881-2-102-1': 'These are coherent states for the GFT field operator, [EQUATION] where [EQUATION]', '1602.05881-2-102-2': 'As can be seen here, the condensate wave function [MATH] is not normalized.', '1602.05881-2-102-3': 'Indeed, its norm captures an important physical quantity, the number of fundamental GFT quanta.', '1602.05881-2-102-4': 'Note that an important property of condensate states is that they are eigenstates of the field operator [MATH], [EQUATION]', '1602.05881-2-102-5': 'In order to restrict the condensate wave function [MATH] to the space of intrinsic geometries of a tetrahedron in 4D (which as already noted above is isomorphic to the space of spatially homogeneous space-times), an additional restriction is imposed upon the condensate wave function.', '1602.05881-2-102-6': 'Besides the right gauge invariance given in [REF], the condensate wave function is also taken to be invariant under diagonal left gauge transformations, [EQUATION]', '1602.05881-2-102-7': 'This choice is rather natural, from a geometric point of view: requiring invariance under diagonal left gauge invariance is equivalent to group-averaging over the relative embedding of the tetrahedron in [MATH], which is not a physically relevant observable.', '1602.05881-2-103-0': 'It is clear that condensate states satisfying the requirement [REF] encode no explicit information about the connectivity of the graphs underlying microscopic states, and therefore also no information about the topology of the space-time itself.', '1602.05881-2-103-1': 'As a result, the appropriate procedure to coarse-grain these states and extract the correct hydrodynamical equations of motion is rather straightforward, but is also less manifest; the coarse-graining is more apparent when working with more general condensate states [CITATION].', '1602.05881-2-103-2': 'While the limitations of the approximation of neglecting the connectivity information should be kept in mind, it is also important to remember that in condensed matter physics the same type of drastic simplification is commonly employed and Gross-Pitaevskii condensate states capture a lot of relevant physics even in their simple form.', '1602.05881-2-104-0': 'Nonetheless, in some settings of physical interest it is to be expected that correlations among GFT quanta may be relevant (in which case quantum fluctuations cannot be neglected), and then the Gross-Pitaevskii approximation will fail.', '1602.05881-2-104-1': 'Indeed, the Gross-Pitaevskii hydrodynamics -as captured by the simple condensate states of the type [REF]- will break down when the interactions between the GFT quanta become sufficiently large.', '1602.05881-2-104-2': 'This can happen either if the coupling constants in front of the interaction terms in the GFT action are big, or when the condensate density (i.e., the modulus of the condensate wave function or equivalently the expectation value of the GFT number operator) grows too large.', '1602.05881-2-104-3': 'We will discuss this point in greater detail in Sec. [REF].', '1602.05881-2-105-0': 'In order to study situations where the interactions become important, it will be necessary to use more complicated condensate states, for example the generalizations defined in [CITATION].', '1602.05881-2-105-1': 'These condensate states contain information about the connectivity of the spin network nodes and hence they are easier to coarse-grain in order to recover the geometry of the resulting space-time; in particular, these generalized condensate states directly provide the topological information of the space-time which is absent in the Gross-Pitaevskii GFT condensate states [REF].', '1602.05881-2-106-0': 'Still, even in the Gross-Pitaevskii approximation the condensate wave function [MATH] encodes both the number of quanta in the state and the distribution over the possible values of the geometric data of homogeneous (and potentially anisotropic) cosmologies.', '1602.05881-2-106-1': 'Therefore, these condensate states provide a natural point of contact with the usual Wheeler-DeWitt picture of quantum cosmology, and have the additional advantages of (i) not requiring any symmetry reduction, and (ii) including the many-body features of the full Hilbert space of the microscopic theory.', '1602.05881-2-107-0': 'Furthermore, for these simple condensate states it is straightforward to extract their effective dynamics.', '1602.05881-2-107-1': 'An exact solution to the quantum equations of motion would satisfy all Schwinger-Dyson equations.', '1602.05881-2-107-2': 'However, here we have made an important approximation by assuming that the relevant state is a condensate state of the simplest type.', '1602.05881-2-107-3': 'Therefore, we can only hope for the Schwinger-Dyson equations to be solved approximately.', '1602.05881-2-107-4': 'For this reason we concentrate on only one Schwinger-Dyson equation - the simplest one which corresponds to the classical equations of motion for the condensate state [MATH], which is [EQUATION]', '1602.05881-2-107-5': 'Note that these equations are non-linear in the condensate wave function and can be non-local on minisuperspace (e.g., if the condensate wave function can be rewritten in terms of minisuperspace variables like the scale factor [MATH], the non-linear terms may couple the condensate wave function at different values of [MATH]).', '1602.05881-2-108-0': 'For the action [REF], the dynamics for the condensate state [REF] corresponding to its classical equations of motion is simply (in the spin representation): [EQUATION] with the interaction term contributing a non-linear term of the order of [MATH], the details of which are unimportant for now.', '1602.05881-2-108-1': 'Also note that there is no sum over [MATH] or [MATH] here.', '1602.05881-2-109-0': 'This equation of motion clearly shows that for the condensate state [MATH], the microscopic dynamics of the GFT quanta can be described hydrodynamically in terms of the collective variable [MATH], the condensate wave function.', '1602.05881-2-109-1': 'The hydrodynamic nature of the resulting effective dynamics is the result of the Schwinger-Dyson equations effectively collapsing, due to the simple form of the state [MATH], to the one-particle correlation function (which is exactly [MATH] in the condensate approximation [REF]).', '1602.05881-2-109-2': 'The continuum nature of the equation arises from the fact that [MATH] is given by an infinite sum over numbers of spin network nodes (and hence implicitly over graphs also), and therefore [MATH] is a non-perturbative state with respect to the Fock vacuum.', '1602.05881-2-110-0': '## The Large Density Case', '1602.05881-2-111-0': 'As has already been mentioned, in the large density regime -i.e., when the state has a large number of quanta- the Gross-Pitaevskii approximation encoded in the simple condensate state [REF] breaks down.', '1602.05881-2-111-1': 'This is because, while solutions to the complete GFT quantum dynamics satisfy the operator equations [REF], the coherent state ansatz [REF] only satisfies the expectation value of that same operator equation and thus necessarily fails to capture enough information about the dynamics in regimes where quantum fluctuations and correlations among the GFT quanta become important.', '1602.05881-2-112-0': 'This will be the case, in particular, if the interaction term is large.', '1602.05881-2-112-1': 'To illustrate this point in a bit more detail, let us consider a simple GFT model whose equation of motion is [EQUATION]', '1602.05881-2-112-2': 'While this is a vacuum GFT model, the results obtained here are rather general and also hold for GFT models with a scalar field.', '1602.05881-2-113-0': 'For the coherent state ansatz [REF], the expectation value of the above equation of motion gives [EQUATION] and it can be seen that the condensate state does not exactly satisfy the quantum equations of motion [REF].', '1602.05881-2-113-1': 'This is not surprising since the condensate ansatz is only expected to provide an approximate solution to the equations of motion.', '1602.05881-2-113-2': 'An estimate of the amplitude of the error can be assessed in a number of ways; in the GFT framework, one possibility is to look at the norm of the quantity [EQUATION]', '1602.05881-2-113-3': 'Using the field equations for [MATH], the state [MATH] can be rewritten as [EQUATION] with [EQUATION]', '1602.05881-2-113-4': 'Then the norm of [MATH] can easily be calculated, giving [EQUATION]', '1602.05881-2-113-5': 'The second term is exactly the result obtained from normal ordering all of the operators in the expectation values.', '1602.05881-2-113-6': 'Therefore, the norm of [MATH] is controlled by the functional of the wave function that is obtained by taking the necessary Wick contractions.', '1602.05881-2-113-7': 'Although the expressions are cumbersome and not particularly illuminating, it is clear that this functional will contain several terms obtained from combining positive powers of the condensate wave function.', '1602.05881-2-113-8': 'When [MATH] becomes large, then the norm of [MATH] could become large as well, in general, and at a much faster rate, given the presence of higher powers.', '1602.05881-2-113-9': 'In this case the Gross-Pitaevskii approximation would clearly break down.', '1602.05881-2-113-10': 'Whether the Gross-Pitaevskii approximation fails for large [MATH] has to be considered on a case by case basis, as it depends on the interplay between the solution and the interaction kernels of the specific GFT model of interest.', '1602.05881-2-114-0': 'This limitation is the analogue of the failure of the dilute gas approximation in the description of the Bose gas whenever the strength of the interactions becomes large with respect to the interatomic distance.', '1602.05881-2-114-1': 'This means that the coherent state approximation can be trusted only in a mesoscopic regime, in which the number of quanta is large enough to admit a continuous geometry approximation but small enough to avoid large deviations from the correct physical state.', '1602.05881-2-114-2': 'In settings where the GFT analogue of the Gross-Pitaevskii ansatz fails, one will have to resort to more involved condensate states (e.g., of the type described in [CITATION]) and/or include the dynamics of fluctuations and their coupling to the background condensate dynamics.', '1602.05881-2-115-0': '## Restriction to Isotropy', '1602.05881-2-116-0': 'In the same spirit of coarse-graining to cosmological observables (rather than symmetry-reduction), we will now impose a further restriction on the form of the condensate wave function so that only isotropic quantities such as the spatial volume or the Hubble rate can be reconstructed from [MATH].', '1602.05881-2-116-1': 'Clearly, this restriction on the type of microscopic states allowed will lead to further simplifications in the effective continuum dynamics.', '1602.05881-2-116-2': 'Note that imposing isotropy on the condensate wave function is a restriction on the macroscopic variable that collectively describes all the microscopic configurations entering the definition of the state - it does not correspond to a symmetry reduction of these microscopic configurations to isotropic ones (this property of the coarse-graining is more apparent for the case of generalized condensate states [CITATION], but also holds in this case.)', '1602.05881-2-117-0': 'Imposing isotropy can be done in a number of different ways.', '1602.05881-2-117-1': 'Since the GFT action for the EPRL spin foam model [REF] is expressed in the spin representation, it is convenient to determine how isotropy is to be imposed in the spin representation.', '1602.05881-2-117-2': "Since spin network nodes can be interpreted as tetrahedra, it is natural to require that the tetrahedra be as 'isotropic' as possible, and therefore we shall define isotropic condensate wave functions to be those that correspond to equilateral tetrahedra (which are the most 'isotropic' configurations).", '1602.05881-2-117-3': 'As a result, an isotropic condensate wave function [MATH] will depend only on the volume of the tetrahedron or, equivalently, the surface area of one of its faces (as well as on the scalar field [MATH])', '1602.05881-2-118-0': 'Since isotropic condensate wave functions are to be those that correspond to equilateral tetrahedra, it is helpful to recall some of their properties.', '1602.05881-2-118-1': 'The classical geometry of an equilateral tetrahedron is entirely described by its four area vectors [MATH], which all have the same norm and are related to each via the discrete subgroup of [MATH] called the tetrahedral group.', '1602.05881-2-118-2': 'Equilateral tetrahedra have two properties that will be relevant for the construction of isotropic condensate wave functions: (i) the areas of the faces are equal, [MATH] and (ii) for given a total surface area [MATH], the volume of the tetrahedron is maximized in the equilateral case.', '1602.05881-2-119-0': 'We can use these two properties to display more clearly what are the true dynamical degrees of freedom that survive in the condensate wave function.', '1602.05881-2-119-1': 'First, using the Peter-Weyl theorem to decompose the wave function into a basis of orthonormal functions given by the Wigner matrices, we see that left and right gauge invariance reduce the wave function to a very simple form: [EQUATION] in terms of linear combinations of pairs of intertwiners, one associated to the left gauge invariance and one to the right closure condition.', '1602.05881-2-119-2': 'The geometric properties are therefore entirely contained in the scalar functions [MATH].', '1602.05881-2-120-0': 'The first geometric condition on the isotropic restriction is encoded in the quantum theory by requiring that [MATH] must have support only on spin network nodes where the four [MATH] on each link are equal, [EQUATION]', '1602.05881-2-120-1': 'The second condition, the requirement that the volume be maximized in equilateral tetrahedra, is imposed by demanding that for a given [MATH] the condensate wave function only has support on the intertwiner that (i) is an eigenvector of the LQG volume operator, and (ii) has the largest eigenvalue.', '1602.05881-2-120-2': 'Denoting this (normalized) intertwiner by [MATH], [EQUATION] where the [MATH] are determined by the above two conditions together with [MATH].', '1602.05881-2-121-0': 'Then, the requirement that the wave function has components only along the equilateral tetrahedra components implies that [EQUATION]', '1602.05881-2-121-1': "Finally, the geometric interpretation of the [MATH] intertwiner (and of the 'left flux operators' also) is not immediately clear.", '1602.05881-2-121-2': 'However, some heuristic arguments do provide some guidance.', '1602.05881-2-121-3': "Since the connectivity in GFTs is imposed (in the spin representation) by requiring that the 'left' angular momentum quantum numbers agree between connected links, it may be possible to interpret the left flux operators as the right flux operators parallel-transported from the node of the spin network to the ends of the spin network links.", '1602.05881-2-121-4': 'Since scalar quantities like the volume should behave trivially under parallel transport, this suggests requiring that the volume eigenstate of [MATH] be equal to that of [MATH], and hence that [MATH].', '1602.05881-2-121-5': 'We leave a more detailed study of this condition for future work.', '1602.05881-2-122-0': 'Then, implementing all of the above conditions, [EQUATION]', '1602.05881-2-122-1': 'With all these restrictions, coming from geometric reasoning and symmetry arguments, only the form of [MATH] remains to be solved for.', '1602.05881-2-122-2': 'As indicated at the beginning of this section, the condensate wave function (at a given [MATH]) is entirely determined by its dependence on just one geometric quantity, the spin [MATH].', '1602.05881-2-122-3': 'For this reason, the only geometric quantities that can be extracted from a condensate wave function of this type are isotropic quantities like the total spatial volume and the Hubble rate, and so [REF] is a natural candidate to extract the homogeneous, isotropic and spatially flat cosmological sector from quantum gravity.', '1602.05881-2-123-0': 'Finally, for the sake of completeness, note that in the group representation an isotropic condensate wave function has the form [EQUATION]', '1602.05881-2-123-1': 'The form of an isotropic condensate wave function in the flux representation can be obtained via a non-commutative Fourier transform.', '1602.05881-2-124-0': '# Effective Cosmological Dynamics of Isotropic Quantum Gravity Condensates', '1602.05881-2-125-0': 'As already outlined in Sec. [REF], in the Gross-Pitaevskii approximation, given a condensate wave function for a specific GFT model, it is a straightforward procedure to obtain the (non-linear) equations of motion for that condensate wave function.', '1602.05881-2-125-1': 'In this section, we will derive the equations of motion for the isotropic condensate wave function [MATH] for a GFT model based on the Lorentzian EPRL spin foam model and with a massless scalar field, whose action is given in [REF].', '1602.05881-2-125-2': 'From these equations of motion it will be possible to extract effective Friedmann equations, expressed in a relational form with respect to the massless scalar field [MATH].', '1602.05881-2-126-0': '## Relational Dynamics', '1602.05881-2-127-0': 'As already explained in Sec. [REF], the equation of motion for a condensate in the Gross-Pitaevskii approximation is given by [REF].', '1602.05881-2-127-1': 'It is worth repeating that the condensate state is expected to be an approximate solution, and therefore not all of the Schwinger-Dyson equations will hold.', '1602.05881-2-127-2': 'Instead, we only impose the first Schwinger-Dyson equation exactly.', '1602.05881-2-127-3': 'For an isotropic condensate wave function of the type [REF], and for the GFT action [REF], based on the EPRL spin foam model and extended in order to include a massless scalar field, this equation has the simple form [EQUATION] which are non-linear in [MATH], and the constants [MATH] and [MATH] are obtained from combinations of the kinetic and interaction terms in the GFT action with the non-[MATH] terms in the condensate wave function [MATH].', '1602.05881-2-127-4': 'To be specific, decomposing the various terms in the action in representations, [EQUATION] using a condensed notation in order to avoid writing explicitly all the tensor indices, and here the [MATH] symbol of [MATH] is that of, e.g., [CITATION].', '1602.05881-2-128-0': 'Note that these equations of motion for the condensate wave function can also be obtained from the action [EQUATION] obtained by replacing the GFT field in the GFT action by the condensate wave function.', '1602.05881-2-128-1': 'Here it is clear that the scalar field [MATH] plays the role of a relational time variable.', '1602.05881-2-129-0': 'The condensate equations of motion depend directly on the details of the GFT action, since these determine in part the coefficients [MATH] and [MATH].', '1602.05881-2-129-1': 'It will be possible to constrain their form by requiring that the Friedmann equation be recovered in an appropriate semi-classical limit.', '1602.05881-2-130-0': 'Crucially, the interaction term does not couple [MATH] with different [MATH].', '1602.05881-2-130-1': 'This is due to the combination of the isotropic restriction and the form of the EPRL vertex amplitude which contain Kronecker deltas [MATH] for all edges that meet in the four-simplex.', '1602.05881-2-130-2': 'Thus, if five equilateral tetrahedra are combined in a four-simplex, and the vertex amplitude is the EPRL one (or one with an analogous property) then it immediately follows that all of the five equilateral tetrahedra must have the same [MATH].', '1602.05881-2-130-3': 'This decoupling does not occur generically, even in the isotropic restriction, for other spin foam models, e.g., those like the Baratin-Oriti model [CITATION] involving more elaborate fusion coefficients.', '1602.05881-2-130-4': "For this reason, the interaction term is 'local' in the spin label since it has the form [MATH] rather than [MATH].", '1602.05881-2-130-5': 'Clearly, this significantly simplifies the equations of motion.', '1602.05881-2-131-0': 'As true in general for GFT condensates, we have thus obtained a quantum cosmology-like equation for a cosmological wave function on the space of (isotropic) homogeneous geometries.', '1602.05881-2-131-1': "This equation is however non-linear, as to be expected in a hydrodynamic context, with the non-linearities effectively encoding the microscopic interactions between the fundamental 'atoms of space', which are also ultimately responsible for the generation of inhomogeneities at both microscopic and macroscopic scales (see also [CITATION] for a similar construction).", '1602.05881-2-132-0': "Before we start analyzing the effective dynamical equations, we point out that, from the symmetries of [MATH], it is obvious that there is a conserved quantity for every [MATH], the 'energy' [MATH] of the condensate wave function [MATH] with respect to the relational time [MATH], [EQUATION]", '1602.05881-2-132-1': 'In addition, in the regime in which the interaction term is small (which is necessary for the Gross-Pitaevskii approximation to hold), the [MATH] charge [MATH] related to the symmetry [MATH] emerges as another conserved quantity [EQUATION]', '1602.05881-2-132-2': 'Note that, following from the definition of the momentum of the massless scalar field, it is easy to check that [MATH] and therefore [MATH] is a conserved quantity also in the limit where the Gross-Pitaevskii approximation holds.', '1602.05881-2-133-0': 'The equation [EQUATION] is exactly the continuity equation in cosmology, for the case of a massless scalar field.', '1602.05881-2-133-1': 'This is a particularly simple example of how the large-scale, coarse-grained effective dynamics can be extracted from the GFT quantum equations of motion for condensate states.', '1602.05881-2-133-2': 'This result is also a first confirmation of the consistency of the identification of the GFT condensate state and an emergent FLRW space-time geometry.', '1602.05881-2-134-0': 'Note that the condition that the interactions be subdominant is required in order to recover the continuity equation for the isotropic condensate state and, as we will show, the Friedmann equations.', '1602.05881-2-134-1': 'While this is to some extent only a technical restriction to a regime where simple condensate states can be trusted, it is not unreasonable from a physical point of view.', '1602.05881-2-134-2': 'One expects that generic interactions would generate correlations between GFT quanta, and there is no reason to expect these to respect any homogeneity condition, but rather to produce inhomogeneities both the microscopic and macroscopic level.', '1602.05881-2-134-3': 'And when inhomogeneities are included in cosmology (even at linear order) the continuity equation is modified.', '1602.05881-2-134-4': 'Note that the heuristic arguments above do not necessarily imply that GFT non-linearities at the level of the hydrodynamic equation encode inhomogeneities (as has been suggested in [CITATION]), but this is an interesting hypothesis to explore, especially considering how similar equations (again inspired by BEC theory) have been obtained as an effective description of inhomogeneities in a non-linear extension of (loop) quantum cosmology [CITATION].', '1602.05881-2-135-0': 'For the remainder of this paper, we will only consider the limit where the interaction term is much smaller than the linear terms.', '1602.05881-2-135-1': 'This is not because the non-linear case is difficult to solve (in fact, for the simple condensate equations of motion considered here, it is relatively straightforward to study the dynamics of the condensate wave function even in the presence of the non-linear term) but rather because in that limit the Gross-Pitaevskii approximation is expected to fail, in the sense that it cannot be justified from a microscopic point of view since the simple condensate state we use here cannot be expected to be a good approximation to a realistic vacuum of the theory, and it is necessary to consider more complex condensate states than [REF].', '1602.05881-2-136-0': 'Therefore, we will study the regime where [MATH] is sufficiently small so that the interaction term is subdominant, but at the same time not so small that the hydrodynamic approximation ceases to make sense: after all, [MATH] corresponds to the average number of GFT quanta at the relational time [MATH], and a large number of quanta is necessary for the hydrodynamic approximation to be valid.', '1602.05881-2-136-1': 'The existence of such a mesoscopic regime depends on the specific form of the GFT action.', '1602.05881-2-136-2': 'To be specific, if the interaction term in the GFT action is sufficiently small compared to the kinetic term, then a mesoscopic regime will exist for some period of relational time.', '1602.05881-2-136-3': 'Indeed, the smaller the interaction term is in relation to the kinetic term, the longer this mesoscopic regime will exist (with respect to the relational clock [MATH]).', '1602.05881-2-136-4': 'The existence of such a mesoscopic regime for appropriate choices of the GFT action has been studied at a phenomenological level in [CITATION].', '1602.05881-2-136-5': 'In the remainder, we will only consider GFT actions whose interaction term is sufficiently small compared to the kinetic term so that an appropriate mesoscopic regime exists.', '1602.05881-2-137-0': 'In this mesoscopic regime, the equation of motion for [MATH] reduces to [EQUATION] with [MATH].', '1602.05881-2-138-0': 'At this point, it is convenient to separate [MATH] into its modulus and phase, [EQUATION] with [MATH] and [MATH] both assumed to be real, and [MATH] to be positive.', '1602.05881-2-138-1': 'From now on, we will drop the argument [MATH], and denote derivatives with respect to [MATH] with primes, e.g., [MATH].', '1602.05881-2-138-2': 'Then, in terms of [MATH] and [MATH], the equation of motion [REF] splits into a real and an imaginary part, which are respectively [EQUATION] and [EQUATION]', '1602.05881-2-138-3': 'The last equation, coming from the imaginary part of [REF], can easily be solved and shows that the combination [MATH] is a constant of the motion, and in fact is precisely the conserved [MATH] charge [REF], [EQUATION]', '1602.05881-2-138-4': "Note that the other conserved charge, the 'GFT energy' for each [MATH], also has a simple form, [EQUATION]", '1602.05881-2-138-5': 'Finally, using [REF], the remaining equation of motion [REF] can be rewritten as [EQUATION] and this has the form of the equation of motion of a particle in a central potential.', '1602.05881-2-138-6': 'In particular, note that the effective potential diverges as [MATH]; this implies that [MATH] remains non-zero at all times (for non-zero [MATH]).', '1602.05881-2-138-7': 'This is what will lead to the resolution of the big-bang and big-crunch singularities in the cosmological space-time, as is explained in detail in the next section, so long as the cosmological dynamics are captured by the above equation.', '1602.05881-2-139-0': 'However, before studying the dynamics in more detail and extracting the equations of motion for geometric quantities, it is important to recall the assumptions that were necessary in order to derive [REF].', '1602.05881-2-139-1': 'First, we have assumed that a cosmological state in quantum gravity is well-approximated by a simple condensate that in particular ignores connectivity information, which is in general a very important set of dynamical degrees of freedom.', '1602.05881-2-139-2': 'However, in the case of isotropic cosmology we expect these degrees of freedom to play a less important role since the only relevant geometric observables are the spatial volume and its conjugate.', '1602.05881-2-139-3': 'Second, we further imposed that the quanta of geometry in the condensate be isotropic, and we are working in the limit where the scalar field [MATH] is assumed to evolve slowly.', '1602.05881-2-139-4': 'Finally, we are considering the regime where the interaction term in [REF] is subdominant, and hence where the [MATH] are sufficiently small.', '1602.05881-2-140-0': 'On the other hand, for there to exist a continuum interpretation of the condensate state as a space-time, there must be a large number of quanta of geometry in the condensate state, which requires the [MATH] to be large.', '1602.05881-2-140-1': '(Also, in order for a consistent continuum geometric interpretation to be valid at least for large total spatial volumes of the universe, a few more conditions are needed, namely that there be a small curvature and a small volume associated to each individual GFT quantum.', '1602.05881-2-140-2': 'These last conditions are not necessary for the mathematical consistency of the condensate approximation, but are necessary to have a clear space-time interpretation for the condensate state.)', '1602.05881-2-141-0': 'A delicate interplay between the values of [MATH] and the coupling constants (and kernels) of the theory is required for the condensate approximation to be valid while at the same time neglecting the interactions.', '1602.05881-2-141-1': 'It is only when all of these assumptions hold that a reliable cosmological interpretation of the condensate state exists and that the effective dynamics extracted here from the full theory can be trusted.', '1602.05881-2-142-0': '## Condensate Friedmann Equations', '1602.05881-2-143-0': 'The effective dynamics of the GFT condensates is (part of) the hydrodynamics of the GFT model we are studying, and is encoded in an equation for the mean field [MATH] (and its complex conjugate) or, in more conventional hydrodynamic form, for a density [MATH] and a phase [MATH], which in turn can be decomposed in terms of modes associated to representations [MATH].', '1602.05881-2-143-1': 'This type of equation has the form of a non-linear extension of a quantum cosmology dynamics, even though the physical interpretation is different.', '1602.05881-2-143-2': 'From this type of equation, just as in (loop) quantum cosmology, it is possible to extract the gravitational dynamics in the form of equations for geometric quantities.', '1602.05881-2-143-3': 'In particular, for homogeneous and isotropic configurations, a natural choice is to derive an effective equation that governs the dynamics of the volume of the universe, coupled to the scalar field.', '1602.05881-2-144-0': 'This can be done in a straightforward fashion in this case starting from the equations of motion for [MATH] obtained in the previous section and relating the spatial volume to the [MATH].', '1602.05881-2-144-1': 'By using the massless scalar field [MATH] as a relational clock, the resulting equations of motion for [MATH] can be compared to the Friedmann equations of cosmology, which are presented in the Appendix [REF].', '1602.05881-2-145-0': 'The quantity of interest here is the total volume of the universe in the condensate state, at a given moment of the relational time [MATH], [EQUATION] where [MATH] is the eigenvalue of the volume operator in canonical loop quantum gravity acting on an equilateral (as defined in Sec. [REF]) four-valent spin network node in the representation [MATH].', '1602.05881-2-145-1': '(Clearly, it follows from the definition of equilateral spin network nodes that [MATH] is the largest eigenvalue of the LQG volume operator possible for a node with all [MATH].)', '1602.05881-2-145-2': 'Note that the scaling mentioned here is approximate, and for a detailed analysis it would be necessary to explicitly calculate [MATH] for each [MATH].', '1602.05881-2-145-3': 'However, this will not be necessary here.', '1602.05881-2-146-0': 'A technical comment is also in order here.', '1602.05881-2-146-1': 'The LQG volume operator depends on the Barbero-Immirzi parameter [MATH], which only appears in spin foam models after the simplicity constraints have been imposed.', '1602.05881-2-146-2': 'In the GFT models based on spin foam models, the simplicity constraints are imposed in the interaction term in the GFT action, whose effect in the equations of motion has been assumed to be negligible.', '1602.05881-2-146-3': 'However, an operator in GFT can only be interpreted as a geometric operator after simplicity has been imposed.', '1602.05881-2-146-4': 'This is why it is important to remember that we are not ignoring the effect of the interaction term but instead we are considering the case where the contribution of the interaction term to the equations of motion is negligible compared to that of the kinetic terms.', '1602.05881-2-146-5': 'The interaction term is nonetheless present and imposes simplicity, but its contribution to the equations of motion of the condensate wave function is negligible and can be ignored.', '1602.05881-2-147-0': 'Now, given [REF], and using the notation of Sec. [REF], [EQUATION] and [EQUATION]', '1602.05881-2-147-1': 'Both [MATH] and [MATH] depend also on the [MATH] interaction term in the equations of motion, but the contribution from the interaction term is assumed to be subdominant in the Gross-Pitaevskii approximation and therefore we neglect these terms here.', '1602.05881-2-148-0': 'From the equations above it follows immediately that the generalised Friedmann equations in terms of the relational time [MATH] are given by [EQUATION] and [EQUATION]', '1602.05881-2-148-1': 'These effective Friedmann equations for the GFT condensate include the correct classical limit (i.e., they reproduce the standard Friedmann equations of general relativity, justifying their name), as shall be shown in Sec. [REF], as well as some quantum corrections coming from the microscopic GFT theory.', '1602.05881-2-148-2': 'Interestingly, some of these corrections have a clear geometric meaning, which shall be discussed shortly.', '1602.05881-2-148-3': 'From these equations, it is possible to solve for the dynamics of the total volume, given some initial state [MATH] at an initial time [MATH].', '1602.05881-2-149-0': 'An important point here is that, for the energy density of the massless scalar field, which is defined in terms of the expectation values of scalar field momentum and volume operators as [EQUATION] to be non-zero, at least one of the [MATH] must be non-zero', '1602.05881-2-150-0': 'The condition that at least one of the [MATH] be non-zero is necessary for the relational dynamics to be well-defined, and also to ensure that the homogeneous and isotropic space-time is an FLRW space-time, not the vacuum Minkowski space-time.', '1602.05881-2-151-0': 'This restriction has important consequences.', '1602.05881-2-151-1': 'Obviously, the condition that at least one of the [MATH] be non-zero is a necessary (although not sufficient) condition for the existence of solutions with a good cosmological interpretation, and also for the consistency of the relational description in the first place.', '1602.05881-2-151-2': 'On the other hand, this is not in itself a necessary condition for the mathematical consistency of the condensate dynamics.', '1602.05881-2-151-3': 'This means that there may be solutions which do not satisfy this condition, but are still mathematically well-defined and within the regime of validity of the condensate hydrodynamics we are studying.', '1602.05881-2-151-4': 'Therefore, this is an additional requirement beyond the assumptions for a condensate which is necessary for the condensate state to be interpreted as a cosmological space-time.', '1602.05881-2-152-0': 'An open question is whether setting all [MATH] (but still having large [MATH]) gives Minkowski space, in which case the condensate state would correspond to a large space-time although there would be no relational dynamics.', '1602.05881-2-152-1': 'We comment further on the vacuum limit in Sec. [REF].', '1602.05881-2-153-0': 'Requiring that the energy density of the massless scalar field be non-vanishing has a very important consequence: since at least one [MATH] must be non-zero to have a solution that can be interpreted as a cosmological space-time, it follows from [REF] that at least one [MATH] will always remain greater than zero.', '1602.05881-2-153-1': 'In turn, since [MATH], it follows that [MATH] will always remain non-zero.', '1602.05881-2-153-2': 'Therefore, we find that for all cosmological solutions, the volume will never become zero.', '1602.05881-2-154-0': 'In this way, the big-bang and big-crunch singularities of classical FLRW space-times that occur generically in general relativity are resolved in the Gross-Pitaevskii GFT condensate states studied here.', '1602.05881-2-154-1': 'The equation of motion for [MATH] [REF] clearly shows that the individual [MATH] will reach a minimal value at which point they will bounce (and it is clear that there is only a single bounce since [MATH] has only one turning point), and thus the cosmological space-time that emerges from the GFT condensate state is that of a bouncing FLRW space-time.', '1602.05881-2-154-2': 'Note that the bounce occurs when the [MATH] are relatively small, at which time the interactions are weakest, and therefore the near-bounce regime is where the Gross-Pitaevskii approximation used here can be most trusted (for the GFT actions considered here for which an appropriate mesoscopic regime exists); for these GFT actions, the approximation that interactions are small will only fail at large volumes far from the bounce.', '1602.05881-2-155-0': 'In order to see exactly how the singularity is resolved, and better understand the nature of the quantum effects causing this resolution, it is necessary to solve our modified Friedmann equations for [MATH] for some initial conditions.', '1602.05881-2-155-1': 'Unfortunately, it is difficult to provide an exact solution to these equations of motion for generic initial conditions, but there are two special cases when an explicit solution can be found.', '1602.05881-2-156-0': '## Classical Limit', '1602.05881-2-157-0': 'As already mentioned, the momentum of the scalar field, defined as the expectation value of the operator [REF] in the condensate state, is given by [MATH] and therefore [MATH] is a conserved quantity: this is exactly the continuity equation for a massless scalar field in an FLRW space-time.', '1602.05881-2-157-1': 'Therefore, the only other requirement in order to verify that the correct semi-classical limit is obtained is to ensure that the correct Friedmann equation is recovered.', '1602.05881-2-158-0': 'The classical limit of the generalised Friedmann equations is obtained when the Hubble rate is small compared to the inverse Planck time, and this will occur at sufficiently large volumes, i.e., when [MATH] and [MATH] (note that the semi-classical limit is not the limit of large volume, but of small space-time curvature; nonetheless, the space-time curvature decreases as the space-time expands and therefore the dominant term in the Friedmann equation at large volumes is also the dominant term when the space-time curvature is small).', '1602.05881-2-158-1': 'As shall be seen in the next section, the terms containing [MATH] and [MATH] can be understood as quantum corrections.', '1602.05881-2-159-0': 'In this limit, the generalised Friedmann equations become [EQUATION] and [EQUATION]', '1602.05881-2-159-1': 'We immediately see from these equations that, in order to recover the classical Friedmann equations of general relativity in terms of the relational time [MATH], which are given in Appendix [REF], (in this specific context where the FLRW space-time emerges as a condensate of isotropic GFT quanta) it is necessary to identify [MATH] for all [MATH].', '1602.05881-2-159-2': 'For these values of [MATH], the GFT condensate dynamics reproduce the classical Friedmann equations of general relativity.', '1602.05881-2-159-3': '(As an aside, note that while it may be possible, at a specific relational instant [MATH], to choose a different set of values for [MATH] that also gives the correct limit, this identification will not be preserved by the dynamics and hence the correct classical Friedmann equations would in this case only be recovered in a small neighbourhood of relational time around [MATH].)', '1602.05881-2-160-0': 'The condition that [MATH] is a requirement on the form of the terms [MATH] and [MATH] that are determined by the GFT action: if [MATH] for some [MATH], then it follows that the correct Friedmann equations are not recovered in the classical limit.', '1602.05881-2-160-1': 'Note also that this should be understood as a definition of [MATH] which arises as a hydrodynamic parameter and it is thus a function of the microscopic GFT parameters, and not as an interpretation of the microscopic parameters.', '1602.05881-2-160-2': 'This is an important conceptual point since this identification has no reason to be valid in a generic regime of the dynamics (e.g., for non-condensate GFT states) and may be different in other settings.', '1602.05881-2-161-0': 'So, if all [MATH], then the generalised Friedmann equations of the GFT condensate become, in the classical limit, [EQUATION] which are exactly the Friedmann equations of general relativity for a spatially flat FLRW space-time with a massless scalar field [MATH], used as a relational time (see Appendix [REF] for details).', '1602.05881-2-162-0': 'The solution to these equations of motion is the standard one of classical general relativity, [EQUATION] as expected, with the sign in the exponent depending on whether the universe is expanding or contracting, and [MATH] depending on the initial conditions.', '1602.05881-2-163-0': '## Single Spin Condensates', '1602.05881-2-164-0': 'The other case where the equations of motion for [MATH] can be solved exactly, and for generic initial conditions, is when only one [MATH] is non-zero, which corresponds to a condensate wave function that is very sharply (infinitely) peaked in [MATH], [EQUATION]', '1602.05881-2-164-1': 'Then the sum over [MATH] in all of the expressions trivializes and an exact solution can be found which includes quantum corrections.', '1602.05881-2-165-0': 'This assumption mirrors the situation that is thought to be relevant in LQC, where there is also the extra assumption that the underlying LQG state consists of a graph with a very large number of nodes and links, and that the spins on all of the links are identical (often chosen to be [MATH]).', '1602.05881-2-165-1': 'It follows that in the LQC picture, a cosmological space-time expands or contracts by modifications to the combinatorial structure of the spin network that consist of adding or removing nodes, rather than by changing the spin labels on the spin network; this is analogous to the volume dynamics extracted from the underlying GFT model where changes in [MATH] correspond to changes in the number of GFT quanta, rather than transitions between GFT quanta coloured by different spin representations.', '1602.05881-2-165-2': "In the limiting case [REF] considered here, the volume dynamics is entirely dictated by the number of GFT quanta via [MATH]; this is essentially identical to the heuristic interpretation suggested by the LQC 'improved dynamics' relating LQC to the underlying LQG spin networks.", '1602.05881-2-165-3': 'Finally, note that the missing connectivity information in the simple GFT condensates considered here does not play any role in LQC either.', '1602.05881-2-166-0': 'Of course, if only one mode [MATH] contributes to the effective dynamics, then the correct classical limit requires a milder condition on the microscopic dynamics to reproduce the classical Friedmann equation with respect to the more general case considered in the previous subsection, namely that [MATH] (and there are no requirements on the other [MATH]).', '1602.05881-2-167-0': 'Therefore, we set [MATH] in the following so that the correct classical limit is ensured.', '1602.05881-2-167-1': 'Now, since [MATH] for all [MATH], only [MATH] is non-zero and [EQUATION]', '1602.05881-2-167-2': 'Given [REF], the total volume is simply given by [EQUATION] and the first modified Friedmann equation simplifies to [EQUATION] which can be rewritten, using the relation for the energy density of a massless scalar field [MATH], as [EQUATION] with [MATH].', '1602.05881-2-167-3': 'It is clear that the first term is the classical limit, and that the second term is a quantum gravity correction.', '1602.05881-2-167-4': 'In addition, from scaling arguments (the Friedmann equation must be invariant under [MATH] and [MATH]) and dimensional analysis, it follows that [MATH].', '1602.05881-2-167-5': 'Since [MATH], it follows that [MATH] and the third term is also a quantum gravity correction to the classical Friedmann equation.', '1602.05881-2-168-0': 'Similarly, the second modified Friedmann equation is [EQUATION]', '1602.05881-2-168-1': 'Since the last term here also vanishes in the limit of [MATH], it is also to be understood as coming from quantum gravity corrections.', '1602.05881-2-169-0': 'By comparing these modified Friedmann equations for the isotropic GFT condensate [REF], under the condition [REF], to the effective Friedmann equations of loop quantum cosmology given in Appendix [REF], it is clear that they are identical in the case that [MATH], while there are differences of order [MATH] if [MATH] is not zero.', '1602.05881-2-169-1': 'Therefore, the condensate states considered here reproduce, within the full GFT theory, a similar type of quantum-corrected cosmological dynamics as the ones found in LQC.', '1602.05881-2-169-2': 'In fact, even the form of [MATH] obtained here is closely analogous to that of the critical energy density of LQC, which according to the heuristic relation between LQC and LQG also goes as [MATH].', '1602.05881-2-170-0': 'These results therefore provide strong support for the qualitative results of LQC, obtained through the assumption of a particular heuristic relation between full loop quantum gravity and the minisuperspace models of LQC.', '1602.05881-2-170-1': 'Remarkably, very similar effective dynamics arise for the GFT condensate states that mirror the heuristic relation that lies at the heart of LQC.', '1602.05881-2-171-0': 'A few differences must be stressed as well.', '1602.05881-2-171-1': 'First, GFT condensate states provide a significantly more general framework than the specific LQG states used to motivate various constructions in LQC; for example, it is possible and even straightforward to consider states with contributions coming from many different [MATH], as is clear from the discussion in Sec. [REF].', '1602.05881-2-171-2': "Another important difference is the presence of a new quantum number [MATH] which corresponds to an 'energy' of the GFT condensate in the spin [MATH] (with respect to the relational time [MATH]), but whose fundamental geometric interpretation remains, for now, unclear.", '1602.05881-2-171-3': 'However, the effect of [MATH] on the dynamics is relatively simple: if [MATH] then the bounce occurs at an energy density greater than [MATH], on the other hand, if [MATH] then the bounce occurs at an energy density less than [MATH].', '1602.05881-2-171-4': 'Importantly, no matter the value of [MATH], a bounce similar to that found in LQC will always occur: the resolution of the big-bang and big-crunch singularities in GFT condensate cosmology is robust to changes in [MATH] (of course, so long as the regime [MATH] falls within the regime of approximations used to obtain the above cosmological dynamics from the full microscopic dynamics).', '1602.05881-2-172-0': 'Note also that these modified Friedmann equations (for single-spin condensates with [MATH]) are also found to arise in a different quantum gravity theory, namely brane-world cosmology scenarios with an extra time dimension [CITATION].', '1602.05881-2-173-0': 'A final comment is in order regarding the space-time interpretation of the GFT condensate.', '1602.05881-2-173-1': 'If there are only a few GFT quanta, it is doubtful that it is possible to associate a continuum space-time to such a quantum state, because the approximation between GFT condensate states and continuum geometries should be expected to fail.', '1602.05881-2-173-2': 'Moreover, there is no reason to expect that the hydrodynamic approximation used here to extract the effective cosmological dynamics would still be viable if only a few fundamental GFT quanta are excited from the Fock vacuum.', '1602.05881-2-173-3': '(Although note that the hydrodynamic equations of motion would formally still apply even in this case.)', '1602.05881-2-173-4': 'Thus, a large number of GFT quanta are necessary for the approximations used here to be valid.', '1602.05881-2-173-5': 'Note however that this is not necessarily problematic in a bouncing cosmological context: as is well known in LQC, the bounce occurs when the space-time curvature nears the Planck curvature scale, not when the volume of the universe is comparable to the Planck volume (another way to put it is that it is the Planck energy density that sets the scale of the bounce, not the Planck volume).', '1602.05881-2-173-6': 'The same is true here: the bounce occurs when the space-time curvature is large, and therefore the space-time can (and typically does) bounce at volumes that are much larger than the Planck volume even though the space-time curvature is Planckian.', '1602.05881-2-173-7': 'When this is the case, there will be a large numbers of fundamental GFT quanta even during the bounce, and therefore, modulo other considerations, there will be room for a space-time interpretation of the GFT condensate at all times, including the bounce point.', '1602.05881-2-174-0': '# The Vacuum Limit', '1602.05881-2-175-0': 'In principle, it is also an interesting problem to study the effective cosmology of GFT condensates, and of quantum gravity more generally, in absence of matter fields.', '1602.05881-2-175-1': 'However, there are several difficulties in doing so (some of which have already been noticed in LQC [CITATION]).', '1602.05881-2-176-0': 'In the present context, the vacuum limit of the FLRW space-time, which should give Minkowksi space, is non-trivial for a number of reasons.', '1602.05881-2-176-1': "The main problem is that, in the absence of a matter field, there is no longer a relational clock available and therefore the issue of defining and using diffeomorphism invariant observables (often subsumed in the label 'problem of time') in quantum gravity must be faced head-on, as one should not expect to have any reliable physics otherwise.", '1602.05881-2-176-2': "In particular, the geometric interpretation of the condensate wave function [MATH] is not clear: if [MATH] is to be interpreted as a three-dimensional spatial slice at the relational 'instant of time' [MATH], then it is not immediately obvious how to extract a four-dimensional space-time, or even a reliable phase space interpretation, from [MATH].", '1602.05881-2-176-3': 'This is because there is no control over which spatial slice it refers to, and any apparent physical feature could be attributed instead to some peculiar slicing of spacetime.', '1602.05881-2-177-0': 'Even starting from the Friedmann equation [REF], it is not obvious how to obtain the vacuum limit.', '1602.05881-2-177-1': 'While the right-hand side of [REF] is easily handled by simply setting [MATH], the left-hand side is much trickier, since the derivative with respect to [MATH] becomes ill-defined.', '1602.05881-2-177-2': 'The fact that this is a singular limit is obvious from the classical theory, where the Friedmann equation is simply [MATH]: there is no straightforward way to take the vacuum limit (at least, not without using the relation [REF] which holds in general relativity but cannot be assumed blindly for a GFT condensate state).', '1602.05881-2-178-0': 'Furthermore, we already know that the Gross-Pitaevskii approximation we use is only the simplest truncation available, which may capture some relevant features of the emergent continuum spacetime, as we have shown, but certainly neglects a lot of important aspects of the theory (e.g., connectivity information).', '1602.05881-2-178-1': 'While one can argue that these simple condensates encode enough information to reproduce the dynamics of homogeneous cosmologies, how much is actually captured can only be checked a posteriori, by studying the emergent cosmological dynamics, as we have done.', '1602.05881-2-178-2': 'Then, recovering the Friedmann equations in an appropriate semi-classical limit provides strong evidence that in fact neglecting some of the microsopic details like the connectivity information of the spin network nodes was a reasonable approximation.', '1602.05881-2-178-3': 'However, in Minkowski space there is no dynamical equation that can be recovered in order to provide this type of check.', '1602.05881-2-179-0': 'An additional difficulty with the Gross-Pitaevskii condensate wave function for a vacuum GFT model of EPRL type -in which case there is no derivative operator on the group manifold in the kinetic term of the GFT action- is that the interactions cannot be negligible.', '1602.05881-2-179-1': 'This is obvious from the equation of motion for the (isotropic) condensate wave function for a vacuum GFT model, which has the general form [EQUATION]', '1602.05881-2-179-2': 'Clearly, for this equation to hold, the amplitude of the interaction term must be the same as that of the linear term.', '1602.05881-2-179-3': 'Therefore, the interactions in this case are necessarily large, so we have to work exactly in the regime where the Gross-Pitaevskii approximation cannot be justified from the microscopic theory.', '1602.05881-2-179-4': 'For the vacuum case, it is then anyway necessary to consider more complicated condensate states, perhaps such as those constructed in [CITATION].', '1602.05881-2-180-0': 'One can still learn some interesting lessons about GFTs and spin foam models from studying these equations.', '1602.05881-2-180-1': 'For example, any solution of these equations is necessarily non-perturbative in the GFT coupling constant (if only one interaction is included), and thus it is not reproducible (nor is its putative physics) via the spin foam expansion.', '1602.05881-2-180-2': 'This is in fact the case for all the classical solutions to GFT models that have been studied to date (see e.g., [CITATION]).', '1602.05881-2-180-3': 'However, for the reasons we have explained above, it is unclear how these formal insights can be turned into physical ones.', '1602.05881-2-181-0': '# Discussion', '1602.05881-2-182-0': 'In this paper, we started from a rather general definition of the EPRL spin foam model for the microscopic dynamics of quantum geometry in the GFT framework (completing the spin foam definition).', '1602.05881-2-182-1': 'We parametrized some of the ambiguities entering the definition of the model, by working with general kinetic kernels, and with the aim of constraining these ambiguities by studying the effective cosmological dynamics they give rise to.', '1602.05881-2-182-2': 'While we focused on one specific model, we expect many of our results to hold more generally, and there are indeed several indications that this is the case.', '1602.05881-2-183-0': 'Then, we explained how to couple quantum geometry with a free massless scalar field and gave the definition of the corresponding extension of the GFT model.', '1602.05881-2-183-1': 'One of the key steps here was enforcing at the GFT level the basic symmetries characterizing the scalar field path integral in the free massless case and then performing a small derivative expansion keeping only the lowest orders.', '1602.05881-2-183-2': 'This gives a relatively simple model that we are confident captures the correct semi-classical continuum physics.', '1602.05881-2-183-3': 'At the same time, this allows us to parametrize some ambiguities by working with more general coefficients encoding the matter-geometry coupling.', '1602.05881-2-183-4': 'This is our first new result.', '1602.05881-2-183-5': 'Beside the intrinsic interest of coupling matter and geometry in a GFT model, it was pivotal for the study of the effective cosmological dynamics since it allows for a formulation of the evolution of the universe in terms of purely diffeomorphism invariant observables, as the theoretical context demands, by using the scalar field as a relational clock; this is also a traditional strategy in quantum cosmology.', '1602.05881-2-183-6': 'Interestingly, the scalar field degrees of freedom also enter the GFT action in the same way in which a local time variable would do in standard quantum field theory.', '1602.05881-2-184-0': 'The next step was to implement a restriction to purely isotropic degrees of freedom for the quantum geometric condensate states to be used for the extraction of cosmological dynamics.', '1602.05881-2-184-1': 'These already implement a notion of homogeneity, as discussed in Sec. [REF], and are thus adapted for the simplest cosmological space-times.', '1602.05881-2-184-2': 'We define what it means to restrict the condensate wave function to isotropic configurations by analysing the simplicial geometry of the GFT states.', '1602.05881-2-184-3': 'Indeed, pure volume information can be most directly extracted by the use of quantum states whose quanta can be described by equilateral tetrahedra.', '1602.05881-2-184-4': 'This is our second, intermediate, result.', '1602.05881-2-185-0': 'Armed with the resulting simple condensate states, their effective cosmological dynamics follow directly from the full quantum dynamics of the model, in the GFT analogue of a Gross-Pitaevskii approximation (amounting to the simplest mean field treatment), and correspond to the hydrodynamics of the GFT model.', '1602.05881-2-185-1': 'Despite the simplicity of the GFT condensate states, the emerging cosmological dynamics are extremely interesting.', '1602.05881-2-185-2': 'The equations of motion are non-linear with respect to the condensate wave function, and they can be interpreted in analogy with a quantum cosmology wave function.', '1602.05881-2-185-3': 'Thus, this equation has the form of an extension of a quantum cosmology dynamics as used, e.g., in loop quantum cosmology, and with analogous geometric variables.', '1602.05881-2-185-4': 'Due to the isotropic restriction on the condensate wave function, the actual form of the equation simplifies greatly and it can be manipulated in a straightforward manner.', '1602.05881-2-186-0': 'By defining appropriate relational volume observables, we obtained two equations governing the dynamics of the volume of the universe as a function of the massless scalar field, and depending on the particular solution of the effective GFT condensate hydrodynamics through the values of [MATH] and [MATH].', '1602.05881-2-186-1': 'These two new quantities correspond to state-dependent conserved charges of the system (in the Gross-Pitaevskii approximation, and when interactions are subdominant).', '1602.05881-2-186-2': 'The two equations take the form of modified Friedmann equations and are our third main result.', '1602.05881-2-187-0': 'Their interpretation is confirmed by their subsequent analysis.', '1602.05881-2-187-1': 'First of all we have shown that in an appropriate semi-classical limit they reduce to the standard Friedmann equations of general relativity.', '1602.05881-2-187-2': 'This is true for any state chosen (among the solutions of the GFT condensate dynamics) and at all times (i.e., for all values of the scalar field) if and only if precise conditions on the initial kinetic kernels of the microscopic GFT models are satisfied.', '1602.05881-2-187-3': "These conditions also provide a definition of Newton's constant [MATH] as a function of the microscopic parameter of the theory, as expected in such a hydrodynamics context.", '1602.05881-2-187-4': 'This is our fourth main result.', '1602.05881-2-188-0': 'Beyond this classical approximation, the theory provides a number of quantum corrections to the classical Friedmann dynamics.', '1602.05881-2-188-1': 'All of them can be computed from first principles starting from the microscopic GFT theory.', '1602.05881-2-188-2': 'We have shown that they share one common feature: the cosmological singularities predicted by the classical theory are generically avoided at least for Gross-Pitaevskii condensate states and a quantum gravity bounce takes their place.', '1602.05881-2-188-3': 'This is our fifth main result.', '1602.05881-2-189-0': 'The scenario suggested by the GFT condensate ansatz for cosmological dynamics is therefore similar to the one found in loop quantum cosmology, which is therefore corroborated by the results found here.', '1602.05881-2-189-1': 'In fact, further evidence in support of LQC is obtained by considering GFT condensate states chosen to only have support on one spin [MATH] - this is the type of state that LQC suggests is relevant for the cosmological sector of LQG.', '1602.05881-2-189-2': 'For this type of condensate state, the resulting effective cosmological dynamics also simplify greatly and can be analyzed in detail, with the result that the effective dynamics are extremely similar to the LQC effective Friedmann equations.', '1602.05881-2-189-3': 'This is our last main result.', '1602.05881-2-190-0': 'Obviously, several approximations were required in order to obtain the results summarized above from the full theory.', '1602.05881-2-190-1': 'As a consequence, these results can only be trusted so long as the GFT state remains within the domain of validity of these approximations.', '1602.05881-2-190-2': 'Let us point them out, for completeness.', '1602.05881-2-191-0': 'First, aside from the choice of the microscopic GFT model itself, we used a small derivative approximation with respect to the scalar field, which is then assumed to be slowly varying throughout the evolution of the universe.', '1602.05881-2-192-0': 'Second, even assuming that the relevant quantum states to study cosmological dynamics within the full GFT theory are condensate states, we used a very simple type of such states, i.e., coherent states of the GFT field operator (i.e., the Gross-Pitaevskii condensate states).', '1602.05881-2-192-1': 'This is a drastic approximation; generic condensate states are much richer.', '1602.05881-2-192-2': 'In particular, we have neglected to include any information about the connectivity of the fundamental spin network degrees of freedom.', '1602.05881-2-192-3': 'While the connectivity information is not required in order to characterize homogeneous and isotropic space-times, this nevertheless represents an important restriction.', '1602.05881-2-192-4': 'For example, it implies that the only information about the topology of the universe is the one that can be inferred from the effective cosmological dynamics itself and that the topology of the space-time cannot be constrained at level of the quantum states.', '1602.05881-2-192-5': 'Still, as is well known from the study of real Bose condensates, even these drastic simplifications in the choice of the condensate states that are considered can provide a wealth of information about the relevant physics.', '1602.05881-2-193-0': 'The restriction to isotropic degrees of freedom is of course an additional approximation.', '1602.05881-2-193-1': 'While it does restrict the GFT condensate states to only depend on the degrees of freedom of interest, it has the disadvantage of being imposed as a starting point and for this reason it may prevent the understanding of some aspects of cosmological dynamics and in particular it makes it impossible to study how an isotropic configuration may be generated from a more general initial state.', '1602.05881-2-194-0': 'Given these approximations in the definition of the GFT model and the choice of the condensate states, we then extracted the effective cosmological dynamics directly from the full quantum dynamics of the microscopic GFT theory.', '1602.05881-2-194-1': 'However, the equation we used is the first Schwinger-Dyson equations, corresponding to the classical GFT equations of motion, and is the simplest formulation of the hydrodynamics of the theory.', '1602.05881-2-194-2': 'Ignoring the other Schwinger-Dyson equations is a truncation of the full quantum dynamics.', '1602.05881-2-195-0': 'Having obtained the equations of motion for the condensate wave function, we further assumed that the GFT interaction terms (corresponding to the non-linear terms in the condensate hydrodynamics) in this equation are subdominant compared to the GFT kinetic terms.', '1602.05881-2-195-1': 'This is the case for small values of the coupling constants (which are subject to the renormalization group flow) and for state densities that are not too large.', '1602.05881-2-195-2': 'This is the direct GFT counterpart of the dilute gas approximation used for standard Bose liquids in the Gross-Pitaevskii approximation.', '1602.05881-2-195-3': 'In fact, this approximation is required for consistency with the previous approximation that connectivity information is negligible, since stronger interactions would imply stronger correlations between the GFT quanta, in which case it would become necessary to take into account the connectivity between the spin network nodes and to work with more complicated condensate states.', '1602.05881-2-196-0': 'A last type of approximation was not explicitly stated since it does not enter directly any of the calculations and it is not in fact required by the formalism itself.', '1602.05881-2-196-1': 'This is the condition that curvatures associated to individual GFT quanta are not too big and that their individual contribution to the total volume is rather small.', '1602.05881-2-196-2': 'This is required for a straightforward interpretation of the corresponding quantum states as associated to a continuum geometry.', '1602.05881-2-196-3': 'At this point, it is not yet clear how strictly this condition should be enforced and at what point it fails.', '1602.05881-2-196-4': 'A more careful inspection of explicit solutions may provide some insight in this direction.', '1602.05881-2-197-0': 'Within the regime of the full theory in which the above approximations hold, our results hold as well and, we believe, provide interesting insights into the cosmological sector of LQG and GFT.', '1602.05881-2-198-0': 'This being said, there remain many open questions to be addressed in order to further corroborate these results and also to develop the path they open.', '1602.05881-2-198-1': 'We shall conclude by outlining some of the open issues and directions for future research.', '1602.05881-2-199-0': 'It is important to repeat the analysis done here for the GFT based on the EPRL spin foam model for another GFT based on a different spin foam model for 4D quantum gravity.', '1602.05881-2-199-1': 'We do not anticipate very different results, though, due to the fact that the main difference between the different GFT models will be encoded in the GFT interaction terms and we have assumed their contributions to the equations of motion of the condensate wave function to be subdominant.', '1602.05881-2-199-2': 'Therefore, a different interaction term would only give different types of quantum corrections to the general cosmological dynamics we have found.', '1602.05881-2-199-3': 'Moreover, we expressed the kinetic terms of the GFT action in a rather general form in order to minimize the reliance on the details of the microscopic model.', '1602.05881-2-199-4': 'However, any difference would be worth investigating.', '1602.05881-2-199-5': 'For example, it is not obvious that models enforcing a weaker relation between [MATH] and [MATH] data than the EPRL model would give an effective equation as simple as the one we obtained in the isotropic restriction, and this may offer interesting insights on the role of Lorentzian structures at the effective cosmological level.', '1602.05881-2-200-0': 'Also concerning the choice of the fundamental GFT model, a more detailed analysis of the coupling of the scalar field to (quantum) geometry is needed involving a careful analysis of the Feynman amplitudes of the coupled GFT model, and in particular it is necessary to go beyond the approximation of small derivatives.', '1602.05881-2-200-1': 'One would like the GFT model coupled to a free massless scalar field to define a proper simplicial path integral for discrete gravity coupled to a scalar field discretized on the same cellular complex corresponding to the GFT Feynman diagram.', '1602.05881-2-200-2': 'This analysis should also suggest how to go beyond the case of a free massless scalar field.', '1602.05881-2-200-3': 'While a scalar field with a non-trivial potential would not be a good global clock, it would be interesting to understand how generic scalar fields should be treated in GFT models.', '1602.05881-2-200-4': 'Moreover, it would be interesting to see how the new terms encoding its potential at the level of the GFT action modify its role as a time variable for the same field theory.', '1602.05881-2-200-5': 'Going even further, it would also be good to include other types of matter fields like Maxwell and Dirac fields, and also to better understand the vacuum case.', '1602.05881-2-201-0': 'Another important next step is to perform a more detailed analysis of the solutions to the cosmological equations we have obtained.', '1602.05881-2-201-1': 'It is only by having at hand explicit solutions of the dynamics that we will be able to study the detailed evolution of geometric observables (and specifically the spatial volume of the space-time as well as the relational Hubble rate) and obtain a detailed picture of the evolution of the universe as dictated by the fundamental GFT.', '1602.05881-2-201-2': 'In addition, this analysis will also allow us to check the exact regimes of validity of the various approximations we employed.', '1602.05881-2-202-0': 'In particular, a careful analysis of this type will provide a better understanding of the quantum corrections to the classical Friedmann equations as predicted by the GFT, and in particular of those coming from the GFT interaction terms (which could perhaps be expressed in terms of geometric observables).', '1602.05881-2-202-1': 'Clearly, the interaction terms will generate quantum gravity corrections to the classical equations of general relativity which will be interesting to study, and in addition, as already explained, the interaction terms encode the differences between different microscopic definitions of the quantum gravity dynamics (i.e., different spin foam models or GFT actions) can be elucidated.', '1602.05881-2-202-2': 'So this is a further reason to study the effect of the interaction term in the GFT action on the emergent cosmological dynamics.', '1602.05881-2-202-3': 'It would also be interesting to derive an effective field theory that captures the same corrections to classical general relativity for homogeneous and isotropic space-times, since this would be a nice way to characterize the type of effective continuum dynamics of geometry that is predicted by the GFT models.', '1602.05881-2-203-0': 'Another important open question is to understand how to encode spatial curvature in the effective dynamics of GFT condensate states.', '1602.05881-2-203-1': 'More specifically, the Friedmann equations recovered for the condensate states studied in this work are, in the classical limit, those for a spatially flat FLRW space-time.', '1602.05881-2-203-2': 'How might it be possible for GFT condensate states to give the Friedmann equation for FLRW space-times with non-vanishing spatial curvature?', '1602.05881-2-203-3': 'There are two obvious possibilities: either by studying more involved condensate states, or by modifying the GFT dynamics.', '1602.05881-2-203-4': 'Let us comment more on the first possibility: perhaps it is necessary to encode the spatial curvature in the connectivity information of the condensate state.', '1602.05881-2-203-5': '(This is consistent with -but certainly not implied by- the fact that in the condensate states considered here where the connectivity information is discarded, the spatially flat FLRW space-time is automatically recovered.)', '1602.05881-2-203-6': 'If this is the case, then in order to properly answer this question it will be necessary to study GFT condensate states where the connectivity information is included in the analysis, e.g., states of the type constructed in [CITATION].', '1602.05881-2-204-0': 'The isotropic restriction on the condensate wave functions could also be lifted.', '1602.05881-2-204-1': 'This would allow for a larger class of operators on the condensate wave function, and could potentially be used in order to study anisotropic (and homogeneous) space-times.', '1602.05881-2-204-2': 'In that case, removing the isotropic restriction could perhaps also give some insight into the transition between isotropic and anisotropic regimes in full quantum gravity.', '1602.05881-2-205-0': 'Finally, there remain two related important open problems concerning extensions of the study of cosmology using GFT condensate states.', '1602.05881-2-205-1': 'The first is to tackle the regime of the dynamics where the GFT interactions cease to be subdominant and have to be included explicitly in the effective cosmological dynamics.', '1602.05881-2-205-2': 'This is not so much a technical issue (since it would be rather straightforward to include the interaction terms in the equations derived in this paper and solve them in the case when the interactions are large), but a conceptual one since the Gross-Pitaevskii approximation fails to be justified by the microscopic theory when interactions become important.', '1602.05881-2-205-3': 'In fact, in the regime where the interaction term is no longer negligible it will likely be necessary to choose a different class of condensate states and, in particular, it may be necessary to use condensate states which take fully into account the connectivity of the underlying fundamental spin network states.', '1602.05881-2-205-4': 'This can be done, but the degree of complexity of the analysis necessary in order to extract the effective cosmological dynamics will be considerably higher than what was needed here (among other complications, a number of graph theoretic issues will need to be addressed).', '1602.05881-2-206-0': 'At a more physical level, the second longer-term direction of research will be to develop a generalization of GFT condensate cosmology in order to include perturbations over the condensate state representing an homogeneous universe, as this would allow for a treatment of cosmological perturbations.', '1602.05881-2-206-1': 'A first step would be to reproduce in the context of the full GFT formalism the basic set-up of the analysis of cosmological perturbations performed in loop quantum cosmology; in particular, the separate universe picture [CITATION] appears rather straightforward to implement.', '1602.05881-2-206-2': 'The real goal, however, is more ambitious: to develop cosmological perturbation theory from first principles in a fundamental theory of quantum gravity, in this case GFTs.', '1602.05881-2-207-0': '# Relational Dynamics in Cosmology', '1602.05881-2-208-0': 'This appendix contains a brief review of the Hamiltonian formalism applied to a spatially flat FLRW space-time, and the resulting Friedmann equations in general relativity and loop quantum cosmology.', '1602.05881-2-208-1': 'As in the main text, we consider the case where the matter content is a minimally coupled massless scalar field [MATH].', '1602.05881-2-209-0': 'In order to avoid divergent integrals in the Hamiltonian formulation, we assume that the topology of the spatial surfaces is [MATH] and that the volume of the 3-torus with respect to the coordinates [MATH] is 1.', '1602.05881-2-210-0': '## General Relativity', '1602.05881-2-211-0': 'The line element for a flat FLRW space-time is [EQUATION] with [MATH] being the lapse and [MATH] the scale factor, and the Hamiltonian constraint -coming from the Einstein-Hilbert action, given the line element [REF]- is given by [EQUATION] where [MATH] is the volume, [MATH] is the variable canonically conjugate to [MATH] with the Poisson bracket [MATH], and [MATH] is the momentum of the scalar field and satisfies [MATH].', '1602.05881-2-212-0': 'The equations of motion are easily derived from the Hamiltonian; the two that are necessary for our purposes are those for [MATH] and [MATH].', '1602.05881-2-212-1': 'For the massless scalar field, [EQUATION] which is clearly monotonic since [MATH] is a constant of motion as [MATH].', '1602.05881-2-212-2': 'Thus, the scalar field can act as a good relational clock.', '1602.05881-2-212-3': 'The dynamics of [MATH] is given by [EQUATION] showing that [MATH] is the Hubble rate in proper time.', '1602.05881-2-212-4': 'Then, using the vanishing of the Hamiltonian constraint [REF] and rewriting the time derivative in terms of the massless scalar field via [REF] gives the first Friedmann equation in terms of the relational clock [MATH], [EQUATION]', '1602.05881-2-212-5': 'It can be checked that the continuity equation, once the energy density and pressure have been expressed as [MATH], is simply [MATH] which is trivially satisfied since [MATH] is a constant of the motion and therefore the remaining relevant equation of motion is the second Friedmann equation obtained by differentiating [REF] with respect to [MATH], giving [EQUATION]', '1602.05881-2-212-6': 'The equations [REF] and [REF] are the two Friedmann equations for a spatially flat FLRW space-time expressed in terms of the relational clock [MATH], which can be directly compared to the Gross-Pitaevski equation of the condensate states studied in this paper.', '1602.05881-2-213-0': '## Loop Quantum Cosmology', '1602.05881-2-214-0': 'Using the non-perturbative quantization techniques of loop quantum gravity, in loop quantum cosmology (LQC) it has been possible to obtain a well-defined quantum theory for FLRW space-times where the big-bang and big-crunch singularities are resolved [CITATION].', '1602.05881-2-214-1': 'An important result in LQC (and in quantum cosmology in general) is that states that are semi-classical (i.e., sharply peaked around a classical solution) in the low curvature regime remain sharply-peaked throughout their evolution, for the reason that observables in the quantum cosmology are global quantities and hence are heavy degrees of freedom [CITATION].', '1602.05881-2-215-0': 'For these semi-classical states, to determine the main features of the quantum dynamics it is sufficient to calculate the evolution of the expectation values of only a few observables, for example the total volume [MATH] and the momentum of the scalar field [MATH].', '1602.05881-2-215-1': 'Interestingly, the dynamics of these observables, for sharply peaked states, are given by a simple modification to the Friedmann equations.', '1602.05881-2-215-2': 'These are called the LQC effective Friedmann equations, and in terms of proper time [MATH] (i.e., for [MATH]) they are [CITATION] [EQUATION] and [EQUATION] where [MATH] is the Hubble rate, a dot denotes a derivative with respect to proper time, and [MATH] is the critical energy density of LQC.', '1602.05881-2-215-3': 'From these equations it is clear that a bounce in the scale factor occurs when the energy density of the matter field equals the critical energy density [MATH], and hence in LQC a contracting universe bounces into an expanding universe, with a quantum gravity bridge linking the two classical branches of the space-time.', '1602.05881-2-216-0': 'Also, from these equations it can easily be checked that the continuity equation is unchanged, [EQUATION]', '1602.05881-2-216-1': 'Recall that for a massless scalar field, [MATH] and, as in the case of general relativity, the continuity equation reduces to [MATH].', '1602.05881-2-217-0': 'In order to make contact with the equations of motion for the GFT condensate, it is necessary to express the LQC effective Friedmann equations in terms of the relational clock [MATH].', '1602.05881-2-217-1': '(Note that the continuity equation [MATH] shows that [MATH] is a good global clock here.)', '1602.05881-2-217-2': 'This can be done via the relation [CITATION] [EQUATION] which is unchanged from the classical relation.', '1602.05881-2-218-0': 'Therefore, the LQC effective Friedmann equations in terms of the relational clock [MATH] are [EQUATION] and [EQUATION]', '1602.05881-2-218-1': 'Interestingly, when expressed in terms of the relational clock [MATH], only the first LQC effective Friedmann equation is modified from its classical form.', '1602.05881-2-218-2': 'This is simply due to what is effectively a choice of the lapse [MATH] when [MATH] is used as a relational clock.', '1602.05881-2-219-0': 'We would like to thank the participants of the "Cosmology from Quantum Gravity" workshop, and in particular Mairi Sakellariadou, Martin Bojowald, Jakub Mielczarek, Steffen Gielen and Andreas Pithis, for helpful discussions.', '1602.05881-2-219-1': 'This work was partially supported by the John Templeton Foundation through the grant PS-GRAV/1401.'}

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['1602.05881-1-44-0', '1602.05881-2-44-0'], ['1602.05881-1-44-1', '1602.05881-2-44-1'], ['1602.05881-1-44-2', '1602.05881-2-44-2'], ['1602.05881-1-212-0', '1602.05881-2-215-0'], ['1602.05881-1-212-1', '1602.05881-2-215-1'], ['1602.05881-1-212-2', '1602.05881-2-215-2'], ['1602.05881-1-212-3', '1602.05881-2-215-3'], ['1602.05881-1-111-0', '1602.05881-2-111-0'], ['1602.05881-1-111-1', '1602.05881-2-111-1'], ['1602.05881-1-29-0', '1602.05881-2-29-0'], ['1602.05881-1-29-1', '1602.05881-2-29-1'], ['1602.05881-1-29-2', '1602.05881-2-29-2'], ['1602.05881-1-29-3', '1602.05881-2-29-3'], ['1602.05881-1-24-0', '1602.05881-2-24-0'], ['1602.05881-1-24-1', '1602.05881-2-24-1'], ['1602.05881-1-24-2', '1602.05881-2-24-2'], ['1602.05881-1-24-3', '1602.05881-2-24-3'], ['1602.05881-1-15-0', '1602.05881-2-15-0'], ['1602.05881-1-15-1', '1602.05881-2-15-1'], ['1602.05881-1-15-2', '1602.05881-2-15-2'], ['1602.05881-1-89-0', '1602.05881-2-89-0'], ['1602.05881-1-89-1', '1602.05881-2-89-1'], ['1602.05881-1-89-2', '1602.05881-2-89-2'], ['1602.05881-1-53-0', '1602.05881-2-53-0'], ['1602.05881-1-18-0', '1602.05881-2-18-0'], ['1602.05881-1-18-1', '1602.05881-2-18-1'], ['1602.05881-1-48-0', '1602.05881-2-48-0'], ['1602.05881-1-86-0', '1602.05881-2-86-0'], ['1602.05881-1-86-1', '1602.05881-2-86-1'], ['1602.05881-1-86-2', '1602.05881-2-86-2'], ['1602.05881-1-31-0', '1602.05881-2-31-0'], ['1602.05881-1-31-1', '1602.05881-2-31-1'], ['1602.05881-1-49-0', '1602.05881-2-49-0'], ['1602.05881-1-62-0', '1602.05881-2-62-0'], ['1602.05881-1-62-1', '1602.05881-2-62-1'], ['1602.05881-1-62-2', '1602.05881-2-62-2'], ['1602.05881-1-136-0', '1602.05881-2-136-0']]

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[]

[['1602.05881-1-0-2', '1602.05881-2-0-2']]

[]

['1602.05881-1-26-0', '1602.05881-2-26-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1602.05881

1512.08934

{'1512.08934-1-0-0': 'We report an efficient program for computing the eigenvalues and symmetry-adapted eigenvectors of very large quaternionic (or Hermitian skew-Hamiltonian) matrices, using which structure-preserving diagonalization of matrices of dimension [MATH] is now routine on a single computer node.', '1512.08934-1-0-1': 'Such matrices appear frequently in relativistic quantum chemistry owing to the time-reversal symmetry.', '1512.08934-1-0-2': 'The implementation is based on a blocked version of the Paige-Van Loan algorithm [D. Kressner, BIT 43, 775 (2003)], which allows us to use the Level 3 BLAS subroutines for most of the computations.', '1512.08934-1-0-3': "Taking advantage of the symmetry, the program is faster by up to a factor of two than state-of-the-art implementations of complex Hermitian diagonalization; diagonalizing a [MATH] matrix took 42.8 (9.5) and 85.6 (12.6) minutes with 1 CPU core (16 CPU cores) using our symmetry-adapted solver and Intel MKL's ZHEEV that is not structure-preserving, respectively.", '1512.08934-1-0-4': 'The source code is publicly available under the FreeBSD license.', '1512.08934-1-1-0': '# Introduction', '1512.08934-1-2-0': 'In relativistic quantum mechanics, the Hamiltonian operator commutes with the time-reversal operator [MATH] in the absence of external magnetic fields, i.e., [EQUATION] which implies that all the physical quantities should belong to one of the two irreducible representations of the symmetry group.', '1512.08934-1-2-1': 'The time-reversal operator is a product of a unitary operator and the complex conjugation operator [CITATION].', '1512.08934-1-2-2': 'In relativistic electronic structure calculations based on the four-component Dirac equation or its two-component approximations [CITATION], many of the matrices, including the Fock matrix and reduced density matrices, have the following structure due to the time-reversal symmetry: [EQUATION] where [MATH] and [MATH] with [MATH] and [MATH].', '1512.08934-1-2-3': 'This structure is generally referred to as (a special case of) skew-Hamiltonian matrices in the applied mathematics literature [CITATION].', '1512.08934-1-2-4': 'Hereafter, we call this symmetry "quaternionic" in this article, because the matrix in Eq. [REF] can also be represented by a [MATH] matrix of quaternions.', '1512.08934-1-2-5': 'Note that the diagonal of [MATH] is zero owing to the skew symmetry, which plays an important role later.', '1512.08934-1-2-6': 'In many of the relativistic quantum chemical algorithms, such as mean-field methods, one must efficiently diagonalize very large matrices of this kind.', '1512.08934-1-3-0': 'Physically, this structure arises because the time reversal operator [MATH] transforms molecular spinors as [EQUATION] in which [MATH] and [MATH] are the spinors in the "[MATH]" and "[MATH]" irreducible representations, respectively; using them, it follows naturally for a time-reversal symmetric operator [MATH] (such as the Fock operator) that [EQUATION] hence the structure in Eq. [REF].', '1512.08934-1-3-1': 'The readers are referred to Ref. [CITATION] for thorough discussions on the time-reversal symmetry in quantum chemistry.', '1512.08934-1-4-0': 'It is trivial to show that when [MATH], with [MATH] and [MATH] being column vectors of length [MATH], is an eigenvector of Eq. [REF], [MATH] is also an eigenvector with the same eigenvalue.', '1512.08934-1-4-1': 'Therefore, the eigenvalue problem for [MATH] can be written as [EQUATION] in which [MATH] and [MATH] is a diagonal matrix whose elements are real.', '1512.08934-1-4-2': 'Since all of the eigenvalues are at least two-fold degenerate, there is a freedom to rotate among the eigenvectors that correspond to the same eigenvalue, i.e., [EQUATION] resulting in a set of eigenvectors without the structure in Eq. [REF].', '1512.08934-1-4-3': 'Here [MATH] denotes a Givens rotation [CITATION] among a pair of degenerate eigenvectors.', '1512.08934-1-4-4': 'The standard Hermitian diagonalization solvers (e.g., ZHEEV) ignore this quaternionic structure and yield eigenvectors with arbitrary [MATH].', '1512.08934-1-5-0': 'In quantum chemical algorithms, however, it is often essential to obtain the eigenvectors of the form in Eq. [REF].', '1512.08934-1-5-1': 'For instance, they are used in complete-active-space self consistent field (CASSCF) algorithms [CITATION] to fix the optimization frame and remove the orbital rotations among degenerate molecular spinors.', '1512.08934-1-5-2': 'Although it is formally possible to symmetrize the eigenvectors after diagonalization, symmetry-adaptation procedures are numerically unstable, especially for large systems, because of the double precision arithmetic used in chemical computations.', '1512.08934-1-5-3': 'Given the recent developments of large-scale two- and four-component relativistic quantum chemistry [CITATION], it is now of practical importance to develop a highly efficient tool for diagonalizing quaternion matrices that preserves the structure.', '1512.08934-1-6-0': 'The conventional approach to obtaining the symmetry-adapted eigenvectors of a quaternionic matrix is to use the quaternion algebra.', '1512.08934-1-6-1': 'By representing the matrix as an [MATH] matrix of quaternions, [EQUATION] with [MATH], a quaternion version of the Householder transformation or Jacobi rotation can be performed; the algorithm has been reported several times in the literature (e.g., Ref. [CITATION] just to name a few).', '1512.08934-1-6-2': 'The quaternion algebra is nevertheless somewhat complicated, and its computation cannot be easily mapped to highly optimized linear algebra libraries such as BLAS and LAPACK.', '1512.08934-1-7-0': 'In this article, we take an alternative route and report a highly efficient implementation of the so-called Paige-Van Loan algorithm.', '1512.08934-1-7-1': 'The Paige-Van Loan algorithm is a generalization of the standard tridiagonalization algorithm to skew-Hamiltonian matrices, a special case of which is a quaternionic matrix.', '1512.08934-1-7-2': 'The blocking scheme proposed by Kressner [CITATION] is used to map most of the computations to the Level 3 BLAS subroutines and to achieve high efficiency.', '1512.08934-1-7-3': 'In what follows, we first review the underlying algorithms, followed by our efficient implementation.', '1512.08934-1-8-0': '# Algorithm', '1512.08934-1-9-0': '## Paige-Van Loan algorithm', '1512.08934-1-10-0': 'The Householder transformation is often used in tridiagonalization of a Hermitian matrix.', '1512.08934-1-10-1': 'It transforms a Hermitian matrix [MATH] as [EQUATION] such that the first column and row of [MATH] is zero except for the first two elements.', '1512.08934-1-10-2': 'The transformation matrix can be computed using the first column of the input matrix [MATH] as [EQUATION] which satisfies [MATH] (the phase of [MATH] is arbitrary).', '1512.08934-1-10-3': 'Hereafter the subscript denotes the iteration number in the algorithm.', '1512.08934-1-10-4': 'By performing this procedure recursively, one obtains a tridiagonal matrix, i.e., [EQUATION]', '1512.08934-1-10-5': 'When a matrix has the skew symmetry [MATH], a similar procedure can be used: [EQUATION]', '1512.08934-1-10-6': 'We note in passing that the transformed matrix has the same symmetry as the original matrix.', '1512.08934-1-11-0': 'The Paige-Van Loan algorithm [CITATION] applies similar transformations to skew-Hamiltonian matrices (including quaternion matrices).', '1512.08934-1-11-1': 'First, a Householder transformation is applied to the off-diagonal blocks as [EQUATION]', '1512.08934-1-11-2': 'Since the symmetry is preserved, the right half of the matrix does not have to be stored and transformed.', '1512.08934-1-11-3': 'The transformation can be written as [EQUATION]', '1512.08934-1-11-4': 'We then use a Givens rotation to clear out the remaining elements in the first column and row in [MATH]: [EQUATION]', '1512.08934-1-11-5': 'The rotation is generated by [EQUATION] where [MATH], while the other elements are identical to the unit matrix.', '1512.08934-1-11-6': 'Finally we perform another Householder transformation for the [MATH] matrix, yielding [EQUATION]', '1512.08934-1-11-7': 'The transformation reads [EQUATION]', '1512.08934-1-11-8': 'Repeating this procedure yields a block-diagonal matrix, whose diagonal blocks are tridiagonal, [EQUATION]', '1512.08934-1-11-9': 'The tridiagonal matrix can be efficiently diagonalized using the ZHBEV subroutine in LAPACK.', '1512.08934-1-11-10': 'In essence, the Paige-Van Loan algorithm takes advantage of the skew symmetry of the off-diagonal blocks and clears them out via successive applications of the Householder transformations and Givens rotations.', '1512.08934-1-11-11': 'It is observed that the operation count of this algorithm is roughly four times that of the diagonalization of [MATH] complex Hermitian matrices (which is about a half of that of [MATH] matrices).', '1512.08934-1-12-0': 'The implementation of this algorithm is relatively straightforward.', '1512.08934-1-12-1': 'For instance, the matlab code that implements the above algorithm has been presented by Loring [CITATION].', '1512.08934-1-12-2': 'However, most of the computations are matrix-vector multiplications and outer-products of vectors, which are computed by the Level 2 BLAS subroutines that have [MATH] floating point operations with [MATH] memory operations.', '1512.08934-1-12-3': 'To attain the efficiency comparable to the state-of-the-art matrix diagonalization algorithms, we ought to introduce a blocked algorithm such that most of the computations are mapped to the Level 3 BLAS subroutines with [MATH] floating point operations and [MATH] memory operations.', '1512.08934-1-13-0': "## Kressner's compact WY-like representation", '1512.08934-1-14-0': "Kressner's compact WY-like representation for structured matrices [CITATION] is reviewed in this section.", '1512.08934-1-14-1': 'In the state-of-the-art implementations of matrix diagonalization (e.g., ZHEEV), the blocked algorithms are used so that most of the computations can be done by the Level 3 BLAS subroutines [CITATION].', '1512.08934-1-14-2': 'These modern algorithms are based on the fact that the product of Householder matrices can be compactly represented by the so-called compact WY representation [CITATION]: [EQUATION] where [MATH] and [MATH] is recursively defined as [EQUATION]', '1512.08934-1-14-3': 'The accumulated transformation matrices for a panel of A are used to transform the rest of the matrix (see Sec. [REF]), leveraging the efficiency of the Level 3 BLAS subroutines.', '1512.08934-1-15-0': 'Kressner has generalized the compact WY representation to the structured (Hamiltonian and skew-Hamiltonian) eigenvalue problems [CITATION].', '1512.08934-1-15-1': 'The product of the transformation matrices are formally written as [EQUATION] where [MATH] is defined in Eqs. [REF] and [REF], and [MATH] is [EQUATION]', '1512.08934-1-15-2': 'Note the complex conjugation of [MATH] in Eq. [REF].', '1512.08934-1-15-3': 'It has been shown in Ref. [CITATION] that [MATH] can be represented in a compact form, [EQUATION] in which the auxiliary matrices have the following structure: [EQUATION] with [MATH].', '1512.08934-1-15-4': 'All other matrices are in [MATH] and upper-triangular.', '1512.08934-1-16-0': 'The WY-like representation can be proven by induction with respect to [MATH] [CITATION], which in turn provides the way of constructing the auxiliary matrices.', '1512.08934-1-16-1': 'The following equations are equivalent to those presented in Ref. [CITATION] except for complex conjugation and minor corrections.', '1512.08934-1-16-2': 'For the first Householder transformation [Eq. [REF]], the updates are [EQUATION] where we used a short-hand notation [MATH] for a row of blocks of [MATH], i.e., [MATH].', '1512.08934-1-16-3': 'Next, we update these matrices for the Givens rotation [Eq. [REF]]: [EQUATION] in which we introduced [MATH] and [MATH] with [MATH] being the rotation angle [see Eq. [REF]].', '1512.08934-1-16-4': '[MATH] is the ([MATH])-th column of the unit matrix.', '1512.08934-1-16-5': 'Subsequently, the second Householder transformation [Eq. [REF]] updates these matrices as [EQUATION]', '1512.08934-1-16-6': 'The matrices can be easily updated in each step by inserting a column and/or a row to the matrices.', '1512.08934-1-16-7': 'Those after the first iteration are [EQUATION] in which it is assumed that the second element of the Householder vectors is set to 1 [Eq. [REF]].', '1512.08934-1-17-0': '## Blocked updates and miscellaneous optimization', '1512.08934-1-18-0': 'In the standard algorithms for Hermitian diagonalization, one can show that [EQUATION] where [MATH].', '1512.08934-1-18-1': 'Efficient programs accumulate [MATH] for a block of columns, or a panel, to update the entire matrix [MATH] with Eq. [REF] at once.', '1512.08934-1-18-2': 'The details are found in Ref. [CITATION].', '1512.08934-1-19-0': 'We use a similar formula, [EQUATION] where [MATH] and [MATH] are in [MATH] and defined as [EQUATION] [MATH] and [MATH] are likewise defined with [MATH].', '1512.08934-1-19-1': "[MATH] and [MATH]'s are computed in a three-step procedure as follows.", '1512.08934-1-19-2': 'After the first Householder transformation, they are [EQUATION]', '1512.08934-1-19-3': 'Note that the auxiliary vector [MATH] is computed in the update of [MATH] and [MATH] in Eq. [REF].', '1512.08934-1-19-4': 'They are transformed by the Givens rotation as [EQUATION]', '1512.08934-1-19-5': 'Finally, after the second Householder transformation, they become [EQUATION] [MATH] and [MATH] are computed similarly.', '1512.08934-1-20-0': 'The block update algorithm is sketched in Figure [REF].', '1512.08934-1-20-1': 'First, we construct [MATH], [MATH], [MATH], and [MATH] using Eqs. [REF]-[REF] while accumulating [MATH], [MATH], [MATH], and [MATH] using Eqs. [REF]-[REF].', '1512.08934-1-20-2': 'The first column is transformed to that in a tridiagonal form during this process.', '1512.08934-1-20-3': 'Next, Eq. [REF] is applied to the second column, transforming it to the corresponding column of [MATH].', '1512.08934-1-20-4': 'We then update all the auxiliary matrices and transform the second column to a tridiagonal form.', '1512.08934-1-20-5': 'This procedure is repeated until we accumulate the matrices for [MATH] columns.', '1512.08934-1-20-6': 'Finally, Eq. [REF] is used to update the rest of the matrix to yield [MATH], which is efficiently performed by the Level 3 BLAS subroutines.', '1512.08934-1-20-7': 'The problem size is then reduced from [MATH] to [MATH].', '1512.08934-1-20-8': 'The above steps are recursively executed to complete the tridiagonalization.', '1512.08934-1-21-0': 'At this point, it is worth noting that several optimizations are possible in the implementation of above formulas.', '1512.08934-1-21-1': 'First, all of the elements in the first row of [MATH] is zero, and therefore, the first row can be removed from the computation.', '1512.08934-1-21-2': "The first rows of [MATH]'s and [MATH]'s can also be omitted.", '1512.08934-1-21-3': '[MATH] is then the first [MATH] columns of the unit matrix, which can be computed easily on the fly without storing it.', '1512.08934-1-21-4': "Second, because all of the elements of [MATH] in Eq. [REF] are identically zero, we do not need to keep [MATH] and [MATH]'s except for the latest column; this column is the only one that contributes to the panel update Eq. [REF] for the untransformed part of the matrix.", '1512.08934-1-21-5': "Therefore we overwrite [MATH] and [MATH]'s in every iteration to reduce the memory requirement.", '1512.08934-1-21-6': 'All of the elements of [MATH] in Eq. [REF] are zero, whereas the contributions from [MATH] in Eq. [REF] are trivial to compute.', '1512.08934-1-21-7': 'Finally, [MATH] and [MATH] (and other quantities) are stored in contiguous memory so that matrix-matrix multiplications can be fused.', '1512.08934-1-21-8': 'The memory requirement for the auxiliary matrices after these optimizations are found to be around [MATH].', '1512.08934-1-22-0': 'Our implementation takes advantage of efficient functions in BLAS and LAPACK as much as possible.', '1512.08934-1-22-1': 'The Householder transformation is generated by ZLARFG, while the Givens rotation parameters are calculated by ZLARTG.', '1512.08934-1-22-2': 'At the end of the algorithm, the [MATH] tridiagonal matrix is diagonalized by ZHBEV.', '1512.08934-1-22-3': 'Matrix-vector and matrix-matrix multiplications are performed by ZTRMV, ZGEMV, and ZGEMM (or ZGEMM3M when the Intel Math Kernel Library is used), respectively.', '1512.08934-1-22-4': 'In the unblocked code, ZGERC and ZGERU are frequently used to calculate the outer product of vectors.', '1512.08934-1-22-5': 'The Level 2 and 3 BLAS subroutines are responsible for shared-memory parallelization in our program.', '1512.08934-1-23-0': 'To our knowledge, this is the first implementation of the blocked Paige-Van Loan algorithm for quaternionic matrices, whereas that for general real (skew-)Hamiltonian matrices has been implemented in the HAPACK package [CITATION].', '1512.08934-1-23-1': 'Though diagonalization of a [MATH] quaternionic matrix can be mapped to diagonalization of a [MATH] skew-Hamiltonian matrix [CITATION], direct solution in complex arithmetic is more efficient because doubling the size increases the cost of diagonalization by a factor of 8, while diagonalization in complex arithmetic is only 3-4 times more expensive than that in real arithmetic.', '1512.08934-1-24-0': '# Timing results', '1512.08934-1-25-0': 'All of the timing data below were obtained on a computer node equipped with two Xeon E5-2650 2.0GHz with the maximum Turbo Boost frequency 2.8GHZ (Sandy Bridge, 8 cores each, 16 cores in total).', '1512.08934-1-25-1': 'Intel Math Kernel Library (version 11.3, 2016.0.109) was used for BLAS and LAPACK functions with threading support.', '1512.08934-1-25-2': 'The compiler used was g++ 5.2.0 with the optimization flags "-O3 -DNDEBUG -mavx."', '1512.08934-1-25-3': 'Figure [REF] summarizes the timing for diagonalizing random matrices with the quaternionic structure using the blocked and unblocked algorithms described above ([MATH] was used).', '1512.08934-1-25-4': "The timing is compared with MKL's ZHEEV, which does not preserve the structure of the eigenvectors.", '1512.08934-1-25-5': 'It is apparent that the use of the blocked algorithm is essential to achieving high efficiency comparable to the state-of-the-art implementation of standard diagonalization.', '1512.08934-1-25-6': "Diagonalizing a [MATH] matrix took 9.5, 33.0, and 12.6 minutes using the blocked and unblocked versions of our code and MKL's ZHEEV using 16 CPU cores.", '1512.08934-1-25-7': 'When only 1 CPU core is used, the wall times were 42.8, 67.7, and 85.6 minutes, respectively.', '1512.08934-1-25-8': "Note that the calculations with 1 CPU core were accelerated by the CPU's Turbo Boost (up to 2.8 GHz).", '1512.08934-1-26-0': 'The shared memory parallelization of our code appears less effective than that in the MKL implementation of ZHEEV, indicating that there is still some room for improvement.', '1512.08934-1-26-1': 'This is in part because our program is only threaded by the underlying BLAS functions and because the above blocked algorithm involves considerably more matrix-vector multiplications in the computation of the WY-like representation than the one used in the standard diagonalization.', '1512.08934-1-26-2': "Although our new implementation does outperform MKL's ZHEEV with 16 CPU cores by 25-30%, the detailed comparison with ZHEEV is not straightforward because of the difference in design of the two programs.", '1512.08934-1-27-0': 'The dependence of the efficiency with respect to the block size [MATH] is presented in Figure [REF] for [MATH] matrices using 16 CPU cores.', '1512.08934-1-27-1': 'The timing improves dramatically till [MATH], has the minimum at around [MATH], and then deteriorates slowly as [MATH] becomes larger.', '1512.08934-1-27-2': 'The initial improvement with respect to [MATH] is attributed to the increased efficiency of the Level 3 BLAS subroutines with the increased panel sizes.', '1512.08934-1-27-3': 'The deterioration with large [MATH] is due to the use of the Level 2 BLAS subroutines within each panel [Eqs. [REF]-[REF] and [REF]-[REF]], although it competes with further efficiency improvement of the Level 3 BLAS subroutines in Eq. [REF].', '1512.08934-1-27-4': 'The optimal [MATH] is expected to be machine dependent.', '1512.08934-1-28-0': '# Conclusions', '1512.08934-1-29-0': 'We have implemented a blocked version of the Paige-Van Loan algorithm into an efficient program, following the earlier work by Kressner [CITATION].', '1512.08934-1-29-1': 'Shared-memory parallelization is performed by the underlying BLAS and LAPACK library.', '1512.08934-1-29-2': "When a single CPU core is used, the elapsed time for structure-preserving diagonalization using this program is about a half of that using ZHEEV without symmetry treatment; with 16 CPU cores, the program still outperforms the MKL's ZHEEV implementation by about 25-30%.", '1512.08934-1-29-3': 'The source code is publicly available under the FreeBSD license [CITATION] so that it can be integrated into any large-scale relativistic quantum chemistry program packages.', '1512.08934-1-29-4': 'The program has been interfaced to the bagel package [CITATION].', '1512.08934-1-29-5': 'The parallel implementation will be investigated in the near future.', '1512.08934-1-30-0': 'This article is part of the special issue in honor of the 60th birthday of Hans Jorgen Aa.', '1512.08934-1-30-1': 'Jensen.', '1512.08934-1-30-2': 'The author thanks Jack Poulson for helpful suggestions and Terry Loring for useful comments.', '1512.08934-1-30-3': 'This work has been supported by the NSF CAREER Award (CHE-1351598).'}

{'1512.08934-2-0-0': 'We report an efficient program for computing the eigenvalues and symmetry-adapted eigenvectors of very large quaternionic (or Hermitian skew-Hamiltonian) matrices, using which structure-preserving diagonalization of matrices of dimension [MATH] is now routine on a single computer node.', '1512.08934-2-0-1': 'Such matrices appear frequently in relativistic quantum chemistry owing to the time-reversal symmetry.', '1512.08934-2-0-2': 'The implementation is based on a blocked version of the Paige-Van Loan algorithm [D. Kressner, BIT 43, 775 (2003)], which allows us to use the Level 3 BLAS subroutines for most of the computations.', '1512.08934-2-0-3': "Taking advantage of the symmetry, the program is faster by up to a factor of two than state-of-the-art implementations of complex Hermitian diagonalization; diagonalizing a [MATH] matrix took 42.8 (9.5) and 85.6 (12.6) minutes with 1 CPU core (16 CPU cores) using our symmetry-adapted solver and Intel MKL's ZHEEV that is not structure-preserving, respectively.", '1512.08934-2-0-4': 'The source code is publicly available under the FreeBSD license.', '1512.08934-2-1-0': '# Introduction', '1512.08934-2-2-0': 'In relativistic quantum mechanics, the Hamiltonian operator commutes with the time-reversal operator [MATH] in the absence of external magnetic fields, i.e., [EQUATION] which implies that all the physical quantities should belong to one of the two irreducible representations of the symmetry group.', '1512.08934-2-2-1': 'The time-reversal operator is a product of a unitary operator and the complex conjugation operator [CITATION].', '1512.08934-2-2-2': 'In relativistic electronic structure calculations based on the four-component Dirac equation or its two-component approximations [CITATION], many of the matrices, including the Fock matrix and reduced density matrices, have the following structure due to the time-reversal symmetry: [EQUATION] where [MATH] and [MATH] with [MATH] and [MATH].', '1512.08934-2-2-3': 'This structure is generally referred to as (a special case of) skew-Hamiltonian matrices in the applied mathematics literature [CITATION].', '1512.08934-2-2-4': 'Hereafter, we call this symmetry "quaternionic" in this article, because the matrix in Eq. [REF] can also be represented by a [MATH] matrix of quaternions.', '1512.08934-2-2-5': 'Note that the diagonal of [MATH] is zero owing to the skew symmetry, which plays an important role later.', '1512.08934-2-2-6': 'In many of the relativistic quantum chemical algorithms, such as mean-field methods, one must efficiently diagonalize very large matrices of this kind.', '1512.08934-2-3-0': 'Physically, this structure arises because the time reversal operator [MATH] transforms molecular spinors as [EQUATION] where [MATH] and [MATH] are said to be Kramers partners; using them, it follows naturally for a time-reversal symmetric operator [MATH] (such as the Fock operator) that [EQUATION] hence the structure in Eq. [REF].', '1512.08934-2-3-1': 'The readers are referred to Ref. [CITATION] for thorough discussions on the time-reversal symmetry in quantum chemistry.', '1512.08934-2-4-0': 'It is trivial to show that when [MATH], with [MATH] and [MATH] being column vectors of length [MATH], is an eigenvector of Eq. [REF], [MATH] is also an eigenvector with the same eigenvalue.', '1512.08934-2-4-1': 'Therefore, the eigenvalue problem for [MATH] can be written as [EQUATION] in which [MATH] and [MATH] is a diagonal matrix whose elements are real.', '1512.08934-2-4-2': 'Since all of the eigenvalues are at least two-fold degenerate, there is a freedom to rotate among the eigenvectors that correspond to the same eigenvalue, i.e., [EQUATION] resulting in a set of eigenvectors without the structure in Eq. [REF].', '1512.08934-2-4-3': 'Here [MATH] denotes a Givens rotation [CITATION] among a pair of degenerate eigenvectors.', '1512.08934-2-4-4': 'The standard Hermitian diagonalization solvers (e.g., ZHEEV) ignore this quaternionic structure and yield eigenvectors with arbitrary [MATH].', '1512.08934-2-5-0': 'In quantum chemical algorithms, however, it is often essential to obtain the eigenvectors of the form in Eq. [REF].', '1512.08934-2-5-1': 'For instance, they are used in complete-active-space self consistent field (CASSCF) algorithms [CITATION] to fix the optimization frame and remove the orbital rotations among degenerate molecular spinors.', '1512.08934-2-5-2': 'Although it is formally possible to symmetrize the eigenvectors after diagonalization, symmetry-adaptation procedures are numerically unstable, especially for large systems, because of the double precision arithmetic used in chemical computations.', '1512.08934-2-5-3': 'Given the recent developments of large-scale two- and four-component relativistic quantum chemistry [CITATION], it is now of practical importance to develop a highly efficient tool for diagonalizing quaternion matrices that preserves the structure.', '1512.08934-2-6-0': 'The conventional approach to obtaining the symmetry-adapted eigenvectors of a quaternionic matrix is to use the quaternion algebra.', '1512.08934-2-6-1': 'By representing the matrix as an [MATH] matrix of quaternions [CITATION], [EQUATION] with [MATH], a quaternion version of the Householder transformation or Jacobi rotation can be performed; the algorithm has been reported several times in the literature (e.g., Ref. [CITATION] just to name a few).', '1512.08934-2-6-2': 'The quaternion algebra is nevertheless somewhat complicated, and its computation cannot be easily mapped to highly optimized linear algebra libraries such as BLAS and LAPACK.', '1512.08934-2-7-0': 'In this article, we take an alternative route and report a highly efficient implementation of the so-called Paige-Van Loan algorithm.', '1512.08934-2-7-1': 'The Paige-Van Loan algorithm is a generalization of the standard tridiagonalization algorithm to skew-Hamiltonian matrices, a special case of which is a quaternionic matrix.', '1512.08934-2-7-2': 'The blocking scheme proposed by Kressner [CITATION] is used to map most of the computations to the Level 3 BLAS subroutines and to achieve high efficiency.', '1512.08934-2-7-3': 'In what follows, we first review the underlying algorithms, followed by our efficient implementation.', '1512.08934-2-8-0': '# Algorithm', '1512.08934-2-9-0': '## Paige-Van Loan algorithm', '1512.08934-2-10-0': 'The Householder transformation is often used in tridiagonalization of a Hermitian matrix.', '1512.08934-2-10-1': 'It transforms a Hermitian matrix [MATH] as [EQUATION] such that the first column and row of [MATH] is zero except for the first two elements.', '1512.08934-2-10-2': 'The transformation matrix can be computed using the first column of the input matrix [MATH] as [EQUATION] which satisfies [MATH] (the phase of [MATH] is arbitrary).', '1512.08934-2-10-3': 'Hereafter the subscript denotes the iteration number in the algorithm.', '1512.08934-2-10-4': 'By performing this procedure recursively, one obtains a tridiagonal matrix, i.e., [EQUATION]', '1512.08934-2-10-5': 'When a matrix has the skew symmetry [MATH], a similar procedure can be used: [EQUATION]', '1512.08934-2-10-6': 'We note in passing that the transformed matrix has the same symmetry as the original matrix.', '1512.08934-2-11-0': 'The Paige-Van Loan algorithm [CITATION] applies similar transformations to skew-Hamiltonian matrices (including quaternion matrices).', '1512.08934-2-11-1': 'First, a Householder transformation is applied to the off-diagonal blocks as [EQUATION]', '1512.08934-2-11-2': 'Since the symmetry is preserved, the right half of the matrix does not have to be stored and transformed.', '1512.08934-2-11-3': 'The transformation can be written as [EQUATION]', '1512.08934-2-11-4': 'We then use a Givens rotation to clear out the remaining elements in the first column and row in [MATH]: [EQUATION]', '1512.08934-2-11-5': 'The rotation is generated by [EQUATION] where [MATH], while the other elements are identical to the unit matrix.', '1512.08934-2-11-6': 'Finally we perform another Householder transformation for the [MATH] matrix, yielding [EQUATION]', '1512.08934-2-11-7': 'The transformation reads [EQUATION]', '1512.08934-2-11-8': 'Repeating this procedure yields a block-diagonal matrix, whose diagonal blocks are tridiagonal, [EQUATION]', '1512.08934-2-11-9': 'The tridiagonal matrix can be efficiently diagonalized using the ZHBEV subroutine in LAPACK.', '1512.08934-2-11-10': 'In essence, the Paige-Van Loan algorithm takes advantage of the skew symmetry of the off-diagonal blocks and clears them out via successive applications of the Householder transformations and Givens rotations.', '1512.08934-2-11-11': 'It is observed that the operation count of this algorithm is roughly four times that of the diagonalization of [MATH] complex Hermitian matrices (which is about a half of that of [MATH] matrices).', '1512.08934-2-12-0': 'The implementation of this algorithm is relatively straightforward.', '1512.08934-2-12-1': 'For instance, the matlab code that implements the above algorithm has been presented by Loring [CITATION].', '1512.08934-2-12-2': 'However, most of the computations are matrix-vector multiplications and outer-products of vectors, which are computed by the Level 2 BLAS subroutines that have [MATH] floating point operations with [MATH] memory operations.', '1512.08934-2-12-3': 'To attain the efficiency comparable to the state-of-the-art matrix diagonalization algorithms, we ought to introduce a blocked algorithm such that most of the computations are mapped to the Level 3 BLAS subroutines with [MATH] floating point operations and [MATH] memory operations.', '1512.08934-2-13-0': "## Kressner's compact WY-like representation", '1512.08934-2-14-0': "Kressner's compact WY-like representation for structured matrices [CITATION] is reviewed in this section.", '1512.08934-2-14-1': 'In the state-of-the-art implementations of matrix diagonalization (e.g., ZHEEV), the blocked algorithms are used so that most of the computations can be done by the Level 3 BLAS subroutines [CITATION].', '1512.08934-2-14-2': 'These modern algorithms are based on the fact that the product of Householder matrices can be compactly represented by the so-called compact WY representation [CITATION]: [EQUATION] where [MATH] and [MATH] are recursively defined as [EQUATION]', '1512.08934-2-14-3': 'The accumulated transformation matrices for a panel of A are used to transform the rest of the matrix (see Sec. [REF]), leveraging the efficiency of the Level 3 BLAS subroutines.', '1512.08934-2-15-0': 'Kressner has generalized the compact WY representation to the structured (Hamiltonian and skew-Hamiltonian) eigenvalue problems [CITATION].', '1512.08934-2-15-1': 'The product of the transformation matrices are formally written as [EQUATION] where [MATH] is defined in Eqs. [REF] and [REF], and [MATH] is [EQUATION]', '1512.08934-2-15-2': 'Note the complex conjugation of [MATH] in Eq. [REF].', '1512.08934-2-15-3': 'It has been shown in Ref. [CITATION] that [MATH] can be represented in a compact form, [EQUATION] in which the auxiliary matrices have the following structure: [EQUATION] with [MATH].', '1512.08934-2-15-4': 'All other matrices are in [MATH] and upper-triangular.', '1512.08934-2-16-0': 'The WY-like representation can be proven by induction with respect to [MATH] [CITATION], which in turn provides the way of constructing the auxiliary matrices.', '1512.08934-2-16-1': 'The following equations are equivalent to those presented in Ref. [CITATION] except for complex conjugation and minor corrections.', '1512.08934-2-16-2': 'For the first Householder transformation [Eq. [REF]], the updates are [EQUATION] where we used a short-hand notation [MATH] for a row of blocks of [MATH], i.e., [MATH].', '1512.08934-2-16-3': 'Next, we update these matrices for the Givens rotation [Eq. [REF]]: [EQUATION] in which we introduced [MATH] and [MATH] with [MATH] being the rotation angle [see Eq. [REF]].', '1512.08934-2-16-4': '[MATH] is the ([MATH])-th column of the unit matrix.', '1512.08934-2-16-5': 'Subsequently, the second Householder transformation [Eq. [REF]] updates these matrices as [EQUATION]', '1512.08934-2-16-6': 'The matrices can be easily updated in each step by inserting a column and/or a row to the matrices.', '1512.08934-2-16-7': 'Those after the first iteration are [EQUATION] in which it is assumed that the second element of the Householder vectors is set to 1 [Eq. [REF]].', '1512.08934-2-17-0': '## Blocked updates and miscellaneous optimization', '1512.08934-2-18-0': 'In the standard algorithms for Hermitian diagonalization, one can show that [EQUATION] where [MATH].', '1512.08934-2-18-1': 'Efficient programs accumulate [MATH] for a block of columns, or a panel, to update the entire matrix [MATH] with Eq. [REF] at once.', '1512.08934-2-18-2': 'The details are found in Ref. [CITATION].', '1512.08934-2-19-0': 'We use a similar formula, [EQUATION] where [MATH] and [MATH] are in [MATH] and defined as [EQUATION] [MATH] and [MATH] are likewise defined with [MATH].', '1512.08934-2-19-1': "[MATH] and [MATH]'s are computed in a three-step procedure as follows.", '1512.08934-2-19-2': 'After the first Householder transformation, they are [EQUATION]', '1512.08934-2-19-3': 'Note that the auxiliary vector [MATH] is computed in the update of [MATH] and [MATH] in Eq. [REF].', '1512.08934-2-19-4': 'They are transformed by the Givens rotation as [EQUATION]', '1512.08934-2-19-5': 'Finally, after the second Householder transformation, they become [EQUATION] [MATH] and [MATH] are computed similarly.', '1512.08934-2-20-0': 'The block update algorithm is sketched in Figure [REF].', '1512.08934-2-20-1': 'First, we construct [MATH], [MATH], [MATH], and [MATH] using Eqs. [REF]-[REF] while accumulating [MATH], [MATH], [MATH], and [MATH] using Eqs. [REF]-[REF].', '1512.08934-2-20-2': 'The first column is transformed to that in a tridiagonal form during this process.', '1512.08934-2-20-3': 'Next, Eq. [REF] is applied to the second column, transforming it to the corresponding column of [MATH].', '1512.08934-2-20-4': 'We then update all the auxiliary matrices and transform the second column to a tridiagonal form.', '1512.08934-2-20-5': 'This procedure is repeated until we accumulate the matrices for [MATH] columns.', '1512.08934-2-20-6': 'Finally, Eq. [REF] is used to update the rest of the matrix to yield [MATH], which is efficiently performed by the Level 3 BLAS subroutines.', '1512.08934-2-20-7': 'The problem size is then reduced from [MATH] to [MATH].', '1512.08934-2-20-8': 'The above steps are recursively executed to complete the tridiagonalization.', '1512.08934-2-21-0': 'At this point, it is worth noting that several optimizations are possible in the implementation of above formulas.', '1512.08934-2-21-1': 'First, all of the elements in the first row of [MATH] is zero, and therefore, the first row can be removed from the computation.', '1512.08934-2-21-2': "The first rows of [MATH]'s and [MATH]'s can also be omitted.", '1512.08934-2-21-3': '[MATH] is then the first [MATH] columns of the unit matrix, which can be computed easily on the fly without storing it.', '1512.08934-2-21-4': "Second, because all of the elements of [MATH] in Eq. [REF] are identically zero, we do not need to keep [MATH] and [MATH]'s except for the latest column; this column is the only one that contributes to the panel update Eq. [REF] for the untransformed part of the matrix.", '1512.08934-2-21-5': "Therefore we overwrite [MATH] and [MATH]'s in every iteration to reduce the memory requirement.", '1512.08934-2-21-6': 'All of the elements of [MATH] in Eq. [REF] are zero, whereas the contributions from [MATH] in Eq. [REF] are trivial to compute.', '1512.08934-2-21-7': 'Finally, [MATH] and [MATH] (and other quantities) are stored in contiguous memory so that matrix-matrix multiplications can be fused.', '1512.08934-2-21-8': 'The memory requirement for the auxiliary matrices after these optimizations are found to be around [MATH].', '1512.08934-2-22-0': 'Our implementation takes advantage of efficient functions in BLAS and LAPACK as much as possible.', '1512.08934-2-22-1': 'The Householder transformation is generated by ZLARFG, while the Givens rotation parameters are calculated by ZLARTG.', '1512.08934-2-22-2': 'At the end of the algorithm, the [MATH] tridiagonal matrix is diagonalized by ZHBEV.', '1512.08934-2-22-3': 'Matrix-vector and matrix-matrix multiplications are performed by ZTRMV, ZGEMV, and ZGEMM (or ZGEMM3M when the Intel Math Kernel Library is used), respectively.', '1512.08934-2-22-4': 'In the unblocked code, ZGERC and ZGERU are frequently used to calculate the outer product of vectors.', '1512.08934-2-22-5': 'The Level 2 and 3 BLAS subroutines are responsible for shared-memory parallelization in our program.', '1512.08934-2-23-0': 'To our knowledge, this is the first implementation of the blocked Paige-Van Loan algorithm for quaternionic matrices, whereas that for general real (skew-)Hamiltonian matrices has been implemented in the HAPACK package [CITATION].', '1512.08934-2-23-1': 'Though diagonalization of a [MATH] quaternionic matrix can be mapped to diagonalization of a [MATH] skew-Hamiltonian matrix [CITATION], direct solution in complex arithmetic is more efficient because doubling the size increases the cost of diagonalization by a factor of 8, while diagonalization in complex arithmetic is only 3-4 times more expensive than that in real arithmetic.', '1512.08934-2-24-0': '# Timing results', '1512.08934-2-25-0': 'All of the timing data below were obtained on a computer node equipped with two Xeon E5-2650 2.0GHz with the maximum Turbo Boost frequency 2.8GHZ (Sandy Bridge, 8 cores each, 16 cores in total).', '1512.08934-2-25-1': 'Intel Math Kernel Library (version 11.3, 2016.0.109) was used for BLAS and LAPACK functions with threading support.', '1512.08934-2-25-2': 'The compiler used was g++ 5.2.0 with the optimization flags "-O3 -DNDEBUG -mavx."', '1512.08934-2-25-3': 'Figure [REF] summarizes the timing for diagonalizing random matrices with the quaternionic structure using the blocked and unblocked algorithms described above ([MATH] was used).', '1512.08934-2-25-4': "The timing is compared with MKL's ZHEEV, which does not preserve the structure of the eigenvectors.", '1512.08934-2-25-5': 'It is apparent that the use of the blocked algorithm is essential to achieving high efficiency comparable to the state-of-the-art implementation of standard diagonalization.', '1512.08934-2-25-6': "Diagonalizing a [MATH] matrix took 9.5, 33.0, and 12.6 minutes using the blocked and unblocked versions of our code and MKL's ZHEEV using 16 CPU cores.", '1512.08934-2-25-7': 'When only 1 CPU core is used, the wall times were 42.8, 67.7, and 85.6 minutes, respectively.', '1512.08934-2-25-8': "Note that the calculations with 1 CPU core were accelerated by the CPU's Turbo Boost (up to 2.8 GHz).", '1512.08934-2-26-0': 'The shared memory parallelization of our code appears less effective than that in the MKL implementation of ZHEEV, indicating that there is still some room for improvement.', '1512.08934-2-26-1': 'This is in part because our program is only threaded by the underlying BLAS functions and because the above blocked algorithm involves considerably more matrix-vector multiplications in the computation of the WY-like representation than the one used in the standard diagonalization.', '1512.08934-2-26-2': "Although our new implementation does outperform MKL's ZHEEV with 16 CPU cores by 25-30%, the detailed comparison with ZHEEV is not straightforward because of the difference in design of the two programs.", '1512.08934-2-27-0': 'The dependence of the efficiency with respect to the block size [MATH] is presented in Figure [REF] for [MATH] matrices using 16 CPU cores.', '1512.08934-2-27-1': 'The timing improves dramatically till [MATH], has the minimum at around [MATH], and then deteriorates slowly as [MATH] becomes larger.', '1512.08934-2-27-2': 'The initial improvement with respect to [MATH] is attributed to the increased efficiency of the Level 3 BLAS subroutines with the increased panel sizes.', '1512.08934-2-27-3': 'The deterioration with large [MATH] is due to the use of the Level 2 BLAS subroutines within each panel [Eqs. [REF]-[REF] and [REF]-[REF]], although it competes with further efficiency improvement of the Level 3 BLAS subroutines in Eq. [REF].', '1512.08934-2-27-4': 'The optimal [MATH] is expected to be machine dependent.', '1512.08934-2-28-0': '# Conclusions', '1512.08934-2-29-0': 'We have implemented a blocked version of the Paige-Van Loan algorithm into an efficient program, following the earlier work by Kressner [CITATION].', '1512.08934-2-29-1': 'Shared-memory parallelization is performed by the underlying BLAS and LAPACK library.', '1512.08934-2-29-2': "When a single CPU core is used, the elapsed time for structure-preserving diagonalization using this program is about a half of that using ZHEEV without symmetry treatment; with 16 CPU cores, the program still outperforms the MKL's ZHEEV implementation by about 25-30%.", '1512.08934-2-29-3': 'The source code is publicly available under the FreeBSD license [CITATION] so that it can be integrated into any large-scale relativistic quantum chemistry program packages.', '1512.08934-2-29-4': 'The program has been interfaced to the bagel package [CITATION].', '1512.08934-2-29-5': 'The parallel implementation will be investigated in the near future.', '1512.08934-2-30-0': 'This article is part of the special issue in honor of the 60th birthday of Hans Jorgen Aa.', '1512.08934-2-30-1': 'Jensen.', '1512.08934-2-30-2': 'The author thanks Jack Poulson for helpful suggestions and Terry Loring for useful comments.', '1512.08934-2-30-3': 'This work has been supported by the NSF CAREER Award (CHE-1351598).', '1512.08934-2-30-4': 'The author is an Alfred P. Sloan Research Fellow.'}

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'1512.08934-3-19-5'], ['1512.08934-2-10-0', '1512.08934-3-10-0'], ['1512.08934-2-10-1', '1512.08934-3-10-1'], ['1512.08934-2-10-3', '1512.08934-3-10-4'], ['1512.08934-2-10-4', '1512.08934-3-10-5'], ['1512.08934-2-10-6', '1512.08934-3-10-7'], ['1512.08934-2-22-0', '1512.08934-3-22-0'], ['1512.08934-2-22-1', '1512.08934-3-22-1'], ['1512.08934-2-22-2', '1512.08934-3-22-2'], ['1512.08934-2-22-3', '1512.08934-3-22-3'], ['1512.08934-2-22-4', '1512.08934-3-22-4'], ['1512.08934-2-22-5', '1512.08934-3-22-5'], ['1512.08934-2-16-0', '1512.08934-3-16-0'], ['1512.08934-2-16-1', '1512.08934-3-16-1'], ['1512.08934-2-16-2', '1512.08934-3-16-2'], ['1512.08934-2-16-3', '1512.08934-3-16-3'], ['1512.08934-2-16-4', '1512.08934-3-16-4'], ['1512.08934-2-16-5', '1512.08934-3-16-5'], ['1512.08934-2-16-6', '1512.08934-3-16-6'], ['1512.08934-2-16-7', '1512.08934-3-16-7'], ['1512.08934-2-0-0', '1512.08934-3-0-0'], ['1512.08934-2-0-1', '1512.08934-3-0-1'], ['1512.08934-2-0-2', '1512.08934-3-0-2'], 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['1512.08934-2-0-3', '1512.08934-3-0-3'], ['1512.08934-2-0-4', '1512.08934-3-0-4'], ['1512.08934-2-12-0', '1512.08934-3-12-0'], ['1512.08934-2-12-1', '1512.08934-3-12-1'], ['1512.08934-2-12-2', '1512.08934-3-12-2'], ['1512.08934-2-12-3', '1512.08934-3-12-3'], ['1512.08934-2-3-0', '1512.08934-3-3-0'], ['1512.08934-2-15-0', '1512.08934-3-15-0'], ['1512.08934-2-15-2', '1512.08934-3-15-2'], ['1512.08934-2-15-3', '1512.08934-3-15-3'], ['1512.08934-2-15-4', '1512.08934-3-15-4'], ['1512.08934-2-21-0', '1512.08934-3-21-0'], ['1512.08934-2-21-1', '1512.08934-3-21-1'], ['1512.08934-2-21-2', '1512.08934-3-21-2'], ['1512.08934-2-21-3', '1512.08934-3-21-3'], ['1512.08934-2-21-4', '1512.08934-3-21-4'], ['1512.08934-2-21-5', '1512.08934-3-21-5'], ['1512.08934-2-21-6', '1512.08934-3-21-6'], ['1512.08934-2-21-7', '1512.08934-3-21-7'], ['1512.08934-2-29-0', '1512.08934-3-29-0'], ['1512.08934-2-29-1', '1512.08934-3-29-1'], ['1512.08934-2-29-2', '1512.08934-3-29-2'], ['1512.08934-2-29-3', '1512.08934-3-29-3'], ['1512.08934-2-29-4', '1512.08934-3-29-4'], ['1512.08934-2-29-5', '1512.08934-3-29-5'], ['1512.08934-2-25-1', '1512.08934-3-25-1'], ['1512.08934-2-25-2', '1512.08934-3-25-2'], ['1512.08934-2-25-3', '1512.08934-3-25-3'], ['1512.08934-2-25-4', '1512.08934-3-25-4'], ['1512.08934-2-25-5', '1512.08934-3-25-5'], ['1512.08934-2-25-6', '1512.08934-3-25-6'], ['1512.08934-2-25-7', '1512.08934-3-25-7'], ['1512.08934-2-25-8', '1512.08934-3-25-8'], ['1512.08934-2-6-0', '1512.08934-3-6-0'], ['1512.08934-2-6-2', '1512.08934-3-6-2'], ['1512.08934-2-27-0', '1512.08934-3-27-0'], ['1512.08934-2-27-1', '1512.08934-3-27-1'], ['1512.08934-2-27-2', '1512.08934-3-27-2'], ['1512.08934-2-27-3', '1512.08934-3-27-3'], ['1512.08934-2-14-0', '1512.08934-3-14-0'], ['1512.08934-2-14-1', '1512.08934-3-14-1'], ['1512.08934-2-14-2', '1512.08934-3-14-2'], ['1512.08934-2-23-0', '1512.08934-3-23-0'], ['1512.08934-2-11-1', '1512.08934-3-11-1'], ['1512.08934-2-11-2', '1512.08934-3-11-2'], ['1512.08934-2-11-3', '1512.08934-3-11-3'], ['1512.08934-2-11-4', '1512.08934-3-11-4'], ['1512.08934-2-11-6', '1512.08934-3-11-6'], ['1512.08934-2-11-7', '1512.08934-3-11-7'], ['1512.08934-2-11-9', '1512.08934-3-11-9'], ['1512.08934-2-11-10', '1512.08934-3-11-10'], ['1512.08934-2-11-11', '1512.08934-3-11-11'], ['1512.08934-2-4-0', '1512.08934-3-4-0'], ['1512.08934-2-4-1', '1512.08934-3-4-1'], ['1512.08934-1-21-6', '1512.08934-2-21-6']]

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[]

[['1512.08934-2-2-0', '1512.08934-3-2-0'], ['1512.08934-2-2-3', '1512.08934-3-2-3'], ['1512.08934-2-10-2', '1512.08934-3-10-2'], ['1512.08934-2-10-2', '1512.08934-3-10-3'], ['1512.08934-2-21-8', '1512.08934-3-21-8'], ['1512.08934-2-27-4', '1512.08934-3-27-4'], ['1512.08934-2-11-0', '1512.08934-3-11-0'], ['1512.08934-2-4-2', '1512.08934-3-4-2'], ['1512.08934-2-4-4', '1512.08934-3-4-3']]

[]

['1512.08934-1-20-1', '1512.08934-1-30-1', '1512.08934-2-20-1', '1512.08934-2-30-1', '1512.08934-3-20-1', '1512.08934-3-30-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1512.08934

{'1512.08934-3-0-0': 'We report an efficient program for computing the eigenvalues and symmetry-adapted eigenvectors of very large quaternionic (or Hermitian skew-Hamiltonian) matrices, using which structure-preserving diagonalization of matrices of dimension [MATH] is now routine on a single computer node.', '1512.08934-3-0-1': 'Such matrices appear frequently in relativistic quantum chemistry owing to the time-reversal symmetry.', '1512.08934-3-0-2': 'The implementation is based on a blocked version of the Paige-Van Loan algorithm [D. Kressner, BIT 43, 775 (2003)], which allows us to use the Level 3 BLAS subroutines for most of the computations.', '1512.08934-3-0-3': "Taking advantage of the symmetry, the program is faster by up to a factor of two than state-of-the-art implementations of complex Hermitian diagonalization; diagonalizing a [MATH] matrix took 42.8 (9.5) and 85.6 (12.6) minutes with 1 CPU core (16 CPU cores) using our symmetry-adapted solver and Intel MKL's ZHEEV that is not structure-preserving, respectively.", '1512.08934-3-0-4': 'The source code is publicly available under the FreeBSD license.', '1512.08934-3-1-0': '# Introduction', '1512.08934-3-2-0': 'In relativistic quantum mechanics, the Hamiltonian operator commutes with the time-reversal operator [MATH] in the absence of external magnetic fields, i.e., [EQUATION] in which [MATH] is a product of a unitary operator and the complex conjugation operator [CITATION].', '1512.08934-3-2-1': 'In Kramers-restricted relativistic electronic structure calculations based on the four-component Dirac equation or its two-component approximations [CITATION], many of the matrices, including the Fock matrix and reduced density matrices, have the following structure due to the time-reversal symmetry and Hermicity: [EQUATION] where [MATH] and [MATH] with [MATH] and [MATH].', '1512.08934-3-2-2': 'In the case of the Fock matrix in two- and four-component calculations, for instance, the values of [MATH] are [MATH] and [MATH], respectively, where [MATH] is the number of the scalar basis functions.', '1512.08934-3-2-3': 'This structure is referred to as Hermitian plus skew-Hamiltonian in the applied mathematics literature [CITATION].', '1512.08934-3-2-4': 'Hereafter, we call this symmetry "quaternionic" in this article, because the matrix in Eq. [REF] can also be represented by an [MATH] matrix of quaternions.', '1512.08934-3-2-5': 'Note that the diagonal of [MATH] is zero owing to the skew symmetry, which plays an important role later.', '1512.08934-3-2-6': 'In many of the relativistic quantum chemical algorithms, such as mean-field methods, one must efficiently diagonalize very large matrices of this kind.', '1512.08934-3-3-0': 'Physically, this structure arises because the time reversal operator [MATH] transforms molecular spinors as [EQUATION] where [MATH] and [MATH] are said to be Kramers partners; using them, it follows naturally for a time-reversal symmetric operator [MATH] (such as the Fock operator) that [EQUATION] hence the structure in Eq. [REF].', '1512.08934-3-3-1': 'The readers are referred to Ref. [CITATION] for thorough discussions on time-reversal symmetry in quantum chemistry.', '1512.08934-3-4-0': 'It is trivial to show that when [MATH], with [MATH] and [MATH] being column vectors of length [MATH], is an eigenvector of Eq. [REF], [MATH] is also an eigenvector with the same eigenvalue.', '1512.08934-3-4-1': 'Therefore, the eigenvalue problem for [MATH] can be written as [EQUATION] in which [MATH] and [MATH] is a diagonal matrix whose elements are real.', '1512.08934-3-4-2': 'However, there is a freedom to rotate among the eigenvectors that correspond to the same eigenvalue and multiply an arbitrary phase factor to each eigenvector, which, in general, result in a set of eigenvectors without the structure in Eq. [REF].', '1512.08934-3-4-3': 'The standard Hermitian diagonalization solvers (e.g., ZHEEV) ignore this quaternionic structure.', '1512.08934-3-5-0': 'In quantum chemical algorithms, however, it is often essential to obtain the eigenvectors of the form in Eq. [REF].', '1512.08934-3-5-1': 'For instance, they are used in complete-active-space self consistent field (CASSCF) algorithms [CITATION] to fix the optimization frame and remove the orbital rotations among degenerate molecular spinors.', '1512.08934-3-5-2': 'Although it is formally possible to symmetrize the eigenvectors after diagonalization, symmetry-adaptation procedures are numerically unstable, especially for large systems, because of the double precision arithmetic used in chemical computations.', '1512.08934-3-5-3': 'Given the recent developments of large-scale two- and four-component relativistic quantum chemistry [CITATION], it is now of practical importance to develop a highly efficient tool for diagonalizing quaternionic matrices that preserves the structure.', '1512.08934-3-6-0': 'The conventional approach to obtaining the symmetry-adapted eigenvectors of a quaternionic matrix is to use the quaternion algebra.', '1512.08934-3-6-1': 'By representing the matrix as an [MATH] matrix of quaternions [CITATION], [EQUATION] with [MATH], a quaternionic version of the Householder transformation or Jacobi rotation can be performed; the algorithm has been reported several times in the literature (e.g., Ref. [CITATION] just to name a few).', '1512.08934-3-6-2': 'The quaternion algebra is nevertheless somewhat complicated, and its computation cannot be easily mapped to highly optimized linear algebra libraries such as BLAS and LAPACK.', '1512.08934-3-7-0': 'In this article, we take an alternative route and report a highly efficient implementation of the so-called Paige-Van Loan algorithm.', '1512.08934-3-7-1': 'The Paige-Van Loan algorithm is a generalization of the standard tridiagonalization algorithm to skew-Hamiltonian matrices, a special case of which is a quaternionic matrix.', '1512.08934-3-7-2': 'The blocking scheme proposed by Kressner [CITATION] is used to map most of the computations to the Level 3 BLAS subroutines and to achieve high efficiency.', '1512.08934-3-7-3': 'In what follows, we first review the underlying algorithms, followed by our efficient implementation.', '1512.08934-3-8-0': '# Algorithm', '1512.08934-3-9-0': '## Paige-Van Loan algorithm', '1512.08934-3-10-0': 'The Householder transformation is often used in tridiagonalization of a Hermitian matrix.', '1512.08934-3-10-1': 'It transforms a Hermitian matrix [MATH] as [EQUATION] such that the first column and row of [MATH] is zero except for the first two elements.', '1512.08934-3-10-2': 'The transformation matrix can be computed using the first column of the input matrix [MATH] as [EQUATION] in which [MATH] denotes a matrix element of [MATH].', '1512.08934-3-10-3': 'The transformation matrix [MATH] satisfies [MATH] (the phase of [MATH] is arbitrary).', '1512.08934-3-10-4': 'Hereafter the subscript denotes the iteration number in the algorithm.', '1512.08934-3-10-5': 'By performing this procedure recursively, one obtains a tridiagonal matrix, i.e., [EQUATION]', '1512.08934-3-10-6': 'When a matrix has skew symmetry [MATH], a similar procedure can be used: [EQUATION]', '1512.08934-3-10-7': 'We note in passing that the transformed matrix has the same symmetry as the original matrix.', '1512.08934-3-11-0': 'The Paige-Van Loan algorithm [CITATION] applies similar transformations to skew-Hamiltonian matrices (see Ref. [CITATION] for its early application to quaternionic matrices).', '1512.08934-3-11-1': 'First, a Householder transformation is applied to the off-diagonal blocks as [EQUATION]', '1512.08934-3-11-2': 'Since the symmetry is preserved, the right half of the matrix does not have to be stored and transformed.', '1512.08934-3-11-3': 'The transformation can be written as [EQUATION]', '1512.08934-3-11-4': 'We then use a Givens rotation to clear out the remaining elements in the first column and row in [MATH]: [EQUATION]', '1512.08934-3-11-5': 'The rotation is generated by [EQUATION] where [MATH], while the other elements of [MATH] are identical to the unit matrix.', '1512.08934-3-11-6': 'Finally we perform another Householder transformation for the [MATH] matrix, yielding [EQUATION]', '1512.08934-3-11-7': 'The transformation reads [EQUATION]', '1512.08934-3-11-8': 'Repeating this procedure yields a block-diagonal matrix, whose diagonal blocks are Hermitian tridiagonal, [EQUATION]', '1512.08934-3-11-9': 'The tridiagonal matrix can be efficiently diagonalized using the ZHBEV subroutine in LAPACK.', '1512.08934-3-11-10': 'In essence, the Paige-Van Loan algorithm takes advantage of the skew symmetry of the off-diagonal blocks and clears them out via successive applications of the Householder transformations and Givens rotations.', '1512.08934-3-11-11': 'It is observed that the operation count of this algorithm is roughly four times that of the diagonalization of [MATH] complex Hermitian matrices (which is about a half of that of [MATH] matrices).', '1512.08934-3-12-0': 'The implementation of this algorithm is relatively straightforward.', '1512.08934-3-12-1': 'For instance, the matlab code that implements the above algorithm has been presented by Loring [CITATION].', '1512.08934-3-12-2': 'However, most of the computations are matrix-vector multiplications and outer-products of vectors, which are computed by the Level 2 BLAS subroutines that have [MATH] floating point operations with [MATH] memory operations.', '1512.08934-3-12-3': 'To attain the efficiency comparable to the state-of-the-art matrix diagonalization algorithms, we ought to introduce a blocked algorithm such that most of the computations are mapped to the Level 3 BLAS subroutines with [MATH] floating point operations and [MATH] memory operations.', '1512.08934-3-13-0': "## Kressner's compact WY-like representation", '1512.08934-3-14-0': "Kressner's compact WY-like representation for structured matrices [CITATION] is reviewed in this section.", '1512.08934-3-14-1': 'In the state-of-the-art implementations of matrix diagonalization (e.g., ZHEEV), the blocked algorithms are used so that most of the computations can be done by the Level 3 BLAS subroutines [CITATION].', '1512.08934-3-14-2': 'These modern algorithms are based on the fact that the product of Householder matrices can be compactly represented by the so-called compact WY representation [CITATION]: [EQUATION] where [MATH] and [MATH] are recursively defined as [EQUATION]', '1512.08934-3-14-3': 'The accumulated transformation matrices for a panel of [MATH] are used to transform the rest of the matrix (see Sec. [REF]), leveraging the efficiency of the Level 3 BLAS subroutines.', '1512.08934-3-15-0': 'Kressner has generalized the compact WY representation to the structured (Hamiltonian and skew-Hamiltonian) eigenvalue problems [CITATION].', '1512.08934-3-15-1': 'The product of the transformation matrices are formally written as [EQUATION] where [MATH] is defined in Eqs. [REF] and [REF], and [MATH] is [EQUATION] with [MATH] and [MATH].', '1512.08934-3-15-2': 'Note the complex conjugation of [MATH] in Eq. [REF].', '1512.08934-3-15-3': 'It has been shown in Ref. [CITATION] that [MATH] can be represented in a compact form, [EQUATION] in which the auxiliary matrices have the following structure: [EQUATION] with [MATH].', '1512.08934-3-15-4': 'All other matrices are in [MATH] and upper-triangular.', '1512.08934-3-16-0': 'The WY-like representation can be proven by induction with respect to [MATH] [CITATION], which in turn provides the way of constructing the auxiliary matrices.', '1512.08934-3-16-1': 'The following equations are equivalent to those presented in Ref. [CITATION] except for complex conjugation and minor corrections.', '1512.08934-3-16-2': 'For the first Householder transformation [Eq. [REF]], the updates are [EQUATION] where we used a short-hand notation [MATH] for a row of blocks of [MATH], i.e., [MATH].', '1512.08934-3-16-3': 'Next, we update these matrices for the Givens rotation [Eq. [REF]]: [EQUATION] in which we introduced [MATH] and [MATH] with [MATH] being the rotation angle [see Eq. [REF]].', '1512.08934-3-16-4': '[MATH] is the ([MATH])-th column of the unit matrix.', '1512.08934-3-16-5': 'Subsequently, the second Householder transformation [Eq. [REF]] updates these matrices as [EQUATION]', '1512.08934-3-16-6': 'The matrices can be easily updated in each step by inserting a column and/or a row to the matrices.', '1512.08934-3-16-7': 'Those after the first iteration are [EQUATION] in which it is assumed that the second element of the Householder vectors is set to 1 [Eq. [REF]].', '1512.08934-3-17-0': '## Blocked updates and miscellaneous optimization', '1512.08934-3-18-0': 'In the standard algorithms for Hermitian diagonalization, one can show that [EQUATION] where [MATH].', '1512.08934-3-18-1': 'Efficient programs accumulate [MATH] for a block of columns, or a panel, to update the entire matrix [MATH] with Eq. [REF] at once.', '1512.08934-3-18-2': 'The details are found in Ref. [CITATION].', '1512.08934-3-19-0': 'We use a similar formula, [EQUATION] where [MATH] and [MATH] are in [MATH] and defined as [EQUATION] [MATH] and [MATH] are likewise defined with [MATH].', '1512.08934-3-19-1': "[MATH] and [MATH]'s are computed in a three-step procedure as follows.", '1512.08934-3-19-2': 'After the first Householder transformation, they are [EQUATION]', '1512.08934-3-19-3': 'Note that the auxiliary vector [MATH] is computed in the update of [MATH] and [MATH] in Eq. [REF].', '1512.08934-3-19-4': 'They are transformed by the Givens rotation as [EQUATION]', '1512.08934-3-19-5': 'Finally, after the second Householder transformation, they become [EQUATION] [MATH] and [MATH] are computed similarly.', '1512.08934-3-20-0': 'The block update algorithm is sketched in Figure [REF].', '1512.08934-3-20-1': 'First, we construct [MATH], [MATH], [MATH], and [MATH] using Eqs. [REF]-[REF] while accumulating [MATH], [MATH], [MATH], and [MATH] using Eqs. [REF]-[REF].', '1512.08934-3-20-2': 'The first column is transformed to that in a tridiagonal form during this process.', '1512.08934-3-20-3': 'Next, Eq. [REF] is applied to the second column, transforming it to the corresponding column of [MATH].', '1512.08934-3-20-4': 'We then update all the auxiliary matrices and transform the second column to a tridiagonal form.', '1512.08934-3-20-5': 'This procedure is repeated until we accumulate the matrices for [MATH] columns.', '1512.08934-3-20-6': 'Finally, Eq. [REF] is used to update the rest of the matrix to yield [MATH], which is efficiently performed by the Level 3 BLAS subroutines.', '1512.08934-3-20-7': 'The problem size is then reduced from [MATH] to [MATH].', '1512.08934-3-20-8': 'The above steps are recursively executed to complete the tridiagonalization.', '1512.08934-3-21-0': 'At this point, it is worth noting that several optimizations are possible in the implementation of above formulas.', '1512.08934-3-21-1': 'First, all of the elements in the first row of [MATH] is zero, and therefore, the first row can be removed from the computation.', '1512.08934-3-21-2': "The first rows of [MATH]'s and [MATH]'s can also be omitted.", '1512.08934-3-21-3': '[MATH] is then the first [MATH] columns of the unit matrix, which can be computed easily on the fly without storing it.', '1512.08934-3-21-4': "Second, because all of the elements of [MATH] in Eq. [REF] are identically zero, we do not need to keep [MATH] and [MATH]'s except for the latest column; this column is the only one that contributes to the panel update Eq. [REF] for the untransformed part of the matrix.", '1512.08934-3-21-5': "Therefore we overwrite [MATH] and [MATH]'s in every iteration to reduce the memory requirement.", '1512.08934-3-21-6': 'All of the elements of [MATH] in Eq. [REF] are zero, whereas the contributions from [MATH] in Eq. [REF] are trivial to compute.', '1512.08934-3-21-7': 'Finally, [MATH] and [MATH] (and other quantities) are stored in contiguous memory so that matrix-matrix multiplications can be fused.', '1512.08934-3-21-8': 'The memory size required for the auxiliary matrices after these optimizations is found to be around [MATH] (note that the optimal cache size for ZHEEV is roughly [MATH]).', '1512.08934-3-22-0': 'Our implementation takes advantage of efficient functions in BLAS and LAPACK as much as possible.', '1512.08934-3-22-1': 'The Householder transformation is generated by ZLARFG, while the Givens rotation parameters are calculated by ZLARTG.', '1512.08934-3-22-2': 'At the end of the algorithm, the [MATH] tridiagonal matrix is diagonalized by ZHBEV.', '1512.08934-3-22-3': 'Matrix-vector and matrix-matrix multiplications are performed by ZTRMV, ZGEMV, and ZGEMM (or ZGEMM3M when the Intel Math Kernel Library is used), respectively.', '1512.08934-3-22-4': 'In the unblocked code, ZGERC and ZGERU are frequently used to calculate the outer product of vectors.', '1512.08934-3-22-5': 'The Level 2 and 3 BLAS subroutines are responsible for shared-memory parallelization in our program.', '1512.08934-3-22-6': 'The program was written in C++.', '1512.08934-3-23-0': 'To our knowledge, this is the first implementation of the blocked Paige-Van Loan algorithm for quaternionic matrices, whereas that for general real (skew-)Hamiltonian matrices has been implemented in the HAPACK package [CITATION].', '1512.08934-3-23-1': 'Though diagonalization of a [MATH] quaternionic matrix can be mapped to diagonalization of a [MATH] skew-Hamiltonian matrix [CITATION], direct solution in complex arithmetic is more efficient because doubling the size increases the cost of diagonalization by a factor of 8, while diagonalization in complex arithmetic is only 3-4 times more expensive than that in real arithmetic (note that matrix multiplication in complex arithmetic is 3 or 4 times more expensive than that in real arithmetic using ZGEMM3M or ZGEMM routines).', '1512.08934-3-24-0': '# Timing results', '1512.08934-3-25-0': 'All of the timing data below were obtained on a computer node equipped with two Xeon E5-2650 2.0 GHz with the maximum Turbo Boost frequency 2.8 GHz (Sandy Bridge, 8 cores each, 16 cores in total).', '1512.08934-3-25-1': 'Intel Math Kernel Library (version 11.3, 2016.0.109) was used for BLAS and LAPACK functions with threading support.', '1512.08934-3-25-2': 'The compiler used was g++ 5.2.0 with the optimization flags "-O3 -DNDEBUG -mavx."', '1512.08934-3-25-3': 'Figure [REF] summarizes the timing for diagonalizing random matrices with the quaternionic structure using the blocked and unblocked algorithms described above ([MATH] was used).', '1512.08934-3-25-4': "The timing is compared with MKL's ZHEEV, which does not preserve the structure of the eigenvectors.", '1512.08934-3-25-5': 'It is apparent that the use of the blocked algorithm is essential to achieving high efficiency comparable to the state-of-the-art implementation of standard diagonalization.', '1512.08934-3-25-6': "Diagonalizing a [MATH] matrix took 9.5, 33.0, and 12.6 minutes using the blocked and unblocked versions of our code and MKL's ZHEEV using 16 CPU cores.", '1512.08934-3-25-7': 'When only 1 CPU core is used, the wall times were 42.8, 67.7, and 85.6 minutes, respectively.', '1512.08934-3-25-8': "Note that the calculations with 1 CPU core were accelerated by the CPU's Turbo Boost (up to 2.8 GHz).", '1512.08934-3-26-0': 'The shared memory parallelization of our code appears less effective than that in the MKL implementation of ZHEEV, indicating that there is still some room for improvement.', '1512.08934-3-26-1': 'This is in part because our program is only threaded by the underlying BLAS functions and because the above blocked algorithm involves considerably more matrix-vector multiplications in the computation of the WY-like representation than the one used in the standard diagonalization.', '1512.08934-3-26-2': "Although our new implementation does outperform MKL's ZHEEV with 16 CPU cores by 25-30%, the detailed comparison with ZHEEV is not straightforward because of the difference in design of the two programs.", '1512.08934-3-27-0': 'The dependence of the efficiency with respect to the block size [MATH] is presented in Figure [REF] for [MATH] matrices using 16 CPU cores.', '1512.08934-3-27-1': 'The timing improves dramatically till [MATH], has the minimum at around [MATH], and then deteriorates slowly as [MATH] becomes larger.', '1512.08934-3-27-2': 'The initial improvement with respect to [MATH] is attributed to the increased efficiency of the Level 3 BLAS subroutines with the increased panel sizes.', '1512.08934-3-27-3': 'The deterioration with large [MATH] is due to the use of the Level 2 BLAS subroutines within each panel [Eqs. [REF]-[REF] and [REF]-[REF]], although it competes with further efficiency improvement of the Level 3 BLAS subroutines in Eq. [REF].', '1512.08934-3-27-4': 'The optimal [MATH] is expected to be machine dependent (e.g., L2 and L3 cache sizes).', '1512.08934-3-28-0': '# Conclusions', '1512.08934-3-29-0': 'We have implemented a blocked version of the Paige-Van Loan algorithm into an efficient program, following the earlier work by Kressner [CITATION].', '1512.08934-3-29-1': 'Shared-memory parallelization is performed by the underlying BLAS and LAPACK library.', '1512.08934-3-29-2': "When a single CPU core is used, the elapsed time for structure-preserving diagonalization using this program is about a half of that using ZHEEV without symmetry treatment; with 16 CPU cores, the program still outperforms the MKL's ZHEEV implementation by about 25-30%.", '1512.08934-3-29-3': 'The source code is publicly available under the FreeBSD license [CITATION] so that it can be integrated into any large-scale relativistic quantum chemistry program packages.', '1512.08934-3-29-4': 'The program has been interfaced to the bagel package [CITATION].', '1512.08934-3-29-5': 'The parallel implementation will be investigated in the near future.', '1512.08934-3-30-0': 'This article is part of the special issue in honor of the 60th birthday of Hans Jorgen Aa.', '1512.08934-3-30-1': 'Jensen.', '1512.08934-3-30-2': 'The author thanks Jack Poulson for helpful suggestions and Terry Loring for useful comments.', '1512.08934-3-30-3': 'This work has been supported by the NSF CAREER Award (CHE-1351598).', '1512.08934-3-30-4': 'The author is an Alfred P. Sloan Research Fellow.'}

1901.03420

{'1901.03420-1-0-0': 'We introduce a physically motivated minimal model for the electronic structure of twisted bilayer graphene (tBLG), which incorporates the crucial role of lattice relaxation.', '1901.03420-1-0-1': 'Our model, based on [MATH] perturbation theory, combines the accuracy of DFT calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models.', '1901.03420-1-0-2': 'The inclusion of relaxation significantly changes the bandstructure at the first magic-angle twist corresponding to flat bands near the Fermi level (the "low-energy" states), and eliminates the appearance of a second magic-angle twist.', '1901.03420-1-0-3': 'We argue that the minimal model for the low-energy states of tBLG consists of ten bands, necessary to capture the changes in electronic states as a function of twist angle.', '1901.03420-1-0-4': 'We also provide information on the nature of these bands through their wavefunctions, which is closely tied to the features of the atomic relaxation.', '1901.03420-1-1-0': 'The discovery of correlated phases in twisted bilayer graphene (tBLG) has generated much interest in this structurally and compositionally rather simple system; it has emerged as a new platform for tunable electronic correlations, and for exploring of the nature of unconventional superconductivity[CITATION].', '1901.03420-1-1-1': 'The challenge in modeling these phenomena from an atomistic perspective is that the actual structure of tBLG near the magic-angle twist ([MATH]) where correlated behavior is observed, consists of a large number of atoms, exceeding [MATH].', '1901.03420-1-1-2': 'To make progress from the theoretical point of view, a minimal model is needed that can capture the essence of single-particle states near the Fermi level ("low-energy" states).', '1901.03420-1-1-3': 'Such a model should reproduce the energy spectrum as a function of their relative twist angle with reasonable accuracy and with the required fidelity in capturing the nature of low-energy states.', '1901.03420-1-1-4': 'The appearance of correlated behavior is related to bands with very low dispersion ("flat" bands) caused by interlayer hybridization between the two Dirac cones from the different layers[CITATION].', '1901.03420-1-2-0': 'Existing models based on DFT calculations [CITATION] or large supercell tight-binding Hamiltonians [CITATION] are too complex to form the basis of a realistic many-body theory.', '1901.03420-1-2-1': 'At the other extreme, simplified continuum models allow for efficient calculations, but are based on heuristic arguments about the nature of the relevant electronic states [CITATION].', '1901.03420-1-2-2': 'An important feature of the physical system is the presence of atomic relaxation near the magic-angle twist, which has significant effects on the low-energy bandstructure [CITATION].', '1901.03420-1-2-3': 'Many simplified models for the flat bands of magic-angle tBLG have been proposed based on symmetry analysis, but they rely on empirical parameterization and are designed for only the magic-angle twist configuration[CITATION], typically ignoring atomic relaxation.', '1901.03420-1-2-4': 'Here, we present an ab initio [MATH] perturbation continuum model for tBLG which accurately accounts for the effects of atomic relaxation.', '1901.03420-1-2-5': 'Our model reproduces the results of DFT-quality tight-binding hamiltonians but at a smaller computational cost and, more importantly, it applies to all twist angles near the magic-angle value.', '1901.03420-1-2-6': 'Using this model, we draw conclusions on the behavior of correlated states at small twist angles, including the interesting result that there are no flat bands in the range of the expected second magic-angle twist ([MATH]).', '1901.03420-1-2-7': 'For reference, we compare our continuum model to the seminal and widely employed [MATH] model of Bitztriter and MacDonald[CITATION] (BMD in the following), and we adopt their dimensionless parameter [MATH] for describing the twist-angle [MATH], where [MATH] is the Fermi velocity, [MATH] is the wave-vector set by the moire length scale and [MATH] is the effective interlayer coupling strength.', '1901.03420-1-3-0': 'Within [MATH] perturbation theory, the set of Bloch states of the two graphene layers is augmented by the addition of interlayer couplings due to the twist-angle induced Umklapp scattering process.', '1901.03420-1-3-1': 'As the low-energy electronic structure of tBLG is dominated by a pair of Dirac cones, the momentum expansion can be carried out about one copy of the cone at a valley K point.', '1901.03420-1-3-2': 'Taking also into account spin degeneracy, each band represents four electronic states in a real system [CITATION].', '1901.03420-1-3-3': 'Here we introduce an expanded ab initio [MATH] model which gives a more complete physical picture of the tBLG system.', '1901.03420-1-3-4': 'Our model has three new key ingredients:', '1901.03420-1-4-0': '(1) relaxation of the bilayer system [CITATION], including the out-of-plane relaxation of different regions as well as the in-plane strain corrections to the Hamiltonian of the individual monolayers;', '1901.03420-1-5-0': '(2) terms beyond the first shell of couplings in the [MATH] continuum model, which are necessary to capture the changes in stacking order at small angles;', '1901.03420-1-6-0': '(3) inclusion of [MATH]-dependent terms, which allow the [MATH] model to reproduce more accurately the particle-hole asymmetry of realistic ab initio bandstructures.', '1901.03420-1-7-0': 'The [MATH] terms are directly computed from an ab initio tight-binding Hamiltonian model [CITATION] for supercells spanning the twist-angle range [MATH].', '1901.03420-1-7-1': 'These terms have smooth dependence on [MATH], allowing for interpolation between the specific twist angles that correspond to finite supercells, to generate a model valid for any desired angle in that range.', '1901.03420-1-7-2': 'We relegate the detailed description of the extended Hamiltonian and the procedure for obtaining the relevant terms of the continuum model to a companion paper [CITATION].', '1901.03420-1-8-0': 'Our continuum model affords a natural interpretation of the electronic structure of tBLG at small twist angles, which is derived directly from the atomic relaxation so we describe this aspect first.', '1901.03420-1-8-1': 'For twist-angle [MATH] smaller than a critical value [MATH], the local atomic structure near the [MATH] and [MATH] stackings of the two layers becomes independent of [MATH].', '1901.03420-1-8-2': 'This creates a pattern of small circular domains of [MATH] stacking and large triangular domains of [MATH] stackings.', '1901.03420-1-8-3': 'Domain walls ([MATH]) of intermediate stacking separate the [MATH] and [MATH] domains and connect the [MATH] regions.', '1901.03420-1-8-4': 'This creates local electronic environments which are locked-in with respect to changing twist-angle for [MATH], where the tBLG system consists of a few fixed elements[CITATION], and only their length scale changes for decreasing twist angle.', '1901.03420-1-8-5': 'These elements are: the [MATH] regions which have a local twist of [MATH], which is independent of the overall twist angle [MATH] between the two layers, the [MATH] and [MATH] regions with negligible local twist.', '1901.03420-1-8-6': 'Moreover, the diameter of the [MATH] regions and the width of the [MATH] regions are approximately equal and remain unchanged for [MATH][CITATION].', '1901.03420-1-8-7': 'These features are shown in Fig. [REF] for [MATH].', '1901.03420-1-9-0': 'The relaxation in tBLG is described by two simultaneous effects.', '1901.03420-1-9-1': 'In-plane relaxation decreases the area of the high stacking energy [MATH] region while it increases that of low stacking energy [MATH] regions.', '1901.03420-1-9-2': 'Out-of-plane relaxation causes corrugation, increasing the vertical separation between the [MATH] regions from the equilibrium distance in [MATH] stacking of 3.35 to 3.59 , a substantial change ([MATH]%).', '1901.03420-1-9-3': 'The reduction in the size of [MATH] stacking can be understood as a minimization of planar stress energy and stacking energies, and has been modeled through various methods [CITATION] leading to a relaxed pattern in agreement with experimental results [CITATION].', '1901.03420-1-9-4': 'The role of vertical relaxation in experimental devices is less understood, as only free-standing tBLG has been modeled.', '1901.03420-1-9-5': 'Experimental tBLG devices are typically encapsulated in hexagonal Boron Nitride, so the actual corrugation may be reduced compared to the free-standing case.', '1901.03420-1-9-6': 'To take this into account, we consider two limits of the vertical relaxation: a "Full" relaxation model (free-standing bilayer result) and a "Flat" model with constant interlayer distance equal to the average of [MATH] and [MATH] interlayer distances (3.47 ).', '1901.03420-1-9-7': 'The magic angle predicted by the fully relaxed model, [MATH], is closer to the experimentally observed value [MATH].', '1901.03420-1-10-0': 'The low-energy electronic states are directly associated with and derive from the presence of the relaxation-induced structural elements described earlier.', '1901.03420-1-10-1': 'A minimal model that can adequately capture all the essential features of the low-energy states consists of 10 bands; a similar model has been introduced on the basis of symmetry arguments[CITATION].', '1901.03420-1-10-2': 'This model comprises three orbitals on a triangular lattice formed by the [MATH] sites, one of [MATH]-like character ([MATH]) and two of [MATH]-like character ([MATH]), three orbitals on a Kagome lattice formed by the domain walls, and four orbitals on a honeycomb lattice, two for each of the [MATH] and [MATH] domains.', '1901.03420-1-10-3': 'Each of the orbitals form sublattice Hamiltonians ([MATH]) with independent on-site energies [MATH] and hopping terms [MATH] and have independent effective chemical potentials [MATH] expressed in terms of [MATH] and [MATH], [EQUATION] with indices [MATH] running over the orbitals in each sublattice, [MATH], and [MATH] and [MATH] for [MATH], [MATH] and [MATH] respectively.', '1901.03420-1-10-4': 'The sublattice Hamiltonians are also coupled to one another through inter-orbital hoppings ([MATH]) so the full Hamiltonian is given by (for a complete description see SM): [EQUATION]', '1901.03420-1-10-5': 'To compare our ab initio [MATH] results to this model, we project non-orthogonal wavefunctions that satisfy the symmetry conditions, shown in Fig. [REF], from band structure calculations.', '1901.03420-1-10-6': 'We fit the model parameters for [MATH], to reproduce the ab initio bands (see SM).', '1901.03420-1-10-7': 'In Fig. [REF]c, the [MATH] for the [MATH] and [MATH] type orbitals reach the critical behavior [MATH] only at the magic angle value [MATH].', '1901.03420-1-10-8': 'The flat bands near the magic angle have [MATH] and [MATH] character (see Fig. [REF]a), showing that the coupling between these states is a necessary ingredient of the model if it is to capture the electronic structure as a function of twist angle.', '1901.03420-1-10-9': 'In particular, the orbital character of the electron and hole bands flips at the magic-angle twist: the hole band has [MATH] character for [MATH] and switches to [MATH] character for [MATH], while the electron band has the reverse character.', '1901.03420-1-10-10': 'We emphasize that the form of these wavefunctions is not sensitive to the twist-angle, and is robust for twist angles within [MATH] of the magic angle.', '1901.03420-1-11-0': 'Two other important parameters in the [MATH] model are the effective interlayer coupling between orbitals of the same sublattice label, [MATH] or [MATH], and that between orbitals of different labels, [MATH] or [MATH].', '1901.03420-1-11-1': 'These nearest-neighbor interlayer couplings have been labeled [MATH] in previous studies and have a simple geometric interpretation: [MATH] is the interlayer electronic coupling at the [MATH] sites and [MATH] is the coupling at [MATH] sites, averaged over the entire moire cell.', '1901.03420-1-11-2': 'The values of these [MATH] parameters depend strongly on the twist angle [MATH].', '1901.03420-1-11-3': 'As the lattice relaxes, the relative size of the [MATH] regions is greatly reduced while that of the [MATH] regions is increased, causing a reduction in the value of [MATH] and a modest increase in the value of [MATH].', '1901.03420-1-11-4': 'This dependence is shown in Fig. [REF]c for the Full and the Flat relaxation models.', '1901.03420-1-11-5': 'The overall [MATH] dependence of the ratio [MATH] is not sensitive to the relaxed height assumption.', '1901.03420-1-11-6': 'The Flat model has a larger ratio as the Full relaxation assumption moves the [MATH] sites closer together (increasing their coupling and the [MATH] value) while moving the [MATH] sites farther apart (reducing their coupling and the [MATH] value).', '1901.03420-1-12-0': 'To elucidate the salient features of the single-particle model, we study three related indicators of the flat-band phenomenon as a function of [MATH]: the Fermi velocity ([MATH]), the bandwidth ([MATH]), and the band gap ([MATH]).', '1901.03420-1-12-1': 'These are shown in Fig [REF].', '1901.03420-1-12-2': 'All three are calculated for both the electron and the hole sides of the flat-band manifold.', '1901.03420-1-12-3': 'The model without relaxation shows large discrepancies between the extrema of the Fermi velocity, gap, and bandwidth, and the electron and hole features have little in common.', '1901.03420-1-12-4': 'The two models (Flat and Full) that include relaxation show more regular dependence on [MATH] and closer correspondence between the electron and hole bands.', '1901.03420-1-12-5': 'The bandwidth for the hole band is always smaller than that of the electron band, and the hole band achieves its minimum twice.', '1901.03420-1-12-6': 'In general, [MATH] does not coincide with bandwidth minima.', '1901.03420-1-12-7': 'We thus draw the important conclusion that the magic-angle is not a single value, but rather a range of [MATH] which spans the extrema in these key features.', '1901.03420-1-12-8': 'This range for the Full relaxed model is [MATH] and [MATH] for the Flat model.', '1901.03420-1-12-9': 'The bandstructures for both models are indistinguishable but occur at different twist-angle values, shifted by [MATH] (the critical value is smaller for the Flat model).', '1901.03420-1-13-0': 'An interesting behavior of the Full relaxed model occurs at the center of the magic-angle regime: although the Dirac cone still has symmetric dispersion near the K-point, the hole band dispersion is such that near the [MATH] point its energy energy is higher than the Fermi level.', '1901.03420-1-13-1': 'Thus the charge neutrality point does not occur at the Dirac point energy.', '1901.03420-1-13-2': 'This effect persists in all of the ab initio [MATH] models (even without relaxation), and is a behavior that can be observed in other tight-binding models in the literature[CITATION] but was not explicitly pointed out before.', '1901.03420-1-13-3': 'For transport measurement, this behavior would result in a range of [MATH] in twist angle where the charge neutrality point of the flat bands does not align with the resistive Dirac-point feature, as well as a reduction in the overall resistivity at the Dirac-point energy.', '1901.03420-1-14-0': 'Another important result of our calculations including atomic relaxation in tBLG is the suppression of the second magic-angle twist, defined as a smaller twist angle at which [MATH] [CITATION].', '1901.03420-1-14-1': 'In Fig. [REF] we show the Fermi velocity as predicted from the BMD model and from our unrelaxed and fully relaxed ab initio [MATH] models.', '1901.03420-1-14-2': 'Although our unrelaxed model shows similar behavior to the BMD model with a second magic angle occurring near [MATH], the inclusion of atomic relaxation removes this feature in near [MATH].', '1901.03420-1-14-3': 'In both the BMD and unrelaxed models, the second magic angle near [MATH] is only defined by [MATH], but there are no extrema in the bandwidth or bandgap at this angle.', '1901.03420-1-14-4': 'This can be understood from a careful study of how the ratio [MATH] affects the stability of the flat-band feature [CITATION].', '1901.03420-1-14-5': 'The Full relaxed model shows a possible dip in the Fermi velocity, corresponding to a possible second magic angle, near [MATH] or [MATH].', '1901.03420-1-14-6': 'However our model needs further refinement to make accurate predictions for very small [MATH], as our current model includes terms only up to the third shell in momentum space.', '1901.03420-1-14-7': 'We are not certain this dip in [MATH] will persist when additional terms are included.', '1901.03420-1-14-8': 'The lattice relaxation in tBLG becomes increasingly sharp on the moire length scale as the twist angle decreases [CITATION], and sharper features in the relaxation introduce additional important couplings in the [MATH] model at larger momenta.', '1901.03420-1-15-0': 'We thank Daniel Massatt, Hoi Chun Po, Alex Kruchkov, Grigory Tarnopolskiy, Pablo Jarillo-Herrero, Hyobin Yoo, Rebecca Engelke, and Philip Kim for useful discussions.', '1901.03420-1-15-1': 'This work was supported by ARO MURI Award W911NF-14-0247 and by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319.', '1901.03420-1-15-2': 'The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.'}

{'1901.03420-2-0-0': 'We introduce a physically motivated minimal model for the electronic structure of twisted bilayer graphene (tBLG), which incorporates the crucial role of lattice relaxation.', '1901.03420-2-0-1': 'Our model, based on [MATH] perturbation theory, combines the accuracy of DFT calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models.', '1901.03420-2-0-2': 'The inclusion of relaxation significantly changes the bandstructure at the first magic-angle twist corresponding to flat bands near the Fermi level (the "low-energy" states), and eliminates the appearance of a second magic-angle twist.', '1901.03420-2-0-3': 'We argue that the minimal model for the low-energy states of tBLG consists of ten bands, necessary to capture the changes in electronic states as a function of twist angle.', '1901.03420-2-0-4': 'We also provide information on the nature of these bands through their wavefunctions, which is closely tied to the features of the atomic relaxation.', '1901.03420-2-1-0': 'The discovery of correlated phases in twisted bilayer graphene (tBLG) has generated much interest in this structurally and compositionally rather simple system; it has emerged as a new platform for tunable electronic correlations, and for exploring of the nature of unconventional superconductivity[CITATION].', '1901.03420-2-1-1': 'The challenge in modeling these phenomena from an atomistic perspective is that the actual structure of tBLG near the magic-angle twist ([MATH]) where correlated behavior is observed, consists of a large number of atoms, exceeding [MATH].', '1901.03420-2-1-2': 'To make progress from the theoretical point of view, a minimal model is needed that can capture the essence of single-particle states near the Fermi level ("low-energy" states).', '1901.03420-2-1-3': 'Such a model should reproduce the energy spectrum as a function of their relative twist angle with reasonable accuracy and with the required fidelity in capturing the nature of low-energy states.', '1901.03420-2-1-4': 'The appearance of correlated behavior is related to bands with very low dispersion ("flat" bands) caused by interlayer hybridization between the two Dirac cones from the different layers[CITATION].', '1901.03420-2-2-0': 'Existing models based on DFT calculations [CITATION] or large supercell tight-binding Hamiltonians [CITATION] are too complex to form the basis of a realistic many-body theory.', '1901.03420-2-2-1': 'At the other extreme, simplified continuum models allow for efficient calculations, but are based on heuristic arguments about the nature of the relevant electronic states [CITATION].', '1901.03420-2-2-2': 'An important feature of the physical system is the presence of atomic relaxation near the magic-angle twist, which has significant effects on the low-energy bandstructure [CITATION].', '1901.03420-2-2-3': 'Many simplified models for the flat bands of magic-angle tBLG have been proposed based on symmetry analysis, but they rely on empirical parameterization and are designed for only the magic-angle twist configuration[CITATION], typically ignoring atomic relaxation.', '1901.03420-2-2-4': 'Here, we present an ab initio [MATH] perturbation continuum model for tBLG which accurately accounts for the effects of atomic relaxation.', '1901.03420-2-2-5': 'Our model reproduces the results of DFT-quality tight-binding hamiltonians but at a smaller computational cost and, more importantly, it applies to all twist angles near the magic-angle value.', '1901.03420-2-2-6': 'Using this model, we draw conclusions on the behavior of correlated states at small twist angles, including the interesting result that there are no flat bands in the range of the expected second magic-angle twist ([MATH]).', '1901.03420-2-2-7': 'For reference, we compare our continuum model to the seminal and widely employed [MATH] model of Bitztriter and MacDonald[CITATION] (BMD in the following), and we adopt their dimensionless parameter [MATH] for describing the twist-angle [MATH], where [MATH] is the Fermi velocity, [MATH] is the wave-vector set by the moire length scale and [MATH] is the effective interlayer coupling strength.', '1901.03420-2-3-0': 'Within [MATH] perturbation theory, the set of Bloch states of the two graphene layers is augmented by the addition of interlayer couplings due to the twist-angle induced Umklapp scattering process.', '1901.03420-2-3-1': 'As the low-energy electronic structure of tBLG is dominated by a pair of Dirac cones, the momentum expansion can be carried out about one copy of the cone at a valley K point.', '1901.03420-2-3-2': 'Taking also into account spin degeneracy, each band represents four electronic states in a real system [CITATION].', '1901.03420-2-3-3': 'Here we introduce an expanded ab initio [MATH] model which gives a more complete physical picture of the tBLG system.', '1901.03420-2-3-4': 'Our model has three new key ingredients:', '1901.03420-2-4-0': '(1) relaxation of the bilayer system [CITATION], including the out-of-plane relaxation of different regions as well as the in-plane strain corrections to the Hamiltonian of the individual monolayers;', '1901.03420-2-5-0': '(2) terms beyond the first shell of couplings in the [MATH] continuum model, which are necessary to capture the changes in stacking order at small angles;', '1901.03420-2-6-0': '(3) inclusion of [MATH]-dependent terms, which allow the [MATH] model to reproduce more accurately the particle-hole asymmetry of realistic ab initio bandstructures.', '1901.03420-2-7-0': 'The [MATH] terms are directly computed from an ab initio tight-binding Hamiltonian model [CITATION] for supercells spanning the twist-angle range [MATH].', '1901.03420-2-7-1': 'These terms have smooth dependence on [MATH], allowing for interpolation between the specific twist angles that correspond to finite supercells, to generate a model valid for any desired angle in that range.', '1901.03420-2-7-2': 'We relegate the detailed description of the extended Hamiltonian and the procedure for obtaining the relevant terms of the continuum model to a companion paper [CITATION].', '1901.03420-2-8-0': 'Our continuum model affords a natural interpretation of the electronic structure of tBLG at small twist angles, which is derived directly from the atomic relaxation so we describe this aspect first.', '1901.03420-2-8-1': 'For twist-angle [MATH] smaller than a critical value [MATH], the local atomic structure near the [MATH] and [MATH] stackings of the two layers becomes independent of [MATH].', '1901.03420-2-8-2': 'This creates a pattern of small circular domains of [MATH] stacking and large triangular domains of [MATH] stackings.', '1901.03420-2-8-3': 'Domain walls ([MATH]) of intermediate stacking separate the [MATH] and [MATH] domains and connect the [MATH] regions.', '1901.03420-2-8-4': 'This creates local electronic environments which are locked-in with respect to changing twist-angle for [MATH], where the tBLG system consists of a few fixed elements[CITATION], and only their length scale changes for decreasing twist angle.', '1901.03420-2-8-5': 'These elements are: the [MATH] regions which have a local twist of [MATH], which is independent of the overall twist angle [MATH] between the two layers, the [MATH] and [MATH] regions with negligible local twist.', '1901.03420-2-8-6': 'Moreover, the diameter of the [MATH] regions and the width of the [MATH] regions are approximately equal and remain unchanged for [MATH][CITATION].', '1901.03420-2-8-7': 'These features are shown in Fig. [REF] for [MATH].', '1901.03420-2-9-0': 'The relaxation in tBLG is described by two simultaneous effects.', '1901.03420-2-9-1': 'In-plane relaxation decreases the area of the high stacking energy [MATH] region while it increases that of low stacking energy [MATH] regions.', '1901.03420-2-9-2': 'Out-of-plane relaxation causes corrugation, increasing the vertical separation between the [MATH] regions from the equilibrium distance in [MATH] stacking of 3.35 to 3.59 , a substantial change ([MATH]%).', '1901.03420-2-9-3': 'The reduction in the size of [MATH] stacking can be understood as a minimization of planar stress energy and stacking energies, and has been modeled through various methods [CITATION] leading to a relaxed pattern in agreement with experimental results [CITATION].', '1901.03420-2-9-4': 'The role of vertical relaxation in experimental devices is less understood, as only free-standing tBLG has been modeled.', '1901.03420-2-9-5': 'Experimental tBLG devices are typically encapsulated in hexagonal Boron Nitride, so the actual corrugation may be reduced compared to the free-standing case.', '1901.03420-2-9-6': 'To take this into account, we consider two limits of the vertical relaxation: a "Full" relaxation model (free-standing bilayer result) and a "Flat" model with constant interlayer distance equal to the average of [MATH] and [MATH] interlayer distances (3.47 ).', '1901.03420-2-9-7': 'The magic angle predicted by the fully relaxed model, [MATH], is closer to the experimentally observed value [MATH].', '1901.03420-2-10-0': 'The low-energy electronic states are directly associated with and derive from the presence of the relaxation-induced structural elements described earlier.', '1901.03420-2-10-1': 'A minimal model that can adequately capture all the essential features of the low-energy states consists of 10 bands; a similar model has been introduced on the basis of symmetry arguments[CITATION].', '1901.03420-2-10-2': 'This model comprises three orbitals on a triangular lattice formed by the [MATH] sites, one of [MATH]-like character ([MATH]) and two of [MATH]-like character ([MATH]), three orbitals on a Kagome lattice formed by the domain walls, and four orbitals on a honeycomb lattice, two for each of the [MATH] and [MATH] domains.', '1901.03420-2-10-3': 'Each of the orbitals form sublattice Hamiltonians ([MATH]) with independent on-site energies [MATH] and hopping terms [MATH] and have independent effective chemical potentials [MATH] expressed in terms of [MATH] and [MATH], [EQUATION] with indices [MATH] running over the orbitals in each sublattice, [MATH], and [MATH] and [MATH] for [MATH], [MATH] and [MATH] respectively.', '1901.03420-2-10-4': 'The sublattice Hamiltonians are also coupled to one another through inter-orbital hoppings ([MATH]) so the full Hamiltonian is given by (for a complete description see SM): [EQUATION]', '1901.03420-2-10-5': 'To compare our ab initio [MATH] results to this model, we project non-orthogonal wavefunctions that satisfy the symmetry conditions, shown in Fig. [REF], from band structure calculations.', '1901.03420-2-10-6': 'We fit the model parameters for [MATH], to reproduce the ab initio bands (see SM).', '1901.03420-2-10-7': 'In Fig. [REF]c, the [MATH] for the [MATH] and [MATH] type orbitals reach the critical behavior [MATH] only at the magic angle value [MATH].', '1901.03420-2-10-8': 'The flat bands near the magic angle have [MATH] and [MATH] character (see Fig. [REF]a), showing that the coupling between these states is a necessary ingredient of the model if it is to capture the electronic structure as a function of twist angle.', '1901.03420-2-10-9': 'In particular, the orbital character of the electron and hole bands flips at the magic-angle twist: the hole band has [MATH] character for [MATH] and switches to [MATH] character for [MATH], while the electron band has the reverse character.', '1901.03420-2-10-10': 'We emphasize that the form of these wavefunctions is not sensitive to the twist-angle, and is robust for twist angles within [MATH] of the magic angle.', '1901.03420-2-11-0': 'Two other important parameters in the [MATH] model are the effective interlayer coupling between orbitals of the same sublattice label, [MATH] or [MATH], and that between orbitals of different labels, [MATH] or [MATH].', '1901.03420-2-11-1': 'These nearest-neighbor interlayer couplings have been labeled [MATH] in previous studies and have a simple geometric interpretation: [MATH] is the interlayer electronic coupling at the [MATH] sites and [MATH] is the coupling at [MATH] sites, averaged over the entire moire cell.', '1901.03420-2-11-2': 'The values of these [MATH] parameters depend strongly on the twist angle [MATH].', '1901.03420-2-11-3': 'As the lattice relaxes, the relative size of the [MATH] regions is greatly reduced while that of the [MATH] regions is increased, causing a reduction in the value of [MATH] and a modest increase in the value of [MATH].', '1901.03420-2-11-4': 'This dependence is shown in Fig. [REF]c for the Full and the Flat relaxation models.', '1901.03420-2-11-5': 'The overall [MATH] dependence of the ratio [MATH] is not sensitive to the relaxed height assumption.', '1901.03420-2-11-6': 'The Flat model has a larger ratio as the Full relaxation assumption moves the [MATH] sites closer together (increasing their coupling and the [MATH] value) while moving the [MATH] sites farther apart (reducing their coupling and the [MATH] value).', '1901.03420-2-12-0': 'To elucidate the salient features of the single-particle model, we study three related indicators of the flat-band phenomenon as a function of [MATH]: the Fermi velocity ([MATH]), the bandwidth ([MATH]), and the band gap ([MATH]).', '1901.03420-2-12-1': 'These are shown in Fig [REF].', '1901.03420-2-12-2': 'All three are calculated for both the electron and the hole sides of the flat-band manifold.', '1901.03420-2-12-3': 'The model without relaxation shows large discrepancies between the extrema of the Fermi velocity, gap, and bandwidth, and the electron and hole features have little in common.', '1901.03420-2-12-4': 'The two models (Flat and Full) that include relaxation show more regular dependence on [MATH] and closer correspondence between the electron and hole bands.', '1901.03420-2-12-5': 'The bandwidth for the hole band is always smaller than that of the electron band, and the hole band achieves its minimum twice.', '1901.03420-2-12-6': 'In general, [MATH] does not coincide with bandwidth minima.', '1901.03420-2-12-7': 'We thus draw the important conclusion that the magic-angle is not a single value, but rather a range of [MATH] which spans the extrema in these key features.', '1901.03420-2-12-8': 'This range for the Full relaxed model is [MATH] and [MATH] for the Flat model.', '1901.03420-2-12-9': 'The bandstructures for both models are indistinguishable but occur at different twist-angle values, shifted by [MATH] (the critical value is smaller for the Flat model).', '1901.03420-2-13-0': 'An interesting behavior of the Full relaxed model occurs at the center of the magic-angle regime: although the Dirac cone still has symmetric dispersion near the K-point, the hole band dispersion is such that near the [MATH] point its energy energy is higher than the Fermi level.', '1901.03420-2-13-1': 'Thus the charge neutrality point does not occur at the Dirac point energy.', '1901.03420-2-13-2': 'This effect persists in all of the ab initio [MATH] models (even without relaxation), and is a behavior that can be observed in other tight-binding models in the literature[CITATION] but was not explicitly pointed out before.', '1901.03420-2-13-3': 'For transport measurement, this behavior would result in a range of [MATH] in twist angle where the charge neutrality point of the flat bands does not align with the resistive Dirac-point feature, as well as a reduction in the overall resistivity at the Dirac-point energy.', '1901.03420-2-14-0': 'Another important result of our calculations including atomic relaxation in tBLG is the suppression of the second magic-angle twist, defined as a smaller twist angle at which [MATH] [CITATION].', '1901.03420-2-14-1': 'In Fig. [REF] we show the Fermi velocity as predicted from the BMD model and from our unrelaxed and fully relaxed ab initio [MATH] models.', '1901.03420-2-14-2': 'Although our unrelaxed model shows similar behavior to the BMD model with a second magic angle occurring near [MATH], the inclusion of atomic relaxation removes this feature in near [MATH].', '1901.03420-2-14-3': 'In both the BMD and unrelaxed models, the second magic angle near [MATH] is only defined by [MATH], but there are no extrema in the bandwidth or bandgap at this angle.', '1901.03420-2-14-4': 'This can be understood from a careful study of how the ratio [MATH] affects the stability of the flat-band feature [CITATION].', '1901.03420-2-14-5': 'The Full relaxed model shows a possible dip in the Fermi velocity, corresponding to a possible second magic angle, near [MATH] or [MATH].', '1901.03420-2-14-6': 'However our model needs further refinement to make accurate predictions for very small [MATH], as our current model includes terms only up to the third shell in momentum space.', '1901.03420-2-14-7': 'We are not certain this dip in [MATH] will persist when additional terms are included.', '1901.03420-2-14-8': 'The lattice relaxation in tBLG becomes increasingly sharp on the moire length scale as the twist angle decreases [CITATION], and sharper features in the relaxation introduce additional important couplings in the [MATH] model at larger momenta.', '1901.03420-2-15-0': 'We thank Daniel Massatt, Hoi Chun Po, Alex Kruchkov, Grigory Tarnopolskiy, Pablo Jarillo-Herrero, Hyobin Yoo, Rebecca Engelke, and Philip Kim for useful discussions.', '1901.03420-2-15-1': 'This work was supported by ARO MURI Award W911NF-14-0247 and by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319.', '1901.03420-2-15-2': 'The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.'}

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[]

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[]

['1901.03420-1-3-4', '1901.03420-1-15-1', '1901.03420-2-3-4', '1901.03420-2-15-1', '1901.03420-3-4-4', '1901.03420-3-18-1', '1901.03420-4-4-4', '1901.03420-4-18-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1901.03420

{'1901.03420-3-0-0': 'We introduce a complete physical model for the single-particle electronic structure of twisted bilayer graphene (tBLG), which incorporates the crucial role of lattice relaxation.', '1901.03420-3-0-1': 'Our model, based on [MATH] perturbation theory, combines the accuracy of DFT calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models.', '1901.03420-3-0-2': 'The inclusion of relaxation significantly changes the bandstructure at the first magic-angle twist corresponding to flat bands near the Fermi level (the "low-energy" states), and eliminates the appearance of a second magic-angle twist.', '1901.03420-3-0-3': 'We show that minimal models for the low-energy states of tBLG can be easily modified to capture the changes in electronic states as a function of twist angle.', '1901.03420-3-1-0': 'The discovery of correlated phases in twisted bilayer graphene (tBLG) has generated much interest in this structurally and compositionally rather simple system; it has emerged as a new platform for tunable electronic correlations, and for exploring of the nature of unconventional superconductivity[CITATION].', '1901.03420-3-1-1': 'The challenge in modeling these phenomena from an atomistic perspective is that the actual structure of tBLG near the magic-angle twist ([MATH]) where correlated behavior is observed, consists of a large number of atoms, exceeding [MATH].', '1901.03420-3-1-2': 'To make progress from the theoretical point of view, a minimal model is needed that can capture the essence of single-particle states near the Fermi level ("low-energy" states).', '1901.03420-3-1-3': 'Such a model should reproduce the energy spectrum as a function of their relative twist angle with reasonable accuracy and with the required fidelity in capturing the nature of low-energy states.', '1901.03420-3-1-4': 'The appearance of correlated behavior is related to bands with very low dispersion ("flat" bands) caused by interlayer hybridization between the two Dirac cones from the different layers[CITATION].', '1901.03420-3-2-0': 'Existing models based on DFT calculations [CITATION] or large supercell tight-binding Hamiltonians [CITATION] are too complex to form the basis of a realistic many-body theory.', '1901.03420-3-2-1': 'At the other extreme, simplified continuum models allow for efficient calculations, but are based on heuristic arguments about the nature of the relevant electronic states [CITATION].', '1901.03420-3-2-2': 'An important feature of the physical system is the presence of atomic relaxation near the magic-angle twist, which has significant effects on the low-energy bandstructure [CITATION].', '1901.03420-3-2-3': 'Many simplified models for the flat bands of magic-angle tBLG have been proposed based on symmetry analysis, but they rely on empirical parameterization and are designed for only the magic-angle twist configuration[CITATION], typically ignoring atomic relaxation.', '1901.03420-3-3-0': 'Here, we present an ab initio [MATH] perturbation continuum model for tBLG which accurately accounts for the effects of atomic relaxation.', '1901.03420-3-3-1': 'Our model reproduces the results of DFT-quality tight-binding hamiltonians but at a smaller computational cost and, more importantly, it applies to all twist angles near the magic-angle value.', '1901.03420-3-3-2': 'Such a single-particle model is a prerequisite for physically meaningful prediction of correlation effects, as the presence of unphysical features in the single-particle band structure causes uncontrolled error in many-body calculations.', '1901.03420-3-3-3': 'We draw new conclusions on the low-energy electronic states at small twist angles, including the interesting result that there are no additional vanishings of the Fermi velocity in the range of the previously expected second and third magic-angles.', '1901.03420-3-3-4': 'For reference, we compare our continuum model to the seminal and widely employed [MATH] model of Bistritzer and MacDonald[CITATION] (BMD in the following), and we adopt their dimensionless parameter [MATH] for describing the twist-angle [MATH], where [MATH] is the Fermi velocity, [MATH] is the wave-vector set by the moire length scale and [MATH] is their effective interlayer coupling strength ([MATH] eV).', '1901.03420-3-4-0': 'Within [MATH] perturbation theory, the set of Bloch states of the two graphene layers is augmented by the addition of interlayer couplings due to the twist-angle induced Umklapp scattering process.', '1901.03420-3-4-1': 'As the low-energy electronic structure of tBLG is dominated by a pair of Dirac cones, the momentum expansion can be carried out about one copy of the cone at a valley K point.', '1901.03420-3-4-2': 'Taking also into account spin degeneracy, each band represents four electronic states in a real system [CITATION].', '1901.03420-3-4-3': 'Here we introduce an expanded ab initio [MATH] model which gives a more complete physical picture of the tBLG system.', '1901.03420-3-4-4': 'Our model has three new key ingredients:', '1901.03420-3-5-0': '(1) relaxation of the bilayer system [CITATION], including the out-of-plane relaxation of different regions as well as the in-plane strain corrections to the Hamiltonian of the individual monolayers;', '1901.03420-3-6-0': '(2) terms beyond the first shell of couplings in the [MATH] continuum model, which are necessary to capture the changes in stacking order at small angles;', '1901.03420-3-7-0': '(3) inclusion of [MATH]-dependent terms, which allow the [MATH] model to reproduce more accurately the particle-hole asymmetry of realistic ab initio bandstructures.', '1901.03420-3-8-0': 'The [MATH] terms are directly computed from an ab initio tight-binding Hamiltonian model [CITATION] for supercells spanning the twist-angle range [MATH].', '1901.03420-3-8-1': 'These terms have smooth dependence on [MATH], allowing for interpolation between the specific twist angles that correspond to finite supercells, to generate a model valid for any desired angle in that range.', '1901.03420-3-8-2': 'We relegate the detailed description of the extended Hamiltonian and the procedure for obtaining the relevant terms of the continuum model to a companion paper [CITATION].', '1901.03420-3-9-0': 'Our continuum model affords a natural interpretation of the electronic structure of tBLG at small twist angles, which is derived directly from the atomic relaxation so we describe this aspect first.', '1901.03420-3-9-1': 'For twist-angle [MATH] smaller than a critical value [MATH], the local atomic structure near the [MATH] and [MATH] stackings of the two layers becomes independent of [MATH].', '1901.03420-3-9-2': 'This creates a pattern of small circular domains of [MATH] stacking and large triangular domains of [MATH] stackings.', '1901.03420-3-9-3': 'Domain walls ([MATH]) of intermediate stacking separate the [MATH] and [MATH] domains and connect the [MATH] regions.', '1901.03420-3-9-4': 'This creates local electronic environments which are locked-in with respect to changing twist-angle for [MATH], where the tBLG system consists of a few fixed elements[CITATION], and only their length scale changes for decreasing twist angle.', '1901.03420-3-9-5': 'These elements are: the [MATH] regions which have a local twist of [MATH], which is independent of the overall twist angle [MATH] between the two layers, the [MATH] and [MATH] regions with negligible local twist.', '1901.03420-3-9-6': 'Moreover, the diameter of the [MATH] regions and the width of the [MATH] regions are approximately equal and remain unchanged for [MATH][CITATION].', '1901.03420-3-9-7': 'These features are shown in Fig. [REF] for [MATH].', '1901.03420-3-10-0': 'The relaxation in tBLG is described by two simultaneous effects.', '1901.03420-3-10-1': 'In-plane relaxation decreases the area of the high stacking energy [MATH] region while it increases that of low stacking energy [MATH] regions.', '1901.03420-3-10-2': 'Out-of-plane relaxation causes corrugation, increasing the vertical separation between the [MATH] regions from the equilibrium distance in [MATH] stacking of 3.35 to 3.59 , a substantial change ([MATH]%).', '1901.03420-3-10-3': 'The reduction in the size of [MATH] stacking can be understood as a minimization of planar stress energy and stacking energies, and has been modeled through various methods [CITATION] leading to a relaxed pattern in agreement with experimental results [CITATION].', '1901.03420-3-10-4': 'The role of vertical relaxation in experimental devices is less understood, as only free-standing tBLG has been modeled.', '1901.03420-3-10-5': 'Experimental tBLG devices are typically encapsulated in hexagonal Boron Nitride, so the actual corrugation may be reduced compared to the free-standing case.', '1901.03420-3-10-6': 'To take this into account, we consider two limits of the vertical relaxation: a "Full" relaxation model (free-standing bilayer result) and a "Flat" model with constant interlayer distance equal to the average of [MATH] and [MATH] interlayer distances (3.47 ).', '1901.03420-3-10-7': 'The magic angle predicted by the fully relaxed model, [MATH], is closer to the angles where correlated phenomena are observed [CITATION].', '1901.03420-3-11-0': 'The low-energy electronic states are directly associated with and derived from the presence of the relaxation-induced structural elements described earlier.', '1901.03420-3-11-1': 'Since the discovery of correlated phases in tBLG, many simplified [MATH]-band models have been proposed for the flat-bands, usually based on localized functions of the BMD model.', '1901.03420-3-11-2': 'One such minimal model consists of 10 bands [CITATION], and we argue that it can accurately capture the electronic effects of the different stacking regions that emerge after relaxation.', '1901.03420-3-11-3': 'This model comprises three orbitals on a triangular lattice formed by the [MATH] sites, one of [MATH]-like character ([MATH]) and two of [MATH]-like character ([MATH]), three orbitals on a Kagome lattice formed by the domain walls, and four orbitals on a honeycomb lattice, two for each of the [MATH] and [MATH] domains.', '1901.03420-3-11-4': 'The full details of the [MATH]-band tight-binding Hamiltonian are provided in the supplementary materials.', '1901.03420-3-12-0': 'To compare our ab initio [MATH] results to the [MATH]-band model, we project non-orthogonal wavefunctions that satisfy the symmetry conditions, shown in Fig. [REF], from band structure calculations.', '1901.03420-3-12-1': 'The form of these wavefunctions is not sensitive to the twist-angle, and is robust for twist angles within [MATH] of the magic angle.', '1901.03420-3-12-2': 'We note that the [MATH] and [MATH] indexing of the [MATH] orbitals describe their symmetry properties over the moire supercell, not their composition in terms of atomic-scale C [MATH] orbitals.', '1901.03420-3-12-3': 'We also fit the parameters of the [MATH]-band tight-binding model for [MATH], to reproduce the bands produced by our continuum model (see SM).', '1901.03420-3-12-4': 'The flat bands near the magic angle have [MATH] and [MATH] character (see Fig. [REF]a), showing that the coupling between these states is a necessary ingredient of the model if it is to capture the electronic structure as a function of twist angle.', '1901.03420-3-12-5': 'In particular, the orbital character of the electron and hole bands at [MATH] flips as one reduces the twisting angle: the hole band has [MATH] character for [MATH] and switches to [MATH] character for [MATH], while the electron band has the reverse character.', '1901.03420-3-12-6': 'As the [MATH] and [MATH] orbitals have opposite [MATH]-plane mirror symmetry eigenvalues ([MATH] and [MATH], respectively), the magic-angle represents a symmetry-protected band inversion.', '1901.03420-3-13-0': 'Two other important parameters in the [MATH] model are the effective interlayer coupling between orbitals of the same sublattice label, [MATH] or [MATH], and that between orbitals of different labels, [MATH] or [MATH].', '1901.03420-3-13-1': 'These nearest-neighbor interlayer couplings have been labeled [MATH] in previous studies and have a simple geometric interpretation: [MATH] is the interlayer electronic coupling at the [MATH] sites and [MATH] is the coupling at [MATH] sites, averaged over the entire moire cell.', '1901.03420-3-13-2': 'The values of these [MATH] parameters depend strongly on the twist angle [MATH].', '1901.03420-3-13-3': 'As the lattice relaxes, the relative size of the [MATH] regions is greatly reduced while that of the [MATH] regions is increased, causing a reduction in the value of [MATH] and a modest increase in the value of [MATH].', '1901.03420-3-13-4': 'This dependence is shown in Fig. [REF]c for the Full and the Flat relaxation models.', '1901.03420-3-13-5': 'The overall [MATH] dependence of the ratio [MATH] is not sensitive to the relaxed height assumption.', '1901.03420-3-13-6': 'The Flat model has a larger ratio as the Full relaxation assumption moves the [MATH] sites closer together (increasing their coupling and the [MATH] value) while moving the [MATH] sites farther apart (reducing their coupling and the [MATH] value).', '1901.03420-3-14-0': 'To elucidate the salient features of the single-particle model, we study three related indicators of the flat-band phenomenon as a function of [MATH]: the Fermi velocity ([MATH]), the bandwidth ([MATH]), and the band gap ([MATH]).', '1901.03420-3-14-1': 'These are shown in Fig [REF].', '1901.03420-3-14-2': 'All three are calculated for both the electron and the hole sides of the flat-band manifold.', '1901.03420-3-14-3': 'The model without relaxation shows large discrepancies between the extrema of the Fermi velocity, gap, and bandwidth, and the electron and hole features have little in common.', '1901.03420-3-14-4': 'The two models (Flat and Full) that include relaxation show more regular dependence on [MATH] and closer correspondence between the electron and hole bands.', '1901.03420-3-14-5': 'The bandwidth for the hole band is always smaller than that of the electron band, and the hole band achieves its minimum twice.', '1901.03420-3-14-6': 'In general, [MATH] does not coincide with bandwidth minima.', '1901.03420-3-14-7': 'We thus draw the important conclusion that the magic-angle is not a single value, but rather a range of [MATH] which spans the extrema in these key features.', '1901.03420-3-14-8': 'In particular, even if an experimental device has a variation in twisting angle over a probed region, if that variation is [MATH] the flat-band models may still be reliable enough to explain correlation effects.', '1901.03420-3-14-9': 'This range for the Full relaxed model is [MATH] and [MATH] for the Flat model.', '1901.03420-3-14-10': 'The bandstructures for both models are similar after accounting for this offset in [MATH].', '1901.03420-3-15-0': 'An interesting behavior of the Full relaxed model occurs at the center of the magic-angle regime: although the Dirac cone still has symmetric dispersion near the K-point, the hole band dispersion is such that near the [MATH] point its energy energy is higher than the Fermi level (see Fig. [REF]).', '1901.03420-3-15-1': 'Thus the charge neutrality point does not occur at the Dirac point energy.', '1901.03420-3-15-2': 'This effect persists in all of our ab initio [MATH] models (even without relaxation), and is a behavior that can be observed in other tight-binding models in the literature[CITATION].', '1901.03420-3-15-3': 'Assuming the bands of tBLG are not perfectly particle-hole symmetric, and that the flat-band regime is defined by a protected [MATH]-tuned band inversion, such a feature is unavoidable.', '1901.03420-3-15-4': 'For transport measurement, this behavior would result in a range of [MATH] in twist angle where the charge neutrality point of the flat bands does not align with the Dirac-point of the moire superlattice, as well as a reduction in the resistivity at the Dirac-point energy due to these other bands near [MATH].', '1901.03420-3-15-5': 'Thus if a clean Dirac-point transport signature is used to assess experimental device quality, this angle-range will be difficult to observe.', '1901.03420-3-16-0': 'Another important result of our calculations including atomic relaxation in tBLG is the suppression of the second magic-angle twist, defined as a smaller twist angle at which [MATH] [CITATION].', '1901.03420-3-16-1': 'In Fig. [REF] we show the Fermi velocity as predicted from the BMD model and from our unrelaxed and fully relaxed ab initio [MATH] models.', '1901.03420-3-16-2': 'Although our unrelaxed model shows similar behavior to the BMD model with a second magic angle occurring near [MATH], the inclusion of atomic relaxation removes this feature in near [MATH].', '1901.03420-3-16-3': 'As the lattice relaxation in tBLG becomes increasingly sharp on the moire length scale as the twist angle decreases [CITATION], these sharper features in the relaxation introduce additional important couplings in the [MATH] model at larger momenta.', '1901.03420-3-16-4': 'Thus to accurately model the electronic structure of tBLG below [MATH] our inclusion of the higher-order [MATH] couplings terms is necessary.', '1901.03420-3-17-0': 'In conclusion, we have presented a [MATH] expansion of the low-energy electronic states of tBLG that can be extended to arbitrary order in pertubation theory.', '1901.03420-3-17-1': "This exact continuum model facilitates a better understanding of the single-particle features of tBLG's flat bands, and provides a solid foundation on which to build correlated models.", '1901.03420-3-17-2': 'We have made this model publicly available in MATLAB, C++, and Python at https://github.com/stcarr/kp_tblg.', '1901.03420-3-18-0': 'We thank Daniel Massatt, Hoi Chun Po, Alex Kruchkov, Grigory Tarnopolskiy, Pablo Jarillo-Herrero, Hyobin Yoo, Rebecca Engelke, and Philip Kim for useful discussions.', '1901.03420-3-18-1': 'This work was supported by ARO MURI Award W911NF-14-0247 and by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319.', '1901.03420-3-18-2': 'The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.'}

{'1901.03420-4-0-0': 'We introduce a complete physical model for the single-particle electronic structure of twisted bilayer graphene (tBLG), which incorporates the crucial role of lattice relaxation.', '1901.03420-4-0-1': 'Our model, based on [MATH] perturbation theory, combines the accuracy of DFT calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models.', '1901.03420-4-0-2': 'The inclusion of relaxation significantly changes the bandstructure at the first magic-angle twist corresponding to flat bands near the Fermi level (the "low-energy" states), and eliminates the appearance of a second magic-angle twist.', '1901.03420-4-0-3': 'We show that minimal models for the low-energy states of tBLG can be easily modified to capture the changes in electronic states as a function of twist angle.', '1901.03420-4-1-0': 'The discovery of correlated phases in twisted bilayer graphene (tBLG) has generated much interest in this structurally and compositionally rather simple system; it has emerged as a new platform for tunable electronic correlations, and for exploring of the nature of unconventional superconductivity[CITATION].', '1901.03420-4-1-1': 'The challenge in modeling these phenomena from an atomistic perspective is that the actual structure of tBLG near the magic-angle twist ([MATH]) where correlated behavior is observed, consists of a large number of atoms, exceeding [MATH].', '1901.03420-4-1-2': 'To make progress from the theoretical point of view, a minimal model is needed that can capture the essence of single-particle states near the Fermi level ("low-energy" states).', '1901.03420-4-1-3': 'Such a model should reproduce the energy spectrum as a function of their relative twist angle with reasonable accuracy and with the required fidelity in capturing the nature of low-energy states.', '1901.03420-4-1-4': 'The appearance of correlated behavior is related to bands with very low dispersion ("flat" bands) caused by interlayer hybridization between the two Dirac cones from the different layers[CITATION].', '1901.03420-4-2-0': 'Existing models based on DFT calculations [CITATION] or large supercell tight-binding Hamiltonians [CITATION] are too complex to form the basis of a realistic many-body theory.', '1901.03420-4-2-1': 'At the other extreme, simplified continuum models allow for efficient calculations, but are based on heuristic arguments about the nature of the relevant electronic states [CITATION].', '1901.03420-4-2-2': 'An important feature of the physical system is the presence of atomic relaxation near the magic-angle twist, which has significant effects on the low-energy bandstructure [CITATION].', '1901.03420-4-2-3': 'Many simplified models for the flat bands of magic-angle tBLG have been proposed based on symmetry analysis, but they rely on empirical parameterization and are designed for only the magic-angle twist configuration[CITATION], typically ignoring atomic relaxation.', '1901.03420-4-3-0': 'Here, we present an ab initio [MATH] perturbation continuum model for tBLG which accurately accounts for the effects of atomic relaxation.', '1901.03420-4-3-1': 'Our model reproduces the results of DFT-quality tight-binding hamiltonians but at a smaller computational cost and, more importantly, it applies to all twist angles near the magic-angle value.', '1901.03420-4-3-2': 'Such a single-particle model is a prerequisite for physically meaningful prediction of correlation effects, as the presence of unphysical features in the single-particle band structure causes uncontrolled error in many-body calculations.', '1901.03420-4-3-3': 'We draw new conclusions on the low-energy electronic states at small twist angles, including the interesting result that there are no additional vanishings of the Fermi velocity in the range of the previously expected second and third magic-angles.', '1901.03420-4-3-4': 'For reference, we compare our continuum model to the seminal and widely employed [MATH] model of Bistritzer and MacDonald[CITATION] (BMD in the following), and we adopt their dimensionless parameter [MATH] for describing the twist-angle [MATH], where [MATH] is the Fermi velocity, [MATH] is the wave-vector set by the moire length scale and [MATH] is their effective interlayer coupling strength ([MATH] eV).', '1901.03420-4-4-0': 'Within [MATH] perturbation theory, the set of Bloch states of the two graphene layers is augmented by the addition of interlayer couplings due to the twist-angle induced Umklapp scattering process.', '1901.03420-4-4-1': 'As the low-energy electronic structure of tBLG is dominated by a pair of Dirac cones, the momentum expansion can be carried out about one copy of the cone at a valley K point.', '1901.03420-4-4-2': 'Taking also into account spin degeneracy, each band represents four electronic states in a real system [CITATION].', '1901.03420-4-4-3': 'Here we introduce an expanded ab initio [MATH] model which gives a more complete physical picture of the tBLG system.', '1901.03420-4-4-4': 'Our model has three new key ingredients:', '1901.03420-4-5-0': '(1) relaxation of the bilayer system [CITATION], including the out-of-plane relaxation of different regions as well as the in-plane strain corrections to the Hamiltonian of the individual monolayers;', '1901.03420-4-6-0': '(2) terms beyond the first shell of couplings in the [MATH] continuum model, which are necessary to capture the changes in stacking order at small angles;', '1901.03420-4-7-0': '(3) inclusion of [MATH]-dependent terms, which allow the [MATH] model to reproduce more accurately the particle-hole asymmetry of realistic ab initio bandstructures.', '1901.03420-4-8-0': 'The [MATH] terms are directly computed from an ab initio tight-binding Hamiltonian model [CITATION] for supercells spanning the twist-angle range [MATH].', '1901.03420-4-8-1': 'These terms have smooth dependence on [MATH], allowing for interpolation between the specific twist angles that correspond to finite supercells, to generate a model valid for any desired angle in that range.', '1901.03420-4-8-2': 'We relegate the detailed description of the extended Hamiltonian and the procedure for obtaining the relevant terms of the continuum model to a companion paper [CITATION].', '1901.03420-4-9-0': 'Our continuum model affords a natural interpretation of the electronic structure of tBLG at small twist angles, which is derived directly from the atomic relaxation so we describe this aspect first.', '1901.03420-4-9-1': 'For twist-angle [MATH] smaller than a critical value [MATH], the local atomic structure near the [MATH] and [MATH] stackings of the two layers becomes independent of [MATH].', '1901.03420-4-9-2': 'This creates a pattern of small circular domains of [MATH] stacking and large triangular domains of [MATH] stackings.', '1901.03420-4-9-3': 'Domain walls ([MATH]) of intermediate stacking separate the [MATH] and [MATH] domains and connect the [MATH] regions.', '1901.03420-4-9-4': 'This creates local electronic environments which are locked-in with respect to changing twist-angle for [MATH], where the tBLG system consists of a few fixed elements[CITATION], and only their length scale changes for decreasing twist angle.', '1901.03420-4-9-5': 'These elements are: the [MATH] regions which have a local twist of [MATH], which is independent of the overall twist angle [MATH] between the two layers, the [MATH] and [MATH] regions with negligible local twist.', '1901.03420-4-9-6': 'Moreover, the diameter of the [MATH] regions and the width of the [MATH] regions are approximately equal and remain unchanged for [MATH][CITATION].', '1901.03420-4-9-7': 'These features are shown in Fig. [REF] for [MATH].', '1901.03420-4-10-0': 'The relaxation in tBLG is described by two simultaneous effects.', '1901.03420-4-10-1': 'In-plane relaxation decreases the area of the high stacking energy [MATH] region while it increases that of low stacking energy [MATH] regions.', '1901.03420-4-10-2': 'Out-of-plane relaxation causes corrugation, increasing the vertical separation between the [MATH] regions from the equilibrium distance in [MATH] stacking of 3.35 to 3.59 , a substantial change ([MATH]%).', '1901.03420-4-10-3': 'The reduction in the size of [MATH] stacking can be understood as a minimization of planar stress energy and stacking energies, and has been modeled through various methods [CITATION] leading to a relaxed pattern in agreement with experimental results [CITATION].', '1901.03420-4-10-4': 'The role of vertical relaxation in experimental devices is less understood, as only free-standing tBLG has been modeled.', '1901.03420-4-10-5': 'Experimental tBLG devices are typically encapsulated in hexagonal Boron Nitride, so the actual corrugation may be reduced compared to the free-standing case.', '1901.03420-4-10-6': 'To take this into account, we consider two limits of the vertical relaxation: a "Full" relaxation model (free-standing bilayer result) and a "Flat" model with constant interlayer distance equal to the average of [MATH] and [MATH] interlayer distances (3.47 ).', '1901.03420-4-10-7': 'The magic angle predicted by the fully relaxed model, [MATH], is closer to the angles where correlated phenomena are observed [CITATION].', '1901.03420-4-11-0': 'The low-energy electronic states are directly associated with and derived from the presence of the relaxation-induced structural elements described earlier.', '1901.03420-4-11-1': 'Since the discovery of correlated phases in tBLG, many simplified [MATH]-band models have been proposed for the flat-bands, usually based on localized functions of the BMD model.', '1901.03420-4-11-2': 'One such minimal model consists of 10 bands [CITATION], and we argue that it can accurately capture the electronic effects of the different stacking regions that emerge after relaxation.', '1901.03420-4-11-3': 'This model comprises three orbitals on a triangular lattice formed by the [MATH] sites, one of [MATH]-like character ([MATH]) and two of [MATH]-like character ([MATH]), three orbitals on a Kagome lattice formed by the domain walls, and four orbitals on a honeycomb lattice, two for each of the [MATH] and [MATH] domains.', '1901.03420-4-11-4': 'The full details of the [MATH]-band tight-binding Hamiltonian are provided in the supplementary materials.', '1901.03420-4-12-0': 'To compare our ab initio [MATH] results to the [MATH]-band model, we project non-orthogonal wavefunctions that satisfy the symmetry conditions, shown in Fig. [REF], from band structure calculations.', '1901.03420-4-12-1': 'The form of these wavefunctions is not sensitive to the twist-angle, and is robust for twist angles within [MATH] of the magic angle.', '1901.03420-4-12-2': 'We note that the [MATH] and [MATH] indexing of the [MATH] orbitals describe their symmetry properties over the moire supercell, not their composition in terms of atomic-scale C [MATH] orbitals.', '1901.03420-4-12-3': 'We also fit the parameters of the [MATH]-band tight-binding model for [MATH], to reproduce the bands produced by our continuum model (see SM).', '1901.03420-4-12-4': 'The flat bands near the magic angle have [MATH] and [MATH] character (see Fig. [REF]a), showing that the coupling between these states is a necessary ingredient of the model if it is to capture the electronic structure as a function of twist angle.', '1901.03420-4-12-5': 'In particular, the orbital character of the electron and hole bands at [MATH] flips as one reduces the twisting angle: the hole band has [MATH] character for [MATH] and switches to [MATH] character for [MATH], while the electron band has the reverse character.', '1901.03420-4-12-6': 'As the [MATH] and [MATH] orbitals have opposite [MATH]-plane mirror symmetry eigenvalues ([MATH] and [MATH], respectively), the magic-angle represents a symmetry-protected band inversion.', '1901.03420-4-13-0': 'Two other important parameters in the [MATH] model are the effective interlayer coupling between orbitals of the same sublattice label, [MATH] or [MATH], and that between orbitals of different labels, [MATH] or [MATH].', '1901.03420-4-13-1': 'These nearest-neighbor interlayer couplings have been labeled [MATH] in previous studies and have a simple geometric interpretation: [MATH] is the interlayer electronic coupling at the [MATH] sites and [MATH] is the coupling at [MATH] sites, averaged over the entire moire cell.', '1901.03420-4-13-2': 'The values of these [MATH] parameters depend strongly on the twist angle [MATH].', '1901.03420-4-13-3': 'As the lattice relaxes, the relative size of the [MATH] regions is greatly reduced while that of the [MATH] regions is increased, causing a reduction in the value of [MATH] and a modest increase in the value of [MATH].', '1901.03420-4-13-4': 'This dependence is shown in Fig. [REF]c for the Full and the Flat relaxation models.', '1901.03420-4-13-5': 'The overall [MATH] dependence of the ratio [MATH] is not sensitive to the relaxed height assumption.', '1901.03420-4-13-6': 'The Flat model has a larger ratio as the Full relaxation assumption moves the [MATH] sites closer together (increasing their coupling and the [MATH] value) while moving the [MATH] sites farther apart (reducing their coupling and the [MATH] value).', '1901.03420-4-14-0': 'To elucidate the salient features of the single-particle model, we study three related indicators of the flat-band phenomenon as a function of [MATH]: the Fermi velocity ([MATH]), the bandwidth ([MATH]), and the band gap ([MATH]).', '1901.03420-4-14-1': 'These are shown in Fig [REF].', '1901.03420-4-14-2': 'All three are calculated for both the electron and the hole sides of the flat-band manifold.', '1901.03420-4-14-3': 'The model without relaxation shows large discrepancies between the extrema of the Fermi velocity, gap, and bandwidth, and the electron and hole features have little in common.', '1901.03420-4-14-4': 'The two models (Flat and Full) that include relaxation show more regular dependence on [MATH] and closer correspondence between the electron and hole bands.', '1901.03420-4-14-5': 'The bandwidth for the hole band is always smaller than that of the electron band, and the hole band achieves its minimum twice.', '1901.03420-4-14-6': 'In general, [MATH] does not coincide with bandwidth minima.', '1901.03420-4-14-7': 'We thus draw the important conclusion that the magic-angle is not a single value, but rather a range of [MATH] which spans the extrema in these key features.', '1901.03420-4-14-8': 'In particular, even if an experimental device has a variation in twisting angle over a probed region, if that variation is [MATH] the flat-band models may still be reliable enough to explain correlation effects.', '1901.03420-4-14-9': 'This range for the Full relaxed model is [MATH] and [MATH] for the Flat model.', '1901.03420-4-14-10': 'The bandstructures for both models are similar after accounting for this offset in [MATH].', '1901.03420-4-15-0': 'An interesting behavior of the Full relaxed model occurs at the center of the magic-angle regime: although the Dirac cone still has symmetric dispersion near the K-point, the hole band dispersion is such that near the [MATH] point its energy energy is higher than the Fermi level (see Fig. [REF]).', '1901.03420-4-15-1': 'Thus the charge neutrality point does not occur at the Dirac point energy.', '1901.03420-4-15-2': 'This effect persists in all of our ab initio [MATH] models (even without relaxation), and is a behavior that can be observed in other tight-binding models in the literature[CITATION].', '1901.03420-4-15-3': 'Assuming the bands of tBLG are not perfectly particle-hole symmetric, and that the flat-band regime is defined by a protected [MATH]-tuned band inversion, such a feature is unavoidable.', '1901.03420-4-15-4': 'For transport measurement, this behavior would result in a range of [MATH] in twist angle where the charge neutrality point of the flat bands does not align with the Dirac-point of the moire superlattice, as well as a reduction in the resistivity at the Dirac-point energy due to these other bands near [MATH].', '1901.03420-4-15-5': 'Thus if a clean Dirac-point transport signature is used to assess experimental device quality, this angle-range will be difficult to observe.', '1901.03420-4-16-0': 'Another important result of our calculations including atomic relaxation in tBLG is the suppression of the second magic-angle twist, defined as a smaller twist angle at which [MATH] [CITATION].', '1901.03420-4-16-1': 'In Fig. [REF] we show the Fermi velocity as predicted from the BMD model and from our unrelaxed and fully relaxed ab initio [MATH] models.', '1901.03420-4-16-2': 'Although our unrelaxed model shows similar behavior to the BMD model with a second magic angle occurring near [MATH], the inclusion of atomic relaxation removes this feature in near [MATH].', '1901.03420-4-16-3': 'As the lattice relaxation in tBLG becomes increasingly sharp on the moire length scale as the twist angle decreases [CITATION], these sharper features in the relaxation introduce additional important couplings in the [MATH] model at larger momenta.', '1901.03420-4-16-4': 'Thus to accurately model the electronic structure of tBLG below [MATH] our inclusion of the higher-order [MATH] couplings terms is necessary.', '1901.03420-4-17-0': 'In conclusion, we have presented a [MATH] expansion of the low-energy electronic states of tBLG that can be extended to arbitrary order in pertubation theory.', '1901.03420-4-17-1': "This exact continuum model facilitates a better understanding of the single-particle features of tBLG's flat bands, and provides a solid foundation on which to build correlated models.", '1901.03420-4-17-2': 'We have made this model publicly available in MATLAB, C++, and Python at https://github.com/stcarr/kp_tblg.', '1901.03420-4-18-0': 'We thank Daniel Massatt, Hoi Chun Po, Alex Kruchkov, Grigory Tarnopolskiy, Pablo Jarillo-Herrero, Hyobin Yoo, Rebecca Engelke, and Philip Kim for useful discussions.', '1901.03420-4-18-1': 'This work was supported by ARO MURI Award W911NF-14-0247 and by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319.', '1901.03420-4-18-2': 'The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.'}

0711.0497

{'0711.0497-1-0-0': 'Recently, Belle Collaboration has reported a resonant state produced in [MATH], which is called [MATH].', '0711.0497-1-0-1': 'This state is charged, so it can not be interpreted as an ordinary charmonium state.', '0711.0497-1-0-2': 'In this paper, we analyze the octet to which this particle belongs and predict the masses of mesons in this octet.', '0711.0497-1-0-3': 'Utilizing flavor SU(3) symmetry, we study production rates in several kinds of [MATH] decays.', '0711.0497-1-0-4': 'The [MATH] and [MATH] decay channels, favored by Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, can have branching ratios of [MATH].', '0711.0497-1-0-5': 'This large branching ratio could be observed at the running [MATH] factories to detect [MATH] particles containing a strange quark.', '0711.0497-1-0-6': 'We also predict large branching ratios of the [MATH] and [MATH]) particle production rates in non-leptonic [MATH] decays and radiative [MATH] decays.', '0711.0497-1-0-7': 'Measurements of these decays at the ongoing [MATH] factories and the forthcoming Large Hadron Collider (LHC)-b experiments are helpful to clarify the mysterious [MATH] particles.', '0711.0497-1-1-0': '# Introduction', '0711.0497-1-2-0': 'Recently, there are many exciting discoveries on new hadron states especially in the hidden-charm sector.', '0711.0497-1-2-1': 'Among these discoveries, the most intriguing one is the new relatively narrow peak named [MATH] found by Belle Collaboration in the invariant mass spectrum of [MATH] in the decay mode [MATH][CITATION].', '0711.0497-1-2-2': 'There is a large branching fraction for the following decay chain: [EQUATION]', '0711.0497-1-2-3': 'Mass and width of this particle are measured as: [EQUATION]', '0711.0497-1-2-4': "The most prominent character is that it is electric charged, that's to say, this new particle can not be described as an ordinary charmonium state or a charmonium-like state such as [MATH].", '0711.0497-1-2-5': 'On the other hand, this particle can decay to [MATH] with a large rate through strong interactions, so it involves at least four quarks [MATH], though we do not have any further detailed information on its inner dynamics at present.', '0711.0497-1-3-0': 'In order to elucidate this particle, many theoretical studies [CITATION] have been put forward.', '0711.0497-1-3-1': 'This meson could be viewed as a genuine tetraquark state with diquark anti-diquark [MATH] content which has a large rate to [MATH] [CITATION].', '0711.0497-1-3-2': 'Moreover based on QCD-string, two different four-quark descriptions are proposed in Ref. [CITATION]: one can be reduced to the ordinary diquark-diquark picture and the other one can not.', '0711.0497-1-3-3': 'Besides this kind of explanation, it has also been identified as the resonance of [MATH] [CITATION] as its mass is close to the thresholds of [MATH] and [MATH].', '0711.0497-1-3-4': 'Within this picture, the authors in Ref. [CITATION] explored the production of [MATH] and [MATH].', '0711.0497-1-3-5': 'Short distance contribution to [MATH] is neglected and the main contribution is from long distance re-scattering effect via [MATH].', '0711.0497-1-3-6': 'With proper parameters, they can successfully explain the much larger production rate of [MATH] than that of [MATH].', '0711.0497-1-3-7': 'In Ref. [CITATION], Bugg took this meson as a [MATH] threshold cusp.', '0711.0497-1-3-8': 'Recently, Qiao also tried to explain this meson with the baryonium picture [CITATION].', '0711.0497-1-3-9': 'Using the technique of QCD sum rules, Lee [MATH] calculated the masses of this particle and its strange partner [MATH] in Ref. [CITATION].', '0711.0497-1-4-0': 'Whether or not these scenarios describe the true dynamics of [MATH], this strange meson indeed plays an important role in the charmonium spectroscopy.', '0711.0497-1-4-1': "In the present note, we do not intend to give an explanation of this meson's structure, but we want to analyze its partners within SU(3) symmetry: the octet to which the meson [MATH] belongs and the corresponding singlet meson.", '0711.0497-1-4-2': 'Up to now, these is no experimental information on these mesons except [MATH].', '0711.0497-1-4-3': 'The decays of [MATH] meson provide a firm potential in searching for these exotic mesons [CITATION], just like the observed decay channel [MATH].', '0711.0497-1-4-4': 'We will investigate the possibilities to detect these [MATH] mesons in [MATH] decays.', '0711.0497-1-4-5': 'In doing this, we will analyze decay amplitudes with the assumption of SU(3) flavor symmetry: to construct effective Hamiltonian using flavor SU(3) meson matrixes.', '0711.0497-1-4-6': 'The decay amplitudes can also be studied by using Feynmann diagrams.', '0711.0497-1-4-7': 'In the discussion of [MATH] production with the graphic technique, we only consider short-distance contributions and neglect soft final state interactions.', '0711.0497-1-4-8': 'Specifically, the considered decays are divided into three categories: CKM allowed non-leptonic decays; CKM suppressed non-leptonic decays; radiative decays.', '0711.0497-1-4-9': 'The first kind of decays have similar branching ratios with the observed [MATH], while the second type of decays is suppressed by about one order in magnitude and we will show that the running [MATH] factories could hardly detect this kind of decays.', '0711.0497-1-4-10': 'Radiative [MATH] decay is a natural filter to exclude the 0-spin mesons and furthermore this kind of process may go through with a sizable branching ratio.', '0711.0497-1-5-0': 'In the next section, we will analyze the octet of [MATH] meson within flavor SU(3) symmetry and try to estimate their masses.', '0711.0497-1-5-1': 'We will construct the effective Hamiltonian using meson matrices and then use them to study the production rates of [MATH] mesons in [MATH] decays.', '0711.0497-1-5-2': 'In Sec. [REF], we will introduce [MATH] meson which consists of three charm quarks, together with a brief discussion on its production in [MATH] decays.', '0711.0497-1-5-3': 'We will summarize this note in the last section.', '0711.0497-1-6-0': '# The octet and the singlet', '0711.0497-1-7-0': 'Just as stated above, [MATH] involves at least four quarks in constituent quark model, and there is an octet which [MATH] belongs to in flavor SU(3) symmetry.', '0711.0497-1-7-1': 'Generally, we can deduce the particles in this octet using group theory: these particles, under the name [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH], are shown in Fig. [REF].', '0711.0497-1-7-2': 'Besides, there exists one singlet meson called [MATH].', '0711.0497-1-7-3': 'In constituent quark model, quark contents of these mesons are listed by: [EQUATION]', '0711.0497-1-7-4': 'In reality, [MATH] quark is slightly heavier than [MATH] quark which is one of the origins for SU(3) symmetry breaking.', '0711.0497-1-7-5': 'Accordingly, the singlet [MATH] can mix with eighth component of the octet [MATH], in analogy with [MATH] and [MATH].', '0711.0497-1-7-6': 'Physical particles, named [MATH] and [MATH], are mixtures of them and can be expressed as: [EQUATION]', '0711.0497-1-7-7': 'The mixing angle [MATH] can be determined through measuring decays of these two particles much accurately in future.', '0711.0497-1-7-8': 'For simplicity, we will assume the mixing is ideal, i.e. [MATH].', '0711.0497-1-7-9': 'In this case, the quark contents are: [EQUATION]', '0711.0497-1-7-10': 'All together, we can use the following meson matrix to describe these mesons: [EQUATION]', '0711.0497-1-7-11': 'With the quark contents given in the above, we are ready to estimate masses of these particles.', '0711.0497-1-7-12': 'Isospin analysis predicts the equal masses for the four mesons with neither open nor hidden strangeness: [MATH].', '0711.0497-1-7-13': 'For the mesons with a strange quark, we have to consider the mass differences between the lighter [MATH] quarks and the heavier [MATH] quark.', '0711.0497-1-7-14': 'One can compare masses of [MATH] and [MATH] to get some information: the mass of [MATH] is [MATH] MeV larger than that of [MATH].', '0711.0497-1-7-15': 'In heavy quark limit [MATH], the light system will not be affected by different heavy quark systems, thus we can simply assume a similar difference for [MATH] mesons which predicts the mass of [MATH] around [MATH] MeV.', '0711.0497-1-7-16': 'Because the mass of newly observed [MATH] meson is not far from the threshold of [MATH], [MATH] meson is regarded as the resonance of [MATH] [CITATION].', '0711.0497-1-7-17': 'Under this mechanism, we could give more precise predictions on the masses for other [MATH] mesons using experimental results for the [MATH] and [MATH] mesons.', '0711.0497-1-7-18': 'Our results are listed in Tab. [REF] and uncertainties in this table are from that of masses of the charmed mesons.', '0711.0497-1-7-19': 'The recent QCD sum rule study predicts the mass by [CITATION]: [EQUATION] which is above the [MATH] and [MATH] threshold by about 160 MeV.', '0711.0497-1-7-20': 'More experimental studies are required to test this description.', '0711.0497-1-8-0': 'Experimentalists have observed the [MATH] particle through the [MATH] with [MATH].', '0711.0497-1-8-1': 'Assuming S-wave decay for [MATH] meson, the quantum numbers can be determined as [MATH].', '0711.0497-1-8-2': "In order to detect the other [MATH] mesons, we have to choose the proper final states to re-construct them, thus the predictions on [MATH]'s strong decays are required.", '0711.0497-1-8-3': 'Using the flavor SU(3) symmetry and [MATH], we also list the strong decays of other [MATH] mesons in Tab. [REF].', '0711.0497-1-8-4': 'With the assumption [MATH], another kind of possible decay modes is [MATH], where [MATH] denotes a light vector meson.', '0711.0497-1-9-0': 'In order to explore the production in [MATH] decays, one can construct the effective Hamiltonian at hadron level using meson matrices [CITATION].', '0711.0497-1-9-1': 'In the following, to construct the related effective Hamiltonian, we will assume the flavor SU(3) symmetry.', '0711.0497-1-9-2': 'In [MATH] decays, the initial state [MATH] forms an SU(3) anti-triplet.', '0711.0497-1-9-3': 'The transition at quark level is either [MATH] or [MATH], which is described by the effective electro-weak Hamiltonian: [EQUATION] where [MATH].', '0711.0497-1-9-4': 'In the following, we will only consider the color allowed contributions because of the large Wilson coefficient [MATH] with the color factor [MATH].', '0711.0497-1-9-5': 'The transition [MATH] is CKM favored: [MATH], while the [MATH] transition is suppressed by [MATH].', '0711.0497-1-9-6': 'The effective electro-weak Hamiltonian given in Eq. ([REF]) can also be written as an SU(3) triplet: [MATH] (i=1 (u), 2 (d), 3(s)), where the only non-zero elements are [MATH] for CKM favored decays [MATH], and [MATH] for CKM suppressed channels [MATH].', '0711.0497-1-9-7': 'The final state mesons can be described by two nonet matrices: [MATH] and [MATH].', '0711.0497-1-9-8': 'The effective Hamiltonian at hadron level could be constructed as: [EQUATION] where the upper index labels rows and the lower labels columns.', '0711.0497-1-10-0': 'The above effective Hamiltonian can be related to Feynmann diagrams with the one-to-one correspondence and the lowest order diagrams are given in Fig. [REF].', '0711.0497-1-10-1': 'The second term in eq. ([REF]) corresponds to the second diagram in Fig. [REF] (called [MATH]-recoiling diagram) in which the spectator light quark in [MATH] meson enters into the heavy [MATH] meson.', '0711.0497-1-10-2': 'If the spectator quark goes to the light meson, we call this kind of diagram (the third one in Fig. [REF]) as the [MATH]-emission diagram which corresponds to the third term in the effective Hamiltonian.', '0711.0497-1-10-3': 'In order to estimate relative sizes of these terms, we have to analyze diagrams at quark level.', '0711.0497-1-10-4': 'Final state mesons move very slowly and thus the gluon generating the [MATH] quark pair is soft: [MATH].', '0711.0497-1-10-5': 'Thus after integrating out high energy scales, decay amplitudes can be expressed as matrix elements of a soft four-quark operator between initial and final states.', '0711.0497-1-10-6': 'The first term in Eq. ([REF]) corresponds to the annihilation diagram (the first one in Fig. [REF]), as flavor indices of [MATH] and [MATH] in this term are contracted with each other.', '0711.0497-1-10-7': 'This kind of diagram is expected to be suppressed in two-body non-leptonic [MATH] decays.', '0711.0497-1-10-8': 'But here since the gluons are soft, decay amplitudes can also be expressed as time-ordered products of a soft four-quark operator and the [MATH] interaction Hamiltonian which contains only soft fields, thus this kind of contribution is comparable with contributions from the second and third terms in Eq. ([REF]).', '0711.0497-1-10-9': 'For SU(3) flavor singlet mesons [MATH] and [MATH], there are additional contributions which are given by the last two terms in Eq. ([REF]).', '0711.0497-1-10-10': 'One kind of typical Feynmann diagram is also shown in Fig. [REF] as the last two diagrams and it is the contribution from the higher Fock states of [MATH] and [MATH].', '0711.0497-1-10-11': 'Even in charmless two-body [MATH] decays [CITATION], this kind of gluonic contribution is sizable.', '0711.0497-1-10-12': 'Here we do not have any implication and thus one can not neglect it with any priori.', '0711.0497-1-11-0': 'With the effective Hamiltonian given in Eq. ([REF]), we give the decay amplitudes for the first kind of non-leptonic [MATH] decay channels in Table [REF].', '0711.0497-1-11-1': 'These decays are induced by the CKM allowed transition [MATH] and go through with a large decay rate (typically the same order with the observed [MATH]).', '0711.0497-1-11-2': 'The flavor SU(3) symmetry implies the following relations for [MATH] decays: [EQUATION] where mass differences and lifetime differences of [MATH] mesons are neglected which can not produce large corrections.', '0711.0497-1-11-3': 'Although all of these decays are expected to go through with large branching fractions, decay rates may differ from each other for distinct coefficients.', '0711.0497-1-11-4': 'Two of the decays in the first line have been observed experimentally, while the possibility to observe the other two channels is a little smaller as the daughter meson [MATH] from [MATH] is relatively more difficult to measure.', '0711.0497-1-11-5': 'The decays in the second line is contributed from the third term of effective Hamiltonian given in Eq. ([REF]), which should also have similar production rates.', '0711.0497-1-11-6': 'Among these four channels, [MATH] and [MATH] can have large branching ratios and the final states ([MATH] or [MATH]) are easily to be measured on the experimental side.', '0711.0497-1-11-7': 'Thus measurements of the [MATH] invariant mass distribution in these two channels are helpful to detect the [MATH] particles and determine relative sizes of [MATH] and [MATH].', '0711.0497-1-11-8': 'The other [MATH] decays are less possible to be measured in the running [MATH] factories as either [MATH] or [MATH] is produced in the final state.', '0711.0497-1-11-9': 'The forthcoming LHC-b experiments and Super-B factories can measurement these decays, together with the [MATH] decays.', '0711.0497-1-12-0': 'For [MATH], the heavy [MATH] quark decays into [MATH], and [MATH] is produced from vacuum.', '0711.0497-1-12-1': 'Subsequently, [MATH], [MATH] and the spectator [MATH] can be transferred into [MATH], and the quarks left form a kaon.', '0711.0497-1-12-2': 'The other [MATH] states can also be produced by selecting a different quark pair [MATH] or changing the [MATH] quark by [MATH] quark.', '0711.0497-1-12-3': 'We give the decay amplitudes for non-leptonic [MATH] decay channels induced by [MATH] transition in Table [REF].', '0711.0497-1-12-4': 'These decays are suppressed by CKM matrix elements [MATH].', '0711.0497-1-12-5': '[MATH] is one example of this kind of decays and the product branching ratio is: [EQUATION] where [MATH] and [MATH] [CITATION].', '0711.0497-1-12-6': 'The uncertainties are from the experimental results for [MATH].', '0711.0497-1-12-7': 'In the above calculation, mass differences and lifetime differences are neglected again.', '0711.0497-1-12-8': 'From the branching ratio for this decay chain, we can see that this kind of process receives strong suppression.', '0711.0497-1-12-9': 'Furthermore, the detection of [MATH] is more difficult than [MATH], thus it could hardly be measured at the present two [MATH] factories.', '0711.0497-1-12-10': 'The relations for the [MATH] decay can be derived similarly using the effective Hamiltonian which are also useful in searching for the [MATH] mesons: [EQUATION]', '0711.0497-1-12-11': 'In [MATH] decays, the spectator is an [MATH] quark, thus the initial state is very simple: a singlet of flavor SU(3) group.', '0711.0497-1-12-12': 'But the effective electro-weak Hamiltonian can form an octet: [MATH].', '0711.0497-1-12-13': 'The effective Hamiltonian at hadron level can be written by: [EQUATION] where the non-zero elements of the transition Hamiltonian are [MATH] for CKM allowed channels [MATH] and [MATH] for CKM suppressed channels [MATH] with a factor [MATH].', '0711.0497-1-12-14': 'The corresponding Feynmann diagrams are given in Fig. [REF].', '0711.0497-1-12-15': 'The coefficients for distinct contributions are given in Tab. [REF].', '0711.0497-1-12-16': 'The CKM matrix element for the decay channels induced by [MATH] is [MATH], which is in the same order with that of [MATH]: [MATH].', '0711.0497-1-12-17': 'Thus without any other suppressions, these [MATH] decays also have similar branching ratios ([MATH]) with [MATH].', '0711.0497-1-12-18': 'The decays in the second part of Tab. [REF] are suppressed by [MATH], which are expected to have smaller decay rates ([MATH]).', '0711.0497-1-12-19': 'Furthermore, the SU(3) symmetry implies the following relations: [EQUATION]', '0711.0497-1-12-20': 'Besides non-leptonic [MATH] decays, [MATH] particles can also be produced in radiative decays, as Fig. [REF] shows.', '0711.0497-1-12-21': 'Two-body radiative decays can serve as a natural filter to exclude spin-0 candidates of Z particle, as the photon can only be transversely polarized.', '0711.0497-1-12-22': 'But in order to predict the production rates, one has to know [MATH] transition form factors.', '0711.0497-1-12-23': 'In the second diagram of Fig. [REF], in order to generate the [MATH], we require at least one hard gluon which will suppress the contribution from this diagram.', '0711.0497-1-12-24': 'Naively thinking, the first diagram will also be suppressed by the off-shell [MATH] quark propagator, but since the emitted photon is not energetic (the total energy release is only about [MATH] GeV), the off-shellness is not large.', '0711.0497-1-12-25': 'This may imply the following radiative [MATH] decays could go through with considerable rates: [EQUATION]', '0711.0497-1-13-0': '# [MATH] Particle', '0711.0497-1-14-0': 'In the above, we have utilized the flavor SU(3) symmetry for light quarks in [MATH] mesons.', '0711.0497-1-14-1': 'One can also replace one or two heavy [MATH] quarks by heavier [MATH] quarks which can predict [MATH] and [MATH] mesons [CITATION].', '0711.0497-1-14-2': 'Another attempt is to replace a light quark by a heavy [MATH] quark.', '0711.0497-1-14-3': 'With this replacement, we obtain three [MATH] states which contain three heavy quarks and a light quark: [MATH], [MATH] and [MATH], together with their charge conjugates.', '0711.0497-1-14-4': 'We have to confess that this replacement may change the internal dynamics, but here we assume the same dynamics with [MATH].', '0711.0497-1-14-5': 'In this case, the masses can be obtained by using mass differences of [MATH] and [MATH] quarks deriving from masses of [MATH] and [MATH].', '0711.0497-1-14-6': 'The rough predictions for masses are around [MATH] and [MATH] MeV.', '0711.0497-1-14-7': 'If these mesons are viewed as the resonance of [MATH] and [MATH], their masses could be predicted as [MATH] MeV, [MATH] MeV and [MATH] MeV.', '0711.0497-1-15-0': 'Because of the large mass of [MATH], these particles only appear in [MATH] meson decays.', '0711.0497-1-15-1': 'The corresponding effective Hamiltonian responsible for non-leptonic decays is [EQUATION] where the first contribution [MATH] comes from the gluonic diagram shown as the first one in Figure [REF]; while the second term [MATH] comes from the [MATH]-recoiling diagram shown as the second one in Figure [REF].', '0711.0497-1-15-2': 'These two contributions give the following amplitudes for [MATH] decays: [EQUATION]', '0711.0497-1-15-3': 'The first 4 decay channels shown in eq. ([REF]-[REF]) are induced by [MATH] at quark level and thus have branching ratios of order [MATH], while the other decays shown in eq. ([REF]-[REF]) are suppressed by [MATH], which have smaller decay rates ([MATH]).', '0711.0497-1-16-0': '# Summary', '0711.0497-1-17-0': 'Belle Collaboration has reported a resonance named [MATH], which consists of at least four quarks in constituent quark model.', '0711.0497-1-17-1': 'In this note, we analyze the octet to which this [MATH] meson belongs and the corresponding singlet meson.', '0711.0497-1-17-2': 'Using the picture that the [MATH] mesons are the resonances of [MATH] and [MATH] mesons, we estimate the masses of these mesons.', '0711.0497-1-17-3': 'We investigate the production in non-leptonic [MATH] decays by constructing the effective Hamiltonian using flavor SU(3) meson matrices.', '0711.0497-1-17-4': 'The transition at quark level is either [MATH] or [MATH], where the former one is CKM favored and the latter is suppressed by [MATH].', '0711.0497-1-17-5': 'Thus the considered non-leptonic decays have either similar branching ratios with the observed decay [MATH]) or smaller branching ratios ([MATH]) as shown in the text.', '0711.0497-1-17-6': 'Utilizing the SU(3) symmetry, we also obtain many relations for various decay channels.', '0711.0497-1-17-7': 'Measurements of the [MATH] invariant mass distribution in [MATH] and [MATH] are helpful to detect the [MATH] particles and determine the relative size of [MATH] and [MATH].', '0711.0497-1-17-8': 'We also study the production rates in non-leptonic [MATH] decays and radiative [MATH] decays in a similar way.', '0711.0497-1-17-9': 'Replacing a light [MATH] quark by a heavy [MATH] quark, we get three states which have three heavy quarks.', '0711.0497-1-17-10': 'The masses and the production in [MATH] decays are also discussed.', '0711.0497-1-17-11': 'Measurements of all these decays at the present [MATH] factories and the forthcoming LHC-b experiments will help us to clarify the new [MATH] particles.'}

{'0711.0497-2-0-0': 'Recently, Belle Collaboration has reported a resonant state produced in [MATH], which is called [MATH].', '0711.0497-2-0-1': 'This state is charged, so it can not be interpreted as an ordinary charmonium state.', '0711.0497-2-0-2': 'In this paper, we analyze the octet to which this particle belongs and predict the masses of mesons in this octet.', '0711.0497-2-0-3': 'Utilizing flavor SU(3) symmetry, we study production rates in several kinds of [MATH] decays.', '0711.0497-2-0-4': 'The [MATH] and [MATH] decay channels, favored by Cabibbo-Kobayashi-Maskawa matrix elements, can have branching ratios of [MATH].', '0711.0497-2-0-5': 'This large branching ratio could be observed at the running [MATH] factories to detect [MATH] particles containing a strange quark.', '0711.0497-2-0-6': 'We also predict large branching ratios of the [MATH] and [MATH]) particle production rates in non-leptonic [MATH] decays and radiative [MATH] decays.', '0711.0497-2-0-7': 'Measurements of these decays at the ongoing [MATH] factories and the forthcoming Large Hadron Collider-b experiments are helpful to clarify the mysterious [MATH] particles.', '0711.0497-2-1-0': '# Introduction', '0711.0497-2-2-0': 'Recently, there are many exciting discoveries on new hadron states especially in the hidden-charm sector.', '0711.0497-2-2-1': 'Among these discoveries, the most intriguing one is the new relatively narrow peak named [MATH] found by Belle Collaboration in the invariant mass spectrum of [MATH] in the decay mode [MATH][CITATION].', '0711.0497-2-2-2': 'There is a large branching fraction for the following decay chain: [EQUATION]', '0711.0497-2-2-3': 'Mass and width of this particle are measured as: [EQUATION]', '0711.0497-2-2-4': "The most prominent characteristic is that it is electric charged, that's to say, this new particle can not be described as an ordinary charmonium state or a charmonium-like state such as [MATH].", '0711.0497-2-2-5': 'On the other hand, this particle can decay to [MATH] with a large rate through strong interactions, so it involves at least four quarks [MATH], though there is not any further detailed information on its inner dynamics at present.', '0711.0497-2-3-0': 'In order to elucidate this particle, many theoretical studies [CITATION] have been put forward.', '0711.0497-2-3-1': 'This meson could be viewed as a genuine tetraquark state with diquark anti-diquark [MATH] content which has a large rate to [MATH] [CITATION].', '0711.0497-2-3-2': 'Moreover based on QCD-string, two different four-quark descriptions are proposed in Ref. [CITATION]: one can be reduced to the ordinary diquark-diquark picture and the other one can not.', '0711.0497-2-3-3': 'Besides this kind of explanation, it has also been identified as the resonance of [MATH] [CITATION] as its mass is close to the thresholds of [MATH] and [MATH].', '0711.0497-2-3-4': 'Within this picture, the authors in Ref. [CITATION] explored the production of [MATH] and [MATH].', '0711.0497-2-3-5': 'Short distance contribution to [MATH] is neglected and the main contribution is from long distance re-scattering effect via [MATH].', '0711.0497-2-3-6': 'With proper parameters, they can successfully explain the much larger production rate of [MATH] than that of [MATH].', '0711.0497-2-3-7': 'In Ref. [CITATION], Bugg took this meson as a [MATH] threshold cusp.', '0711.0497-2-3-8': 'Recently, Qiao also tried to explain this meson with the baryonium picture [CITATION].', '0711.0497-2-3-9': 'Using the technique of QCD sum rules, Lee [MATH] calculated the masses of this particle and its strange partner [MATH] in Ref. [CITATION].', '0711.0497-2-4-0': 'Whether or not these scenarios describe the true dynamics of [MATH], this strange meson indeed plays an important role in the charmonium spectroscopy.', '0711.0497-2-4-1': "In the present paper, we do not intend to give an explanation of this meson's structure, but we want to analyze its partners within SU(3) symmetry: the octet to which the meson [MATH] belongs and the corresponding singlet meson.", '0711.0497-2-4-2': 'Up to now, these is no experimental information on these mesons except [MATH].', '0711.0497-2-4-3': 'The decays of [MATH] meson provide a firm potential in searching for these exotic mesons [CITATION], just like the observed decay channel [MATH].', '0711.0497-2-4-4': 'We will investigate the possibilities to detect these [MATH] mesons in [MATH] decays.', '0711.0497-2-4-5': 'In doing this, we will analyze decay amplitudes with the assumption of SU(3) flavor symmetry: to construct effective Hamiltonian using flavor SU(3) meson matrixes.', '0711.0497-2-4-6': 'The decay amplitudes can also be studied by using Feynmann diagrams.', '0711.0497-2-4-7': 'In the discussion of [MATH] production with the graphic technique, we only consider short-distance contributions and neglect soft final state interactions.', '0711.0497-2-4-8': 'Specifically, the considered decays are divided into three categories: Cabibbo-Kobayashi-Maskawa (CKM) allowed non-leptonic decays; CKM suppressed non-leptonic decays; radiative decays.', '0711.0497-2-4-9': 'The first kind of decays have similar branching ratios with the observed [MATH], while the second type of decays is suppressed by about one order in magnitude and we will show that the running [MATH] factories could hardly detect this kind of decays.', '0711.0497-2-4-10': 'Radiative [MATH] decay is a natural filter to exclude the 0-spin mesons and furthermore this kind of process may go through with a sizable branching ratio.', '0711.0497-2-5-0': 'In the next section, we will analyze the octet of [MATH] meson within flavor SU(3) symmetry and try to estimate their masses.', '0711.0497-2-5-1': 'We will construct the effective Hamiltonian using meson matrices and then use them to study the production rates of [MATH] mesons in [MATH] decays.', '0711.0497-2-5-2': 'In Sec. [REF], we will introduce [MATH] meson which consists of three charm quarks, together with a brief discussion on its production in [MATH] decays.', '0711.0497-2-5-3': 'We will summarize this note in the last section.', '0711.0497-2-6-0': '# The octet and the singlet', '0711.0497-2-7-0': 'Just as stated above, [MATH] involves at least four quarks in constituent quark model, and there is an octet which [MATH] belongs to in flavor SU(3) symmetry.', '0711.0497-2-7-1': 'Generally, we can deduce the particles in this octet using group theory: these particles, under the name [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH], are shown in Fig. [REF].', '0711.0497-2-7-2': 'Besides, there exists one singlet meson called [MATH].', '0711.0497-2-7-3': 'In constituent quark model, quark contents of these mesons are listed by: [EQUATION]', '0711.0497-2-7-4': 'In reality, [MATH] quark is slightly heavier than [MATH] quark which is one of the origins for SU(3) symmetry breaking.', '0711.0497-2-7-5': 'Accordingly, the singlet [MATH] can mix with eighth component of the octet [MATH], in analogy with [MATH] and [MATH].', '0711.0497-2-7-6': 'Physical particles, named [MATH] and [MATH], are mixtures of them and can be expressed as: [EQUATION]', '0711.0497-2-7-7': 'The mixing angle [MATH] can be determined through measuring decays of these two particles in future.', '0711.0497-2-7-8': 'For simplicity, we will assume the mixing is ideal, i.e. [MATH].', '0711.0497-2-7-9': 'In this case, the quark contents are: [EQUATION]', '0711.0497-2-7-10': 'All together, one can use the following meson matrix to describe these mesons: [EQUATION]', '0711.0497-2-7-11': 'With the quark contents given in the above, we are ready to estimate masses of these particles.', '0711.0497-2-7-12': 'Isospin analysis predicts the equal masses for the four mesons with neither open nor hidden strangeness: [MATH].', '0711.0497-2-7-13': 'For the mesons with a strange quark, the mass differences between the lighter [MATH] quarks and the heavier [MATH] quark are required.', '0711.0497-2-7-14': 'One can compare masses of [MATH] and [MATH] to get some information: the mass of [MATH] is [MATH] MeV larger than that of [MATH].', '0711.0497-2-7-15': 'In heavy quark limit [MATH], the light system will not be affected by different heavy quark systems, thus we can simply assume a similar difference for [MATH] mesons which predicts the mass of [MATH] around [MATH] MeV.', '0711.0497-2-7-16': 'Because the mass of newly observed [MATH] meson is not far from the threshold of [MATH], [MATH] meson is regarded as the resonance of [MATH] [CITATION].', '0711.0497-2-7-17': 'Under this mechanism, we could give more precise predictions on the masses for other [MATH] mesons using experimental results for the [MATH] and [MATH] mesons.', '0711.0497-2-7-18': 'Our results are listed in Tab. [REF] and uncertainties in this table are from that of masses of the charmed mesons.', '0711.0497-2-7-19': 'In the heavy quark limit, mesons with the same light system can be related to each other.', '0711.0497-2-7-20': 'But if the [MATH] particles are viewed as tetra-quark states, the effective strange quark mass in [MATH] could be different from that in the usual mesons as the light systems in the two kinds of particles are different.', '0711.0497-2-7-21': 'If [MATH] mesons are described as molecules, probably they would not belong to a full SU(3) nonet and the predicted masses may not be suitable.', '0711.0497-2-7-22': 'Currently, there is no better solution and we will use this assumption in the present study.', '0711.0497-2-7-23': 'The recent QCD sum rule study predicts the mass by [CITATION]: [EQUATION] which is above the [MATH] and [MATH] threshold by about 160 MeV.', '0711.0497-2-7-24': 'More experimental studies are required to test this description.', '0711.0497-2-8-0': 'Experimentalists have observed the [MATH] particle through the [MATH] with [MATH].', '0711.0497-2-8-1': 'Assuming S-wave decay for [MATH] meson, the quantum numbers can be determined as [MATH] [CITATION].', '0711.0497-2-8-2': "In order to detect the other [MATH] mesons, experimentalists will choose the proper final states to re-construct them, thus the predictions on [MATH]'s strong decays are required.", '0711.0497-2-8-3': 'Using the flavor SU(3) symmetry and [MATH], we also list the strong decays of other [MATH] mesons in Tab. [REF].', '0711.0497-2-8-4': 'With the assumption [MATH], another kind of possible decay modes is [MATH] [CITATION], where [MATH] denotes a light vector meson.', '0711.0497-2-9-0': 'In order to explore the production in [MATH] decays, one can construct the effective Hamiltonian at hadron level using meson matrices [CITATION].', '0711.0497-2-9-1': 'In the following, to construct the related effective Hamiltonian, we will assume the flavor SU(3) symmetry.', '0711.0497-2-9-2': 'In [MATH] decays, the initial state [MATH] forms an SU(3) anti-triplet.', '0711.0497-2-9-3': 'The transition at quark level is either [MATH] or [MATH], which is described by the effective electro-weak Hamiltonian: [EQUATION] where [MATH].', '0711.0497-2-9-4': '[MATH] and [MATH] are color indices.', '0711.0497-2-9-5': 'The transition [MATH] is CKM favored: [MATH], while the [MATH] transition is suppressed by [MATH].', '0711.0497-2-9-6': 'To construct the effective Hamiltonian at hadron level, only the flavor structures needs to be concerned.', '0711.0497-2-9-7': 'The effective electro-weak Hamiltonian given in Eq. ([REF]) can also be written as an SU(3) triplet: [MATH] (i=1 (u), 2 (d), 3(s)), where the only non-zero elements are [MATH] for CKM favored decays [MATH], and [MATH] for CKM suppressed channels [MATH].', '0711.0497-2-9-8': 'The final state mesons can be described by two nonet matrices: [MATH] and [MATH].', '0711.0497-2-9-9': 'The effective Hamiltonian at hadron level could be constructed as: [EQUATION] where the upper index labels rows and the lower labels columns.', '0711.0497-2-10-0': 'The above effective Hamiltonian can be related to Feynmann diagrams with the one-to-one correspondence and the lowest order diagrams are given in Fig. [REF].', '0711.0497-2-10-1': 'The second term in eq. ([REF]) corresponds to the second diagram in Fig. [REF] (called [MATH]-recoiling diagram) in which the spectator light quark in [MATH] meson enters into the heavy [MATH] meson.', '0711.0497-2-10-2': 'If the spectator quark goes to the light meson, we call this kind of diagram (the third one in Fig. [REF]) as the [MATH]-emission diagram which corresponds to the third term in the effective Hamiltonian.', '0711.0497-2-10-3': 'In order to estimate relative sizes of these terms, we have to analyze diagrams at quark level.', '0711.0497-2-10-4': 'Final state mesons move very slowly and thus the gluon generating the [MATH] quark pair is soft: [MATH].', '0711.0497-2-10-5': 'Thus after integrating out high energy scales, decay amplitudes can be expressed as matrix elements of a soft four-quark operator between initial and final states.', '0711.0497-2-10-6': 'The first term in Eq. ([REF]) corresponds to the annihilation diagram (the first one in Fig. [REF]), as flavor indices of [MATH] and [MATH] in this term are contracted with each other.', '0711.0497-2-10-7': 'This kind of diagram is expected to be suppressed in two-body non-leptonic [MATH] decays.', '0711.0497-2-10-8': 'But here since the gluons are soft, decay amplitudes can also be expressed as time-ordered products of a soft four-quark operator and the [MATH] interaction Hamiltonian which contains only soft fields, thus this kind of contribution is comparable with contributions from the second and third terms in Eq. ([REF]).', '0711.0497-2-10-9': 'For SU(3) flavor singlet mesons [MATH] and [MATH], there are additional contributions which are given by the last two terms in Eq. ([REF]).', '0711.0497-2-10-10': 'One kind of typical Feynmann diagram is also shown in Fig. [REF] as the last two diagrams and it is the contribution from the higher Fock states of [MATH] and [MATH].', '0711.0497-2-10-11': 'Even in charmless two-body [MATH] decays [CITATION], this kind of gluonic contribution is sizable.', '0711.0497-2-10-12': 'Here we do not have any implication and thus one can not neglect it with any a priori.', '0711.0497-2-11-0': 'With the effective Hamiltonian given in Eq. ([REF]), we give the decay amplitudes for the first kind of non-leptonic [MATH] decay channels in Table [REF].', '0711.0497-2-11-1': 'These decays are induced by the CKM allowed transition [MATH] and go through with a large decay rate (typically the same order with the observed [MATH]).', '0711.0497-2-11-2': 'The flavor SU(3) symmetry implies the following relations for [MATH] decays: [EQUATION] where mass differences and lifetime differences of [MATH] mesons are neglected which can not produce large corrections.', '0711.0497-2-11-3': 'Although all of these decays are expected to go through with large branching fractions, decay rates may differ from each other for distinct coefficients.', '0711.0497-2-11-4': 'Two of the decays in the first line have been observed experimentally, while the possibility to observe the other two channels is a little smaller as the daughter meson [MATH] from [MATH] is relatively more difficult to measure.', '0711.0497-2-11-5': 'The decays in the second line is contributed from the third term of effective Hamiltonian given in Eq. ([REF]), which should also have similar production rates.', '0711.0497-2-11-6': 'Among these four channels, [MATH] and [MATH] can have large branching ratios and the final states ([MATH] or [MATH]) are easily to be measured on the experimental side.', '0711.0497-2-11-7': 'Thus measurements of the [MATH] invariant mass distribution in these two channels are helpful to detect the [MATH] particles and determine relative sizes of [MATH] and [MATH].', '0711.0497-2-11-8': 'The other [MATH] decays are less possible to be measured in the running [MATH] factories as either [MATH] or [MATH] is produced in the final state.', '0711.0497-2-11-9': 'The forthcoming LHC-b experiments and Super-B factories can measure these decays, together with the [MATH] decays.', '0711.0497-2-12-0': 'For [MATH], the heavy [MATH] quark decays into [MATH], and [MATH] is produced from vacuum.', '0711.0497-2-12-1': 'Subsequently, [MATH], [MATH] and the spectator [MATH] can be transferred into [MATH], and the quarks left form a kaon.', '0711.0497-2-12-2': 'The other [MATH] states can also be produced by selecting a different quark pair [MATH] or changing the [MATH] quark by [MATH] quark.', '0711.0497-2-12-3': 'We give the decay amplitudes for non-leptonic [MATH] decay channels induced by [MATH] transition in Table [REF].', '0711.0497-2-12-4': 'These decays are suppressed by CKM matrix elements [MATH].', '0711.0497-2-12-5': '[MATH] is one example of this kind of decays and the product branching ratio is: [EQUATION] where [MATH] and [MATH] [CITATION].', '0711.0497-2-12-6': 'The uncertainties are from the experimental results for [MATH].', '0711.0497-2-12-7': 'In the above calculation, mass differences and lifetime differences are neglected again.', '0711.0497-2-12-8': 'From the branching ratio for this decay chain, we can see that this kind of process receives strong suppression.', '0711.0497-2-12-9': 'Furthermore, the detection of [MATH] is more difficult than [MATH], thus it could hardly be measured at the present two [MATH] factories.', '0711.0497-2-12-10': 'The relations for the [MATH] decay can be derived similarly using the effective Hamiltonian which are also useful in searching for the [MATH] mesons: [EQUATION]', '0711.0497-2-12-11': 'As pointed out in ref. [CITATION], the study of charmonium like states production in [MATH] decays is easier.', '0711.0497-2-12-12': 'Here we also consider the Z(4430) particle production in [MATH] decays.', '0711.0497-2-12-13': 'In this case, the spectator is a [MATH] quark, thus the initial state is very simple: a singlet of flavor SU(3) group.', '0711.0497-2-12-14': 'But the effective electro-weak Hamiltonian can form an octet: [MATH].', '0711.0497-2-12-15': 'The effective Hamiltonian at hadron level can be written by: [EQUATION] where the non-zero elements of the transition Hamiltonian are [MATH] for CKM allowed channels [MATH] and [MATH] for CKM suppressed channels [MATH] with a factor [MATH].', '0711.0497-2-12-16': 'The corresponding Feynmann diagrams are given in Fig. [REF].', '0711.0497-2-12-17': 'The coefficients for distinct contributions are given in Tab. [REF].', '0711.0497-2-12-18': 'The CKM matrix element for the decay channels induced by [MATH] is [MATH], which is in the same order with that of [MATH]: [MATH].', '0711.0497-2-12-19': 'Thus without any other suppressions, these [MATH] decays also have similar branching ratios ([MATH]) with [MATH].', '0711.0497-2-12-20': 'The decays in the second part of Tab. [REF] are suppressed by [MATH], which are expected to have smaller decay rates ([MATH]).', '0711.0497-2-12-21': 'Furthermore, the SU(3) symmetry implies the following relations: [EQUATION]', '0711.0497-2-12-22': 'Besides non-leptonic [MATH] decays, [MATH] particles can also be produced in radiative decays, as Fig. [REF] shows.', '0711.0497-2-12-23': 'Two-body radiative decays can serve as a natural filter to exclude spin-0 candidates of Z particles, as the photon can only be transversely polarized.', '0711.0497-2-12-24': 'But in order to predict the production rates, one has to know [MATH] transition form factors.', '0711.0497-2-12-25': 'In the second diagram of Fig. [REF], in order to generate the [MATH], we require at least one hard gluon which will suppress the contribution from this diagram.', '0711.0497-2-12-26': 'Naively thinking, the first diagram will also be suppressed by the off-shell [MATH] quark propagator, but since the emitted photon is not energetic (the total energy release is only about [MATH] GeV), the off-shellness is not large.', '0711.0497-2-12-27': 'This may imply the following radiative [MATH] decays could go through with considerable rates: [EQUATION]', '0711.0497-2-13-0': '# [MATH] Particle', '0711.0497-2-14-0': 'In the above, we have utilized the flavor SU(3) symmetry for light quarks in [MATH] mesons.', '0711.0497-2-14-1': 'One can also replace one or two heavy [MATH] quarks by heavier [MATH] quarks which can predict [MATH] and [MATH] mesons [CITATION].', '0711.0497-2-14-2': 'Another attempt is to replace a light quark by a heavy [MATH] quark.', '0711.0497-2-14-3': 'With this replacement, we obtain three [MATH] states which contain three heavy quarks and a light quark: [MATH], [MATH] and [MATH], together with their charge conjugates.', '0711.0497-2-14-4': 'We have to confess that this replacement may change the internal dynamics, but here we assume the same dynamics with [MATH].', '0711.0497-2-14-5': 'In this case, the masses can be obtained by using mass differences of [MATH] and [MATH] quarks deriving from masses of [MATH] and [MATH].', '0711.0497-2-14-6': 'The rough predictions for masses are around [MATH] and [MATH] MeV.', '0711.0497-2-14-7': 'If these mesons are viewed as the resonance of [MATH] and [MATH], their masses could be predicted as [MATH] MeV, [MATH] MeV and [MATH] MeV.', '0711.0497-2-15-0': 'Because of the large mass of [MATH], these particles only appear in [MATH] meson decays.', '0711.0497-2-15-1': 'The corresponding effective Hamiltonian responsible for non-leptonic decays is [EQUATION] where the first contribution [MATH] comes from the gluonic diagram shown as the first one in Figure [REF]; while the second term [MATH] comes from the [MATH]-recoiling diagram shown as the second one in Figure [REF].', '0711.0497-2-15-2': 'These two contributions give the following amplitudes for [MATH] decays: [EQUATION]', '0711.0497-2-15-3': 'The first 4 decay channels shown in eq. ([REF]-[REF]) are induced by [MATH] at quark level and thus have branching ratios of order [MATH], while the other decays shown in eq. ([REF]-[REF]) are suppressed by [MATH], which have smaller decay rates ([MATH]).', '0711.0497-2-16-0': '# Summary', '0711.0497-2-17-0': 'Belle Collaboration has reported a resonance named [MATH], which consists of at least four quarks in constituent quark model.', '0711.0497-2-17-1': 'In this note, we analyze the octet to which this [MATH] meson belongs and the corresponding singlet meson.', '0711.0497-2-17-2': 'Using the picture that the [MATH] mesons are the resonances of [MATH] and [MATH] mesons, we estimate the masses of these mesons.', '0711.0497-2-17-3': 'We investigate the production in non-leptonic [MATH] decays by constructing the effective Hamiltonian using flavor SU(3) meson matrices.', '0711.0497-2-17-4': 'The transition at quark level is either [MATH] or [MATH], where the former one is CKM favored and the latter is suppressed by [MATH].', '0711.0497-2-17-5': 'Thus the considered non-leptonic decays have either similar branching ratios with the observed decay [MATH]) or smaller branching ratios ([MATH]) as shown in the text.', '0711.0497-2-17-6': 'Utilizing the SU(3) symmetry, we also obtain many relations for various decay channels.', '0711.0497-2-17-7': 'Measurements of the [MATH] invariant mass distribution in [MATH] and [MATH] are helpful to detect the [MATH] particles and determine the relative size of [MATH] and [MATH].', '0711.0497-2-17-8': 'We also study the production rates in non-leptonic [MATH] decays and radiative [MATH] decays in a similar way.', '0711.0497-2-17-9': 'Replacing a light [MATH] quark by a heavy [MATH] quark, we get three states which have three heavy quarks.', '0711.0497-2-17-10': 'The masses and the production in [MATH] decays are also discussed.', '0711.0497-2-17-11': 'Measurements of all these decays at the present [MATH] factories and the forthcoming LHC-b experiments will help us to clarify the new [MATH] particles.'}

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[]

[['0711.0497-1-7-13', '0711.0497-2-7-13']]

[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0711.0497

1801.03109

{'1801.03109-1-0-0': "We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures.", '1801.03109-1-0-1': 'In particular, we show that the range of a nonatomic quantum probability measure is a weak[MATH]-closed convex set of quantum effects (positive operators whose eigenvalues are less than or equal to one).', '1801.03109-1-0-2': "To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).", '1801.03109-1-1-0': '# Introduction', '1801.03109-1-2-0': "Lyapunov's theorem (Corollary [REF] below) states that if [MATH] are finite positive nonatomic measures on a measurable space [MATH], then the set of points in [MATH] of the form [MATH], with [MATH] ranging over the measurable subsets of [MATH], is closed and convex.", '1801.03109-1-2-1': 'In particular, the range of a single finite positive nonatomic measure is closed and convex.', '1801.03109-1-2-2': 'Our goal is to extend this result into the quantum, or operator-valued, setting.', '1801.03109-1-3-0': 'In quantum mechanics, a quantum system is described by a Hilbert space [MATH].', '1801.03109-1-3-1': 'Positive operator-valued measures (POVMs) and quantum probability measures (POVMs that sum to the identity) are studied in quantum information theory in order to fully describe measurements of a quantum system [CITATION].', '1801.03109-1-3-2': 'The more general notion of operator-valued measures (OVMs) has been studied independently of quantum information theory; see [CITATION].', '1801.03109-1-4-0': 'Denote the C[MATH]-algebra of all bounded operators acting on [MATH] by [MATH] and recall that the predual of [MATH] is [MATH], the ideal of trace class operators.', '1801.03109-1-4-1': 'In the case when [MATH] is finite-dimensional, [MATH] and [MATH] coincide as sets, and so often in the finite-dimensional setting authors do not distinguish between the two.', '1801.03109-1-4-2': 'Unless stated otherwise, we take [MATH] to be infinite-dimensional herein.', '1801.03109-1-4-3': 'Denote the set of all positive operators of unit trace by [MATH]; this is a convex subset of [MATH].', '1801.03109-1-4-4': 'General elements of [MATH], often denoted by [MATH], are called states or density operators and play a critical role in quantum information theory.', '1801.03109-1-4-5': 'Finally, let [MATH] be a Hausdorff space and [MATH] a [MATH]-algebra of subsets of [MATH] (often [MATH] is taken to be the [MATH]-algebra of Borel sets of [MATH] as it will be in the next section).', '1801.03109-1-5-0': '[CITATION] A map [MATH] is an operator-valued measure (OVM) if it is weakly countably additive: for every countable collection [MATH] with [MATH] for [MATH] we have [EQUATION] where the convergence on the right side of the equation above is with respect to the ultraweak topology of [MATH].', '1801.03109-1-5-1': 'An OVM [MATH] is [(i)] bounded if [MATH], self-adjoint if [MATH], for all [MATH],', '1801.03109-1-6-0': 'positive if [MATH], for all [MATH], spectral if [MATH], for all [MATH]', '1801.03109-1-7-0': 'Moreover, an OVM [MATH] is called a positive operator-valued measure (POVM) if it is positive (note that this necessarily implies [MATH] is bounded), a positive operator-valued probability measure or quantum probability measure if it is positive and [MATH], and is called a projection-valued measure (PVM) if it is self-adjoint and spectral (PVMs show up as an important subclass of POVMs in quantum information theory).', '1801.03109-1-8-0': 'We briefly recall some useful definitions from classical measure theory.', '1801.03109-1-8-1': 'Let [MATH] be a set, [MATH] a [MATH]-algebra over [MATH], and [MATH] a measure on the measurable space [MATH].', '1801.03109-1-8-2': 'An atom for [MATH] is a set [MATH] of nonzero measure such that, for any subset [MATH], either [MATH] or [MATH].', '1801.03109-1-8-3': 'A measure [MATH] is atomic if every set of nonzero measure contains an atom.', '1801.03109-1-8-4': 'A measure [MATH] is nonatomic if it has no atoms; that is, if every set of nonzero measure has a subset of different nonzero measure.', '1801.03109-1-8-5': 'Note that a measure that is not atomic is not necessarily nonatomic, and vice-versa.', '1801.03109-1-8-6': 'A classical partial order on measures is that of absolute continuity: if [MATH] are measures on [MATH], then [MATH] is absolutely continuous with respect to [MATH], denoted by [MATH], if [MATH] for all [MATH] for which [MATH].', '1801.03109-1-9-0': 'The definitions of atomic, nonatomic, and absolutely continuous can be readily adapted, mutatis mutandis, to OVMs; in fact, absolute continuity can be used to compare OVMs from the same space [MATH] into different Hilbert spaces, in particular one can compare an OVM with a (classical) measure.', '1801.03109-1-10-0': 'Given an OVM [MATH], for every state [MATH], we can associate to [MATH] the induced complex measure [MATH] on [MATH] defined by [EQUATION].', '1801.03109-1-10-1': 'If [MATH] is a quantum probability measure, then the complex measure [MATH] can be interpreted as mapping the measurement event [MATH] to the corresponding measurement statistics; the probability that event [MATH] is measured by the quantum probability measure [MATH] when the system [MATH] is in state [MATH] is [MATH].', '1801.03109-1-11-0': 'Note that [MATH] is countably additive since [MATH] is weakly countably additive.', '1801.03109-1-12-0': 'It is not difficult to see that [MATH] and [MATH] are mutually absolutely continuous for any full-rank [MATH].', '1801.03109-1-13-0': 'Assume [MATH] is separable and let [MATH] be an orthonormal basis.', '1801.03109-1-13-1': 'As in [CITATION], denote [MATH] the complex measure [MATH].', '1801.03109-1-13-2': 'Now, for any full-rank density operator [MATH] we have [MATH].', '1801.03109-1-13-3': 'Thus, by the classical Radon-Nikodym theorem, there is a unique [MATH] such that [EQUATION] [CITATION] Let [MATH] be a POVM and [MATH] be a full-rank density operator.', '1801.03109-1-13-4': 'The Radon-Nikodym derivative of [MATH] with respect to [MATH] is defined to be [EQUATION].', '1801.03109-1-13-5': 'If [MATH] is a valid quantum random variable (in the sense that it takes every [MATH] to a bounded operator), then [MATH] is said to exist.', '1801.03109-1-14-0': 'If [MATH] is into a finite-dimensional Hilbert space then [MATH] always exists (see [CITATION] for further analysis into the finite-dimensional setting).', '1801.03109-1-14-1': 'However, for infinite dimensions this derivative is potentially into the unbounded operators and does not exist in such cases.', '1801.03109-1-15-0': 'If [MATH] exists for one full-rank density operator [MATH], then it exists for all full-rank [MATH] [CITATION].', '1801.03109-1-16-0': 'In Section [REF] we define quantum random variables, the integral of such a function against a POVM, and concepts in relation to such functions such as essential support, essential range, and essentially bounded.', '1801.03109-1-16-1': "In Section [REF] we present our main results extending Lyapunov's theorem to positive operator-valued measures.", '1801.03109-1-16-2': "In particular, Corollary [REF] shows that the range of a nonatomic quantum probability measure is a weak[MATH]-closed convex set, which can be viewed as the quantum analogue to Lyapunov's theorem.", '1801.03109-1-17-0': '# Integration and essentially bounded functions', '1801.03109-1-18-0': 'A Borel measurable function [MATH] between the [MATH]-algebra generated by the open sets of [MATH] and the [MATH]-algebra generated by the open sets of [MATH] is a quantum random variable.', '1801.03109-1-19-0': 'Equivalently, [MATH] is a quantum random variable if and only if [EQUATION] are Borel measurable functions for every state [MATH].', '1801.03109-1-20-0': 'A quantum random variable [MATH] is [(i)] bounded if [MATH],', '1801.03109-1-21-0': 'self-adjoint if [MATH], for all [MATH], positive if [MATH], for all [MATH].', '1801.03109-1-22-0': 'Observe that the complex-valued functions [MATH] are measurable and are thus (classical) random variables.', '1801.03109-1-22-1': 'See [CITATION] for more detailed analyses of quantum random variables.', '1801.03109-1-23-0': 'We are interested in integrating a quantum random variable with respect to a positive operator-valued measure: [CITATION] Let [MATH] be a POVM such that [MATH] exists.', '1801.03109-1-23-1': 'A positive quantum random variable [MATH] is [MATH]-integrable if the function [EQUATION] is [MATH]-integrable for every state [MATH].', '1801.03109-1-24-0': 'If [MATH] is a self-adjoint quantum random variable, then [MATH] defined by [EQUATION] are positive quantum random variables [CITATION].', '1801.03109-1-24-1': 'An arbitrary quantum random variable [MATH] is then said to be [MATH]-integrable if and only if [MATH] and [MATH] are [MATH]-integrable.', '1801.03109-1-25-0': '[CITATION] Let [MATH] be a POVM such that [MATH] exists.', '1801.03109-1-25-1': 'If [MATH] is a [MATH]-integrable quantum random variable then the integral of [MATH] with respect to [MATH], denoted [MATH], is implicitly defined by the formula [EQUATION].', '1801.03109-1-26-0': 'This definition of the integral of a quantum random variable with respect to a POVM was first defined in the finite-dimensional case in [CITATION].', '1801.03109-1-26-1': 'The infinite case was shown to work in [CITATION] and one should note that showing that the above definition works takes some proving.', '1801.03109-1-27-0': 'A good sign that this is a reasonable definition is that for every set [MATH] we have that [EQUATION].', '1801.03109-1-27-1': 'For a detailed discussion of this integral and a proof of the above consult [CITATION].', '1801.03109-1-28-0': 'Let [MATH] be an OVM and let [MATH] be a quantum random variable.', '1801.03109-1-28-1': 'Then the essential support of [MATH] is [EQUATION].', '1801.03109-1-29-0': 'Let [MATH] be an POVM and let [MATH] be a quantum random variable.', '1801.03109-1-29-1': 'Then the essential range of [MATH] is the set [MATH] of all [MATH] for which [MATH] for every neighbourhood [MATH] of [MATH].', '1801.03109-1-30-0': 'A standard classical result is that the essential range of a measurable function is equal to the intersection of the closure of the image of the function over all measurable sets whose complement has measure zero.', '1801.03109-1-30-1': 'We show this is also the case in the setting of quantum random variables.', '1801.03109-1-30-2': 'Although the proof is a straightforward generalization of the classical setting (see [CITATION]), we include it here for completeness and rigor.', '1801.03109-1-31-0': 'Let [MATH] be a quantum random variable.', '1801.03109-1-31-1': 'Then [EQUATION].', '1801.03109-1-32-0': 'Let [MATH] and suppose [MATH] is such that [MATH].', '1801.03109-1-32-1': 'This implies [MATH].', '1801.03109-1-32-2': 'Indeed, if that were not the case, then there would be an open set [MATH] containing [MATH] such that [MATH], and so [MATH]; however, [MATH] (since [MATH]), a contradiction.', '1801.03109-1-32-3': 'It follows that [MATH] for every [MATH] is such that [MATH].', '1801.03109-1-33-0': 'Conversely, let [MATH].', '1801.03109-1-33-1': 'Then [MATH] for some neighbourhood [MATH] of [MATH].', '1801.03109-1-33-2': 'Let [MATH].', '1801.03109-1-33-3': 'This implies [MATH].', '1801.03109-1-33-4': 'Indeed, if that were not the case, then [MATH], but then [MATH], a contradiction.', '1801.03109-1-33-5': 'It follows that [MATH] for every [MATH] such that [MATH].', '1801.03109-1-34-0': 'Again, analogous to the classical setting, we define the essential supremum of a quantum random variable.', '1801.03109-1-35-0': 'Let [MATH] be a quantum random variable.', '1801.03109-1-35-1': 'The essential supremum of [MATH] is the quantity [EQUATION]', '1801.03109-1-35-2': 'If [MATH] then [MATH] is essentially bounded and [EQUATION].', '1801.03109-1-36-0': 'Let [MATH] be a POVM such that [MATH] exists for any full-rank density operator [MATH].', '1801.03109-1-36-1': '[CITATION] tells us that every quantum random variable that is essentially bounded with respect to [MATH] is [MATH]-integrable.', '1801.03109-1-36-2': 'Denote the set of all such quantum random variables by [MATH].', '1801.03109-1-37-0': 'As in the classical case, for [MATH] let [MATH] which is a seminorm on [MATH].', '1801.03109-1-37-1': 'Quotienting by the ideal of functions that vanish under this seminorm we get [MATH].', '1801.03109-1-37-2': 'Because [MATH] and [MATH] are mutually absolutely continuous they give the same set of essentially bounded quantum random variables from [MATH] to [MATH] by the second part of the definition of essentially bounded.', '1801.03109-1-37-3': 'This implies that [EQUATION] is a Banach space, actually a von Neumann algebra (and so the tensor product is unique by [CITATION]).', '1801.03109-1-37-4': "In the same section of Takesaki's book there is the following predual identity [EQUATION].", '1801.03109-1-37-5': 'Thus [MATH] carries a weak[MATH]-topology.', '1801.03109-1-37-6': 'To be clear, on sums of simple tensors these isomorphisms are both given by [EQUATION].', '1801.03109-1-37-7': 'So we consider both [MATH] and its predual as sets of equivalence classes of functions from [MATH] into [MATH] or [MATH], respectively.', '1801.03109-1-38-0': 'Integration is a very important concept in our analysis and so, for a given POVM [MATH] such that [MATH] exists for any full-rank density operator [MATH], we define [MATH] to be [EQUATION]', '1801.03109-1-38-1': 'This integral operator is called the quantum expected value of [MATH] relative to the POVM [MATH] in [CITATION] and denoted [MATH].', '1801.03109-1-39-0': 'Let [MATH] be a POVM such that [MATH] exists.', '1801.03109-1-39-1': 'Then [MATH] is weak[MATH]-weak[MATH]-continuous.', '1801.03109-1-40-0': 'For every [MATH] we first need to establish that [EQUATION] is an element of [MATH].', '1801.03109-1-40-1': 'To this end, consider the bounded linear functional [MATH], [MATH] which is precisely integration against [MATH] in the first component and taking the trace against [MATH] in the second component of [MATH].', '1801.03109-1-40-2': 'In particular, the span of these functionals is dense in [MATH].', '1801.03109-1-41-0': 'We then have [EQUATION] by Corollary 2.9 of [CITATION] since [MATH] is essentially bounded.', '1801.03109-1-41-1': 'Therefore, [EQUATION].', '1801.03109-1-42-0': 'Now suppose [MATH] in [MATH].', '1801.03109-1-42-1': 'By the above, every bounded linear functional of [MATH] is given as [MATH] for [MATH].', '1801.03109-1-42-2': 'Then [EQUATION] from which it follows that [MATH].', '1801.03109-1-43-0': 'A very interesting problem that should be studied is whether this integration map [MATH] for a nonatomic POVM is injective.', '1801.03109-1-43-1': 'For instance, the map [MATH] is not injective when [MATH] is finite-dimensional.', '1801.03109-1-43-2': 'This is easily seen by cardinality alone since [MATH] being nonatomic implies that [MATH] is uncountable which certainly implies that [MATH] is infinite-dimensional while [MATH] is not.', '1801.03109-1-43-3': 'There are a few other technical conditions that will insure that [MATH] is not injective but they do not provide a complete answer.', '1801.03109-1-43-4': 'We thus state it here as an open problem:', '1801.03109-1-44-0': 'If [MATH] is nonatomic then is [MATH] not injective?', '1801.03109-1-45-0': '# Operator-valued Lyapunov theorem', '1801.03109-1-46-0': 'We are now in a position to prove the main theorem of this paper.', '1801.03109-1-47-0': 'Let [MATH] be a nonatomic POVM such that [MATH] exists and let [MATH] be as above.', '1801.03109-1-47-1': 'Then the set [MATH] is a weak[MATH]-compact convex set in [MATH].', '1801.03109-1-48-0': 'The proof follows the same general structure as in the classical setting (see [CITATION] or [CITATION]).', '1801.03109-1-48-1': 'Let [EQUATION]', '1801.03109-1-48-2': 'It is well known from the classical case that the set on the left side of the tensor product is weak[MATH]-closed and thus weak[MATH]-compact (being in its unit ball).', '1801.03109-1-48-3': 'Hence, [MATH] is weak[MATH]-compact as well.', '1801.03109-1-49-0': 'It is easy to see that the map [MATH] defined in equation ([REF]) is affine with respect to quantum random variables [MATH] (although it is not affine with respect to POVMs [MATH]).', '1801.03109-1-49-1': 'Since [MATH] is weak[MATH]-weak[MATH]-continuous by Theorem [REF] we have that [MATH] is weak[MATH]-compact and convex.', '1801.03109-1-50-0': 'One should note here that [MATH] and contains [MATH] and [MATH] but it does not have to equal the operator interval [MATH].', '1801.03109-1-50-1': 'For instance, [MATH] would equal [MATH] if [MATH] where [MATH] is Lebesgue measure on [MATH].', '1801.03109-1-51-0': 'Pick [MATH].', '1801.03109-1-51-1': 'The set [MATH] is the weak[MATH]-continuous inverse image of a weak[MATH]-closed set, and so [MATH] is weak[MATH]-closed.', '1801.03109-1-51-2': 'Clearly [MATH] is convex as well.', '1801.03109-1-51-3': 'By the Krein-Milman theorem, there exists [MATH].', '1801.03109-1-51-4': 'We claim that [MATH] for some [MATH].', '1801.03109-1-51-5': 'This will complete the proof since we already have that [MATH], giving [MATH] is weak[MATH]-compact and convex.', '1801.03109-1-52-0': 'Now, we prove the claim by contradiction: suppose [MATH] is not a characteristic function; that is, [MATH] in [MATH].', '1801.03109-1-52-1': 'Thus, the essential range of [MATH] contains some [MATH] with [MATH].', '1801.03109-1-52-2': 'Let [MATH] and define [MATH].', '1801.03109-1-52-3': 'Note that [MATH] is open and [MATH] but [MATH].', '1801.03109-1-52-4': 'We then have that [MATH] is a Borel set and [MATH].', '1801.03109-1-53-0': 'By the non-atomicity of [MATH] there exist disjoint sets [MATH] such that [MATH].', '1801.03109-1-53-1': 'Define a quantum random variable [MATH] by [EQUATION].', '1801.03109-1-53-2': 'Thus, [MATH].', '1801.03109-1-54-0': 'Hence [MATH].', '1801.03109-1-54-1': 'Since [MATH], this contradicts the assumption that [MATH], and the claim is proven.', '1801.03109-1-55-0': 'Let [MATH] be nonatomic POVMs such that [MATH] exists for [MATH].', '1801.03109-1-55-1': 'Then the set of points in [MATH] of the form [MATH], with [MATH] ranging over all [MATH], is a weak[MATH]-closed convex set.', '1801.03109-1-56-0': 'Define [MATH] as [MATH] on [MATH].', '1801.03109-1-56-1': 'It is not hard to see that [MATH] is a nonatomic POVM and [MATH] exists.', '1801.03109-1-56-2': 'The result follows.', '1801.03109-1-57-0': 'Note that if [MATH] is a discrete set, then the range of [MATH] cannot be convex, and so the above corollary shows that nonatomic POVMs such that [MATH] exists for [MATH] have fundamentally different range spaces compared to atomic POVMs.', '1801.03109-1-58-0': "Lyapunov's Convexity Theorem follows as a one-dimensional consequence of Corollary [REF]:", '1801.03109-1-59-0': '[CITATION] Let [MATH] be finite positive nonatomic measures on [MATH].', '1801.03109-1-59-1': 'Then the set of points in [MATH] of the form [MATH], with [MATH] ranging over all [MATH], is a closed convex set.', '1801.03109-1-60-0': 'Our methods rely on the existence of the Radon-Nikodym derivative [MATH] (which, as we mentioned before, always exists when [MATH] is finite-dimensional).', '1801.03109-1-60-1': "In the absence of this derivative there are easy examples where one can see that Lyapunov's theorem will still hold but we do not have a proof of the general situation.", '1801.03109-1-60-2': 'One such case is the following:', '1801.03109-1-61-0': 'Let [MATH] be a set of mutually singular nonatomic probability measures on the measure space [MATH].', '1801.03109-1-61-1': 'Define [MATH] and so [MATH] is a nonatomic quantum probability measure.', '1801.03109-1-61-2': 'By the same reasoning as [CITATION] it is easy to see that [MATH] does not exist (just take [MATH]).', '1801.03109-1-62-0': 'Let [MATH] and by the mutually singular assumption we have that [MATH].', '1801.03109-1-62-1': "For any infinite tuple [MATH] with [MATH] we can find by Lyapunov's theorem above that there exists [MATH] such that [MATH] and [MATH].", '1801.03109-1-62-2': 'Thus, for [MATH] we have [MATH].', '1801.03109-1-62-3': 'Therefore, [MATH] is a weak[MATH]-closed and convex subset in the operator interval [MATH].'}

{'1801.03109-2-0-0': "We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures.", '1801.03109-2-0-1': 'In particular, we show that the range of a nonatomic quantum probability measure is a weak[MATH]-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration.', '1801.03109-2-0-2': "To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).", '1801.03109-2-1-0': '# Introduction', '1801.03109-2-2-0': "Lyapunov's theorem ([CITATION], and Corollary [REF] below) states that if [MATH], [MATH], [MATH], [MATH] are finite positive nonatomic measures on a measurable space [MATH], then the set of points in [MATH] of the form [MATH], with [MATH] ranging over the measurable subsets of [MATH], is closed and convex.", '1801.03109-2-2-1': 'In particular, the range of a single finite positive nonatomic measure is closed and convex.', '1801.03109-2-2-2': 'Our goal is to extend this result into the quantum, or operator-valued, setting.', '1801.03109-2-3-0': 'In quantum mechanics, a quantum system is described by a Hilbert space [MATH].', '1801.03109-2-3-1': 'Positive operator-valued measures (POVMs) and quantum probability measures (POVMs that sum to the identity) are studied in quantum information theory in order to fully describe measurements of a quantum system [CITATION].', '1801.03109-2-3-2': 'The more general notion of operator-valued measures (OVMs) has been studied independently of quantum information theory; see [CITATION].', '1801.03109-2-4-0': 'Denote the C[MATH]-algebra of all bounded operators acting on [MATH] by [MATH] and recall that the predual of [MATH] is [MATH], the ideal of trace class operators.', '1801.03109-2-4-1': 'In the case when [MATH] is finite-dimensional, [MATH] and [MATH] coincide as sets, and so often in the finite-dimensional setting authors do not distinguish between the two.', '1801.03109-2-4-2': 'Unless stated otherwise, we take [MATH] to be infinite-dimensional herein.', '1801.03109-2-4-3': 'Denote the set of all positive operators of unit trace by [MATH]; this is a convex subset of [MATH].', '1801.03109-2-4-4': 'General elements of [MATH], often denoted by [MATH], are called states or density operators and play a critical role in quantum information theory.', '1801.03109-2-4-5': 'Finally, let [MATH] be a Hausdorff space and [MATH] a [MATH]-algebra of subsets of [MATH] (often [MATH] is taken to be the [MATH]-algebra of Borel sets of [MATH] as it will be in the next section).', '1801.03109-2-5-0': '[CITATION] A map [MATH] is an operator-valued measure (OVM) if it is weakly countably additive: for every countable collection [MATH] with [MATH] for [MATH] we have [EQUATION] where the convergence on the right side of the equation above is with respect to the ultraweak topology of [MATH].', '1801.03109-2-5-1': 'An OVM [MATH] is [(i)] bounded if [MATH], self-adjoint if [MATH], for all [MATH],', '1801.03109-2-6-0': 'positive if [MATH], for all [MATH], spectral if [MATH], for all [MATH]', '1801.03109-2-7-0': 'Moreover, an OVM [MATH] is called a positive operator-valued measure (POVM) if it is positive (note that this necessarily implies [MATH] is bounded), a positive operator-valued probability measure or quantum probability measure if it is positive and [MATH] (the identity operator in [MATH], and is called a projection-valued measure (PVM) if it is self-adjoint and spectral (PVMs show up as an important subclass of POVMs in quantum information theory).', '1801.03109-2-8-0': 'We briefly recall some useful definitions from classical measure theory.', '1801.03109-2-8-1': 'Let [MATH] be a set, [MATH] a [MATH]-algebra over [MATH], and [MATH] a measure on the measurable space [MATH].', '1801.03109-2-8-2': 'An atom for [MATH] is a set [MATH] of nonzero measure such that, for any subset [MATH], either [MATH] or [MATH].', '1801.03109-2-8-3': 'A measure [MATH] is atomic if every set of nonzero measure contains an atom.', '1801.03109-2-8-4': 'A measure [MATH] is nonatomic if it has no atoms; that is, if every set of nonzero measure has a subset of different nonzero measure.', '1801.03109-2-8-5': 'Note that a measure that is not atomic is not necessarily nonatomic, and vice-versa.', '1801.03109-2-8-6': 'A classical partial order on measures is that of absolute continuity: if [MATH] are measures on [MATH], then [MATH] is absolutely continuous with respect to [MATH], denoted by [MATH], if [MATH] for all [MATH] for which [MATH].', '1801.03109-2-9-0': 'The definitions of atomic, nonatomic, and absolutely continuous can be readily adapted, mutatis mutandis, to OVMs; in fact, absolute continuity can be used to compare OVMs from the same space [MATH] into different Hilbert spaces, in particular one can compare an OVM with a (classical) measure.', '1801.03109-2-10-0': 'Given an OVM [MATH], for every state [MATH], we can associate to [MATH] the induced complex measure [MATH] on [MATH] defined by [EQUATION].', '1801.03109-2-10-1': 'If [MATH] is a quantum probability measure, then the complex measure [MATH] can be interpreted as mapping the measurement event [MATH] to the corresponding measurement statistics; the probability that event [MATH] is measured by the quantum probability measure [MATH] when the system [MATH] is in state [MATH] is [MATH].', '1801.03109-2-11-0': 'Note that [MATH] is countably additive since [MATH] is weakly countably additive.', '1801.03109-2-12-0': 'It is not difficult to see that [MATH] and [MATH] are mutually absolutely continuous for any full-rank [MATH].', '1801.03109-2-13-0': 'Assume [MATH] is separable and let [MATH] be an orthonormal basis.', '1801.03109-2-13-1': 'As in [CITATION], denote [MATH] the complex measure [MATH].', '1801.03109-2-13-2': 'Now, for any full-rank density operator [MATH] we have [MATH].', '1801.03109-2-13-3': 'Thus, by the classical Radon-Nikodym theorem, there is a unique [MATH] such that [EQUATION] [CITATION] Let [MATH] be a POVM and [MATH] be a full-rank density operator.', '1801.03109-2-13-4': 'The Radon-Nikodym derivative of [MATH] with respect to [MATH] is defined to be [EQUATION] where [MATH] are the matrix units.', '1801.03109-2-13-5': 'If [MATH] is a valid quantum random variable (in the sense that it takes every [MATH] to a bounded operator), then [MATH] is said to exist.', '1801.03109-2-14-0': 'If [MATH] is into a finite-dimensional Hilbert space then [MATH] always exists (see [CITATION] for further analysis into the finite-dimensional setting).', '1801.03109-2-14-1': 'However, for infinite dimensions this derivative is potentially into the unbounded operators and does not exist in such cases.', '1801.03109-2-15-0': 'If [MATH] exists for one full-rank density operator [MATH], then it exists for all full-rank [MATH] [CITATION].', '1801.03109-2-16-0': 'The following is a truly infinite-dimensional example, in the sense that it is not merely finite-dimensional living in infinite dimensions, where the Radon-Nikodym derivative exists.', '1801.03109-2-17-0': 'Let [MATH] be Lebesgue measure on [MATH] and let [MATH] be restrictions of [MATH] to the intervals [MATH], respectively.', '1801.03109-2-17-1': 'Define [MATH] and so [MATH] is a nonatomic POVM.', '1801.03109-2-17-2': 'Taking [MATH], we can see that [EQUATION] where [MATH] is the [MATH]-matrix unit.', '1801.03109-2-17-3': 'Hence, the Radon-Nikodym derivative exists.', '1801.03109-2-18-0': "An important condition that arises in the study of Lyapunov's theorem (both the operator-valued version herein and the classical version) is a condition on the non-injectivity of integration.", '1801.03109-2-18-1': 'This will be described in more detail at the end of Section 2.', '1801.03109-2-19-0': 'Lyapunov himself found a counterexample to the convexity theorem when there is an infinite number of classical measures [CITATION].', '1801.03109-2-19-1': '[CITATION] also contains counterexamples.', '1801.03109-2-19-2': 'These examples all fail because of injectivity.', '1801.03109-2-20-0': 'In Section [REF] we define quantum random variables, the integral of such a function against a POVM, and concepts in relation to such functions such as essential support, essential range, and essentially bounded.', '1801.03109-2-20-1': "In Section [REF] we present our main results extending Lyapunov's theorem to positive operator-valued measures.", '1801.03109-2-20-2': "In particular, Corollary [REF] shows that the range of a nonatomic quantum probability measure is a weak[MATH]-closed convex set, which can be viewed as the quantum analogue to Lyapunov's theorem.", '1801.03109-2-21-0': '# Integration and essentially bounded functions', '1801.03109-2-22-0': 'A Borel measurable function [MATH] between the [MATH]-algebra generated by the open sets of [MATH] and the [MATH]-algebra generated by the open sets of [MATH] is a quantum random variable.', '1801.03109-2-23-0': 'Equivalently, [MATH] is a quantum random variable if and only if [EQUATION] are Borel measurable functions for every state [MATH].', '1801.03109-2-24-0': 'A quantum random variable [MATH] is [(i)] bounded if [MATH],', '1801.03109-2-25-0': 'self-adjoint if [MATH], for all [MATH], positive if [MATH], for all [MATH].', '1801.03109-2-26-0': 'Observe that the complex-valued functions [MATH] are measurable and are thus (classical) random variables.', '1801.03109-2-26-1': 'See [CITATION] for more detailed analyses of quantum random variables.', '1801.03109-2-27-0': 'We are interested in integrating a quantum random variable with respect to a positive operator-valued measure: [CITATION] Let [MATH] be a POVM such that [MATH] exists.', '1801.03109-2-27-1': 'A positive quantum random variable [MATH] is [MATH]-integrable if the function [EQUATION] is [MATH]-integrable for every state [MATH].', '1801.03109-2-28-0': 'If [MATH] is a self-adjoint quantum random variable, then [MATH] defined by [EQUATION] are positive quantum random variables [CITATION].', '1801.03109-2-28-1': 'An arbitrary quantum random variable [MATH] is then said to be [MATH]-integrable if and only if [MATH] and [MATH] are [MATH]-integrable.', '1801.03109-2-29-0': '[CITATION] Let [MATH] be a POVM such that [MATH] exists.', '1801.03109-2-29-1': 'If [MATH] is a [MATH]-integrable quantum random variable then the integral of [MATH] with respect to [MATH], denoted [MATH], is implicitly defined by the formula [EQUATION].', '1801.03109-2-30-0': 'This definition of the integral of a quantum random variable with respect to a POVM was first defined in the finite-dimensional case in [CITATION].', '1801.03109-2-30-1': 'The infinite case was shown to work in [CITATION] and one should note that showing that the above definition works takes some proving.', '1801.03109-2-31-0': 'A good sign that this is a reasonable definition is that for every set [MATH] we have that [EQUATION].', '1801.03109-2-31-1': 'For a detailed discussion of this integral and a proof of the above consult [CITATION].', '1801.03109-2-32-0': 'Let [MATH] be a POVM and let [MATH] be a quantum random variable.', '1801.03109-2-32-1': 'Then the essential support of [MATH] is [EQUATION].', '1801.03109-2-33-0': 'Let [MATH] be an POVM and let [MATH] be a quantum random variable.', '1801.03109-2-33-1': 'Then the essential range of [MATH] is the set [MATH] of all [MATH] for which [MATH] for every neighbourhood [MATH] of [MATH].', '1801.03109-2-34-0': 'A standard classical result is that the essential range of a measurable function is equal to the intersection of the closure of the image of the function over all measurable sets whose complement has measure zero.', '1801.03109-2-34-1': 'We show this is also the case in the setting of quantum random variables.', '1801.03109-2-34-2': 'Although the proof is a straightforward generalization of the classical setting (see [CITATION]), we include it here for completeness and rigor.', '1801.03109-2-35-0': 'Let [MATH] be a quantum random variable.', '1801.03109-2-35-1': 'Then [EQUATION].', '1801.03109-2-36-0': 'Let [MATH] and suppose [MATH] is such that [MATH].', '1801.03109-2-36-1': 'This implies [MATH].', '1801.03109-2-36-2': 'Indeed, if that were not the case, then there would be an open set [MATH] containing [MATH] such that [MATH], and so [MATH]; however, [MATH] (since [MATH]), a contradiction.', '1801.03109-2-36-3': 'It follows that [MATH] for every [MATH] is such that [MATH].', '1801.03109-2-37-0': 'Conversely, let [MATH].', '1801.03109-2-37-1': 'Then [MATH] for some neighbourhood [MATH] of [MATH].', '1801.03109-2-37-2': 'Let [MATH].', '1801.03109-2-37-3': 'This implies [MATH].', '1801.03109-2-37-4': 'Indeed, if that were not the case, then [MATH], but then [MATH], a contradiction.', '1801.03109-2-37-5': 'It follows that [MATH] for every [MATH] such that [MATH].', '1801.03109-2-38-0': 'Again, analogous to the classical setting, we define the essential supremum of a quantum random variable.', '1801.03109-2-39-0': 'Let [MATH] be a quantum random variable.', '1801.03109-2-39-1': 'The essential supremum of [MATH] is the quantity [EQUATION]', '1801.03109-2-39-2': 'If [MATH] then [MATH] is essentially bounded and [EQUATION].', '1801.03109-2-40-0': 'Let [MATH] be a POVM such that [MATH] exists for any full-rank density operator [MATH].', '1801.03109-2-40-1': '[CITATION] tells us that every quantum random variable that is essentially bounded with respect to [MATH] is [MATH]-integrable.', '1801.03109-2-40-2': 'Denote the set of all such quantum random variables by [MATH].', '1801.03109-2-41-0': 'As in the classical case, for [MATH] let [MATH] which is a seminorm on [MATH].', '1801.03109-2-41-1': 'Quotienting by the ideal of functions that vanish under this seminorm we get [MATH].', '1801.03109-2-41-2': 'Because [MATH] and [MATH] are mutually absolutely continuous they give the same set of essentially bounded quantum random variables from [MATH] to [MATH] by the second part of the definition of essentially bounded.', '1801.03109-2-41-3': 'This implies that [EQUATION] is a Banach space, actually a von Neumann algebra (and so the tensor product is unique by [CITATION]).', '1801.03109-2-41-4': 'In the same section of the previous reference, there is the following predual identity [EQUATION].', '1801.03109-2-41-5': 'Thus [MATH] carries a weak[MATH]-topology.', '1801.03109-2-41-6': 'To be clear, on sums of simple tensors these isomorphisms are both given by [EQUATION].', '1801.03109-2-41-7': 'So we consider both [MATH] and its predual as sets of equivalence classes of functions from [MATH] into [MATH] or [MATH], respectively.', '1801.03109-2-42-0': 'Integration is a very important concept in our analysis and so, for a given POVM [MATH] such that [MATH] exists for any full-rank density operator [MATH], we define [MATH] to be [EQUATION]', '1801.03109-2-42-1': 'This integral operator is called the quantum expected value of [MATH] relative to the POVM [MATH] in [CITATION] and denoted [MATH].', '1801.03109-2-43-0': 'Let [MATH] be a POVM such that [MATH] exists.', '1801.03109-2-43-1': 'Then [MATH] is weak[MATH]-weak[MATH]-continuous.', '1801.03109-2-44-0': 'For every [MATH] we first need to establish that [EQUATION] is an element of [MATH].', '1801.03109-2-44-1': 'To this end, consider the bounded linear functional [MATH], [MATH] which is precisely integration against [MATH] in the first component and taking the trace against [MATH] in the second component of [MATH].', '1801.03109-2-44-2': 'In particular, the span of these functionals is dense in [MATH].', '1801.03109-2-45-0': 'We then have [EQUATION] by Corollary 2.9 of [CITATION] since [MATH] is essentially bounded.', '1801.03109-2-45-1': 'Therefore, [EQUATION].', '1801.03109-2-46-0': 'Now suppose [MATH] in [MATH].', '1801.03109-2-46-1': 'By the above, every bounded linear functional of [MATH] is given as [MATH] for [MATH].', '1801.03109-2-46-2': 'Then [EQUATION] from which it follows that [MATH].', '1801.03109-2-47-0': 'A very natural problem that arises from the study of integration against a POVM is that of whether [MATH] is injective or not.', '1801.03109-2-47-1': 'More specifically, for a POVM [MATH] we say that integration against [MATH] is classically non-injective if for every [MATH] such that [MATH] we have that [EQUATION] is non-injective.', '1801.03109-2-48-0': 'It is immediate that integration against any measure with an atom fails this condition.', '1801.03109-2-48-1': 'By cardinality alone it is also immediate that integration against a non-atomic finite-dimensional POVM is classically non-injective.', '1801.03109-2-49-0': 'Such a condition is quite essential to proving a convexity result as shown by an example of Uhl [CITATION].', '1801.03109-2-50-0': '[Uhl] Let [MATH] be Lebesgue measure on [MATH] and [MATH].', '1801.03109-2-50-1': 'Define the POVM [MATH] by [EQUATION].', '1801.03109-2-50-2': 'Then the range space [MATH] is not convex.', '1801.03109-2-50-3': 'Note that it can be proven that this non-atomic measure is not classicaly non-injective.', '1801.03109-2-51-0': '# Operator-valued Lyapunov theorem', '1801.03109-2-52-0': 'We are now in a position to prove the main theorem of this paper.', '1801.03109-2-53-0': 'Let [MATH] be a nonatomic POVM such that [MATH] exists and let [MATH] be as above.', '1801.03109-2-53-1': 'If integration against [MATH] is classically non-injective then the set [MATH] is a weak[MATH]-compact convex set in [MATH].', '1801.03109-2-54-0': 'The proof follows the same general structure as in the classical setting (see [CITATION] or [CITATION]).', '1801.03109-2-54-1': 'Let [EQUATION]', '1801.03109-2-54-2': 'It is well known from the classical case that the set on the left side of the tensor product is weak[MATH]-closed and thus weak[MATH]-compact (being in its unit ball).', '1801.03109-2-54-3': 'Hence, [MATH] is weak[MATH]-compact as well.', '1801.03109-2-55-0': 'It is easy to see that the map [MATH] defined in equation ([REF]) is affine with respect to quantum random variables [MATH] (although it is not affine with respect to POVMs [MATH]).', '1801.03109-2-55-1': 'Since [MATH] is weak[MATH]-weak[MATH]-continuous by Theorem [REF] we have that [MATH] is weak[MATH]-compact and convex.', '1801.03109-2-56-0': 'One should note here that [MATH] and contains [MATH] and [MATH] but it does not have to equal the operator interval [MATH].', '1801.03109-2-56-1': 'For instance, [MATH] would equal [MATH] if [MATH] where [MATH] is Lebesgue measure on [MATH].', '1801.03109-2-57-0': 'Pick [MATH].', '1801.03109-2-57-1': 'The set [MATH] is the weak[MATH]-continuous inverse image of a weak[MATH]-closed set, and so [MATH] is weak[MATH]-closed.', '1801.03109-2-57-2': 'Clearly [MATH] is convex as well.', '1801.03109-2-57-3': 'By the Krein-Milman theorem, there exists [MATH].', '1801.03109-2-57-4': 'We claim that [MATH] for some [MATH].', '1801.03109-2-57-5': 'This will complete the proof since we already have that [MATH], giving [MATH] is weak[MATH]-compact and convex.', '1801.03109-2-58-0': 'Now, we prove the claim by contradiction: suppose [MATH] is not a characteristic function; that is, [MATH] in [MATH].', '1801.03109-2-58-1': 'Thus, the essential range of [MATH] contains some [MATH] with [MATH].', '1801.03109-2-58-2': 'Let [MATH] and define [MATH].', '1801.03109-2-58-3': 'Note that [MATH] is open and [MATH] but [MATH].', '1801.03109-2-58-4': 'We then have that [MATH] is a Borel set and [MATH].', '1801.03109-2-59-0': 'So [MATH] is a measurable set with [MATH], because [MATH] is in the essential range of [MATH].', '1801.03109-2-59-1': 'By the fact that integration against [MATH] is classically non-injective there exists a [MATH] such that [MATH].', '1801.03109-2-59-2': 'One can consider [MATH] on [MATH] and hence [MATH].', '1801.03109-2-59-3': 'Moreover, [MATH] and so [MATH] contradicting the assumption that [MATH], and the claim is proven.', '1801.03109-2-60-0': 'Let [MATH] be nonatomic POVMs such that [MATH] exists for [MATH].', '1801.03109-2-60-1': 'If integration against [MATH] is classically non-injective then the set of points in [MATH] of the form [MATH], with [MATH] ranging over all [MATH], is a weak[MATH]-closed convex set.', '1801.03109-2-61-0': 'Define [MATH] as [MATH] on [MATH].', '1801.03109-2-61-1': 'It is not hard to see that [MATH] is a nonatomic POVM and [MATH] exists.', '1801.03109-2-61-2': 'The result follows from the last theorem since integration by [MATH] is classically non-invertible.', '1801.03109-2-62-0': 'Note that if [MATH] is a discrete set, then the range of [MATH] cannot be convex, and so the above corollary shows that nonatomic POVMs such that [MATH] exists for [MATH] have fundamentally different range spaces compared to atomic POVMs.', '1801.03109-2-63-0': "Lyapunov's Convexity Theorem follows as a one-dimensional consequence of Corollary [REF]:", '1801.03109-2-64-0': '[CITATION] Let [MATH] be finite positive nonatomic measures on [MATH].', '1801.03109-2-64-1': 'Then the set of points in [MATH] of the form [MATH], with [MATH] ranging over all [MATH], is a closed convex set.', '1801.03109-2-65-0': 'Our methods rely on the existence of the Radon-Nikodym derivative [MATH] (which, as we mentioned before, always exists when [MATH] is finite-dimensional).', '1801.03109-2-65-1': "In the absence of this derivative there are easy examples where one can see that Lyapunov's theorem will still hold but we do not have a proof of the general situation.", '1801.03109-2-65-2': 'One such case is the following:', '1801.03109-2-66-0': 'Let [MATH] be a set of mutually singular nonatomic probability measures on the measure space [MATH].', '1801.03109-2-66-1': 'Define [MATH] and so [MATH] is a nonatomic quantum probability measure.', '1801.03109-2-66-2': 'By the same reasoning as [CITATION] it is easy to see that [MATH] does not exist (just take [MATH]).', '1801.03109-2-67-0': 'Let [MATH] and by the mutually singular assumption we have that [MATH].', '1801.03109-2-67-1': "For any infinite tuple [MATH] with [MATH] we can find by Lyapunov's theorem above that there exists [MATH] such that [MATH] and [MATH].", '1801.03109-2-67-2': 'Thus, for [MATH] we have [MATH].', '1801.03109-2-67-3': 'Therefore, [MATH] is a weak[MATH]-closed and convex subset in the operator interval [MATH].'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1801.03109

cond-mat-0601677

{'cond-mat-0601677-1-0-0': 'We develop a general scheme of deriving boundary conditions for spin-charge coupled transport in disordered systems with spin-orbit interactions.', 'cond-mat-0601677-1-0-1': 'To illustrate the application of the method, we explicitly derive boundary conditions for spin diffusion in the Rashba model, using a hard wall potential as a model of the boundary.', 'cond-mat-0601677-1-0-2': 'Due to the surface spin precession, the boundary conditions are non-trivial and contain terms, which couple different components of the spin density.', 'cond-mat-0601677-1-0-3': 'The corresponding solution of the diffusion equation shows an electrically induced spin accumulation near the edges (spin Hall effect).', 'cond-mat-0601677-1-0-4': 'We argue that boundary conditions and the corresponding observable spin accumulation generally depend on the nature of the boundary and therefore the spin Hall effect in a spin-orbit coupled system can be viewed as a non-universal edge phenomenon, not directly related to any bulk linear-response quantity.', 'cond-mat-0601677-1-1-0': 'There has been a lot of recent interest in a phenomenon dubbed the spin Hall effect [CITATION].', 'cond-mat-0601677-1-1-1': 'Experimentally this effect manifests itself as an equilibrium spin accumulation near the edges, when an electric field is applied parallel to a Hall bar of a spin-orbit coupled electron system.', 'cond-mat-0601677-1-1-2': 'This kind of electric-field-induced spin polarization has been observed in recent experiments [CITATION].', 'cond-mat-0601677-1-1-3': 'This phenomenon has a potential for being technologically important, as it allows one to electrically manipulate the spin degree of freedom in non-magnetic systems.', 'cond-mat-0601677-1-1-4': 'There are still many controversies regarding the physical origin of the spin accumulation observed in real experiment (intrinsic [CITATION] vs. extrinsic [CITATION]).', 'cond-mat-0601677-1-1-5': 'But even within a given theoretical model (e.g., a two-dimensional electron gas with Rashba or Dresselhaus spin-orbit interactions or Luttinger model), there is still no consensus on the correct description of the spin Hall effect [CITATION].', 'cond-mat-0601677-1-2-0': 'A common theoretical approach to the intrinsic spin Hall effect is as follows: One introduces the notion of a spin current, which is an operator intuitively related to spin transport.', 'cond-mat-0601677-1-2-1': 'Using the Kubo formula or any other equivalent method, one then calculates the spin-current autocorrelator and the corresponding linear response quantity, called the "spin Hall conductivity" [CITATION].', 'cond-mat-0601677-1-2-2': "In clean systems, it appears to be related to an elegant topological Berry's phase structure [CITATION], which may exist in multi-band systems.", 'cond-mat-0601677-1-2-3': 'In a disordered system, the notion of the Fermi surface Berry\'s phase is unclear, but the "spin Hall conductivity" is well-defined.', 'cond-mat-0601677-1-2-4': 'It is believed that this quantity is connected to the spin accumulation in the spin-Hall experiment.', 'cond-mat-0601677-1-2-5': 'While this theoretical approach is mathematically well-defined and may lead to technically challenging problems [CITATION], its relation to experiment remains unclear.', 'cond-mat-0601677-1-2-6': 'The problem is that in spin-orbit coupled systems, the spin is not conserved and thus the definition of the spin current is ambiguous [CITATION].', 'cond-mat-0601677-1-2-7': 'A non-equilibrium spin density in a region relaxes not only due to the flux through the boundaries of the region, but also due to spin precession.', 'cond-mat-0601677-1-2-8': 'Therefore, the spin current is not easily measurable.', 'cond-mat-0601677-1-3-0': 'To describe realistic experiment, it is preferable to use another route without involving the notion of a spin current.', 'cond-mat-0601677-1-3-1': 'In disordered systems, this can be done with the help of a kinetic equation [CITATION] or a spin-charge coupled diffusion equation [CITATION].', 'cond-mat-0601677-1-3-2': 'The diffusion equation provides a complete and physically meaningful description of spin and charge diffusive transport in terms of position and time dependent spin and charge densities.', 'cond-mat-0601677-1-3-3': 'However, the explicit solution of the diffusion equation (which is generally a complicated set of partial differential equations) strongly depends on boundary conditions.', 'cond-mat-0601677-1-3-4': 'While for usual charge transport, boundary conditions are often obvious and follow from conservation laws, this is not the case in spin-orbit coupled systems.', 'cond-mat-0601677-1-3-5': 'Again, the problem is that the spin is not conserved and thus there is spin precession at the edges.', 'cond-mat-0601677-1-3-6': 'This spin dynamics is very sensitive to the actual boundary (depending on the physical situation, the boundary could be modeled as a hard wall, WKB potential, rough surface, etc.).', 'cond-mat-0601677-1-3-7': 'Since the bulk diffusion equation describes spin/charge dynamics only at the length-scales much larger than the mean-free path [MATH], it can not capture the behavior of the spin in the immediate vicinity of the sample edges.', 'cond-mat-0601677-1-3-8': 'Due to the aforementioned boundary spin precession, the "trivial boundary conditions" (i.e., boundary conditions, which one would have, if the spin had been conserved: [MATH], where [MATH] is the unit vector normal to the sample edge and [MATH] is the spin density) acquire corrections proportional to the spin-orbit coupling.', 'cond-mat-0601677-1-3-9': 'These corrections are very important, as they select a unique solution of the diffusion equation and determine the observable spin accumulation.', 'cond-mat-0601677-1-4-0': 'The derivation of boundary conditions is technically a difficult task.', 'cond-mat-0601677-1-4-1': 'Generally, one needs to determine the behavior of the system in the ballistic region near the boundary and match it with the diffusive behavior in the bulk.', 'cond-mat-0601677-1-4-2': "It requires the knowledge of the exact [MATH]-matrix or the boundary Green's function.", 'cond-mat-0601677-1-4-3': 'We should point out that there is an extensive literature on the related issues in the context of the diffusion of light in a random medium [CITATION] and neutron diffusion [CITATION].', 'cond-mat-0601677-1-4-4': 'In these problems, the general form of boundary conditions is obvious due to conservation laws and only numerical coefficients are unknown.', 'cond-mat-0601677-1-4-5': 'There exists a general method of deriving these numerical coefficients.', 'cond-mat-0601677-1-4-6': 'But there are just a very few models in which exact analytical results are available (a notable example is the Milne problem [CITATION], which describes a boundary separating diffusive and ballistic regions).', 'cond-mat-0601677-1-4-7': 'We note that in the context of spin diffusion in spin-orbit coupled systems, even the qualitative form of boundary conditions is not known as there are no obvious conservation laws, which would provide guidance.', 'cond-mat-0601677-1-4-8': 'Even though in the bulk, one has conserved quantities labeled by the band index (e.g., chirality in the Rashba model), the boundary generally does not respect these conservation laws and mixes up different bands.', 'cond-mat-0601677-1-5-0': 'In this Letter we formulate a general method of deriving boundary conditions in spin-orbit coupled systems.', 'cond-mat-0601677-1-5-1': 'Using this proposed method, we derive boundary conditions for the Rashba model [CITATION] in the leading order with respect to the spin-orbit coupling, which is assumed small.', 'cond-mat-0601677-1-5-2': 'We show that there are corrections to the "trivial boundary conditions:" In leading order, we find that at the boundary, the spin component perpendicular to the plane of the two-dimensional gas gets coupled to the in-plane spin component perpendicular to the boundary.', 'cond-mat-0601677-1-5-3': 'We argue that the spin accumulation in the spin Hall effect setup is determined by the combination of the bulk spin-charge coupling and the boundary effects.', 'cond-mat-0601677-1-6-0': 'Let us consider a disordered electron system occupying a region [MATH].', 'cond-mat-0601677-1-6-1': "One can derive the following integral equation for the diffuson [CITATION], which determines the dynamics of the charge density ([MATH]) and the spin densities ([MATH], [MATH]): [EQUATION] where [MATH] is the density of states per spin at the Fermi surface, [MATH] is the scattering time, the Latin indices label charge ([MATH]) and spin ([MATH], [MATH], and [MATH]) degrees of freedom, and [MATH] is the diffuson, which is the Green's function of the spin-charge coupled diffusion equation.", 'cond-mat-0601677-1-6-2': "The kernel in Eq. ([REF]) reads [EQUATION] where [MATH], [MATH] is the unity matrix, [MATH] are the Pauli matrices, and [MATH] are disorder-averaged retarded/advanced electron Green's functions, which include the effect of the boundary (here and below we denote the boundary Green's functions as [MATH] and the bulk Green's functions as [MATH]).", 'cond-mat-0601677-1-6-3': 'They depend exponentially on distances via factors of the following types, [MATH], where [MATH] is the mean-free path.', 'cond-mat-0601677-1-6-4': "Deep in the bulk, the edge effects are exponentially small and the bulk Green's functions can be used, [MATH].", 'cond-mat-0601677-1-6-5': 'Performing a gradient expansion of the bulk polarizability (i.e., taking the limit [MATH]), one obtains the spin-charge coupled diffusion equation.', 'cond-mat-0601677-1-6-6': 'Generally, it has the following form: [EQUATION] where [MATH] are the charge/spin densities, [MATH] are the diffusion coefficients, [MATH] are the relaxation times, and [MATH] are the spin-spin and spin-charge couplings.', 'cond-mat-0601677-1-7-0': "Note that the integral equation ([REF]) does not require boundary conditions (it already contains this information through the boundary Green's function), but the differential diffusion equation ([REF]) does.", 'cond-mat-0601677-1-7-1': 'To derive these boundary conditions one needs to perform the gradient expansion near the boundary [MATH].', 'cond-mat-0601677-1-7-2': 'Let us consider the problem in half-space [MATH].', 'cond-mat-0601677-1-7-3': 'To find the diffusion equation near the boundary, we take the limit [MATH].', 'cond-mat-0601677-1-7-4': 'This limit should be understood in the following sense: [MATH], i.e., we consider a point, which is near the boundary as compared to the diffusion length-scale [MATH], but still far from it as compared to the ballistic length scale described by the Fermi wave-length.', 'cond-mat-0601677-1-7-5': 'Let us first consider the case of zero spin-orbit interactions.', 'cond-mat-0601677-1-7-6': "If the boundary is a hard wall potential, then the boundary Green's function can be obtained using the method of mirror images [EQUATION] where [MATH], [MATH], and [MATH] is the mirror with respect to the sample boundary [in the given case [MATH] and [MATH]].", 'cond-mat-0601677-1-7-7': 'For the sake of concreteness, let us assume that the dimensionality [MATH] (all qualitative arguments are independent of dimensionality).', 'cond-mat-0601677-1-7-8': "In two dimensions, the Green's function has the following form [EQUATION]", 'cond-mat-0601677-1-7-9': 'We note here that the frequency dependence of the diffusion kernel is irrelevant for the boundary condition problem and will be omitted from now on.', 'cond-mat-0601677-1-7-10': 'When the method of mirror images applies, the polarizability kernel ([REF]) has the following general structure [MATH] [where [MATH] implies the "reflected" Green\'s function, i.e, the second term in Eq. ([REF])].', 'cond-mat-0601677-1-7-11': 'The key observation is that due to the fast-oscillating factor [MATH] and our choice of the point [MATH]), the cross terms [MATH] in the expression for [MATH] become negligible in the integral over [MATH] and can be omitted.', 'cond-mat-0601677-1-7-12': 'The remaining integral can be easily calculated and we find (in the "trivial" case of no spin-orbit coupling): [EQUATION] where [MATH] is a dimensionless coefficient, which in two dimensions is equal to [MATH].', 'cond-mat-0601677-1-7-13': 'From Eqs. ([REF]) and ([REF]), we recover the familiar boundary condition [MATH], which is simply the consequence of the charge and spin conservation.', 'cond-mat-0601677-1-8-0': 'In a system with spin-orbit interaction, the method of images does not apply because the "reflected" Green\'s function does not satisfy the corresponding Schrodinger equation.', 'cond-mat-0601677-1-8-1': 'This is a major complication, which never occurs in the problems usually studied in the literature in this context [CITATION].', 'cond-mat-0601677-1-8-2': 'In addition, the matrix structure of the boundary diffusion kernel becomes essential.', 'cond-mat-0601677-1-8-3': "There are generally two ways to obtain the boundary Green's function (we have explicitly verified that both approaches lead to identical results in the Rashba model [CITATION]): (i) The first is to solve the Schrodinger equation near the boundary and express the Green's function using the corresponding wave-functions as a basis; (ii) The second is to find the Green's functions perturbatively, using the mirror-image result ([REF]) as the leading approximation: [EQUATION] where [MATH] is the spin-orbit interaction operator as it appears in the Hamiltonian.", 'cond-mat-0601677-1-8-4': "The resulting Green's function may be quite complicated even in the simplest cases.", 'cond-mat-0601677-1-8-5': "However, the diffusion kernel always contains products of the following types: [EQUATION] where [MATH]'s and [MATH]'s are some functions, which do not oscillate on a Fermi wavelength scale.", 'cond-mat-0601677-1-8-6': 'Again, in the integral of Eq. ([REF]), the cross-terms [MATH] become negligible due to the fast-oscillations and can be omitted, but the other terms survive.', 'cond-mat-0601677-1-8-7': "One can define the following matrix, which determines boundary conditions [EQUATION] where the correction to polarizability is determined by the Green's function ([REF]) and Eq. ([REF]) and the limit is understood in the sense [MATH].", 'cond-mat-0601677-1-8-8': 'If the boundary is an impenetrable wall, general boundary conditions take the form [EQUATION] where [MATH], [MATH] is a unit vector normal to the boundary, [MATH] is a number ([MATH]), and [MATH] is a dimensionless matrix defined above.', 'cond-mat-0601677-1-8-9': 'We note that if the boundary is different from a hard wall, the general method does not change but the results do even in the lowest order.', 'cond-mat-0601677-1-8-10': 'For instance, if the boundary is characterized by an isotropic reflection coefficient [MATH], the zeroth order boundary condition becomes [MATH] (see also Ref. [[CITATION]], which discusses the [MATH] case).', 'cond-mat-0601677-1-9-0': 'To illustrate the application of our method, let us consider the Rashba model [CITATION].', 'cond-mat-0601677-1-9-1': 'In this case, the spin-orbit interaction operator has the form [MATH], where [MATH] is the Rashba spin-orbit coupling and [MATH].', 'cond-mat-0601677-1-9-2': 'The differential diffusion equation has been derived Ref. [[CITATION]] and reads [EQUATION]', 'cond-mat-0601677-1-9-3': 'The spin-spin coupling in the Rashba model is [MATH]; I.e., it is linear in the spin-orbit parameter [MATH] and universal, in the sense that it does not depend on the details of the electronic spectrum and can be calculated in the [MATH]-approximation [CITATION].', 'cond-mat-0601677-1-9-4': 'However, the spin-charge coupling is non-universal, since it strongly depends on the electronic spectrum and the high-energy cut-off (if the spectrum is quadratic [MATH] and [MATH], otherwise).', 'cond-mat-0601677-1-9-5': 'We should point out however that these results hold only in the semiclassical diffusion approximation, i.e., only in the leading order with respect to the inverse conductance [MATH], which is an independent parameter of the model.', 'cond-mat-0601677-1-9-6': 'Apparently there are contributions to the spin-charge coupling of order [MATH], which are coming from the ballistic parts of the trajectories .', 'cond-mat-0601677-1-9-7': 'They originate, in particular, from diagrams with crossed impurity lines, which are omitted in the diffusion approximation (we note that there is no direct relation between [MATH] and the spin Hall conductivity, which vanishes to all orders [CITATION]).', 'cond-mat-0601677-1-9-8': 'These quantum corrections are expected to be parametrically larger than the semiclassical diffusive terms, whose smallness is due to accidental cancellations specific to the Rashba model.', 'cond-mat-0601677-1-9-9': 'In what follows, we will treat the spin-charge coupling terms in the diffusion equation ([REF], [REF]) as non-universal phenomenological parameters.', 'cond-mat-0601677-1-10-0': 'The diffusion equation ([REF], [REF]) must be supplemented by boundary conditions.', 'cond-mat-0601677-1-10-1': 'We consider the problem in leading approximation with respect to the spin-orbit coupling parameter [MATH] and the inverse conductance.', 'cond-mat-0601677-1-10-2': "First, we use Eqs. ([REF]) and ([REF]) to derive a correction to the polarizability kernel [EQUATION] and [EQUATION] where the boundary Green's functions are defined by Eqs. ([REF]) and ([REF]) and we introduced the following function [EQUATION]", 'cond-mat-0601677-1-10-3': 'This equation contains four terms (we can symbolically denote them as [MATH], [MATH], [MATH], and [MATH]).', 'cond-mat-0601677-1-10-4': 'Let us analyze the first one, the other terms can be calculated in a similar manner: [EQUATION]', 'cond-mat-0601677-1-10-5': 'To calculate this integral it is convenient to consider a new coordinate system [MATH] in which the points [MATH] and [MATH] have the coordinates [MATH] and [MATH] respectively ([MATH]).', 'cond-mat-0601677-1-10-6': 'Next, we introduce the elliptic coordinates [MATH], with the points [MATH] and [MATH] in the foci of the ellipses: [MATH] and [MATH] being the angle between the vector [MATH] and the [MATH] axis.', 'cond-mat-0601677-1-10-7': 'Using these elliptic coordinates and Eq. ([REF]), we write the integral ([REF]) in the following form [EQUATION]', 'cond-mat-0601677-1-10-8': 'We note that the limits of integration over [MATH] are generally non-trivial due to the constraint [MATH].', 'cond-mat-0601677-1-10-9': "It is not possible to calculate the Green's function in terms of elementary functions for generic [MATH] and [MATH].", 'cond-mat-0601677-1-10-10': 'However, it is possible to evaluate analytically the correction to the diffusion kernel [see Eqs. ([REF]), ([REF]), and ([REF])] and the boundary matrix ([REF]).', 'cond-mat-0601677-1-10-11': 'Indeed from Eqs. ([REF]), ([REF]), and ([REF]), we find that due to the fast-oscillating exponents, only a few terms survive the integration over [MATH] in the leading approximation with respect to the large conductance [MATH].', 'cond-mat-0601677-1-10-12': 'These remaining terms contain combinations of the following types [MATH], which constrain the parameter [MATH] in ([REF]) to be close to unity.', 'cond-mat-0601677-1-10-13': 'This allows one to calculate the integral ([REF]) asymptotically and we find in the leading order the following results for the boundary matrix ([REF]) [EQUATION] where the last equation implies that the boundary spin-charge coupling is small with respect to the inverse conductance and vanishes in the framework of the diffusion approximation and the accuracy of the method.', 'cond-mat-0601677-1-10-14': 'We note that corrections of this order exist only for the spin density component parallel to the edge.', 'cond-mat-0601677-1-10-15': 'We also emphasize that the smallness of boundary and bulk spin-charge coupling terms is specific to the Rashba model, which is in some sense pathological as it contains accidental cancellations.', 'cond-mat-0601677-1-10-16': 'In a generic model, the spin-spin and spin-charge couplings are expected to be of the same order.', 'cond-mat-0601677-1-11-0': 'From Eqs. ([REF]) and ([REF]), we find the following boundary conditions in the leading approximation with respect to the spin-orbit coupling and the inverse conductance: [EQUATION] where [MATH] is defined after Eq. ([REF]).', 'cond-mat-0601677-1-11-1': 'These boundary conditions can be generalized to describe any form of a hard-wall boundary (provided that the typical length-scales are much larger than the mean-free path).', 'cond-mat-0601677-1-11-2': 'To find such a general form of boundary conditions, one should do the following replacements: [MATH] and [MATH], where [MATH] is the unit vector normal to the segment of the boundary and [MATH] is the corresponding component of the spin density.', 'cond-mat-0601677-1-11-3': 'However, if the boundary is a smooth potential with the length-scale in the [MATH]-direction being [MATH], then boundary conditions would be different and strongly dependent on the ratios of [MATH] and [MATH] [CITATION].', 'cond-mat-0601677-1-11-4': 'Also, the boundary potential would lead to an additional boundary spin-orbit coupling via the standard relativistic effect.', 'cond-mat-0601677-1-11-5': 'Since the actual solution of the diffusion equation is selected by boundary conditions, the corresponding spin accumulation would be determined by the geometry of the boundary and the spin-orbit coupling at the edges.', 'cond-mat-0601677-1-12-0': 'As an example, we calculate the accumulation of the [MATH]-component of the spin in the spin Hall setup with boundary conditions defined by Eqs. ([REF]) and ([REF]) (i.e., we consider a strip of width [MATH] with an electric field [MATH] applied parallel to the strip boundaries, [MATH]).', 'cond-mat-0601677-1-12-1': 'We do it perturbatively with respect to [MATH].', 'cond-mat-0601677-1-12-2': 'From the diffusion equation ([REF],[REF]) it follows that the zeroth order approximation leads to a uniform density of the [MATH]-component of the spin, [MATH], where [MATH] is the electronic density of states per spin.', 'cond-mat-0601677-1-12-3': 'Corrections to boundary conditions ([REF]) lead to a non-zero spin accumulation of the [MATH]-component near the edges of the sample: [MATH], where [MATH] is the spin diffusion length and [MATH].', 'cond-mat-0601677-1-12-4': 'We reiterate that the spin accumulation may be different if the boundary potential is different from a hard wall.', 'cond-mat-0601677-1-12-5': 'In this sense, the spin Hall effect is generally a non-universal phenomenon, which depends on the structure of the sample edges.', 'cond-mat-0601677-1-13-0': 'V.M.G. and A.A.B. are grateful to Leon Balents, Emmanuel Rashba, and Yaroslav Tserkovnyak for helpful discussions.', 'cond-mat-0601677-1-13-1': 'A.A.B. is supported by the NSF under grant DMR02-33773.', 'cond-mat-0601677-1-13-2': 'S.D.S. is supported by US-ONR, NSF, and LPS.'}

{'cond-mat-0601677-2-0-0': 'We develop a general scheme of deriving boundary conditions for spin-charge coupled transport in disordered systems with spin-orbit interactions.', 'cond-mat-0601677-2-0-1': 'To illustrate the application of the method, we explicitly derive boundary conditions for spin diffusion in the Rashba model, using a hard wall potential as a model of the boundary.', 'cond-mat-0601677-2-0-2': 'Due to the surface spin precession, the boundary conditions are non-trivial and contain terms, which couple different components of the spin density.', 'cond-mat-0601677-2-0-3': 'The corresponding solution of the diffusion equation shows an electrically induced spin accumulation near the edges (spin Hall effect).', 'cond-mat-0601677-2-0-4': 'We argue that boundary conditions and the corresponding observable spin accumulation generally depend on the nature of the boundary and therefore the spin Hall effect in a spin-orbit coupled system can be viewed as a non-universal edge phenomenon, not directly related to any bulk linear-response quantity.', 'cond-mat-0601677-2-1-0': 'There has been a lot of recent interest in a phenomenon dubbed the spin Hall effect [CITATION].', 'cond-mat-0601677-2-1-1': 'Experimentally this effect manifests itself as an equilibrium spin accumulation near the edges, when an electric field is applied parallel to a Hall bar of a spin-orbit coupled electron system.', 'cond-mat-0601677-2-1-2': 'This kind of electric-field-induced spin polarization has been observed in recent experiments [CITATION].', 'cond-mat-0601677-2-1-3': 'This phenomenon has a potential for being technologically important, as it allows one to electrically manipulate the spin degree of freedom in non-magnetic systems.', 'cond-mat-0601677-2-1-4': 'There are still many controversies regarding the physical origin of the spin accumulation observed in real experiment (intrinsic [CITATION] vs. extrinsic [CITATION]).', 'cond-mat-0601677-2-1-5': 'But even within a given theoretical model (e.g., a two-dimensional electron gas with Rashba or Dresselhaus spin-orbit interactions or Luttinger model), there is still no consensus on the correct description of the spin Hall effect [CITATION].', 'cond-mat-0601677-2-2-0': 'A common theoretical approach to the intrinsic spin Hall effect is as follows: One introduces the notion of a spin current, which is an operator intuitively related to spin transport.', 'cond-mat-0601677-2-2-1': 'Using the Kubo formula or any other equivalent method, one then calculates the spin-current autocorrelator and the corresponding linear response quantity, called the "spin Hall conductivity" [CITATION].', 'cond-mat-0601677-2-2-2': "In clean systems, it appears to be related to an elegant topological Berry's phase structure [CITATION], which may exist in multi-band systems.", 'cond-mat-0601677-2-2-3': 'In a disordered system, the notion of the Fermi surface Berry\'s phase is unclear, but the "spin Hall conductivity" is well-defined.', 'cond-mat-0601677-2-2-4': 'It is believed that this quantity is connected to the spin accumulation in the spin-Hall experiment.', 'cond-mat-0601677-2-2-5': 'While this theoretical approach is mathematically well-defined and may lead to technically challenging problems [CITATION], its relation to experiment remains unclear.', 'cond-mat-0601677-2-2-6': 'The problem is that in spin-orbit coupled systems, the spin is not conserved and thus the definition of the spin current is ambiguous [CITATION].', 'cond-mat-0601677-2-2-7': 'A non-equilibrium spin density in a region relaxes not only due to the flux through the boundaries of the region, but also due to spin precession.', 'cond-mat-0601677-2-2-8': 'Therefore, the spin current is not easily measurable.', 'cond-mat-0601677-2-3-0': 'To describe realistic experiment, it is preferable to use another route without involving the notion of a spin current.', 'cond-mat-0601677-2-3-1': 'In disordered systems, this can be done with the help of a kinetic equation [CITATION] or a spin-charge coupled diffusion equation [CITATION].', 'cond-mat-0601677-2-3-2': 'The diffusion equation provides a complete and physically meaningful description of spin and charge diffusive transport in terms of position and time dependent spin and charge densities.', 'cond-mat-0601677-2-3-3': 'However, the explicit solution of the diffusion equation (which is generally a complicated set of partial differential equations) strongly depends on boundary conditions.', 'cond-mat-0601677-2-3-4': 'While for usual charge transport, boundary conditions are often obvious and follow from conservation laws, this is not the case in spin-orbit coupled systems.', 'cond-mat-0601677-2-3-5': 'Again, the problem is that the spin is not conserved and thus there is spin precession at the edges.', 'cond-mat-0601677-2-3-6': 'This spin dynamics is very sensitive to the actual boundary (depending on the physical situation, the boundary could be modeled as a hard wall, WKB potential, rough surface, etc.).', 'cond-mat-0601677-2-3-7': 'Since the bulk diffusion equation describes spin/charge dynamics only at the length-scales much larger than the mean-free path [MATH], it can not capture the behavior of the spin in the immediate vicinity of the sample edges.', 'cond-mat-0601677-2-3-8': 'Due to the aforementioned boundary spin precession, the "trivial boundary conditions" (i.e., boundary conditions, which one would have, if the spin had been conserved: [MATH], where [MATH] is the unit vector normal to the sample edge and [MATH] is the spin density) acquire corrections proportional to the spin-orbit coupling.', 'cond-mat-0601677-2-3-9': 'These corrections are very important, as they select a unique solution of the diffusion equation and determine the observable spin accumulation.', 'cond-mat-0601677-2-4-0': 'The derivation of boundary conditions is technically a difficult task.', 'cond-mat-0601677-2-4-1': 'Generally, one needs to determine the behavior of the system in the ballistic region near the boundary and match it with the diffusive behavior in the bulk.', 'cond-mat-0601677-2-4-2': "It requires the knowledge of the exact [MATH]-matrix or the boundary Green's function.", 'cond-mat-0601677-2-4-3': 'We should point out that there is an extensive literature on the related issues in the context of the diffusion of light in a random medium [CITATION] and neutron diffusion [CITATION].', 'cond-mat-0601677-2-4-4': 'In these problems, the general form of boundary conditions is obvious due to conservation laws and only numerical coefficients are unknown.', 'cond-mat-0601677-2-4-5': 'There exists a general method of deriving these numerical coefficients.', 'cond-mat-0601677-2-4-6': 'But there are just a very few models in which exact analytical results are available (a notable example is the Milne problem [CITATION], which describes a boundary separating diffusive and ballistic regions).', 'cond-mat-0601677-2-4-7': 'We note that in the context of spin diffusion in spin-orbit coupled systems, even the qualitative form of boundary conditions is not known as there are no obvious conservation laws, which would provide guidance.', 'cond-mat-0601677-2-4-8': 'Even though in the bulk, one has conserved quantities labeled by the band index (e.g., chirality in the Rashba model), the boundary generally does not respect these conservation laws and mixes up different bands.', 'cond-mat-0601677-2-5-0': 'In this Letter we formulate a general method of deriving boundary conditions in spin-orbit coupled systems.', 'cond-mat-0601677-2-5-1': 'Using this proposed method, we derive boundary conditions for the Rashba model [CITATION] in the leading order with respect to the spin-orbit coupling, which is assumed small.', 'cond-mat-0601677-2-5-2': 'We show that there are corrections to the "trivial boundary conditions:" In leading order, we find that at the boundary, the spin component perpendicular to the plane of the two-dimensional gas gets coupled to the in-plane spin component perpendicular to the boundary.', 'cond-mat-0601677-2-5-3': 'We argue that spin accumulation in the spin Hall effect setup is determined by the combination of the bulk spin-charge coupling and boundary effects.', 'cond-mat-0601677-2-6-0': 'Let us consider a disordered electron system occupying a region [MATH].', 'cond-mat-0601677-2-6-1': "One can derive the following integral equation for the diffuson [CITATION], which determines the dynamics of the charge density ([MATH]) and the spin densities ([MATH], [MATH]): [EQUATION] where [MATH] is the density of states per spin at the Fermi surface, [MATH] is the scattering time, the Latin indices label charge ([MATH]) and spin ([MATH], [MATH], and [MATH]) degrees of freedom, and [MATH] is the diffuson, which is the Green's function of the spin-charge coupled diffusion equation.", 'cond-mat-0601677-2-6-2': "The kernel in Eq. ([REF]) reads [EQUATION] where [MATH], [MATH] is the unity matrix, [MATH] are the Pauli matrices, and [MATH] are disorder-averaged retarded/advanced electron Green's functions, which include the effect of the boundary (here and below we denote the boundary Green's functions as [MATH] and the bulk Green's functions as [MATH]).", 'cond-mat-0601677-2-6-3': 'They depend exponentially on distances via factors of the following types, [MATH], where [MATH] is the mean-free path.', 'cond-mat-0601677-2-6-4': "Deep in the bulk, the edge effects are exponentially small and the bulk Green's functions can be used, [MATH].", 'cond-mat-0601677-2-6-5': 'Performing a gradient expansion of the bulk polarizability (i.e., taking the limit [MATH]), one obtains the spin-charge coupled diffusion equation.', 'cond-mat-0601677-2-6-6': 'Generally, it has the following form: [EQUATION] where [MATH] are the charge/spin densities, [MATH] are the diffusion coefficients, [MATH] are the relaxation times, and [MATH] are the spin-spin and spin-charge couplings.', 'cond-mat-0601677-2-7-0': "Note that the integral equation ([REF]) does not require boundary conditions (it already contains this information through the boundary Green's function), but the differential diffusion equation ([REF]) does.", 'cond-mat-0601677-2-7-1': 'To derive these boundary conditions one needs to perform the gradient expansion near the boundary [MATH].', 'cond-mat-0601677-2-7-2': 'Let us consider the problem in half-space [MATH].', 'cond-mat-0601677-2-7-3': 'To find the diffusion equation near the boundary, we take the limit [MATH].', 'cond-mat-0601677-2-7-4': 'This limit should be understood in the following sense: [MATH], i.e., we consider a point, which is near the boundary as compared to the diffusion length-scale [MATH], but still far from it as compared to the ballistic length scale described by the Fermi wave-length.', 'cond-mat-0601677-2-7-5': 'Let us first consider the case of zero spin-orbit interactions.', 'cond-mat-0601677-2-7-6': "If the boundary is a hard wall potential, then the boundary Green's function can be obtained using the method of mirror images [EQUATION] where [MATH], [MATH], and [MATH] is the mirror with respect to the sample boundary [in the given case [MATH] and [MATH]].", 'cond-mat-0601677-2-7-7': 'For the sake of concreteness, let us assume that the dimensionality [MATH] (all qualitative arguments are independent of dimensionality).', 'cond-mat-0601677-2-7-8': "In two dimensions, the Green's function has the following form [EQUATION]", 'cond-mat-0601677-2-7-9': 'We note here that the frequency dependence of the diffusion kernel is irrelevant for the boundary condition problem and will be omitted from now on.', 'cond-mat-0601677-2-7-10': 'When the method of mirror images applies, the polarizability kernel ([REF]) has the following general structure [MATH] [where [MATH] implies the "reflected" Green\'s function, i.e, the second term in Eq. ([REF])].', 'cond-mat-0601677-2-7-11': 'The key observation is that due to the fast-oscillating factor [MATH] and our choice of the point [MATH]), the cross terms [MATH] in the expression for [MATH] become negligible in the integral over [MATH] and can be omitted.', 'cond-mat-0601677-2-7-12': 'The remaining integral can be easily calculated and we find (in the "trivial" case of no spin-orbit coupling): [EQUATION] where [MATH] is a dimensionless coefficient, which in two dimensions is equal to [MATH].', 'cond-mat-0601677-2-7-13': 'From Eqs. ([REF]) and ([REF]), we recover the familiar boundary condition [MATH], which is simply the consequence of the charge and spin conservation.', 'cond-mat-0601677-2-8-0': 'In a system with spin-orbit interaction, the method of images does not apply because the "reflected" Green\'s function does not satisfy the corresponding Schrodinger equation.', 'cond-mat-0601677-2-8-1': 'This is a major complication, which never occurs in the problems usually studied in the literature in this context [CITATION].', 'cond-mat-0601677-2-8-2': 'In addition, the matrix structure of the boundary diffusion kernel becomes essential.', 'cond-mat-0601677-2-8-3': "There are generally two ways to obtain the boundary Green's function (we have explicitly verified that both approaches lead to identical results in the Rashba model [CITATION]): (i) The first is to solve the Schrodinger equation near the boundary and express the Green's function using the corresponding wave-functions as a basis; (ii) The second is to find the Green's functions perturbatively, using the mirror-image result ([REF]) as the leading approximation: [EQUATION] where [MATH] is the spin-orbit interaction operator as it appears in the Hamiltonian.", 'cond-mat-0601677-2-8-4': 'We note that disorder corrections to the spin-orbit vertex can be shown to be small as long as [MATH].', 'cond-mat-0601677-2-8-5': "The resulting Green's function ([REF]) may be quite complicated even in the simplest cases.", 'cond-mat-0601677-2-8-6': "However, the diffusion kernel always contains products of the following types: [EQUATION] where [MATH]'s and [MATH]'s are some functions, which do not oscillate on a Fermi wavelength scale.", 'cond-mat-0601677-2-8-7': 'Again, in the integral of Eq. ([REF]), the cross-terms [MATH] become negligible due to the fast-oscillations and can be omitted, but the other terms survive.', 'cond-mat-0601677-2-8-8': "One can define the following matrix, which determines boundary conditions [EQUATION] where the correction to polarizability is determined by the Green's function ([REF]) and Eq. ([REF]) and the limit is understood in the sense [MATH].", 'cond-mat-0601677-2-8-9': 'If the boundary is an impenetrable wall, general boundary conditions take the form [EQUATION] where [MATH], [MATH] is a unit vector normal to the boundary, [MATH] is a number ([MATH]), and [MATH] is a dimensionless matrix defined above.', 'cond-mat-0601677-2-8-10': 'We note that if the boundary is different from a hard wall, the general method does not change but the results do even in the lowest order.', 'cond-mat-0601677-2-8-11': 'For instance, if the boundary is characterized by an isotropic reflection coefficient [MATH], the zeroth order boundary condition becomes [MATH] (see also Ref. [[CITATION]], which discusses the [MATH] case).', 'cond-mat-0601677-2-9-0': 'To illustrate the application of our method, let us consider the Rashba model [CITATION].', 'cond-mat-0601677-2-9-1': 'In this case, the spin-orbit interaction operator has the form [MATH], where [MATH] is the Rashba spin-orbit coupling and [MATH].', 'cond-mat-0601677-2-9-2': 'The differential diffusion equation was derived in Ref. [[CITATION]] and reads [EQUATION]', 'cond-mat-0601677-2-9-3': 'The spin-spin coupling in the Rashba model is [MATH] and the spin relaxation rates are [MATH].', 'cond-mat-0601677-2-9-4': 'These parameters are universal in the sense that they do not depend on the details of the electronic spectrum and can be calculated in the [MATH]-approximation [CITATION].', 'cond-mat-0601677-2-9-5': 'However, the spin-charge coupling is not universal, since it strongly depends on the electronic spectrum and the high-energy cut-off (if the spectrum is quadratic, [MATH]).', 'cond-mat-0601677-2-9-6': 'We should point out however that this result holds only in the semiclassical diffusion approximation, i.e., only in the leading order with respect to the inverse conductance [MATH], which is an independent parameter of the model.', 'cond-mat-0601677-2-9-7': 'Apparently there may exist contributions to the spin-charge coupling of order [MATH].', 'cond-mat-0601677-2-9-8': 'They originate from diagrams with crossed impurity lines, which are omitted in the diffusion approximation (we note that there is no direct relation between [MATH] and the spin Hall conductivity, which vanishes to all orders [CITATION]).', 'cond-mat-0601677-2-9-9': 'These quantum corrections are expected to be parametrically larger than the semiclassical diffusive terms, whose smallness is due to accidental cancellations specific to the Rashba model.', 'cond-mat-0601677-2-9-10': 'In what follows, we will treat the spin-charge coupling terms in the diffusion equation ([REF], [REF]) as non-universal phenomenological parameters.', 'cond-mat-0601677-2-10-0': 'The diffusion equation ([REF], [REF]) must be supplemented by boundary conditions.', 'cond-mat-0601677-2-10-1': 'We consider the problem in leading approximation with respect to the spin-orbit coupling parameter [MATH] and the inverse conductance.', 'cond-mat-0601677-2-10-2': "First, we use Eqs. ([REF]) and ([REF]) to derive a correction to the polarizability kernel [EQUATION] and [EQUATION] where the boundary Green's functions are defined by Eqs. ([REF]) and ([REF]) and we introduced the following function [EQUATION]", 'cond-mat-0601677-2-10-3': 'This equation contains four terms (we can symbolically denote them as [MATH], [MATH], [MATH], and [MATH]).', 'cond-mat-0601677-2-10-4': 'Let us analyze the first one, the other terms can be calculated in a similar manner: [EQUATION]', 'cond-mat-0601677-2-10-5': 'To calculate this integral it is convenient to consider a new coordinate system [MATH] in which the points [MATH] and [MATH] have the coordinates [MATH] and [MATH] respectively ([MATH]).', 'cond-mat-0601677-2-10-6': 'Next, we introduce the elliptic coordinates [MATH], with the points [MATH] and [MATH] in the foci of the ellipses: [MATH] and [MATH] being the angle between the vector [MATH] and the [MATH] axis.', 'cond-mat-0601677-2-10-7': 'Using these elliptic coordinates and Eq. ([REF]), we write the integral ([REF]) in the following form [EQUATION]', 'cond-mat-0601677-2-10-8': 'We note that the limits of integration over [MATH] are generally non-trivial due to the constraint [MATH].', 'cond-mat-0601677-2-10-9': "It is not possible to calculate the Green's function in terms of elementary functions for generic [MATH] and [MATH].", 'cond-mat-0601677-2-10-10': 'However, it is possible to evaluate analytically the correction to the diffusion kernel [see Eqs. ([REF]), ([REF]), and ([REF])] and the boundary matrix ([REF]).', 'cond-mat-0601677-2-10-11': 'Indeed from Eqs. ([REF]), ([REF]), and ([REF]), we find that due to the fast-oscillating exponents, only a few terms survive the integration over [MATH] in the leading approximation with respect to the large conductance [MATH].', 'cond-mat-0601677-2-10-12': 'These remaining terms contain combinations of the following types [MATH], which constrain the parameter [MATH] in ([REF]) to be close to unity.', 'cond-mat-0601677-2-10-13': 'This allows one to calculate the integral ([REF]) asymptotically and we find in the leading order the following results for the boundary matrix ([REF]) [EQUATION] where the last equation implies that the boundary spin-charge coupling is small with respect to the inverse conductance and vanishes in the framework of the diffusion approximation and the accuracy of the method.', 'cond-mat-0601677-2-10-14': 'We note that corrections of this order exist only for the spin density component parallel to the edge.', 'cond-mat-0601677-2-10-15': 'We also emphasize that the smallness of boundary and bulk spin-charge coupling terms is specific to the Rashba model, which is in some sense pathological as it contains accidental cancellations.', 'cond-mat-0601677-2-10-16': 'In a generic model, the spin-spin and spin-charge couplings are expected to be of the same order.', 'cond-mat-0601677-2-11-0': 'From Eqs. ([REF]) and ([REF]), we find the following boundary conditions in the leading approximation with respect to the spin-orbit coupling and the inverse conductance: [EQUATION] where [MATH] is defined after Eq. ([REF]).', 'cond-mat-0601677-2-11-1': 'These boundary conditions can be generalized to describe any form of a hard-wall boundary (provided that the typical length-scales are much larger than the mean-free path).', 'cond-mat-0601677-2-11-2': 'To find such a general form of boundary conditions, one should do the following replacements: [MATH] and [MATH], where [MATH] is the unit vector normal to the segment of the boundary and [MATH] is the corresponding component of the spin density.', 'cond-mat-0601677-2-11-3': 'However, if the boundary is a smooth potential with the length-scale in the [MATH]-direction being [MATH], then boundary conditions would be different and strongly dependent on the ratios of [MATH] and [MATH] [CITATION].', 'cond-mat-0601677-2-11-4': 'Also, the boundary potential would lead to an additional boundary spin-orbit coupling via the standard relativistic effect.', 'cond-mat-0601677-2-11-5': 'Since the actual solution of the diffusion equation is selected by boundary conditions, the corresponding spin accumulation would be determined by the geometry of the boundary and the spin-orbit coupling at the edges.', 'cond-mat-0601677-2-12-0': 'As an example, we consider a two-dimensional gas of Rashba electrons occupying a half-space [MATH] and in the presence of an electric field parallel to the edge.', 'cond-mat-0601677-2-12-1': 'It is convenient to rewrite the diffusion equation ([REF], [REF]) in terms of the rescaled variables: [MATH] and [MATH], where [MATH] is the dimensionless distance (measured in the units of the spin-diffusion length) and [MATH] is the uniform spin density due to the spin-charge coupling.', 'cond-mat-0601677-2-12-2': 'In these variables, the diffusion equation takes the following form [EQUATION]', 'cond-mat-0601677-2-12-3': 'Boundary conditions are [MATH] and [MATH].', 'cond-mat-0601677-2-12-4': 'We see that these equations and boundary conditions do not contain any dimensional parameters.', 'cond-mat-0601677-2-12-5': 'The solution of the differential equation ([REF], [REF]) is straightforward and we find [EQUATION] and [EQUATION] where [MATH] is an eigenvalue of the diffusion equation ([REF], [REF]).', 'cond-mat-0601677-2-12-6': 'Since the corresponding differential operator is non-Hermitian, the eigenvalues are complex.', 'cond-mat-0601677-2-12-7': 'This leads to oscillations of the spin density near the boundary with the period [MATH].', 'cond-mat-0601677-2-12-8': 'We should point out that there definitely exists another type of small-lengthscale oscillations (with the period [MATH]), which are superimposed onto the regular behavior ([REF], [REF]).', 'cond-mat-0601677-2-12-9': 'However, these Friedel oscillations are beyond the reach of the diffusion approximation.', 'cond-mat-0601677-2-12-10': 'From Eqs. ([REF], [REF]) and Fig. 1, we see that the spin density rapidly decays as we move away from the boundary.', 'cond-mat-0601677-2-12-11': 'The corresponding lengthscale is [MATH].', 'cond-mat-0601677-2-13-0': 'The spin accumulation ([REF], [REF]) is an exact solution of the diffusion equation ([REF], [REF]), which satisfies the correct boundary conditions ([REF], [REF]).', 'cond-mat-0601677-2-13-1': 'However we reiterate that the results may change if the boundary potential is different from a hard wall: The eigenvalues, which determine the density profile, are robust and universal, but the overall amplitude and phase shifts are boundary-specific.', 'cond-mat-0601677-2-13-2': 'Therefore, the spin Hall effect is generally a non-universal phenomenon, which depends on the structure of the sample edges.', 'cond-mat-0601677-2-14-0': 'V.M.G. and A.A.B. are grateful to Leon Balents, Emmanuel Rashba, and Yaroslav Tserkovnyak for helpful discussions.', 'cond-mat-0601677-2-14-1': 'A.A.B. is supported by the NSF under grant DMR02-33773.', 'cond-mat-0601677-2-14-2': 'S.D.S. is supported by US-ONR, NSF, and LPS.'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/', '4': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0601677

{'cond-mat-0601677-3-0-0': 'We develop a general scheme of deriving boundary conditions for spin-charge coupled transport in disordered systems with spin-orbit interactions.', 'cond-mat-0601677-3-0-1': 'To illustrate the application of the method, we explicitly derive boundary conditions for spin diffusion in the Rashba model.', 'cond-mat-0601677-3-0-2': 'Due to the surface spin precession, the boundary conditions are non-trivial and contain terms, which couple different components of the spin density.', 'cond-mat-0601677-3-0-3': 'We argue that boundary conditions and the corresponding electric-field-induced spin accumulation generally depend on the nature of the boundary and therefore the spin Hall effect in a spin-orbit coupled system can be viewed as a non-universal edge phenomenon.', 'cond-mat-0601677-3-1-0': 'There has been a lot of recent interest in a phenomenon dubbed the spin Hall effect [CITATION].', 'cond-mat-0601677-3-1-1': 'Experimentally this effect manifests itself as an equilibrium spin accumulation near the edges, when an electric field is applied parallel to a Hall bar of a spin-orbit (SO) coupled electron system.', 'cond-mat-0601677-3-1-2': 'This kind of electric-field-induced spin polarization has been observed in recent experiments [CITATION].', 'cond-mat-0601677-3-1-3': 'This phenomenon has a potential for being technologically important, as it allows one to electrically manipulate the spin degree of freedom in non-magnetic systems.', 'cond-mat-0601677-3-1-4': 'There are still many controversies regarding the physical origin of the spin accumulation observed in real experiment (intrinsic [CITATION] vs. extrinsic [CITATION]).', 'cond-mat-0601677-3-1-5': 'But even within a given theoretical model there is still no consensus on the correct description of the spin Hall effect [CITATION].', 'cond-mat-0601677-3-2-0': 'A common theoretical approach to the intrinsic spin Hall effect is as follows: One introduces the notion of a spin current, which is an operator intuitively related to spin transport.', 'cond-mat-0601677-3-2-1': 'Using the Kubo formula, one then calculates the spin-current autocorrelator and the corresponding linear response quantity, called the "spin Hall conductivity" [CITATION].', 'cond-mat-0601677-3-2-2': "In clean systems, it appears to be related to an elegant topological Berry's phase structure [CITATION].", 'cond-mat-0601677-3-2-3': 'In a disordered system, the notion of the Fermi surface Berry\'s phase is unclear, but the "spin Hall conductivity" is well-defined.', 'cond-mat-0601677-3-2-4': 'It is believed that this quantity is connected to spin accumulation in the spin-Hall experiment.', 'cond-mat-0601677-3-2-5': 'While this theoretical approach is mathematically well-defined and may lead to technically challenging problems [CITATION], its relation to experiment remains unclear.', 'cond-mat-0601677-3-2-6': 'The problem is that in SO coupled systems, the spin is not conserved and thus the definition of the spin current is ambiguous [CITATION].', 'cond-mat-0601677-3-3-0': 'To describe realistic experiment, it is preferable to use another route without involving the notion of a spin current.', 'cond-mat-0601677-3-3-1': 'In disordered systems, this can be done with the help of a kinetic equation [CITATION] or a spin-charge coupled diffusion equation [CITATION].', 'cond-mat-0601677-3-3-2': 'The diffusion equation provides a complete and physically meaningful description of spin and charge diffusive transport in terms of position and time dependent spin and charge densities.', 'cond-mat-0601677-3-3-3': 'However, the explicit solution of the diffusion equation strongly depends on boundary conditions (BC).', 'cond-mat-0601677-3-3-4': 'While for usual charge transport, BC are often obvious and follow from conservation laws, this is not the case in SO coupled systems.', 'cond-mat-0601677-3-3-5': 'The problem is that the spin is not conserved and thus there is spin precession at the edges.', 'cond-mat-0601677-3-3-6': 'This spin dynamics is very sensitive to the actual boundary (depending on the physical situation, the boundary could be modeled as a hard wall, WKB potential, rough surface, etc.).', 'cond-mat-0601677-3-3-7': 'Since the bulk diffusion equation describes spin/charge dynamics only at the length-scales much larger than the mean-free path [MATH], it can not capture the behavior of the spin in the immediate vicinity of the sample edges.', 'cond-mat-0601677-3-3-8': 'Due to the aforementioned boundary spin precession, the "trivial BC" (i.e., BC, which one would have, if the spin had been conserved) acquire corrections proportional to the SO coupling.', 'cond-mat-0601677-3-3-9': 'These corrections are very important, as they select a unique solution of the diffusion equation and determine the observable spin accumulation.', 'cond-mat-0601677-3-4-0': 'In this Letter we formulate a general method of deriving boundary conditions in SO coupled systems.', 'cond-mat-0601677-3-4-1': 'Using this proposed method, we derive BC for the Rashba model [CITATION] in the leading order with respect to the SO coupling, which is assumed small.', 'cond-mat-0601677-3-4-2': 'We show that there are corrections to the "trivial BC:" In leading order, we find that at the boundary, the spin component perpendicular to the plane of the two-dimensional gas gets coupled to the in-plane spin component perpendicular to the boundary.', 'cond-mat-0601677-3-4-3': 'We argue that spin accumulation in the spin Hall effect setup is determined by the combination of the bulk spin-charge coupling and boundary effects.', 'cond-mat-0601677-3-5-0': 'Let us consider a disordered electron system occupying a region [MATH].', 'cond-mat-0601677-3-5-1': "One can derive the following integral equation for the diffuson [CITATION], which determines the dynamics of the charge density ([MATH]) and the spin densities ([MATH], [MATH]): [EQUATION] where [MATH] is the density of states per spin at the Fermi surface, [MATH] is the scattering time, the Latin indices label charge ([MATH]) and spin ([MATH], [MATH], and [MATH]) degrees of freedom, and [MATH] is the diffuson, which is the Green's function of the spin-charge coupled diffusion equation.", 'cond-mat-0601677-3-5-2': "The kernel in Eq. ([REF]) reads [EQUATION] where [MATH], [MATH] is the unity matrix, [MATH] are the Pauli matrices, and [MATH] are disorder-averaged retarded/advanced electron Green's functions, which include the effect of the boundary (here and below we denote the boundary Green's functions as [MATH] and the bulk Green's functions as [MATH]).", 'cond-mat-0601677-3-5-3': 'They depend exponentially on distances via factors of the following types, [MATH], where [MATH] is the mean-free path.', 'cond-mat-0601677-3-5-4': "Deep in the bulk, the edge effects are exponentially small and the bulk Green's functions can be used, [MATH].", 'cond-mat-0601677-3-5-5': 'Performing a gradient expansion of the bulk polarizability (i.e., taking the limit [MATH]), one obtains the spin-charge coupled diffusion equation.', 'cond-mat-0601677-3-5-6': 'Generally, it has the following form: [EQUATION] where [MATH] are the charge/spin densities, [MATH] are the diffusion coefficients, [MATH] are the relaxation times, and [MATH] are the spin-spin and spin-charge couplings.', 'cond-mat-0601677-3-6-0': "Note that the integral equation ([REF]) does not require boundary conditions (it already contains this information through the boundary Green's function), but the differential diffusion equation ([REF]) does.", 'cond-mat-0601677-3-6-1': 'To derive these BC one needs to perform the gradient expansion near the boundary [MATH].', 'cond-mat-0601677-3-6-2': 'Let us consider the problem in half-space [MATH].', 'cond-mat-0601677-3-6-3': 'To find the diffusion equation near the boundary, we take the limit [MATH].', 'cond-mat-0601677-3-6-4': 'This limit should be understood in the following sense: [MATH], i.e., we consider a point, which is near the boundary as compared to the diffusion length-scale [MATH], but still far from it as compared to the ballistic length scale described by the Fermi wave-length.', 'cond-mat-0601677-3-6-5': 'Let us first consider the case of zero SO interactions.', 'cond-mat-0601677-3-6-6': "If the boundary is a hard wall potential, then the boundary Green's function can be obtained using the method of mirror images [EQUATION] where [MATH], [MATH], and [MATH] is the mirror image with respect to the sample boundary.", 'cond-mat-0601677-3-6-7': 'For the sake of concreteness, let us assume that the dimensionality [MATH] (all qualitative arguments are independent of dimensionality).', 'cond-mat-0601677-3-6-8': "In two dimensions, the Green's function has the following form [EQUATION]", 'cond-mat-0601677-3-6-9': 'We note here that the frequency dependence of the diffusion kernel is irrelevant for the boundary condition problem and will be omitted from now on.', 'cond-mat-0601677-3-6-10': 'When the method of mirror images applies, the polarizability kernel ([REF]) has the following general structure [MATH] [where [MATH] implies the "reflected" Green\'s function, i.e, the second term in Eq. ([REF])].', 'cond-mat-0601677-3-6-11': 'The key observation is that due to the fast-oscillating factor [MATH] and our choice of the point [MATH]), the cross terms [MATH] in the expression for [MATH] become negligible in the integral over [MATH] and can be omitted.', 'cond-mat-0601677-3-6-12': 'The remaining integral can be easily calculated and we find (in the "trivial" case of no SO coupling): [EQUATION] where [MATH] is a dimensionless coefficient, which in two dimensions is equal to [MATH].', 'cond-mat-0601677-3-6-13': 'From Eqs. ([REF]) and ([REF]), we recover the familiar boundary condition [MATH], which is simply the consequence of the charge and spin conservation.', 'cond-mat-0601677-3-7-0': 'In a system with SO interaction, the method of images does not apply because the "reflected" Green\'s function does not satisfy the corresponding Schrodinger equation.', 'cond-mat-0601677-3-7-1': 'This is a major complication, which never occurs in the problems usually studied in the literature in this context [CITATION].', 'cond-mat-0601677-3-7-2': 'In addition, the matrix structure of the boundary diffusion kernel becomes essential.', 'cond-mat-0601677-3-7-3': "There are generally two ways to obtain the boundary Green's function: (i) The first is to solve the Schrodinger equation near the boundary and express the Green's function using the corresponding wave-functions as a basis; (ii) The second is to find the Green's functions perturbatively, using the mirror-image result ([REF]) as the leading approximation: [EQUATION] where [MATH] is the SO interaction operator as it appears in the Hamiltonian.", 'cond-mat-0601677-3-7-4': 'We note that disorder corrections to the SO vertex can be shown to be small as long as [MATH].', 'cond-mat-0601677-3-7-5': "The resulting Green's function ([REF]) may be quite complicated even in the simplest cases.", 'cond-mat-0601677-3-7-6': "However, the diffusion kernel always contains products of the following types: [EQUATION] where [MATH]'s and [MATH]'s are some functions, which do not oscillate on a Fermi wavelength scale.", 'cond-mat-0601677-3-7-7': 'Again, in the integral of Eq. ([REF]), the cross-terms [MATH] become negligible due to the fast-oscillations and can be omitted, but the other terms survive.', 'cond-mat-0601677-3-7-8': "One can define the following matrix, which determines BC [EQUATION] where the correction to polarizability is determined by the Green's function ([REF]) and Eq. ([REF]) and the limit is understood in the sense [MATH].", 'cond-mat-0601677-3-7-9': 'If the boundary is an impenetrable wall, general BC take the form [EQUATION] where [MATH], [MATH] is a unit vector normal to the boundary, [MATH] is a number ([MATH]), and [MATH] is a dimensionless matrix defined above.', 'cond-mat-0601677-3-7-10': 'If an external field is present one may need to consider another term in the BC, which has the meaning of the boundary spin-charge coupling [MATH].', 'cond-mat-0601677-3-7-11': 'In this case the boundary condition becomes: [EQUATION]', 'cond-mat-0601677-3-7-12': 'We note that if the boundary is different from a hard wall, the general method does not change but the results do even in the lowest order.', 'cond-mat-0601677-3-7-13': 'For instance, if the boundary is characterized by an isotropic reflection coefficient [MATH], the zeroth order boundary condition becomes [MATH].', 'cond-mat-0601677-3-7-14': 'If the reflection coefficient is not too close to unity, one can use the following leading order boundary conditions [EQUATION]', 'cond-mat-0601677-3-7-15': 'This boundary condition implies that an excess charge or spin density can not exist at the boundary if the boundary is penetrable.', 'cond-mat-0601677-3-7-16': 'The excess density leaks through the edge.', 'cond-mat-0601677-3-8-0': 'To illustrate the application of our method, let us consider the Rashba model [CITATION] in the leading order in SO coupling.', 'cond-mat-0601677-3-8-1': 'In this case, the SO interaction operator has the form [MATH], where [MATH] is the Rashba SO coupling and [MATH].', 'cond-mat-0601677-3-8-2': 'The differential diffusion equation was derived in Ref. [[CITATION]] and reads [EQUATION]', 'cond-mat-0601677-3-8-3': 'The spin-spin coupling in the Rashba model is [MATH] and the spin relaxation rates are [MATH].', 'cond-mat-0601677-3-8-4': 'These parameters are universal in the sense that they do not depend on the details of the electronic spectrum and can be calculated in the [MATH]-approximation [CITATION].', 'cond-mat-0601677-3-8-5': 'However, the spin-charge coupling is not universal, since it strongly depends on the electronic spectrum and the high-energy cut-off (if the spectrum is quadratic, [MATH]).', 'cond-mat-0601677-3-8-6': 'We should point out however that this result holds only in the semiclassical diffusion approximation, i.e., only in the leading order with respect to the inverse conductance [MATH], which is an independent parameter of the model.', 'cond-mat-0601677-3-8-7': 'There may exist contributions to the spin-charge coupling of order [MATH].', 'cond-mat-0601677-3-8-8': 'They originate from diagrams with crossed impurity lines, which are omitted in the diffusion approximation (we note that there is no direct relation between [MATH] and the spin Hall conductivity, which vanishes to all orders [CITATION]).', 'cond-mat-0601677-3-8-9': 'These quantum corrections can be neglected only if [MATH].', 'cond-mat-0601677-3-8-10': 'This is a very strong constraint, which is not always satisfied in experimentally studied systems.', 'cond-mat-0601677-3-8-11': 'In what follows, we will treat the spin-charge coupling terms in the diffusion equation ([REF], [REF]) as non-universal phenomenological parameters.', 'cond-mat-0601677-3-9-0': 'The diffusion equation ([REF], [REF]) must be supplemented by BC.', 'cond-mat-0601677-3-9-1': 'We consider the problem in leading approximation with respect to the SO coupling parameter [MATH] and the inverse conductance.', 'cond-mat-0601677-3-9-2': "First, we use Eqs. ([REF]) and ([REF]) to derive a correction to the polarizability kernel [EQUATION] and [EQUATION] where the boundary Green's functions are defined by Eqs. ([REF]) and ([REF]) and we introduced the following function [MATH].", 'cond-mat-0601677-3-9-3': 'This function contains four terms (we can symbolically denote them as [MATH], [MATH], [MATH], and [MATH]).', 'cond-mat-0601677-3-9-4': 'Let us analyze the first one, the other terms can be calculated in a similar manner: To calculate the corresponding integral, we use a new coordinate system [MATH] in which the points [MATH] and [MATH] have the coordinates [MATH] and [MATH] respectively ([MATH]).', 'cond-mat-0601677-3-9-5': 'Next, we introduce the elliptic coordinates [MATH], with the points [MATH] and [MATH] in the foci of the ellipses: [MATH] and [MATH] being the angle between the vector [MATH] and the [MATH] axis.', 'cond-mat-0601677-3-9-6': 'Using these elliptic coordinates and Eq. ([REF]), we write the integral [MATH] in the following form [EQUATION]', 'cond-mat-0601677-3-9-7': 'We note that the limits of integration over [MATH] are generally non-trivial due to the constraint [MATH].', 'cond-mat-0601677-3-9-8': 'It is possible to evaluate analytically the correction to the diffusion kernel [see Eqs. ([REF]), ([REF]), and ([REF])] and the boundary matrix ([REF]).', 'cond-mat-0601677-3-9-9': 'Indeed, from Eqs. ([REF]), ([REF]), and ([REF]), we find that due to the fast-oscillating exponents, only a few terms survive the integration over [MATH].', 'cond-mat-0601677-3-9-10': 'These remaining terms contain combinations of the following types [MATH], which constrain the parameter [MATH] in ([REF]) to be close to unity.', 'cond-mat-0601677-3-9-11': 'We find in the leading order the following results for the boundary matrix ([REF]) [EQUATION] where the last equation implies that the boundary spin-charge coupling is small with respect to the inverse conductance and vanishes in the framework of the diffusion approximation and the accuracy of the method.', 'cond-mat-0601677-3-10-0': 'From Eqs. ([REF]) and ([REF]), we find the following BC in the leading approximation with respect to the SO coupling and the inverse conductance: [EQUATION] where [MATH] is defined after Eq. ([REF]).', 'cond-mat-0601677-3-10-1': 'These BC can be generalized to describe any form of a hard-wall boundary.', 'cond-mat-0601677-3-10-2': 'To find such a general form of BC, one should do the following replacements: [MATH] and [MATH], where [MATH] is the unit vector normal to the segment of the boundary and [MATH] is the corresponding component of the spin density.', 'cond-mat-0601677-3-10-3': 'However, if the boundary is a smooth potential with the length-scale in the [MATH]-direction being [MATH], then BC would be different and strongly dependent on the ratios of [MATH] and [MATH] [CITATION].', 'cond-mat-0601677-3-10-4': 'Also, the boundary potential would lead to an additional boundary SO coupling, which would strongly affect the BC.', 'cond-mat-0601677-3-11-0': 'We note here that in the next-to-leading order with respect to the SO interaction, two new effects appear: Boundary spin-charge coupling and boundary spin relaxation (the former is parametrically larger than the latter due to an additional smallness of the spin density).', 'cond-mat-0601677-3-11-1': 'To this order boundary conditions take the form: [MATH] (where [MATH] and [MATH] are constants [MATH]).', 'cond-mat-0601677-3-11-2': 'We should note that the boundary spin-charge coupling term can be shown to be strongly dependent on the electronic spectrum and the physics far from the Fermi line and as such is non-universal and can not be calculated in the [MATH]-approximation used in this paper.', 'cond-mat-0601677-3-12-0': 'We now solve the diffusion equation for a two-dimensional gas of Rashba electrons occupying a half-space [MATH] and in the presence of an electric field parallel to the edge.', 'cond-mat-0601677-3-12-1': 'We consider two types of boundary: a transparent boundary and a hard wall boundary.', 'cond-mat-0601677-3-12-2': 'It is convenient to rewrite the diffusion equation ([REF], [REF]) in terms of the rescaled variables: [MATH] and [MATH], where [MATH] is the dimensionless distance and [MATH] is the uniform spin density due to the spin-charge coupling.', 'cond-mat-0601677-3-12-3': 'In these variables, the diffusion equation takes the following form [EQUATION]', 'cond-mat-0601677-3-12-4': 'Boundary conditions for a transparent boundary are [MATH] and [MATH] and for a hard-wall boundary [MATH] and [MATH], where [MATH] is a non-universal spectrum-dependent parameter.', 'cond-mat-0601677-3-12-5': 'We note that if the latter is zero, one has to take into account the next-to-leading order boundary spin relaxation terms, which should determine the observable spin density.', 'cond-mat-0601677-3-12-6': 'The solution of the differential equation ([REF], [REF]) is straightforward and we find for the transparent boundary (e.g., ballistic contacts) [EQUATION] and for the hard-wall boundary [EQUATION] where [MATH] is an eigenvalue of the diffusion equation ([REF], [REF]).', 'cond-mat-0601677-3-12-7': 'Since the corresponding differential operator is non-Hermitian, the eigenvalues are complex.', 'cond-mat-0601677-3-12-8': 'This leads to oscillations of the spin density near the boundary with the period [MATH].', 'cond-mat-0601677-3-12-9': 'We should point out that there definitely exists other types of small-lengthscale oscillations (with the period [MATH]), which are superimposed onto the regular behavior ([REF], [REF]).', 'cond-mat-0601677-3-12-10': 'However, these Friedel oscillations are beyond the reach of the diffusion approximation.', 'cond-mat-0601677-3-12-11': 'From Eqs. ([REF], [REF]), we see that the spin density rapidly decays as we move away from the boundary.', 'cond-mat-0601677-3-12-12': 'The corresponding lengthscale is [MATH].', 'cond-mat-0601677-3-13-0': 'The authors are grateful to L. Balents, H.-A.Engel, A.G. Malshukov, E.I. Rashba, and Ya.', 'cond-mat-0601677-3-13-1': 'Tserkovnyak for helpful discussions.', 'cond-mat-0601677-3-13-2': 'A.A.B. is supported by the NSF under grant DMR02-33773.', 'cond-mat-0601677-3-13-3': 'S.D.S. is supported by US-ONR, NSF, and LPS.'}

{'cond-mat-0601677-4-0-0': 'We develop a general scheme of deriving boundary conditions for spin-charge coupled transport in disordered systems with spin-orbit interactions.', 'cond-mat-0601677-4-0-1': 'To illustrate the application of the method, we explicitly derive boundary conditions for spin diffusion in the Rashba model.', 'cond-mat-0601677-4-0-2': 'Due to the surface spin precession, the boundary conditions are non-trivial and contain terms, which couple different components of the spin density.', 'cond-mat-0601677-4-0-3': 'We argue that boundary conditions and the corresponding electric-field-induced spin accumulation generally depend on the nature of the boundary and therefore the spin Hall effect in a spin-orbit coupled system can be viewed as a non-universal edge phenomenon.', 'cond-mat-0601677-4-1-0': '# Introduction', 'cond-mat-0601677-4-2-0': 'There has been a lot of recent interest in a phenomenon dubbed the spin Hall effect [CITATION].', 'cond-mat-0601677-4-2-1': 'Experimentally this effect manifests itself as an equilibrium spin accumulation near the edges, when an electric field is applied parallel to a Hall bar of a spin-orbit coupled electron system.', 'cond-mat-0601677-4-2-2': 'This kind of electric-field-induced spin polarization has been observed in recent experiments [CITATION].', 'cond-mat-0601677-4-2-3': 'This phenomenon has a potential for being technologically important, as it allows one to electrically manipulate the spin degree of freedom in non-magnetic systems.', 'cond-mat-0601677-4-2-4': 'There are still many controversies regarding the physical origin of the spin accumulation observed in real experiment (intrinsic [CITATION] vs. extrinsic [CITATION]).', 'cond-mat-0601677-4-2-5': 'But even within a given theoretical model (e.g., a two-dimensional electron gas with Rashba or Dresselhaus spin-orbit interactions or Luttinger model), there is still no consensus on the correct description of the spin Hall effect [CITATION].', 'cond-mat-0601677-4-3-0': 'A common theoretical approach to the intrinsic spin Hall effect is as follows: One introduces the notion of a spin current, which is an operator intuitively related to spin transport.', 'cond-mat-0601677-4-3-1': 'Using the Kubo formula, one then calculates the spin-current autocorrelator and the corresponding linear response quantity, called the "spin Hall conductivity" [CITATION].', 'cond-mat-0601677-4-3-2': "In clean systems, it appears to be related to an elegant topological Berry's phase structure [CITATION].", 'cond-mat-0601677-4-3-3': 'In a disordered system, the notion of the Fermi surface Berry\'s phase is unclear, but the "spin Hall conductivity" is well-defined.', 'cond-mat-0601677-4-3-4': 'It is believed that this quantity is connected to spin accumulation in the spin-Hall experiment.', 'cond-mat-0601677-4-3-5': 'While this theoretical approach is mathematically well-defined and may lead to technically challenging problems [CITATION], its relation to experiment remains unclear.', 'cond-mat-0601677-4-3-6': 'The problem is that in spin-orbit coupled systems, the spin is not conserved and thus the definition of the spin current is ambiguous [CITATION].', 'cond-mat-0601677-4-3-7': 'A non-equilibrium spin density in a region relaxes not only due to the flux through the boundaries of the region, but also due to spin precession.', 'cond-mat-0601677-4-3-8': 'Therefore, the spin current is not easily measurable.', 'cond-mat-0601677-4-4-0': 'To describe realistic experiment, it is preferable to use another route without involving the notion of a spin current.', 'cond-mat-0601677-4-4-1': 'In disordered systems, this can be done with the help of a kinetic equation [CITATION] or a spin-charge coupled diffusion equation [CITATION].', 'cond-mat-0601677-4-4-2': 'The diffusion equation provides a complete and physically meaningful description of spin and charge diffusive transport in terms of position and time dependent spin and charge densities.', 'cond-mat-0601677-4-4-3': 'However, the explicit solution of the diffusion equation (which is generally a complicated set of partial differential equations) strongly depends on boundary conditions [CITATION].', 'cond-mat-0601677-4-4-4': 'While for usual charge transport, boundary conditions are often obvious and follow from conservation laws, this is not the case in spin-orbit coupled systems.', 'cond-mat-0601677-4-4-5': 'The problem is that the spin is not conserved and thus there is spin precession at the edges.', 'cond-mat-0601677-4-4-6': 'This spin dynamics is very sensitive to the actual boundary (depending on the physical situation, the boundary could be modelled as a hard wall, WKB potential, rough surface, etc.).', 'cond-mat-0601677-4-4-7': 'Since the bulk diffusion equation describes spin/charge dynamics only at the length-scales much larger than the mean-free path [MATH], it can not capture the behavior of the spin in the immediate vicinity of the sample edges.', 'cond-mat-0601677-4-4-8': 'Due to the aforementioned boundary spin precession, the "trivial boundary conditions" (i.e., boundary conditions, which one would have, if the spin had been conserved) acquire corrections proportional to the spin-orbit coupling.', 'cond-mat-0601677-4-4-9': 'These corrections are very important, as they select a unique solution of the diffusion equation and determine the observable spin accumulation.', 'cond-mat-0601677-4-5-0': 'The derivation of boundary conditions is technically a difficult task.', 'cond-mat-0601677-4-5-1': 'Generally, one needs to determine the behavior of the system in the ballistic region near the boundary and match it with the diffusive behavior in the bulk.', 'cond-mat-0601677-4-5-2': "It requires the knowledge of the exact [MATH]-matrix or the boundary Green's function.", 'cond-mat-0601677-4-5-3': 'We should point out that there exists an extensive literature on the related issues in the context of the diffusion of light in a random medium [CITATION] and neutron diffusion [CITATION].', 'cond-mat-0601677-4-5-4': 'In these problems, the general form of boundary conditions is obvious due to conservation laws and only numerical coefficients are unknown.', 'cond-mat-0601677-4-5-5': 'There exists a general method of deriving these numerical coefficients.', 'cond-mat-0601677-4-5-6': 'But there are just a very few models in which exact analytical results are available (a notable example is the Milne problem [CITATION], which describes a boundary separating diffusive and ballistic regions).', 'cond-mat-0601677-4-5-7': 'We note that in the context of spin diffusion in spin-orbit coupled systems, even the qualitative form of boundary conditions is not known as there are no obvious conservation laws, which would provide guidance.', 'cond-mat-0601677-4-5-8': 'Even though in the bulk, one has conserved quantities labeled by the band index (e.g., chirality in the Rashba model), the boundary generally does not respect these conservation laws and mixes up different bands.', 'cond-mat-0601677-4-6-0': 'In this paper we formulate a general method of deriving boundary conditions in spin-orbit coupled systems.', 'cond-mat-0601677-4-6-1': 'Using this proposed method, we derive boundary conditions for the Rashba model [CITATION] in the leading order with respect to the spin-orbit coupling, which is assumed small.', 'cond-mat-0601677-4-6-2': 'We show that there are corrections to the "trivial boundary conditions:" In leading order, we find that at the boundary, the spin component perpendicular to the plane of the two-dimensional gas gets coupled to the in-plane spin component perpendicular to the boundary.', 'cond-mat-0601677-4-6-3': 'We argue that spin accumulation in the spin Hall effect setup is determined by the combination of the bulk spin-charge coupling and boundary effects.', 'cond-mat-0601677-4-7-0': '# General method of deriving boundary conditions', 'cond-mat-0601677-4-8-0': 'In this section we formulate a general method of deriving boundary conditions applicable to any spin-orbit coupled system with a boundary.', 'cond-mat-0601677-4-8-1': 'Let us consider a disordered electron system occupying a region [MATH].', 'cond-mat-0601677-4-8-2': "One can derive the following integral equation for the diffuson [CITATION], which determines the dynamics of the charge density ([MATH]) and the spin densities ([MATH], [MATH]): [EQUATION] where [MATH] is the density of states per spin at the Fermi surface, [MATH] is the scattering time, the Latin indices label charge ([MATH]) and spin ([MATH], [MATH], and [MATH]) degrees of freedom, and [MATH] is the diffuson, which is the Green's function of the spin-charge coupled diffusion equation.", 'cond-mat-0601677-4-8-3': "The kernel in Eq. ([REF]) reads [EQUATION] where [MATH], [MATH] is the unity matrix, [MATH] are the Pauli matrices, and [MATH] are disorder-averaged retarded/advanced electron Green's functions, which include the effect of the boundary (here and below we denote the boundary Green's functions as [MATH] and the bulk Green's functions as [MATH]).", 'cond-mat-0601677-4-8-4': 'They depend exponentially on distances via factors of the following types, [MATH], where [MATH] is the mean-free path.', 'cond-mat-0601677-4-8-5': "Deep in the bulk, the edge effects are exponentially small and the bulk Green's functions can be used, [MATH].", 'cond-mat-0601677-4-8-6': 'Performing a gradient expansion of the bulk polarizability (i.e., taking the limit [MATH]), one obtains the spin-charge coupled diffusion equation.', 'cond-mat-0601677-4-8-7': 'Generally, it has the following form: [EQUATION] where [MATH] are the charge/spin densities, [MATH] are the diffusion coefficients, [MATH] are the relaxation times, and [MATH] are the spin-spin and spin-charge couplings.', 'cond-mat-0601677-4-9-0': "Note that the integral equation ([REF]) does not require boundary conditions (it already contains this information through the boundary Green's function), but the differential diffusion equation ([REF]) does.", 'cond-mat-0601677-4-9-1': 'To derive these boundary conditions one needs to perform the gradient expansion near the boundary [MATH].', 'cond-mat-0601677-4-9-2': 'Let us consider the problem in half-space [MATH].', 'cond-mat-0601677-4-9-3': 'To find the diffusion equation near the boundary, we take the limit [MATH].', 'cond-mat-0601677-4-9-4': 'This limit should be understood in the following sense: [MATH], i.e., we consider a point, which is near the boundary as compared to the diffusion length-scale [MATH], but still far from it as compared to the ballistic length scale described by the Fermi wave-length.', 'cond-mat-0601677-4-9-5': 'Let us first consider the case of zero spin-orbit interactions.', 'cond-mat-0601677-4-9-6': "If the boundary is a hard wall potential, then the boundary Green's function can be obtained using the method of mirror images [EQUATION] where [MATH], [MATH], and [MATH] is the mirror image with respect to the sample boundary.", 'cond-mat-0601677-4-9-7': 'For the sake of concreteness, let us assume that the dimensionality [MATH] (all qualitative arguments are independent of dimensionality).', 'cond-mat-0601677-4-9-8': "In two dimensions, the Green's function has the following form [EQUATION]", 'cond-mat-0601677-4-9-9': 'We note here that the frequency dependence of the diffusion kernel is irrelevant for the boundary condition problem and will be omitted from now on.', 'cond-mat-0601677-4-9-10': 'When the method of mirror images applies, the polarizability kernel ([REF]) has the following general structure [MATH] [where [MATH] implies the "reflected" Green\'s function, i.e, the second term in Eq. ([REF])].', 'cond-mat-0601677-4-9-11': 'The key observation is that due to the fast-oscillating factor [MATH] and our choice of the point [MATH]), the cross terms [MATH] in the expression for [MATH] become negligible in the integral over [MATH] and can be omitted.', 'cond-mat-0601677-4-9-12': 'The remaining integral can be easily calculated and we find (in the "trivial" case of no spin-orbit coupling): [EQUATION] where [MATH] is a dimensionless coefficient, which in two dimensions is equal to [MATH].', 'cond-mat-0601677-4-9-13': 'From Eqs. ([REF]) and ([REF]), we recover the familiar boundary condition [MATH], which is simply the consequence of the charge and spin conservation.', 'cond-mat-0601677-4-10-0': 'In a system with spin-orbit interaction, the method of images does not apply because the "reflected" Green\'s function does not satisfy the corresponding Schrodinger equation.', 'cond-mat-0601677-4-10-1': 'This is a major complication, which never occurs in the problems usually studied in the literature in this context [CITATION].', 'cond-mat-0601677-4-10-2': 'In addition, the matrix structure of the boundary diffusion kernel becomes essential.', 'cond-mat-0601677-4-10-3': "There are generally two ways to obtain the boundary Green's function (we have explicitly verified that both approaches lead to identical perturbative results in the leading order [CITATION]): (i) The first is to solve the Schrodinger equation near the boundary and express the Green's function using the corresponding wave-functions as a basis; (ii) The second is to find the Green's functions perturbatively, using the mirror-image result ([REF]) as the leading approximation: [EQUATION] where [MATH] is the spin-orbit interaction operator as it appears in the Hamiltonian.", 'cond-mat-0601677-4-10-4': 'We note that disorder corrections to the spin-orbit vertex can be shown to be small as long as [MATH].', 'cond-mat-0601677-4-10-5': "The resulting Green's function ([REF]) may be quite complicated even in the simplest cases.", 'cond-mat-0601677-4-10-6': "However, the diffusion kernel always contains products of the following types: [EQUATION] where [MATH]'s and [MATH]'s are some functions, which do not oscillate on a Fermi wavelength scale.", 'cond-mat-0601677-4-10-7': 'Again, in the integral of Eq. ([REF]), the cross-terms [MATH] become negligible due to the fast-oscillations and can be omitted, but the other terms survive.', 'cond-mat-0601677-4-10-8': "One can define the following matrix, which determines boundary conditions [EQUATION] where the correction to polarizability is determined by the Green's function ([REF]) and Eq. ([REF]) and the limit is understood in the sense [MATH].", 'cond-mat-0601677-4-10-9': 'If the boundary is an impenetrable wall, general boundary conditions take the form [EQUATION] where [MATH], [MATH] is a unit vector normal to the boundary, [MATH] is a number ([MATH]), and [MATH] is a dimensionless matrix defined above.', 'cond-mat-0601677-4-10-10': 'If an external field is present one may need to consider another term in the boundary conditions, which has the meaning of the boundary spin-charge coupling [CITATION] [EQUATION]', 'cond-mat-0601677-4-10-11': 'In this case the boundary condition becomes: [EQUATION]', 'cond-mat-0601677-4-10-12': 'We note that if the boundary is not impenetrable, the general method does not change but the results do even in the lowest order.', 'cond-mat-0601677-4-10-13': 'For instance, if the boundary is characterized by an isotropic reflection coefficient [MATH], the zeroth order boundary condition becomes [MATH].', 'cond-mat-0601677-4-10-14': 'If the reflection coefficient is not too close to unity, the right-hand side of the latter equation can be considered small and one can use the following leading order boundary conditions [EQUATION]', 'cond-mat-0601677-4-10-15': 'This boundary condition implies that an excess charge or spin density can not exist at the boundary if the boundary is penetrable.', 'cond-mat-0601677-4-10-16': 'The excess density leaks through the edge (see also Ref. [[CITATION]]).', 'cond-mat-0601677-4-11-0': 'We note that strictly speaking the matching point at the boundary [which we set to zero in Eqs. ([REF]) and ([REF])] can not be obtained exactly within the method described here, but all dimensional parameters and numerical coefficients are robust.', 'cond-mat-0601677-4-11-1': 'To determine the correct matching point on a lengthscale of order [MATH], one actually needs to solve exactly the integral equation, which is possible only in very special cases [e.g., the Milne problem (i.e., R=0 and no spin-orbit coupling)].', 'cond-mat-0601677-4-12-0': '# Application to the Rashba model', 'cond-mat-0601677-4-13-0': 'To illustrate the application of our method, let us consider the Rashba model [CITATION] in the leading order in spin-orbit coupling.', 'cond-mat-0601677-4-13-1': 'In this case, the spin-orbit interaction operator has the form [MATH], where [MATH] is the Rashba spin-orbit coupling and [MATH].', 'cond-mat-0601677-4-13-2': 'The differential diffusion equation was derived in Ref. [[CITATION]] and reads [EQUATION]', 'cond-mat-0601677-4-13-3': 'The spin-spin coupling in the Rashba model is [MATH] and the spin relaxation times are [MATH].', 'cond-mat-0601677-4-13-4': 'These parameters are universal in the sense that they do not depend on the details of the electronic spectrum and can be calculated in the [MATH]-approximation [CITATION].', 'cond-mat-0601677-4-13-5': 'However, the spin-charge coupling is not universal, since it strongly depends on the electronic spectrum and the high-energy cut-off (if the spectrum is quadratic, [MATH]).', 'cond-mat-0601677-4-13-6': 'We should point out however that this result holds only in the semiclassical diffusion approximation, i.e., only in the leading order with respect to the inverse conductance [MATH], which is an independent parameter of the model.', 'cond-mat-0601677-4-13-7': 'There may exist contributions to the spin-charge coupling of order [MATH].', 'cond-mat-0601677-4-13-8': 'They originate from diagrams with crossed impurity lines, which are omitted in the diffusion approximation (we note that there is no direct relation between [MATH] and the spin Hall conductivity, which vanishes to all orders [CITATION]).', 'cond-mat-0601677-4-13-9': 'These quantum corrections can be neglected only if [MATH].', 'cond-mat-0601677-4-13-10': 'This is a very strong constraint, which is not always satisfied in experimentally studied systems.', 'cond-mat-0601677-4-13-11': 'In what follows, we will treat the spin-charge coupling terms in the diffusion equation ([REF], [REF]) as non-universal phenomenological parameters and assume that the above-mentioned constraint is satisfied.', 'cond-mat-0601677-4-14-0': 'The diffusion equation ([REF], [REF]) must be supplemented by boundary conditions.', 'cond-mat-0601677-4-14-1': 'We consider the problem in leading approximation with respect to the spin-orbit coupling parameter [MATH] and the inverse conductance.', 'cond-mat-0601677-4-14-2': "First, we use Eqs. ([REF]) and ([REF]) to derive a correction to the polarizability kernel [EQUATION] and [EQUATION] where the boundary Green's functions are defined by Eqs. ([REF]) and ([REF]) and we introduced the following function [EQUATION].", 'cond-mat-0601677-4-14-3': 'This function contains four terms (we can symbolically denote them as [MATH], [MATH], [MATH], and [MATH]).', 'cond-mat-0601677-4-14-4': 'Let us analyze the first one, the other terms can be calculated in a similar manner: To calculate the corresponding integral, we use a new coordinate system [MATH] in which the points [MATH] and [MATH] have the coordinates [MATH] and [MATH] respectively ([MATH]).', 'cond-mat-0601677-4-14-5': 'Next, we introduce the elliptic coordinates [MATH], with the points [MATH] and [MATH] in the foci of the ellipses: [MATH] and [MATH] being the angle between the vector [MATH] and the [MATH] axis.', 'cond-mat-0601677-4-14-6': 'Using these elliptic coordinates and Eq. ([REF]), we write the integral [MATH] in the following form [EQUATION]', 'cond-mat-0601677-4-14-7': 'We note that the limits of integration over [MATH] are generally non-trivial due to the constraint [MATH].', 'cond-mat-0601677-4-14-8': 'It is possible to evaluate analytically the correction to the diffusion kernel [see Eqs. ([REF]), ([REF]), and ([REF])] and the boundary matrix ([REF]).', 'cond-mat-0601677-4-14-9': 'Indeed, from Eqs. ([REF]), ([REF]), and ([REF]), we find that due to the fast-oscillating exponents, only a few terms survive the integration over [MATH].', 'cond-mat-0601677-4-14-10': 'These remaining terms contain combinations of the following types [MATH], which constrain the parameter [MATH] in ([REF]) to be close to unity.', 'cond-mat-0601677-4-14-11': 'We find in the leading order the following results for the boundary matrix ([REF]) [EQUATION] where the last equation implies that the coupling between the spin and charge densities is small with respect to the inverse conductance and vanishes in the framework of the diffusion approximation and the accuracy of the method.', 'cond-mat-0601677-4-15-0': 'From Eqs. ([REF]) and ([REF]), we find the following boundary conditions in the leading approximation with respect to the spin-orbit coupling and the inverse conductance: [EQUATION] where [MATH] is defined after Eq. ([REF]).', 'cond-mat-0601677-4-15-1': 'These boundary conditions can be generalized to describe any form of a hard-wall boundary.', 'cond-mat-0601677-4-15-2': 'To find such a general form of boundary conditions, one should do the following replacements: [MATH] and [MATH], where [MATH] is the unit vector normal to the segment of the boundary and [MATH] is the corresponding component of the spin density.', 'cond-mat-0601677-4-15-3': 'However, if the boundary is a smooth potential with the length-scale in the [MATH]-direction being [MATH], then boundary conditions would be different and strongly dependent on the ratios of [MATH] and [MATH] [CITATION].', 'cond-mat-0601677-4-15-4': 'Also, the boundary potential would lead to an additional boundary spin-orbit coupling, which would strongly affect the boundary conditions.', 'cond-mat-0601677-4-16-0': 'We note here that in the next-to-leading order with respect to the spin-orbit interaction, two new effects appear: Boundary spin-charge coupling and boundary spin relaxation (the former is parametrically larger than the latter due to an additional smallness of the spin density).', 'cond-mat-0601677-4-16-1': 'To this order boundary conditions take the form: [EQUATION] where [MATH] and [MATH] are constants [MATH].', 'cond-mat-0601677-4-16-2': 'We should note that the boundary spin-charge coupling term can be shown to be strongly dependent on the electronic spectrum and the physics far from the Fermi line and as such is non-universal and can not be calculated in the [MATH]-approximation used in this paper.', 'cond-mat-0601677-4-17-0': '# Solution of the diffusion equation', 'cond-mat-0601677-4-18-0': 'We now solve the diffusion equation for a two-dimensional gas of Rashba electrons occupying a half-space [MATH] and in the presence of an electric field parallel to the edge.', 'cond-mat-0601677-4-18-1': 'We consider two types of edges: a partially or totally transparent interface and an impenetrable boundary.', 'cond-mat-0601677-4-18-2': 'It is convenient to rewrite the diffusion equation ([REF], [REF]) in terms of the rescaled variables: [MATH] and [MATH], where [MATH] is the dimensionless distance and [MATH] is the uniform spin density due to the spin-charge coupling.', 'cond-mat-0601677-4-18-3': 'In these variables, the diffusion equation takes the following form [EQUATION]', 'cond-mat-0601677-4-18-4': 'Boundary conditions for a transparent boundary are [MATH] and [MATH] and for a hard-wall boundary [MATH] and [MATH], where [MATH] is a non-universal spectrum-dependent parameter.', 'cond-mat-0601677-4-18-5': 'We note that if the latter is zero, one has to take into account the next-to-leading order boundary spin relaxation terms, which should determine the observable spin density.', 'cond-mat-0601677-4-18-6': 'The solution of the differential equation ([REF], [REF]) is straightforward and we find for the transparent boundary (e.g., ballistic contacts) [EQUATION] and [EQUATION] and for the hard-wall boundary [EQUATION] and [EQUATION] where [MATH] is an eigenvalue of the diffusion equation ([REF], [REF]).', 'cond-mat-0601677-4-18-7': 'Since the corresponding differential operator is non-Hermitian, the eigenvalues are complex.', 'cond-mat-0601677-4-18-8': 'This leads to oscillations of the spin density near the boundary with the period [MATH].', 'cond-mat-0601677-4-18-9': 'We should point out that there definitely exists other types of small-lengthscale oscillations (with the period [MATH]), which are superimposed onto the regular behavior ([REF] - [REF]).', 'cond-mat-0601677-4-18-10': 'However, these Friedel oscillations are beyond the reach of the diffusion approximation.', 'cond-mat-0601677-4-18-11': 'From Eqs. ([REF] - [REF]), we see that the spin density rapidly decays as we move away from the boundary.', 'cond-mat-0601677-4-18-12': 'The corresponding lengthscale is [MATH].', 'cond-mat-0601677-4-19-0': '# Summary', 'cond-mat-0601677-4-20-0': 'In this paper, we have developed a general method of deriving boundary conditions for spin diffusion equation.', 'cond-mat-0601677-4-20-1': "The method involves a gradient expansion of the boundary Green's function, which takes into account the behavior of the electron wave-functions on ballistic lengthscales near the edges.", 'cond-mat-0601677-4-20-2': 'Using the proposed method, we have found a general form of boundary conditions in the Rashba model for two types of edges: a transparent interface and a hard wall.', 'cond-mat-0601677-4-21-0': 'We have found an exact solution of the diffusion equation satisfying these two types of boundary conditions.', 'cond-mat-0601677-4-21-1': 'The spin accumulation ([REF] - [REF]) is shown to oscillate and decay away from the boundary with the corresponding lengthscales being determined by the eigenvalues of the bulk diffusion equation.', 'cond-mat-0601677-4-21-2': 'However, we argue that even small changes in the boundary potential would lead to a different solution of the diffusion equation: The eigenvalues, which determine the density profile, are robust and universal, but the overall amplitude and phase shifts are boundary-specific.', 'cond-mat-0601677-4-21-3': 'Therefore, the spin Hall effect is generally a non-universal phenomenon, which depends on the structure of the sample edges.', 'cond-mat-0601677-4-22-0': 'The authors are grateful to L. Balents, H.-A.Engel, A.G. Malshukov, E.I. Rashba, and Ya.', 'cond-mat-0601677-4-22-1': 'Tserkovnyak for helpful discussions.', 'cond-mat-0601677-4-22-2': 'A.A.B. is supported by the NSF under grant DMR02-33773.', 'cond-mat-0601677-4-22-3': 'S.D.S. is supported by US-ONR, NSF, and LPS.'}

1607.08725

{'1607.08725-1-0-0': 'The vanilla attention-based neural machine translation has achieved promising performance because of its capability in leveraging varying-length source annotations.', '1607.08725-1-0-1': 'However, this model still suffers from failures in long sentence translation, for its incapability in capturing long-term dependencies.', '1607.08725-1-0-2': 'In this paper, we propose a novel recurrent neural machine translation (RNMT), which not only preserves the ability to model varying-length source annotations but also better captures long-term dependencies.', '1607.08725-1-0-3': 'Instead of the conventional attention mechanism, RNMT employs a recurrent neural network to extract the context vector, where the target-side previous hidden state serves as its initial state, and the source annotations serve as its inputs.', '1607.08725-1-0-4': 'We refer to this new component as contexter.', '1607.08725-1-0-5': 'As the encoder, contexter and decoder in our model are all derivable recurrent neural networks, our model can still be trained end-to-end on large-scale corpus via stochastic algorithms.', '1607.08725-1-0-6': 'Experiments on Chinese-English translation tasks demonstrate the superiority of our model to attention-based neural machine translation, especially on long sentences.', '1607.08725-1-0-7': 'Besides, further analysis of the contexter revels that our model can implicitly reflect the alignment to source sentence.', '1607.08725-1-1-0': '# Introduction', '1607.08725-1-2-0': 'Recently, the neural machine translation (NMT) enhanced with attention mechanism has achieved very promising results on multiple language pairs, including English-French [CITATION], English-German [CITATION] and Chinese-English [CITATION], etc.', '1607.08725-1-2-1': 'Typically, NMT models build upon an encoder-decoder framework, where the encoder reads and encodes a source language sentence into distributed vectors, from which the decoder generates its corresponding target language sentence word by word [CITATION].', '1607.08725-1-2-2': 'In order to fully leverage the varying-length source annotations according to target-side states, Bahdanau et al. [CITATION] introduce the attention mechanism, which makes current NMT beyond the early sequence-to-sequence learning [CITATION].', '1607.08725-1-3-0': 'However, current NMT still suffers from undesirable failures in long sentence translation.', '1607.08725-1-3-1': 'The main reason for this lies in that the attention mechanism only models a relative weight for each source annotation separately, but ignores the potential internal dependencies among these annotations, especially the long term dependencies.', '1607.08725-1-3-2': 'Consequently, some words, even clauses, are often mistakenly untranslated or over-translated, which is referred to as the coverage problem [CITATION].', '1607.08725-1-3-3': 'Borrowing from conventional SMT, Tu et al. [CITATION] propose to maintain a coverage vector as a guidance to deal with those untranslated and over-translated words and clauses.', '1607.08725-1-3-4': 'In spite of their success, their model still suffers from the inability in modeling dependencies inside the source annotations.', '1607.08725-1-4-0': 'In this paper, we propose a recurrent neural machine translation (RNMT) to solve the dependency problem.', '1607.08725-1-4-1': 'As recurrent neural network (RNN) can not only deal with varying-length inputs, but also capture long-term dependencies inside these inputs [CITATION], we replace the attention mechanism with a RNN, or more concretely, the GRU model [CITATION].', '1607.08725-1-4-2': 'As the GRU aims at dynamically generating a context vector for a target word being generated, we refer to this component as contexter in our model.', '1607.08725-1-4-3': 'Figure [REF] shows the overall architecture.', '1607.08725-1-4-4': 'Particularly, we treat the target-side previous hidden state [MATH] as the initial state [MATH] for the contexter, and feed all the hidden states produced by the encoder as its inputs.', '1607.08725-1-4-5': 'This enables the output of our contexter to vary according to the target-side translation history.', '1607.08725-1-4-6': 'We investigate two different methods for the output of our contexter: mean-pooling and last state.', '1607.08725-1-4-7': 'The former has been verified in various classification problems, while the latter has often been assumed to encode the whole input information.', '1607.08725-1-4-8': 'No matter which output, our model is still a unified neural network that can be trained end-to-end via stochastic algorithms.', '1607.08725-1-5-0': 'We compare the proposed model against the vanilla attention mechanism on Chinese-English translation tasks.', '1607.08725-1-5-1': 'The empirical results show that our model yields significantly better translation performance in terms of both BLEU and NIST scores, especially on long sentences (with a gain of up to 4.0 BLEU points) where long-term dependencies are heavily required.', '1607.08725-1-5-2': 'Besides, we further analyze the gates in GRU and find that our model can reflect the alignment to source sentence implicitly.', '1607.08725-1-6-0': 'Our major contribution lies in the following three aspects:', '1607.08725-1-7-0': '# Background', '1607.08725-1-8-0': 'In this section, we briefly review the attention-based NMT which directly estimates the conditional probability [MATH] given a collection of source and target sentence pairs [MATH].', '1607.08725-1-8-1': 'We explain the model with regards to the encoder, the decoder and the attention mechanism.', '1607.08725-1-8-2': 'The former two are shared in our model, and shown in Figure [REF] (a) and (c) respectively.', '1607.08725-1-8-3': 'The latter, however, is replaced with GRU in our model.', '1607.08725-1-8-4': 'We start from the encoder.', '1607.08725-1-9-0': 'Encoder.', '1607.08725-1-9-1': 'The encoder is a bidirectional RNN that composes of a forward and a backward RNN.', '1607.08725-1-9-2': 'The forward RNN reads the source sentence [MATH] from left to right while the backward RNN reads in the opposite direction (see the parallel arrows in Figure [REF] (a)): [EQUATION] where [MATH] is the embedding for source word [MATH], and [MATH] are the hidden states generated in two directions.', '1607.08725-1-9-3': 'Following previous work [CITATION], we set the encoding function [MATH] to a GRU.', '1607.08725-1-10-0': 'The source annotations [MATH] are obtained by concatenating the forward and backward hidden states: [EQUATION]', '1607.08725-1-10-1': 'In this way, each annotation [MATH] encodes information about the [MATH]-th word with respect to all the other surrounding words in the source sentence.', '1607.08725-1-11-0': 'Decoder.', '1607.08725-1-11-1': 'The decoder is a forward RNN, whose initial hidden state [MATH] is initialized according to the source-side last hidden state [MATH].', '1607.08725-1-11-2': 'Given the previous target word [MATH], the [MATH]-th hidden state of the decoder [MATH] and the context vector [MATH], the decoder calculates the conditional probability of the [MATH]-th target word as follows (see yellow lines in Figure [REF] (c)): [EQUATION] where [MATH] is a non-linear function, [MATH] is obtained by the attention mechanism and [MATH] is calculated as follows: [EQUATION]', '1607.08725-1-11-3': 'Here, [MATH] and [MATH].', '1607.08725-1-11-4': '[MATH] is, again, the GRU.', '1607.08725-1-11-5': 'The decoder thus utilizes Eq. ([REF]) and ([REF]) alternatively to generate each word in the target sentence until reaching a predefined end token.', '1607.08725-1-12-0': 'Attention Mechanism.', '1607.08725-1-12-1': 'The attention mechanism aims at extracting a context vector dynamically according to the previous decoder state.', '1607.08725-1-12-2': 'Formally, [EQUATION]', '1607.08725-1-12-3': 'The non-linear function [MATH] defines this process.', '1607.08725-1-12-4': 'Typically, it first computes a relevance weight [MATH] of the [MATH]-th annotation for the [MATH]-th target word via a feed forward neural network that takes as input [MATH] and the previous decoder state [MATH]: [EQUATION] where the relevance score [MATH] is computed as follows, [EQUATION]', '1607.08725-1-12-5': 'Intuitively, the weights [MATH] act as the word alignment, and quantify how much each source annotation contributes to the word prediction at each time step.', '1607.08725-1-12-6': 'The context vector [MATH] is thus calculated as the weighted summation of [MATH]: [EQUATION]', '1607.08725-1-12-7': 'More details could be found in [CITATION].', '1607.08725-1-13-0': 'Because of the ability in modeling source context, attention-based NMT goes beyond the sequence to sequence model.', '1607.08725-1-13-1': 'However, as we can see from Eq. ([REF]), the relevance score only models the relation between [MATH] and [MATH], which ignores the dependencies among [MATH] completely.', '1607.08725-1-13-2': 'As a result, attention-based NMT often has poor performance for long sentence translation.', '1607.08725-1-13-3': 'We elaborate how to solve this problem in next section.', '1607.08725-1-14-0': '# Recurrent Neural Machine Translation', '1607.08725-1-15-0': 'The overall architecture of our model is shown in Figure [REF], where the encoder and decoder is essentially the same as attention-based NMT.', '1607.08725-1-15-1': 'Instead of the attention mechanism, we propose a contexter (Figure [REF] (b)) as its alternate to extract the context vector.', '1607.08725-1-15-2': 'The mathematic form of our contexter is the same as Eq. ([REF]).', '1607.08725-1-16-0': 'We employ the RNN enhanced with GRU to this end because of its capacity in handling long-term dependencies in sequential data.', '1607.08725-1-16-1': 'The reason why we do not use the vanilla RNN and the Long Short-Term Memory is that the former often suffers from the vanishing and exploding gradient problem, while the latter is usually computational inefficient.', '1607.08725-1-17-0': 'Concretely, the [MATH]-th GRU unit in our contexter consists of a reset gate [MATH], a update gate [MATH] and a hidden state [MATH], [EQUATION] where, [EQUATION] [MATH] are weight matrices and [MATH] are bias terms.', '1607.08725-1-17-1': 'In this way, our model explicitly integrates the dependencies among source annotations into the hidden state [MATH].', '1607.08725-1-17-2': 'Intuitively, at each time step for target word prediction, our contexter rereads the source sentence together with current history decoding information, and then extracts important source words for the translation.', '1607.08725-1-17-3': 'This, to some extent, is consistent with human behavior.', '1607.08725-1-18-0': 'We explore two approaches to extract the context vector from hidden states of the contexter [MATH], namely mean-pooling and last state.', '1607.08725-1-18-1': 'The former uses the average of [MATH] as the context vector, which has been demonstrated in various classification tasks: [EQUATION]', '1607.08725-1-18-2': 'The latter uses the last contexter state, i.e. [MATH], which is often assumed to have encoded all the input information.', '1607.08725-1-18-3': 'We conduct experiments to check their effectiveness respectively.', '1607.08725-1-19-0': 'Model Training We train the encoder, the decoder and the contexter to maximize the conditional log-likelihood of the training data: [EQUATION] where [MATH] is the bilingual training corpus.', '1607.08725-1-19-1': 'As all the components in our model are derivable, we can optimize the model parameters [MATH] using standard stochastic algorithms.', '1607.08725-1-20-0': '# Experiments', '1607.08725-1-21-0': 'We conducted a series of experiments on the NIST Chinese-English translation tasks to evaluate the effectiveness of the proposed model.', '1607.08725-1-22-0': '## Setup', '1607.08725-1-23-0': 'Datasets and Evaluation Metrics Our training data is a combination of LDC2003E14, LDC2004T07, LDC2005T06, LDC2005T10 and LDC2004T08 (Hong Kong Hansards/Laws/News), which contain 2.9M sentence pairs, with 80.9M Chinese words and 86.4M English words respectively.', '1607.08725-1-23-1': 'We used the NIST 2005 (1082 sentences) dataset as our development set, and the NIST 2002 (878 sentences), 2003 (919 sentences), 2004 (1788 sentences), 2006 (1664 sentences), 2008 (1357 sentences) datasets as our test sets.', '1607.08725-1-23-2': 'We employed the case-insensitive BLEU-4 metric [CITATION] and the NIST score [CITATION] to evaluate translation quality.', '1607.08725-1-23-3': 'We performed paired bootstrap sampling [CITATION] for significance test using the script in Moses.', '1607.08725-1-24-0': 'Baselines and Settings We compared our model against two state-of-the-art SMT and NMT systems:', '1607.08725-1-25-0': 'For the Moses, we trained a 4-gram language model on the Xinhua section of the English Gigaword corpus (306M words) using the SRILM toolkit with modified Kneser-Ney smoothing.', '1607.08725-1-25-1': 'All other parameters were kept as the default settings.', '1607.08725-1-25-2': 'Besides, we used all source and target words in the training corpus for this system.', '1607.08725-1-26-0': 'For the GroundHog, we set the maximum length of training sentences to be 50 words, and preserved the most frequent 30K words as the source and target vocabulary respectively, which covers 98.9% and 99.2% of the text respectively.', '1607.08725-1-26-1': 'All other words were represented by a specific token "UNK" as the default implementation.', '1607.08725-1-26-2': 'Following Bahdanau et al. [CITATION], we set [MATH], [MATH].', '1607.08725-1-26-3': 'All other settings are the same as the default configuration (for RNNSearch-50).', '1607.08725-1-26-4': 'We used the Adadelta algorithm for optimization, with a batch size of [MATH].', '1607.08725-1-26-5': 'The model parameters were selected according to the maximum BLEU points on the development set.', '1607.08725-1-26-6': 'Besides, during decoding, we used the beam-search algorithm, and set the beam size to 10.', '1607.08725-1-27-0': 'For the RNMT, we initialized its parameters with the RNNSearch in GroundHog.', '1607.08725-1-27-1': 'The settings of our model are the same as that of GroundHog, except for the dimension of our contexter which is 1000 in our model, but 2000 in GroundHog.', '1607.08725-1-27-2': 'We implemented our RNMT based on the code of GroundHog.', '1607.08725-1-28-0': 'All NMT systems are trained on a GeForce GTX TITAN X.', '1607.08725-1-28-1': 'In one hour, the GroundHog system processes about 1087 batches, while our RNMT processes 300 batches.', '1607.08725-1-28-2': 'It takes about two weeks to train our model on the training data.', '1607.08725-1-29-0': '## Translation Results', '1607.08725-1-30-0': 'Our first experiment checks whether RNMT can obtain satisfactory performance on the test sets.', '1607.08725-1-30-1': 'Table [REF] summarizes the experiment results.', '1607.08725-1-30-2': 'As the strongest conventional translation system, Moses achieves very promising results in terms of both BLUE and NIST (especially it achieves the best results on NIST).', '1607.08725-1-30-3': 'This is because Moses not only preserves all the words in vocabulary, but also leverages a large monolingual corpus for language model training.', '1607.08725-1-30-4': 'In contrast, NMT systems only use a limited vocabulary due to the computation limitation.', '1607.08725-1-30-5': 'The performance of GroundHog can be regarded as comparable with Moses.', '1607.08725-1-30-6': 'Replacing the attention mechanism with our contexter, our model achieves significant improvements in terms of both BLEU and NIST.', '1607.08725-1-30-7': 'Specifically, our model outperforms the Moses 1.31 BLEU points, and the GroundHog 1.79 and 0.36 BLEU and NIST points respectively on the whole test set.', '1607.08725-1-30-8': 'As the only difference between our RNMT and the GroundHog is the contexter, these results demonstrate that the recurrent contexter surpasses the attention mechanism.', '1607.08725-1-31-0': 'It is interesting to observe that RNMT with last state yields better performance than that with mean-pooling, with a gain of 1.0 and 0.21 BLEU and NIST points respectively on the whole test set.', '1607.08725-1-31-1': 'The reason behind is not completely understood, but can be reflected, to some extent, through the visualization of the contexter in the next subsection.', '1607.08725-1-31-2': 'Basically, this result suggests that last state is more suitable for machine translation.', '1607.08725-1-32-0': 'We argue that our model is able to model long-term dependencies, which is superior to the GroundHog.', '1607.08725-1-32-1': 'To verify this point, we carried out two additional experiments.', '1607.08725-1-33-0': 'Our second experiment testifies whether RNMT has better translation quality on long sentences in the original NIST test sets.', '1607.08725-1-33-1': 'To this end, we divide our test sets into 6 disjoint groups according to the length of source sentences, each of which has 601, 1918, 1803, 1196, 633 and 455 sentences respectively.', '1607.08725-1-33-2': 'Figure [REF] illustrates the overall results.', '1607.08725-1-33-3': 'We found that as the length of source sentence excels a certain threshold (here from the 5 group to 6 group), the performance of NMT systems drops sharply, around 10 and 4 BLEU and NIST points respectively.', '1607.08725-1-33-4': 'This indicates that long sentence translation is a common yet serious challenge for NMT systems.', '1607.08725-1-33-5': 'However, compared with the GroundHog, our RNMT model behaves more robust to long sentences, with a gain of 3.69 and 1.37 BLEU and NIST points respectively on the final group.', '1607.08725-1-33-6': 'This further demonstrates that our model indeed deals better with long sentences than the attention-based NMT.', '1607.08725-1-34-0': 'Our third experiment verifies the findings in our second experiment by evaluating our model on a synthetic test data whose source sentence tends to be much longer.', '1607.08725-1-34-1': 'We observed that the NIST test sets are organized in different documents.', '1607.08725-1-34-2': 'Therefore, we concatenated the original two neighboring sentences in one document as a new sentence for evaluation.', '1607.08725-1-34-3': 'In this way, we have the same number of evaluation dataset, but the source and target sentences in the new dataset are much longer.', '1607.08725-1-34-4': 'The average sentence length for each dataset is shown in Table [REF].', '1607.08725-1-34-5': 'Notice that an important property of the new sentence is that it is semantic coherent, thus reasoning inside the sentence is required for correct translation.', '1607.08725-1-35-0': 'Table [REF] gives the translation results.', '1607.08725-1-35-1': 'We can observe that our RNMT especially with last state outperforms the GroudHog significantly, which gains about 3 and 1 BLEU and NIST points respectively.', '1607.08725-1-35-2': 'In some dataset, our model yields above 4 BLEU points.', '1607.08725-1-35-3': 'Besides, both of our model achieve improvement over the GroudHog no matter which evaluation metric.', '1607.08725-1-35-4': 'This result, without any doubt, demonstrates that our model is capable of dealing with long sentence translation.', '1607.08725-1-36-0': '## Contexter Visualization', '1607.08725-1-37-0': 'A common challenge for deep learning is to understand what has happened inside the model.', '1607.08725-1-37-1': 'Although our model achieves better performance, we are also curious about the internal mechanism of our contexter.', '1607.08725-1-37-2': 'To this end, we provided a method to visualize the contexter qualitatively.', '1607.08725-1-38-0': "Let's revisit the Eq. ([REF]).", '1607.08725-1-38-1': 'The GRU consists of two gates: reset gate [MATH] and update gate [MATH].', '1607.08725-1-38-2': 'Intuitively, the reset gate controls how much information flow from previous state into current state, while the update gate decides how much information should come from current input.', '1607.08725-1-38-3': 'That is, the reset gate can reflect the dependencies among the source annotation, while the update gate can qualify how much each source annotation contributes to the word prediction.', '1607.08725-1-38-4': 'Therefore, we define two metrics to visualize our contexter: [EQUATION] where [MATH] because of the [MATH] in Eq. ([REF]).', '1607.08725-1-39-0': 'The heatmaps of [MATH] and [MATH] in our RNMT with mean-pooling and last state for a same bilingual sentence pair are shown in Figure [REF].', '1607.08725-1-39-1': 'We find that the update gates differ from the reset gates significantly, which suggests that these two kinds of gates learn different aspects of the source annotations.', '1607.08725-1-39-2': 'With regards to the update gate, we find that it looks somewhat like the word alignments, with each block revealing a translation relation from the source to target.', '1607.08725-1-39-3': 'From this perspective, the last state has a better alignment intuition than the mean-pooling.', '1607.08725-1-40-0': 'With respect to the reset gate, we find that it prefers to be opposite to the update gates.', '1607.08725-1-40-1': 'For example, the leading diagonal of update gates is white, while that of reset gates is black.', '1607.08725-1-40-2': 'Besides, the blocks for aligned word pairs seem to be turn points, which tend to have relatively smaller reset values.', '1607.08725-1-40-3': 'This is desirable, since previous hidden states on these points are blocked, while the current hidden states are enabled to flow into the final context vector.', '1607.08725-1-41-0': 'All these indicate that the GRU in our contexter indeed learns something related to translation.', '1607.08725-1-41-1': 'Particularly, the contexter implicitly reflects the alignment to source sentence via the reset and update gates.', '1607.08725-1-41-2': 'From this perspective, we argue that the ability of automatic word alignment is important for machine translation.', '1607.08725-1-42-0': '# Related Work', '1607.08725-1-43-0': 'Our work brings together two tightly related line of research: recurrent neural network and neural machine translation.', '1607.08725-1-43-1': 'Although previous work have already testified the effectiveness of the former for the encoder and decoder in the latter, the introduction for the contexter, to the best of our knowledge, has never been investigated before.', '1607.08725-1-44-0': 'Recurrent Neural Network takes a variable-length sequence of tokens as its input, and processes each token recursively while maintaining an internal hidden state.', '1607.08725-1-44-1': 'Typically, the hidden state is able to encode appealing context information for the history tokens.', '1607.08725-1-44-2': 'However, conventional RNNs often suffer from the vanishing and the exploding gradient problems during training [CITATION], which results in difficulties in capturing long-term dependencies.', '1607.08725-1-44-3': 'To solve this problem, Hochreiter and Schmidhuber Hochreiter:1997:LSM:1246443.1246450 propose the well-known Long Short-Term Memory (LSTM) which explicitly brings in a memory cell to memorize long-term dependencies together with input, forget and output gates for information controlling.', '1607.08725-1-44-4': 'Considering the inefficiency of LSTM in physical computation, Chung et al. journals/corr/ChungGCB14 further simplify this architecture, and introduce the gated recurrent unit (GRU) where only a reset and a update gate remains.', '1607.08725-1-44-5': 'Both LSTM and GRU have been demonstrated efficient in modeling long-term dependencies through several empirical evaluations [CITATION].', '1607.08725-1-45-0': 'As our goal is to model the long-term dependencies inside source annotations, we borrow the spirit of RNNs and employ the GRU as our contexter in our model.', '1607.08725-1-46-0': 'Neural Machine Translation, unlike conventional statistical machine translation, transforms a source sentence directly into its corresponding target reference via a single, large yet trainable neural network.', '1607.08725-1-46-1': 'In an early stage, the NMT model follows a sequence-to-sequence manner, where a multilayered LSTM encoder encodes the source sentence into a fixed vector, from which a multilayered LSTM decoder generates its target reference word by word until reaching a special end token [CITATION].', '1607.08725-1-46-2': 'A major limitation of this model, however, is that it forces the source-side information within a single vector.', '1607.08725-1-46-3': 'To settle this problem, Bahdanau et al. [CITATION] develop the attention mechanism, which is able to generate target word according to previous target-side hidden state and all the source annotations.', '1607.08725-1-47-0': 'Our work closely relates to the attention-based NMT in the following two aspects: 1) our work borrows the same framework from the attention-based NMT, and 2) our work is also inspired by the basic motivation of the attention mechanism.', '1607.08725-1-47-1': 'Despite of its success, current attention-based NMT model still suffers from failures in long sentence translation, e.g. the under-translation problem.', '1607.08725-1-47-2': 'Tu et al. DBLP:journals/corr/TuLLLL16 argue that it is the wrong attention that results in these failures, and propose to maintain a coverage vector that encourages NMT to pay more attention to untranslated words and less attention to translated word.', '1607.08725-1-47-3': 'In contrast, Cohn et al. DBLP:journals/corr/CohnHVYDH16 and Feng et al. DBLP:journals/corr/FengLLZ16 explore the "fertility" for the attention mechanism and constrain it with the alignment scores and previous attentional context respectively.', '1607.08725-1-47-4': 'Cheng et al. DBLP:journals/corr/ChengSHHWSL15 propose to encourage the agreement on word alignment matrices produced by the source-to-target and target-to-source bidirectional translation models.', '1607.08725-1-47-5': 'Nevertheless, what they mainly focus on is the word alignment, rather than the drawback of attention in modeling dependencies among source annotations.', '1607.08725-1-47-6': 'We, instead, make up this drawback by replacing the attention mechanism with a more powerful neural network, i.e. GRU.', '1607.08725-1-48-0': '# Conclusion and Future Work', '1607.08725-1-49-0': 'In this paper, we have presented a recurrent neural machine translation model that replaces the attention mechanism with a gated recurrent unit (GRU).', '1607.08725-1-49-1': 'In this way, our model is able to not only leverage the varying-length source annotations, but also handle long-term dependencies among these annotations.', '1607.08725-1-49-2': 'Experiment results demonstrate the ability of our model in translating long sentences, and further analysis of the GRU reveals that our model implicitly captures the word alignment.', '1607.08725-1-50-0': 'In the future, we would like to explore two interesting directions: 1) Despite of the effectiveness of our model, the GRU still suffers from computation inefficiency.', '1607.08725-1-50-1': 'We will try to optimize its speed.', '1607.08725-1-50-2': '2) We will explore other neural architectures as the contexter, such as deep neural network, convolutional neural network, bidirectional recurrent neural networks, etc.'}

{'1607.08725-2-0-0': 'The vanilla sequence-to-sequence learning (seq2seq) reads and encodes a source sequence into a fixed-length vector only once, suffering from its insufficiency in modeling structural correspondence between the source and target sequence.', '1607.08725-2-0-1': 'Instead of handling this insufficiency with a linearly weighted attention mechanism, in this paper, we propose to use a recurrent neural network (RNN) as an alternative (Cseq2seq-I).', '1607.08725-2-0-2': 'During decoding, Cseq2seq-I cyclically feeds the previous decoding state back to the encoder as the initial state of the RNN, and reencodes source representations to produce context vectors.', '1607.08725-2-0-3': 'We surprisingly find that the introduced RNN succeeds in dynamically detecting translation-related source tokens according to the partial target sequence.', '1607.08725-2-0-4': 'Based on this finding, we further hypothesize that the partial target sequence can act as a feedback to improve the understanding of the source sequence.', '1607.08725-2-0-5': 'To test this hypothesis, we propose cyclic sequence-to-sequence learning (Cseq2seq-II) which differs from the seq2seq only in the reintroduction of previous decoding state into the same encoder.', '1607.08725-2-0-6': 'We further perform parameter sharing on Cseq2seq-II to reduce parameter redundancy and enhance regularization.', '1607.08725-2-0-7': 'In particular, we share the weights of the encoder and decoder, and two target-side word embeddings, making Cseq2seq-II equivalent to a single conditional RNN model, with 31% parameters pruned but even better performance.', '1607.08725-2-0-8': 'Cseq2seq-II not only preserves the simplicity of seq2seq but also yields comparable and promising results on machine translation tasks.', '1607.08725-2-0-9': 'Experiments on Chinese-English and English-German translation show that Cseq2seq achieves significant and consistent improvements over seq2seq and is as competitive as the attention-based seq2seq model.', '1607.08725-2-1-0': '# Introduction', '1607.08725-2-2-0': 'Due to its unified model architecture and efficient end-to-end training schema, sequence-to-sequence learning (seq2seq) has attracted increasing attention from different communities and achieved very promising performance on various difficult problems, especially on machine translation tasks in which neural machine translation (NMT) has achieved state-of-the-art results on multiple languages [CITATION].', '1607.08725-2-2-1': 'Current NMT models originate from the vanilla seq2seq, where an encoder encodes a source sequence into a fixed-length vector, from which a decoder generates its corresponding target sequence token by token [CITATION].', '1607.08725-2-3-0': 'Unfortunately, this fixed-length vector is not sufficient to model the structural correspondence between the source and target sequence, resulting in unsatisfactory performance [CITATION].', '1607.08725-2-3-1': 'To address this issue, Bahdanau et al. [CITATION] introduce an attention mechanism (see Figure [REF]) which, as a bridge, makes the encoder and the decoder couple more tightly.', '1607.08725-2-3-2': 'In spite of its success, the underlying attention network is a weighted sum of encoded source representations, which is relatively simple and shallow.', '1607.08725-2-4-0': 'In this paper, we propose to use a recurrent neural network (RNN) as an alternative to the attention network.', '1607.08725-2-4-1': 'The reasons are twofold: 1) RNN is able to handle variable-length inputs naturally [CITATION] and 2) RNN, specifically the GRU model [CITATION], is capable of capturing complex and nonlinear dependencies inside its inputs.', '1607.08725-2-4-2': 'As shown in Figure [REF] which we refer to as Cseq2seq-I, we treat the target-side previous decoding state [MATH] as the initial state [MATH] of the RNN, and feed all encoded source representations as its inputs.', '1607.08725-2-4-3': 'This enables the output of the RNN to change in accordance with the target-side translation history.', '1607.08725-2-4-4': 'Surprisingly, in practice we find that this RNN succeeds in dynamically detecting translation-related source tokens at each decoding step.', '1607.08725-2-5-0': 'Motivated by the success of Cseq2seq-I, we hypothesize that the partial target sequence can act as a feedback to improve the understanding of the source sequence, based on which we propose cyclic sequence-to-sequence learning (Cseq2seq-II, Figure [REF]).', '1607.08725-2-5-1': 'Cseq2seq-II extends the vanilla seq2seq by merely reintroducing the previous decoding state into the encoder during decoding.', '1607.08725-2-5-2': 'In this way, the information from the decoder state is able to update or amend the understanding of the source sequence so that information flow between the encoder and the decoder becomes two-way.', '1607.08725-2-5-3': 'Thus, Cseq2seq-II can fully leverage what has been translated before and then guide the encoding procedure to focus more on those untranslated source tokens.', '1607.08725-2-5-4': 'This mixed understanding and translating behavior would benefit our model especially when the sequence is very long (e.g., a long sentence or a document).', '1607.08725-2-6-0': 'Partially inspired by the recently proposed zero-shot translation [CITATION] where different languages use the same encoder and decoder, for Cseq2seq-II, we further share parameters between the encoder and decoder as well as between two sorts of target-side word embeddings.', '1607.08725-2-6-1': 'This significantly prunes out 31% parameters with even better performance, making our Cseq2seq-II equivalent to a single conditional RNN model.', '1607.08725-2-6-2': 'Cseq2seq-II is not only as simple as seq2seq in terms of architecture and training, but also efficient in inducing target-sequence-related source semantics.', '1607.08725-2-6-3': 'Even without the attention mechanism, Cseq2seq-II is still capable of capturing what should be translated in the next step as demonstrated in our experiments.', '1607.08725-2-7-0': 'We conduct experiments on machine translation tasks over a variety of language pairs, including NIST Chinese-English and WMT English-German.', '1607.08725-2-7-1': 'The experimental results show that Cseq2seq achieves significant improvements over the vanilla seq2seq and obtains competitive results against the popular attention-based NMT.', '1607.08725-2-8-0': '# Background: Attention-based Seq2seq', '1607.08725-2-9-0': 'This section briefly reviews the attention-based seq2seq which directly models the conditional translation probability [MATH] of a target sequence [MATH] given its corresponding source sequence [MATH] as follows: [EQUATION] where [MATH] are tokens in the source and target sequence respectively, and [MATH] is a partial translation.', '1607.08725-2-9-1': 'To predict the [MATH]-th target token [MATH], attention-based seq2seq employs a non-linear neural network: [EQUATION] where [MATH] is the weight matrix for target token prediction ([MATH] is the size of target vocabulary), [MATH] is the embedding of previous generated target token [MATH], [MATH] is the [MATH]-th target-side hidden state (see Figure [REF]), [MATH] is the translation-guided context vector and [MATH] refers to a multilayer perceptron that projects the inputs into a [MATH]-dimensional semantic space.', '1607.08725-2-9-2': 'In this work, we treat [MATH] as another type of target-side word embedding in addition to [MATH].', '1607.08725-2-10-0': 'Attention-based seq2seq uses a recurrent neural decoder to perform the above mentioned prediction: [EQUATION]', '1607.08725-2-10-1': 'The vanilla attention-based seq2seq directly uses a GRU to model [MATH].', '1607.08725-2-10-2': 'However, exploiting the information of [MATH] before reading from the representation of source sequence should be beneficial, which has already supported in the latest dl4mt system.', '1607.08725-2-10-3': 'Therefore, we design [MATH] in two hierarchies: [EQUATION]', '1607.08725-2-10-4': 'The decoder thus utilizes Eq. ([REF]) and ([REF]) alternatively to generate each token in the target sequence until reaching a predefined end token.', '1607.08725-2-11-0': 'The context vector [MATH] is generated by the attention mechanism, a weighted sum of the encoded source representations: [EQUATION] [MATH] is the attention weight.', '1607.08725-2-11-1': 'Intuitively, it measures how well the output at position [MATH] matches the inputs around position [MATH].', '1607.08725-2-11-2': '[MATH] is the encoded source representation at position [MATH], which is calculated via a bidirectional recurrent encoder: [EQUATION] where [MATH] is the embedding for source token [MATH], and [MATH] are the hidden states generated in two directions.', '1607.08725-2-11-3': 'Following Bahdanau et al. [CITATION], we set the encoding function [MATH] to a GRU.', '1607.08725-2-12-0': '# The Model', '1607.08725-2-13-0': 'In this section, we elaborate Cseq2seq-I and Cseq2seq-II.', '1607.08725-2-13-1': 'Both models use the same decoder as the attention-based seq2seq does.', '1607.08725-2-13-2': 'Therefore we focus on their differences in generating context vectors.', '1607.08725-2-14-0': '## Cseq2seq-I', '1607.08725-2-15-0': 'As we can see from Eq. ([REF]), the network for estimating the attention weights is relatively shallow, which may be inappropriate for dealing with complex nonlinear dependencies during decoding.', '1607.08725-2-15-1': 'We, instead, propose to use a RNN as the alternate to obtain context vectors.', '1607.08725-2-15-2': 'Particularly, we employ GRU as our recurrent unit (see Figure [REF]): [EQUATION]', '1607.08725-2-15-3': 'Notice [MATH].', '1607.08725-2-15-4': 'Intuitively, the decoding state [MATH] denotes the partial translation, which prompts the RNN to reread the source sequence so as to put more emphasis on translation-relevant source tokens.', '1607.08725-2-15-5': 'This new information further guides the generation of the next target token.', '1607.08725-2-16-0': 'Comparing against the attention mechanism, the RNN uses deeper layers to encode two different sets of complex nonlinear dependencies: 1) the dependencies among encoded source representations [MATH], and 2) the dependencies between target translation history [MATH] and source representations [MATH].', '1607.08725-2-16-1': 'All these dependencies are modeled with various nonlinear neural nets inside GRU, enabling our model to capture structural correspondence in-between the source and target sequences, which is especially beneficial for translation tasks.', '1607.08725-2-17-0': 'Context Vector We explore two approaches to compute the context vector from the hidden states [MATH], namely mean-pooling and last-state.', '1607.08725-2-17-1': 'The former uses the average of [MATH] as the context vector, which has proven effective in various classification tasks: [MATH].', '1607.08725-2-17-2': 'The latter uses the last hidden state, which is often assumed to have encoded all input information: [MATH].', '1607.08725-2-17-3': 'We conduct experiments to evaluate their effectiveness.', '1607.08725-2-18-0': '## Cseq2seq-II', '1607.08725-2-19-0': 'The success of Cseq2seq-I inspires us that the partial target sequence is able to prompt the RNN to detect translation-relevant source tokens so as to improve the understanding of the source sequence.', '1607.08725-2-19-1': 'We further propose Cseq2seq-II.', '1607.08725-2-19-2': 'Notice that in the vanilla seq2seq, the context vector [MATH] is always a fixed-length vector: [EQUATION]', '1607.08725-2-19-3': 'Therefore, [MATH] has no relation with its generated partial translation.', '1607.08725-2-19-4': 'This is undesirable since compressing a source sequence into a single vector with a fixed dimensionality is insufficient to capture the varying translation context.', '1607.08725-2-19-5': 'Cseq2seq-II solves this problem by regarding the previous decoding hidden state [MATH] as its translation context and using this state to initialize the same encoder to produce a translation-guided context vector: [EQUATION] where [MATH] denotes a one-layer feed-forward neural network.', '1607.08725-2-19-6': 'This cyclic encoder is initialized with the translation-aware decoder state.', '1607.08725-2-19-7': 'As a result, the encoder knows the current partial translation and what has been translated so far such that it can learn what should be translated next through controlling different gates to pay more attention to specific source tokens.', '1607.08725-2-20-0': 'We empirically find that the initialization of decoder state [MATH] in Cseq2seq-II is very important.', '1607.08725-2-20-1': 'Following Sutskever et al. [CITATION], we use the same encoder but in a reverse direction to build [MATH]: [EQUATION]', '1607.08725-2-20-2': 'In this way, the model introduces more short-term dependencies in the data that make the optimization problem much easier [CITATION].', '1607.08725-2-20-3': 'Notice that the [MATH] in Eq. ([REF]) and Eq. ([REF]) are the same encoder using the same parameters but operate in different directions.', '1607.08725-2-21-0': '# Parameter Sharing for Cseq2seq-II', '1607.08725-2-22-0': 'Deep neural models often suffer from over-parameterization, requiring large storage for a huge number of parameters [CITATION].', '1607.08725-2-22-1': 'In this paper, we explore two types of parameter sharing for Cseq2seq-II:', '1607.08725-2-23-0': 'This can bring the following two advantages: 1) sharing model parameters naturally reduces the parameter redundancy, and accordingly eases the requirement for physical memory; 2) forcing the parameters of different component to be the same encourages special model regularization, e.g. SWord enforces the NMT to not only capture syntactic and semantic properties of words but also utilize these properties to rank the candidate words for generating next word.', '1607.08725-2-23-1': 'After sharing these parameters, there are only a source word embedding matrix [MATH], a target word embedding matrix [MATH], two feed-forward neural nets ([MATH] in Eq. ([REF]) and ([REF])) and a conditional GRU [MATH].', '1607.08725-2-23-2': 'All together, our Cseq2seq-II is equivalent to use a single fully conditional RNN to perform the sequence-to-sequence learning tasks.', '1607.08725-2-24-0': '# Experiments', '1607.08725-2-25-0': 'We conducted a series of experiments on Chinese-English (Zh-En) and English-German (En-De) translation task to examine the effectiveness of the proposed model.', '1607.08725-2-26-0': '## Setup', '1607.08725-2-27-0': 'Datasets and Evaluation For Zh-En, we collected 1.25M sentence pairs from LDC corpora as our training data, which contains 27.9M Chinese words and 34.5M English words.', '1607.08725-2-27-1': 'We used the NIST 2005 (1082 sentences) dataset as our development set, and the NIST 2002 (878 sentences), 2003 (919 sentences), 2004 (1788 sentences), 2006 (1664 sentences), 2008 (1357 sentences) datasets as our test sets.', '1607.08725-2-28-0': 'For En-De, we used the same WMT14 training data consisting of 4.5M sentence pairs with 116M English words and 110M German words [CITATION].', '1607.08725-2-28-1': 'We used newstest2013 (3000 sentences) as our development set, and the newstest2014 (2737 sentences) and newstest2017 (3004 sentences) as the test sets.', '1607.08725-2-29-0': 'We measured the translation quality using case-insensitive (Zh-En) and case-sensitive (En-De) BLEU-4 metric [CITATION], and performed paired bootstrap sampling [CITATION] for significance test.', '1607.08725-2-30-0': 'Baselines and Settings We compared our model against three popular SMT and NMT systems:', '1607.08725-2-31-0': 'Training Settings We set the vocabulary size to 30K (Zh-En) and 40K (En-De).', '1607.08725-2-31-1': 'Byte pair encoding compression algorithm is used for En-De to handle the rich morphology issue [CITATION].', '1607.08725-2-31-2': "Words that don't appear in the vocabulary were mapped to a special token UNK.", '1607.08725-2-31-3': 'We trained all NMT models with sentences of length up to 50 (Zh-En) and 80 (En-De) words in the training data, and set [MATH], [MATH].', '1607.08725-2-31-4': 'We employed the Adadelta algorithm for optimization, with a batch size of [MATH] and gradient norm as [MATH].', '1607.08725-2-31-5': 'The initial learning rate was set to 1.0, and halved after each epoch.', '1607.08725-2-31-6': 'During decoding, we used the beam-search algorithm with a beam size of 10.', '1607.08725-2-32-0': 'For all Cseq2seq models, we trained them from scratch, i.e. randomly initialized its parameters as in other NMT models.', '1607.08725-2-32-1': 'We implemented Cseq2seq based on the code of dl4mt.', '1607.08725-2-33-0': 'Complexity Analysis Suppose the representation of a source sentence is [MATH]-dimensional with a length of [MATH].', '1607.08725-2-33-1': 'At each decoding step, both RNNSearch and Cseq2seq have the same computational complexity of [MATH].', '1607.08725-2-33-2': 'The difference of them lies at the required number of sequential operation, where Cseq2seq needs [MATH] operations compared with [MATH] for RNNSearch.', '1607.08725-2-33-3': 'We further run all these models with Theano on GeForce GTX 1080 to check the practical effects.', '1607.08725-2-33-4': 'Within one hour, RNNSearch processes about 2800 batches, while our Cseq2seq-I and Cseq2seq-II process about 800 and 1000 batches respectively.', '1607.08725-2-34-0': '## Results on Chinese-English Translation', '1607.08725-2-35-0': 'Table [REF] summarizes the experiment results in terms of BLEU score.', '1607.08725-2-35-1': 'We find that the all NMT models achieve very promising results except the vanilla Seq2seq. This is mainly because that the vanilla Seq2seq model heavily relies on multi-layered deep encoders and decoders which are beyond our research scope.', '1607.08725-2-35-2': 'Our implemented RNNSearch model outperforms the Moses up to 3 points in total.', '1607.08725-2-35-3': 'Replacing the attention mechanism with GRU-based RNN model, Cseq2seq-I obtains another improvements of 1.94 points with the last-state strategy, and yields the best results across all test sets.', '1607.08725-2-35-4': 'Even without the attention mechanism, all Cseq2seq-II models outperform both RNNSearch and Moses system.', '1607.08725-2-35-5': 'All these improvements are significant.', '1607.08725-2-36-0': 'Last-state vs. Mean-pooling for Cseq2seq-I.', '1607.08725-2-36-1': 'It is interesting to observe that Cseq2seq-I with the last-state strategy yields significantly better performance than that with the strategy of mean-pooling, with a gain of 2.62 BLEU points in total.', '1607.08725-2-36-2': 'The reason behind is not completely clear, but can be revealed, to some extent, through the visualization of the GRU gates, where the mean-pooling operation may destroy the learned dependencies inside source sequences and in-between source and target sequences.', '1607.08725-2-36-3': 'This suggests that the last-state strategy for estimating context vectors is more suitable for Cseq2seq-I.', '1607.08725-2-37-0': 'Whether can the partial target sequence act as a feedback to improve the understanding of the source sequence?', '1607.08725-2-37-1': 'Obviously, Cseq2seq-II obtains very competitive translation results against the RNNSearch, and significantly outperforms the Seq2seq by 18.15 BLEU points.', '1607.08725-2-37-2': 'Based on the fact that Cseq2seq-II only differs from Seq2seq in the feedback of the partial target sequence, we argue that the answer should be yes.', '1607.08725-2-37-3': 'As can be seen in the gates analysis, the partial target sequence guides the encoder to focus more on the translation-relevant source tokens.', '1607.08725-2-37-4': 'During decoding, this feedback help to improve the understanding of the source sentence toward adequate translation.', '1607.08725-2-38-0': 'Effects of Parameter Sharing for Cseq2seq-II.', '1607.08725-2-38-1': 'To achieve similar results, RNNSearch uses 89.67M parameters, and Cseq2seq-I uses 90.05M parameters, while our Cseq2seq-II uses only 75.19M parameters (almost the same as that of Seq2seq 74.19M), with around 15M parameters pruned naturally.', '1607.08725-2-38-2': 'This illustrates the ability of Cseq2seq-II in using less parameters to model training data.', '1607.08725-2-39-0': 'We then performed parameter sharing to verify whether there are parameters redundant in Cseq2seq-II.', '1607.08725-2-39-1': 'Firstly, we shared GRU parameters, which pruned 4.87M parameters.', '1607.08725-2-39-2': 'As shown in Table [REF], "Cseq2seq-II + SGRU" achieves 34.85 BLEU points in total, with a slight gain of 0.24 BLEU points.', '1607.08725-2-39-3': 'This suggests that different languages can indeed share their recurrent parameters, not only across the encoder or decoder [CITATION], but also between the encoder and decoder.', '1607.08725-2-39-4': 'Secondly, we shared word embeddings, which pruned 18.6M parameters.', '1607.08725-2-39-5': 'Under this setting, "Cseq2seq-II + SWord" achieves 34.89 BLEU points, a gain of 0.28 BLEU points over Cseq2seq-II, indicating the two target-side word embeddings indeed share very similar syntactic properties and can be typed together.', '1607.08725-2-39-6': 'Finally, we combined the above two approaches, leading to a total of 23.47M parameters being pruned.', '1607.08725-2-39-7': 'With only 51.72M parameters, "Cseq2seq-II + SGRU + SWord" achieves 34.89 BLEU points, yielding 0.28 BLEU points improvement.', '1607.08725-2-39-8': 'In other words, we pruned 31% parameters in Cseq2seq-II with no performance loss, which will benefit applications in memory-limited devices.', '1607.08725-2-40-0': 'Effects of Cseq2seq on Long Sequences.', '1607.08725-2-40-1': 'Since RNN is able to model complex dependencies and is well known to encode long sequences, we carried out another group of experiments.', '1607.08725-2-41-0': 'Our second set of experiments testify whether Cseq2seq has better translation quality on long sentences in the original test sets.', '1607.08725-2-41-1': 'To this end, we divide our test sets into 6 disjoint groups according to the length of source sentences, each of which has 680, 1923, 1839, 1189, 597 and 378 sentences respectively.', '1607.08725-2-41-2': 'Figure [REF] illustrates the overall results.', '1607.08725-2-41-3': 'We find that as the length of source sentence exceeds a certain threshold (here over 50), the performance of NMT systems drops sharply, around 8 BLEU points.', '1607.08725-2-41-4': 'This indicates that long sentence translation is still a serious challenge for attention-based NMT systems, which resonates with findings of Tu et al. [CITATION] and Bentivogli et al. [CITATION].', '1607.08725-2-41-5': 'However, compared with RNNSearch, Cseq2seq behaves more robust on translating long sentences, and all Cseq2seq models outperform RNNSearch on the longest sentences.', '1607.08725-2-41-6': 'Another notable observation is that Cseq2seq-I with the mean-pooling strategy generates worse results than the RNNSearch, but it outperforms the RNNSearch model on the longest sentence groups, a gain of 1.64 BLEU points.', '1607.08725-2-41-7': 'This further demonstrates that our model deals better with long sentences than the attention-based NMT.', '1607.08725-2-42-0': '## Results on English-German Translation', '1607.08725-2-43-0': 'Table [REF] shows the translation results.', '1607.08725-2-43-1': 'We report both tokenized and detokenized BLEU scores, where the latter computes the evaluation metric using an in-built tokenizer and is consistent with WMT official BLEU results.', '1607.08725-2-43-2': 'In particular, the winning system in WMT14 [CITATION], a phrase-based system enhanced with a strong language model trained on huge monolingual text, yields a detokenized BLEU score of 20.70, which is slightly lower than our RNNSearch baseline (20.78).', '1607.08725-2-43-3': 'After being equipped with our proposed recurrent mechanism, both Cseq2seq-I and Cseq2seq-II achieve significant improvements over the RNNSearch.', '1607.08725-2-43-4': 'Especially, Cseq2seq-II (LS) produces a tokenized BLEU score of 23.05, outperforming the RNNSearch with a substantial margin of 2.18 BLEU points.', '1607.08725-2-43-5': 'In addition, we have observed a similar phenomenon as in Chinese-English translation when model parameters are shared.', '1607.08725-2-43-6': 'This can be shown in the "Cseq2seq-II + SWord" which achieves a tokenized BLEU score of 22.54 with a gain of 0.78 BLEU points over the original Cseq2seq-II.', '1607.08725-2-43-7': 'And this can be also observed on WMT17 test set.', '1607.08725-2-43-8': 'All these indicate that 1) reducing the parameter redundancy benefits the translation performance and 2) our model can generalize to different language pairs with different-scale training corpus.', '1607.08725-2-44-0': '## GRU Gate Analysis', '1607.08725-2-45-0': 'We are curious about the working mechanism inside Cseq2seq. To this end, we introduce a method to analyze the cyclic GRU RNN quantitatively.', '1607.08725-2-46-0': 'The GRU consists of two gates: reset gate [MATH] and update gate [MATH].', '1607.08725-2-46-1': 'Intuitively, the reset gate controls the amount of information flowing from a previous state into a current state, while the update gate determines how much information should come from an input.', '1607.08725-2-46-2': 'That is, the reset gate can reflect the dependencies among source token representations, while the update gate can qualify how much each source token representation contributes to the target token prediction.', '1607.08725-2-46-3': 'Therefore, we define two metrics to analyze our cyclic RNN: [MATH], where [MATH] due to the constraint of [MATH] in the gate function.', '1607.08725-2-47-0': 'We visualize the heatmaps of [MATH] and [MATH] in Cseq2seq-I with mean-pooling and last-state for the same bilingual sentence pair in Figure [REF].', '1607.08725-2-47-1': 'We find that the update gates differ from the reset gates significantly, which suggests that these two gates learn different aspects of a sentence pair.', '1607.08725-2-47-2': 'In the update gate heatmap, it is very interesting to observe that the update gate can somewhat capture word alignments, with each pixel revealing a translation relation from the source to target language.', '1607.08725-2-47-3': 'In this perspective, the last-state strategy has a better capability of learning alignments than the mean-pooling strategy.', '1607.08725-2-47-4': 'Further quantitative analysis of the update gate on sufficiently many sentence pairs show that this gate is able to learn such translation attention.', '1607.08725-2-47-5': 'As for the reset gate, we find that it prefers to be opposite to the update gate.', '1607.08725-2-47-6': 'For example, the leading diagonals of the update gate heatmaps are white (value close to 1), while those of the reset gate heatmaps are black (value close 0).', '1607.08725-2-47-7': 'Additionally, the pixels for aligned word pairs seem to be turn points, which tend to have relatively small reset values.', '1607.08725-2-47-8': 'This is desirable, since previous hidden states on these points are blocked, while current hidden states are enabled to flow into final context vectors.', '1607.08725-2-48-0': 'Secondly, we treated the update gate as word alignments, and evaluated its quality in terms of alignment error rate (AER) following Tu et al. [CITATION].', '1607.08725-2-48-1': 'We show the evaluation results in Table [REF].', '1607.08725-2-48-2': 'Consistent with the translation results, Cseq2seq-I with last-state strategy achieves the best alignment score.', '1607.08725-2-48-3': 'Notice that Cseq2seq does not aim at inducing high-quality word alignments.', '1607.08725-2-48-4': 'Instead, these AER results yielded by Cseq2seq demonstrate that the GRU gates inside Cseq2seq indeed learns what should be translated during translation.', '1607.08725-2-49-0': '# Related Work', '1607.08725-2-50-0': 'We focus on the advancements of sequence-to-sequence learning in NMT.', '1607.08725-2-50-1': 'Sutskever et al. DBLP:journals/corr/SutskeverVL14 and Cho et al. cho-EtAl:2014:EMNLP2014 propose the vanilla seq2seq model that only includes an encoder and a decoder.', '1607.08725-2-50-2': 'Soon, Bahdanau et al. DBLP:journals/corr/BahdanauCB14 find that the fixed-length vector encoded in seq2seq is a bottleneck in improving its performance, and propose the attention-based seq2seq to automatically (soft-)search for parts of a source sequence that are relevant to predicting a target token.', '1607.08725-2-50-3': 'Luong et al. luong-pham-manning:2015:EMNLP further propose several different architectures, namely global and local model, for the attention mechanism so that the NMT can focus on more relevant parts on the source side during decoding.', '1607.08725-2-50-4': 'Different from these models, Cseq2seq-I uses a RNN as an alternative to the attention mechanism to model complex and highly nonlinear dependencies inside source sequences and in-between source and target sequences.', '1607.08725-2-50-5': 'Additionally, Cseq2seq-II significantly extends the seq2seq by dynamically rereading a source sentence at each decoding step.', '1607.08725-2-50-6': 'This cyclic encoding schema is very novel and efficient.', '1607.08725-2-51-0': 'Cseq2seq is also related with memory networks [CITATION] where the source sequence can be regarded as the read-only memory, and Cseq2seq uses a recurrent encoder to perform the reading operation (no writing operation).', '1607.08725-2-51-1': 'This differs significantly from the recent work of Wang et al. [CITATION] and Meng et al. [CITATION].', '1607.08725-2-51-2': 'The former leverages an additional fixed-length memory to enhance the capacity of the decoder alone, while the latter treats the encoded source-side representations as an interactive memory.', '1607.08725-2-51-3': 'Both models perform both reading and writing operations, which make them more complex in training and design.', '1607.08725-2-51-4': 'From the view of memory networks, using recurrent models as the reading operation, to the best of our knowledge, has never been investigated before.', '1607.08725-2-52-0': 'The idea of sharing the parameters of RNNs and word embeddings is partially motivated by the work of zero-shot translation [CITATION].', '1607.08725-2-52-1': 'They find that different language pairs can share the same encoder or decoder without changing any structure of the NMT model, but the parameters of the encoder and decoder has to be separated.', '1607.08725-2-52-2': 'Different from their work, however, we share the parameters of the encoder and decoder, which to the best of our knowledge, is the first attempt.', '1607.08725-2-52-3': 'Our experimental results demonstrate its effectiveness, and also resonate with the findings of these zero-shot translations.', '1607.08725-2-53-0': '# Conclusion and Future Work', '1607.08725-2-54-0': 'In this paper, we have presented the cyclic sequence-to-sequence learning framework that leverages RNN to detect translation-relevant source tokens during decoding according to the partial target sequence.', '1607.08725-2-54-1': 'We observe that: 1) Cseq2seq can learn correspondences between the source and target language in an implicit way without any attention networks; 2) The encoder and decoder can share their RNN parameters without performance loss; 3) Cseq2seq is more effective than the traditional attention-based encoder-decoder in both Chinese-English and English-German machine translation.', '1607.08725-2-55-0': 'In the future, we would like to test our approach on other sequence-to-sequence learning tasks, e.g. neural conversation system.', '1607.08725-2-55-1': 'Additionally, our model seems to be a non-proficient child translator who rereads the entire source sentence each time when a target word is generated.', '1607.08725-2-55-2': 'Rereading the whole source sentence for each target word is not necessary for a skilled translator.', '1607.08725-2-55-3': 'The source sentence will be reread only when cognitive difficulties occur during translation.', '1607.08725-2-55-4': 'We would like to model this process automatically using new neural models.', '1607.08725-2-55-5': 'Finally, we are also interested in applying our model to simultaneous translation.'}

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['1607.08725-1-6-0', '1607.08725-1-9-0', '1607.08725-1-11-0', '1607.08725-1-11-3', '1607.08725-1-12-0', '1607.08725-1-12-2', '1607.08725-1-23-1', '1607.08725-1-24-0', '1607.08725-2-15-3', '1607.08725-2-22-1', '1607.08725-2-27-1', '1607.08725-2-30-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1607.08725

0807.1690

{'0807.1690-1-0-0': 'We consider the infinite family of Feynman graphs known as the "banana graphs" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern-Schwartz-MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes.', '0807.1690-1-0-1': 'We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs.', '0807.1690-1-0-2': 'We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.', '0807.1690-1-1-0': '# Introduction', '0807.1690-1-2-0': 'Since the extensive study of [CITATION] revealed the systematic appearance of multiple zeta values as the result of Feynman diagram computations in perturbative quantum field theory, the question of finding a direct relation between Feynman diagrams and periods of motives has become a rich field of investigation.', '0807.1690-1-2-1': 'The formulation of Feynman integrals that seems most suitable for an algebro-geometric approach is the one involving Schwinger and Feynman parameters, as in that form the integral acquires directly an interpretation as a period of an algebraic variety, namely the complement of a hypersurface in a projective space constructed out of the combinatorial information of a graph.', '0807.1690-1-2-2': 'These graph hypersurfaces and the corresponding periods have been investigated in the algebro-geometric perspective in the recent work of Bloch-Esnault-Kreimer ([CITATION], [CITATION]) and more recently, from the point of view of Hodge theory, in [CITATION] and [CITATION].', '0807.1690-1-2-3': 'In particular, the question of whether only motives of mixed Tate type would arise in the quantum field theory context is still unsolved.', '0807.1690-1-2-4': 'Despite the general result of [CITATION], which shows that the graph hypersurfaces are general enough from the motivic point of view to generate the Grothendieck ring of varieties, the particular results of [CITATION] and [CITATION] point to the fact that, even though the varieties themselves are very general, the part of the cohomology that supports the period of interest to quantum field theory might still be of the mixed Tate form.', '0807.1690-1-3-0': 'One complication involved in the algebro-geometric computations with graph hypersurfaces is the fact that these are typically singular, with a singular locus of small codimension.', '0807.1690-1-3-1': 'It becomes then an interesting question in itself to estimate how singular the graph hypersurfaces are, across certain families of Feynman graphs (the half open ladder graphs, the wheels with spokes, the banana graphs etc.).', '0807.1690-1-3-2': 'Since the main goal is to describe what happens at the motivic level, one wants to have invariants that detect how singular the hypersurface is and that are also somehow adapted to its decomposition in the Grothendieck ring of motives.', '0807.1690-1-3-3': 'In this paper we concentrate on a particular example and illustrate some general methods for computing such invariants based on the theory of characteristic classes of singular varieties.', '0807.1690-1-4-0': 'Part of the purpose of the present paper is to familiarize physicists working in perturbative quantum field theory with some techniques of algebraic geometry that are useful in the analysis of graph hypersurfaces.', '0807.1690-1-4-1': 'Thus, we try as mush as possible to spell out everything in detail and recall the necessary background.', '0807.1690-1-5-0': 'In [REF], we begin by recalling the general form of the parametric Feynman integrals for a scalar field theory and the construction of the associated projective graph hypersurface.', '0807.1690-1-5-1': 'We recall the relation between the graph hypersurface of a planar graph and that of the dual graph via the standard Cremona transformation.', '0807.1690-1-5-2': 'We then present the specific example of the infinite family of "banana graphs".', '0807.1690-1-5-3': 'We formulate a positivity conjecture for the characteristic classes of graph hypersurfaces.', '0807.1690-1-6-0': 'For the convenience of the reader, we recall in [REF] some general facts and results, both about the Grothendieck ring of varieties and motives, and about the theory of characteristic classes of singular algebraic varieties.', '0807.1690-1-6-1': 'We outline the similarities and differences between these constructions.', '0807.1690-1-7-0': 'In [REF] we give the explicit computation of the classes in the Grothendieck ring of the hypersurfaces of the banana graphs.', '0807.1690-1-7-1': 'We conclude with a general remark on the relation between the class of the hypersurface of a planar graph and that of a dual graph.', '0807.1690-1-8-0': 'In [REF] we obtain an explicit formula for the Chern-Schwartz-MacPherson classes of the hypersurfaces of the banana graphs.', '0807.1690-1-8-1': 'We first prove a general pullback formula for these classes, which is necessary in order to compute the contribution to the CSM class of the complement of the algebraic simplex in the graph hypersurface.', '0807.1690-1-8-2': 'The formula is then obtained by assembling the contribution of the intersection with the algebraic simplex and of its complement via inclusion-exclusion, as in the case of the classes in the Grothendieck ring.', '0807.1690-1-9-0': 'We give then, in [REF], a formula for the CSM classes of cones on hypersurfaces and use them to obtain formulae for graph hypersurfaces obtained from known one by simple operations on the graphs, such as doubling or splitting an edge, and attaching single-edge loops or trees to vertices.', '0807.1690-1-10-0': 'Finally, in [REF], we look at the deformations of ordinary [MATH] theory to a noncommutative spacetime given by a Moyal space.', '0807.1690-1-10-1': 'We look at the ribbon graphs that correspond to the original banana graphs in this noncommutative quantum field theory.', '0807.1690-1-10-2': 'We explain the relation between the graph hypersurfaces of the noncommutative theory and of the original commutative one.', '0807.1690-1-10-3': 'We show by an explicit computation of CSM classes that in noncommutative QFT the positivity conjecture fails for non-planar ribbon graphs.'}

{'0807.1690-2-0-0': 'We consider the infinite family of Feynman graphs known as the "banana graphs" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern-Schwartz-MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes.', '0807.1690-2-0-1': 'We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs.', '0807.1690-2-0-2': 'We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.', '0807.1690-2-1-0': '# Introduction', '0807.1690-2-2-0': 'Since the extensive study of [CITATION] revealed the systematic appearance of multiple zeta values as the result of Feynman diagram computations in perturbative quantum field theory, the question of finding a direct relation between Feynman diagrams and periods of motives has become a rich field of investigation.', '0807.1690-2-2-1': 'The formulation of Feynman integrals that seems most suitable for an algebro-geometric approach is the one involving Schwinger and Feynman parameters, as in that form the integral acquires directly an interpretation as a period of an algebraic variety, namely the complement of a hypersurface in a projective space constructed out of the combinatorial information of a graph.', '0807.1690-2-2-2': 'These graph hypersurfaces and the corresponding periods have been investigated in the algebro-geometric perspective in the recent work of Bloch-Esnault-Kreimer ([CITATION], [CITATION]) and more recently, from the point of view of Hodge theory, in [CITATION] and [CITATION].', '0807.1690-2-2-3': 'In particular, the question of whether only motives of mixed Tate type would arise in the quantum field theory context is still unsolved.', '0807.1690-2-2-4': 'Despite the general result of [CITATION], which shows that the graph hypersurfaces are general enough from the motivic point of view to generate the Grothendieck ring of varieties, the particular results of [CITATION] and [CITATION] point to the fact that, even though the varieties themselves are very general, the part of the cohomology that supports the period of interest to quantum field theory might still be of the mixed Tate form.', '0807.1690-2-3-0': 'One complication involved in the algebro-geometric computations with graph hypersurfaces is the fact that these are typically singular, with a singular locus of small codimension.', '0807.1690-2-3-1': 'It becomes then an interesting question in itself to estimate how singular the graph hypersurfaces are, across certain families of Feynman graphs (the half open ladder graphs, the wheels with spokes, the banana graphs etc.).', '0807.1690-2-3-2': 'Since the main goal is to describe what happens at the motivic level, one wants to have invariants that detect how singular the hypersurface is and that are also somehow adapted to its decomposition in the Grothendieck ring of motives.', '0807.1690-2-3-3': 'In this paper we concentrate on a particular example and illustrate some general methods for computing such invariants based on the theory of characteristic classes of singular varieties.', '0807.1690-2-4-0': 'Part of the purpose of the present paper is to familiarize physicists working in perturbative quantum field theory with some techniques of algebraic geometry that are useful in the analysis of graph hypersurfaces.', '0807.1690-2-4-1': 'Thus, we try as mush as possible to spell out everything in detail and recall the necessary background.', '0807.1690-2-5-0': 'In [REF], we begin by recalling the general form of the parametric Feynman integrals for a scalar field theory and the construction of the associated projective graph hypersurface.', '0807.1690-2-5-1': 'We recall the relation between the graph hypersurface of a planar graph and that of the dual graph via the standard Cremona transformation.', '0807.1690-2-5-2': 'We then present the specific example of the infinite family of "banana graphs".', '0807.1690-2-5-3': 'We formulate a positivity conjecture for the characteristic classes of graph hypersurfaces.', '0807.1690-2-6-0': 'For the convenience of the reader, we recall in [REF] some general facts and results, both about the Grothendieck ring of varieties and motives, and about the theory of characteristic classes of singular algebraic varieties.', '0807.1690-2-6-1': 'We outline the similarities and differences between these constructions.', '0807.1690-2-7-0': 'In [REF] we give the explicit computation of the classes in the Grothendieck ring of the hypersurfaces of the banana graphs.', '0807.1690-2-7-1': 'We conclude with a general remark on the relation between the class of the hypersurface of a planar graph and that of a dual graph.', '0807.1690-2-8-0': 'In [REF] we obtain an explicit formula for the Chern-Schwartz-MacPherson classes of the hypersurfaces of the banana graphs.', '0807.1690-2-8-1': 'We first prove a general pullback formula for these classes, which is necessary in order to compute the contribution to the CSM class of the complement of the algebraic simplex in the graph hypersurface.', '0807.1690-2-8-2': 'The formula is then obtained by assembling the contribution of the intersection with the algebraic simplex and of its complement via inclusion-exclusion, as in the case of the classes in the Grothendieck ring.', '0807.1690-2-9-0': 'We give then, in [REF], a formula for the CSM classes of cones on hypersurfaces and use them to obtain formulae for graph hypersurfaces obtained from known one by simple operations on the graphs, such as doubling or splitting an edge, and attaching single-edge loops or trees to vertices.', '0807.1690-2-10-0': 'Finally, in [REF], we look at the deformations of ordinary [MATH] theory to a noncommutative spacetime given by a Moyal space.', '0807.1690-2-10-1': 'We look at the ribbon graphs that correspond to the original banana graphs in this noncommutative quantum field theory.', '0807.1690-2-10-2': 'We explain the relation between the graph hypersurfaces of the noncommutative theory and of the original commutative one.', '0807.1690-2-10-3': 'We show by an explicit computation of CSM classes that in noncommutative QFT the positivity conjecture fails for non-planar ribbon graphs.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0807.1690

1303.5493

{'1303.5493-1-0-0': '# Introduction', '1303.5493-1-1-0': 'The concept of fractional exclusion statistics (FES) was introduced by Haldane in Ref. [CITATION] and the statistical mechanics of FES systems was formulated by several authors, employing different methods [CITATION].', '1303.5493-1-2-0': 'The FES was applied to quasiparticle excitations at the lowest Landau level in the fractional quantum Hall effect, spinon excitations in a spin-[MATH] quantum antiferomagnet [CITATION], Bose and Fermi systems described in the thermodynamic Bethe ansatz [CITATION], excitations [CITATION] or motifs of spins [CITATION] in spin chains, elementary volumes obtained by coarse-graining in the phase-space of a system [CITATION], interacting particles described in the mean-field approximation [CITATION].', '1303.5493-1-2-1': 'Thermodynamic properties and correlations in generic FES systems in different numbers of dimensions or interacting with external fields have also been calculated [CITATION].', '1303.5493-1-2-2': 'In Refs. [CITATION] it was shown that all the systems with the same, constant density of states and diagonal, constant FES parameters (FES is manifested within each species) have the same entropy and heat capacity.', '1303.5493-1-2-3': 'This property extends the Bose-Fermi thermodynamic equivalence in 2D systems discovered long time ago [CITATION], to the FES systems [CITATION].', '1303.5493-1-3-0': 'A stochastic method for the simulation of the time evolution of FES systems was introduced in Ref. [CITATION] as a generalization of a similar method used for Bose and Fermi systems [CITATION] whereas the relatively recent experimental realization of the Fermi degeneracy in cold atomic gases have renewed the interest in the theoretical investigation of non-ideal Fermi systems at low temperatures and their interpretation as ideal FES systems [CITATION].', '1303.5493-1-4-0': 'The general FES formalism was amended to include the change of the FES parameters at the change of the particle species [CITATION].', '1303.5493-1-4-1': 'This amendment allows the general implementation of FES as a method for the description of interacting particle systems as ideal (quasi)particle systems [CITATION].', '1303.5493-1-5-0': 'In this paper we apply FES to a system of particles with general particle-particle interaction, [MATH], in an arbitrary external potential, [MATH], described in the quasiclassical approximation.', '1303.5493-1-5-1': 'We define quasiparticle energies which determine our FES parameters and using these we calculate the FES equilibrium particle distribution.', '1303.5493-1-5-2': 'We compare these results with the standard Fermi liquid theory (FLT) calculations.', '1303.5493-1-6-0': 'The structure of the paper is as follows.', '1303.5493-1-6-1': 'In the next section we introduce our model and calculate the equilibrium particle population in the FLT approach.', '1303.5493-1-6-2': 'In the following section we implement the FES description by introducing new quasiparticle energies and the definition of species.', '1303.5493-1-6-3': 'These species are related by the FES parameters, which we calculate.', '1303.5493-1-6-4': 'Using the FES parameters and quasiparticle energies we write the equations for the equilibrium particle populations.', '1303.5493-1-6-5': 'By making the correspondence between the FLT and the FES quasiparticle energies corresponding to the same single-particle states, we show that the FES equations are satisfied by the FLT populations.', '1303.5493-1-6-6': 'This proves that the FES formalism is consistent and suitable for the description of such interacting particle systems and that the FES quasiparticle populations may be calculated using the correspondence between the quasiparticle energies.', '1303.5493-1-7-0': 'The interacting particle system may be described as an ideal FES gas in a rather general formalism.', '1303.5493-1-8-0': '# The model and the FLT approach', '1303.5493-1-9-0': 'We analyze a quasiclassical system of particles (bosons or fermions) with a generic interaction, in an external potential and [MATH]-dimensional space.', '1303.5493-1-9-1': 'We assume that the particle-particle interaction, [MATH], depends only on the distance between the particles, whereas the external potential is [MATH].', '1303.5493-1-9-2': 'If we denote by [MATH] the energies of the non-interacting particles, in the continuous limit the total energy of the system is [EQUATION] where [MATH] is the total volume of the system and [MATH] is the particle population of the level [MATH] in the elementary volume [MATH].', '1303.5493-1-9-3': 'In an elementary volume, [MATH], centered at [MATH], the density of states (DOS) along the [MATH] axis is [MATH].', '1303.5493-1-9-4': 'The logarithm of the grandcanonical partition function is [EQUATION] where the upper and lower signs correspond to fermions and bosons, respectively.', '1303.5493-1-10-0': 'The standard way to obtain the thermodynamics of the system is to maximize [MATH] with respect to [MATH].', '1303.5493-1-10-1': "In this way we obtain the equilibrium populations, [EQUATION] where [EQUATION] are the Landau's quasiparticle energies.", '1303.5493-1-11-0': 'The equilibrium distribution of particles, [MATH], is determined by solving self-consistently Eqs. ([REF]) and ([REF]).', '1303.5493-1-11-1': 'Plugging [MATH] back into [MATH] ([REF]) and [MATH] ([REF]), one can calculate the total energy of the system, the grandcanonical potential and from these, the other thermodynamic quantities.', '1303.5493-1-11-2': 'We observe that the sum of the quasiparticle energies is [EQUATION] which is different from the total energy [MATH].', '1303.5493-1-12-0': '# The FES formalism', '1303.5493-1-13-0': 'In the FES, the quasiparticle energies are defined in such a way that the total energy of the system is equal to the sum of quasiparticle energies [CITATION].', '1303.5493-1-13-1': 'For this purpose we define new quasiparticle energies, [EQUATION]', '1303.5493-1-13-2': 'From Eq. ([REF]) we obtain the DOS along the [MATH] axis, [EQUATION]', '1303.5493-1-13-3': 'To apply FES we split the system into imaginary elementary volumes, [MATH] (centered at [MATH]), and in each such volume the quasiparticle energy axis, [MATH], is split into elementary intervals, [MATH] (centered at [MATH]).', '1303.5493-1-13-4': 'Each such [MATH]-dimensional elementary volume, [MATH], represents a FES species.', '1303.5493-1-14-0': 'From Eqs. ([REF]) and ([REF]) and applying the procedure of Ref. [CITATION] we obtain the FES parameters [EQUATION]', '1303.5493-1-14-1': 'The partition function is the same as the one in Eq. ([REF]), with [MATH], by definition [CITATION].', '1303.5493-1-14-2': 'Maximizing [MATH] with respect to the population along the [MATH] axis, we get the equations for the equilibrium particle population [CITATION] [EQUATION]', '1303.5493-1-14-3': 'Now we can check the equivalence between the FES and the FLT descriptions of the system by substituting Eq. ([REF]) into Eq. ([REF]).', '1303.5493-1-14-4': 'By doing so we recover the relation between [MATH] and [MATH], [EQUATION] which is in accordance with the definitions ([REF]) and ([REF]).', '1303.5493-1-15-0': 'We illustrate our model and the relation between the FES and the FLT descriptions on a one dimensional system of fermions with repulsive Coulomb interactions - [MATH], in the absence of external fields.', '1303.5493-1-15-1': 'The length of the system, [MATH], is discretized as in Ref. [CITATION] into [MATH] equal elementary segments, [MATH], where [MATH].', '1303.5493-1-15-2': 'In each such elementary "volume" the DOS is taken to be constant, [MATH], and on the [MATH] axis we define [MATH] equal consecutive segments between 0 and [MATH], [MATH], where [MATH] - we choose [MATH] such that [MATH] for any [MATH].', '1303.5493-1-15-3': 'In this way we obtain [MATH] species of particles, [MATH], identified also by a double index, [MATH] [CITATION].', '1303.5493-1-15-4': 'To avoid the singularity at the origin of the interaction potential, we consider a cut-off distance of [MATH] - the choice of the cut-off does not affect the procedure.', '1303.5493-1-16-0': 'The total number of particles in the system is [MATH], where [MATH] is the Fermi energy in the noninteracting system.', '1303.5493-1-16-1': 'We set the energy scale of the system by fixing [MATH] and [MATH].', '1303.5493-1-17-0': 'Figure [REF](a) shows the quasiparticle density of states in the FES description, [MATH] (Eq. [REF]).', '1303.5493-1-17-1': "The density of Landau's quasiparticle states may be calculated also from Eq. ([REF]) and we obtain [MATH], which is different from zero only for [MATH].", '1303.5493-1-17-2': 'The minimum value of [MATH] ([REF]) is [MATH].', '1303.5493-1-17-3': 'The position dependent [MATH] is plotted in Fig. [REF] (b).', '1303.5493-1-18-0': 'Figure [REF](c) shows the populations, [MATH] ([REF]).', '1303.5493-1-18-1': "To obtain the Landau's populations ([REF]), we simply shift the quasiparticle energy from [MATH] ([REF]) to [MATH] ([REF]) and [MATH] becomes a Fermi distribution in [MATH] for any [MATH] and with the same [MATH] as for the FES distribution.", '1303.5493-1-18-2': 'The difference between the distributions [MATH] in different volumes, [MATH], being the range of [MATH], as shown in Fig. [REF](b).', '1303.5493-1-19-0': 'The particle density, [MATH], is represented in Fig. [REF](d).', '1303.5493-1-19-1': 'Due to the symmetry of the problem, we have pair-wise identical populations, [MATH] and [MATH], for any [MATH] and [MATH].', '1303.5493-1-19-2': 'The largest deviations in the particle density occurs for the extremal species, placed at both ends of the 1D box as an effect of repulsive interactions.', '1303.5493-1-20-0': 'We observe from Eqs. ([REF]), ([REF]) and from Fig. [REF] that the range of [MATH] is [MATH] in any elementary volume, [MATH].', '1303.5493-1-20-1': 'This property of the FES representation allows for particle transport along the system at any energy, [MATH] or [MATH].', '1303.5493-1-20-2': 'On the other hand, the range of [MATH] is [MATH], with [MATH] for any [MATH].', '1303.5493-1-20-3': 'The presence of [MATH] creates a non-constant energy gap (Fig. [REF]b) for the quasi-particle populations and the transport of low energy quasiparticles from one end of the system to the other may occur only through tunneling or higher order processes which involve rearrangements of quasiparticles above the energy gap.', '1303.5493-1-20-4': 'In the FES description these rearrangements are taken care of implicitly by the formalism.', '1303.5493-1-21-0': '# Conclusions', '1303.5493-1-22-0': 'We have formulated an approach by which a system of particles with general particle-particle interaction, [MATH], in an [MATH]-dimensional space and external potential [MATH] is described as an ideal gas of fractional exclusion statistics (FES).', '1303.5493-1-22-1': 'We have given the equations for the calculation of the FES parameters and equilibrium populations.', '1303.5493-1-23-0': "The FES approach have been compared with the Landau's Fermi liquid formalism and we have shown that although there are differences in the definitions of certain quantities like the quasiparticle energies, the physical results are the same.", '1303.5493-1-23-1': 'The main difference between the two formalisms is that in the FES approach the quasiparticle energies are independent of the populations of other quasiparticle states and therefore the FES gas is "ideal", with the total energy of the gas being equal to the sum of the quasiparticle energies.', '1303.5493-1-24-0': 'We have exemplified our procedure on a one-dimensional system of fermions with repulsive Coulomb interaction for which we calculated the main microscopic parameters, like the quasiparticle energies, quasiparticle density of states and energy levels populations.', '1303.5493-1-24-1': 'For each of these quantities we discussed the similarities and differences between the FES and the FLT approaches.', '1303.5493-1-25-0': 'One practical consequence that appear from our calculations is that the solution of the FES integral equations may eventually be calculated easier by solving self-consistently the FLT equations for population and quasiparticle energies, ([REF]) and ([REF]).', '1303.5493-1-26-0': 'Another consequence is that while in the FLT formulation the quasiparticle energies may form an energy gap at the lowest end of the spectrum due to the particle-particle interaction, in the FES description such an energy gap does not exist.', '1303.5493-1-26-1': 'For this reason the low energy particle tunneling in FLT may be described as higher order processes, involving rearrangements of particles on the higher energy levels, wheres in the FES these rearrangements are taken into account directly by the manifestation of the exclusion statistics.', '1303.5493-1-27-0': 'Enlightening discussions with Francesca Gulminelli and her hospitality during the visit of DVA at the LPC Caen are gratefully acknowledged.', '1303.5493-1-27-1': 'The work was supported by the Romanian National Authority for Scientific Research CNCS-UEFISCDI projects PN-II-ID-PCE-2011-3-0960 and PN09370102/2009.', '1303.5493-1-27-2': 'The travel support from the Romania-JINR Dubna collaboration project Titeica-Markov are gratefully acknowledged.'}

{'1303.5493-2-0-0': 'We explore the connections between the description of interacting particles systems in terms of fractional exclusion statistics (FES) and other many-body methods used for the same purpose.', '1303.5493-2-0-1': 'We consider a system of particles with generic, particle-particle interaction in the quasi-classical limit and in the mean-field approximation.', '1303.5493-2-0-2': 'We define the FES quasiparticle energies, we calculate the FES parameters of the system and we deduce the equations for the equilibrium particle populations.', '1303.5493-2-0-3': 'The FES gas is "ideal", in the sense that the quasiparticle energies do not depend on the other quasiparticle levels populations and the sum of the quasiparticle energies is equal to the total energy of the system.', '1303.5493-2-0-4': 'We prove that this FES formalism is equivalent to the semi-classical or Thomas Fermi (TF) limit of the self-consistent mean-field theory and the FES quasiparticle populations may be calculated from the Landau quasiparticle populations by making the correspondence between the FES and the Landau quasiparticle energies.', '1303.5493-2-0-5': 'The FES provides a natural semi-classical ideal gas description of the interacting particle gas in a generic external potential and in any number of dimensions.', '1303.5493-2-1-0': 'The concept of fractional exclusion statistics (FES) was introduced by Haldane in Ref. [CITATION] and the statistical mechanics of FES systems was formulated by several authors, employing different methods [CITATION].', '1303.5493-2-1-1': 'The FES have been a very prolific concept and was applied to both, quantum and classical systems (see e.g. [CITATION]),', '1303.5493-2-2-0': 'A stochastic method for the simulation of the time evolution of FES systems was introduced in Ref. [CITATION] as a generalization of a similar method used for Bose and Fermi systems [CITATION], whereas the relatively recent experimental realization of the Fermi degeneracy in cold atomic gases have renewed the interest in the theoretical investigation of non-ideal Fermi systems at low temperatures and their interpretation as ideal FES systems [CITATION].', '1303.5493-2-3-0': 'The general FES formalism was amended to include the change of the FES parameters at the change of the particle species [CITATION].', '1303.5493-2-3-1': 'This amendment allows the general implementation of FES as a method for the description of interacting particle systems as ideal (quasi)particle systems [CITATION].', '1303.5493-2-3-2': 'However, the level of approximation of such description and the connections with other many-particle methods are not completely clear.', '1303.5493-2-4-0': 'In Refs. [CITATION] the FES was applied to describe Bose gases with local (delta-function) interaction in two-dimensional traps in the TF limit.', '1303.5493-2-4-1': 'In this paper we generalize this result to apply FES to generic Bose and Fermi system of particles with generic two-body interaction, [MATH], in an arbitrary external potential in the Landau or TF semi-classical limit in any number of dimensions.', '1303.5493-2-4-2': 'We define quasiparticle energies which determine our FES parameters and using these we calculate the FES equilibrium particle distribution.', '1303.5493-2-4-3': "Moreover, we calculate the particle distribution also starting from the mean-field description by defining Landau type of quasiparticles and we show that the two descriptions are equivalent and the populations are identical, provided that we make the mapping between the FES and the the Landau's quasiparticle energies.", '1303.5493-2-5-0': 'The structure of the paper is as follows.', '1303.5493-2-5-1': "We introduce our model and calculate Landau's equilibrium particle population in the TF approach.", '1303.5493-2-5-2': 'Then we implement the FES description by introducing new quasiparticle energies and the definition of species.', '1303.5493-2-5-3': 'These species are related by the FES parameters, which we calculate.', '1303.5493-2-5-4': 'Using the FES parameters and quasiparticle energies we write the equations for the equilibrium particle populations.', '1303.5493-2-5-5': "By making the correspondence between the Landau's and the FES quasiparticle energies, we show that the FES equations are satisfied by the Landau's populations.", '1303.5493-2-5-6': "This proves that the FES formalism is consistent and suitable for the description of such interacting particle systems and that the FES quasiparticle populations may be calculated by the Landau's approach, using the correspondence between the quasiparticle energies.", '1303.5493-2-6-0': 'Let us consider a system of interacting particles described by the generic Hamiltonian: [EQUATION] where the indexes [MATH] denote the single particle states, and [MATH] are creation (annihilation) operators obeying commutations rules which define the quantum statistics of the system (Bose or Fermi).', '1303.5493-2-6-1': 'We assume that the particle-particle interaction, [MATH], depends only on the distance between the particles, whereas the external potential is [MATH].', '1303.5493-2-6-2': 'In the mean-field approximation the total energy of the system can be written as: [EQUATION] where [MATH] are single-particle kinetic energies, [MATH], and [MATH] are antisymmetrized (symmetrized) matrix elements for fermions (bosons).', '1303.5493-2-6-3': 'The upper and lower signs in the superscripts of the occupation numbers, [MATH], stand for fermions and bosons, respectively.', '1303.5493-2-7-0': 'At the thermodynamic limit, the finite temperature properties of the system can be accessed via the grandcanonical partition sum defined by the mean-field one-body entropy as: [EQUATION]', '1303.5493-2-7-1': 'Maximizing this function with respect to the single particle occupations gives the equilibrium particle populations, [EQUATION]', '1303.5493-2-7-2': "The quantities, [EQUATION] are the Landau's quasiparticle energies.", '1303.5493-2-8-0': 'If we assume a large number of particles, and a sufficiently slowly varying external potential (the quasiclassical limit), we can divide the system into macroscopic cells where the external field is locally constant [MATH], and apply the thermodynamic limit in each cell.', '1303.5493-2-8-1': 'In this limit, single-particle energies are continuous variables and it is meaningful to introduce the density of states (DOS), [MATH].', '1303.5493-2-8-2': 'In the same limit, [MATH] .', '1303.5493-2-8-3': 'Then eq. ([REF]) becomes [EQUATION]', '1303.5493-2-8-4': 'The Thomas-Fermi (or Landau) theory amounts to extend this mean-field formalism to a finite inhomogeneous system employing a semi-classical limit for the kinetic energy.', '1303.5493-2-8-5': 'In that case the quasi-particle energies read [EQUATION]', '1303.5493-2-8-6': 'We observe that the sum of the quasiparticle energies [EQUATION] is not equal to the total mean-field energy [MATH], because of the well-known double counting of the interactions.', '1303.5493-2-9-0': 'Now we want to show that the semi-classical Thomas-Fermi results can be reproduced by an ideal system, provided that such a system obeys FES.', '1303.5493-2-9-1': 'In the ideal system the total energy is equal to the sum of quasiparticle energies, which are independent of the populations.', '1303.5493-2-9-2': 'The new quasiparticle energies are [EQUATION] and we can immediately check that [EQUATION]', '1303.5493-2-9-3': 'From Eq. ([REF]) we obtain the DOS along the [MATH] axis, [EQUATION] where we can clearly see that FES will arise, as the density of states depends on the occupations.', '1303.5493-2-9-4': 'The FES parameters are calculated according to Ref. [CITATION].', '1303.5493-2-9-5': 'To this aim we split the system into imaginary elementary volumes, [MATH] (centered at [MATH]), and in each such volume the quasiparticle energy axis, [MATH], is split into elementary intervals, [MATH] (centered at [MATH]).', '1303.5493-2-9-6': 'Each such [MATH]-dimensional elementary volume, [MATH], represents a FES species.', '1303.5493-2-10-0': 'From Eqs. ([REF]) and ([REF]) and applying the procedure of Ref. [CITATION] we obtain the FES parameters [EQUATION]', '1303.5493-2-10-1': 'The presence of non-zero FES parameters is a proof that the system obeys FES.', '1303.5493-2-10-2': 'We now turn to show that the whole thermodynamics can be equivalently calculated in the TF and in the FES approaches.', '1303.5493-2-11-0': 'The partition function is the same as the one in Eq. ([REF]), with the summation taken over the species, [MATH] [CITATION].', '1303.5493-2-11-1': 'Indeed a maximization of Eq. ([REF]) corresponds to the entropy maximization under the condition of independent particles, and under the constraint that the total energy is [MATH] and the total particle number [MATH].', '1303.5493-2-12-0': 'Maximizing [MATH] with respect to the population along the [MATH] axis, we get the equations for the equilibrium particle population [CITATION] [EQUATION]', '1303.5493-2-12-1': 'Now we can check the equivalence between the FES and the TF descriptions of the system by substituting Eq. ([REF]) into Eq. ([REF]).', '1303.5493-2-12-2': 'By doing so we recover the relation between [MATH] and [MATH], [EQUATION] which is in accordance with the definitions ([REF]) and ([REF]).', '1303.5493-2-13-0': 'We illustrate our model and the relation between the FES and the TF descriptions on a one dimensional system of fermions with repulsive Coulomb interactions - [MATH], in the absence of external fields.', '1303.5493-2-13-1': 'The length of the system, [MATH], is discretized as in Ref. [CITATION] into [MATH] equal elementary segments, [MATH], where [MATH].', '1303.5493-2-13-2': 'In each such elementary "volume" the DOS is taken to be constant, [MATH], and on the [MATH] axis we define [MATH] equal consecutive segments between 0 and [MATH], [MATH], where [MATH] - we choose [MATH] such that [MATH] for any [MATH].', '1303.5493-2-13-3': 'In this way we obtain [MATH] species of particles, [MATH], identified also by a double index, [MATH] [CITATION].', '1303.5493-2-13-4': 'To avoid the singularity at the origin of the interaction potential, we consider a cut-off distance of [MATH] - the choice of the cut-off does not affect the procedure.', '1303.5493-2-14-0': 'The total number of particles in the system is [MATH], where [MATH] is the Fermi energy in the noninteracting system.', '1303.5493-2-14-1': 'We set the energy scale of the system by fixing [MATH] and [MATH].', '1303.5493-2-15-0': 'Figure [REF](a) shows the quasiparticle density of states in the FES description, [MATH] (Eq. [REF]).', '1303.5493-2-15-1': "The density of Landau's quasiparticle states may be calculated also as [MATH] and we obtain [MATH], which is different from zero only for [MATH].", '1303.5493-2-15-2': 'The minimum value of [MATH] ([REF]) is [MATH].', '1303.5493-2-15-3': 'The position dependent [MATH] is plotted in Fig. [REF] (b).', '1303.5493-2-16-0': 'Figure [REF](c) shows the populations, [MATH] ([REF]).', '1303.5493-2-16-1': "To obtain the Landau's populations ([REF]), we simply shift the quasiparticle energy from [MATH] ([REF]) to [MATH] ([REF]) and [MATH] becomes a Fermi distribution in [MATH] for any [MATH] and with the same [MATH] as for the FES distribution.", '1303.5493-2-16-2': 'For example one may substitute [MATH] into Eq. ([REF]) and obtain the same population for the species with lowest energy, [MATH], presented in the FES description in Fig. [REF](c).', '1303.5493-2-17-0': 'The particle density, [MATH], is represented in Fig. [REF](d).', '1303.5493-2-17-1': 'Due to the symmetry of the problem, we have pair-wise identical populations, [MATH] and [MATH], for any [MATH] and [MATH].', '1303.5493-2-17-2': 'The largest deviations in the particle density occurs for the extremal species, placed at both ends of the 1D box as an effect of repulsive interactions.', '1303.5493-2-18-0': 'We observe from Eqs. ([REF]), ([REF]) and from Fig. [REF] that because [MATH], the range of [MATH] is [MATH] in any elementary volume, [MATH], whereas the range of [MATH] is [MATH], with [MATH].', '1303.5493-2-19-0': 'In conclusion we have formulated an approach by which a system of quantum particles with general particle-particle interaction, [MATH], in an [MATH]-dimensional space and external potential [MATH] is described in the quasiclassical limit as an ideal gas of fractional exclusion statistics (FES).', '1303.5493-2-19-1': 'We have given the equations for the calculation of the FES parameters and equilibrium populations.', '1303.5493-2-20-0': 'The FES approach have been compared with the Thomas-Fermi (TF) formalism and we have shown that although there are differences in the definitions of certain quantities like the quasiparticle energies, the physical results are the same.', '1303.5493-2-20-1': 'The main difference between the two formalisms is that in the FES approach the quasiparticle energies are independent of the populations of other quasiparticle states and therefore the FES gas is "ideal", with the total energy of the gas being equal to the sum of the quasiparticle energies.', '1303.5493-2-21-0': 'We have exemplified our procedure on a one-dimensional system of fermions with repulsive Coulomb interaction for which we calculated the main microscopic parameters, like the quasiparticle energies, quasiparticle density of states and energy levels populations.', '1303.5493-2-21-1': 'For each of these quantities we discussed the similarities and differences between the FES and the TF approaches.', '1303.5493-2-22-0': 'One practical consequence that appear from our calculations is that the solution of the FES integral equations may eventually be calculated easier by solving self-consistently the TF equations for population and quasiparticle energies, ([REF]) and ([REF]).', '1303.5493-2-23-0': 'Another consequence is that while in the TF formulation the quasiparticle energies may form an energy gap at the lowest end of the spectrum due to the particle-particle interaction, in the FES description such an energy gap does not exist.', '1303.5493-2-24-0': 'In conclusion, by establishing the equivalence between the self-consistent mean-field theory and the FES approach we show that in general a quasi-classical interacting systems can be mapped onto an ideal FES system.', '1303.5493-2-25-0': 'The work was supported by the Romanian National Authority for Scientific Research CNCS-UEFISCDI projects PN-II-ID-PCE-2011-3-0960 and PN09370102/2009.', '1303.5493-2-25-1': 'The travel support from the Romania-JINR Dubna collaboration project Titeica-Markov are gratefully acknowledged.'}

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'1303.5493-4-20-1'], ['1303.5493-3-36-1', '1303.5493-4-36-1'], ['1303.5493-3-21-0', '1303.5493-4-21-0'], ['1303.5493-3-21-1', '1303.5493-4-21-1'], ['1303.5493-3-21-2', '1303.5493-4-21-2'], ['1303.5493-3-21-3', '1303.5493-4-21-3'], ['1303.5493-3-34-1', '1303.5493-4-34-1'], ['1303.5493-3-28-0', '1303.5493-4-28-0'], ['1303.5493-3-17-1', '1303.5493-4-17-1'], ['1303.5493-3-5-0', '1303.5493-4-5-0'], ['1303.5493-3-5-1', '1303.5493-4-5-1'], ['1303.5493-3-5-2', '1303.5493-4-5-2'], ['1303.5493-3-5-5', '1303.5493-4-5-5'], ['1303.5493-3-5-7', '1303.5493-4-5-7'], ['1303.5493-3-5-8', '1303.5493-4-5-8'], ['1303.5493-3-8-0', '1303.5493-4-8-0'], ['1303.5493-3-16-0', '1303.5493-4-16-0'], ['1303.5493-3-16-1', '1303.5493-4-16-1'], ['1303.5493-3-16-2', '1303.5493-4-16-2'], ['1303.5493-3-16-3', '1303.5493-4-16-3'], ['1303.5493-3-14-0', '1303.5493-4-14-0'], ['1303.5493-3-23-0', '1303.5493-4-23-0'], ['1303.5493-3-23-1', '1303.5493-4-23-1'], ['1303.5493-3-13-0', '1303.5493-4-13-0'], ['1303.5493-3-13-1', 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[]

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[['1303.5493-1-11-2', '1303.5493-2-8-6']]

['1303.5493-2-1-1', '1303.5493-2-18-0', '1303.5493-3-24-0', '1303.5493-3-26-0', '1303.5493-3-30-0', '1303.5493-4-26-0', '1303.5493-4-30-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1303.5493

{'1303.5493-3-0-0': 'We describe a mean field interacting particle system in any number of dimensions and generic external potential as an ideal gas with fractional exclusion statistics (FES).', '1303.5493-3-0-1': 'We define the FES quasiparticle energies, we calculate the FES parameters of the system and we deduce the equations for the equilibrium particle populations.', '1303.5493-3-0-2': 'The FES gas is "ideal", in the sense that the quasiparticle energies do not depend on the other quasiparticle levels populations and the sum of the quasiparticle energies is equal to the total energy of the system.', '1303.5493-3-0-3': 'We prove that the FES formalism is equivalent to the semi-classical or Thomas Fermi (TF) limit of the self-consistent mean-field theory and the FES quasiparticle populations may be calculated from the Landau quasiparticle populations by making the correspondence between the FES and the Landau quasiparticle energies.', '1303.5493-3-0-4': 'The FES provides a natural semi-classical ideal gas description of the interacting particle gas.', '1303.5493-3-1-0': '# Introduction', '1303.5493-3-2-0': 'Quantum (Bose or Fermi) statistics is a very basic concept in physics, directly arising from the indiscernability principle of the microscopic world.', '1303.5493-3-2-1': 'However effective models employing different statistics have proved to be very useful to describe some selected aspects of complex interacting microscopic systems.', '1303.5493-3-2-2': 'As an example, an ideal Fermi gas with Fermi-Dirac statistics can be described in certain approximations by a Boltzmann distribution with repulsive interaction, while a bosonic ideal gas, by an attractive potential [CITATION].', '1303.5493-3-2-3': 'On more general grounds, the statistical mechanics of fractional exclusion statistics (FES) quasi-particle systems was formulated by several authors [CITATION], based on the FES concept introduced by Haldane in Ref. [CITATION] This has been a prolific concept and was applied to both quantum and classical systems (see e.g. [CITATION]).', '1303.5493-3-3-0': 'A stochastic method for the simulation of the time evolution of FES systems was introduced in Ref. [CITATION] as a generalization of a similar method used for Bose and Fermi systems [CITATION], whereas the relatively recent experimental realization of the Fermi degeneracy in cold atomic gases have renewed the interest in the theoretical investigation of non-ideal Fermi systems at low temperatures and their interpretation as ideal FES systems [CITATION].', '1303.5493-3-4-0': 'The general FES formalism was amended to include the change of the FES parameters at the change of the particle species [CITATION].', '1303.5493-3-4-1': 'This amendment allows the general implementation of FES as a method for the description of interacting particle systems as ideal (quasi)particle gases [CITATION].', '1303.5493-3-4-2': 'However, the level of approximation of such description and the connections with other many-particle methods are not yet clear.', '1303.5493-3-5-0': 'A rigorous connection between the FES and the Bethe ansatz equations for an exactly solvable model was worked out in Refs. [CITATION] in the one-dimensional case.', '1303.5493-3-5-1': 'In higher dimensions, one can expect that the statistics and interaction would not be transmutable in general because interaction and exchange positions between two particles can occur separately.', '1303.5493-3-5-2': 'However indications exist that a mapping might exist between ideal FES and interacting particles with regular quantum statistics in the quasi-particle semi-classical approximation given by the self-consistent mean-field theory.', '1303.5493-3-5-3': 'Indeed, in Refs. [CITATION] the FES was applied to describe Bose gases with local (delta-function) interaction in two-dimensional traps in the Thomas Fermi limit.', '1303.5493-3-5-4': 'In this paper we generalize this result to apply FES to generic Bose and Fermi system of particles with generic two-body interaction, [MATH], in an arbitrary external potential in the Landau or TF semi-classical limit in any number of dimensions.', '1303.5493-3-5-5': 'We define quasiparticle energies which determine our FES parameters and using these we calculate the FES equilibrium particle distribution.', '1303.5493-3-5-6': "Moreover, we calculate the particle distribution also starting from the mean-field description by defining Landau type of quasiparticles and we show that the two descriptions are equivalent, i.e. the populations are identical, provided that we make the mapping between the FES and the Landau's quasiparticle energies.", '1303.5493-3-5-7': 'This equivalence proves that our FES description of the interacting particle system corresponds to a self-consistent mean-field approximation.', '1303.5493-3-5-8': 'Such an approximation, though certainly not adequate to fully describe strongly interacting quantum systems, is the basis of a number of highly predictive theoretical methods in many-body physics, from Landau Fermi liquid theory to density functional methods in correlated electron or nucleon systems.', '1303.5493-3-6-0': 'The structure of the paper is as follows.', '1303.5493-3-6-1': "In Section [REF] we introduce our model and calculate Landau's equilibrium particle population in the TF approach.", '1303.5493-3-6-2': 'In Section [REF] we implement the FES description by introducing new quasiparticle energies and the definition of species.', '1303.5493-3-6-3': 'These species are related by the FES parameters, which we calculate.', '1303.5493-3-6-4': 'Using the FES parameters and quasiparticle energies we write the equations for the equilibrium particle populations.', '1303.5493-3-6-5': "By making the correspondence between the Landau's and the FES quasiparticle energies, we show that the FES equations are satisfied by the Landau's populations.", '1303.5493-3-6-6': "This proves that the FES formalism is consistent and suitable for the description of such interacting particle systems and that the FES quasiparticle populations may be calculated by the Landau's approach, using the correspondence between the quasiparticle energies.", '1303.5493-3-6-7': 'In Sections [REF] and [REF] we show some numerical and analytical examples, respectively, whereas in Section [REF] we give the conclusions.', '1303.5493-3-7-0': "# The model in the Landau's approach", '1303.5493-3-8-0': 'Let us consider a system of [MATH] interacting particles described by the generic Hamiltonian: [EQUATION] where the indexes [MATH] denote the single particle states, and [MATH] are creation (annihilation) operators obeying commutations rules which define the quantum statistics of the system (Bose or Fermi).', '1303.5493-3-8-1': 'We assume that the particle-particle interaction, [MATH], depends only on the distance between the particles, whereas the external potential is [MATH].', '1303.5493-3-8-2': 'In the mean-field approximation the total energy of the system can be written as: [EQUATION] where [MATH] are single-particle kinetic energies, [MATH], and [MATH] are antisymmetrized (symmetrized) matrix elements for fermions (bosons).', '1303.5493-3-8-3': 'The upper and lower signs in the superscripts of the occupation numbers, [MATH], stand for fermions and bosons, respectively.', '1303.5493-3-9-0': 'At the thermodynamic limit, the finite temperature properties of the system can be accessed via the grandcanonical partition sum defined by the mean-field one-body entropy as: [EQUATION]', '1303.5493-3-9-1': "Maximizing this function with respect to the single particle occupations gives the equilibrium particle populations, [EQUATION] where the quantities, [MATH] are the Landau's quasiparticle energies.", '1303.5493-3-10-0': 'If we assume a large number of particles, and a sufficiently slowly varying external potential, we can employ the Thomas-Fermi (or Landau) theory which amounts to extend this mean-field formalism to a finite inhomogeneous system employing a semi-classical limit for the kinetic energy.', '1303.5493-3-10-1': 'We can divide the system into macroscopic cells, [MATH], centered at [MATH], where the external field is locally constant and apply the thermodynamic limit in each cell.', '1303.5493-3-10-2': 'The single-particle energies are continuous variables and it is meaningful to introduce the density of states (DOS) in each cell as [MATH].', '1303.5493-3-10-3': 'Then the quasi-particle energies read [EQUATION]', '1303.5493-3-10-4': 'We observe that the sum of the quasiparticle energies [EQUATION] is not equal to the total mean-field energy [MATH], because of the well-known double counting of the interactions.', '1303.5493-3-11-0': '# The FES formalism', '1303.5493-3-12-0': 'Now we want to show that the semi-classical Thomas-Fermi results can be reproduced by an ideal system, provided that such a system obeys FES.', '1303.5493-3-12-1': 'A FES system consists of a countable number of species, denoted here by an index, [MATH] or [MATH] (we use capital letters to denote species).', '1303.5493-3-12-2': 'Each species, [MATH], contains [MATH] available single-particle states and [MATH] particles, each of them of energy [MATH] and chemical potential [MATH].', '1303.5493-3-12-3': 'If the particles are bosons, [MATH] represents also the number of states in the species.', '1303.5493-3-12-4': 'If the particles are fermions, the number of single-particle states in the species [MATH] is [MATH].', '1303.5493-3-12-5': 'The FES character of the system consists in the fact that if the number of particles in one species, for example in species [MATH], changes by [MATH], then the number of states in any other species, [MATH], changes as [MATH], for bosons, or [MATH], for fermions.', '1303.5493-3-12-6': 'The parameters [MATH] are called the FES parameters.', '1303.5493-3-12-7': 'The total number of micro-configurations in the system is [EQUATION] for fermions; the products in Eqs. ([REF]) are taken over all the species in the system.', '1303.5493-3-12-8': 'Using Eqs. ([REF]) and the fact that all the particles in a species have the same energy and chemical potential, we write the partition functions, [EQUATION] where we used the notations [EQUATION]', '1303.5493-3-12-9': 'In Eqs. ([REF]) we made the usual simplifying assumption that all the chemical potentials in all the species are the same, i.e. [MATH] for all [MATH].', '1303.5493-3-12-10': 'This assumption is justified by the fact that in the present application all the particles are identical and, as we shall see below, a species represents a single particle energy interval.', '1303.5493-3-13-0': 'The equilibrium populations are obtained by maximizing [MATH] and [MATH] with respect to [MATH], taking into account the variation of the number of states in the species with the number of particles.', '1303.5493-3-13-1': 'We obtain the equations [EQUATION] where [MATH] and [MATH] [CITATION].', '1303.5493-3-14-0': 'A fermionic system may be transformed into a bosonic system if we define [MATH] and [MATH], which leads to [MATH].', '1303.5493-3-15-0': 'In the ideal system the total energy is equal to the sum of quasiparticle energies, which are independent of the populations.', '1303.5493-3-15-1': 'We define new quasiparticle energies-the FES quasiparticle energies-as [EQUATION] and we can immediately check that [EQUATION]', '1303.5493-3-15-2': 'Equation ([REF]) shows that the total energy can be obtained as a simple sum of quasi-particles energies, meaning that these latter can be viewed as an ideal gas.', '1303.5493-3-15-3': 'This surprising result for an interacting system is well known in the context of FES.', '1303.5493-3-15-4': 'From Eq. ([REF]) we obtain the DOS along the [MATH] axis, [EQUATION]', '1303.5493-3-15-5': 'Although [MATH] depends explicitly on [MATH] ([REF]), we can transfer this dependence to the statistical interaction through the FES parameters which leaves the quasiparticle gas an ideal gas, as explained for example in Ref. [CITATION].', '1303.5493-3-15-6': 'The FES parameters are calculated similarly to Ref. [CITATION].', '1303.5493-3-15-7': 'To this aim in each such volume, [MATH], the quasiparticle energy axis, [MATH], is split into elementary intervals, [MATH], centered at [MATH].', '1303.5493-3-15-8': 'Each [MATH]-dimensional elementary volume, [MATH], represents a FES species, as indicated in Fig. [REF].', '1303.5493-3-15-9': 'Between species with the same energy, [MATH] [arrow (a) in Fig. [REF]], but located in different volumes, [MATH] and [MATH], we have the FES parameters [EQUATION]', '1303.5493-3-15-10': 'The presence of non-zero FES parameters is a proof that the system obeys FES.', '1303.5493-3-16-0': 'We now turn to show that the whole thermodynamics can be equivalently calculated in the TF and in the FES approaches.', '1303.5493-3-16-1': 'Plugging the FES parameters ([REF]) and the quasiparticle energies ([REF]) into the FES equations ([REF]) we obtain [EQUATION]', '1303.5493-3-16-2': 'Now we can check the equivalence between the FES and the TF descriptions of the system by substituting Eq. ([REF]) into Eq. ([REF]).', '1303.5493-3-16-3': 'By doing so we recover the relation between [MATH] and [MATH], [EQUATION] which is in accordance with the definitions ([REF]) and ([REF]).', '1303.5493-3-17-0': 'Equations ([REF]) and ([REF]), as well as Eqs. ([REF]) and ([REF]) have sense only if the integrals in these expressions converge.', '1303.5493-3-17-1': 'This is a limitation of the mean-field formalism.', '1303.5493-3-18-0': '# Numerical example', '1303.5493-3-19-0': 'We illustrate our model and the relation between the FES and the TF descriptions on a one dimensional system of fermions with repulsive Coulomb interactions, [MATH], in the absence of external fields.', '1303.5493-3-19-1': 'The length of the system, [MATH], is discretized as in Ref. [CITATION] into [MATH] equal elementary segments, [MATH], where [MATH].', '1303.5493-3-19-2': 'In each such elementary "volume" the DOS is taken to be constant, [MATH], and on the [MATH] axis we define [MATH] equal consecutive segments between 0 and [MATH], [MATH], where [MATH] - we choose [MATH] such that [MATH] for any [MATH].', '1303.5493-3-19-3': 'In this way we obtain [MATH] species of particles, [MATH], identified also by a double index, [MATH] [CITATION].', '1303.5493-3-19-4': 'To avoid the singularity at the origin of the interaction potential, we consider a cut-off distance of [MATH].', '1303.5493-3-20-0': 'The total number of particles in the system is [MATH], where [MATH] is the Fermi energy in the noninteracting system.', '1303.5493-3-20-1': 'We set the energy scale of the system by fixing [MATH] and [MATH].', '1303.5493-3-21-0': 'Figure [REF](a) shows the quasiparticle density of states in the FES description, [MATH] (Eq. [REF]).', '1303.5493-3-21-1': "The density of Landau's quasiparticle states may be calculated also as [MATH] and we obtain [MATH], which is different from zero only for [MATH].", '1303.5493-3-21-2': 'The minimum value of [MATH] ([REF]) is [MATH].', '1303.5493-3-21-3': 'The position dependent [MATH] is plotted in Fig. [REF] (b).', '1303.5493-3-22-0': 'Figure [REF](c) shows the populations, [MATH] ([REF]).', '1303.5493-3-22-1': "To obtain the Landau's populations ([REF]), we simply shift the quasiparticle energy from [MATH] ([REF]) to [MATH] ([REF]) and [MATH] becomes a Fermi distribution in [MATH] for any [MATH] and with the same [MATH] as for the FES distribution.", '1303.5493-3-22-2': 'For example one may substitute [MATH] into Eq. ([REF]) and obtain the same population for the species with lowest energy, [MATH], presented in the FES description in Fig. [REF](c).', '1303.5493-3-23-0': 'The particle density, [MATH], is represented in Fig. [REF](d).', '1303.5493-3-23-1': 'Due to the symmetry of the problem, we have pair-wise identical populations, [MATH] and [MATH], for any [MATH] and [MATH].', '1303.5493-3-23-2': 'The largest deviations in the particle density occurs for the extremal species, placed at both ends of the 1D box as an effect of repulsive interactions.', '1303.5493-3-24-0': 'We observe from Eqs. ([REF]), ([REF]) and from Fig. [REF] that because [MATH], the range of [MATH] is [MATH] in any elementary volume, [MATH], whereas the range of [MATH] is [MATH], with [MATH].', '1303.5493-3-25-0': '# Analytical examples', '1303.5493-3-26-0': 'Calogero-Sutherland model in a one-dimensional harmonic trap.', '1303.5493-3-27-0': 'In Ref. [CITATION] Murthy and Shankar analyzed the Calogero-Sutherland model (CSM), i.e. a 1D system of fermions in a harmonic potential of frequency [MATH], with inverse square law particle-particle interaction potential, [MATH], [EQUATION] where [MATH] is the fixed total number of particles and we take, like in Ref. [CITATION], [MATH], [MATH] being the particle mass.', '1303.5493-3-27-1': 'Such a system is not solvable in the TF approximation, but its spectrum is exactly known [CITATION]: [EQUATION] where [MATH], [MATH] is an integer, [MATH] is the occupation number, and [MATH].', '1303.5493-3-27-2': 'The energy ([REF]) is of mean-field type ([REF]) and from this point the formalism of Section [REF] may be applied straightforwardly, with [MATH], independent of [MATH].', '1303.5493-3-27-3': 'The quasiparticle energies are [CITATION] [EQUATION] and the species are small intervals, [MATH], centered at [MATH], along the quasiparticle energy axis.', '1303.5493-3-27-4': 'In this way, from ([REF]) and observing that [MATH], we get the "diagonal" FES parameters, [EQUATION]', '1303.5493-3-27-5': 'From ([REF]) we get [MATH] for any [MATH], since the DOS is constant in this case.', '1303.5493-3-28-0': 'Finally, Eq. ([REF]) is not applicable to this system since we work with single-particle states extended over the whole system and not in the TF approximation.', '1303.5493-3-28-1': 'In conclusion we can write in general that [MATH].', '1303.5493-3-29-0': 'This result is identical with that of Murthy and Shankar, considering that we calculate [MATH] in the fermionic picture, whereas [MATH] of Ref. [CITATION] was calculated in the bosonic picture.', '1303.5493-3-29-1': "The two [MATH]'s should satisfy the relation [MATH] (see Section [REF]), which is correct.", '1303.5493-3-30-0': 'Calogero-Sutherland model on a ring.', '1303.5493-3-31-0': 'FES may be applied not only in the energy space, but also in the (quasi)momentum space [CITATION].', '1303.5493-3-31-1': 'Following [CITATION] (and keeping [MATH]) for a CSM system of [MATH] particles on a ring of length [MATH], the equation for the asymptotic momentum, [MATH], is [EQUATION]', '1303.5493-3-31-2': 'The sum is taken over all the particles in the system, [MATH] is an integer, and [MATH] is the phase shift due to the particle-particle interaction.', '1303.5493-3-31-3': 'The total number of particles, momentum and energy of the system are [EQUATION] respectively.', '1303.5493-3-31-4': 'Since [MATH] takes integer values, Eq. ([REF]) leads to a density of states along the [MATH] axis [CITATION], [EQUATION]', '1303.5493-3-31-5': 'The species are defined as intervals [MATH], centered at [MATH], along the momentum axis, and the FES parameters are [EQUATION] where [MATH], [MATH] is the occupation of the state with asymptotic momentum [MATH].', '1303.5493-3-31-6': 'If the interaction is like in the previous example, [MATH], then [MATH], with [MATH] being the sign of [MATH], and we obtain again [MATH] [CITATION].', '1303.5493-3-32-0': 'The problem may be transferred from the momentum space to the quasiparticle energy space.', '1303.5493-3-32-1': 'The total energy of the system is [MATH] ([REF]) and the quasiparticle energy is [EQUATION]', '1303.5493-3-32-2': 'The species [MATH] along the [MATH] axis are mapped into species [MATH] along the [MATH] axis.', '1303.5493-3-32-3': 'To each species [MATH] it corresponds two species, [MATH] and [MATH], symmetric with respect to the origin on the [MATH] axis - that is, if [MATH] is centered at [MATH], then [MATH] is centered at [MATH] and [MATH].', '1303.5493-3-32-4': 'Therefore every two symmetric species, [MATH] and [MATH] are combined into one energy species, [MATH].', '1303.5493-3-32-5': 'If the dimensions of the species on the [MATH] axis are [MATH] and [MATH], respectively, then the dimension of the species [MATH] is [MATH].', '1303.5493-3-32-6': 'A similar relation holds for the particle numbers - [MATH].', '1303.5493-3-32-7': 'The FES parameters in the [MATH] space are obtained by applying the rules of Ref. [CITATION].', '1303.5493-3-32-8': 'If we calculate the [MATH], which connects the species [MATH] to the species [MATH], then the following relations have to be satisfied: [EQUATION] where [MATH] corresponds to the intervals [MATH] and [MATH] on the [MATH] axis, whereas [MATH] corresponds to the intervals [MATH] and [MATH].', '1303.5493-3-32-9': 'If [MATH], then [EQUATION] like in the case of CSM in harmonic trap.', '1303.5493-3-33-0': '# Conclusions', '1303.5493-3-34-0': 'In conclusion we have formulated an approach by which a system of quantum particles with general particle-particle interaction, [MATH], in an [MATH]-dimensional space and external potential [MATH] is described in the quasiclassical limit as an ideal gas of fractional exclusion statistics (FES).', '1303.5493-3-34-1': 'We have given the equations for the calculation of the FES parameters and equilibrium populations.', '1303.5493-3-35-0': 'The FES approach has been compared with the Thomas-Fermi (TF) formalism and we have shown that although there are differences in the definitions of certain quantities like the quasiparticle energies, the physical results are the same.', '1303.5493-3-35-1': 'The main difference between the two formalisms is that in the FES approach the quasiparticle energies are independent of the populations of other quasiparticle states and therefore the FES gas is "ideal", with the total energy of the gas being equal to the sum of the quasiparticle energies, whereas in the TF approach the quasiparticles are interacting and the energy of the quasiparticle gas is not equal to the energy of the system.', '1303.5493-3-36-0': 'We have exemplified our procedure on a one-dimensional system of fermions with repulsive Coulomb interaction for which we calculated the main microscopic parameters, like the quasiparticle energies, quasiparticle density of states and energy levels populations.', '1303.5493-3-36-1': 'For each of these quantities we discussed the similarities and differences between the FES and the TF approaches.', '1303.5493-3-36-2': 'We also applied our procedure on the one-dimensional Calogero-Sutherland model (CSM) which is well studied in the literature [CITATION] and proved that the results are consistent.', '1303.5493-3-37-0': 'One practical consequence that appear from our calculations is that the solution of the FES integral equations may eventually be calculated easier by solving self-consistently the TF equations for population and quasiparticle energies, ([REF]) and ([REF]).', '1303.5493-3-38-0': 'Another consequence is that while in the TF formulation the quasiparticle energies may form an energy gap at the lowest end of the spectrum due to the particle-particle interaction, in the FES description such an energy gap does not exist.', '1303.5493-3-39-0': 'By establishing the equivalence between the self-consistent mean-field theory and the FES approach we show that in general a quasi-classical interacting system can be mapped onto an ideal FES system.'}

{'1303.5493-4-0-0': 'We describe a mean field interacting particle system in any number of dimensions and in a generic external potential as an ideal gas with fractional exclusion statistics (FES).', '1303.5493-4-0-1': 'We define the FES quasiparticle energies, we calculate the FES parameters of the system and we deduce the equations for the equilibrium particle populations.', '1303.5493-4-0-2': 'The FES gas is "ideal," in the sense that the quasiparticle energies do not depend on the other quasiparticle levels populations and the sum of the quasiparticle energies is equal to the total energy of the system.', '1303.5493-4-0-3': 'We prove that the FES formalism is equivalent to the semi-classical or Thomas Fermi limit of the self-consistent mean-field theory and the FES quasiparticle populations may be calculated from the Landau quasiparticle populations by making the correspondence between the FES and the Landau quasiparticle energies.', '1303.5493-4-0-4': 'The FES provides a natural semi-classical ideal gas description of the interacting particle gas.', '1303.5493-4-1-0': '# Introduction', '1303.5493-4-2-0': 'Quantum (Bose or Fermi) statistics is a very basic concept in physics, directly arising from the indiscernability principle of the microscopic world.', '1303.5493-4-2-1': 'However effective models employing different statistics have proved to be very useful to describe some selected aspects of complex interacting microscopic systems.', '1303.5493-4-2-2': 'As an example, an ideal Fermi gas with Fermi-Dirac statistics can be described in certain approximations by a Boltzmann distribution with repulsive interaction, while a bosonic ideal gas, by an attractive potential [CITATION].', '1303.5493-4-2-3': 'On more general grounds, the statistical mechanics of fractional exclusion statistics (FES) quasi-particle systems was formulated by several authors [CITATION], based on the FES concept introduced by Haldane in Ref. [CITATION].', '1303.5493-4-2-4': 'This has been a prolific concept and was applied to both quantum and classical systems (see e.g. [CITATION]).', '1303.5493-4-3-0': 'A stochastic method for the simulation of the time evolution of FES systems was introduced in Ref. [CITATION] as a generalization of a similar method used for Bose and Fermi systems [CITATION], whereas the relatively recent experimental realization of the Fermi degeneracy in cold atomic gases has renewed the interest in the theoretical investigation of non-ideal Fermi systems at low temperatures and their interpretation as ideal FES systems [CITATION].', '1303.5493-4-4-0': 'The general FES formalism was amended to include the change of the FES parameters at the change of the particle species [CITATION].', '1303.5493-4-4-1': 'This amendment allows the general implementation of FES as a method for the description of interacting particle systems as ideal (quasi)particle gases [CITATION].', '1303.5493-4-4-2': 'However, the level of approximation of such a description and the connections with other many-particle methods are not yet clear.', '1303.5493-4-5-0': 'A rigorous connection between the FES and the Bethe ansatz equations for an exactly solvable model was worked out in Refs. [CITATION] in the one-dimensional case.', '1303.5493-4-5-1': 'In higher dimensions, one can expect that the statistics and interaction would not be transmutable in general because interaction and exchange positions between two particles can occur separately.', '1303.5493-4-5-2': 'However indications exist that a mapping might exist between ideal FES and interacting particles with regular quantum statistics in the quasi-particle semi-classical approximation given by the self-consistent mean-field theory.', '1303.5493-4-5-3': 'Indeed, in Refs. [CITATION] the FES was applied to describe Bose gases with local ([MATH]-function) interaction in two-dimensional traps in the Thomas Fermi (TF) limit.', '1303.5493-4-5-4': 'In this paper we generalize this result to apply FES to Bose and Fermi systems of particles with generic two-body interactions in arbitrary external potentials in the Landau or TF semiclassical limit in any number of dimensions.', '1303.5493-4-5-5': 'We define quasiparticle energies which determine our FES parameters and using these we calculate the FES equilibrium particle distribution.', '1303.5493-4-5-6': "Moreover, we calculate the particle distribution also starting from the mean-field description by defining the Landau type of quasiparticles, and we show that the two descriptions are equivalent, i.e. the populations are identical, provided that we make the mapping between the FES and Landau's quasiparticle energies.", '1303.5493-4-5-7': 'This equivalence proves that our FES description of the interacting particle system corresponds to a self-consistent mean-field approximation.', '1303.5493-4-5-8': 'Such an approximation, though certainly not adequate to fully describe strongly interacting quantum systems, is the basis of a number of highly predictive theoretical methods in many-body physics, from Landau Fermi liquid theory to density functional methods in correlated electron or nucleon systems.', '1303.5493-4-6-0': 'The structure of the paper is as follows.', '1303.5493-4-6-1': "In Section [REF] we introduce our model and calculate Landau's equilibrium particle population in the TF approach.", '1303.5493-4-6-2': 'In Section [REF] we implement the FES description by using alternative quasiparticle energies and a definition of species.', '1303.5493-4-6-3': 'These species are related by the FES parameters, which we calculate.', '1303.5493-4-6-4': 'Using the FES parameters and quasiparticle energies we write the equations for the equilibrium particle populations.', '1303.5493-4-6-5': "By making the correspondence between Landau's and the FES quasiparticle energies, we show that the FES equations are satisfied by Landau's populations.", '1303.5493-4-6-6': "This proves that the FES formalism is consistent and suitable for the description of such interacting particle systems and that the FES quasiparticle populations may be calculated by Landau's approach, using the correspondence between the quasiparticle energies.", '1303.5493-4-6-7': 'In Sections [REF] and [REF] we show some numerical and analytical examples, respectively, whereas in Section [REF] we give the conclusions.', '1303.5493-4-7-0': "# The model in Landau's approach", '1303.5493-4-8-0': 'Let us consider a system of [MATH] interacting particles described by the generic Hamiltonian: [EQUATION] where the indexes [MATH] denote the single particle states, and [MATH] are creation (annihilation) operators obeying commutations rules which define the quantum statistics of the system (Bose or Fermi).', '1303.5493-4-8-1': 'We assume that the particle-particle interaction [MATH] depends only on the distance between the particles, whereas the external potential is [MATH].', '1303.5493-4-8-2': 'In the mean-field approximation the total energy of the system can be written as [EQUATION] where [MATH] are single-particle kinetic energies, [MATH], and [MATH] are antisymmetrized (symmetrized) matrix elements for fermions (bosons).', '1303.5493-4-8-3': 'The upper and lower signs in the superscripts of the occupation numbers [MATH] stand for fermions and bosons, respectively.', '1303.5493-4-9-0': 'At the thermodynamic limit, the finite temperature properties of the system can be accessed via the grandcanonical partition sum defined by the mean-field one-body entropy as [EQUATION]', '1303.5493-4-9-1': "Maximizing this function with respect to the single particle occupations gives the equilibrium particle populations, [EQUATION] where the quantities [MATH] are Landau's quasiparticle energies.", '1303.5493-4-10-0': 'If we assume a large number of particles, and a sufficiently slowly varying external potential, we can employ the Thomas-Fermi (or Landau) theory which amounts to extending this mean-field formalism to a finite inhomogeneous system employing a semi-classical limit for the kinetic energy.', '1303.5493-4-10-1': 'We can divide the system into macroscopic cells, [MATH], centered at [MATH], where the external field is locally constant and apply the thermodynamic limit in each cell.', '1303.5493-4-10-2': 'The single-particle energies are continuous variables and it is meaningful to introduce the density of states (DOS) in each cell as [MATH].', '1303.5493-4-10-3': 'Then the quasi-particle energies read [EQUATION]', '1303.5493-4-10-4': 'We observe that the sum of the quasiparticle energies [EQUATION] is not equal to the total mean-field energy [MATH], because of the well-known double counting of the interactions.', '1303.5493-4-11-0': '# The FES formalism', '1303.5493-4-12-0': 'Now we want to show that the semi-classical Thomas-Fermi results can be reproduced by an ideal system, provided that such a system obeys FES.', '1303.5493-4-12-1': 'A FES system consists of a countable number of species, denoted here by an index, [MATH] or [MATH] (we use capital letters to denote species).', '1303.5493-4-12-2': 'Each species [MATH] contains [MATH] available single-particle states and [MATH] particles, each of them of energy [MATH] and chemical potential [MATH].', '1303.5493-4-12-3': 'If the particles are bosons, [MATH] represents also the number of states in the species.', '1303.5493-4-12-4': 'If the particles are fermions, the number of single-particle states in the species [MATH] is [MATH].', '1303.5493-4-12-5': 'The FES character of the system consists of the fact that if the number of particles in one species, for example, in species [MATH], changes by [MATH], then the number of states in any other species [MATH] changes as [MATH] for bosons, or [MATH] for fermions.', '1303.5493-4-12-6': 'The parameters [MATH] are called the FES parameters.', '1303.5493-4-12-7': 'The total number of micro-configurations in the system is [EQUATION] for fermions; the products in Eqs. ([REF]) are taken over all the species in the system.', '1303.5493-4-12-8': 'Using Eqs. ([REF]) and the fact that all the particles in a species have the same energy and chemical potential, we write the partition functions, [EQUATION] where we used the notations [EQUATION]', '1303.5493-4-12-9': 'In Eqs. ([REF]) we made the usual simplifying assumption that all the chemical potentials in all the species are the same, i.e. [MATH] for all [MATH].', '1303.5493-4-12-10': 'This assumption is justified by the fact that in the present application all the particles are identical and, as we shall see below, a species represents a single particle energy interval.', '1303.5493-4-13-0': 'The equilibrium populations are obtained by maximizing [MATH] and [MATH] with respect to [MATH], taking into account the variation of the number of states in the species with the number of particles.', '1303.5493-4-13-1': 'We obtain the equations [EQUATION] where [MATH] and [MATH] [CITATION].', '1303.5493-4-14-0': 'A fermionic system may be transformed into a bosonic system if we define [MATH] and [MATH], which leads to [MATH].', '1303.5493-4-15-0': 'In the ideal system the total energy is equal to the sum of quasiparticle energies, which are independent of the populations.', '1303.5493-4-15-1': 'We define alternative quasiparticle energies-the FES quasiparticle energies-as [EQUATION] and we can immediately check that [EQUATION]', '1303.5493-4-15-2': 'Equation ([REF]) shows that the total energy can be obtained as a simple sum of quasi-particles energies, meaning that these latter can be viewed as an ideal gas.', '1303.5493-4-15-3': 'This surprising result for an interacting system is well known in the context of FES.', '1303.5493-4-15-4': 'From Eq. ([REF]) we obtain the DOS along the [MATH] axis, [EQUATION]', '1303.5493-4-15-5': 'Although [MATH] depends explicitly on [MATH] ([REF]), we can transfer this dependence to the statistical interaction through the FES parameters which leaves the quasiparticle gas an ideal gas, as explained, for example, in Ref. [CITATION].', '1303.5493-4-15-6': 'The FES parameters are calculated similarly to Refs. [CITATION].', '1303.5493-4-15-7': 'To this aim in each such volume [MATH] the quasiparticle energy axis [MATH] is split into elementary intervals [MATH] centered at [MATH].', '1303.5493-4-15-8': 'Each [MATH]-dimensional elementary volume, [MATH], represents a FES species, as indicated in Fig. [REF].', '1303.5493-4-15-9': 'Between species with the same energy, [MATH] [arrow (a) in Fig. [REF]], but located in different volumes, [MATH] and [MATH], we have the FES parameters [EQUATION]', '1303.5493-4-15-10': 'The presence of non-zero FES parameters is a proof that the system obeys FES.', '1303.5493-4-16-0': 'We now turn to show that the whole thermodynamics can be equivalently calculated in the TF and in the FES approaches.', '1303.5493-4-16-1': 'Plugging the FES parameters ([REF]) and the quasiparticle energies ([REF]) into the FES equations ([REF]) we obtain [EQUATION]', '1303.5493-4-16-2': 'Now we can check the equivalence between the FES and the TF descriptions of the system by substituting Eq. ([REF]) into Eq. ([REF]).', '1303.5493-4-16-3': 'By doing so we recover the relation between [MATH] and [MATH], [EQUATION] which is in accordance with the definitions ([REF]) and ([REF]).', '1303.5493-4-17-0': 'Equations ([REF]) and ([REF]), as well as Eqs. ([REF]) and ([REF]), make sense only if the integrals in these expressions converge.', '1303.5493-4-17-1': 'This is a limitation of the mean-field formalism.', '1303.5493-4-18-0': '# Numerical example', '1303.5493-4-19-0': 'We illustrate our model and the relation between the FES and the TF descriptions on a one dimensional system of fermions with repulsive Coulomb interactions, [MATH], in the absence of external fields.', '1303.5493-4-19-1': 'The length of the system, [MATH], is discretized as in Ref. [CITATION] into [MATH] equal elementary segments, [MATH], where [MATH].', '1303.5493-4-19-2': 'In each such elementary "volume" the DOS is taken to be constant, [MATH], and on the [MATH] axis we define [MATH] equal consecutive segments between 0 and [MATH], [MATH], where [MATH] - we choose [MATH] such that [MATH] for any [MATH].', '1303.5493-4-19-3': 'In this way we obtain [MATH] species of particles, [MATH], identified also by a double index, [MATH] [CITATION].', '1303.5493-4-19-4': 'To avoid the singularity at the origin of the interaction potential, we consider a cut-off distance of [MATH].', '1303.5493-4-20-0': 'The total number of particles in the system is [MATH], where [MATH] is the Fermi energy in the noninteracting system.', '1303.5493-4-20-1': 'We set the energy scale of the system by fixing [MATH] and [MATH].', '1303.5493-4-21-0': 'Figure [REF](a) shows the quasiparticle density of states in the FES description, [MATH] (Eq. [REF]).', '1303.5493-4-21-1': "The density of Landau's quasiparticle states may be calculated also as [MATH] and we obtain [MATH], which is different from zero only for [MATH].", '1303.5493-4-21-2': 'The minimum value of [MATH] ([REF]) is [MATH].', '1303.5493-4-21-3': 'The position dependent [MATH] is plotted in Fig. [REF] (b).', '1303.5493-4-22-0': 'Figure [REF](c) shows the populations, [MATH] ([REF]).', '1303.5493-4-22-1': "To obtain Landau's populations ([REF]), we simply shift the quasiparticle energy from [MATH] ([REF]) to [MATH] ([REF]) and [MATH] becomes a Fermi distribution in [MATH] for any [MATH] and with the same [MATH] as for the FES distribution.", '1303.5493-4-22-2': 'For example one may substitute [MATH] into Eq. ([REF]) and obtain the same population for the species with the lowest energy, [MATH], presented in the FES description in Fig. [REF](c).', '1303.5493-4-23-0': 'The particle density, [MATH], is represented in Fig. [REF](d).', '1303.5493-4-23-1': 'Due to the symmetry of the problem, we have pair-wise identical populations, [MATH] and [MATH], for any [MATH] and [MATH].', '1303.5493-4-23-2': 'The largest deviations in the particle density occur for the extremal species, placed at both ends of the one-dimensional (1D) box as an effect of repulsive interactions.', '1303.5493-4-24-0': 'We observe from Eqs. ([REF]) and ([REF]) and from Fig. [REF] that because [MATH], the range of [MATH] is [MATH] in any elementary volume [MATH], whereas the range of [MATH] is [MATH], with [MATH].', '1303.5493-4-25-0': '# Analytical examples', '1303.5493-4-26-0': 'Calogero-Sutherland model in a one-dimensional harmonic trap.', '1303.5493-4-27-0': 'In Ref. [CITATION] Murthy and Shankar analyzed the Calogero-Sutherland model (CSM), i.e. a 1D system of fermions in a harmonic potential of frequency [MATH], with inverse square law particle-particle interaction potential, [MATH], [EQUATION] where [MATH] is the fixed total number of particles and we take, like in Ref. [CITATION], [MATH], [MATH] being the particle mass.', '1303.5493-4-27-1': 'Such a system is not solvable in the TF approximation, but its spectrum is exactly known [CITATION]: [EQUATION] where [MATH], [MATH] is an integer, [MATH] is the occupation number, and [MATH].', '1303.5493-4-27-2': 'The energy ([REF]) is of mean-field type ([REF]) and from this point the formalism of Section [REF] may be applied straightforwardly, with [MATH], independent of [MATH].', '1303.5493-4-27-3': 'The quasiparticle energies are [CITATION] [EQUATION] and the species are small intervals [MATH] centered at [MATH], along the quasiparticle energy axis.', '1303.5493-4-27-4': 'In this way, from ([REF]) and observing that [MATH], we get the "diagonal" FES parameters, [EQUATION]', '1303.5493-4-27-5': 'From ([REF]) we get [MATH] for any [MATH], since the DOS is constant in this case.', '1303.5493-4-28-0': 'Finally, Eq. ([REF]) is not applicable to this system since we work with single-particle states extended over the whole system and not in the TF approximation.', '1303.5493-4-28-1': 'In conclusion we can write, in general, that [MATH].', '1303.5493-4-29-0': 'This result is identical with that of Murthy and Shankar, considering that we calculate [MATH] in the fermionic picture, whereas [MATH] of Ref. [CITATION] was calculated in the bosonic picture.', '1303.5493-4-29-1': "The two [MATH]'s should satisfy the relation [MATH] (see Section [REF]), which is correct.", '1303.5493-4-30-0': 'Calogero-Sutherland model on a ring.', '1303.5493-4-31-0': 'FES may be applied not only in the energy space, but also in the (quasi)momentum space [CITATION].', '1303.5493-4-31-1': 'Following Ref. [CITATION] (and keeping [MATH]) for a CSM system of [MATH] particles on a ring of length [MATH], the equation for the asymptotic momentum [MATH] is [EQUATION]', '1303.5493-4-31-2': 'The sum is taken over all the particles in the system, [MATH] is an integer, and [MATH] is the phase shift due to the particle-particle interaction.', '1303.5493-4-31-3': 'The total number of particles, the momentum, and the energy of the system are [EQUATION] respectively.', '1303.5493-4-31-4': 'Since [MATH] takes integer values, Eq. ([REF]) leads to a density of states along the [MATH] axis [CITATION], [EQUATION]', '1303.5493-4-31-5': 'The species are defined as intervals [MATH], centered at [MATH], along the momentum axis, and the FES parameters are [EQUATION] where [MATH], and [MATH] is the occupation of the state with asymptotic momentum [MATH].', '1303.5493-4-31-6': 'If the interaction is like in the previous example, [MATH], then [MATH], with [MATH] being the sign of [MATH], and we obtain again [MATH] [CITATION].', '1303.5493-4-32-0': 'The problem may be transferred from the momentum space to the quasiparticle energy space.', '1303.5493-4-32-1': 'The total energy of the system is [MATH] ([REF]) and the quasiparticle energy is [EQUATION]', '1303.5493-4-32-2': 'The species [MATH] along the [MATH] axis are mapped into species [MATH] along the [MATH] axis.', '1303.5493-4-32-3': 'To each species [MATH], there corresponds two species, [MATH] and [MATH], symmetric with respect to the origin on the [MATH] axis - that is, if [MATH] is centered at [MATH], then [MATH] is centered at [MATH] and [MATH].', '1303.5493-4-32-4': 'Therefore every two symmetric species, [MATH] and [MATH] are combined into one energy species, [MATH].', '1303.5493-4-32-5': 'If the dimensions of the species on the [MATH] axis are [MATH] and [MATH], respectively, then the dimension of the species [MATH] is [MATH].', '1303.5493-4-32-6': 'A similar relation holds for the particle numbers: [MATH].', '1303.5493-4-32-7': 'The FES parameters in the [MATH] space are obtained by applying the rules of Ref. [CITATION].', '1303.5493-4-32-8': 'If we calculate the [MATH], which connects the species [MATH] to the species [MATH], then the following relations have to be satisfied: [EQUATION] where [MATH] corresponds to the intervals [MATH] and [MATH] on the [MATH] axis, whereas [MATH] corresponds to the intervals [MATH] and [MATH].', '1303.5493-4-32-9': 'If [MATH], then [EQUATION] like in the case of the CSM in a harmonic trap.', '1303.5493-4-33-0': '# Conclusions', '1303.5493-4-34-0': 'In conclusion we have formulated an approach by which a system of quantum particles with general particle-particle interaction [MATH] in an [MATH]-dimensional space and external potential [MATH] is described in the quasiclassical limit as an ideal gas of FES.', '1303.5493-4-34-1': 'We have given the equations for the calculation of the FES parameters and equilibrium populations.', '1303.5493-4-35-0': 'The FES approach has been compared with the TF formalism and we have shown that although there are differences in the definitions of certain quantities like the quasiparticle energies, the physical results are the same.', '1303.5493-4-35-1': 'The main difference between the two formalisms is that in the FES approach the quasiparticle energies are independent of the populations of other quasiparticle states and therefore the FES gas is "ideal", with the total energy of the gas being equal to the sum of the quasiparticle energies, whereas in the TF approach the quasiparticles are interacting and the energy of the quasiparticle gas is not equal to the energy of the system.', '1303.5493-4-36-0': 'We have exemplified our procedure on a one-dimensional system of fermions with repulsive Coulomb interaction for which we calculated the main microscopic parameters, like the quasiparticle energies, quasiparticle density of states, and energy levels populations.', '1303.5493-4-36-1': 'For each of these quantities we discussed the similarities and differences between the FES and the TF approaches.', '1303.5493-4-36-2': 'We also applied our procedure on the one-dimensional CSM which is well studied in the literature [CITATION] and proved that the results are consistent.', '1303.5493-4-37-0': 'One practical consequence that appears from our calculations is that the solution of the FES integral equations may eventually be calculated easier by solving self-consistently the TF equations for population and quasiparticle energies, ([REF]) and ([REF]).', '1303.5493-4-38-0': 'Another consequence is that while in the TF formulation the quasiparticle energies may form an energy gap at the lowest end of the spectrum due to the particle-particle interaction, in the FES description such an energy gap does not exist.', '1303.5493-4-39-0': 'By establishing the equivalence between the self-consistent mean-field theory and the FES approach we show that in general a quasi-classical interacting system can be mapped onto an ideal FES system.'}

0904.4352

{'0904.4352-1-0-0': 'Kingman Cheung[MATH], Otto C. W. Kong[MATH], and Jae Sik Lee[MATH] Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300', '0904.4352-1-1-0': '[MATH]Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300', '0904.4352-1-2-0': '[MATH]Division of Quantum Phases Devices, Konkuk University, Seoul 143-701, Korea', '0904.4352-1-3-0': '[MATH]Department of Physics and Center for Mathematics and Theoretical Physics,', '0904.4352-1-4-0': 'National Central University, Chung-Li, Taiwan 32054', '0904.4352-1-5-0': 'ABSTRACT', '0904.4352-1-6-0': 'We study the electric dipole moment (EDM) and the anomalous magnetic dipole moment (MDM) of the muon in the CP-violating Minimal Supersymmetric extension of the Standard Model (MSSM).', '0904.4352-1-6-1': 'We take into account the contributions from the chargino- and neutralino-mediated one-loop graphs and the dominant two-loop Higgs-mediated Barr-Zee diagrams.', '0904.4352-1-6-2': 'We improve earlier calculations by incorporating CP-violating Higgs-boson mixing effects and the resummed threshold corrections to the Yukawa couplings of the charged leptons as well as that of the bottom quark.', '0904.4352-1-6-3': 'The analytic correlation between the muon EDM and MDM is explicitly presented at one- and two-loop levels and, through several numerical examples, we illustrate its dependence on the source of the dominant contributions.', '0904.4352-1-6-4': 'We have implemented the analytic expressions for the muon EDM and MDM in an updated version of the public code CPsuperH2.0.', '0904.4352-1-7-0': '# Introduction', '0904.4352-1-8-0': 'The anomalous magnetic dipole moment of the muon, [MATH], has provided one of the most sensitive test grounds for the validity of the Standard Model (SM) [CITATION]: [EQUATION]', '0904.4352-1-8-1': 'At the same time, it also provides an important constraint on new physics with precise SM predictions available.', '0904.4352-1-8-2': 'The SM prediction consists of QED, electroweak (EW), and hadronic contributions.', '0904.4352-1-8-3': 'The hadronic contribution is further decomposed into leading-order (LO) part and higher-order vacuum polarization (VP) and light-by-light (LBL) parts [CITATION] : [EQUATION]', '0904.4352-1-8-4': 'Equations ([REF]) and ([REF]) suggest that there is currently a [MATH] discrepancy between the experimental result and the SM prediction, which can be attributed to possible contributions from physics beyond the SM : [EQUATION]', '0904.4352-1-8-5': 'One of the most appealing scenarios for physics beyond the SM is augmented with a softly broken supersymmetry (SUSY) around the TeV scale.', '0904.4352-1-8-6': 'The supersymmetric contributions to [MATH] from such models are known up to two-loop level.', '0904.4352-1-8-7': 'The one-loop results can be found in [CITATION] and the two-loop results in [CITATION].', '0904.4352-1-8-8': 'It is well-known that the dominant two-loop contribution comes from Higgs-mediated Barr-Zee diagrams [CITATION].', '0904.4352-1-8-9': 'The error associated with the known SUSY contributions is estimated to be [MATH] [CITATION], which is smaller than half of the current experimental and SM theoretical ones.', '0904.4352-1-9-0': 'On the other hand, the SUSY augmented models can contain additional CP-violating phases beyond the SM Cabibbo-Kobayashi-Maskawa (CKM) phase leading to sizable EDMs [CITATION].', '0904.4352-1-9-1': 'The current limit on the muon EDM [CITATION] is [EQUATION] which is much weaker than the constraints from the non-observation of the Thallium [CITATION], neutron [CITATION], and Mercury [CITATION] EDMs: [EQUATION]', '0904.4352-1-9-2': 'Nevertheless, if the muon EDM experiment in the future can achieve the projected sensitivity [CITATION] [EQUATION] the precision of the experiment will be comparable to that of the current Thallium EDM experiment.', '0904.4352-1-10-0': 'In this paper, we study the correlation between the muon EDM and MDM in the CP-violating MSSM [CITATION].', '0904.4352-1-10-1': 'We present the relation between the one-loop chargino- and neutralino-mediated EDM and MDM of the muon.', '0904.4352-1-10-2': 'We also derive an analytic relation between them in the two-loop contributions from the dominant Higgs-mediated Barr-Zee diagrams.', '0904.4352-1-10-3': 'We improve the earlier results by including CP-violating Higgs-boson mixing effects in the Barr-Zee diagrams and resumming the threshold corrections to the muon, tau, and bottom-quark Yukawa couplings in the one- and two-loop graphs.', '0904.4352-1-10-4': 'We then focus our numerical studies on three types of scenarios in which [MATH] the muon EDM and MDM are dominated by the one-loop contributions, [MATH] the lightest Higgs boson is mostly CP odd and lighter than [MATH] 50 GeV, and [MATH] the dominant contributions to the muon EDM and MDM come from the two-loop Barr-Zee graphs.', '0904.4352-1-11-0': 'For the presentation of our analytic results, we follow the conventions and notations of CPsuperH [CITATION], especially for the masses and mixing matrices of the neutral Higgs bosons and SUSY particles.', '0904.4352-1-11-1': 'The layout of the paper is as follows.', '0904.4352-1-11-2': 'Section 2 presents formulas relevant to the one-loop contributions to the muon EDM and MDM from chargino- and neutralino-mediated diagrams.', '0904.4352-1-11-3': 'Non-holomorphic threshold effects on the muon Yukawa coupling have been appropriately resummed.', '0904.4352-1-11-4': 'In Section 3, we present analytic results for the Higgs-mediated two-loop Barr-Zee diagrams.', '0904.4352-1-11-5': 'For this, most importantly, the resummed threshold corrections to the tau and bottom-quark Yukawa couplings and the CP-violating Higgs-boson mixing effects have been incorporated.', '0904.4352-1-11-6': 'In Section 4, we present some numerical examples, depending on the source of dominant contributions to the muon EDM and MDM.', '0904.4352-1-11-7': 'We summarize our findings in Section 5.', '0904.4352-1-12-0': '# One-Loop EDMs and MDMs of charged leptons', '0904.4352-1-13-0': 'The relevant interaction Lagrangian of the spin-1/2 lepton with EDM [MATH] and MDM [MATH] is given by [EQUATION] where [MATH].', '0904.4352-1-13-1': 'The EDM and MDM amplitudes are given by [EQUATION]', '0904.4352-1-13-2': 'Generic interactions of charginos [MATH] or neutralinos [MATH], collectively denoted by [MATH], with a lepton [MATH] and a slepton (sneutrino) [MATH]) are given by [EQUATION]', '0904.4352-1-13-3': 'The Lagrangian for the interactions of [MATH] and [MATH] with the photon field [MATH] is [EQUATION]', '0904.4352-1-13-4': 'The diagrams in Fig. [REF] induce the MDM and EDM of the lepton [MATH] as follows: [EQUATION] where [EQUATION]', '0904.4352-1-13-5': 'We have checked that our analytic expressions for the one-loop MDM and EDM agree with those given in, for example, Ref. [CITATION] and [CITATION] with [MATH] and [MATH].', '0904.4352-1-13-6': 'Note [MATH], [MATH] and [MATH].', '0904.4352-1-13-7': 'Finally, in Eq. ([REF]), the chargino-lepton-sneutrino couplings are given by [EQUATION] and the neutralino-lepton-slepton couplings are given by [EQUATION] with [MATH] and [MATH].', '0904.4352-1-14-0': 'There are non-holomorphic threshold corrections to the Yukawa couplings [MATH] which appear in the chargino and neutralino couplings [CITATION].', '0904.4352-1-14-1': 'These corrections become significant at large [MATH] and we resum these effects by redefining the Yukawa couplings as follows: [EQUATION]', '0904.4352-1-14-2': 'In the presence of the CP phases, [MATH] takes the form [CITATION] [EQUATION] where the one-loop function is [EQUATION]', '0904.4352-1-14-3': 'These effects on the muon MDM have been considered in Ref. [CITATION] only in the CP conserving case.', '0904.4352-1-15-0': '# Barr-Zee Graphs', '0904.4352-1-16-0': 'The dominant two-loop contributions come from the Barr-Zee diagrams mediated by neutral Higgs-boson exchanges, see Fig. [REF].', '0904.4352-1-16-1': 'Here, we consider loops of third-generation fermions and sfermions, charged Higgs bosons, and charginos.', '0904.4352-1-16-2': 'The diagrams can be evaluated first by one-loop computation of the [MATH]-[MATH]-[MATH] vertex.', '0904.4352-1-16-3': 'Thanks to gauge invariance the effective vertex takes the form [EQUATION] where [MATH] is the incoming four-momentum of the external photon, [MATH]) the four-momentum of the incoming (outgoing) lepton, and [MATH] the four-momentum of the internal photon going out of the upper loop.', '0904.4352-1-16-4': 'Note that [MATH].', '0904.4352-1-16-5': 'Explicitly, keeping only the terms linear in the external momenta, the fermionic contributions are given by [EQUATION] with the generic [MATH]-[MATH]-[MATH] interaction: [MATH].', '0904.4352-1-16-6': 'The color factor [MATH] for quarks and 1 for leptons and charginos.', '0904.4352-1-16-7': 'On the other hand, the sfermion loops contribute only to the scalar form factor as [EQUATION] with the generic [MATH]-[MATH]-[MATH] interaction: [MATH].', '0904.4352-1-17-0': 'Together with the interaction Lagrangian for the couplings of the neutral-Higgs bosons and photon to charged leptons [EQUATION] where [MATH] (as in the CPsuperH convention), the Barr-Zee amplitude is given by [EQUATION] where again we keep only the terms linear in the external momenta.', '0904.4352-1-17-1': 'We note the numerator of the integrand is proportional to [EQUATION]', '0904.4352-1-17-2': 'We observe that the first term gives the MDM while the second one gives the EDM of the lepton [MATH].', '0904.4352-1-17-3': 'Consequently, the Barr-Zee contributions to the MDM and EDM are related by', '0904.4352-1-18-0': '[EQUATION] with normalizations of the MDM and EDM of the lepton [MATH] as given by Eq. ([REF]).', '0904.4352-1-18-1': 'For the dipole moment diagrams with chirality flip inside the loop, the MDM and EDM parts correspond directly to real and imaginary parts of the overall amplitude.', '0904.4352-1-18-2': 'The above relation is a recasting of that statement in terms of the effective scalar and pseudoscalar couplings of the generally CP-mixed Higgs states involved.', '0904.4352-1-18-3': 'To give the MDM result explicitly, with [MATH], we have [EQUATION] where [MATH] and the two-loop functions [MATH], [MATH], and [MATH] are [EQUATION]', '0904.4352-1-18-4': 'For genuine SUSY contributions, the embedded SM contribution should be subtracted.', '0904.4352-1-18-5': 'We estimate the SM contributions as [EQUATION] where [MATH].', '0904.4352-1-18-6': 'We find that [MATH] when [MATH] GeV and it decreases as [MATH] increases.', '0904.4352-1-18-7': 'In our numerical analysis, we safely neglect [MATH].', '0904.4352-1-19-0': '# Numerical Analysis', '0904.4352-1-20-0': 'Adding up all the contributions considered in the previous sections and neglecting [MATH], the supersymmetric contribution to the muon MDM is given by [CITATION] [EQUATION] where the large QED logarithm takes into account the renormalization-group (RG) evolution of [MATH] from the SUSY scale down to the muon-mass scale.', '0904.4352-1-20-1': 'The logarithmic correction amounts to -7 % and -9 % for [MATH] GeV and 1000 GeV, respectively.', '0904.4352-1-21-0': '## A typical scenario where 1 loop dominates', '0904.4352-1-22-0': 'We first consider a typical scenario in which the dominant contributions come from the one-loop chargino and neutralino diagrams [CITATION].', '0904.4352-1-22-1': 'We set [EQUATION]', '0904.4352-1-22-2': 'The common scale [MATH] and [MATH] are varied.', '0904.4352-1-22-3': 'For CP phases, we first consider the two values for [MATH] while taking vanishing CP phases for the gaugino mass and [MATH] parameters: [MATH].', '0904.4352-1-22-4': 'The remaining relevant parameters are fixed as [EQUATION]', '0904.4352-1-22-5': 'In Fig. [REF], we show the one-loop contributions from the chargino and neutralino diagrams (upper) and the two-loop contributions from the Barr-Zee graphs (lower) to the SUSY muon MDM as functions of [MATH] for several values of [MATH].', '0904.4352-1-22-6': 'The left frames are for [MATH] and the right ones for [MATH].', '0904.4352-1-22-7': 'In both cases, the two-loop and one-loop contributions have same signs for most regions of [MATH].', '0904.4352-1-22-8': 'The one-loop and two-loop contributions drops rapidly as [MATH] increases.', '0904.4352-1-22-9': 'The case of [MATH] gives the correct sign and, for example, we have [MATH]>[MATH][MATH] at [MATH] GeV when [MATH]>[MATH][MATH].', '0904.4352-1-22-10': 'For [MATH], [MATH] is negative and the considered regions of [MATH] are not allowed because data prefers a positive [MATH].', '0904.4352-1-22-11': 'Note that for [MATH], the [MATH]-enhanced threshold corrections can turn the [MATH]-quark Yukawa coupling non-perturbative.', '0904.4352-1-22-12': 'This happens when [MATH] (or [MATH]) is sufficiently large, as for the case at hand.', '0904.4352-1-22-13': 'It is shown by the termination of the curves in the right panels of Fig. [REF].', '0904.4352-1-22-14': 'Our results are in good agreement with existing ones in literature.', '0904.4352-1-23-0': 'Figure [REF] shows the SUSY muon MDM (upper) and EDM (lower) as functions of [MATH] taking [MATH] (left) and [MATH] (right).', '0904.4352-1-23-1': 'We have taken [MATH] GeV, [MATH], and the other parameters the same as in Fig. [REF].', '0904.4352-1-23-2': 'In both the MDM and EDM, we observe that the dominant contribution is coming from the one-loop chargino diagrams.', '0904.4352-1-23-3': 'The subleading contribution from the neutralino diagrams is about 5 to 10 times smaller and likely has an opposite sign with respect to the dominant chargino contribution.', '0904.4352-1-23-4': 'The contributions from the Higgs-mediated Barr-Zee diagrams are negligible.', '0904.4352-1-23-5': 'Numerically, we have [MATH] and [MATH]<[MATH][MATH].', '0904.4352-1-23-6': 'We clearly see the (shifted) cosine and sine functional forms of the MDM and EDM, respectively, as to be anticipated from the relations given in Eq. ([REF]).', '0904.4352-1-23-7': 'In the upper frames, the horizontal band is the experimental [MATH]-[MATH]-allowed region, [MATH]; see Eq. ([REF]).', '0904.4352-1-23-8': 'In the lower frames the region is overlayed with thick dots along the thick solid line.', '0904.4352-1-23-9': 'We note the chosen parameter set is compatible with the experimental data only for the non-trivial values of [MATH] around [MATH] and [MATH], resulting in large EDM of about [MATH]cm, which can be easily observed once the projected sensitivity of [MATH] can be achieved.', '0904.4352-1-24-0': '## CPX scenario', '0904.4352-1-25-0': 'Next we consider the CPX scenario[CITATION]: [EQUATION]', '0904.4352-1-25-1': 'Taking [MATH], we fixed [MATH], [MATH] TeV, [MATH] GeV with [MATH], and [MATH].', '0904.4352-1-25-2': 'For our analysis, the most relevant feature of the scenario is that the combined searches of the four LEP collaborations reported two allowed regions where the lightest Higgs boson [MATH] can be very light for moderate values of [MATH]<[MATH][MATH]<[MATH][MATH] [CITATION]: [EQUATION]', '0904.4352-1-25-3': 'On the other hand, a lower limit on the lightest Higgs boson, [MATH]>[MATH][MATH] GeV, is available from the bottomonium decay [MATH] [CITATION].', '0904.4352-1-25-4': 'Figure [REF] shows [MATH] and [MATH] in the CPX scenario as functions of [MATH] taking [MATH].', '0904.4352-1-25-5': 'When [MATH]<[MATH][MATH] GeV, the one-loop contributions to [MATH] are negligible compared to the Higgs-mediated two-loop contributions.', '0904.4352-1-25-6': 'The sign of [MATH] is plus ([MATH]) since it is dominated by the bottom-quark and tau-lepton loops mediated by [MATH] which is almost the CP odd state; see Eq. ([REF]).', '0904.4352-1-25-7': 'However, it is still difficult to achieve [MATH]>[MATH][MATH] only with a mostly CP-odd Higgs boson as light as [MATH] 8 GeV.', '0904.4352-1-25-8': 'For the EDM, the one- and two-loop contributions are comparable and tend to cancel each other.', '0904.4352-1-26-0': '## An extreme scenario', '0904.4352-1-27-0': 'Finally, we consider a scenario in which the one-loop neutralino and chargino contributions are suppressed while the two-loop Barr-Zee contributions dominate.', '0904.4352-1-27-1': 'This scenario is characterized by large [MATH], a light charged Higgs boson, very heavy smuons and muon sneutrinos, and very large [MATH] and [MATH] parameters.', '0904.4352-1-27-2': 'Explicitly, we have chosen [EQUATION] while varying [EQUATION] where [MATH] and we have taken [MATH].', '0904.4352-1-27-3': 'Note [MATH] is fixed in this scenario and the results are almost independent of [MATH].', '0904.4352-1-28-0': 'Figure [REF] shows the regions where [MATH] (blue) and [MATH] (blue[MATH]red) in the [MATH] and [MATH] plane for several values of [MATH], taking [MATH].', '0904.4352-1-28-1': 'The unshaded regions are not theoretically allowed and we have [MATH] GeV in the over-shaded regions (yellow), for example, in the upper-left corner in the upper-left frame.', '0904.4352-1-28-2': 'We found that [MATH] is always lighter than 100 GeV when [MATH] (lower-left) because the resummed threshold corrections modify the bottom-quark Yukawa coupling significantly in this case.', '0904.4352-1-28-3': 'We note the region with larger [MATH] is more preferred.', '0904.4352-1-29-0': 'Figure [REF] shows the dependence on [MATH] of [MATH], [MATH], and the masses of the lighter stau and the lightest Higgs boson.', '0904.4352-1-29-1': 'In the upper-left frame, the horizontal band is the experimental [MATH]-[MATH] region of [MATH].', '0904.4352-1-29-2': 'We found [MATH]<[MATH][MATH]<[MATH][MATH] and [MATH] can make [MATH] consistent with the experimental value for [MATH] TeV and 10.5 TeV, respectively.', '0904.4352-1-29-3': 'In the upper-right frame, the 1-[MATH] region is overlayed with blank boxes along the dashed ([MATH] TeV) and dash-dotted ([MATH] TeV) lines.', '0904.4352-1-29-4': 'We have [MATH]<[MATH][MATH]cm and [MATH]cm for [MATH] TeV and 10.5 TeV, respectively.', '0904.4352-1-29-5': 'We observe that the larger [MATH] results in the lighter staus as shown in the lower-left frame.', '0904.4352-1-29-6': 'This leads to a larger [MATH] as the dominant contribution in this case comes from the Higgs-mediated stau Barr-Zee graphs, as will be shown later.', '0904.4352-1-29-7': 'When [MATH] TeV, [MATH] becomes lighter than 100 GeV for [MATH]<[MATH][MATH] and [MATH]>[MATH][MATH], as shown in the lower-right frame.', '0904.4352-1-29-8': 'When [MATH] TeV, [MATH]>[MATH][MATH] GeV.', '0904.4352-1-30-0': 'Figure [REF] shows the dependence of [MATH] on [MATH] (upper-left) and [MATH] (upper-right) for several values of [MATH] and [MATH], respectively, taking [MATH] and [MATH].', '0904.4352-1-30-1': 'In the lower frames we also show the dependence of the mass of the lighter stau [MATH].', '0904.4352-1-30-2': 'Again we observe that large [MATH] and [MATH] can easily make [MATH] consistent with the current [MATH].', '0904.4352-1-31-0': 'Figure [REF] shows various constituent as well as the total two-loop Barr-Zee contributions to the muon MDM [MATH], as functions of [MATH] taking [MATH] TeV, [MATH] and [MATH].', '0904.4352-1-31-1': 'The thick line is for the total and the thin lines are for the constituent eight contributions (see Fig. [REF]).', '0904.4352-1-31-2': 'For smaller [MATH], the dominant contribution comes from the charged Higgs boson loop.', '0904.4352-1-31-3': 'This is because the [MATH]-[MATH]-[MATH] couplings have loop-induced enhancement from large [MATH] and [MATH] [CITATION].', '0904.4352-1-31-4': 'The possibility of such a significant contribution of the (photon-)Barr-Zee diagram with a (closed) charged Higgs boson loop has apparently not been noticed before.', '0904.4352-1-31-5': 'In fact, the particular diagram is typically not included in analyses at more or less the same level of numerical precision to ours presented here.', '0904.4352-1-31-6': 'As [MATH] grows, however, the contribution from the stau loops is enhanced and thus becomes dominant.', '0904.4352-1-31-7': 'The contribution from the sbottom loop is not significant because the resummed threshold corrections suppress the bottom-quark Yukawa coupling for the chosen parameter set.', '0904.4352-1-32-0': 'The scenario likely seems too contrived to some readers.', '0904.4352-1-32-1': 'However, we are presenting it here mainly to illustrate the significant roles of the various two-loop Barr-Zee contributions in some region of the parameter space.', '0904.4352-1-32-2': 'While the dominance of the latter group reduces as one moves away from the extreme corner of the parameter space, its significance maintains over a substantial region.', '0904.4352-1-32-3': 'For instance, we show the dependence of [MATH] on [MATH] and [MATH] in Fig. [REF], i.e. the effect of bringing back the smuon mass from the heavy limit.', '0904.4352-1-32-4': 'In particular, the Higgs-mediated two-loop Barr-Zee contributions are shown to be actually dominate over the one-loop contributions even when the smuon mass parameters get down to as low as 1 TeV (left), for the full range of [MATH] values (right).', '0904.4352-1-33-0': 'Before closing this section, we comment on the relation between the muon and electron EDMs.', '0904.4352-1-33-1': 'The most important one-loop contribution [MATH] and that from the two-loop [MATH] have somewhat different features.', '0904.4352-1-33-2': 'The Barr-Zee diagrams for the muon and the electron are identical except for the muon and the electron lines themselves.', '0904.4352-1-33-3': 'Hence, we have the robust relation [EQUATION]', '0904.4352-1-33-4': 'For the case of [MATH], however, it is sensitive to the flavor dependence of the soft SUSY breaking terms.', '0904.4352-1-33-5': 'In most of the models on the origin of the soft SUSY breaking terms available in the literature, those of the first two generations are more or less the same.', '0904.4352-1-33-6': 'Explicitly, [EQUATION].', '0904.4352-1-33-7': 'That does give us a relation [EQUATION] which is particularly sensitive to the CP phases of the [MATH] and [MATH] parameters.', '0904.4352-1-33-8': 'To summarize, one may consider special cases with ([MATH]) similar or universal soft terms (for electron and muon) or ([MATH]) electron EDM dominated by the two-loop Barr-Zee graphs, [MATH], possibly due to much heavier selectron masses.', '0904.4352-1-33-9': 'Depending on situations, [MATH] or [MATH] is very strongly constrained by the Thallium EDM as [EQUATION] where we used [MATH], [MATH] and [MATH]cm [CITATION].', '0904.4352-1-33-10': 'Note that these limits are model dependent and the muon EDM should be measured independently of the Thallium EDM.', '0904.4352-1-34-0': '# Conclusions', '0904.4352-1-35-0': 'We have studied in detail various supersymmetric contributions to muon MDM and EDM, including one-loop chargino and neutralino diagrams and dominant two-loop Barr-Zee diagrams.', '0904.4352-1-35-1': 'In general, the one-loop contributions dominate over the two-loop contributions, however there are interesting regions of the model parameter space where the two-loop Barr-Zee diagrams are the major source of contributions to muon MDM and/or EDM.', '0904.4352-1-35-2': 'The model parameter space is huge.', '0904.4352-1-35-3': 'It is not feasible for us to present and discuss here the numerical results for more than a few cases of interest.', '0904.4352-1-35-4': 'We try to pick cases that can illustrate the essential and interesting features, and leave it mostly to the readers to project onto the parameter space regions in between, or considered otherwise to be of special interest.', '0904.4352-1-35-5': 'We illustrate numerically 3 scenarios under various choices of soft parameters: (i) the one when the one-loop contributions dominate, (ii) the CPX in which the Barr-Zee dominates but the overall sizes of MDM and EDM are small, and (iii) a more exotic one in which the Barr-Zee dominates and the overall sizes of MDM and EDM are large.', '0904.4352-1-35-6': 'We have also shown interesting relations between the MDM and EDM.', '0904.4352-1-35-7': 'For the case when one-loop contributions dominate the MDM and EDM are just, respectively, the (shifted) cosines and sines of the phase of the parameters involved.', '0904.4352-1-35-8': 'Existing experimentally preferred range of MDM already predicts an interesting range of EDM, which can be further tested in the future muon EDM experiments.', '0904.4352-1-35-9': 'For the case the MDM is dominated by two-loop Barr-Zee contributions the MDM and EDM can be connected by the relation in Eq. ([REF]) - a result of the more complicated Higgs sector phase structure.', '0904.4352-1-36-0': 'The CPX scenario may still allow a light Higgs boson after taking into account all the existing search limits.', '0904.4352-1-36-1': 'Potentially, the light Higgs boson could give large enhancement to muon MDM.', '0904.4352-1-36-2': 'However, after imposing the lower limit on [MATH] the resulting MDM is always less than [MATH], which is smaller than the experimentally favored value.', '0904.4352-1-37-0': 'The last scenario associated with large [MATH], large [MATH], heavy smuons and muon sneutrinos, and large [MATH] but light charged Higgs boson and stau is rather interesting.', '0904.4352-1-37-1': 'It suppresses the one-loop contributions but the two-loop contributions are large enough to explain the muon MDM data, which are dominated by charged Higgs boson and stau at smaller and larger [MATH], respectively.', '0904.4352-1-37-2': 'For large enough [MATH] the MDM data can be accommodated easily.', '0904.4352-1-37-3': 'The particular interesting role of the charged Higgs boson escaped earlier studies.', '0904.4352-1-37-4': "We have also illustrated that major features of the scenario persist over a region of the parameter space with milder conditions - in particular, more 'regular' smuon masses.", '0904.4352-1-38-0': 'In addition, we offer the following comments.', '0904.4352-1-39-0': 'We have included the threshold corrections to Yukawa couplings.', '0904.4352-1-39-1': 'In particular, the bottom-quark Yukawa can receive large corrections at large [MATH] with large [MATH] and [MATH].', '0904.4352-1-39-2': 'For [MATH]>[MATH][MATH] the threshold corrections can make the [MATH]-quark Yukawa coupling turn non-perturbative when [MATH] (or [MATH]) is sufficiently large.', '0904.4352-1-40-0': 'The one-loop contributions to MDM and EDM can vary as shifted cosines and sines of the phase of the parameters.', '0904.4352-1-41-0': 'As shown in Fig. [REF], the prediction for EDM is of order [MATH] within the allowed range of MDM.', '0904.4352-1-41-1': 'It is about [MATH] orders of magnitude below the current limit, but will be within reach of future muon EDM experiments [CITATION].', '0904.4352-1-42-0': 'The CPX scenario may still allow a [MATH] as light as a few to tens of GeVs.', '0904.4352-1-42-1': 'It could be searched in the subsequent decay of the [MATH], where [MATH] is the SM-like Higgs boson.', '0904.4352-1-42-2': 'The contribution of [MATH] to the muon EDM has a right sign but it may not be large enough to accommodate [MATH] after taking account of the constraint from the bottomonium decay [MATH].', '0904.4352-1-43-0': 'The last scenario that we studied is characterized by a very light stau, a light bino and wino, a light [MATH], and a light charged Higgs boson.', '0904.4352-1-43-1': 'The predicted EDM is from 0 to [MATH].', '0904.4352-1-43-2': 'Experimental searches for this scenario at the LHC will be a lot of tau leptons in the final state because of lightness of stau.', '0904.4352-1-44-0': 'The parameter space compatible with the [MATH] value required is generally extended by allowing CP phases.', '0904.4352-1-44-1': 'For example, in Fig. [REF]: the CP-conserving cases ([MATH] and [MATH]) are excluded.'}

{'0904.4352-2-0-0': 'Kingman Cheung[MATH], Otto C. W. Kong[MATH], and Jae Sik Lee[MATH] Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300', '0904.4352-2-1-0': '[MATH]Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300', '0904.4352-2-2-0': '[MATH]Division of Quantum Phases Devices, Konkuk University, Seoul 143-701, Korea', '0904.4352-2-3-0': '[MATH]Department of Physics and Center for Mathematics and Theoretical Physics,', '0904.4352-2-4-0': 'National Central University, Chung-Li, Taiwan 32054', '0904.4352-2-5-0': 'ABSTRACT', '0904.4352-2-6-0': 'We study the electric dipole moment (EDM) and the anomalous magnetic dipole moment (MDM) of the muon in the CP-violating Minimal Supersymmetric extension of the Standard Model (MSSM).', '0904.4352-2-6-1': 'We take into account the contributions from the chargino- and neutralino-mediated one-loop graphs and the dominant two-loop Higgs-mediated Barr-Zee diagrams.', '0904.4352-2-6-2': 'We improve earlier calculations by incorporating CP-violating Higgs-boson mixing effects and the resummed threshold corrections to the Yukawa couplings of the charged leptons as well as that of the bottom quark.', '0904.4352-2-6-3': 'The analytic correlation between the muon EDM and MDM is explicitly presented at one- and two-loop levels and, through several numerical examples, we illustrate its dependence on the source of the dominant contributions.', '0904.4352-2-6-4': 'We have implemented the analytic expressions for the muon EDM and MDM in an updated version of the public code CPsuperH2.0.', '0904.4352-2-7-0': '# Introduction', '0904.4352-2-8-0': 'The anomalous magnetic dipole moment of the muon, [MATH], has provided one of the most sensitive test grounds for the validity of the Standard Model (SM) [CITATION]: [EQUATION]', '0904.4352-2-8-1': 'At the same time, it also provides an important constraint on new physics with precise SM predictions available.', '0904.4352-2-8-2': 'The SM prediction consists of QED, electroweak (EW), and hadronic contributions.', '0904.4352-2-8-3': 'The hadronic contribution is further decomposed into leading-order (LO) part and higher-order vacuum polarization (VP) and light-by-light (LBL) parts [CITATION] : [EQUATION]', '0904.4352-2-8-4': 'Equations ([REF]) and ([REF]) suggest that there is currently a [MATH] discrepancy between the experimental result and the SM prediction, which can be attributed to possible contributions from physics beyond the SM : [EQUATION]', '0904.4352-2-8-5': 'One of the most appealing scenarios for physics beyond the SM is augmented with a softly broken supersymmetry (SUSY) around the TeV scale.', '0904.4352-2-8-6': 'The supersymmetric contributions to [MATH] from such models are known up to two-loop level.', '0904.4352-2-8-7': 'The one-loop results can be found in [CITATION] and the two-loop results in [CITATION].', '0904.4352-2-8-8': 'It is well-known that the dominant two-loop contribution comes from Higgs-mediated Barr-Zee diagrams [CITATION].', '0904.4352-2-8-9': 'The error associated with the known SUSY contributions is estimated to be [MATH] [CITATION], which is smaller than half of the current experimental and SM theoretical ones.', '0904.4352-2-9-0': 'On the other hand, the SUSY augmented models can contain additional CP-violating phases beyond the SM Cabibbo-Kobayashi-Maskawa (CKM) phase leading to sizable EDMs [CITATION].', '0904.4352-2-9-1': 'The current limit on the muon EDM [CITATION] is [EQUATION] which is much weaker than the constraints from the non-observation of the Thallium [CITATION], neutron [CITATION], and Mercury [CITATION] EDMs: [EQUATION]', '0904.4352-2-9-2': 'Nevertheless, if the muon EDM experiment in the future can achieve the projected sensitivity [CITATION] [EQUATION] the precision of the experiment will be comparable to that of the current Thallium EDM experiment.', '0904.4352-2-10-0': 'In this paper, we study the correlation between the muon EDM and MDM in the CP-violating MSSM [CITATION].', '0904.4352-2-10-1': 'We present the relation between the one-loop chargino- and neutralino-mediated EDM and MDM of the muon.', '0904.4352-2-10-2': 'We also derive an analytic relation between them in the two-loop contributions from the dominant Higgs-mediated Barr-Zee diagrams.', '0904.4352-2-10-3': 'We improve the earlier results by including CP-violating Higgs-boson mixing effects in the Barr-Zee diagrams and resumming the threshold corrections to the muon, tau, and bottom-quark Yukawa couplings in the one- and two-loop graphs.', '0904.4352-2-10-4': 'We then focus our numerical studies on three types of scenarios in which [MATH] the muon EDM and MDM are dominated by the one-loop contributions, [MATH] the lightest Higgs boson is mostly CP odd and lighter than [MATH] 50 GeV, and [MATH] the dominant contributions to the muon EDM and MDM come from the two-loop Barr-Zee graphs.', '0904.4352-2-11-0': 'For the presentation of our analytic results, we follow the conventions and notations of CPsuperH [CITATION], especially for the masses and mixing matrices of the neutral Higgs bosons and SUSY particles.', '0904.4352-2-11-1': 'The layout of the paper is as follows.', '0904.4352-2-11-2': 'Section 2 presents formulas relevant to the one-loop contributions to the muon EDM and MDM from chargino- and neutralino-mediated diagrams.', '0904.4352-2-11-3': 'Non-holomorphic threshold effects on the muon Yukawa coupling have been appropriately resummed.', '0904.4352-2-11-4': 'In Section 3, we present analytic results for the Higgs-mediated two-loop Barr-Zee diagrams.', '0904.4352-2-11-5': 'For this, most importantly, the resummed threshold corrections to the tau and bottom-quark Yukawa couplings and the CP-violating Higgs-boson mixing effects have been incorporated.', '0904.4352-2-11-6': 'In Section 4, we present some numerical examples, depending on the source of dominant contributions to the muon EDM and MDM.', '0904.4352-2-11-7': 'We summarize our findings in Section 5.', '0904.4352-2-12-0': '# One-Loop EDMs and MDMs of charged leptons', '0904.4352-2-13-0': 'The relevant interaction Lagrangian of the spin-1/2 lepton with EDM [MATH] and MDM [MATH] is given by [EQUATION] where [MATH].', '0904.4352-2-13-1': 'The EDM and MDM amplitudes are given by [EQUATION]', '0904.4352-2-13-2': 'Generic interactions of charginos [MATH] or neutralinos [MATH], collectively denoted by [MATH], with a lepton [MATH] and a slepton (sneutrino) [MATH]) are given by [EQUATION]', '0904.4352-2-13-3': 'The Lagrangian for the interactions of [MATH] and [MATH] with the photon field [MATH] is [EQUATION]', '0904.4352-2-13-4': 'The diagrams in Fig. [REF] induce the MDM and EDM of the lepton [MATH] as follows: [EQUATION] where [EQUATION]', '0904.4352-2-13-5': 'We have checked that our analytic expressions for the one-loop MDM and EDM agree with those given in, for example, Ref. [CITATION] and [CITATION] with [MATH] and [MATH].', '0904.4352-2-13-6': 'Note [MATH], [MATH] and [MATH].', '0904.4352-2-13-7': 'Finally, in Eq. ([REF]), the chargino-lepton-sneutrino couplings are given by [EQUATION] and the neutralino-lepton-slepton couplings are given by [EQUATION] with [MATH] and [MATH].', '0904.4352-2-14-0': 'There are non-holomorphic threshold corrections to the Yukawa couplings [MATH] which appear in the chargino and neutralino couplings [CITATION].', '0904.4352-2-14-1': 'These corrections become significant at large [MATH] and we resum these effects by redefining the Yukawa couplings as follows: [EQUATION]', '0904.4352-2-14-2': 'In the presence of the CP phases, [MATH] takes the form [CITATION] [EQUATION] where the one-loop function is [EQUATION]', '0904.4352-2-14-3': 'These effects on the muon MDM have been considered in Ref. [CITATION] only in the CP conserving case.', '0904.4352-2-15-0': '# Barr-Zee Graphs', '0904.4352-2-16-0': 'The dominant two-loop contributions come from the Barr-Zee diagrams mediated by neutral Higgs-boson exchanges, see Fig. [REF].', '0904.4352-2-16-1': 'Here, we consider loops of third-generation fermions and sfermions, charged Higgs bosons, and charginos.', '0904.4352-2-16-2': 'The diagrams can be evaluated first by one-loop computation of the [MATH]-[MATH]-[MATH] vertex.', '0904.4352-2-16-3': 'Thanks to gauge invariance the effective vertex takes the form [EQUATION] where [MATH] is the incoming four-momentum of the external photon, [MATH]) the four-momentum of the incoming (outgoing) lepton, and [MATH] the four-momentum of the internal photon going out of the upper loop.', '0904.4352-2-16-4': 'Note that [MATH].', '0904.4352-2-16-5': 'Explicitly, keeping only the terms linear in the external momenta, the fermionic contributions are given by [EQUATION] with the generic [MATH]-[MATH]-[MATH] interaction: [MATH].', '0904.4352-2-16-6': 'The color factor [MATH] for quarks and 1 for leptons and charginos.', '0904.4352-2-16-7': 'On the other hand, the sfermion loops contribute only to the scalar form factor as [EQUATION] with the generic [MATH]-[MATH]-[MATH] interaction: [MATH].', '0904.4352-2-17-0': 'Together with the interaction Lagrangian for the couplings of the neutral-Higgs bosons and photon to charged leptons [EQUATION] where [MATH] (as in the CPsuperH convention), the Barr-Zee amplitude is given by [EQUATION] where again we keep only the terms linear in the external momenta.', '0904.4352-2-17-1': 'We note the numerator of the integrand is proportional to [EQUATION]', '0904.4352-2-17-2': 'We observe that the first term gives the MDM while the second one gives the EDM of the lepton [MATH].', '0904.4352-2-17-3': 'Consequently, the Barr-Zee contributions to the MDM and EDM are related by', '0904.4352-2-18-0': '[EQUATION] with normalizations of the MDM and EDM of the lepton [MATH] as given by Eq. ([REF]).', '0904.4352-2-18-1': 'For the dipole moment diagrams with chirality flip inside the loop, the MDM and EDM parts correspond directly to real and imaginary parts of the overall amplitude.', '0904.4352-2-18-2': 'The above relation is a recasting of that statement in terms of the effective scalar and pseudoscalar couplings of the generally CP-mixed Higgs states involved.', '0904.4352-2-18-3': 'To give the MDM result explicitly, with [MATH], we have [EQUATION] where [MATH] and the two-loop functions [MATH], [MATH], and [MATH] are [EQUATION]', '0904.4352-2-18-4': 'For genuine SUSY contributions, the embedded SM contribution should be subtracted.', '0904.4352-2-18-5': 'We estimate the SM contributions as [EQUATION] where [MATH].', '0904.4352-2-18-6': 'We find that [MATH] when [MATH] GeV and it decreases as [MATH] increases.', '0904.4352-2-18-7': 'In our numerical analysis, we safely neglect [MATH].', '0904.4352-2-19-0': '# Numerical Analysis', '0904.4352-2-20-0': 'Adding up all the contributions considered in the previous sections and neglecting [MATH], the supersymmetric contribution to the muon MDM is given by [CITATION] [EQUATION] where the large QED logarithm takes into account the renormalization-group (RG) evolution of [MATH] from the SUSY scale down to the muon-mass scale.', '0904.4352-2-20-1': 'The logarithmic correction amounts to -7 % and -9 % for [MATH] GeV and 1000 GeV, respectively.', '0904.4352-2-21-0': '## A typical scenario where 1 loop dominates', '0904.4352-2-22-0': 'We first consider a typical scenario in which the dominant contributions come from the one-loop chargino and neutralino diagrams [CITATION].', '0904.4352-2-22-1': 'We set [EQUATION]', '0904.4352-2-22-2': 'The common scale [MATH] and [MATH] are varied.', '0904.4352-2-22-3': 'For CP phases, we first consider the two values for [MATH] while taking vanishing CP phases for the gaugino mass and [MATH] parameters: [MATH].', '0904.4352-2-22-4': 'The remaining relevant parameters are fixed as [EQUATION]', '0904.4352-2-22-5': 'In Fig. [REF], we show the one-loop contributions from the chargino and neutralino diagrams (upper) and the two-loop contributions from the Barr-Zee graphs (lower) to the SUSY muon MDM as functions of [MATH] for several values of [MATH].', '0904.4352-2-22-6': 'The left frames are for [MATH] and the right ones for [MATH].', '0904.4352-2-22-7': 'In both cases, the two-loop and one-loop contributions have same signs for most regions of [MATH].', '0904.4352-2-22-8': 'The one-loop and two-loop contributions drops rapidly as [MATH] increases.', '0904.4352-2-22-9': 'The case of [MATH] gives the correct sign and, for example, we have [MATH]>[MATH][MATH] at [MATH] GeV when [MATH]>[MATH][MATH].', '0904.4352-2-22-10': 'For [MATH], [MATH] is negative and the considered regions of [MATH] are not allowed because data prefers a positive [MATH].', '0904.4352-2-22-11': 'Note that for [MATH], the [MATH]-enhanced threshold corrections can turn the [MATH]-quark Yukawa coupling non-perturbative.', '0904.4352-2-22-12': 'This happens when [MATH] (or [MATH]) is sufficiently large, as for the case at hand.', '0904.4352-2-22-13': 'It is shown by the termination of the curves in the right panels of Fig. [REF].', '0904.4352-2-22-14': 'Our results are in good agreement with existing ones in literature.', '0904.4352-2-23-0': 'Figure [REF] shows the SUSY muon MDM (upper) and EDM (lower) as functions of [MATH] taking [MATH] (left) and [MATH] (right).', '0904.4352-2-23-1': 'We have taken [MATH] GeV, [MATH], and the other parameters the same as in Fig. [REF].', '0904.4352-2-23-2': 'In both the MDM and EDM, we observe that the dominant contribution is coming from the one-loop chargino diagrams.', '0904.4352-2-23-3': 'The subleading contribution from the neutralino diagrams is about 5 to 10 times smaller and likely has an opposite sign with respect to the dominant chargino contribution.', '0904.4352-2-23-4': 'The contributions from the Higgs-mediated Barr-Zee diagrams are negligible.', '0904.4352-2-23-5': 'Numerically, we have [MATH] and [MATH]<[MATH][MATH].', '0904.4352-2-23-6': 'We clearly see the (shifted) cosine and sine functional forms of the MDM and EDM, respectively, as to be anticipated from the relations given in Eq. ([REF]).', '0904.4352-2-23-7': 'In the upper frames, the horizontal band is the experimental [MATH]-[MATH]-allowed region, [MATH]; see Eq. ([REF]).', '0904.4352-2-23-8': 'In the lower frames the region is overlayed with thick dots along the thick solid line.', '0904.4352-2-23-9': 'We note the chosen parameter set is compatible with the experimental data only for the non-trivial values of [MATH] around [MATH] and [MATH], resulting in large EDM of about [MATH]cm, which can be easily observed once the projected sensitivity of [MATH] can be achieved.', '0904.4352-2-24-0': '## CPX scenario', '0904.4352-2-25-0': 'Next we consider the CPX scenario[CITATION]: [EQUATION]', '0904.4352-2-25-1': 'Taking [MATH], we fixed [MATH], [MATH] TeV, [MATH] GeV with [MATH], and [MATH].', '0904.4352-2-25-2': 'For our analysis, the most relevant feature of the scenario is that the combined searches of the four LEP collaborations reported two allowed regions where the lightest Higgs boson [MATH] can be very light for moderate values of [MATH]<[MATH][MATH]<[MATH][MATH] [CITATION]: [EQUATION]', '0904.4352-2-25-3': 'On the other hand, a lower limit on the lightest Higgs boson, [MATH]>[MATH][MATH] GeV, is available from the bottomonium decay [MATH] [CITATION].', '0904.4352-2-25-4': 'Figure [REF] shows [MATH] and [MATH] in the CPX scenario as functions of [MATH] taking [MATH].', '0904.4352-2-25-5': 'When [MATH]<[MATH][MATH] GeV, the one-loop contributions to [MATH] are negligible compared to the Higgs-mediated two-loop contributions.', '0904.4352-2-25-6': 'The sign of [MATH] is plus ([MATH]) since it is dominated by the bottom-quark and tau-lepton loops mediated by [MATH] which is almost the CP odd state; see Eq. ([REF]).', '0904.4352-2-25-7': 'However, it is still difficult to achieve [MATH]>[MATH][MATH] only with a mostly CP-odd Higgs boson as light as [MATH] 8 GeV.', '0904.4352-2-25-8': 'For the EDM, the one- and two-loop contributions are comparable and tend to cancel each other.', '0904.4352-2-26-0': '## An extreme scenario', '0904.4352-2-27-0': 'Finally, we consider a scenario in which the one-loop neutralino and chargino contributions are suppressed while the two-loop Barr-Zee contributions dominate.', '0904.4352-2-27-1': 'This scenario is characterized by large [MATH], a light charged Higgs boson, very heavy smuons and muon sneutrinos, and very large [MATH] and [MATH] parameters.', '0904.4352-2-27-2': 'Explicitly, we have chosen [EQUATION] while varying [EQUATION] where [MATH] and we have taken [MATH].', '0904.4352-2-27-3': 'Note [MATH] is fixed in this scenario and the results are almost independent of [MATH].', '0904.4352-2-28-0': 'Figure [REF] shows the regions where [MATH] (blue) and [MATH] (blue[MATH]red) in the [MATH] and [MATH] plane for several values of [MATH], taking [MATH].', '0904.4352-2-28-1': 'The unshaded regions are not theoretically allowed and we have [MATH] GeV in the over-shaded regions (yellow), for example, in the upper-left corner in the upper-left frame.', '0904.4352-2-28-2': 'We found that [MATH] is always lighter than 100 GeV when [MATH] (lower-left) because the resummed threshold corrections modify the bottom-quark Yukawa coupling significantly in this case.', '0904.4352-2-28-3': 'We note the region with larger [MATH] is more preferred.', '0904.4352-2-29-0': 'Figure [REF] shows the dependence on [MATH] of [MATH], [MATH], and the masses of the lighter stau and the lightest Higgs boson.', '0904.4352-2-29-1': 'In the upper-left frame, the horizontal band is the experimental [MATH]-[MATH] region of [MATH].', '0904.4352-2-29-2': 'We found [MATH]<[MATH][MATH]<[MATH][MATH] and [MATH] can make [MATH] consistent with the experimental value for [MATH] TeV and 10.5 TeV, respectively.', '0904.4352-2-29-3': 'In the upper-right frame, the 1-[MATH] region is overlayed with blank boxes along the dashed ([MATH] TeV) and dash-dotted ([MATH] TeV) lines.', '0904.4352-2-29-4': 'We have [MATH]<[MATH][MATH]cm and [MATH]cm for [MATH] TeV and 10.5 TeV, respectively.', '0904.4352-2-29-5': 'We observe that the larger [MATH] results in the lighter staus as shown in the lower-left frame.', '0904.4352-2-29-6': 'This leads to a larger [MATH] as the dominant contribution in this case comes from the Higgs-mediated stau Barr-Zee graphs, as will be shown later.', '0904.4352-2-29-7': 'When [MATH] TeV, [MATH] becomes lighter than 100 GeV for [MATH]<[MATH][MATH] and [MATH]>[MATH][MATH], as shown in the lower-right frame.', '0904.4352-2-29-8': 'When [MATH] TeV, [MATH]>[MATH][MATH] GeV.', '0904.4352-2-30-0': 'Figure [REF] shows the dependence of [MATH] on [MATH] (upper-left) and [MATH] (upper-right) for several values of [MATH] and [MATH], respectively, taking [MATH] and [MATH].', '0904.4352-2-30-1': 'In the lower frames we also show the dependence of the mass of the lighter stau [MATH].', '0904.4352-2-30-2': 'Again we observe that large [MATH] and [MATH] can easily make [MATH] consistent with the current [MATH].', '0904.4352-2-31-0': 'Figure [REF] shows various constituent as well as the total two-loop Barr-Zee contributions to the muon MDM [MATH], as functions of [MATH] taking [MATH] TeV, [MATH] and [MATH].', '0904.4352-2-31-1': 'The thick line is for the total and the thin lines are for the constituent eight contributions (see Fig. [REF]).', '0904.4352-2-31-2': 'For smaller [MATH], the dominant contribution comes from the charged Higgs boson loop.', '0904.4352-2-31-3': 'This is because the [MATH]-[MATH]-[MATH] couplings have loop-induced enhancement from large [MATH] and [MATH] [CITATION].', '0904.4352-2-31-4': 'The possibility of such a significant contribution of the (photon-)Barr-Zee diagram with a (closed) charged Higgs boson loop has apparently not been noticed before.', '0904.4352-2-31-5': 'In fact, the particular diagram is typically not included in analyses at more or less the same level of numerical precision to ours presented here.', '0904.4352-2-31-6': 'As [MATH] grows, however, the contribution from the stau loops is enhanced and thus becomes dominant.', '0904.4352-2-31-7': 'The contribution from the sbottom loop is not significant because the resummed threshold corrections suppress the bottom-quark Yukawa coupling for the chosen parameter set.', '0904.4352-2-32-0': 'The scenario likely seems too contrived to some readers.', '0904.4352-2-32-1': 'However, we are presenting it here mainly to illustrate the significant roles of the various two-loop Barr-Zee contributions in some region of the parameter space.', '0904.4352-2-32-2': 'While the dominance of the latter group reduces as one moves away from the extreme corner of the parameter space, its significance maintains over a substantial region.', '0904.4352-2-32-3': 'For instance, we show the dependence of [MATH] on [MATH] and [MATH] in Fig. [REF], i.e. the effect of bringing back the smuon mass from the heavy limit.', '0904.4352-2-32-4': 'In particular, the Higgs-mediated two-loop Barr-Zee contributions are shown to be actually dominate over the one-loop contributions even when the smuon mass parameters get down to as low as 1 TeV (left), for the full range of [MATH] values (right).', '0904.4352-2-33-0': 'Before closing this section, we comment on the relation between the muon and electron EDMs.', '0904.4352-2-33-1': 'The most important one-loop contribution [MATH] and that from the two-loop [MATH] have somewhat different features.', '0904.4352-2-33-2': 'The Barr-Zee diagrams for the muon and the electron are identical except for the muon and the electron lines themselves.', '0904.4352-2-33-3': 'Hence, we have the robust relation [EQUATION]', '0904.4352-2-33-4': 'For the case of [MATH], however, it is sensitive to the flavor dependence of the soft SUSY breaking terms.', '0904.4352-2-33-5': 'In most of the models on the origin of the soft SUSY breaking terms available in the literature, those of the first two generations are more or less the same.', '0904.4352-2-33-6': 'Explicitly, [EQUATION].', '0904.4352-2-33-7': 'That does give us a relation [EQUATION] which is particularly sensitive to the CP phases of the [MATH] and [MATH] parameters.', '0904.4352-2-33-8': 'To summarize, one may consider special cases with ([MATH]) similar or universal soft terms (for electron and muon) or ([MATH]) electron EDM dominated by the two-loop Barr-Zee graphs, [MATH], possibly due to much heavier selectron masses.', '0904.4352-2-33-9': 'Depending on situations, [MATH] or [MATH] is very strongly constrained by the Thallium EDM as [EQUATION] where we used [MATH], [MATH] and [MATH]cm [CITATION].', '0904.4352-2-33-10': 'Note that these limits are model dependent and the muon EDM should be measured independently of the Thallium EDM.', '0904.4352-2-34-0': '# Conclusions', '0904.4352-2-35-0': 'We have studied in detail various supersymmetric contributions to muon MDM and EDM, including one-loop chargino and neutralino diagrams and dominant two-loop Barr-Zee diagrams.', '0904.4352-2-35-1': 'In general, the one-loop contributions dominate over the two-loop contributions, however there are interesting regions of the model parameter space where the two-loop Barr-Zee diagrams are the major source of contributions to muon MDM and/or EDM.', '0904.4352-2-35-2': 'The model parameter space is huge.', '0904.4352-2-35-3': 'It is not feasible for us to present and discuss here the numerical results for more than a few cases of interest.', '0904.4352-2-35-4': 'We try to pick cases that can illustrate the essential and interesting features, and leave it mostly to the readers to project onto the parameter space regions in between, or considered otherwise to be of special interest.', '0904.4352-2-35-5': 'We illustrate numerically 3 scenarios under various choices of soft parameters: (i) the one when the one-loop contributions dominate, (ii) the CPX in which the Barr-Zee dominates but the overall sizes of MDM and EDM are small, and (iii) a more exotic one in which the Barr-Zee dominates and the overall sizes of MDM and EDM are large.', '0904.4352-2-35-6': 'We have also shown interesting relations between the MDM and EDM.', '0904.4352-2-35-7': 'For the case when one-loop contributions dominate the MDM and EDM are just, respectively, the (shifted) cosines and sines of the phase of the parameters involved.', '0904.4352-2-35-8': 'Existing experimentally preferred range of MDM already predicts an interesting range of EDM, which can be further tested in the future muon EDM experiments.', '0904.4352-2-35-9': 'For the case the MDM is dominated by two-loop Barr-Zee contributions the MDM and EDM can be connected by the relation in Eq. ([REF]) - a result of the more complicated Higgs sector phase structure.', '0904.4352-2-36-0': 'The CPX scenario may still allow a light Higgs boson after taking into account all the existing search limits.', '0904.4352-2-36-1': 'Potentially, the light Higgs boson could give large enhancement to muon MDM.', '0904.4352-2-36-2': 'However, after imposing the lower limit on [MATH] the resulting MDM is always less than [MATH], which is smaller than the experimentally favored value.', '0904.4352-2-37-0': 'The last scenario associated with large [MATH], large [MATH], heavy smuons and muon sneutrinos, and large [MATH] but light charged Higgs boson and stau is rather interesting.', '0904.4352-2-37-1': 'It suppresses the one-loop contributions but the two-loop contributions are large enough to explain the muon MDM data, which are dominated by charged Higgs boson and stau at smaller and larger [MATH], respectively.', '0904.4352-2-37-2': 'For large enough [MATH] the MDM data can be accommodated easily.', '0904.4352-2-37-3': 'The particular interesting role of the charged Higgs boson escaped earlier studies.', '0904.4352-2-37-4': "We have also illustrated that major features of the scenario persist over a region of the parameter space with milder conditions - in particular, more 'regular' smuon masses.", '0904.4352-2-38-0': 'In addition, we offer the following comments.', '0904.4352-2-39-0': 'We have included the threshold corrections to Yukawa couplings.', '0904.4352-2-39-1': 'In particular, the bottom-quark Yukawa can receive large corrections at large [MATH] with large [MATH] and [MATH].', '0904.4352-2-39-2': 'For [MATH]>[MATH][MATH] the threshold corrections can make the [MATH]-quark Yukawa coupling turn non-perturbative when [MATH] (or [MATH]) is sufficiently large.', '0904.4352-2-40-0': 'The one-loop contributions to MDM and EDM can vary as shifted cosines and sines of the phase of the parameters.', '0904.4352-2-41-0': 'As shown in Fig. [REF], the prediction for EDM is of order [MATH] within the allowed range of MDM.', '0904.4352-2-41-1': 'It is about [MATH] orders of magnitude below the current limit, but will be within reach of future muon EDM experiments [CITATION].', '0904.4352-2-42-0': 'The CPX scenario may still allow a [MATH] as light as a few to tens of GeVs.', '0904.4352-2-42-1': 'It could be searched in the subsequent decay of the [MATH], where [MATH] is the SM-like Higgs boson.', '0904.4352-2-42-2': 'The contribution of [MATH] to the muon EDM has a right sign but it may not be large enough to accommodate [MATH] after taking account of the constraint from the bottomonium decay [MATH].', '0904.4352-2-43-0': 'The last scenario that we studied is characterized by a very light stau, a light bino and wino, a light [MATH], and a light charged Higgs boson.', '0904.4352-2-43-1': 'The predicted EDM is from 0 to [MATH].', '0904.4352-2-43-2': 'Experimental searches for this scenario at the LHC will be a lot of tau leptons in the final state because of lightness of stau.', '0904.4352-2-44-0': 'The parameter space compatible with the [MATH] value required is generally extended by allowing CP phases.', '0904.4352-2-44-1': 'For example, in Fig. [REF]: the CP-conserving cases ([MATH] and [MATH]) are excluded.'}

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[]

[]

[]

['0904.4352-1-1-0', '0904.4352-1-2-0', '0904.4352-1-3-0', '0904.4352-1-4-0', '0904.4352-1-5-0', '0904.4352-1-13-6', '0904.4352-1-16-4', '0904.4352-1-22-1', '0904.4352-1-23-5', '0904.4352-1-29-8', '0904.4352-1-33-6', '0904.4352-1-38-0', '0904.4352-2-1-0', '0904.4352-2-2-0', '0904.4352-2-3-0', '0904.4352-2-4-0', '0904.4352-2-5-0', '0904.4352-2-13-6', '0904.4352-2-16-4', '0904.4352-2-22-1', '0904.4352-2-23-5', '0904.4352-2-29-8', '0904.4352-2-33-6', '0904.4352-2-38-0', '0904.4352-3-1-0', '0904.4352-3-2-0', '0904.4352-3-3-0', '0904.4352-3-4-0', '0904.4352-3-5-0', '0904.4352-3-13-6', '0904.4352-3-16-4', '0904.4352-3-22-1', '0904.4352-3-23-5', '0904.4352-3-29-8', '0904.4352-3-33-6', '0904.4352-3-38-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0904.4352

{'0904.4352-3-0-0': 'Kingman Cheung[MATH], Otto C. W. Kong[MATH], and Jae Sik Lee[MATH] Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300', '0904.4352-3-1-0': '[MATH]Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300', '0904.4352-3-2-0': '[MATH]Division of Quantum Phases Devices, Konkuk University, Seoul 143-701, Korea', '0904.4352-3-3-0': '[MATH]Department of Physics and Center for Mathematics and Theoretical Physics,', '0904.4352-3-4-0': 'National Central University, Chung-Li, Taiwan 32054', '0904.4352-3-5-0': 'ABSTRACT', '0904.4352-3-6-0': 'We study the electric dipole moment (EDM) and the anomalous magnetic dipole moment (MDM) of the muon in the CP-violating Minimal Supersymmetric extension of the Standard Model (MSSM).', '0904.4352-3-6-1': 'We take into account the contributions from the chargino- and neutralino-mediated one-loop graphs and the dominant two-loop Higgs-mediated Barr-Zee diagrams.', '0904.4352-3-6-2': 'We improve earlier calculations by incorporating CP-violating Higgs-boson mixing effects and the resummed threshold corrections to the Yukawa couplings of the charged leptons as well as that of the bottom quark.', '0904.4352-3-6-3': 'The analytic correlation between the muon EDM and MDM is explicitly presented at one- and two-loop levels and, through several numerical examples, we illustrate its dependence on the source of the dominant contributions.', '0904.4352-3-6-4': 'We have implemented the analytic expressions for the muon EDM and MDM in an updated version of the public code CPsuperH2.0.', '0904.4352-3-7-0': '# Introduction', '0904.4352-3-8-0': 'The anomalous magnetic dipole moment of the muon, [MATH], has provided one of the most sensitive test grounds for the validity of the Standard Model (SM) [CITATION]: [EQUATION]', '0904.4352-3-8-1': 'At the same time, it also provides an important constraint on new physics with precise SM predictions available.', '0904.4352-3-8-2': 'The SM prediction consists of QED, electroweak (EW), and hadronic contributions.', '0904.4352-3-8-3': 'The hadronic contribution is further decomposed into leading-order (LO) part and higher-order vacuum polarization (VP) and light-by-light (LBL) parts [CITATION] : [EQUATION]', '0904.4352-3-8-4': 'Equations ([REF]) and ([REF]) suggest that there is currently a [MATH] discrepancy between the experimental result and the SM prediction, which can be attributed to possible contributions from physics beyond the SM : [EQUATION]', '0904.4352-3-8-5': 'One of the most appealing scenarios for physics beyond the SM is augmented with a softly broken supersymmetry (SUSY) around the TeV scale.', '0904.4352-3-8-6': 'The supersymmetric contributions to [MATH] from such models are known up to dominant two-loop contributions.', '0904.4352-3-8-7': 'The one-loop results can be found in [CITATION] and the two-loop results in [CITATION].', '0904.4352-3-8-8': 'It is well-known that the dominant two-loop contribution comes from Higgs-mediated Barr-Zee diagrams [CITATION].', '0904.4352-3-8-9': 'The error associated with the known SUSY contributions is estimated to be [MATH] [CITATION], which is smaller than half of the current experimental and SM theoretical ones.', '0904.4352-3-9-0': 'On the other hand, the SUSY augmented models can contain additional CP-violating phases beyond the SM Cabibbo-Kobayashi-Maskawa (CKM) phase leading to sizable EDMs [CITATION].', '0904.4352-3-9-1': 'The current limit on the muon EDM [CITATION] is [EQUATION] which is much weaker than the constraints from the non-observation of the Thallium [CITATION], neutron [CITATION], and Mercury [CITATION] EDMs: [EQUATION]', '0904.4352-3-9-2': 'Nevertheless, if the muon EDM experiment in the future can achieve the projected sensitivity [CITATION] [EQUATION] the precision of the experiment will be comparable to that of the current Thallium EDM experiment.', '0904.4352-3-10-0': 'In this paper, we study the correlation between the muon EDM and MDM in the CP-violating MSSM [CITATION].', '0904.4352-3-10-1': 'We present the relation between the one-loop chargino- and neutralino-mediated EDM and MDM of the muon.', '0904.4352-3-10-2': 'We also derive an analytic relation between them in the two-loop contributions from the dominant Higgs-mediated Barr-Zee diagrams.', '0904.4352-3-10-3': 'We improve the earlier results by including CP-violating Higgs-boson mixing effects in the Barr-Zee diagrams and resumming the threshold corrections to the muon, tau, and bottom-quark Yukawa couplings in the one- and two-loop graphs.', '0904.4352-3-10-4': 'We then focus our numerical studies on three types of scenarios in which [MATH] the muon EDM and MDM are dominated by the one-loop contributions, [MATH] the lightest Higgs boson is mostly CP odd and lighter than [MATH] 50 GeV, and [MATH] the dominant contributions to the muon EDM and MDM come from the two-loop Barr-Zee graphs.', '0904.4352-3-11-0': 'For the presentation of our analytic results, we follow the conventions and notations of CPsuperH [CITATION], especially for the masses and mixing matrices of the neutral Higgs bosons and SUSY particles.', '0904.4352-3-11-1': 'The layout of the paper is as follows.', '0904.4352-3-11-2': 'Section 2 presents formulas relevant to the one-loop contributions to the muon EDM and MDM from chargino- and neutralino-mediated diagrams.', '0904.4352-3-11-3': 'Non-holomorphic threshold effects on the muon Yukawa coupling have been appropriately resummed.', '0904.4352-3-11-4': 'In Section 3, we present analytic results for the Higgs-mediated two-loop Barr-Zee diagrams.', '0904.4352-3-11-5': 'For this, most importantly, the resummed threshold corrections to the tau and bottom-quark Yukawa couplings and the CP-violating Higgs-boson mixing effects have been incorporated.', '0904.4352-3-11-6': 'In Section 4, we present some numerical examples, depending on the source of dominant contributions to the muon EDM and MDM.', '0904.4352-3-11-7': 'We summarize our findings in Section 5.', '0904.4352-3-12-0': '# One-Loop EDMs and MDMs of charged leptons', '0904.4352-3-13-0': 'The relevant interaction Lagrangian of the spin-1/2 lepton with EDM [MATH] and MDM [MATH] is given by [EQUATION] where [MATH].', '0904.4352-3-13-1': 'The EDM and MDM amplitudes are given by [EQUATION]', '0904.4352-3-13-2': 'Generic interactions of charginos [MATH] or neutralinos [MATH], collectively denoted by [MATH], with a lepton [MATH] and a slepton (sneutrino) [MATH]) are given by [EQUATION]', '0904.4352-3-13-3': 'The Lagrangian for the interactions of [MATH] and [MATH] with the photon field [MATH] is [EQUATION]', '0904.4352-3-13-4': 'The diagrams in Fig. [REF] induce the MDM and EDM of the lepton [MATH] as follows: [EQUATION] where [EQUATION]', '0904.4352-3-13-5': 'We have checked that our analytic expressions for the one-loop MDM and EDM agree with those given in, for example, Ref. [CITATION] and [CITATION] with [MATH] and [MATH].', '0904.4352-3-13-6': 'Note [MATH], [MATH] and [MATH].', '0904.4352-3-13-7': 'Finally, in Eq. ([REF]), the chargino-lepton-sneutrino couplings are given by [EQUATION] and the neutralino-lepton-slepton couplings are given by [EQUATION] with [MATH] and [MATH].', '0904.4352-3-14-0': 'There are non-holomorphic threshold corrections to the Yukawa couplings [MATH] which appear in the chargino and neutralino couplings [CITATION].', '0904.4352-3-14-1': 'These corrections become significant at large [MATH] and we resum these effects by redefining the Yukawa couplings as follows: [EQUATION]', '0904.4352-3-14-2': 'In the presence of the CP phases, [MATH] takes the form [CITATION] [EQUATION] where the one-loop function is [EQUATION]', '0904.4352-3-14-3': 'These effects on the muon MDM have been considered in Ref. [CITATION] only in the CP conserving case.', '0904.4352-3-15-0': '# Barr-Zee Graphs', '0904.4352-3-16-0': 'The dominant two-loop contributions come from the Barr-Zee diagrams mediated by neutral Higgs-boson exchanges, see Fig. [REF].', '0904.4352-3-16-1': 'Here, we consider loops of third-generation fermions and sfermions, charged Higgs bosons, and charginos.', '0904.4352-3-16-2': 'The diagrams can be evaluated first by one-loop computation of the [MATH]-[MATH]-[MATH] vertex.', '0904.4352-3-16-3': 'Thanks to gauge invariance the effective vertex takes the form [EQUATION] where [MATH] is the incoming four-momentum of the external photon, [MATH]) the four-momentum of the incoming (outgoing) lepton, and [MATH] the four-momentum of the internal photon going out of the upper loop.', '0904.4352-3-16-4': 'Note that [MATH].', '0904.4352-3-16-5': 'Explicitly, keeping only the terms linear in the external momenta, the fermionic contributions are given by [EQUATION] with the generic [MATH]-[MATH]-[MATH] interaction: [MATH].', '0904.4352-3-16-6': 'The color factor [MATH] for quarks and 1 for leptons and charginos.', '0904.4352-3-16-7': 'On the other hand, the sfermion loops contribute only to the scalar form factor as [EQUATION] with the generic [MATH]-[MATH]-[MATH] interaction: [MATH].', '0904.4352-3-17-0': 'Together with the interaction Lagrangian for the couplings of the neutral-Higgs bosons and photon to charged leptons [EQUATION] where [MATH] (as in the CPsuperH convention), the Barr-Zee amplitude is given by [EQUATION] where again we keep only the terms linear in the external momenta.', '0904.4352-3-17-1': 'We note the numerator of the integrand is proportional to [EQUATION]', '0904.4352-3-17-2': 'We observe that the first term gives the MDM while the second one gives the EDM of the lepton [MATH].', '0904.4352-3-17-3': 'Consequently, the Barr-Zee contributions to the MDM and EDM are related by', '0904.4352-3-18-0': '[EQUATION] with normalizations of the MDM and EDM of the lepton [MATH] as given by Eq. ([REF]).', '0904.4352-3-18-1': 'For the dipole moment diagrams with chirality flip inside the loop, the MDM and EDM parts correspond directly to real and imaginary parts of the overall amplitude.', '0904.4352-3-18-2': 'The above relation is a recasting of that statement in terms of the effective scalar and pseudoscalar couplings of the generally CP-mixed Higgs states involved.', '0904.4352-3-18-3': 'To give the MDM result explicitly, with [MATH], we have [EQUATION] where [MATH] and the two-loop functions [MATH], [MATH], and [MATH] are [EQUATION]', '0904.4352-3-18-4': 'For genuine SUSY contributions, the embedded SM contribution should be subtracted.', '0904.4352-3-18-5': 'We estimate the SM contributions as [EQUATION] where [MATH].', '0904.4352-3-18-6': 'We find that [MATH] when [MATH] GeV and it decreases as [MATH] increases.', '0904.4352-3-18-7': 'In our numerical analysis, we safely neglect [MATH].', '0904.4352-3-19-0': '# Numerical Analysis', '0904.4352-3-20-0': 'Adding up all the contributions considered in the previous sections and neglecting [MATH], the supersymmetric contribution to the muon MDM is given by [CITATION] [EQUATION] where the large QED logarithm takes into account the renormalization-group (RG) evolution of [MATH] from the SUSY scale down to the muon-mass scale.', '0904.4352-3-20-1': 'The logarithmic correction amounts to -7 % and -9 % for [MATH] GeV and 1000 GeV, respectively.', '0904.4352-3-21-0': '## A typical scenario where 1 loop dominates', '0904.4352-3-22-0': 'We first consider a typical scenario in which the dominant contributions come from the one-loop chargino and neutralino diagrams [CITATION].', '0904.4352-3-22-1': 'We set [EQUATION]', '0904.4352-3-22-2': 'The common scale [MATH] and [MATH] are varied.', '0904.4352-3-22-3': 'For CP phases, we first consider the two values for [MATH] while taking vanishing CP phases for the gaugino mass and [MATH] parameters: [MATH].', '0904.4352-3-22-4': 'The remaining relevant parameters are fixed as [EQUATION]', '0904.4352-3-22-5': 'In Fig. [REF], we show the one-loop contributions from the chargino and neutralino diagrams (upper) and the two-loop contributions from the Barr-Zee graphs (lower) to the SUSY muon MDM as functions of [MATH] for several values of [MATH].', '0904.4352-3-22-6': 'The left frames are for [MATH] and the right ones for [MATH].', '0904.4352-3-22-7': 'In both cases, the two-loop and one-loop contributions have same signs for most regions of [MATH].', '0904.4352-3-22-8': 'The one-loop and two-loop contributions drops rapidly as [MATH] increases.', '0904.4352-3-22-9': 'The case of [MATH] gives the correct sign and, for example, we have [MATH]>[MATH][MATH] at [MATH] GeV when [MATH]>[MATH][MATH].', '0904.4352-3-22-10': 'For [MATH], [MATH] is negative and the considered regions of [MATH] are not allowed because data prefers a positive [MATH].', '0904.4352-3-22-11': 'Note that for [MATH], the [MATH]-enhanced threshold corrections can turn the [MATH]-quark Yukawa coupling non-perturbative.', '0904.4352-3-22-12': 'This happens when [MATH] (or [MATH]) is sufficiently large, as for the case at hand.', '0904.4352-3-22-13': 'It is shown by the termination of the curves in the right panels of Fig. [REF].', '0904.4352-3-22-14': 'Our results are in good agreement with existing ones in literature.', '0904.4352-3-23-0': 'Figure [REF] shows the SUSY muon MDM (upper) and EDM (lower) as functions of [MATH] taking [MATH] (left) and [MATH] (right).', '0904.4352-3-23-1': 'We have taken [MATH], [MATH] GeV, [MATH], and the other parameters the same as in Fig. [REF].', '0904.4352-3-23-2': 'In both the MDM and EDM, we observe that the dominant contribution is coming from the one-loop chargino diagrams.', '0904.4352-3-23-3': 'The subleading contribution from the neutralino diagrams is about 5 to 10 times smaller and likely has an opposite sign with respect to the dominant chargino contribution.', '0904.4352-3-23-4': 'The contributions from the Higgs-mediated Barr-Zee diagrams are negligible.', '0904.4352-3-23-5': 'Numerically, we have [MATH] and [MATH]<[MATH][MATH].', '0904.4352-3-23-6': 'We clearly see the (shifted) cosine and sine functional forms of the MDM and EDM, respectively, as to be anticipated from the relations given in Eq. ([REF]).', '0904.4352-3-23-7': 'In the upper frames, the horizontal band is the experimental [MATH]-[MATH]-allowed region, [MATH]; see Eq. ([REF]).', '0904.4352-3-23-8': 'In the lower frames the region is overlayed with thick dots along the thick solid line.', '0904.4352-3-23-9': 'We note the chosen parameter set is compatible with the experimental data only for the non-trivial values of [MATH] around [MATH] and [MATH], resulting in large EDM of about [MATH]cm, which can be easily observed once the projected sensitivity of [MATH] can be achieved.', '0904.4352-3-24-0': '## CPX scenario', '0904.4352-3-25-0': 'Next we consider the CPX scenario[CITATION]: [EQUATION]', '0904.4352-3-25-1': 'Taking [MATH], we fixed [MATH], [MATH] TeV, [MATH] GeV with [MATH], and [MATH].', '0904.4352-3-25-2': 'For our analysis, the most relevant feature of the scenario is that the combined searches of the four LEP collaborations reported two allowed regions where the lightest Higgs boson [MATH] can be very light for moderate values of [MATH]<[MATH][MATH]<[MATH][MATH] [CITATION]: [EQUATION]', '0904.4352-3-25-3': 'On the other hand, a lower limit on the lightest Higgs boson, [MATH]>[MATH][MATH] GeV, is available from the bottomonium decay [MATH] [CITATION].', '0904.4352-3-25-4': 'Figure [REF] shows [MATH] and [MATH] in the CPX scenario as functions of [MATH] taking [MATH].', '0904.4352-3-25-5': 'When [MATH]<[MATH][MATH] GeV, the one-loop contributions to [MATH] are negligible compared to the Higgs-mediated two-loop contributions.', '0904.4352-3-25-6': 'The sign of [MATH] is plus ([MATH]) since it is dominated by the bottom-quark and tau-lepton loops mediated by [MATH] which is almost the CP odd state; see Eq. ([REF]).', '0904.4352-3-25-7': 'However, it is still difficult to achieve [MATH]>[MATH][MATH] only with a mostly CP-odd Higgs boson as light as [MATH] 8 GeV.', '0904.4352-3-25-8': 'For the EDM, the one- and two-loop contributions are comparable and tend to cancel each other.', '0904.4352-3-26-0': '## An extreme scenario', '0904.4352-3-27-0': 'Finally, we consider a scenario in which the one-loop neutralino and chargino contributions are suppressed while the two-loop Barr-Zee contributions dominate.', '0904.4352-3-27-1': 'This scenario is characterized by large [MATH], a light charged Higgs boson, very heavy smuons and muon sneutrinos, and very large [MATH] and [MATH] parameters.', '0904.4352-3-27-2': 'Explicitly, we have chosen [EQUATION] while varying [EQUATION] where [MATH] and we have taken [MATH].', '0904.4352-3-27-3': 'Note [MATH] is fixed in this scenario and the results are almost independent of [MATH].', '0904.4352-3-28-0': 'Figure [REF] shows the regions where [MATH] (blue) and [MATH] (blue[MATH]red) in the [MATH] and [MATH] plane for several values of [MATH], taking [MATH].', '0904.4352-3-28-1': 'The unshaded regions are not theoretically allowed and we have [MATH] GeV in the over-shaded regions (yellow), for example, in the upper-left corner in the upper-left frame.', '0904.4352-3-28-2': 'We found that [MATH] is always lighter than 100 GeV when [MATH] (lower-left) because the resummed threshold corrections modify the bottom-quark Yukawa coupling significantly in this case.', '0904.4352-3-28-3': 'We note the region with larger [MATH] is more preferred.', '0904.4352-3-29-0': 'Figure [REF] shows the dependence on [MATH] of [MATH], [MATH], and the masses of the lighter stau and the lightest Higgs boson.', '0904.4352-3-29-1': 'In the upper-left frame, the horizontal band is the experimental [MATH]-[MATH] region of [MATH].', '0904.4352-3-29-2': 'We found [MATH]<[MATH][MATH]<[MATH][MATH] and [MATH] can make [MATH] consistent with the experimental value for [MATH] TeV and 10.5 TeV, respectively.', '0904.4352-3-29-3': 'In the upper-right frame, the 1-[MATH] region is overlayed with blank boxes along the dashed ([MATH] TeV) and dash-dotted ([MATH] TeV) lines.', '0904.4352-3-29-4': 'We have [MATH]<[MATH][MATH]cm and [MATH]cm for [MATH] TeV and 10.5 TeV, respectively.', '0904.4352-3-29-5': 'We observe that the larger [MATH] results in the lighter staus as shown in the lower-left frame.', '0904.4352-3-29-6': 'This leads to a larger [MATH] as the dominant contribution in this case comes from the Higgs-mediated stau Barr-Zee graphs, as will be shown later.', '0904.4352-3-29-7': 'When [MATH] TeV, [MATH] becomes lighter than 100 GeV for [MATH]<[MATH][MATH] and [MATH]>[MATH][MATH], as shown in the lower-right frame.', '0904.4352-3-29-8': 'When [MATH] TeV, [MATH]>[MATH][MATH] GeV.', '0904.4352-3-30-0': 'Figure [REF] shows the dependence of [MATH] on [MATH] (upper-left) and [MATH] (upper-right) for several values of [MATH] and [MATH], respectively, taking [MATH] and [MATH].', '0904.4352-3-30-1': 'In the lower frames we also show the dependence of the mass of the lighter stau [MATH].', '0904.4352-3-30-2': 'Again we observe that large [MATH] and [MATH] can easily make [MATH] consistent with the current [MATH].', '0904.4352-3-31-0': 'Figure [REF] shows various constituent as well as the total two-loop Barr-Zee contributions to the muon MDM [MATH], as functions of [MATH] taking [MATH] TeV, [MATH] and [MATH].', '0904.4352-3-31-1': 'The thick line is for the total and the thin lines are for the constituent eight contributions (see Fig. [REF]).', '0904.4352-3-31-2': 'For smaller [MATH], the dominant contribution comes from the charged Higgs boson loop.', '0904.4352-3-31-3': 'This is because the [MATH]-[MATH]-[MATH] couplings have loop-induced enhancement from large [MATH] and [MATH] [CITATION].', '0904.4352-3-31-4': 'The possibility of such a significant contribution of the (photon-)Barr-Zee diagram with a (closed) charged Higgs boson loop has apparently not been noticed before.', '0904.4352-3-31-5': 'In fact, the particular diagram is typically not included in analyses at more or less the same level of numerical precision to ours presented here.', '0904.4352-3-31-6': 'As [MATH] grows, however, the contribution from the stau loops is enhanced and thus becomes dominant.', '0904.4352-3-31-7': 'The contribution from the sbottom loop is not significant because the resummed threshold corrections suppress the bottom-quark Yukawa coupling for the chosen parameter set.', '0904.4352-3-32-0': 'The scenario likely seems too contrived to some readers.', '0904.4352-3-32-1': 'However, we are presenting it here mainly to illustrate the significant roles of the various two-loop Barr-Zee contributions in some region of the parameter space.', '0904.4352-3-32-2': 'While the dominance of the latter group reduces as one moves away from the extreme corner of the parameter space, its significance maintains over a substantial region.', '0904.4352-3-32-3': 'For instance, we show the dependence of [MATH] on [MATH] and [MATH] in Fig. [REF], i.e. the effect of bringing back the smuon mass from the heavy limit.', '0904.4352-3-32-4': 'In particular, the Higgs-mediated two-loop Barr-Zee contributions are shown to be actually dominate over the one-loop contributions even when the smuon mass parameters get down to as low as 1 TeV (left), for the full range of [MATH] values (right).', '0904.4352-3-33-0': 'Before closing this section, we comment on the relation between the muon and electron EDMs.', '0904.4352-3-33-1': 'The most important one-loop contribution [MATH] and that from the two-loop [MATH] have somewhat different features.', '0904.4352-3-33-2': 'The Barr-Zee diagrams for the muon and the electron are identical except for the muon and the electron lines themselves.', '0904.4352-3-33-3': 'Hence, we have the robust relation [EQUATION]', '0904.4352-3-33-4': 'For the case of [MATH], however, it is sensitive to the flavor dependence of the soft SUSY breaking terms.', '0904.4352-3-33-5': 'In most of the models on the origin of the soft SUSY breaking terms available in the literature, those of the first two generations are more or less the same.', '0904.4352-3-33-6': 'Explicitly, [EQUATION].', '0904.4352-3-33-7': 'That does give us a relation [EQUATION] which is particularly sensitive to the CP phases of the [MATH] and [MATH] parameters.', '0904.4352-3-33-8': 'To summarize, one may consider special cases with ([MATH]) similar or universal soft terms (for electron and muon) or ([MATH]) electron EDM dominated by the two-loop Barr-Zee graphs, [MATH], possibly due to much heavier selectron masses.', '0904.4352-3-33-9': 'Depending on situations, [MATH] or [MATH] is very strongly constrained by the Thallium EDM as [EQUATION] where we used [MATH], [MATH] and [MATH]cm [CITATION].', '0904.4352-3-33-10': 'Note that these limits are model dependent and the muon EDM should be measured independently of the Thallium EDM.', '0904.4352-3-34-0': '# Conclusions', '0904.4352-3-35-0': 'We have studied in detail various supersymmetric contributions to muon MDM and EDM, including one-loop chargino and neutralino diagrams and dominant two-loop Barr-Zee diagrams.', '0904.4352-3-35-1': 'In general, the one-loop contributions dominate over the two-loop contributions, however there are interesting regions of the model parameter space where the two-loop Barr-Zee diagrams are the major source of contributions to muon MDM and/or EDM.', '0904.4352-3-35-2': 'The model parameter space is huge.', '0904.4352-3-35-3': 'It is not feasible for us to present and discuss here the numerical results for more than a few cases of interest.', '0904.4352-3-35-4': 'We try to pick cases that can illustrate the essential and interesting features, and leave it mostly to the readers to project onto the parameter space regions in between, or considered otherwise to be of special interest.', '0904.4352-3-35-5': 'We illustrate numerically 3 scenarios under various choices of soft parameters: (i) the one when the one-loop contributions dominate, (ii) the CPX in which the Barr-Zee dominates but the overall sizes of MDM and EDM are small, and (iii) a more exotic one in which the Barr-Zee dominates and the overall sizes of MDM and EDM are large.', '0904.4352-3-35-6': 'We have also shown interesting relations between the MDM and EDM.', '0904.4352-3-35-7': 'For the case when one-loop contributions dominate the MDM and EDM are just, respectively, the (shifted) cosines and sines of the phase of the parameters involved.', '0904.4352-3-35-8': 'Existing experimentally preferred range of MDM already predicts an interesting range of EDM, which can be further tested in the future muon EDM experiments.', '0904.4352-3-35-9': 'For the case the MDM is dominated by two-loop Barr-Zee contributions the MDM and EDM can be connected by the relation in Eq. ([REF]) - a result of the more complicated Higgs sector phase structure.', '0904.4352-3-36-0': 'The CPX scenario may still allow a light Higgs boson after taking into account all the existing search limits.', '0904.4352-3-36-1': 'Potentially, the light Higgs boson could give large enhancement to muon MDM.', '0904.4352-3-36-2': 'However, after imposing the lower limit on [MATH] the resulting MDM is always less than [MATH], which is smaller than the experimentally favored value.', '0904.4352-3-37-0': 'The last scenario associated with large [MATH], large [MATH], heavy smuons and muon sneutrinos, and large [MATH] but light charged Higgs boson and stau is rather interesting.', '0904.4352-3-37-1': 'It suppresses the one-loop contributions but the two-loop contributions are large enough to explain the muon MDM data, which are dominated by charged Higgs boson and stau at smaller and larger [MATH], respectively.', '0904.4352-3-37-2': 'For large enough [MATH] the MDM data can be accommodated easily.', '0904.4352-3-37-3': 'The particular interesting role of the charged Higgs boson escaped earlier studies.', '0904.4352-3-37-4': "We have also illustrated that major features of the scenario persist over a region of the parameter space with milder conditions - in particular, more 'regular' smuon masses.", '0904.4352-3-38-0': 'In addition, we offer the following comments.', '0904.4352-3-39-0': 'We have included the threshold corrections to Yukawa couplings.', '0904.4352-3-39-1': 'In particular, the bottom-quark Yukawa can receive large corrections at large [MATH] with large [MATH] and [MATH].', '0904.4352-3-39-2': 'For [MATH]>[MATH][MATH] the threshold corrections can make the [MATH]-quark Yukawa coupling turn non-perturbative when [MATH] (or [MATH]) is sufficiently large.', '0904.4352-3-40-0': 'The one-loop contributions to MDM and EDM can vary as shifted cosines and sines of the phase of the parameters.', '0904.4352-3-41-0': 'As shown in Fig. [REF], the prediction for EDM is of order [MATH] within the allowed range of MDM.', '0904.4352-3-41-1': 'It is about [MATH] orders of magnitude below the current limit, but will be within reach of future muon EDM experiments [CITATION].', '0904.4352-3-42-0': 'The CPX scenario may still allow a [MATH] as light as a few to tens of GeVs.', '0904.4352-3-42-1': 'It could be searched in the subsequent decay of the [MATH], where [MATH] is the SM-like Higgs boson.', '0904.4352-3-42-2': 'The contribution of [MATH] to the muon EDM has a right sign but it may not be large enough to accommodate [MATH] after taking account of the constraint from the bottomonium decay [MATH].', '0904.4352-3-43-0': 'The last scenario that we studied is characterized by a very light stau, a light bino and wino, a light [MATH], and a light charged Higgs boson.', '0904.4352-3-43-1': 'The predicted EDM is from 0 to [MATH].', '0904.4352-3-43-2': 'Experimental searches for this scenario at the LHC will be a lot of tau leptons in the final state because of lightness of stau.', '0904.4352-3-44-0': 'The parameter space compatible with the [MATH] value required is generally extended by allowing CP phases.', '0904.4352-3-44-1': 'For example, in Fig. [REF]: the CP-conserving cases ([MATH] and [MATH]) are excluded.'}

1305.6259

{'1305.6259-1-0-0': 'english RNA-seq allows detection and precise quantification of transcripts, provides comprehensive understanding of exon/intron boundaries, aids discovery of alternatively spliced isoforms and fusion transcripts along with measurement of allele-specific expression.', '1305.6259-1-0-1': 'Researchers interested in studying and constructing transcriptomes, especially for non-model species, often face the conundrum of choosing from a number of available de novo and genome-guided assemblers.', '1305.6259-1-0-2': 'A comprehensive comparative study is required to assess and evaluate their efficiency and sensitivity for transcript assembly, reconstruction and recovery.', '1305.6259-1-0-3': 'None of the popular assembly tools in use today achieves requisite sensitivity, specificity or recovery of full-length transcripts on its own.', '1305.6259-1-0-4': 'Hence, it is imperative that methods be developed in order to augment assemblies generated from multiple tools, with minimal compounding of error.', '1305.6259-1-0-5': 'Here, we present an approach to combinatorially augment transciptome assembly based on a rigorous comparative study of popular de novo and genome-guided transcriptome assembly tools.', '1305.6259-1-1-0': 'englishIntroduction', '1305.6259-1-2-0': 'english High-throughput technology has changed our understanding of many facets of biology, from diseases to plant genetics , and to synthetic biology .', '1305.6259-1-2-1': 'The advent of DNA microarrays in the 90s ushered an era of high-throughput genome-wide gene expression profiling studies .', '1305.6259-1-2-2': 'DNA microarray, although a powerful technique, is dependent on gene annotation, and hence genome sequence information.', '1305.6259-1-2-3': 'This is circumvented by RNA sequencing (RNA-seq), which uses next-generation sequencing instruments.', '1305.6259-1-2-4': 'This shift from the semi-quantitative, hybridization-based approaches, as in DNA microarrays, to the quantitative, sequencing-based approaches has tremendously facilitated gene expression analysis.', '1305.6259-1-2-5': 'RNA-seq experiments yield additional information on transcriptome characterization and quantification, including strand-specificity, mapping of fusion transcripts, small RNA identification and alternate splicing information .', '1305.6259-1-3-0': 'english A number of tools have been developed for transcriptome assembly in the past.', '1305.6259-1-3-1': 'They are classified under de novo and genome-guided assembly categories.', '1305.6259-1-3-2': 'Trinity , SOAPdenovo-Trans , Oases and Trans-ABySS are de novo tools, while TopHat-Cufflinks, and Genome guided Trinity fall under the second category.', '1305.6259-1-3-3': 'De novo assembly tools create short contigs from overlapping reads, which are extended based on insert size estimates.', '1305.6259-1-3-4': 'They could be based on either construction of de Bruijn graphs (using k-mers for short reads) or use overlap-layout-consensus (for longer reads) algorithms .', '1305.6259-1-3-5': 'In Genome guided Trinity, the reads are aligned to the genome and partitioned into read clusters, which are individually assembled using Trinity.', '1305.6259-1-3-6': 'TopHat-Cufflinks is an aligner cum assembly pipeline, involving spliced read alignment to the genome and subsequent assembly of these aligned reads into transcripts.', '1305.6259-1-3-7': 'The genome template and read pairing are utilized here to produce full-length or near full-length transcripts.', '1305.6259-1-3-8': 'Studies have identified k-mer size as an important parameter for short-read transcriptome assembly quality.', '1305.6259-1-3-9': 'The theoretical calculation on assembling transcriptomes based on k-mer size (a lower or higher k-mer length for low- or high-expressed genes respectively) has also been backed by experimental evidence .', '1305.6259-1-3-10': 'The k-mer length depends on the genome complexity, sequencing depth and sequencing error .', '1305.6259-1-3-11': 'One can take a decision on k-mer length depending on desired transcript diversity vs continuity of a given transcriptome.', '1305.6259-1-4-0': 'english Varying expression levels among genes, presence of homologues and spliced isoforms increase the complexity in transcriptome assembly.', '1305.6259-1-4-1': 'Availability of genome sequence can aid the performance of transcriptome assembly process.', '1305.6259-1-4-2': 'In its absence, genomes of closely related organisms can be used for transcriptome assemblies .', '1305.6259-1-4-3': 'Both de novo and genome-guided approaches have their own strengths and one can think of combining data from both to achieve higher sensitivity.', '1305.6259-1-4-4': 'However, the process of combining assemblies from multiple assemblers is not error free and may produce false assemblies.', '1305.6259-1-4-5': 'In the current paper, we first describe a detailed analysis of existing de novo and genome-guided tools and then determine the best combination of tools that can be used to augment the process of assembly while keeping the false assemblies to a minimum.', '1305.6259-1-5-0': 'englishMaterials and Methods', '1305.6259-1-6-0': 'englishSimulating RNA-seq reads', '1305.6259-1-7-0': 'english The Arabidopsis thaliana (TAIR10) complete genome sequence, co-ordinates for genes and transposons were downloaded from ftp://ftp.arabidopsis.org/home/tair/Genes/ftp://ftp.arabidopsis.org/home/tair/Genes/TAIR10_genome_releaseenglishTAIR10genomerelease and the GFF file was parsed to obtain exonic coordinates.', '1305.6259-1-7-1': 'These were used to simulate Illumina-like RNA-seq reads using Flux-simulator (FS-nightly-build1.1.1-20121119021549) with the following options supplied within a parameter file (Supplementary Methods): NBMOLECULES 5000000 , SIZEDISTRIBUTION N(300,30) , READNUMBER 4000000 , READLENGTH 76 , PAIREDEND YES , ERRFILE 76.', '1305.6259-1-7-2': 'The resultant Illumina-like reads were split into 2 fastq files corresponding to read1 and read2 using a Python script from the Galaxy tool suite (Supplementary Methods).', '1305.6259-1-8-0': 'englishZebrafish transcriptome data', '1305.6259-1-9-0': 'english RNA-seq reads for the zebrafish, Danio rerio, (2dpf) embryo were downloaded from SRA (ERR003998) .', '1305.6259-1-9-1': 'The zebrafish genome (Zv9) downloaded from Ensemble was used for genome-guided transcriptome assembly.', '1305.6259-1-9-2': 'The transcripts for hox gene cluster (listed in Table 2 in Corredor-Adamez et al. 2005) were downloaded from ENSEMBL, and used to identify hox-related genes from the assemblies through Megablast.', '1305.6259-1-9-3': 'We focused on read-covered shared and unique regions of the hox gene cluster.', '1305.6259-1-10-0': 'englishRead assembly', '1305.6259-1-11-0': 'english The simulated and Danio rerio reads were assembled using four de novo (Trinity, version r2012-06-08; Trans-ABySS, version 1.3.2; Oases, version 0.2.08; SOAPdenovo-Trans, version 1.0 for simulated data and version 1.02 for Danio rerio) and two genome-guided (Tophat1, version 2.0.4 that uses Bowtie1 , version 0.12.8 and Cufflinks, version 2.0.0; Genome guided Trinity, version r2012-10-05 that uses GSNAP , version 2012-07-20 [V3] for simulated data and version 2013-03-31 [V5]) for zebrafish data in the process of transcriptome assembly pipelines.', '1305.6259-1-11-1': 'We used Trinity and Genome guided Trinity with a fixed kmer size of 25bp, Trans-ABySS on ABySS (version 1.3.3) multi-kmer assemblies (ranging from 20-64bp), Oases on Velvet (version 1.2.07) multi-kmer assemblies (every alternate kmer ranging from 19-71bp) and SOAPdenovo-Trans with a fixed kmer size of 23bp.', '1305.6259-1-11-2': 'We tested the Tophat2 (version 2.0.7 that uses Bowtie version 2.0.5)-Cufflinks pipeline on the simulated data but did not observe any difference in the assembly statistics compared to the Tophat1-Cufflinks pipeline that uses Bowtie1.', '1305.6259-1-11-3': 'All assemblers were run using default parameters (details in Supplementary Methods).', '1305.6259-1-11-4': 'However, we fixed the parameter for minimum length of assembled fragment as 76bp (equal to the length of the read).', '1305.6259-1-11-5': 'In the case of SOAPdenovo-Trans, contigs were used instead of scaffolds for all downstream analyses, as the minimum length cutoff could not be set for scaffolds.', '1305.6259-1-12-0': 'english In order to test whether we could augment the assembly of Trinity-predicted transcript fragments with those from Tophat1-Cufflinks, we used Megablast to map the Trinity transcript fragments against the Tophat1-Cufflinks transcripts, and then augmented Trinity assembly with those transcript fragments unique to Tophat1-Cufflinks assembly (see Supplementary Methods for details).', '1305.6259-1-13-0': 'englishRedundancy removal using CD-HIT-EST', '1305.6259-1-14-0': 'english We used CD-HIT-EST (version 4.5.4) to remove redundancy in each assembly.', '1305.6259-1-14-1': 'It retained the longest sequence out of a cluster of sequences that shared at least 95% sequence similarity based on a word size of 8.', '1305.6259-1-14-2': 'Further details on the options used to run CD-HIT-EST are provided in Supplementary Methods.', '1305.6259-1-15-0': 'englishModel Assembly', '1305.6259-1-16-0': 'english We defined all read-covered transcript regions (TAIR10 for simulated data and hox gene cluster for Danio rerio) as Model Assembly or MA as previously described in the report by Mundry et al 2012 .', '1305.6259-1-17-0': 'english Calculation of N50 and N[MATH]50 statistics for an assembly', '1305.6259-1-18-0': 'english N50 is defined as the minimum contig length for 50% of the assembly, after sorting the contigs in the descending order of their lengths.', '1305.6259-1-18-1': 'We calculated N50 values for all assemblies, MA and the TAIR10 simulated transcripts, pre- and post-CD-HIT-EST.', '1305.6259-1-18-2': 'We also calculated the N50 values for each assembler, while taking the MA cumulative size (bp) as the denominator instead of the respective assembly sizes.', '1305.6259-1-18-3': 'We termed these N50 values as the N[MATH]50.', '1305.6259-1-19-0': 'englishMapping assemblies to MA using Megablast', '1305.6259-1-20-0': 'english The assembled fragments (query) were mapped against the MA fragments (subject) using Megablast (blast+ version 2.2.26) with default parameters (Supplementary Methods).', '1305.6259-1-20-1': 'The Megablast hits were parsed in order to either maximize query coverage (to compute mis-assembly statistics) or maximize subject coverage (to compute MA recovery statistics).', '1305.6259-1-20-2': 'This was done to choose the best hits and discard the partially overlapping hits when the unique coverage was lower than or equal to 10bp.', '1305.6259-1-20-3': 'The scripts used to remove nested and partial overlaps from Megablast hits are provided in Supplementary Methods.', '1305.6259-1-21-0': 'englishExpression level bin categories', '1305.6259-1-22-0': 'english We estimated the average per nucleotide coverage (pnc) for all MA fragments, based on their read support as Number of reads multiplied by Read Length divided by MA fragment length.', '1305.6259-1-22-1': 'The MA fragments were then categorized into 8 expression level bins, B1 to B8, the pnc for each being: 1 for B1, 1 = 2 for B2, 2 = 3 for B3, 3 = 4 for B4, 4 = 5 for B5, 5 =10 for B6, 10 = 30 for B7, and 30 for B8.', '1305.6259-1-22-2': 'We chose denser sampling for the lower pnc values and sparser sampling for the higher pnc values since we observed the distribution of MA fragments to be denser in the lower pnc categories.', '1305.6259-1-23-0': 'englishRecovery of isoforms', '1305.6259-1-24-0': 'english Using simulated data, we obtained the MA equivalent for the exonic regions of isoform-bearing genes, and performed a Megablast search of the assemblies against it (Supplementary Methods).', '1305.6259-1-24-1': 'The Megablast hits were parsed to maximize subject coverage, after removing nested and partial overlaps (same as described earlier).', '1305.6259-1-24-2': 'For all assemblers, we calculated the number of exons recovered per isoform and the length recovery of each exon.', '1305.6259-1-25-0': 'englishResults', '1305.6259-1-26-0': 'english We compared the performance of assemblers using a variety of assembly-based parameters (numbers and lengths of assembled fragments, N50, N[MATH]50 and extent of redundancy) and mapping parameters (mapping based recovery of MA fragments (numbers and lengths), mis-assembly, reliance on pnc) for isoforms and non-isoforms, and shared and unique transcript regions.', '1305.6259-1-27-0': 'englishAssembly statistics', '1305.6259-1-28-0': 'english We simulated reads for the exonic regions of the Arabidopsis thaliana TAIR10 genome (see Methods for details) and obtained Illumina-like paired-end 76bp reads covering 15,532 transcripts.', '1305.6259-1-28-1': 'The transcript regions contiguously covered by reads were termed as Model Assembly (MA) fragments and were used as a valid reference for mapping the assemblies.', '1305.6259-1-28-2': 'A given transcript, therefore, comprised of one or more MA fragments, the shortest being 76 bp in length.', '1305.6259-1-28-3': 'We obtained 70,382 MA fragments from the 15,532 TAIR10 transcripts.', '1305.6259-1-28-4': 'The reads were assembled using Trinity, Trans-ABySS, Oases, SOAPdenovo-Trans, Tophat1-Cufflinks and Genome guided Trinity (see Methods for details) with default parameters and a minimum of 76bp assembled fragment length.', '1305.6259-1-28-5': 'We compared the numbers of assembled fragments, the minimum and maximum fragment lengths, their respective length frequency distributions, the N50 and N[MATH]50 statistics for the six assemblers, before and after eliminating redundancy in the assembly using CD-HIT-EST.', '1305.6259-1-29-0': 'english The total number of assembled fragments varied widely across the six assemblers (Table 1).', '1305.6259-1-29-1': 'We observed a 4- and 1.5-fold reduction in assembly size post redundancy removal for Trans-ABySS and Oases, respectively (Table 1).', '1305.6259-1-29-2': 'The shortest fragment reported by all assemblers was 76bp (as fixed by the minimum reporting length threshhold).', '1305.6259-1-29-3': 'The range of fragments at the long end varied across assemblers with the longest for Tophat1-Cufflinks at 10,502 bp (same as the maximum MA fragment length, Table 1).', '1305.6259-1-29-4': 'This was expected since Tophat1-Cufflinks is a genome-guided assembler that allows recovery of full length or near full length transcripts.', '1305.6259-1-29-5': 'For Trinity, genome guided Trinity and SOAPdenovo-Trans, we observed no difference pre- and post-redundancy removal across the entire distribution of assembled fragment lengths (Figure 1).', '1305.6259-1-29-6': 'However, Trans-ABySS resulted in a higher number of longer redundant assembled fragments (Figure 1) and interestingly a low number of shorter assembled fragments.', '1305.6259-1-29-7': 'Like the results from the frequency distribution statistics for long- and short-assembled fragments, Trinity, genome guided Trinity and SOAPdenovo-Trans yielded N50 values closer to that of MA.', '1305.6259-1-29-8': 'However, Trans-ABySS, Oases and TopHat1-Cufflinks yielded N50 numbers that were much higher than that of MA.', '1305.6259-1-29-9': 'After eliminating redundancy, the N50 values were not affected for most assemblers except for Trans-ABySS (Table 1).', '1305.6259-1-29-10': 'In contrast, the N_(MA)50 values, which are not dependent on the size of any assembly, were lower than that of the MA, for Trinity, Genome guided Trinity and SOAPdenovo-Trans, both before and after redundancy removal.', '1305.6259-1-29-11': 'For Trans-ABySS and Oases, the N_(MA)50 values were relatively higher, and were reduced 3 and 1.5-fold respectively, after redundancy removal.', '1305.6259-1-29-12': 'The N_(MA)50 values were higher than that of the MA for Tophat1-Cufflinks, both before and after redundancy removal.', '1305.6259-1-30-0': 'english MA fragments from TAIR10 transcripts corresponding to transposable elements and transcript isoforms, as per the GFF annotation, were not included in the expression level based binning.', '1305.6259-1-30-1': 'We excluded these elements since they have stretches of identical sequences, which posed a problem in assigning the pnc index and reliable mapping to the correct isoform and/or transposable element.', '1305.6259-1-30-2': 'Out of a total of 70,382 MA fragments, we were thus left with 46,805 fragments, which still had certain level of sub-sequence similarity, presumably arising from gene paralogs and SSRs.', '1305.6259-1-30-3': 'The numbers of MA fragments obtained in different expression bins were, 15,706 in B1 (the lowest expression bin), 9,722 in B2, 5,205 in B3, 2,797 in B4, 1,820 in B5, 4,522 in B6, 4,862 in B7 and 2,171 in B8 (the highest expression bin).', '1305.6259-1-31-0': 'englishMA recovery statistics', '1305.6259-1-32-0': 'english The assembled transcript fragments from all assemblers were mapped to the MA using Megablast to assign common identifiers for comparisons.', '1305.6259-1-32-1': 'We compared the numbers and lengths of MA fragments recovered within expression bins, B1 to B8, using the best hits from Megablast.', '1305.6259-1-32-2': 'Trinity detected the highest number of MA fragments in B1 followed by Genome guided Trinity and SOAPdenovo-Trans (Figure 2).', '1305.6259-1-32-3': 'However, the number of unique transcripts detected was highest for Tophat1-Cufflinks, followed by Trinity for the lowest expression bin B1 (Figure 2).', '1305.6259-1-32-4': 'The extent of overlapping between Trinity and either Genome guided Trinity or SOAPdenovo-Trans was 30% for the lowest expression bin B1.', '1305.6259-1-32-5': 'The numbers detected were highest for Trinity for all expression bins (B1-B8) with other assemblers showing recovery with increase in expression.', '1305.6259-1-32-6': 'In comparison to Trinity, for Oases and Trans-ABySS the detection sensitity increased with increased expression but the overall number of recovered MA fragments remained lower in all expression bins (Figure 2).', '1305.6259-1-32-7': 'Tophat1-Cufflinks showed a drop in its detection sensitivity from B1 to B2, and a steady rise thereafter.', '1305.6259-1-32-8': 'For all other assemblers, with the exception of Trans-ABySS, we observed an increase in higher-order intersections proportional to the decrease in unique detections and lower order intersections across bins B2-B8 (Figure 2).', '1305.6259-1-33-0': 'english We observed that longer MA fragments tend to have higher read support, thus assigning them to the higher expression level bins (Supplementary Figure 1).', '1305.6259-1-33-1': 'The shorter MA fragments, however, got assigned to all expression level bins, as they had low-to-high read support (Supplementary Figure 1).', '1305.6259-1-33-2': 'Therefore, we decided to compare the length recovery across assemblers, for each expression level bin, in different MA fragment length categories (Figure 3).', '1305.6259-1-33-3': 'For the shortest length category, 76bp, the median length recovery was close to 100%, across all expression bins for all assemblers, with the exception of Trans-ABySS (Figure 3).', '1305.6259-1-33-4': 'The outlier trend was highly variable across assemblers, and clustered closer to the median for SOAPdenovo-Trans, followed by Trinity and Genome guided Trinity.', '1305.6259-1-33-5': 'For Trinity, Trans-ABySS, Oases and Genome guided Trinity, we observed the outliers clustering around 40 length recovery (30bp) for the MA fragments in the lowest expression bin that matched the word size for Megablast (28bp).', '1305.6259-1-33-6': 'The outliers in the lowest expression bin for Tophat1-Cufflinks and those in the highest expression bins for Oases spanned all the way from 40 to the median.', '1305.6259-1-33-7': 'For the subsequent length categories, we observed a gradual increase in the median length recovery from as low as 20 to all the way upto 100, with an increase in the expression levels.', '1305.6259-1-33-8': 'This gradual increase was also seen, to a minor extent, for the same bin across MA length categories, for most assemblers.', '1305.6259-1-33-9': 'Tophat1-Cufflinks showed a median recovery of 100% across all expression bins and all MA fragment length categories.', '1305.6259-1-33-10': 'The pattern of length recovery was similar for Trinity, SOAPdenovo-Trans and Genome guided Trinity across all the expression bins and MA fragment length categories.', '1305.6259-1-33-11': 'Overall, these three assemblers outperformed Trans-ABySS and Oases in terms of higher median MA fragment lengh recovery and tighter distribution around the median (Figure 3).', '1305.6259-1-34-0': 'englishMis-assembly statistics', '1305.6259-1-35-0': 'english We used Megablast-based mapping to evaluate the accuracy of assembled fragments, and determine whether each assembled fragment belonged to a single MA source or was a chimera across multiple sources (mis-assembled).', '1305.6259-1-35-1': 'We classified the assembled fragments into three categories, =90%, between 60-90%, and 60% based on the extent of their lengths mapped to any single MA fragment.', '1305.6259-1-35-2': 'They were further classified into various assembled length categories to check their relationship with assembly quality, before and after removing redundancy.', '1305.6259-1-35-3': 'Trans-ABySS and Oases showed relatively higher number of mis-assembled fragments (between 60-90%, and 60% mapping) in the 200-400bp and 500-600bp ranges respectively (Figure 4).', '1305.6259-1-35-4': 'In the case of Trans-ABySS, the extent of mis-assembly decreased after removing redundancy, whereas in the case of Oases, it went up.', '1305.6259-1-35-5': 'We did not observe any difference in the extent of mis-assembly, pre- and post-redundancy removal, for any other assembler (Figure 4).', '1305.6259-1-35-6': 'The extent of mis-assembly was over-estimated with Tophat1-Cufflinks as it surpassed the length of the MA fragment (Table 1).', '1305.6259-1-35-7': 'For Trinity, SOAPdenovo-Trans and Genome guided Trinity, there was a clear trend of decreasing mis-assembly with increasing assembled fragment length (Figure 4).', '1305.6259-1-35-8': 'The degree of mis-assembly was lesser for SOAPdenovo-Trans than for Trinity or Genome guided Trinity for the shorter fragments.', '1305.6259-1-36-0': 'englishRecovery of isoforms', '1305.6259-1-37-0': 'english Next, we compared the extent of recovery of known isoforms from the transcripts assembled with various assemblers.', '1305.6259-1-37-1': 'We had a total of 24,846 exons in the MA fragments corresponding to 2,970 isoforms.', '1305.6259-1-37-2': 'We mapped assembled fragments from all assemblers to these exons, using Megablast, and maximized exon coverage.', '1305.6259-1-37-3': 'Trinity, followed by Genome guided Trinity and SOAPdenovo-Trans had the highest numbers of isoforms in the 80-100% length recovery category (Figure 5).', '1305.6259-1-37-4': 'We observed a correlation between the median pnc for each recovery category and the numbers of exons recovered per isoform.', '1305.6259-1-37-5': 'The median pnc was in the range of 2-3 for unrecovered isoforms and 17-23 for those which were fully recovered or close to fully recovered by all assemblers.', '1305.6259-1-37-6': 'Trans-ABySS and Oases, which showed a relatively lower recovery, correlated with a higher median pnc, suggesting a higher threshold of pnc needed for good recovery by those assemblers (Figure 5).', '1305.6259-1-38-0': 'english In order to test whether our understanding of isoform recovery from simulated reads holds true for a real dataset, we measured the recovery of shared and unique assembled transcripts for the shared and unique regions of the hox gene cluster in zebrafish (see Methods for details).', '1305.6259-1-38-1': 'We observed that across assemblers, Trinity, followed by Genome guided Trinity and SOAPdenovo-Trans recovered most and closer to full length transcripts (Figures 6-8).', '1305.6259-1-38-2': 'Tophat1-Cufflinks recovered mostly full-length transcripts, but fewer in numbers.', '1305.6259-1-38-3': 'Trans-ABySS and Oases recovered fewer and truncated transcripts.', '1305.6259-1-38-4': 'The length recovery (%) by all assemblers except Tophat1-Cufflinks displayed a dependency on pnc which was more obvious in the recovery of shared regions of transcripts, as expected due to a wider range of read depth (Figures 6-7).', '1305.6259-1-39-0': 'englishAugmenting transcriptome assembly', '1305.6259-1-40-0': 'english Since each assembler produced a set of unique transcripts or fragments, we proceeded towards augmenting englishthe assemblies, one with another.', '1305.6259-1-40-1': 'We ruled out most combinations of assemblers, and chose Trinity and Tophat1- Cufflinks, as Trinity produced most number of transcript fragments and TopHat-Cufflinks the most number of full-length transcripts.', '1305.6259-1-40-2': 'After aligning the Trinity transcript fragments to the Tophat1-Cufflinks transcripts, we obtained a cumulative size increase of 1.37 Mbp (20% of original Trinity assembly size), unique to Tophat1-Cufflinks,.', '1305.6259-1-40-3': 'We observed an increase of 1,377 in number of MA fragments and 5,23,127nt in total cumulative length recovered after augmenting Trinity transcript fragments with Tophat1-Cufflinks-specific transcript regions.', '1305.6259-1-40-4': 'Relaxing the stringency of Megablast word size, from 28 to lower, would have increased our MA length recovery further, at the expense of losing isoform reconstruction capability and sensitivity to variation in read-depth, maintained by the fragmented structure of Trinity transcriptome assembly.', '1305.6259-1-40-5': 'We further observed that the median length recovery for the augmented Trinity was overall better than Trinity by itself (Figure 9).', '1305.6259-1-40-6': 'For transcripts longer than 1500nt, this augmentation yielded full-length transcripts across all expression levels (Figure 9).', '1305.6259-1-40-7': 'Even for transcripts that are of sizes around 500nt, we saw the advantage of augmentation, at least for recovering more transcripts that are towards full length.', '1305.6259-1-40-8': 'The outliers were fewer in augmented Trinity than Tophat1-Cufflinks in all length categories.', '1305.6259-1-40-9': 'The improved recovery with augmented englishTrinity proved that an integrative approach was useful, particularly when one has access to genome in addition to the RNA-seq reads.', '1305.6259-1-41-0': 'englishDiscussion', '1305.6259-1-42-0': 'english RNA-seq using next-generation sequencing is a powerful technology to understand the transcriptome of an organism.', '1305.6259-1-42-1': 'Although, genome guided assemblers like Tophat-Cufflinks can assemble full length transcripts, most de novo approaches and even Genome guided Trinity, where the genome is used only to partition the RNA-seq reads, are useful in detecting novel transcripts.', '1305.6259-1-42-2': 'In our comparative study, we found that each assembler produced a set of unique transcripts or fragments, especially at lower levels of expression (Figure 2).', '1305.6259-1-42-3': 'Hence, we started by asking a question whether one can obtain a better transcriptome assembly when augmenting the de novo assembly with that from a genome-guided approach.', '1305.6259-1-42-4': 'While this was a reasonable question to ask, the presence of multiple tools, and hence errors associated with each of them, compounded the problem.', '1305.6259-1-42-5': 'Therefore, we started by finding out the efficiency of individual tools, the ones that are popularly used by the community, with the hope that we could choose from assemblers and combine the results from them without compounding additional errors in the final assembly.', '1305.6259-1-43-0': 'english A lower threshold for minimum fragment length allows one to retain valid assemblies (as demonstrated in Figure 3) at the expense of increasing errors.', '1305.6259-1-43-1': 'However, in our observation, the errors are comparatively lesser than the number of valid assemblies (Figure 4).', '1305.6259-1-43-2': 'The extent of mis-assembly by different assemblers was different.', '1305.6259-1-43-3': 'Oases and Trans-ABySS resulted in more mis-assemblies than the other tools (Figure 4).', '1305.6259-1-43-4': 'We suspected that a large number of redundant transcripts produced by Oases and Trans-ABySS were possibly mis-assembled.', '1305.6259-1-43-5': 'Interestingly, however, when we compared the mis-assembly statistics before and after redundancy removal, we found that Oases had more mis-assembly in the non-redundant regions of the transcriptome than Trans-ABySS (Figure 4) pointing towards assembly of more number of chimeric transcripts.', '1305.6259-1-44-0': 'english englishDue to higher sub-sequence similarity in isoforms, which contain more shared regions than non-isoforms, the chances of mis-assembly are greater.', '1305.6259-1-44-1': 'The shared regions among transcripts also pose additional difficulty in discerning their true identity at the time of estimating recovery using mapping.', '1305.6259-1-44-2': 'We analyzed the shared and unique regions within isoforms separately in order to distinguish their recovery and underlying pnc patterns.', '1305.6259-1-44-3': 'englishAll measures of pnc used for comparisons in this paper are based on the reads mapped to the MA as we were left with the option of remapping the reads to the assemblies in order to calculate the expression level, i.e. pnc in the absence of read tracking during transcriptome assembly.', '1305.6259-1-44-4': 'Re-mapping of reads is typically done using aligners that might result in multiple hits.', '1305.6259-1-44-5': 'This compels the user to arbitrarily assign the pnc to a transcript fragment, which may or may not be the same as the pnc estimated by an assembler.', '1305.6259-1-44-6': 'With simulated reads, where we knew the actual pnc of MA fragments, relationship between the actual (simulated) and estimated pnc using Megablast could be correlated.', '1305.6259-1-44-7': 'As expected, we found this to be true (Supplementary Figure 2).', '1305.6259-1-44-8': 'Based on this, we expected that the estimated pnc of transcript fragments by any assembler to be positively correlated with the actual pnc.', '1305.6259-1-44-9': 'In addition to the simulated dataset, we used the zebrafish hox gene cluster for transcript recovery analysis.', '1305.6259-1-44-10': 'Vertebrate Hox englishgenes are known to be involved during development, are arranged in sets of uninterrupted clusters, and are in most cases expressed in a collinear fashion making it an ideal gene cluster candidate for our assembly analysis.', '1305.6259-1-44-11': 'We used RNA-seq reads from zebrafish to perform the assembly and derived assembled fragments related to the hox gene cluster to perform recovery analysis.', '1305.6259-1-44-12': 'We expected the shared regions had sequencing reads at a greater read depth than the unique regions.', '1305.6259-1-44-13': 'Indeed, we found this to be reflected in terms of a greater median pnc for the shared regions in the 50% recovery category (Figures 6 and 7).', '1305.6259-1-45-0': 'english englishN50 is an assembly attribute widely used to compare the quality of a transcriptome assembly.', '1305.6259-1-45-1': 'In the absence of the knowledge of actual length distribution in the sequenced dataset (transcripts sequenced), it is generally assumed that a higher N50 is correlated with a better assembly length-recovery/contiguity.', '1305.6259-1-45-2': 'We know that errors and redundancy in the transcriptome assembly affect its total size.', '1305.6259-1-45-3': 'In addition, the lengths of assembly fragments were also variable (e.g. Tophat1-Cufflinks produced more full length transcripts and Trinity produced more number of small fragments than the rest of the assemblers).', '1305.6259-1-45-4': 'Therefore, basing a quality metric on the total assembly size to benchmark the assembler, like what the N50 does, can be highly inaccurate.', '1305.6259-1-45-5': 'Instead, basing the quality metric on the expected size of the transcriptome (NT50) or regions of the transcriptome that is covered by sequencing reads, as N_(MA)50 suggests in simulated reads, tend to provide a more accurate benchmark.', '1305.6259-1-45-6': 'Indeed, we found that the N_(MA)50 to accurately reflect the recovery quotient of an assembler (Table 1).', '1305.6259-1-45-7': 'The reduction in N_(MA)50 for well-performing assemblers (Trinity, Genome guided Trinity and SOAPdenovo-Trans) when compared to that for the MA, was in proportion to the relative reduction in their assembly numbers in comparison to the number of MA fragments (Table 1).', '1305.6259-1-45-8': 'In addition to N_(MA)50, we found that the median, inter-quartile range, minimum, maximum and outliers for transcript assembly length were more useful in describing a transcriptome assembly.', '1305.6259-1-46-0': 'english englishMis-assembly can occur as a result of sub-sequence similarity within reads, which manifests as highly branched nodes in a de Bruijn graph.', '1305.6259-1-46-1': 'This subsequence similarity, along with mismatches/errors in sequencing reads, can also cause spurious blast hits that are seen as outliers observed in the box and whisker plot for MA recovery, represented in Figure 3.', '1305.6259-1-46-2': 'Spurious blast hits due to the former tend to span from Megablast word size of 28bp onwards.', '1305.6259-1-46-3': 'We observed fewer outliers of the first kind in multi-kmer based approaches like Oases and Trans-ABySS.', '1305.6259-1-46-4': 'SOAPdenovo-Trans also appears to contain fewer spurious blast hits, especially for the 76bp MA length category.', '1305.6259-1-46-5': 'It is known that SOAPdenovo algorithm chooses to discard a repetitive node if there is unequal read support on edges to and from that node, and only build parallel paths carrying the node if there are equal number of reads on either edges .', '1305.6259-1-46-6': 'Based on our observations of low redundancy in mapped assemblies in the 76bp MA length category (Figure 3) and higher % of assembly mapping correctly to a single MA fragment in the lower assembled fragment length categories (Figure 4), it is possible that SOAPdenovo-Trans discarded repetitive sequences from the assembly for these length categories.', '1305.6259-1-46-7': 'TransABySS, prior to eliminating redundancy, showed a higher frequency for assembled fragments lengths 200bp (Figure 1A).', '1305.6259-1-46-8': 'The frequency of assembled fragments in this length range was much lower, post CD-HIT-EST (Figure 1B), suggesting the sequence redundancy in this range.', '1305.6259-1-46-9': 'Hence, we believe that there was relationship between redundancy and recovery of assembled fragments in our study.', '1305.6259-1-47-0': 'english In our study, we found Trinity to perform best in transcript recovery across all expression levels (Figures 2 and 3).', '1305.6259-1-47-1': 'The median length recovery by Tophat1-Cufflinks was always 100% (Figure 3).', '1305.6259-1-47-2': 'We chose to combine the assemblies from the Tophat1-Cufflinks assembly pipeline, that resulted in 1300 more transcript fragements corresponding to an cumulative increase of nearly 0.5million nt in length.', '1305.6259-1-47-3': 'This increase in sensitivity and specificity of the augmented assembly (Figure 10) compared to any one tool alone makes it an ideal process for transcriptome assembly augmentation.', '1305.6259-1-47-4': 'This process resulted in an augmented assembly with even greater sensitivity and only minimal compromising its specificity maintaining the expression-based fragmented assembly structure of Trinity post- augmentation.', '1305.6259-1-47-5': 'The improved recovery post-augmentation proved that an integrative approach may be employed in recovering more transcript fragments, particularly when one has access to genome assembly in addition to RNA-seq reads.', '1305.6259-1-48-0': 'english'}

{'1305.6259-2-0-0': 'english RNA-seq allows detection and precise quantification of transcripts, provides comprehensive understanding of exon/intron boundaries, aids discovery of alternatively spliced isoforms and fusion transcripts along with measurement of allele-specific expression.', '1305.6259-2-0-1': 'Researchers interested in studying and constructing transcriptomes, especially for non-model species, often face the conundrum of choosing from a number of available de novo and genome-guided assemblers.', '1305.6259-2-0-2': 'A comprehensive comparative study is required to assess and evaluate their efficiency and sensitivity for transcript assembly, reconstruction and recovery.', '1305.6259-2-0-3': 'None of the popular assembly tools in use today achieves requisite sensitivity, specificity or recovery of full-length transcripts on its own.', '1305.6259-2-0-4': 'Hence, it is imperative that methods be developed in order to augment assemblies generated from multiple tools, with minimal compounding of error.', '1305.6259-2-0-5': 'Here, we present an approach to combinatorially augment transciptome assembly based on a rigorous comparative study of popular de novo and genome-guided transcriptome assembly tools.', '1305.6259-2-1-0': 'englishIntroduction', '1305.6259-2-2-0': 'english High-throughput technology has changed our understanding of many facets of biology, from diseases to plant genetics , and to synthetic biology .', '1305.6259-2-2-1': 'The advent of DNA microarrays in the 90s ushered an era of high-throughput genome-wide gene expression profiling studies .', '1305.6259-2-2-2': 'DNA microarray, although a powerful technique, is dependent on gene annotation, and hence genome sequence information.', '1305.6259-2-2-3': 'This is circumvented by RNA sequencing (RNA-seq), which uses next-generation sequencing instruments.', '1305.6259-2-2-4': 'This shift from the semi-quantitative, hybridization-based approaches, as in DNA microarrays, to the quantitative, sequencing-based approaches has tremendously facilitated gene expression analysis.', '1305.6259-2-2-5': 'RNA-seq experiments yield additional information on transcriptome characterization and quantification, including strand-specificity, mapping of fusion transcripts, small RNA identification and alternate splicing information .', '1305.6259-2-3-0': 'english A number of tools have been developed for transcriptome assembly in the past.', '1305.6259-2-3-1': 'They are classified under de novo and genome-guided assembly categories.', '1305.6259-2-3-2': 'Trinity , SOAPdenovo-Trans , Oases and Trans-ABySS are de novo tools, while TopHat-Cufflinks, and Genome guided Trinity fall under the second category.', '1305.6259-2-3-3': 'De novo assembly tools create short contigs from overlapping reads, which are extended based on insert size estimates.', '1305.6259-2-3-4': 'They could be based on either construction of de Bruijn graphs (using k-mers for short reads) or use overlap-layout-consensus (for longer reads) algorithms .', '1305.6259-2-3-5': 'In Genome guided Trinity, the reads are aligned to the genome and partitioned into read clusters, which are individually assembled using Trinity.', '1305.6259-2-3-6': 'TopHat-Cufflinks is an aligner cum assembly pipeline, involving spliced read alignment to the genome and subsequent assembly of these aligned reads into transcripts.', '1305.6259-2-3-7': 'The genome template and read pairing are utilized here to produce full-length or near full-length transcripts.', '1305.6259-2-3-8': 'Studies have identified k-mer size as an important parameter for short-read transcriptome assembly quality.', '1305.6259-2-3-9': 'The theoretical calculation on assembling transcriptomes based on k-mer size (a lower or higher k-mer length for low- or high-expressed genes respectively) has also been backed by experimental evidence .', '1305.6259-2-3-10': 'The k-mer length depends on the genome complexity, sequencing depth and sequencing error .', '1305.6259-2-3-11': 'One can take a decision on k-mer length depending on desired transcript diversity vs continuity of a given transcriptome.', '1305.6259-2-4-0': 'english Varying expression levels among genes, presence of homologues and spliced isoforms increase the complexity in transcriptome assembly.', '1305.6259-2-4-1': 'Availability of genome sequence can aid the performance of transcriptome assembly process.', '1305.6259-2-4-2': 'In its absence, genomes of closely related organisms can be used for transcriptome assemblies .', '1305.6259-2-4-3': 'Both de novo and genome-guided approaches have their own strengths and one can think of combining data from both to achieve higher sensitivity.', '1305.6259-2-4-4': 'However, the process of combining assemblies from multiple assemblers is not error free and may produce false assemblies.', '1305.6259-2-4-5': 'In the current paper, we first describe a detailed analysis of existing de novo and genome-guided tools and then determine the best combination of tools that can be used to augment the process of assembly while keeping the false assemblies to a minimum.', '1305.6259-2-5-0': 'englishMaterials and Methods', '1305.6259-2-6-0': 'englishSimulating RNA-seq reads', '1305.6259-2-7-0': 'english The Arabidopsis thaliana (TAIR10) complete genome sequence, co-ordinates for genes and transposons were downloaded from ftp://ftp.arabidopsis.org/home/tair/Genes/ftp://ftp.arabidopsis.org/home/tair/Genes/TAIR10_genome_releaseenglishTAIR10genomerelease and the GFF file was parsed to obtain exonic coordinates.', '1305.6259-2-7-1': 'These were used to simulate Illumina-like RNA-seq reads using Flux-simulator (FS-nightly-build1.1.1-20121119021549) with the following options supplied within a parameter file (Supplementary Methods): NBMOLECULES 5000000 , SIZEDISTRIBUTION N(300,30) , READNUMBER 4000000 , READLENGTH 76 , PAIREDEND YES , ERRFILE 76.', '1305.6259-2-7-2': 'The resultant Illumina-like reads were split into 2 fastq files corresponding to read1 and read2 using a Python script from the Galaxy tool suite (Supplementary Methods).', '1305.6259-2-8-0': 'englishZebrafish transcriptome data', '1305.6259-2-9-0': 'english RNA-seq reads for the zebrafish, Danio rerio, (2dpf) embryo were downloaded from SRA (ERR003998) .', '1305.6259-2-9-1': 'The zebrafish genome (Zv9) downloaded from Ensemble was used for genome-guided transcriptome assembly.', '1305.6259-2-9-2': 'The transcripts for hox gene cluster (listed in Table 2 in Corredor-Adamez et al. 2005) were downloaded from ENSEMBL, and used to identify hox-related genes from the assemblies through Megablast.', '1305.6259-2-9-3': 'We focused on read-covered shared and unique regions of the hox gene cluster.', '1305.6259-2-10-0': 'englishRead assembly', '1305.6259-2-11-0': 'english The simulated and Danio rerio reads were assembled using four de novo (Trinity, version r2012-06-08; Trans-ABySS, version 1.3.2; Oases, version 0.2.08; SOAPdenovo-Trans, version 1.0 for simulated data and version 1.02 for Danio rerio) and two genome-guided (Tophat1, version 2.0.4 that uses Bowtie1 , version 0.12.8 and Cufflinks, version 2.0.0; Genome guided Trinity, version r2012-10-05 that uses GSNAP , version 2012-07-20 [V3] for simulated data and version 2013-03-31 [V5]) for zebrafish data in the process of transcriptome assembly pipelines.', '1305.6259-2-11-1': 'We used Trinity and Genome guided Trinity with a fixed kmer size of 25bp, Trans-ABySS on ABySS (version 1.3.3) multi-kmer assemblies (ranging from 20-64bp), Oases on Velvet (version 1.2.07) multi-kmer assemblies (every alternate kmer ranging from 19-71bp) and SOAPdenovo-Trans with a fixed kmer size of 23bp.', '1305.6259-2-11-2': 'We tested the Tophat2 (version 2.0.7 that uses Bowtie version 2.0.5)-Cufflinks pipeline on the simulated data but did not observe any difference in the assembly statistics compared to the Tophat1-Cufflinks pipeline that uses Bowtie1.', '1305.6259-2-11-3': 'All assemblers were run using default parameters (details in Supplementary Methods).', '1305.6259-2-11-4': 'However, we fixed the parameter for minimum length of assembled fragment as 76bp (equal to the length of the read).', '1305.6259-2-11-5': 'In the case of SOAPdenovo-Trans, contigs were used instead of scaffolds for all downstream analyses, as the minimum length cutoff could not be set for scaffolds.', '1305.6259-2-12-0': 'english In order to test whether we could augment the assembly of Trinity-predicted transcript fragments with those from Tophat1-Cufflinks, we used Megablast to map the Trinity transcript fragments against the Tophat1-Cufflinks transcripts, and then augmented Trinity assembly with those transcript fragments unique to Tophat1-Cufflinks assembly (see Supplementary Methods for details).', '1305.6259-2-13-0': 'englishRedundancy removal using CD-HIT-EST', '1305.6259-2-14-0': 'english We used CD-HIT-EST (version 4.5.4) to remove redundancy in each assembly.', '1305.6259-2-14-1': 'It retained the longest sequence out of a cluster of sequences that shared at least 95% sequence similarity based on a word size of 8.', '1305.6259-2-14-2': 'Further details on the options used to run CD-HIT-EST are provided in Supplementary Methods.', '1305.6259-2-15-0': 'englishModel Assembly', '1305.6259-2-16-0': 'english We defined all read-covered transcript regions (TAIR10 for simulated data and hox gene cluster for Danio rerio) as Model Assembly or MA as previously described in the report by Mundry et al 2012 .', '1305.6259-2-17-0': 'english Calculation of N50 and N[MATH]50 statistics for an assembly', '1305.6259-2-18-0': 'english N50 is defined as the minimum contig length for 50% of the assembly, after sorting the contigs in the descending order of their lengths.', '1305.6259-2-18-1': 'We calculated N50 values for all assemblies, MA and the TAIR10 simulated transcripts, pre- and post-CD-HIT-EST.', '1305.6259-2-18-2': 'We also calculated the N50 values for each assembler, while taking the MA cumulative size (bp) as the denominator instead of the respective assembly sizes.', '1305.6259-2-18-3': 'We termed these N50 values as the N[MATH]50.', '1305.6259-2-19-0': 'englishMapping assemblies to MA using Megablast', '1305.6259-2-20-0': 'english The assembled fragments (query) were mapped against the MA fragments (subject) using Megablast (blast+ version 2.2.26) with default parameters (Supplementary Methods).', '1305.6259-2-20-1': 'The Megablast hits were parsed in order to either maximize query coverage (to compute mis-assembly statistics) or maximize subject coverage (to compute MA recovery statistics).', '1305.6259-2-20-2': 'This was done to choose the best hits and discard the partially overlapping hits when the unique coverage was lower than or equal to 10bp.', '1305.6259-2-20-3': 'The scripts used to remove nested and partial overlaps from Megablast hits are provided in Supplementary Methods.', '1305.6259-2-21-0': 'englishExpression level bin categories', '1305.6259-2-22-0': 'english We estimated the average per nucleotide coverage (pnc) for all MA fragments, based on their read support as Number of reads multiplied by Read Length divided by MA fragment length.', '1305.6259-2-22-1': 'The MA fragments were then categorized into 8 expression level bins, B1 to B8, the pnc for each being: 1 for B1, 1 = 2 for B2, 2 = 3 for B3, 3 = 4 for B4, 4 = 5 for B5, 5 =10 for B6, 10 = 30 for B7, and 30 for B8.', '1305.6259-2-22-2': 'We chose denser sampling for the lower pnc values and sparser sampling for the higher pnc values since we observed the distribution of MA fragments to be denser in the lower pnc categories.', '1305.6259-2-23-0': 'englishRecovery of isoforms', '1305.6259-2-24-0': 'english Using simulated data, we obtained the MA equivalent for the exonic regions of isoform-bearing genes, and performed a Megablast search of the assemblies against it (Supplementary Methods).', '1305.6259-2-24-1': 'The Megablast hits were parsed to maximize subject coverage, after removing nested and partial overlaps (same as described earlier).', '1305.6259-2-24-2': 'For all assemblers, we calculated the number of exons recovered per isoform and the length recovery of each exon.', '1305.6259-2-25-0': 'englishResults', '1305.6259-2-26-0': 'english We compared the performance of assemblers using a variety of assembly-based parameters (numbers and lengths of assembled fragments, N50, N[MATH]50 and extent of redundancy) and mapping parameters (mapping based recovery of MA fragments (numbers and lengths), mis-assembly, reliance on pnc) for isoforms and non-isoforms, and shared and unique transcript regions.', '1305.6259-2-27-0': 'englishAssembly statistics', '1305.6259-2-28-0': 'english We simulated reads for the exonic regions of the Arabidopsis thaliana TAIR10 genome (see Methods for details) and obtained Illumina-like paired-end 76bp reads covering 15,532 transcripts.', '1305.6259-2-28-1': 'The transcript regions contiguously covered by reads were termed as Model Assembly (MA) fragments and were used as a valid reference for mapping the assemblies.', '1305.6259-2-28-2': 'A given transcript, therefore, comprised of one or more MA fragments, the shortest being 76 bp in length.', '1305.6259-2-28-3': 'We obtained 70,382 MA fragments from the 15,532 TAIR10 transcripts.', '1305.6259-2-28-4': 'The reads were assembled using Trinity, Trans-ABySS, Oases, SOAPdenovo-Trans, Tophat1-Cufflinks and Genome guided Trinity (see Methods for details) with default parameters and a minimum of 76bp assembled fragment length.', '1305.6259-2-28-5': 'We compared the numbers of assembled fragments, the minimum and maximum fragment lengths, their respective length frequency distributions, the N50 and N[MATH]50 statistics for the six assemblers, before and after eliminating redundancy in the assembly using CD-HIT-EST.', '1305.6259-2-29-0': 'english The total number of assembled fragments varied widely across the six assemblers (Table 1).', '1305.6259-2-29-1': 'We observed a 4- and 1.5-fold reduction in assembly size post redundancy removal for Trans-ABySS and Oases, respectively (Table 1).', '1305.6259-2-29-2': 'The shortest fragment reported by all assemblers was 76bp (as fixed by the minimum reporting length threshhold).', '1305.6259-2-29-3': 'The range of fragments at the long end varied across assemblers with the longest for Tophat1-Cufflinks at 10,502 bp (same as the maximum MA fragment length, Table 1).', '1305.6259-2-29-4': 'This was expected since Tophat1-Cufflinks is a genome-guided assembler that allows recovery of full length or near full length transcripts.', '1305.6259-2-29-5': 'For Trinity, genome guided Trinity and SOAPdenovo-Trans, we observed no difference pre- and post-redundancy removal across the entire distribution of assembled fragment lengths (Figure 1).', '1305.6259-2-29-6': 'However, Trans-ABySS resulted in a higher number of longer redundant assembled fragments (Figure 1) and interestingly a low number of shorter assembled fragments.', '1305.6259-2-29-7': 'Like the results from the frequency distribution statistics for long- and short-assembled fragments, Trinity, genome guided Trinity and SOAPdenovo-Trans yielded N50 values closer to that of MA.', '1305.6259-2-29-8': 'However, Trans-ABySS, Oases and TopHat1-Cufflinks yielded N50 numbers that were much higher than that of MA.', '1305.6259-2-29-9': 'After eliminating redundancy, the N50 values were not affected for most assemblers except for Trans-ABySS (Table 1).', '1305.6259-2-29-10': 'In contrast, the N_(MA)50 values, which are not dependent on the size of any assembly, were lower than that of the MA, for Trinity, Genome guided Trinity and SOAPdenovo-Trans, both before and after redundancy removal.', '1305.6259-2-29-11': 'For Trans-ABySS and Oases, the N_(MA)50 values were relatively higher, and were reduced 3 and 1.5-fold respectively, after redundancy removal.', '1305.6259-2-29-12': 'The N_(MA)50 values were higher than that of the MA for Tophat1-Cufflinks, both before and after redundancy removal.', '1305.6259-2-30-0': '& englishTAIR10 transcripts & englishModel Assembly (MA) & englishTrinity & englishTrans-ABySS & englishOases & englishSOAPdenovo-Trans & englishTophat1-Cufflinks & englishGenome guided Trinity', '1305.6259-2-31-0': 'english Total transcripts or fragments & english 15532 & english 70382 & english 45978 & english 65594 & english 52533 & english 39533 & english 8671 & english 46064', '1305.6259-2-32-0': 'english Transcriptome size (bp) & english 2.61E+07 & english 1.67E+07 & english 1.27E+07 & english 6.62E+07 & english 4.21E+07 & english 1.02E+07 & english 1.15E+07 & english 1.27E+07', '1305.6259-2-33-0': 'english N50 (bp) & english 1952 & english 720 & english 684 & english 1390 & english 1453 & english 617 & english 1624 & english 646', '1305.6259-2-34-0': 'english N(MA)50 (bp) & english 1952 & english 720 & english 264 & english 2802 & english 2519 & english 114 & english 1136 & english 254', '1305.6259-2-35-0': 'english Median transcript/fragment length (bp) & english 1456 & english 85 & english 115 & english 804 & english 523 & english 103 & english 1124 & english 121', '1305.6259-2-36-0': 'english Min.', '1305.6259-2-36-1': 'transcript/fragment length (bp) & english 201 & english 76 & english 76 & english 76 & english 76 & english 76 & english 76 & english 76', '1305.6259-2-37-0': 'english Max.', '1305.6259-2-37-1': 'transcript/fragment length (bp)', '1305.6259-2-38-0': 'englishPost-CDHit-EST & english 14623 & english 10501 & english 6888 & english 7326 & english 8128 & english 6300 & english 10502 & english 7833', '1305.6259-2-39-0': '& englishTAIR10 transcripts & englishModel Assembly (MA) & englishTrinity & englishTrans-ABySS & englishOases & englishSOAPdenovo-Trans & englishTophat1-Cufflinks & englishGenome guided Trinity', '1305.6259-2-40-0': 'english Total transcripts or fragments & english 14615 & english 62528 & english 45281 & english 16060 & english 28051 & english 39471 & english 8395 & english 42968', '1305.6259-2-41-0': 'english Transcriptome size (bp) & english 2.45E+07 & english 1.56E+07 & english 1.21E+07 & english 1.15E+07 & english 1.85E+07 & english 1.02E+07 & english 1.11E+07 & english 1.20E+07', '1305.6259-2-42-0': 'english N50 (bp) & english 1951 & english 788 & english 627 & english 1194 & english 1485 & english 617 & english 1621 & english 661', '1305.6259-2-43-0': 'english N(MA)50 (bp) & english 1951 & english 788 & english 260 & english 798 & english 1680 & english 155 & english 1197 & english 273', '1305.6259-2-44-0': 'english Median transcript/fragment length (bp) & english 1451 & english 87 & english 113 & english 477 & english 216 & english 103 & english 1121 & english 120', '1305.6259-2-45-0': 'english Min.', '1305.6259-2-45-1': 'transcript/fragment length (bp) & english 201 & english 76 & english 76 & english 76 & english 76 & english 76 & english 76 & english 76', '1305.6259-2-46-0': 'english Max.', '1305.6259-2-46-1': 'transcript/fragment length (bp) & english 14623 & english 10501 & english 6888 & english 7326 & english 8128 & english 6300 & english 10502 & english 7833', '1305.6259-2-47-0': 'englishMapping-based statistics', '1305.6259-2-48-0': 'english MA fragments from TAIR10 transcripts corresponding to transposable elements and transcript isoforms, as per the GFF annotation, were not included in the expression level based binning.', '1305.6259-2-48-1': 'We excluded these elements since they have stretches of identical sequences, which posed a problem in assigning the pnc index and reliable mapping to the correct isoform and/or transposable element.', '1305.6259-2-48-2': 'Out of a total of 70,382 MA fragments, we were thus left with 46,805 fragments, which still had certain level of sub-sequence similarity, presumably arising from gene paralogs and SSRs.', '1305.6259-2-48-3': 'The numbers of MA fragments obtained in different expression bins were, 15,706 in B1 (the lowest expression bin), 9,722 in B2, 5,205 in B3, 2,797 in B4, 1,820 in B5, 4,522 in B6, 4,862 in B7 and 2,171 in B8 (the highest expression bin).', '1305.6259-2-49-0': 'englishMA recovery statistics', '1305.6259-2-50-0': 'english The assembled transcript fragments from all assemblers were mapped to the MA using Megablast to assign common identifiers for comparisons.', '1305.6259-2-50-1': 'We compared the numbers and lengths of MA fragments recovered within expression bins, B1 to B8, using the best hits from Megablast.', '1305.6259-2-50-2': 'Trinity detected the highest number of MA fragments in B1 followed by Genome guided Trinity and SOAPdenovo-Trans (Figure 2).', '1305.6259-2-50-3': 'However, the number of unique transcripts detected was highest for Tophat1-Cufflinks, followed by Trinity for the lowest expression bin B1 (Figure 2).', '1305.6259-2-50-4': 'The extent of overlapping between Trinity and either Genome guided Trinity or SOAPdenovo-Trans was 30% for the lowest expression bin B1.', '1305.6259-2-50-5': 'The numbers detected were highest for Trinity for all expression bins (B1-B8) with other assemblers showing recovery with increase in expression.', '1305.6259-2-50-6': 'In comparison to Trinity, for Oases and Trans-ABySS the detection sensitity increased with increased expression but the overall number of recovered MA fragments remained lower in all expression bins (Figure 2).', '1305.6259-2-50-7': 'Tophat1-Cufflinks showed a drop in its detection sensitivity from B1 to B2, and a steady rise thereafter.', '1305.6259-2-50-8': 'For all other assemblers, with the exception of Trans-ABySS, we observed an increase in higher-order intersections proportional to the decrease in unique detections and lower order intersections across bins B2-B8 (Figure 2).', '1305.6259-2-51-0': 'english We observed that longer MA fragments tend to have higher read support, thus assigning them to the higher expression level bins (Supplementary Figure 1).', '1305.6259-2-51-1': 'The shorter MA fragments, however, got assigned to all expression level bins, as they had low-to-high read support (Supplementary Figure 1).', '1305.6259-2-51-2': 'Therefore, we decided to compare the length recovery across assemblers, for each expression level bin, in different MA fragment length categories (Figure 3).', '1305.6259-2-51-3': 'For the shortest length category, 76bp, the median length recovery was close to 100%, across all expression bins for all assemblers, with the exception of Trans-ABySS (Figure 3).', '1305.6259-2-51-4': 'The outlier trend was highly variable across assemblers, and clustered closer to the median for SOAPdenovo-Trans, followed by Trinity and Genome guided Trinity.', '1305.6259-2-51-5': 'For Trinity, Trans-ABySS, Oases and Genome guided Trinity, we observed the outliers clustering around 40 length recovery (30bp) for the MA fragments in the lowest expression bin that matched the word size for Megablast (28bp).', '1305.6259-2-51-6': 'The outliers in the lowest expression bin for Tophat1-Cufflinks and those in the highest expression bins for Oases spanned all the way from 40 to the median.', '1305.6259-2-51-7': 'For the subsequent length categories, we observed a gradual increase in the median length recovery from as low as 20 to all the way upto 100, with an increase in the expression levels.', '1305.6259-2-51-8': 'This gradual increase was also seen, to a minor extent, for the same bin across MA length categories, for most assemblers.', '1305.6259-2-51-9': 'Tophat1-Cufflinks showed a median recovery of 100% across all expression bins and all MA fragment length categories.', '1305.6259-2-51-10': 'The pattern of length recovery was similar for Trinity, SOAPdenovo-Trans and Genome guided Trinity across all the expression bins and MA fragment length categories.', '1305.6259-2-51-11': 'Overall, these three assemblers outperformed Trans-ABySS and Oases in terms of higher median MA fragment lengh recovery and tighter distribution around the median (Figure 3).', '1305.6259-2-52-0': 'englishMis-assembly statistics', '1305.6259-2-53-0': 'english We used Megablast-based mapping to evaluate the accuracy of assembled fragments, and determine whether each assembled fragment belonged to a single MA source or was a chimera across multiple sources (mis-assembled).', '1305.6259-2-53-1': 'We classified the assembled fragments into three categories, =90%, between 60-90%, and 60% based on the extent of their lengths mapped to any single MA fragment.', '1305.6259-2-53-2': 'They were further classified into various assembled length categories to check their relationship with assembly quality, before and after removing redundancy.', '1305.6259-2-53-3': 'Trans-ABySS and Oases showed relatively higher number of mis-assembled fragments (between 60-90%, and 60% mapping) in the 200-400bp and 500-600bp ranges respectively (Figure 4).', '1305.6259-2-53-4': 'In the case of Trans-ABySS, the extent of mis-assembly decreased after removing redundancy, whereas in the case of Oases, it went up.', '1305.6259-2-53-5': 'We did not observe any difference in the extent of mis-assembly, pre- and post-redundancy removal, for any other assembler (Figure 4).', '1305.6259-2-53-6': 'The extent of mis-assembly was over-estimated with Tophat1-Cufflinks as it surpassed the length of the MA fragment (Table 1).', '1305.6259-2-53-7': 'For Trinity, SOAPdenovo-Trans and Genome guided Trinity, there was a clear trend of decreasing mis-assembly with increasing assembled fragment length (Figure 4).', '1305.6259-2-53-8': 'The degree of mis-assembly was lesser for SOAPdenovo-Trans than for Trinity or Genome guided Trinity for the shorter fragments.', '1305.6259-2-54-0': 'englishRecovery of isoforms', '1305.6259-2-55-0': 'english Next, we compared the extent of recovery of known isoforms from the transcripts assembled with various assemblers.', '1305.6259-2-55-1': 'We had a total of 24,846 exons in the MA fragments corresponding to 2,970 isoforms.', '1305.6259-2-55-2': 'We mapped assembled fragments from all assemblers to these exons, using Megablast, and maximized exon coverage.', '1305.6259-2-55-3': 'Trinity, followed by Genome guided Trinity and SOAPdenovo-Trans had the highest numbers of isoforms in the 80-100% length recovery category (Figure 5).', '1305.6259-2-55-4': 'We observed a correlation between the median pnc for each recovery category and the numbers of exons recovered per isoform.', '1305.6259-2-55-5': 'The median pnc was in the range of 2-3 for unrecovered isoforms and 17-23 for those which were fully recovered or close to fully recovered by all assemblers.', '1305.6259-2-55-6': 'Trans-ABySS and Oases, which showed a relatively lower recovery, correlated with a higher median pnc, suggesting a higher threshold of pnc needed for good recovery by those assemblers (Figure 5).', '1305.6259-2-56-0': 'english In order to test whether our understanding of isoform recovery from simulated reads holds true for a real dataset, we measured the recovery of shared and unique assembled transcripts for the shared and unique regions of the hox gene cluster in zebrafish (see Methods for details).', '1305.6259-2-56-1': 'We observed that across assemblers, Trinity, followed by Genome guided Trinity and SOAPdenovo-Trans recovered most and closer to full length transcripts (Figures 6-8).', '1305.6259-2-56-2': 'Tophat1-Cufflinks recovered mostly full-length transcripts, but fewer in numbers.', '1305.6259-2-56-3': 'Trans-ABySS and Oases recovered fewer and truncated transcripts.', '1305.6259-2-56-4': 'The length recovery (%) by all assemblers except Tophat1-Cufflinks displayed a dependency on pnc which was more obvious in the recovery of shared regions of transcripts, as expected due to a wider range of read depth (Figures 6-7).', '1305.6259-2-57-0': 'englishAugmenting transcriptome assembly', '1305.6259-2-58-0': 'english Since each assembler produced a set of unique transcripts or fragments, we proceeded towards augmenting englishthe assemblies, one with another.', '1305.6259-2-58-1': 'We ruled out most combinations of assemblers, and chose Trinity and Tophat1- Cufflinks, as Trinity produced most number of transcript fragments and TopHat-Cufflinks the most number of full-length transcripts.', '1305.6259-2-58-2': 'After aligning the Trinity transcript fragments to the Tophat1-Cufflinks transcripts, we obtained a cumulative size increase of 1.37 Mbp (20% of original Trinity assembly size), unique to Tophat1-Cufflinks,.', '1305.6259-2-58-3': 'We observed an increase of 1,377 in number of MA fragments and 5,23,127nt in total cumulative length recovered after augmenting Trinity transcript fragments with Tophat1-Cufflinks-specific transcript regions.', '1305.6259-2-58-4': 'Relaxing the stringency of Megablast word size, from 28 to lower, would have increased our MA length recovery further, at the expense of losing isoform reconstruction capability and sensitivity to variation in read-depth, maintained by the fragmented structure of Trinity transcriptome assembly.', '1305.6259-2-58-5': 'We further observed that the median length recovery for the augmented Trinity was overall better than Trinity by itself (Figure 9).', '1305.6259-2-58-6': 'For transcripts longer than 1500nt, this augmentation yielded full-length transcripts across all expression levels (Figure 9).', '1305.6259-2-58-7': 'Even for transcripts that are of sizes around 500nt, we saw the advantage of augmentation, at least for recovering more transcripts that are towards full length.', '1305.6259-2-58-8': 'The outliers were fewer in augmented Trinity than Tophat1-Cufflinks in all length categories.', '1305.6259-2-58-9': 'The improved recovery with augmented englishTrinity proved that an integrative approach was useful, particularly when one has access to genome in addition to the RNA-seq reads.', '1305.6259-2-59-0': 'englishDiscussion', '1305.6259-2-60-0': 'english RNA-seq using next-generation sequencing is a powerful technology to understand the transcriptome of an organism.', '1305.6259-2-60-1': 'Although, genome guided assemblers like Tophat-Cufflinks can assemble full length transcripts, most de novo approaches and even Genome guided Trinity, where the genome is used only to partition the RNA-seq reads, are useful in detecting novel transcripts.', '1305.6259-2-60-2': 'In our comparative study, we found that each assembler produced a set of unique transcripts or fragments, especially at lower levels of expression (Figure 2).', '1305.6259-2-60-3': 'Hence, we started by asking a question whether one can obtain a better transcriptome assembly when augmenting the de novo assembly with that from a genome-guided approach.', '1305.6259-2-60-4': 'While this was a reasonable question to ask, the presence of multiple tools, and hence errors associated with each of them, compounded the problem.', '1305.6259-2-60-5': 'Therefore, we started by finding out the efficiency of individual tools, the ones that are popularly used by the community, with the hope that we could choose from assemblers and combine the results from them without compounding additional errors in the final assembly.', '1305.6259-2-61-0': 'english A lower threshold for minimum fragment length allows one to retain valid assemblies (as demonstrated in Figure 3) at the expense of increasing errors.', '1305.6259-2-61-1': 'However, in our observation, the errors are comparatively lesser than the number of valid assemblies (Figure 4).', '1305.6259-2-61-2': 'The extent of mis-assembly by different assemblers was different.', '1305.6259-2-61-3': 'Oases and Trans-ABySS resulted in more mis-assemblies than the other tools (Figure 4).', '1305.6259-2-61-4': 'We suspected that a large number of redundant transcripts produced by Oases and Trans-ABySS were possibly mis-assembled.', '1305.6259-2-61-5': 'Interestingly, however, when we compared the mis-assembly statistics before and after redundancy removal, we found that Oases had more mis-assembly in the non-redundant regions of the transcriptome than Trans-ABySS (Figure 4) pointing towards assembly of more number of chimeric transcripts.', '1305.6259-2-62-0': 'english englishDue to higher sub-sequence similarity in isoforms, which contain more shared regions than non-isoforms, the chances of mis-assembly are greater.', '1305.6259-2-62-1': 'The shared regions among transcripts also pose additional difficulty in discerning their true identity at the time of estimating recovery using mapping.', '1305.6259-2-62-2': 'We analyzed the shared and unique regions within isoforms separately in order to distinguish their recovery and underlying pnc patterns.', '1305.6259-2-62-3': 'englishAll measures of pnc used for comparisons in this paper are based on the reads mapped to the MA as we were left with the option of remapping the reads to the assemblies in order to calculate the expression level, i.e. pnc in the absence of read tracking during transcriptome assembly.', '1305.6259-2-62-4': 'Re-mapping of reads is typically done using aligners that might result in multiple hits.', '1305.6259-2-62-5': 'This compels the user to arbitrarily assign the pnc to a transcript fragment, which may or may not be the same as the pnc estimated by an assembler.', '1305.6259-2-62-6': 'With simulated reads, where we knew the actual pnc of MA fragments, relationship between the actual (simulated) and estimated pnc using Megablast could be correlated.', '1305.6259-2-62-7': 'As expected, we found this to be true (Supplementary Figure 2).', '1305.6259-2-62-8': 'Based on this, we expected that the estimated pnc of transcript fragments by any assembler to be positively correlated with the actual pnc.', '1305.6259-2-62-9': 'In addition to the simulated dataset, we used the zebrafish hox gene cluster for transcript recovery analysis.', '1305.6259-2-62-10': 'Vertebrate Hox englishgenes are known to be involved during development, are arranged in sets of uninterrupted clusters, and are in most cases expressed in a collinear fashion making it an ideal gene cluster candidate for our assembly analysis.', '1305.6259-2-62-11': 'We used RNA-seq reads from zebrafish to perform the assembly and derived assembled fragments related to the hox gene cluster to perform recovery analysis.', '1305.6259-2-62-12': 'We expected the shared regions had sequencing reads at a greater read depth than the unique regions.', '1305.6259-2-62-13': 'Indeed, we found this to be reflected in terms of a greater median pnc for the shared regions in the 50% recovery category (Figures 6 and 7).', '1305.6259-2-63-0': 'english englishN50 is an assembly attribute widely used to compare the quality of a transcriptome assembly.', '1305.6259-2-63-1': 'In the absence of the knowledge of actual length distribution in the sequenced dataset (transcripts sequenced), it is generally assumed that a higher N50 is correlated with a better assembly length-recovery/contiguity.', '1305.6259-2-63-2': 'We know that errors and redundancy in the transcriptome assembly affect its total size.', '1305.6259-2-63-3': 'In addition, the lengths of assembly fragments were also variable (e.g. Tophat1-Cufflinks produced more full length transcripts and Trinity produced more number of small fragments than the rest of the assemblers).', '1305.6259-2-63-4': 'Therefore, basing a quality metric on the total assembly size to benchmark the assembler, like what the N50 does, can be highly inaccurate.', '1305.6259-2-63-5': 'Instead, basing the quality metric on the expected size of the transcriptome (NT50) or regions of the transcriptome that is covered by sequencing reads, as N_(MA)50 suggests in simulated reads, tend to provide a more accurate benchmark.', '1305.6259-2-63-6': 'Indeed, we found that the N_(MA)50 to accurately reflect the recovery quotient of an assembler (Table 1).', '1305.6259-2-63-7': 'The reduction in N_(MA)50 for well-performing assemblers (Trinity, Genome guided Trinity and SOAPdenovo-Trans) when compared to that for the MA, was in proportion to the relative reduction in their assembly numbers in comparison to the number of MA fragments (Table 1).', '1305.6259-2-63-8': 'In addition to N_(MA)50, we found that the median, inter-quartile range, minimum, maximum and outliers for transcript assembly length were more useful in describing a transcriptome assembly.', '1305.6259-2-64-0': 'english englishMis-assembly can occur as a result of sub-sequence similarity within reads, which manifests as highly branched nodes in a de Bruijn graph.', '1305.6259-2-64-1': 'This subsequence similarity, along with mismatches/errors in sequencing reads, can also cause spurious blast hits that are seen as outliers observed in the box and whisker plot for MA recovery, represented in Figure 3.', '1305.6259-2-64-2': 'Spurious blast hits due to the former tend to span from Megablast word size of 28bp onwards.', '1305.6259-2-64-3': 'We observed fewer outliers of the first kind in multi-kmer based approaches like Oases and Trans-ABySS.', '1305.6259-2-64-4': 'SOAPdenovo-Trans also appears to contain fewer spurious blast hits, especially for the 76bp MA length category.', '1305.6259-2-64-5': 'It is known that SOAPdenovo algorithm chooses to discard a repetitive node if there is unequal read support on edges to and from that node, and only build parallel paths carrying the node if there are equal number of reads on either edges .', '1305.6259-2-64-6': 'Based on our observations of low redundancy in mapped assemblies in the 76bp MA length category (Figure 3) and higher % of assembly mapping correctly to a single MA fragment in the lower assembled fragment length categories (Figure 4), it is possible that SOAPdenovo-Trans discarded repetitive sequences from the assembly for these length categories.', '1305.6259-2-64-7': 'TransABySS, prior to eliminating redundancy, showed a higher frequency for assembled fragments lengths 200bp (Figure 1A).', '1305.6259-2-64-8': 'The frequency of assembled fragments in this length range was much lower, post CD-HIT-EST (Figure 1B), suggesting the sequence redundancy in this range.', '1305.6259-2-64-9': 'Hence, we believe that there was relationship between redundancy and recovery of assembled fragments in our study.', '1305.6259-2-64-10': 'english In our study, we found Trinity to perform best in transcript recovery across all expression levels (Figures 2 and 3).', '1305.6259-2-64-11': 'The median length recovery by Tophat1-Cufflinks was always 100% (Figure 3).', '1305.6259-2-64-12': 'We chose to combine the Trinity assemblies with that from the Tophat1-Cufflinks assembly pipeline.', '1305.6259-2-64-13': 'This resulted in 1300 more transcript fragements corresponding to an cumulative increase of nearly 0.5million nt in length.', '1305.6259-2-64-14': 'The process resulted in an augmented assembly with even greater sensitivity and only minimally compromising its specificity, while maintaining the expression-based fragmented assembly structure of Trinity post- augmentation (Figure 10).', '1305.6259-2-64-15': 'The improved recovery post-augmentation proved that an integrative approach may be employed in recovering more transcript fragments, particularly when one has access to genome assembly in addition to RNA-seq reads.', '1305.6259-2-65-0': 'english'}

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'1305.6259-2-61-0'], ['1305.6259-1-43-1', '1305.6259-2-61-1'], ['1305.6259-1-43-2', '1305.6259-2-61-2'], ['1305.6259-1-43-3', '1305.6259-2-61-3'], ['1305.6259-1-43-4', '1305.6259-2-61-4'], ['1305.6259-1-43-5', '1305.6259-2-61-5'], ['1305.6259-1-9-0', '1305.6259-2-9-0'], ['1305.6259-1-9-1', '1305.6259-2-9-1'], ['1305.6259-1-9-2', '1305.6259-2-9-2'], ['1305.6259-1-9-3', '1305.6259-2-9-3'], ['1305.6259-1-18-0', '1305.6259-2-18-0'], ['1305.6259-1-18-1', '1305.6259-2-18-1'], ['1305.6259-1-18-2', '1305.6259-2-18-2'], ['1305.6259-1-18-3', '1305.6259-2-18-3'], ['1305.6259-1-44-0', '1305.6259-2-62-0'], ['1305.6259-1-44-1', '1305.6259-2-62-1'], ['1305.6259-1-44-2', '1305.6259-2-62-2'], ['1305.6259-1-44-3', '1305.6259-2-62-3'], ['1305.6259-1-44-4', '1305.6259-2-62-4'], ['1305.6259-1-44-5', '1305.6259-2-62-5'], ['1305.6259-1-44-6', '1305.6259-2-62-6'], ['1305.6259-1-44-7', '1305.6259-2-62-7'], ['1305.6259-1-44-8', '1305.6259-2-62-8'], ['1305.6259-1-44-9', '1305.6259-2-62-9'], ['1305.6259-1-44-10', 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[]

[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1305.6259

1901.09346

{'1901.09346-1-0-0': 'We introduce the concrete autoencoder, an end-to-end differentiable method for global feature selection, which efficiently identifies a subset of the most informative features and simultaneously learns a neural network to reconstruct the input data from the selected features.', '1901.09346-1-0-1': 'Our method is unsupervised, and is based on using a concrete selector layer as the encoder and using a standard neural network as the decoder.', '1901.09346-1-0-2': 'During the training phase, the temperature of the concrete selector layer is gradually decreased, which encourages a user-specified number of discrete features to be learned.', '1901.09346-1-0-3': 'During test time, the selected features can be used with the decoder network to reconstruct the remaining input features.', '1901.09346-1-0-4': 'We evaluate concrete autoencoders on a variety of datasets, where they significantly outperform state-of-the-art methods for feature selection and data reconstruction.', '1901.09346-1-0-5': 'In particular, on a large-scale gene expression dataset, the concrete autoencoder selects a small subset of genes whose expression levels can be use to impute the expression levels of the remaining genes.', '1901.09346-1-0-6': 'In doing so, it improves on the current widely-used expert-curated L1000 landmark genes, potentially reducing measurement costs by 20%.', '1901.09346-1-0-7': 'The concrete autoencoder can be implemented by adding just a few lines of code to a standard autoencoder.', '1901.09346-1-1-0': '# Introduction', '1901.09346-1-2-0': 'High-dimensional datasets often pose a challenge for machine learning algorithms.', '1901.09346-1-2-1': 'Feature selection methods aim to reduce dimensionality of data by identifying the subset of relevant features in a dataset.', '1901.09346-1-2-2': 'A large number of algorithms have been proposed for feature selection in both supervised and unsupervised settings .', '1901.09346-1-2-3': 'These methods provide insight into the relationship between features in complex data and can simplify the process of training downstream models.', '1901.09346-1-2-4': 'Feature selection is particularly useful if the data with the full set of features is expensive or difficult to collect, as it can eliminate the need to measure irrelevant or redundant features.', '1901.09346-1-3-0': 'As a motivating example, consider a dataset that consists of the expression of various genes across tissue samples.', '1901.09346-1-3-1': 'Such "omics" measurements are increasingly carried out to fully characterize biological samples at the individual and single-cell level .', '1901.09346-1-3-2': 'Yet it remains expensive to conduct all of the biological assays that are needed to characterize such samples.', '1901.09346-1-3-3': 'It is natural to ask: what are the most important features in this dataset?', '1901.09346-1-3-4': 'Are there redundant features that do not need to measured?', '1901.09346-1-3-5': 'The idea of only measuring a subset of biological features and then reconstructing the remaining features is not new; in fact, this line of thinking has motivated the identification of the landmark genes, also known as the L1000 genes, which are a small subset of the over 20,000 human genes.', '1901.09346-1-3-6': 'The expression levels of the L1000 are strongly correlated with the expression levels of other genes, and thus this subset can be measured cheaply and then used to impute the remaining gene expression levels .', '1901.09346-1-4-0': 'The problem of feature selection is different from the more general problem of dimensionality reduction.', '1901.09346-1-4-1': 'Standard techniques for dimensionality reduction, such as principal components analysis and autoencoders , are be able to represent data with fewer dimensions while preserving maximal variance or minimizing reconstruction loss.', '1901.09346-1-4-2': 'However, such methods do not select a set of features present in the original dataset, and thus cannot be directly used to eliminate redundant features and reduce experimental costs.', '1901.09346-1-4-3': 'We emphasize the unsupervised nature of this problem: specific prediction tasks may not be known ahead of time, and thus it is important to develop methods that can identify a subset of features while allowing imputation of the remaining features with minimal distortion for arbitrary downstream tasks.', '1901.09346-1-5-0': 'In this paper, we propose a new end-to-end method to perform feature subset selection and imputation that leverages the power of deep autoencoders for discrete feature selection.', '1901.09346-1-5-1': 'Our method, the concrete autoencoder, uses a relaxation of the discrete distribution, the Concrete distribution , and the reparametrization trick to differentiate through an arbitrary (e.g. reconstruction) loss and select input features to minimize this loss.', '1901.09346-1-5-2': 'A visual example of results from our method is shown in Fig. [REF], where the concrete autoencoder selects the 20 most informative pixels (out of a total of 784) on the MNIST dataset, and reconstructs the original images with high accuracy.', '1901.09346-1-6-0': 'We test concrete autoencoders on a variety of datasets, and find that they generally outperform state-of-the-art methods for feature selection and data reconstruction.', '1901.09346-1-6-1': 'We have made the code for our algorithm and experiments available on a public repository.', '1901.09346-1-7-0': 'Related Works Feature selection methods are generally divided into three classes: filter, wrapper, and embedded methods.', '1901.09346-1-7-1': 'Filter methods rank the features of the dataset using statistical tests, such as the variance, in order to select features that individually maximize the desired criteria .', '1901.09346-1-7-2': 'Filter methods generally do not consider interactions between features and do not provide a way to impute the remaining features; one must train a separate algorithm to do so.', '1901.09346-1-7-3': 'Wrapper methods select subsets of features that maximize a objective function, which is optimized over the choice of input features using a black-box optimization method, such as sequential search or genetic algorithms .', '1901.09346-1-7-4': 'Since wrapper methods evaluate subsets of features, they are able to detect potential relationships between features, but usually at the expense of increased computation time.', '1901.09346-1-7-5': 'Embedded methods also consider relationships between features but generally do so more efficiently as they incorporate feature selection into the learning phase of another algorithm.', '1901.09346-1-7-6': 'A well-known example is the Lasso , which can be used to select features for regression by varying the strength of [MATH] regularization.', '1901.09346-1-8-0': 'Many embedded unsupervised feature selection algorithms use regularization as the means to select discrete features.', '1901.09346-1-8-1': 'The popular UDFS algorithm [CITATION] incorporates [MATH] regularization on a set of weights applied to the input to select features most useful for local discriminative analysis.', '1901.09346-1-8-2': 'Similarly, the MCFS algorithm uses regularization to solve for the features which preserve the clustering structure in the data.', '1901.09346-1-8-3': 'The recently-proposed AEFS algorithm also uses [MATH] regularization on the weights of the encoder that maps the input data to a latent space and optimizes these weights for their ability to reconstruct the original input.', '1901.09346-1-9-0': 'In this paper, we select discrete features using an embedded method but without resorting to regularization.', '1901.09346-1-9-1': 'Rather, we use a relaxation of the discrete random variables, the Concrete distribution , which allows a low-variance estimate of the gradient through discrete stochastic nodes.', '1901.09346-1-9-2': 'By using Concrete random variables, we can directly parametrize the selection of input features, and differentiate through the parameters.', '1901.09346-1-9-3': 'As we show through experiments in Section [REF], this leads to lower reconstruction errors on real-world datasets compared to the aforementioned regularization-based methods.', '1901.09346-1-10-0': '# Problem Formulation', '1901.09346-1-11-0': 'We now describe the problem of global feature selection.', '1901.09346-1-11-1': 'Although global feature selection is relevant for both unsupervised and supervised settings, we describe here the unsupervised case, which is the primary focus of this paper, and defer discussion of the supervised case to Appendix [REF].', '1901.09346-1-12-0': 'Consider a data-generating probability distribution [MATH] over a [MATH]-dimensional space.', '1901.09346-1-12-1': 'The goal is to learn a subset [MATH] of features of specified size [MATH] and also learn a reconstruction function [MATH], such that the expected loss between the reconstructed sample [MATH] and the original sample [MATH] is minimized, where [MATH] consists of those elements [MATH] such that [MATH].', '1901.09346-1-12-2': 'In other words, we would like to optimize [EQUATION]', '1901.09346-1-12-3': 'In practice, we do not know [MATH]; rather we have [MATH] samples, generally assumed to be drawn i.i.d. from [MATH].', '1901.09346-1-12-4': 'These samples can be represented in a data matrix [MATH], and so the goal becomes choosing [MATH] columns of [MATH] such that sub-matrix [MATH], defined analogously to [MATH], can be used to reconstruct the original matrix [MATH].', '1901.09346-1-12-5': 'Let us overload [MATH] to mean the matrix that results from applying [MATH] to each of the rows of [MATH] and stacking the resulting outputs.', '1901.09346-1-12-6': 'We seek to minimize the empirical reconstruction error: [EQUATION] where [MATH] denotes the Frobenius norm of the matrix.', '1901.09346-1-12-7': 'The principal difficulty in solving ([REF]) is the optimization over the discrete set of features [MATH], whose choices grow exponentially in [MATH].', '1901.09346-1-12-8': 'Thus, even for simple choices of [MATH], such as linear regression, the optimization problem in ([REF]) is NP-hard to solve [CITATION].', '1901.09346-1-13-0': 'Furthermore, the complexity of [MATH] can significantly affect reconstruction error and even the choice of [MATH].', '1901.09346-1-13-1': 'More expressive and non-linear choices for [MATH], such as neural networks, will naturally allow for lower reconstruction error, potentially at the expense of a more difficult optimization problem.', '1901.09346-1-13-2': 'We seek to develop a method that can approximate the solution for any given class of functions [MATH], from linear regression to deep fully-connected neural networks.', '1901.09346-1-14-0': 'Finally, we note that the choice of mean-squared reconstruction error as the metric for optimization in ([REF]) and ([REF]) stems from the fact that it is a smooth differentiable function that serves as a proxy for many downstream analyses, such as clustering performance and classification accuracy.', '1901.09346-1-14-1': 'However, other differentiable metrics, such as a variational approximation of the mutual information between [MATH] and [MATH], may be considered as well .', '1901.09346-1-15-0': '# Proposed Method', '1901.09346-1-16-0': 'The concrete autoencoder is an adaption of the standard autoencoder for discrete feature selection.', '1901.09346-1-16-1': 'Instead of using series of fully-connected layers for the encoder, we propose a concrete selector layer with a user-specified number of nodes, [MATH].', '1901.09346-1-16-2': 'This layer selects stochastic linear combinations of input features during training, which converge to a discrete set of [MATH] features by the end of training and during test time.', '1901.09346-1-17-0': 'The way in which input features are combined depends on the temperature of this layer, which we modulate using a simple annealing schedule.', '1901.09346-1-17-1': 'As the temperature of the layer approaches zero, the layer selects [MATH] individual input features.', '1901.09346-1-17-2': 'The decoder of a concrete autoencoder, which serves as the reconstruction function, is the same as that of a standard autoencoder: a neural network whose architecture can be set by the user based on dataset size and complexity.', '1901.09346-1-17-3': 'In effect, then, the concrete autoencoder is a method for selecting a discrete set of features that are optimized for an arbitrarily-complex reconstruction function.', '1901.09346-1-17-4': 'We describe the ingredients for our method in more detail in the next two subsections.', '1901.09346-1-18-0': '## Concrete Selector Layer', '1901.09346-1-19-0': 'The concrete selector layer is based on Concrete random variables .', '1901.09346-1-19-1': 'A Concrete random variable can be sampled to produce a continuous relaxation of the one-hot vector.', '1901.09346-1-19-2': 'The extent to which the one-hot vector is relaxed is controlled by a temperature parameter [MATH].', '1901.09346-1-19-3': 'To sample a Concrete random variable in [MATH] dimensions with parameters [MATH] and [MATH], one first samples a [MATH]-dimensional vector of i.i.d. samples from a Gumbel distribution , [MATH].', '1901.09346-1-19-4': 'Then each element of the sample [MATH] from the Concrete distribution is defined as: [EQUATION] where [MATH] refers to the [MATH] element in a particular sample vector.', '1901.09346-1-19-5': 'In the limit [MATH], the concrete random variable smoothly approaches the discrete distribution, outputting one hot vectors with [MATH] with probability [MATH].', '1901.09346-1-19-6': 'The desirable aspect of the Concrete random variable is that it allows for differentiation with respect to its parameters [MATH] via the reparametrization trick .', '1901.09346-1-20-0': 'We use Concrete random variables to select input features in the following way.', '1901.09346-1-20-1': 'For each of the [MATH] nodes in the concrete selector layer, we sample a [MATH]-dimensional Concrete random variable [MATH] (note that the superscript here indexes the node in the selector layer, whereas the subscript earlier referred to the element in the vector).', '1901.09346-1-20-2': 'The [MATH] node in the selector layer [MATH] outputs [MATH].', '1901.09346-1-20-3': 'This is, in general, a weighted linear combination of the input features, but notice that when [MATH], each node in the concrete selector layer outputs exactly one of the input features.', '1901.09346-1-20-4': 'After the network is trained, during test time, we thus replace the concrete selector layer with a discrete [MATH] layer in which the output of the [MATH] neuron is [MATH].', '1901.09346-1-21-0': 'We randomly initialize [MATH] to small positive values, to encourage the selector layer to stochastically explore different linear combinations of input features.', '1901.09346-1-21-1': 'However, as the network is trained, the values of [MATH] become more sparse, as the network becomes more confident in particular choices of input features, reducing the stochasticity in selected features.', '1901.09346-1-21-2': 'The concrete autoencoder architecture is shown in Fig. [REF](a) and the pseudocode for training in Fig. [REF](b).', '1901.09346-1-22-0': '## Annealing Schedule', '1901.09346-1-23-0': 'The temperature of Concrete random variables in the concrete selector layer has a significant affect on the output of the nodes.', '1901.09346-1-23-1': 'If the temperature is held high, the concrete selector layer continuously outputs a linear combination of features.', '1901.09346-1-23-2': 'On the contrary, if the temperature is held low, the concrete selector layer is not able to explore different combinations of features and converges to a poor local minimum.', '1901.09346-1-23-3': 'Neither fixed temperature allows the concrete selector layer to converge to informative features.', '1901.09346-1-24-0': 'Instead, we propose a simple annealing schedule that sets the temperature for all of the concrete variables, initially beginning with a high temperature [MATH] and gradually decaying the temperature until a final temperate [MATH] at each epoch according to a first-order exponential decay: [MATH] where [MATH] is the temperature at epoch number [MATH], and [MATH] is the total number of epochs.', '1901.09346-1-24-1': 'We compare various methods for setting the temperature of the concrete selector nodes in Fig. [REF].', '1901.09346-1-24-2': 'We find that this annealing schedule allows the concrete selector layer to effectively stochastically explore combinations of features in the initial phases of training, while in the later stages of training, the lowered temperature allows the the network to converge to informative individual features.', '1901.09346-1-25-0': '# Experiments', '1901.09346-1-26-0': 'In this section, we carry out experiments to compare the performance of concrete autoencoders to other feature subset selections on standard public datasets.', '1901.09346-1-26-1': 'For all of the experiments, we use Adam optimizer with a learning rate of [MATH].', '1901.09346-1-26-2': 'The initial temperature of the concrete autoencoder [MATH] was set to [MATH] and the final temperature [MATH] to [MATH].', '1901.09346-1-26-3': 'We trained the concrete autoencoder until until the mean of the concrete samples exceeded 0.99.', '1901.09346-1-26-4': 'For UDFS and AEFS, we swept values of each regularization hyperparameter and report the results with optimal hyperparameters according to mean squared error for reconstruction for each method.', '1901.09346-1-27-0': 'Furthermore, since the reconstruction [MATH] can overfit to patterns particular to the training set, we divide each dataset randomly into train, validation, and test datasets according to a 72-8-20 split.', '1901.09346-1-27-1': 'We use the training set to learn the parameters of the concrete autoencoders, the validation set to select optimal hyperparameters, and the test set to evaluate generalization performance, which we report below.', '1901.09346-1-28-0': 'We compare concrete autoencoders to many of the unsupervised feature selection methods mentioned in Related Works including UDFS, MCFS, and AEFS.', '1901.09346-1-28-1': 'We also include principal feature analysis (PFA), proposed by [CITATION], which is a popular method for selecting discrete features based on PCA, as well as a spectral method, the Laplacian score .', '1901.09346-1-28-2': 'Where available, we made use of scikit-feature implementation of each method .', '1901.09346-1-28-3': 'In our experiments, we also include, as upper bounds on performance, dimensionality reduction methods that are not restricted to choosing individual features.', '1901.09346-1-28-4': 'In experiments with linear decoders, we use PCA and in experiments with non-linear decoders, we use equivalent autoencoders.', '1901.09346-1-28-5': 'The methods, because they allow [MATH] combinations of features, bound the performance of any feature selection techniqe.', '1901.09346-1-29-0': 'We evaluate these methods on a number of datasets (the sizes of the datasets can be found in Table [REF]):', '1901.09346-1-30-0': 'MNIST and MNIST-Fashion consist of 28-by-28 grayscale images of hand-written digits and clothing items, respectively.', '1901.09346-1-30-1': 'We choose these datasets because they are widely known in the machine learning community.', '1901.09346-1-30-2': 'Although these are image datasets, the objects in each image are centered, which means we can meaningfully treat each 784 pixels in the image as a separate feature.', '1901.09346-1-31-0': 'ISOLET consists of preprocessed speech data of people speaking the names of the letters in the English alphabet.', '1901.09346-1-31-1': 'This dataset is widely used as a benchmark in the feature selection literature.', '1901.09346-1-31-2': 'Each feature is one of the 617 quantities produced as a result of preprocessing, including spectral coefficients and sonorant features.', '1901.09346-1-32-0': 'COIL-20 consists of centered grayscale images of 20 objects.', '1901.09346-1-32-1': 'Images of the objects were taken at pose intervals of 5 degrees amounting to 72 images for each object.', '1901.09346-1-32-2': 'During preprocessing, the images were resized to produce 20-by-20 images, with each feature being one of the 400 pixels.', '1901.09346-1-33-0': 'Smartphone Dataset for Human Activity Recognition consists of sensor data collected from a smartphone mounted on subjects while they performed several activities such as walking upstairs, standing and laying.', '1901.09346-1-33-1': 'Each feature represents one of the 561 raw or processed quantities from the sensors on the phone.', '1901.09346-1-34-0': 'Mice Protein Dataset consists of protein expression levels measured in the cortex of normal and trisomic mice who had been exposed to different experimental conditions.', '1901.09346-1-34-1': 'Each feature is the expression level of one protein.', '1901.09346-1-35-0': 'GEO Dataset consists of gene expression profiles measured from a large number of samples through a microarray-based platform.', '1901.09346-1-35-1': 'Each of the 10,463 features represents the expression level of one gene.', '1901.09346-1-36-0': 'To evaluate the various feature selection methods, we examine two metrics, both reported on a hold-out test set:', '1901.09346-1-37-0': 'Reconstruction error: We extract the [MATH] selected features.', '1901.09346-1-37-1': 'We pass the resulting matrix [MATH] through the reconstruction function [MATH] that we have trained.', '1901.09346-1-37-2': 'We measure the Frobenius norm between the original and reconstructed test matrices [MATH], normalized by the number of features [MATH].', '1901.09346-1-38-0': 'Classification accuracy: We extract the [MATH] features selected by the method.', '1901.09346-1-38-1': 'We then pass the resulting matrix [MATH] to an extremely randomized trees classifier , a variant of random forests that has been used with feature selection methods in prior literature .', '1901.09346-1-38-2': 'We measure the accuracy between the predicted labels and true labels, which are available for each of the datasets.', '1901.09346-1-38-3': 'Note that the labels are only used for training the classifier and not for the feature selection.', '1901.09346-1-39-0': '## Concrete Autoencoders (Non-Linear Decoder)', '1901.09346-1-40-0': 'First, we constructed a concrete autoencoder with a non-linear decoder architecture consisting of one hidden layer with [MATH] neurons, with [MATH] being the number of selected features.', '1901.09346-1-40-1': 'We performed a series of experiments with the ISOLET dataset, which is widely used as a benchmark in prior feature selection literature.', '1901.09346-1-40-2': 'We benchmarked each feature selection method (besides UDFS whose run-time was prohibitive) with varying numbers of of features (from [MATH] to [MATH]), measuring the reconstruction error using a 1-hidden-layer neural network as well as classification accuracy.', '1901.09346-1-40-3': 'The number of neurons in the hidden layer of the reconstruction network was varied within [MATH], and the network with the highest validation accuracy was selected and measured on the test set.', '1901.09346-1-41-0': 'To control for the performance of the reconstruction network, we trained each reconstruction network for the same number of epochs, 200.', '1901.09346-1-41-1': 'For the concrete autoencoder, we did not use the decoder that was learned during training, but re-trained the reconstruction networks from scratch.', '1901.09346-1-41-2': 'Our resulting classification accuracies and reconstruction error on each dataset are shown in Fig. [REF].', '1901.09346-1-41-3': 'We find that the concrete autoencoder consistently outperformed other feature selection methods on the ISOLET dataset.', '1901.09346-1-42-0': '## Concrete Autoencoders (Linear Decoder)', '1901.09346-1-43-0': 'Next, we carried out a series of experiments in which we compared concrete autoencoders with linear decoders to the other methods using linear regression as the reconstruction function.', '1901.09346-1-43-1': 'Since linear regression can be trained easily to convergence, this allowed us to isolate the effect of using the concrete selector layer for feature selection, and allowed us to train on a wider variety of datasets with less risk of overfitting.', '1901.09346-1-43-2': 'We selected a fixed number [MATH] of features with each method, with the exception of the Mice Protein Dataset, for which we used [MATH] due to its small size.', '1901.09346-1-44-0': 'After selecting the features using concrete autoencoder and the other feature selection methods, we trained a standard linear regressor with no regularization to impute the original features.', '1901.09346-1-44-1': 'The resulting reconstruction errors on a hold-out test set are shown in Table [REF].', '1901.09346-1-44-2': 'We also used the selected features to measure classification accuracies, which are shown in Table [REF] in Appendix [REF].', '1901.09346-1-44-3': 'On almost all datasets, we found that the concrete autoencoder continued to have the lowest reconstruction error and a high classification accuracy.', '1901.09346-1-45-0': '## Interpreting Related Features', '1901.09346-1-46-0': 'An added benefit of using the concrete selector layer is that it allows the user to not only identify the most informative features for reconstruction, but also identify sets of related features through examination of the learned Concrete parameters [MATH].', '1901.09346-1-46-1': 'Because the concrete selector layer samples the input features stochastically based on [MATH], any of the features with the large values in the vector [MATH] may be selected, and are thus likely to be correlated to one another.', '1901.09346-1-47-0': 'In Fig. [REF], we show how this can reveal related features by visualizing the top 3 pixels with the highest values in the [MATH] vector for each of the 20 concrete selector nodes on the MNIST digits.', '1901.09346-1-47-1': 'We notice that the pixels that are selected by each node are spatially close to one another, which agrees with intuitive notions of related features, as neighboring pixel values are likely to be correlated in handwritten digits.', '1901.09346-1-47-2': 'These patterns are even more striking when generated for individual classes of digits; in that case, the set of correlated pixels may even suggest the direction of the stroke when the digit was written (see Appendix [REF] for more details).', '1901.09346-1-47-3': 'Such analysis may be carried out more generally to find sets of related features, such as sets of related genes in a gene expression dataset.', '1901.09346-1-48-0': '## Case Study: L1000 Gene Expression', '1901.09346-1-49-0': 'We now turn to a large-scale test of the concrete autoencoder: examining whether we can improve gene expression inference.', '1901.09346-1-49-1': 'Gene expression inference arises from an important problem in molecular biology: characterizing the state of cells in different biological conditions.', '1901.09346-1-49-2': 'In particular, the response of cells to diseases, mutations, and drugs is often characterized by the measurement of gene expression patterns .', '1901.09346-1-50-0': 'However, measuring all of the genes expressed in a human cell can be expensive, and thus researchers have looked to computational methods to reduce the cost and time required for gene expression profiling.', '1901.09346-1-50-1': 'In particular, researchers from the LINCS Project found that, because gene expression is correlated in different conditions, a set of roughly a thousand carefully-chosen genes can capture most of the gene expression information in the entire human transcriptome .', '1901.09346-1-50-2': 'It is thus possible to use a linear regression model trained on genome-wide gene expression to infer the gene expression values of the remaining genes.', '1901.09346-1-50-3': 'More recently, [CITATION] showed that it is possible to leverage the representation power of neural networks to improve the accuracy of gene expression inference in an approach they referred to as D-GEX.', '1901.09346-1-51-0': 'Here, we ask whether it is possible to use concrete autoencoders to determine a good subset of genes, perhaps as an alternative to the landmark genes, without utilizing any prior biological knowledge of gene networks or gene function.', '1901.09346-1-51-1': 'We relied only on a large dataset of gene expression data, from which we aim to select the most informative features.', '1901.09346-1-52-0': 'We used the version of the GEO dataset used in the D-GEX paper, and followed the same preprocessing scheme to obtain a dataset of sample size 112,171 and dimensionality 10,463 genes.', '1901.09346-1-52-1': 'We then randomly partitioned the dataset in a manner similar to that performed by [CITATION]: as a result, the training set had 88,807 samples, the validation set had 11,101, and the test set had 12,263.', '1901.09346-1-52-2': 'We then considered 3 kinds of reconstruction functions: in the simplest case, we considered multitarget linear regression, and we also implemented neural networks with 1 and 2 hidden layers.', '1901.09346-1-52-3': 'See Appendix [REF] for the architecture of the networks.', '1901.09346-1-53-0': 'First, we trained a concrete autoencoder to select 943 using only a linear regression decoder.', '1901.09346-1-53-1': 'An analysis of the selected genes showed very little overlap with the landmark genes: only 90 of the CAE-selected 943 genes were among the landmark genes.', '1901.09346-1-53-2': 'For consistency with the D-GEX results, we used this same set of 943 genes, selected by a concrete autoencoder with a linear decoder, with all of our reconstruction networks.', '1901.09346-1-54-0': 'We trained each reconstruction networks to impute all of the original 10,463 genes.', '1901.09346-1-54-1': 'We measured the reconstruction error on a hold-out test set that was used neither to train the concrete autoencoder nor the reconstruction functions.', '1901.09346-1-54-2': 'Our results are summarized in Fig. [REF](a), where we plot the mean-squared error of imputation.', '1901.09346-1-54-3': 'We show that not only is it possible to use concrete autoencoders to perform gene selection gene expression inference in a differentiable, end-to-end manner on large-scale datasets, doing so improves the performance of the gene expression imputation on a holdout test set of gene expression by around 3% for each architecture, which is significant as the L1000 landmark genes were expert curated using a combination of computational prediction with domain knowledge, and is a very strong benchmark and is widely used in genomics.', '1901.09346-1-55-0': 'Next, we investigated whether it would be possible to obtain similar accuracies as to the landmark genes while using a smaller set of CAE-selected genes.', '1901.09346-1-55-1': 'We trained concrete autoencoders from scratch using [MATH], using the same architecture described in [REF] and using a linear regression decoder.', '1901.09346-1-55-2': 'We found that using linear regression as the reconstruction function, we could obtain reconstruction MSEs as low as the landmark genes, while using only about 750 genes, which represents roughly a [MATH] reduction in the number of genes measured, potentially saving substantial experimental cost.', '1901.09346-1-55-3': 'Results are shown in Fig. [REF](b).', '1901.09346-1-56-0': '# Discussion', '1901.09346-1-57-0': 'In this paper, we have proposed a new method for differentiable, end-to-end feature selection via backpropagation.', '1901.09346-1-57-1': 'At its core, the concrete autoencoder uses Concrete random variables and the reparametrization trick to allow gradients to flow through a layer that stochastically selects discrete input features.', '1901.09346-1-57-2': 'The stochasticity of the concrete autoencoder allows it to efficiently explore and converge to a subset of input features of specified size that minimizes a particular loss, as described in Section [REF].', '1901.09346-1-57-3': 'The learned parameters can be further probed to allow the anlayst to interpret related features, as demonstrated in Section [REF].', '1901.09346-1-57-4': 'This makes concrete autoencoders different from many competing methods, which rely on regularization to encourage sparse feature selection.', '1901.09346-1-58-0': 'We show via experiments on a variety of public datasets that concrete autoencoders effectively minimize the reconstruction error and maximize classification accuracy using selected features.', '1901.09346-1-58-1': 'In the six public datasets that we tested concrete autoencoders, we found that concrete autoencoders outperformed many different complex feature selection methods.', '1901.09346-1-58-2': 'This remains the case even when we reduced the decoder layer to be a single linear layer, showing that the concrete selector node is useful even when selecting input features that minimize the loss when using a linear regression as the reconstruction function.', '1901.09346-1-59-0': 'Because the concrete autoencoder is an adaptation of the standard autoencoder, it scales easily to datasets with many samples or high dimensionality.', '1901.09346-1-59-1': 'We demonstrated this in section [REF] using a gene expression dataset with more than 100,000 samples and 10,000 features, where the features selected by the concrete autoencoder outperformed the state-of-the-art gene subset.', '1901.09346-1-59-2': 'Furthermore, because of its general formulation, the concrete autoencoder can be easily extended in many ways.', '1901.09346-1-59-3': 'For example, it possible to use concrete autoencoders in a supervised manner - to select a set of features that minimize a cross-entropy loss, for example, rather than a reconstruction loss.', '1901.09346-1-59-4': 'More details and examples of this approach are provided in Appendix [REF].', '1901.09346-1-59-5': 'Another possible extension is to attach different costs to selecting different features, for example if certain features represent tests or assays that are much more expensive than others.', '1901.09346-1-59-6': 'Such as cost may be incorporated into the loss function and allow the analyst to trade off cost for accuracy.', '1901.09346-1-60-0': 'Advantages of the concrete autoencoder include its generality and ease of use.', '1901.09346-1-60-1': 'Implementing the architecture in popular machine learning frameworks requires only modifying a few lines of code from a standard autoencoder.', '1901.09346-1-60-2': 'Furthermore, the runtime of the concrete autoencoder is similar to that of the standard autoencoder and improves with hardware acceleration and parallelization techniques commonplace in deep learning.', '1901.09346-1-60-3': 'The only additional hyperparameters of the concrete autoencoder are the initial and final temperatures used in the annealing schedule.', '1901.09346-1-60-4': 'We find that the default values used in this paper work well for a variety of datasets.', '1901.09346-1-61-0': 'Concrete autoencoders, like the other feature selection methods we compared with in this paper, do not provide p-values or statistical significance quantification.', '1901.09346-1-61-1': 'Features discovered through concrete autoencoders should be validated through hypothesis testing or additional analysis using relevant domain knowledge.', '1901.09346-1-61-2': 'We believe that the concrete autoencoder can be of particular use in simplifying assays and experiments that measure a large number of related quantities, such as medical lab tests and genotype sequencing.'}

{'1901.09346-2-0-0': 'We introduce the concrete autoencoder, an end-to-end differentiable method for global feature selection, which efficiently identifies a subset of the most informative features and simultaneously learns a neural network to reconstruct the input data from the selected features.', '1901.09346-2-0-1': 'Our method is unsupervised, and is based on using a concrete selector layer as the encoder and using a standard neural network as the decoder.', '1901.09346-2-0-2': 'During the training phase, the temperature of the concrete selector layer is gradually decreased, which encourages a user-specified number of discrete features to be learned.', '1901.09346-2-0-3': 'During test time, the selected features can be used with the decoder network to reconstruct the remaining input features.', '1901.09346-2-0-4': 'We evaluate concrete autoencoders on a variety of datasets, where they significantly outperform state-of-the-art methods for feature selection and data reconstruction.', '1901.09346-2-0-5': 'In particular, on a large-scale gene expression dataset, the concrete autoencoder selects a small subset of genes whose expression levels can be use to impute the expression levels of the remaining genes.', '1901.09346-2-0-6': 'In doing so, it improves on the current widely-used expert-curated L1000 landmark genes, potentially reducing measurement costs by 20%.', '1901.09346-2-0-7': 'The concrete autoencoder can be implemented by adding just a few lines of code to a standard autoencoder.', '1901.09346-2-1-0': '# Introduction', '1901.09346-2-2-0': 'High-dimensional datasets often pose a challenge for machine learning algorithms.', '1901.09346-2-2-1': 'Feature selection methods aim to reduce dimensionality of data by identifying the subset of relevant features in a dataset.', '1901.09346-2-2-2': 'A large number of algorithms have been proposed for feature selection in both supervised and unsupervised settings .', '1901.09346-2-2-3': 'These methods provide insight into the relationship between features in complex data and can simplify the process of training downstream models.', '1901.09346-2-2-4': 'Feature selection is particularly useful if the data with the full set of features is expensive or difficult to collect, as it can eliminate the need to measure irrelevant or redundant features.', '1901.09346-2-3-0': 'As a motivating example, consider a dataset that consists of the expression of various genes across tissue samples.', '1901.09346-2-3-1': 'Such "omics" measurements are increasingly carried out to fully characterize biological samples at the individual and single-cell level .', '1901.09346-2-3-2': 'Yet it remains expensive to conduct all of the biological assays that are needed to characterize such samples.', '1901.09346-2-3-3': 'It is natural to ask: what are the most important features in this dataset?', '1901.09346-2-3-4': 'Are there redundant features that do not need to be measured?', '1901.09346-2-3-5': 'The idea of only measuring a subset of biological features and then reconstructing the remaining features is not new; in fact, this line of thinking has motivated the identification of the landmark genes, also known as the L1000 genes, which are a small subset of the over 20,000 human genes.', '1901.09346-2-3-6': 'The expression levels of the L1000 are strongly correlated with the expression levels of other genes, and thus this subset can be measured cheaply and then used to impute the remaining gene expression levels .', '1901.09346-2-4-0': 'The problem of feature selection is different from the more general problem of dimensionality reduction.', '1901.09346-2-4-1': 'Standard techniques for dimensionality reduction, such as principal components analysis and autoencoders , are be able to represent data with fewer dimensions while preserving maximal variance or minimizing reconstruction loss.', '1901.09346-2-4-2': 'However, such methods do not select a set of features present in the original dataset, and thus cannot be directly used to eliminate redundant features and reduce experimental costs.', '1901.09346-2-4-3': 'We emphasize the unsupervised nature of this problem: specific prediction tasks may not be known ahead of time, and thus it is important to develop methods that can identify a subset of features while allowing imputation of the remaining features with minimal distortion for arbitrary downstream tasks.', '1901.09346-2-5-0': 'In this paper, we propose a new end-to-end method to perform feature subset selection and imputation that leverages the power of deep autoencoders for discrete feature selection.', '1901.09346-2-5-1': 'Our method, the concrete autoencoder, uses a relaxation of the discrete distribution, the Concrete distribution , and the reparametrization trick to differentiate through an arbitrary (e.g. reconstruction) loss and select input features to minimize this loss.', '1901.09346-2-5-2': 'A visual example of results from our method is shown in Fig. [REF], where the concrete autoencoder selects the 20 most informative pixels (out of a total of 784) on the MNIST dataset, and reconstructs the original images with high accuracy.', '1901.09346-2-6-0': 'We test concrete autoencoders on a variety of datasets, and find that they generally outperform state-of-the-art methods for feature selection and data reconstruction.', '1901.09346-2-6-1': 'We have made the code for our algorithm and experiments available on a public repository.', '1901.09346-2-7-0': 'Related Works Feature selection methods are generally divided into three classes: filter, wrapper, and embedded methods.', '1901.09346-2-7-1': 'Filter methods rank the features of the dataset using statistical tests, such as the variance, in order to select features that individually maximize the desired criteria .', '1901.09346-2-7-2': 'Filter methods generally do not consider interactions between features and do not provide a way to impute the remaining features; one must train a separate algorithm to do so.', '1901.09346-2-7-3': 'Wrapper methods select subsets of features that maximize a objective function, which is optimized over the choice of input features using a black-box optimization method, such as sequential search or genetic algorithms .', '1901.09346-2-7-4': 'Since wrapper methods evaluate subsets of features, they are able to detect potential relationships between features, but usually at the expense of increased computation time.', '1901.09346-2-7-5': 'Embedded methods also consider relationships between features but generally do so more efficiently as they incorporate feature selection into the learning phase of another algorithm.', '1901.09346-2-7-6': 'A well-known example is the Lasso , which can be used to select features for regression by varying the strength of [MATH] regularization.', '1901.09346-2-8-0': 'Many embedded unsupervised feature selection algorithms use regularization as the means to select discrete features.', '1901.09346-2-8-1': 'The popular UDFS algorithm [CITATION] incorporates [MATH] regularization on a set of weights applied to the input to select features most useful for local discriminative analysis.', '1901.09346-2-8-2': 'Similarly, the MCFS algorithm uses regularization to solve for the features which preserve the clustering structure in the data.', '1901.09346-2-8-3': 'The recently-proposed AEFS algorithm also uses [MATH] regularization on the weights of the encoder that maps the input data to a latent space and optimizes these weights for their ability to reconstruct the original input.', '1901.09346-2-9-0': 'In this paper, we select discrete features using an embedded method but without resorting to regularization.', '1901.09346-2-9-1': 'Rather, we use a relaxation of the discrete random variables, the Concrete distribution , which allows a low-variance estimate of the gradient through discrete stochastic nodes.', '1901.09346-2-9-2': 'By using Concrete random variables, we can directly parametrize the selection of input features, and differentiate through the parameters.', '1901.09346-2-9-3': 'As we show through experiments in Section [REF], this leads to lower reconstruction errors on real-world datasets compared to the aforementioned regularization-based methods.', '1901.09346-2-10-0': '# Problem Formulation', '1901.09346-2-11-0': 'We now describe the problem of global feature selection.', '1901.09346-2-11-1': 'Although global feature selection is relevant for both unsupervised and supervised settings, we describe here the unsupervised case, which is the primary focus of this paper, and defer discussion of the supervised case to Appendix [REF].', '1901.09346-2-12-0': 'Consider a data-generating probability distribution [MATH] over a [MATH]-dimensional space.', '1901.09346-2-12-1': 'The goal is to learn a subset [MATH] of features of specified size [MATH] and also learn a reconstruction function [MATH], such that the expected loss between the reconstructed sample [MATH] and the original sample [MATH] is minimized, where [MATH] consists of those elements [MATH] such that [MATH].', '1901.09346-2-12-2': 'In other words, we would like to optimize [EQUATION]', '1901.09346-2-12-3': 'In practice, we do not know [MATH]; rather we have [MATH] samples, generally assumed to be drawn i.i.d. from [MATH].', '1901.09346-2-12-4': 'These samples can be represented in a data matrix [MATH], and so the goal becomes choosing [MATH] columns of [MATH] such that sub-matrix [MATH], defined analogously to [MATH], can be used to reconstruct the original matrix [MATH].', '1901.09346-2-12-5': 'Let us overload [MATH] to mean the matrix that results from applying [MATH] to each of the rows of [MATH] and stacking the resulting outputs.', '1901.09346-2-12-6': 'We seek to minimize the empirical reconstruction error: [EQUATION] where [MATH] denotes the Frobenius norm of the matrix.', '1901.09346-2-12-7': 'The principal difficulty in solving ([REF]) is the optimization over the discrete set of features [MATH], whose choices grow exponentially in [MATH].', '1901.09346-2-12-8': 'Thus, even for simple choices of [MATH], such as linear regression, the optimization problem in ([REF]) is NP-hard to solve [CITATION].', '1901.09346-2-13-0': 'Furthermore, the complexity of [MATH] can significantly affect reconstruction error and even the choice of [MATH].', '1901.09346-2-13-1': 'More expressive and non-linear choices for [MATH], such as neural networks, will naturally allow for lower reconstruction error, potentially at the expense of a more difficult optimization problem.', '1901.09346-2-13-2': 'We seek to develop a method that can approximate the solution for any given class of functions [MATH], from linear regression to deep fully-connected neural networks.', '1901.09346-2-14-0': 'Finally, we note that the choice of mean-squared reconstruction error as the metric for optimization in ([REF]) and ([REF]) stems from the fact that it is a smooth differentiable function that serves as a proxy for many downstream analyses, such as clustering performance and classification accuracy.', '1901.09346-2-14-1': 'However, other differentiable metrics, such as a variational approximation of the mutual information between [MATH] and [MATH], may be considered as well .', '1901.09346-2-15-0': '# Proposed Method', '1901.09346-2-16-0': 'The concrete autoencoder is an adaption of the standard autoencoder for discrete feature selection.', '1901.09346-2-16-1': 'Instead of using series of fully-connected layers for the encoder, we propose a concrete selector layer with a user-specified number of nodes, [MATH].', '1901.09346-2-16-2': 'This layer selects stochastic linear combinations of input features during training, which converge to a discrete set of [MATH] features by the end of training and during test time.', '1901.09346-2-17-0': 'The way in which input features are combined depends on the temperature of this layer, which we modulate using a simple annealing schedule.', '1901.09346-2-17-1': 'As the temperature of the layer approaches zero, the layer selects [MATH] individual input features.', '1901.09346-2-17-2': 'The decoder of a concrete autoencoder, which serves as the reconstruction function, is the same as that of a standard autoencoder: a neural network whose architecture can be set by the user based on dataset size and complexity.', '1901.09346-2-17-3': 'In effect, then, the concrete autoencoder is a method for selecting a discrete set of features that are optimized for an arbitrarily-complex reconstruction function.', '1901.09346-2-17-4': 'We describe the ingredients for our method in more detail in the next two subsections.', '1901.09346-2-18-0': '## Concrete Selector Layer', '1901.09346-2-19-0': 'The concrete selector layer is based on Concrete random variables .', '1901.09346-2-19-1': 'A Concrete random variable can be sampled to produce a continuous relaxation of the one-hot vector.', '1901.09346-2-19-2': 'The extent to which the one-hot vector is relaxed is controlled by a temperature parameter [MATH].', '1901.09346-2-19-3': 'To sample a Concrete random variable in [MATH] dimensions with parameters [MATH] and [MATH], one first samples a [MATH]-dimensional vector of i.i.d. samples from a Gumbel distribution , [MATH].', '1901.09346-2-19-4': 'Then each element of the sample [MATH] from the Concrete distribution is defined as: [EQUATION] where [MATH] refers to the [MATH] element in a particular sample vector.', '1901.09346-2-19-5': 'In the limit [MATH], the concrete random variable smoothly approaches the discrete distribution, outputting one hot vectors with [MATH] with probability [MATH].', '1901.09346-2-19-6': 'The desirable aspect of the Concrete random variable is that it allows for differentiation with respect to its parameters [MATH] via the reparametrization trick .', '1901.09346-2-20-0': 'We use Concrete random variables to select input features in the following way.', '1901.09346-2-20-1': 'For each of the [MATH] nodes in the concrete selector layer, we sample a [MATH]-dimensional Concrete random variable [MATH] (note that the superscript here indexes the node in the selector layer, whereas the subscript earlier referred to the element in the vector).', '1901.09346-2-20-2': 'The [MATH] node in the selector layer [MATH] outputs [MATH].', '1901.09346-2-20-3': 'This is, in general, a weighted linear combination of the input features, but notice that when [MATH], each node in the concrete selector layer outputs exactly one of the input features.', '1901.09346-2-20-4': 'After the network is trained, during test time, we thus replace the concrete selector layer with a discrete [MATH] layer in which the output of the [MATH] neuron is [MATH].', '1901.09346-2-21-0': 'We randomly initialize [MATH] to small positive values, to encourage the selector layer to stochastically explore different linear combinations of input features.', '1901.09346-2-21-1': 'However, as the network is trained, the values of [MATH] become more sparse, as the network becomes more confident in particular choices of input features, reducing the stochasticity in selected features.', '1901.09346-2-21-2': 'The concrete autoencoder architecture is shown in Fig. [REF](a) and the pseudocode for training in Fig. [REF](b).', '1901.09346-2-22-0': '## Annealing Schedule', '1901.09346-2-23-0': 'The temperature of Concrete random variables in the concrete selector layer has a significant affect on the output of the nodes.', '1901.09346-2-23-1': 'If the temperature is held high, the concrete selector layer continuously outputs a linear combination of features.', '1901.09346-2-23-2': 'On the contrary, if the temperature is held low, the concrete selector layer is not able to explore different combinations of features and converges to a poor local minimum.', '1901.09346-2-23-3': 'Neither fixed temperature allows the concrete selector layer to converge to informative features.', '1901.09346-2-24-0': 'Instead, we propose a simple annealing schedule that sets the temperature for all of the concrete variables, initially beginning with a high temperature [MATH] and gradually decaying the temperature until a final temperate [MATH] at each epoch according to a first-order exponential decay: [MATH] where [MATH] is the temperature at epoch number [MATH], and [MATH] is the total number of epochs.', '1901.09346-2-24-1': 'We compare various methods for setting the temperature of the concrete selector nodes in Fig. [REF].', '1901.09346-2-24-2': 'We find that this annealing schedule allows the concrete selector layer to effectively stochastically explore combinations of features in the initial phases of training, while in the later stages of training, the lowered temperature allows the the network to converge to informative individual features.', '1901.09346-2-25-0': '# Experiments', '1901.09346-2-26-0': 'In this section, we carry out experiments to compare the performance of concrete autoencoders to other feature subset selections on standard public datasets.', '1901.09346-2-26-1': 'For all of the experiments, we use Adam optimizer with a learning rate of [MATH].', '1901.09346-2-26-2': 'The initial temperature of the concrete autoencoder [MATH] was set to [MATH] and the final temperature [MATH] to [MATH].', '1901.09346-2-26-3': 'We trained the concrete autoencoder until until the mean of the concrete samples exceeded 0.99.', '1901.09346-2-26-4': 'For UDFS and AEFS, we swept values of each regularization hyperparameter and report the results with optimal hyperparameters according to mean squared error for reconstruction for each method.', '1901.09346-2-27-0': 'Furthermore, since the reconstruction [MATH] can overfit to patterns particular to the training set, we divide each dataset randomly into train, validation, and test datasets according to a 72-8-20 split.', '1901.09346-2-27-1': 'We use the training set to learn the parameters of the concrete autoencoders, the validation set to select optimal hyperparameters, and the test set to evaluate generalization performance, which we report below.', '1901.09346-2-28-0': 'We compare concrete autoencoders to many of the unsupervised feature selection methods mentioned in Related Works including UDFS, MCFS, and AEFS.', '1901.09346-2-28-1': 'We also include principal feature analysis (PFA), proposed by [CITATION], which is a popular method for selecting discrete features based on PCA, as well as a spectral method, the Laplacian score .', '1901.09346-2-28-2': 'Where available, we made use of scikit-feature implementation of each method .', '1901.09346-2-28-3': 'In our experiments, we also include, as upper bounds on performance, dimensionality reduction methods that are not restricted to choosing individual features.', '1901.09346-2-28-4': 'In experiments with linear decoders, we use PCA and in experiments with non-linear decoders, we use equivalent autoencoders.', '1901.09346-2-28-5': 'The methods, because they allow [MATH] combinations of features, bound the performance of any feature selection techniqe.', '1901.09346-2-29-0': 'We evaluate these methods on a number of datasets (the sizes of the datasets can be found in Table [REF]):', '1901.09346-2-30-0': 'MNIST and MNIST-Fashion consist of 28-by-28 grayscale images of hand-written digits and clothing items, respectively.', '1901.09346-2-30-1': 'We choose these datasets because they are widely known in the machine learning community.', '1901.09346-2-30-2': 'Although these are image datasets, the objects in each image are centered, which means we can meaningfully treat each 784 pixels in the image as a separate feature.', '1901.09346-2-31-0': 'ISOLET consists of preprocessed speech data of people speaking the names of the letters in the English alphabet.', '1901.09346-2-31-1': 'This dataset is widely used as a benchmark in the feature selection literature.', '1901.09346-2-31-2': 'Each feature is one of the 617 quantities produced as a result of preprocessing, including spectral coefficients and sonorant features.', '1901.09346-2-32-0': 'COIL-20 consists of centered grayscale images of 20 objects.', '1901.09346-2-32-1': 'Images of the objects were taken at pose intervals of 5 degrees amounting to 72 images for each object.', '1901.09346-2-32-2': 'During preprocessing, the images were resized to produce 20-by-20 images, with each feature being one of the 400 pixels.', '1901.09346-2-33-0': 'Smartphone Dataset for Human Activity Recognition consists of sensor data collected from a smartphone mounted on subjects while they performed several activities such as walking upstairs, standing and laying.', '1901.09346-2-33-1': 'Each feature represents one of the 561 raw or processed quantities from the sensors on the phone.', '1901.09346-2-34-0': 'Mice Protein Dataset consists of protein expression levels measured in the cortex of normal and trisomic mice who had been exposed to different experimental conditions.', '1901.09346-2-34-1': 'Each feature is the expression level of one protein.', '1901.09346-2-35-0': 'GEO Dataset consists of gene expression profiles measured from a large number of samples through a microarray-based platform.', '1901.09346-2-35-1': 'Each of the 10,463 features represents the expression level of one gene.', '1901.09346-2-36-0': 'To evaluate the various feature selection methods, we examine two metrics, both reported on a hold-out test set:', '1901.09346-2-37-0': 'Reconstruction error: We extract the [MATH] selected features.', '1901.09346-2-37-1': 'We pass the resulting matrix [MATH] through the reconstruction function [MATH] that we have trained.', '1901.09346-2-37-2': 'We measure the Frobenius norm between the original and reconstructed test matrices [MATH], normalized by the number of features [MATH].', '1901.09346-2-38-0': 'Classification accuracy: We extract the [MATH] features selected by the method.', '1901.09346-2-38-1': 'We then pass the resulting matrix [MATH] to an extremely randomized trees classifier , a variant of random forests that has been used with feature selection methods in prior literature .', '1901.09346-2-38-2': 'We measure the accuracy between the predicted labels and true labels, which are available for each of the datasets.', '1901.09346-2-38-3': 'Note that the labels are only used for training the classifier and not for the feature selection.', '1901.09346-2-39-0': '## Concrete Autoencoders (Non-Linear Decoder)', '1901.09346-2-40-0': 'First, we constructed a concrete autoencoder with a non-linear decoder architecture consisting of one hidden layer with [MATH] neurons, with [MATH] being the number of selected features.', '1901.09346-2-40-1': 'We performed a series of experiments with the ISOLET dataset, which is widely used as a benchmark in prior feature selection literature.', '1901.09346-2-40-2': 'We benchmarked each feature selection method (besides UDFS whose run-time was prohibitive) with varying numbers of of features (from [MATH] to [MATH]), measuring the reconstruction error using a 1-hidden-layer neural network as well as classification accuracy.', '1901.09346-2-40-3': 'The number of neurons in the hidden layer of the reconstruction network was varied within [MATH], and the network with the highest validation accuracy was selected and measured on the test set.', '1901.09346-2-41-0': 'To control for the performance of the reconstruction network, we trained each reconstruction network for the same number of epochs, 200.', '1901.09346-2-41-1': 'For the concrete autoencoder, we did not use the decoder that was learned during training, but re-trained the reconstruction networks from scratch.', '1901.09346-2-41-2': 'Our resulting classification accuracies and reconstruction error on each dataset are shown in Fig. [REF].', '1901.09346-2-41-3': 'We find that the concrete autoencoder consistently outperformed other feature selection methods on the ISOLET dataset.', '1901.09346-2-42-0': '## Concrete Autoencoders (Linear Decoder)', '1901.09346-2-43-0': 'Next, we carried out a series of experiments in which we compared concrete autoencoders with linear decoders to the other methods using linear regression as the reconstruction function.', '1901.09346-2-43-1': 'Since linear regression can be trained easily to convergence, this allowed us to isolate the effect of using the concrete selector layer for feature selection, and allowed us to train on a wider variety of datasets with less risk of overfitting.', '1901.09346-2-43-2': 'We selected a fixed number [MATH] of features with each method, with the exception of the Mice Protein Dataset, for which we used [MATH] due to its small size.', '1901.09346-2-44-0': 'After selecting the features using concrete autoencoder and the other feature selection methods, we trained a standard linear regressor with no regularization to impute the original features.', '1901.09346-2-44-1': 'The resulting reconstruction errors on a hold-out test set are shown in Table [REF].', '1901.09346-2-44-2': 'We also used the selected features to measure classification accuracies, which are shown in Table [REF] in Appendix [REF].', '1901.09346-2-44-3': 'On almost all datasets, we found that the concrete autoencoder continued to have the lowest reconstruction error and a high classification accuracy.', '1901.09346-2-45-0': '## Interpreting Related Features', '1901.09346-2-46-0': 'An added benefit of using the concrete selector layer is that it allows the user to not only identify the most informative features for reconstruction, but also identify sets of related features through examination of the learned Concrete parameters [MATH].', '1901.09346-2-46-1': 'Because the concrete selector layer samples the input features stochastically based on [MATH], any of the features with the large values in the vector [MATH] may be selected, and are thus likely to be correlated to one another.', '1901.09346-2-47-0': 'In Fig. [REF], we show how this can reveal related features by visualizing the top 3 pixels with the highest values in the [MATH] vector for each of the 20 concrete selector nodes on the MNIST digits.', '1901.09346-2-47-1': 'We notice that the pixels that are selected by each node are spatially close to one another, which agrees with intuitive notions of related features, as neighboring pixel values are likely to be correlated in handwritten digits.', '1901.09346-2-47-2': 'These patterns are even more striking when generated for individual classes of digits; in that case, the set of correlated pixels may even suggest the direction of the stroke when the digit was written (see Appendix [REF] for more details).', '1901.09346-2-47-3': 'Such analysis may be carried out more generally to find sets of related features, such as sets of related genes in a gene expression dataset.', '1901.09346-2-48-0': '## Case Study: L1000 Gene Expression', '1901.09346-2-49-0': 'We now turn to a large-scale test of the concrete autoencoder: examining whether we can improve gene expression inference.', '1901.09346-2-49-1': 'Gene expression inference arises from an important problem in molecular biology: characterizing the state of cells in different biological conditions.', '1901.09346-2-49-2': 'In particular, the response of cells to diseases, mutations, and drugs is often characterized by the measurement of gene expression patterns .', '1901.09346-2-50-0': 'However, measuring all of the genes expressed in a human cell can be expensive, and thus researchers have looked to computational methods to reduce the cost and time required for gene expression profiling.', '1901.09346-2-50-1': 'In particular, researchers from the LINCS Project found that, because gene expression is correlated in different conditions, a set of roughly a thousand carefully-chosen genes can capture most of the gene expression information in the entire human transcriptome .', '1901.09346-2-50-2': 'It is thus possible to use a linear regression model trained on genome-wide gene expression to infer the gene expression values of the remaining genes.', '1901.09346-2-50-3': 'More recently, [CITATION] showed that it is possible to leverage the representation power of neural networks to improve the accuracy of gene expression inference in an approach they referred to as D-GEX.', '1901.09346-2-51-0': 'Here, we ask whether it is possible to use concrete autoencoders to determine a good subset of genes, perhaps as an alternative to the landmark genes, without utilizing any prior biological knowledge of gene networks or gene function.', '1901.09346-2-51-1': 'We relied only on a large dataset of gene expression data, from which we aim to select the most informative features.', '1901.09346-2-52-0': 'We used the version of the GEO dataset used in the D-GEX paper, and followed the same preprocessing scheme to obtain a dataset of sample size 112,171 and dimensionality 10,463 genes.', '1901.09346-2-52-1': 'We then randomly partitioned the dataset in a manner similar to that performed by [CITATION]: as a result, the training set had 88,807 samples, the validation set had 11,101, and the test set had 12,263.', '1901.09346-2-52-2': 'We then considered 3 kinds of reconstruction functions: in the simplest case, we considered multitarget linear regression, and we also implemented neural networks with 1 and 2 hidden layers.', '1901.09346-2-52-3': 'See Appendix [REF] for the architecture of the networks.', '1901.09346-2-53-0': 'First, we trained a concrete autoencoder to select 943 using only a linear regression decoder.', '1901.09346-2-53-1': 'An analysis of the selected genes showed very little overlap with the landmark genes: only 90 of the CAE-selected 943 genes were among the landmark genes.', '1901.09346-2-53-2': 'For consistency with the D-GEX results, we used this same set of 943 genes, selected by a concrete autoencoder with a linear decoder, with all of our reconstruction networks.', '1901.09346-2-54-0': 'We trained each reconstruction networks to impute all of the original 10,463 genes.', '1901.09346-2-54-1': 'We measured the reconstruction error on a hold-out test set that was used neither to train the concrete autoencoder nor the reconstruction functions.', '1901.09346-2-54-2': 'Our results are summarized in Fig. [REF](a), where we plot the mean-squared error of imputation.', '1901.09346-2-54-3': 'We show that not only is it possible to use concrete autoencoders to perform gene selection gene expression inference in a differentiable, end-to-end manner on large-scale datasets, doing so improves the performance of the gene expression imputation on a holdout test set of gene expression by around 3% for each architecture, which is significant as the L1000 landmark genes were expert curated using a combination of computational prediction with domain knowledge, and is a very strong benchmark and is widely used in genomics.', '1901.09346-2-55-0': 'Next, we investigated whether it would be possible to obtain similar accuracies as to the landmark genes while using a smaller set of CAE-selected genes.', '1901.09346-2-55-1': 'We trained concrete autoencoders from scratch using [MATH], using the same architecture described in Appendix [REF] and using a linear regression decoder.', '1901.09346-2-55-2': 'We found that using linear regression as the reconstruction function, we could obtain reconstruction MSEs about as low as the landmark genes, using only 750 genes, which represents roughly a [MATH] reduction in the number of genes measured, potentially saving substantial experimental costs.', '1901.09346-2-55-3': 'These results are illustrated in Fig. [REF](b).', '1901.09346-2-56-0': '# Discussion', '1901.09346-2-57-0': 'In this paper, we have proposed a new method for differentiable, end-to-end feature selection via backpropagation.', '1901.09346-2-57-1': 'At its core, the concrete autoencoder uses Concrete random variables and the reparametrization trick to allow gradients to flow through a layer that stochastically selects discrete input features.', '1901.09346-2-57-2': 'The stochasticity of the concrete autoencoder allows it to efficiently explore and converge to a subset of input features of specified size that minimizes a particular loss, as described in Section [REF].', '1901.09346-2-57-3': 'The learned parameters can be further probed to allow the anlayst to interpret related features, as demonstrated in Section [REF].', '1901.09346-2-57-4': 'This makes concrete autoencoders different from many competing methods, which rely on regularization to encourage sparse feature selection.', '1901.09346-2-58-0': 'We show via experiments on a variety of public datasets that concrete autoencoders effectively minimize the reconstruction error and maximize classification accuracy using selected features.', '1901.09346-2-58-1': 'In the six public datasets that we tested concrete autoencoders, we found that concrete autoencoders outperformed many different complex feature selection methods.', '1901.09346-2-58-2': 'This remains the case even when we reduced the decoder layer to be a single linear layer, showing that the concrete selector node is useful even when selecting input features that minimize the loss when using a linear regression as the reconstruction function.', '1901.09346-2-59-0': 'Because the concrete autoencoder is an adaptation of the standard autoencoder, it scales easily to datasets with many samples or high dimensionality.', '1901.09346-2-59-1': 'We demonstrated this in section [REF] using a gene expression dataset with more than 100,000 samples and 10,000 features, where the features selected by the concrete autoencoder outperformed the state-of-the-art gene subset.', '1901.09346-2-59-2': 'Furthermore, because of its general formulation, the concrete autoencoder can be easily extended in many ways.', '1901.09346-2-59-3': 'For example, it possible to use concrete autoencoders in a supervised manner - to select a set of features that minimize a cross-entropy loss, for example, rather than a reconstruction loss.', '1901.09346-2-59-4': 'More details and examples of this approach are provided in Appendix [REF].', '1901.09346-2-59-5': 'Another possible extension is to attach different costs to selecting different features, for example if certain features represent tests or assays that are much more expensive than others.', '1901.09346-2-59-6': 'Such as cost may be incorporated into the loss function and allow the analyst to trade off cost for accuracy.', '1901.09346-2-60-0': 'Advantages of the concrete autoencoder include its generality and ease of use.', '1901.09346-2-60-1': 'Implementing the architecture in popular machine learning frameworks requires only modifying a few lines of code from a standard autoencoder.', '1901.09346-2-60-2': 'Furthermore, the runtime of the concrete autoencoder is similar to that of the standard autoencoder and improves with hardware acceleration and parallelization techniques commonplace in deep learning.', '1901.09346-2-60-3': 'The only additional hyperparameters of the concrete autoencoder are the initial and final temperatures used in the annealing schedule.', '1901.09346-2-60-4': 'We find that the default values used in this paper work well for a variety of datasets.', '1901.09346-2-61-0': 'Concrete autoencoders, like the other feature selection methods we compared with in this paper, do not provide p-values or statistical significance quantification.', '1901.09346-2-61-1': 'Features discovered through concrete autoencoders should be validated through hypothesis testing or additional analysis using relevant domain knowledge.', '1901.09346-2-61-2': 'We believe that the concrete autoencoder can be of particular use in simplifying assays and experiments that measure a large number of related quantities, such as medical lab tests and genotype sequencing.'}

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[]

[['1901.09346-1-55-3', '1901.09346-2-55-3']]

[]

['1901.09346-1-29-0', '1901.09346-1-36-0', '1901.09346-2-29-0', '1901.09346-2-36-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1901.09346

1703.07052

{'1703.07052-1-0-0': 'Background: The accurate determination of atomic final states following nuclear [MATH] decay plays an important role in many experiments.', '1703.07052-1-0-1': 'In particular, the charge state distributions of ions following nuclear [MATH] decay are important for determinations of the [MATH] angular correlation with improved precision.', '1703.07052-1-1-0': 'Purpose: Our measurement aims at providing benchmarks to test theoretical calculations.', '1703.07052-1-2-0': 'Method: The kinematics of Li[MATH] ions produced following the [MATH] decay of [MATH] within an electric field were measured using [MATH] atoms in the metastable [MATH] and in the [MATH] states confined by a magneto-optical trap.', '1703.07052-1-2-1': 'The electron shake-off probabilities were deduced including their dependence on ion energy.', '1703.07052-1-3-0': 'Results: We find significant discrepancies on the fractions of Li ions in the different charge states with respect to a recent calculation and discuss a plausible explanation.', '1703.07052-1-3-1': 'We also point out that there is no explanation for a large discrepancy between the same calculation and a previous measurement of the Li-ion energy dependence of the charge distribution from decays of the electronic ground state.', '1703.07052-1-4-0': '# Introduction', '1703.07052-1-5-0': 'Atomic and molecular degrees of freedom can play an important role in precision nuclear beta-decay experiments.', '1703.07052-1-5-1': 'For example, in measurements of the [MATH] spectrum near the end-point from molecular tritium, the final electronic state distribution [CITATION] affects the determination of the mass of anti-neutrinos; in precision determinations of the [MATH] angular correlation, the molecular binding of the trapped atoms can produce sizable deviations in measurements of recoil spectra [CITATION].', '1703.07052-1-6-0': 'The beta decay of [MATH] presents a good opportunity to determine the [MATH] correlation.', '1703.07052-1-6-1': 'Because of the large endpoint and the relatively light mass of the nucleus, the [MATH] correlation has a significant effect on the kinematics of the charged particles from the decay.', '1703.07052-1-6-2': 'A measurement performed in 1963 [CITATION] was one of several landmark experiments that determined the [MATH] nature of the weak interaction.', '1703.07052-1-6-3': 'Fundamental measurements of this kind have renewed interest in the context of searching for hints of new physics as deviations from the expectations based on the Standard Model [CITATION].', '1703.07052-1-6-4': 'An ongoing experiment is aiming at a measurement of the [MATH] angular correlation in [MATH] decay with improved precision [CITATION].', '1703.07052-1-6-5': 'In this experiment, the momentum of the recoil ion emitted from a cold and dilute cloud of laser-cooled [MATH] atoms confined in a magneto-optical trap is determined through a full kinematics reconstruction in a strong electric field.', '1703.07052-1-6-6': 'Due to the sudden change in nuclear charge, the electrons do not always find the corresponding orbits in the Li atom and can be shaken off.', '1703.07052-1-6-7': 'Thus, the [MATH] ion can have electric charges between [MATH] and [MATH].', '1703.07052-1-6-8': 'The fraction in a given charge state not only depends on the overlap of the initial electronic wave function and the final continuum states, but also on the ion energy, so a proper extraction of the [MATH] correlation coefficient requires understanding the shake-off effect.', '1703.07052-1-7-0': 'Because the electronic wave functions for He can be calculated with high accuracy, the case of [MATH] also presents a nice benchmark to test aspects of the calculations that are relevant for other problems, like the role of electron-electron interactions, the use of the sudden approximation, and methods for calculating charge distributions after the shake-off process.', '1703.07052-1-7-1': 'Several calculations have been performed for [MATH] [CITATION].', '1703.07052-1-7-2': 'A confirmation of the calculated fraction of [MATH] was recently performed with the hydrogen-like system of [MATH] ions [CITATION].', '1703.07052-1-7-3': 'Here, we address the case of the [MATH] neutral atom, which presents additional ingredients associated with the two electrons.', '1703.07052-1-8-0': 'The charge distribution of Li ions from the decay of [MATH] in its electronic ground state was measured by Carlson et al. [CITATION].', '1703.07052-1-8-1': 'Table [REF] shows a comparison to the most recent calculation of Schulhoff and Drake [CITATION].', '1703.07052-1-8-2': 'As can be observed, there are significant discrepancies for Li-ion fractions in the different charge states.', '1703.07052-1-8-3': 'The main aim of the calculations in Ref. [CITATION] was to study the dependence of charge distribution on the Li-ion energy, but a clear prediction is also given for the overall fractions in different charge states from [MATH], both in its ground and metastable states.', '1703.07052-1-9-0': 'We report here the first measurement of the electron emission probabilities following the [MATH] decay of [MATH] atoms confined via a magneto-optical trap working between the metastable [MATH] and the [MATH] states.', '1703.07052-1-9-1': 'We present data with both the trapping lasers off (pure [MATH]) and on (approximate 50%/50% mixture of [MATH] and [MATH]) and compare with the calculations of Schulhoff and Drake [CITATION] assuming decay from the [MATH] state.', '1703.07052-1-10-0': '# Experimental method', '1703.07052-1-11-0': 'The [MATH] atoms were produced via the [MATH] reaction.', '1703.07052-1-11-1': 'Up to 2[MATH] atoms per second were produced by bombarding a lithium target with an 18 MeV deuteron beam, delivered by the tandem Van de Graaff accelerator at the University of Washington [CITATION].', '1703.07052-1-12-0': 'The [MATH] atoms were pumped into a RF-discharge tube where a fraction ([MATH]) of the [MATH]He atoms were brought to the [MATH] metastable atomic state.', '1703.07052-1-12-1': 'The forward-going metastable atoms were then transversely-cooled, slowed by a Zeeman slower, and trapped in a magneto-optical trap (MOT) [CITATION], all based on resonant excitation of the [MATH] to [MATH] electronic transition via 1083 nm laser light.', '1703.07052-1-12-2': 'Due to the small efficiency for pumping [MATH] atoms to the metastable state there was a considerable amount of ground-state [MATH] in the chamber hosting the MOT.', '1703.07052-1-12-3': 'To reduce events from non-trapped [MATH] atoms, the atoms in the MOT were periodically pushed by a laser beam into a measurement chamber through a 5 mm-diameter 30-mm-long aperture tube for differential pumping and recaptured by a second MOT.', '1703.07052-1-12-4': 'This [MATH] trap contained over 2100 atoms on average.', '1703.07052-1-13-0': 'During the measurement, the trapping lasers of this MOT were alternatively switched on and off with a [MATH] duty cycle and a period of 100 [MATH]s.', '1703.07052-1-13-1': 'At the beginning of each off-cycle of 50 [MATH]s, the [MATH]50 fraction of the atoms previously excited to the [MATH] level quickly decay back to the [MATH] level within the 100 ns lifetime of the excited state.', '1703.07052-1-13-2': 'Time correlation of the decay events with the switching cycle thus allowed us to isolate decays from [MATH]He purely in the [MATH] state (laser off) from decays with an [MATH]50 admixture of the [MATH] excited state (laser on).', '1703.07052-1-13-3': 'Meanwhile, the fast switching provided sufficient confinement of the atom cloud.', '1703.07052-1-14-0': 'We studied the charge-state distributions of the recoil [MATH] ions by analyzing their time-of-flight (TOF) and energy spectra, which required detecting the [MATH] particle and the recoil ion in coincidence.', '1703.07052-1-14-1': 'The configuration of the detection chamber is shown in Fig. [REF].', '1703.07052-1-15-0': 'A [MATH]-telescope, consisting of a multi-wire proportional chamber (MWPC) and a scintillation detector, is placed above the trap.', '1703.07052-1-15-1': 'The scintillation detector measures the energy of the [MATH] particle ([MATH]).', '1703.07052-1-15-2': 'The MWPC detects the entrance position of the particles and strongly suppresses [MATH]-ray backgrounds triggering the scintillator when applying an appropriate coincidence gate between the two [MATH] detectors.', '1703.07052-1-15-3': 'The MWPC runs with 1 atm [MATH] by volume) gas and is separated from the MOT vacuum by a [MATH] thick 3.81-cm diameter beryllium window.', '1703.07052-1-16-0': 'A Micro Channel Plate (MCP) detector [CITATION] is placed below the trap for detecting recoil ions and determining their hit positions with a resolution of 190 [MATH]m (FWHM).', '1703.07052-1-16-1': 'Electrodes are installed in-between the [MATH]-telescope and the MCP detector to create an electric field of [MATH] and accelerate the recoil ions emitted from the trap towards the MCP detector.', '1703.07052-1-16-2': 'This enhances the ion-collection solid angle so that [MATH] of the [MATH] and 100 of the [MATH] and [MATH] ions are collected within the 75 mm diameter active area of the MCP.', '1703.07052-1-16-3': 'It also allows for the ions to have enough energy to trigger the MCP detector with a maximal detection efficiency ([MATH]50), independent of their initial charge state and recoil energy.', '1703.07052-1-17-0': 'The TOF measurement of the recoil ion is started by a scintillator signal and stopped by a MCP signal.', '1703.07052-1-17-1': 'The resolution of the TOF measurement is 820 ps (FWHM).', '1703.07052-1-17-2': '[MATH] ions in different charge states have different accelerations in the applied electric field, and are thus partially separated in TOF as shown in Fig. [REF].', '1703.07052-1-17-3': 'The overlaps between charge states can be avoided by applying a high enough threshold to the [MATH]-energy as indicated by the horizontal dashed lines.', '1703.07052-1-18-0': 'To fully reconstruct the initial energy of the recoil ion, the MOT position must also be determined.', '1703.07052-1-18-1': 'We ionize the trapped [MATH]He atoms periodically (at [MATH]20 Hz) using a 2-ns pulsed beam of ultraviolet (337 nm) nitrogen laser that has sufficient photon energy to photo-ionize the [MATH] state but not the [MATH] state.', '1703.07052-1-18-2': 'The vertical coordinate (along the electric field direction) of the MOT is determined through the TOF of the photo-ions with respect to the laser pulse.', '1703.07052-1-19-0': 'A unique feature of the trapping of metastable [MATH] atoms is that it allows for monitoring the horizontal shape of the MOT via "Penning-ions".', '1703.07052-1-19-1': 'The latter are generated by collisions between neutral atoms in the non-perfect vacuum and metastable [MATH] atoms.', '1703.07052-1-19-2': 'The uncertainty of the start position of each [MATH] recoil ion is limited in this data set by the size of the MOT which is 1.4 mm (FWHM).', '1703.07052-1-19-3': 'With the precisely measured MOT position, electric field strength, recoil-ion TOF and hit positions, the initial momenta of recoil ions are determined.', '1703.07052-1-19-4': 'The detector responses are calibrated daily, and the electric field and MOT position are monitored while the data are taken.', '1703.07052-1-19-5': 'The details of the construction, calibration and performance of the detector system are described in Ref. [CITATION].', '1703.07052-1-20-0': 'Decays from non-trapped [MATH]He atoms in the decay chamber and scattering of [MATH] particles prior to their detection generate an undesired background.', '1703.07052-1-20-1': 'To suppress this background, we reconstructed the momentum of the emitted anti-neutrino using the measured [MATH]-momentum and ion-momentum, and evaluated the [MATH]-value (the total kinetic energy released from the decay), which should be 3.5 MeV.', '1703.07052-1-20-2': 'Decays outside the MOT yield incorrect [MATH]-values because the reconstruction assumes the events originate at the MOT.', '1703.07052-1-20-3': 'Our [MATH]-value cut ([MATH]-cut) accepted events with [MATH], so that [MATH]% of the events from non-trapped decays were eliminated while the total data loss for events from trapped decays was less than 0.1% [CITATION].', '1703.07052-1-20-4': 'The width of this cut was limited by the energy resolution in our scintillator.', '1703.07052-1-20-5': 'In order to determine the contribution of the remaining events from non-trapped decays, we introduced large amounts of ground-state [MATH] atoms via a bypass pipe.', '1703.07052-1-21-0': 'Fig. [REF] shows a comparison of the TOF spectra.', '1703.07052-1-21-1': 'In Fig. [REF] the data from the non-trapped [MATH] is normalized and overlaid with the data taken with the trap on.', '1703.07052-1-21-2': 'The shortest TOF for a [MATH] ion emitted from the MOT is [MATH] ns which is the TOF leading edge for [MATH].', '1703.07052-1-21-3': 'Events with TOF[MATH] must be from non-trapped [MATH]He decays occurring closer to the MCP.', '1703.07052-1-21-4': 'Therefore, the normalization was chosen to match the TOF spectra without the [MATH]-cut in the TOF region from 10 ns to 110 ns.', '1703.07052-1-21-5': 'The same normalization factor was applied to the TOF and ion-energy spectra after the [MATH]-cut (shown in Fig. [REF]) and then these spectra were subtracted from those from the corresponding data run to remove the remaining events from untrapped [MATH] atoms.', '1703.07052-1-22-0': '# Data Analysis and Results', '1703.07052-1-23-0': 'As apparent from the data shown in Fig. [REF] and Fig. [REF], we clearly observed a [MATH]% shake-off fraction yielding [MATH] ions, however, we could not observe [MATH] ions above background and can therefore only set an upper limit.', '1703.07052-1-23-1': 'In the following, we will treat these two cases separately.', '1703.07052-1-24-0': 'We first address the total fraction of [MATH] ions ([MATH]), regardless of their energies.', '1703.07052-1-24-1': '[MATH] is the ratio [MATH] , where [MATH] is the number of observed [MATH] ions, and [MATH] is the number of observed [MATH] ions in all charge states.', '1703.07052-1-24-2': 'It is important to choose an appropriate [MATH] threshold to ensure a correct determination of [MATH].', '1703.07052-1-24-3': 'As shown in Fig. [REF], some of the [MATH] events with small [MATH] are lost because they have too high transverse momentum to hit the active area of the MCP.', '1703.07052-1-24-4': 'Therefore, the [MATH] threshold should be set high enough so that [MATH] ions in all charge states above the [MATH] threshold fall in the active area of the MCP.', '1703.07052-1-24-5': 'We chose 1 MeV as the [MATH] threshold, and under this condition the event loss due to the finite size of the MCP fiducial area is then less than 0.1% according to the Monte Carlo simulation.', '1703.07052-1-24-6': 'However, an [MATH] threshold at 1 MeV is not high enough to separate [MATH] from [MATH] ions.', '1703.07052-1-24-7': 'Therefore, to obtain the correct [MATH], we determined the number of events below the leading edge ([MATH]) of [MATH] TOF spectrum, and then corrected the [MATH] counts to include the counts overlapping with [MATH] events using a Monte Carlo simulation.', '1703.07052-1-24-8': 'Given the [MATH] threshold at 1 MeV, the percentage of [MATH] ions that are below [MATH] over all [MATH] ions is 85.5.', '1703.07052-1-24-9': '(Alternatively, it is also possible to choose an [MATH] threshold at 2.1 MeV so that the [MATH] ions are completely separated from the [MATH] ions, but such a high threshold rules out [MATH]75 of the data, yielding poor statistics so it was not adopted.)', '1703.07052-1-25-0': 'The TOF spectrum with the [MATH]-cut for the laser-off data is plotted in Fig. [REF].', '1703.07052-1-25-1': 'The event counts in the [MATH] TOF region (TOF[MATH]) are dominated by the background generated by the non-trapped [MATH]He decays.', '1703.07052-1-25-2': 'A summary of results is shown in Table. [REF].', '1703.07052-1-25-3': 'The measurements are consistent with [MATH] within 1 standard deviations, and the 90% confidence levels are calculated only in the physical region where the count of events are greater than 0.', '1703.07052-1-26-0': 'Next, we studied the fraction of [MATH] ions and its dependence on the recoil-ion energy.', '1703.07052-1-26-1': 'In this study, the initial energy of the recoil ions ([MATH]) are reconstructed for each event, so it is necessary to determine the charge state of each ion with no ambiguity.', '1703.07052-1-26-2': 'Therefore, the minimal threshold on [MATH], 1.5 MeV, was applied so that [MATH] ions are completely separated from [MATH] in TOF as shown by the lower of the two horizontal red dashed lines in Fig. [REF].', '1703.07052-1-26-3': 'As discussed above, the upper limit for the probability of having a [MATH] ion is at the [MATH] level, approximately three orders of magnitude lower than the probability of having a [MATH] ion.', '1703.07052-1-26-4': 'Therefore, we neglected [MATH] ions in this study assuming that all events below the leading edge ([MATH] ns) of [MATH] TOF spectrum correspond to [MATH] ions.', '1703.07052-1-26-5': 'The [MATH] spectra for [MATH] and [MATH] with their corresponding normalized backgrounds are plotted in Fig. [REF].', '1703.07052-1-27-0': 'Theoretically, the probability distribution functions (PDFs) of [MATH] in the two charge states are [EQUATION] where [MATH] indicates the charge state, and [MATH] is the probability of having an ion in such charge state.', '1703.07052-1-27-1': '[MATH] represents the part of the probability distribution function that is independent of charge state and depends only on the [MATH]-decay dynamics, i.e. the [MATH] correlation coefficient [MATH].', '1703.07052-1-27-2': 'The detection efficiency, [MATH], depends on the detection geometry, detector response functions and event reconstruction parameters and conditions.', '1703.07052-1-27-3': 'The expected energy dependence for [MATH] is [CITATION]: [EQUATION] where [MATH] and [MATH] are parameters.', '1703.07052-1-27-4': 'Taking the ratio of PDF[MATH] and PDF[MATH], one gets [EQUATION]', '1703.07052-1-27-5': 'The [MATH] functions in the numerator and the denominator cancel each other and [MATH] no longer depends on [MATH] explicitly.', '1703.07052-1-27-6': 'With the [MATH] keV threshold there is no event loss due to MCP fiducial area for both charge states [MATH] and [MATH].', '1703.07052-1-27-7': 'In principle, the ion detection efficiency [MATH] depends on the charge state because [MATH] and [MATH] have very different final energies ([MATH]13 keV versus [MATH]26 keV), and their position distributions on the MCP are also different.', '1703.07052-1-27-8': 'However, the difference in gain of the MCP for these two charge states is only 2.5%.', '1703.07052-1-27-9': 'Due to a very low MCP charge threshold, it results in less than [MATH] efficiency difference.', '1703.07052-1-27-10': 'Therefore, we made the assumption that [MATH], so they cancel each other in Eq. ([REF]).', '1703.07052-1-27-11': '(The systematic uncertainty generated by this approximation was studied using Monte Carlo simulations, assuming that there is still a 0.82% (RMS) variation of the MCP efficiency over the surface of the MCP.)', '1703.07052-1-27-12': 'Because [MATH] ions are neglected, the sum of [MATH] and [MATH] is 1, and thus Eq. ([REF]) becomes [EQUATION]', '1703.07052-1-27-13': 'We take the ratio of the measured [MATH] spectra (with background subtraction applied) for the charge states [MATH] and [MATH], and fit the ratio histogram to Eq. ([REF]), as shown in Fig. [REF].', '1703.07052-1-27-14': 'The fit region is chosen to be from 0.1 keV to 1.1 keV, in order to make the systematic uncertainties introduced by the assumptions made above negligible compared to the statistical uncertainty.', '1703.07052-1-27-15': '[MATH] and [MATH] are fit parameters, and the fit results are listed in Table [REF].', '1703.07052-1-27-16': 'We also obtain the fractions of [MATH] and [MATH] regardless of their energies by counting the total number of events above and below [MATH].', '1703.07052-1-27-17': 'The results of the fractions for all three charge states of [MATH] ions are summarized and compared to theoretical calculations [CITATION] in Table [REF].', '1703.07052-1-27-18': 'All the fractions determined in this experiment are obtained with an [MATH] threshold.', '1703.07052-1-27-19': 'Applying an [MATH] threshold, which is inevitable in this experiment, changes the energy spectrum of the recoil ions, and thus affects the fractions of [MATH] ions in different charge states due to their dependences on the ion energies.', '1703.07052-1-27-20': 'Therefore, in Table [REF] the theoretical fractions are calculated based on the results of Ref. [CITATION] and applied with the same [MATH] threshold as in our experiment.', '1703.07052-1-28-0': 'In order to understand the systematic shifts of [MATH] and [MATH] caused by approximations like assuming [MATH] and [MATH] to be identical, we ran Monte Carlo simulations with the values of [MATH] and [MATH] from Ref. [CITATION].', '1703.07052-1-28-1': 'In the simulations, we modeled the experimental parameters, such as the detector geometry and response functions, and the efficiencies and electric field, as close as possible to the experimental setup.', '1703.07052-1-28-2': 'The simulated data were processed in the same way as that used for the experimental data.', '1703.07052-1-28-3': 'There is no significant deviation of the extracted [MATH] value from its input value, while the extracted [MATH] deviates from its input by [MATH].', '1703.07052-1-28-4': 'Implementation of a 0.82% spatial variation of the MCP efficiency does not result in significant deviations of [MATH] and [MATH].', '1703.07052-1-28-5': 'We also studied how the fit values of [MATH] and [MATH] change with respect to parameters used in the ion-energy reconstruction, and the systematic uncertainties associated with these parameters.', '1703.07052-1-28-6': 'The corresponding non-negligible systematic uncertainties of [MATH] and [MATH] are listed in Table [REF], and in total are smaller than the statistical uncertainties listed in Table [REF].', '1703.07052-1-28-7': 'Note that these systematic uncertainties are all related to the ion-energy calculation, so they do not affect the determination of the energy-integrated charge-state fraction listed in Table [REF].', '1703.07052-1-29-0': '# Discussion', '1703.07052-1-30-0': 'A comparison of our measurements to calculations for the decay of [MATH] from its atomic metastable state is shown in Table [REF].', '1703.07052-1-30-1': 'The [MATH] and [MATH] fractions measured in this experiment have small ([MATH]%) but significant ([MATH]) discrepancies with the theoretical calculations of Ref. [CITATION].', '1703.07052-1-30-2': 'The calculation for the [MATH] decays from the atomic ground state similarly over-predicts the [MATH] fraction measured by Carlson et al. [CITATION] as shown in Table [REF].', '1703.07052-1-30-3': 'It is possible that there is a missing consideration in the calculation that systematically leads to a higher [MATH] fraction for both initial atomic states.', '1703.07052-1-30-4': 'In Ref. [CITATION] the charge state of the [MATH] ion is determined solely by the final energy of the two orbital electrons, [MATH], with respect to the ionization energies for 1 and 2 electrons, [MATH] and [MATH]: [EQUATION]', '1703.07052-1-30-5': 'In considering possible sources for the discrepancy we note, for example, that the condition [MATH] could be met without double ionization if one of the electrons takes away a significant fraction of the energy as kinetic energy, leaving the other electron bound.', '1703.07052-1-30-6': 'This could lead to the systematic overestimation of the [MATH] probability as observed.', '1703.07052-1-31-0': 'In contrast to the integrated charge-state fractions, the ion-energy dependencies of the [MATH] ion charge-state fractions (the [MATH] parameters) are of concern for the determinations of the [MATH] correlation coefficient.', '1703.07052-1-31-1': 'The experiment of Ref. [CITATION] was performed with the same method and apparatus used in Ref. [CITATION] to determine the [MATH] correlation coefficient.', '1703.07052-1-31-2': 'As shown in Table [REF], there is a significant discrepancy between the [MATH] parameters measured by Carlson et al. and the ones calculated in Ref. [CITATION].', '1703.07052-1-31-3': 'While we find a plausible explanation for some of the differences between theory and experiment for the overall Li-ion fractions, as stated above, it is more difficult to understand how the [MATH] factors could be a factor of [MATH] smaller in the calculations.', '1703.07052-1-31-4': 'The [MATH] values measured by Carlson et al. are close to a naive prediction ignoring a cancellation that takes place between [MATH] and [MATH] final state configurations shown in detail in Ref. [CITATION].', '1703.07052-1-31-5': 'Johnson et al. were aware of the cancellation and the fact that their measured [MATH] factors disagreed with the more accurate calculation [CITATION], but at the time they reported concerns about the difficulty of including a large enough set of states in their calculation.', '1703.07052-1-31-6': 'There is an implicit suggestion that the discrepancy should not be taken seriously because the calculation was incomplete.', '1703.07052-1-31-7': 'No such concerns exist for the recent calculation of Schulhoff and Drake [CITATION], so we conclude that this suggests either an unaccounted-for experimental issue or a failure of the framework for the calculation.', '1703.07052-1-31-8': 'Changing the [MATH] parameters from their measured value to zero affects the determination of [MATH] only by [MATH]% which is smaller than the [MATH]% uncertainty claimed by Ref. [CITATION].', '1703.07052-1-31-9': 'In the present context of trying to achieve more precise determinations, the issue is more important.', '1703.07052-1-32-0': 'Our results for the [MATH] and [MATH] parameters for the [MATH]He decays from the atomic metastable state and the corresponding theoretical calculations are shown in Table [REF].', '1703.07052-1-32-1': "Unfortunately, our results don't have the statistical power to claim a precision test of the calculations, in particular for the [MATH] parameter.", '1703.07052-1-32-2': 'We also calculated the parameters [MATH] and [MATH] for decays from initial atomic state [MATH], based on the [MATH] and [MATH] values in Table [REF] for the laser-on case (mixture of 50% [MATH] and 50% [MATH]) and laser-off case (pure [MATH]).', '1703.07052-1-32-3': 'The results together with the final results for the [MATH] initial states are summarized in Table [REF].', '1703.07052-1-32-4': 'The results for these two initial states are not significantly different from each other.', '1703.07052-1-33-0': 'Additionally, in measurements of the [MATH] correlation, where the TOF spectrum is fitted to templates generated by Monte Carlo simulations and the charge state groups are not completely separated in TOF, both the [MATH] and [MATH] parameters can affect the fit result.', '1703.07052-1-33-1': 'We have used a Monte Carlo simulations to study how the uncertainties of [MATH] and [MATH] translate into the uncertainty of the [MATH] correlation coefficient.', '1703.07052-1-33-2': 'Based on the experimental uncertainties listed in Table [REF], [MATH] results in a 0.3 relative uncertainty of [MATH], and [MATH] results in a 0.6 relative uncertainty.', '1703.07052-1-33-3': 'Therefore, the results from this paper are sufficient for an experiment aiming at determining [MATH] to 1.', '1703.07052-1-33-4': 'Because the charge-state analysis and the [MATH] analysis can use the same data set, the uncertainties of [MATH] and [MATH] will be improved as more data are taken to achieve better than 1 uncertainties on [MATH].', '1703.07052-1-34-0': '# Conclusions', '1703.07052-1-35-0': 'We have measured the [MATH] ion charge-state fractions for [MATH] decays from atomic metastable state [MATH].', '1703.07052-1-35-1': 'The overall fractions for [MATH] and [MATH] (Table [REF]) show small but significant disagreement with the recent theoretical calculation of Ref. [CITATION].', '1703.07052-1-35-2': 'We discuss a plausible explanation.', '1703.07052-1-36-0': 'We also point out that there is no satisfactory explanation for a large discrepancy between the same calculation and the results of Carlson et al.[CITATION] for the Li-ion energy dependence of the fractions from decays of the electronic ground state, suggesting either an unaccounted-for experimental issue or a failure of the framework for the calculation.', '1703.07052-1-37-0': 'The [MATH] and [MATH] parameters in the ion-energy dependent charge-state fraction expression (Eq. ([REF])) were also determined (Table. [REF]), and the precision is sufficient for a determination of [MATH] with 1% relative precision.', '1703.07052-1-37-1': 'The precisions of the [MATH] and [MATH] parameters can be improved as more data for the [MATH] correlation measurement are taken.', '1703.07052-1-38-0': 'This work is supported by the Department of Energy, Office of Nuclear Physics, under contract numbers DE-AC02-06CH11357 and DE-FG02-97ER41020.', '1703.07052-1-38-1': 'This work is also supported in part by the U.S. National Science Foundation under Grant No. PHY-11-02511.', '1703.07052-1-38-2': 'We thank Gordon Drake for many useful discussions.'}

{'1703.07052-2-0-0': 'Background: The accurate determination of atomic final states following nuclear [MATH] decay plays an important role in many experiments.', '1703.07052-2-0-1': 'In particular, the charge state distributions of ions following nuclear [MATH] decay are important for determinations of the [MATH] angular correlation with improved precision.', '1703.07052-2-1-0': 'Purpose: Our measurement aims at providing benchmarks to test theoretical calculations.', '1703.07052-2-2-0': 'Method: The kinematics of Li[MATH] ions produced following the [MATH] decay of [MATH] within an electric field were measured using [MATH] atoms in the metastable [MATH] and in the [MATH] states confined by a magneto-optical trap.', '1703.07052-2-2-1': 'The electron shake-off probabilities were deduced including their dependence on ion energy.', '1703.07052-2-3-0': 'Results: We find significant discrepancies on the fractions of Li ions in the different charge states with respect to a recent calculation and discuss a plausible explanation.', '1703.07052-2-3-1': 'We also point out that there is no explanation for a large discrepancy between the same calculation and a previous measurement of the Li-ion energy dependence of the charge distribution from decays of the electronic ground state.', '1703.07052-2-4-0': '# Introduction', '1703.07052-2-5-0': 'Atomic and molecular degrees of freedom can play an important role in precision nuclear beta-decay experiments.', '1703.07052-2-5-1': 'For example, in measurements of the [MATH] spectrum near the end-point from molecular tritium, the final electronic state distribution [CITATION] affects the determination of the mass of anti-neutrinos; in precision determinations of the [MATH] angular correlation, the molecular binding of the trapped atoms can produce sizable deviations in measurements of recoil spectra [CITATION].', '1703.07052-2-6-0': 'The beta decay of [MATH] presents a good opportunity to determine the [MATH] correlation.', '1703.07052-2-6-1': 'Because of the large endpoint and the relatively light mass of the nucleus, the [MATH] correlation has a significant effect on the kinematics of the charged particles from the decay.', '1703.07052-2-6-2': 'A measurement performed in 1963 [CITATION] was one of several landmark experiments that determined the [MATH] nature of the weak interaction.', '1703.07052-2-6-3': 'Fundamental measurements of this kind have renewed interest in the context of searching for hints of new physics as deviations from the expectations based on the Standard Model [CITATION].', '1703.07052-2-6-4': 'An ongoing experiment is aiming at a measurement of the [MATH] angular correlation in [MATH] decay with improved precision [CITATION].', '1703.07052-2-6-5': 'In this experiment, the momentum of the recoil ion emitted from a cold and dilute cloud of laser-cooled [MATH] atoms confined in a magneto-optical trap (MOT) [CITATION] is determined through a full kinematics reconstruction in a strong electric field.', '1703.07052-2-6-6': 'Due to the sudden change in nuclear charge, the electrons do not always find the corresponding orbits in the Li atom and can be shaken off.', '1703.07052-2-6-7': 'Thus, the [MATH] ion can have electric charges between [MATH] and [MATH].', '1703.07052-2-6-8': 'The fraction in a given charge state not only depends on the overlap of the initial electronic wave function and the final continuum states, but also on the ion energy, so a proper extraction of the [MATH] correlation coefficient requires understanding the shake-off effect.', '1703.07052-2-7-0': 'Because the electronic wave functions for helium can be calculated with high accuracy, the case of [MATH] also presents a nice benchmark to test aspects of the calculations that are relevant for other problems, like the role of electron-electron interactions, the use of the sudden approximation, and methods for calculating charge distributions after the shake-off process.', '1703.07052-2-7-1': 'Several calculations have been performed for [MATH] [CITATION].', '1703.07052-2-7-2': 'A confirmation of the calculated fraction of [MATH] was recently performed with the hydrogen-like system of [MATH] ions [CITATION].', '1703.07052-2-7-3': 'Here, we address the case of the [MATH] neutral atom, which presents additional ingredients associated with the two electrons.', '1703.07052-2-8-0': 'The charge distribution of Li ions from the decay of [MATH] in its electronic ground state was measured by Carlson et al. [CITATION].', '1703.07052-2-8-1': 'Table [REF] shows a comparison to the most recent calculation of Schulhoff and Drake [CITATION].', '1703.07052-2-8-2': 'As can be observed, there are significant discrepancies for Li-ion fractions in the different charge states.', '1703.07052-2-8-3': 'The main aim of the calculations in Ref. [CITATION] was to study the dependence of charge distribution on the Li-ion energy, but a clear prediction is also given for the overall fractions in different charge states from [MATH], both in its ground and metastable states.', '1703.07052-2-9-0': 'We report here the first measurement of the electron emission probabilities following the [MATH] decay of [MATH] atoms confined via a magneto-optical trap working between the metastable [MATH] and the [MATH] states.', '1703.07052-2-9-1': 'We present data with both the trapping lasers off (pure [MATH]) and on (approximate 50%/50% mixture of [MATH] and [MATH]) and compare with the calculations of Schulhoff and Drake [CITATION] assuming decay from the [MATH] state.', '1703.07052-2-10-0': '# Experimental method', '1703.07052-2-11-0': 'The [MATH] atoms were produced via the [MATH] reaction.', '1703.07052-2-11-1': 'Up to 2[MATH] atoms per second were produced by bombarding a lithium target with an 18 MeV deuteron beam, delivered by the tandem Van de Graaff accelerator at the University of Washington [CITATION].', '1703.07052-2-12-0': 'The [MATH] atoms were pumped into a RF-discharge tube where a fraction ([MATH]) of the [MATH]He atoms were brought to the [MATH] metastable atomic state.', '1703.07052-2-12-1': 'The forward-going metastable atoms were then transversely-cooled, slowed by a Zeeman slower, and trapped in a magneto-optical trap, all based on resonant excitation of the [MATH] to [MATH] electronic transition via 1083 nm laser light.', '1703.07052-2-12-2': 'Due to the small efficiency for pumping [MATH] atoms to the metastable state there was a considerable amount of ground-state [MATH] in the chamber hosting the MOT.', '1703.07052-2-12-3': 'To reduce events from non-trapped [MATH] atoms, the atoms in the MOT were periodically pushed by a laser beam into a measurement chamber through a 5 mm-diameter 30-mm-long aperture tube for differential pumping and recaptured by a second MOT.', '1703.07052-2-12-4': 'This [MATH] trap contained over 2100 atoms on average.', '1703.07052-2-13-0': 'During the measurement, the trapping lasers of this MOT were alternatively switched on and off with a [MATH] duty cycle and a period of 100 [MATH]s.', '1703.07052-2-13-1': 'At the beginning of each off-cycle of 50 [MATH]s, the [MATH]50 fraction of the atoms previously excited to the [MATH] level quickly decay back to the [MATH] level within the 100 ns lifetime of the excited state.', '1703.07052-2-13-2': 'Time correlation of the decay events with the switching cycle thus allowed us to isolate decays from [MATH]He purely in the [MATH] state (laser off) from decays with an [MATH]50 admixture of the [MATH] excited state (laser on).', '1703.07052-2-13-3': 'Meanwhile, the fast switching provided sufficient confinement of the atom cloud.', '1703.07052-2-14-0': 'We studied the charge-state distributions of the recoil [MATH] ions by analyzing their time-of-flight (TOF) and energy spectra, which required detecting the [MATH] particle and the recoil ion in coincidence.', '1703.07052-2-14-1': 'The configuration of the detection chamber is shown in Fig. [REF].', '1703.07052-2-15-0': 'A [MATH]-telescope, consisting of a multi-wire proportional chamber (MWPC) and a scintillation detector, is placed above the trap.', '1703.07052-2-15-1': 'The scintillation detector measures the energy of the [MATH] particle ([MATH]).', '1703.07052-2-15-2': 'The MWPC detects the entrance position of the particles and strongly suppresses [MATH]-ray backgrounds triggering the scintillator when applying an appropriate coincidence gate between the two [MATH] detectors.', '1703.07052-2-15-3': 'The MWPC runs with 1 atm [MATH] by volume) gas and is separated from the MOT vacuum by a [MATH] thick 3.81-cm diameter beryllium window.', '1703.07052-2-16-0': 'A Micro Channel Plate (MCP) detector [CITATION] is placed below the trap for detecting recoil ions and determining their hit positions with a resolution of 190 [MATH]m (FWHM).', '1703.07052-2-16-1': 'Electrodes are installed in-between the [MATH]-telescope and the MCP detector to create an electric field of [MATH] and accelerate the recoil ions emitted from the trap towards the MCP detector.', '1703.07052-2-16-2': 'This enhances the ion-collection solid angle so that [MATH] of the [MATH] and 100 of the [MATH] and [MATH] ions are collected within the 75 mm diameter active area of the MCP.', '1703.07052-2-16-3': 'It also allows for the ions to have enough energy to trigger the MCP detector with a maximal detection efficiency ([MATH]50), independent of their initial charge state and recoil energy.', '1703.07052-2-17-0': 'The TOF measurement of the recoil ion is started by a scintillator signal and stopped by a MCP signal.', '1703.07052-2-17-1': 'The resolution of the TOF measurement is 820 ps (FWHM).', '1703.07052-2-17-2': '[MATH] ions in different charge states have different accelerations in the applied electric field, and are thus partially separated in TOF as shown in Fig. [REF].', '1703.07052-2-17-3': 'The overlaps between charge states can be avoided by applying a high enough threshold to the [MATH]-energy as indicated by the horizontal dashed lines.', '1703.07052-2-18-0': 'To fully reconstruct the initial energy of the recoil ion, the MOT position must also be determined.', '1703.07052-2-18-1': 'We ionize the trapped [MATH]He atoms periodically (at [MATH]20 Hz) using a 2-ns pulsed beam of ultraviolet (337 nm) nitrogen laser that has sufficient photon energy to photo-ionize the [MATH] state but not the [MATH] state.', '1703.07052-2-18-2': 'The vertical coordinate (along the electric field direction) of the MOT is determined through the TOF of the photo-ions with respect to the laser pulse.', '1703.07052-2-19-0': 'A unique feature of the trapping of metastable [MATH] atoms is that it allows for monitoring the horizontal shape of the MOT via "Penning-ions".', '1703.07052-2-19-1': 'The latter are generated by collisions between neutral atoms in the non-perfect vacuum and metastable [MATH] atoms.', '1703.07052-2-19-2': 'The uncertainty of the start position of each [MATH] recoil ion is limited in this data set by the size of the MOT which is 1.4 mm (FWHM).', '1703.07052-2-19-3': 'With the precisely measured MOT position, electric field strength, recoil-ion TOF and hit positions, the initial momenta of recoil ions are determined.', '1703.07052-2-19-4': 'The detector responses are calibrated daily, and the electric field and MOT position are monitored while the data are taken.', '1703.07052-2-19-5': 'The details of the construction, calibration and performance of the detector system are described in Ref. [CITATION].', '1703.07052-2-20-0': 'Decays from non-trapped [MATH]He atoms in the decay chamber and scattering of [MATH] particles prior to their detection generate an undesired background.', '1703.07052-2-20-1': 'To suppress this background, we reconstructed the momentum of the emitted anti-neutrino using the measured [MATH]-momentum and ion-momentum, and evaluated the [MATH]-value (the total kinetic energy released from the decay), which should be 3.5 MeV.', '1703.07052-2-20-2': 'Decays outside the MOT yield incorrect [MATH]-values because the reconstruction assumes the events originate at the MOT.', '1703.07052-2-20-3': 'Our [MATH]-value cut ([MATH]-cut) accepted events with [MATH], so that [MATH]% of the events from non-trapped decays were eliminated while the total data loss for events from trapped decays was less than 0.1% [CITATION].', '1703.07052-2-20-4': 'The width of this cut was limited by the energy resolution in our scintillator.', '1703.07052-2-20-5': 'In order to determine the contribution of the remaining events from non-trapped decays, we introduced large amounts of ground-state [MATH] atoms via a bypass pipe.', '1703.07052-2-21-0': 'Fig. [REF] shows a comparison of the TOF spectra.', '1703.07052-2-21-1': 'In Fig. [REF] the data from the non-trapped [MATH] is normalized and overlaid with the data taken with the trap on.', '1703.07052-2-21-2': 'The shortest TOF for a [MATH] ion emitted from the MOT is [MATH] ns which is the TOF leading edge for [MATH].', '1703.07052-2-21-3': 'Events with TOF[MATH] must be from non-trapped [MATH]He decays occurring closer to the MCP.', '1703.07052-2-21-4': 'Therefore, the normalization was chosen to match the TOF spectra without the [MATH]-cut in the TOF region from 10 ns to 110 ns.', '1703.07052-2-21-5': 'The same normalization factor was applied to the TOF and ion-energy spectra after the [MATH]-cut (shown in Fig. [REF]) and then these spectra were subtracted from those from the corresponding data run to remove the remaining events from untrapped [MATH] atoms.', '1703.07052-2-22-0': '# Data Analysis and Results', '1703.07052-2-23-0': 'As apparent from the data shown in Fig. [REF] and Fig. [REF], we clearly observed a [MATH]% shake-off fraction yielding [MATH] ions, however, we could not observe [MATH] ions above background and can therefore only set an upper limit.', '1703.07052-2-23-1': 'In the following, we will treat these two cases separately.', '1703.07052-2-24-0': 'We first address the total fraction of [MATH] ions ([MATH]), regardless of their energies.', '1703.07052-2-24-1': '[MATH] is the ratio [MATH] , where [MATH] is the number of observed [MATH] ions, and [MATH] is the number of observed [MATH] ions in all charge states.', '1703.07052-2-24-2': 'It is important to choose an appropriate [MATH] threshold to ensure a correct determination of [MATH].', '1703.07052-2-24-3': 'As shown in Fig. [REF], some of the [MATH] events with small [MATH] are lost because they have too high transverse momentum to hit the active area of the MCP.', '1703.07052-2-24-4': 'Therefore, the [MATH] threshold should be set high enough so that [MATH] ions in all charge states above the [MATH] threshold fall in the active area of the MCP.', '1703.07052-2-24-5': 'We chose 1 MeV as the [MATH] threshold, and under this condition the event loss due to the finite size of the MCP fiducial area is then less than 0.1% according to the Monte Carlo simulation.', '1703.07052-2-24-6': 'However, an [MATH] threshold at 1 MeV is not high enough to separate [MATH] from [MATH] ions.', '1703.07052-2-24-7': 'Therefore, to obtain the correct [MATH], we determined the number of events below the leading edge ([MATH]) of [MATH] TOF spectrum, and then corrected the [MATH] counts to include the counts overlapping with [MATH] events using a Monte Carlo simulation.', '1703.07052-2-24-8': 'Given the [MATH] threshold at 1 MeV, the percentage of [MATH] ions that are below [MATH] over all [MATH] ions is 85.5.', '1703.07052-2-24-9': '(Alternatively, it is also possible to choose an [MATH] threshold at 2.1 MeV so that the [MATH] ions are completely separated from the [MATH] ions, but such a high threshold rules out [MATH]75 of the data, yielding poor statistics so it was not adopted.)', '1703.07052-2-25-0': 'The TOF spectrum with the [MATH]-cut for the laser-off data is plotted in Fig. [REF].', '1703.07052-2-25-1': 'The event counts in the [MATH] TOF region (TOF[MATH]) are dominated by the background generated by the non-trapped [MATH]He decays.', '1703.07052-2-25-2': 'A summary of results is shown in Table. [REF].', '1703.07052-2-25-3': 'The measurements are consistent with [MATH] within 1 standard deviations, and the 90% confidence levels are calculated only in the physical region where the count of events are greater than 0.', '1703.07052-2-26-0': 'Next, we studied the fraction of [MATH] ions and its dependence on the recoil-ion energy.', '1703.07052-2-26-1': 'In this study, the initial energy of the recoil ions ([MATH]) are reconstructed for each event, so it is necessary to determine the charge state of each ion with no ambiguity.', '1703.07052-2-26-2': 'Therefore, the minimal threshold on [MATH], 1.5 MeV, was applied so that [MATH] ions are completely separated from [MATH] in TOF as shown by the lower of the two horizontal red dashed lines in Fig. [REF].', '1703.07052-2-26-3': 'As discussed above, the upper limit for the probability of having a [MATH] ion is at the [MATH] level, approximately three orders of magnitude lower than the probability of having a [MATH] ion.', '1703.07052-2-26-4': 'Therefore, we neglected [MATH] ions in this study assuming that all events below the leading edge ([MATH] ns) of [MATH] TOF spectrum correspond to [MATH] ions.', '1703.07052-2-26-5': 'The [MATH] spectra for [MATH] and [MATH] with their corresponding normalized backgrounds are plotted in Fig. [REF].', '1703.07052-2-27-0': 'Theoretically, the probability distribution functions (PDFs) of [MATH] in the two charge states are [EQUATION] where [MATH] indicates the charge state, and [MATH] is the probability of having an ion in such charge state.', '1703.07052-2-27-1': '[MATH] represents the part of the probability distribution function that is independent of charge state and depends only on the [MATH]-decay dynamics, i.e. the [MATH] correlation coefficient [MATH].', '1703.07052-2-27-2': 'The detection efficiency, [MATH], depends on the detection geometry, detector response functions and event reconstruction parameters and conditions.', '1703.07052-2-27-3': 'The expected energy dependence for [MATH] is [CITATION]: [EQUATION] where [MATH] and [MATH] are parameters.', '1703.07052-2-27-4': 'Taking the ratio of PDF[MATH] and PDF[MATH], one gets [EQUATION]', '1703.07052-2-27-5': 'The [MATH] functions in the numerator and the denominator cancel each other and [MATH] no longer depends on [MATH] explicitly.', '1703.07052-2-27-6': 'With the [MATH] keV threshold there is no event loss due to MCP fiducial area for both charge states [MATH] and [MATH].', '1703.07052-2-27-7': 'In principle, the ion detection efficiency [MATH] depends on the charge state because [MATH] and [MATH] have very different final energies ([MATH]13 keV versus [MATH]26 keV), and their position distributions on the MCP are also different.', '1703.07052-2-27-8': 'However, the difference in gain of the MCP for these two charge states is only 2.5%.', '1703.07052-2-27-9': 'Due to a very low MCP charge threshold, it results in less than [MATH] efficiency difference.', '1703.07052-2-27-10': 'Therefore, we made the assumption that [MATH], so they cancel each other in Eq. ([REF]).', '1703.07052-2-27-11': '(The systematic uncertainty generated by this approximation was studied using Monte Carlo simulations including the measured 0.82% (RMS) variation of the MCP efficiency over the surface of the MCP.)', '1703.07052-2-27-12': 'Because [MATH] ions are neglected, the sum of [MATH] and [MATH] is 1, and thus Eq. ([REF]) becomes [EQUATION]', '1703.07052-2-27-13': 'We take the ratio of the measured [MATH] spectra (with background subtraction applied) for the charge states [MATH] and [MATH], and fit the ratio histogram to Eq. ([REF]), as shown in Fig. [REF].', '1703.07052-2-27-14': 'The fit region is chosen to be from 0.1 keV to 1.1 keV, in order to make the systematic uncertainties introduced by the assumptions made above negligible compared to the statistical uncertainty.', '1703.07052-2-27-15': '[MATH] and [MATH] are fit parameters, and the fit results are listed in Table [REF].', '1703.07052-2-27-16': 'We also obtain the fractions of [MATH] and [MATH] regardless of their energies by counting the total number of events above and below [MATH].', '1703.07052-2-27-17': 'The results of the fractions for all three charge states of [MATH] ions are summarized and compared to theoretical calculations [CITATION] in Table [REF].', '1703.07052-2-27-18': 'All the fractions determined in this experiment are obtained with an [MATH] threshold.', '1703.07052-2-27-19': 'Applying an [MATH] threshold, which is inevitable in this experiment, changes the energy spectrum of the recoil ions, and thus affects the fractions of [MATH] ions in different charge states due to their dependences on the ion energies.', '1703.07052-2-27-20': 'Therefore, in Table [REF] the theoretical fractions are calculated based on the results of Ref. [CITATION] and applied with the same [MATH] threshold as in our experiment.', '1703.07052-2-28-0': 'In order to understand the systematic shifts of [MATH] and [MATH] caused by approximations like assuming [MATH] and [MATH] to be identical, we ran Monte Carlo simulations with the values of [MATH] and [MATH] from Ref. [CITATION].', '1703.07052-2-28-1': 'In the simulations, we modeled the experimental parameters, such as the detector geometry and response functions, and the efficiencies and electric field, as close as possible to the experimental setup.', '1703.07052-2-28-2': 'The simulated data were processed in the same way as that used for the experimental data.', '1703.07052-2-28-3': 'There is no significant deviation of the extracted [MATH] value from its input value, while the extracted [MATH] deviates from its input by [MATH].', '1703.07052-2-28-4': 'Implementation of the 0.82% spatial variation of the MCP efficiency does not result in significant deviations of [MATH] and [MATH].', '1703.07052-2-28-5': 'We also studied how the fit values of [MATH] and [MATH] change with respect to parameters used in the ion-energy reconstruction, and the systematic uncertainties associated with these parameters.', '1703.07052-2-28-6': 'The corresponding non-negligible systematic uncertainties of [MATH] and [MATH] are listed in Table [REF], and in total are smaller than the statistical uncertainties listed in Table [REF].', '1703.07052-2-28-7': 'Note that these systematic uncertainties are all related to the ion-energy calculation, so they do not affect the determination of the energy-integrated charge-state fraction listed in Table [REF].', '1703.07052-2-29-0': '# Discussion', '1703.07052-2-30-0': 'A comparison of our measurements to calculations for the decay of [MATH] from its atomic metastable state is shown in Table [REF].', '1703.07052-2-30-1': 'The [MATH] and [MATH] fractions measured in this experiment have small ([MATH]%) but significant ([MATH] for [MATH] and [MATH] for [MATH]) discrepancies with the theoretical calculations of Ref. [CITATION].', '1703.07052-2-30-2': 'The calculation for the [MATH] decays from the atomic ground state similarly over-predicts the [MATH] fraction measured by Carlson et al. [CITATION] as shown in Table [REF].', '1703.07052-2-30-3': 'It is possible that there is a missing consideration in the calculation that systematically leads to a higher [MATH] fraction for both initial atomic states.', '1703.07052-2-30-4': 'In Ref. [CITATION] the charge state of the [MATH] ion is determined solely by the final energy of the two orbital electrons, [MATH], with respect to the ionization energies for 1 and 2 electrons, [MATH] and [MATH]: [EQUATION]', '1703.07052-2-30-5': 'In considering possible sources for the discrepancy we note, for example, that the condition [MATH] could be met without double ionization if one of the electrons takes away a significant fraction of the energy as kinetic energy, leaving the other electron bound.', '1703.07052-2-30-6': 'This could lead to the systematic overestimation of the [MATH] probability as observed.', '1703.07052-2-31-0': 'In contrast to the integrated charge-state fractions, the ion-energy dependencies of the [MATH] ion charge-state fractions (the [MATH] parameters) are of concern for the determinations of the [MATH] correlation coefficient.', '1703.07052-2-31-1': 'The experiment of Ref. [CITATION] was performed with the same method and apparatus used in Ref. [CITATION] to determine the [MATH] correlation coefficient.', '1703.07052-2-31-2': 'As shown in Table [REF], there is a significant discrepancy between the [MATH] parameters measured by Carlson et al. and the ones calculated in Ref. [CITATION].', '1703.07052-2-31-3': 'While we find a plausible explanation for some of the differences between theory and experiment for the overall Li-ion fractions, as stated above, it is more difficult to understand how the [MATH] factors could be a factor of [MATH] smaller in the calculations.', '1703.07052-2-31-4': 'The [MATH] values measured by Carlson et al. are close to a naive prediction ignoring a cancellation that takes place between [MATH] and [MATH] final state configurations shown in detail in Ref. [CITATION].', '1703.07052-2-31-5': 'Johnson et al. were aware of the cancellation and the fact that their measured [MATH] factors disagreed with the more accurate calculation [CITATION], but at the time they reported concerns about the difficulty of including a large enough set of states in their calculation.', '1703.07052-2-31-6': 'There is an implicit suggestion that the discrepancy should not be taken seriously because the calculation was incomplete.', '1703.07052-2-31-7': 'No such concerns exist for the recent calculation of Schulhoff and Drake [CITATION], so we conclude that this suggests either an unaccounted-for experimental issue or a failure of the framework for the calculation.', '1703.07052-2-31-8': 'Changing the [MATH] parameters from their measured value to zero affects the determination of [MATH] only by [MATH]% which is smaller than the [MATH]% uncertainty claimed by Ref. [CITATION].', '1703.07052-2-31-9': 'In the present context of trying to achieve more precise determinations, the issue is more important.', '1703.07052-2-32-0': 'Our results for the [MATH] and [MATH] parameters for the [MATH]He decays from the atomic metastable state and the corresponding theoretical calculations are shown in Table [REF].', '1703.07052-2-32-1': "Unfortunately, our results don't have the statistical power to claim a precision test of the calculations, in particular for the [MATH] parameter.", '1703.07052-2-32-2': 'We also calculated the parameters [MATH] and [MATH] for decays from initial atomic state [MATH], based on the [MATH] and [MATH] values in Table [REF] for the laser-on case (mixture of 50% [MATH] and 50% [MATH]) and laser-off case (pure [MATH]).', '1703.07052-2-32-3': 'The results together with the final results for the [MATH] initial states are summarized in Table [REF].', '1703.07052-2-32-4': 'The results for these two initial states are not significantly different from each other.', '1703.07052-2-33-0': 'Additionally, in measurements of the [MATH] correlation, where the TOF spectrum is fitted to templates generated by Monte Carlo simulations and the charge state groups are not completely separated in TOF, both the [MATH] and [MATH] parameters can affect the fit result.', '1703.07052-2-33-1': 'We have used a Monte Carlo simulations to study how the uncertainties of [MATH] and [MATH] translate into the uncertainty of the [MATH] correlation coefficient.', '1703.07052-2-33-2': 'Based on the experimental uncertainties listed in Table [REF], [MATH] results in a 0.3 relative uncertainty of [MATH], and [MATH] results in a 0.6 relative uncertainty.', '1703.07052-2-33-3': 'Therefore, the results from this paper are sufficient for an experiment aiming at determining [MATH] to 1.', '1703.07052-2-33-4': 'Because the charge-state analysis and the [MATH] analysis can use the same data set, the uncertainties of [MATH] and [MATH] will be improved as more data are taken to achieve better than 1 uncertainties on [MATH].', '1703.07052-2-34-0': '# Conclusions', '1703.07052-2-35-0': 'We have measured the [MATH] ion charge-state fractions for [MATH] decays from atomic metastable state [MATH].', '1703.07052-2-35-1': 'The overall fractions for [MATH] and [MATH] (Table [REF]) show small but significant disagreement with the recent theoretical calculation of Ref. [CITATION].', '1703.07052-2-35-2': 'We discuss a plausible explanation.', '1703.07052-2-36-0': 'We also point out that there is no satisfactory explanation for a large discrepancy between the same calculation and the results of Carlson et al.[CITATION] for the Li-ion energy dependence of the fractions from decays of the electronic ground state, suggesting either an unaccounted-for experimental issue or a failure of the framework for the calculation.', '1703.07052-2-37-0': 'The [MATH] and [MATH] parameters in the ion-energy dependent charge-state fraction expression (Eq. ([REF])) were also determined (Table. [REF]), and the precision is sufficient for a determination of [MATH] with 1% relative precision.', '1703.07052-2-37-1': 'The precisions of the [MATH] and [MATH] parameters can be improved as more data for the [MATH] correlation measurement are taken.', '1703.07052-2-38-0': 'This work is supported by the Department of Energy, Office of Nuclear Physics, under contract numbers DE-AC02-06CH11357 and DE-FG02-97ER41020.', '1703.07052-2-38-1': 'This work is also supported in part by the U.S. National Science Foundation under Grant No. PHY-11-02511.', '1703.07052-2-38-2': 'We thank Gordon Drake for many useful discussions.'}

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'1703.07052-2-6-4'], ['1703.07052-1-6-6', '1703.07052-2-6-6'], ['1703.07052-1-6-7', '1703.07052-2-6-7'], ['1703.07052-1-6-8', '1703.07052-2-6-8'], ['1703.07052-1-27-0', '1703.07052-2-27-0'], ['1703.07052-1-27-1', '1703.07052-2-27-1'], ['1703.07052-1-27-2', '1703.07052-2-27-2'], ['1703.07052-1-27-3', '1703.07052-2-27-3'], ['1703.07052-1-27-4', '1703.07052-2-27-4'], ['1703.07052-1-27-5', '1703.07052-2-27-5'], ['1703.07052-1-27-6', '1703.07052-2-27-6'], ['1703.07052-1-27-7', '1703.07052-2-27-7'], ['1703.07052-1-27-8', '1703.07052-2-27-8'], ['1703.07052-1-27-9', '1703.07052-2-27-9'], ['1703.07052-1-27-10', '1703.07052-2-27-10'], ['1703.07052-1-27-12', '1703.07052-2-27-12'], ['1703.07052-1-27-13', '1703.07052-2-27-13'], ['1703.07052-1-27-14', '1703.07052-2-27-14'], ['1703.07052-1-27-15', '1703.07052-2-27-15'], ['1703.07052-1-27-16', '1703.07052-2-27-16'], ['1703.07052-1-27-17', '1703.07052-2-27-17'], ['1703.07052-1-27-18', '1703.07052-2-27-18'], ['1703.07052-1-27-19', '1703.07052-2-27-19'], ['1703.07052-1-27-20', '1703.07052-2-27-20'], ['1703.07052-1-31-0', '1703.07052-2-31-0'], ['1703.07052-1-31-1', '1703.07052-2-31-1'], ['1703.07052-1-31-2', '1703.07052-2-31-2'], ['1703.07052-1-31-3', '1703.07052-2-31-3'], ['1703.07052-1-31-4', '1703.07052-2-31-4'], ['1703.07052-1-31-5', '1703.07052-2-31-5'], ['1703.07052-1-31-6', '1703.07052-2-31-6'], ['1703.07052-1-31-7', '1703.07052-2-31-7'], ['1703.07052-1-31-8', '1703.07052-2-31-8'], ['1703.07052-1-31-9', '1703.07052-2-31-9'], ['1703.07052-1-23-0', '1703.07052-2-23-0'], ['1703.07052-1-23-1', '1703.07052-2-23-1'], ['1703.07052-1-33-0', '1703.07052-2-33-0'], ['1703.07052-1-33-1', '1703.07052-2-33-1'], ['1703.07052-1-33-2', '1703.07052-2-33-2'], ['1703.07052-1-33-3', '1703.07052-2-33-3'], ['1703.07052-1-33-4', '1703.07052-2-33-4'], ['1703.07052-1-25-0', '1703.07052-2-25-0'], ['1703.07052-1-25-1', '1703.07052-2-25-1'], ['1703.07052-1-25-2', '1703.07052-2-25-2'], ['1703.07052-1-25-3', '1703.07052-2-25-3'], ['1703.07052-1-20-0', '1703.07052-2-20-0'], ['1703.07052-1-20-1', '1703.07052-2-20-1'], ['1703.07052-1-20-2', '1703.07052-2-20-2'], ['1703.07052-1-20-3', '1703.07052-2-20-3'], ['1703.07052-1-20-4', '1703.07052-2-20-4'], ['1703.07052-1-20-5', '1703.07052-2-20-5'], ['1703.07052-1-17-0', '1703.07052-2-17-0'], ['1703.07052-1-17-1', '1703.07052-2-17-1'], ['1703.07052-1-17-2', '1703.07052-2-17-2'], ['1703.07052-1-17-3', '1703.07052-2-17-3'], ['1703.07052-1-16-0', '1703.07052-2-16-0'], ['1703.07052-1-16-1', '1703.07052-2-16-1'], ['1703.07052-1-16-2', '1703.07052-2-16-2'], ['1703.07052-1-16-3', '1703.07052-2-16-3'], ['1703.07052-1-1-0', '1703.07052-2-1-0'], ['1703.07052-1-2-0', '1703.07052-2-2-0'], ['1703.07052-1-2-1', '1703.07052-2-2-1'], ['1703.07052-1-12-0', '1703.07052-2-12-0'], ['1703.07052-1-12-2', '1703.07052-2-12-2'], ['1703.07052-1-12-3', '1703.07052-2-12-3'], ['1703.07052-1-12-4', '1703.07052-2-12-4'], ['1703.07052-1-9-0', '1703.07052-2-9-0'], ['1703.07052-1-9-1', '1703.07052-2-9-1'], ['1703.07052-1-18-0', '1703.07052-2-18-0'], ['1703.07052-1-18-1', '1703.07052-2-18-1'], ['1703.07052-1-18-2', '1703.07052-2-18-2'], ['1703.07052-1-30-0', '1703.07052-2-30-0'], ['1703.07052-1-30-2', '1703.07052-2-30-2'], ['1703.07052-1-30-3', '1703.07052-2-30-3'], ['1703.07052-1-30-4', '1703.07052-2-30-4'], ['1703.07052-1-30-5', '1703.07052-2-30-5'], ['1703.07052-1-30-6', '1703.07052-2-30-6'], ['1703.07052-1-13-0', '1703.07052-2-13-0'], ['1703.07052-1-13-1', '1703.07052-2-13-1'], ['1703.07052-1-13-2', '1703.07052-2-13-2'], ['1703.07052-1-13-3', '1703.07052-2-13-3'], ['1703.07052-1-15-0', '1703.07052-2-15-0'], ['1703.07052-1-15-1', '1703.07052-2-15-1'], ['1703.07052-1-15-2', '1703.07052-2-15-2'], ['1703.07052-1-15-3', '1703.07052-2-15-3'], ['1703.07052-1-7-1', '1703.07052-2-7-1'], ['1703.07052-1-7-2', '1703.07052-2-7-2'], ['1703.07052-1-7-3', '1703.07052-2-7-3'], ['1703.07052-1-36-0', '1703.07052-2-36-0'], ['1703.07052-1-32-0', '1703.07052-2-32-0'], ['1703.07052-1-32-1', '1703.07052-2-32-1'], ['1703.07052-1-32-2', '1703.07052-2-32-2'], ['1703.07052-1-32-3', '1703.07052-2-32-3'], ['1703.07052-1-32-4', '1703.07052-2-32-4'], ['1703.07052-1-0-0', '1703.07052-2-0-0'], ['1703.07052-1-0-1', '1703.07052-2-0-1'], ['1703.07052-1-28-0', '1703.07052-2-28-0'], ['1703.07052-1-28-1', '1703.07052-2-28-1'], ['1703.07052-1-28-2', '1703.07052-2-28-2'], ['1703.07052-1-28-3', '1703.07052-2-28-3'], ['1703.07052-1-28-5', '1703.07052-2-28-5'], ['1703.07052-1-28-6', '1703.07052-2-28-6'], ['1703.07052-1-28-7', '1703.07052-2-28-7'], ['1703.07052-1-8-0', '1703.07052-2-8-0'], ['1703.07052-1-8-1', '1703.07052-2-8-1'], ['1703.07052-1-8-2', '1703.07052-2-8-2'], ['1703.07052-1-8-3', '1703.07052-2-8-3'], ['1703.07052-1-37-0', '1703.07052-2-37-0'], ['1703.07052-1-37-1', '1703.07052-2-37-1'], ['1703.07052-1-35-0', '1703.07052-2-35-0'], ['1703.07052-1-35-1', '1703.07052-2-35-1'], ['1703.07052-1-35-2', '1703.07052-2-35-2'], ['1703.07052-1-5-0', '1703.07052-2-5-0'], ['1703.07052-1-5-1', '1703.07052-2-5-1'], ['1703.07052-1-19-0', '1703.07052-2-19-0'], ['1703.07052-1-19-1', '1703.07052-2-19-1'], ['1703.07052-1-19-2', '1703.07052-2-19-2'], ['1703.07052-1-19-3', '1703.07052-2-19-3'], ['1703.07052-1-19-4', '1703.07052-2-19-4'], ['1703.07052-1-19-5', '1703.07052-2-19-5'], ['1703.07052-1-3-0', '1703.07052-2-3-0'], ['1703.07052-1-3-1', '1703.07052-2-3-1'], ['1703.07052-1-38-0', '1703.07052-2-38-0'], ['1703.07052-1-38-2', '1703.07052-2-38-2'], ['1703.07052-1-11-0', '1703.07052-2-11-0'], ['1703.07052-1-11-1', '1703.07052-2-11-1'], ['1703.07052-1-21-0', '1703.07052-2-21-0'], ['1703.07052-1-21-1', '1703.07052-2-21-1'], ['1703.07052-1-21-2', '1703.07052-2-21-2'], ['1703.07052-1-21-3', '1703.07052-2-21-3'], ['1703.07052-1-21-4', '1703.07052-2-21-4'], ['1703.07052-1-21-5', '1703.07052-2-21-5'], ['1703.07052-2-6-0', '1703.07052-3-3-0'], ['1703.07052-2-6-1', '1703.07052-3-3-1'], ['1703.07052-2-6-3', '1703.07052-3-3-3'], ['1703.07052-2-6-4', '1703.07052-3-3-4'], ['1703.07052-2-6-5', '1703.07052-3-3-5'], ['1703.07052-2-6-6', '1703.07052-3-3-6'], ['1703.07052-2-6-7', '1703.07052-3-3-7'], ['1703.07052-2-6-8', '1703.07052-3-3-8'], ['1703.07052-2-11-0', '1703.07052-3-8-0'], ['1703.07052-2-11-1', '1703.07052-3-8-1'], ['1703.07052-2-13-0', '1703.07052-3-10-0'], ['1703.07052-2-13-1', '1703.07052-3-10-1'], ['1703.07052-2-13-2', '1703.07052-3-10-2'], ['1703.07052-2-13-3', '1703.07052-3-10-3'], ['1703.07052-2-36-0', '1703.07052-3-33-0'], ['1703.07052-2-12-0', '1703.07052-3-9-0'], ['1703.07052-2-12-1', '1703.07052-3-9-1'], ['1703.07052-2-12-2', '1703.07052-3-9-2'], ['1703.07052-2-12-3', '1703.07052-3-9-3'], ['1703.07052-2-12-4', '1703.07052-3-9-4'], ['1703.07052-2-27-0', '1703.07052-3-24-0'], ['1703.07052-2-27-1', '1703.07052-3-24-1'], ['1703.07052-2-27-2', '1703.07052-3-24-2'], ['1703.07052-2-27-3', '1703.07052-3-24-3'], ['1703.07052-2-27-5', '1703.07052-3-24-5'], ['1703.07052-2-27-6', '1703.07052-3-24-6'], ['1703.07052-2-27-7', '1703.07052-3-24-7'], ['1703.07052-2-27-8', '1703.07052-3-24-8'], ['1703.07052-2-27-9', '1703.07052-3-24-9'], ['1703.07052-2-27-10', '1703.07052-3-24-10'], ['1703.07052-2-27-11', '1703.07052-3-24-11'], ['1703.07052-2-27-12', '1703.07052-3-24-12'], ['1703.07052-2-27-13', '1703.07052-3-24-13'], ['1703.07052-2-27-14', '1703.07052-3-24-14'], ['1703.07052-2-27-15', '1703.07052-3-24-15'], ['1703.07052-2-27-16', '1703.07052-3-24-16'], ['1703.07052-2-27-17', '1703.07052-3-24-17'], ['1703.07052-2-27-18', '1703.07052-3-24-18'], ['1703.07052-2-27-19', '1703.07052-3-24-19'], ['1703.07052-2-27-20', '1703.07052-3-24-20'], ['1703.07052-2-9-0', '1703.07052-3-6-0'], ['1703.07052-2-9-1', '1703.07052-3-6-1'], ['1703.07052-2-15-0', '1703.07052-3-12-0'], ['1703.07052-2-15-1', '1703.07052-3-12-1'], ['1703.07052-2-15-2', '1703.07052-3-12-2'], ['1703.07052-2-15-3', '1703.07052-3-12-3'], ['1703.07052-2-25-0', '1703.07052-3-22-0'], ['1703.07052-2-25-1', '1703.07052-3-22-1'], ['1703.07052-2-25-2', '1703.07052-3-22-2'], ['1703.07052-2-25-3', '1703.07052-3-22-3'], ['1703.07052-2-28-0', '1703.07052-3-25-0'], ['1703.07052-2-28-1', '1703.07052-3-25-1'], ['1703.07052-2-28-2', '1703.07052-3-25-2'], ['1703.07052-2-28-3', '1703.07052-3-25-3'], ['1703.07052-2-28-4', '1703.07052-3-25-4'], ['1703.07052-2-28-5', '1703.07052-3-25-5'], ['1703.07052-2-28-6', '1703.07052-3-25-6'], ['1703.07052-2-28-7', '1703.07052-3-25-7'], ['1703.07052-2-37-0', '1703.07052-3-34-0'], ['1703.07052-2-37-1', '1703.07052-3-34-1'], ['1703.07052-2-8-0', '1703.07052-3-5-0'], ['1703.07052-2-8-1', '1703.07052-3-5-1'], ['1703.07052-2-8-2', '1703.07052-3-5-2'], ['1703.07052-2-8-3', '1703.07052-3-5-3'], ['1703.07052-2-5-0', '1703.07052-3-2-0'], ['1703.07052-2-23-1', '1703.07052-3-20-1'], ['1703.07052-2-35-0', '1703.07052-3-32-0'], ['1703.07052-2-35-1', '1703.07052-3-32-1'], ['1703.07052-2-35-2', '1703.07052-3-32-2'], ['1703.07052-2-16-0', '1703.07052-3-13-0'], ['1703.07052-2-16-1', '1703.07052-3-13-1'], ['1703.07052-2-16-2', '1703.07052-3-13-2'], ['1703.07052-2-16-3', '1703.07052-3-13-3'], ['1703.07052-2-32-0', '1703.07052-3-29-0'], ['1703.07052-2-32-1', '1703.07052-3-29-1'], ['1703.07052-2-32-2', '1703.07052-3-29-2'], ['1703.07052-2-32-3', '1703.07052-3-29-3'], ['1703.07052-2-32-4', '1703.07052-3-29-4'], ['1703.07052-2-19-0', '1703.07052-3-16-0'], ['1703.07052-2-19-1', '1703.07052-3-16-1'], ['1703.07052-2-19-2', '1703.07052-3-16-2'], ['1703.07052-2-19-3', '1703.07052-3-16-3'], ['1703.07052-2-19-4', '1703.07052-3-16-4'], ['1703.07052-2-19-5', '1703.07052-3-16-5'], ['1703.07052-2-21-0', '1703.07052-3-18-0'], ['1703.07052-2-21-1', '1703.07052-3-18-1'], ['1703.07052-2-21-2', '1703.07052-3-18-2'], ['1703.07052-2-21-3', '1703.07052-3-18-3'], ['1703.07052-2-21-4', '1703.07052-3-18-4'], ['1703.07052-2-21-5', '1703.07052-3-18-5'], ['1703.07052-2-20-0', '1703.07052-3-17-0'], ['1703.07052-2-20-1', '1703.07052-3-17-1'], ['1703.07052-2-20-2', '1703.07052-3-17-2'], ['1703.07052-2-20-3', '1703.07052-3-17-3'], ['1703.07052-2-20-4', '1703.07052-3-17-4'], ['1703.07052-2-20-5', '1703.07052-3-17-5'], ['1703.07052-2-14-0', '1703.07052-3-11-0'], ['1703.07052-2-14-1', '1703.07052-3-11-1'], ['1703.07052-2-26-0', '1703.07052-3-23-0'], ['1703.07052-2-26-1', '1703.07052-3-23-1'], ['1703.07052-2-26-2', '1703.07052-3-23-2'], ['1703.07052-2-26-3', '1703.07052-3-23-3'], ['1703.07052-2-26-4', '1703.07052-3-23-4'], ['1703.07052-2-26-5', '1703.07052-3-23-5'], ['1703.07052-2-24-0', '1703.07052-3-21-0'], ['1703.07052-2-24-1', '1703.07052-3-21-1'], ['1703.07052-2-24-2', '1703.07052-3-21-2'], ['1703.07052-2-24-3', '1703.07052-3-21-3'], ['1703.07052-2-24-4', '1703.07052-3-21-4'], ['1703.07052-2-24-5', '1703.07052-3-21-5'], ['1703.07052-2-24-6', '1703.07052-3-21-6'], ['1703.07052-2-24-7', '1703.07052-3-21-7'], ['1703.07052-2-24-8', '1703.07052-3-21-8'], ['1703.07052-2-24-9', '1703.07052-3-21-9'], ['1703.07052-2-17-0', '1703.07052-3-14-0'], ['1703.07052-2-17-1', '1703.07052-3-14-1'], ['1703.07052-2-17-2', '1703.07052-3-14-2'], 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['1703.07052-2-17-3', '1703.07052-3-14-3'], ['1703.07052-2-33-0', '1703.07052-3-30-0'], ['1703.07052-2-33-1', '1703.07052-3-30-1'], ['1703.07052-2-33-2', '1703.07052-3-30-2'], ['1703.07052-2-33-3', '1703.07052-3-30-3'], ['1703.07052-2-33-4', '1703.07052-3-30-4'], ['1703.07052-2-31-0', '1703.07052-3-28-0'], ['1703.07052-2-31-1', '1703.07052-3-28-1'], ['1703.07052-2-31-2', '1703.07052-3-28-2'], ['1703.07052-2-31-3', '1703.07052-3-28-3'], ['1703.07052-2-31-4', '1703.07052-3-28-4'], ['1703.07052-2-31-5', '1703.07052-3-28-5'], ['1703.07052-2-31-6', '1703.07052-3-28-6'], ['1703.07052-2-31-7', '1703.07052-3-28-7'], ['1703.07052-2-31-8', '1703.07052-3-28-8'], ['1703.07052-2-31-9', '1703.07052-3-28-9'], ['1703.07052-2-30-0', '1703.07052-3-27-0'], ['1703.07052-2-30-2', '1703.07052-3-27-2'], ['1703.07052-2-30-3', '1703.07052-3-27-3'], ['1703.07052-2-30-4', '1703.07052-3-27-4'], ['1703.07052-2-30-5', '1703.07052-3-27-5'], ['1703.07052-2-30-6', '1703.07052-3-27-6'], ['1703.07052-2-7-1', '1703.07052-3-4-5'], ['1703.07052-2-7-2', '1703.07052-3-4-6'], ['1703.07052-2-7-3', '1703.07052-3-4-7'], ['1703.07052-2-38-0', '1703.07052-3-35-0'], ['1703.07052-2-38-2', '1703.07052-3-35-2'], ['1703.07052-2-18-0', '1703.07052-3-15-0'], ['1703.07052-2-18-1', '1703.07052-3-15-1'], ['1703.07052-2-18-2', '1703.07052-3-15-2'], ['1703.07052-2-0-1', '1703.07052-3-0-1'], ['1703.07052-2-2-1', '1703.07052-3-0-5']]

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[]

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[]

['1703.07052-1-38-1', '1703.07052-2-38-1', '1703.07052-3-35-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1703.07052

{'1703.07052-3-0-0': 'The accurate determination of atomic final states following nuclear [MATH] decay plays an important role in several experiments.', '1703.07052-3-0-1': 'In particular, the charge state distributions of ions following nuclear [MATH] decay are important for determinations of the [MATH] angular correlation with improved precision.', '1703.07052-3-0-2': 'Beyond the hydrogenic cases, the decay of neutral [MATH] presents the simplest case.', '1703.07052-3-0-3': 'Our measurement aims at providing benchmarks to test theoretical calculations.', '1703.07052-3-0-4': 'The kinematics of Li[MATH] ions produced following the [MATH] decay of [MATH] within an electric field were measured using [MATH] atoms in the metastable [MATH] and in the [MATH] states confined by a magneto-optical trap.', '1703.07052-3-0-5': 'The electron shake-off probabilities were deduced including their dependence on ion energy.', '1703.07052-3-0-6': 'We find significant discrepancies on the fractions of Li ions in the different charge states with respect to a recent calculation.', '1703.07052-3-1-0': '# Introduction', '1703.07052-3-2-0': 'Atomic and molecular degrees of freedom can play an important role in precision nuclear beta-decay experiments.', '1703.07052-3-2-1': 'In nuclear beta decays, two energetic leptons (an electron and an anti-neutrino) are emitted, while the daughter nucleus recoils.', '1703.07052-3-2-2': 'The nucleus is usually in an atom or molecule, and the sudden change of its charge and its recoiling motion may cause electron excitations, shake-offs and molecular excitations.', '1703.07052-3-2-3': 'The final state of the recoil ion affects the shape of the [MATH]-energy spectrum.', '1703.07052-3-2-4': 'For example, in measurements of the [MATH]-energy spectrum near the end-point from molecular tritium, the final electronic state distribution [CITATION] can affect the determination of the mass of anti-neutrinos.', '1703.07052-3-2-5': 'The helicity properties of the weak interaction imply correlations between the momenta of the outgoing particles [CITATION].', '1703.07052-3-2-6': 'Thus, precise measurements of the [MATH] angular correlation can be used to search for new interactions[CITATION].', '1703.07052-3-2-7': 'In such experiments the [MATH] angular correlation coefficient is deduced from the kinematics of the recoil ion, which can depend on the molecular binding[CITATION].', '1703.07052-3-3-0': 'The beta decay of [MATH] presents a good opportunity to determine the [MATH] correlation.', '1703.07052-3-3-1': 'Because of the large endpoint and the relatively light mass of the nucleus, the [MATH] correlation has a significant effect on the kinematics of the charged particles from the decay.', '1703.07052-3-3-2': 'A measurement performed in 1963 [CITATION] was one of several landmark experiments that determined the [MATH] nature of the weak interaction: the charged weak currents are of vector and axial vector type.', '1703.07052-3-3-3': 'Fundamental measurements of this kind have renewed interest in the context of searching for hints of new physics as deviations from the expectations based on the Standard Model [CITATION].', '1703.07052-3-3-4': 'An ongoing experiment is aiming at a measurement of the [MATH] angular correlation in [MATH] decay with improved precision [CITATION].', '1703.07052-3-3-5': 'In this experiment, the momentum of the recoil ion emitted from a cold and dilute cloud of laser-cooled [MATH] atoms confined in a magneto-optical trap (MOT) [CITATION] is determined through a full kinematics reconstruction in a strong electric field.', '1703.07052-3-3-6': 'Due to the sudden change in nuclear charge, the electrons do not always find the corresponding orbits in the Li atom and can be shaken off.', '1703.07052-3-3-7': 'Thus, the [MATH] ion can have electric charges between [MATH] and [MATH].', '1703.07052-3-3-8': 'The fraction in a given charge state not only depends on the overlap of the initial electronic wave function and the final continuum states, but also on the ion energy, so a proper extraction of the [MATH] correlation coefficient requires understanding the shake-off effect.', '1703.07052-3-4-0': 'Quantitative comparisons of charge distributions have been presented for two heavier systems, [MATH]Ar [CITATION] and [MATH]Na [CITATION].', '1703.07052-3-4-1': 'The calculations in [MATH]Na did not take into account a potentially important cancellation factor in the recoil-energy-dependent fraction that we describe below.', '1703.07052-3-4-2': 'The comparisons in [MATH]Ar showed agreement at the 1% level.', '1703.07052-3-4-3': 'As we indicate below the [MATH]He system invites for higher precision comparisons, because the electronic wave functions for helium can be calculated with high accuracy.', '1703.07052-3-4-4': 'The case of [MATH] also presents a nice benchmark to test aspects of the calculations that are relevant for other problems, like the role of electron-electron interactions, the use of the sudden approximation, and methods for calculating charge distributions after the shake-off process.', '1703.07052-3-4-5': 'Several calculations have been performed for [MATH] [CITATION].', '1703.07052-3-4-6': 'A confirmation of the calculated fraction of [MATH] was recently performed with the hydrogen-like system of [MATH] ions [CITATION].', '1703.07052-3-4-7': 'Here, we address the case of the [MATH] neutral atom, which presents additional ingredients associated with the two electrons.', '1703.07052-3-5-0': 'The charge distribution of Li ions from the decay of [MATH] in its electronic ground state was measured by Carlson et al. [CITATION].', '1703.07052-3-5-1': 'Table [REF] shows a comparison to the most recent calculation of Schulhoff and Drake [CITATION].', '1703.07052-3-5-2': 'As can be observed, there are significant discrepancies for Li-ion fractions in the different charge states.', '1703.07052-3-5-3': 'The main aim of the calculations in Ref. [CITATION] was to study the dependence of charge distribution on the Li-ion energy, but a clear prediction is also given for the overall fractions in different charge states from [MATH], both in its ground and metastable states.', '1703.07052-3-6-0': 'We report here the first measurement of the electron emission probabilities following the [MATH] decay of [MATH] atoms confined via a magneto-optical trap working between the metastable [MATH] and the [MATH] states.', '1703.07052-3-6-1': 'We present data with both the trapping lasers off (pure [MATH]) and on (approximate 50%/50% mixture of [MATH] and [MATH]) and compare with the calculations of Schulhoff and Drake [CITATION] assuming decay from the [MATH] state.', '1703.07052-3-7-0': '# Experimental method', '1703.07052-3-8-0': 'The [MATH] atoms were produced via the [MATH] reaction.', '1703.07052-3-8-1': 'Up to 2[MATH] atoms per second were produced by bombarding a lithium target with an 18 MeV deuteron beam, delivered by the tandem Van de Graaff accelerator at the University of Washington [CITATION].', '1703.07052-3-9-0': 'The [MATH] atoms were pumped into a RF-discharge tube where a fraction ([MATH]) of the [MATH]He atoms were brought to the [MATH] metastable atomic state.', '1703.07052-3-9-1': 'The forward-going metastable atoms were then transversely-cooled, slowed by a Zeeman slower, and trapped in a magneto-optical trap, all based on resonant excitation of the [MATH] to [MATH] electronic transition via 1083 nm laser light.', '1703.07052-3-9-2': 'Due to the small efficiency for pumping [MATH] atoms to the metastable state there was a considerable amount of ground-state [MATH] in the chamber hosting the MOT.', '1703.07052-3-9-3': 'To reduce events from non-trapped [MATH] atoms, the atoms in the MOT were periodically pushed by a laser beam into a measurement chamber through a 5 mm-diameter 30-mm-long aperture tube for differential pumping and recaptured by a second MOT.', '1703.07052-3-9-4': 'This [MATH] trap contained over 2100 atoms on average.', '1703.07052-3-10-0': 'During the measurement, the trapping lasers of this MOT were alternatively switched on and off with a [MATH] duty cycle and a period of 100 [MATH]s.', '1703.07052-3-10-1': 'At the beginning of each off-cycle of 50 [MATH]s, the [MATH]50 fraction of the atoms previously excited to the [MATH] level quickly decay back to the [MATH] level within the 100 ns lifetime of the excited state.', '1703.07052-3-10-2': 'Time correlation of the decay events with the switching cycle thus allowed us to isolate decays from [MATH]He purely in the [MATH] state (laser off) from decays with an [MATH]50 admixture of the [MATH] excited state (laser on).', '1703.07052-3-10-3': 'Meanwhile, the fast switching provided sufficient confinement of the atom cloud.', '1703.07052-3-11-0': 'We studied the charge-state distributions of the recoil [MATH] ions by analyzing their time-of-flight (TOF) and energy spectra, which required detecting the [MATH] particle and the recoil ion in coincidence.', '1703.07052-3-11-1': 'The configuration of the detection chamber is shown in Fig. [REF].', '1703.07052-3-12-0': 'A [MATH]-telescope, consisting of a multi-wire proportional chamber (MWPC) and a scintillation detector, is placed above the trap.', '1703.07052-3-12-1': 'The scintillation detector measures the energy of the [MATH] particle ([MATH]).', '1703.07052-3-12-2': 'The MWPC detects the entrance position of the particles and strongly suppresses [MATH]-ray backgrounds triggering the scintillator when applying an appropriate coincidence gate between the two [MATH] detectors.', '1703.07052-3-12-3': 'The MWPC runs with 1 atm [MATH] by volume) gas and is separated from the MOT vacuum by a [MATH] thick 3.81-cm diameter beryllium window.', '1703.07052-3-13-0': 'A Micro Channel Plate (MCP) detector [CITATION] is placed below the trap for detecting recoil ions and determining their hit positions with a resolution of 190 [MATH]m (FWHM).', '1703.07052-3-13-1': 'Electrodes are installed in-between the [MATH]-telescope and the MCP detector to create an electric field of [MATH] and accelerate the recoil ions emitted from the trap towards the MCP detector.', '1703.07052-3-13-2': 'This enhances the ion-collection solid angle so that [MATH] of the [MATH] and 100 of the [MATH] and [MATH] ions are collected within the 75 mm diameter active area of the MCP.', '1703.07052-3-13-3': 'It also allows for the ions to have enough energy to trigger the MCP detector with a maximal detection efficiency ([MATH]50), independent of their initial charge state and recoil energy.', '1703.07052-3-14-0': 'The TOF measurement of the recoil ion is started by a scintillator signal and stopped by a MCP signal.', '1703.07052-3-14-1': 'The resolution of the TOF measurement is 820 ps (FWHM).', '1703.07052-3-14-2': '[MATH] ions in different charge states have different accelerations in the applied electric field, and are thus partially separated in TOF as shown in Fig. [REF].', '1703.07052-3-14-3': 'The overlaps between charge states can be avoided by applying a high enough threshold to the [MATH]-energy as indicated by the horizontal dashed lines.', '1703.07052-3-15-0': 'To fully reconstruct the initial energy of the recoil ion, the MOT position must also be determined.', '1703.07052-3-15-1': 'We ionize the trapped [MATH]He atoms periodically (at [MATH]20 Hz) using a 2-ns pulsed beam of ultraviolet (337 nm) nitrogen laser that has sufficient photon energy to photo-ionize the [MATH] state but not the [MATH] state.', '1703.07052-3-15-2': 'The vertical coordinate (along the electric field direction) of the MOT is determined through the TOF of the photo-ions with respect to the laser pulse.', '1703.07052-3-16-0': 'A unique feature of the trapping of metastable [MATH] atoms is that it allows for monitoring the horizontal shape of the MOT via "Penning-ions".', '1703.07052-3-16-1': 'The latter are generated by collisions between neutral atoms in the non-perfect vacuum and metastable [MATH] atoms.', '1703.07052-3-16-2': 'The uncertainty of the start position of each [MATH] recoil ion is limited in this data set by the size of the MOT which is 1.4 mm (FWHM).', '1703.07052-3-16-3': 'With the precisely measured MOT position, electric field strength, recoil-ion TOF and hit positions, the initial momenta of recoil ions are determined.', '1703.07052-3-16-4': 'The detector responses are calibrated daily, and the electric field and MOT position are monitored while the data are taken.', '1703.07052-3-16-5': 'The details of the construction, calibration and performance of the detector system are described in Ref. [CITATION].', '1703.07052-3-17-0': 'Decays from non-trapped [MATH]He atoms in the decay chamber and scattering of [MATH] particles prior to their detection generate an undesired background.', '1703.07052-3-17-1': 'To suppress this background, we reconstructed the momentum of the emitted anti-neutrino using the measured [MATH]-momentum and ion-momentum, and evaluated the [MATH]-value (the total kinetic energy released from the decay), which should be 3.5 MeV.', '1703.07052-3-17-2': 'Decays outside the MOT yield incorrect [MATH]-values because the reconstruction assumes the events originate at the MOT.', '1703.07052-3-17-3': 'Our [MATH]-value cut ([MATH]-cut) accepted events with [MATH], so that [MATH]% of the events from non-trapped decays were eliminated while the total data loss for events from trapped decays was less than 0.1% [CITATION].', '1703.07052-3-17-4': 'The width of this cut was limited by the energy resolution in our scintillator.', '1703.07052-3-17-5': 'In order to determine the contribution of the remaining events from non-trapped decays, we introduced large amounts of ground-state [MATH] atoms via a bypass pipe.', '1703.07052-3-18-0': 'Fig. [REF] shows a comparison of the TOF spectra.', '1703.07052-3-18-1': 'In Fig. [REF] the data from the non-trapped [MATH] is normalized and overlaid with the data taken with the trap on.', '1703.07052-3-18-2': 'The shortest TOF for a [MATH] ion emitted from the MOT is [MATH] ns which is the TOF leading edge for [MATH].', '1703.07052-3-18-3': 'Events with TOF[MATH] must be from non-trapped [MATH]He decays occurring closer to the MCP.', '1703.07052-3-18-4': 'Therefore, the normalization was chosen to match the TOF spectra without the [MATH]-cut in the TOF region from 10 ns to 110 ns.', '1703.07052-3-18-5': 'The same normalization factor was applied to the TOF and ion-energy spectra after the [MATH]-cut (shown in Fig. [REF]) and then these spectra were subtracted from those from the corresponding data run to remove the remaining events from untrapped [MATH] atoms.', '1703.07052-3-19-0': '# Data Analysis and Results', '1703.07052-3-20-0': 'As apparent from the data shown in Fig. [REF] and Fig. [REF], we clearly observe some shake-off fraction yielding [MATH] ions, however, we do not observe [MATH] ions above background and can therefore only set an upper limit.', '1703.07052-3-20-1': 'In the following, we will treat these two cases separately.', '1703.07052-3-21-0': 'We first address the total fraction of [MATH] ions ([MATH]), regardless of their energies.', '1703.07052-3-21-1': '[MATH] is the ratio [MATH] , where [MATH] is the number of observed [MATH] ions, and [MATH] is the number of observed [MATH] ions in all charge states.', '1703.07052-3-21-2': 'It is important to choose an appropriate [MATH] threshold to ensure a correct determination of [MATH].', '1703.07052-3-21-3': 'As shown in Fig. [REF], some of the [MATH] events with small [MATH] are lost because they have too high transverse momentum to hit the active area of the MCP.', '1703.07052-3-21-4': 'Therefore, the [MATH] threshold should be set high enough so that [MATH] ions in all charge states above the [MATH] threshold fall in the active area of the MCP.', '1703.07052-3-21-5': 'We chose 1 MeV as the [MATH] threshold, and under this condition the event loss due to the finite size of the MCP fiducial area is then less than 0.1% according to the Monte Carlo simulation.', '1703.07052-3-21-6': 'However, an [MATH] threshold at 1 MeV is not high enough to separate [MATH] from [MATH] ions.', '1703.07052-3-21-7': 'Therefore, to obtain the correct [MATH], we determined the number of events below the leading edge ([MATH]) of [MATH] TOF spectrum, and then corrected the [MATH] counts to include the counts overlapping with [MATH] events using a Monte Carlo simulation.', '1703.07052-3-21-8': 'Given the [MATH] threshold at 1 MeV, the percentage of [MATH] ions that are below [MATH] over all [MATH] ions is 85.5.', '1703.07052-3-21-9': '(Alternatively, it is also possible to choose an [MATH] threshold at 2.1 MeV so that the [MATH] ions are completely separated from the [MATH] ions, but such a high threshold rules out [MATH]75 of the data, yielding poor statistics so it was not adopted.)', '1703.07052-3-22-0': 'The TOF spectrum with the [MATH]-cut for the laser-off data is plotted in Fig. [REF].', '1703.07052-3-22-1': 'The event counts in the [MATH] TOF region (TOF[MATH]) are dominated by the background generated by the non-trapped [MATH]He decays.', '1703.07052-3-22-2': 'A summary of results is shown in Table. [REF].', '1703.07052-3-22-3': 'The measurements are consistent with [MATH] within 1 standard deviations, and the 90% confidence levels are calculated only in the physical region where the count of events are greater than 0.', '1703.07052-3-23-0': 'Next, we studied the fraction of [MATH] ions and its dependence on the recoil-ion energy.', '1703.07052-3-23-1': 'In this study, the initial energy of the recoil ions ([MATH]) are reconstructed for each event, so it is necessary to determine the charge state of each ion with no ambiguity.', '1703.07052-3-23-2': 'Therefore, the minimal threshold on [MATH], 1.5 MeV, was applied so that [MATH] ions are completely separated from [MATH] in TOF as shown by the lower of the two horizontal red dashed lines in Fig. [REF].', '1703.07052-3-23-3': 'As discussed above, the upper limit for the probability of having a [MATH] ion is at the [MATH] level, approximately three orders of magnitude lower than the probability of having a [MATH] ion.', '1703.07052-3-23-4': 'Therefore, we neglected [MATH] ions in this study assuming that all events below the leading edge ([MATH] ns) of [MATH] TOF spectrum correspond to [MATH] ions.', '1703.07052-3-23-5': 'The [MATH] spectra for [MATH] and [MATH] with their corresponding normalized backgrounds are plotted in Fig. [REF].', '1703.07052-3-24-0': 'Theoretically, the probability distribution functions (PDFs) of [MATH] in the two charge states are [EQUATION] where [MATH] indicates the charge state, and [MATH] is the probability of having an ion in such charge state.', '1703.07052-3-24-1': '[MATH] represents the part of the probability distribution function that is independent of charge state and depends only on the [MATH]-decay dynamics, i.e. the [MATH] correlation coefficient [MATH].', '1703.07052-3-24-2': 'The detection efficiency, [MATH], depends on the detection geometry, detector response functions and event reconstruction parameters and conditions.', '1703.07052-3-24-3': 'The expected energy dependence for [MATH] is [CITATION]: [EQUATION] where [MATH] and [MATH] are parameters.', '1703.07052-3-24-4': 'Taking the ratio of [MATH] and [MATH], one gets [EQUATION]', '1703.07052-3-24-5': 'The [MATH] functions in the numerator and the denominator cancel each other and [MATH] no longer depends on [MATH] explicitly.', '1703.07052-3-24-6': 'With the [MATH] keV threshold there is no event loss due to MCP fiducial area for both charge states [MATH] and [MATH].', '1703.07052-3-24-7': 'In principle, the ion detection efficiency [MATH] depends on the charge state because [MATH] and [MATH] have very different final energies ([MATH]13 keV versus [MATH]26 keV), and their position distributions on the MCP are also different.', '1703.07052-3-24-8': 'However, the difference in gain of the MCP for these two charge states is only 2.5%.', '1703.07052-3-24-9': 'Due to a very low MCP charge threshold, it results in less than [MATH] efficiency difference.', '1703.07052-3-24-10': 'Therefore, we made the assumption that [MATH], so they cancel each other in Eq. ([REF]).', '1703.07052-3-24-11': '(The systematic uncertainty generated by this approximation was studied using Monte Carlo simulations including the measured 0.82% (RMS) variation of the MCP efficiency over the surface of the MCP.)', '1703.07052-3-24-12': 'Because [MATH] ions are neglected, the sum of [MATH] and [MATH] is 1, and thus Eq. ([REF]) becomes [EQUATION]', '1703.07052-3-24-13': 'We take the ratio of the measured [MATH] spectra (with background subtraction applied) for the charge states [MATH] and [MATH], and fit the ratio histogram to Eq. ([REF]), as shown in Fig. [REF].', '1703.07052-3-24-14': 'The fit region is chosen to be from 0.1 keV to 1.1 keV, in order to make the systematic uncertainties introduced by the assumptions made above negligible compared to the statistical uncertainty.', '1703.07052-3-24-15': '[MATH] and [MATH] are fit parameters, and the fit results are listed in Table [REF].', '1703.07052-3-24-16': 'We also obtain the fractions of [MATH] and [MATH] regardless of their energies by counting the total number of events above and below [MATH].', '1703.07052-3-24-17': 'The results of the fractions for all three charge states of [MATH] ions are summarized and compared to theoretical calculations [CITATION] in Table [REF].', '1703.07052-3-24-18': 'All the fractions determined in this experiment are obtained with an [MATH] threshold.', '1703.07052-3-24-19': 'Applying an [MATH] threshold, which is inevitable in this experiment, changes the energy spectrum of the recoil ions, and thus affects the fractions of [MATH] ions in different charge states due to their dependences on the ion energies.', '1703.07052-3-24-20': 'Therefore, in Table [REF] the theoretical fractions are calculated based on the results of Ref. [CITATION] and applied with the same [MATH] threshold as in our experiment.', '1703.07052-3-25-0': 'In order to understand the systematic shifts of [MATH] and [MATH] caused by approximations like assuming [MATH] and [MATH] to be identical, we ran Monte Carlo simulations with the values of [MATH] and [MATH] from Ref. [CITATION].', '1703.07052-3-25-1': 'In the simulations, we modeled the experimental parameters, such as the detector geometry and response functions, and the efficiencies and electric field, as close as possible to the experimental setup.', '1703.07052-3-25-2': 'The simulated data were processed in the same way as that used for the experimental data.', '1703.07052-3-25-3': 'There is no significant deviation of the extracted [MATH] value from its input value, while the extracted [MATH] deviates from its input by [MATH].', '1703.07052-3-25-4': 'Implementation of the 0.82% spatial variation of the MCP efficiency does not result in significant deviations of [MATH] and [MATH].', '1703.07052-3-25-5': 'We also studied how the fit values of [MATH] and [MATH] change with respect to parameters used in the ion-energy reconstruction, and the systematic uncertainties associated with these parameters.', '1703.07052-3-25-6': 'The corresponding non-negligible systematic uncertainties of [MATH] and [MATH] are listed in Table [REF], and in total are smaller than the statistical uncertainties listed in Table [REF].', '1703.07052-3-25-7': 'Note that these systematic uncertainties are all related to the ion-energy calculation, so they do not affect the determination of the energy-integrated charge-state fraction listed in Table [REF].', '1703.07052-3-26-0': '# Discussion', '1703.07052-3-27-0': 'A comparison of our measurements to calculations for the decay of [MATH] from its atomic metastable state is shown in Table [REF].', '1703.07052-3-27-1': 'The [MATH] and [MATH] fractions measured in this experiment have small ([MATH]%) but significant ([MATH] for [MATH] and [MATH]) discrepancies with the theoretical calculations of Ref. [CITATION].', '1703.07052-3-27-2': 'The calculation for the [MATH] decays from the atomic ground state similarly over-predicts the [MATH] fraction measured by Carlson et al. [CITATION] as shown in Table [REF].', '1703.07052-3-27-3': 'It is possible that there is a missing consideration in the calculation that systematically leads to a higher [MATH] fraction for both initial atomic states.', '1703.07052-3-27-4': 'In Ref. [CITATION] the charge state of the [MATH] ion is determined solely by the final energy of the two orbital electrons, [MATH], with respect to the ionization energies for 1 and 2 electrons, [MATH] and [MATH]: [EQUATION]', '1703.07052-3-27-5': 'In considering possible sources for the discrepancy we note, for example, that the condition [MATH] could be met without double ionization if one of the electrons takes away a significant fraction of the energy as kinetic energy, leaving the other electron bound.', '1703.07052-3-27-6': 'This could lead to the systematic overestimation of the [MATH] probability as observed.', '1703.07052-3-28-0': 'In contrast to the integrated charge-state fractions, the ion-energy dependencies of the [MATH] ion charge-state fractions (the [MATH] parameters) are of concern for the determinations of the [MATH] correlation coefficient.', '1703.07052-3-28-1': 'The experiment of Ref. [CITATION] was performed with the same method and apparatus used in Ref. [CITATION] to determine the [MATH] correlation coefficient.', '1703.07052-3-28-2': 'As shown in Table [REF], there is a significant discrepancy between the [MATH] parameters measured by Carlson et al. and the ones calculated in Ref. [CITATION].', '1703.07052-3-28-3': 'While we find a plausible explanation for some of the differences between theory and experiment for the overall Li-ion fractions, as stated above, it is more difficult to understand how the [MATH] factors could be a factor of [MATH] smaller in the calculations.', '1703.07052-3-28-4': 'The [MATH] values measured by Carlson et al. are close to a naive prediction ignoring a cancellation that takes place between [MATH] and [MATH] final state configurations shown in detail in Ref. [CITATION].', '1703.07052-3-28-5': 'Johnson et al. were aware of the cancellation and the fact that their measured [MATH] factors disagreed with the more accurate calculation [CITATION], but at the time they reported concerns about the difficulty of including a large enough set of states in their calculation.', '1703.07052-3-28-6': 'There is an implicit suggestion that the discrepancy should not be taken seriously because the calculation was incomplete.', '1703.07052-3-28-7': 'No such concerns exist for the recent calculation of Schulhoff and Drake [CITATION], so we conclude that this suggests either an unaccounted-for experimental issue or a failure of the framework for the calculation.', '1703.07052-3-28-8': 'Changing the [MATH] parameters from their measured value to zero affects the determination of [MATH] only by [MATH]% which is smaller than the [MATH]% uncertainty claimed by Ref. [CITATION].', '1703.07052-3-28-9': 'In the present context of trying to achieve more precise determinations, the issue is more important.', '1703.07052-3-29-0': 'Our results for the [MATH] and [MATH] parameters for the [MATH]He decays from the atomic metastable state and the corresponding theoretical calculations are shown in Table [REF].', '1703.07052-3-29-1': "Unfortunately, our results don't have the statistical power to claim a precision test of the calculations, in particular for the [MATH] parameter.", '1703.07052-3-29-2': 'We also calculated the parameters [MATH] and [MATH] for decays from initial atomic state [MATH], based on the [MATH] and [MATH] values in Table [REF] for the laser-on case (mixture of 50% [MATH] and 50% [MATH]) and laser-off case (pure [MATH]).', '1703.07052-3-29-3': 'The results together with the final results for the [MATH] initial states are summarized in Table [REF].', '1703.07052-3-29-4': 'The results for these two initial states are not significantly different from each other.', '1703.07052-3-30-0': 'Additionally, in measurements of the [MATH] correlation, where the TOF spectrum is fitted to templates generated by Monte Carlo simulations and the charge state groups are not completely separated in TOF, both the [MATH] and [MATH] parameters can affect the fit result.', '1703.07052-3-30-1': 'We have used a Monte Carlo simulations to study how the uncertainties of [MATH] and [MATH] translate into the uncertainty of the [MATH] correlation coefficient.', '1703.07052-3-30-2': 'Based on the experimental uncertainties listed in Table [REF], [MATH] results in a 0.3 relative uncertainty of [MATH], and [MATH] results in a 0.6 relative uncertainty.', '1703.07052-3-30-3': 'Therefore, the results from this paper are sufficient for an experiment aiming at determining [MATH] to 1.', '1703.07052-3-30-4': 'Because the charge-state analysis and the [MATH] analysis can use the same data set, the uncertainties of [MATH] and [MATH] will be improved as more data are taken to achieve better than 1 uncertainties on [MATH].', '1703.07052-3-31-0': '# Conclusions', '1703.07052-3-32-0': 'We have measured the [MATH] ion charge-state fractions for [MATH] decays from atomic metastable state [MATH].', '1703.07052-3-32-1': 'The overall fractions for [MATH] and [MATH] (Table [REF]) show small but significant disagreement with the recent theoretical calculation of Ref. [CITATION].', '1703.07052-3-32-2': 'We discuss a plausible explanation.', '1703.07052-3-33-0': 'We also point out that there is no satisfactory explanation for a large discrepancy between the same calculation and the results of Carlson et al.[CITATION] for the Li-ion energy dependence of the fractions from decays of the electronic ground state, suggesting either an unaccounted-for experimental issue or a failure of the framework for the calculation.', '1703.07052-3-34-0': 'The [MATH] and [MATH] parameters in the ion-energy dependent charge-state fraction expression (Eq. ([REF])) were also determined (Table. [REF]), and the precision is sufficient for a determination of [MATH] with 1% relative precision.', '1703.07052-3-34-1': 'The precisions of the [MATH] and [MATH] parameters can be improved as more data for the [MATH] correlation measurement are taken.', '1703.07052-3-35-0': 'This work is supported by the Department of Energy, Office of Nuclear Physics, under contract numbers DE-AC02-06CH11357 and DE-FG02-97ER41020.', '1703.07052-3-35-1': 'This work is also supported in part by the U.S. National Science Foundation under Grant No. PHY-11-02511.', '1703.07052-3-35-2': 'We thank Gordon Drake for many useful discussions.'}

1606.05823

{'1606.05823-1-0-0': 'We study a system of two Higgs bound state, interacting via a real scalar Dark Matter mediating field, without imposing [MATH] symmetry on the DM sector of the postulated Lagrangian.', '1606.05823-1-0-1': 'The variational method in the Hamiltonian formalism of QFT is used to derive relativistic wave equations for the two-Higgs system, using a truncated Fock-space trial state.', '1606.05823-1-0-2': 'Approximate solutions of the 2-body relativistic coupled integral equations are presented, and conditions for the existence of Higgs bound states is examined in a broad parameter space of DM mass and coupling constants.', '1606.05823-1-1-0': '# Introduction', '1606.05823-1-2-0': 'Astrophysical observations of gravitational effects at all length scales of the universe, including our Milky Way galaxy, indicate the existence of Dark Matter (DM) [CITATION].', '1606.05823-1-2-1': 'To date, DM has been observed only through its gravitational influence such as rotation curves of galaxies, mismatch of estimated mass and luminous matter in galaxy clusters, lensing of galactic Supernovae of Type Ia and Cosmic Microwave Background images, etc.', '1606.05823-1-2-2': "A major open question is the nature of DM, since to date, there are no direct measurements of DM's electroweak, or any non-gravitational, interactions.", '1606.05823-1-3-0': 'One conjecture is that DM is made up non-baryonic particles called WIMPs (Weakly Interacting Massive Particles).', '1606.05823-1-3-1': 'Experimental efforts at detecting DM, including WIMPs, are ongoing [CITATION].', '1606.05823-1-3-2': 'The literature on this subject is extensive.', '1606.05823-1-3-3': 'A brief, useful overview of DM and efforts to detect them is given in ref. [CITATION].', '1606.05823-1-3-4': 'An outstanding question is how DM interacts with Standard Model particles, including the Higgs particles.', '1606.05823-1-3-5': 'There have been a number of theoretical models that address this problem (see, for example, references [CITATION] and citations therein).', '1606.05823-1-3-6': 'Recently, Petersson et al. have proposed a Supersymmetric generalization of the Standard Model in which the Higgs boson can disintegrate into a photon and DM particles [CITATION].', '1606.05823-1-3-7': 'The possibility of Higgs bound states ("Higgsonium") has been examined well before the experimental detection of the Higgs boson (cf. references [CITATION] to [CITATION] ).', '1606.05823-1-3-8': 'In these prior studies, the domain of the Higgs mass and coupling strength for which Higgsonium binding might occur, were investigated.', '1606.05823-1-3-9': 'These 2-Higgs bound-state results (due to Higgs self coupling) are somewhat moot, now that the Standard Model Higgs mass has been observed to be [MATH] GeV.', '1606.05823-1-4-0': 'The present work is concerned with the interactions of Higgs particles with DM particles.', '1606.05823-1-4-1': 'In particular we shall examine the possibility of two-body Higgs bound states due a mediating DM field.', '1606.05823-1-4-2': 'To our knowledge, this is the first investigation of Higgs relativistic bound state formation, using a general DM Lagrangian, via a singlet scalar DM channel.', '1606.05823-1-5-0': '# The model', '1606.05823-1-6-0': 'In this work the Dark Matter is assumed to be a spinless, singlet, massive scalar field [MATH], of mass [MATH], coupled with the Higgs field [MATH], which is taken to be the Higgs sector of the Standard Model.', '1606.05823-1-6-1': 'Models with scalar singlet DM have been studied previously (see for e.g. references [CITATION] [CITATION]).', '1606.05823-1-6-2': 'In this study, we shall consider the interaction of the Higgs particles with DM only.', '1606.05823-1-6-3': 'The interaction terms will be taken to be of a general form consistent with classical stability and renormalizability, that is the postulated Lagrangian density of this model is ([MATH]) [EQUATION] where [MATH], [MATH], [MATH], [MATH] and [MATH] are positive coupling constants; [MATH] being dimensionless, and [MATH], [MATH], having dimensions of mass.', '1606.05823-1-6-4': 'Aside from the Higgs sector coupling constants [MATH] and [MATH], all the others are presently unknown and will be treated to be adjustable.', '1606.05823-1-6-5': 'In order to decrease the coupling-constant parameter space we shall set [MATH] for this paper.', '1606.05823-1-7-0': 'The proposed Lagrangian ([REF]) allows for the decay of a very heavy single DM particle into two Higgs, provided that [MATH], where [MATH], and [MATH] are the DM and Higgs masses respectively.', '1606.05823-1-8-0': 'It is shown below that in the domain where the effective Higgs-DM dimensionless coupling constant varies from [MATH]-[MATH], two Higgs bound states are formed only if [MATH], which excludes DM decay into the Higgs.', '1606.05823-1-9-0': 'Models in which the DM is represented by a scalar with an assumed [MATH] symmetry, have been investigated by many authors, e.g. [CITATION].', '1606.05823-1-9-1': 'Our Lagrangian, however, does not impose a [MATH] symmetry, since the DM particle is represented by a spinless real scalar - thus allowing for trilinear terms, such as [MATH], in which the dimensionful coupling constant is similar to that of the Higgs sector of the DM.', '1606.05823-1-9-2': 'Our choice is motivated by the fact that the popularity of the [MATH] symmetry has largely resulted in the avoidance of studying the simplest case of a real scalar DM with no additional internal degrees of freedom.', '1606.05823-1-9-3': 'In an article on scalar DM, Rodejohann and Yaguna [CITATION] have noted that ".', '1606.05823-1-9-4': 'use of discrete symmetries is questionable not only due to its lack of motivation, but also because they are expected to be broken by gravitational effects at the Planck scale, including dark matter decay and likely destroying the feasibility of such models.', '1606.05823-1-9-5': 'That is why it is often implicitly assumed that such a discrete symmetries are actually the remnants of additional gauge or flavor symmetries present at a higher scale, thereby delegating the problem to a framework larger than the model under consideration."', '1606.05823-1-9-6': 'In another recent paper, Cirelli [CITATION] notes that "The "stabilization symmetry" has become such a household tool for the model builder that often he/she does not even spend time arguing about it: when in a hurry, just say that you add a [MATH] symmetry and move on."', '1606.05823-1-10-0': '# Quantization and Hamiltonian Formalism', '1606.05823-1-11-0': 'Upon canonical quantization (in the interaction picture) the classical fields [MATH] become operators [EQUATION] where [MATH], [MATH], [MATH] and [MATH].', '1606.05823-1-11-1': 'The DM and Higgs operators [MATH] and [MATH] satisfy the usual commutation rules, [EQUATION] and all others vanish.', '1606.05823-1-12-0': 'In the Hamiltonian formalism of QFT, the equations to be solved are [EQUATION] where [MATH] and [MATH] are the energy-momentum operator and corresponding eigenvalues.', '1606.05823-1-12-1': 'The [MATH] (energy) component of equation ([REF]) is generally impossible to solve and this applies to the present model.', '1606.05823-1-12-2': 'Thus, approximate solutions need to be obtained.', '1606.05823-1-12-3': 'We shall use the variational method, which is applicable to strongly coupled systems.', '1606.05823-1-12-4': 'This method is based on the principle [EQUATION] where [MATH] is normal ordered, and [MATH] is a suitable trial state.', '1606.05823-1-12-5': 'The subscript [MATH]=0 in equation ([REF]) indicates transformation to the Schrodinger picture, which is convenient for bound states.', '1606.05823-1-13-0': '# Trial state and channel equations', '1606.05823-1-14-0': 'For a system of two Higgs particles interacting via the DM field, the simplest trial state that yields tractable, non-trivial results is [EQUATION] where [MATH] denotes Higgs and [MATH] Dark Matter, and [MATH], [MATH] are variational channel wave functions to be determined.', '1606.05823-1-14-1': 'We shall work in the rest frame in which [MATH], which implies that [MATH] in [MATH] and [MATH] in [MATH].', '1606.05823-1-14-2': 'Substituting the expression ([REF]) into equation ([REF]), evaluating the indicated matrix elements and performing the variations, leads to the following equation for the channel trial functions [MATH] and [MATH]: [EQUATION] and [EQUATION]', '1606.05823-1-14-3': 'It is not possible to obtain exact analytic solutions of the coupled, relativistic equations ([REF]) and ([REF]), so we shall resort to obtaining approximate variational-perturbative solutions.', '1606.05823-1-14-4': 'In lowest order approximation we take [MATH] in equation ([REF]), whereupon equation ([REF]) simplifies to [EQUATION]', '1606.05823-1-14-5': 'Substituting the expression ([REF]) into equation ([REF]), the latter, in the rest frame (i.e. the total momentum [MATH]), simplifies to the single relativistic momentum-space equation [EQUATION] where [MATH], and [MATH] is a dimensionless coupling constant.', '1606.05823-1-14-6': 'The interaction in the integral equation ([REF]) is represented by the kernel, which is a relativistic generalization of the potential.', '1606.05823-1-14-7': 'Our principal interest is to determine the conditions under which the two-Higgs system can form bound states due to a Dark Matter mediating field.', '1606.05823-1-14-8': 'For this purpose it is sufficient to study ground states, for which the wave functions are spherically symmetric.', '1606.05823-1-14-9': 'We shall examine the [MATH] parameter space, where [MATH] is the (unknown) DM particle mass, and [MATH] (or, equivalently, [MATH]), is the (similarly unknown) dimensionless coupling constant.', '1606.05823-1-15-0': '# Approximate solutions and results', '1606.05823-1-16-0': 'Unfortunately the relativistic equation ([REF]) is not analytically solvable even for spherically symmetric states, so approximate variational solutions will be obtained.', '1606.05823-1-16-1': 'Variational approximations, as is well known, are only as good as the trial states that are used.', '1606.05823-1-16-2': 'For our purposes it will be sufficient to use simple ground state trial functions, particularly in light of the large parameter space to be examined.', '1606.05823-1-16-3': 'We shall obtain approximate variational solutions of ([REF]) for the Higgs-Higgs ground state, using the spherically symmetric trial wave function [EQUATION] where [MATH], [MATH] and [MATH] is an adjustable parameter, whose value is chosen so that the [MATH] is a least upper bound to the unknown exact mass of the two-Higgs bound state.', '1606.05823-1-16-4': 'It should be noted that for non-relativistic Higgs particles, i.e. in the limit [MATH], (equivalently, [MATH]), equation ([REF]) simplifies to [EQUATION] where [MATH].', '1606.05823-1-16-5': 'This is recognized to be the momentum space representation of the non-relativistic Schrodinger equation for the relative motion of two Higgs particles interacting via an attractive Yukawa potential [MATH].', '1606.05823-1-16-6': 'As is well known, this non-relativistic equation is not analytically solvable for [MATH] though in the limit of massless Dark Matter particles, [MATH], the solutions are the familiar Hydrogenic ones with [MATH] and [MATH] for the ground state.', '1606.05823-1-17-0': 'The variationally obtained relativistic two-Higgs ground-state rest masses [MATH] are listed in Table 1 for various values of the dimensionless coupling constant [MATH] and for various values of the DM mass [MATH].', '1606.05823-1-17-1': 'All results are given in units of the Higgs mass [MATH], i.e. [MATH] is [MATH] , [MATH] is [MATH] is [MATH]).', '1606.05823-1-17-2': 'We also list the previously obtained non-relativistic limit results for comparison purposes [CITATION].', '1606.05823-1-17-3': 'These relativistic and non relativistic results are also plotted in Figure 1 and in Figure 2.', '1606.05823-1-17-4': 'Table 1.', '1606.05823-1-17-5': 'Variational relativistic two Higgs rest energy [MATH] for various values of the DM mass [MATH] (both in units of the Higgs mass [MATH]) and for various values of the dimensionless coupling constant [MATH].', '1606.05823-1-17-6': 'The quantity [MATH] is the optimal value of the variational scale parameter [MATH] (in units of the Higgs mass [MATH]).', '1606.05823-1-17-7': 'We also give the non-relativistic result [MATH] for comparison purposes.', '1606.05823-1-18-0': 'Note that the relativistic values of the bound two-Higgs rest mass [MATH] are significantly higher (i.e. the binding energy is significantly lower) than the non-relativistic values, and the difference grows with increasing values of the coupling constant [MATH].', '1606.05823-1-18-1': 'This underscores the importance of using a relativistic description.', '1606.05823-1-19-0': '# Concluding remarks', '1606.05823-1-20-0': 'Our results show that two-Higgs bound states can be formed due to a massive spinless real scalar Dark Matter mediating field, over a broad range of the effective dimensionless coupling constant [MATH] and DM mass [MATH], namely [MATH] and [MATH], i.e. decay of the DM particle into two Higgs particles is not possible.', '1606.05823-1-20-1': 'Of course, these bound states are actually quasi-bound states since the Higgs particle has a very short lifetime and so the two-Higgs bound system is also short lived.', '1606.05823-1-20-2': 'Such quasi-bound states are expected to manifest themselves as resonances in the scattering cross section in DM on DM collisions.', '1606.05823-1-20-3': 'This is analogous to bound states of positronium, which are also short-lived quasi-bound states of an electron-positron system that manifest themselves as resonances in photon-photon scattering [CITATION].', '1606.05823-1-20-4': 'In the present work we do not study [MATH] on [MATH] elastic scattering states, that is solutions of equation ([REF]) with [MATH], and the effect of the coupling constants other than [MATH] are not sampled.', '1606.05823-1-20-5': 'Their effect on the binding energies can be evaluated perturbatively or by expanding the number of Fock components in the trial state [MATH].', '1606.05823-1-20-6': 'In any case, their effect would not eliminate the binding, since the interactions in this model are overall attractive.'}

{'1606.05823-2-0-0': 'We study a system of two Higgs bound state, interacting via a real scalar Dark Matter mediating field, without imposing [MATH] symmetry on the DM sector of the postulated Lagrangian.', '1606.05823-2-0-1': 'The variational method in the Hamiltonian formalism of QFT is used to derive relativistic wave equations for the two-Higgs system, using a truncated Fock-space trial state.', '1606.05823-2-0-2': 'Approximate solutions of the 2-body relativistic coupled integral equations are presented, and conditions for the existence of Higgs bound states is examined in a broad parameter space of DM mass and coupling constants.', '1606.05823-2-1-0': '# Introduction', '1606.05823-2-2-0': 'Astrophysical observations of gravitational effects at all length scales of the universe, including our Milky Way galaxy, indicate the existence of Dark Matter (DM) [CITATION].', '1606.05823-2-2-1': 'To date, DM has been observed only through its gravitational influence such as rotation curves of galaxies, mismatch of estimated mass and luminous matter in galaxy clusters, lensing of galactic Supernovae of Type Ia and Cosmic Microwave Background images, etc.', '1606.05823-2-2-2': "A major open question is the nature of DM, since to date, there are no direct measurements of DM's electroweak, or any non-gravitational, interactions.", '1606.05823-2-3-0': 'One conjecture is that DM is made up non-baryonic particles called WIMPs (Weakly Interacting Massive Particles).', '1606.05823-2-3-1': 'Experimental efforts at detecting DM, including WIMPs, are ongoing [CITATION].', '1606.05823-2-3-2': 'The literature on this subject is extensive.', '1606.05823-2-3-3': 'A brief, useful overview of DM and efforts to detect them is given in ref. [CITATION].', '1606.05823-2-3-4': 'An outstanding question is how DM interacts with Standard Model particles, including the Higgs particles.', '1606.05823-2-3-5': 'There have been a number of theoretical models that address this problem (see, for example, references [CITATION] and citations therein).', '1606.05823-2-3-6': 'Recently, Petersson et al. have proposed a Supersymmetric generalization of the Standard Model in which the Higgs boson can disintegrate into a photon and DM particles [CITATION].', '1606.05823-2-3-7': 'The possibility of Higgs bound states ("Higgsonium") has been examined well before the experimental detection of the Higgs boson (cf. references [CITATION] to [CITATION] ).', '1606.05823-2-3-8': 'In these prior studies, the domain of the Higgs mass and coupling strength for which Higgsonium binding might occur, were investigated.', '1606.05823-2-3-9': 'These 2-Higgs bound-state results (due to Higgs self coupling) are somewhat moot, now that the Standard Model Higgs mass has been observed to be [MATH] GeV.', '1606.05823-2-4-0': 'The present work is concerned with the interactions of Higgs particles with DM particles.', '1606.05823-2-4-1': 'In particular we shall examine the possibility of two-body Higgs bound states due a mediating DM field.', '1606.05823-2-4-2': 'To our knowledge, this is the first investigation of Higgs relativistic bound state formation, using a general DM Lagrangian, via a singlet scalar DM channel.', '1606.05823-2-5-0': '# The model', '1606.05823-2-6-0': 'In this work the Dark Matter is assumed to be a spinless, singlet, massive scalar field [MATH], of mass [MATH], coupled with the Higgs field [MATH], which is taken to be the Higgs sector of the Standard Model.', '1606.05823-2-6-1': 'Models with scalar singlet DM have been studied previously (see for e.g. references [CITATION] [CITATION]).', '1606.05823-2-6-2': 'In this study, we shall consider the interaction of the Higgs particles with DM only.', '1606.05823-2-6-3': 'The interaction terms will be taken to be of a general form consistent with classical stability and renormalizability, that is the postulated Lagrangian density of this model is ([MATH]) [EQUATION] where [MATH], [MATH], [MATH], [MATH] and [MATH] are positive coupling constants; [MATH] being dimensionless, and [MATH], [MATH], having dimensions of mass.', '1606.05823-2-6-4': 'Aside from the Higgs sector coupling constants [MATH] and [MATH], all the others are presently unknown and will be treated to be adjustable.', '1606.05823-2-7-0': 'Note that we have not included a [MATH] term in our postulated Lagrangian ([REF]) becase such a mixing term would contain a coupling constant [MATH] with dimensions of [MATH], which is unusual and would allow for instability of the DM.', '1606.05823-2-8-0': 'The proposed Lagrangian ([REF]) does allows for the decay of a very heavy single DM particle into two Higgs, provided that [MATH], where [MATH], and [MATH] are the DM and Higgs masses respectively.', '1606.05823-2-9-0': 'In order to decrease the coupling-constant parameter space we shall set [MATH] for this paper.', '1606.05823-2-10-0': 'It is shown below that in the domain where the effective Higgs-DM dimensionless coupling constant varies from [MATH]-[MATH], two Higgs bound states are formed only if [MATH], which excludes DM decay into the Higgs.', '1606.05823-2-11-0': 'Models in which the DM is represented by a scalar with an assumed [MATH] symmetry, have been investigated by many authors, e.g. [CITATION].', '1606.05823-2-11-1': 'Our Lagrangian, however, does not impose a [MATH] symmetry, since the DM particle is represented by a spinless real scalar - thus allowing for trilinear terms, such as [MATH], in which the dimensionful coupling constant is similar to that of the Higgs sector of the DM.', '1606.05823-2-11-2': 'Our choice is motivated by the fact that the popularity of the [MATH] symmetry has largely resulted in the avoidance of studying the simplest case of a real scalar DM with no additional internal degrees of freedom.', '1606.05823-2-11-3': 'In an article on scalar DM, Rodejohann and Yaguna [CITATION] have noted that ".', '1606.05823-2-11-4': 'use of discrete symmetries is questionable not only due to its lack of motivation, but also because they are expected to be broken by gravitational effects at the Planck scale, including dark matter decay and likely destroying the feasibility of such models.', '1606.05823-2-11-5': 'That is why it is often implicitly assumed that such a discrete symmetries are actually the remnants of additional gauge or flavor symmetries present at a higher scale, thereby delegating the problem to a framework larger than the model under consideration."', '1606.05823-2-11-6': 'In another recent paper, Cirelli [CITATION] notes that "The "stabilization symmetry" has become such a household tool for the model builder that often he/she does not even spend time arguing about it: when in a hurry, just say that you add a [MATH] symmetry and move on."', '1606.05823-2-12-0': '# Quantization and Hamiltonian Formalism', '1606.05823-2-13-0': 'Upon canonical quantization (in the interaction picture) the classical fields [MATH] become operators [EQUATION] where [MATH], [MATH], [MATH] and [MATH].', '1606.05823-2-13-1': 'The DM and Higgs operators [MATH] and [MATH] satisfy the usual commutation rules, [EQUATION] and all others vanish.', '1606.05823-2-14-0': 'In the Hamiltonian formalism of QFT, the equations to be solved are [EQUATION] where [MATH] and [MATH] are the energy-momentum operator and corresponding eigenvalues.', '1606.05823-2-14-1': 'The [MATH] (energy) component of equation ([REF]) is generally impossible to solve and this applies to the present model.', '1606.05823-2-14-2': 'Thus, approximate solutions need to be obtained.', '1606.05823-2-14-3': 'We shall use the variational method, which is applicable to strongly coupled systems.', '1606.05823-2-14-4': 'This method is based on the principle [EQUATION] where [MATH] is normal ordered, and [MATH] is a suitable trial state.', '1606.05823-2-14-5': 'The subscript [MATH]=0 in equation ([REF]) indicates transformation to the Schrodinger picture, which is convenient for bound states.', '1606.05823-2-15-0': '# Trial state and channel equations', '1606.05823-2-16-0': 'For a system of two Higgs particles interacting via the DM field, the simplest trial state that yields tractable, non-trivial results is [EQUATION] where [MATH] denotes Higgs and [MATH] Dark Matter, and [MATH], [MATH] are variational channel wave functions to be determined.', '1606.05823-2-16-1': 'We shall work in the rest frame in which [MATH], which implies that [MATH] in [MATH] and [MATH] in [MATH].', '1606.05823-2-16-2': 'Substituting the expression ([REF]) into equation ([REF]), evaluating the indicated matrix elements and performing the variations, leads to the following equation for the channel trial functions [MATH] and [MATH]: [EQUATION] and [EQUATION]', '1606.05823-2-16-3': 'It is not possible to obtain exact analytic solutions of the coupled, relativistic equations ([REF]) and ([REF]), so we shall resort to obtaining approximate variational-perturbative solutions.', '1606.05823-2-16-4': 'In lowest order approximation we take [MATH] in equation ([REF]), whereupon equation ([REF]) simplifies to [EQUATION]', '1606.05823-2-16-5': 'Substituting the expression ([REF]) into equation ([REF]), the latter, in the rest frame (i.e. the total momentum [MATH]), simplifies to the single relativistic momentum-space equation [EQUATION] where [MATH], and [MATH] is a dimensionless coupling constant.', '1606.05823-2-16-6': 'The interaction in the integral equation ([REF]) is represented by the kernel, which is a relativistic generalization of the potential.', '1606.05823-2-16-7': 'Our principal interest is to determine the conditions under which the two-Higgs system can form bound states due to a Dark Matter mediating field.', '1606.05823-2-16-8': 'For this purpose it is sufficient to study ground states, for which the wave functions are spherically symmetric.', '1606.05823-2-16-9': 'We shall examine the [MATH] parameter space, where [MATH] is the (unknown) DM particle mass, and [MATH] (or, equivalently, [MATH]), is the (similarly unknown) dimensionless coupling constant.', '1606.05823-2-17-0': '# Approximate solutions and results', '1606.05823-2-18-0': 'Unfortunately the relativistic equation ([REF]) is not analytically solvable even for spherically symmetric states, so approximate variational solutions will be obtained.', '1606.05823-2-18-1': 'Variational approximations, as is well known, are only as good as the trial states that are used.', '1606.05823-2-18-2': 'For our purposes it will be sufficient to use simple ground state trial functions, particularly in light of the large parameter space to be examined.', '1606.05823-2-18-3': 'We shall obtain approximate variational solutions of ([REF]) for the Higgs-Higgs ground state, using the spherically symmetric trial wave function [EQUATION] where [MATH], [MATH] and [MATH] is an adjustable parameter, whose value is chosen so that the [MATH] is a least upper bound to the unknown exact mass of the two-Higgs bound state.', '1606.05823-2-18-4': 'It should be noted that for non-relativistic Higgs particles, i.e. in the limit [MATH], (equivalently, [MATH]), equation ([REF]) simplifies to [EQUATION] where [MATH].', '1606.05823-2-18-5': 'This is recognized to be the momentum space representation of the non-relativistic Schrodinger equation for the relative motion of two Higgs particles interacting via an attractive Yukawa potential [MATH].', '1606.05823-2-18-6': 'As is well known, this non-relativistic equation is not analytically solvable for [MATH] though in the limit of massless Dark Matter particles, [MATH], the solutions are the familiar Hydrogenic ones with [MATH] and [MATH] for the ground state.', '1606.05823-2-19-0': 'The variationally obtained relativistic two-Higgs ground-state rest masses [MATH] are listed in Table 1 for various values of the dimensionless coupling constant [MATH] and for various values of the DM mass [MATH].', '1606.05823-2-19-1': 'All results are given in units of the Higgs mass [MATH], i.e. [MATH] is [MATH] , [MATH] is [MATH] is [MATH]).', '1606.05823-2-20-0': 'We also list the previously obtained non-relativistic limit results for comparison purposes [CITATION].', '1606.05823-2-20-1': 'These relativistic and non relativistic results are also plotted in Figure 1 and in Figure 2.', '1606.05823-2-20-2': 'Table 1.', '1606.05823-2-20-3': 'Variational relativistic two Higgs rest energy [MATH] for various values of the DM mass [MATH] (both in units of the Higgs mass [MATH]) and for various values of the dimensionless coupling constant [MATH].', '1606.05823-2-20-4': 'The quantity [MATH] is the optimal value of the variational scale parameter [MATH] (in units of the Higgs mass [MATH]).', '1606.05823-2-20-5': 'We also give the non-relativistic result [MATH] for comparison purposes.', '1606.05823-2-21-0': 'It is evident that the relativistic values of the bound two-Higgs rest mass [MATH] are significantly higher (i.e. the binding energy is significantly lower) than the non-relativistic values, and the difference grows with increasing values of the coupling constant [MATH].', '1606.05823-2-21-1': 'This underscores the importance of using a relativistic description.', '1606.05823-2-22-0': '# Concluding remarks', '1606.05823-2-23-0': 'Our results show that two-Higgs bound states can be formed due to a massive spinless real scalar Dark Matter mediating field, over a broad range of the effective dimensionless coupling constant [MATH] and DM mass [MATH], namely [MATH] and [MATH], i.e. decay of the DM particle into two Higgs particles is not possible.', '1606.05823-2-23-1': 'Note that the indicated domain of the effective dimentionless coupling constant [MATH] does not require that [MATH] be large since [MATH] is proportional to [MATH].', '1606.05823-2-24-0': 'Of course, these bound states are actually quasi-bound states since the Higgs particle has a very short lifetime and so the two-Higgs bound system is also short lived.', '1606.05823-2-24-1': 'Such quasi-bound states are expected to manifest themselves as resonances in the scattering cross section in DM on DM collisions.', '1606.05823-2-24-2': 'This is analogous to bound states of positronium, which are also short-lived quasi-bound states of an electron-positron system that manifest themselves as resonances in photon-photon scattering [CITATION].', '1606.05823-2-24-3': 'In the present work we do not study [MATH] on [MATH] elastic scattering states, that is solutions of equation ([REF]) with [MATH].', '1606.05823-2-24-4': 'Also, the effect of the coupling constants other than [MATH] are not sampled in this work.', '1606.05823-2-24-5': 'Their effect on the binding energies can be evaluated perturbatively or by expanding the number of Fock components in the trial state [MATH].', '1606.05823-2-24-6': 'In any case, their effect would not eliminate the binding, since the interactions in this model are overall attractive.', '1606.05823-2-24-7': 'We shall report on the effect of interaction terms of the Lagrangian ([REF]) with coupling constants other than [MATH] in subsequent work.'}

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['1606.05823-1-6-0', '1606.05823-2-6-0'], ['1606.05823-1-6-1', '1606.05823-2-6-1'], ['1606.05823-1-6-2', '1606.05823-2-6-2'], ['1606.05823-1-6-3', '1606.05823-2-6-3'], ['1606.05823-1-6-4', '1606.05823-2-6-4'], ['1606.05823-1-6-5', '1606.05823-2-9-0'], ['1606.05823-1-17-0', '1606.05823-2-19-0'], ['1606.05823-1-17-1', '1606.05823-2-19-1'], ['1606.05823-1-17-2', '1606.05823-2-20-0'], ['1606.05823-1-17-3', '1606.05823-2-20-1'], ['1606.05823-1-17-5', '1606.05823-2-20-3'], ['1606.05823-1-17-6', '1606.05823-2-20-4'], ['1606.05823-1-17-7', '1606.05823-2-20-5']]

[['1606.05823-1-18-0', '1606.05823-2-21-0'], ['1606.05823-1-7-0', '1606.05823-2-8-0'], ['1606.05823-2-19-1', '1606.05823-3-20-1']]

[]

[['1606.05823-1-20-4', '1606.05823-2-24-3'], ['1606.05823-1-20-4', '1606.05823-2-24-4']]

[]

['1606.05823-1-17-4', '1606.05823-2-20-2', '1606.05823-3-21-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1606.05823

{'1606.05823-3-0-0': 'We study a system of two Higgs bound state, interacting via a real scalar Dark Matter mediating field, without imposing [MATH] symmetry on the DM sector of the postulated Lagrangian.', '1606.05823-3-0-1': 'The variational method in the Hamiltonian formalism of QFT is used to derive relativistic wave equations for the two-Higgs system, using a truncated Fock-space trial state.', '1606.05823-3-0-2': 'Approximate solutions of the 2-body relativistic coupled integral equations are presented, and conditions for the existence of Higgs bound states is examined in a broad parameter space of DM mass and coupling constants.', '1606.05823-3-1-0': 'Keywords: Scalar Dark Matter, Higgs, bound state, Quantum Field Theory, variational method.', '1606.05823-3-2-0': '# Introduction', '1606.05823-3-3-0': 'Astrophysical observations of gravitational effects at all length scales of the universe, including our Milky Way galaxy, indicate the existence of Dark Matter (DM) [CITATION].', '1606.05823-3-3-1': 'To date, DM has been observed only through its gravitational influence such as rotation curves of galaxies, mismatch of estimated mass and luminous matter in galaxy clusters, lensing of galactic Supernovae of Type Ia and Cosmic Microwave Background images, etc.', '1606.05823-3-3-2': "A major open question is the nature of DM, since to date, there are no direct measurements of DM's electroweak, or any non-gravitational, interactions.", '1606.05823-3-4-0': 'One conjecture is that DM is made up non-baryonic particles called WIMPs (Weakly Interacting Massive Particles).', '1606.05823-3-4-1': 'Experimental efforts at detecting DM, including WIMPs, are ongoing [CITATION].', '1606.05823-3-4-2': 'The literature on this subject is extensive.', '1606.05823-3-4-3': 'A brief, useful overview of DM and efforts to detect them is given in ref. [CITATION].', '1606.05823-3-4-4': 'An outstanding question is how DM interacts with Standard Model particles, including the Higgs particles.', '1606.05823-3-4-5': 'There have been a number of theoretical models that address this problem (see, for example, references [CITATION] and citations therein).', '1606.05823-3-4-6': 'Recently, Petersson et al. have proposed a Supersymmetric generalization of the Standard Model in which the Higgs boson can disintegrate into a photon and DM particles [CITATION].', '1606.05823-3-4-7': 'The possibility of Higgs bound states ("Higgsonium") has been examined well before the experimental detection of the Higgs boson (cf. references [CITATION] to [CITATION] ).', '1606.05823-3-4-8': 'In these prior studies, the domain of the Higgs mass and coupling strength for which Higgsonium binding might occur, were investigated.', '1606.05823-3-4-9': 'These 2-Higgs bound-state results (due to Higgs self coupling) are somewhat moot, now that the Standard Model Higgs mass has been observed to be [MATH] GeV.', '1606.05823-3-5-0': 'The present work is concerned with the interactions of Higgs particles with DM particles.', '1606.05823-3-5-1': 'In particular we shall examine the possibility of two-body Higgs bound states due a mediating DM field.', '1606.05823-3-5-2': 'To our knowledge, this is the first investigation of Higgs relativistic bound state formation, using a general DM Lagrangian, via a singlet scalar DM channel.', '1606.05823-3-6-0': '# The model', '1606.05823-3-7-0': 'In this work the Dark Matter is assumed to be a spinless, singlet, massive scalar field [MATH], of mass [MATH], coupled with the Higgs field [MATH], which is taken to be the Higgs sector of the Standard Model.', '1606.05823-3-7-1': 'Models with scalar singlet DM have been studied previously (see for e.g. references [CITATION] [CITATION]).', '1606.05823-3-7-2': 'In this study, we shall consider the interaction of the Higgs particles with DM only.', '1606.05823-3-7-3': 'The interaction terms will be taken to be of a general form consistent with classical stability and renormalizability, that is the postulated Lagrangian density of this model is ([MATH]) [EQUATION] where [MATH], [MATH], [MATH], [MATH] and [MATH] are positive coupling constants; [MATH] being dimensionless, and [MATH], [MATH], having dimensions of mass.', '1606.05823-3-7-4': 'Aside from the Higgs sector coupling constants [MATH] and [MATH], all the others are presently unknown and will be treated to be adjustable.', '1606.05823-3-8-0': 'Note that we have not included a [MATH] term in our postulated Lagrangian ([REF]) becase such a mixing term would contain a coupling constant [MATH] with dimensions of [MATH], which is unusual and would allow for instability of the DM.', '1606.05823-3-9-0': 'The proposed Lagrangian ([REF]) does allows for the decay of a very heavy single DM particle into two Higgs, provided that [MATH], where [MATH], and [MATH] are the DM and Higgs masses respectively.', '1606.05823-3-10-0': 'In order to decrease the coupling-constant parameter space we shall set [MATH] for this paper.', '1606.05823-3-11-0': 'It is shown below that in the domain where the effective Higgs-DM dimensionless coupling constant varies from [MATH]-[MATH], two Higgs bound states are formed only if [MATH], which excludes DM decay into the Higgs.', '1606.05823-3-12-0': 'Models in which the DM is represented by a scalar with an assumed [MATH] symmetry, have been investigated by many authors, e.g. [CITATION].', '1606.05823-3-12-1': 'Our Lagrangian, however, does not impose a [MATH] symmetry, since the DM particle is represented by a spinless real scalar - thus allowing for trilinear terms, such as [MATH], in which the dimensionful coupling constant is similar to that of the Higgs sector of the DM.', '1606.05823-3-12-2': 'Our choice is motivated by the fact that the popularity of the [MATH] symmetry has largely resulted in the avoidance of studying the simplest case of a real scalar DM with no additional internal degrees of freedom.', '1606.05823-3-12-3': 'In an article on scalar DM, Rodejohann and Yaguna [CITATION] have noted that ".', '1606.05823-3-12-4': 'use of discrete symmetries is questionable not only due to its lack of motivation, but also because they are expected to be broken by gravitational effects at the Planck scale, including dark matter decay and likely destroying the feasibility of such models.', '1606.05823-3-12-5': 'That is why it is often implicitly assumed that such a discrete symmetries are actually the remnants of additional gauge or flavor symmetries present at a higher scale, thereby delegating the problem to a framework larger than the model under consideration."', '1606.05823-3-12-6': 'In another recent paper, Cirelli [CITATION] notes that "The "stabilization symmetry" has become such a household tool for the model builder that often he/she does not even spend time arguing about it: when in a hurry, just say that you add a [MATH] symmetry and move on."', '1606.05823-3-13-0': '# Quantization and Hamiltonian Formalism', '1606.05823-3-14-0': 'Upon canonical quantization (in the interaction picture) the classical fields [MATH] become operators [EQUATION] where [MATH], [MATH], [MATH] and [MATH].', '1606.05823-3-14-1': 'The DM and Higgs operators [MATH] and [MATH] satisfy the usual commutation rules, [EQUATION] and all others vanish.', '1606.05823-3-15-0': 'In the Hamiltonian formalism of QFT, the equations to be solved are [EQUATION] where [MATH] and [MATH] are the energy-momentum operator and corresponding eigenvalues.', '1606.05823-3-15-1': 'The [MATH] (energy) component of equation ([REF]) is generally impossible to solve and this applies to the present model.', '1606.05823-3-15-2': 'Thus, approximate solutions need to be obtained.', '1606.05823-3-15-3': 'We shall use the variational method, which is applicable to strongly coupled systems.', '1606.05823-3-15-4': 'This method is based on the principle [EQUATION] where [MATH] is normal ordered, and [MATH] is a suitable trial state.', '1606.05823-3-15-5': 'The subscript [MATH]=0 in equation ([REF]) indicates transformation to the Schrodinger picture, which is convenient for bound states.', '1606.05823-3-16-0': '# Trial state and channel equations', '1606.05823-3-17-0': 'For a system of two Higgs particles interacting via the DM field, the simplest trial state that yields tractable, non-trivial results is [EQUATION] where [MATH] denotes Higgs and [MATH] Dark Matter, and [MATH], [MATH] are variational channel wave functions to be determined.', '1606.05823-3-17-1': 'We shall work in the rest frame in which [MATH], which implies that [MATH] in [MATH] and [MATH] in [MATH].', '1606.05823-3-17-2': 'Substituting the expression ([REF]) into equation ([REF]), evaluating the indicated matrix elements and performing the variations, leads to the following equation for the channel trial functions [MATH] and [MATH]: [EQUATION] and [EQUATION]', '1606.05823-3-17-3': 'It is not possible to obtain exact analytic solutions of the coupled, relativistic equations ([REF]) and ([REF]), so we shall resort to obtaining approximate variational-perturbative solutions.', '1606.05823-3-17-4': 'In lowest order approximation we take [MATH] in equation ([REF]), whereupon equation ([REF]) simplifies to [EQUATION]', '1606.05823-3-17-5': 'Substituting the expression ([REF]) into equation ([REF]), the latter, in the rest frame (i.e. the total momentum [MATH]), simplifies to the single relativistic momentum-space equation [EQUATION] where [MATH], and [MATH] is a dimensionless coupling constant.', '1606.05823-3-17-6': 'The interaction in the integral equation ([REF]) is represented by the kernel, which is a relativistic generalization of the potential.', '1606.05823-3-17-7': 'Our principal interest is to determine the conditions under which the two-Higgs system can form bound states due to a Dark Matter mediating field.', '1606.05823-3-17-8': 'For this purpose it is sufficient to study ground states, for which the wave functions are spherically symmetric.', '1606.05823-3-17-9': 'We shall examine the [MATH] parameter space, where [MATH] is the (unknown) DM particle mass, and [MATH] (or, equivalently, [MATH]), is the (similarly unknown) dimensionless coupling constant.', '1606.05823-3-18-0': '# Approximate solutions and results', '1606.05823-3-19-0': 'Unfortunately the relativistic equation ([REF]) is not analytically solvable even for spherically symmetric states, so approximate variational solutions will be obtained.', '1606.05823-3-19-1': 'Variational approximations, as is well known, are only as good as the trial states that are used.', '1606.05823-3-19-2': 'For our purposes it will be sufficient to use simple ground state trial functions, particularly in light of the large parameter space to be examined.', '1606.05823-3-19-3': 'We shall obtain approximate variational solutions of ([REF]) for the Higgs-Higgs ground state, using the spherically symmetric trial wave function [EQUATION] where [MATH], [MATH] and [MATH] is an adjustable parameter, whose value is chosen so that the [MATH] is a least upper bound to the unknown exact mass of the two-Higgs bound state.', '1606.05823-3-19-4': 'It should be noted that for non-relativistic Higgs particles, i.e. in the limit [MATH], (equivalently, [MATH]), equation ([REF]) simplifies to [EQUATION] where [MATH].', '1606.05823-3-19-5': 'This is recognized to be the momentum space representation of the non-relativistic Schrodinger equation for the relative motion of two Higgs particles interacting via an attractive Yukawa potential [MATH].', '1606.05823-3-19-6': 'As is well known, this non-relativistic equation is not analytically solvable for [MATH] though in the limit of massless Dark Matter particles, [MATH], the solutions are the familiar Hydrogenic ones with [MATH] and [MATH] for the ground state.', '1606.05823-3-20-0': 'The variationally obtained relativistic two-Higgs ground-state rest masses [MATH] are listed in Table 1 for various values of the dimensionless coupling constant [MATH] and for various values of the DM mass [MATH].', '1606.05823-3-20-1': 'All results are given in units of the Higgs mass [MATH], i.e. [MATH] is [MATH], [MATH] is [MATH], [MATH] is [MATH].', '1606.05823-3-21-0': 'We also list the previously obtained non-relativistic limit results for comparison purposes [CITATION].', '1606.05823-3-21-1': 'These relativistic and non relativistic results are also plotted in Figure 1 and in Figure 2.', '1606.05823-3-21-2': 'Table 1.', '1606.05823-3-21-3': 'Variational relativistic two Higgs rest energy [MATH] for various values of the DM mass [MATH] (both in units of the Higgs mass [MATH]) and for various values of the dimensionless coupling constant [MATH].', '1606.05823-3-21-4': 'The quantity [MATH] is the optimal value of the variational scale parameter [MATH] (in units of the Higgs mass [MATH]).', '1606.05823-3-21-5': 'We also give the non-relativistic result [MATH] for comparison purposes.', '1606.05823-3-22-0': 'It is evident that the relativistic values of the bound two-Higgs rest mass [MATH] are significantly higher (i.e. the binding energy is significantly lower) than the non-relativistic values, and the difference grows with increasing values of the coupling constant [MATH].', '1606.05823-3-22-1': 'This underscores the importance of using a relativistic description.', '1606.05823-3-23-0': '# Concluding remarks', '1606.05823-3-24-0': 'Our results show that two-Higgs bound states can be formed due to a massive spinless real scalar Dark Matter mediating field, over a broad range of the effective dimensionless coupling constant [MATH] and DM mass [MATH], namely [MATH] and [MATH], i.e. decay of the DM particle into two Higgs particles is not possible.', '1606.05823-3-24-1': 'Note that the indicated domain of the effective dimentionless coupling constant [MATH] does not require that [MATH] be large since [MATH] is proportional to [MATH].', '1606.05823-3-25-0': 'Of course, these bound states are actually quasi-bound states since the Higgs particle has a very short lifetime and so the two-Higgs bound system is also short lived.', '1606.05823-3-25-1': 'Such quasi-bound states are expected to manifest themselves as resonances in the scattering cross section in DM on DM collisions.', '1606.05823-3-25-2': 'This is analogous to bound states of positronium, which are also short-lived quasi-bound states of an electron-positron system that manifest themselves as resonances in photon-photon scattering [CITATION].', '1606.05823-3-25-3': 'In the present work we do not study [MATH] on [MATH] elastic scattering states, that is solutions of equation ([REF]) with [MATH].', '1606.05823-3-25-4': 'Also, the effect of the coupling constants other than [MATH] are not sampled in this work.', '1606.05823-3-25-5': 'Their effect on the binding energies can be evaluated perturbatively or by expanding the number of Fock components in the trial state [MATH].', '1606.05823-3-25-6': 'In any case, their effect would not eliminate the binding, since the interactions in this model are overall attractive.', '1606.05823-3-25-7': 'We shall report on the effect of interaction terms of the Lagrangian ([REF]) with coupling constants other than [MATH] in subsequent work.'}

hep-ph-0512177

{'hep-ph-0512177-1-0-0': 'We extend our previous determination of the thermodynamic pressure of the Standard Model so that the result can be applied down to temperatures corresponding to the electroweak crossover.', 'hep-ph-0512177-1-0-1': 'This requires a further resummation which can be cleanly organised within the effective theory framework.', 'hep-ph-0512177-1-0-2': 'The result allows for a precise determination of the expansion rate of the Universe for temperatures around the electroweak crossover.', 'hep-ph-0512177-1-1-0': 'diagrams', 'hep-ph-0512177-1-2-0': 'style_def dash_dot expr p = save dpp, k; numeric dpp, k; dpp = ceiling (pixlen (p, 10) / (1.5*dash_len)) / length p; k=0; forever: exitif k+.33 > dpp*length(p); cdraw point k/dpp of p .', 'hep-ph-0512177-1-2-1': 'point (k+.33)/dpp of p; exitif k+.67 > dpp*length(p); cdrawdot point (k+.67)/dpp of p; k := k+1; endfor enddef;', 'hep-ph-0512177-1-3-0': 'style_def dash_dot_arrow expr p = draw_dash_dot p; cfill (arrow p); enddef;', 'hep-ph-0512177-1-4-0': '# Introduction', 'hep-ph-0512177-1-5-0': 'In a recent paper [CITATION] we calculated the pressure of the standard model at high temperatures to three loops, or to order [MATH] in the coupling constants.', 'hep-ph-0512177-1-5-1': 'That work followed a long series of calculations dedicated to understanding the perturbative expansion of the pressure of gauge field theories at high temperatures.', 'hep-ph-0512177-1-5-2': 'Especially, within QCD such calculations have been important: the computation of the coefficients of the expansion in [MATH] at high temperature has a long history and the result is known today up to the last perturbatively calculable term of order [MATH] [CITATION], marking an endpoint to an impressive set of computations carried out in [CITATION].', 'hep-ph-0512177-1-5-3': 'An optimal approach to calculating those coefficients is the effective theory method [CITATION], based on asymptotic freedom and on separating the relevant mass scales: [MATH], the electric scale [MATH] and the magnetic scale [MATH].', 'hep-ph-0512177-1-6-0': 'An obvious drawback of those calculations within QCD is that they cannot be extended to study the QCD phase transition since the coupling grows large and thus any perturbative calculations become unreliable.', 'hep-ph-0512177-1-6-1': 'That is not the case for the electroweak sector of the standard model.', 'hep-ph-0512177-1-6-2': 'The Landau pole related to the weak interactions corresponds to a length scale [MATH] m and therefore the confining effects can be expected to be negligible.', 'hep-ph-0512177-1-6-3': 'As a result, calculating the properties of the electroweak phase transition using, for example, perturbative 1-loop [CITATION] and 2-loop [CITATION] effective potential calculations was possible for small Higgs masses.', 'hep-ph-0512177-1-6-4': 'A better approach, well defined for large Higgs masses as well, is to perturbatively match the full 4-dimensional theory to an effective 3-dimensional theory [CITATION] and then to numerically solve the phase diagram from the effective theory using lattice Monte Carlo techniques [CITATION].', 'hep-ph-0512177-1-6-5': 'Such studies show that the phase diagram has a first order line which ends in a 2nd order critical point of Ising universality class [CITATION].', 'hep-ph-0512177-1-6-6': 'For experimentally allowed Higgs masses the transition is a crossover.', 'hep-ph-0512177-1-6-7': 'Similar techniques have been used to solve the phase diagram also when the external U([MATH]) magnetic field [CITATION] or the chemical potentials related to the baryon and lepton numbers [CITATION] are nonzero.', 'hep-ph-0512177-1-6-8': 'Solving the phase diagram with numerical studies of the full 4-dimensional theory has also been achieved[CITATION].', 'hep-ph-0512177-1-7-0': 'Although the properties of the electroweak phase transition are known today, computation of the pressure of the theory has been missing.', 'hep-ph-0512177-1-7-1': 'In [CITATION] we performed this calculation when the temperature of the system is high.', 'hep-ph-0512177-1-7-2': 'In the present paper we will extend that result to temperatures close to the critical temperature of the electroweak phase transition.', 'hep-ph-0512177-1-7-3': 'Critical temperature and the phase transition should be understood in perturbation theory framework, where there is always a first order phase transition.', 'hep-ph-0512177-1-7-4': 'We will use these terms throughout this paper, although the actual transition is just a crossover.', 'hep-ph-0512177-1-8-0': 'The present computation requires a reorganization of the effective theories.', 'hep-ph-0512177-1-8-1': 'Close to the transition the fundamental scalar (which drives the transition) becomes light with respect to the electric scale and thus in a consistent calculation we have to formulate an additional effective theory by integrating out the adjoint scalars.', 'hep-ph-0512177-1-8-2': 'The remaining theory contains just the fundamental scalar and the gauge fields.', 'hep-ph-0512177-1-8-3': 'The pressure will then be composed of three parts: the contribution from fermions and the nonzero Matsubara modes of bosons [MATH], from the adjoint scalars [MATH] and [MATH], [MATH] and finally from the fundamental scalar [MATH] where [MATH] and [MATH].', 'hep-ph-0512177-1-8-4': 'At each step of the calculation there will be [MATH] poles from ultraviolet and infrared divergences which will not cancel until all the contributions are summed together.', 'hep-ph-0512177-1-9-0': 'The paper is organized as follows: in section [REF] we explicitly define the theory we are working with, fix various conventions and briefly review the method of dimensional reduction as applied to the present case.', 'hep-ph-0512177-1-9-1': 'Sections [REF] and [REF] contain the essential calculations and in section [REF] we discuss the result.', 'hep-ph-0512177-1-10-0': '# Basic setting', 'hep-ph-0512177-1-11-0': 'We consider the [MATH] standard model with [MATH] families of fermions and [MATH] fundamental scalar doublets, and evaluate the pressure of this theory at temperatures slightly above the electroweak phase transition.', 'hep-ph-0512177-1-11-1': 'The theory is specified by the Euclidean action (in the units [MATH]) [EQUATION] where [EQUATION]', 'hep-ph-0512177-1-11-2': 'Here [MATH], [MATH] and [MATH] are gauge bosons of weak-, hyper- and strong interactions, respectively; [MATH] is the fundamental scalar doublet; [MATH] and [MATH] are the left-handed lepton doublets and the right-handed lepton singlets (wrt. weak charge), and [MATH], [MATH] and [MATH] are the left-handed quark doublets and the right-handed up and down -type quark singlets.', 'hep-ph-0512177-1-11-3': 'The Yukawa coupling is taken into account for the top quark only.', 'hep-ph-0512177-1-11-4': 'Summation over different families is assumed.', 'hep-ph-0512177-1-11-5': 'Also, [MATH] in dimensional regularization, [MATH].', 'hep-ph-0512177-1-11-6': 'The gamma matrices are defined in Euclidean space so that [MATH], [MATH] and [MATH].', 'hep-ph-0512177-1-11-7': 'The color indices are [MATH] for the weak interaction and [MATH] for the strong interaction.', 'hep-ph-0512177-1-11-8': 'The different group theory factors for SU([MATH]) with generators [MATH] are defined as:', 'hep-ph-0512177-1-12-0': '0.51 [EQUATION] 0.46 [EQUATION]', 'hep-ph-0512177-1-12-1': 'For SU(2) with [MATH] they are [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0512177-1-13-0': 'The theory in Eq. ([REF]) contains six couplings that run with the renormalization scale: gauge couplings [MATH], [MATH] and [MATH], the fundamental scalar quartic self-coupling [MATH] and its mass parameter [MATH], and [MATH].', 'hep-ph-0512177-1-13-1': 'We fix the values of these couplings at the scale [MATH] according to their tree-level relation with different physical parameters as in [CITATION], where the 1-loop running of the parameters is also given.', 'hep-ph-0512177-1-13-2': 'We employ a power counting rule [MATH] and assume the temperature to be such that [MATH].', 'hep-ph-0512177-1-14-0': 'The physical observable we are studying is the pressure, defined by [EQUATION]', 'hep-ph-0512177-1-14-1': 'It is normalized such that the (real part of the) pressure of the symmetric phase vanishes at [MATH].', 'hep-ph-0512177-1-14-2': 'Other interesting variables, such as entropy and energy densities [MATH] and [MATH], can then be evaluated using standard thermodynamic relations, [MATH], [MATH].', 'hep-ph-0512177-1-14-3': 'The purpose is to calculate the pressure up to, and including, order [MATH], employing the power counting rules above.', 'hep-ph-0512177-1-14-4': 'For high temperatures, [MATH], this calculation was performed in [CITATION].', 'hep-ph-0512177-1-14-5': 'However, close to the phase transition, which is often the most interesting region, our earlier computation is not valid, since the (thermal) fundamental scalar mass is much smaller than the Debye masses and introduces another hierarchy of scales which needs to be sorted out.', 'hep-ph-0512177-1-14-6': 'The ratio of the two mass scales at different temperatures is shown in Fig. [REF], which illustrates that there is a range of temperatures in which [MATH].', 'hep-ph-0512177-1-14-7': 'The expression used for [MATH] in Fig. [REF] is Eq. ([REF]), renormalized in the [MATH] scheme and [MATH] set to 0.', 'hep-ph-0512177-1-15-0': 'It is well known that simply evaluating all 1-, 2- and 3-loop vacuum diagrams in perturbation theory fails because of infrared divergences.', 'hep-ph-0512177-1-15-1': 'Instead, one has to separate the contributions of different scales into successive effective theories [CITATION], where all the large scales are integrated out one by one.', 'hep-ph-0512177-1-15-2': 'First, we integrate [EQUATION] where [MATH] contains only the static Matsubara modes of the gauge bosons and of the fundamental scalar (Higgs) field.', 'hep-ph-0512177-1-15-3': 'The contributions of the nonzero Matsubara modes and fermions to the pressure show up as the matching constant [MATH] (App.', 'hep-ph-0512177-1-15-4': '[REF]) and in the parameters of [MATH] (App.', 'hep-ph-0512177-1-15-5': '[REF]).', 'hep-ph-0512177-1-15-6': 'The spatial (magnetic) gauge field components remain massless, while the temporal component gets a thermal mass [MATH].', 'hep-ph-0512177-1-15-7': 'The theory defined by [MATH] can then be viewed as a 3d gauge theory with adjoint and fundamental scalar fields.', 'hep-ph-0512177-1-16-0': 'The effective theory thus obtained still contains contributions from several scales, [MATH], [MATH] and the magnetic scale [MATH].', 'hep-ph-0512177-1-16-1': 'We restrict ourselves to temperatures close to the phase transition, where [MATH] is small compared to the Debye masses and assume [MATH], motivated by Eq. ([REF]).', 'hep-ph-0512177-1-16-2': 'We integrate out the scale [MATH], i.e. the fields [MATH] and [MATH], and are left with [EQUATION]', 'hep-ph-0512177-1-16-3': 'This is a 3d gauge theory with a fundamental scalar field.', 'hep-ph-0512177-1-16-4': 'The largest mass scale in this theory is the fundamental scalar mass [MATH], so this theory contributes at [MATH] or higher order, and we need to evaluate only 1- and 2-loop vacuum diagrams from this theory.', 'hep-ph-0512177-1-16-5': 'The potential infrared problems can be isolated to another effective theory by integrating out the fundamental scalar, [EQUATION]', 'hep-ph-0512177-1-16-6': 'The remaining effective theory contains only the (massless) spatial gauge fields.', 'hep-ph-0512177-1-16-7': 'Therefore the only mass scale of the theory is provided by the 3d gauge coupling and is of the order [MATH] and consequently the contribution of this theory to the pressure is of the order [MATH].', 'hep-ph-0512177-1-16-8': 'The final result of our calculation can then be written as [EQUATION] where [MATH] can be taken from [CITATION].', 'hep-ph-0512177-1-16-9': 'One-loop quark diagrams are included in [MATH], so they must be subtracted from [MATH].', 'hep-ph-0512177-1-16-10': 'The pressure [MATH] was computed in [CITATION], and is given in Appendix [REF], whereas [MATH] and [MATH] are computed in Sections [REF] and [REF], respectively.', 'hep-ph-0512177-1-17-0': '# Calculation of the pressure [MATH]', 'hep-ph-0512177-1-18-0': 'In this section we integrate over the scale [MATH].', 'hep-ph-0512177-1-18-1': 'This means that we need to integrate out the adjoint scalars [MATH] and [MATH], leaving an effective theory with only gauge bosons and the fundamental scalar field.', 'hep-ph-0512177-1-19-0': 'The 3d theory of gauge fields with fundamental and adjoint scalars is given by [EQUATION] where the parameters were computed in [CITATION] to required order and are listed in Appendix [REF].', 'hep-ph-0512177-1-20-0': 'To find out the contribution of the adjoint scalars to the pressure we need to calculate all the vacuum diagrams in the theory ([REF]) containing these fields.', 'hep-ph-0512177-1-20-1': 'Since both [MATH] and [MATH] are massive, there are no infrared divergences related to this integration.', 'hep-ph-0512177-1-20-2': 'The mass of the fundamental scalar [MATH] is parametrically smaller ([MATH]) so we treat it as a perturbation and expand the integrands in [MATH].', 'hep-ph-0512177-1-20-3': 'In fact, the leading [MATH] order is sufficient to our computations.', 'hep-ph-0512177-1-20-4': 'The required diagrams are shown in Fig. [REF] and are evaluated in [CITATION].', 'hep-ph-0512177-1-21-0': 'The result, with [MATH] and [MATH], reads [EQUATION]', 'hep-ph-0512177-1-21-1': 'The divergences and the scale dependence in [MATH] cancel against those in [MATH], and the ones in [MATH] against the sunset diagram in [MATH], Eq. ([REF]).', 'hep-ph-0512177-1-22-0': '# Calculation of the pressure [MATH]', 'hep-ph-0512177-1-23-0': 'The theory containing just the gauge fields and the fundamental scalar is defined by [EQUATION] where the field strength tensors [MATH] and [MATH] and the covariant derivative [MATH] are defined as before (with couplings [MATH] and [MATH]).', 'hep-ph-0512177-1-23-1': 'The integration from [MATH] to [MATH] will not introduce any new corrections to the couplings to get the pressure to the required order and therefore [MATH].', 'hep-ph-0512177-1-23-2': 'The scalar mass does, however, get an additional contribution (calculated in [CITATION] apart from the term [MATH] which is needed in the present calculation) coming from the tadpole diagrams with adjoint scalars at the loops: [EQUATION]', 'hep-ph-0512177-1-23-3': 'Since the pressure is to leading order given by [MATH] and since we assume the temperature to be such that [MATH], we see that corrections of order [MATH] to the scalar mass would contribute to pressure at order [MATH] and we will neglect those.', 'hep-ph-0512177-1-23-4': 'Similarly, we can neglect the [MATH] corrections to [MATH].', 'hep-ph-0512177-1-23-5': 'The running of the mass [MATH] (and hence of [MATH] as well), which sets in at the order [MATH] does not therefore influence the present calculation as it did the high temperature calculation.', 'hep-ph-0512177-1-23-6': 'Furthermore, even though [MATH] is assumed to be of the order [MATH] near the phase transition, the expression for [MATH] does contain individual terms of the order [MATH] which, when summed up, effectively cancel to order [MATH] (assuming proper temperature region).', 'hep-ph-0512177-1-23-7': 'The individual terms of order [MATH] must, however, be taken carefully into account when cancelling the [MATH] poles in the pressure.', 'hep-ph-0512177-1-24-0': 'The pressure is obtained by calculating the diagrams in Fig. [REF].', 'hep-ph-0512177-1-24-1': 'Three-loop diagrams would contribute to order [MATH] and are neglected now.', 'hep-ph-0512177-1-24-2': 'The result is [EQUATION]', 'hep-ph-0512177-1-24-3': 'The [MATH] poles cancel against those coming from [MATH] and [MATH].', 'hep-ph-0512177-1-25-0': '# Numerical results', 'hep-ph-0512177-1-26-0': 'In this section we will consider some numerical implications of the result, given by Eq. ([REF]).', 'hep-ph-0512177-1-26-1': 'The convergence and the scale dependence of the perturbative expansion were already studied in [CITATION] so now we will just concentrate on the difference between the results of that paper and the present paper, which are significant when the fundamental scalar becomes light.', 'hep-ph-0512177-1-26-2': 'The Higgs mass is taken to be [MATH] GeV and the W mass [MATH] GeV.', 'hep-ph-0512177-1-27-0': 'It is instructive to first look at a simpler SU(2) + fundamental Higgs model for which the perturbative expansion is better behaved.', 'hep-ph-0512177-1-27-1': 'In Fig. [REF] we have plotted the pressure of this theory in the temperature region where the fundamental scalar becomes much lighter than the adjoint scalars (see Fig. [REF]) using both the calculation valid at high temperatures ([MATH]) and the calculation valid near the phase transition ([MATH]), normalized to [MATH] (ideal gas pressure of SU(2) gauge bosons and a massless fundamental scalar).', 'hep-ph-0512177-1-27-2': 'It can be immediately noticed that there is a temperature region where the high temperature calculation is not well behaved but becomes singular.', 'hep-ph-0512177-1-27-3': 'This is due to the terms of the type [MATH] in the expansion of [MATH] which become singular when [MATH].', 'hep-ph-0512177-1-27-4': 'In the present calculation which takes into account that [MATH] such terms will not appear and therefore [MATH] is seen to be well behaved.', 'hep-ph-0512177-1-27-5': 'It is worth noting, however, that the temperature region where the singular behavior is manifest is very narrow.', 'hep-ph-0512177-1-27-6': 'That is due to the small gauge coupling: the singular terms are of the order [MATH] and their contribution to the pressure vanishes fast as one moves away from the singular point.', 'hep-ph-0512177-1-27-7': 'More specifically, the singular terms behave as: [EQUATION] where [MATH] is the location of the singularity and [MATH].', 'hep-ph-0512177-1-27-8': 'Since the curve behaves as [MATH] near the singularity and since the prefactor is so small, the contribution from the singularity becomes small very fast as [MATH] is increased.', 'hep-ph-0512177-1-28-0': 'If one excludes this narrow region around the singularity, the pressure [MATH] is smaller than the pressure [MATH] roughly by constant[MATH].', 'hep-ph-0512177-1-28-1': 'The relative difference between [MATH] and [MATH] is small, [MATH] but it nevertheless is of the same order of magnitude as differences in the pressure at different orders in perturbation theory and hence it is important to take this effect into account when calculating the pressure to high accuracy.', 'hep-ph-0512177-1-28-2': 'This is related to the reorganization of the effective theories.', 'hep-ph-0512177-1-28-3': 'Keeping the fundamental scalar mass, as defined in Eq. ([REF]), unexpanded when inserted to the expression for the pressure [MATH], Eq. ([REF]), effectively resums all the ring diagrams of the type shown in Fig. [REF] to the pressure.', 'hep-ph-0512177-1-28-4': 'The high temperature calculation, however, corresponds to expanding this mass in terms of [MATH] in the expression for the pressure and to keeping just the lowest order terms of the expansion.', 'hep-ph-0512177-1-28-5': 'At high temperatures this expansion can be made, since [MATH] but close to the phase transition [MATH] and the expansion is not possible.', 'hep-ph-0512177-1-28-6': 'Moreover, even though differences in the pressure when calculated from the high temperature theory as compared to the correct calculation near the phase transition are small, this is not necesserily the case for other interesting variables such as the critical temperature and to obtain these correctly, one needs to use the theory that takes into account that the fundamental scalar becomes light.', 'hep-ph-0512177-1-29-0': 'Although we do not study the specific features of the phase transition, which are already well established [CITATION], we nevertheless plot the pressure of the broken phase as well in Fig. [REF] to order [MATH] to indicate the temperature where the phase transition takes place.', 'hep-ph-0512177-1-29-1': 'It is given by [EQUATION] where [MATH], [MATH] and [MATH] are the zero temperature masses of the particles, and [MATH] (the expectation value of the Higgs field) is such that [MATH].', 'hep-ph-0512177-1-29-2': 'The critical temperature below which the pressure of the broken phase is bigger than the pressure of the symmetric phase, indicating that the symmetry is spontanously broken, is, for [MATH] GeV and [MATH] GeV, about [MATH] GeV.', 'hep-ph-0512177-1-30-0': 'In Fig. [REF] we have plotted the pressures [MATH] and [MATH] for the full standard model, normalized to [MATH].', 'hep-ph-0512177-1-30-1': 'A similar structure can be seen to appear here: the high temperature calculation of the pressure developes a singularity when [MATH], but the temperature region where this is of importance is very narrow, while the calculation which takes into account that near the phase transition [MATH] is seen to behave well.', 'hep-ph-0512177-1-30-2': 'Excluding the singularity, the pressures are again observed to differ roughly by a constant[MATH], but the difference is not in this case even as large as in the SU(2) + Higgs theory, especially when compared to the corrections introduced by each new order of perturbation theory.', 'hep-ph-0512177-1-30-3': 'This reflects the fact that the fundamental scalar carries only a small part of the total number of degrees of freedom and hence changes in the scalar sector of the standard model do not lead to significant changes in the pressure.', 'hep-ph-0512177-1-31-0': '# Conclusions', 'hep-ph-0512177-1-32-0': 'In this paper we have extended the calculation of the pressure of the standard model at high temperatures carried out in [CITATION] to the case when the temperature of the system is close to the critical temperature of the electroweak phase transition [MATH].', 'hep-ph-0512177-1-32-1': 'The previous calculation was not consistent in that region since it assumed that the fundamental scalar has a mass comparable to the masses of the adjoint scalars while, in fact, near the phase transition the fundamental scalar becomes light.', 'hep-ph-0512177-1-32-2': 'This inconsistency manifests itself as an unphysical singularity in the pressure if one tries to apply the result of the previous calculation to temperatures near [MATH].', 'hep-ph-0512177-1-32-3': 'The calculation performed in this paper is done in a consistant manner, taking into account the lightness of the fundamental scalar near the phase transition and no such unphysical singularities remain in the final result.', 'hep-ph-0512177-1-33-0': 'It is possible to go still a bit further in the perturbative expansion of the pressure, the next term would be of the order [MATH].', 'hep-ph-0512177-1-33-1': 'Its calculation would require a two-loop matching of the fundamental scalar mass (order [MATH]) and a three-loop calculation of the pressure [MATH], both of which are in principle manageable.', 'hep-ph-0512177-1-33-2': 'Furthermore, one could determine the final perturbatively calculable term [MATH] by calculating the vacuum energy densities of the three dimensional effective theories.', 'hep-ph-0512177-1-33-3': 'However, since the convergence of the perturbative expansion is rather fast (as noted in [CITATION]), especially for the SU(2) + Higgs theory, the calculation of those terms would not lead to large numerical differences in the pressure.'}

{'hep-ph-0512177-2-0-0': 'We extend our previous determination of the thermodynamic pressure of the Standard Model so that the result can be applied down to temperatures corresponding to the electroweak crossover.', 'hep-ph-0512177-2-0-1': 'This requires a further resummation which can be cleanly organised within the effective theory framework.', 'hep-ph-0512177-2-0-2': 'The result allows for a precise determination of the expansion rate of the Universe for temperatures around the electroweak crossover.', 'hep-ph-0512177-2-1-0': 'diagrams', 'hep-ph-0512177-2-2-0': 'style_def dash_dot expr p = save dpp, k; numeric dpp, k; dpp = ceiling (pixlen (p, 10) / (1.5*dash_len)) / length p; k=0; forever: exitif k+.33 > dpp*length(p); cdraw point k/dpp of p .', 'hep-ph-0512177-2-2-1': 'point (k+.33)/dpp of p; exitif k+.67 > dpp*length(p); cdrawdot point (k+.67)/dpp of p; k := k+1; endfor enddef;', 'hep-ph-0512177-2-3-0': 'style_def dash_dot_arrow expr p = draw_dash_dot p; cfill (arrow p); enddef;', 'hep-ph-0512177-2-4-0': '# Introduction', 'hep-ph-0512177-2-5-0': 'In a recent paper [CITATION] we calculated the pressure of the standard model at high temperatures to three loops, or to order [MATH] in the coupling constants.', 'hep-ph-0512177-2-5-1': 'That work followed a long series of calculations dedicated to understanding the perturbative expansion of the pressure of gauge field theories at high temperatures.', 'hep-ph-0512177-2-5-2': 'Especially, within QCD such calculations have been important: the computation of the coefficients of the expansion in [MATH] at high temperature has a long history and the result is known today up to the last perturbatively calculable term of order [MATH] [CITATION], marking an endpoint to an impressive set of computations carried out in [CITATION].', 'hep-ph-0512177-2-5-3': 'An optimal approach to calculating those coefficients is the effective theory method [CITATION], based on asymptotic freedom and on separating the relevant mass scales: [MATH], the electric scale [MATH] and the magnetic scale [MATH].', 'hep-ph-0512177-2-6-0': 'An obvious drawback of those calculations within QCD is that they cannot be extended to study the QCD phase transition since the coupling grows large and thus any perturbative calculations become unreliable.', 'hep-ph-0512177-2-6-1': 'That is not the case for the electroweak sector of the standard model.', 'hep-ph-0512177-2-6-2': 'The Landau pole related to the weak interactions corresponds to a length scale [MATH] m and therefore the confining effects can be expected to be negligible.', 'hep-ph-0512177-2-6-3': 'As a result, calculating the properties of the electroweak phase transition using, for example, perturbative 1-loop [CITATION] and 2-loop [CITATION] effective potential calculations was possible for small Higgs masses.', 'hep-ph-0512177-2-6-4': 'A better approach, well defined for large Higgs masses as well, is to perturbatively match the full 4-dimensional theory to an effective 3-dimensional theory [CITATION] and then to numerically solve the phase diagram from the effective theory using lattice Monte Carlo techniques [CITATION].', 'hep-ph-0512177-2-6-5': 'Such studies show that the phase diagram has a first order line which ends in a 2nd order critical point of Ising universality class [CITATION].', 'hep-ph-0512177-2-6-6': 'For experimentally allowed Higgs masses the transition is a crossover.', 'hep-ph-0512177-2-6-7': 'Similar techniques have been used to solve the phase diagram also when the external U([MATH]) magnetic field [CITATION] or the chemical potentials related to the baryon and lepton numbers [CITATION] are nonzero.', 'hep-ph-0512177-2-6-8': 'Solving the phase diagram with numerical studies of the full 4-dimensional theory has also been achieved[CITATION].', 'hep-ph-0512177-2-7-0': 'Although the properties of the electroweak phase transition are known today, computation of the pressure of the theory has been missing.', 'hep-ph-0512177-2-7-1': 'However, the ongoing and future measurements of cosmic microwave background radiation (WMAP and Planck) allow for a precise determination of, for example, the WIMP relic density, which depends on the equation of state of matter in the early universe.', 'hep-ph-0512177-2-7-2': 'A precise determination of the pressure is needed to match the accuracy of the observations [CITATION].', 'hep-ph-0512177-2-7-3': 'In [CITATION] we performed this calculation when the temperature of the system is high.', 'hep-ph-0512177-2-7-4': 'In the present paper we will extend that result to temperatures close to the critical temperature of the electroweak phase transition.', 'hep-ph-0512177-2-7-5': 'Critical temperature and the phase transition should be understood in perturbation theory framework, where there is always a first order phase transition.', 'hep-ph-0512177-2-7-6': 'We will use these terms throughout this paper, although the actual transition is just a crossover.', 'hep-ph-0512177-2-8-0': 'The present computation requires a reorganization of the effective theories.', 'hep-ph-0512177-2-8-1': 'Close to the transition the fundamental scalar (which drives the transition) becomes light with respect to the electric scale and thus in a consistent calculation we have to formulate an additional effective theory by integrating out the adjoint scalars.', 'hep-ph-0512177-2-8-2': 'The remaining theory contains just the fundamental scalar and the gauge fields.', 'hep-ph-0512177-2-8-3': 'The pressure will then be composed of three parts: the contribution from fermions and the nonzero Matsubara modes of bosons [MATH], from the adjoint scalars [MATH] and [MATH], [MATH] and finally from the fundamental scalar [MATH] where [MATH] and [MATH].', 'hep-ph-0512177-2-8-4': 'At each step of the calculation there will be [MATH] poles from ultraviolet and infrared divergences which will not cancel until all the contributions are summed together.', 'hep-ph-0512177-2-9-0': 'The paper is organized as follows: in section [REF] we explicitly define the theory we are working with, fix various conventions and briefly review the method of dimensional reduction as applied to the present case.', 'hep-ph-0512177-2-9-1': 'Sections [REF] and [REF] contain the essential calculations and in section [REF] we discuss the result.', 'hep-ph-0512177-2-10-0': '# Basic setting', 'hep-ph-0512177-2-11-0': 'We consider the [MATH] standard model with [MATH] families of fermions and [MATH] fundamental scalar doublets, and evaluate the pressure of this theory at temperatures slightly above the electroweak phase transition.', 'hep-ph-0512177-2-11-1': 'The theory is specified by the Euclidean action (in the units [MATH]) [EQUATION] where [EQUATION]', 'hep-ph-0512177-2-11-2': 'Here [MATH], [MATH] and [MATH] are gauge bosons of weak-, hyper- and strong interactions, respectively; [MATH] is the fundamental scalar doublet; [MATH] and [MATH] are the left-handed lepton doublets and the right-handed lepton singlets (wrt. weak charge), and [MATH], [MATH] and [MATH] are the left-handed quark doublets and the right-handed up and down -type quark singlets.', 'hep-ph-0512177-2-11-3': 'The Yukawa coupling is taken into account for the top quark only.', 'hep-ph-0512177-2-11-4': 'Summation over different families is assumed.', 'hep-ph-0512177-2-11-5': 'Also, [MATH] in dimensional regularization, [MATH].', 'hep-ph-0512177-2-11-6': 'The gamma matrices are defined in Euclidean space so that [MATH], [MATH] and [MATH].', 'hep-ph-0512177-2-11-7': 'The color indices are [MATH] for the weak interaction and [MATH] for the strong interaction.', 'hep-ph-0512177-2-11-8': 'The different group theory factors for SU([MATH]) with generators [MATH] are defined as:', 'hep-ph-0512177-2-12-0': '0.51 [EQUATION] 0.46 [EQUATION]', 'hep-ph-0512177-2-12-1': 'For SU(2) with [MATH] they are [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0512177-2-13-0': 'The theory in Eq. ([REF]) contains six couplings that run with the renormalization scale: gauge couplings [MATH], [MATH] and [MATH], the fundamental scalar quartic self-coupling [MATH] and its mass parameter [MATH], and [MATH].', 'hep-ph-0512177-2-13-1': 'We fix the values of these couplings at the scale [MATH] according to their tree-level relation with different physical parameters as in [CITATION], where the 1-loop running of the parameters is also given.', 'hep-ph-0512177-2-13-2': 'We employ a power counting rule [MATH] and assume the temperature to be such that [MATH].', 'hep-ph-0512177-2-14-0': 'The physical observable we are studying is the pressure, defined by [EQUATION]', 'hep-ph-0512177-2-14-1': 'It is normalized such that the (real part of the) pressure of the symmetric phase vanishes at [MATH].', 'hep-ph-0512177-2-14-2': 'Other interesting variables, such as entropy and energy densities [MATH] and [MATH], can then be evaluated using standard thermodynamic relations, [MATH], [MATH].', 'hep-ph-0512177-2-14-3': 'The purpose is to calculate the pressure up to, and including, order [MATH], employing the power counting rules above.', 'hep-ph-0512177-2-14-4': 'For high temperatures, [MATH], this calculation was performed in [CITATION].', 'hep-ph-0512177-2-14-5': 'However, close to the phase transition, which is often the most interesting region, our earlier computation is not valid, since the (thermal) fundamental scalar mass is much smaller than the Debye masses and introduces another hierarchy of scales which needs to be sorted out.', 'hep-ph-0512177-2-14-6': 'The ratio of the two mass scales at different temperatures is shown in Fig. [REF], which illustrates that there is a range of temperatures in which [MATH].', 'hep-ph-0512177-2-14-7': 'The expression used for [MATH] in Fig. [REF] is Eq. ([REF]), renormalized in the [MATH] scheme and [MATH] set to 0.', 'hep-ph-0512177-2-15-0': 'It is well known that simply evaluating all 1-, 2- and 3-loop vacuum diagrams in perturbation theory fails because of infrared divergences.', 'hep-ph-0512177-2-15-1': 'Instead, one has to separate the contributions of different scales into successive effective theories [CITATION], where all the large scales are integrated out one by one.', 'hep-ph-0512177-2-15-2': 'First, we integrate [EQUATION] where [MATH] contains only the static Matsubara modes of the gauge bosons and of the fundamental scalar (Higgs) field.', 'hep-ph-0512177-2-15-3': 'The contributions of the nonzero Matsubara modes and fermions to the pressure show up as the matching constant [MATH] (App.', 'hep-ph-0512177-2-15-4': '[REF]) and in the parameters of [MATH] (App.', 'hep-ph-0512177-2-15-5': '[REF]).', 'hep-ph-0512177-2-15-6': 'The spatial (magnetic) gauge field components remain massless, while the temporal component gets a thermal mass [MATH].', 'hep-ph-0512177-2-15-7': 'The theory defined by [MATH] can then be viewed as a 3d gauge theory with adjoint and fundamental scalar fields.', 'hep-ph-0512177-2-16-0': 'The effective theory thus obtained still contains contributions from several scales, [MATH], [MATH] and the magnetic scale [MATH].', 'hep-ph-0512177-2-16-1': 'We restrict ourselves to temperatures close to the phase transition, where [MATH] is small compared to the Debye masses and assume [MATH], motivated by Eq. ([REF]).', 'hep-ph-0512177-2-16-2': 'We integrate out the scale [MATH], i.e. the fields [MATH] and [MATH], and are left with [EQUATION]', 'hep-ph-0512177-2-16-3': 'This is a 3d gauge theory with a fundamental scalar field.', 'hep-ph-0512177-2-16-4': 'The largest mass scale in this theory is the fundamental scalar mass [MATH], so this theory contributes at [MATH] or higher order, and we need to evaluate only 1- and 2-loop vacuum diagrams from this theory.', 'hep-ph-0512177-2-16-5': 'The potential infrared problems can be isolated to another effective theory by integrating out the fundamental scalar, [EQUATION]', 'hep-ph-0512177-2-16-6': 'The remaining effective theory contains only the (massless) spatial gauge fields.', 'hep-ph-0512177-2-16-7': 'Therefore the only mass scale of the theory is provided by the 3d gauge coupling and is of the order [MATH] and consequently the contribution of this theory to the pressure is of the order [MATH].', 'hep-ph-0512177-2-16-8': 'The final result of our calculation can then be written as [EQUATION] where [MATH] can be taken from [CITATION].', 'hep-ph-0512177-2-16-9': 'One-loop quark diagrams are included in [MATH], so they must be subtracted from [MATH].', 'hep-ph-0512177-2-16-10': 'The pressure [MATH] was computed in [CITATION], and is given in Appendix [REF], whereas [MATH] and [MATH] are computed in Sections [REF] and [REF], respectively.', 'hep-ph-0512177-2-17-0': '# Calculation of the pressure [MATH]', 'hep-ph-0512177-2-18-0': 'In this section we integrate over the scale [MATH].', 'hep-ph-0512177-2-18-1': 'This means that we need to integrate out the adjoint scalars [MATH] and [MATH], leaving an effective theory with only gauge bosons and the fundamental scalar field.', 'hep-ph-0512177-2-19-0': 'The 3d theory of gauge fields with fundamental and adjoint scalars is given by [EQUATION] where the parameters were computed in [CITATION] to required order and are listed in Appendix [REF].', 'hep-ph-0512177-2-20-0': 'To find out the contribution of the adjoint scalars to the pressure we need to calculate all the vacuum diagrams in the theory ([REF]) containing these fields.', 'hep-ph-0512177-2-20-1': 'Since both [MATH] and [MATH] are massive, there are no infrared divergences related to this integration.', 'hep-ph-0512177-2-20-2': 'The mass of the fundamental scalar [MATH] is parametrically smaller ([MATH]) so we treat it as a perturbation and expand the integrands in [MATH].', 'hep-ph-0512177-2-20-3': 'In fact, the leading [MATH] order is sufficient to our computations.', 'hep-ph-0512177-2-20-4': 'The required diagrams are shown in Fig. [REF] and are evaluated in [CITATION].', 'hep-ph-0512177-2-21-0': 'The result, with [MATH] and [MATH], reads [EQUATION]', 'hep-ph-0512177-2-21-1': 'The divergences and the scale dependence in [MATH] cancel against those in [MATH], and the ones in [MATH] against the sunset diagram in [MATH], Eq. ([REF]).', 'hep-ph-0512177-2-22-0': '# Calculation of the pressure [MATH]', 'hep-ph-0512177-2-23-0': 'The theory containing just the gauge fields and the fundamental scalar is defined by [EQUATION] where the field strength tensors [MATH] and [MATH] and the covariant derivative [MATH] are defined as before (with couplings [MATH] and [MATH]).', 'hep-ph-0512177-2-23-1': 'The integration from [MATH] to [MATH] will not introduce any new corrections to the couplings to get the pressure to the required order and therefore [MATH].', 'hep-ph-0512177-2-23-2': 'The scalar mass does, however, get an additional contribution (calculated in [CITATION] apart from the term [MATH] which is needed in the present calculation) coming from the tadpole diagrams with adjoint scalars at the loops: [EQUATION]', 'hep-ph-0512177-2-23-3': 'Since the pressure is to leading order given by [MATH] and since we assume the temperature to be such that [MATH], we see that corrections of order [MATH] to the scalar mass would contribute to pressure at order [MATH] and we will neglect those.', 'hep-ph-0512177-2-23-4': 'Similarly, we can neglect the [MATH] corrections to [MATH].', 'hep-ph-0512177-2-23-5': 'The running of the mass [MATH] (and hence of [MATH] as well), which sets in at the order [MATH] does not therefore influence the present calculation as it did the high temperature calculation.', 'hep-ph-0512177-2-23-6': 'Furthermore, even though [MATH] is assumed to be of the order [MATH] near the phase transition, the expression for [MATH] does contain individual terms of the order [MATH] which, when summed up, effectively cancel to order [MATH] (assuming proper temperature region).', 'hep-ph-0512177-2-23-7': 'The individual terms of order [MATH] must, however, be taken carefully into account when cancelling the [MATH] poles in the pressure.', 'hep-ph-0512177-2-24-0': 'The pressure is obtained by calculating the diagrams in Fig. [REF].', 'hep-ph-0512177-2-24-1': 'Three-loop diagrams would contribute to order [MATH] and are neglected now.', 'hep-ph-0512177-2-24-2': 'The result is [EQUATION]', 'hep-ph-0512177-2-24-3': 'The [MATH] poles cancel against those coming from [MATH] and [MATH].', 'hep-ph-0512177-2-25-0': '# Numerical results', 'hep-ph-0512177-2-26-0': 'In this section we will consider some numerical implications of the result, given by Eq. ([REF]).', 'hep-ph-0512177-2-26-1': 'The convergence and the scale dependence of the perturbative expansion were already studied in [CITATION] so now we will just concentrate on the difference between the results of that paper and the present paper, which are significant when the fundamental scalar becomes light.', 'hep-ph-0512177-2-26-2': 'The Higgs mass is taken to be [MATH] GeV and the W mass [MATH] GeV.', 'hep-ph-0512177-2-27-0': 'It is instructive to first look at a simpler SU(2) + fundamental Higgs model for which the perturbative expansion is better behaved.', 'hep-ph-0512177-2-27-1': 'In Fig. [REF] we have plotted the pressure of this theory in the temperature region where the fundamental scalar becomes much lighter than the adjoint scalars (see Fig. [REF]) using both the calculation valid at high temperatures ([MATH]) and the calculation valid near the phase transition ([MATH]), normalized to [MATH] (ideal gas pressure of SU(2) gauge bosons and a massless fundamental scalar).', 'hep-ph-0512177-2-27-2': 'It can be immediately noticed that there is a temperature region where the high temperature calculation is not well behaved but becomes singular.', 'hep-ph-0512177-2-27-3': 'This is due to the terms of the type [MATH] in the expansion of [MATH] which become singular when [MATH].', 'hep-ph-0512177-2-27-4': 'In the present calculation which takes into account that [MATH] such terms will not appear and therefore [MATH] is seen to be well behaved.', 'hep-ph-0512177-2-27-5': 'It is worth noting, however, that the temperature region where the singular behavior is manifest is very narrow.', 'hep-ph-0512177-2-27-6': 'That is due to the small gauge coupling: the singular terms are of the order [MATH] and their contribution to the pressure vanishes fast as one moves away from the singular point.', 'hep-ph-0512177-2-27-7': 'More specifically, the singular terms behave as: [EQUATION] where [MATH] is the location of the singularity and [MATH].', 'hep-ph-0512177-2-27-8': 'Since the curve behaves as [MATH] near the singularity and since the prefactor is so small, the contribution from the singularity becomes small very fast as [MATH] is increased.', 'hep-ph-0512177-2-28-0': 'If one excludes this narrow region around the singularity, the pressure [MATH] is smaller than the pressure [MATH] roughly by constant[MATH].', 'hep-ph-0512177-2-28-1': 'The relative difference between [MATH] and [MATH] is small, [MATH] but it nevertheless is of the same order of magnitude as differences in the pressure at different orders in perturbation theory and hence it is important to take this effect into account when calculating the pressure to high accuracy.', 'hep-ph-0512177-2-28-2': 'This is related to the reorganization of the effective theories.', 'hep-ph-0512177-2-28-3': 'Keeping the fundamental scalar mass, as defined in Eq. ([REF]), unexpanded when inserted to the expression for the pressure [MATH], Eq. ([REF]), effectively resums all the ring diagrams of the type shown in Fig. [REF] to the pressure.', 'hep-ph-0512177-2-28-4': 'The high temperature calculation, however, corresponds to expanding this mass in terms of [MATH] in the expression for the pressure and to keeping just the lowest order terms of the expansion.', 'hep-ph-0512177-2-28-5': 'At high temperatures this expansion can be made, since [MATH] but close to the phase transition [MATH] and the expansion is not possible.', 'hep-ph-0512177-2-28-6': 'Moreover, even though differences in the pressure when calculated from the high temperature theory as compared to the correct calculation near the phase transition are small, this is not necesserily the case for other interesting variables such as the critical temperature and to obtain these correctly, one needs to use the theory that takes into account that the fundamental scalar becomes light.', 'hep-ph-0512177-2-29-0': 'Although we do not study the specific features of the phase transition, which are already well established [CITATION], we nevertheless plot the pressure of the broken phase as well in Fig. [REF] to order [MATH] to indicate the temperature where the phase transition takes place.', 'hep-ph-0512177-2-29-1': 'In the previous sections the pressure was calculated in the symmetric phase and thus the result cannot be extended to the broken symmetry phase as such.', 'hep-ph-0512177-2-29-2': 'However, we can make use of the effective potential calculations [CITATION] and write [MATH], where the effective potential is normalized so that [MATH].', 'hep-ph-0512177-2-29-3': 'The pressure in the broken phase to [MATH] is then given by [EQUATION] where [MATH], [MATH] and [MATH] are the zero temperature masses of the particles, and [MATH] (the expectation value of the Higgs field) is such that [MATH].', 'hep-ph-0512177-2-29-4': 'The critical temperature below which the pressure of the broken phase is bigger than the pressure of the symmetric phase, indicating that the symmetry is spontanously broken, is, for [MATH] GeV and [MATH] GeV, about [MATH] GeV.', 'hep-ph-0512177-2-30-0': 'In Fig. [REF] we have plotted the pressures [MATH] and [MATH] for the full standard model, normalized to [MATH].', 'hep-ph-0512177-2-30-1': 'A similar structure can be seen to appear here: the high temperature calculation of the pressure developes a singularity when [MATH], but the temperature region where this is of importance is very narrow, while the calculation which takes into account that near the phase transition [MATH] is seen to behave well.', 'hep-ph-0512177-2-30-2': 'Excluding the singularity, the pressures are again observed to differ roughly by a constant[MATH], but the difference is not in this case even as large as in the SU(2) + Higgs theory, especially when compared to the corrections introduced by each new order of perturbation theory.', 'hep-ph-0512177-2-30-3': 'This reflects the fact that the fundamental scalar carries only a small part of the total number of degrees of freedom and hence changes in the scalar sector of the standard model do not lead to significant changes in the pressure.', 'hep-ph-0512177-2-31-0': '# Conclusions', 'hep-ph-0512177-2-32-0': 'In this paper we have extended the calculation of the pressure of the standard model at high temperatures carried out in [CITATION] to the case when the temperature of the system is close to the critical temperature of the electroweak phase transition [MATH].', 'hep-ph-0512177-2-32-1': 'The previous calculation was not consistent in that region since it assumed that the fundamental scalar has a mass comparable to the masses of the adjoint scalars while, in fact, near the phase transition the fundamental scalar becomes light.', 'hep-ph-0512177-2-32-2': 'This inconsistency manifests itself as an unphysical singularity in the pressure if one tries to apply the result of the previous calculation to temperatures near [MATH].', 'hep-ph-0512177-2-32-3': 'The calculation performed in this paper is done in a consistant manner, taking into account the lightness of the fundamental scalar near the phase transition and no such unphysical singularities remain in the final result.', 'hep-ph-0512177-2-33-0': 'It is possible to go still a bit further in the perturbative expansion of the pressure, the next term would be of the order [MATH].', 'hep-ph-0512177-2-33-1': 'Its calculation would require a two-loop matching of the fundamental scalar mass (order [MATH]) and a three-loop calculation of the pressure [MATH], both of which are in principle manageable.', 'hep-ph-0512177-2-33-2': 'Furthermore, one could determine the final perturbatively calculable term [MATH] by calculating the vacuum energy densities of the three dimensional effective theories.', 'hep-ph-0512177-2-33-3': 'However, since the convergence of the perturbative expansion is rather fast (as noted in [CITATION]), especially for the SU(2) + Higgs theory, the calculation of those terms would not lead to large numerical differences in the pressure.'}

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'hep-ph-0512177-2-16-9'], ['hep-ph-0512177-1-16-10', 'hep-ph-0512177-2-16-10'], ['hep-ph-0512177-1-13-0', 'hep-ph-0512177-2-13-0'], ['hep-ph-0512177-1-13-1', 'hep-ph-0512177-2-13-1'], ['hep-ph-0512177-1-13-2', 'hep-ph-0512177-2-13-2'], ['hep-ph-0512177-1-23-0', 'hep-ph-0512177-2-23-0'], ['hep-ph-0512177-1-23-1', 'hep-ph-0512177-2-23-1'], ['hep-ph-0512177-1-23-2', 'hep-ph-0512177-2-23-2'], ['hep-ph-0512177-1-23-3', 'hep-ph-0512177-2-23-3'], ['hep-ph-0512177-1-23-4', 'hep-ph-0512177-2-23-4'], ['hep-ph-0512177-1-23-5', 'hep-ph-0512177-2-23-5'], ['hep-ph-0512177-1-23-6', 'hep-ph-0512177-2-23-6'], ['hep-ph-0512177-1-23-7', 'hep-ph-0512177-2-23-7'], ['hep-ph-0512177-1-8-0', 'hep-ph-0512177-2-8-0'], ['hep-ph-0512177-1-8-1', 'hep-ph-0512177-2-8-1'], ['hep-ph-0512177-1-8-2', 'hep-ph-0512177-2-8-2'], ['hep-ph-0512177-1-8-3', 'hep-ph-0512177-2-8-3'], ['hep-ph-0512177-1-8-4', 'hep-ph-0512177-2-8-4'], ['hep-ph-0512177-1-3-0', 'hep-ph-0512177-2-3-0'], ['hep-ph-0512177-1-2-0', 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[['hep-ph-0512177-1-29-1', 'hep-ph-0512177-2-29-3']]

[]

[]

[]

['hep-ph-0512177-1-1-0', 'hep-ph-0512177-1-2-1', 'hep-ph-0512177-1-11-8', 'hep-ph-0512177-1-12-0', 'hep-ph-0512177-1-12-1', 'hep-ph-0512177-1-15-5', 'hep-ph-0512177-2-1-0', 'hep-ph-0512177-2-2-1', 'hep-ph-0512177-2-11-8', 'hep-ph-0512177-2-12-0', 'hep-ph-0512177-2-12-1', 'hep-ph-0512177-2-15-5']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0512177

quant-ph-9903094

{'quant-ph-9903094-1-0-0': 'We describe an experiment in which a mirror is cooled by the radiation pressure of light.', 'quant-ph-9903094-1-0-1': 'A high-finesse optical cavity with a mirror coated on a mechanical resonator is used as an optomechanical sensor of the Brownian motion of the mirror.', 'quant-ph-9903094-1-0-2': 'A feedback mechanism controls this motion via the radiation pressure of a laser beam reflected on the mirror.', 'quant-ph-9903094-1-0-3': 'We have observed either a cooling or a heating of the mirror, depending on the gain of the feedback loop.', 'quant-ph-9903094-1-1-0': 'PACS : 05.40.', 'quant-ph-9903094-1-1-1': 'Jc, 04.80.', 'quant-ph-9903094-1-1-2': 'Nn, 42.50.', 'quant-ph-9903094-1-1-3': 'Lc Thermal noise is a basic limit for many very sensitive optical measurements such as interferometric gravitational-wave detection [CITATION].', 'quant-ph-9903094-1-1-4': 'Brownian motion of suspended mirrors can be decomposed into suspension and internal thermal noises.', 'quant-ph-9903094-1-1-5': 'The latter is due to thermally induced deformations of the mirror surface and constitutes the major limitation of gravitational-wave detectors in the intermediate frequency domain[CITATION].', 'quant-ph-9903094-1-1-6': 'Observation and control of this noise have thus become an important issue in precision measurements[CITATION].', 'quant-ph-9903094-1-1-7': 'In order to reduce thermal noise effects, it is not always possible to lower the temperature and other techniques have been proposed such as a feedback control of a macroscopic oscillator[CITATION].', 'quant-ph-9903094-1-2-0': 'In this letter we report the first experimental observation of the cooling of a mirror by feedback control via radiation pressure.', 'quant-ph-9903094-1-2-1': 'The basic principle of the experiment is to detect the Brownian motion of the mirror with an optomechanical sensor and then to freeze the motion by applying an electronically controlled radiation pressure on the mirror.', 'quant-ph-9903094-1-2-2': 'Some mechanical effects of light on macroscopic objects have already been observed, such as the dissipative effects of electromagnetic radiation[CITATION], the optical bistability and mirror confinement in a cavity induced by radiation pressure[CITATION], or the regulation of the mechanical response of a microcantilever by feedback via the photothermal force[CITATION].', 'quant-ph-9903094-1-2-3': 'In our experiment the radiation pressure is driven by the feedback loop in such a way that a viscous force is applied to the mirror.', 'quant-ph-9903094-1-2-4': 'It thus plays a role somewhat similar to the one in optical molasses for atoms.', 'quant-ph-9903094-1-3-0': 'The cooling mechanism can be understood from the experimental setup shown in figure [REF].', 'quant-ph-9903094-1-3-1': 'The mirror is used as the rear mirror of a single-ended Fabry-Perot cavity.', 'quant-ph-9903094-1-3-2': 'The phase of the field reflected by the cavity is very sensitive to changes in the cavity length[CITATION].', 'quant-ph-9903094-1-3-3': 'For a resonant cavity, a displacement [MATH] of the rear mirror induces a phase shift [MATH] of the reflected field on the order of [EQUATION] where [MATH] is the quantum noise of the phase of the incident beam, [MATH] is the cavity finesse and [MATH] is the optical wavelength.', 'quant-ph-9903094-1-3-4': 'The Brownian motion of the mirror can be detected by measuring the phase of the reflected field, provided that the cavity finesse is high enough[CITATION].', 'quant-ph-9903094-1-4-0': 'To cool the mirror we use an auxiliary beam derived from the laser and reflected from the back on the mirror.', 'quant-ph-9903094-1-4-1': 'This beam is intensity-modulated by an acousto-optic modulator driven by the feedback loop so that a modulated radiation pressure is applied to the mirror.', 'quant-ph-9903094-1-4-2': 'The resulting motion can be described by its Fourier transform [MATH] at frequency [MATH] which is proportional to the applied forces [EQUATION] where [MATH] is the mechanical susceptibility of the mirror.', 'quant-ph-9903094-1-4-3': 'If we assume that the mechanical response is harmonic, this susceptibility has a lorentzian shape [EQUATION] characterized by a mass [MATH], a resonance frequency [MATH] and a damping [MATH] related to the quality factor [MATH] of the mechanical resonance by [MATH].', 'quant-ph-9903094-1-5-0': 'The first force [MATH] in eq. ([REF]) is a Langevin force responsible for the Brownian motion of the mirror.', 'quant-ph-9903094-1-5-1': 'At thermal equilibrium its spectrum [MATH] is related to the mechanical susceptibility by the fluctuation-dissipation theorem [EQUATION] where [MATH] is the temperature.', 'quant-ph-9903094-1-5-2': 'The resulting thermal noise spectrum [MATH] of the mirror motion has a lorentzian shape centered at frequency [MATH] and of width [MATH].', 'quant-ph-9903094-1-6-0': 'The second force [MATH] in eq. ([REF]) is the radiation pressure exerted by the auxiliary laser beam and modulated by the feedback loop.', 'quant-ph-9903094-1-6-1': 'Neglecting the quantum noise [MATH] in the control signal [MATH] (eq. [REF]), this force is proportional to the displacement [MATH] of the mirror.', 'quant-ph-9903094-1-6-2': 'We choose the feedback gain in such a way that the radiation pressure is proportional to the speed [MATH] of the mirror [EQUATION] where [MATH] is related to the electronic gain.', 'quant-ph-9903094-1-6-3': 'The radiation pressure exerted by the auxiliary laser beam thus corresponds to an additional viscous force for the mirror.', 'quant-ph-9903094-1-6-4': 'The resulting motion is given by [EQUATION]', 'quant-ph-9903094-1-6-5': 'This equation is similar to the one obtained without feedback (eq. [REF] with [MATH]) except that the radiation pressure changes the damping without adding any fluctuations.', 'quant-ph-9903094-1-6-6': 'The noise spectrum [MATH] of the mirror motion still has a lorentzian shape but with a different width [MATH] and a different height.', 'quant-ph-9903094-1-6-7': 'The variation of height can be characterized by the amplitude noise reduction [MATH] at resonance frequency [EQUATION]', 'quant-ph-9903094-1-6-8': 'The resulting motion is then equivalent to a thermal equilibrium at a different temperature [MATH] given by [EQUATION]', 'quant-ph-9903094-1-6-9': 'One can either reduce or increase this effective temperature, depending on the sign of the gain [MATH].', 'quant-ph-9903094-1-7-0': 'The model presented here is of course oversimplified.', 'quant-ph-9903094-1-7-1': 'In particular the mirror motion depends on many internal acoustic modes and a description as a single harmonic oscillator may not be appropriate.', 'quant-ph-9903094-1-7-2': 'We have also neglected the quantum noise [MATH] in the feedback loop which would induce a lower limit for the effective temperature.', 'quant-ph-9903094-1-7-3': 'This simplified model is however satisfactory to interpret the main features of our current experimental results.', 'quant-ph-9903094-1-7-4': 'In particular we focus in the following on the fundamental acoustic mode of the mirror which has a harmonic response.', 'quant-ph-9903094-1-8-0': 'We now describe in detail our experiment.', 'quant-ph-9903094-1-8-1': 'The mirror is coated on the plane side of a small plano-convex mechanical resonator made of silica (figure [REF]).', 'quant-ph-9903094-1-8-2': 'The coating has been made at the Institut de Physique Nucleaire de Lyon on a 1.5-mm thick substrate with a diameter of 14 mm and a curvature radius of the convex side of 100 mm.', 'quant-ph-9903094-1-8-3': 'Internal acoustic modes correspond to gaussian modes confined around the central axis of the resonator[CITATION].', 'quant-ph-9903094-1-8-4': 'The fundamental mode studied in this paper is a compression mode with a waist equal to 3.4 mm, a resonance frequency close to 2 MHz and an effective mass of 30 mg[CITATION].', 'quant-ph-9903094-1-9-0': 'The front mirror of the high-finesse cavity has a curvature radius of 1 meter and a typical transmission of 50 ppm (Newport high-finesse SuperMirror).', 'quant-ph-9903094-1-9-1': 'This mirror is held at a fixed distance of 1 mm from the back mirror by a rigid cylinder.', 'quant-ph-9903094-1-9-2': 'We have measured the following parameters of the cavity : free spectral range = 141 GHz, cavity bandwidth = 1.9 MHz, beam waist = 90 [MATH]m.', 'quant-ph-9903094-1-9-3': 'These values correspond to a finesse [MATH] of 37000.', 'quant-ph-9903094-1-10-0': 'The light entering the cavity is provided by a titane-sapphire laser working at 810 nm and frequency-locked to a stable external cavity.', 'quant-ph-9903094-1-10-1': 'The frequency is also locked to a resonance of the high-finesse cavity by monitoring the residual light transmitted by the rear mirror via a control of the external cavity length.', 'quant-ph-9903094-1-10-2': 'The laser intensity is actively stabilized and a mode cleaner reduces the astigmatism of the beam.', 'quant-ph-9903094-1-10-3': 'The light power incident on the cavity is equal to 100 [MATH]W with a mode matching of 98%.', 'quant-ph-9903094-1-11-0': 'The phase of the field reflected by the high-finesse cavity is measured by homodyne detection.', 'quant-ph-9903094-1-11-1': 'The reflected field is mixed on two photodiodes with a 10-mW local oscillator derived from the incident beam.', 'quant-ph-9903094-1-11-2': 'A servoloop monitors the length of the local oscillator arm so that the reflected field is in quadrature with the local oscillator.', 'quant-ph-9903094-1-11-3': 'The difference between the two photocurrents is then proportional to the phase fluctuations [MATH] of the reflected field.', 'quant-ph-9903094-1-11-4': 'This signal is sent both to the feedback loop and to a spectrum analyzer.', 'quant-ph-9903094-1-12-0': 'The feedback loop consists of an amplifier with variable gain and phase which drives the acousto-optic modulator.', 'quant-ph-9903094-1-12-1': 'The 500-mW auxiliary beam is uncoupled from the high-finesse cavity by a frequency shift of the acousto-optic modulator (200 MHz) and by a tilt angle of 10[MATH] with respect to the cavity axis.', 'quant-ph-9903094-1-12-2': 'We have checked that this beam has no spurious effect on the homodyne detection.', 'quant-ph-9903094-1-13-0': 'A band-pass filter centered at the fundamental resonance frequency of the mirror is also inserted in the feedback loop to reduce its saturation.', 'quant-ph-9903094-1-13-1': 'For large gains, the radiation pressure [MATH] can become of the same order as the Langevin force [MATH] and must be restricted in frequency in order to get a finite variance.', 'quant-ph-9903094-1-13-2': 'The electronic filter has a quality factor of 200 and limits the efficiency of the feedback loop to a bandwidth of 9 kHz around the fundamental resonance frequency.', 'quant-ph-9903094-1-14-0': 'Figure [REF] shows the phase noise spectrum of the reflected field obtained by an average of 1000 scans of the spectrum analyzer with a resolution bandwidth of 10 Hz.', 'quant-ph-9903094-1-14-1': 'Curve (a) is obtained at room temperature without feedback.', 'quant-ph-9903094-1-14-2': 'It reproduces the thermal noise spectrum [MATH] of the mirror which is concentrated around the fundamental resonance frequency (1858.9 kHz) with a width [MATH] of 45 Hz (mechanical quality factor [MATH]).', 'quant-ph-9903094-1-14-3': 'The spectrum is normalized to the shot-noise level and it clearly appears that the thermal noise is much larger than the quantum noise of the measurement[CITATION].', 'quant-ph-9903094-1-15-0': 'Curves (b) to (d) are obtained with feedback for increasing electronic gains.', 'quant-ph-9903094-1-15-1': 'The phase of the amplifier is adjusted to maximize the correction at resonance.', 'quant-ph-9903094-1-15-2': 'From eqs. ([REF]) and ([REF]) this corresponds to a global imaginary gain for the loop and to a purely viscous radiation pressure force.', 'quant-ph-9903094-1-15-3': 'The control of the mirror motion is clearly visible on those curves.', 'quant-ph-9903094-1-15-4': 'The thermal peak is strongly reduced while its width is increased.', 'quant-ph-9903094-1-15-5': 'The amplitude noise reduction [MATH] at resonance is larger than 20 for large gains (eq. [REF]).', 'quant-ph-9903094-1-16-0': 'The effective temperature [MATH] can be deduced from the variance [MATH] of the mirror motion which is equal to the integral of the spectrum [MATH].', 'quant-ph-9903094-1-16-1': 'From eqs. ([REF]), ([REF]) and ([REF]) one gets the usual relation for a harmonic oscillator at thermal equilibrium [EQUATION]', 'quant-ph-9903094-1-16-2': 'The decrease of the area of the thermal peak observed in figure [REF] thus corresponds to a cooling of the mirror.', 'quant-ph-9903094-1-17-0': 'Figure [REF] shows the effect of feedback for a reverse gain ([MATH]).', 'quant-ph-9903094-1-17-1': 'Noise spectra are obtained by an average of 500 scans with a resolution bandwidth of 1 Hz.', 'quant-ph-9903094-1-17-2': 'Curve (b) exhibits a strong increase of the thermal peak which now corresponds to a heating of the mirror.', 'quant-ph-9903094-1-17-3': 'The feedback also reduces the damping from [MATH] to [MATH], thus increasing the quality factor of the resonance.', 'quant-ph-9903094-1-17-4': 'We have obtained a maximum effective quality factor of [MATH]), limited by the saturation of the feedback loop.', 'quant-ph-9903094-1-18-0': 'It is finally instructive to study the efficiency of the cooling or heating mechanism with respect to the gain of the feedback loop.', 'quant-ph-9903094-1-18-1': 'Figure [REF] shows the variation of the damping [MATH], of the amplitude noise reduction [MATH] at resonance and of the cooling factor [MATH], as a function of the feedback gain.', 'quant-ph-9903094-1-18-2': 'These parameters are derived from the experimental spectra by lorentzian fits which give the width and the area of the thermal peak, the latter being related to the effective temperature by eq. ([REF]).', 'quant-ph-9903094-1-18-3': 'To measure the feedback gain, we detect the intensity of the auxiliary beam after reflection on the mirror.', 'quant-ph-9903094-1-18-4': 'The ratio between the modulation spectrum of this intensity at frequency [MATH] and the noise spectrum [MATH] is proportional to the gain [MATH].', 'quant-ph-9903094-1-18-5': 'This measurement takes into account any nonlinearity of the gain due to a possible saturation of the acousto-optic modulator.', 'quant-ph-9903094-1-19-0': 'As expected from eqs. ([REF]) and ([REF]), the damping and the amplitude noise reduction [MATH] have a linear dependence with the gain, as well for cooling ([MATH]) as for heating ([MATH]).', 'quant-ph-9903094-1-19-1': 'The straight line in figure [REF] is in excellent agreement with experimental data and allows to normalize the gain [MATH] to the damping [MATH], as this has been done in the figure.', 'quant-ph-9903094-1-20-0': 'This figure also shows that large cooling factors [MATH] can be obtained.', 'quant-ph-9903094-1-20-1': 'This cooling factor does not however evolve linearly with the gain as it would be expected from eq. ([REF]).', 'quant-ph-9903094-1-20-2': 'This is due to the presence of a background thermal noise visible in figure [REF].', 'quant-ph-9903094-1-20-3': 'This noise is related to all other acoustic modes of the mirror and to the thermal noise of the coupling mirror of the cavity.', 'quant-ph-9903094-1-20-4': 'The feedback loop has not the same effect on this noise and on the fundamental thermal peak.', 'quant-ph-9903094-1-20-5': 'The solid curve in figure [REF] corresponds to a theoretical model in which the background noise is assumed to be unchanged by the feedback.', 'quant-ph-9903094-1-20-6': 'As a consequence, only the fondamental mode is cooled at a temperature [MATH] whereas all other modes stay in thermal equilibrium at the initial temperature [MATH].', 'quant-ph-9903094-1-20-7': 'The resulting cooling factor [MATH] is in excellent agreement with experimental data.', 'quant-ph-9903094-1-21-0': 'In conclusion, we have observed a cooling of the fundamental acoustic mode of a mirror.', 'quant-ph-9903094-1-21-1': 'The radiation pressure exerted by the feedback loop corresponds to a viscous force which increases the damping of the mirror without adding thermal fluctuations.', 'quant-ph-9903094-1-21-2': 'The Brownian motion is reduced around the fundamental resonance frequency by a factor 20.', 'quant-ph-9903094-1-21-3': 'For large gains, the thermal peak of the fundamental mode becomes of the same order as the background thermal noise and no global thermal equilibrium is reached.', 'quant-ph-9903094-1-21-4': 'As far as an effective temperature can be defined for the fundamental mode, we have obtained a reduction of this temperature by a factor 40.', 'quant-ph-9903094-1-22-0': 'We have also observed a heating of the mirror for reverse feedback gains.', 'quant-ph-9903094-1-22-1': 'The thermal peak is then sharpened and we have obtained an effective quality factor 50 times larger than the one without feedback.', 'quant-ph-9903094-1-23-0': 'The cooling mechanism demonstrated in this paper may be useful to increase the sensitivity of gravitational-wave interferometers.', 'quant-ph-9903094-1-23-1': 'The main difficulty is to freeze the thermal noise without changing the effect of the signal.', 'quant-ph-9903094-1-23-2': 'We propose in figure [REF] a possible scheme to control the thermal noise of one mirror of the interferometer.', 'quant-ph-9903094-1-23-3': 'A cavity performs a local measurement of the mirror motion which is fed back to the mirror via the radiation pressure of a laser beam.', 'quant-ph-9903094-1-23-4': 'The coupling mirror of the cavity is a small plano-convex mirror with a high mechanical resonance frequency and a low background thermal noise at low frequency.', 'quant-ph-9903094-1-23-5': 'As a consequence the cavity measures the thermal noise of the mirror of the interferometer.', 'quant-ph-9903094-1-23-6': 'For a short cavity, this measurement is not sensitive to a gravitational wave and the cooling reduces the thermal noise without changing the response of the interferometer.', 'quant-ph-9903094-1-23-7': 'To increase the sensitivity of the interferometer at the gravitational-wave frequencies, it is necessary to control the background thermal noise at low frequency instead of controling the thermal peak at resonance.', 'quant-ph-9903094-1-23-8': 'The principle of the cooling mechanism is however the same as the one demonstrated in this paper as far as the applied radiation pressure can compensate the Langevin force associated with thermal fluctuations.', 'quant-ph-9903094-1-24-0': 'Dorsel83 A. Dorsel, J.D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys.', 'quant-ph-9903094-1-25-0': 'Meystre85 P. Meystre, E.M. Wright, J.D. McCullen, E. Vignes, J. Opt.', 'quant-ph-9903094-1-26-0': 'Hadjar99 Y. Hadjar, P.F. Cohadon, C.G. Aminoff, M. Pinard, A. Heidmann, quant-ph/9901056', 'quant-ph-9903094-1-27-0': 'Pinard99 M. Pinard, Y. Hadjar, A. Heidmann, quant-ph/9901057, to be published in Eur.'}

{'quant-ph-9903094-2-0-0': 'We describe an experiment in which a mirror is cooled by the radiation pressure of light.', 'quant-ph-9903094-2-0-1': 'A high-finesse optical cavity with a mirror coated on a mechanical resonator is used as an optomechanical sensor of the Brownian motion of the mirror.', 'quant-ph-9903094-2-0-2': 'A feedback mechanism controls this motion via the radiation pressure of a laser beam reflected on the mirror.', 'quant-ph-9903094-2-0-3': 'We have observed either a cooling or a heating of the mirror, depending on the gain of the feedback loop.', 'quant-ph-9903094-2-1-0': 'PACS : 42.50.', 'quant-ph-9903094-2-1-1': 'Lc, 04.80.', 'quant-ph-9903094-2-1-2': 'Nn, 05.40.', 'quant-ph-9903094-2-1-3': 'Jc Thermal noise is a basic limit for many very sensitive optical measurements such as interferometric gravitational-wave detection[CITATION].', 'quant-ph-9903094-2-1-4': 'Brownian motion of suspended mirrors can be decomposed into suspension and internal thermal noises.', 'quant-ph-9903094-2-1-5': 'The latter is due to thermally induced deformations of the mirror surface and constitutes the major limitation of gravitational-wave detectors in the intermediate frequency domain[CITATION].', 'quant-ph-9903094-2-1-6': 'Observation and control of this noise have thus become an important issue in precision measurements[CITATION].', 'quant-ph-9903094-2-1-7': 'In order to reduce thermal noise effects, it is not always possible to lower the temperature and other techniques have been proposed such as feedback control[CITATION].', 'quant-ph-9903094-2-2-0': 'In this letter we report the first experimental observation of the cooling of a mirror by feedback control.', 'quant-ph-9903094-2-2-1': 'The principle of the experiment is to detect the Brownian motion of the mirror with an optomechanical sensor and then to freeze the motion by applying an electronically controlled radiation pressure on the mirror.', 'quant-ph-9903094-2-2-2': 'Mechanical effects of light on macroscopic objects have already been observed, such as the dissipative effects of electromagnetic radiation[CITATION], the optical bistability and mirror confinement in a cavity induced by radiation pressure[CITATION], or the regulation of the mechanical response of a microcantilever by feedback via the photothermal force[CITATION].', 'quant-ph-9903094-2-2-3': 'In our experiment the radiation pressure is driven by the feedback loop in such a way that a viscous force is applied to the mirror.', 'quant-ph-9903094-2-2-4': 'It thus plays a role somewhat similar to the one in optical molasses for atoms.', 'quant-ph-9903094-2-3-0': 'The cooling mechanism can be understood from the experimental setup shown in figure [REF].', 'quant-ph-9903094-2-3-1': 'The mirror is used as the rear mirror of a single-ended Fabry-Perot cavity.', 'quant-ph-9903094-2-3-2': 'The phase of the field reflected by the cavity is very sensitive to changes in the cavity length[CITATION].', 'quant-ph-9903094-2-3-3': 'For a resonant cavity, a displacement [MATH] of the rear mirror induces a phase shift [MATH] of the reflected field on the order of [EQUATION] where [MATH] is the cavity finesse and [MATH] is the optical wavelength.', 'quant-ph-9903094-2-3-4': 'This signal is superimposed to the quantum phase noise of the reflected beam.', 'quant-ph-9903094-2-3-5': 'Provided that the cavity finesse is high enough, this quantum noise is negligible and the Brownian motion of the mirror can be detected by measuring the phase of the reflected field[CITATION].', 'quant-ph-9903094-2-4-0': 'To cool the mirror we use an auxiliary laser beam reflected from the back on the mirror.', 'quant-ph-9903094-2-4-1': 'This beam is intensity-modulated by an acousto-optic modulator driven by the feedback loop so that a modulated radiation pressure is applied to the mirror.', 'quant-ph-9903094-2-4-2': 'The resulting motion can be described by its Fourier transform [MATH] at frequency [MATH] which is proportional to the applied forces [EQUATION] where [MATH] is the mechanical susceptibility of the mirror.', 'quant-ph-9903094-2-4-3': 'If we assume that the mechanical response is harmonic, this susceptibility has a lorentzian shape [EQUATION] characterized by a mass [MATH], a resonance frequency [MATH] and a damping [MATH] related to the quality factor [MATH] of the mechanical resonance by [MATH].', 'quant-ph-9903094-2-5-0': 'The force [MATH] in eq. ([REF]) is a Langevin force responsible for the Brownian motion of the mirror.', 'quant-ph-9903094-2-5-1': 'At thermal equilibrium its spectrum [MATH] is related to the mechanical susceptibility by the fluctuation-dissipation theorem [EQUATION] where [MATH] is the temperature.', 'quant-ph-9903094-2-5-2': 'The resulting thermal noise spectrum [MATH] of the mirror motion has a lorentzian shape centered at frequency [MATH] and of width [MATH].', 'quant-ph-9903094-2-6-0': 'The second force [MATH] in eq. ([REF]) is the radiation pressure exerted by the auxiliary laser beam and modulated by the feedback loop.', 'quant-ph-9903094-2-6-1': 'Neglecting the quantum phase noise in the control signal, this force is proportional to the displacement [MATH] of the mirror (eq. [REF]).', 'quant-ph-9903094-2-6-2': 'We choose the feedback gain in such a way that the radiation pressure is proportional to the speed [MATH] of the mirror [EQUATION] where [MATH] is related to the electronic gain.', 'quant-ph-9903094-2-6-3': 'The radiation pressure exerted by the auxiliary laser beam thus corresponds to an additional viscous force for the mirror.', 'quant-ph-9903094-2-6-4': 'The resulting motion is given by [EQUATION]', 'quant-ph-9903094-2-6-5': 'This equation is similar to the one obtained without feedback (eq. [REF] with [MATH]) except that the radiation pressure changes the damping without adding any fluctuations.', 'quant-ph-9903094-2-6-6': 'The noise spectrum [MATH] of the mirror motion still has a lorentzian shape but with a different width [MATH] and a different height.', 'quant-ph-9903094-2-6-7': 'The variation of height can be characterized by the amplitude noise reduction [MATH] at resonance frequency [EQUATION]', 'quant-ph-9903094-2-6-8': 'The resulting motion is then equivalent to a thermal equilibrium at a different temperature [MATH] which can be either reduced or increased depending on the sign of the gain [MATH] [EQUATION]', 'quant-ph-9903094-2-6-9': 'We now describe our experiment.', 'quant-ph-9903094-2-6-10': 'The mirror is coated on the plane side of a small plano-convex mechanical resonator made of silica (figure [REF]).', 'quant-ph-9903094-2-6-11': 'The coating has been made at the Institut de Physique Nucleaire de Lyon on a 1.5-mm thick substrate with a diameter of 14 mm and a curvature radius of the convex side of 100 mm.', 'quant-ph-9903094-2-6-12': 'Internal acoustic modes correspond to gaussian modes confined around the central axis of the resonator[CITATION].', 'quant-ph-9903094-2-6-13': 'The fundamental mode studied in this paper is a compression mode with a waist equal to 3.4 mm and a resonance frequency close to 2 MHz[CITATION].', 'quant-ph-9903094-2-6-14': 'The front mirror of the cavity is a Newport high-finesse SuperMirror (curvature radius = 1 m, transmission = 50 ppm) held at 1 mm from the back mirror.', 'quant-ph-9903094-2-6-15': 'We have measured the following parameters of the cavity : free spectral range = 141 GHz, cavity bandwidth = 1.9 MHz, beam waist = 90 [MATH]m.', 'quant-ph-9903094-2-6-16': 'These values correspond to a finesse [MATH] of 37000.', 'quant-ph-9903094-2-7-0': 'The light entering the cavity is provided by a titane-sapphire laser working at 810 nm and frequency-locked to a stable external cavity which is locked to a resonance of the high-finesse cavity by monitoring the residual light transmitted by the rear mirror.', 'quant-ph-9903094-2-7-1': 'The beam is intensity-stabilized and spatially filtered by a mode cleaner.', 'quant-ph-9903094-2-7-2': 'One gets a 100-[MATH]W incident beam on the high-finesse cavity with a mode matching of 98%.', 'quant-ph-9903094-2-7-3': 'The phase of the field reflected by the cavity is measured by homodyne detection.', 'quant-ph-9903094-2-7-4': 'The reflected field is mixed with a 10-mW local oscillator and a servoloop monitors the length of the local oscillator arm so that we measure the phase fluctuations of the reflected field.', 'quant-ph-9903094-2-7-5': 'This signal is sent both to the feedback loop and to a spectrum analyzer.', 'quant-ph-9903094-2-8-0': 'The feedback loop consists of an amplifier with variable gain and phase which drives the acousto-optic modulator.', 'quant-ph-9903094-2-8-1': 'The 500-mW auxiliary beam is uncoupled from the high-finesse cavity by a frequency shift of the acousto-optic modulator (200 MHz) and by a tilt angle of 10[MATH] with respect to the cavity axis.', 'quant-ph-9903094-2-8-2': 'We have checked that this beam has no spurious effect on the homodyne detection.', 'quant-ph-9903094-2-8-3': 'A band-pass filter centered at the fundamental resonance frequency of the mirror is also inserted in the feedback loop to reduce its saturation.', 'quant-ph-9903094-2-8-4': 'For large gains, the radiation pressure [MATH] can become of the same order as the Langevin force [MATH] and it must be restricted in frequency in order to get a finite variance.', 'quant-ph-9903094-2-8-5': 'The electronic filter has a quality factor of 200 and limits the efficiency of the feedback loop to a bandwidth of 9 kHz around the fundamental resonance frequency.', 'quant-ph-9903094-2-9-0': 'Figure [REF] shows the phase noise spectrum of the reflected field obtained by an average of 1000 scans of the spectrum analyzer with a resolution bandwidth of 10 Hz.', 'quant-ph-9903094-2-9-1': 'Curve (a) is obtained at room temperature without feedback.', 'quant-ph-9903094-2-9-2': 'It reproduces the thermal noise spectrum [MATH] of the mirror which is concentrated around the fundamental resonance frequency (1858.9 kHz) with a width [MATH] of 45 Hz (mechanical quality factor [MATH]).', 'quant-ph-9903094-2-9-3': 'The spectrum is normalized to the shot-noise level and it clearly appears that the thermal noise is much larger than the quantum phase noise[CITATION].', 'quant-ph-9903094-2-10-0': 'Curves (b) to (d) are obtained with feedback for increasing electronic gains.', 'quant-ph-9903094-2-10-1': 'The phase of the amplifier is adjusted to maximize the correction at resonance.', 'quant-ph-9903094-2-10-2': 'From eqs. ([REF]) and ([REF]) this corresponds to a global imaginary gain for the loop and to a purely viscous radiation pressure force.', 'quant-ph-9903094-2-10-3': 'The control of the mirror motion is clearly visible on those curves.', 'quant-ph-9903094-2-10-4': 'The thermal peak is strongly reduced while its width is increased.', 'quant-ph-9903094-2-10-5': 'The amplitude noise reduction [MATH] at resonance is larger than 20 for large gains.', 'quant-ph-9903094-2-11-0': 'The effective temperature [MATH] can be deduced from the variance [MATH] of the mirror motion which is equal to the integral of the spectrum [MATH].', 'quant-ph-9903094-2-11-1': 'From eqs. ([REF]), ([REF]) and ([REF]) one gets the usual relation for a harmonic oscillator at thermal equilibrium [EQUATION]', 'quant-ph-9903094-2-11-2': 'The decrease of the area of the thermal peak observed in figure [REF] thus corresponds to a cooling of the mirror.', 'quant-ph-9903094-2-12-0': 'Figure [REF] shows the effect of feedback for a reverse gain ([MATH]).', 'quant-ph-9903094-2-12-1': 'Noise spectra are obtained by an average of 500 scans with a resolution bandwidth of 1 Hz.', 'quant-ph-9903094-2-12-2': 'Curve (b) exhibits a strong increase of the thermal peak which now corresponds to a heating of the mirror.', 'quant-ph-9903094-2-12-3': 'The feedback also reduces the damping from [MATH] to [MATH], thus increasing the quality factor of the resonance.', 'quant-ph-9903094-2-12-4': 'We have obtained a maximum effective quality factor of [MATH]), limited by the saturation of the feedback loop.', 'quant-ph-9903094-2-13-0': 'It is instructive to study the efficiency of the cooling or heating mechanism with respect to the gain of the feedback loop.', 'quant-ph-9903094-2-13-1': 'Figure [REF] shows the variation of the damping [MATH], of the amplitude noise reduction [MATH] at resonance and of the cooling factor [MATH], as a function of the feedback gain.', 'quant-ph-9903094-2-13-2': 'These parameters are derived from the experimental spectra by lorentzian fits which give the width and the area of the thermal peak, the latter being related to the effective temperature by eq. ([REF]).', 'quant-ph-9903094-2-13-3': 'To measure the feedback gain, we detect the intensity of the auxiliary beam after reflection on the mirror.', 'quant-ph-9903094-2-13-4': 'The ratio between the modulation spectrum of this intensity at frequency [MATH] and the noise spectrum [MATH] is proportional to the gain [MATH].', 'quant-ph-9903094-2-13-5': 'This measurement takes into account any nonlinearity of the gain due to a possible saturation of the acousto-optic modulator.', 'quant-ph-9903094-2-14-0': 'As expected from eqs. ([REF]) and ([REF]), the damping and the amplitude noise reduction [MATH] have a linear dependence with the gain, as well for cooling ([MATH]) as for heating ([MATH]).', 'quant-ph-9903094-2-14-1': 'The straight line in figure [REF] is in excellent agreement with experimental data and allows to normalize the gain [MATH] to the damping [MATH], as this has been done in the figure.', 'quant-ph-9903094-2-15-0': 'This figure also shows that large cooling factors [MATH] can be obtained.', 'quant-ph-9903094-2-15-1': 'This cooling factor does not however evolve linearly with the gain as it would be expected from eq. ([REF]).', 'quant-ph-9903094-2-15-2': 'This is due to the presence of a background thermal noise visible in figure [REF].', 'quant-ph-9903094-2-15-3': 'This noise is related to all other acoustic modes of the mirror and to the thermal noise of the coupling mirror of the cavity.', 'quant-ph-9903094-2-15-4': 'The feedback loop has not the same effect on this noise and on the fundamental thermal peak.', 'quant-ph-9903094-2-15-5': 'The solid curve in figure [REF] corresponds to a theoretical model in which the background noise is assumed to be unchanged by the feedback.', 'quant-ph-9903094-2-15-6': 'As a consequence, only the fondamental mode is cooled at a temperature [MATH] whereas all other modes stay in thermal equilibrium at the initial temperature [MATH].', 'quant-ph-9903094-2-15-7': 'The resulting cooling factor [MATH] is in excellent agreement with experimental data.', 'quant-ph-9903094-2-16-0': 'The cooling mechanism is not limited to the mechanical resonance frequencies.', 'quant-ph-9903094-2-16-1': 'Figure [REF] shows the cooling obtained at frequencies well below the fundamental resonance frequency.', 'quant-ph-9903094-2-16-2': 'The electronic filter is now centered around 800 kHz and the feedback loop reduces the background thermal noise (curve a).', 'quant-ph-9903094-2-16-3': 'The width of the noise reduction (curve b) is related to the filter bandwidth.', 'quant-ph-9903094-2-16-4': 'The phase and the gain of the feedback loop has been adjusted since the electronic gain [MATH] has now to be compared to the real part [MATH] of the inverse of the mechanical susceptibility at low frequency (eq. [REF]).', 'quant-ph-9903094-2-16-5': 'Note that the amplitude of the radiation pressure exerted by the auxiliary laser beam is however approximately the same as in the resonant case.', 'quant-ph-9903094-2-16-6': 'Large noise reduction is actually obtained when the radiation pressure [MATH] is on the order of the Langevin force [MATH] whose amplitude is independent of frequency (eq. [REF]).', 'quant-ph-9903094-2-17-0': 'In conclusion, we have observed a thermal noise reduction by a factor 20 near the fundamental resonance frequency.', 'quant-ph-9903094-2-17-1': 'The radiation pressure exerted by the feedback loop corresponds to a viscous force which increases the damping of the mirror without adding thermal fluctuations.', 'quant-ph-9903094-2-17-2': 'For large gains, the thermal peak of the fundamental mode becomes of the same order as the background thermal noise and no global thermal equilibrium is reached.', 'quant-ph-9903094-2-17-3': 'As far as an effective temperature can be defined for the fundamental mode, we have obtained a reduction of this temperature by a factor 40.', 'quant-ph-9903094-2-17-4': 'We have also observed a heating of the mirror for reverse feedback gains, and a cooling of the background thermal noise at low frequencies.', 'quant-ph-9903094-2-18-0': 'The cooling mechanism demonstrated in this paper may be useful to increase the sensitivity of gravitational-wave interferometers.', 'quant-ph-9903094-2-18-1': 'The main difficulty is to freeze the thermal noise without changing the effect of the signal.', 'quant-ph-9903094-2-18-2': 'We propose in figure [REF] a possible scheme to control the thermal noise of one mirror of the interferometer.', 'quant-ph-9903094-2-18-3': 'A cavity performs a local measurement of the mirror motion which is fed back to the mirror via the radiation pressure of a laser beam.', 'quant-ph-9903094-2-18-4': 'The coupling mirror of the cavity is a small plano-convex mirror with a high mechanical resonance frequency and a low background thermal noise at low frequency.', 'quant-ph-9903094-2-18-5': 'As a consequence the cavity measures the thermal noise of the mirror of the interferometer.', 'quant-ph-9903094-2-18-6': 'For a short cavity, this measurement is not sensitive to a gravitational wave and the cooling can reduce the background thermal noise at the gravitational-wave frequencies without changing the response of the interferometer.', 'quant-ph-9903094-2-19-0': 'Dorsel83 A. Dorsel, J.D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys.', 'quant-ph-9903094-2-20-0': 'Hadjar99 Y. Hadjar, P.F. Cohadon, C.G. Aminoff, M. Pinard, A. Heidmann, quant-ph/9901056', 'quant-ph-9903094-2-21-0': 'Pinard99 M. Pinard, Y. Hadjar, A. Heidmann, quant-ph/9901057, to be published in Eur.'}

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[]

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[]

['quant-ph-9903094-1-1-0', 'quant-ph-9903094-1-1-1', 'quant-ph-9903094-1-1-2', 'quant-ph-9903094-2-1-0', 'quant-ph-9903094-2-1-1', 'quant-ph-9903094-2-1-2']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/quant-ph/9903094

1905.01133

{'1905.01133-1-0-0': 'Estimating the galaxy two-point correlation function using a split random catalog', '1905.01133-1-1-0': 'The two-point correlation function of the galaxy distribution is a key cosmological observable that allows us to constrain the dynamical and geometrical state of our Universe.', '1905.01133-1-1-1': 'To measure the correlation function we need to know both the galaxy positions and the expected galaxy density field.', '1905.01133-1-1-2': 'The expected field is commonly specified using a Monte-Carlo sampling of the volume covered by the survey and, to minimize additional sampling errors, this random catalog has to be much larger than the data catalog.', '1905.01133-1-1-3': 'Correlation function estimators compare data-data pair counts to data-random and random-random pair counts, where random-random pairs usually dominate the computational cost.', '1905.01133-1-1-4': 'Future redshift surveys will deliver spectroscopic catalogs of tens of millions of galaxies.', '1905.01133-1-1-5': 'Given the large number of random objects required to guarantee sub-percent accuracy, it is of paramount importance to improve the efficiency of the algorithm without degrading its precision.', '1905.01133-1-1-6': 'We show both analytically and numerically that splitting the random catalog into a number of subcatalogs of the same size as the data catalog when calculating random-random pairs, and excluding pairs across different subcatalogs provides the optimal error at fixed computational cost.', '1905.01133-1-1-7': 'For a random catalog fifty times larger than the data catalog, this reduces the computation time by a factor of more than ten without affecting estimator variance or bias.', '1905.01133-1-2-0': 'Two-point correlation function with split random catalog', '1905.01133-1-3-0': '# Introduction', '1905.01133-1-4-0': 'The spatial distribution of luminous matter in the Universe is a key diagnostic for studying cosmological models and the physical processes involved in the assembly of structure.', '1905.01133-1-4-1': 'In particular, light from galaxies is a robust tracer of the overall matter distribution, whose statistical properties can be predicted by cosmological models.', '1905.01133-1-4-2': 'Two-point correlation statistics are very effective tools for compressing the cosmological information encoded in the spatial distribution of the mass in the Universe.', '1905.01133-1-4-3': 'In particular, the two-point correlation function in configuration space has emerged as one of the most popular cosmological probes.', '1905.01133-1-4-4': 'Its success stems from the presence of characterized features that can be identified, measured and effectively compared to theoretical models to extract clean cosmological information.', '1905.01133-1-5-0': 'One such feature is baryon acoustic oscillations (BAO), which imprint a characteristic scale in the two-point correlation that can be used as a standard ruler.', '1905.01133-1-5-1': 'After the first detection in the two-point correlation function of SDSS DR3 and 2dFGRS galaxy catalogs , the BAO signal has been identified, with different degrees of statistical significance, and used to constrain the expansion history of the Universe in many spectroscopic galaxy samples (see e.g., [CITATION]).', '1905.01133-1-5-2': 'Several of these studies did not focus on the BAO feature only but also analyzed the anisotropies in the 2-point correlation function induced by the peculiar velocities , the so-called redshift space distortions (RSD), and by assigning cosmology-dependent distances to the observed redshifts (the [CITATION] test).', '1905.01133-1-5-3': 'For RSD analyses see also, e.g., [CITATION].', '1905.01133-1-6-0': 'Methods to estimate the galaxy two-point correlation function (2PCF) [MATH] from survey data are based on its definition as excess probability of finding a galaxy pair.', '1905.01133-1-6-1': 'One counts from the data ([MATH]) catalog the number [MATH] of pairs of galaxies with separation [MATH], where [MATH] is a bin of separation vectors, and compares it to the number of pairs [MATH] in a corresponding randomly generated ([MATH]) catalog and to the number of data-random pairs [MATH].', '1905.01133-1-6-2': 'The bin may be a 1D ([MATH]), 2D, or a 3D bin.', '1905.01133-1-6-3': 'In the 1D case, [MATH] is the length of the separation vector and [MATH] is the width of the bin.', '1905.01133-1-6-4': "From here on 'separation [MATH]' means that the separation falls in this bin.", '1905.01133-1-7-0': 'Several estimators of the 2PCF have been proposed by [CITATION], [CITATION], [CITATION], and [CITATION], building on the original [CITATION] proposal.', '1905.01133-1-7-1': 'These correspond to different combinations of the [MATH], [MATH], and [MATH] counts to obtain a 2PCF estimate [MATH].', '1905.01133-1-7-2': '(See [CITATION] and [CITATION] for more estimators.)', '1905.01133-1-7-3': 'The Landy-Szalay estimator () := ((DD()RR() - DR()RR() + 1', '1905.01133-1-8-0': "(we call this method 'standard LS' in the following) is the most commonly used, since it provides the minimum variance when [MATH] and is unbiased in the limit [MATH].", '1905.01133-1-8-1': 'Here [MATH] is the size (number of objects) of the data catalog and [MATH] is the size of the random catalog.', '1905.01133-1-8-2': 'We define [MATH].', '1905.01133-1-8-3': 'To minimize random error from the random catalog, [MATH] should be used.', '1905.01133-1-8-4': '(For a different approach, see [CITATION].)', '1905.01133-1-9-0': 'One is usually interested in [MATH] only up to some [MATH] (the maximum separation in the survey), so pairs with larger separations can be skipped.', '1905.01133-1-9-1': 'Efficient implementations of the LS estimator involve pre-ordering of the catalogs, through kd-tree, chain-mesh, or other algorithms to facilitate this.', '1905.01133-1-9-2': 'The computational cost is then roughly proportional to the actual number of pairs with separation [MATH].', '1905.01133-1-10-0': 'The correlation function is small for large separations, and in cosmological surveys [MATH] is large enough that for most pairs [MATH].', '1905.01133-1-10-1': 'Then the fraction [MATH] of [MATH] pairs with [MATH] is not very different from the fraction of [MATH] or [MATH] pairs with [MATH].', '1905.01133-1-10-2': 'The computational cost is dominated by the part proportional to the total number of pairs needed, [MATH], which in turn is dominated by the [MATH] pairs as [MATH].', '1905.01133-1-10-3': 'The smaller number of [MATH] pairs contribute much more to the error of the estimate than the large number of [MATH] pairs, whereas the cost is dominated by [MATH].', '1905.01133-1-10-4': 'Thus a significant saving of computation time with an insignificant loss of accuracy may be achieved by counting only a subset of [MATH] pairs, while still counting the full set (up to [MATH]) of [MATH] pairs.', '1905.01133-1-11-0': 'A good way to achieve this is to use many ([MATH]) small (i.e, low-density) [MATH] catalogs instead of one large (high-density) catalog , or, equivalently, splitting an already generated large [MATH] catalog into [MATH] small ones for the calculation of [MATH] pairs while using the full [MATH] catalog for the [MATH] counts.', '1905.01133-1-11-1': 'One might also consider obtaining a similar cost saving by diluting (subsampling) the [MATH] catalog for [MATH] counts, but, as we show below, this is not a good idea.', '1905.01133-1-11-2': "We refer to these two cost saving methods as 'split' and 'dilution'.", '1905.01133-1-12-0': "In this work we derive theoretically the additional covariance and bias due to the size and treatment of the [MATH] catalog; test these predictions numerically with mock catalogs representative of next generation datasets, such as the spectroscopic galaxy samples that will be obtained by the future Euclid satellite mission ; and show that the 'split' method, while reducing the computational cost by a large factor, retains the advantages of the LS estimator.", '1905.01133-1-13-0': 'We follow the approach of [CITATION], but generalize it in a number of ways: In particular, since we focus on the effect of the random catalog, we do not work in the limit [MATH].', '1905.01133-1-13-1': 'Also, we calculate covariances, not just variances, and make fewer approximations (see Sect. [REF]).', '1905.01133-1-14-0': 'The layout of the paper is as follows.', '1905.01133-1-14-1': 'In Sect. [REF] we derive theoretical results for bias and covariance.', '1905.01133-1-14-2': 'In Sect. [REF] we focus on the split LS estimator and its optimization.', '1905.01133-1-14-3': 'In Sect. [REF] we test the different estimators with mock catalogs.', '1905.01133-1-14-4': 'Finally, we discuss the results and present our conclusions in Sect. [REF].', '1905.01133-1-15-0': '# Theoretical results: bias and covariance', '1905.01133-1-16-0': '## General derivation', '1905.01133-1-17-0': 'We follow the derivation and notations in [CITATION] but extend to the case that includes random counts covariance.We consider the survey volume as divided into [MATH] microcells (very small subvolumes) and work in the limit [MATH], so that no two objects will ever be located within the same microcell.', '1905.01133-1-18-0': 'Define [MATH], [MATH], [MATH] to represent the relative deviation of the [MATH], [MATH], [MATH] counts from their expectation values (mean values over an infinite number of independent realizations): DD() & =: & DD() [1+()] ,', '1905.01133-1-19-0': 'DR() & =: & DR() [1+()] ,', '1905.01133-1-20-0': 'RR() & =: & RR() [1+()] .', '1905.01133-1-21-0': 'By definition [MATH].', '1905.01133-1-22-0': 'The expectation values for the pair counts are: DD()& = & (,', '1905.01133-1-23-0': 'DR()& = & () ,', '1905.01133-1-24-0': 'RR()& = & (.', '1905.01133-1-25-0': 'where [MATH] is the correlation function normalized to the actual number density of galaxies in the survey and () := 2K^2 _i<j^K ^ij() is the fraction of microcell pairs with separation [MATH].', '1905.01133-1-25-1': 'Here [MATH] if [MATH] falls in the [MATH]-bin, otherwise [MATH].', '1905.01133-1-26-0': 'The expectation value of the LS estimator [REF] is & = & (1+)1+1+- 21+1++ 1', '1905.01133-1-27-0': '& & + (+ 2.', '1905.01133-1-28-0': 'A finite [MATH] catalog thus introduces a (small) bias.', '1905.01133-1-28-1': "This expression is calculated to second order in [MATH], [MATH], and [MATH] (we denote equality to second order in them by '[MATH]').", '1905.01133-1-28-2': 'Since data and random catalogs are independent, [MATH].', '1905.01133-1-29-0': 'For the covariance we get', '1905.01133-1-30-0': '&Cov[(_1),(_2)] (_1)(_2)- (_1)(_2)', '1905.01133-1-31-0': '& (1+_1)(1+_2)_1_2+ 4_1_2+ (1-_1)(1-_2)_1_2', '1905.01133-1-32-0': '&- 2(1+_1)_1_2- 2(1+_2)_1_2', '1905.01133-1-33-0': '&- 2(1-_1)_1_2- 2(1-_2)_1_2.', '1905.01133-1-34-0': 'Terms with [MATH] represent additional variance due to finite [MATH], and are new compared to [CITATION].', '1905.01133-1-34-1': 'Also [MATH] collects additional contribution from variations in the random field.', '1905.01133-1-34-2': 'The cross terms [MATH] and [MATH], instead, depend linearly on the random field, and average to the [MATH] result.', '1905.01133-1-34-3': 'The additional contribution due to finite [MATH] is thus', '1905.01133-1-35-0': '&Cov[(_1),(_2)] 4_1_2+ (1-_1)(1-_2)_1_2', '1905.01133-1-36-0': '&- 2(1-_1)_1_2- 2(1-_2)_1_2.', '1905.01133-1-37-0': "The deviations' covariances are obtained from _1_2& = & DD_1 DD_2 - DD_1DD_2DD_1DD_2 ,", '1905.01133-1-38-0': '_1_2& = &DD_1 DR_2 - DD_1 DR_2 DD_1 DR_2 etc.', '1905.01133-1-39-0': 'where [MATH] etc. and [MATH] etc. are shorthand notations for [MATH] etc. and [MATH] etc.', '1905.01133-1-40-0': '## Quadruplets, triplets, approximations', '1905.01133-1-41-0': 'We denote by _12 := (_1,_2) := 1K^3^_ijk^ik_1^jk_2 the fraction of ordered microcell triplets, where [MATH] and [MATH].', '1905.01133-1-41-1': 'The notation [MATH] means that only terms where all indices (microcells) are different are included.', '1905.01133-1-41-2': 'Here [MATH] is of the same magnitude as [MATH], but larger.', '1905.01133-1-42-0': 'We give examples how the [MATH] etc. in [REF] are calculated in Appendix [REF].', '1905.01133-1-42-1': 'These covariances involve expectation values [MATH], where [MATH] is the number of objects (0 or 1) in microcell [MATH] etc., and only cases where the four microcells are separated pairwise by [MATH] and [MATH] are included.', '1905.01133-1-42-2': 'If all 4 microcells [MATH], [MATH], [MATH], [MATH] are different, we call this case a quadruplet; it consists of two pairs with separations [MATH] and [MATH].', '1905.01133-1-42-3': 'If two of the indices, i.e., microcells, are equal, we have a triplet, with a center cell (the equal indices) and two end cells separated from the center by [MATH] and [MATH].', '1905.01133-1-43-0': 'We make the following three approximations:', '1905.01133-1-44-0': 'For microcell quadruplets the correlations between unconnected cells are approximated by zero on average.', '1905.01133-1-44-1': 'Three-point correlations vanish.', '1905.01133-1-44-2': 'The part of 4-point correlations that does not arise from the 2-point correlations vanishes.', '1905.01133-1-45-0': 'With approximations 2 and 3, we have for the expectation value of a galaxy triplet n_in_jn_k1+_ij+_jk+_ik , where [MATH], and for a quadruplet n_in_jn_kn_l1+_ij+_jk+_ik+_il+_jl+_kl+_ij_kl+_ik_jl+_il_jk .', '1905.01133-1-46-0': "We denote results based on these three approximations by '[MATH]'.", '1905.01133-1-46-1': 'Approximation 1 is good as long as the survey size is large compared to [MATH].', '1905.01133-1-46-2': 'It allows dropping other terms than [MATH] in [REF].', '1905.01133-1-46-3': 'Approximations 2 and 3 hold for Gaussian density fluctuations, but in the realistic cosmological situation they are not good: the presence of the higher-order correlations makes the estimation of the covariance of [MATH] estimators a difficult problem.', '1905.01133-1-46-4': 'However, this difficulty applies only to the contribution of the data to the covariance.', '1905.01133-1-46-5': 'The key point in this work is that while our theoretical result for the total covariance does not hold in a realistic situation (it is an underestimate), our results for the difference in estimator covariance due to different treatments of the random catalog hold well.', '1905.01133-1-47-0': 'In addition to working in the limit [MATH]), [CITATION] considered only 1D bins and the case where [MATH] (i.e., variances, not covariances) and made also a fourth approximation: for triplets (which in this case have legs of equal length) they approximated the correlation between the end cells (whose separation in this case varies between [MATH] and [MATH]) by [MATH].', '1905.01133-1-47-1': 'We denote the mean value of the correlation between triplet end cells (separated from the triplet center by [MATH] and [MATH]) by [MATH].', '1905.01133-1-47-2': '(For our plots in Sect. [REF] we make a similar approximation to it as LandySzalay.)', '1905.01133-1-47-3': 'Also, [CITATION] calculated just to first order in [MATH], whereas we do not make this approximation.', '1905.01133-1-47-4': '[CITATION] considered also covariances, and included the effect of 3-point and 4-point correlations, but worked in the limit [MATH]).', '1905.01133-1-48-0': '## Poisson, edge, and [MATH] terms', '1905.01133-1-49-0': 'Following the definition of [MATH] and [MATH] in [CITATION], we define t_12 & := & 1[_12_1_2-1] ,', '1905.01133-1-50-0': '_12 & := & 1[_12_1_2-1] = t_12 ,', '1905.01133-1-51-0': 'p_12 & := & 2(_12(1+_1)_1 - 2_12_1_2 + 1] ,', '1905.01133-1-52-0': '_12 & := & 1[_12_1 - 2_12_1_2 + 1] ,', '1905.01133-1-53-0': '_12 & := & 2(_12_1 - 2_12_1_2 + 1] ,', '1905.01133-1-54-0': 'q_12 & := & 1_12_1_2 = t_12 + 1 ,', '1905.01133-1-55-0': '_12 & := & 1_12_1_2 = _12 + 1 .', '1905.01133-1-56-0': 'For their diagonals ([MATH]) we write [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH].', '1905.01133-1-56-1': 'Thus [MATH], [MATH] etc. (We use superscripts for the matrices, e.g. [MATH] and subscripts for their diagonals, e.g. [MATH].)', '1905.01133-1-57-0': 'We get (see Appendix [REF]) for the standard LS', '1905.01133-1-58-0': '(1+_1)(1+_2)_1_2 (1+_1)(1+_2)(4t_12 + p_12)', '1905.01133-1-59-0': '+ 4(12-_1_2)q_12 ,', '1905.01133-1-60-0': '_1_2 t_12 + _12 + _12 + _12_12 ,', '1905.01133-1-61-0': '_1_2 = 4_12 + _12 ,', '1905.01133-1-62-0': '_1_2 2t_12 , _1_2 = 2_12 ,and _1_2 = 0 .', '1905.01133-1-63-0': 'Thus only [MATH] and [MATH] are affected by [MATH] (in our approximation its effect cancels in [MATH]).', '1905.01133-1-63-1': 'The results for [MATH], [MATH], and [MATH] are exact.', '1905.01133-1-63-2': 'The result for [MATH] involves all three approximations mentioned above, [MATH] involves approximations 1 and 2, and [MATH] involves approximation 1.', '1905.01133-1-64-0': "We call [MATH], [MATH] and [MATH] 'Poisson' terms and [MATH] and [MATH] 'edge' terms (the difference between [MATH] and [MATH] is due to edge effects).", '1905.01133-1-64-1': 'While the Poisson terms are strongly diagonal dominated, the edge terms are not.', '1905.01133-1-64-2': 'Since [MATH], the [MATH] terms are much larger than the edge terms, but they get multiplied by [MATH] or [MATH]', '1905.01133-1-65-0': 'In the limit [MATH]: [MATH], [MATH], [MATH]; [MATH] and also [MATH] are unaffected.', '1905.01133-1-66-0': 'We see that [MATH]-[MATH] and [MATH]-[MATH] correlations arise from edge effects.', '1905.01133-1-66-1': 'If we increase the density of data or random objects, the Poisson terms decrease as [MATH] but the edge terms decrease only as [MATH] so the edge effects are more important for a higher density of objects.', '1905.01133-1-67-0': 'Doubling the bin size (combining neighboring bins) doubles [MATH] but makes [MATH] four times as large, since also triplets where one leg was in one of the original smaller bins and the other leg in the other are now included.', '1905.01133-1-67-1': 'Thus the ratio [MATH] and [MATH] are not affected, but the dominant term in [MATH], [MATH] is halved.', '1905.01133-1-67-2': 'Edge effects are thus more important for larger bins.', '1905.01133-1-68-0': '## Results for the standard Landy-Szalay estimator', '1905.01133-1-69-0': 'Inserting the results for [MATH] etc. to Eqs. ([REF]) and ([REF]), we get that the expectation value of the standard LS estimator [REF] is = + (+ 4.', '1905.01133-1-70-0': 'This holds also for large [MATH] and in the presence of 3-point and 4-point correlations.', '1905.01133-1-70-1': 'A finite [MATH] catalog thus introduces a bias [MATH]; the edge ([MATH]) part of the bias cancels in the [MATH] limit.', '1905.01133-1-71-0': 'For the covariance we get', '1905.01133-1-72-0': '&Cov[(_1),(_2)] (_1)(_2)- (_1)(_2)', '1905.01133-1-73-0': '& (1+_1)(1+_2)p_12 + 4_12 + (1-_1)(1-_2)_12', '1905.01133-1-74-0': '&+ 4_1_2(t_12+_12)', '1905.01133-1-75-0': '&+ 412-_1_2)q_12 + 4_12_12 .', '1905.01133-1-76-0': 'Because of the approximations made, this result for the covariance does not apply to the realistic cosmological case; not even for large separations [MATH], where [MATH] is small, since large correlations at small [MATH] increase the covariance also at large [MATH].', '1905.01133-1-76-1': 'However, this concerns only [MATH] and [MATH].', '1905.01133-1-76-2': 'Our focus here is on the additional covariance due to the size and handling of the random catalog, which for standard LS is', '1905.01133-1-77-0': '&Cov[(_1),(_2)] 4_1_2+ (1-_1)(1-_2)_1_2', '1905.01133-1-78-0': '&- 2(1-_1)_1_2- 2(1-_2)_1_2', '1905.01133-1-79-0': '& 4_12 + (1-_1)(1-_2)_12 +4_1_2_12 + 4_12_12 .', '1905.01133-1-80-0': 'To zeroth order in [MATH] the covariance is given by the Poisson terms and the edge terms cancel to first order in [MATH].', '1905.01133-1-80-1': 'This is the property for which the standard LS estimator was designed.', '1905.01133-1-80-2': 'To first order in [MATH], the [MATH] terms contribute.', '1905.01133-1-80-3': 'This [MATH] contribution involves the triplet correlation [MATH], which, depending on the form of [MATH], may be larger than [MATH] or [MATH].', '1905.01133-1-81-0': 'If we try to save cost by using a diluted random catalog with [MATH] for [MATH] pairs, [MATH] is replaced by [MATH] with [MATH] in place of [MATH], but [MATH] and [MATH] are unaffected, so that the edge terms involving randoms no longer cancel.', '1905.01133-1-81-1': 'In Sec. [REF] we see that this is a large effect.', '1905.01133-1-81-2': 'Therefore one should not use dilution.', '1905.01133-1-82-0': '# Split random catalog', '1905.01133-1-83-0': 'In the split method one has, instead of one large random catalog [MATH], [MATH] independent smaller [MATH] catalogs of size [MATH].', '1905.01133-1-83-1': 'Their union [MATH] has size [MATH].', '1905.01133-1-83-2': "The pair counts [MATH] and [MATH] are calculated as DR() := _DR^() and RR'() := _R^R^() , i.e., pairs across different [MATH] catalogs are not included in [MATH].", '1905.01133-1-83-3': 'The total number of pairs in [MATH] is [MATH].', '1905.01133-1-83-4': '[MATH] equals its value in standard LS.', '1905.01133-1-84-0': "The split Landy-Szalay estimator is _split() := ((DD()RR'() - DR()RR'() + 1 .", '1905.01133-1-84-1': 'Compared to standard LS, [MATH], [MATH], and [MATH] are unaffected.', '1905.01133-1-84-2': "We construct [MATH], [MATH], and [MATH] from the standard LS results, bearing in mind that the random catalog is a union of independent catalogs, arriving at _1_2' &=& 2_12 ,", '1905.01133-1-85-0': '&=& 4_12 + _12 , where _12 := ((12 12 .', '1905.01133-1-85-1': 'The first is the same as in standard LS and dilution, but the second differs both from standard LS and from dilution, since it involves both [MATH] and [MATH].', '1905.01133-1-86-0': 'For the expectation value we get _split = + (+ 4,', '1905.01133-1-87-0': 'so that the bias is [MATH].', '1905.01133-1-87-1': 'In the limit [MATH] the edge part cancels, leaving only the Poisson term.', '1905.01133-1-88-0': 'The covariance is', '1905.01133-1-89-0': '&Cov[_split(_1),_split(_2)]', '1905.01133-1-90-0': '& (1+_1)(1+_2)p_12 + 4_12 + (1-_1)(1-_2)_12', '1905.01133-1-91-0': '&+ 4_1_2(t_12+_12) .', '1905.01133-1-92-0': 'The change in the covariance compared to the standard LS method is', '1905.01133-1-93-0': '&Cov[^split_1,^split_2] - Cov[^LS_1,^LS_2]', '1905.01133-1-94-0': '&= (1-_1)(1-_2)(_12-_12) ,', '1905.01133-1-95-0': 'which again applies in the realistic cosmological situation.', '1905.01133-1-95-1': 'This is our main result: in the split method the edge effects cancel and the bias and covariance are the same as for standard LS, except that the Poisson term [MATH] from [MATH] is replaced with the larger [MATH].', '1905.01133-1-96-0': '## Optimizing computational cost and variance of the split method', '1905.01133-1-97-0': 'The bias is small compared to variance in our application, so we focus on variance as the figure of merit.', '1905.01133-1-97-1': 'The computational cost should be roughly proportional to ^2(1+2+^2) =: ^2c ,', '1905.01133-1-98-0': 'and the additional variance due to finite [MATH] catalog in the [MATH] limit becomes var (2+^2)p =: vp.', '1905.01133-1-99-0': 'Here [MATH] and [MATH] are fixed by the survey and the requested [MATH] binning, but we can vary [MATH] and [MATH] in the search for the optimal computational method.', '1905.01133-1-99-1': "In the above we defined the 'cost' and 'variance' factors [MATH] and [MATH].", '1905.01133-1-100-0': 'We may ask two questions:', '1905.01133-1-101-0': 'For a fixed level of variance [MATH], which combination of [MATH] and [MATH] minimizes computational cost [MATH]?', '1905.01133-1-101-1': 'For a fixed computational cost [MATH], which combination of [MATH] and [MATH] minimizes the variance [MATH]?', '1905.01133-1-102-0': 'The answer to both questions is = c = 1+3andv = 3 .', '1905.01133-1-103-0': 'Thus the optimal version of the split method is the natural one where [MATH].', '1905.01133-1-103-1': 'In this case the additional variance in the [MATH] limit becomes var (2 + )p and the computational cost factor [MATH] becomes (1 + 2 + )^2 , so that [MATH] pairs contribute twice as much as [MATH] pairs to the variance and also twice as much computational cost is invested in them.', '1905.01133-1-103-2': 'The memory requirement for the random catalog is then the same as for the data catalog.', '1905.01133-1-103-3': 'The cost saving estimate above is optimistic, since the computation involves some overhead not proportional to the number of pairs.', '1905.01133-1-104-0': 'For small scales, where [MATH], the situation is different.', '1905.01133-1-104-1': 'The larger density of [MATH] pairs due to the correlation requires a larger density of the [MATH] catalog for the additional variance from it not to be relatively larger.', '1905.01133-1-104-2': 'From Eq. [REF] we see that the balance of the [MATH] and the [MATH] contributions is different for large [MATH] (the [MATH] term vs. the other terms).', '1905.01133-1-104-3': 'We may consider recomputing [MATH] for the small scales, using a smaller [MATH] and a larger [MATH] catalog.', '1905.01133-1-104-4': "Considering just the Poisson terms ([MATH] and [MATH] or [MATH]) with a 'representative' [MATH] value, [REF] and [REF] become [MATH] and [MATH] which modifies the above result (Eq. [REF]) for the optimal choice of [MATH] and [MATH] to = , i.e. = .", '1905.01133-1-104-5': 'This result is just indicative, since it assumed a constant [MATH] for [MATH].', '1905.01133-1-104-6': 'Especially it does not apply for [MATH], since then the approximation of ignoring the [MATH] and [MATH] terms in [REF] is not good.', '1905.01133-1-105-0': '# Tests on mock catalogs', '1905.01133-1-106-0': '## Minerva simulations and methodology', '1905.01133-1-107-0': 'The Minerva mocks are a set of 300 cosmological mocks produced with [MATH]-body simulations , stored at five output redshifts [MATH].', '1905.01133-1-107-1': 'The cosmology is flat [MATH]CDM with [MATH], and we use the [MATH] outputs.', '1905.01133-1-107-2': 'The mocks have [MATH] objects ("halos" found by a friends-of-friend algorithm) in a box of [MATH] cubed.', '1905.01133-1-108-0': 'To model the survey geometry of a redshift bin with [MATH] at [MATH], we placed the observer at comoving distance [MATH] from the center of the cube and selected from the cube a shell [MATH]-[MATH] from the observer.', '1905.01133-1-108-1': 'The comoving thickness of the shell is [MATH].', '1905.01133-1-108-2': 'The resulting mock sub-catalogs have [MATH] and are representative of the galaxy number density of the future Euclid spectroscopic galaxy catalog.', '1905.01133-1-109-0': 'We ignore peculiar velocities, i.e. we perform our analysis in real space.', '1905.01133-1-109-1': 'Therefore we consider results for the 1D 2PCF [MATH].', '1905.01133-1-109-2': 'We estimated [MATH] up to [MATH] using [MATH] bins.', '1905.01133-1-110-0': 'We chose standard LS with [MATH] as the reference method.', '1905.01133-1-110-1': 'In the following, LS without further qualification refers to this.', '1905.01133-1-110-2': 'The random catalog was generated separately for each shell mock to measure their contribution to the variance.', '1905.01133-1-110-3': 'For one of the random catalogs we calculated also triplets to obtain the edge effect quantity [MATH].', '1905.01133-1-111-0': 'While dilution can already be discarded on theoretical grounds, we show results obtained for it; since they provide the scale for edge effects demonstrating the importance of eliminating them with a careful choice of method.', '1905.01133-1-111-1': 'For the dilution and split methods we used [MATH] also, and tried out dilution fractions [MATH] and split factors [MATH] (chosen to have pairwise similar computational costs).', '1905.01133-1-111-2': 'In addition, we considered also standard LS with [MATH], which has the same number of [MATH] pairs as [MATH] and [MATH], but only half the number of [MATH] pairs; and standard LS with [MATH] to demonstrate the effect of a small [MATH].', '1905.01133-1-112-0': 'The code used to estimate the 2PCF implements a highly optimized pair-counting method, specifically designed for the search of object pairs in a given range of separations, imported from the CosmoBolognaLib, a large set of free software C++/python libraries for cosmological calculations .', '1905.01133-1-113-0': '## Variance and bias', '1905.01133-1-114-0': 'In Fig. [REF] we show the mean (over the 300 mock shells) estimated correlation function and the scatter (square root of the variance) of the estimates using the LS, split, and dilution methods; our theoretical approximate result for the scatter for LS; and our theoretical result for bias for the different methods.', '1905.01133-1-115-0': 'The theoretical result for the scatter is shown with and without the [MATH] terms, which include the triplet correlation [MATH], for which we used here the approximation [MATH].', '1905.01133-1-115-1': 'This behaves as expected, i.e., underestimates the variance, since we neglected the higher-order correlations in the [MATH] catalog.', '1905.01133-1-115-2': 'Nevertheless, it (see the dash-dotted line in Fig. [REF]) has similar features as the measured variance (dashed lines).', '1905.01133-1-116-0': 'Consider now the variance differences (from standard LS with [MATH]), for which our theoretical results should be accurate.', '1905.01133-1-116-1': 'Fig. [REF] compares the measured variance difference to the theoretical result.', '1905.01133-1-116-2': 'For the diluted estimators and LS with [MATH] the measured result agrees with theory, although clearly the measurement with just 300 mocks is rather noisy.', '1905.01133-1-116-3': 'For the split estimators and LS with [MATH] the difference is too small to be appreciated with 300 mocks, but at least the measurement does not disagree with the theoretical result.', '1905.01133-1-117-0': 'In Fig. [REF] we show the relative theoretical increase in scatter compared to the best possible case, which is LS in the limit [MATH].', '1905.01133-1-117-1': 'Since we do not have a valid theoretical result for the total scatter, we estimate it by by subtracting the theoretical difference from LS with [MATH] from the measured variance of the latter.', '1905.01133-1-118-0': 'At scales [MATH] the theoretical prediction is about the same for dilution and split and neither method looks promising for [MATH] where [MATH].', '1905.01133-1-118-1': 'This suggests that for optimizing cost and accuracy, a different method should be used for small scales than large scales.', '1905.01133-1-118-2': 'The number of [MATH] pairs with small separations is much less, so for the small-scale computation there is no need to restrict the computation to a subset of [MATH] pairs, or alternatively, one can afford to increase [MATH].', '1905.01133-1-118-3': 'For the small scales, we may consider the split method with increased [MATH] as an alternative to standard LS.', '1905.01133-1-118-4': 'We have the same number of pairs to compute as in the reference LS case, if we use [MATH] and [MATH].', '1905.01133-1-118-5': 'We added this case to Fig. [REF].', '1905.01133-1-118-6': 'It seems to perform better than LS at intermediate scales, but for the smallest scales LS has the smaller variance.', '1905.01133-1-118-7': 'This is in line with our conclusion in Sect. [REF] that when [MATH], it is not optimal to split the [MATH] catalog into this small subsets.', '1905.01133-1-119-0': 'We also compared the differences in the mean estimate from the different estimators to our theoretical results on the bias differences (see Fig. [REF]), but the theoretical bias differences are much smaller than the expected error of the mean from 300 mocks; and we just confirm that the differences we see are consistent with the error of the mean and thus consistent with the true bias being much smaller as predicted by theory.', '1905.01133-1-119-1': 'We did tests also with completely random ([MATH]) mocks, and with a large number ([MATH]) of mocks confirmed the theoretical bias result for the different estimators in this case.', '1905.01133-1-119-2': 'Since the bias is too small to be interesting we do not report these results in more detail here.', '1905.01133-1-120-0': 'However, note that for the estimation of the 2D 2PCF and its multipoles, the 2D bins will contain a smaller number of objects than the 1D bins of these test runs and therefore the bias is larger.', '1905.01133-1-120-1': 'Using the theoretical results [REF] or [REF] the bias can be removed afterwards with accuracy depending on how well we know the true [MATH].', '1905.01133-1-121-0': '## Computation time and variance', '1905.01133-1-122-0': 'The test runs were made using a single full 24-core node for each run.', '1905.01133-1-122-1': 'Table [REF] shows the mean computation time and mean estimator variance for different [MATH] ranges for the different cases we tested.', '1905.01133-1-122-2': 'Of these [MATH] ranges, the [MATH]-[MATH] is maybe the most interesting, since it contains the important BAO scale.', '1905.01133-1-122-3': 'Thus we plot the mean variance at this range vs mean computation time in Fig. [REF], together with our theoretical predictions.', '1905.01133-1-122-4': 'The theoretical estimate for the computation time for other dilution fractions and split factors is (1+24.75d^2) 306s and(1+24.75/^2) 306s (assuming [MATH]).', '1905.01133-1-122-5': 'For standard LS with other random catalog sizes the computation time estimate is (1+2+^2) 3.03s .', '1905.01133-1-123-0': '# Conclusions', '1905.01133-1-124-0': 'The computational time of the standard Landy-Szalay estimator is dominated by the [MATH] pairs, but, except at small scales where correlations are large, these make a negligible contribution to the expected error, compared to the contribution from the [MATH] and [MATH] pairs.', '1905.01133-1-124-1': 'Thus a substantial saving of computation time with an insignificant loss of accuracy can be achieved by counting a smaller subset of [MATH] pairs.', '1905.01133-1-125-0': 'We considered two ways to reduce the number of [MATH] pairs, dilution and split.', '1905.01133-1-125-1': 'In dilution, only a subset of the [MATH] catalog is used for [MATH] pairs.', '1905.01133-1-125-2': 'In split, the [MATH] catalog is split into a number of smaller subcatalogs, and only pairs within each subcatalog are counted.', '1905.01133-1-125-3': 'We derived theoretical results for the additional estimator covariance and bias due to the finite size of the random catalog for these different variants of the LS estimator, extending in many ways the original results by [CITATION], who worked in the limit of an infinite random catalog.', '1905.01133-1-125-4': 'We tested our results using 300 mock data catalogs, representative of the [MATH]-[MATH] redshift range of the future Euclid survey.', '1905.01133-1-125-5': 'The split method maintains the property the Landy-Szalay estimator was designed for, cancellation of edge effects in bias and variance (for [MATH]), whereas dilution loses this cancellation and therefore should not be used.', '1905.01133-1-126-0': 'For small scales, where correlations are large, one should not reduce [MATH] counts as much.', '1905.01133-1-126-1': 'The natural dividing line is the scale [MATH] where [MATH].', '1905.01133-1-126-2': 'Interestingly, the difference in bias and covariance between the different estimators (split, dilution, and LS) vanishes when [MATH].', '1905.01133-1-126-3': 'We recommend the natural version of the split method, [MATH], for large scales where [MATH].', '1905.01133-1-126-4': 'This leads to a saving in computation time by more than a factor of 10 (assuming [MATH]) with a negligible effect on variance and bias.', '1905.01133-1-126-5': 'For small scales, where [MATH], one should consider using a larger random catalog and one can use either the standard LS method or the split method with a more modest split factor.', '1905.01133-1-126-6': 'Because the number of pairs with these small separations is much smaller, the computation time is not a similar issue as for large separations.', '1905.01133-1-127-0': 'The results of our analysis will have an impact also on the computationally more demanding task of covariance matrix estimation.', '1905.01133-1-127-1': 'However, since in that case the exact computational cost is determined by the balance of data and randoms that does not need to be the same as for the individual 2-point correlation estimate, we postpone a quantitative analysis to a future, dedicated study.', '1905.01133-1-127-2': 'The same kind of methods can be applied to higher-order statistics (3-point and 4-point correlation functions) to speed up their estimation .', '1905.01133-1-128-0': 'We thank Will Percival and Cristiano Porciani for useful discussions.', '1905.01133-1-128-1': 'The 2PCF computations were done at the Euclid Science Data Center Finland (SDC-FI, urn:nbn:fi:research-infras-2016072529), for whose computational resources we thank CSC - IT Center for Science, the Finnish Grid and Cloud Computing Infrastructure (FGCI, urn:nbn:fi:research-infras-2016072533), and the Academy of Finland grant 292882.', '1905.01133-1-128-2': 'This work was supported by the Academy of Finland grant 295113.', '1905.01133-1-128-3': 'VL was supported by the Jenny and Antti Wihuri Foundation, AV by the Vaisala Foundation, AS by the Magnus Ehrnrooth Foundation, and JV by the Finnish Cultural Foundation.', '1905.01133-1-128-4': 'We also acknowledge travel support from the Jenny and Antti Wihuri Foundation.', '1905.01133-1-128-5': "VA acknowledges funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 749348.", '1905.01133-1-128-6': 'FM acknowledges the grants ASI n.I/023/12/0 "Attivit\'a relative alla fase B2/C per la missione Euclid" and PRIN MIUR 2015 "Cosmology and Fundamental Physics: illuminating the Dark Universe with Euclid".', '1905.01133-1-129-0': '# Derivation examples', '1905.01133-1-130-0': 'From the expression [REF] for the deviation covariances we obtain for them', '1905.01133-1-131-0': '(1+_1)(1+_2)_1_2 4 (1+_1)(1+_2)[_12_1_2 - 1]', '1905.01133-1-132-0': '+ 2(1+_1)(_12_1 - 2(1+_2)_12_1_2 + (1+_2) ]', '1905.01133-1-133-0': '+ 4((12-_1_2)_12_1_2 ,', '1905.01133-1-134-0': '_1_2 1 [_12_1_2-1] + [_12_1_2-1]', '1905.01133-1-135-0': '+ 1 - 2_12_1_2 + _12_1 + _12_12_1_2 ,', '1905.01133-1-136-0': '_1_2 = 2( 2(_12_1_2 - 1] + _12_1 - 1 ,', '1905.01133-1-137-0': '_1_2 2[_12_1_2 - 1] ,', '1905.01133-1-138-0': '_1_2 = 2[_12_1_2 - 1] ,', '1905.01133-1-139-0': '_1_2 = 0 .', '1905.01133-1-140-0': 'This is Eq. [REF] before inserting the quantities defined in [REF].', '1905.01133-1-141-0': 'As examples of how these are derived, following the method presented in [CITATION], we give three of the cases here: [MATH] (common for all the variants of LS), and [MATH] and [MATH] for the split method.', '1905.01133-1-141-1': 'The rest are calculated in a similar manner.', '1905.01133-1-142-0': 'To derive the [MATH] appearing in the [MATH] of [REF] we start from DD_1 DD_2 = _i<j^K _k<l^Kn_in_jn_kn_l^ij(_1)^kl(_2) ,', '1905.01133-1-143-0': 'where both [MATH] and [MATH] sum over all microcell pairs; [MATH] is the number of galaxies in microcell [MATH].', '1905.01133-1-143-1': 'There are three different cases for the terms [MATH] depending on how many indices are equal ([MATH] and [MATH] for all of them).', '1905.01133-1-144-0': 'The first case (quadruplets, i.e., two pairs of microcells) is when [MATH] are all different.', '1905.01133-1-144-1': 'We denote by [MATH] this part of the sum.', '1905.01133-1-144-2': 'There are [MATH] (we work in the limit [MATH]) such terms and they have & & n_in_jn_kn_l', '1905.01133-1-145-0': '& = & K - 1K - 1- 2K- 2- 3K- 3(1+_i)(1+_j)(1+_k)(1+_l)', '1905.01133-1-146-0': '& = & ( - 1)( - 2)( - 3)K^4[1+_i_j+_k_l+_i_k+', '1905.01133-1-147-0': '& & +_i_l+_j_k+_j_l+', '1905.01133-1-148-0': '& & + _i_j_k+_i_j_l+_i_k_l+_j_k_l+_i_j_k_l]', '1905.01133-1-149-0': '& = & ( - 1)( - 2)( -3 )K^4[1+(_ij)+(_kl)+(_ik)+', '1905.01133-1-150-0': '& & + (_il)+(_jk)+(_jl)+', '1905.01133-1-151-0': '& & + (_i,_j,_k)+(_i,_k,_l)+(_i,_j,_l)+(_j,_k,_l)+', '1905.01133-1-152-0': '& & + (_ij)(_kl)+(_ik)(_jl)+(_il)(_jk)', '1905.01133-1-153-0': '& & +(_i,_j,_k,_l)] ,', '1905.01133-1-154-0': 'where [MATH]) is the density perturbation (relative to the actual mean density of galaxies in the survey), [MATH] is the three-point correlation, and [MATH] is the connected (i.e., the part that does not arise from the two-point correlation) four-point correlation.', '1905.01133-1-154-1': 'The fraction of microcell quadruplets (pairs of pairs) that satisfy [MATH] and [MATH] is [MATH].', '1905.01133-1-154-2': 'In the limit of large [MATH] the number of other index quadruplets is negligible compared to those where all indices have different values, so we have ^_ijkl ^ij_1^kl_2 = K(K-1)(K-2)(K-3)4_1_2 = K^44_1_2.', '1905.01133-1-155-0': 'For the connected pairs [MATH] and [MATH] we have [MATH] and [MATH].', '1905.01133-1-156-0': 'The second case (triplets of microcells) is when [MATH] or [MATH] is equal to [MATH] or [MATH].', '1905.01133-1-156-1': 'We denote this part of the sum by ^*_ijkn_in_jn_k_1^ik_2^jk .', '1905.01133-1-156-2': 'It turns out that it goes over all ordered combinations of [MATH], where [MATH] are all different, exactly once, so there are [MATH] such terms (triplets).', '1905.01133-1-156-3': 'Here & & n_in_jn_k', '1905.01133-1-157-0': '& = & K - 1K - 1 - 2K - 2(1+_i)(1+_j)(1+_k)', '1905.01133-1-158-0': '& = & ( - 1)( - 2)K^3[1 +_i_k+_j_k+_i_j+ _i_j_k]', '1905.01133-1-159-0': '& = & ( - 1)( - 2)K^3[1+(_ik)+(_jk)+(_ij)+', '1905.01133-1-160-0': '& & +(_i,_j,_k)]', '1905.01133-1-161-0': 'and ^_ijk^ik_1^jk_2 = K^3_12 .', '1905.01133-1-162-0': 'The third case (pairs of microcells) is when [MATH] and [MATH].', '1905.01133-1-162-1': 'This part of the sum becomes _i<j n_in_j^ij_1^ij_2 , where n_in_j = (K^2[1+(_ij)] and _i<j^ij_1^ij_2 = _12K^22_1 , i.e., it vanishes unless the two bins are the same, [MATH].', '1905.01133-1-163-0': 'We apply now the three approximations listed in Sect. [REF]: 1) [MATH] in [REF]; 2) [MATH] in [REF] and [REF]; 3) [MATH] in [REF].', '1905.01133-1-163-1': 'We obtain & & DD_1 DD_2', '1905.01133-1-164-0': '& & (+', '1905.01133-1-165-0': '& & + (12)_12 +', '1905.01133-1-166-0': '& & + (12(1+_1)_1 and using [MATH] from [REF] we arrive at the [MATH] result given in Eq. [REF].', '1905.01133-1-167-0': "For the split method, we give the calculation for & = & RR'_1 DR_2 - RR'_1 DR_2 RR'_1 DR_2 ,", '1905.01133-1-168-0': "& = & RR'_1 RR'_2 - RR'_1RR'_2RR'_1RR'_2 below.", '1905.01133-1-169-0': "First the [MATH]: RR'_1 DR_2 = _R^R^_1 DR_2 = R^R^_1 DR_2, where R^R^_1 DR_2 = _i<j_kls_is_jn_kr_l^ij_1^kl_2 , [MATH] is the number (0 or 1) of [MATH] objects in microcell [MATH], and [MATH] is the number of [MATH] objects in microcell [MATH].", '1905.01133-1-170-0': 'There are [MATH] quadruplet terms, for which (see [REF]) [MATH], and s_is_jn_kr_l = (K^4 .', '1905.01133-1-171-0': 'Triplets where [MATH] or [MATH] is equal to [MATH] have [MATH], since [MATH] always (we cannot have two objects, from [MATH] and [MATH], in the same cell).', '1905.01133-1-171-1': 'There are [MATH] triplet terms where [MATH] or [MATH] is equal to [MATH]: for them [MATH], and s_is_ln_kr_l = s_is_ln_k = (K^3 (if [MATH] then also [MATH] since [MATH]).', '1905.01133-1-172-0': 'Pairs where [MATH] or [MATH] have [MATH], since again we cannot have two different objects in the same cell.', '1905.01133-1-172-1': "Thus RR'_1 DR_2 = (+ (12 and we get that = 2[_12_1_2 - 1] = 2_12 , which is equal to the [MATH] of standard LS.", '1905.01133-1-173-0': "Then the [MATH]: RR'_1 RR'_2 = __R^R^_1R^R^_2, where there are [MATH] terms with [MATH] giving R^R^_1 R^R^_2 = R^R^_1R^R^_2 = ()^2( and [MATH] terms with [MATH] giving R^R^_1 R^R^_2& = & (+", '1905.01133-1-174-0': '& & + (12', '1905.01133-1-175-0': '& & + (12_1 .', '1905.01133-1-175-1': "Adding these up gives RR'_1 RR'_2 & = & (+", '1905.01133-1-176-0': '& & + (12 +', '1905.01133-1-177-0': '& & + (12_1 and & = & 2( 2(_12_1_2 - 1] + _12_1 - 1', '1905.01133-1-178-0': '& = & 4_12 + _12 .', '1905.01133-1-178-1': 'This differs both from standard LS and from dilution, since it involves both [MATH] and [MATH].'}

{'1905.01133-2-0-0': 'Estimating the galaxy two-point correlation function using a split random catalog', '1905.01133-2-1-0': 'The two-point correlation function of the galaxy distribution is a key cosmological observable that allows us to constrain the dynamical and geometrical state of our Universe.', '1905.01133-2-1-1': 'To measure the correlation function we need to know both the galaxy positions and the expected galaxy density field.', '1905.01133-2-1-2': 'The expected field is commonly specified using a Monte-Carlo sampling of the volume covered by the survey and, to minimize additional sampling errors, this random catalog has to be much larger than the data catalog.', '1905.01133-2-1-3': 'Correlation function estimators compare data-data pair counts to data-random and random-random pair counts, where random-random pairs usually dominate the computational cost.', '1905.01133-2-1-4': 'Future redshift surveys will deliver spectroscopic catalogs of tens of millions of galaxies.', '1905.01133-2-1-5': 'Given the large number of random objects required to guarantee sub-percent accuracy, it is of paramount importance to improve the efficiency of the algorithm without degrading its precision.', '1905.01133-2-1-6': 'We show both analytically and numerically that splitting the random catalog into a number of subcatalogs of the same size as the data catalog when calculating random-random pairs and excluding pairs across different subcatalogs provides the optimal error at fixed computational cost.', '1905.01133-2-1-7': 'For a random catalog fifty times larger than the data catalog, this reduces the computation time by a factor of more than ten without affecting estimator variance or bias.', '1905.01133-2-2-0': 'Two-point correlation function with split random catalog', '1905.01133-2-3-0': '# Introduction', '1905.01133-2-4-0': 'The spatial distribution of luminous matter in the Universe is a key diagnostic for studying cosmological models and the physical processes involved in the assembly of structure.', '1905.01133-2-4-1': 'In particular, light from galaxies is a robust tracer of the overall matter distribution, whose statistical properties can be predicted by cosmological models.', '1905.01133-2-4-2': 'Two-point correlation statistics are very effective tools for compressing the cosmological information encoded in the spatial distribution of the mass in the Universe.', '1905.01133-2-4-3': 'In particular, the two-point correlation function in configuration space has emerged as one of the most popular cosmological probes.', '1905.01133-2-4-4': 'Its success stems from the presence of characterized features that can be identified, measured, and effectively compared to theoretical models to extract clean cosmological information.', '1905.01133-2-5-0': 'One such feature is baryon acoustic oscillations (BAOs), which imprint a characteristic scale in the two-point correlation function that can be used as a standard ruler.', '1905.01133-2-5-1': 'After the first detection in the two-point correlation function of SDSS DR3 and 2dFGRS galaxy catalogs , the BAO signal was identified, with different degrees of statistical significance, and has since been used to constrain the expansion history of the Universe in many spectroscopic galaxy samples (see e.g., [CITATION] [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]).', '1905.01133-2-5-2': 'Several of these studies did not focus on the BAO feature only but also analyzed the anisotropies in the two-point correlation function induced by the peculiar velocities , the so-called redshift space distortions (RSD), and by assigning cosmology-dependent distances to the observed redshifts (the [CITATION] test).', '1905.01133-2-5-3': 'For RSD analyses, see also, for example, [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION].', '1905.01133-2-6-0': 'Methods to estimate the galaxy two-point correlation function (2PCF) [MATH] from survey data are based on its definition as the excess probability of finding a galaxy pair.', '1905.01133-2-6-1': 'One counts from the data ([MATH]) catalog the number [MATH] of pairs of galaxies with separation [MATH], where [MATH] is a bin of separation vectors, and compares it to the number of pairs [MATH] in a corresponding randomly generated ([MATH]) catalog and to the number of data-random pairs [MATH].', '1905.01133-2-6-2': 'The bin may be a 1D ([MATH]), 2D, or a 3D bin.', '1905.01133-2-6-3': 'In the 1D case, [MATH] is the length of the separation vector and [MATH] is the width of the bin.', '1905.01133-2-6-4': "From here on, 'separation [MATH]' indicates that the separation falls in this bin.", '1905.01133-2-7-0': 'Several estimators of the 2PCF have been proposed by [CITATION], [CITATION], [CITATION], and [CITATION], building on the original [CITATION] proposal.', '1905.01133-2-7-1': 'These correspond to different combinations of the [MATH], [MATH], and [MATH] counts to obtain a 2PCF estimate [MATH]; see [CITATION] and [CITATION] for more estimators.', '1905.01133-2-7-2': 'The Landy-Szalay estimator () := ((DD()RR() - DR()RR() + 1', '1905.01133-2-8-0': ",(we call this method 'standard LS' in the following) is the most commonly used, since it provides the minimum variance when [MATH] and is unbiased in the limit [MATH].", '1905.01133-2-8-1': 'Here [MATH] is the size (number of objects) of the data catalog and [MATH] is the size of the random catalog.', '1905.01133-2-8-2': 'We define [MATH].', '1905.01133-2-8-3': 'To minimize random error from the random catalog, [MATH] should be used (for a different approach, see [CITATION]).', '1905.01133-2-9-0': 'One is usually interested in [MATH] only up to some [MATH] (the maximum separation in the survey), and therefore pairs with larger separations can be skipped.', '1905.01133-2-9-1': 'Efficient implementations of the LS estimator involve pre-ordering of the catalogs through kd-tree, chain-mesh, or other algorithms to facilitate this.', '1905.01133-2-9-2': 'The computational cost is then roughly proportional to the actual number of pairs with separation [MATH].', '1905.01133-2-10-0': 'The correlation function is small for large separations, and in cosmological surveys [MATH] is large enough so that for most pairs [MATH].', '1905.01133-2-10-1': 'The fraction [MATH] of [MATH] pairs with [MATH] is therefore not very different from the fraction of [MATH] or [MATH] pairs with [MATH].', '1905.01133-2-10-2': 'The computational cost is dominated by the part proportional to the total number of pairs needed, [MATH], which in turn is dominated by the [MATH] pairs as [MATH].', '1905.01133-2-10-3': 'The smaller number of [MATH] pairs contribute much more to the error of the estimate than the large number of [MATH] pairs, whereas the cost is dominated by [MATH].', '1905.01133-2-10-4': 'Thus, a significant saving of computation time with an insignificant loss of accuracy may be achieved by counting only a subset of [MATH] pairs, while still counting the full set (up to [MATH]) of [MATH] pairs.', '1905.01133-2-11-0': 'A good way to achieve this is to use many small (i.e, low-density) [MATH] catalogs instead of one large (high-density) catalog , or, equivalently, to split an already generated large [MATH] catalog into [MATH] small ones for the calculation of [MATH] pairs while using the full [MATH] catalog for the [MATH] counts.', '1905.01133-2-11-1': 'This method has been used by some practitioners (e.g., [CITATION]), but this is usually not documented in the literature.', '1905.01133-2-11-2': 'One might also consider obtaining a similar cost saving by diluting (subsampling) the [MATH] catalog for [MATH] counts, but, as we show below, this is not a good idea.', '1905.01133-2-11-3': "We refer to these two cost-saving methods as 'split' and 'dilution'.", '1905.01133-2-12-0': "In this work we theoretically derive the additional covariance and bias due to the size and treatment of the [MATH] catalog; test these predictions numerically with mock catalogs representative of next-generation datasets, such as the spectroscopic galaxy samples that will be obtained by the future Euclid satellite mission ; and show that the 'split' method, while reducing the computational cost by a large factor, retains the advantages of the LS estimator.", '1905.01133-2-13-0': 'We follow the approach of Landy Szalay (1993; hereafter, LS93), but generalize it in a number of ways: In particular, since we focus on the effect of the random catalog, we do not work in the limit [MATH].', '1905.01133-2-13-1': 'Also, we calculate covariances, not just variances, and make fewer approximations (see Sect. [REF]).', '1905.01133-2-14-0': 'The layout of the paper is as follows.', '1905.01133-2-14-1': 'In Sect. [REF] we derive theoretical results for bias and covariance.', '1905.01133-2-14-2': 'In Sect. [REF] we focus on the split LS estimator and its optimization.', '1905.01133-2-14-3': 'In Sect. [REF] we test the different estimators with mock catalogs.', '1905.01133-2-14-4': 'Finally, we discuss the results and present our conclusions in Sect. [REF].', '1905.01133-2-15-0': '# Theoretical results: bias and covariance', '1905.01133-2-16-0': '## General derivation', '1905.01133-2-17-0': 'We follow the derivation and notations in LS93 but extend to the case that includes random counts covariance.', '1905.01133-2-17-1': 'We consider the survey volume as divided into [MATH] microcells (very small subvolumes) and work in the limit [MATH], which means that no two objects will ever be located within the same microcell.', '1905.01133-2-18-0': 'Here, [MATH], [MATH], and [MATH] represent the relative deviation of the [MATH], [MATH], and [MATH] counts from their expectation values (mean values over an infinite number of independent realizations): DD() & =: & DD() [1+()] ,', '1905.01133-2-19-0': 'DR() & =: & DR() [1+()] ,', '1905.01133-2-20-0': 'RR() & =: & RR() [1+()] .', '1905.01133-2-21-0': 'By definition [MATH].', '1905.01133-2-21-1': 'The factors [MATH], [MATH], and [MATH] represent fluctuations in the pair counts, which arise as a result of a Poisson process.', '1905.01133-2-21-2': 'As long as the mean pair counts per bin are large ([MATH]) the relative fluctuations will be small.', '1905.01133-2-21-3': 'We calculate up to second order in [MATH], [MATH], and [MATH], and ignore the higher-order terms (in the limit [MATH], [MATH], so LS93 set [MATH] at this point).', '1905.01133-2-22-0': 'The expectation values for the pair counts are: DD()& = & (,', '1905.01133-2-23-0': 'DR()& = & () ,', '1905.01133-2-24-0': 'RR()& = & (,', '1905.01133-2-25-0': 'where [MATH] is the correlation function normalized to the actual number density of galaxies in the survey and () := 2K^2 _i<j^K ^ij() is the fraction of microcell pairs with separation [MATH].', '1905.01133-2-25-1': 'Here [MATH] if [MATH] falls in the [MATH]-bin, otherwise it is equal to zero.', '1905.01133-2-26-0': 'The expectation value of the LS estimator [REF] is & = & (1+)1+1+- 21+1++ 1', '1905.01133-2-27-0': '& & + (+ 2.', '1905.01133-2-28-0': 'A finite [MATH] catalog thus introduces a (small) bias.', '1905.01133-2-28-1': '(In LS93, [MATH], so the estimator is unbiased in this limit.)', '1905.01133-2-28-2': "This expression is calculated to second order in [MATH], [MATH], and [MATH] (we denote equality to second order by '[MATH]').", '1905.01133-2-28-3': 'Calculation to higher order is beyond the scope of this work.', '1905.01133-2-28-4': 'Since data and random catalogs are independent, [MATH].', '1905.01133-2-29-0': 'We introduce shorthand notations [MATH] for [MATH], [MATH] for [MATH], and similarly for other terms.', '1905.01133-2-30-0': 'For the covariance we get', '1905.01133-2-31-0': '&Cov[(_1),(_2)] (_1)(_2)- (_1)(_2)', '1905.01133-2-32-0': '& (1+_1)(1+_2)_1_2+ 4_1_2+ (1-_1)(1-_2)_1_2', '1905.01133-2-33-0': '&- 2(1+_1)_1_2- 2(1+_2)_1_2', '1905.01133-2-34-0': '&- 2(1-_1)_1_2- 2(1-_2)_1_2.', '1905.01133-2-35-0': 'Terms with [MATH] represent additional variance due to finite [MATH], and are new compared to those of LS93.', '1905.01133-2-35-1': 'Also, [MATH] collects an additional contribution, which we denote by [MATH], from variations in the random field (see Sect. [REF]).', '1905.01133-2-35-2': 'The cross terms [MATH] and [MATH], instead depend linearly on the random field, and average to the [MATH] result.', '1905.01133-2-35-3': 'The additional contribution due to finite [MATH] is thus', '1905.01133-2-36-0': '&Cov[(_1),(_2)] 4_1_2+ (1-_1)(1-_2)_1_2', '1905.01133-2-37-0': '&- 2(1-_1)_1_2- 2(1-_2)_1_2.', '1905.01133-2-38-0': 'From [REF], DD_1 DD_2 = DD_1DD_2(1+_1_2) ,and so on, so that the covariances of the deviations are obtained from _1_2& = & DD_1 DD_2 - DD_1DD_2DD_1DD_2 ,', '1905.01133-2-39-0': '_1_2& = & DR_1 DR_2 - DR_1DR_2DR_1DR_2 ,', '1905.01133-2-40-0': '_1_2& = & RR_1 RR_2 - RR_1RR_2RR_1RR_2 ,', '1905.01133-2-41-0': '_1_2& = &DD_1 DR_2 - DD_1 DR_2 DD_1 DR_2 ,', '1905.01133-2-42-0': '_1_2& = & DR_1 RR_2 - DR_1 RR_2 DR_1 RR_2 ,', '1905.01133-2-43-0': '_1_2& = & DD_1 RR_2 - DD_1 RR_1 DD_1 RR_2 = 0.', '1905.01133-2-44-0': '## Quadruplets, triplets, and approximations', '1905.01133-2-45-0': 'We use _12 := (_1,_2) := 1K^3^_ijk^ik_1^jk_2 to denote the fraction of ordered microcell triplets, where [MATH] and [MATH].', '1905.01133-2-45-1': 'The notation [MATH] means that only terms where all indices (microcells) are different are included.', '1905.01133-2-45-2': 'Here [MATH] is of the same magnitude as [MATH] but is larger.', '1905.01133-2-46-0': 'Appendix A gives examples of how the [MATH] and so on in [REF] are calculated.', '1905.01133-2-46-1': 'These covariances involve expectation values [MATH], where [MATH] is the number of objects (0 or 1) in microcell [MATH] and so on, and only cases where the four microcells are separated pairwise by [MATH] and [MATH] are included.', '1905.01133-2-46-2': 'If all four microcells [MATH], [MATH], [MATH], and [MATH] are different, we call this case a quadruplet; it consists of two pairs with separations [MATH] and [MATH].', '1905.01133-2-46-3': 'If two of the indices, that is, microcells, are equal, we have a triplet with a center cell (the equal indices) and two end cells separated from the center by [MATH] and [MATH].', '1905.01133-2-47-0': 'We make the following three approximations:', '1905.01133-2-48-0': 'For microcell quadruplets, the correlations between unconnected cells are approximated by zero on average.', '1905.01133-2-48-1': 'Three-point correlations vanish.', '1905.01133-2-48-2': 'The part of four-point correlations that does not arise from the two-point correlations vanishes.', '1905.01133-2-49-0': 'With approximations (2) and (3), we have for the expectation value of a galaxy triplet n_in_jn_k1+_ij+_jk+_ik , where [MATH], and for a quadruplet n_in_jn_kn_l1+_ij+_jk+_ik+_il+_jl+_kl+_ij_kl+_ik_jl+_il_jk .', '1905.01133-2-50-0': "We use '[MATH]' to denote results based on these three approximations.", '1905.01133-2-50-1': 'Approximation (1) is good as long as the survey size is large compared to [MATH].', '1905.01133-2-50-2': 'It allows us to drop terms other than [MATH] in [REF].', '1905.01133-2-50-3': 'Approximations (2) and (3) hold for Gaussian density fluctuations, but in the realistic cosmological situation they are not good: the presence of the higher-order correlations makes the estimation of the covariance of [MATH] estimators a difficult problem.', '1905.01133-2-50-4': 'However, this difficulty applies only to the contribution of the data to the covariance, that is, to the part that does not depend on the size and treatment of the random catalog.', '1905.01133-2-50-5': 'The key point in this work is that while our theoretical result for the total covariance does not hold in a realistic situation (it is an underestimate), our results for the difference in estimator covariance due to different treatments of the random catalog hold well.', '1905.01133-2-51-0': 'In addition to working in the limit [MATH]), LS93 considered only 1D bins and the case where [MATH] (i.e., variances, not covariances) and made also a fourth approximation: for triplets (which in this case have legs of equal length) they approximated the correlation between the end cells (whose separation in this case varies between [MATH] and [MATH]) by [MATH].', '1905.01133-2-51-1': 'We use [MATH] to denote the mean value of the correlation between triplet end cells (separated from the triplet center by [MATH] and [MATH]).', '1905.01133-2-51-2': '(For our plots in Sect. [REF] we make a similar approximation of [MATH] as LandySzalay, see Sect. [REF].)', '1905.01133-2-51-3': 'Also, LS93 only calculated to first order in [MATH], whereas we do not make this approximation.', '1905.01133-2-51-4': '[CITATION] also considered covariances, and included the effect of three-point and four-point correlations, but worked in the limit [MATH]).', '1905.01133-2-52-0': '## Poisson, edge, and [MATH] terms', '1905.01133-2-53-0': 'After calculating all the [MATH] and so on (see Appendix [REF]), [REF] becomes', '1905.01133-2-54-0': '(1+_1)(1+_2)_1_2 4 (1+_1)(1+_2)[_12_1_2 - 1]', '1905.01133-2-55-0': '+ 2(1+_1)(_12_1 - 2(1+_2)_12_1_2 + (1+_2) ]', '1905.01133-2-56-0': '+ 4((12-_1_2)_12_1_2 ,', '1905.01133-2-57-0': '_1_2 1 [_12_1_2-1] + [_12_1_2-1]', '1905.01133-2-58-0': '+ 1 - 2_12_1_2 + _12_1 + _12_12_1_2 ,', '1905.01133-2-59-0': '_1_2 = 2( 2(_12_1_2 - 1] + _12_1 - 1 ,', '1905.01133-2-60-0': '_1_2 2[_12_1_2 - 1] ,', '1905.01133-2-61-0': '_1_2 = 2[_12_1_2 - 1] ,', '1905.01133-2-62-0': '_1_2 = 0,', '1905.01133-2-63-0': 'for the standard LS estimator.', '1905.01133-2-64-0': 'Following the definition of [MATH] and [MATH] in LS93, we define t_12 & := & 1[_12_1_2-1] ,', '1905.01133-2-65-0': '_12 & := & 1[_12_1_2-1] = t_12 ,', '1905.01133-2-66-0': 'p_12 & := & 2(_12(1+_1)_1 - 2_12_1_2 + 1] ,', '1905.01133-2-67-0': '_12 & := & 1[_12_1 - 2_12_1_2 + 1] ,', '1905.01133-2-68-0': '_12 & := & 2(_12_1 - 2_12_1_2 + 1] ,', '1905.01133-2-69-0': 'q_12 & := & 1_12_1_2 = t_12 + 1 ,', '1905.01133-2-70-0': '_12 & := & 1_12_1_2 = _12 + 1 .', '1905.01133-2-71-0': 'For their diagonals ([MATH]), we write [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH].', '1905.01133-2-71-1': 'Thus, [MATH], [MATH] and so on.', '1905.01133-2-71-2': '(We use superscripts for the matrices, e.g., [MATH] and subscripts for their diagonals, e.g., [MATH].)', '1905.01133-2-72-0': 'Using these definitions, [REF] becomes', '1905.01133-2-73-0': '(1+_1)(1+_2)_1_2 (1+_1)(1+_2)(4t_12 + p_12)', '1905.01133-2-74-0': '+ 4(12-_1_2)q_12 ,', '1905.01133-2-75-0': '_1_2 t_12 + _12 + _12 + _12_12 ,', '1905.01133-2-76-0': '_1_2 = 4_12 + _12 ,', '1905.01133-2-77-0': '_1_2 2t_12 , _1_2 = 2_12 ,and _1_2 = 0 .', '1905.01133-2-78-0': 'The new part in [MATH] due to finite size of the random catalog is _1_2 _12 + _12 + _12_12 .', '1905.01133-2-79-0': 'Thus only [MATH] and [MATH] are affected by [MATH] (in our approximation its effect cancels in [MATH]).', '1905.01133-2-79-1': 'The results for [MATH], [MATH], and [MATH] are exact.', '1905.01133-2-79-2': 'The result for [MATH] involves all three approximations mentioned above, [MATH] involves approximations (1) and (2), and [MATH] involves approximation (1).', '1905.01133-2-80-0': "We refer to [MATH], [MATH], and [MATH] as 'Poisson' terms and [MATH] and [MATH] as 'edge' terms (the difference between [MATH] and [MATH] is due to edge effects).", '1905.01133-2-80-1': 'While the Poisson terms are strongly diagonal dominated, the edge terms are not.', '1905.01133-2-80-2': 'Since [MATH], the [MATH] terms are much larger than the edge terms, but they get multiplied by [MATH] or [MATH].', '1905.01133-2-80-3': 'In the limit [MATH]: [MATH], [MATH], [MATH]; [MATH] and also [MATH] are unaffected.', '1905.01133-2-81-0': 'We see that [MATH]-[MATH] and [MATH]-[MATH] correlations arise from edge effects.', '1905.01133-2-81-1': 'If we increase the density of data or random objects, the Poisson terms decrease as [MATH] but the edge terms decrease only as [MATH] so the edge effects are more important for a higher density of objects.', '1905.01133-2-82-0': 'Doubling the bin size (combining neighboring bins) doubles [MATH] but makes [MATH] four times as large, since triplets where one leg was in one of the original smaller bins and the other leg was in the other bin are now also included.', '1905.01133-2-82-1': 'Thus, the ratio [MATH] and [MATH] are not affected, but the dominant term in [MATH], [MATH] is halved.', '1905.01133-2-82-2': 'Edge effects are thus more important for larger bins.', '1905.01133-2-83-0': '## Results for the standard Landy-Szalay estimator', '1905.01133-2-84-0': 'Inserting the results for [MATH] and so on into Eqs. ([REF]) and ([REF]), we get that the expectation value of the standard LS estimator [REF] is = + (+ 4.', '1905.01133-2-85-0': 'This holds also for large [MATH] and in the presence of three-point and four-point correlations.', '1905.01133-2-85-1': 'A finite [MATH] catalog thus introduces a bias [MATH]; the edge ([MATH]) part of the bias cancels in the [MATH] limit.', '1905.01133-2-86-0': 'For the covariance we get', '1905.01133-2-87-0': '&Cov[(_1),(_2)] (_1)(_2)- (_1)(_2)', '1905.01133-2-88-0': '& (1+_1)(1+_2)p_12 + 4_12 + (1-_1)(1-_2)_12', '1905.01133-2-89-0': '&+ 4_1_2(t_12+_12)', '1905.01133-2-90-0': '&+ 412-_1_2)q_12 + 4_12_12 .', '1905.01133-2-91-0': 'Because of the approximations made, this result for the covariance does not apply to the realistic cosmological case; not even for large separations [MATH], where [MATH] is small, since large correlations at small [MATH] increase the covariance also at large [MATH].', '1905.01133-2-91-1': 'However, this concerns only [MATH] and [MATH].', '1905.01133-2-91-2': 'Our focus here is on the additional covariance due to the size and handling of the random catalog, which for standard LS is', '1905.01133-2-92-0': '&Cov[(_1),(_2)] 4_1_2+ (1-_1)(1-_2)_1_2', '1905.01133-2-93-0': '&- 2(1-_1)_1_2- 2(1-_2)_1_2', '1905.01133-2-94-0': '& 4_12 + (1-_1)(1-_2)_12 +4_1_2_12 + 4_12_12 .', '1905.01133-2-95-0': 'To zeroth order in [MATH] the covariance is given by the Poisson terms and the edge terms cancel to first order in [MATH].', '1905.01133-2-95-1': 'This is the property for which the standard LS estimator was designed.', '1905.01133-2-95-2': 'To first order in [MATH], the [MATH] terms contribute.', '1905.01133-2-95-3': 'This [MATH] contribution involves the triplet correlation [MATH], which, depending on the form of [MATH], may be larger than [MATH] or [MATH].', '1905.01133-2-96-0': 'If we try to save cost by using a diluted random catalog with [MATH] for [MATH] pairs, [MATH] is replaced by [MATH] with [MATH] in place of [MATH], but [MATH] and [MATH] are unaffected, so that the edge terms involving randoms no longer cancel.', '1905.01133-2-96-1': 'In Sect. [REF] we see that this is a large effect.', '1905.01133-2-96-2': 'Therefore, one should not use dilution.', '1905.01133-2-97-0': '# Split random catalog', '1905.01133-2-98-0': '## Bias and covariance for the split method', '1905.01133-2-99-0': 'In the split method one has [MATH] independent smaller [MATH] catalogs of size [MATH] instead of one large random catalog [MATH].', '1905.01133-2-99-1': 'Their union, [MATH] has a size of [MATH].', '1905.01133-2-99-2': "The pair counts [MATH] and [MATH] are calculated as DR() := _DR^() and RR'() := _R^R^() , that is, pairs across different [MATH] catalogs are not included in [MATH].", '1905.01133-2-99-3': 'The total number of pairs in [MATH] is [MATH].', '1905.01133-2-99-4': 'Here, [MATH] is equal to its value in standard LS.', '1905.01133-2-100-0': "The split Landy-Szalay estimator is _split() := ((DD()RR'() - DR()RR'() + 1 .", '1905.01133-2-100-1': 'Compared to standard LS, [MATH], [MATH], and [MATH] are unaffected.', '1905.01133-2-100-2': "We construct [MATH], [MATH], and [MATH] from the standard LS results, bearing in mind that the random catalog is a union of independent catalogs, arriving at _1_2' &=& 2_12 ,", '1905.01133-2-101-0': '&=& 4_12 + _12 , where _12 := ((12 12 .', '1905.01133-2-101-1': 'The first is the same as in standard LS and dilution, but the second differs both from standard LS and from dilution, since it involves both [MATH] and [MATH].', '1905.01133-2-102-0': 'For the expectation value we get _split = + (+ 4,', '1905.01133-2-103-0': 'so that the bias is [MATH].', '1905.01133-2-103-1': 'In the limit [MATH] the edge part cancels, leaving only the Poisson term.', '1905.01133-2-104-0': 'The covariance is', '1905.01133-2-105-0': '&Cov[_split(_1),_split(_2)]', '1905.01133-2-106-0': '& (1+_1)(1+_2)p_12 + 4_12 + (1-_1)(1-_2)_12', '1905.01133-2-107-0': '&+ 4_1_2(t_12+_12) .', '1905.01133-2-108-0': 'The change in the covariance compared to the standard LS method is', '1905.01133-2-109-0': '&Cov[^split_1,^split_2] - Cov[^LS_1,^LS_2]', '1905.01133-2-110-0': '&= (1-_1)(1-_2)(_12-_12) ,', '1905.01133-2-111-0': 'which again applies in the realistic cosmological situation.', '1905.01133-2-111-1': 'Our main result is that in the split method the edge effects cancel and the bias and covariance are the same as for standard LS, except that the Poisson term [MATH] from [MATH] is replaced with the larger [MATH].', '1905.01133-2-112-0': '## Optimizing computational cost and variance of the split method', '1905.01133-2-113-0': 'The bias is small compared to variance in our application (see Fig. [REF] for the theoretical result and Fig. [REF] for an attempted bias measurement), and therefore we focus on variance as the figure of merit.', '1905.01133-2-113-1': 'The computational cost should be roughly proportional to ^2(1+2+^2) =: ^2c ,', '1905.01133-2-114-0': 'and the additional variance due to finite [MATH] catalog in the [MATH] limit becomes var (2+^2)p =: vp.', '1905.01133-2-115-0': 'Here, [MATH] and [MATH] are fixed by the survey and the requested [MATH] binning, but we can vary [MATH] and [MATH] in the search for the optimal computational method.', '1905.01133-2-115-1': "In the above we defined the 'cost' and 'variance' factors [MATH] and [MATH].", '1905.01133-2-116-0': 'We may ask two questions:', '1905.01133-2-117-0': 'For a fixed level of variance [MATH], which combination of [MATH] and [MATH] minimizes computational cost [MATH]?', '1905.01133-2-117-1': 'For a fixed computational cost [MATH], which combination of [MATH] and [MATH] minimizes the variance [MATH]?', '1905.01133-2-118-0': 'The answer to both questions is = c = 1+3andv = 3 .', '1905.01133-2-119-0': 'Thus, the optimal version of the split method is the natural one where [MATH].', '1905.01133-2-119-1': 'In this case the additional variance in the [MATH] limit becomes var (2 + )p ,and the computational cost factor [MATH] becomes (1 + 2 + )^2 , meaning that [MATH] pairs contribute twice as much as [MATH] pairs to the variance and also twice as much computational cost is invested in them.', '1905.01133-2-119-2': 'The memory requirement for the random catalog is then the same as for the data catalog.', '1905.01133-2-119-3': 'The cost saving estimate above is optimistic, since the computation involves some overhead not proportional to the number of pairs.', '1905.01133-2-120-0': 'For small scales, where [MATH], the situation is different.', '1905.01133-2-120-1': 'The greater density of [MATH] pairs due to the correlation requires a greater density of the [MATH] catalog so that the additional variance from it is not greater.', '1905.01133-2-120-2': 'From Eq. [REF] we see that the balance of the [MATH] and the [MATH] contributions is different for large [MATH] (the [MATH] term vs. the other terms).', '1905.01133-2-120-3': 'We may consider recomputing [MATH] for the small scales using a smaller [MATH] and a larger [MATH] catalog.', '1905.01133-2-120-4': "Considering just the Poisson terms ([MATH] and [MATH] or [MATH]) with a 'representative' [MATH] value, [REF] and [REF] become [MATH] and [MATH] which modifies the above result (Eq. [REF]) for the optimal choice of [MATH] and [MATH] to = , that is, = .", '1905.01133-2-120-5': 'This result is only indicative, since it assumes a constant [MATH] for [MATH].', '1905.01133-2-120-6': 'In particular, it does not apply for [MATH], because then the approximation of ignoring the [MATH] and [MATH] terms in [REF] is not good.', '1905.01133-2-121-0': '# Tests on mock catalogs', '1905.01133-2-122-0': '## Minerva simulations and methodology', '1905.01133-2-123-0': 'The Minerva mocks are a set of 300 cosmological mocks produced with [MATH]-body simulations , stored at five output redshifts [MATH].', '1905.01133-2-123-1': 'The cosmology is flat [MATH]CDM with [MATH], and we use the [MATH] outputs.', '1905.01133-2-123-2': 'The mocks have [MATH] objects ("halos" found by a friends-of-friend algorithm) in a box of [MATH] cubed.', '1905.01133-2-124-0': 'To model the survey geometry of a redshift bin with [MATH] at [MATH], we placed the observer at comoving distance [MATH] from the center of the cube and selected from the cube a shell [MATH]-[MATH] from the observer.', '1905.01133-2-124-1': 'The comoving thickness of the shell is [MATH].', '1905.01133-2-124-2': 'The resulting mock sub-catalogs have [MATH] and are representative of the galaxy number density of the future Euclid spectroscopic galaxy catalog.', '1905.01133-2-125-0': 'We ignore peculiar velocities, that is, we perform our analysis in real space.', '1905.01133-2-125-1': 'Therefore, we consider results for the 1D 2PCF [MATH].', '1905.01133-2-125-2': 'We estimated [MATH] up to [MATH] using [MATH] bins.', '1905.01133-2-126-0': 'We chose standard LS with [MATH] as the reference method.', '1905.01133-2-126-1': 'In the following, LS without further qualification refers to this.', '1905.01133-2-126-2': 'The random catalog was generated separately for each shell mock to measure their contribution to the variance.', '1905.01133-2-126-3': 'For one of the random catalogs we calculated also triplets to obtain the edge effect quantity [MATH].', '1905.01133-2-127-0': 'While dilution can already be discarded on theoretical grounds, we show results obtained using dilution, since these results provide the scale for edge effects demonstrating the importance of eliminating them with a careful choice of method.', '1905.01133-2-127-1': 'For the dilution and split methods we also used [MATH] , and tried out dilution fractions [MATH] and split factors [MATH] (chosen to have similar pairwise computational costs).', '1905.01133-2-127-2': 'In addition, we considered standard LS with [MATH], which has the same number of [MATH] pairs as [MATH] and [MATH], but only half the number of [MATH] pairs; and standard LS with [MATH] to demonstrate the effect of a small [MATH].', '1905.01133-2-128-0': 'The code used to estimate the 2PCF implements a highly optimized pair-counting method, specifically designed for the search of object pairs in a given range of separations.', '1905.01133-2-128-1': 'In particular, the code provides two alternative counting methods, the chain-mesh and the kd-tree.', '1905.01133-2-128-2': 'Both methods measure the exact number of object pairs in separation bins, without any approximation.', '1905.01133-2-128-3': 'However, since they implement different algorithms to search for pairs, they perform differently at different scales, both in terms of CPU time and memory usage.', '1905.01133-2-128-4': 'Overall, the efficiency of the two methods depends on the ratio between the scale range of the searching region and the maximum separation between the objects in the catalog.', '1905.01133-2-129-0': 'The kd-tree method first constructs a space-partitioning data structure that is filled with catalog objects.', '1905.01133-2-129-1': 'The minimum and maximum separations probed by the objects are kept in the data structure and are used to prune object pairs with separations outside the range of interest.', '1905.01133-2-129-2': 'The tree pair search is performed through the dual-tree method in which cross-pairs between two dual trees are explored.', '1905.01133-2-129-3': 'This is an improvement in terms of exploration time over the single-tree method.', '1905.01133-2-130-0': 'On the other hand, in the chain-mesh method the catalog is divided in cubic cells of equal size, and the indexes of the objects in each cell are stored in vectors.', '1905.01133-2-130-1': 'To avoid counting object pairs with separations outside the interest range, the counting is performed only on the cells in a selected range of distances from each object.', '1905.01133-2-130-2': 'The chain-mesh algorithm has been imported from the CosmoBolognaLib, a large set of free software C++/python libraries for cosmological calculations .', '1905.01133-2-131-0': 'For our test runs we used the chain-mesh method.', '1905.01133-2-132-0': '## Variance and bias', '1905.01133-2-133-0': 'In Fig. [REF] we show the mean (over the 300 mock shells) estimated correlation function and the scatter (square root of the variance) of the estimates using the LS, split, and dilution methods; our theoretical approximate result for the scatter for LS; and our theoretical result for bias for the different methods.', '1905.01133-2-134-0': 'The theoretical result for the scatter is shown with and without the [MATH] terms, which include the triplet correlation [MATH], for which we used here the approximation [MATH].', '1905.01133-2-134-1': 'This behaves as expected, that is, it underestimates the variance, since we neglected the higher-order correlations in the [MATH] catalog.', '1905.01133-2-134-2': 'Nevertheless, it (see the dash-dotted line in Fig. [REF]) has similar features to the measured variance (dashed lines).', '1905.01133-2-135-0': 'In Fig. [REF] we plot the diagonals of the [MATH], [MATH], and [MATH] quantities.', '1905.01133-2-135-1': 'This shows how their relative importance changes with separation scale.', '1905.01133-2-135-2': 'It also confirms that our initial assumption on small relative fluctuations is valid in this simulation case.', '1905.01133-2-136-0': 'Consider now the variance differences (from standard LS with [MATH]), for which our theoretical results should be accurate.', '1905.01133-2-136-1': 'Figure [REF] compares the measured variance difference to the theoretical result.', '1905.01133-2-136-2': 'For the diluted estimators and LS with [MATH] the measured result agrees with theory, although clearly the measurement with just 300 mocks is rather noisy.', '1905.01133-2-136-3': 'For the split estimators and LS with [MATH] the difference is too small to be appreciated with 300 mocks, but at least the measurement does not disagree with the theoretical result.', '1905.01133-2-137-0': 'In Fig. [REF] we show the relative theoretical increase in scatter compared to the best possible case, which is LS in the limit [MATH].', '1905.01133-2-137-1': 'Since we do not have a valid theoretical result for the total scatter, we estimate it by subtracting the theoretical difference from LS with [MATH] from the measured variance of the latter.', '1905.01133-2-138-0': 'At scales [MATH] the theoretical prediction is about the same for dilution and split and neither method looks promising for [MATH] where [MATH].', '1905.01133-2-138-1': 'This suggests that for optimizing cost and accuracy, a different method should be used for smaller scales than that used for large scales.', '1905.01133-2-138-2': 'The number of [MATH] pairs with small separations is much less.', '1905.01133-2-138-3': 'Therefore, for the small-scale computation there is no need to restrict the computation to a subset of [MATH] pairs, or alternatively, one can afford to increase [MATH].', '1905.01133-2-138-4': 'For the small scales, we may consider the split method with increased [MATH] as an alternative to standard LS.', '1905.01133-2-138-5': 'We have the same number of pairs to compute as in the reference LS case, if we use [MATH] and [MATH].', '1905.01133-2-138-6': 'We added this case to Fig. [REF].', '1905.01133-2-138-7': 'It seems to perform better than LS at intermediate scales, but for the smallest scales LS has the smaller variance.', '1905.01133-2-138-8': 'This is in line with our conclusion in Sect. [REF], which is that when [MATH], it is not optimal to split the [MATH] catalog into small subsets.', '1905.01133-2-139-0': 'We also compared the differences in the mean estimate from the different estimators to our theoretical results on the bias differences (see Fig. [REF]), but the theoretical bias differences are much smaller than the expected error of the mean from 300 mocks; and we simply confirm that the differences we see are consistent with the error of the mean and thus consistent with the true bias being much smaller.', '1905.01133-2-139-1': 'We also performed tests with completely random ([MATH]) mocks, and with a large number ([MATH]) of mocks confirmed the theoretical bias result for the different estimators in this case.', '1905.01133-2-139-2': 'Since the bias is too small to be interesting we do not report these results in more detail here.', '1905.01133-2-140-0': 'However, we note that for the estimation of the 2D 2PCF and its multipoles, the 2D bins will contain a smaller number of objects than the 1D bins of these test runs and therefore the bias is larger.', '1905.01133-2-140-1': 'Using the theoretical results [REF] or [REF] the bias can be removed afterwards with accuracy depending on how well we know the true [MATH].', '1905.01133-2-141-0': '## Computation time and variance', '1905.01133-2-142-0': 'The test runs were made using a single full 24-core node for each run.', '1905.01133-2-142-1': 'Table [REF] shows the mean computation time and mean estimator variance for different [MATH] ranges for the different cases we tested.', '1905.01133-2-142-2': 'Of these [MATH] ranges, the [MATH]-[MATH] is perhaps the most interesting, since it contains the important BAO scale.', '1905.01133-2-142-3': 'Therefore, we plot the mean variance at this range versus mean computation time in Fig. [REF], together with our theoretical predictions.', '1905.01133-2-142-4': 'The theoretical estimate for the computation time for other dilution fractions and split factors is (1+24.75d^2) 306s and(1+24.75/^2) 306s ,assuming [MATH].', '1905.01133-2-142-5': 'For standard LS with other random catalog sizes, the computation time estimate is (1+2+^2) 3.03s .', '1905.01133-2-143-0': '# Conclusions', '1905.01133-2-144-0': 'The computational time of the standard Landy-Szalay estimator is dominated by the [MATH] pairs.', '1905.01133-2-144-1': 'However, except at small scales where correlations are large, these make a negligible contribution to the expected error compared to the contribution from the [MATH] and [MATH] pairs.', '1905.01133-2-144-2': 'Therefore, a substantial saving of computation time with an insignificant loss of accuracy can be achieved by counting a smaller subset of [MATH] pairs.', '1905.01133-2-145-0': 'We considered two ways to reduce the number of [MATH] pairs: dilution and split.', '1905.01133-2-145-1': 'In dilution, only a subset of the [MATH] catalog is used for [MATH] pairs.', '1905.01133-2-145-2': 'In split, the [MATH] catalog is split into a number of smaller subcatalogs, and only pairs within each subcatalog are counted.', '1905.01133-2-145-3': 'We derived theoretical results for the additional estimator covariance and bias due to the finite size of the random catalog for these different variants of the LS estimator, extending in many ways the original results by [CITATION], who worked in the limit of an infinite random catalog.', '1905.01133-2-145-4': 'We tested our results using 300 mock data catalogs, representative of the [MATH]-[MATH] redshift range of the future Euclid survey.', '1905.01133-2-145-5': 'The split method maintains the property the Landy-Szalay estimator was designed for, namely cancelation of edge effects in bias and variance (for [MATH]), whereas dilution loses this cancellation and therefore should not be used.', '1905.01133-2-146-0': 'For small scales, where correlations are large, one should not reduce [MATH] counts as much.', '1905.01133-2-146-1': 'The natural dividing line is the scale [MATH] where [MATH].', '1905.01133-2-146-2': 'Interestingly, the difference in bias and covariance between the different estimators (split, dilution, and LS) vanishes when [MATH].', '1905.01133-2-146-3': 'We recommend the natural version of the split method, [MATH], for large scales where [MATH].', '1905.01133-2-146-4': 'This leads to a saving in computation time by more than a factor of ten (assuming [MATH]) with a negligible effect on variance and bias.', '1905.01133-2-146-5': 'For small scales, where [MATH], one should consider using a larger random catalog and one can use either the standard LS method or the split method with a more modest split factor.', '1905.01133-2-146-6': 'Because the number of pairs with these small separations is much smaller, the computation time is not a similar issue as for large separations.', '1905.01133-2-147-0': 'The results of our analysis will have an impact also on the computationally more demanding task of covariance matrix estimation.', '1905.01133-2-147-1': 'However, since in that case the exact computational cost is determined by the balance of data catalogs and random catalogs, which does not need to be the same as for the individual two-point correlation estimate, we postpone a quantitative analysis to a future, dedicated study.', '1905.01133-2-147-2': 'The same kind of methods can be applied to higher-order statistics (three-point and four-point correlation functions) to speed up their estimation .', '1905.01133-2-148-0': 'We thank Will Percival and Cristiano Porciani for useful discussions.', '1905.01133-2-148-1': 'The 2PCF computations were done at the Euclid Science Data Center Finland (SDC-FI, urn:nbn:fi:research-infras-2016072529), for whose computational resources we thank CSC - IT Center for Science, the Finnish Grid and Cloud Computing Infrastructure (FGCI, urn:nbn:fi:research-infras-2016072533), and the Academy of Finland grant 292882.', '1905.01133-2-148-2': 'This work was supported by the Academy of Finland grant 295113.', '1905.01133-2-148-3': 'VL was supported by the Jenny and Antti Wihuri Foundation, AV by the Vaisala Foundation, AS by the Magnus Ehrnrooth Foundation, and JV by the Finnish Cultural Foundation.', '1905.01133-2-148-4': 'We also acknowledge travel support from the Jenny and Antti Wihuri Foundation.', '1905.01133-2-148-5': "VA acknowledges funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 749348.", '1905.01133-2-148-6': 'FM acknowledges the grants ASI n.I/023/12/0 "Attivit\'a relative alla fase B2/C per la missione Euclid" and PRIN MIUR 2015 "Cosmology and Fundamental Physics: illuminating the Dark Universe with Euclid".', '1905.01133-2-149-0': '# Derivation examples', '1905.01133-2-150-0': 'As examples of how the variances of the different deviations, [MATH] and so on, are derived, following the method presented in LS93, we give three of the cases here: [MATH] (common for all the variants of LS), and [MATH] and [MATH] for the split method.', '1905.01133-2-150-1': 'The rest are calculated in a similar manner.', '1905.01133-2-151-0': 'To derive the [MATH] appearing in the [MATH] of [REF] we start from DD_1 DD_2 = _i<j^K _k<l^Kn_in_jn_kn_l^ij(_1)^kl(_2) ,', '1905.01133-2-152-0': 'where both [MATH] and [MATH] sum over all microcell pairs; [MATH] is the number of galaxies in microcell [MATH].', '1905.01133-2-152-1': 'There are three different cases for the terms [MATH] depending on how many indices are equal ([MATH] and [MATH] for all of them).', '1905.01133-2-153-0': 'The first case (quadruplets, i.e., two pairs of microcells) is when [MATH] are all different.', '1905.01133-2-153-1': 'We use [MATH] to denote this part of the sum.', '1905.01133-2-153-2': 'There are [MATH] (we work in the limit [MATH]) such terms and they have & & n_in_jn_kn_l', '1905.01133-2-154-0': '& = & K - 1K - 1- 2K- 2- 3K- 3(1+_i)(1+_j)(1+_k)(1+_l)', '1905.01133-2-155-0': '& = & ( - 1)( - 2)( - 3)K^4[1+_i_j+_k_l+_i_k+', '1905.01133-2-156-0': '& & +_i_l+_j_k+_j_l+', '1905.01133-2-157-0': '& & + _i_j_k+_i_j_l+_i_k_l+_j_k_l+_i_j_k_l]', '1905.01133-2-158-0': '& = & ( - 1)( - 2)( -3 )K^4[1+(_ij)+(_kl)+(_ik)+', '1905.01133-2-159-0': '& & + (_il)+(_jk)+(_jl)+', '1905.01133-2-160-0': '& & + (_i,_j,_k)+(_i,_k,_l)+(_i,_j,_l)+(_j,_k,_l)+', '1905.01133-2-161-0': '& & + (_ij)(_kl)+(_ik)(_jl)+(_il)(_jk)', '1905.01133-2-162-0': '& & +(_i,_j,_k,_l)] ,', '1905.01133-2-163-0': 'where [MATH]) is the density perturbation (relative to the actual mean density of galaxies in the survey), [MATH] is the three-point correlation, and [MATH] is the connected (i.e., the part that does not arise from the two-point correlation) four-point correlation.', '1905.01133-2-163-1': 'The fraction of microcell quadruplets (pairs of pairs) that satisfy [MATH] and [MATH] is [MATH].', '1905.01133-2-163-2': 'In the limit of large [MATH] the number of other index quadruplets is negligible compared to those where all indices have different values, so we have ^_ijkl ^ij_1^kl_2 = K(K-1)(K-2)(K-3)4_1_2 = K^44_1_2.', '1905.01133-2-164-0': 'For the connected pairs [MATH] and [MATH] we have [MATH] and [MATH].', '1905.01133-2-165-0': 'The second case (triplets of microcells) is when [MATH] or [MATH] is equal to [MATH] or [MATH].', '1905.01133-2-165-1': 'We denote this part of the sum by ^*_ijkn_in_jn_k_1^ik_2^jk .', '1905.01133-2-165-2': 'It turns out that it goes over all ordered combinations of [MATH], where [MATH] are all different, exactly once, so there are [MATH] such terms (triplets).', '1905.01133-2-165-3': 'Here & & n_in_jn_k', '1905.01133-2-166-0': '& = & K - 1K - 1 - 2K - 2(1+_i)(1+_j)(1+_k)', '1905.01133-2-167-0': '& = & ( - 1)( - 2)K^3[1 +_i_k+_j_k+_i_j+ _i_j_k]', '1905.01133-2-168-0': '& = & ( - 1)( - 2)K^3[1+(_ik)+(_jk)+(_ij)+', '1905.01133-2-169-0': '& & +(_i,_j,_k)]', '1905.01133-2-170-0': ',and ^_ijk^ik_1^jk_2 = K^3_12 .', '1905.01133-2-171-0': 'The third case (pairs of microcells) is when [MATH] and [MATH].', '1905.01133-2-171-1': 'This part of the sum becomes _i<j n_in_j^ij_1^ij_2 , where n_in_j = (K^2[1+(_ij)] ,and _i<j^ij_1^ij_2 = _12K^22_1 , that is, the sum vanishes unless the two bins are the same, [MATH].', '1905.01133-2-172-0': 'We now apply the three approximations listed in Sect. [REF]: 1) [MATH] in [REF]; 2) [MATH] in [REF] and [REF]; 3) [MATH] in [REF].', '1905.01133-2-172-1': 'We obtain & & DD_1 DD_2', '1905.01133-2-173-0': '& & (+', '1905.01133-2-174-0': '& & + (12)_12 +', '1905.01133-2-175-0': '& & + (12(1+_1)_1 ,and using [MATH] from [REF] we arrive at the [MATH] result given in Eq. [REF].', '1905.01133-2-176-0': "For the split method, we give the calculation for & = & RR'_1 DR_2 - RR'_1 DR_2 RR'_1 DR_2 ,", '1905.01133-2-177-0': "& = & RR'_1 RR'_2 - RR'_1RR'_2RR'_1RR'_2 below.", '1905.01133-2-178-0': "First the [MATH]: RR'_1 DR_2 = _R^R^_1 DR_2 = R^R^_1 DR_2, where R^R^_1 DR_2 = _i<j_kls_is_jn_kr_l^ij_1^kl_2 , [MATH] is the number (0 or 1) of [MATH] objects in microcell [MATH], and [MATH] is the number of [MATH] objects in microcell [MATH].", '1905.01133-2-179-0': 'There are [MATH] quadruplet terms, for which (see [REF]) [MATH], and s_is_jn_kr_l = (K^4 .', '1905.01133-2-180-0': 'Triplets where [MATH] or [MATH] is equal to [MATH] have [MATH], since [MATH] always (we cannot have two objects, from [MATH] and [MATH], in the same cell).', '1905.01133-2-180-1': 'There are [MATH] triplet terms where [MATH] or [MATH] is equal to [MATH]: for them [MATH], and s_is_ln_kr_l = s_is_ln_k = (K^3 ,where if [MATH] then also [MATH] since [MATH].', '1905.01133-2-181-0': 'Pairs where [MATH] or [MATH] have [MATH], since again we cannot have two different objects in the same cell.', '1905.01133-2-181-1': "Thus RR'_1 DR_2 = (+ (12 ,and we obtain that = 2[_12_1_2 - 1] = 2_12 , which is equal to the [MATH] of standard LS.", '1905.01133-2-182-0': "Then the [MATH]: RR'_1 RR'_2 = __R^R^_1R^R^_2, where there are [MATH] terms with [MATH] giving R^R^_1 R^R^_2 = R^R^_1R^R^_2 = ()^2( ,and [MATH] terms with [MATH] giving R^R^_1 R^R^_2& = & (+", '1905.01133-2-183-0': '& & + (12', '1905.01133-2-184-0': '& & + (12_1 .', '1905.01133-2-184-1': "Adding these up gives RR'_1 RR'_2 & = & (+", '1905.01133-2-185-0': '& & + (12 +', '1905.01133-2-186-0': '& & + (12_1 and & = & 2( 2(_12_1_2 - 1] + _12_1 - 1', '1905.01133-2-187-0': '& = & 4_12 + _12 .', '1905.01133-2-187-1': 'This differs both from standard LS and from dilution, since it involves both [MATH] and [MATH].'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1905.01133

1512.06824

{'1512.06824-1-0-0': 'We discuss non-standard interpretations of the [MATH] GeV diphoton excess recently reported by the ATLAS and CMS Collaborations which do not involve a new, relatively broad, resonance with a mass near [MATH] GeV.', '1512.06824-1-0-1': 'Instead, we consider the sequential cascade decay of a much heavier, possibly quite narrow, resonance into two photons along with one or more invisible particles.', '1512.06824-1-0-2': 'The resulting diphoton invariant mass signal is generically rather broad, as suggested by the data.', '1512.06824-1-0-3': 'We examine three specific event topologies - the "antler", the "sandwich", and the 2-step cascade decay, and show that they all can provide a good fit to the observed published data.', '1512.06824-1-0-4': 'In each case, we delineate the preferred mass parameter space selected by the best fit.', '1512.06824-1-0-5': 'In spite of the presence of invisible particles in the final state, the measured missing transverse energy is moderate, due to its anti-correlation with the diphoton invariant mass.', '1512.06824-1-0-6': 'We comment on the future prospects of discriminating with higher statistics between our scenarios, as well as from more conventional interpretations.', '1512.06824-1-1-0': 'Introduction.', '1512.06824-1-1-1': 'Recently, the ATLAS and CMS Collaborations have reported first results with data obtained at the Large Hadron Collider (LHC) operating at 13 TeV.', '1512.06824-1-1-2': 'The data shows an intriguing excess in the inclusive diphoton final state [CITATION].', '1512.06824-1-1-3': 'The ATLAS Collaboration further reported that about 15 events in the diphoton invariant mass distribution are observed above the standard model expectation at 3.9[MATH] local significance (2.3[MATH] global significance) with 3.2 fb[MATH] of data.', '1512.06824-1-1-4': 'The excess appears as a bump at [MATH] 750 GeV with a relatively broad width [MATH] 45 GeV, resulting in [MATH] [CITATION].', '1512.06824-1-1-5': 'Similar results are reported by the CMS Collaboration for 2.6 fb[MATH] of data - there are about 10 excess events at a local significance of 2.6[MATH] (2.0[MATH]) assuming a narrow (wide) width [CITATION].', '1512.06824-1-1-6': 'The anomalous events are not accompanied by significant missing energy, or jet or lepton multiplicity.', '1512.06824-1-1-7': 'The required cross section for the excess is [MATH] 10 fb at 13 TeV, and so far no indication of a similar excess has been observed in other channels.', '1512.06824-1-2-0': 'While waiting for the definitive verdict on this anomaly from additional LHC data, it is fun to speculate on new physics scenarios which are consistent with the current data.', '1512.06824-1-2-1': 'Given that the excess is observed in the diphoton invariant mass spectrum, the most straightforward interpretation would involve the production of a resonance with mass near 750 GeV, which decays directly to two photons.', '1512.06824-1-2-2': 'The relative broadness of the observed feature in turn would imply that this resonance has a relatively large width, creating some tension with its non-observation in other channels.', '1512.06824-1-2-3': 'Since the initial announcement, many models along those lines have been proposed [CITATION].', '1512.06824-1-3-0': 'In this letter, we entertain a different interpretation of the diphoton excess in the context of a sequential cascade decay of a much heavier, possibly quite narrow, resonance, resulting in a final state with two photons and one or two invisible particles.', '1512.06824-1-3-1': 'Three specific examples of such simplified model event topologies are exhibited in Fig. [REF]: an "antler" topology [CITATION] in Fig. [REF][MATH], a "sandwich" topology [CITATION] in Fig. [REF][MATH] and a 2-step cascade decay in Fig. [REF][MATH].', '1512.06824-1-3-2': 'In such scenarios, the resulting diphoton invariant mass is typically characterized by a somewhat broad distribution, which eliminates the necessity of an intrinsically broad resonance.', '1512.06824-1-3-3': 'Furthermore, the peak of the diphoton mass distribution is found near the upper kinematic endpoint, making it likely that the first signal events will be seen at large invariant mass, while the low mass tail remains buried under the steeply falling standard model background.', '1512.06824-1-3-4': 'Interestingly, for signal events with required extreme values of the diphoton mass, the missing transverse momentum turns out to be rather moderate, due to its anti-correlation with the diphoton mass.', '1512.06824-1-3-5': 'Given the small signal statistics ([MATH] events) such cascade decays may easily fake the standard diphoton resonance signature, and deserve further scrutiny.', '1512.06824-1-4-0': 'Diphoton invariant mass spectrum.', '1512.06824-1-4-1': 'We first review the diphoton invariant mass distributions corresponding to the above-mentioned three event topologies from Fig. [REF].', '1512.06824-1-4-2': 'The differential distribution of the diphoton invariant mass [MATH] [EQUATION] is known analytically (see, e.g., [CITATION]) and is simply a function of the unknown masses [MATH], [MATH] and [MATH].', '1512.06824-1-4-3': 'The kinematic endpoint (henceforth denoted as [MATH]) is defined as the maximum value of [MATH] allowing a non-zero [MATH]: [EQUATION]', '1512.06824-1-4-4': 'Ignoring for the moment spin correlations and assuming pure phase space distributions, the shape in the case of the antler topology of Fig. [REF][MATH] is given by [CITATION] [EQUATION] where the endpoint [MATH] and the parameter [MATH] are defined in terms of the mass parameters as [EQUATION]', '1512.06824-1-4-5': 'It is well-known that the distribution in Eq. [REF] shows a kinematic cusp at its peak at [MATH].', '1512.06824-1-5-0': 'The corresponding shape for the sandwich topology is given by the same expression ([REF]), only this time [MATH] and [MATH] are defined as follows [CITATION]: [EQUATION]', '1512.06824-1-5-1': 'Again, the kinematic cusp is located at [MATH].', '1512.06824-1-6-0': 'Finally, the two-step cascade decay has the well-known triangular shape [EQUATION] where the distribution extends up to [EQUATION]', '1512.06824-1-6-1': 'Fig. [REF] displays unit-normalized distributions for the theoretical results ([REF]) and ([REF]) (black solid lines) overlayed with Monte Carlo simulation results (red histograms) conducted with a phase space generator in ROOT [CITATION] for all three event topologies.', '1512.06824-1-6-2': 'The left panel of Fig. [REF] shows the two identical cases of Fig. [REF](a) and Fig. [REF](b), where we choose the mass spectra so that the location of the cusp is at 95% of the kinematic endpoint.', '1512.06824-1-6-3': 'The right panel in Fig. [REF] corresponds to Fig. [REF](c).', '1512.06824-1-6-4': 'We observe that in all cases, the distributions are characterized by a relatively broad peak near the kinematic endpoint, and a continuously falling tail to lower values of [MATH].', '1512.06824-1-6-5': 'Given that the standard model background distribution for [MATH] is a very steeply falling function, the low [MATH] tail can be easily hidden in the background, and the only feature of the distributions in Fig. [REF] which would be visible in the early data is the peak itself.', '1512.06824-1-7-0': 'Data analysis.', '1512.06824-1-7-1': 'Given the analytical results ([REF]-[REF]) from the previous section, we now try to fit the three models from Fig. [REF] to the background-subtracted data reported by the ATLAS Collaboration [CITATION].', '1512.06824-1-7-2': 'The results are shown in Fig. [REF], where the data points are represented by black dots, while the best-fitted model (assuming pure phase space) is shown with the red solid curve.', '1512.06824-1-7-3': 'We then perform an ordinary weighted [MATH] fit.', '1512.06824-1-8-0': 'For the antler and sandwich cases, we obtain a [MATH] value of 13.04 with 22 degrees of freedom (i.e., 25 data points subtracted by the 3 fitting parameters [MATH], [MATH], and the normalization [MATH]).', '1512.06824-1-8-1': 'The reduced [MATH] of 0.59 shows that the relevant model reproduces the data fairly well, given the small signal statistics.', '1512.06824-1-8-2': 'The best-fitted values for [MATH] and [MATH] are [EQUATION] where the reported errors are [MATH] statistical uncertainties from the fit.', '1512.06824-1-8-3': 'Due to the set of cuts applied in the ATLAS analysis to suppress the standard model backgrounds, the resulting signal distributions could be distorted.', '1512.06824-1-8-4': 'In order to account for those effects, we generate events with a mass spectrum accommodating the best-fit [MATH] and [MATH] from ([REF]), impose the selection cuts as in Ref. [CITATION], and numerically reconstruct the signal template using the surviving events.', '1512.06824-1-8-5': 'Since the antler and the sandwich scenarios have, in principle, different cut-sensitivity, we show the corresponding distributions with the blue dot-dashed and green dashed curves in the upper panel of Fig. [REF].', '1512.06824-1-9-0': 'Moving on to the two-step cascade scenario of Fig. [REF](c), we obtain the [MATH] value of 13.04 with 23 degrees of freedom(i.e., 25 data points subtracted by the 2 fitting parameters [MATH] and [MATH]).', '1512.06824-1-9-1': 'The reduced [MATH] of 0.57 shows that this model also describes the data fairly well.', '1512.06824-1-9-2': 'The best-fitted value for [MATH] is [EQUATION] where again the quoted errors are 1[MATH] statistical uncertainties from the fit.', '1512.06824-1-9-3': 'As in the previous case, the signal distribution after cuts is shown by the blue dot-dashed curve in the lower panel of Fig. [REF].', '1512.06824-1-10-0': 'Discussions and outlook.', '1512.06824-1-10-1': 'Since the number of experimentally measurable parameters for the antler topology is two (namely, [MATH] and [MATH]) [CITATION], the underlying mass spectrum is not fully determined.', '1512.06824-1-10-2': 'However, a phenomenologically motivated scenario is the case where the decay is symmetric, i.e., [MATH] and [MATH].', '1512.06824-1-10-3': 'We then have three input mass parameters, two of which can be given as functions over the third mass, using the measured values for [MATH] and [MATH].', '1512.06824-1-10-4': 'Taking the mass of [MATH] as a free parameter, we find that [MATH] and [MATH] can be expressed as follows: [EQUATION]', '1512.06824-1-10-5': 'The upper-left panel of Fig. [REF] displays the corresponding 1[MATH] mass ranges for the [MATH] (blue region and curves) and [MATH] (red region and curves) particles as a function of the [MATH] mass.', '1512.06824-1-11-0': 'For the sandwich topology of Fig. [REF](b), we can similarly reduce the number of input mass degrees of freedom by considering the simple case of [MATH] as a well-motivated phenomenological scenario.', '1512.06824-1-11-1': 'Then, using the measurements ([REF]), we can predict the masses of two of the unknown particles, say [MATH] and [MATH], as a function of the other two, [MATH] and [MATH], as shown in the middle left panel of Fig. [REF].', '1512.06824-1-12-0': 'Finally, for the two-step cascade topology of Fig. [REF](c), only one parameter, Eq. ([REF]), can be measured from the data.', '1512.06824-1-12-1': 'This provides one relation among the three unknown masses [MATH], [MATH] and [MATH], which is depicted in the bottom right panel of Fig. [REF].', '1512.06824-1-13-0': 'As mentioned earlier, the interesting events contributing to the excess do not seem to be accompanied by substantial missing transverse momentum.', '1512.06824-1-13-1': 'On the other hand, the three scenarios considered here have invisible particles in the final state, and one might naively expect that they would be in contradiction with the data.', '1512.06824-1-13-2': 'However, note that the events contributing to the excess are typically populated near the kinematic endpoint (see Figs. [REF] and [REF]).', '1512.06824-1-13-3': 'For such events, the typical angular separation (in the laboratory frame) between the two photons are anticipated to be large.', '1512.06824-1-13-4': 'Now consider the antler event topology of Fig. [REF](a), where each particle [MATH] decays into a photon and a [MATH].', '1512.06824-1-13-5': "If the photons are almost back-to-back, then so must be the two [MATH]'s, yielding a relatively small net missing transverse momentum.", '1512.06824-1-13-6': 'This inverse correlation between [MATH] and [MATH] is shown in the right panels of Fig. [REF], where for completeness we also provide similar temperature plots for the sandwich and the two-step cascade decay topologies.', '1512.06824-1-13-7': 'Clearly, once more data is accumulated, the missing transverse momentum will eventually be a good discriminator between the conventional resonance scenario (in which no missing momentum is expected) and the cascade decay scenario involving invisible particles.', '1512.06824-1-14-0': 'An alternative handle to discriminate among these competing interpretations is provided by the photon energy spectrum.', '1512.06824-1-14-1': 'In the case of the resonance scenario with a large decay with, the photon energy spectrum has a symmetric distribution with a peak at half the mass of the heavy resonance.', '1512.06824-1-14-2': 'On the other hand, the (symmetric) antler topology develops a box-type energy distribution; [EQUATION] with [MATH].', '1512.06824-1-14-3': 'For the other two cascade scenarios (i.e., the sandwich and the 2-step cascade decay), the photon spectrum depends on the scenario of particle [MATH] production.', '1512.06824-1-14-4': 'With low statistics we assume the production of [MATH] near the threshold from Parton Distribution Function (PDF) suppression, the energy spectrum of the photon near the [MATH] will have a peak at [EQUATION] and the energy of the photon which is far from the [MATH] will develop a box type energy distribution.', '1512.06824-1-14-5': 'The photon energy spectra for the different scenarios are contrasted in Fig. [REF].', '1512.06824-1-15-0': 'Finally, we discuss the potential impact of spin correlations on our analysis.', '1512.06824-1-15-1': 'It is well-known that the overall shape of invariant mass distributions can be distorted by the introduction of non-trivial spin correlations [CITATION].', '1512.06824-1-15-2': 'One could then populate most of signal events in a (relatively) narrow region around the peak, which would further improve the relevant fit.', '1512.06824-1-15-3': 'Denoting [MATH] as the distribution in the presence of spin correlations, for the antler and sandwich event topologies we can write [CITATION] [EQUATION]', '1512.06824-1-15-4': 'Here [MATH] and [MATH]) represent coefficients encoding the underlying spin information.', '1512.06824-1-15-5': 'For the decay topology in Fig. [REF][MATH], the relevant expression is given by the first line of Eq. [REF] [CITATION]: [EQUATION] and the presence of the additional terms beyond ([REF]) can also favorably sculpt the distribution in the vicinity of the peak.', '1512.06824-1-16-0': 'In conclusion, we investigated the nature of the anomalous excesses reported by the ATLAS and CMS Collaborations in terms of cascade decay topologies from a heavy, possibly quite narrow, resonance.', '1512.06824-1-16-1': 'Our scenarios can generically accommodate a (relatively) large width of the peak accompanied with a (relatively) small missing transverse momentum.', '1512.06824-1-16-2': 'The presence of invisible particles in the final state opens the door for discovery of not just a new particle beyond the standard model, but possibly of the dark matter.', '1512.06824-1-16-3': 'We also discussed the potential of distinguishing the competing interpretations with more data, using the missing transverse momentum and photon energy distributions.', '1512.06824-1-16-4': 'We eagerly await the resolution of this puzzle with new data from the LHC.'}

{'1512.06824-2-0-0': 'We discuss non-standard interpretations of the [MATH] GeV diphoton excess recently reported by the ATLAS and CMS Collaborations which do not involve a new, relatively broad, resonance with a mass near [MATH] GeV.', '1512.06824-2-0-1': 'Instead, we consider the sequential cascade decay of a much heavier, possibly quite narrow, resonance into two photons along with one or more invisible particles.', '1512.06824-2-0-2': 'The resulting diphoton invariant mass signal is generically rather broad, as suggested by the data.', '1512.06824-2-0-3': 'We examine three specific event topologies - the "antler", the "sandwich", and the 2-step cascade decay, and show that they all can provide a good fit to the observed published data.', '1512.06824-2-0-4': 'In each case, we delineate the preferred mass parameter space selected by the best fit.', '1512.06824-2-0-5': 'In spite of the presence of invisible particles in the final state, the measured missing transverse energy is moderate, due to its anti-correlation with the diphoton invariant mass.', '1512.06824-2-0-6': 'We comment on the future prospects of discriminating with higher statistics between our scenarios, as well as from more conventional interpretations.', '1512.06824-2-1-0': 'Introduction.', '1512.06824-2-1-1': 'Recently, the ATLAS and CMS Collaborations have reported first results with data obtained at the Large Hadron Collider (LHC) operating at 13 TeV.', '1512.06824-2-1-2': 'The data shows an intriguing excess in the inclusive diphoton final state [CITATION].', '1512.06824-2-1-3': 'The ATLAS Collaboration further reported that about 15 events in the diphoton invariant mass distribution are observed above the Standard Model (SM) expectation at 3.9[MATH] local significance (2.3[MATH] global significance) with 3.2 fb[MATH] of data.', '1512.06824-2-1-4': 'The excess appears as a bump at [MATH] 750 GeV with a relatively broad width [MATH] 45 GeV, resulting in [MATH] [CITATION].', '1512.06824-2-1-5': 'Similar results are reported by the CMS Collaboration for 2.6 fb[MATH] of data - there are about 10 excess events at a local significance of 2.6[MATH] (2.0[MATH]) assuming a narrow (wide) width [CITATION].', '1512.06824-2-1-6': 'The anomalous events are not accompanied by significant missing energy, or jet or lepton multiplicity.', '1512.06824-2-1-7': 'The required cross section for the excess is [MATH] 10 fb at 13 TeV, and so far no indication of a similar excess has been observed in other channels.', '1512.06824-2-2-0': 'While waiting for the definitive verdict on this anomaly from additional LHC data, it is fun to speculate on new physics scenarios which are consistent with the current data.', '1512.06824-2-2-1': 'Since the excess was seen in the diphoton invariant mass spectrum, the most straightforward interpretation would involve the production of a resonance with mass near 750 GeV, which decays directly to two photons.', '1512.06824-2-2-2': 'The relative broadness of the observed feature in turn would imply that this resonance has a relatively large width, creating some tension with its non-observation in other channels.', '1512.06824-2-2-3': 'Since the initial announcement, many models along those lines have been proposed [CITATION].', '1512.06824-2-3-0': 'In this letter, we entertain a different interpretation of the diphoton excess in the context of a sequential cascade decay of a much heavier, possibly quite narrow, resonance, resulting in a final state with two photons and one or two invisible particles.', '1512.06824-2-3-1': 'Three specific examples of such simplified model event topologies are exhibited in Fig. [REF]: an "antler" topology [CITATION] in Fig. [REF][MATH], a "sandwich" topology [CITATION] in Fig. [REF][MATH] and a 2-step cascade decay in Fig. [REF][MATH].', '1512.06824-2-3-2': 'In such scenarios, the resulting diphoton invariant mass is typically characterized by a somewhat broad distribution, which eliminates the necessity of an intrinsically broad resonance.', '1512.06824-2-3-3': 'Furthermore, the peak of the diphoton mass distribution is found near the upper kinematic endpoint, making it likely that the first signal events will be seen at large invariant mass, while the low mass tail remains buried under the steeply falling SM background.', '1512.06824-2-3-4': 'Interestingly, for signal events with the required extreme values of the diphoton mass, the missing transverse momentum turns out to be rather moderate, due to its anti-correlation with the diphoton mass.', '1512.06824-2-3-5': 'Given the small signal statistics ([MATH] events) such cascade decays may easily fake the standard diphoton resonance signature, and deserve further scrutiny.', '1512.06824-2-4-0': 'Diphoton invariant mass spectrum.', '1512.06824-2-4-1': 'We first review the diphoton invariant mass distributions corresponding to the above-mentioned three event topologies from Fig. [REF].', '1512.06824-2-4-2': 'The differential distribution of the diphoton invariant mass [MATH] [EQUATION] is known analytically (see, e.g., [CITATION]) and is simply a function of the unknown masses [MATH], [MATH] and [MATH].', '1512.06824-2-4-3': 'The kinematic endpoint (henceforth denoted as [MATH]) is defined as the maximum value of [MATH] allowing a non-zero [MATH], i.e., [MATH].', '1512.06824-2-5-0': 'Ignoring for the moment spin correlations and assuming pure phase space distributions, the shape in the case of the antler topology of Fig. [REF][MATH] is given by [CITATION] [EQUATION] where the endpoint [MATH] and the parameter [MATH] are defined in terms of the mass parameters as [EQUATION]', '1512.06824-2-5-1': 'The corresponding shape for the sandwich topology is given by the same expression ([REF]), only this time [MATH] and [MATH] are defined as follows [CITATION]: [EQUATION]', '1512.06824-2-5-2': 'Finally, the two-step cascade decay has the well-known triangular shape [EQUATION] where the distribution extends up to [EQUATION]', '1512.06824-2-6-0': 'Fig. [REF] displays unit-normalized distributions for the theoretical results ([REF]) and ([REF]) (black dashed lines) overlayed with Monte Carlo simulation results (solid line histograms) obtained with the phase space generator in ROOT [CITATION], for all three event topologies.', '1512.06824-2-6-1': 'The left panel of Fig. [REF] shows the two identical cases of Fig. [REF](a) and Fig. [REF](b).', '1512.06824-2-6-2': 'Here we take five representative [MATH] values to demonstrate the dependence of the peak position on [MATH] - for small enough values of [MATH], the peak can be arbitrarily close to the endpoint.', '1512.06824-2-6-3': 'The right panel in Fig. [REF] corresponds to Fig. [REF](c).', '1512.06824-2-6-4': 'In all cases (assuming small enough values of [MATH] in ([REF])), the distributions are characterized by a relatively broad peak near the kinematic endpoint, and a continuously falling tail to lower values of [MATH].', '1512.06824-2-6-5': 'Given that the SM background distribution for [MATH] is a very steeply falling function, the low [MATH] tail can be easily hidden in the background, and the only feature of the distributions in Fig. [REF] which would be visible in the early data is the peak itself.', '1512.06824-2-7-0': 'Data analysis.', '1512.06824-2-7-1': 'Given the analytical results ([REF]-[REF]) from the previous section, we now try to fit the three models from Fig. [REF] to the [MATH] spectrum data reported by the ATLAS Collaboration [CITATION] (black dots in Fig. [REF]).', '1512.06824-2-7-2': 'To describe the background portion in the data, we introduce the same background model function as in [CITATION]: [EQUATION] where [MATH] and [MATH] are fit parameters to be determined by data and [MATH] with [MATH].', '1512.06824-2-7-3': 'We then perform likelihood fits using combined signal [MATH] background templates using [MATH] from Eqs. ([REF], [REF]) and [MATH] from Eq. [REF].', '1512.06824-2-7-4': 'Our fit results for the case of Fig. [REF](a-b) (Fig. [REF](c)) are shown in the upper (lower) panel of Fig. [REF].', '1512.06824-2-7-5': 'The red solid curves represent the best-fit models (i.e., signal[MATH]background), while the red dashed (blue solid) curves describe their background (signal) components.', '1512.06824-2-7-6': 'To estimate parameter errors more carefully with low statistics, we generate 10,000 pseudo data sets via random samplings of the data point in each bin, assuming a Poisson distribution with the mean value set to its original one.', '1512.06824-2-7-7': 'We then conduct the same fit procedure explained above for all pseudo data sets and obtain distributions of fitted model parameters, from which we extract mean values and 1[MATH] confidence intervals for them, along with reduced [MATH] distributions.', '1512.06824-2-8-0': 'For the antler and sandwich cases, the relevant reduced [MATH] distribution yields a mean (median) value of 0.98 (0.93) with the Gaussian width around 1, indicating that our fitting template accommodates pseudo data samples well enough.', '1512.06824-2-8-1': 'The extracted best-fit parameter values and their 1[MATH] errors are [EQUATION]', '1512.06824-2-8-2': 'Due to the set of cuts applied in the ATLAS analysis to suppress the SM backgrounds, the resulting signal distributions could be distorted.', '1512.06824-2-8-3': 'In order to account for those effects, we (here and henceforth) take [MATH]-induced single production of scalar [MATH] of [MATH] TeV with the LHC13, and let it decay according to a mass spectrum accommodating the best-fit [MATH] and [MATH] from Eq. ([REF]).', '1512.06824-2-8-4': 'We then impose the selection cuts as in Ref. [CITATION], and numerically reconstruct the signal template using the surviving events.', '1512.06824-2-8-5': 'Since the antler and the sandwich scenarios have, in principle, different cut-sensitivity, we show the corresponding distributions with the blue (antler) and green (sandwich) dashed curves in the upper panel of Fig. [REF].', '1512.06824-2-9-0': 'For the two-step cascade scenario, the relevant reduced [MATH] distribution shows a mean (median) value of 0.69 (0.67) with the Gaussian width around 0.5, indicating that this model also reproduces the data well enough.', '1512.06824-2-9-1': 'The best-fit value for [MATH] and its 1[MATH] error are reported as [EQUATION]', '1512.06824-2-9-2': 'As before, the signal distribution after cuts is shown by the blue dashed curve in the lower panel of Fig. [REF].', '1512.06824-2-10-0': 'Discussions and outlook.', '1512.06824-2-10-1': 'Since the number of experimentally measurable parameters for the antler topology is two (namely, [MATH] and [MATH]) [CITATION], the underlying mass spectrum is not fully determined.', '1512.06824-2-10-2': 'However, a phenomenologically motivated scenario is the case where the decay is symmetric, i.e., [MATH] and [MATH].', '1512.06824-2-10-3': 'We then have three input mass parameters, two of which can be given as functions of the third mass, using the measured values for [MATH] and [MATH].', '1512.06824-2-10-4': 'Taking [MATH] as a free parameter, we find that [MATH] and [MATH] can be expressed as follows: [EQUATION]', '1512.06824-2-10-5': 'The upper-left panel of Fig. [REF] displays the corresponding 1[MATH] mass ranges for the [MATH] (blue region and curves) and [MATH] (red region and curves) particles as a function of [MATH].', '1512.06824-2-11-0': 'For the sandwich topology of Fig. [REF](b), we can similarly reduce the number of input mass degrees of freedom by considering the simple case of [MATH] as a well-motivated phenomenological scenario.', '1512.06824-2-11-1': 'Then, using the measurements ([REF]), we can predict the masses of two of the unknown particles, say [MATH] and [MATH], as a function of the other two, [MATH] and [MATH], as shown in the middle left panel of Fig. [REF] ([MATH] only for illustration).', '1512.06824-2-12-0': 'Finally, for the two-step cascade topology of Fig. [REF](c), only one parameter, Eq. ([REF]), can be measured from the data.', '1512.06824-2-12-1': 'This provides one relation among the three unknown masses [MATH], [MATH] and [MATH], which is depicted in the bottom left panel of Fig. [REF].', '1512.06824-2-13-0': 'As mentioned earlier, the interesting events contributing to the excess do not seem to be accompanied by substantial missing transverse momentum.', '1512.06824-2-13-1': 'On the other hand, the three scenarios considered here have invisible particles in the final state, and one might naively expect that they would be in contradiction with the data.', '1512.06824-2-13-2': 'However, note that the events contributing to the excess are typically populated near the kinematic endpoint (see Figs. [REF] and [REF]).', '1512.06824-2-13-3': 'For such events, the typical angular separation (in the laboratory frame) between the two photons is anticipated to be large.', '1512.06824-2-13-4': 'Now consider the antler event topology of Fig. [REF](a), where each particle [MATH] decays into a photon and a [MATH].', '1512.06824-2-13-5': "If the photons are almost back-to-back, then so must be the two [MATH]'s, yielding a relatively small net missing transverse momentum.", '1512.06824-2-13-6': 'This inverse correlation between [MATH] and [MATH] is shown in the right panels of Fig. [REF], where for completeness we also provide similar temperature plots for the sandwich and the two-step cascade decay topologies.', '1512.06824-2-13-7': 'In the left panel of Fig. [REF], we show [MATH] distributions of simulated events around [MATH] GeV (near the measured peak position) with the same mass spectra as in the right panels of Fig. [REF].', '1512.06824-2-13-8': 'Again, the selection cuts adopted in Ref. [CITATION] have been imposed.', '1512.06824-2-13-9': 'We clearly see that the antler scenario favors small [MATH] as expected.', '1512.06824-2-13-10': 'Certainly, once more data is accumulated, the missing transverse momentum will eventually be a good discriminator between the conventional resonance scenario (in which no [MATH] is expected) and a cascade decay scenario involving invisible particles.', '1512.06824-2-14-0': 'An alternative handle to discriminate among these competing interpretations is provided by the photon energy spectrum.', '1512.06824-2-14-1': 'In the conventional case of a single resonance scenario with a large decay width [CITATION], the photon energy spectrum has a unique peak at half the resonance mass [CITATION], and the peak may show a sharp kink structure if the heavy resonance is singly produced [CITATION].', '1512.06824-2-14-2': 'On the other hand, the energy distribution for the (symmetric) antler scenario develops a peak at [EQUATION] which is, in principle, different from the peak in the [MATH] distribution.', '1512.06824-2-14-3': 'For the other two cases, the corresponding photon spectrum could develop a double-bump structure depending on the underlying mass spectrum [CITATION].', '1512.06824-2-14-4': 'These expectations are summarized in the right panel of Fig. [REF].', '1512.06824-2-15-0': 'Finally, we note the potential impact of spin correlations on our analysis.', '1512.06824-2-15-1': 'It is well-known that the overall shape of invariant mass distributions can be distorted by the introduction of non-trivial spin correlations [CITATION].', '1512.06824-2-15-2': 'One could then repopulate most of the signal events in a (relatively) narrow region around the peak, which would further improve the fit.', '1512.06824-2-15-3': 'Let [MATH] be the relevant [MATH] distribution in the presence of spin correlations.', '1512.06824-2-15-4': 'For the antler and sandwich cases, one can write [CITATION] [EQUATION]', '1512.06824-2-15-5': 'Here [MATH] and [MATH]) represent coefficients encoding the underlying spin information.', '1512.06824-2-15-6': 'For the decay topology in Fig. [REF](c), the relevant expression is given by the first line of Eq. [REF] [CITATION]: [EQUATION] and the presence of the additional terms beyond Eq. ([REF]) can also favorably sculpt the distribution in the vicinity of the peak.', '1512.06824-2-16-0': 'In conclusion, we investigated the nature of the anomalous excesses reported by the ATLAS and CMS Collaborations in terms of cascade decay topologies from a heavy, possibly quite narrow, resonance.', '1512.06824-2-16-1': 'Our scenarios can generically accommodate a (relatively) large width of the peak accompanied with a (relatively) small missing transverse momentum.', '1512.06824-2-16-2': 'The presence of invisible particles in the final state opens the door for discovery of not just a new particle beyond the Standard Model, but possibly of the dark matter.', '1512.06824-2-16-3': 'We also discussed the potential of distinguishing the competing interpretations with more data, using the missing transverse momentum and photon energy distributions.', '1512.06824-2-16-4': 'We eagerly await the resolution of this puzzle with new data from the LHC.'}

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[]

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[['1512.06824-2-13-3', '1512.06824-3-14-5']]

['1512.06824-1-1-0', '1512.06824-1-7-0', '1512.06824-1-10-0', '1512.06824-2-1-0', '1512.06824-2-7-0', '1512.06824-2-10-0', '1512.06824-3-1-0', '1512.06824-3-7-0', '1512.06824-3-8-0', '1512.06824-3-11-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1512.06824

{'1512.06824-3-0-0': 'We discuss non-standard interpretations of the [MATH] GeV diphoton excess recently reported by the ATLAS and CMS Collaborations which do not involve a new, relatively broad, resonance with a mass near [MATH] GeV.', '1512.06824-3-0-1': 'Instead, we consider the sequential cascade decay of a much heavier, possibly quite narrow, resonance into two photons along with one or more additional particles.', '1512.06824-3-0-2': 'The resulting diphoton invariant mass signal is generically rather broad, as suggested by the data.', '1512.06824-3-0-3': 'We examine three specific event topologies - the "antler", the "sandwich", and the 2-step cascade decay, and show that they all can provide a good fit to the observed published data.', '1512.06824-3-0-4': 'In each case, we delineate the preferred mass parameter space selected by the best fit.', '1512.06824-3-0-5': 'In spite of the presence of extra particles in the final state, the measured diphoton [MATH] spectrum is moderate, due to its anti-correlation with the diphoton invariant mass.', '1512.06824-3-0-6': 'We comment on the future prospects of discriminating with higher statistics between our scenarios, as well as from more conventional interpretations.', '1512.06824-3-1-0': 'Introduction.', '1512.06824-3-1-1': 'Recently, the ATLAS and CMS Collaborations have reported first results with data obtained at the Large Hadron Collider (LHC) operating at 13 TeV.', '1512.06824-3-1-2': 'The data shows an intriguing excess in the inclusive diphoton final state [CITATION].', '1512.06824-3-1-3': 'The ATLAS Collaboration further reported that about 15 events in the diphoton invariant mass distribution are observed above the Standard Model (SM) expectation at 3.9[MATH] local significance (2.3[MATH] global significance) with 3.2 fb[MATH] of data.', '1512.06824-3-1-4': 'The excess appears as a bump at [MATH] 750 GeV with a relatively broad width [MATH] 45 GeV, resulting in [MATH] [CITATION].', '1512.06824-3-1-5': 'Similar results are reported by the CMS Collaboration for 2.6 fb[MATH] of data - there are about 10 excess events at a local significance of 2.6[MATH] (2.0[MATH]) assuming a narrow (wide) width [CITATION].', '1512.06824-3-1-6': 'The anomalous events are not accompanied by significant extra activity, e.g. missing transverse energy [MATH] [CITATION].', '1512.06824-3-1-7': 'The required cross section for the excess is [MATH] 10 fb at 13 TeV, and so far no indication of a similar excess has been observed in other channels.', '1512.06824-3-2-0': 'While waiting for the definitive verdict on this anomaly from additional LHC data, it is fun to speculate on new physics scenarios which are consistent with the current data.', '1512.06824-3-2-1': 'Since the excess was seen in the diphoton invariant mass spectrum, the most straightforward interpretation would involve the production of a resonance with mass near 750 GeV, which decays directly to two photons.', '1512.06824-3-2-2': 'The relative broadness of the observed feature would imply that this resonance has a relatively large width, creating some tension with its non-observation in other channels.', '1512.06824-3-2-3': 'Since the initial announcement, many models along those lines have been proposed, e.g. in the context of extended Higgs sectors [CITATION], supersymmetry [CITATION], extra dimensions [CITATION], strong dynamics [CITATION], or effective field theory [CITATION].', '1512.06824-3-3-0': 'In this letter, we entertain a different interpretation of the diphoton excess in the context of a sequential cascade decay of a much heavier, possibly quite narrow, resonance, resulting in a final state with two photons and one or two additional particles (see also [CITATION]).', '1512.06824-3-3-1': 'Three specific examples of such simplified model event topologies are exhibited in Fig. [REF]: an "antler" topology [CITATION] in Fig. [REF][MATH], a "sandwich" topology [CITATION] in Fig. [REF][MATH] and a 2-step cascade decay in Fig. [REF][MATH].', '1512.06824-3-3-2': 'In such scenarios, the resulting diphoton invariant mass [MATH] is typically characterized by a somewhat broad distribution, which eliminates the necessity of an intrinsically broad resonance.', '1512.06824-3-3-3': 'Furthermore, the peak of the [MATH] distribution is found near the upper kinematic endpoint, making it likely that the first signal events will be seen at large invariant mass, while the low mass tail remains buried under the steeply falling SM background.', '1512.06824-3-3-4': 'Interestingly, for signal events with the required extreme values of [MATH], the transverse momentum of the diphoton system [MATH] turns out to be rather moderate, due to its anti-correlation with the diphoton mass [MATH].', '1512.06824-3-3-5': 'Given the small signal statistics ([MATH] events) such cascade decays may easily fake the standard diphoton resonance signature, and deserve further scrutiny.', '1512.06824-3-3-6': 'We note that this observation is not restricted to the diphoton channel, but is quite general and applicable to any inclusive resonance search in a two-body final state.', '1512.06824-3-4-0': 'Diphoton invariant mass spectrum.', '1512.06824-3-4-1': 'We first review the diphoton invariant mass distributions corresponding to the above-mentioned three event topologies from Fig. [REF].', '1512.06824-3-4-2': 'The differential distribution of the diphoton invariant mass [MATH] [EQUATION] is known analytically (see, e.g., [CITATION]) and is simply a function of the unknown masses [MATH], [MATH] and [MATH].', '1512.06824-3-4-3': 'The kinematic endpoint (henceforth denoted as [MATH]) is defined as the maximum value of [MATH] allowing a non-zero [MATH], i.e., [MATH].', '1512.06824-3-5-0': 'Ignoring for the moment spin correlations and assuming pure phase space distributions, the shape in the case of the antler topology of Fig. [REF][MATH] is given by [CITATION] [EQUATION] where the endpoint [MATH] and the parameter [MATH] are defined in terms of the mass parameters as [EQUATION]', '1512.06824-3-5-1': 'The corresponding shape for the sandwich topology is given by the same expression ([REF]), only this time [MATH] and [MATH] are defined as follows [CITATION]: [EQUATION]', '1512.06824-3-5-2': 'In both cases, for small enough values of [MATH], the peak location [MATH] can be arbitrarily close to the endpoint [MATH].', '1512.06824-3-6-0': 'Finally, the two-step cascade decay has the well-known triangular shape [EQUATION] where the distribution extends up to [EQUATION]', '1512.06824-3-6-1': 'For all three event topologies in Fig. [REF] (assuming small enough values of [MATH] in ([REF])), the distributions are characterized by a relatively broad peak near the kinematic endpoint, and a continuously falling tail to lower values of [MATH].', '1512.06824-3-6-2': 'Given that the SM background distribution for [MATH] is a very steeply falling function, the low [MATH] tail can be easily hidden in the background, and the only feature of the signal distribution which would be visible in the early data is the peak itself.', '1512.06824-3-7-0': 'Signal models.', '1512.06824-3-7-1': 'For the numerical studies below we choose the following signal models realizing the topologies of Fig. [REF].', '1512.06824-3-7-2': 'In the antler topology of Fig. [REF](a), the particle [MATH], [MATH]) is a scalar (fermion, fermion), and the fermion coupling to the photon is vector-like, [MATH], where [MATH] is the photon field strength tensor.', '1512.06824-3-7-3': 'For the sandwich topology of Fig. [REF](b), particle [MATH] is a heavy [MATH] vector boson with field strength tensor [MATH], which couples to a scalar [MATH] as [MATH], while [MATH] and [MATH] are fermions with vector-like couplings to photons as before.', '1512.06824-3-7-4': 'Finally, in the two-step cascade decay of Fig. [REF](c), [MATH] and [MATH] are vector particles coupling to a scalar [MATH] as above.', '1512.06824-3-7-5': 'In all three cases, the diphoton invariant mass distribution is given by the analytical results of the previous section [CITATION].', '1512.06824-3-8-0': 'Data analysis.', '1512.06824-3-8-1': 'Given the analytical results ([REF]-[REF]) from the previous section, we now try to fit the three models from Fig. [REF] to the [MATH] spectrum data reported by the ATLAS Collaboration [CITATION] (black dots in Fig. [REF]).', '1512.06824-3-8-2': 'To describe the background portion in the data, we introduce the same background model function as in [CITATION]: [EQUATION] where [MATH] and [MATH] are fit parameters to be determined by data and [MATH] with [MATH].', '1512.06824-3-8-3': 'We then perform likelihood fits using combined signal [MATH] background templates using [MATH] from Eqs. ([REF], [REF]) and [MATH] from Eq. [REF].', '1512.06824-3-8-4': 'Our fit results for the case of Fig. [REF](a-b) (Fig. [REF](c)) are shown in the upper (lower) panel of Fig. [REF].', '1512.06824-3-8-5': 'The red solid curves represent the best-fit models (i.e., signal[MATH]background), while the red dashed (blue solid) curves describe their background (signal) components.', '1512.06824-3-8-6': 'To estimate parameter errors more carefully with low statistics, we generate 10,000 pseudo data sets via random samplings of the data point in each bin, assuming a Poisson distribution with its mean value parameter set to the number of events in each bin reported by the ATLAS collaboration (any zero bins in the real data were always sampled to be 0).', '1512.06824-3-8-7': 'We then conduct the same fit procedure explained above for all pseudo data sets and obtain distributions of fitted model parameters, from which we extract mean values and 1[MATH] confidence intervals, along with reduced [MATH] distributions.', '1512.06824-3-9-0': 'For the antler and sandwich cases, the relevant reduced [MATH] distribution yields a mean (median) value of 0.98 (0.93) with the Gaussian width around 1, indicating that our fitting template accommodates pseudo data samples well enough.', '1512.06824-3-9-1': 'The extracted best-fit parameter values and their 1[MATH] errors are [EQUATION]', '1512.06824-3-9-2': 'Due to the set of cuts applied in the ATLAS analysis to suppress the SM backgrounds, the resulting signal distributions could be distorted.', '1512.06824-3-9-3': 'In order to account for those effects, we simulate single production of particle [MATH] at LHC13 with ^MadGraph5_aMC@NLO^ [CITATION], followed by Pythia 6.4 [CITATION] and Delphes 3 [CITATION].', '1512.06824-3-9-4': 'We take [MATH] TeV and the remaining masses are chosen in accordance with the best-fit [MATH] and [MATH] from Eq. ([REF]).', '1512.06824-3-9-5': 'Since the antler and the sandwich scenarios have, in principle, different cut-sensitivity, we show the corresponding distributions with the blue (antler) and green (sandwich) dashed curves in the upper panel of Fig. [REF].', '1512.06824-3-10-0': 'For the two-step cascade scenario, the relevant reduced [MATH] distribution shows a mean (median) value of 0.69 (0.67) with the Gaussian width around 0.5, indicating that this model also reproduces the data well enough.', '1512.06824-3-10-1': 'The best-fit value for [MATH] and its 1[MATH] error are reported as [EQUATION]', '1512.06824-3-10-2': 'As before, the signal distribution after cuts is shown by the blue dashed curve in the lower panel of Fig. [REF].', '1512.06824-3-11-0': 'Discussions and outlook.', '1512.06824-3-11-1': 'Since the number of experimentally measurable parameters for the antler topology is two (namely, [MATH] and [MATH]) [CITATION], the underlying mass spectrum is not fully determined.', '1512.06824-3-11-2': 'However, a phenomenologically motivated scenario is the case where the decay is symmetric, i.e., [MATH] and [MATH].', '1512.06824-3-11-3': 'We then have three input mass parameters, two of which can be given as functions of the third mass, using the measured values for [MATH] and [MATH].', '1512.06824-3-11-4': 'Taking [MATH] as a free parameter, we find that [MATH] and [MATH] can be expressed as follows: [EQUATION]', '1512.06824-3-11-5': 'The upper-left panel of Fig. [REF] displays the corresponding 1[MATH] mass ranges for the [MATH] (blue region and curves) and [MATH] (red region and curves) particles as a function of [MATH].', '1512.06824-3-12-0': 'For the sandwich topology of Fig. [REF](b), we can similarly reduce the number of input mass degrees of freedom by considering the simple case of [MATH] as a well-motivated phenomenological scenario.', '1512.06824-3-12-1': 'Then, using the measurements ([REF]), we can predict the masses of two of the unknown particles, say [MATH] and [MATH], as a function of the other two, [MATH] and [MATH], as shown in the middle left panel of Fig. [REF] ([MATH] only for illustration).', '1512.06824-3-13-0': 'Finally, for the two-step cascade topology of Fig. [REF](c), only one parameter, Eq. ([REF]), can be measured from the data.', '1512.06824-3-13-1': 'This provides one relation among the three unknown masses [MATH], [MATH] and [MATH], which is depicted in the bottom left panel of Fig. [REF].', '1512.06824-3-14-0': 'We have seen that the cascade event topologies from Fig. [REF] can provide a good fit to the diphoton invariant mass spectrum in Fig. [REF].', '1512.06824-3-14-1': 'It is therefore natural to ask, what other kinematic variables of the diphoton system can be used to test our hypothesis.', '1512.06824-3-14-2': 'One such possibility is the transverse momentum of the diphoton system, [MATH], since it is sensitive to other objects recoiling against the two photons.', '1512.06824-3-14-3': 'However, there exists an inverse correlation between the two diphoton kinematic variables, [MATH] and [MATH], as illustrated in the right panels of Fig. [REF]: events with extreme values of [MATH] have relatively small [MATH] and vice versa.', '1512.06824-3-14-4': 'The anti-correlation trend is especially pronounced for the antler event topology, as demonstrated in the left panel of Fig. [REF], where we show the [MATH] distribution of simulated events near the bump, [MATH] GeV, for 3.2 fb[MATH] of data with the ATLAS selection cuts [CITATION].', '1512.06824-3-14-5': "For such events, the typical angular separation (in the laboratory frame) between the two photons is anticipated to be large, and if the photons are almost back-to-back, then so must be the two [MATH]'s, yielding a relatively small net [MATH].", '1512.06824-3-14-6': 'Fig. [REF] is consistent with Ref. [CITATION] and shows that the signal events lead to a rather featureless tail in the [MATH] distribution.', '1512.06824-3-14-7': 'With the accumulation of more data, [MATH] will eventually be a good discriminator between the conventional resonance scenario (with relatively soft [MATH]) and the cascade decay scenarios considered here.', '1512.06824-3-15-0': 'Another handle to discriminate among the competing interpretations of Fig. [REF] is provided by the photon energy spectrum.', '1512.06824-3-15-1': 'In the conventional case of a single resonance with a large decay width [CITATION], the photon energy spectrum has a single peak at half the resonance mass [CITATION], which may show a sharp kink structure if the heavy resonance is singly produced [CITATION].', '1512.06824-3-15-2': 'On the other hand, the energy distribution for the (symmetric) antler scenario develops a peak at a different position, [EQUATION]', '1512.06824-3-15-3': 'For the other two cases, the corresponding photon spectrum could develop a double-bump structure depending on the underlying mass spectrum [CITATION].', '1512.06824-3-15-4': 'These expectations are summarized in the right panel of Fig. [REF].', '1512.06824-3-16-0': 'As the excess was observed in the inclusive diphoton channel [CITATION], we have focused our attention primarily on the kinematics of the diphoton system itself.', '1512.06824-3-16-1': 'Of course, more exclusive studies could target the detector signatures of the additional particles [MATH].', '1512.06824-3-16-2': 'For example, if the particles [MATH] are stable and weakly interacting, they will be invisible in the detector and cause missing transverse energy [MATH].', '1512.06824-3-16-3': 'The predicted [MATH] distribution would be similar to the [MATH] distribution shown in Fig. [REF], and at this point seems to be disfavored by the data [CITATION] (another constraint would be provided by the inclusive diphoton plus [MATH] search for new physics [CITATION]).', '1512.06824-3-16-4': 'Of course, the particles [MATH] could be visible, or further decay visibly themselves.', '1512.06824-3-16-5': 'The exact nature of their signatures (and kinematic distributions) is rather model-dependent and beyond the scope of this letter.', '1512.06824-3-17-0': 'Finally, we note the potential impact of spin correlations on our analysis.', '1512.06824-3-17-1': 'It is well-known that the overall shape of invariant mass distributions can be distorted by the introduction of non-trivial spin correlations [CITATION].', '1512.06824-3-17-2': 'One could then repopulate most of the signal events in a (relatively) narrow region around the peak, which would further improve the fit.', '1512.06824-3-17-3': 'Let [MATH] be the relevant [MATH] distribution in the presence of spin correlations.', '1512.06824-3-17-4': 'For the antler and sandwich cases, one can write [CITATION] [EQUATION]', '1512.06824-3-17-5': 'Here [MATH] and [MATH]) represent coefficients encoding the underlying spin information.', '1512.06824-3-17-6': 'For the decay topology in Fig. [REF](c), the relevant expression is given by the first line of Eq. [REF] [CITATION]: [EQUATION] and the presence of the additional terms beyond Eq. ([REF]) can also favorably sculpt the distribution in the vicinity of the peak.', '1512.06824-3-18-0': 'In conclusion, we investigated the nature of the anomalous excesses reported by the ATLAS and CMS Collaborations in terms of cascade decay topologies from a heavy, possibly quite narrow, resonance.', '1512.06824-3-18-1': 'Our scenarios can generically accommodate a (relatively) large width of the peak accompanied with a (relatively) small diphoton transverse momentum.', '1512.06824-3-18-2': 'We also discussed the potential of distinguishing the competing interpretations with more data, using the diphoton transverse momentum and photon energy distributions.', '1512.06824-3-18-3': 'We eagerly await the resolution of this puzzle with new data from the LHC.'}

1311.0582

{'1311.0582-1-0-0': 'We propose the complex-plane generalization of a powerful algebraic sequence acceleration algorithm, the Method of Weighted Averages (MWA), to guarantee exponential-cum-algebraic convergence of Fourier and Fourier-Hankel (F-H) integral transforms.', '1311.0582-1-0-1': 'This "complex-plane" MWA, effected via a linear-path detour in the complex plane, results in rapid, absolute convergence of field/potential solutions in multi-layered environments regardless of the source-observer geometry and anisotropy/loss of the media present.', '1311.0582-1-0-2': 'In this work, we first introduce a new integration path used to evaluate the field contribution arising from the radiation spectra.', '1311.0582-1-0-3': 'Subsequently, we (1) exhibit the foundational relations behind the complex-plane extension to a general Levin-type sequence convergence accelerator, (2) specialize this analysis to one member of the Levin transform family (the MWA), (3) address and circumvent restrictions, arising for two-dimensional integrals associated with wave dynamics problems, through minimal complex-plane detour restrictions and a novel partition of the integration domain, (4) develop and compare two formulations based on standard/real-axis MWA variants, and (5) present validation results and convergence characteristics for one of these two formulations.', '1311.0582-1-1-0': '# Introduction', '1311.0582-1-2-0': 'In many application areas concerning time-harmonic electromagnetic (EM) fields, one encounters environments containing media of varying and arbitrary anisotropy whose inhomogeneity can be approximated as multi-layered in nature.', '1311.0582-1-2-1': 'Examples include geophysical prospection [CITATION], plasma physics [CITATION], antenna design [CITATION], optical field control [CITATION], microwave remote sensing [CITATION], ground-penetrating radar [CITATION], and microwave circuits [CITATION], among others.', '1311.0582-1-2-2': 'Such applications regularly encounter integrals of the form [EQUATION] and/or [EQUATION] which express space-domain field/potential functions as Fourier and Fourier-Hankel (F-H) integral transforms (resp.)', '1311.0582-1-3-0': 'In many practical applications, these integrals must often be rapidly evaluated for a wide range of longitudinal and transverse source-observer separation geometries [MATH] (e.g. for potential or field profile reconstruction).', '1311.0582-1-3-1': 'However, when using standard integration paths that run on/close to the real axis such as (1) the classic Sommerfeld Integration Path (SIP) [CITATION] and (2) paths detouring around the branch points, branch cuts, and poles followed by real-axis integration [CITATION], the convergence rate of these integrals is strongly dependent upon the transverse ([MATH]) and longitudinal ([MATH]) separations.', '1311.0582-1-3-2': "[MATH] determines the rapidity of the integrand's oscillation due to the Fourier and/or Hankel kernels in [REF]-[REF], with rising [MATH] leading to an integrand that traditionally requires increasingly finer sampling to limit spatial aliasing and thus leads to undesirably long computation times.", '1311.0582-1-3-3': "Furthermore, the longitudinal separation [MATH] governs the rate at which the evanescent spectrum's field contribution decays with increasing transverse wave number magnitudes, with rising [MATH] effecting more rapid decay (and hence faster convergence) [CITATION].", '1311.0582-1-3-4': 'On the other hand, as [MATH] the convergence rate lessens, with the limiting case [MATH] yielding integrals of the form [EQUATION] and [EQUATION] that lead to divergent results when numerically evaluated, using these standard paths, without convergence acceleration.', '1311.0582-1-4-0': 'See Figure [REF] for typical application scenarios wherein these standard paths either succeed or fail to deliver accurate field results.', '1311.0582-1-4-1': 'Observing Figure [REF], one immediately realizes that devising an evaluation method for these integrals exhibiting robustness with respect to all ranges of [MATH] and medium classes (e.g. isotropic, uniaxial, biaxial) is highly desirable.', '1311.0582-1-4-2': 'This robustness criterion inherently excludes fundamentally approximate methods such as image and asymptotic methods due to their geometry-specific applicability and lack of rigorous error control [CITATION].', '1311.0582-1-4-3': 'As a result, to reliably ensure accurate field results for arbitrary environmental medium composition/source-observer geometry combinations, we choose a direct numerical integration method.', '1311.0582-1-5-0': 'In this vein, one option involves pairing standard integration methods with (real-axis path based) algebraic convergence acceleration techniques such as the standard MWA which, based on published numerical results, successfully imparts algebraic convergence acceleration even when [MATH] [CITATION].', '1311.0582-1-5-1': 'However, it is desirable to (1) guarantee absolute, exponential convergence in the classical/Riemann sense for any [MATH] separation geometry (in contrast to only guaranteeing algebraic convergence in the Abel sense when [MATH] [CITATION]) and (2) endow error control to the evanescent-zone field contribution associated with the tail integral, whose relative importance (compared to the radiation-zone contribution) to the field solution grows as [MATH] decreases, to ensure that both the radiation-zone and evanescent-zone contributions are accurately evaluated.', '1311.0582-1-5-2': 'To this end, we propose a novel numerical integration method, representing a complex-plane generalization of a specific member of the "scalar Levin-type sequence transform" (SLST) family [CITATION] (i.e. the MWA), that:', '1311.0582-1-6-0': 'bends the "extrapolation region"/tail [CITATION] integration path sections to guarantee absolute, exponential convergence of integrals like [REF]-[REF], imparts added, robust algebraic convergence acceleration to the tail integrals, which compounds with the exponential convergence acceleration to effect absolute, exponential-cum-algebraic convergence, via use of a linear path bend combined with our novel, complex-plane generalization of the MWA [CITATION], adjusts the detour bend angles to account for the presence of branch points, branch cuts, and poles (summarily referred to here as "critical points"), and addresses the added challenges associated with evaluating two-dimensional integral transforms arising as solutions to the wave equation in planar-stratified environments lacking azimuthal symmetry.', '1311.0582-1-7-0': 'We note that other path deformation techniques, such as the Steepest Descent Path (SDP) and one comprising the enclosure of the first/fourth quadrants of the [MATH] plane involving an imaginary-axis integration, have been investigated and used [CITATION].', '1311.0582-1-7-1': 'However, we seek a robust integration method, valid for all [MATH] geometries, that obviates having to separately account for discrete poles while possessing applicability to multi-layered environments containing media with arbitrary anisotropy and loss.', '1311.0582-1-7-2': 'Thus while our method may result in longer solution times versus above-mentioned methods, it touts general applicability and minimal necessary book-keeping as its defining virtues.', '1311.0582-1-8-0': 'Furthermore, a robust detour path within the pre-extrapolation region [CITATION] maintaining a near-constant separation between the path and critical points near/on the real axis would be preferred over more traditional paths used with the MWA [CITATION].', '1311.0582-1-8-1': "To address this, the paper's second contribution entails a trapezoidal integration path paired with adaptive [MATH] refinement.", '1311.0582-1-9-0': 'In Section [REF] we present and discuss our revision to the radiation-zone integration path.', '1311.0582-1-9-1': 'In Sections [REF] and [REF] we develop the detoured linear integration path and complex-plane generalization to SLST for efficiently evaluating the tail sections of [REF]-[REF], as well as exhibit and compare two possible candidate formulations to implement the resulting modified-MWA.', '1311.0582-1-9-2': 'These developments are formulated in the context of two-dimensional integrals such as [REF] to simultaneously address herein their additional issues versus one-dimensional integrals.', '1311.0582-1-9-3': 'However, the formulation applies equally to one-dimensional F-H transforms like [REF] appearing in field/potential computations within cylindrically- and (azimuthal-symmetric) planar-stratified environments and, after converting the Fourier-Bessel (F-B) transform to a F-H transform [CITATION], to F-B transforms as well.', '1311.0582-1-9-4': 'Section [REF] presents validation results using one of the two new formulations.', '1311.0582-1-9-5': 'In Section [REF] we present a study on the convergence characteristics of our algorithm as concerning the same formulation used to generate the results in Section [REF].', '1311.0582-1-9-6': 'Finally, Section [REF] contains our concluding remarks.', '1311.0582-1-10-0': 'In the ensuing discussion, we assume appropriate transformations to the material tensors and source vector have already been performed to effect a coordinate rotation such that in the resultant (azimuthal-rotated) coordinate frame, within which all integration is performed, one has [MATH] .', '1311.0582-1-10-1': 'Discussed in detail at the end of Section [REF], this is done to guarantee absolute convergence and maximize exponential decay of both the [MATH] and [MATH] integrals.', '1311.0582-1-11-0': '# Pre-Extrapolation Region Path Revision', '1311.0582-1-12-0': 'First we discuss the parameterization and initial sub-division of the [MATH] plane pre-extrapolation region; discussion of the [MATH] plane follows identically due to our assuming [MATH].', '1311.0582-1-12-1': 'Applying a parameterization similar to that in [CITATION], define [MATH] as the points on the Re[MATH]] axis within which one detours, [MATH] as the maximum height of the trapezoid-shaped detour, and [MATH] as the points on the Re[MATH]] axis within which one adaptively integrates (see Figure [REF]).', '1311.0582-1-13-0': 'To compute [MATH], first define [MATH] as the magnitude of the real part of the global "effective" refractive index among all the layers (see [CITATION] on computing [MATH]).', '1311.0582-1-13-1': 'One then computes [MATH] analogously to [CITATION] and sets [MATH], where [MATH] is a user-defined pre-extrapolation region magnification constant.', '1311.0582-1-13-2': 'Next, define [MATH], [MATH], [MATH], and [MATH], where [MATH] and [MATH].', '1311.0582-1-13-3': 'Now compute the following pre-extrapolation region integration path parameters [CITATION]: [EQUATION] where Int[MATH] converts its argument to an integer via fractional truncation.', '1311.0582-1-13-4': 'Now parameterize the pre-extrapolation region integration path, for Re[MATH], as [EQUATION] for the trapezoidal contour (used to integrate up to [MATH]) combined with a real-axis path to integrate within the section [MATH].', '1311.0582-1-13-5': 'An analogous parameterization holds for the Re[MATH] pre-extrapolation region path.', '1311.0582-1-13-6': 'Note that [MATH] is independent of [MATH] and thus can be computed prior to integration, unlike other commonly used detours.', '1311.0582-1-13-7': "This is the trapezoidal path's second benefit in addition to that mentioned in footnote [REF].", '1311.0582-1-14-0': 'Now we splice the regions [MATH] and [MATH] each into [MATH] regions, where [MATH] is calculated as follows.', '1311.0582-1-14-1': 'First define [EQUATION] as the largest magnitude assumed by [MATH] along the trapezoidal path, [MATH] as the user-defined maximum allowed magnitude change of [MATH] between two sampling points, and [MATH] and [MATH] as two user-defined parameters.', '1311.0582-1-14-2': 'Subsequently, define the quantities [EQUATION] which are used to yield [MATH]=Int(1+[MATH]/[MATH]=1,2) with the corresponding final result [MATH].', '1311.0582-1-14-3': 'Note that this method of parameterizing the pre-extrapolation region path is empirical in nature and based on the pessimistic assumption of equidistant sampling [CITATION].', '1311.0582-1-15-0': '# Extrapolation Region Path Revision', '1311.0582-1-16-0': 'The MWA, initially constructed in [CITATION] with further variants developed in [CITATION] and [CITATION], has also demonstrated the ability to accelerate convergence of infinite-range Fourier double-integrals in high-loss, planar-stratified environments containing anisotropic media [CITATION].', '1311.0582-1-16-1': 'However, due to the highly oscillatory behavior of the mixed-domain integrand in integrals such as [REF] arising from the Fourier kernels [MATH] and [MATH] when one has large [MATH] and [MATH] (resp.)', '1311.0582-1-16-2': ', the solution times (in our experience) became inordinately long.', '1311.0582-1-16-3': 'Therefore, it would be desirable to also deform the [MATH] and [MATH] plane "extrapolation" region contours to lend additional exponential decay via these two kernels, thereby dramatically accelerating convergence of the Fourier tail integrals and guaranteeing their absolute convergence even in the "worst-case" scenario [MATH].', '1311.0582-1-16-4': 'A cursory analysis reveals an apparent severe drawback, however: one can no longer employ the MWA, which was derived assuming a real axis integration path [CITATION].', '1311.0582-1-16-5': "However, choosing a linear deformed path retains the MWA's algebraic convergence acceleration, as we show below.", '1311.0582-1-16-6': 'For this analysis, take [MATH]) as the inner (outer) integration variable.', '1311.0582-1-17-0': 'We first exhibit the foundational relations needed to implement the complex-plane extension to a general SLST followed by exhibiting the specific case arising from modeling the tail integral truncation error using the function family stipulated in the "Mosig-Michalski Algorithm" (MMA) [CITATION].', '1311.0582-1-17-1': 'Subsequently, we naively compute the optimal extrapolation region path detour angles without consideration for', '1311.0582-1-18-0': 'the presence of critical points in the [MATH] and [MATH] planes and two-dimensional integrals, associated with wave propagation phenomena, imparting a transitory nature to these critical points in the [MATH] plane (i.e. their locations now depend on the fixed [MATH] value for which the [MATH] integral is evaluated).', '1311.0582-1-19-0': 'To address the first concern, we pessimistically estimate the locations of critical points and reduce the [MATH] plane departure angle of the deformed paths to ensure these features are not crossed.', '1311.0582-1-19-1': 'To address the latter concern, we (1) adjust the departure angles of the [MATH] plane integration path and (2) partition the [MATH] integration domain to ensure that the critical points', '1311.0582-1-20-0': 'possess real parts with magnitude decaying as [MATH] increases, leading to a bounded pre-extrapolation region, and do not extend into the second/fourth quadrants, as this would require a) tracking their locations and b) adjusting the [MATH] integration path, both of which would become functions of [MATH] and lead to a non-robust integration path.', '1311.0582-1-21-0': 'For simplicity, the analysis developing the complex-plane SLST generalization assumes isotropic planar layers.', '1311.0582-1-21-1': 'Fix [MATH] at some (generally complex) value [MATH] and assume [MATH] ; we see then that the inner integral of [REF] writes as [EQUATION] where [MATH] is the up-going mode propagation constant, which for our time convention has positive imaginary part.', '1311.0582-1-21-2': "Assuming [MATH] extrapolation intervals are used [CITATION], the linear path detour used in the [MATH] integration path's extrapolation region is parameterized as [EQUATION] where one defines [MATH], [MATH], [MATH], and [MATH] as real-valued.", '1311.0582-1-21-3': '[MATH] is assumed large enough to ensure that we have sufficiently detoured past any critical points near the real axis [CITATION].', '1311.0582-1-21-4': 'Now recall that plane wave propagation in a homogeneous, unbounded, isotropic medium with wave number [MATH] is governed by the dispersion relation [MATH] [CITATION].', '1311.0582-1-21-5': 'For large [MATH] this relation becomes [EQUATION]', '1311.0582-1-21-6': 'Next, assuming that (asymptotically) [MATH] [CITATION], one can substitute this series expression into the extrapolation region section of [REF] to obtain [EQUATION] where "ext" denotes the [MATH] plane extrapolation region integration path section and the [MATH] comprise a family of functions used to asymptotically model [MATH] and the truncation error (discussed below) [CITATION].', '1311.0582-1-21-7': 'Setting [MATH], [MATH], [MATH], and [MATH], [REF] becomes the union of [REF] and [REF]: [EQUATION] with the respective truncation error integrals of [REF]-[REF] manifesting as [EQUATION]', '1311.0582-1-21-8': 'Performing a change of variables on [REF]-[REF] subsequently yields the following relations [EQUATION]', '1311.0582-1-21-9': 'Next, one evaluates [REF]-[REF] for [MATH] different values of [MATH] (e.g. [MATH]), truncates these [MATH] relations after the [MATH] error series term (i.e. retain the first [MATH] series terms), defines the [MATH]th truncation error series coefficient pair as [MATH] (e.g. see [REF]-[REF] below), and solves the corresponding [MATH]-order system to estimate [MATH] and [MATH].', '1311.0582-1-21-10': 'This procedure represents the complex-plane SLST generalization, applicable to the sequence of [MATH] successive "cumulative tail integral" estimates [CITATION], to accelerate evaluation of [MATH] and [MATH].', '1311.0582-1-22-0': 'Let us now examine the specific case of modeling [MATH] using the family of [MATH]-parameterized functions [MATH] for Re[MATH] and [MATH] for Re[MATH].', '1311.0582-1-22-1': 'Performing Maclaurin expansions of the [MATH] and [MATH], retaining only their respective zeroth-order expansion terms, setting [MATH] and [MATH], and defining [EQUATION] yields a pair of expressions capturing the dominant behavior of the truncation error [MATH]: [EQUATION] which comprises the complex-path generalization, as concerning infinite-range Fourier integrals, to the error expression developed in Section 2 of [CITATION].', '1311.0582-1-22-2': 'The corresponding truncation error expressions associated with F-H transforms like [REF] follow in analogous fashion.', '1311.0582-1-23-0': 'With the foundational expressions available, we now seek to maximize exponential convergence acceleration of [REF] through a suitable choice of the detour departure angles [MATH] and [MATH].', '1311.0582-1-23-1': 'Differentiating the exponent expressions [MATH] and [MATH] with respect to [MATH] and [MATH] (resp.)', '1311.0582-1-23-2': 'and setting the resulting expressions equal to zero leads us to initially (naively) choose [MATH], which (asymptotically) corresponds to the path of most rapid exponential decay or (equivalently) the Constant Phase Path (CPP).', '1311.0582-1-23-3': 'This detour angle choice can be likened to a compromise between the so-called "[MATH]-transmission representation" and "radial transmission representation" [CITATION] of the space-domain field, which were discussed therein in the context of F-H and F-B transforms.', '1311.0582-1-23-4': 'Identical expressions hold for the [MATH] plane detour departure angles [MATH] and [MATH].', '1311.0582-1-23-5': 'Next we consider the detour constraints imposed by these two phenomena.', '1311.0582-1-24-0': 'To this end, first define the branch point as [MATH] and temporarily assume that the [MATH] path was chosen so that [MATH] plane critical points neither manifest in the second/fourth quadrants nor migrate towards Re[[MATH]][MATH] with increasing [MATH].', '1311.0582-1-24-1': 'Recalling the effective refractive indices [MATH]) for layer [MATH] and how we subsequently computed [MATH] [CITATION], set [MATH] equal to either (1) the angle between [MATH] and the [MATH]th layer\'s [MATH]th "effective wave number" [MATH], if [MATH], or (2) [MATH] if [MATH].', '1311.0582-1-24-2': 'Then, [MATH] is updated as [MATH].', '1311.0582-1-24-3': 'No critical points are located in the second/fourth quadrants by assuming (for simplicity) the absence of "double-negative"/meta-material and active/gain media.', '1311.0582-1-24-4': 'Therefore, we do not have to constrain [MATH].', '1311.0582-1-24-5': 'However, these calculations can be readily adjusted to appropriately constrain both [MATH] and [MATH] if such media are present so that our assuming their absence represents a trivial constraint in our methodology.', '1311.0582-1-25-0': 'Now we justify the assumptions above about the [MATH] path, and constrain it to avoid the two issues stated earlier regarding two-dimensional integral transforms arising as the solution to wave-dynamics problems in planar-stratified environments lacking azimuthal symmetry.', '1311.0582-1-25-1': 'To this end, for some arbitrary [MATH] value along the [MATH] plane integration path first expand the branch point [MATH] as [EQUATION] and recall that the radiation branch cut is jointly defined by the conditions Im[MATH] and Re[MATH] [CITATION].', '1311.0582-1-25-2': 'To ensure that critical points in the first (third) quadrant of the [MATH] plane do not migrate towards Re[[MATH]]=[MATH] (Re[[MATH]]=[MATH]) for large [MATH], one must ensure that asymptotically Re[MATH] as [MATH].', '1311.0582-1-25-3': 'Observing the real part of [REF], we see that one must constrain [MATH] and [MATH] to the interval [MATH].', '1311.0582-1-25-4': 'Furthermore, to prevent critical points from migrating into the second/fourth [MATH] plane quadrants, we require that Im[MATH] as [MATH].', '1311.0582-1-25-5': "Observing the imaginary part of [REF] and noting in the region [MATH] that [MATH], a cursory analysis suggests that one cannot safely choose a non-zero value of [MATH] without risking this migration, which would force one to dynamically re-define the [MATH] integral's pre-extrapolation region path, now a function of [MATH], to ensure that one (1) encloses all the quadrant one critical points that migrated into quadrant four while (2) avoiding the encirclement of quadrant three critical points that migrated into quadrant two.", '1311.0582-1-26-0': 'As a result, it appears that one must set the additional, more restrictive constraint [MATH], which in theory may lead to an outer integral exhibiting monotonic-divergent behavior when [MATH] [CITATION].', '1311.0582-1-26-1': 'However, this limitation can be overcome via clever partition of the two-dimensional integration domain; see Figure [REF], which summarizes the proposed partition.', '1311.0582-1-26-2': "Integrating first over Regions I, IIa, and IIb in Figure [REF] followed by integrating in Region III, which encompasses the intersection of the [MATH] and [MATH] plane extrapolation regions, renders the Region III integration's result immune to the migration of critical points into the second/fourth quadrants.", '1311.0582-1-26-3': 'This is because one had already stipulated a domain partitioning and completed integration over Regions I, IIa, and IIb.', '1311.0582-1-27-0': 'We conclude that so long as one conforms to the restrictions [MATH] and [MATH], one can detour in all four spectral "quadrants" [MATH], [MATH], [MATH], and [MATH] (see Figure [REF]) through which the stipulated integration path proceeds.', '1311.0582-1-27-1': "Indeed, our proposed partition of the [MATH] integration domain ensures that for any [MATH] geometry, the double-integral [REF] exhibits absolute-convergent behavior in the classical/Riemann sense and thus, by Fubini's theorem [CITATION], can be evaluated using an iterated integral whose solution is independent of the order of integration.", '1311.0582-1-28-0': 'This discussion also brings to light the benefit of our starting assumption in this analysis, made at the end of Section [REF], that [MATH]: a compromise is reached that ensures exponential-cum-algebraic convergence of both the [MATH] and [MATH] integrals throughout the integration domain.', '1311.0582-1-28-1': 'As an alternative we could have, for example, performed an azimuthal rotation such that [MATH] and [MATH] to maximize convergence acceleration of the [MATH] integral.', '1311.0582-1-28-2': 'However, when [MATH], the [MATH] integral may exhibit monotonic-divergent behavior.', '1311.0582-1-28-3': 'In contrast to oscillatory-divergent behavior [CITATION], the MWA variants (including the generalized version developed herein) cannot curb monotonic-divergent behavior due to the lack of oscillations that must be present for the MWA to "average out" the oscillatory-divergent sequence of cumulative tail integral estimates to obtain a final, convergent result.', '1311.0582-1-29-0': 'Note that for integration in Regions IIa, IIb, and III in Figure [REF], one performs a separate integration and extrapolation of the individual [MATH] and/or [MATH] half-tail integral sections.', '1311.0582-1-29-1': "This is in contrast to the method developed in [CITATION] wherein we folded the half-tail integrals in the [MATH] half-planes to yield a cosine or sine oscillatory kernel, based on assuming spectral symmetry in the environment's plane wave reflection/transmission properties, prior to performing tail integral extrapolation along the positive Re[[MATH]] axis (and similarly for the [MATH] plane).", '1311.0582-1-29-2': 'Our present method, in bending both half-tail [MATH] paths into the upper-half [MATH] plane, forbids such folding due to the now-absent lack of reflection symmetry (about the Im[[MATH]] axis) with respect to the two halves of the extrapolation region path.', '1311.0582-1-29-3': 'The resulting penalty paid in using the complex-plane MWA manifests in having to use twice the number of weight sets versus when one can perform half-tail integral folding followed by cumulative tail integral sequence extrapolation, leading to increased memory requirement and computation time in regards to procuring the MWA weight sets.', '1311.0582-1-29-4': 'As a practical consideration, then, we wish to reduce the number of extrapolation weight sets that must be evaluated.', '1311.0582-1-29-5': 'To this end, we take two steps to halve this number to the six weight sets originally required when performing folding followed by extrapolation.', '1311.0582-1-29-6': 'First, we set [MATH] and [MATH].', '1311.0582-1-29-7': 'Second, we make the approximation (for each field component) that the asymptotic monomial power dependence on both [MATH] and [MATH] [CITATION] equals the average monomial power dependencies on [MATH] and [MATH].', '1311.0582-1-29-8': 'For example, if we determine the integrand for one field component has asymptotic monomial dependencies of [MATH] and [MATH], then we take (as both our [MATH] and [MATH] monomial dependence factors) [MATH], where Nint([MATH]) converts its argument to an integer via rounding.', '1311.0582-1-29-9': 'Furthermore, to ensure stability of the accelerator weight expressions and minimize aliasing effects due to inordinately long extrapolation region intervals, we neglect the integrand oscillation due to [MATH] in the exponential kernels of [REF]-[REF].', '1311.0582-1-29-10': 'This allows one to update the truncation error estimates [MATH] as [MATH] when [MATH] [CITATION].', '1311.0582-1-30-0': 'Beyond the concern of weight computation stability, we also ignore the phase variation associated with [MATH] due to [MATH], in general, being ill-defined.', '1311.0582-1-30-1': 'Indeed, in (1) an anisotropic homogeneous environment or (2) a stratified environment containing isotropic and/or anisotropic media, several phenomena typically obfuscate a univocal, clear definition for the effective longitudinal distance traversed by the characteristic plane wave fields when traveling from [MATH] in layer [MATH] to [MATH] in layer [MATH].', '1311.0582-1-30-2': 'These are (1) multi-bounce within slab layers, (2) the layer and (for anisotropic media) mode dependence of the longitudinal propagation constants, (3) interface reflections in layer [MATH] causing both up-going and down-going modal fields (four total modes in general) to contribute to the observed field at [MATH], and (4) inter-mode coupling at the interfaces.', '1311.0582-1-30-3': 'In fact these four considerations, along with the inherently asymptotic nature of the CPP parameterization and the constraints associated with critical points/two-dimensional integrals addressed above, lead one in practice to not integrate exactly along the CPP.', '1311.0582-1-30-4': 'As a result, one typically finds the integrands of extrapolation-region integrals still exhibiting undesirable residual oscillation due to the complex exponential factors.', '1311.0582-1-30-5': 'While, for [MATH], one still always has a non-zero detour angle for both the [MATH] and [MATH] extrapolation region paths, these practical considerations are what demand the inclusion of an algebraic convergence accelerator like the MWA that exactly acts upon the very types of oscillatory integrals that will typically result.', '1311.0582-1-30-6': 'Therefore, while one does not typically realize the ideal situation of maximized exponential convergence acceleration (the strongest acceleration theoretically available here outside of the SDP), we largely mitigate this pitfall with the robust algebraic acceleration afforded by the MWA, which is agnostic to the environment/source-observer scenario (so long as [MATH]).', '1311.0582-1-30-7': 'Indeed, for an order-[MATH] MWA method used (see below) one realizes a reduction in truncation error between [MATH]) and [MATH]) [CITATION].', '1311.0582-1-31-0': '# Revised Accelerator Weight Computation', '1311.0582-1-32-0': 'The MWA, both in its form as the MMA [CITATION] and its more recent variant the new/"revisited" MWA [CITATION], each offer different, desirable attributes.', '1311.0582-1-32-1': 'The latter version offers a straightforward methodology to unambiguously define arbitrary-order accelerator weight sets and recursively compute higher-order weight sets upon demand.', '1311.0582-1-32-2': 'While we showed previously [CITATION] for the MMA how one can reduce the FLOP count involving the cumulative integrals themselves, the weight computations (1) depended on whether the series of successive extrapolation region sub-interval integrals exhibited oscillating or monotone behavior [CITATION], and (2) the FLOP count to compute the weight sets rapidly grows for successive weight sets, placing a practical limit on obtainable accuracy in the weights (and thus the estimated tail integral) due to roundoff error accumulation in the computed weights.', '1311.0582-1-32-3': 'On the other hand, the computation of the new MWA weight sets is (1) a numerically unstable process rapidly leading to numerical overflow (when using the form exhibited in [CITATION]) and (2) directly linked to procuring the estimated tail integral [CITATION], which is the solution to a highly ill-conditioned linear system (shown in Section [REF]), which previously led us to use the MMA in [CITATION].', '1311.0582-1-32-4': "Nevertheless, both flavors of MWA offer useful mathematical developments for the weights that are couched in the framework of SLST, using a family of functions in a series representation to model the spectral portion of the mixed-domain Green's Function and resulting tail integral truncation error.", '1311.0582-1-32-5': 'For the MMA, the proposed series [CITATION] [EQUATION] is intuitive in its form, and we confirm below the validity (and in fact optimality) of using this approximating series by a straightforward mathematical analysis.', '1311.0582-1-32-6': 'It will be shown that this optimality arises due to the error modeling series (see [REF]-[REF] above and [CITATION]) being entirely consistent with the closed-form expression of the truncation error both in the absence and (via linear superposition) presence of stratified inhomogeneity.', '1311.0582-1-32-7': "The practical consequence of this function family's modeling optimality manifests in minimizing the number of cumulative tail integral estimates required to accurately estimate [MATH] and [MATH], as demonstrated in Section [REF].", '1311.0582-1-33-0': 'In summary, we seek a revised, complex-plane MWA that combines the best aspects of both the MMA [CITATION] and more recent MWA variant [CITATION], in tandem with incorporating the added exponential convergence acceleration afforded by bending the extrapolation region integration path, to effect robust and powerful field solution convergence acceleration.', '1311.0582-1-33-1': 'To this end, in this section we (1) analyze and justify using [REF] as the approximating series for [MATH] and (2) exhibit and compare two proposed formulations for implementing the complex-plane generalization of the MWA, using the new "remainder estimates" [MATH] [CITATION] and the asymptotic series expansion [REF].', '1311.0582-1-34-0': '## The Optimal Error-Modeling Function Family', '1311.0582-1-35-0': 'Herein we examine the inner spectral ([MATH]) integral for some fixed [MATH] in the region Re[MATH].', '1311.0582-1-35-1': 'Furthermore, assume [MATH] has an asymptotic [MATH] monomial dependence of [MATH] [CITATION].', '1311.0582-1-35-2': 'One then has the asymptotic truncation error [EQUATION] which just equals [REF] with the asymptotic series expansion for [MATH] replaced by the dominant series term [MATH].', '1311.0582-1-35-3': 'Next note that [MATH] has a closed-form, convergent solution for [MATH]; setting [MATH] and [MATH], one obtains [EQUATION] and so on for other values of [MATH] .', '1311.0582-1-35-4': 'Examining the asymptotic limit for these three illustrative cases, we find: [EQUATION] where [MATH] represents the set of positive natural numbers.', '1311.0582-1-35-5': 'Similarly, one expects that the reflected/transmitted field terms will also have an asymptotic monomial dependence [CITATION].', '1311.0582-1-35-6': 'For example, consider a two-layer, planar-stratified environment containing isotropic media.', '1311.0582-1-35-7': 'The TE[MATH]/TM[MATH] reflection and transmission coefficients for a plane wave, incident from half-space number one upon half-space number two, write as [CITATION] [EQUATION] where [EQUATION]', '1311.0582-1-35-8': 'Indeed, we see that for large [MATH] the reflection/transmission coefficients have a monomial power dependence [MATH].', '1311.0582-1-36-0': 'Pulling out the dominant monomial term [MATH] in [REF], setting the [MATH]th cumulative remainder estimate as [MATH], and invoking superposition (see footnote [REF]), one now has [MATH], which recovers the dominant contribution to the complex-path extension [REF] of the error expression derived (using [REF]) in [CITATION].', '1311.0582-1-36-1': 'The same procedure shown above, using instead [MATH], [MATH], and [MATH], can be repeated for the Re[MATH] tail integral to obtain a dual set of expressions that recover [REF].', '1311.0582-1-36-2': 'Based on this analysis, when proposing two revised MWA methods we will use the [MATH]-parameterized function families [MATH] and [MATH] to model the tail integral truncation error.', '1311.0582-1-37-0': '## Two Proposed Formulations', '1311.0582-1-38-0': 'For the first formulation we take inspiration from [CITATION].', '1311.0582-1-38-1': 'To this end, for the Re[MATH] tail integral first define [MATH] and [MATH] as two input cumulative tail integral estimates and the under-determined linear system, with respect to which the non-truncated tail integral [MATH] is defined, as [CITATION] [EQUATION] whose equations are subsequently truncated after the [MATH] term [CITATION].', '1311.0582-1-38-2': 'This truncation yields a second-order linear system solved for an improved estimate [MATH] of [MATH] that is free of the [MATH] term in its truncation error series [CITATION]: [EQUATION]', '1311.0582-1-38-3': 'Similarly, using [MATH]) cumulative tail integral estimates [MATH] to eliminate the first [MATH] terms of [MATH], one has for the [MATH]th truncated linear equation ([MATH].', '1311.0582-1-38-4': 'Subsequently, one procures the weights via solving the associated order-([MATH]) linear system for the best [MATH] estimate (i.e. [MATH]), whose solution implicitly contains the expressions for the weights [CITATION].', '1311.0582-1-38-5': 'However, obtaining all desired weight tier sets by directly solving the associated linear systems (1) is very costly and (2) possibly exacerbates weight accuracy degradation due to the poor conditioning of these systems (see below).', '1311.0582-1-38-6': 'Instead, one can obtain closed-form solutions to the weight sets using the methodology outlined in [CITATION] as adapted to our choice of (1) error-modeling functions [MATH] and (2) truncation error estimates [MATH].', '1311.0582-1-38-7': 'In [CITATION] it was assumed that [MATH] asymptotically exhibited a monomial power dependence of the form [MATH] being some constant), with the obvious consequence that [MATH] corresponds to a new function asymptotically behaving as [MATH] .', '1311.0582-1-38-8': 'Rearranging the order-([MATH]) linear system thus yields a similar (but not yet identical) system to equation (22) in [CITATION]: [EQUATION]', '1311.0582-1-38-9': 'Noting that the weight sets in [CITATION] were computed for arbitrary monomial power dependence [MATH], one can cross-multiply the [MATH] factors in the [MATH] across the respective rows of [REF] to obtain an analogous system, where now the [MATH] factors in the modified form of [REF] represent (up to a constant) successive [MATH] derivatives of [MATH] evaluated at [MATH] .', '1311.0582-1-38-10': 'Having now matched our linear system to [CITATION], the [MATH]th weight ([MATH]) for the tier-([MATH]), complex-plane generalization of the new MWA writes as [CITATION] [EQUATION] with the expression for our best tail integral estimate given as [EQUATION]', '1311.0582-1-38-11': 'The expressions for the [MATH] corresponding to Re[MATH] tail integral follows analogously.', '1311.0582-1-38-12': 'Furthermore, one expects that with a different choice of [MATH], this derivation can be repeated to develop complex-plane extensions to other SLST algorithmic members.', '1311.0582-1-39-0': 'From an analytic standpoint, the derivation of the weights for this formulation is complete.', '1311.0582-1-39-1': 'However, despite the analytic form of the new MWA weights shown in [REF] and [CITATION], in a finite-precision, numerical implementation this casting leads to arithmetic overflow.', '1311.0582-1-39-2': 'This drawback, along with the numerically unstable means to recursively update the weights to procure higher-order weight sets, can be easily remedied as follows:', '1311.0582-1-40-0': 'Starting at some tier-[MATH] weight set (e.g. set [MATH]), multiply all the weights by [MATH].', '1311.0582-1-40-1': 'This ensures that the weights remain bounded for all [MATH] and [MATH].', '1311.0582-1-40-2': 'To subsequently obtain a tier-[MATH] weight set from the tier-[MATH] set:', '1311.0582-1-41-0': 'Set [MATH].', '1311.0582-1-41-1': 'For the remaining [MATH] weights, set [MATH]', '1311.0582-1-42-0': 'where [EQUATION]', '1311.0582-1-42-1': 'The second proposed formulation employs the MMA [CITATION], as extended to facilitate adaptive tail integral evaluation [CITATION], in conjunction with our complex extension to the truncation error estimates [MATH].', '1311.0582-1-42-2': 'The formulae to compute arbitrary-order weight sets is given in [CITATION], while the method to recursively find higher-order weight sets is exhibited in [CITATION].', '1311.0582-1-42-3': 'Therefore, the reader is referred to these two references for the elementary details.', '1311.0582-1-43-0': 'Between these two formulations, we opt to implement and show validation results for the second formulation based on the MMA.', '1311.0582-1-43-1': "This is because of the first formulation's poor suitability for an adaptive tail integral evaluation scheme, which in turn is due to increasingly higher-order weight sets being the solutions to increasingly ill-conditioned linear systems.", '1311.0582-1-43-2': "Even though we now have available the analytically recast, numerically stable, closed-form expressions for the first formulation's weights and their update scheme (which obviates any potential algorithmic instability exacerbating computed weight errors), the relative accuracy of the computed weights is still fundamentally capped by the linear system's conditioning.", '1311.0582-1-43-3': 'To illustrate the ill-conditioning of the weight computation, we show in Figure [REF] below, for four different [MATH] geometries, the two-norm condition number [MATH] [CITATION] of [REF] as a function of its rank [MATH] :', '1311.0582-1-44-0': '[MATH]m [MATH]m [MATH]m [MATH]m', '1311.0582-1-45-0': 'To confirm that the system matrix ill-conditioning is not due to the complex-plane generalization of the new MWA, in Figure [REF] we show the two-norm condition number for [MATH] while in Figure [REF] we show, for the same four [MATH] geometries, the conditioning for [MATH] (i.e. as if we performed the standard, real-axis MWA from [CITATION]).', '1311.0582-1-46-0': 'One readily observes from Figure [REF] that accurate weight computation is unrealistic as [MATH] increases; in fact, the situation is downright prohibitive for an adaptive MWA implementation (e.g. [CITATION]).', '1311.0582-1-46-1': 'Even for the best-conditioned geometry (i.e. [MATH]m), one cannot realistically expect even a single digit of precision in the weights for [MATH] equalling or exceeding approximately seven and ten in Figures [REF] and [REF] (resp.)', '1311.0582-1-46-2': ', as can be seen from the intersection of the corresponding curves in Figures [REF]-[REF] with the solid horizontal curve corresponding to Log[MATH].', '1311.0582-1-46-3': 'As a result, we choose the second proposed MWA formulation, based on the MMA [CITATION], for computing validation results in Section [REF].', '1311.0582-1-46-4': 'Based on our previous work using the standard, real-axis MMA [CITATION] for environments containing high loss and conductively-uniaxial layers, one can expect its success in again producing high-precision results.', '1311.0582-1-46-5': 'Indeed, the validation results in Section [REF] speak to this effect.', '1311.0582-1-47-0': '# Results and Discussion', '1311.0582-1-48-0': 'In this section we exhibit validation results in scenarios involving the modeling of induction sondes for geophysical prospection of hydrocarbons (i.e. induction well logging [CITATION]).', '1311.0582-1-48-1': 'Previously, we demonstrated numerous simulated resistivity logs pertaining to environments containing a combination of isotropic and reciprocal, electrically uniaxial media [CITATION] as probed by longitudinally-oriented induction sondes [CITATION].', '1311.0582-1-48-2': 'For those case studies, the adaptive, real-axis MMA was successfully incorporated into our algorithm to yield high-precision results exhibiting excellent agreement with data from previous literature [CITATION].', '1311.0582-1-49-0': 'Herein, we exhibit a case study involving a near-horizontal tool orientation where the tool axis dip angle [MATH], tool axis strike angle [MATH] , and source-observer separation [MATH]m, corresponding to a source-observer depth separation [MATH]mm.', '1311.0582-1-49-1': 'Consequently, this study serves to validate the efficacy of our new algorithm and its ability to impart absolute, exponential-cum-algebraic convergence on Fourier double-integrals like [REF] even for the traditionally prohibitive regime [MATH].', '1311.0582-1-49-2': 'Furthermore, to exemplify the general-purpose nature of our new algorithm in regards to the media present, we generate synthetic resistivity logs for a two-layer, planar-stratified environment containing reciprocal, electrically biaxial media.', '1311.0582-1-49-3': 'In this scenario, wherein all four characteristic plane wave modes in the anisotropic layer containing [MATH] can (in general) contribute to the observed field, the definition of an exact "[MATH]" and thus CPP is ill-defined (see Section [REF]).', '1311.0582-1-49-4': "Therefore, this set of results also justifies our retaining the MWA's robust environment/source-observer geometry convergence acceleration characteristic, yielding an overall robust and rapid electromagnetic field solution method.", '1311.0582-1-50-0': 'Note that save for Figure [REF], there exists strong agreement across the full logging path in each plot.', '1311.0582-1-50-1': 'Even for Figure [REF], with some discrepancy in the upper half-space [MATH], overall there is strong qualitative agreement and (in the bottom half-space) quantitative agreement too.', '1311.0582-1-51-0': '# Convergence Characteristics', '1311.0582-1-52-0': "To characterize our numerical formulation's ability to converge towards the field solution, we present two case studies concerning the [MATH]-directed electric field component [MATH] produced by a [MATH]-directed electric dipole radiating at [MATH]=2MHz in free space.", '1311.0582-1-52-1': 'The first case comprises a benign scenario in which [MATH]m, while the second case represents a very challenging scenario wherein [MATH]m and [MATH]m.', '1311.0582-1-52-2': "The latter scenario's prohibitive challenges, when using a standard numerical integration method, are that the integrand (1) oscillates on the order of [MATH] times more rapidly than the integrand in case one and (2) exhibits absolutely no exponential decay due to the annihilation of the exp([MATH])-type factors.", '1311.0582-1-52-3': 'If we were to use a traditional numerical integration methodology, we emphasize that one would obtain a divergent result.', '1311.0582-1-53-0': 'For each case, we present results related to the Region III field contribution (see Figure [REF]).', '1311.0582-1-53-1': 'Since one cannot obtain a closed-form solution to this field contribution, reference field values from which one measures relative accuracy must be appropriately chosen; their computation details are provided in Figure [REF] below.', '1311.0582-1-53-2': 'As in [CITATION], we assume the integrand is well-behaved in Region III and thus do not perform adaptive interval sub-division.', '1311.0582-1-53-3': 'Instead, we set the [MATH] and [MATH] plane extrapolation region interval lengths as per Section [REF] and examine the accuracy versus (1) the number of extrapolation region intervals employed ([MATH]) and (2) the Legendre-Gauss quadrature order used ([MATH]) to integrate each interval.', '1311.0582-1-54-0': 'Our comments on the relative importance of aliasing and truncation error are analogous to [CITATION]: Up to approximately [MATH] the truncation error dominates the total relative error, while using more than approximately [MATH] or 7 intervals effects no noticeable decrease in the error for a fixed [MATH].', '1311.0582-1-54-1': 'Beyond this point aliasing error dominates the total relative error, which is evidenced by the error decreasing versus increasing [MATH] but remaining flat versus increasing [MATH].', '1311.0582-1-54-2': "However, we notice the following two remarkable characteristics about the algorithm's convergence for case two:", '1311.0582-1-55-0': "The [MATH] curve reaches within 25dB of case one's [MATH] curve despite representing a scenario wherein the field solution would ordinarily have diverged using standard numerical integration techniques.", '1311.0582-1-55-1': 'Despite this case representing a far more prohibitive scenario (if traditionally evaluated) versus case two presented in [CITATION], wherein [MATH]m, at [MATH] the [MATH] curve here levels off at an error approximately 23dB lower than its case two counterpart in [CITATION].', '1311.0582-1-56-0': 'Note that (akin to, and for the same reasons stated in, [CITATION]) relative errors below -150dB were coerced to -150dB.', '1311.0582-1-57-0': 'Since the pre-extrapolation region formulation in this paper is not radically different from that in [CITATION], we expect similar convergence characteristics when using the trapezoidal detour (versus those presented in [CITATION]) and thus omit the Region I convergence study for brevity.', '1311.0582-1-57-1': "Furthermore, the Region IIa/IIb convergence studies are omitted as well since the field convergence results would be affected by the algorithm's handling of both the pre-extrapolation and extrapolation region sections of the [MATH] and [MATH] plane integration paths.", '1311.0582-1-57-2': 'Equivalently, presenting information on the Region I and Region III field convergence characteristics sheds insight into the Region IIa/IIb convergence characteristics.', '1311.0582-1-57-3': 'This is because if the respective algorithms handling the Region I and Region III integrations robustly yield accurate, rapidly convergent results, one can expect similar behavior for the Region IIa/IIb results.', '1311.0582-1-58-0': '# Conclusion', '1311.0582-1-59-0': 'In this work, we have presented a novel integration scheme composed of (1) a complex-plane, adaptive/error-controlling extension to the standard real-axis MMA in conjunction with (2) a more robust pre-extrapolation region integration path to effect fast, absolute, and exponential-cum-algebraic convergence of Fourier- and F-H-type integral transforms such as [REF]-[REF].', '1311.0582-1-59-1': 'Due to combining the detour with the MMA and its robust algebraic convergence acceleration characteristic, this is indeed the case irrespective of the source-observer geometry and loss/anisotropy characteristics of the stratified media present.', '1311.0582-1-59-2': 'Furthermore, this is accomplished without the added complication of having to separately account for slab/interface mode contributions whose poles may be crossed when otherwise deforming to more well-known, rapidly-convergent paths such as the SDP [CITATION], resulting in a numerically robust and easily-implemented integration methodology.', '1311.0582-1-60-0': 'The algorithm\'s ability to accurately simulate the observed fields for classically "worst-case" scenarios [MATH], and that too in complex, planar-stratified environments containing biaxial-conductive media, has been verified through numerous validation checks against [CITATION].', '1311.0582-1-60-1': "Finally, the algorithm's convergence characteristics in the strongly-evanescent spectral zone have been explored, analyzed, and shown to be superior compared to an older methodology exhibited in [CITATION] that was based on an adaptive extension to the real-axis MMA.", '1311.0582-1-61-0': "We conclude that the present algorithm's robustness with respect to source-observer geometries and medium types present, as well as its straight-forward nature and ease of implementation, makes it very useful for the analysis of electromagnetic wave propagation and scattering in multi-layered environments containing media of arbitrary anisotropy and loss."}

{'1311.0582-2-0-0': 'We propose the complex-plane generalization of a powerful algebraic sequence acceleration algorithm, the Method of Weighted Averages (MWA), to guarantee exponential-cum-algebraic convergence of Fourier and Fourier-Hankel (F-H) integral transforms.', '1311.0582-2-0-1': 'This "complex-plane" MWA, effected via a linear-path detour in the complex plane, results in rapid, absolute convergence of field/potential solutions in multi-layered environments regardless of the source-observer geometry and anisotropy/loss of the media present.', '1311.0582-2-0-2': 'In this work, we first introduce a new integration path used to evaluate the field contribution arising from the radiation spectra.', '1311.0582-2-0-3': 'Subsequently, we (1) exhibit the foundational relations behind the complex-plane extension to a general Levin-type sequence convergence accelerator, (2) specialize this analysis to one member of the Levin transform family (the MWA), (3) address and circumvent restrictions, arising for two-dimensional integrals associated with wave dynamics problems, through minimal complex-plane detour restrictions and a novel partition of the integration domain, (4) develop and compare two formulations based on standard/real-axis MWA variants, and (5) present validation results and convergence characteristics for one of these two formulations.', '1311.0582-2-1-0': '# Introduction', '1311.0582-2-2-0': 'In many application areas concerning time-harmonic electromagnetic (EM) fields, one encounters environments containing media of varying and arbitrary anisotropy whose inhomogeneity can be approximated as multi-layered in nature.', '1311.0582-2-2-1': 'Examples include geophysical prospection [CITATION], plasma physics [CITATION], antenna design [CITATION], optical field control [CITATION], microwave remote sensing [CITATION], ground-penetrating radar [CITATION], and microwave circuits [CITATION], among others.', '1311.0582-2-2-2': 'Such applications regularly encounter integrals of the form [EQUATION] and/or [EQUATION] which express space-domain field/potential functions as Fourier and Fourier-Hankel (F-H) integral transforms (resp.)', '1311.0582-2-3-0': 'In many practical applications, these integrals must often be rapidly evaluated for a wide range of longitudinal and transverse source-observer separation geometries [MATH] (e.g. for potential or field profile reconstruction).', '1311.0582-2-3-1': 'However, when using standard integration paths that run on/close to the real axis such as (1) the classic Sommerfeld Integration Path (SIP) [CITATION] and (2) paths detouring around the branch points, branch cuts, and poles followed by real-axis integration [CITATION], the convergence rate of these integrals is strongly dependent upon the transverse ([MATH]) and longitudinal ([MATH]) separations.', '1311.0582-2-3-2': "[MATH] determines the rapidity of the integrand's oscillation due to the Fourier and/or Hankel kernels in [REF]-[REF], with rising [MATH] leading to an integrand that traditionally requires increasingly finer sampling to limit spatial aliasing and thus leads to undesirably long computation times.", '1311.0582-2-3-3': "Furthermore, the longitudinal separation [MATH] governs the rate at which the evanescent spectrum's field contribution decays with increasing transverse wave number magnitudes, with rising [MATH] effecting more rapid decay (and hence faster convergence) [CITATION].", '1311.0582-2-3-4': 'On the other hand, as [MATH] the convergence rate lessens, with the limiting case [MATH] yielding integrals of the form [EQUATION] and [EQUATION] that lead to divergent results when numerically evaluated, using these standard paths, without convergence acceleration.', '1311.0582-2-4-0': 'See Figure [REF] for typical application scenarios wherein these standard paths either succeed or fail to deliver accurate field results.', '1311.0582-2-4-1': 'Observing Figure [REF], one immediately realizes that devising an evaluation method for these integrals exhibiting robustness with respect to all ranges of [MATH] and medium classes (e.g. isotropic, uniaxial, biaxial) is highly desirable.', '1311.0582-2-4-2': 'This robustness criterion inherently excludes fundamentally approximate methods such as image and asymptotic methods due to their geometry-specific applicability and lack of rigorous error control [CITATION].', '1311.0582-2-4-3': 'As a result, to reliably ensure accurate field results for arbitrary environmental medium composition/source-observer geometry combinations, we choose a direct numerical integration method.', '1311.0582-2-5-0': 'In this vein, one option involves pairing standard integration methods with (real-axis path based) algebraic convergence acceleration techniques such as the standard MWA which, based on published numerical results, successfully imparts algebraic convergence acceleration even when [MATH] [CITATION].', '1311.0582-2-5-1': 'However, it is desirable to (1) guarantee absolute, exponential convergence in the classical/Riemann sense for any [MATH] separation geometry (in contrast to only guaranteeing algebraic convergence in the Abel sense when [MATH] [CITATION]) and (2) endow error control to the evanescent-zone field contribution associated with the tail integral, whose relative importance (compared to the radiation-zone contribution) to the field solution grows as [MATH] decreases, to ensure that both the radiation-zone and evanescent-zone contributions are accurately evaluated.', '1311.0582-2-5-2': 'To this end, we propose a novel numerical integration method, representing a complex-plane generalization of a specific member of the "scalar Levin-type sequence transform" (SLST) family [CITATION] (i.e. the MWA), that:', '1311.0582-2-6-0': 'bends the "extrapolation region"/tail [CITATION] integration path sections to guarantee absolute, exponential convergence of integrals like [REF]-[REF], imparts added, robust algebraic convergence acceleration to the tail integrals, which compounds with the exponential convergence acceleration to effect absolute, exponential-cum-algebraic convergence, via use of a linear path bend combined with our novel, complex-plane generalization of the MWA [CITATION], adjusts the detour bend angles to account for the presence of branch points, branch cuts, and poles (summarily referred to here as "critical points"), and addresses the added challenges associated with evaluating two-dimensional integral transforms arising as solutions to the wave equation in planar-stratified environments lacking azimuthal symmetry.', '1311.0582-2-7-0': 'We note that other path deformation techniques, such as the Steepest Descent Path (SDP) and one comprising the enclosure of the first/fourth quadrants of the [MATH] plane involving an imaginary-axis integration, have been investigated and used [CITATION].', '1311.0582-2-7-1': 'However, we seek a robust integration method, valid for all [MATH] geometries, that obviates having to separately account for discrete poles while possessing applicability to multi-layered environments containing media with arbitrary anisotropy and loss.', '1311.0582-2-7-2': 'Thus while our method may result in longer solution times versus above-mentioned methods, it touts general applicability and minimal necessary book-keeping as its defining virtues.', '1311.0582-2-8-0': 'Furthermore, a robust detour path within the pre-extrapolation region [CITATION] maintaining a near-constant separation between the path and critical points near/on the real axis would be preferred over more traditional paths used with the MWA [CITATION].', '1311.0582-2-8-1': "To address this, the paper's second contribution entails a trapezoidal integration path paired with adaptive [MATH] refinement.", '1311.0582-2-9-0': 'In Section [REF] we present and discuss our revision to the radiation-zone integration path.', '1311.0582-2-9-1': 'In Sections [REF] and [REF] we develop the detoured linear integration path and complex-plane generalization to SLST for efficiently evaluating the tail sections of [REF]-[REF], as well as exhibit and compare two possible candidate formulations to implement the resulting modified-MWA.', '1311.0582-2-9-2': 'These developments are formulated in the context of two-dimensional integrals such as [REF] to simultaneously address herein their additional issues versus one-dimensional integrals.', '1311.0582-2-9-3': 'However, the formulation applies equally to one-dimensional F-H transforms like [REF] appearing in field/potential computations within cylindrically- and (azimuthal-symmetric) planar-stratified environments and, after converting the Fourier-Bessel (F-B) transform to a F-H transform [CITATION], to F-B transforms as well.', '1311.0582-2-9-4': 'Section [REF] presents validation results using one of the two new formulations.', '1311.0582-2-9-5': 'In Section [REF] we present a study on the convergence characteristics of our algorithm as concerning the same formulation used to generate the results in Section [REF].', '1311.0582-2-9-6': 'Finally, Section [REF] contains our concluding remarks.', '1311.0582-2-10-0': 'In the ensuing discussion, we assume appropriate transformations to the material tensors and source vector have already been performed to effect a coordinate rotation such that in the resultant (azimuthal-rotated) coordinate frame, within which all integration is performed, one has [MATH] .', '1311.0582-2-10-1': 'Discussed in detail at the end of Section [REF], this is done to guarantee absolute convergence and maximize exponential decay of both the [MATH] and [MATH] integrals.', '1311.0582-2-11-0': '# Pre-Extrapolation Region Path Revision', '1311.0582-2-12-0': 'First we discuss the parameterization and initial sub-division of the [MATH] plane pre-extrapolation region; discussion of the [MATH] plane follows identically due to our assuming [MATH].', '1311.0582-2-12-1': 'Applying a parameterization similar to that in [CITATION], define [MATH] as the points on the Re[MATH]] axis within which one detours, [MATH] as the maximum height of the trapezoid-shaped detour, and [MATH] as the points on the Re[MATH]] axis within which one adaptively integrates (see Figure [REF]).', '1311.0582-2-13-0': 'To compute [MATH], first define [MATH] as the magnitude of the real part of the global "effective" refractive index among all the layers (see [CITATION] on computing [MATH]).', '1311.0582-2-13-1': 'One then computes [MATH] analogously to [CITATION] and sets [MATH], where [MATH] is a user-defined pre-extrapolation region magnification constant.', '1311.0582-2-13-2': 'Next, define [MATH], [MATH], [MATH], and [MATH], where [MATH] and [MATH].', '1311.0582-2-13-3': 'Now compute the following pre-extrapolation region integration path parameters [CITATION]: [EQUATION] where Int[MATH] converts its argument to an integer via fractional truncation.', '1311.0582-2-13-4': 'Now parameterize the pre-extrapolation region integration path, for Re[MATH], as [EQUATION] for the trapezoidal contour (used to integrate up to [MATH]) combined with a real-axis path to integrate within the section [MATH].', '1311.0582-2-13-5': 'An analogous parameterization holds for the Re[MATH] pre-extrapolation region path.', '1311.0582-2-13-6': 'Note that [MATH] is independent of [MATH] and thus can be computed prior to integration, unlike other commonly used detours.', '1311.0582-2-13-7': "This is the trapezoidal path's second benefit in addition to that mentioned in footnote [REF].", '1311.0582-2-14-0': 'Now we splice the regions [MATH] and [MATH] each into [MATH] regions, where [MATH] is calculated as follows.', '1311.0582-2-14-1': 'First define [EQUATION] as the largest magnitude assumed by [MATH] along the trapezoidal path, [MATH] as the user-defined maximum allowed magnitude change of [MATH] between two sampling points, and [MATH] and [MATH] as two user-defined parameters.', '1311.0582-2-14-2': 'Subsequently, define the quantities [EQUATION] which are used to yield [MATH]=Int(1+[MATH]/[MATH]=1,2) with the corresponding final result [MATH].', '1311.0582-2-14-3': 'Note that this method of parameterizing the pre-extrapolation region path is empirical in nature and based on the pessimistic assumption of equidistant sampling [CITATION].', '1311.0582-2-15-0': '# Extrapolation Region Path Revision', '1311.0582-2-16-0': 'The MWA, initially constructed in [CITATION] with further variants developed in [CITATION] and [CITATION], has also demonstrated the ability to accelerate convergence of infinite-range Fourier double-integrals in high-loss, planar-stratified environments containing anisotropic media [CITATION].', '1311.0582-2-16-1': 'However, due to the highly oscillatory behavior of the mixed-domain integrand in integrals such as [REF] arising from the Fourier kernels [MATH] and [MATH] when one has large [MATH] and [MATH] (resp.)', '1311.0582-2-16-2': ', the solution times (in our experience) became inordinately long.', '1311.0582-2-16-3': 'Therefore, it would be desirable to also deform the [MATH] and [MATH] plane "extrapolation" region contours to lend additional exponential decay via these two kernels, thereby dramatically accelerating convergence of the Fourier tail integrals and guaranteeing their absolute convergence even in the "worst-case" scenario [MATH].', '1311.0582-2-16-4': 'A cursory analysis reveals an apparent severe drawback, however: one can no longer employ the MWA, which was derived assuming a real axis integration path [CITATION].', '1311.0582-2-16-5': "However, choosing a linear deformed path retains the MWA's algebraic convergence acceleration, as we show below.", '1311.0582-2-16-6': 'For this analysis, take [MATH]) as the inner (outer) integration variable.', '1311.0582-2-17-0': 'We first exhibit the foundational relations needed to implement the complex-plane extension to a general SLST followed by exhibiting the specific case arising from modeling the tail integral truncation error using the function family stipulated in the "Mosig-Michalski Algorithm" (MMA) [CITATION].', '1311.0582-2-17-1': 'Subsequently, we naively compute the optimal extrapolation region path detour angles without consideration for', '1311.0582-2-18-0': 'the presence of critical points in the [MATH] and [MATH] planes and two-dimensional integrals, associated with wave propagation phenomena, imparting a transitory nature to these critical points in the [MATH] plane (i.e. their locations now depend on the fixed [MATH] value for which the [MATH] integral is evaluated).', '1311.0582-2-19-0': 'To address the first concern, we pessimistically estimate the locations of critical points and reduce the [MATH] plane departure angle of the deformed paths to ensure these features are not crossed.', '1311.0582-2-19-1': 'To address the latter concern, we (1) adjust the departure angles of the [MATH] plane integration path and (2) partition the [MATH] integration domain to ensure that the critical points', '1311.0582-2-20-0': 'possess real parts with magnitude decaying as [MATH] increases, leading to a bounded pre-extrapolation region, and do not extend into the second/fourth quadrants, as this would require a) tracking their locations and b) adjusting the [MATH] integration path, both of which would become functions of [MATH] and lead to a non-robust integration path.', '1311.0582-2-21-0': 'For simplicity, the analysis developing the complex-plane SLST generalization assumes isotropic planar layers.', '1311.0582-2-21-1': 'Fix [MATH] at some (generally complex) value [MATH] and assume [MATH] ; we see then that the inner integral of [REF] writes as [EQUATION] where [MATH] is the up-going mode propagation constant, which for our time convention has positive imaginary part.', '1311.0582-2-21-2': "Assuming [MATH] extrapolation intervals are used [CITATION], the linear path detour used in the [MATH] integration path's extrapolation region is parameterized as [EQUATION] where one defines [MATH], [MATH], [MATH], and [MATH] as real-valued.", '1311.0582-2-21-3': '[MATH] is assumed large enough to ensure that we have sufficiently detoured past any critical points near the real axis [CITATION].', '1311.0582-2-21-4': 'Now recall that plane wave propagation in a homogeneous, unbounded, isotropic medium with wave number [MATH] is governed by the dispersion relation [MATH] [CITATION].', '1311.0582-2-21-5': 'For large [MATH] this relation becomes [EQUATION]', '1311.0582-2-21-6': 'Next, assuming that (asymptotically) [MATH] [CITATION], one can substitute this series expression into the extrapolation region section of [REF] to obtain [EQUATION] where "ext" denotes the [MATH] plane extrapolation region integration path section and the [MATH] comprise a family of functions used to asymptotically model [MATH] and the truncation error (discussed below) [CITATION].', '1311.0582-2-21-7': 'Setting [MATH], [MATH], [MATH], and [MATH], [REF] becomes the union of [REF] and [REF]: [EQUATION] with the respective truncation error integrals of [REF]-[REF] manifesting as [EQUATION]', '1311.0582-2-21-8': 'Performing a change of variables on [REF]-[REF] subsequently yields the following relations [EQUATION]', '1311.0582-2-21-9': 'Next, one evaluates [REF]-[REF] for [MATH] different values of [MATH] (e.g. [MATH]), truncates these [MATH] relations after the [MATH] error series term (i.e. retain the first [MATH] series terms), defines the [MATH]th truncation error series coefficient pair as [MATH] (e.g. see [REF]-[REF] below), and solves the corresponding [MATH]-order system to estimate [MATH] and [MATH].', '1311.0582-2-21-10': 'This procedure represents the complex-plane SLST generalization, applicable to the sequence of [MATH] successive "cumulative tail integral" estimates [CITATION], to accelerate evaluation of [MATH] and [MATH].', '1311.0582-2-22-0': 'Let us now examine the specific case of modeling [MATH] using the family of [MATH]-parameterized functions [MATH] for Re[MATH] and [MATH] for Re[MATH].', '1311.0582-2-22-1': 'Performing Maclaurin expansions of the [MATH] and [MATH], retaining only their respective zeroth-order expansion terms, setting [MATH] and [MATH], and defining [EQUATION] yields a pair of expressions capturing the dominant behavior of the truncation error [MATH]: [EQUATION] which comprises the complex-path generalization, as concerning infinite-range Fourier integrals, to the error expression developed in Section 2 of [CITATION].', '1311.0582-2-22-2': 'The corresponding truncation error expressions associated with F-H transforms like [REF] follow in analogous fashion.', '1311.0582-2-23-0': 'With the foundational expressions available, we now seek to maximize exponential convergence acceleration of [REF] through a suitable choice of the detour departure angles [MATH] and [MATH].', '1311.0582-2-23-1': 'Differentiating the exponent expressions [MATH] and [MATH] with respect to [MATH] and [MATH] (resp.)', '1311.0582-2-23-2': 'and setting the resulting expressions equal to zero leads us to initially (naively) choose [MATH], which (asymptotically) corresponds to the path of most rapid exponential decay or (equivalently) the Constant Phase Path (CPP).', '1311.0582-2-23-3': 'This detour angle choice can be likened to a compromise between the so-called "[MATH]-transmission representation" and "radial transmission representation" [CITATION] of the space-domain field, which were discussed therein in the context of F-H and F-B transforms.', '1311.0582-2-23-4': 'Identical expressions hold for the [MATH] plane detour departure angles [MATH] and [MATH].', '1311.0582-2-23-5': 'Next we consider the detour constraints imposed by these two phenomena.', '1311.0582-2-24-0': 'To this end, first define the branch point as [MATH] and temporarily assume that the [MATH] path was chosen so that [MATH] plane critical points neither manifest in the second/fourth quadrants nor migrate towards Re[[MATH]][MATH] with increasing [MATH].', '1311.0582-2-24-1': 'Recalling the effective refractive indices [MATH]) for layer [MATH] and how we subsequently computed [MATH] [CITATION], set [MATH] equal to either (1) the angle between [MATH] and the [MATH]th layer\'s [MATH]th "effective wave number" [MATH], if [MATH], or (2) [MATH] if [MATH].', '1311.0582-2-24-2': 'Then, [MATH] is updated as [MATH].', '1311.0582-2-24-3': 'No critical points are located in the second/fourth quadrants by assuming (for simplicity) the absence of "double-negative"/meta-material and active/gain media.', '1311.0582-2-24-4': 'Therefore, we do not have to constrain [MATH].', '1311.0582-2-24-5': 'However, these calculations can be readily adjusted to appropriately constrain both [MATH] and [MATH] if such media are present so that our assuming their absence represents a trivial constraint in our methodology.', '1311.0582-2-25-0': 'Now we justify the assumptions above about the [MATH] path, and constrain it to avoid the two issues stated earlier regarding two-dimensional integral transforms arising as the solution to wave-dynamics problems in planar-stratified environments lacking azimuthal symmetry.', '1311.0582-2-25-1': 'To this end, for some arbitrary [MATH] value along the [MATH] plane integration path first expand the branch point [MATH] as [EQUATION] and recall that the radiation branch cut is jointly defined by the conditions Im[MATH] and Re[MATH] [CITATION].', '1311.0582-2-25-2': 'To ensure that critical points in the first (third) quadrant of the [MATH] plane do not migrate towards Re[[MATH]]=[MATH] (Re[[MATH]]=[MATH]) for large [MATH], one must ensure that asymptotically Re[MATH] as [MATH].', '1311.0582-2-25-3': 'Observing the real part of [REF], we see that one must constrain [MATH] and [MATH] to the interval [MATH].', '1311.0582-2-25-4': 'Furthermore, to prevent critical points from migrating into the second/fourth [MATH] plane quadrants, we require that Im[MATH] as [MATH].', '1311.0582-2-25-5': "Observing the imaginary part of [REF] and noting in the region [MATH] that [MATH], a cursory analysis suggests that one cannot safely choose a non-zero value of [MATH] without risking this migration, which would force one to dynamically re-define the [MATH] integral's pre-extrapolation region path, now a function of [MATH], to ensure that one (1) encloses all the quadrant one critical points that migrated into quadrant four while (2) avoiding the encirclement of quadrant three critical points that migrated into quadrant two.", '1311.0582-2-26-0': 'As a result, it appears that one must set the additional, more restrictive constraint [MATH], which in theory may lead to an outer integral exhibiting monotonic-divergent behavior when [MATH] [CITATION].', '1311.0582-2-26-1': 'However, this limitation can be overcome via clever partition of the two-dimensional integration domain; see Figure [REF], which summarizes the proposed partition.', '1311.0582-2-26-2': "Integrating first over Regions I, IIa, and IIb in Figure [REF] followed by integrating in Region III, which encompasses the intersection of the [MATH] and [MATH] plane extrapolation regions, renders the Region III integration's result immune to the migration of critical points into the second/fourth quadrants.", '1311.0582-2-26-3': 'This is because one had already stipulated a domain partitioning and completed integration over Regions I, IIa, and IIb.', '1311.0582-2-27-0': 'We conclude that so long as one conforms to the restrictions [MATH] and [MATH], one can detour in all four spectral "quadrants" [MATH], [MATH], [MATH], and [MATH] (see Figure [REF]) through which the stipulated integration path proceeds.', '1311.0582-2-27-1': "Indeed, our proposed partition of the [MATH] integration domain ensures that for any [MATH] geometry, the double-integral [REF] exhibits absolute-convergent behavior in the classical/Riemann sense and thus, by Fubini's theorem [CITATION], can be evaluated using an iterated integral whose solution is independent of the order of integration.", '1311.0582-2-28-0': 'This discussion also brings to light the benefit of our starting assumption in this analysis, made at the end of Section [REF], that [MATH]: a compromise is reached that ensures exponential-cum-algebraic convergence of both the [MATH] and [MATH] integrals throughout the integration domain.', '1311.0582-2-28-1': 'As an alternative we could have, for example, performed an azimuthal rotation such that [MATH] and [MATH] to maximize convergence acceleration of the [MATH] integral.', '1311.0582-2-28-2': 'However, when [MATH], the [MATH] integral may exhibit monotonic-divergent behavior.', '1311.0582-2-28-3': 'In contrast to oscillatory-divergent behavior [CITATION], the MWA variants (including the generalized version developed herein) cannot curb monotonic-divergent behavior due to the lack of oscillations that must be present for the MWA to "average out" the oscillatory-divergent sequence of cumulative tail integral estimates to obtain a final, convergent result.', '1311.0582-2-29-0': 'Note that for integration in Regions IIa, IIb, and III in Figure [REF], one performs a separate integration and extrapolation of the individual [MATH] and/or [MATH] half-tail integral sections.', '1311.0582-2-29-1': "This is in contrast to the method developed in [CITATION] wherein we folded the half-tail integrals in the [MATH] half-planes to yield a cosine or sine oscillatory kernel, based on assuming spectral symmetry in the environment's plane wave reflection/transmission properties, prior to performing tail integral extrapolation along the positive Re[[MATH]] axis (and similarly for the [MATH] plane).", '1311.0582-2-29-2': 'Our present method, in bending both half-tail [MATH] paths into the upper-half [MATH] plane, forbids such folding due to the now-absent lack of reflection symmetry (about the Im[[MATH]] axis) with respect to the two halves of the extrapolation region path.', '1311.0582-2-29-3': 'The resulting penalty paid in using the complex-plane MWA manifests in having to use twice the number of weight sets versus when one can perform half-tail integral folding followed by cumulative tail integral sequence extrapolation, leading to increased memory requirement and computation time in regards to procuring the MWA weight sets.', '1311.0582-2-29-4': 'As a practical consideration, then, we wish to reduce the number of extrapolation weight sets that must be evaluated.', '1311.0582-2-29-5': 'To this end, we take two steps to halve this number to the six weight sets originally required when performing folding followed by extrapolation.', '1311.0582-2-29-6': 'First, we set [MATH] and [MATH].', '1311.0582-2-29-7': 'Second, we make the approximation (for each field component) that the asymptotic monomial power dependence on both [MATH] and [MATH] [CITATION] equals the average monomial power dependencies on [MATH] and [MATH].', '1311.0582-2-29-8': 'For example, if we determine the integrand for one field component has asymptotic monomial dependencies of [MATH] and [MATH], then we take (as both our [MATH] and [MATH] monomial dependence factors) [MATH], where Nint([MATH]) converts its argument to an integer via rounding.', '1311.0582-2-29-9': 'Furthermore, to ensure stability of the accelerator weight expressions and minimize aliasing effects due to inordinately long extrapolation region intervals, we neglect the integrand oscillation due to [MATH] in the exponential kernels of [REF]-[REF].', '1311.0582-2-29-10': 'This allows one to update the truncation error estimates [MATH] as [MATH] when [MATH] [CITATION].', '1311.0582-2-30-0': 'Beyond the concern of weight computation stability, we also ignore the phase variation associated with [MATH] due to [MATH], in general, being ill-defined.', '1311.0582-2-30-1': 'Indeed, in (1) an anisotropic homogeneous environment or (2) a stratified environment containing isotropic and/or anisotropic media, several phenomena typically obfuscate a univocal, clear definition for the effective longitudinal distance traversed by the characteristic plane wave fields when traveling from [MATH] in layer [MATH] to [MATH] in layer [MATH].', '1311.0582-2-30-2': 'These are (1) multi-bounce within slab layers, (2) the layer and (for anisotropic media) mode dependence of the longitudinal propagation constants, (3) interface reflections in layer [MATH] causing both up-going and down-going modal fields (four total modes in general) to contribute to the observed field at [MATH], and (4) inter-mode coupling at the interfaces.', '1311.0582-2-30-3': 'In fact these four considerations, along with the inherently asymptotic nature of the CPP parameterization and the constraints associated with critical points/two-dimensional integrals addressed above, lead one in practice to not integrate exactly along the CPP.', '1311.0582-2-30-4': 'As a result, one typically finds the integrands of extrapolation-region integrals still exhibiting undesirable residual oscillation due to the complex exponential factors.', '1311.0582-2-30-5': 'While, for [MATH], one still always has a non-zero detour angle for both the [MATH] and [MATH] extrapolation region paths, these practical considerations are what demand the inclusion of an algebraic convergence accelerator like the MWA that exactly acts upon the very types of oscillatory integrals that will typically result.', '1311.0582-2-30-6': 'Therefore, while one does not typically realize the ideal situation of maximized exponential convergence acceleration (the strongest acceleration theoretically available here outside of the SDP), we largely mitigate this pitfall with the robust algebraic acceleration afforded by the MWA, which is agnostic to the environment/source-observer scenario (so long as [MATH]).', '1311.0582-2-30-7': 'Indeed, for an order-[MATH] MWA method used (see below) one realizes a reduction in truncation error between [MATH]) and [MATH]) [CITATION].', '1311.0582-2-31-0': '# Revised Accelerator Weight Computation', '1311.0582-2-32-0': 'The MWA, both in its form as the MMA [CITATION] and its more recent variant the new/"revisited" MWA [CITATION], each offer different, desirable attributes.', '1311.0582-2-32-1': 'The latter version offers a straightforward methodology to unambiguously define arbitrary-order accelerator weight sets and recursively compute higher-order weight sets upon demand.', '1311.0582-2-32-2': 'While we showed previously [CITATION] for the MMA how one can reduce the FLOP count involving the cumulative integrals themselves, the weight computations (1) depended on whether the series of successive extrapolation region sub-interval integrals exhibited oscillating or monotone behavior [CITATION], and (2) the FLOP count to compute the weight sets rapidly grows for successive weight sets, placing a practical limit on obtainable accuracy in the weights (and thus the estimated tail integral) due to roundoff error accumulation in the computed weights.', '1311.0582-2-32-3': 'On the other hand, the computation of the new MWA weight sets is (1) a numerically unstable process rapidly leading to numerical overflow (when using the form exhibited in [CITATION]) and (2) directly linked to procuring the estimated tail integral [CITATION], which is the solution to a highly ill-conditioned linear system (shown in Section [REF]), which previously led us to use the MMA in [CITATION].', '1311.0582-2-32-4': "Nevertheless, both flavors of MWA offer useful mathematical developments for the weights that are couched in the framework of SLST, using a family of functions in a series representation to model the spectral portion of the mixed-domain Green's Function and resulting tail integral truncation error.", '1311.0582-2-32-5': 'For the MMA, the proposed series [CITATION] [EQUATION] is intuitive in its form, and we confirm below the validity (and in fact optimality) of using this approximating series by a straightforward mathematical analysis.', '1311.0582-2-32-6': 'It will be shown that this optimality arises due to the error modeling series (see [REF]-[REF] above and [CITATION]) being entirely consistent with the closed-form expression of the truncation error both in the absence and (via linear superposition) presence of stratified inhomogeneity.', '1311.0582-2-32-7': "The practical consequence of this function family's modeling optimality manifests in minimizing the number of cumulative tail integral estimates required to accurately estimate [MATH] and [MATH], as demonstrated in Section [REF].", '1311.0582-2-33-0': 'In summary, we seek a revised, complex-plane MWA that combines the best aspects of both the MMA [CITATION] and more recent MWA variant [CITATION], in tandem with incorporating the added exponential convergence acceleration afforded by bending the extrapolation region integration path, to effect robust and powerful field solution convergence acceleration.', '1311.0582-2-33-1': 'To this end, in this section we (1) analyze and justify using [REF] as the approximating series for [MATH] and (2) exhibit and compare two proposed formulations for implementing the complex-plane generalization of the MWA, using the new "remainder estimates" [MATH] [CITATION] and the asymptotic series expansion [REF].', '1311.0582-2-34-0': '## The Optimal Error-Modeling Function Family', '1311.0582-2-35-0': 'Herein we examine the inner spectral ([MATH]) integral for some fixed [MATH] in the region Re[MATH].', '1311.0582-2-35-1': 'Furthermore, assume [MATH] has an asymptotic [MATH] monomial dependence of [MATH] [CITATION].', '1311.0582-2-35-2': 'One then has the asymptotic truncation error [EQUATION] which just equals [REF] with the asymptotic series expansion for [MATH] replaced by the dominant series term [MATH].', '1311.0582-2-35-3': 'Next note that [MATH] has a closed-form, convergent solution for [MATH]; setting [MATH] and [MATH], one obtains [EQUATION] and so on for other values of [MATH] .', '1311.0582-2-35-4': 'Examining the asymptotic limit for these three illustrative cases, we find: [EQUATION] where [MATH] represents the set of positive natural numbers.', '1311.0582-2-35-5': 'Similarly, one expects that the reflected/transmitted field terms will also have an asymptotic monomial dependence [CITATION].', '1311.0582-2-35-6': 'For example, consider a two-layer, planar-stratified environment containing isotropic media.', '1311.0582-2-35-7': 'The TE[MATH]/TM[MATH] reflection and transmission coefficients for a plane wave, incident from half-space number one upon half-space number two, write as [CITATION] [EQUATION] where [EQUATION]', '1311.0582-2-35-8': 'Indeed, we see that for large [MATH] the reflection/transmission coefficients have a monomial power dependence [MATH].', '1311.0582-2-36-0': 'Pulling out the dominant monomial term [MATH] in [REF], setting the [MATH]th cumulative remainder estimate as [MATH], and invoking superposition (see footnote [REF]), one now has [MATH], which recovers the dominant contribution to the complex-path extension [REF] of the error expression derived (using [REF]) in [CITATION].', '1311.0582-2-36-1': 'The same procedure shown above, using instead [MATH], [MATH], and [MATH], can be repeated for the Re[MATH] tail integral to obtain a dual set of expressions that recover [REF].', '1311.0582-2-36-2': 'Based on this analysis, when proposing two revised MWA methods we will use the [MATH]-parameterized function families [MATH] and [MATH] to model the tail integral truncation error.', '1311.0582-2-37-0': '## Two Proposed Formulations', '1311.0582-2-38-0': 'For the first formulation we take inspiration from [CITATION].', '1311.0582-2-38-1': 'To this end, for the Re[MATH] tail integral first define [MATH] and [MATH] as two input cumulative tail integral estimates and the under-determined linear system, with respect to which the non-truncated tail integral [MATH] is defined, as [CITATION] [EQUATION] whose equations are subsequently truncated after the [MATH] term [CITATION].', '1311.0582-2-38-2': 'This truncation yields a second-order linear system solved for an improved estimate [MATH] of [MATH] that is free of the [MATH] term in its truncation error series [CITATION]: [EQUATION]', '1311.0582-2-38-3': 'Similarly, using [MATH]) cumulative tail integral estimates [MATH] to eliminate the first [MATH] terms of [MATH], one has for the [MATH]th truncated linear equation ([MATH].', '1311.0582-2-38-4': 'Subsequently, one procures the weights via solving the associated order-([MATH]) linear system for the best [MATH] estimate (i.e. [MATH]), whose solution implicitly contains the expressions for the weights [CITATION].', '1311.0582-2-38-5': 'However, obtaining all desired weight tier sets by directly solving the associated linear systems (1) is very costly and (2) possibly exacerbates weight accuracy degradation due to the poor conditioning of these systems (see below).', '1311.0582-2-38-6': 'Instead, one can obtain closed-form solutions to the weight sets using the methodology outlined in [CITATION] as adapted to our choice of (1) error-modeling functions [MATH] and (2) truncation error estimates [MATH].', '1311.0582-2-38-7': 'In [CITATION] it was assumed that [MATH] asymptotically exhibited a monomial power dependence of the form [MATH] being some constant), with the obvious consequence that [MATH] corresponds to a new function asymptotically behaving as [MATH] .', '1311.0582-2-38-8': 'Rearranging the order-([MATH]) linear system thus yields a similar (but not yet identical) system to equation (22) in [CITATION]: [EQUATION]', '1311.0582-2-38-9': 'Noting that the weight sets in [CITATION] were computed for arbitrary monomial power dependence [MATH], one can cross-multiply the [MATH] factors in the [MATH] across the respective rows of [REF] to obtain an analogous system, where now the [MATH] factors in the modified form of [REF] represent (up to a constant) successive [MATH] derivatives of [MATH] evaluated at [MATH] .', '1311.0582-2-38-10': 'Having now matched our linear system to [CITATION], the [MATH]th weight ([MATH]) for the tier-([MATH]), complex-plane generalization of the new MWA writes as [CITATION] [EQUATION] with the expression for our best tail integral estimate given as [EQUATION]', '1311.0582-2-38-11': 'The expressions for the [MATH] corresponding to Re[MATH] tail integral follows analogously.', '1311.0582-2-38-12': 'Furthermore, one expects that with a different choice of [MATH], this derivation can be repeated to develop complex-plane extensions to other SLST algorithmic members.', '1311.0582-2-39-0': 'From an analytic standpoint, the derivation of the weights for this formulation is complete.', '1311.0582-2-39-1': 'However, despite the analytic form of the new MWA weights shown in [REF] and [CITATION], in a finite-precision, numerical implementation this casting leads to arithmetic overflow.', '1311.0582-2-39-2': 'This drawback, along with the numerically unstable means to recursively update the weights to procure higher-order weight sets, can be easily remedied as follows:', '1311.0582-2-40-0': 'Starting at some tier-[MATH] weight set (e.g. set [MATH]), multiply all the weights by [MATH].', '1311.0582-2-40-1': 'This ensures that the weights remain bounded for all [MATH] and [MATH].', '1311.0582-2-40-2': 'To subsequently obtain a tier-[MATH] weight set from the tier-[MATH] set:', '1311.0582-2-41-0': 'Set [MATH].', '1311.0582-2-41-1': 'For the remaining [MATH] weights, set [MATH]', '1311.0582-2-42-0': 'where [EQUATION]', '1311.0582-2-42-1': 'The second proposed formulation employs the MMA [CITATION], as extended to facilitate adaptive tail integral evaluation [CITATION], in conjunction with our complex extension to the truncation error estimates [MATH].', '1311.0582-2-42-2': 'The formulae to compute arbitrary-order weight sets is given in [CITATION], while the method to recursively find higher-order weight sets is exhibited in [CITATION].', '1311.0582-2-42-3': 'Therefore, the reader is referred to these two references for the elementary details.', '1311.0582-2-43-0': 'Between these two formulations, we opt to implement and show validation results for the second formulation based on the MMA.', '1311.0582-2-43-1': "This is because of the first formulation's poor suitability for an adaptive tail integral evaluation scheme, which in turn is due to increasingly higher-order weight sets being the solutions to increasingly ill-conditioned linear systems.", '1311.0582-2-43-2': "Even though we now have available the analytically recast, numerically stable, closed-form expressions for the first formulation's weights and their update scheme (which obviates any potential algorithmic instability exacerbating computed weight errors), the relative accuracy of the computed weights is still fundamentally capped by the linear system's conditioning.", '1311.0582-2-43-3': 'To illustrate the ill-conditioning of the weight computation, we show in Figure [REF] below, for four different [MATH] geometries, the two-norm condition number [MATH] [CITATION] of [REF] as a function of its rank [MATH] :', '1311.0582-2-44-0': '[MATH]m [MATH]m [MATH]m [MATH]m', '1311.0582-2-45-0': 'To confirm that the system matrix ill-conditioning is not due to the complex-plane generalization of the new MWA, in Figure [REF] we show the two-norm condition number for [MATH] while in Figure [REF] we show, for the same four [MATH] geometries, the conditioning for [MATH] (i.e. as if we performed the standard, real-axis MWA from [CITATION]).', '1311.0582-2-46-0': 'One readily observes from Figure [REF] that accurate weight computation is unrealistic as [MATH] increases; in fact, the situation is downright prohibitive for an adaptive MWA implementation (e.g. [CITATION]).', '1311.0582-2-46-1': 'Even for the best-conditioned geometry (i.e. [MATH]m), one cannot realistically expect even a single digit of precision in the weights for [MATH] equalling or exceeding approximately seven and ten in Figures [REF] and [REF] (resp.)', '1311.0582-2-46-2': ', as can be seen from the intersection of the corresponding curves in Figures [REF]-[REF] with the solid horizontal curve corresponding to Log[MATH].', '1311.0582-2-46-3': 'As a result, we choose the second proposed MWA formulation, based on the MMA [CITATION], for computing validation results in Section [REF].', '1311.0582-2-46-4': 'Based on our previous work using the standard, real-axis MMA [CITATION] for environments containing high loss and conductively-uniaxial layers, one can expect its success in again producing high-precision results.', '1311.0582-2-46-5': 'Indeed, the validation results in Section [REF] speak to this effect.', '1311.0582-2-47-0': '# Results and Discussion', '1311.0582-2-48-0': 'In this section we exhibit validation results in scenarios involving the modeling of induction sondes for geophysical prospection of hydrocarbons (i.e. induction well logging [CITATION]).', '1311.0582-2-48-1': 'Previously, we demonstrated numerous simulated resistivity logs pertaining to environments containing a combination of isotropic and reciprocal, electrically uniaxial media [CITATION] as probed by longitudinally-oriented induction sondes [CITATION].', '1311.0582-2-48-2': 'For those case studies, the adaptive, real-axis MMA was successfully incorporated into our algorithm to yield high-precision results exhibiting excellent agreement with data from previous literature [CITATION].', '1311.0582-2-49-0': 'Herein, we exhibit a case study involving a near-horizontal tool orientation where the tool axis dip angle [MATH], tool axis strike angle [MATH] , and source-observer separation [MATH]m, corresponding to a source-observer depth separation [MATH]mm.', '1311.0582-2-49-1': 'Consequently, this study serves to validate the efficacy of our new algorithm and its ability to impart absolute, exponential-cum-algebraic convergence on Fourier double-integrals like [REF] even for the traditionally prohibitive regime [MATH].', '1311.0582-2-49-2': 'Furthermore, to exemplify the general-purpose nature of our new algorithm in regards to the media present, we generate synthetic resistivity logs for a two-layer, planar-stratified environment containing reciprocal, electrically biaxial media.', '1311.0582-2-49-3': 'In this scenario, wherein all four characteristic plane wave modes in the anisotropic layer containing [MATH] can (in general) contribute to the observed field, the definition of an exact "[MATH]" and thus CPP is ill-defined (see Section [REF]).', '1311.0582-2-49-4': "Therefore, this set of results also justifies our retaining the MWA's robust environment/source-observer geometry convergence acceleration characteristic, yielding an overall robust and rapid electromagnetic field solution method.", '1311.0582-2-50-0': 'Note that save for Figure [REF], there exists strong agreement across the full logging path in each plot.', '1311.0582-2-50-1': 'Even for Figure [REF], with some discrepancy in the upper half-space [MATH], overall there is strong qualitative agreement and (in the bottom half-space) quantitative agreement too.', '1311.0582-2-51-0': '# Convergence Characteristics', '1311.0582-2-52-0': "To characterize our numerical formulation's ability to converge towards the field solution, we present two case studies concerning the [MATH]-directed electric field component [MATH] produced by a [MATH]-directed electric dipole radiating at [MATH]=2MHz in free space.", '1311.0582-2-52-1': 'The first case comprises a benign scenario in which [MATH]m, while the second case represents a very challenging scenario wherein [MATH]m and [MATH]m.', '1311.0582-2-52-2': "The latter scenario's prohibitive challenges, when using a standard numerical integration method, are that the integrand (1) oscillates on the order of [MATH] times more rapidly than the integrand in case one and (2) exhibits absolutely no exponential decay due to the annihilation of the exp([MATH])-type factors.", '1311.0582-2-52-3': 'If we were to use a traditional numerical integration methodology, we emphasize that one would obtain a divergent result.', '1311.0582-2-53-0': 'For each case, we present results related to the Region III field contribution (see Figure [REF]).', '1311.0582-2-53-1': 'Since one cannot obtain a closed-form solution to this field contribution, reference field values from which one measures relative accuracy must be appropriately chosen; their computation details are provided in Figure [REF] below.', '1311.0582-2-53-2': 'As in [CITATION], we assume the integrand is well-behaved in Region III and thus do not perform adaptive interval sub-division.', '1311.0582-2-53-3': 'Instead, we set the [MATH] and [MATH] plane extrapolation region interval lengths as per Section [REF] and examine the accuracy versus (1) the number of extrapolation region intervals employed ([MATH]) and (2) the Legendre-Gauss quadrature order used ([MATH]) to integrate each interval.', '1311.0582-2-54-0': 'Our comments on the relative importance of aliasing and truncation error are analogous to [CITATION]: Up to approximately [MATH] the truncation error dominates the total relative error, while using more than approximately [MATH] or 7 intervals effects no noticeable decrease in the error for a fixed [MATH].', '1311.0582-2-54-1': 'Beyond this point aliasing error dominates the total relative error, which is evidenced by the error decreasing versus increasing [MATH] but remaining flat versus increasing [MATH].', '1311.0582-2-54-2': "However, we notice the following two remarkable characteristics about the algorithm's convergence for case two:", '1311.0582-2-55-0': "The [MATH] curve reaches within 25dB of case one's [MATH] curve despite representing a scenario wherein the field solution would ordinarily have diverged using standard numerical integration techniques.", '1311.0582-2-55-1': 'Despite this case representing a far more prohibitive scenario (if traditionally evaluated) versus case two presented in [CITATION], wherein [MATH]m, at [MATH] the [MATH] curve here levels off at an error approximately 23dB lower than its case two counterpart in [CITATION].', '1311.0582-2-56-0': 'Note that (akin to, and for the same reasons stated in, [CITATION]) relative errors below -150dB were coerced to -150dB.', '1311.0582-2-57-0': 'Since the pre-extrapolation region formulation in this paper is not radically different from that in [CITATION], we expect similar convergence characteristics when using the trapezoidal detour (versus those presented in [CITATION]) and thus omit the Region I convergence study for brevity.', '1311.0582-2-57-1': "Furthermore, the Region IIa/IIb convergence studies are omitted as well since the field convergence results would be affected by the algorithm's handling of both the pre-extrapolation and extrapolation region sections of the [MATH] and [MATH] plane integration paths.", '1311.0582-2-57-2': 'Equivalently, presenting information on the Region I and Region III field convergence characteristics sheds insight into the Region IIa/IIb convergence characteristics.', '1311.0582-2-57-3': 'This is because if the respective algorithms handling the Region I and Region III integrations robustly yield accurate, rapidly convergent results, one can expect similar behavior for the Region IIa/IIb results.', '1311.0582-2-58-0': '# Conclusion', '1311.0582-2-59-0': 'In this work, we have presented a novel integration scheme composed of (1) a complex-plane, adaptive/error-controlling extension to the standard real-axis MMA in conjunction with (2) a more robust pre-extrapolation region integration path to effect fast, absolute, and exponential-cum-algebraic convergence of Fourier- and F-H-type integral transforms such as [REF]-[REF].', '1311.0582-2-59-1': 'Due to combining the detour with the MMA and its robust algebraic convergence acceleration characteristic, this is indeed the case irrespective of the source-observer geometry and loss/anisotropy characteristics of the stratified media present.', '1311.0582-2-59-2': 'Furthermore, this is accomplished without the added complication of having to separately account for slab/interface mode contributions whose poles may be crossed when otherwise deforming to more well-known, rapidly-convergent paths such as the SDP [CITATION], resulting in a numerically robust and easily-implemented integration methodology.', '1311.0582-2-60-0': 'The algorithm\'s ability to accurately simulate the observed fields for classically "worst-case" scenarios [MATH], and that too in complex, planar-stratified environments containing biaxial-conductive media, has been verified through numerous validation checks against [CITATION].', '1311.0582-2-60-1': "Finally, the algorithm's convergence characteristics in the strongly-evanescent spectral zone have been explored, analyzed, and shown to be superior compared to an older methodology exhibited in [CITATION] that was based on an adaptive extension to the real-axis MMA.", '1311.0582-2-61-0': "We conclude that the present algorithm's robustness with respect to source-observer geometries and medium types present, as well as its straight-forward nature and ease of implementation, makes it very useful for the analysis of electromagnetic wave propagation and scattering in multi-layered environments containing media of arbitrary anisotropy and loss."}

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[]

[]

[]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1311.0582

{'1311.0582-3-0-0': 'We propose the complex-plane generalization of a powerful algebraic sequence acceleration algorithm, the Method of Weighted Averages (MWA), to guarantee exponential-cum-algebraic convergence of Fourier and Fourier-Hankel (F-H) integral transforms.', '1311.0582-3-0-1': 'This "complex-plane" MWA, effected via a linear-path detour in the complex plane, results in rapid, absolute convergence of field/potential solutions in multi-layered environments regardless of the source-observer geometry and anisotropy/loss of the media present.', '1311.0582-3-0-2': 'In this work, we first introduce a new integration path used to evaluate the field contribution arising from the radiation spectra.', '1311.0582-3-0-3': 'Subsequently, we (1) exhibit the foundational relations behind the complex-plane extension to a general Levin-type sequence convergence accelerator, (2) specialize this analysis to one member of the Levin transform family (the MWA), (3) address and circumvent restrictions, arising for two-dimensional integrals associated with wave dynamics problems, through minimal complex-plane detour restrictions and a novel partition of the integration domain, (4) develop and compare two formulations based on standard/real-axis MWA variants, and (5) present validation results and convergence characteristics for one of these two formulations.', '1311.0582-3-1-0': '# Introduction', '1311.0582-3-2-0': 'In many application areas concerning time-harmonic electromagnetic (EM) fields, one encounters environments containing media of varying and arbitrary anisotropy whose inhomogeneity can be approximated as multi-layered in nature.', '1311.0582-3-2-1': 'Examples include geophysical prospection [CITATION], plasma physics [CITATION], antenna design [CITATION], optical field control [CITATION], microwave remote sensing [CITATION], ground-penetrating radar [CITATION], and microwave circuits [CITATION], among others.', '1311.0582-3-2-2': 'Such applications regularly encounter integrals of the form [EQUATION] and/or [EQUATION] which express space-domain field/potential functions as Fourier and Fourier-Hankel (F-H) integral transforms (resp.)', '1311.0582-3-3-0': 'In many practical applications, these integrals must often be rapidly evaluated for a wide range of longitudinal and transverse source-observer separation geometries [MATH] (e.g. for potential or field profile reconstruction).', '1311.0582-3-3-1': 'However, when using standard integration paths that run on/close to the real axis such as (1) the classic Sommerfeld Integration Path (SIP) [CITATION] and (2) paths detouring around the branch points, branch cuts, and poles followed by real-axis integration [CITATION], the convergence rate of these integrals is strongly dependent upon the transverse ([MATH]) and longitudinal ([MATH]) separations.', '1311.0582-3-3-2': "[MATH] determines the rapidity of the integrand's oscillation due to the Fourier and/or Hankel kernels in [REF]-[REF], with rising [MATH] leading to an integrand that traditionally requires increasingly finer sampling to limit spatial aliasing and thus leads to undesirably long computation times.", '1311.0582-3-3-3': "Furthermore, the longitudinal separation [MATH] governs the rate at which the evanescent spectrum's field contribution decays with increasing transverse wave number magnitudes, with rising [MATH] effecting more rapid decay (and hence faster convergence) [CITATION].", '1311.0582-3-3-4': 'On the other hand, as [MATH] the convergence rate lessens, with the limiting case [MATH] yielding integrals of the form [EQUATION] and [EQUATION] that lead to divergent results when numerically evaluated, using these standard paths, without convergence acceleration.', '1311.0582-3-4-0': 'See Figure [REF] for typical application scenarios wherein these standard paths either succeed or fail to deliver accurate field results.', '1311.0582-3-4-1': 'Observing Figure [REF], one immediately realizes that devising an evaluation method for these integrals exhibiting robustness with respect to all ranges of [MATH] and medium classes (e.g. isotropic, uniaxial, biaxial) is highly desirable.', '1311.0582-3-4-2': 'This robustness criterion inherently excludes fundamentally approximate methods such as image and asymptotic methods due to their geometry-specific applicability and lack of rigorous error control [CITATION].', '1311.0582-3-4-3': 'As a result, to reliably ensure accurate field results for arbitrary environmental medium composition/source-observer geometry combinations, we choose a direct numerical integration method.', '1311.0582-3-5-0': 'In this vein, one option involves pairing standard integration methods with (real-axis path based) algebraic convergence acceleration techniques such as the standard MWA which, based on published numerical results, successfully imparts algebraic convergence acceleration even when [MATH] [CITATION].', '1311.0582-3-5-1': 'However, it is desirable to (1) guarantee absolute, exponential convergence in the classical/Riemann sense for any [MATH] separation geometry (in contrast to only guaranteeing algebraic convergence in the Abel sense when [MATH] [CITATION]) and (2) endow error control to the evanescent-zone field contribution associated with the tail integral, whose relative importance (compared to the radiation-zone contribution) to the field solution grows as [MATH] decreases, to ensure that both the radiation-zone and evanescent-zone contributions are accurately evaluated.', '1311.0582-3-5-2': 'To this end, we propose a novel numerical integration method, representing a complex-plane generalization of a specific member of the "scalar Levin-type sequence transform" (SLST) family [CITATION] (i.e. the MWA), that:', '1311.0582-3-6-0': 'bends the "extrapolation region"/tail [CITATION] integration path sections to guarantee absolute, exponential convergence of integrals like [REF]-[REF], imparts added, robust algebraic convergence acceleration to the tail integrals, which compounds with the exponential convergence acceleration to effect absolute, exponential-cum-algebraic convergence, via use of a linear path bend combined with our novel, complex-plane generalization of the MWA [CITATION], adjusts the detour bend angles to account for the presence of branch points, branch cuts, and poles (summarily referred to here as "critical points"), and addresses the added challenges associated with evaluating two-dimensional integral transforms arising as solutions to the wave equation in planar-stratified environments lacking azimuthal symmetry.', '1311.0582-3-7-0': 'We note that other path deformation techniques, such as the Steepest Descent Path (SDP) and one comprising the enclosure of the first/fourth quadrants of the [MATH] plane involving an imaginary-axis integration, have been investigated and used [CITATION].', '1311.0582-3-7-1': 'However, we seek a robust integration method, valid for all [MATH] geometries, that obviates having to separately account for discrete poles while possessing applicability to multi-layered environments containing media with arbitrary anisotropy and loss.', '1311.0582-3-7-2': 'Thus while our method may result in longer solution times versus above-mentioned methods, it touts general applicability and minimal necessary book-keeping as its defining virtues.', '1311.0582-3-8-0': 'Furthermore, a robust detour path within the pre-extrapolation region [CITATION] maintaining a near-constant separation between the path and critical points near/on the real axis would be preferred over more traditional paths used with the MWA [CITATION].', '1311.0582-3-8-1': "To address this, the paper's second contribution entails a trapezoidal integration path paired with adaptive [MATH] refinement.", '1311.0582-3-9-0': 'In Section [REF] we present and discuss our revision to the radiation-zone integration path.', '1311.0582-3-9-1': 'In Sections [REF] and [REF] we develop the detoured linear integration path and complex-plane generalization to SLST for efficiently evaluating the tail sections of [REF]-[REF], as well as exhibit and compare two possible candidate formulations to implement the resulting modified-MWA.', '1311.0582-3-9-2': 'These developments are formulated in the context of two-dimensional integrals such as [REF] to simultaneously address herein their additional issues versus one-dimensional integrals.', '1311.0582-3-9-3': 'However, the formulation applies equally to one-dimensional F-H transforms like [REF] appearing in field/potential computations within cylindrically- and (azimuthal-symmetric) planar-stratified environments and, after converting the Fourier-Bessel (F-B) transform to a F-H transform [CITATION], to F-B transforms as well.', '1311.0582-3-9-4': 'Section [REF] presents validation results using one of the two new formulations.', '1311.0582-3-9-5': 'In Section [REF] we present a study on the convergence characteristics of our algorithm as concerning the same formulation used to generate the results in Section [REF].', '1311.0582-3-9-6': 'Finally, Section [REF] contains our concluding remarks.', '1311.0582-3-10-0': 'In the ensuing discussion, we assume appropriate transformations to the material tensors and source vector have already been performed to effect a coordinate rotation such that in the resultant (azimuthal-rotated) coordinate frame, within which all integration is performed, one has [MATH] .', '1311.0582-3-10-1': 'Discussed in detail at the end of Section [REF], this is done to guarantee absolute convergence and maximize exponential decay of both the [MATH] and [MATH] integrals.', '1311.0582-3-11-0': '# Pre-Extrapolation Region Path Revision', '1311.0582-3-12-0': 'First we discuss the parameterization and initial sub-division of the [MATH] plane pre-extrapolation region; discussion of the [MATH] plane follows identically due to our assuming [MATH].', '1311.0582-3-12-1': 'Applying a parameterization similar to that in [CITATION], define [MATH] as the points on the Re[MATH]] axis within which one detours, [MATH] as the maximum height of the trapezoid-shaped detour, and [MATH] as the points on the Re[MATH]] axis within which one adaptively integrates (see Figure [REF]).', '1311.0582-3-13-0': 'To compute [MATH], first define [MATH] as the magnitude of the real part of the global "effective" refractive index among all the layers (see [CITATION] on computing [MATH]).', '1311.0582-3-13-1': 'One then computes [MATH] analogously to [CITATION] and sets [MATH], where [MATH] is a user-defined pre-extrapolation region magnification constant.', '1311.0582-3-13-2': 'Next, define [MATH], [MATH], [MATH], and [MATH], where [MATH] and [MATH].', '1311.0582-3-13-3': 'Now compute the following pre-extrapolation region integration path parameters [CITATION]: [EQUATION] where Int[MATH] converts its argument to an integer via fractional truncation.', '1311.0582-3-13-4': 'Now parameterize the pre-extrapolation region integration path, for Re[MATH], as [EQUATION] for the trapezoidal contour (used to integrate up to [MATH]) combined with a real-axis path to integrate within the section [MATH].', '1311.0582-3-13-5': 'An analogous parameterization holds for the Re[MATH] pre-extrapolation region path.', '1311.0582-3-13-6': 'Note that [MATH] is independent of [MATH] and thus can be computed prior to integration, unlike other commonly used detours.', '1311.0582-3-13-7': "This is the trapezoidal path's second benefit in addition to that mentioned in footnote [REF].", '1311.0582-3-14-0': 'Now we splice the regions [MATH] and [MATH] each into [MATH] regions, where [MATH] is calculated as follows.', '1311.0582-3-14-1': 'First define [EQUATION] as the largest magnitude assumed by [MATH] along the trapezoidal path, [MATH] as the user-defined maximum allowed magnitude change of [MATH] between two sampling points, and [MATH] and [MATH] as two user-defined parameters.', '1311.0582-3-14-2': 'Subsequently, define the quantities [EQUATION] which are used to yield [MATH]=Int(1+[MATH]/[MATH]=1,2) with the corresponding final result [MATH].', '1311.0582-3-14-3': 'Note that this method of parameterizing the pre-extrapolation region path is empirical in nature and based on the pessimistic assumption of equidistant sampling [CITATION].', '1311.0582-3-15-0': '# Extrapolation Region Path Revision', '1311.0582-3-16-0': 'The MWA, initially constructed in [CITATION] with further variants developed in [CITATION] and [CITATION], has also demonstrated the ability to accelerate convergence of infinite-range Fourier double-integrals in high-loss, planar-stratified environments containing anisotropic media [CITATION].', '1311.0582-3-16-1': 'However, due to the highly oscillatory behavior of the mixed-domain integrand in integrals such as [REF] arising from the Fourier kernels [MATH] and [MATH] when one has large [MATH] and [MATH] (resp.)', '1311.0582-3-16-2': ', the solution times (in our experience) became inordinately long.', '1311.0582-3-16-3': 'Therefore, it would be desirable to also deform the [MATH] and [MATH] plane "extrapolation" region contours to lend additional exponential decay via these two kernels, thereby dramatically accelerating convergence of the Fourier tail integrals and guaranteeing their absolute convergence even in the "worst-case" scenario [MATH].', '1311.0582-3-16-4': 'A cursory analysis reveals an apparent severe drawback, however: one can no longer employ the MWA, which was derived assuming a real axis integration path [CITATION].', '1311.0582-3-16-5': "However, choosing a linear deformed path retains the MWA's algebraic convergence acceleration, as we show below.", '1311.0582-3-16-6': 'For this analysis, take [MATH]) as the inner (outer) integration variable.', '1311.0582-3-17-0': 'We first exhibit the foundational relations needed to implement the complex-plane extension to a general SLST followed by exhibiting the specific case arising from modeling the tail integral truncation error using the function family stipulated in the "Mosig-Michalski Algorithm" (MMA) [CITATION].', '1311.0582-3-17-1': 'Subsequently, we naively compute the optimal extrapolation region path detour angles without consideration for', '1311.0582-3-18-0': 'the presence of critical points in the [MATH] and [MATH] planes and two-dimensional integrals, associated with wave propagation phenomena, imparting a transitory nature to these critical points in the [MATH] plane (i.e. their locations now depend on the fixed [MATH] value for which the [MATH] integral is evaluated).', '1311.0582-3-19-0': 'To address the first concern, we pessimistically estimate the locations of critical points and reduce the [MATH] plane departure angle of the deformed paths to ensure these features are not crossed.', '1311.0582-3-19-1': 'To address the latter concern, we (1) adjust the departure angles of the [MATH] plane integration path and (2) partition the [MATH] integration domain to ensure that the critical points', '1311.0582-3-20-0': 'possess real parts with magnitude decaying as [MATH] increases, leading to a bounded pre-extrapolation region, and do not extend into the second/fourth quadrants, as this would require a) tracking their locations and b) adjusting the [MATH] integration path, both of which would become functions of [MATH] and lead to a non-robust integration path.', '1311.0582-3-21-0': 'For simplicity, the analysis developing the complex-plane SLST generalization assumes isotropic planar layers.', '1311.0582-3-21-1': 'Fix [MATH] at some (generally complex) value [MATH] and assume [MATH] ; we see then that the inner integral of [REF] writes as [EQUATION] where [MATH] is the up-going mode propagation constant, which for our time convention has positive imaginary part.', '1311.0582-3-21-2': "Assuming [MATH] extrapolation intervals are used [CITATION], the linear path detour used in the [MATH] integration path's extrapolation region is parameterized as [EQUATION] where one defines [MATH], [MATH], [MATH], and [MATH] as real-valued.", '1311.0582-3-21-3': '[MATH] is assumed large enough to ensure that we have sufficiently detoured past any critical points near the real axis [CITATION].', '1311.0582-3-21-4': 'Now recall that plane wave propagation in a homogeneous, unbounded, isotropic medium with wave number [MATH] is governed by the dispersion relation [MATH] [CITATION].', '1311.0582-3-21-5': 'For large [MATH] this relation becomes [EQUATION]', '1311.0582-3-21-6': 'Next, assuming that (asymptotically) [MATH] [CITATION], one can substitute this series expression into the extrapolation region section of [REF] to obtain [EQUATION] where "ext" denotes the [MATH] plane extrapolation region integration path section and the [MATH] comprise a family of functions used to asymptotically model [MATH] and the truncation error (discussed below) [CITATION].', '1311.0582-3-21-7': 'Setting [MATH], [MATH], [MATH], and [MATH], [REF] becomes the union of [REF] and [REF]: [EQUATION] with the respective truncation error integrals of [REF]-[REF] manifesting as [EQUATION]', '1311.0582-3-21-8': 'Performing a change of variables on [REF]-[REF] subsequently yields the following relations [EQUATION]', '1311.0582-3-21-9': 'Next, one evaluates [REF]-[REF] for [MATH] different values of [MATH] (e.g. [MATH]), truncates these [MATH] relations after the [MATH] error series term (i.e. retain the first [MATH] series terms), defines the [MATH]th truncation error series coefficient pair as [MATH] (e.g. see [REF]-[REF] below), and solves the corresponding [MATH]-order system to estimate [MATH] and [MATH].', '1311.0582-3-21-10': 'This procedure represents the complex-plane SLST generalization, applicable to the sequence of [MATH] successive "cumulative tail integral" estimates [CITATION], to accelerate evaluation of [MATH] and [MATH].', '1311.0582-3-22-0': 'Let us now examine the specific case of modeling [MATH] using the family of [MATH]-parameterized functions [MATH] for Re[MATH] and [MATH] for Re[MATH].', '1311.0582-3-22-1': 'Performing Maclaurin expansions of the [MATH] and [MATH], retaining only their respective zeroth-order expansion terms, setting [MATH] and [MATH], and defining [EQUATION] yields a pair of expressions capturing the dominant behavior of the truncation error [MATH]: [EQUATION] which comprises the complex-path generalization, as concerning infinite-range Fourier integrals, to the error expression developed in Section 2 of [CITATION].', '1311.0582-3-22-2': 'The corresponding truncation error expressions associated with F-H transforms like [REF] follow in analogous fashion.', '1311.0582-3-23-0': 'With the foundational expressions available, we now seek to maximize exponential convergence acceleration of [REF] through a suitable choice of the detour departure angles [MATH] and [MATH].', '1311.0582-3-23-1': 'Differentiating the exponent expressions [MATH] and [MATH] with respect to [MATH] and [MATH] (resp.)', '1311.0582-3-23-2': 'and setting the resulting expressions equal to zero leads us to initially (naively) choose [MATH], which (asymptotically) corresponds to the path of most rapid exponential decay or (equivalently) the Constant Phase Path (CPP).', '1311.0582-3-23-3': 'This detour angle choice can be likened to a compromise between the so-called "[MATH]-transmission representation" and "radial transmission representation" [CITATION] of the space-domain field, which were discussed therein in the context of F-H and F-B transforms.', '1311.0582-3-23-4': 'Identical expressions hold for the [MATH] plane detour departure angles [MATH] and [MATH].', '1311.0582-3-23-5': 'Next we consider the detour constraints imposed by these two phenomena.', '1311.0582-3-24-0': 'To this end, first define the branch point as [MATH] and temporarily assume that the [MATH] path was chosen so that [MATH] plane critical points neither manifest in the second/fourth quadrants nor migrate towards Re[[MATH]][MATH] with increasing [MATH].', '1311.0582-3-24-1': 'Recalling the effective refractive indices [MATH]) for layer [MATH] and how we subsequently computed [MATH] [CITATION], set [MATH] equal to either (1) the angle between [MATH] and the [MATH]th layer\'s [MATH]th "effective wave number" [MATH], if [MATH], or (2) [MATH] if [MATH].', '1311.0582-3-24-2': 'Then, [MATH] is updated as [MATH].', '1311.0582-3-24-3': 'No critical points are located in the second/fourth quadrants by assuming (for simplicity) the absence of "double-negative"/meta-material and active/gain media.', '1311.0582-3-24-4': 'Therefore, we do not have to constrain [MATH].', '1311.0582-3-24-5': 'However, these calculations can be readily adjusted to appropriately constrain both [MATH] and [MATH] if such media are present so that our assuming their absence represents a trivial constraint in our methodology.', '1311.0582-3-25-0': 'Now we justify the assumptions above about the [MATH] path, and constrain it to avoid the two issues stated earlier regarding two-dimensional integral transforms arising as the solution to wave-dynamics problems in planar-stratified environments lacking azimuthal symmetry.', '1311.0582-3-25-1': 'To this end, for some arbitrary [MATH] value along the [MATH] plane integration path first expand the branch point [MATH] as [EQUATION] and recall that the radiation branch cut is jointly defined by the conditions Im[MATH] and Re[MATH] [CITATION].', '1311.0582-3-25-2': 'To ensure that critical points in the first (third) quadrant of the [MATH] plane do not migrate towards Re[[MATH]]=[MATH] (Re[[MATH]]=[MATH]) for large [MATH], one must ensure that asymptotically Re[MATH] as [MATH].', '1311.0582-3-25-3': 'Observing the real part of [REF], we see that one must constrain [MATH] and [MATH] to the interval [MATH].', '1311.0582-3-25-4': 'Furthermore, to prevent critical points from migrating into the second/fourth [MATH] plane quadrants, we require that Im[MATH] as [MATH].', '1311.0582-3-25-5': "Observing the imaginary part of [REF] and noting in the region [MATH] that [MATH], a cursory analysis suggests that one cannot safely choose a non-zero value of [MATH] without risking this migration, which would force one to dynamically re-define the [MATH] integral's pre-extrapolation region path, now a function of [MATH], to ensure that one (1) encloses all the quadrant one critical points that migrated into quadrant four while (2) avoiding the encirclement of quadrant three critical points that migrated into quadrant two.", '1311.0582-3-26-0': 'As a result, it appears that one must set the additional, more restrictive constraint [MATH], which in theory may lead to an outer integral exhibiting monotonic-divergent behavior when [MATH] [CITATION].', '1311.0582-3-26-1': 'However, this limitation can be overcome via clever partition of the two-dimensional integration domain; see Figure [REF], which summarizes the proposed partition.', '1311.0582-3-26-2': "Integrating first over Regions I, IIa, and IIb in Figure [REF] followed by integrating in Region III, which encompasses the intersection of the [MATH] and [MATH] plane extrapolation regions, renders the Region III integration's result immune to the migration of critical points into the second/fourth quadrants.", '1311.0582-3-26-3': 'This is because one had already stipulated a domain partitioning and completed integration over Regions I, IIa, and IIb.', '1311.0582-3-27-0': 'We conclude that so long as one conforms to the restrictions [MATH] and [MATH], one can detour in all four spectral "quadrants" [MATH], [MATH], [MATH], and [MATH] (see Figure [REF]) through which the stipulated integration path proceeds.', '1311.0582-3-27-1': "Indeed, our proposed partition of the [MATH] integration domain ensures that for any [MATH] geometry, the double-integral [REF] exhibits absolute-convergent behavior in the classical/Riemann sense and thus, by Fubini's theorem [CITATION], can be evaluated using an iterated integral whose solution is independent of the order of integration.", '1311.0582-3-28-0': 'This discussion also brings to light the benefit of our starting assumption in this analysis, made at the end of Section [REF], that [MATH]: a compromise is reached that ensures exponential-cum-algebraic convergence of both the [MATH] and [MATH] integrals throughout the integration domain.', '1311.0582-3-28-1': 'As an alternative we could have, for example, performed an azimuthal rotation such that [MATH] and [MATH] to maximize convergence acceleration of the [MATH] integral.', '1311.0582-3-28-2': 'However, when [MATH], the [MATH] integral may exhibit monotonic-divergent behavior.', '1311.0582-3-28-3': 'In contrast to oscillatory-divergent behavior [CITATION], the MWA variants (including the generalized version developed herein) cannot curb monotonic-divergent behavior due to the lack of oscillations that must be present for the MWA to "average out" the oscillatory-divergent sequence of cumulative tail integral estimates to obtain a final, convergent result.', '1311.0582-3-29-0': 'Note that for integration in Regions IIa, IIb, and III in Figure [REF], one performs a separate integration and extrapolation of the individual [MATH] and/or [MATH] half-tail integral sections.', '1311.0582-3-29-1': "This is in contrast to the method developed in [CITATION] wherein we folded the half-tail integrals in the [MATH] half-planes to yield a cosine or sine oscillatory kernel, based on assuming spectral symmetry in the environment's plane wave reflection/transmission properties, prior to performing tail integral extrapolation along the positive Re[[MATH]] axis (and similarly for the [MATH] plane).", '1311.0582-3-29-2': 'Our present method, in bending both half-tail [MATH] paths into the upper-half [MATH] plane, forbids such folding due to the now-absent lack of reflection symmetry (about the Im[[MATH]] axis) with respect to the two halves of the extrapolation region path.', '1311.0582-3-29-3': 'The resulting penalty paid in using the complex-plane MWA manifests in having to use twice the number of weight sets versus when one can perform half-tail integral folding followed by cumulative tail integral sequence extrapolation, leading to increased memory requirement and computation time in regards to procuring the MWA weight sets.', '1311.0582-3-29-4': 'As a practical consideration, then, we wish to reduce the number of extrapolation weight sets that must be evaluated.', '1311.0582-3-29-5': 'To this end, we take two steps to halve this number to the six weight sets originally required when performing folding followed by extrapolation.', '1311.0582-3-29-6': 'First, we set [MATH] and [MATH].', '1311.0582-3-29-7': 'Second, we make the approximation (for each field component) that the asymptotic monomial power dependence on both [MATH] and [MATH] [CITATION] equals the average monomial power dependencies on [MATH] and [MATH].', '1311.0582-3-29-8': 'For example, if we determine the integrand for one field component has asymptotic monomial dependencies of [MATH] and [MATH], then we take (as both our [MATH] and [MATH] monomial dependence factors) [MATH], where Nint([MATH]) converts its argument to an integer via rounding.', '1311.0582-3-29-9': 'Furthermore, to ensure stability of the accelerator weight expressions and minimize aliasing effects due to inordinately long extrapolation region intervals, we neglect the integrand oscillation due to [MATH] in the exponential kernels of [REF]-[REF].', '1311.0582-3-29-10': 'This allows one to update the truncation error estimates [MATH] as [MATH] when [MATH] [CITATION].', '1311.0582-3-30-0': 'Beyond the concern of weight computation stability, we also ignore the phase variation associated with [MATH] due to [MATH], in general, being ill-defined.', '1311.0582-3-30-1': 'Indeed, in (1) an anisotropic homogeneous environment or (2) a stratified environment containing isotropic and/or anisotropic media, several phenomena typically obfuscate a univocal, clear definition for the effective longitudinal distance traversed by the characteristic plane wave fields when traveling from [MATH] in layer [MATH] to [MATH] in layer [MATH].', '1311.0582-3-30-2': 'These are (1) multi-bounce within slab layers, (2) the layer and (for anisotropic media) mode dependence of the longitudinal propagation constants, (3) interface reflections in layer [MATH] causing both up-going and down-going modal fields (four total modes in general) to contribute to the observed field at [MATH], and (4) inter-mode coupling at the interfaces.', '1311.0582-3-30-3': 'In fact these four considerations, along with the inherently asymptotic nature of the CPP parameterization and the constraints associated with critical points/two-dimensional integrals addressed above, lead one in practice to not integrate exactly along the CPP.', '1311.0582-3-30-4': 'As a result, one typically finds the integrands of extrapolation-region integrals still exhibiting undesirable residual oscillation due to the complex exponential factors.', '1311.0582-3-30-5': 'While, for [MATH], one still always has a non-zero detour angle for both the [MATH] and [MATH] extrapolation region paths, these practical considerations are what demand the inclusion of an algebraic convergence accelerator like the MWA that exactly acts upon the very types of oscillatory integrals that will typically result.', '1311.0582-3-30-6': 'Therefore, while one does not typically realize the ideal situation of maximized exponential convergence acceleration (the strongest acceleration theoretically available here outside of the SDP), we largely mitigate this pitfall with the robust algebraic acceleration afforded by the MWA, which is agnostic to the environment/source-observer scenario (so long as [MATH]).', '1311.0582-3-30-7': 'Indeed, for an order-[MATH] MWA method used (see below) one realizes a reduction in truncation error between [MATH]) and [MATH]) [CITATION].', '1311.0582-3-31-0': '# Revised Accelerator Weight Computation', '1311.0582-3-32-0': 'The MWA, both in its form as the MMA [CITATION] and its more recent variant the new/"revisited" MWA [CITATION], each offer different, desirable attributes.', '1311.0582-3-32-1': 'The latter version offers a straightforward methodology to unambiguously define arbitrary-order accelerator weight sets and recursively compute higher-order weight sets upon demand.', '1311.0582-3-32-2': 'While we showed previously [CITATION] for the MMA how one can reduce the FLOP count involving the cumulative integrals themselves, the weight computations (1) depended on whether the series of successive extrapolation region sub-interval integrals exhibited oscillating or monotone behavior [CITATION], and (2) the FLOP count to compute the weight sets rapidly grows for successive weight sets, placing a practical limit on obtainable accuracy in the weights (and thus the estimated tail integral) due to roundoff error accumulation in the computed weights.', '1311.0582-3-32-3': 'On the other hand, the computation of the new MWA weight sets is (1) a numerically unstable process rapidly leading to numerical overflow (when using the form exhibited in [CITATION]) and (2) directly linked to procuring the estimated tail integral [CITATION], which is the solution to a highly ill-conditioned linear system (shown in Section [REF]), which previously led us to use the MMA in [CITATION].', '1311.0582-3-32-4': "Nevertheless, both flavors of MWA offer useful mathematical developments for the weights that are couched in the framework of SLST, using a family of functions in a series representation to model the spectral portion of the mixed-domain Green's Function and resulting tail integral truncation error.", '1311.0582-3-32-5': 'For the MMA, the proposed series [CITATION] [EQUATION] is intuitive in its form, and we confirm below the validity (and in fact optimality) of using this approximating series by a straightforward mathematical analysis.', '1311.0582-3-32-6': 'It will be shown that this optimality arises due to the error modeling series (see [REF]-[REF] above and [CITATION]) being entirely consistent with the closed-form expression of the truncation error both in the absence and (via linear superposition) presence of stratified inhomogeneity.', '1311.0582-3-32-7': "The practical consequence of this function family's modeling optimality manifests in minimizing the number of cumulative tail integral estimates required to accurately estimate [MATH] and [MATH], as demonstrated in Section [REF].", '1311.0582-3-33-0': 'In summary, we seek a revised, complex-plane MWA that combines the best aspects of both the MMA [CITATION] and more recent MWA variant [CITATION], in tandem with incorporating the added exponential convergence acceleration afforded by bending the extrapolation region integration path, to effect robust and powerful field solution convergence acceleration.', '1311.0582-3-33-1': 'To this end, in this section we (1) analyze and justify using [REF] as the approximating series for [MATH] and (2) exhibit and compare two proposed formulations for implementing the complex-plane generalization of the MWA, using the new "remainder estimates" [MATH] [CITATION] and the asymptotic series expansion [REF].', '1311.0582-3-34-0': '## The Optimal Error-Modeling Function Family', '1311.0582-3-35-0': 'Herein we examine the inner spectral ([MATH]) integral for some fixed [MATH] in the region Re[MATH].', '1311.0582-3-35-1': 'Furthermore, assume [MATH] has an asymptotic [MATH] monomial dependence of [MATH] [CITATION].', '1311.0582-3-35-2': 'One then has the asymptotic truncation error [EQUATION] which just equals [REF] with the asymptotic series expansion for [MATH] replaced by the dominant series term [MATH].', '1311.0582-3-35-3': 'Next note that [MATH] has a closed-form, convergent solution for [MATH]; setting [MATH] and [MATH], one obtains [EQUATION] and so on for other values of [MATH] .', '1311.0582-3-35-4': 'Examining the asymptotic limit for these three illustrative cases, we find: [EQUATION] where [MATH] represents the set of positive natural numbers.', '1311.0582-3-35-5': 'Similarly, one expects that the reflected/transmitted field terms will also have an asymptotic monomial dependence [CITATION].', '1311.0582-3-35-6': 'For example, consider a two-layer, planar-stratified environment containing isotropic media.', '1311.0582-3-35-7': 'The TE[MATH]/TM[MATH] reflection and transmission coefficients for a plane wave, incident from half-space number one upon half-space number two, write as [CITATION] [EQUATION] where [EQUATION]', '1311.0582-3-35-8': 'Indeed, we see that for large [MATH] the reflection/transmission coefficients have a monomial power dependence [MATH].', '1311.0582-3-36-0': 'Pulling out the dominant monomial term [MATH] in [REF], setting the [MATH]th cumulative remainder estimate as [MATH], and invoking superposition (see footnote [REF]), one now has [MATH], which recovers the dominant contribution to the complex-path extension [REF] of the error expression derived (using [REF]) in [CITATION].', '1311.0582-3-36-1': 'The same procedure shown above, using instead [MATH], [MATH], and [MATH], can be repeated for the Re[MATH] tail integral to obtain a dual set of expressions that recover [REF].', '1311.0582-3-36-2': 'Based on this analysis, when proposing two revised MWA methods we will use the [MATH]-parameterized function families [MATH] and [MATH] to model the tail integral truncation error.', '1311.0582-3-37-0': '## Two Proposed Formulations', '1311.0582-3-38-0': 'For the first formulation we take inspiration from [CITATION].', '1311.0582-3-38-1': 'To this end, for the Re[MATH] tail integral first define [MATH] and [MATH] as two input cumulative tail integral estimates and the under-determined linear system, with respect to which the non-truncated tail integral [MATH] is defined, as [CITATION] [EQUATION] whose equations are subsequently truncated after the [MATH] term [CITATION].', '1311.0582-3-38-2': 'This truncation yields a second-order linear system solved for an improved estimate [MATH] of [MATH] that is free of the [MATH] term in its truncation error series [CITATION]: [EQUATION]', '1311.0582-3-38-3': 'Similarly, using [MATH]) cumulative tail integral estimates [MATH] to eliminate the first [MATH] terms of [MATH], one has for the [MATH]th truncated linear equation ([MATH].', '1311.0582-3-38-4': 'Subsequently, one procures the weights via solving the associated order-([MATH]) linear system for the best [MATH] estimate (i.e. [MATH]), whose solution implicitly contains the expressions for the weights [CITATION].', '1311.0582-3-38-5': 'However, obtaining all desired weight tier sets by directly solving the associated linear systems (1) is very costly and (2) possibly exacerbates weight accuracy degradation due to the poor conditioning of these systems (see below).', '1311.0582-3-38-6': 'Instead, one can obtain closed-form solutions to the weight sets using the methodology outlined in [CITATION] as adapted to our choice of (1) error-modeling functions [MATH] and (2) truncation error estimates [MATH].', '1311.0582-3-38-7': 'In [CITATION] it was assumed that [MATH] asymptotically exhibited a monomial power dependence of the form [MATH] being some constant), with the obvious consequence that [MATH] corresponds to a new function asymptotically behaving as [MATH] .', '1311.0582-3-38-8': 'Rearranging the order-([MATH]) linear system thus yields a similar (but not yet identical) system to equation (22) in [CITATION]: [EQUATION]', '1311.0582-3-38-9': 'Noting that the weight sets in [CITATION] were computed for arbitrary monomial power dependence [MATH], one can cross-multiply the [MATH] factors in the [MATH] across the respective rows of [REF] to obtain an analogous system, where now the [MATH] factors in the modified form of [REF] represent (up to a constant) successive [MATH] derivatives of [MATH] evaluated at [MATH] .', '1311.0582-3-38-10': 'Having now matched our linear system to [CITATION], the [MATH]th weight ([MATH]) for the tier-([MATH]), complex-plane generalization of the new MWA writes as [CITATION] [EQUATION] with the expression for our best tail integral estimate given as [EQUATION]', '1311.0582-3-38-11': 'The expressions for the [MATH] corresponding to Re[MATH] tail integral follows analogously.', '1311.0582-3-38-12': 'Furthermore, one expects that with a different choice of [MATH], this derivation can be repeated to develop complex-plane extensions to other SLST algorithmic members.', '1311.0582-3-39-0': 'From an analytic standpoint, the derivation of the weights for this formulation is complete.', '1311.0582-3-39-1': 'However, despite the analytic form of the new MWA weights shown in [REF] and [CITATION], in a finite-precision, numerical implementation this casting leads to arithmetic overflow.', '1311.0582-3-39-2': 'This drawback, along with the numerically unstable means to recursively update the weights to procure higher-order weight sets, can be easily remedied as follows:', '1311.0582-3-40-0': 'Starting at some tier-[MATH] weight set (e.g. set [MATH]), multiply all the weights by [MATH].', '1311.0582-3-40-1': 'This ensures that the weights remain bounded for all [MATH] and [MATH].', '1311.0582-3-40-2': 'To subsequently obtain a tier-[MATH] weight set from the tier-[MATH] set:', '1311.0582-3-41-0': 'Set [MATH].', '1311.0582-3-41-1': 'For the remaining [MATH] weights, set [MATH]', '1311.0582-3-42-0': 'where [EQUATION]', '1311.0582-3-42-1': 'The second proposed formulation employs the MMA [CITATION], as extended to facilitate adaptive tail integral evaluation [CITATION], in conjunction with our complex extension to the truncation error estimates [MATH].', '1311.0582-3-42-2': 'The formulae to compute arbitrary-order weight sets is given in [CITATION], while the method to recursively find higher-order weight sets is exhibited in [CITATION].', '1311.0582-3-42-3': 'Therefore, the reader is referred to these two references for the elementary details.', '1311.0582-3-43-0': 'Between these two formulations, we opt to implement and show validation results for the second formulation based on the MMA.', '1311.0582-3-43-1': "This is because of the first formulation's poor suitability for an adaptive tail integral evaluation scheme, which in turn is due to increasingly higher-order weight sets being the solutions to increasingly ill-conditioned linear systems.", '1311.0582-3-43-2': "Even though we now have available the analytically recast, numerically stable, closed-form expressions for the first formulation's weights and their update scheme (which obviates any potential algorithmic instability exacerbating computed weight errors), the relative accuracy of the computed weights is still fundamentally capped by the linear system's conditioning.", '1311.0582-3-43-3': 'To illustrate the ill-conditioning of the weight computation, we show in Figure [REF] below, for four different [MATH] geometries, the two-norm condition number [MATH] [CITATION] of [REF] as a function of its rank [MATH] :', '1311.0582-3-44-0': '[MATH]m [MATH]m [MATH]m [MATH]m', '1311.0582-3-45-0': 'To confirm that the system matrix ill-conditioning is not due to the complex-plane generalization of the new MWA, in Figure [REF] we show the two-norm condition number for [MATH] while in Figure [REF] we show, for the same four [MATH] geometries, the conditioning for [MATH] (i.e. as if we performed the standard, real-axis MWA from [CITATION]).', '1311.0582-3-46-0': 'One readily observes from Figure [REF] that accurate weight computation is unrealistic as [MATH] increases; in fact, the situation is downright prohibitive for an adaptive MWA implementation (e.g. [CITATION]).', '1311.0582-3-46-1': 'Even for the best-conditioned geometry (i.e. [MATH]m), one cannot realistically expect even a single digit of precision in the weights for [MATH] equalling or exceeding approximately seven and ten in Figures [REF] and [REF] (resp.)', '1311.0582-3-46-2': ', as can be seen from the intersection of the corresponding curves in Figures [REF]-[REF] with the solid horizontal curve corresponding to Log[MATH].', '1311.0582-3-46-3': 'As a result, we choose the second proposed MWA formulation, based on the MMA [CITATION], for computing validation results in Section [REF].', '1311.0582-3-46-4': 'Based on our previous work using the standard, real-axis MMA [CITATION] for environments containing high loss and conductively-uniaxial layers, one can expect its success in again producing high-precision results.', '1311.0582-3-46-5': 'Indeed, the validation results in Section [REF] speak to this effect.', '1311.0582-3-47-0': '# Results and Discussion', '1311.0582-3-48-0': 'In this section we exhibit validation results in scenarios involving the modeling of induction sondes for geophysical prospection of hydrocarbons (i.e. induction well logging [CITATION]).', '1311.0582-3-48-1': 'Previously, we demonstrated numerous simulated resistivity logs pertaining to environments containing a combination of isotropic and reciprocal, electrically uniaxial media [CITATION] as probed by longitudinally-oriented induction sondes [CITATION].', '1311.0582-3-48-2': 'For those case studies, the adaptive, real-axis MMA was successfully incorporated into our algorithm to yield high-precision results exhibiting excellent agreement with data from previous literature [CITATION].', '1311.0582-3-49-0': 'Herein, we exhibit a case study involving a near-horizontal tool orientation where the tool axis dip angle [MATH], tool axis strike angle [MATH] , and source-observer separation [MATH]m, corresponding to a source-observer depth separation [MATH]mm.', '1311.0582-3-49-1': 'Consequently, this study serves to validate the efficacy of our new algorithm and its ability to impart absolute, exponential-cum-algebraic convergence on Fourier double-integrals like [REF] even for the traditionally prohibitive regime [MATH].', '1311.0582-3-49-2': 'Furthermore, to exemplify the general-purpose nature of our new algorithm in regards to the media present, we generate synthetic resistivity logs for a two-layer, planar-stratified environment containing reciprocal, electrically biaxial media.', '1311.0582-3-49-3': 'In this scenario, wherein all four characteristic plane wave modes in the anisotropic layer containing [MATH] can (in general) contribute to the observed field, the definition of an exact "[MATH]" and thus CPP is ill-defined (see Section [REF]).', '1311.0582-3-49-4': "Therefore, this set of results also justifies our retaining the MWA's robust environment/source-observer geometry convergence acceleration characteristic, yielding an overall robust and rapid electromagnetic field solution method.", '1311.0582-3-50-0': 'Note that save for Figure [REF], there exists strong agreement across the full logging path in each plot.', '1311.0582-3-50-1': 'Even for Figure [REF], with some discrepancy in the upper half-space [MATH], overall there is strong qualitative agreement and (in the bottom half-space) quantitative agreement too.', '1311.0582-3-51-0': '# Convergence Characteristics', '1311.0582-3-52-0': "To characterize our numerical formulation's ability to converge towards the field solution, we present two case studies concerning the [MATH]-directed electric field component [MATH] produced by a [MATH]-directed electric dipole radiating at [MATH]=2MHz in free space.", '1311.0582-3-52-1': 'The first case comprises a benign scenario in which [MATH]m, while the second case represents a very challenging scenario wherein [MATH]m and [MATH]m.', '1311.0582-3-52-2': "The latter scenario's prohibitive challenges, when using a standard numerical integration method, are that the integrand (1) oscillates on the order of [MATH] times more rapidly than the integrand in case one and (2) exhibits absolutely no exponential decay due to the annihilation of the exp([MATH])-type factors.", '1311.0582-3-52-3': 'If we were to use a traditional numerical integration methodology, we emphasize that one would obtain a divergent result.', '1311.0582-3-53-0': 'For each case, we present results related to the Region III field contribution (see Figure [REF]).', '1311.0582-3-53-1': 'Since one cannot obtain a closed-form solution to this field contribution, reference field values from which one measures relative accuracy must be appropriately chosen; their computation details are provided in Figure [REF] below.', '1311.0582-3-53-2': 'As in [CITATION], we assume the integrand is well-behaved in Region III and thus do not perform adaptive interval sub-division.', '1311.0582-3-53-3': 'Instead, we set the [MATH] and [MATH] plane extrapolation region interval lengths as per Section [REF] and examine the accuracy versus (1) the number of extrapolation region intervals employed ([MATH]) and (2) the Legendre-Gauss quadrature order used ([MATH]) to integrate each interval.', '1311.0582-3-54-0': 'Our comments on the relative importance of aliasing and truncation error are analogous to [CITATION]: Up to approximately [MATH] the truncation error dominates the total relative error, while using more than approximately [MATH] or 7 intervals effects no noticeable decrease in the error for a fixed [MATH].', '1311.0582-3-54-1': 'Beyond this point aliasing error dominates the total relative error, which is evidenced by the error decreasing versus increasing [MATH] but remaining flat versus increasing [MATH].', '1311.0582-3-54-2': "However, we notice the following two remarkable characteristics about the algorithm's convergence for case two:", '1311.0582-3-55-0': "The [MATH] curve reaches within 25dB of case one's [MATH] curve despite representing a scenario wherein the field solution would ordinarily have diverged using standard numerical integration techniques.", '1311.0582-3-55-1': 'Despite this case representing a far more prohibitive scenario (if traditionally evaluated) versus case two presented in [CITATION], wherein [MATH]m, at [MATH] the [MATH] curve here levels off at an error approximately 23dB lower than its case two counterpart in [CITATION].', '1311.0582-3-56-0': 'Note that (akin to, and for the same reasons stated in, [CITATION]) relative errors below -150dB were coerced to -150dB.', '1311.0582-3-57-0': 'Since the pre-extrapolation region formulation in this paper is not radically different from that in [CITATION], we expect similar convergence characteristics when using the trapezoidal detour (versus those presented in [CITATION]) and thus omit the Region I convergence study for brevity.', '1311.0582-3-57-1': "Furthermore, the Region IIa/IIb convergence studies are omitted as well since the field convergence results would be affected by the algorithm's handling of both the pre-extrapolation and extrapolation region sections of the [MATH] and [MATH] plane integration paths.", '1311.0582-3-57-2': 'Equivalently, presenting information on the Region I and Region III field convergence characteristics sheds insight into the Region IIa/IIb convergence characteristics.', '1311.0582-3-57-3': 'This is because if the respective algorithms handling the Region I and Region III integrations robustly yield accurate, rapidly convergent results, one can expect similar behavior for the Region IIa/IIb results.', '1311.0582-3-58-0': '# Conclusion', '1311.0582-3-59-0': 'In this work, we have presented a novel integration scheme composed of (1) a complex-plane, adaptive/error-controlling extension to the standard real-axis MMA in conjunction with (2) a more robust pre-extrapolation region integration path to effect fast, absolute, and exponential-cum-algebraic convergence of Fourier- and F-H-type integral transforms such as [REF]-[REF].', '1311.0582-3-59-1': 'Due to combining the detour with the MMA and its robust algebraic convergence acceleration characteristic, this is indeed the case irrespective of the source-observer geometry and loss/anisotropy characteristics of the stratified media present.', '1311.0582-3-59-2': 'Furthermore, this is accomplished without the added complication of having to separately account for slab/interface mode contributions whose poles may be crossed when otherwise deforming to more well-known, rapidly-convergent paths such as the SDP [CITATION], resulting in a numerically robust and easily-implemented integration methodology.', '1311.0582-3-60-0': 'The algorithm\'s ability to accurately simulate the observed fields for classically "worst-case" scenarios [MATH], and that too in complex, planar-stratified environments containing biaxial-conductive media, has been verified through numerous validation checks against [CITATION].', '1311.0582-3-60-1': "Finally, the algorithm's convergence characteristics in the strongly-evanescent spectral zone have been explored, analyzed, and shown to be superior compared to an older methodology exhibited in [CITATION] that was based on an adaptive extension to the real-axis MMA.", '1311.0582-3-61-0': "We conclude that the present algorithm's robustness with respect to source-observer geometries and medium types present, as well as its straight-forward nature and ease of implementation, makes it very useful for the analysis of electromagnetic wave propagation and scattering in multi-layered environments containing media of arbitrary anisotropy and loss."}

1809.09341

{'1809.09341-1-0-0': 'In this paper, we show that any smooth one-parameter deformations of a strictly convex integrable billiard table [MATH] preserving the integrability near the boundary have to be tangent to a finite dimensional space passing through [MATH].', '1809.09341-1-1-0': 'Dedicated to Yulij Ilyashenko on his 75th birthday', '1809.09341-1-2-0': '# Introduction', '1809.09341-1-3-0': 'A billiard system ([CITATION]) consists by the inertial motions of a point mass inside a fixed domain and the elastic reflections at the boundary.', '1809.09341-1-3-1': 'Let [MATH] be a strictly convex domain in [MATH] with [MATH] boundary [MATH], with [MATH].', '1809.09341-1-3-2': 'The phase space [MATH] of the induced billiard system is a (topological) cylinder formed by the pair [MATH], with [MATH] being a foot point on [MATH] and [MATH] being an inward unit vector.', '1809.09341-1-3-3': 'The billiard ball map [MATH] takes [MATH] to [MATH], where [MATH] is the position on the boundary [MATH], where the trajectory of the point mass starting at [MATH] with velocity [MATH] first hits, and [MATH] is the reflected velocity, according to the standard reflection law of light: the angle of incidence is equal to the angle of reflection.', '1809.09341-1-3-4': 'For a systematic introduction to the billiard dynamics, see e.g. [CITATION].', '1809.09341-1-4-0': 'A smooth convex curve [MATH] is called a caustic, if whenever a trajectory is tangent to it, then they remain tangent after each reflection.', '1809.09341-1-4-1': 'Notice that each convex caustic [MATH] corresponds to an invariant curve of the associated billiard map [MATH] and, hence, has a well-defined rotation number.', '1809.09341-1-4-2': 'If the union of all the caustics form a set with non-empty interior, then we call the billiard table integrable.', '1809.09341-1-4-3': 'The famous Birkhoff conjecture ([CITATION]) claims that every integrable billiard table has a circle or an ellipse as its boundary.', '1809.09341-1-4-4': 'Though much attention it has attracted, this conjecture remains open, and only a few partial progresses were obtained.', '1809.09341-1-4-5': 'As far as our understanding of integrable billiards is concerned, the most important related results are 1) a theorem ([CITATION]) by Bialy which asserts that if the phase space of a billiard map is almost everywhere foliated by non-null homotopic invariant curves, then the corresponding billiard table is a disk; 2) a result ([CITATION]) by Innami, in which he showed that if a strictly convex billiard table admits a sequence of smooth convex caustics with rotation numbers converge to 1/2, then its boundary has to be an ellipse; 3) a result ([CITATION]) by Delshams and Ramirez-Ros in which they study entire perturbations of elliptic billiards and prove that any nontrivial symmetric perturbations of the elliptic billiard is not integrable (see also [CITATION]); 4) and the more recent works ([CITATION]) by Kaloshin et al., justifying a perturbative version of the Birkhoff conjecture for billiard tables with boundary close to ellipses, assuming integrability near the boundary.', '1809.09341-1-5-0': 'In this work, we study deformation of a strictly convex integrable billiard table, which may not be closed to an ellipse.', '1809.09341-1-5-1': 'Let us introduce some notions of this paper.', '1809.09341-1-6-0': '(i) We say that [MATH] is an integrable rational caustic for the billiard system in [MATH] if the corresponding (non-contractible) invariant curve consists of periodic points; in particular, the corresponding rotation number is rational.', '1809.09341-1-7-0': '(ii) Let [MATH].', '1809.09341-1-7-1': 'If the billiard system induced by [MATH] admits integrable rational caustics of rotation number [MATH] for all [MATH], we say that [MATH] is [MATH]-rationally integrable.', '1809.09341-1-8-0': 'Let [MATH] denote the union of all smooth convex caustics of the billiard in [MATH]; if the interior of [MATH] contains caustics of rotation numbers [MATH] for all [MATH], then [MATH] is [MATH]-rationally integrable.', '1809.09341-1-8-1': 'See [CITATION].', '1809.09341-1-9-0': 'The main result of this work is the following:', '1809.09341-1-10-0': 'Let [MATH] be a strictly convex [MATH]-smooth ([MATH]) domain that is [MATH]-rationally integrable.', '1809.09341-1-10-1': 'Then there is [MATH] and, in the space of strictly convex [MATH]-smooth domains, a [MATH]-dimensional space [MATH] passing through [MATH] such that any smooth deformation [MATH] with [MATH] being [MATH]-rationally integrable is tangent to [MATH].', '1809.09341-1-11-0': 'For the proof, we only need the preservation of integrable caustics with rotation numbers [MATH], [MATH].', '1809.09341-1-11-1': 'Our approach here is inspired by those in [CITATION].', '1809.09341-1-11-2': 'Namely, we first derive the necessary annihilation conditions (see Proposition [REF]) for the infinitesimal deformation function (see [REF]) of the integrable deformation, then using these constraints we construct an operator (see [REF]) which is invertible for suitable [MATH] (see Theorem [REF]), and finally from the invertibility of the operator, we conclude that the infinitesimal deformation function must belong to certain linear space of finite dimension.', '1809.09341-1-12-0': 'The most illuminating example of this theorem is the integrable deformations of a domain with an ellipse as its boundary.', '1809.09341-1-12-1': 'The domain enclosed by an ellipse is 2-rationally integrable.', '1809.09341-1-12-2': 'Due to [CITATION], any smooth one parameter family of deformations of this domain, preserving the [MATH]-integrability, are consisted of a family of domains with ellipses as their boundary, belonging to a [MATH]-dimensional space.', '1809.09341-1-13-0': 'Theorem [REF] can be viewed as a finite-dimensional reduction for integrable deformations.', '1809.09341-1-13-1': 'More explicit bounds on the dimension [MATH] is subjected to future development.', '1809.09341-1-14-0': '# Necessary conditions for the preservation of caustics', '1809.09341-1-15-0': 'From now on, we restrict ourself to strictly convex [MATH]-domains with [MATH].', '1809.09341-1-15-1': 'Consider a one-parameter smooth deformation [MATH], [MATH], of the strictly convex domain [MATH], preserving the existence of an integrable caustics with rotation number [MATH], [MATH].', '1809.09341-1-15-2': 'Let [MATH] be a parametrization of [MATH].', '1809.09341-1-15-3': 'As in [CITATION], we define the infinitesimal deformation function [EQUATION] where [MATH] is the usual scalar product in [MATH] and [MATH] is the outgoing unit normal vector to [MATH] at the point [MATH].', '1809.09341-1-15-4': 'Note that [MATH] is continuous in [MATH], and [MATH] for each [MATH].', '1809.09341-1-16-0': 'Let [MATH] be a periodic orbit of the billiard map induced by [MATH], where the starting point is [MATH], that is [MATH], and [MATH] is the angle between the trajectory and the tangent line of [MATH] at [MATH].', '1809.09341-1-16-1': 'For any [MATH]-smooth function [MATH], we define [EQUATION].', '1809.09341-1-16-2': 'The function [MATH] is a constant with respect to [MATH].', '1809.09341-1-16-3': 'In particular, [EQUATION].', '1809.09341-1-17-0': 'Let [MATH] be the perimeter of the periodic orbit [MATH].', '1809.09341-1-17-1': 'From [CITATION] we have that [EQUATION].', '1809.09341-1-17-2': 'Since [MATH] is a one-parameter family preserving the existence of an integrable caustic with rotation number [MATH], we have that [MATH] is a function independent of [MATH].', '1809.09341-1-17-3': 'So does [MATH].', '1809.09341-1-17-4': 'Moreover, [MATH].', '1809.09341-1-18-0': 'The above statement is true for any parametrization of the boundary [MATH].', '1809.09341-1-18-1': 'Now we fix [MATH] and choose the Lazutkin parametrization [MATH] ([CITATION]), which is particularly convenient when dealing with nearly glancing orbits (e.g. periodic orbits with rotation number [MATH], when [MATH] is large).', '1809.09341-1-18-2': 'Let [MATH] be the length parameter of the boundary [MATH] and [MATH] is the radii of curvature of [MATH] at [MATH].', '1809.09341-1-18-3': 'Note that [MATH] is [MATH], since [MATH] is [MATH].', '1809.09341-1-18-4': 'Then the Lazutkin parametrization of the boundary [MATH] is given as follows: [EQUATION].', '1809.09341-1-18-5': 'We introduce the Lazutkin density: [EQUATION] when we denote [MATH], the radius of curvature in the Lazutkin parametrization.', '1809.09341-1-18-6': 'The following statement was obtained in [CITATION].', '1809.09341-1-19-0': 'Assume [MATH].', '1809.09341-1-19-1': 'There exist constant [MATH] and [MATH]-periodic functions [MATH] and [MATH] with [MATH] such that for each [MATH], there exist [MATH]-periodic functions [MATH], [MATH], [MATH], [MATH] such that [EQUATION] and for any periodic orbit [MATH] with rotation number [MATH], we have [EQUATION].', '1809.09341-1-19-2': 'Moreover, if [MATH] denotes the angle of reflection of the trajectory at the [MATH]-th collision, we have [EQUATION] and [EQUATION] where [EQUATION] (i) Notice that [MATH]', '1809.09341-1-20-0': '(ii) The functions [MATH] and [MATH] satisfy [MATH] (see Lemma [REF]).', '1809.09341-1-21-0': '(iii) The function [EQUATION] where [MATH] (see Lemma [REF]).', '1809.09341-1-22-0': 'Now denoting [MATH], we use Lemma [REF] to write the quantity [EQUATION] more explicitly,', '1809.09341-1-23-0': '[EQUATION].', '1809.09341-1-23-1': 'For any [MATH]-function [MATH], we consider two linear operators: [EQUATION] and [EQUATION]', '1809.09341-1-23-2': 'By definition, we have [EQUATION]', '1809.09341-1-23-3': 'For any [MATH], let [EQUATION] be its Fourier series.', '1809.09341-1-23-4': 'For [MATH], define a subspace [MATH] as [EQUATION] with [EQUATION] where [MATH].', '1809.09341-1-23-5': 'The space [MATH] is a (separable) Banach space.', '1809.09341-1-23-6': 'We have [MATH], since [MATH].', '1809.09341-1-23-7': 'So the linear operators [MATH] and [MATH] are well defined on [MATH].', '1809.09341-1-24-0': 'Let us introduce another (separable) Banach space [MATH], [EQUATION] equipped with the norm [EQUATION].', '1809.09341-1-24-1': 'For some [MATH], we define the following linear map, [EQUATION] where [EQUATION].', '1809.09341-1-24-2': 'Let [MATH] be the number such that [EQUATION].', '1809.09341-1-24-3': 'Observe that [MATH].', '1809.09341-1-24-4': 'The our main theorem is implied by the following statement:', '1809.09341-1-25-0': 'For any [MATH] such that [MATH], there exists [MATH] such that for each [MATH], we have that the linear map [EQUATION] is invertible.', '1809.09341-1-26-0': 'Proof of Theorem [REF].', '1809.09341-1-26-1': 'Fix [MATH].', '1809.09341-1-26-2': 'Let [MATH] be one parameter [MATH]-rationally integrable deformations of the domain [MATH], with [MATH].', '1809.09341-1-26-3': 'Then by Theorem [REF] and Proposition [REF] we have that [EQUATION] where [MATH] is a finite dimensional subspace of [MATH].', '1809.09341-1-26-4': 'Since [MATH], and [MATH] is a nowhere vanishing function because of the fact that [MATH] is strictly convex, we have that the infinitesimal deformation function [MATH] belongs to the set [MATH], which is a finite dimensional linear subspace of [MATH].', '1809.09341-1-26-5': 'Hence the assertion of Theorem [REF] follows.', '1809.09341-1-26-6': '[MATH]', '1809.09341-1-27-0': 'In the rest of this paper, we focus on the proof of Theorem [REF].', '1809.09341-1-27-1': 'In Section [REF], we obtain estimates for the linear operators [MATH] and [MATH], applying to the base functions [MATH] and [MATH].', '1809.09341-1-27-2': 'Then in Section [REF] we complete the proof of Theorem [REF].', '1809.09341-1-28-0': '# Estimates for the operators [MATH] and [MATH]', '1809.09341-1-29-0': 'In this section we study the linear operator [MATH] and [MATH], which are defined in [REF] and [REF].', '1809.09341-1-30-0': 'We first consider [MATH] with [MATH].', '1809.09341-1-30-1': 'Then, [EQUATION] where we have used the equality [MATH] and [MATH].', '1809.09341-1-30-2': 'Note that for [MATH], then [MATH] could be replaced by [MATH].', '1809.09341-1-30-3': 'Let us set [EQUATION] and [EQUATION].', '1809.09341-1-30-4': 'Clearly, [EQUATION].', '1809.09341-1-30-5': 'For [MATH], the following simple inequality is enough for our purpose here, [EQUATION] where [MATH] depends only on the [MATH]-norm of [MATH] and [MATH].', '1809.09341-1-31-0': 'Now let us study [MATH].', '1809.09341-1-31-1': 'If [MATH], we have that [EQUATION] where [MATH] depends only on the [MATH]-norm of [MATH].', '1809.09341-1-32-0': 'For [MATH], let us consider the function [EQUATION] and its Fourier series [EQUATION].', '1809.09341-1-32-1': 'Then [EQUATION].', '1809.09341-1-32-2': 'Now we estimate each quantity.', '1809.09341-1-32-3': 'Let us begin with [MATH], [EQUATION]', '1809.09341-1-32-4': 'For the first term, using integration by part, we have [EQUATION].', '1809.09341-1-32-5': 'Similarly, for the second term in [REF], we have [EQUATION].', '1809.09341-1-32-6': 'Therefore, [EQUATION] where the constant [MATH] depends only on [MATH]-norm of [MATH].', '1809.09341-1-33-0': 'For the quantity [MATH],', '1809.09341-1-34-0': '[EQUATION]', '1809.09341-1-34-1': 'Hence there exists [MATH] depending only on [MATH]-norm of [MATH] such that if [MATH] for all [MATH], then [EQUATION] and if [MATH] for some [MATH], then [EQUATION].', '1809.09341-1-34-2': 'So if [MATH] for some [MATH], then we have [EQUATION].', '1809.09341-1-34-3': 'If [MATH] for some [MATH] and [MATH], [EQUATION]', '1809.09341-1-34-4': 'Similarly, we have [EQUATION].', '1809.09341-1-34-5': 'Then, if [MATH] for some [MATH], we have [EQUATION] and if [MATH] for some [MATH] and [MATH], we have [EQUATION].', '1809.09341-1-34-6': 'Therefore, for [MATH], we have that', '1809.09341-1-35-0': 'If [MATH], [MATH].', '1809.09341-1-35-1': 'If [MATH], we have [EQUATION]', '1809.09341-1-35-2': 'If [MATH] for some [MATH], [EQUATION]', '1809.09341-1-35-3': 'If [MATH] for some [MATH] and [MATH], [EQUATION]', '1809.09341-1-35-4': 'Now consider [MATH].', '1809.09341-1-35-5': 'Then [EQUATION]', '1809.09341-1-35-6': 'Then similar to [MATH], we have that', '1809.09341-1-36-0': 'If [MATH], then [EQUATION]', '1809.09341-1-36-1': 'If [MATH], then [EQUATION]', '1809.09341-1-36-2': 'If [MATH], then [EQUATION].', '1809.09341-1-37-0': 'We end this section with a straightforward estimate for the operator [MATH].', '1809.09341-1-38-0': 'There exists a constant [MATH] which depends only on the [MATH]-norm of [MATH], [MATH] and [MATH], such that for [MATH] or [MATH], [EQUATION]', '1809.09341-1-39-0': '# proof of theorem [REF]', '1809.09341-1-40-0': 'In this section we prove Theorem [REF], using the results in Section [REF].', '1809.09341-1-40-1': 'Now we fix [MATH].', '1809.09341-1-41-0': 'The operator [MATH] defined in [REF] can be identified as a pair formed by two infinite matrices [EQUATION] where for [MATH], [EQUATION] and for [MATH], [MATH], [MATH], [EQUATION].', '1809.09341-1-41-1': 'For [EQUATION] we have that [EQUATION] with [EQUATION] and [EQUATION].', '1809.09341-1-41-2': 'Then we have the norm of [MATH] as a linear operator from [MATH] to [MATH] is [EQUATION].', '1809.09341-1-41-3': 'To show that [MATH] is an invertible operator, we consider the operator [MATH], where [EQUATION] if [MATH] is given as in [REF].', '1809.09341-1-41-4': 'Clearly, if we show for [MATH] large enough, [EQUATION] then [MATH] is an invertible linear operator from [MATH] to [MATH].', '1809.09341-1-42-0': 'From Lemmas [REF], [REF] and [REF], we know that for [MATH], [EQUATION] and there exists [MATH] such that if [MATH], [EQUATION] if [MATH], [MATH],', '1809.09341-1-43-0': '[EQUATION] and if [MATH] for some [MATH] and [MATH], [EQUATION].', '1809.09341-1-43-1': 'Now, letting [MATH] be a constant to be determined later and using for [MATH] that the simple estimate [EQUATION] we have that [EQUATION].', '1809.09341-1-43-2': 'We simplified the right hand side of the above inequality and get [EQUATION]', '1809.09341-1-43-3': 'Now we estimate each term on the right hand side.', '1809.09341-1-44-0': 'For the first term, since [MATH], we have that [EQUATION].', '1809.09341-1-44-1': 'For second term, we have [EQUATION]', '1809.09341-1-44-2': 'For the third term, [EQUATION]', '1809.09341-1-44-3': 'For the fourth term, [EQUATION]', '1809.09341-1-44-4': 'We want the exponents on [MATH] of the right hand side of [REF], [REF] and [REF] to be negative, that is, [EQUATION]', '1809.09341-1-44-5': 'Therefore, we need [EQUATION]', '1809.09341-1-44-6': 'Therefore if [EQUATION] then with the choice of [MATH], [EQUATION] all the inequalities in [REF] are satisfied, that is, all the exponents on [MATH] are negative.', '1809.09341-1-44-7': 'Hence, we have that for each [MATH], then there exists [MATH], [EQUATION]', '1809.09341-1-44-8': 'So for [MATH] large enough, [EQUATION] which implies the invertibility of [MATH] as an operator from [MATH] to [MATH].', '1809.09341-1-44-9': 'Thus, we finish the proof of Theorem [REF].', '1809.09341-1-45-0': '# Higher order constraints', '1809.09341-1-46-0': 'In this section we derive additional constraints on the infinitesimal deformation function when the smooth deformation preserves both the integrable caustics with rotation numbers [MATH] and [MATH] is an odd number).', '1809.09341-1-46-1': 'The following lemma is a slight modification of the results in [CITATION].', '1809.09341-1-46-2': 'There exists [MATH] such that for each odd number [MATH], there exist [MATH]-periodic functions [MATH], [MATH], [MATH], [MATH], such that [EQUATION] and for any periodic orbit [MATH] with rotation number [MATH], we have [EQUATION].', '1809.09341-1-46-3': 'Moreover, if [MATH] denotes the angle of reflection of the trajectory at the [MATH]-th collision, we have [EQUATION] and [EQUATION].', '1809.09341-1-46-4': 'Here the function [MATH] and [MATH] are the same as in Lemma [REF], and [EQUATION].', '1809.09341-1-47-0': 'Then from Proposition [REF], we have [EQUATION].', '1809.09341-1-47-1': 'Now we consider the quantity [EQUATION].', '1809.09341-1-47-2': 'Then we have [EQUATION]', '1809.09341-1-47-3': 'Therefore, we have [EQUATION].', '1809.09341-1-47-4': 'Notice that [EQUATION].', '1809.09341-1-47-5': 'Then we obtain a new constraint on the infinitesimal function [EQUATION]', '1809.09341-1-47-6': 'In the same way, for even numbers of the form [MATH], with [MATH] is also an even number, we could study the higher order constraints from the preservation of integrable caustics with rotation numbers [MATH] and [MATH], and obtain similar restriction on the infinitesimal deformation function.', '1809.09341-1-47-7': 'These additional conditions on the infinitesimal deformation function [MATH] might lead to further reduction of dimension.', '1809.09341-1-48-0': '# Relation between [MATH], [MATH], and curvature [MATH]', '1809.09341-1-49-0': 'In this section we sketch the proofs of the explicit formulae for functions [MATH] and [MATH] in term of the curvature of the boundary, as mentioned in Remark [REF].', '1809.09341-1-49-1': 'Though these formulae are not used in an essential way in this work, we believe they are important for future researches.', '1809.09341-1-50-0': 'The following relation between [MATH] and [MATH] holds true: [EQUATION]', '1809.09341-1-50-1': 'Consider the billiard map in the Lazutkin coordinates (see, e.g. [CITATION]): [EQUATION] where [MATH] and [MATH].', '1809.09341-1-51-0': 'The map [MATH] preserves the area form [MATH].', '1809.09341-1-51-1': 'Therefore, for [MATH] [EQUATION].', '1809.09341-1-51-2': 'Compute the determinant of the linearisation: [EQUATION]', '1809.09341-1-51-3': 'Substitute [EQUATION].', '1809.09341-1-51-4': 'This implies that [EQUATION]', '1809.09341-1-51-5': 'From Lemma [REF], we have that [EQUATION].', '1809.09341-1-51-6': 'Then [EQUATION].', '1809.09341-1-51-7': 'For the [MATH]-component, we have [EQUATION] and [EQUATION].', '1809.09341-1-51-8': 'Comparing it with the relation two formulas above and the relation between [MATH] and [MATH] we get [EQUATION].', '1809.09341-1-51-9': 'Substitute into the Lazutkin formula for the [MATH]-component [EQUATION].', '1809.09341-1-51-10': 'This leads to the following equality [EQUATION]', '1809.09341-1-51-11': 'The assertion of the lemma is proved.', '1809.09341-1-52-0': 'The following lemma was communicated to the authors by Qiaoling Wei.', '1809.09341-1-53-0': 'We have the following [EQUATION] where [MATH].', '1809.09341-1-54-0': 'Recall that Lazutkin coordinate is given by [MATH] where [EQUATION] where [MATH].', '1809.09341-1-54-1': 'The billiard ball map [MATH] in Taylor expansion is ([CITATION]) [EQUATION] with [EQUATION].', '1809.09341-1-54-2': 'Through straightforward calculations, we have [EQUATION].', '1809.09341-1-54-3': 'Notice that and [EQUATION] and [EQUATION].', '1809.09341-1-54-4': 'We get that [EQUATION].', '1809.09341-1-54-5': 'On the other hand, [EQUATION].', '1809.09341-1-54-6': 'Combining with [MATH], we can conclude with the assertion of the lemma.'}

{'1809.09341-2-0-0': 'In this paper, we show that any smooth one-parameter deformations of a strictly convex integrable billiard table [MATH] preserving the integrability near the boundary have to be tangent to a finite dimensional space passing through [MATH].', '1809.09341-2-1-0': 'Dedicated to Yulij Ilyashenko on his 75th birthday', '1809.09341-2-2-0': '# Introduction', '1809.09341-2-3-0': 'A billiard system ([CITATION]) consists by the inertial motions of a point mass inside a fixed domain and the elastic reflections at the boundary.', '1809.09341-2-3-1': 'Let [MATH] be a strictly convex domain in [MATH] with [MATH] boundary [MATH], with [MATH].', '1809.09341-2-3-2': 'The phase space [MATH] of the induced billiard system is a (topological) cylinder formed by the pair [MATH], with [MATH] being a foot point on [MATH] and [MATH] being an inward unit vector.', '1809.09341-2-3-3': 'The billiard ball map [MATH] takes [MATH] to [MATH], where [MATH] is the position on the boundary [MATH], where the trajectory of the point mass starting at [MATH] with velocity [MATH] first hits, and [MATH] is the reflected velocity, according to the standard reflection law of light: the angle of incidence is equal to the angle of reflection.', '1809.09341-2-3-4': 'For a systematic introduction to the billiard dynamics, see e.g. [CITATION].', '1809.09341-2-4-0': 'A smooth convex curve [MATH] is called a caustic, if whenever a trajectory is tangent to it, then they remain tangent after each reflection.', '1809.09341-2-4-1': 'Notice that each convex caustic [MATH] corresponds to an invariant curve of the associated billiard map [MATH] and, hence, has a well-defined rotation number.', '1809.09341-2-4-2': 'If the union of all the caustics form a set with non-empty interior, then we call the billiard table integrable.', '1809.09341-2-4-3': 'The famous Birkhoff conjecture ([CITATION]) claims that every integrable billiard table has a circle or an ellipse as its boundary.', '1809.09341-2-4-4': 'Though much attention it has attracted, this conjecture remains open, and only a few partial progresses were obtained.', '1809.09341-2-4-5': 'As far as our understanding of integrable billiards is concerned, the most important related results are 1) a theorem ([CITATION]) by Bialy which asserts that if the phase space of a billiard map is almost everywhere foliated by non-null homotopic invariant curves, then the corresponding billiard table is a disk; 2) a result ([CITATION]) by Innami, in which he showed that if a strictly convex billiard table admits a sequence of smooth convex caustics with rotation numbers converge to 1/2, then its boundary has to be an ellipse; 3) a result ([CITATION]) by Delshams and Ramirez-Ros in which they study entire perturbations of elliptic billiards and prove that any nontrivial symmetric perturbations of the elliptic billiard is not integrable (see also [CITATION]); 4) and the more recent works ([CITATION]) by Kaloshin et al., justifying a perturbative version of the Birkhoff conjecture for billiard tables with boundary close to ellipses, assuming integrability near the boundary.', '1809.09341-2-5-0': 'In this work, we study deformation of a strictly convex integrable billiard table, which may not be closed to an ellipse.', '1809.09341-2-5-1': 'Let us introduce some notions of this paper.', '1809.09341-2-6-0': '(i) We say that [MATH] is an integrable rational caustic for the billiard system in [MATH] if the corresponding (non-contractible) invariant curve consists of periodic points; in particular, the corresponding rotation number is rational.', '1809.09341-2-7-0': '(ii) Let [MATH].', '1809.09341-2-7-1': 'If the billiard system induced by [MATH] admits integrable rational caustics of rotation number [MATH] for all [MATH], we say that [MATH] is [MATH]-rationally integrable.', '1809.09341-2-8-0': 'Let [MATH] denote the union of all smooth convex caustics of the billiard in [MATH]; if the interior of [MATH] contains caustics of rotation numbers [MATH] for all [MATH], then [MATH] is [MATH]-rationally integrable.', '1809.09341-2-8-1': 'See [CITATION].', '1809.09341-2-9-0': 'The main result of this work is the following:', '1809.09341-2-10-0': 'Let [MATH] be a strictly convex [MATH]-smooth ([MATH]) domain that is [MATH]-rationally integrable.', '1809.09341-2-10-1': 'Then there is [MATH] and, in the space of strictly convex [MATH]-smooth domains, a [MATH]-dimensional space [MATH] passing through [MATH] such that any smooth deformation [MATH] with [MATH] being [MATH]-rationally integrable is tangent to [MATH].', '1809.09341-2-11-0': 'For the proof, we only need the preservation of integrable caustics with rotation numbers [MATH], [MATH].', '1809.09341-2-11-1': 'Our approach here is inspired by those in [CITATION].', '1809.09341-2-11-2': 'Namely, we first derive the necessary annihilation conditions (see Proposition [REF]) for the infinitesimal deformation function (see [REF]) of the integrable deformation, then using these constraints we construct an operator (see [REF]) which is invertible for suitable [MATH] (see Theorem [REF]), and finally from the invertibility of the operator, we conclude that the infinitesimal deformation function must belong to certain linear space of finite dimension.', '1809.09341-2-12-0': 'The most illuminating example of this theorem is the integrable deformations of a domain with an ellipse as its boundary.', '1809.09341-2-12-1': 'The domain enclosed by an ellipse is 2-rationally integrable.', '1809.09341-2-12-2': 'Due to [CITATION], any smooth one parameter family of deformations of this domain, preserving the [MATH]-integrability, are consisted of a family of domains with ellipses as their boundary, belonging to a [MATH]-dimensional space.', '1809.09341-2-13-0': 'Theorem [REF] can be viewed as a finite-dimensional reduction for integrable deformations.', '1809.09341-2-13-1': 'More explicit bounds on the dimension [MATH] is subjected to future development.', '1809.09341-2-14-0': '# Necessary conditions for the preservation of caustics', '1809.09341-2-15-0': 'From now on, we restrict ourself to strictly convex [MATH]-domains with [MATH].', '1809.09341-2-15-1': 'Consider a one-parameter smooth deformation [MATH], [MATH], of the strictly convex domain [MATH], preserving the existence of an integrable caustic with rotation number [MATH], [MATH].', '1809.09341-2-15-2': 'Let [MATH] be a parametrization of [MATH].', '1809.09341-2-15-3': 'As in [CITATION], we define the infinitesimal deformation function [EQUATION] where [MATH] is the usual scalar product in [MATH] and [MATH] is the outgoing unit normal vector to [MATH] at the point [MATH].', '1809.09341-2-15-4': 'Note that [MATH] is continuous in [MATH], and [MATH] for each [MATH].', '1809.09341-2-16-0': 'Let [MATH] be a periodic orbit of the billiard map induced by [MATH], where the starting point is [MATH], that is [MATH], and [MATH] is the angle between the trajectory and the tangent line of [MATH] at [MATH].', '1809.09341-2-16-1': 'For any [MATH]-smooth function [MATH], we define [EQUATION].', '1809.09341-2-16-2': 'The function [MATH] is a constant with respect to [MATH].', '1809.09341-2-16-3': 'In particular, [EQUATION].', '1809.09341-2-17-0': 'Let [MATH] be the perimeter of the periodic orbit [MATH].', '1809.09341-2-17-1': 'From [CITATION] we have that [EQUATION].', '1809.09341-2-17-2': 'Since [MATH] is a one-parameter family preserving the existence of an integrable caustic with rotation number [MATH], we have that [MATH] is a function independent of [MATH].', '1809.09341-2-17-3': 'So does [MATH].', '1809.09341-2-17-4': 'Moreover, [MATH].', '1809.09341-2-18-0': 'The above statement is true for any parametrization of the boundary [MATH].', '1809.09341-2-18-1': 'Now we fix [MATH] and choose the Lazutkin parametrization [MATH] ([CITATION]), which is particularly convenient when dealing with nearly glancing orbits (e.g. periodic orbits with rotation number [MATH], when [MATH] is large).', '1809.09341-2-18-2': 'Let [MATH] be the length parameter of the boundary [MATH] and [MATH] is the radii of curvature of [MATH] at [MATH].', '1809.09341-2-18-3': 'Note that [MATH] is [MATH], since [MATH] is [MATH].', '1809.09341-2-18-4': 'Then the Lazutkin parametrization of the boundary [MATH] is given as follows: [EQUATION].', '1809.09341-2-18-5': 'We introduce the Lazutkin density: [EQUATION] when we denote [MATH], the radius of curvature in the Lazutkin parametrization.', '1809.09341-2-18-6': 'The following statement was obtained in [CITATION].', '1809.09341-2-19-0': 'Assume [MATH].', '1809.09341-2-19-1': 'There exist constant [MATH] and [MATH]-periodic functions [MATH] and [MATH] such that for each [MATH], there exist [MATH]-periodic functions [MATH], [MATH], [MATH], [MATH] such that [EQUATION] and for any periodic orbit [MATH] with rotation number [MATH], we have [EQUATION].', '1809.09341-2-19-2': 'Moreover, if [MATH] denotes the angle of reflection of the trajectory at the [MATH]-th collision, we have [EQUATION] and [EQUATION] where [EQUATION] (i) Notice that [MATH]', '1809.09341-2-20-0': '(ii) The functions [MATH] and [MATH] satisfy [EQUATION].', '1809.09341-2-20-1': 'See Lemma [REF].', '1809.09341-2-21-0': 'Now denoting [MATH], we use Lemma [REF] to write the quantity [EQUATION] more explicitly,', '1809.09341-2-22-0': '[EQUATION].', '1809.09341-2-22-1': 'For any [MATH]-function [MATH], we consider two linear operators: [EQUATION] and [EQUATION]', '1809.09341-2-22-2': 'By definition, we have [EQUATION]', '1809.09341-2-22-3': 'For any [MATH], let [EQUATION] be its Fourier series.', '1809.09341-2-22-4': 'For [MATH], define a subspace [MATH] as [EQUATION] with [EQUATION] where [MATH].', '1809.09341-2-22-5': 'The space [MATH] is a (separable) Banach space.', '1809.09341-2-22-6': 'We have [MATH], since [MATH].', '1809.09341-2-22-7': 'So the linear operators [MATH] and [MATH] are well defined on [MATH].', '1809.09341-2-23-0': 'Let us introduce another (separable) Banach space [MATH], [EQUATION] equipped with the norm [EQUATION].', '1809.09341-2-23-1': 'For some [MATH], we define the following linear map, [EQUATION] where [EQUATION].', '1809.09341-2-23-2': 'Let [MATH] be the number such that [EQUATION].', '1809.09341-2-23-3': 'Observe that [MATH].', '1809.09341-2-23-4': 'The our main theorem is implied by the following statement:', '1809.09341-2-24-0': 'For any [MATH] such that [MATH], there exists [MATH] such that for each [MATH], we have that the linear map [EQUATION] is invertible.', '1809.09341-2-25-0': 'Proof of Theorem [REF].', '1809.09341-2-25-1': 'Fix [MATH].', '1809.09341-2-25-2': 'Let [MATH] be one parameter [MATH]-rationally integrable deformations of the domain [MATH], with [MATH].', '1809.09341-2-25-3': 'Then by Theorem [REF] and Proposition [REF] we have that [EQUATION] where [MATH] is a finite dimensional subspace of [MATH].', '1809.09341-2-25-4': 'Since [MATH], and [MATH] is a nowhere vanishing function because of the fact that [MATH] is strictly convex, we have that the infinitesimal deformation function [MATH] belongs to the set [MATH], which is a finite dimensional linear subspace of [MATH].', '1809.09341-2-25-5': 'Hence the assertion of Theorem [REF] follows.', '1809.09341-2-25-6': '[MATH]', '1809.09341-2-26-0': 'In the rest of this paper, we focus on the proof of Theorem [REF].', '1809.09341-2-26-1': 'In Section [REF], we obtain estimates for the linear operators [MATH] and [MATH], applying to the base functions [MATH] and [MATH].', '1809.09341-2-26-2': 'Then in Section [REF] we complete the proof of Theorem [REF].', '1809.09341-2-27-0': '# Estimates for the operators [MATH] and [MATH]', '1809.09341-2-28-0': 'In this section we study the linear operator [MATH] and [MATH], which are defined in [REF] and [REF].', '1809.09341-2-29-0': 'We first consider [MATH] with [MATH].', '1809.09341-2-29-1': 'Then, [EQUATION] where we have used [MATH].', '1809.09341-2-29-2': 'Note that for [MATH], then [MATH] could be replaced by [MATH].', '1809.09341-2-29-3': 'Let us set [EQUATION] and [EQUATION].', '1809.09341-2-29-4': 'Clearly, [EQUATION].', '1809.09341-2-29-5': 'For [MATH], the following simple inequality is enough for our purpose here, [EQUATION] where [MATH] depends only on the [MATH]-norm of [MATH], [MATH] and [MATH].', '1809.09341-2-30-0': 'Now let us study [MATH].', '1809.09341-2-30-1': 'If [MATH], we have that [EQUATION] where [MATH] depends only on the [MATH]-norm of [MATH].', '1809.09341-2-31-0': 'For [MATH], let us consider the function [EQUATION] and its Fourier series [EQUATION].', '1809.09341-2-31-1': 'Then [EQUATION].', '1809.09341-2-31-2': 'Now we estimate each quantity.', '1809.09341-2-31-3': 'Let us begin with [MATH], [EQUATION]', '1809.09341-2-31-4': 'For the first term, using integration by part, we have [EQUATION].', '1809.09341-2-31-5': 'Similarly, for the second term in [REF], we have [EQUATION].', '1809.09341-2-31-6': 'Therefore, [EQUATION] where the constant [MATH] depends only on [MATH]-norm of [MATH].', '1809.09341-2-32-0': 'For the quantity [MATH],', '1809.09341-2-33-0': '[EQUATION]', '1809.09341-2-33-1': 'Hence there exists [MATH] depending only on [MATH]-norm of [MATH] such that if [MATH] for all [MATH], then [EQUATION] and if [MATH] for some [MATH], then [EQUATION].', '1809.09341-2-33-2': 'So if [MATH] for some [MATH], then we have [EQUATION].', '1809.09341-2-33-3': 'If [MATH] for some [MATH] and [MATH], [EQUATION]', '1809.09341-2-33-4': 'Similarly, we have [EQUATION].', '1809.09341-2-33-5': 'Then, if [MATH] for some [MATH], we have [EQUATION] and if [MATH] for some [MATH] and [MATH], we have [EQUATION].', '1809.09341-2-33-6': 'Therefore, for [MATH], we have that', '1809.09341-2-34-0': 'If [MATH], [MATH].', '1809.09341-2-34-1': 'If [MATH], we have [EQUATION]', '1809.09341-2-34-2': 'If [MATH] for some [MATH], [EQUATION]', '1809.09341-2-34-3': 'If [MATH] for some [MATH] and [MATH], [EQUATION]', '1809.09341-2-34-4': 'Now consider [MATH].', '1809.09341-2-34-5': 'Then [EQUATION]', '1809.09341-2-34-6': 'Then similar to [MATH], we have that', '1809.09341-2-35-0': 'If [MATH], then [EQUATION]', '1809.09341-2-35-1': 'If [MATH], then [EQUATION]', '1809.09341-2-35-2': 'If [MATH], then [EQUATION].', '1809.09341-2-36-0': 'We end this section with a straightforward estimate for the operator [MATH].', '1809.09341-2-37-0': 'There exists a constant [MATH] which depends only on the [MATH]-norm of [MATH], [MATH] and [MATH], such that for [MATH] or [MATH], [EQUATION]', '1809.09341-2-38-0': '# proof of theorem [REF]', '1809.09341-2-39-0': 'In this section we prove Theorem [REF], using the results in Section [REF].', '1809.09341-2-39-1': 'Now we fix [MATH].', '1809.09341-2-40-0': 'The operator [MATH] defined in [REF] can be identified as a pair formed by two infinite matrices [EQUATION] where for [MATH], [EQUATION] and for [MATH], [MATH], [MATH], [EQUATION].', '1809.09341-2-40-1': 'For [EQUATION] we have that [EQUATION] with [EQUATION] and [EQUATION].', '1809.09341-2-40-2': 'Then we have the norm of [MATH] as a linear operator from [MATH] to [MATH] is [EQUATION].', '1809.09341-2-40-3': 'To show that [MATH] is an invertible operator, we consider the operator [MATH], where [EQUATION] if [MATH] is given as in [REF].', '1809.09341-2-40-4': 'Clearly, if we show for [MATH] large enough, [EQUATION] then [MATH] is an invertible linear operator from [MATH] to [MATH].', '1809.09341-2-41-0': 'From Lemmas [REF], [REF] and [REF], we know that for [MATH], [EQUATION] and there exists [MATH] such that if [MATH], [EQUATION] if [MATH], [MATH],', '1809.09341-2-42-0': '[EQUATION] and if [MATH] for some [MATH] and [MATH], [EQUATION].', '1809.09341-2-42-1': 'Now, letting [MATH] be a constant to be determined later and using for [MATH] that the simple estimate [EQUATION] we have that [EQUATION].', '1809.09341-2-42-2': 'We simplified the right hand side of the above inequality and get [EQUATION]', '1809.09341-2-42-3': 'Now we estimate each term on the right hand side.', '1809.09341-2-43-0': 'For the first term, since [MATH], we have that [EQUATION].', '1809.09341-2-43-1': 'For second term, we have [EQUATION]', '1809.09341-2-43-2': 'For the third term, [EQUATION]', '1809.09341-2-43-3': 'For the fourth term, [EQUATION]', '1809.09341-2-43-4': 'We want the exponents on [MATH] of the right hand side of [REF], [REF] and [REF] to be negative, that is, [EQUATION]', '1809.09341-2-43-5': 'Therefore, we need [EQUATION]', '1809.09341-2-43-6': 'Therefore if [EQUATION] then with the choice of [MATH], [EQUATION] all the inequalities in [REF] are satisfied, that is, all the exponents on [MATH] are negative.', '1809.09341-2-43-7': 'Hence, we have that for each [MATH], then there exists [MATH], [EQUATION]', '1809.09341-2-43-8': 'So for [MATH] large enough, [EQUATION] which implies the invertibility of [MATH] as an operator from [MATH] to [MATH].', '1809.09341-2-43-9': 'Thus, we finish the proof of Theorem [REF].', '1809.09341-2-44-0': '# Higher order constraints', '1809.09341-2-45-0': 'In this section we derive additional constraints on the infinitesimal deformation function when the smooth deformation preserves both the integrable caustics with rotation numbers [MATH] and [MATH] is an odd number).', '1809.09341-2-45-1': 'The following lemma is a slight modification of the results in [CITATION].', '1809.09341-2-45-2': 'There exists [MATH] such that for each odd number [MATH], there exist [MATH]-periodic functions [MATH], [MATH], [MATH], [MATH], such that [EQUATION] and for any periodic orbit [MATH] with rotation number [MATH], we have [EQUATION].', '1809.09341-2-45-3': 'Moreover, if [MATH] denotes the angle of reflection of the trajectory at the [MATH]-th collision, we have [EQUATION] and [EQUATION].', '1809.09341-2-45-4': 'Here the function [MATH] and [MATH] are the same as in Lemma [REF], and [EQUATION].', '1809.09341-2-46-0': 'Then from Proposition [REF], we have [EQUATION].', '1809.09341-2-46-1': 'Now we consider the quantity [EQUATION].', '1809.09341-2-46-2': 'Then we have [EQUATION]', '1809.09341-2-46-3': 'Therefore, we have [EQUATION].', '1809.09341-2-46-4': 'Notice that [EQUATION]', '1809.09341-2-46-5': 'Then we obtain a new constraint on the infinitesimal deformation function [EQUATION]', '1809.09341-2-46-6': 'In the same way, for even numbers of the form [MATH], with [MATH] is also an even number, we could study the higher order constraints from the preservation of integrable caustics with rotation numbers [MATH] and [MATH], and obtain similar restriction on the infinitesimal deformation function.', '1809.09341-2-46-7': 'These additional conditions on the infinitesimal deformation function [MATH] might lead to further reduction of dimension.', '1809.09341-2-47-0': '# Relation between [MATH], [MATH], and the radii of the curvature [MATH]', '1809.09341-2-48-0': 'In this section we sketch the proof of the explicit relation for functions [MATH], [MATH] and the radii of the curvature of the boundary, as mentioned in Remark [REF].', '1809.09341-2-48-1': 'Though this formula is not used in this work, we hope it would be useful for future research.', '1809.09341-2-49-0': 'The following relation between [MATH], [MATH] and the radii of the curvature holds true: [EQUATION] where [MATH].', '1809.09341-2-50-0': 'Consider the billiard map in the Lazutkin coordinates (see, e.g. [CITATION]) [EQUATION] where [MATH].', '1809.09341-2-50-1': 'Then the billiard map is written as [EQUATION] where [MATH].', '1809.09341-2-51-0': 'The billiard ball map [MATH] in Taylor expansion is ([CITATION]) [EQUATION] with [EQUATION].', '1809.09341-2-51-1': 'Through straightforward calculations, in the Lazutkin parametrization, with [MATH] and [MATH]we have [EQUATION].', '1809.09341-2-51-2': 'Since [MATH], we have [EQUATION] where [MATH] is read as [MATH].', '1809.09341-2-51-3': 'Therefore we have [EQUATION].', '1809.09341-2-52-0': 'From Lemma [REF], for the period orbit [MATH] with rotation number [MATH], we have that for [MATH], [EQUATION].', '1809.09341-2-52-1': 'Then [EQUATION].', '1809.09341-2-52-2': 'For the [MATH]-component, recalling that [MATH], we have [EQUATION].', '1809.09341-2-52-3': 'Denote [EQUATION].', '1809.09341-2-52-4': 'For the [MATH]-component, we also have [EQUATION].', '1809.09341-2-52-5': 'This leads to the following equality [EQUATION]', '1809.09341-2-52-6': 'That is [EQUATION]'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1809.09341

1501.06848

{'1501.06848-1-0-0': 'This paper presents a nonparametric diffusion modeling approach for forecasting partially observed noisy turbulent modes.', '1501.06848-1-0-1': 'The proposed forecast model uses a basis of smooth functions (constructed with the diffusion maps algorithm) to represent probability densities, so that the forecast model becomes a linear map in this basis.', '1501.06848-1-0-2': 'We estimate this linear map by exploiting a previously established rigorous connection between the discrete time shift map and the semi-group solution associated to the backward Kolmogorov equation.', '1501.06848-1-0-3': 'In order to smooth the noisy data, we apply diffusion maps to a delay embedding of the noisy data, which also helps to account for the interactions between the observed and unobserved modes.', '1501.06848-1-0-4': 'We show that this delay embedding biases the geometry of the data in a way which extracts the most predictable component of the dynamics.', '1501.06848-1-0-5': 'The resulting model approximates the semigroup solutions of the generator of the underlying dynamics in the limit of large data and in the observation noise limit.', '1501.06848-1-0-6': 'We will show numerical examples on a wide-range of well-studied turbulent modes, including the Fourier modes of the energy conserving Truncated Burgers-Hopf (TBH) model, the Lorenz-96 model in weakly chaotic to fully turbulent regimes, and the barotropic modes of a quasi-geostrophic model with baroclinic instabilities.', '1501.06848-1-0-7': 'In these examples, forecasting skills of the nonparametric diffusion model are compared to a wide-range of stochastic parametric modeling approaches, which account for the nonlinear interactions between the observed and unobserved modes with white and colored noises.', '1501.06848-1-1-0': '# Introduction', '1501.06848-1-2-0': 'A long-standing issue in modeling turbulent dynamics is the so-called turbulent closure problem (see e.g.[CITATION]) where the goal is to find a set of effective equations up to a finite resolution which represent the fundamental features of the turbulent dynamics.', '1501.06848-1-2-1': 'In other words, we have fewer equations than unknowns to predict the coarse grained variables of interest.', '1501.06848-1-2-2': 'Common closure approaches typically use physical insights to choose a parametric ansatz to represent the feedback from the unresolved scales (see e.g., [CITATION] for various closure approximations for passive scalar turbulence problems).', '1501.06848-1-2-3': 'Alternatively, many stochastic modeling approaches have been successful in predicting geophysical turbulence [CITATION].', '1501.06848-1-2-4': 'While these parametric modeling approaches were typically derived from the first principles through some asymptotic analysis, they also use some physical insight to choose an appropriate ansatz to represent the unresolved scales.', '1501.06848-1-3-0': 'Despite these successes, the parametric modeling approaches have practical issues due to model error when the necessary physical insights are not known.', '1501.06848-1-3-1': 'If the parametric model (or ansatz) is not chosen appropriately, one can end up with a model with poor predictive skills (or even with solutions which diverge catastrophically) even when the parameters can be obtained by a standard regression fitting procedure [CITATION].', '1501.06848-1-3-2': 'Moreover, even when an appropriate parametric form is chosen, specifying the parameters from noisy observations of the physical variables can be nontrivial since the parameters are typically not directly observed.', '1501.06848-1-3-3': 'Indeed, it was shown that an appropriate parameterization scheme is crucial for accurate filtering and climatology (which represents the long term forecast) even when the parametric forms are appropriately chosen [CITATION].', '1501.06848-1-4-0': 'Recently, a nonparametric modeling approach for forecasting the evolution of the probability density of low-dimensional dynamics was introduced in [CITATION].', '1501.06848-1-4-1': 'The key idea behind this modeling approach is to use a basis of smooth functions (constructed with the diffusion maps algorithm [CITATION]) to represent probability densities, so that the forecast model becomes a linear map in this basis.', '1501.06848-1-4-2': 'Numerically, this linear map is estimated by exploiting a rigorous connection between the discrete time shift map and semi-group solution associated to the backward Kolmogorov equation.', '1501.06848-1-4-3': 'In [CITATION], it was shown that the resulting model estimates the semigroup solutions of the generator of the underlying dynamics in the limit of large data.', '1501.06848-1-4-4': 'In this paper, we test this nonparametric modeling approach as a method of forecasting noisy observed Fourier modes of a selection of well-studied high-dimensional dynamical systems in various turbulent regimes.', '1501.06848-1-4-5': 'The novel aspect of this paper is that we consider building a forecasting model given only finite amount of noisy training data set as in practical situation, in contrast to the work in [CITATION] which assumed the data set is noise-free.', '1501.06848-1-5-0': 'We should note that the method of [CITATION] could be applied to noisy signals, however the nonparametric model would implicitly include the noise in the model which would limit the forecast skill compared to treating the noise as a separate process.', '1501.06848-1-5-1': "Treating the noise as a separate process requires first learning the model from the noisy training data set, and then generating 'clean' initial conditions for forecasting from the noisy observations.", '1501.06848-1-5-2': 'In [CITATION] it was shown that applying diffusion maps to the delay-embedded data reduced the noise in the training data.', '1501.06848-1-5-3': 'Building upon the work in [CITATION], we apply the theory of [CITATION] to show that building the nonparametric model using the delay-embedded data biases the geometry of the data in a way which extracts the most predictable component of the dynamics.', '1501.06848-1-5-4': 'Having built a nonparametric model from the noisy training data, the next step is to determine the initial state for forecasting from a noisy observation.', '1501.06848-1-5-5': 'To find the initial state, we introduce a Bayesian filtering method which iteratively assimilates each observation in order to find the probability density of the current state given all the previous observations.', '1501.06848-1-6-0': 'One interesting question which we address here is whether it is possible to build a skillful nonparametric forecasting model for a turbulent mode given only a small amount of noisy training data, when the true dynamics are solutions of a high-dimensional dynamical systems with chaotic behavior.', '1501.06848-1-6-1': 'This question arises because the nonparametric model has a practical limitation in terms of modeling dynamics with high-dimensional attractors, namely: it will require an immense amount of data to unwind the attractors.', '1501.06848-1-6-2': 'Moreover, even given a sufficiently large data set, the required computational power would be a limiting factor since the diffusion maps algorithm requires storing and computing eigenvectors of an [MATH] matrix, where [MATH] is the number of data points.', '1501.06848-1-6-3': 'Given a small data set, such that we cannot unwind the full attractor of the high-dimensional attractor, we attempt to circumvent the curse-of-dimensionality by decomposing the data into Fourier modes.', '1501.06848-1-6-4': 'Of course, the standard theory of embedology [CITATION] suggests that the delay-embedding of a single Fourier mode would reconstruct the entire high-dimensional attractor, which would again be inaccessible to our nonparametric model due to dimensionality.', '1501.06848-1-6-5': 'However, while the full attractor is reconstructed in a topological sense, the geometry of the reconstructed attractor is dramatically altered.', '1501.06848-1-6-6': 'The nonparametric model of [CITATION] leverages this biased geometry so that each Fourier mode gives a separate component of the dynamics.', '1501.06848-1-7-0': 'The remainder of this paper is organized as follows.', '1501.06848-1-7-1': 'In Section 2, we introduce the problems under consideration and establish the necessary background, including a brief overview of the nonparametric modeling approach introduced in [CITATION] as well as discussing of how the theory of [CITATION] is applied to mitigate the effect of noise on the model.', '1501.06848-1-7-2': 'We conclude Section 2 by introducing the iterative Bayesian filter which we use to generate initial conditions for forecasting with the nonparametric model.', '1501.06848-1-7-3': 'In Section 3, we numerically compare the nonparametric model to a physics constrained nonlinear regression model [CITATION] for predicting an energy-conserving turbulent mode.', '1501.06848-1-7-4': 'In Section 4, we numerically compare the nonparametric model with various parametric models, including the persistence model, autoregressive models of order-1 (MSM [CITATION]) and order-3 [CITATION], and the perfect model, for predicting Fourier modes of Lorenz-96 model in various chaotic regimes.', '1501.06848-1-7-5': 'In Section 5, we numerically compare the nonparametric model with a stochastic model with additive and multiplicative noises (SPEKF model [CITATION]) for predicting the barotropic modes of a geostrophic turbulence.', '1501.06848-1-7-6': 'We close this paper with a short summary in Section 5.', '1501.06848-1-8-0': '# Nonparametric diffusion modeling', '1501.06848-1-9-0': 'Let [MATH] be the solutions of an ergodic system of nonlinear PDEs, [EQUATION] where [MATH] denotes nonlinear differential operators, for smooth initial conditions [MATH] and periodic boundary conditions on a non-dimensionalized periodic domain [MATH].', '1501.06848-1-9-1': 'To simplify the exposition, set [MATH] without loss of generality.', '1501.06848-1-9-2': 'Here, the solutions of [REF] can be described by the infinite Fourier series, [EQUATION] where the Fourier modes [MATH] can be utilized in analyzing [REF].', '1501.06848-1-10-0': 'Define [MATH] whose components are the solutions of [REF] at time [MATH] realized at grid point [MATH], such that [MATH].', '1501.06848-1-10-1': 'Our goal is to construct a forecasting model for the discrete Fourier coefficients, [EQUATION] given the corresponding partially observed modes, [MATH] where the [MATH]-th component of [MATH], [EQUATION] is the solution of [REF], corrupted by i.i.d. Gaussian noise, at the grid point [MATH] and time step [MATH].', '1501.06848-1-10-2': 'Notice that [MATH] forms an orthonormal basis of [MATH] with respect to the inner product defined in [REF].', '1501.06848-1-10-3': 'One should also note that [MATH] where [MATH] and [MATH] [CITATION].', '1501.06848-1-11-0': 'Given the noisy time series of a single mode [MATH], our goal is to use this noisy data set to train the nonparametric model and to generate initial conditions for forecasting.', '1501.06848-1-11-1': 'In Section [REF], we provide an overview of the construction of the nonparametric model introduced in [CITATION] for fully observed data without observation noise.', '1501.06848-1-11-2': 'In Section [REF], we show that by applying a delay embedding we can compensate for the partially observed noisy data, [MATH].', '1501.06848-1-11-3': 'Finally, in Section [REF], we introduce a simple Bayesian filtering method for generating initial conditions for forecasting from the noisy observations [MATH].', '1501.06848-1-12-0': '## Nonparametric forecasting model', '1501.06848-1-13-0': 'In this section, we review the nonparametric diffusion forecasting model introduced in [CITATION], assuming the data set consists of full observations of the dynamical system with no observation noise.', '1501.06848-1-13-1': 'Consider a system of SDEs, [EQUATION] where [MATH] is a vector field, [MATH] is a diffusion tensor, and [MATH] is an i.i.d. Wiener process; all defined on a manifold [MATH].', '1501.06848-1-13-2': 'Assume that the dynamical system governed by [REF] is ergodic and that it has a density function [MATH] which converges to the invariant measure [MATH], where [MATH] denotes the Fokker-Planck (or forward) operator and [MATH] denotes an initial density.', '1501.06848-1-14-0': 'Given a time series [MATH], where [MATH] , our goal is to approximate [MATH] without knowing or estimating [MATH] and [MATH].', '1501.06848-1-14-1': 'Instead, we will directly estimate the semigroup solution [MATH] associated to the generator [MATH] of [REF] by projecting the density onto an appropriate basis for [MATH].', '1501.06848-1-14-2': 'In particular, we choose eigenfunctions [MATH] of the generator [MATH] of a stochastically forced gradient flow system, [EQUATION] where the potential function [MATH] is defined by the invariant measure of the full system [REF] so that the invariant measure of [REF] is [MATH].', '1501.06848-1-14-3': 'This choice of basis is motivated by several considerations.', '1501.06848-1-14-4': 'First, we can estimate the eigenfunctions [MATH] by the diffusion maps algorithm for data lying on a compact manifold [CITATION] and on a non-compact manifold [CITATION].', '1501.06848-1-14-5': 'In this paper, we will use the variable bandwidth kernel introduced in [CITATION] to construct these eigenfunctions since the sampling measure of the data set may be arbitrarily small, which would imply that the data does not lie on a compact manifold.', '1501.06848-1-14-6': 'Second, by projecting the density function on this basis of eigenfunctions, we can accurately forecast each projected component by a discrete representation of the semigroup solution [MATH].', '1501.06848-1-14-7': 'We now show how to construct the discrete representation of [MATH] using the "shift operator".', '1501.06848-1-15-0': 'Let [MATH] be the eigenfunctions of the generator [MATH] of the gradient system in [REF]; these eigenfunctions are orthonormal under [MATH] in [MATH] and the integral here (and in all of the inner product defined below) is with respect to the volume form [MATH] which [MATH] inherits from the ambient space.', '1501.06848-1-15-1': 'Note that [MATH] is the generator of gradient system in [REF] and it is the adjoint of the Fokker-Planck operator [MATH] with respect to inner-product [MATH] in [MATH].', '1501.06848-1-15-2': 'One can show that [MATH] are eigenfunctions of [MATH] which are orthonormal with respect to inner-product [MATH] in [MATH].', '1501.06848-1-15-3': 'Given an initial density [MATH], we can write the forecast density [MATH] as, [EQUATION]', '1501.06848-1-15-4': 'Setting [MATH] in [REF] we find, [EQUATION] where we define [MATH].', '1501.06848-1-15-5': 'Substituting [REF] into [REF], we obtain, [EQUATION]', '1501.06848-1-15-6': 'The key idea of the non-parametric forecasting algorithm in [CITATION] is to approximate [MATH] by replacing the semi-group solution [MATH] with the discrete time shift operator [MATH], which is defined as [MATH] for any [MATH].', '1501.06848-1-15-7': 'In [CITATION], it was shown that [MATH] is a stochastic estimate of [MATH] and the error due to the stochastic terms can be minimized by projecting [MATH] on the basis [MATH].', '1501.06848-1-15-8': 'Indeed it was shown that [MATH] is an unbiased estimate of [MATH], meaning that [MATH].', '1501.06848-1-15-9': 'Furthermore, assuming that [MATH] are independent samples of [MATH], one can show that the error of this estimate is of order [MATH], which means that one can apply this approximation for any sampling time [MATH] given a sufficiently large data set [MATH].', '1501.06848-1-15-10': 'By the ergodicity property, the largest eigenvalue of [MATH] should be equal to 1 (with no complex component), numerically however, we sometimes find that [MATH] has some eigenvalues slightly larger than one, due to the finite number of samples in the Monte-Carlo integrals.', '1501.06848-1-15-11': 'If this occurs, we can easily ensure the stability of the diffusion forecast by dividing any eigenvalue with norm greater than 1 so that is has norm equal to one.', '1501.06848-1-15-12': 'However, In our numerical experiments below this issue rarely occurs.', '1501.06848-1-16-0': 'We should also note that if [REF] is a gradient flow as in [REF], then [MATH], where [MATH] are the eigenvalues of [MATH], which can be obtained directly from the diffusion maps algorithm [CITATION].', '1501.06848-1-16-1': 'Moreover, the matrix [MATH] becomes diagonal (due to the orthonormality of [MATH]) so that the diffusion maps algorithm estimates [MATH] directly and the shift operator approximation is not required.', '1501.06848-1-16-2': 'See [CITATION] for various uncertainty quantification applications for this special case.', '1501.06848-1-17-0': 'To conclude, the non-parametric forecasting algorithm (which we refer to as the diffusion forecast) for a given initial density [MATH] is performed as follows:', '1501.06848-1-18-0': 'Learning phase: Given a data set [MATH], apply the diffusion maps algorithm [CITATION] to obtain eigenvectors [MATH], whose [MATH]-th component, [MATH], approximates the eigenfunction [MATH] of the gradient flow [REF] evaluated at the data point [MATH].', '1501.06848-1-18-1': 'We implement the diffusion maps algorithm with a variable bandwidth kernel, see the Appendix for the step-by-step algorithm to obtain these eigenfunctions.', '1501.06848-1-18-2': 'Also, see the supplementary material of [CITATION] for a short overview of [CITATION].', '1501.06848-1-18-3': 'Initial conditions: Represent the initial density [MATH] in this basis as, [EQUATION] which is numerically obtainable from the last inner product as a Monte-Carlo integral since we have [MATH] evaluated on the data set [MATH] which are samples of [MATH].', '1501.06848-1-18-4': 'Forecasting: Apply a Monte-Carlo integral to approximate the shift operator [MATH] in coordinate basis [MATH], [EQUATION]', '1501.06848-1-18-5': 'We then approximate [MATH] with [MATH] such that the diffusion forecast is given by [EQUATION] where in practice, these sums are truncated at a finite number, [MATH], of eigenfunctions.', '1501.06848-1-18-6': 'With this truncation, the coefficient [MATH] is the [MATH]-th component of a matrix-vector multiplication, [MATH], of an [MATH] matrix [MATH] and an [MATH]-dimensional vector [MATH].', '1501.06848-1-19-0': '## Time-lagged embedding', '1501.06848-1-20-0': 'The diffusion forecast algorithm of Section [REF] assumes that the data [MATH] are sampled directly from the full dynamical system [REF].', '1501.06848-1-20-1': 'That is, the full system [MATH] in [REF] is equivalent to the dynamical system for [MATH] and the turbulent dynamics considered here will be very high dimensional.', '1501.06848-1-20-2': 'For such high dimensional problems, exploring the entire attractor would require a prohibitively large data set; larger than will typically be available in applications.', '1501.06848-1-20-3': 'Instead, we attempt to build a low-dimensional model for each mode [MATH] individually.', '1501.06848-1-20-4': "An individual mode [MATH] is simply a linear projection of the full system [MATH], and moreover it is an 'observation' of the spatial state [MATH].", '1501.06848-1-20-5': 'While we could apply the diffusion forecast to the one dimensional time series [MATH], this would ignore the dependence of the [MATH]-th mode on all the other modes.', '1501.06848-1-20-6': 'The fundamental problem with building a one-dimensional stochastic model for [MATH] is that any interaction with the other modes will result in (among other changes to the model) an inflated stochastic component in the closed model for [MATH] (see [CITATION] for a rigorous example).', '1501.06848-1-20-7': 'Inflating the stochastic component of the model for [MATH] will severely limit the predictive skill of the nonparametric model.', '1501.06848-1-20-8': 'On the other hand, if we include other modes in the nonparametric model, this would increase the dimensionality and not all of the information in the other modes would be relevant to forecasting the [MATH]-th mode.', '1501.06848-1-20-9': 'Instead, we will apply the delay coordinate reconstruction of [CITATION] to implicitly recover only the missing components of the dynamics which are important for forecasting each [MATH].', '1501.06848-1-20-10': 'Moreover, we will apply the theory of [CITATION] to show that the delay coordinate reconstruction projects the missing components of the dynamics onto the component most important for the prediction of [MATH].', '1501.06848-1-20-11': 'Finally, in practice we will only have access to a noisy data set [MATH], and the theory of [CITATION] shows that the delay coordinate reconstruction also reduces the influence of the noise.', '1501.06848-1-21-0': 'Given a time series [MATH] of the [MATH]-th mode, we construct the delay embedding coordinates, [EQUATION].', '1501.06848-1-21-1': 'The theory of embedology shows that if the [MATH]-th mode is a generic observation of the full system [REF], then for sufficiently many lags, [MATH], there exists a diffeomorphism [MATH] such that [MATH].', '1501.06848-1-21-2': 'This statement was first shown for deterministic dynamics on smooth attractors in [CITATION] and subsequently generalized to fractal attractors in [CITATION] and then to non-autonomous systems in [CITATION] and stochastic systems in [CITATION].', '1501.06848-1-21-3': 'The existence of the diffeomorphism [MATH] says that topologically the attractor of the delay reconstruction [MATH] is equivalent to the full dynamics [MATH] and so we have reconstructed the hidden variables in the evolution of the [MATH]-th mode.', '1501.06848-1-21-4': 'However, the theory of embedology only concerns the topology of the attractor, whereas the basis of eigenfunctions [MATH] will depend on the geometry of the delay reconstruction [MATH].', '1501.06848-1-22-0': 'In [CITATION] it was shown that the delay reconstruction severely biases the geometry in the reconstructed coordinates [MATH], so that in the limit of infinite delays, [MATH], the dynamics were projected on the most stable component of the dynamics.', '1501.06848-1-22-1': 'Consider the [MATH]-th mode [MATH] as an observation of the state [MATH], where the observation function is given by [MATH].', '1501.06848-1-22-2': 'Notice that the derivative of the observation function is [MATH] where the [MATH] occurs in the [MATH]-th entry.', '1501.06848-1-22-3': 'Moreover, the previous value of the [MATH]-th mode, [MATH], can also be considered an observation of [MATH] where the observation function is given by [MATH] and the map [MATH] is given by the reverse time evolution for the discrete time step [MATH].', '1501.06848-1-22-4': 'Interpretating each [MATH] as an observation of the full state [MATH] at time [MATH] shows that the time-delay embedding [MATH] is an observation of [MATH] with observation function [MATH].', '1501.06848-1-22-5': 'For [MATH] sufficiently large and assuming that the [MATH]-th mode is a generic observable on the manifold, the observation [MATH] will have a full rank derivative.', '1501.06848-1-22-6': 'Our goal now is to examine the change in metric induced by this derivative by seeing how it acts on tangent vectors on the attractor.', '1501.06848-1-22-7': 'Let [MATH] be tangent vectors on the attractor and let [MATH] and [MATH] where [MATH] are the transformed tangent vectors in the new geometry given by the time-delay embedding.', '1501.06848-1-22-8': 'Then the inner product [MATH] in the delay embedding space is [EQUATION] where [MATH] is the reversed shift map which takes [MATH] to [MATH].', '1501.06848-1-22-9': 'If [MATH] and [MATH] are in the [MATH]-th Oseledets space, with Lyapunov exponent [MATH], then the inner product reduces to, [MATH].', '1501.06848-1-22-10': 'This shows that the most stable Oseledets space will dominate the geometry in the embedding since [MATH] will be most negative (largest in absolute value of the negative Lyapunov exponents) in the most stable direction.', '1501.06848-1-23-0': 'The bias introduced into the geometry by the delay embedding has several advantages for forecasting.', '1501.06848-1-23-1': 'First, the most stable components of the dynamics are those that have the most predictability.', '1501.06848-1-23-2': 'Since numerically we will not be able to represent the whole attractor, the variable bandwidth kernel will implicitly project away the small components of the geometry.', '1501.06848-1-23-3': 'By amplifying the stable components we insure that the most desirable aspects of the dynamics will be well represented in the discrete representation of the geometry.', '1501.06848-1-23-4': 'Secondly, when applied to the noisy data, [MATH], the noise will correspond to an unstable component, and so the delay embedding geometry will de-emphasize the noise component of the data.', '1501.06848-1-23-5': 'Finally, note that each mode, [MATH], will capture a different aspect of the most stable components of the dynamics since [MATH] corresponds to the projection of the vectors [MATH] into the [MATH]-th coordinate direction.', '1501.06848-1-23-6': 'Thus, the delay embedded modes [MATH] represent orthogonal components of the most stable Oseledets directions of the dynamics, which in turn are orthogonal to the noise in the limit of large [MATH].', '1501.06848-1-24-0': 'We now briefly demonstrate the effect of the biased geometry on a simple example.', '1501.06848-1-24-1': 'We will generate a stochastic dynamics on the unit circle by first numerically integrating the one-dimensional SDE, [EQUATION] and then mapping the intrinsic variable [MATH] onto the unit circle embedded in [MATH] by the map [MATH].', '1501.06848-1-24-2': 'In this example we use discrete time step [MATH] to produce 10000 samples [MATH] of the system [REF] which are mapped into the plane as [MATH].', '1501.06848-1-24-3': 'We then generate noisy observations [MATH] where [MATH] are independently sampled from a two dimensional Gaussian distribution with mean zero and covariance matrix [MATH].', '1501.06848-1-25-0': 'Our goal in this example is to show how the delay-embedding can reduce the influence of noise on a diffusion model by biasing the geometry.', '1501.06848-1-25-1': 'We use the first 5000 noisy data points [MATH] to train multiple diffusion models, each time applying a different number of delays before building the diffusion model.', '1501.06848-1-25-2': 'We also train a diffusion model using the clean data set [MATH].', '1501.06848-1-25-3': 'For each diffusion model, we apply the Bayesian filter developed in Section [REF] below to generate initial conditions for forecasting from the noisy observations [MATH] in the verification period [MATH].', '1501.06848-1-25-4': 'Each initial condition in the verification period is forecast for a total of 500 forecast steps (50 time units) and the RMSE between the forecast and the truth is averaged over the verification period.', '1501.06848-1-25-5': 'The RMSE as a function of the number of forecast steps for each model is shown in Figure [REF], along with the forecast of the [MATH] coordinate at the 50 step lead time, compared to the true value of [MATH].', '1501.06848-1-26-0': 'While no amount of delays is able to match the diffusion model built using the clean data set, the improvement in forecast skill is significant as the number of delays in increases.', '1501.06848-1-26-1': 'We should note that since this model is intrinsically one dimensional, the most stable component of the dynamics represents the entire dynamical system.', '1501.06848-1-26-2': 'This implies that projecting on the most stable component can only improve the forecasting skill for this example.', '1501.06848-1-26-3': 'For high-dimensional examples we suspect that as the number of delays becomes very high, the projection onto the most stable component will remove information which is important to the forecast.', '1501.06848-1-26-4': 'Therefore, in general we expect to see a tradeoff where small numbers of delays can remove the influence of noise, but extremely large numbers of delays may actually degrade the forecast by projecting away valuable components of the dynamics.', '1501.06848-1-27-0': '## Bayesian filter', '1501.06848-1-28-0': 'In Step 2 of the forecasting algorithm described in Section [REF], we assume that the initial distributions for forecasting are given.', '1501.06848-1-28-1': 'In practice, however, we only have the noisy data [MATH], and the goal of this section is to develop a numerical method to determine the initial distribution.', '1501.06848-1-28-2': 'We will use a Bayesian filter to iteratively combine the information from the noisy data with the diffusion forecast.', '1501.06848-1-28-3': 'For the remainder of the paper we will be working with a single mode [MATH] and so we will drop the subscript [MATH] for convenience.', '1501.06848-1-29-0': 'If the observations were continuous (i.e. for [MATH] very small), we could filter the noisy observations [MATH] using the Zakai equation projected onto the basis [MATH], and this was the approach taken in [CITATION].', '1501.06848-1-29-1': 'However, since the observation time may be long, we will take a different approach here and apply a Bayesian update at discrete finite time step.', '1501.06848-1-29-2': 'Explicitly, we want to find the posterior density [MATH] by updating the previous posterior density [MATH] with the current noisy observation [MATH].', '1501.06848-1-30-0': 'Our Bayesian filter will follow the standard predictor-corrector approach.', '1501.06848-1-30-1': 'Given an initial density at the previous time, [MATH], the first step is to compute the prior forecast density with our nonparametric model, [EQUATION] where the coefficients [MATH] represent the posterior density at time [MATH], projected in diffusion coordinates, [MATH].', '1501.06848-1-30-2': 'Since the noisy observation [MATH] has the form [MATH] where [MATH], the likelihood function for the noisy observation is, [EQUATION].', '1501.06848-1-30-3': 'We can now assimilate the observation [MATH] by using Bayes law to combine the prior forecast and the likelihood function above as follows, [EQUATION]', '1501.06848-1-30-4': 'By computing the product [REF] we can find the desired posterior at time [MATH], up to the normalization factor.', '1501.06848-1-30-5': 'We estimate the normalization factor, [MATH], as a Monte-Carlo integral, [EQUATION] and dividing the product in [REF] by [MATH] we recover the posterior at time [MATH].', '1501.06848-1-31-0': 'In order to initialize this iterative procedure, we spin up the filtering process for few assimilation cycles starting from the invariant measure [MATH].', '1501.06848-1-31-1': 'To obtain initial density [MATH], which is used for forecasting starting at time [MATH], we run this Bayesian filter, starting from [MATH] to allow for [MATH] steps of spin-up with the invariant distribution as the initial density, [MATH].', '1501.06848-1-31-2': 'In our numerical simulations below, we used a very short spin-up time of [MATH] steps.', '1501.06848-1-32-0': '# Forecasting an energy-conserving turbulent mode', '1501.06848-1-33-0': 'As our first numerical example, we consider forecasting the first Fourier component of the Truncated Burgers-Hopf (TBH) model.', '1501.06848-1-33-1': 'The TBH model is essentially a Fourier Galerkin approximation to the inviscid Burgers equation which has intrinsic stochastic dynamics with strong evidence of ergodicity and mixing for large enough degrees of freedom [CITATION].', '1501.06848-1-33-2': 'Moreover, it was shown that this model conserves energy and has a family of Gaussian invariant measures with mean zero and constant equipartition energy.', '1501.06848-1-33-3': 'The TBH model is described by the following quadratically nonlinear equation for the complex Fourier modes, [MATH], [MATH]: [EQUATION]', '1501.06848-1-33-4': 'We simulate the truth, following [CITATION], by setting [MATH] and numerically integrating the TBH model with a pseudo-spectral method combined with the fourth-order Runge-Kutta time discretization with small enough time step [MATH] to conserve energy with small relative error.', '1501.06848-1-34-0': 'Our choice to model only the first mode is motivated by the fact that this mode has the longest autocorrelation time [CITATION] and, thus, the largest statistical memory of the dynamics.', '1501.06848-1-34-1': 'This fact implies that designing reduced models for predicting this mode alone can be difficult since one has to account for the nontrivial memory terms resulting from the interactions between the first mode and remaining unresolved modes.', '1501.06848-1-34-2': 'For this test case, it was shown that these memory terms can be modeled with a stochastic parameterization that involves a physics constrained multi-level regression parametric model [CITATION].', '1501.06848-1-34-3': 'In particular, the proposed stochastic parametric model of [CITATION] for this specific problem is given by the following equation, [EQUATION]', '1501.06848-1-34-4': 'This choice of stochastic model can be obtained by truncating the TBH model up to the first two modes, in order to respect the energy conservation, [MATH], of the quadratic terms.', '1501.06848-1-34-5': 'One then adds linear dissipation terms to the first two modes and a colored noise [MATH] to represent the unresolved (or truncated) modes, where [MATH] is the standard physicists notation for white noise.', '1501.06848-1-35-0': 'In [CITATION], they applied an adaptive Kalman filtering method to estimate the parameters [MATH] in [REF] from noisy observations of [MATH] at time step [MATH] with observations noise variance of [MATH] without knowing [MATH].', '1501.06848-1-35-1': 'It was shown in [CITATION] that the resulting model produces remarkably accurate equilibrium statistics when compared to the true signal in a long-time integration of the model in [REF].', '1501.06848-1-35-2': 'Here, we will compare the forecasting skill of this parametric model and that of the proposed nonparametric model discussed in Section [REF].', '1501.06848-1-36-0': 'Given [MATH] noisy observations [MATH], at discrete time step [MATH] (10 times longer than the one used to parameterize [REF]), we use the first 10000 noisy data points, [MATH], to train the diffusion model and the remaining [MATH] data points to verify the prediction skill.', '1501.06848-1-36-1': 'In this numerical experiment, the delay embedding is applied with an arbitrarily chosen lag [MATH] and the nonparametric model is resolved with [MATH] modes.', '1501.06848-1-36-2': 'To see whether this lag size is large enough to smooth out the noise, yet small enough that it does not project away the important unresolved modes that we want to model, we also include an experiment in which we train the diffusion model with the perfect (meaning the noise-free) data set.', '1501.06848-1-36-3': 'To diagnose the prediction skill, we use the standard Root-Mean-Squared Error (RMSE) and pattern correlation skill defined as follows, [EQUATION] where [MATH] denotes the truth and [MATH] denotes the forecast.', '1501.06848-1-36-4': 'Also, [MATH] in [REF] and [REF] denotes the empirical time average over the verification period.', '1501.06848-1-36-5': 'For this example, of course [MATH] and [MATH] are the mean forecasts resulting from either the an ensemble forecast of the parametric model in [REF] with a relatively small ensemble size of [MATH] or our nonparametric diffusion model.', '1501.06848-1-37-0': 'In Figure [REF], we show the forecasting skill in terms of the measures in [REF] and [REF].', '1501.06848-1-37-1': 'We also show the equilibrium statistical error in the top panel and the true autocorrelation function in the bottom panel (dashes) which can be used to diagnose the predictability limit in the following sense: The equilibrium statistical error is a measure of the average error between the true solution at a randomly selected time and the statistical mean of the steady state; the autocorrelation function shows the rate at which a randomly selected delta function initial condition approaches the invariant measure.', '1501.06848-1-37-2': 'Notice that the parametric model gives a relatively accurate forecasting skill, with a pattern correlation which decays exactly like the true autocorrelation function and with error saturating at the invariant measure around a forecast lead time of 1.2.', '1501.06848-1-37-3': 'This result gives us confidence that the parametric model in [REF] is a reasonable reduced model which can be used to predict this time series.', '1501.06848-1-37-4': 'On the other hand, the forecasting skill of the non-parametric model, although not as good as the parametric model, is nearly as accurate in terms of the rate of decay of the RMSE and correlation.', '1501.06848-1-37-5': 'First, notice that the forecasting skill of the diffusion model constructed from the noisy data does not differ significantly from that of the noise-free data, the latter having only slightly better skill.', '1501.06848-1-37-6': 'This suggests that the choice of lag [MATH] is appropriate in this experiment; [MATH] is large enough to reduce the influence of the noise, but not so large as to project away important information from the reconstructed components of the dynamics.', '1501.06848-1-37-7': 'Notice also that the error (and correlation) at the initial time, resulting from the Bayesian filtering, is not as good as the parametric model.', '1501.06848-1-37-8': 'This initial error is due to the error in the reconstruction of the density which results from the projection onto the finite dimensional space spanned by [MATH].', '1501.06848-1-37-9': 'In particular, since the [MATH] are very smooth functions (see Section [REF]), densities which are very close to delta functions will suffer from significant Gibbs phenomenon in the reconstruction.', '1501.06848-1-37-10': 'In Figure [REF], we compare the mean of the diffusion forecast density to the truth at various forecasting lead times for the time interval [MATH] within the verification period.', '1501.06848-1-37-11': 'Notice that the diffusion forecast has reasonable skill while being less noisy than the forecast produced by the parametric model.', '1501.06848-1-38-0': 'From the numerical results, we conclude that the diffusion forecast is relatively skillful, especially considering that the nonparametric model is effectively a black box and has no knowledge of the true equation whatsoever, whereas the parametric model in [REF] was designed using knowledge of the equations of the true dynamics.', '1501.06848-1-39-0': '# Forecasting weakly chaotic to fully turbulent modes of Lorenz-96 model', '1501.06848-1-40-0': 'As our second example, we consider Fourier modes of the Lorenz-96 (L96) model [CITATION] in various chaotic regimes.', '1501.06848-1-40-1': 'The Lorenz-96 model is defined by the following forced-dissipative nonlinear system of ODEs, [EQUATION]', '1501.06848-1-40-2': 'The right hand side of [REF] consists of an energy conserving quadratic nonlinear advective-like term, a linear dissipation, and a forcing parameter [MATH].', '1501.06848-1-40-3': 'Following [CITATION], we resolve the L96 model at 40 equally spaced grid points with a periodic boundary (represented by the subscript [MATH] being taken modulo 40) in order to mimic weather wave patterns on a midlatitude belt.', '1501.06848-1-40-4': 'The statistical behavior of the Fourier modes of this model has been analyzed in [CITATION]; as the forcing parameter [MATH] increases, the model becomes fully turbulent with Gaussian-like modal distributions.', '1501.06848-1-40-5': 'In our numerical experiments we will consider three different values of [MATH] following [CITATION].', '1501.06848-1-40-6': 'As reported in [CITATION], for [MATH], the system is weakly chaotic with largest Lyapunov exponent [MATH] and the dimension of the expanding subspace of the attractor is [MATH].', '1501.06848-1-40-7': 'For [MATH], which is the original choice in [CITATION], the system is strongly chaotic with [MATH] and [MATH].', '1501.06848-1-40-8': 'For [MATH], the system is "fully turbulent" with [MATH] and [MATH].', '1501.06848-1-40-9': 'We should also note that when [MATH], the L96 model has a long memory depth with relatively slow decaying time, this is manifested visibly as a regular dominant westward propagating "weather-like" wave pattern of wavenumber-8.', '1501.06848-1-40-10': 'The memory depth becomes shorter as [MATH] increases and the "weather-like" wave pattern becomes less obvious.', '1501.06848-1-41-0': 'In the following numerical experiments we will examine the forecasting skill of the nonparametric modeling approach on a selection of Fourier modes, including those with high and low energies as well as large and small correlation times.', '1501.06848-1-41-1': 'For diagnostic purpose, we compare the diffusion forecast on these modes with forecasts from various stochastic modeling approaches that were designed as cheap filter models: the Mean Stochastic Model (MSM) [CITATION] and the stable constrained autoregressive model of order-3 (AR3) [CITATION].', '1501.06848-1-42-0': 'The key idea of MSM is to model each Fourier mode as a complex valued linear stochastic model with additive white noise, [EQUATION] where [MATH] is a complex valued Wiener noise with variance [MATH].', '1501.06848-1-42-1': 'The parameters [MATH] and [MATH] are determined by matching the analytical expression for the variance and correlation time at the statistical equilibrium state of the MSM model in [REF] with the corresponding empirically estimated statistics from the data (see Chapter 12 of [CITATION] for a detailed formulation).', '1501.06848-1-42-2': 'In our implementation below, we will obtain these parameters from the statistics of the noisy dataset, [MATH], during the training period.', '1501.06848-1-42-3': 'To generate initial conditions for the forecast, we apply a one-dimensional Kalman filter in order to account for the observation noise in [MATH], and we assume that the noise observation variance, [MATH], is known.', '1501.06848-1-43-0': 'The stable and consistent AR3 model was introduced in [CITATION] as a cheap linear model for filtering turbulent modes with long memory depth, such as the L96 model with [MATH].', '1501.06848-1-43-1': 'The standard AR3 model is given by, [EQUATION]', '1501.06848-1-43-2': 'The stability of this model is sensitive to the choice of sampling time [MATH], especially when a standard linear regression is used to determine the model parameters.', '1501.06848-1-43-3': 'In [CITATION], it was shown that one can obtain accurate filtered mean estimates in a linear filtering problem with truth generated from [REF] if the parameters [MATH] are chosen to satisfy a set of linear algebraic consistency conditions which depends on [MATH] of [REF] and the sampling time [MATH] (see [CITATION] for details).', '1501.06848-1-43-4': 'For nonlinear problems, a stable model together with these consistency constraints can be determined by an algebraic method proposed in [CITATION] and this AR3 model is what we use in the examples below.', '1501.06848-1-43-5': 'In particular, we will enforce the consistency condition for the AR3 model using the parameter [MATH] obtained from the MSM fit of the noisy data at the training period.', '1501.06848-1-43-6': 'To generate initial conditions for the forecast, we apply a three-dimensional Kalman filter to account for the noise observed [MATH], assuming that the noise observation variance, [MATH], is known (see e.g. [CITATION] for the details of the AR filter).', '1501.06848-1-44-0': 'In our comparison, we also include the forecast of the perfect model and a simple persistence model for diagnostic purpose.', '1501.06848-1-44-1': 'In the perfect model case, we implement a standard ensemble Kalman filter method [CITATION] for the full spatial model [REF] using 80 ensemble members, double the dimension of the L96 model.', '1501.06848-1-44-2': 'These analysis ensemble provides the initial conditions for an ensemble forecast, again using the true model [REF], and the Fourier modes are then computed from the ensemble mean.', '1501.06848-1-44-3': 'Finally, we also include the persistence model which fixes the noisy observation, [MATH], at the initial time [MATH] as the forecast for each lead time.', '1501.06848-1-45-0': 'In our numerical simulations, the true time series are generated by integrating the L96 model with the fourth-order Runge-Kutta method with time step [MATH].', '1501.06848-1-45-1': 'Noisy observations are generated by adding random samples of a Gaussian mean zero and variance [MATH] to the true solutions at every grid point at each observation time step [MATH] as in [REF], resulting in a noise variance of [MATH] on each Fourier mode.', '1501.06848-1-45-2': 'Given a time series of length 15000 consisting of a single Fourier mode of the noisy observations, we use the first 5000 data points to train the three models (diffusion, MSM, and AR3) and we compute the prediction skill measures on the remaining 10000 data points.', '1501.06848-1-45-3': 'For the diffusion forecast model, we used [MATH] lags in the time-delay embedding, and the number of diffusion modes was chosen to be [MATH].', '1501.06848-1-45-4': 'As in the previous numerical example, we measure the prediction skill in terms of the RMSE and pattern correlation as defined in [REF] and [REF], respectively.', '1501.06848-1-46-0': 'In Figure [REF], we show the RMSE and pattern correlation for forecasting three different Fourier modes of the L96 model in a weakly chaotic regime with [MATH]: mode-8 carries the largest energy with a relatively short correlation time; mode-13 carries low energy with a relatively long correlation time; and mode-18 carries low energy with a relatively short correlation time.', '1501.06848-1-46-1': 'Note that the ratios between the observation noise variance [MATH] and the energy of the modes are very different for these three modes; [MATH] for mode-8, [MATH] for mode-13, and [MATH] for mode-18.', '1501.06848-1-46-2': 'Therefore, we expect that the noise does not affect the forecasting skill of modes-8 and 13 as much as that of mode-18.', '1501.06848-1-46-3': 'To see this, for each mode we include a separate diffusion forecast in which we train the model with perfect (noise-free) data set.', '1501.06848-1-46-4': 'Notice that the difference between the forecasting skill of the diffusion model trained with noisy data (solid, black) and that trained with noise-free data (dashes, black) are insignificant for mode-8, still small for mode-13, and somewhat noticeable for mode-18.', '1501.06848-1-46-5': 'As discussed in Section [REF], we expect that it may be possible to improve the estimate for mode-18 by choosing a larger number of lags, [MATH], but at the same time one should not choose [MATH] too large, since this could project away important components of the dynamics.', '1501.06848-1-47-0': 'Notice that the perfect model provides the best forecast in all modes, as expected.', '1501.06848-1-47-1': 'The diffusion forecast performs very well on the energetic mode-8, outperforming the other reduced stochastic models which have very little predictive skill.', '1501.06848-1-47-2': 'On mode-13, the diffusion forecast skill is similar to the other reduced stochastic models, and in the very short term forecast none of these models has significantly more skill than the trivial persistence forecast.', '1501.06848-1-47-3': 'The MSM and AR3 are not very different; AR3 is slightly better on modes with long correlation time, as expected.', '1501.06848-1-47-4': 'The persistence model always produces severely biased forecasts in the long run, since fixing a particular observation will always be worse than predicting the statistical mean in a long term forecast.', '1501.06848-1-47-5': 'On mode-18, the MSM and persistence model is slightly better than diffusion forecast (learned from noisy data) in the short term, but then the diffusion forecast performs much better in the intermediate and long-term predictions.', '1501.06848-1-47-6': 'On this mode, the AR3 performs the worst.', '1501.06848-1-48-0': 'In Figures [REF] and [REF], we compare the forecast of each model to the truth for forecast lead times of 1 and 3 model time units respectively.', '1501.06848-1-48-1': 'This comparison is made for mode-8 with [MATH] for verification time steps between [MATH] and [MATH].', '1501.06848-1-48-2': 'Notice that the diffusion model forecasting skill is very similar to that of the true model for both lead times 1 and 3, while the persistence model forecasts are clearly out-of-phase.', '1501.06848-1-48-3': 'On the other hand, both the MSM and AR3 models significantly underestimate the amplitude of the signals and in the long term they simply forecast zero, as expected since these forecasts represent the mean of linear unbiased autoregressive models.', '1501.06848-1-48-4': 'We also report the forecast skills of the more chaotic regime with [MATH] and the fully turbulent regime with [MATH] in Figure [REF].', '1501.06848-1-48-5': 'In these cases, the observation noise becomes negligible since the ratio between the observation noise error variance and the energy of this mode decreases as [MATH] increases, namely, [MATH] for [MATH] and [MATH] for [MATH].', '1501.06848-1-48-6': 'Based on the RMSE and pattern correlation measures, these results suggest that the diffusion model still produces the most skillful reduced model compared to the MSM, AR3, and persistence models.', '1501.06848-1-48-7': 'While not shown, we also compared the forecast skills on the less energetic modes and found that the forecasting skill of the diffusion, MSM, and AR3 models are more or less comparable, as was found on the corresponding modes shown in Figure [REF].', '1501.06848-1-48-8': 'We note that other stochastic reduced models exists [CITATION] which would certainly outperform the simple MSM and AR3 models, however those parametric models are derived using knowledge of the true form of the equations [REF].', '1501.06848-1-48-9': 'Clearly, none of the black box modeling forecasts are close to the perfect model forecast, however the diffusion forecast does offer significant improvement in skill over the other non-physically designed statistical parameterizations considered here.', '1501.06848-1-49-0': 'The numerical experiments in this section suggest that the diffusion model is most skillful when modeling the highly energetic modes.', '1501.06848-1-49-1': 'We should note that as [MATH] increases, although mode-8 is still the most energetic mode, the energy is less dominant relative to the other modes in these turbulent regimes [CITATION].', '1501.06848-1-49-2': 'We suspect that this is the main reason that the relative forecasting skill deteriorates when [MATH] is larger, in addition to the more quickly decaying correlation functions in the fully turbulent regimes.', '1501.06848-1-50-0': '# Forecasting geophysical turbulent modes', '1501.06848-1-51-0': 'In this section, we consider forecasting turbulent modes of the quasigeostrophic model, a prototypical model for midlatitude atmosphere and oceanic dynamics [CITATION].', '1501.06848-1-51-1': 'We consider a two-layer quasigeostrophic (QG) model which is externally forced by a mean shear with streamfunctions [EQUATION] such that it exhibits baroclinic instabilities; the properties of the turbulent cascade has been extensively discussed in this setting (see e.g., [CITATION] and citations in [CITATION]).', '1501.06848-1-52-0': 'The governing equations for the two-layer QG model with a flat bottom, rigid lid, and equal depth [MATH] are given by, [EQUATION] where [MATH] denotes the perturbed streamfunction, subscript 1 corresponds to the upper layer and subscript 2 corresponds to the bottom layer.', '1501.06848-1-52-1': 'In [REF]-[REF], [MATH] is the meridional gradient of the Coriolis parameter; [MATH] is the Ekman bottom drag coefficient; [MATH] is a Jacobian function which acts as nonlinear advection; [MATH] is the zonal mean shear, selected so that the QG equations exhibit baroclinic instability with a turbulent cascade; [MATH] is the hyperviscosity coefficient, chosen so that [MATH] filters out the energy buildup on smaller scales when finite discretization is enforced.', '1501.06848-1-52-2': 'The perturbed QG potential vorticity [MATH] is defined for each layer by [EQUATION]', '1501.06848-1-52-3': "The parameter [MATH] gives the wavenumber associated with radius of deformation, or Rossby radius, [MATH] (the scale at which Earth's rotation becomes significant to the dynamics of the system).", '1501.06848-1-53-0': 'In our numerical simulations, the true signal is generated by resolving ([REF])-([REF]) with [MATH] Fourier modes, which corresponds to the [MATH] grid points.', '1501.06848-1-53-1': 'With this resolution, the numerical integration of this turbulent system only needs slightly more than 30,000 state variables because one can always compute [MATH] from [MATH] and vice versa, via ([REF]).', '1501.06848-1-53-2': 'This model has two important nondimensional parameters: [MATH] where [MATH] is the horizontal non-dimensionalized velocity scale and [MATH] is the horizontal domain size in both directions (we choose [MATH]), and [MATH] which is inversely proportional to the deformation radius [CITATION].', '1501.06848-1-53-3': 'As in [CITATION], we will consider two cases: one with a deformation radius [MATH] such that [MATH] which roughly mimics a turbulent jet in the midlatitude atmosphere and the other case will have a deformation radius [MATH] such that [MATH] which roughly mimics a turbulent jet in the ocean.', '1501.06848-1-53-4': 'For the ocean case, the QG model in [REF]-[REF] is numerically very stiff since the term that involves [MATH] is large.', '1501.06848-1-54-0': 'The large-scale components of this turbulent system are barotropic and for the two-layer model with equal depth, the barotropic streamfunction is defined as an average between the two layers, [MATH] [CITATION].', '1501.06848-1-54-1': 'In the atmospheric regime with [MATH], the barotropic flow [MATH] is dominated by a strong zonal jet while in the ocean case with [MATH] there are transitions between zonal flow and large-scale Rossby waves (e.g., see [CITATION]).', '1501.06848-1-55-0': 'In our numerical examples, we build a diffusion model for the two-dimensional Fourier modes [MATH] of the barotropic streamfunction, [MATH].', '1501.06848-1-55-1': 'Following the experiments in [CITATION], our choice to only model this mode is mainly due to the fact that small-scale observations of a turbulent jet are typically not available, especially in the ocean.', '1501.06848-1-55-2': 'For diagnostic purposes, we compare the diffusion model to a simple stochastic model with combined white and colored additive and multiplicative noises that was designed as a test filter model for stochastic parameterization in the presence of model error [CITATION].', '1501.06848-1-55-3': 'The governing equation of this model is given as follows, [EQUATION] where variable [MATH] models the Fourier mode of [MATH], dropping the horizontal and vertical wave components [MATH] to simplify the notation.', '1501.06848-1-55-4': 'The equation governing [MATH] represents the interactions between this resolved mode and the remaining unresolved modes by several noise terms.', '1501.06848-1-55-5': 'The first noise term, [MATH], is a real valued multiplicative noise.', '1501.06848-1-55-6': 'The second noise term is [MATH], which is an additive complex valued noise governed by the Ornstein-Uhlenbeck mean reverting SDE.', '1501.06848-1-55-7': 'The final noise term is [MATH], which is an additive white noise.', '1501.06848-1-55-8': 'In [REF], the stochastic terms, [MATH], are complex valued and [MATH] are real valued Wiener processes.', '1501.06848-1-55-9': 'We should note that while the simple system in [REF] is nonlinear (due to the multiplicative term [MATH]), it has analytical statistical solutions [CITATION] which we will utilize in our numerical experiments here.', '1501.06848-1-56-0': 'Implementing a Kalman update on the analytical mean and covariance solutions of [REF], is a filtering method known as SPEKF [CITATION], which stands for Stochastic Parameterized Extended Kalman Filter.', '1501.06848-1-56-1': 'In our implementation below, we will use SPEKF to generate initial conditions for our forecast at the verification times.', '1501.06848-1-56-2': 'In our experiments, we consider two different choices of parameters, [MATH], in [REF]: The original set of parameters which was empirically chosen to optimize the filtered solutions in [CITATION] and we will refer to this forecasting strategy as SPEKF.', '1501.06848-1-56-3': 'The second choice is to use the adaptive Kalman filtering method [CITATION] (which was also used in Section [REF] above) in order to extract these parameters from noisy the data set [MATH] in the training period.', '1501.06848-1-56-4': 'The adaptive Kalman filter will starting with the parameters set from the SPEKF as initial guess.', '1501.06848-1-56-5': 'We refer to this forecasting strategy, with parameters determined adaptively, as SPEKF-QR.', '1501.06848-1-57-0': 'In our numerical experiments, we generate a time series of 9000 data points at discrete time step [MATH] and added Gaussian noise in the physical space at 36 regularly spaced grid points in the spatial domain as in [CITATION] with noise variance [MATH].', '1501.06848-1-57-1': 'In Fourier modes, the observation error variance is [MATH]; in Figure [REF], we show [MATH] relative to the variance of the 12 modes corresponding to the 36 regularly spaced observed grid points for both the atmospheric and ocean regimes.', '1501.06848-1-57-2': 'We use the first 5000 noisy data points to train the diffusion and SPEKF-QR models and the remaining 4000 data points to compute the forecasting skill measures.', '1501.06848-1-57-3': 'For the diffusion forecast model, we used [MATH] lags in the time-delay embedding and [MATH] diffusion modes.', '1501.06848-1-58-0': 'In Figures [REF] and [REF], we show the RMSE and pattern correlation for the first two energetic modes in the atmospheric and ocean regimes, respectively.', '1501.06848-1-58-1': 'For the atmospheric regime, the first two energetic modes correspond to the two-dimensional horizontal wave numbers [MATH] and [MATH] with explained variances of about 65% and 73%, respectively, so the behavior is dominated by the zonal jet in the horizontal direction [CITATION].', '1501.06848-1-58-2': 'In this regime, notice that the SPEKF-QR produces slightly better forecasts compared to the diffusion model in terms of RMSE and pattern correlation.', '1501.06848-1-58-3': 'On the other hand, the SPEKF model, with parameters empirically chosen to optimize the filter accuracy, produces lower forecasting skill after 10 unit times.', '1501.06848-1-58-4': 'For the ocean case, the first two energetic modes correspond to the two-dimensional horizontal wave numbers [MATH] and [MATH] with explained variance of about 50% and 97%, respectively, so the behavior is dominated by the Rossby mode [MATH] and the zonal jet mode [MATH] [CITATION].', '1501.06848-1-58-5': 'In this case, notice that for the first mode both the SPEKF-QR forecasts and the SPEKF forecasts are unstable to the point that the RMSE diverge far beyond the invariant measure.', '1501.06848-1-58-6': 'For the second mode, the SPEKF-QR diverges while the SPEKF produces accurate forecasts.', '1501.06848-1-58-7': 'The instability in SPEKF-QR and SPEKF were due to the fact that the SPEKF model is statistically stable only if certain algebraic constraints are satisfied (see [CITATION] for the precise definition of these algebraic constraints) and these constraints were not imposed in choosing the SPEKF parameters and also not imposed in the adaptive parameterization scheme which is used to choose the parameters in SPEKF-QR.', '1501.06848-1-58-8': 'In contrast, the nonparametric model is always stable and reasonably skillful in terms of RMSE and pattern correlation.', '1501.06848-1-59-0': 'In Figure [REF], we show the forecasts of the real component of the second most energetic mode for the ocean case with [MATH] at various lead times 0-5 for the verification period [MATH].', '1501.06848-1-59-1': 'Notice that the SPEKF-QR forecasts are very noisy, whereas the diffusion forecast and SPEKF forecast are reasonably accurate.', '1501.06848-1-59-2': 'In Figures [REF]-[REF], we show the spatial reconstruction of the barotropic streamfunction [MATH], based on separately modeling 12 Fourier modes corresponding to the 36 regularly spaced sparse noisy observations in the physical space.', '1501.06848-1-59-3': 'We should point out that while the observation noise is relatively small compared to the two most energetic modes (see Figure [REF]), the less energetic modes are completely overwhelmed by the noise in this numerical experiment and the results below are reported without tuning the lagged embedding dimension, [MATH].', '1501.06848-1-59-4': 'Nevertheless, we find that for the atmospheric case (see Figure [REF]), the dominant turbulent zonal jet structure of the streamfunction is recovered by all the methods at lead times 4 and 8; the spatial structure of the streamfunctions produced by the diffusion model appears closer to the truth compared to those from the parametric SPEKF-QR and SPEKF models.', '1501.06848-1-59-5': 'For the ocean case (see Figure [REF], notice that the diffusion model is the only one that produces accurate forecasts.', '1501.06848-1-60-0': 'From the simulations in this section on a nontrivial prototypical example of midlatitude geophysical turbulent flows, we see that the proposed non-parametric diffusion models always produce stable forecasts.', '1501.06848-1-60-1': 'Of course, one can always resolve this instability issue by carefully choosing a different parametric form, for example the physics constrained model in [REF], in which the stability constraint was implicitly imposed in the parametric equations.', '1501.06848-1-60-2': 'Our point here is to show that if the parametric form for the reduced stochastic model is not chosen appropriately, then one can easily run into these instability issues.', '1501.06848-1-60-3': 'In contrast, this issue is not encountered in the non-parametric modeling.', '1501.06848-1-60-4': 'We also see that the diffusion models can recover the non-trivial structure of streamfunctions despite the fact that each Fourier mode is modeled separately and some of the low energy modes are completely swamped by the observation noise.', '1501.06848-1-60-5': 'This encouraging result empirically suggests that predicting the first two most energetic modes that are dominant can overcome the least accurate models of the low energetic modes.', '1501.06848-1-61-0': '# Summary', '1501.06848-1-62-0': 'In this paper, we apply the nonparametric diffusion modeling approach proposed in [CITATION] to forecast Fourier modes of turbulent dynamical systems given finite set of noisy observations of these modes.', '1501.06848-1-62-1': 'Motivated by the results in [CITATION], we build the nonparametric model in the delay embedding coordinates in order to account for the interaction between the resolved and unresolved modes as well as to smooth out the noisy observations.', '1501.06848-1-62-2': 'While our theoretical derivation suggests that too large of embedding space dimension, [MATH], may project away the reconstructed unresolved modes, we empirically verify that larger [MATH] helps to reduce the influence of the observation noise and improves the diffusion model forecasts.', '1501.06848-1-62-3': 'The choice of [MATH] should not be so large as to project away the unresolved modes, but large enough to smooth out the noise in the training data set.', '1501.06848-1-62-4': 'To initiate each forecast, we apply a simple Bayesian filtering method which accounts for the information of the noisy observations up to the current time.', '1501.06848-1-62-5': 'We compare the forecasting skills of the proposed nonparametric model with various modeling approaches including the persistence model, linear autoregressive models of order-1 (MSM) and order-3, the SPEKF model which involves combined additive and multiplicative noise terms, the physics constrained nonlinear regression models which use multi-level colored noises, and the perfect model on various examples.', '1501.06848-1-62-6': 'We should note that these numerical results were obtained without any careful tunings of size of the training data set, [MATH], and the number of eigenfunctions, [MATH] and all the remaining nuisance parameters in the nonparametric model were automatically tuned from the data set as described in [CITATION].', '1501.06848-1-62-7': 'The numerical results empirically suggest that the diffusion model gives more accurate forecasts for modes that are highly energetic with slow decaying correlation functions, which is a characteristic of weakly turbulence.', '1501.06848-1-63-0': 'One important thing to take away from this paper is that although the proposed nonparametric model seems to be competitive with the standard parametric models chosen here, we do not claim that this equation-free approach is the model to use for everything.', '1501.06848-1-63-1': 'If one knows the physics of the underlying dynamics (or the appropriate reduced parametric models, such as equation [REF] for modeling the first mode of the TBH example in Section [REF] above), then one should use the physics-based model rather than the equation-free diffusion models.', '1501.06848-1-63-2': 'Indeed, it was shown rigorously in a simple setup that optimal filtering and accurate statistical prediction can be achieved with an appropriate stochastic parametric modeling of the unresolved scales [CITATION].', '1501.06848-1-63-3': 'However, there are at least two valuable aspects of the nonparametric model we develop here.', '1501.06848-1-63-4': 'First, one can use this model to diagnose whether his/her modeling approach is appropriate; we expect that appropriate physics-based models should beat this black box approach.', '1501.06848-1-63-5': 'Of course when an appropriate, physically motivated model is not available, then this approach will often outperform adhoc parametric models.', '1501.06848-1-63-6': 'Second, in practice it is difficult to guess the appropriate choice of parametric models even if some physics-based knowledge has been imposed in deriving a reduced model (e.g., in turbulent closure problems, one typically uses parameters to represent some small-scale processes or some physical processes which are not well understood).', '1501.06848-1-63-7': 'In this situation, we propose to extract the time series of these processes (if possible) and to subsequently build nonparametric models for such processes.', '1501.06848-1-63-8': 'This idea is what we refer to as the semi-parametric modeling approach and is introduced in the companion paper [CITATION].'}

{'1501.06848-2-0-0': 'In this paper, we apply a recently developed nonparametric modeling approach, the "diffusion forecast", to predict the time-evolution of Fourier modes of turbulent dynamical systems.', '1501.06848-2-0-1': 'While the diffusion forecasting method assumes the availability of a noise-free training data set observing the full state space of the dynamics, in real applications we often have only partial observations which are corrupted by noise.', '1501.06848-2-0-2': 'To alleviate these practical issues, following the theory of embedology, the diffusion model is built using the delay-embedding coordinates of the data.', '1501.06848-2-0-3': 'We show that this delay embedding biases the geometry of the data in a way which extracts the most stable component of the dynamics and reduces the influence of independent additive observation noise.', '1501.06848-2-0-4': 'The resulting diffusion forecast model approximates the semigroup solutions of the generator of the underlying dynamics in the limit of large data and when the observation noise vanishes.', '1501.06848-2-0-5': 'As in any standard forecasting problem, the forecasting skill depends crucially on the accuracy of the initial conditions.', '1501.06848-2-0-6': 'We introduce a novel Bayesian method for filtering the discrete-time noisy observations which works with the diffusion forecast to determine the forecast initial densities.', '1501.06848-2-1-0': 'Numerically, we compare this nonparametric approach with standard stochastic parametric models on a wide-range of well-studied turbulent modes, including the Lorenz-96 model in weakly chaotic to fully turbulent regimes and the barotropic modes of a quasi-geostrophic model with baroclinic instabilities.', '1501.06848-2-1-1': 'We show that when the only available data is the low-dimensional set of noisy modes that are being modeled, the diffusion forecast is indeed competitive to the perfect model.', '1501.06848-2-2-0': '# Introduction', '1501.06848-2-3-0': 'A long-standing issue in modeling turbulent dynamics is the so-called turbulent closure problem (see e.g.[CITATION]) where the goal is to find a set of effective equations to represent low-order statistics of the coarse-grained variables of interest.', '1501.06848-2-3-1': 'The main difficulty of this problem is largely due to the infinite dimensionality and nontrivial coupling of the governing equations of the statistics.', '1501.06848-2-3-2': 'In order to predict a few lower-order statistics of some resolved variables, common closure approaches were developed using physical insights to choose a parametric ansatz to represent the feedback from the unresolved scales (see e.g., [CITATION] for various closure approximations for predicting passive scalar turbulence and [CITATION] for various stochastic modeling approaches for predicting geophysical turbulence).', '1501.06848-2-4-0': 'Despite these successes, the parametric modeling approaches have practical issues due to model error when the necessary physical insights are not known.', '1501.06848-2-4-1': 'If the parametric model (or ansatz) is not chosen appropriately, one can end up with a model with poor predictive skills (or even with solutions which diverge catastrophically) even when the parameters can be obtained by a standard regression fitting procedure [CITATION].', '1501.06848-2-4-2': 'Moreover, even when an appropriate parametric form is chosen, specifying the parameters from noisy observations of the physical variables can be nontrivial since the parameters are typically not directly observed.', '1501.06848-2-4-3': 'Indeed, it was shown that an appropriate parameterization scheme is crucial for accurate filtering and equilibrium statistical prediction even when the parametric forms are appropriately chosen [CITATION].', '1501.06848-2-5-0': 'Recently, a nonparametric modeling approach, called the diffusion forecast, for predicting the evolution of the probability density of low-dimensional dynamical system was introduced in [CITATION].', '1501.06848-2-5-1': 'The approach of [CITATION] can be intuitively viewed as extending the standard nonparametric statistical models (such as kernel density estimates) which are used to estimate time-independent densities [CITATION].', '1501.06848-2-5-2': 'The key idea behind the diffusion forecast is to use a basis of smooth functions to represent probability densities, so that the forecast model becomes a linear map in this basis.', '1501.06848-2-5-3': 'Numerically, this linear map is estimated by exploiting a rigorous connection between the discrete time shift map and semi-group solution associated to the backward Kolmogorov equation.', '1501.06848-2-5-4': 'In [CITATION], it was shown that the resulting model estimates the semigroup solutions of the generator of the underlying dynamics in the limit of large data.', '1501.06848-2-5-5': 'Moreover, the smooth basis is defined on the training data set, using the diffusion maps algorithm [CITATION], which means that the data requirements only depend on the intrinsic dimensionality of the dynamics.', '1501.06848-2-6-0': 'In this paper, we test this nonparametric modeling approach as a method of forecasting noisy observations of Fourier modes from a selection of well-studied high-dimensional dynamical systems in various turbulent regimes.', '1501.06848-2-6-1': 'A novel aspect of this paper is that we consider building a forecasting model given a noisy training data set consisting of partial observations of the dynamics, as is common in practical applications, in contrast to the work in [CITATION] which used noiseless full observations to train the diffusion forecasting model.', '1501.06848-2-6-2': 'A key ingredient for solving initial value problems in any forecasting problem is accurate initial conditions.', '1501.06848-2-6-3': 'While initial conditions were assumed to be given in [CITATION], in this paper, we introduce a novel Bayesian filtering method to iteratively assimilate each observation and find the initial probability densities given all of the past noisy observations up to the corresponding initial time.', '1501.06848-2-7-0': 'We should note that the diffusion forecasting method [CITATION] could be naively applied to signals corrupted by observation noise, however the resulting nonparametric model would implicitly include the observation noise in the model, which would limit the forecast skill compared to treating the noise as a separate process.', '1501.06848-2-7-1': "Treating the noise as a separate process requires first learning the 'correct' model from the noisy training data set, and then generating 'clean' initial conditions for forecasting from the noisy observations.", '1501.06848-2-7-2': 'In [CITATION] it was shown that applying diffusion maps to the delay-embedded data reduces the influence of the noise on the diffusion maps basis.', '1501.06848-2-7-3': 'Building upon the work in [CITATION], we apply the theory of [CITATION] to show that building the nonparametric model using the delay-embedded data biases the geometry of the data in a way which extracts the most predictable component of the dynamics.', '1501.06848-2-7-4': 'We extend the theory of [CITATION] by giving a rigorous justification for the reduction of the influence of independent additive observation noise on the resulting diffusion forecast model.', '1501.06848-2-8-0': 'One interesting question which we address here is whether it is possible to build a skillful nonparametric forecasting model for a turbulent mode given only a small amount of noisy training data, when the true dynamics are solutions of a high-dimensional dynamical system with chaotic behavior.', '1501.06848-2-8-1': 'This question arises because the nonparametric model has a practical limitation in terms of modeling dynamics with high-dimensional attractors, namely: it will require an immense amount of data to unwind the attractors since the required data increases exponentially as a function of the dimension of the attractor.', '1501.06848-2-8-2': 'Moreover, even given a sufficiently large data set, the required computational power would be a limiting factor since the diffusion maps algorithm requires storing and computing eigenvectors of a sparse [MATH] matrix, where [MATH] is the number of data points.', '1501.06848-2-8-3': 'Constrained by a small data set, the curse-of-dimensionality implies that we cannot unwind the full high-dimensional attractor.', '1501.06848-2-8-4': 'We attempt to circumvent the curse-of-dimensionality by decomposing the data into Fourier modes in the hope that delay reconstruction of each mode projects onto a different component of the dynamics.', '1501.06848-2-8-5': 'We do not claim that the Fourier decomposition can completely resolve this issue but we will numerically demonstrate that the Fourier decomposition will map an isotropic turbulent field in the spatial domain (which implies that each spatial component is as predictable as any other spatial component) to a coordinate system in which some modes are more predictable than others.', '1501.06848-2-8-6': 'Of course, the standard theory of embedology [CITATION] suggests that the delay-embedding of a single Fourier mode would reconstruct the entire high-dimensional attractor, which would again be inaccessible to our nonparametric model due to the curse-of-dimensionality.', '1501.06848-2-8-7': 'This would suggest that nothing could be gained by building separate models based on delay-embedding of each mode.', '1501.06848-2-8-8': 'However, the full attractors reconstructed from each mode are only equivalent in a topological sense, and the geometries of these reconstructed attractors are dramatically different.', '1501.06848-2-8-9': 'The biased geometry influences the nonparametric model of [CITATION] through the use of the diffusion map algorithm which is known to preserve the geometry which the data inherits from the embedding space [CITATION].', '1501.06848-2-8-10': 'The diffusion maps algorithm preserves the biased geometry of the delay embedding as was shown in [CITATION]; and we will see that this biased geometry projects the full dynamical system onto the most stable components of the dynamics in the direction of the chosen observations.', '1501.06848-2-8-11': 'When we apply the nonparametric model of [CITATION] using the basis arising from this biased geometry, we find improved forecasting skill and robustness to observation noise.', '1501.06848-2-9-0': 'The remainder of this paper is organized as follows.', '1501.06848-2-9-1': 'In Section 2, we introduce the problems under consideration and establish the necessary background, including a brief overview of the nonparametric modeling approach introduced in [CITATION] as well as a discussion on how the theory of [CITATION] is applied to mitigate the effect of noise on the model.', '1501.06848-2-9-2': 'We conclude Section 2 by introducing the iterative Bayesian filter which we use to generate initial conditions for forecasting with the nonparametric model.', '1501.06848-2-9-3': 'In Section 3, we numerically compare predicting Fourier modes of the Lorenz-96 model in various chaotic regimes using the nonparametric model with the persistence model, perfect model, and various parametric models, including the autoregressive models of order-1 (MSM [CITATION]).', '1501.06848-2-9-4': 'In Section 4, we numerically compare the nonparametric model with a stochastic model with additive and multiplicative noises (SPEKF model [CITATION]) in predicting the barotropic modes of a geostrophic turbulence.', '1501.06848-2-9-5': 'We close this paper with a short summary in Section 5.', '1501.06848-2-10-0': '# Nonparametric diffusion modeling', '1501.06848-2-11-0': 'Let [MATH] be solutions of an ergodic system of nonlinear PDEs, [EQUATION] where [MATH] denotes nonlinear differential operators, for smooth initial conditions [MATH] and periodic boundary conditions on a non-dimensionalized periodic domain [MATH].', '1501.06848-2-11-1': 'To simplify the exposition, set [MATH] without loss of generality.', '1501.06848-2-11-2': 'Here, the solutions of [REF] can be described by the infinite Fourier series, [EQUATION] where the Fourier modes [MATH] can be utilized in analyzing [REF].', '1501.06848-2-12-0': 'Define [MATH] whose components are the solutions of [REF] at time [MATH] realized at grid point [MATH], such that [MATH].', '1501.06848-2-12-1': 'Our goal is to construct a forecasting model for the discrete Fourier coefficients, [EQUATION] given the corresponding observed modes, [MATH] where the [MATH]-th component of [MATH], [EQUATION] is the solution of [REF], corrupted by i.i.d. Gaussian noise, at the grid point [MATH] and time step [MATH].', '1501.06848-2-12-2': 'Notice that [MATH] forms an orthonormal basis of [MATH] with respect to the inner product defined in [REF].', '1501.06848-2-12-3': 'One should also note that [MATH] where [MATH] and [MATH] [CITATION].', '1501.06848-2-13-0': 'Given the noisy time series of modes [MATH], our goal is to use this noisy data set to train the nonparametric model and to generate initial conditions for forecasting modes [MATH] at time index [MATH].', '1501.06848-2-13-1': 'Particularly, we will consider [MATH] to be a set containing a single mode or containing three modes in our numerical experiments.', '1501.06848-2-13-2': 'In Section [REF], we provide an overview of the construction of the nonparametric model introduced in [CITATION] for fully observed data (all modes) without observation noise.', '1501.06848-2-13-3': 'In Section [REF], we show that by applying a delay embedding on noisy data, [MATH], we can improve the forecasting skill of the underlying mode, [MATH].', '1501.06848-2-13-4': 'Finally, in Section [REF], we introduce a simple Bayesian filtering method for generating initial conditions for forecasting [MATH] from the noisy observations [MATH].', '1501.06848-2-14-0': '## Nonparametric forecasting model', '1501.06848-2-15-0': 'In this section, we review the nonparametric diffusion forecasting model introduced in [CITATION], assuming the data set consists of full observations of the dynamical system with no observation noise.', '1501.06848-2-15-1': 'Consider a system of SDEs, [EQUATION] where [MATH] is a vector field, [MATH] is a diffusion tensor, and [MATH] is an i.i.d. Wiener process; all defined on a manifold [MATH].', '1501.06848-2-15-2': 'Assume that the dynamical system governed by [REF] is ergodic on [MATH] with an invariant measure with density function [MATH].', '1501.06848-2-15-3': 'We also assume that the evolution of a density function is given by [MATH] which converges to the invariant measure [MATH], where [MATH] denotes the Fokker-Planck (or forward) operator and [MATH] denotes an initial density.', '1501.06848-2-15-4': 'The approach we will describe below is nonparametric in the sense that it does not assume any parametric structure on the dynamics, [MATH], [MATH], the distribution, [MATH], or even the manifold, [MATH], which will all be implicitly represented in the model.', '1501.06848-2-15-5': 'However, this does not mean that the method does not have parameters, and these parameters are roughly analogous to the bin size in a histogram.', '1501.06848-2-16-0': 'Given a time series [MATH], our goal is to approximate [MATH] without knowing or even explicitly estimating [MATH].', '1501.06848-2-16-1': 'Instead, we will directly estimate the semigroup solution [MATH] associated to the generator [MATH] of [REF] by projecting the density onto an appropriate basis for [MATH].', '1501.06848-2-16-2': 'In particular, we choose eigenfunctions [MATH] of the generator [MATH] of a stochastically forced gradient system, [EQUATION] where the potential function [MATH] is defined by the invariant measure of the full system [REF] so that the invariant measure of [REF] is [MATH].', '1501.06848-2-16-3': 'This choice of basis is motivated by several considerations.', '1501.06848-2-16-4': 'First, we can estimate the eigenfunctions [MATH] by the diffusion maps algorithm for data lying on a compact manifold [CITATION] and on a non-compact manifold [CITATION].', '1501.06848-2-16-5': 'In this paper, we will use the variable bandwidth kernel introduced in [CITATION] to construct these eigenfunctions since the sampling measure of the data set may be arbitrarily small, which would imply that the data does not lie on a compact manifold.', '1501.06848-2-16-6': 'Second, by projecting the density function on this basis of eigenfunctions, we can accurately forecast each projected component by a discrete representation of the semigroup solution [MATH], where [MATH].', '1501.06848-2-16-7': 'We now show how to construct the discrete representation of [MATH] using the "shift operator".', '1501.06848-2-17-0': 'Let [MATH] be the eigenfunctions of the generator [MATH] of the gradient system in [REF]; these eigenfunctions are orthonormal under [MATH] in [MATH] and the integral here (and in all of the inner product defined below) is with respect to the volume form [MATH] which [MATH] inherits from the ambient space.', '1501.06848-2-17-1': 'Note that [MATH] is the generator of gradient system in [REF] and it is the adjoint of the Fokker-Planck operator [MATH] with respect to inner-product [MATH] in [MATH].', '1501.06848-2-17-2': 'One can show that [MATH] are eigenfunctions of [MATH] which are orthonormal with respect to inner-product [MATH] in [MATH].', '1501.06848-2-17-3': 'Given an initial density [MATH], we can write the forecast density [MATH] as, [EQUATION]', '1501.06848-2-17-4': 'Setting [MATH] in [REF] we find, [EQUATION] where we define [MATH].', '1501.06848-2-17-5': 'Substituting [REF] into [REF], we obtain, [EQUATION]', '1501.06848-2-17-6': 'The key idea of the non-parametric forecasting algorithm in [CITATION] is to approximate [MATH] by replacing the semi-group solution [MATH] with the discrete time shift operator [MATH], which is defined as [MATH] for any [MATH].', '1501.06848-2-17-7': 'In [CITATION], it was shown that [MATH] is a stochastic estimate of [MATH] and the error due to the stochastic terms can be minimized by projecting [MATH] on the basis [MATH].', '1501.06848-2-17-8': 'Indeed it was shown that [MATH] is an unbiased estimate of [MATH], meaning that [MATH].', '1501.06848-2-17-9': 'Furthermore, assuming that [MATH] are independent samples of [MATH], one can show that the error of this estimate is of order [MATH], which means that one can apply this approximation for any sampling time [MATH] given a sufficiently large data set [MATH].', '1501.06848-2-18-0': 'Notice that for any [MATH], we have: [EQUATION]', '1501.06848-2-18-1': 'From the ergodicity property of [REF], one can deduce that the largest eigenvalue of [MATH] is equal to 1 with constant eigenfunction, [MATH].', '1501.06848-2-18-2': 'Setting [MATH] in [REF], we have, [EQUATION] which implies that [MATH] or in compact form, [MATH], where [MATH], so [MATH] is [MATH] on the first component and zero otherwise.', '1501.06848-2-18-3': 'Assuming that the data is sampled from [MATH], we just deduced that the largest eigenvalue of [MATH] should be equal to 1 (with no complex component).', '1501.06848-2-18-4': 'Numerically however, we sometimes find that [MATH] has some eigenvalues slightly larger than one, due to the finite number of samples in the Monte-Carlo integrals.', '1501.06848-2-18-5': 'If this occurs, we can easily ensure the stability of the diffusion forecast by dividing any eigenvalue with norm greater than 1 so that is has norm equal to one.', '1501.06848-2-18-6': 'We note that in our numerical experiments below this issue rarely occurs.', '1501.06848-2-18-7': 'The ease with which we can guarantee the stability of this nonparametric model is an advantage over many parametric methods.', '1501.06848-2-19-0': 'We should also note that if [REF] is a gradient system as in [REF], then [MATH], where [MATH] are the eigenvalues of [MATH], which can be obtained directly from the diffusion maps algorithm [CITATION].', '1501.06848-2-19-1': 'Moreover, the matrix [MATH] becomes diagonal (due to the orthonormality of [MATH]) so that the diffusion maps algorithm estimates [MATH] directly and the shift operator approximation is not required.', '1501.06848-2-19-2': 'See [CITATION] for various uncertainty quantification applications for this special case.', '1501.06848-2-20-0': 'To conclude, the non-parametric forecasting algorithm (which we refer to as the diffusion forecast) for a given initial density [MATH] is performed as follows:', '1501.06848-2-21-0': 'Learning phase: Given a data set [MATH], apply the diffusion maps algorithm [CITATION] to obtain eigenvectors [MATH], whose [MATH]-th component, [MATH], approximates the eigenfunction [MATH] of the gradient flow [REF] evaluated at the data point [MATH].', '1501.06848-2-21-1': 'We implement the diffusion maps algorithm with a variable bandwidth kernel, see the Appendix for the step-by-step algorithm to obtain these eigenfunctions.', '1501.06848-2-21-2': 'Also, see the supplementary material of [CITATION] for a short overview of [CITATION].', '1501.06848-2-21-3': 'Initial conditions: Represent the initial density [MATH] in this basis as, [EQUATION] which is numerically obtainable from the last inner product as a Monte-Carlo integral since we have [MATH] evaluated on the data set [MATH] which are samples of [MATH].', '1501.06848-2-21-4': 'Forecasting: Apply a Monte-Carlo integral to approximate the shift operator [MATH] in coordinate basis [MATH], [EQUATION]', '1501.06848-2-21-5': 'We then approximate [MATH] with [MATH] such that the diffusion forecast is given by [EQUATION] where in practice, these sums are truncated at a finite number, [MATH], of eigenfunctions.', '1501.06848-2-21-6': 'With this truncation, the coefficient [MATH] is the [MATH]-th component of a matrix-vector multiplication, [MATH], of an [MATH] matrix [MATH] and an [MATH]-dimensional vector [MATH].', '1501.06848-2-22-0': '## Time-lagged embedding', '1501.06848-2-23-0': 'The diffusion forecast algorithm of Section [REF] assumes that the data [MATH] are sampled directly from the full dynamical system [REF].', '1501.06848-2-23-1': 'That is, the full system [MATH] in [REF] is equivalent to the dynamical system for [MATH] and the turbulent dynamics considered here will be very high dimensional.', '1501.06848-2-23-2': 'For such high dimensional problems, exploring the entire attractor would require a prohibitively large data set; larger than will typically be available in applications.', '1501.06848-2-23-3': 'Instead, we attempt to build a low-dimensional model for each mode [MATH] individually.', '1501.06848-2-23-4': "An individual mode [MATH] is simply a linear projection of the full system [MATH], and moreover it is an 'observation' of the spatial state [MATH].", '1501.06848-2-23-5': 'While we could apply the diffusion forecast to the one dimensional time series [MATH], this would ignore the dependence of the [MATH]-th mode on all the other modes.', '1501.06848-2-23-6': 'The fundamental problem with building a one-dimensional stochastic model for [MATH] is that any interaction with the other modes will result in (among other changes to the model) an inflated stochastic component in the closed model for [MATH] (see [CITATION] for a rigorous example).', '1501.06848-2-23-7': 'Inflating the stochastic component of the model for [MATH] will severely limit the predictive skill of the nonparametric model.', '1501.06848-2-23-8': 'On the other hand, if we include other modes in the nonparametric model, this would increase the dimensionality and not all of the information in the other modes would be relevant to forecasting the [MATH]-th mode.', '1501.06848-2-23-9': 'Instead, we will apply the delay coordinate reconstruction of [CITATION] to implicitly recover only the missing components of the dynamics which are important for forecasting each [MATH].', '1501.06848-2-23-10': 'Moreover, we will apply the theory of [CITATION] to show that the delay coordinate reconstruction projects the missing components of the dynamics onto the most stable component of [MATH].', '1501.06848-2-23-11': 'Finally, in practice we will only have access to a noisy data set [MATH], and the theory of [CITATION] suggests that the delay coordinate reconstruction also reduces the influence of the observations noise, which we will now make rigorous.', '1501.06848-2-24-0': 'Given a time series [MATH] of the [MATH]-th mode, we construct the delay embedding coordinates, [EQUATION].', '1501.06848-2-24-1': 'The theory of embedology shows that if the [MATH]-th mode is a generic observation of the full system [REF], then for sufficiently many lags, [MATH], there exists a diffeomorphism [MATH] such that [MATH].', '1501.06848-2-24-2': 'This statement was first shown for deterministic dynamics on smooth attractors in [CITATION] and subsequently generalized to fractal attractors in [CITATION] and then to non-autonomous systems in [CITATION] and stochastic systems in [CITATION].', '1501.06848-2-24-3': 'The existence of the diffeomorphism [MATH] says that topologically the attractor of the delay reconstruction [MATH] is equivalent to the full dynamics [MATH] and so we have reconstructed the hidden variables in the evolution of the [MATH]-th mode.', '1501.06848-2-24-4': 'However, the theory of embedology only concerns the topology of the attractor, whereas the basis of eigenfunctions [MATH] that are estimated from the diffusion maps algorithm will depend on the geometry of the delay reconstruction [MATH].', '1501.06848-2-25-0': 'In [CITATION] it was shown that the delay reconstruction severely biases the geometry in the reconstructed coordinates [MATH], so that in the limit of infinite delays, [MATH], the dynamics are projected on the most stable component of the dynamics.', '1501.06848-2-25-1': 'Consider the [MATH]-th mode [MATH] as an observation of the state [MATH], where the observation function is given by, [EQUATION].', '1501.06848-2-25-2': 'Notice that the derivative of the observation function is [MATH] where the [MATH] occurs in the [MATH]-th entry.', '1501.06848-2-25-3': 'Moreover, the previous value of the [MATH]-th mode, [MATH], can also be considered an observation of [MATH] where the observation function is given by [MATH] and the map [MATH] is given by the reverse time evolution for the discrete time step [MATH].', '1501.06848-2-25-4': 'Interpreting each [MATH] as an observation of the full state [MATH] at time [MATH] shows that the time-delay embedding [MATH] is an observation of [MATH] with observation function [MATH] as follows, [EQUATION] where [MATH] is the reversed shift map which takes [MATH] to [MATH].', '1501.06848-2-25-5': 'For [MATH] sufficiently large and assuming that the [MATH]-th mode is a generic observable on the manifold, the observation [MATH] will have a full rank derivative [CITATION].', '1501.06848-2-25-6': 'Our goal now is to examine the change in metric induced by this derivative by seeing how it acts on tangent vectors on the attractor.', '1501.06848-2-25-7': 'Let [MATH] be tangent vectors on the attractor and let [MATH] and [MATH] where [MATH] are the transformed tangent vectors in the new geometry given by the time-delay embedding.', '1501.06848-2-25-8': 'Then the inner product [MATH] in the delay embedding space is [EQUATION] where we used the fact that [MATH] and the subscript-[MATH] denotes the [MATH]th component of the corresponding vector.', '1501.06848-2-25-9': 'If [MATH] and [MATH] are in the [MATH]-th Oseledets space, with Lyapunov exponent [MATH], then the inner product reduces to, [MATH].', '1501.06848-2-25-10': 'This shows that the most stable Oseledets space will dominate the geometry in the embedding since [MATH] will be most negative (largest in absolute value of the negative Lyapunov exponents) in the most stable direction.', '1501.06848-2-26-0': 'The bias introduced into the geometry by the delay embedding has several advantages for forecasting.', '1501.06848-2-26-1': 'First, the most stable components of the dynamics are the easiest to predict since trajectories converge very quickly when measured in these components.', '1501.06848-2-26-2': 'Given an initial condition which is a small perturbation of the truth in a stable direction, the perturbation will decrease in the forecast, reducing the effect of the initial perturbation.', '1501.06848-2-26-3': 'Since numerically we will not be able to represent the whole attractor, the variable bandwidth kernel will implicitly project away the small components of the geometry.', '1501.06848-2-26-4': 'By amplifying the stable components we insure that the most desirable aspects of the dynamics will be well represented in the discrete representation of the geometry.', '1501.06848-2-26-5': 'Secondly, when applied to the noisy data, [MATH], the noise will correspond to an unstable component, and so the delay embedding geometry will de-emphasize the noise component of the data.', '1501.06848-2-26-6': 'Formally, consider a noisy observation [MATH] where [MATH] are independent identically distributed random variables independent of [MATH] with [MATH].', '1501.06848-2-26-7': 'When we compute the inner product of noisy vectors in the delay embedding space we find the relative difference, [EQUATION].', '1501.06848-2-26-8': 'Since the noise [MATH] is independent and independent of [MATH] all the terms inside the sum above have expected value of zero, so by the law of large numbers, in the limit as [MATH] we find that for [MATH], [EQUATION] assuming that [MATH] is nonzero as [MATH].', '1501.06848-2-26-9': 'Since we preserve all the pairwise inner product in the embedding space, this implies that we preserve the metric up to a constant.', '1501.06848-2-26-10': 'This shows how the additive observation noise has a smaller impact on the delay embedding than the original embedding, especially for large [MATH].', '1501.06848-2-27-0': 'Finally, note that the delay reconstruction of each mode, [MATH], will capture a different aspect of the most stable components of the dynamics since [MATH] corresponds to the projection of the vectors [MATH] into the [MATH]-th coordinate direction.', '1501.06848-2-27-1': 'Thus, the delay embedded modes [MATH] represent orthogonal components of the most stable Oseledets directions of the dynamics, which in turn are orthogonal to the additive observation noise in the limit of large [MATH].', '1501.06848-2-28-0': 'We now demonstrate the effect of the biased geometry on a simple example.', '1501.06848-2-28-1': 'We will generate a stochastic dynamics on the unit circle by first numerically integrating the one-dimensional SDE, [EQUATION] and then mapping the intrinsic variable [MATH] onto the unit circle embedded in [MATH] by the map [MATH].', '1501.06848-2-28-2': 'In this example we use discrete time step [MATH] to produce 10000 samples [MATH] of the system [REF] which are mapped into the plane as [MATH].', '1501.06848-2-28-3': 'We then generate noisy observations [MATH] where the observation noise [MATH] are independently sampled from a two dimensional Gaussian distribution with mean zero and covariance matrix [MATH].', '1501.06848-2-28-4': 'We chose this example to be very predictable given the cleaning training data [MATH], and the small stochastic component is only included so that even the perfect model can only predict for a finite time.', '1501.06848-2-28-5': 'Notice that the observation noise in this example is very large, as shown in Figure [REF], for example the variance of the noise in the x-coordinate is approximately 32% of the variance of the true signal.', '1501.06848-2-29-0': 'Our goal in this example is to show how the delay-embedding can reduce the influence of observation noise on a diffusion model by biasing the geometry.', '1501.06848-2-29-1': 'We use the first 5000 noisy data points [MATH] to train multiple diffusion models, each time applying a different number of delays before building the diffusion model.', '1501.06848-2-29-2': 'We also train a diffusion model using the clean data set [MATH].', '1501.06848-2-29-3': 'For each diffusion model, we apply the Bayesian filter developed in Section [REF] below to generate initial conditions for forecasting from the noisy observations [MATH] in the verification period [MATH].', '1501.06848-2-29-4': 'Each initial condition in the verification period is forecast for a total of 500 forecast steps (50 time units) and the RMSE between the forecast and the truth is averaged over the verification period.', '1501.06848-2-29-5': 'The RMSE as a function of the number of forecast steps for each model is shown in Figure [REF], along with the forecast of the [MATH] coordinate at the 50 step lead time, compared to the true value of [MATH].', '1501.06848-2-30-0': 'While no amount of delays is able to match the diffusion model built using the clean data set, the improvement in forecast skill is significant as the number of delays increases.', '1501.06848-2-30-1': 'We should note that since this model is intrinsically one dimensional, the most stable component of the dynamics represents the entire dynamical system.', '1501.06848-2-30-2': 'This implies that projecting on the most stable component can only improve the forecasting skill for this example.', '1501.06848-2-30-3': 'For high-dimensional examples, as the number of delays becomes very high, the projection onto the most stable component will remove information which is important to the forecast, leading to reduced forecast skill as shown in Figure [REF] which we will discuss in Section 3 below.', '1501.06848-2-30-4': 'Therefore, in general we expect to see a tradeoff where small numbers of delays can remove the influence of noise, but extremely large numbers of delays may actually degrade the forecast by projecting away valuable components of the dynamics.', '1501.06848-2-31-0': '## Bayesian filter', '1501.06848-2-32-0': 'In Step 2 of the forecasting algorithm described in Section [REF], we assume that the initial distributions for forecasting are given.', '1501.06848-2-32-1': 'In practice, however, we only have the noisy data [MATH], and the goal of this section is to develop a numerical method to determine the initial distribution of the corresponding modes, [MATH].', '1501.06848-2-32-2': 'We will introduce a Bayesian filter to iteratively combine the information from the noisy data with the diffusion forecast.', '1501.06848-2-32-3': 'We should note that the noisy observations which we use for forecasting are always taken out-of-sample, meaning they are separate from the training data.', '1501.06848-2-33-0': 'If the observations were continuous (i.e. for [MATH]), we could filter the noisy observations [MATH] using the Zakai equation projected onto the basis [MATH], and this was the approach taken in [CITATION].', '1501.06848-2-33-1': 'However, since the observation time may be long, we will take a different approach here and apply a Bayesian update at discrete finite time step.', '1501.06848-2-33-2': 'Explicitly, we want to find the posterior density [MATH] by updating the previous posterior density [MATH] with the current noisy observation [MATH].', '1501.06848-2-34-0': 'Our Bayesian filter will follow the standard predictor-corrector approach.', '1501.06848-2-34-1': 'Given an initial density at the previous time, [MATH], the first step is to compute the prior forecast density with our nonparametric model, [EQUATION] where the coefficients [MATH] represent the posterior density at time [MATH], projected in diffusion coordinates, [MATH].', '1501.06848-2-34-2': 'Since the noisy observation [MATH] has the form [MATH] where [MATH], the likelihood function for the noisy observation is, [EQUATION].', '1501.06848-2-34-3': 'We can now assimilate the observation [MATH] by using Bayes law to combine the prior forecast and the likelihood function above as follows, [EQUATION]', '1501.06848-2-34-4': 'By computing the product [REF] we can find the desired posterior at time [MATH], up to the normalization factor.', '1501.06848-2-34-5': 'We estimate the normalization factor, [MATH], as a Monte-Carlo integral, [EQUATION] and dividing the product in [REF] by [MATH] we recover the posterior density at time [MATH].', '1501.06848-2-35-0': 'In order to initialize this iterative procedure, we spin up the filtering process for few assimilation cycles starting from the invariant measure [MATH].', '1501.06848-2-35-1': 'To obtain initial density [MATH], which is used for forecasting starting at time [MATH], we run this Bayesian filter, starting from [MATH] to allow for [MATH] steps of spin-up with the invariant distribution as the initial density, [MATH].', '1501.06848-2-35-2': 'In our numerical simulations below, we used a very short spin-up time of [MATH] steps.', '1501.06848-2-36-0': 'The method developed above requires only that the observation noise is independent and that the likelihood function function [MATH] is known.', '1501.06848-2-36-1': 'The only assumption on the posterior density is that it is well approximated in the truncated basis [MATH], which intuitively implies that the posterior density is smooth and is not very highly oscillatory.', '1501.06848-2-37-0': '# Forecasting weakly chaotic to fully turbulent modes of Lorenz-96 model', '1501.06848-2-38-0': 'As our test example, we consider Fourier modes of the Lorenz-96 (L96) model [CITATION] in various chaotic regimes.', '1501.06848-2-38-1': 'The Lorenz-96 model is defined by the following forced-dissipative nonlinear system of ODEs, [EQUATION]', '1501.06848-2-38-2': 'The right hand side of [REF] consists of an energy conserving quadratic advective-like term, a linear dissipation, and a forcing parameter [MATH].', '1501.06848-2-38-3': 'Following [CITATION], we resolve the L96 model at 40 equally spaced grid points with a periodic boundary (represented by the subscript [MATH] being taken modulo 40) in order to mimic weather wave patterns on a midlatitude belt.', '1501.06848-2-38-4': 'The statistical behavior of the Fourier modes of this model has been analyzed in [CITATION]; as the forcing parameter [MATH] increases, the model becomes fully turbulent with Gaussian-like modal distributions.', '1501.06848-2-38-5': 'In our numerical experiments we will consider three different values of [MATH] following [CITATION].', '1501.06848-2-38-6': 'As reported in [CITATION], for [MATH], the system is weakly chaotic with largest Lyapunov exponent [MATH] and the dimension of the expanding subspace of the attractor is [MATH].', '1501.06848-2-38-7': 'For [MATH], which is the original choice in [CITATION], the system is strongly chaotic with [MATH] and [MATH].', '1501.06848-2-38-8': 'For [MATH], the system is "fully turbulent" with [MATH] and [MATH].', '1501.06848-2-38-9': 'We should also note that when [MATH], the L96 model has a longer "memory" in the sense that it has a relatively slow decaying time (which is defined as the integral of autocorrelation function, see e.g., [CITATION]), this is manifested visibly as a regular dominant westward propagating "weather-like" wave pattern of wavenumber-8.', '1501.06848-2-38-10': 'The memory becomes shorter as [MATH] increases and the "weather-like" wave pattern becomes less obvious.', '1501.06848-2-39-0': '## Diagnostic models', '1501.06848-2-40-0': 'In our numerical experiments we will examine the forecasting skill of the nonparametric modeling approach on a selection of Fourier modes, including those with high and low energies as well as large and small correlation times.', '1501.06848-2-40-1': 'To diagnose the forecasting skill, we compare the diffusion forecast on these modes with forecasts from standard statistical approaches when the underlying dynamics are unknown as well as with the perfect model.', '1501.06848-2-40-2': 'In particular, we will consider:', '1501.06848-2-41-0': '### Persistence model', '1501.06848-2-42-0': 'The simplest nonparametric model for predicting the future when the underlying dynamics are not known is by setting the current observation, [MATH], as the forecast at each lead time.', '1501.06848-2-43-0': '### Linear autoregressive models', '1501.06848-2-44-0': 'A popular statistical modeling approach when the underlying dynamics are not known is to fit the data to a class of linear autoregressive models.', '1501.06848-2-44-1': 'In this paper, we consider two models from such a parametric modeling approach that were designed previously as cheap filter models: the Mean Stochastic Model (MSM) [CITATION] and the stable constrained autoregressive model of order-3 (AR3) [CITATION].', '1501.06848-2-45-0': 'The key idea of MSM is to model each Fourier mode as a complex valued linear stochastic model with additive white noise, [EQUATION] where [MATH] is a complex valued Wiener noise with variance [MATH].', '1501.06848-2-45-1': 'The parameters [MATH] and [MATH] are determined by matching the analytical expression for the variance and correlation time at the statistical equilibrium state of the MSM model in [REF] with the corresponding empirically estimated statistics from the data (see Chapter 12 of [CITATION] for a detailed formulation).', '1501.06848-2-45-2': 'In our implementation below, we will obtain these parameters from the statistics of the noisy dataset, [MATH], during the training period.', '1501.06848-2-45-3': 'To generate initial conditions for the forecast, we apply a one-dimensional Kalman filter in order to account for the observation noise in [MATH], and we assume that the noise observation variance, [MATH], is known.', '1501.06848-2-46-0': 'The stable and consistent AR3 model was introduced in [CITATION] as a cheap linear model for filtering turbulent modes with long memory, such as the L96 model with [MATH].', '1501.06848-2-46-1': 'The standard AR3 model is given by, [EQUATION]', '1501.06848-2-46-2': 'The stability of this model is sensitive to the choice of sampling time [MATH], especially when a standard linear regression is used to determine the model parameters.', '1501.06848-2-46-3': 'In [CITATION], it was shown that one can obtain accurate filtered mean estimates in a linear filtering problem with truth generated from [REF] if the parameters [MATH] are chosen to satisfy a set of linear algebraic consistency conditions which depends on [MATH] of [REF] and the sampling time [MATH] (see [CITATION] for details).', '1501.06848-2-46-4': 'For nonlinear problems, a stable model together with these consistency constraints can be determined by an algebraic method proposed in [CITATION] and this AR3 model is what we use in the examples below.', '1501.06848-2-46-5': 'In particular, we will enforce the consistency condition for the AR3 model using the parameter [MATH] obtained from the MSM fit of the noisy data at the training period.', '1501.06848-2-46-6': 'To generate initial conditions for the forecast, we apply a three-dimensional Kalman filter to account for the noise observed [MATH], assuming that the observation noise variance, [MATH], is known (see e.g. [CITATION] for the details of the AR filter).', '1501.06848-2-47-0': '### Perfect model', '1501.06848-2-48-0': 'Finally, we also include the forecast of the perfect model in [REF].', '1501.06848-2-48-1': 'To generate the initial conditions, we implement a standard ensemble Kalman filter method [CITATION] with 80 ensemble members, double the dimension of the L96 model.', '1501.06848-2-48-2': 'The perfect model experiment with full data refers to forecasting skill where the initial conditions are determined by filtering noisy observations at all 40 grid points.', '1501.06848-2-49-0': 'We also show forecasting skill with a perfect model experiment given only observations of noisy modes [MATH].', '1501.06848-2-49-1': 'In this scenario, the ensemble Kalman filter estimates the initial conditions at unobserved modes which will be needed for integrating the full model; obviously, if every mode is observed, we are back to the perfect model experiment with full data.', '1501.06848-2-49-2': 'Comparing this scenario with the diffusion forecast constructed from the same [MATH] provides the most objective comparison between the full model and the nonparametric diffusion model.', '1501.06848-2-49-3': 'In the example below, we will consider examples where [MATH] consists of a single mode and the three most energetic modes, [MATH].', '1501.06848-2-50-0': '## Experimental design', '1501.06848-2-51-0': 'In our numerical simulations, the true time series are generated by integrating the L96 model with the fourth-order Runge-Kutta method with time step [MATH].', '1501.06848-2-51-1': 'Noisy observations are generated by adding independent random samples of a Gaussian with mean zero and variance [MATH] to the true solutions at every grid point at each observation time step [MATH] as in [REF], resulting in a noise variance of [MATH] on each Fourier mode.', '1501.06848-2-51-2': 'Given a time series of length 15000, we use the first 5000 data points to train the three models (diffusion, MSM, and AR3) and we compute the prediction skill on the remaining [MATH] data points.', '1501.06848-2-51-3': 'To diagnose the prediction skill, we use the standard Root-Mean-Squared Error (RMSE) and anomaly pattern correlation skill defined as follows, [EQUATION] where [MATH] denotes the truth and [MATH] denotes the forecast at lead time [MATH].', '1501.06848-2-51-4': 'As defined in [CITATION], the anomalies are defined with respect to the climatology which is empirically estimated by time average of the truth, [MATH].', '1501.06848-2-51-5': 'In our evaluation below, we will evaluate the forecast skill on Fourier modes [MATH], where [MATH]; the climatological means of these nonzero wave numbers are zero.', '1501.06848-2-51-6': 'We will also evaluate these measures on the 40-dimensional spatial domain solutions and in this case, the climatological mean [MATH] is nonzero in our example.', '1501.06848-2-52-0': 'The nonparametric modeling method has two nuisance parameters, the number of lags [MATH] in the time-delay embedding and the number of diffusion modes, [MATH].', '1501.06848-2-52-1': 'As discussed in Section [REF] a large number of lags can reduce the influence of independent identically distributed observation noise and also biases the geometry towards the stable component of the dynamics.', '1501.06848-2-52-2': 'In the example in Section [REF] we saw that a large number of delays could significantly reduce the influence of observation noise on a one-dimensional dynamical system.', '1501.06848-2-52-3': 'In Figure [REF] we show that for higher-dimensional dynamical systems, there is a limit to the number of lags which can be used to reduce the influence of noise.', '1501.06848-2-52-4': 'This limit is explained by the projection of the geometry onto the most stable component of the dynamics, which for large lags will practically eliminate less stable components which also contain information necessary for forecasting.', '1501.06848-2-52-5': 'The number of lags that can be usefully included is limited by the size of the most stable Lyapunov exponent, since this controls the exponential bias of the delay-embedding towards the most stable component as shown in Section [REF].', '1501.06848-2-52-6': 'Notice that in Figure [REF], mode-18 appears to be more sensitive to the number of lags because the noise is much larger as a percentage of the variance.', '1501.06848-2-52-7': 'In each case the standard deviation of the observation noise is [MATH], and the standard deviation of the mode is shown by the RMSE of the corresponding invariant measure.', '1501.06848-2-52-8': 'In each case the optimal number of lags is between [MATH] and [MATH], which is consistent with each mode being a generic observation of the same Lorenz-96 dynamical system, and hence each mode has the same most stable Lyapunov exponent.', '1501.06848-2-52-9': 'In the remaining numerical experiments in this section, we show results with [MATH] lags in the time-delay embedding.', '1501.06848-2-53-0': 'The other nuisance parameter is the choice of the number of diffusion modes, [MATH].', '1501.06848-2-53-1': 'In Figure [REF] we show the forecasting skill for various choices of [MATH] for an individual mode-8 for the case of weak turbulence [MATH].', '1501.06848-2-53-2': 'Notice that while the initial errors increase as fewer modes are used, the forecast skill at later times are not very different.', '1501.06848-2-53-3': 'For deterministic problems, our experience suggests that larger [MATH] help improve the prediction skill.', '1501.06848-2-53-4': 'For stochastic problems, however, the error estimate between [MATH] and [MATH] derived in [CITATION] suggests that a large data set is required to obtain accurate estimation of the eigenfunctions associated with these high modes.', '1501.06848-2-53-5': 'At the moment, a systematic method for choosing [MATH] is still under investigation and for the remainder of this paper, we simply set [MATH] for every mode.', '1501.06848-2-53-6': 'Figures [REF] and [REF] indicate that the diffusion forecast is not very sensitive to these nuisance parameters for turbulent modes, and these parameters can be tuned using cross-validation on the training data set.', '1501.06848-2-53-7': 'Also, there is clearly some improvement with appropriate choice of lag [MATH] relative to no delay coordinate embedding.', '1501.06848-2-53-8': 'Even with such empirical choices of parameters ([MATH] and [MATH]), we will see that the forecasting skill is still competitive to those of the perfect model when only the modeled modes are observed.', '1501.06848-2-54-0': '## Single-mode observations', '1501.06848-2-55-0': 'In Figure [REF], we show the RMSE and pattern correlation for forecasting three different Fourier modes of the L96 model in a weakly chaotic regime with [MATH]: mode-8 carries the largest energy with a relatively short correlation time; mode-13 carries low energy with a relatively long correlation time; and mode-18 carries low energy with a relatively short correlation time.', '1501.06848-2-55-1': 'Note that the ratios between the observation noise variance [MATH] and the energy of the modes are very different for these three modes; [MATH] for mode-8, [MATH] for mode-13, and [MATH] for mode-18.', '1501.06848-2-55-2': 'Therefore, we expect that the noise does not affect the forecasting skill of modes-8 and 13 as much as that of mode-18.', '1501.06848-2-56-0': 'As expected, the perfect model with full observation provides the best forecast on all modes.', '1501.06848-2-56-1': 'The diffusion forecast performs very well on all modes, outperforming the perfect model when given equal information, meaning only observations of the corresponding individual mode (which is referred as "perfect sparse obs" in Figure [REF]) on modes 8 and 18.', '1501.06848-2-56-2': 'For these modes, both autoregressive models (MSM, AR3) have very little predictive skill.', '1501.06848-2-56-3': 'On mode-13, the forecasting skill of all four models (AR-3, MSM, perfect sparse obs, and diffusion models) are not very different.', '1501.06848-2-56-4': 'The persistence model always produces severely biased forecasts in the long run, since fixing a particular observation will always be worse than predicting the statistical mean in a long term forecast.', '1501.06848-2-57-0': 'In Figures [REF], we compare the forecast time series of each model to the truth for forecast lead time of 2 model time units.', '1501.06848-2-57-1': 'This comparison is made for mode-8 with [MATH] for verification time steps between [MATH] and [MATH].', '1501.06848-2-57-2': 'Notice that the diffusion model forecast underestimates many of the peaks slightly whereas the perfect model initialized with the corresponding mode (perfect partial obs) overestimates many of the peaks.', '1501.06848-2-57-3': 'The persistence model forecasts are clearly out-of-phase.', '1501.06848-2-57-4': 'On the other hand, both the MSM and AR3 models significantly underestimate the amplitude of the signals and in the long term they simply forecast zero, as expected since these forecasts represent the mean of linear unbiased autoregressive models.', '1501.06848-2-58-0': 'We also report the forecast skills of the more chaotic regime with [MATH] in Figure [REF] and the fully turbulent regime with [MATH] in Figure [REF].', '1501.06848-2-58-1': 'Based on the RMSE and pattern correlation measures, these results suggest that the diffusion model still produces the most skillful forecasts compared to the MSM, AR3, the perfect model partial obs and persistence models on mode-8.', '1501.06848-2-58-2': 'On higher wave numbers, the forecasting skill diminishes as [MATH] increases and the advantage of the nonparametric model over the parametric models become negligible.', '1501.06848-2-59-0': 'The numerical experiments in this section suggest that the diffusion model is competitive with the perfect model when the available data is the corresponding noisy observations of the single mode that is being modeled.', '1501.06848-2-59-1': 'We also find that the diffusion model is most skillful when modeling the highly energetic modes.', '1501.06848-2-59-2': 'We should note that as [MATH] increases, although mode-8 is still the most energetic mode, the energy is less dominant relative to the other modes in these turbulent regimes [CITATION].', '1501.06848-2-59-3': 'We suspect that this is one of the causes of degradation in the forecasting skill when [MATH] is larger, in addition to the more quickly decaying correlation functions in the fully turbulent regimes.', '1501.06848-2-60-0': '## Multi-modes observations', '1501.06848-2-61-0': 'Now, we consider forecasting the three most energetic modes, [MATH].', '1501.06848-2-61-1': 'There are two strategies that can be adopted in the diffusion modeling: 1) Learning the 3-modes diffusion model directly; 2) Concatenate the three independent one-mode diffusion models (we refer to this strategy as Concatenated DM or CDM).', '1501.06848-2-61-2': 'Obviously, strategy 1) requires more data for accurate estimation since the dimensionality of the underlying dynamics that are being modeled increases.', '1501.06848-2-61-3': 'In our numerical experiment, however, we do not increase the amount of data, instead we use the same 5000 data points that were used in building the individual one-mode diffusion models.', '1501.06848-2-62-0': 'In Figure [REF], we see that while the perfect model given full data is still the best model, when only these three modes data are available, the forecasting skill of the perfect model is comparable to the concatenated three-modes diffusion model.', '1501.06848-2-62-1': 'The concatenated diffusion models (CDM) is slightly better than the three-modes DM since the underlying dynamics that are being modeled increases in CDM.', '1501.06848-2-62-2': 'In this case, these nonparametric approaches significantly supersede the persistence model.', '1501.06848-2-63-0': 'Finally, we also report the overall performance of the diffusion forecast given data of all Fourier modes in Figure [REF].', '1501.06848-2-63-1': 'Since constructing the full 40-dimensional diffusion model is beyond the computational capacity, we just show results where we concatenate all 21 independently constructed one-mode diffusion models.', '1501.06848-2-63-2': 'In this verification, the quantifying RMSE [REF] and pattern correlation [REF] are measured in the spatial domain.', '1501.06848-2-63-3': 'While the diffusion forecast is not expected to be comparable to the perfect model given full observation data, one can see that it still produces some skillful forecast beyond the persistence model and the skill diminishes as [MATH] increases.', '1501.06848-2-64-0': 'From these numerical experiments, one can see that if the only available data is an individual mode or few modes that are being predicted, the forecasting skill of the diffusion model is indeed competitive with that of the perfect model.', '1501.06848-2-64-1': 'The degrading performance of the perfect model is due to inaccurate estimation of initial conditions at unobserved modes.', '1501.06848-2-64-2': 'When the full data is available, this issue disappears in the perfect model experiment.', '1501.06848-2-64-3': 'Given all the observations, the nonparametric diffusion model does not have capability to represent the full nontrivial interactions of the dynamics that can only be achieved given the full model with accurate initial conditions.', '1501.06848-2-64-4': 'This degradation is totally expected since the nonparametric model is constructed with a short time series, and as we mentioned in the introduction, even if a large data set is available, the nonparametric model is subject to the curse-of-dimensionality.', '1501.06848-2-65-0': '# Forecasting geophysical turbulent modes', '1501.06848-2-66-0': 'In this section, we consider forecasting turbulent modes of the quasigeostrophic model, a prototypical model for midlatitude atmosphere and oceanic dynamics [CITATION].', '1501.06848-2-66-1': 'We consider a two-layer quasigeostrophic (QG) model which is externally forced by a mean shear with streamfunctions [EQUATION] such that it exhibits baroclinic instabilities; the properties of the turbulent cascade has been extensively discussed in this setting (see e.g., [CITATION] and citations in [CITATION]).', '1501.06848-2-67-0': 'The governing equations for the two-layer QG model with a flat bottom, rigid lid, and equal depth [MATH] are given by, [EQUATION] where [MATH] denotes the perturbed streamfunction, subscript 1 corresponds to the upper layer and subscript 2 corresponds to the bottom layer.', '1501.06848-2-67-1': 'In [REF]-[REF], [MATH] is the meridional gradient of the Coriolis parameter; [MATH] is the Ekman bottom drag coefficient; [MATH] is a Jacobian function which acts as nonlinear advection; [MATH] is the zonal mean shear, selected so that the QG equations exhibit baroclinic instability with a turbulent cascade; [MATH] is the hyperviscosity coefficient, chosen so that [MATH] filters out the energy buildup on smaller scales when finite discretization is enforced.', '1501.06848-2-67-2': 'The perturbed QG potential vorticity [MATH] is defined for each layer by [EQUATION]', '1501.06848-2-67-3': "The parameter [MATH] gives the wavenumber associated with radius of deformation, or Rossby radius, [MATH] (the scale at which Earth's rotation becomes significant to the dynamics of the system).", '1501.06848-2-68-0': 'In our numerical simulations, the true signal is generated by resolving ([REF])-([REF]) with [MATH] Fourier modes, which corresponds to the [MATH] grid points.', '1501.06848-2-68-1': 'This model has two important nondimensional parameters: [MATH] where [MATH] is the horizontal non-dimensionalized velocity scale and [MATH] is the horizontal domain size in both directions (we choose [MATH]), and [MATH] which is inversely proportional to the deformation radius [CITATION].', '1501.06848-2-68-2': 'As in [CITATION], we will consider two cases: one with a deformation radius [MATH] such that [MATH] which roughly mimics a turbulent jet in the midlatitude atmosphere and the other case will have a deformation radius [MATH] such that [MATH] which roughly mimics a turbulent jet in the ocean.', '1501.06848-2-68-3': 'For the ocean case, the QG model in [REF]-[REF] is numerically very stiff since the term that involves [MATH] is large.', '1501.06848-2-69-0': 'The large-scale components of this turbulent system are barotropic and for the two-layer model with equal depth, the barotropic streamfunction is defined as an average between the two layers, [MATH] [CITATION].', '1501.06848-2-69-1': 'In the atmospheric regime with [MATH], the barotropic flow [MATH] is dominated by a strong zonal jet while in the ocean case with [MATH] there are transitions between zonal flow and large-scale Rossby waves (e.g., see [CITATION]).', '1501.06848-2-70-0': 'In our numerical examples, we build a diffusion model for the two-dimensional Fourier modes [MATH] of the barotropic streamfunction, [MATH].', '1501.06848-2-70-1': 'Following the experiments in [CITATION], our choice to only model this mode is mainly due to the fact that small-scale observations of a turbulent jet are typically not available, especially in the ocean.', '1501.06848-2-70-2': 'For diagnostic purposes, we compare the diffusion model to a simple stochastic model with combined white and colored additive and multiplicative noises that was designed as a test filter model for stochastic parameterization in the presence of model error [CITATION].', '1501.06848-2-70-3': 'The governing equation of this model is given as follows, [EQUATION] where variable [MATH] models the Fourier mode of [MATH], dropping the horizontal and vertical wave components [MATH] to simplify the notation.', '1501.06848-2-70-4': 'The equation governing [MATH] represents the interactions between this resolved mode and the remaining unresolved modes by several noise terms.', '1501.06848-2-70-5': 'The first noise term, [MATH], is a real valued multiplicative noise.', '1501.06848-2-70-6': 'The second noise term is [MATH], which is an additive complex valued noise governed by the Ornstein-Uhlenbeck mean reverting SDE.', '1501.06848-2-70-7': 'The final noise term is [MATH], which is an additive white noise.', '1501.06848-2-70-8': 'In [REF], the stochastic terms, [MATH], are complex valued and [MATH] are real valued Wiener processes.', '1501.06848-2-70-9': 'We should note that while the simple system in [REF] is nonlinear (due to the multiplicative term [MATH]), it has analytical statistical solutions [CITATION] which we will utilize in our numerical experiments here.', '1501.06848-2-71-0': 'In [CITATION], they implemented a Kalman update on the analytical mean and covariance solutions of [REF] and called this filtering method SPEKF [CITATION], which stands for Stochastic Parameterized Extended Kalman Filter.', '1501.06848-2-71-1': 'In our implementation below, we will use SPEKF to generate initial conditions for our forecast at the verification times.', '1501.06848-2-71-2': 'As in any standard parametric modeling approach, one of the main issue is in choosing parameters [MATH] in [REF] to produce high forecasting skill.', '1501.06848-2-71-3': 'We use parameters that are tuned to optimize the filtered solutions as in [CITATION].', '1501.06848-2-71-4': 'We found that this set of parameters produce better forecast compared to that obtained by an adaptive estimation method such as [CITATION] (results are not shown).', '1501.06848-2-71-5': 'Even though the parameters are empirically chosen, we shall see that the forecasting skill of the SPEKF model can be competitive with the diffusion forecast in some regimes.', '1501.06848-2-72-0': 'In our numerical experiments, we generate a time series of 9000 data points at discrete time step [MATH] and added Gaussian noise in the physical space at 36 regularly spaced grid points in the spatial domain as in [CITATION] with noise variance [MATH].', '1501.06848-2-72-1': 'In Fourier space, the observation error variance is [MATH]; in Figure [REF], we show [MATH] relative to the variance of the 12 modes corresponding to the 36 regularly spaced observed grid points for both the atmospheric and ocean regimes.', '1501.06848-2-72-2': 'We use the first 5000 noisy data points to train the diffusion model and the remaining 4000 data points to compute the forecasting skill measures.', '1501.06848-2-72-3': 'For the diffusion forecast model, we empirically choose [MATH] for the atmospheric regime and [MATH] for the ocean regime.', '1501.06848-2-72-4': 'In all experiments, we use [MATH] diffusion modes.', '1501.06848-2-72-5': 'Note that one can compare the results with the perfect model forecasts, filtering the same 36 observations to generate initial conditions.', '1501.06848-2-72-6': 'In previous work [CITATION] it was shown that while the initial conditions for the atmospheric regime can be accurately specified with an appropriate EnKF scheme, in the ocean regime the initial conditions are not easily estimated due to numerical stiffness.', '1501.06848-2-72-7': 'Since finding accurate initial conditions for the perfect model is not the main point this paper, we neglect showing the perfect model experiment in this section.', '1501.06848-2-72-8': 'Instead, we just compare it with the truth which readily represents the best forecasting scenario.', '1501.06848-2-73-0': 'In Figures [REF], we show the RMSE and pattern correlation for the first two energetic modes in the atmospheric regime.', '1501.06848-2-73-1': 'Here, the first two energetic modes correspond to the two-dimensional horizontal wave numbers [MATH] and [MATH] with explained variances of about 65% and 73%, respectively, so the behavior is dominated by the zonal jet in the horizontal direction [CITATION].', '1501.06848-2-73-2': 'In this regime, notice that the persistence model produces high forecasting skill with a very long correlation for the first mode.', '1501.06848-2-73-3': 'However, the persistence model is not useful in forecasting the second mode.', '1501.06848-2-73-4': 'In contrast, the parametric SPEKF model produces biased long-time forecast on the first mode but more accurate forecast on the second mode.', '1501.06848-2-73-5': 'In this regime, the diffusion forecasting skill is quite robust for both modes with best RMSE and PC scores.', '1501.06848-2-73-6': 'In Figure [REF], we show the corresponding forecast time series of the real component of these two most energetic modes at lead times 0, 4, and 8 model time units.', '1501.06848-2-73-7': 'Notice that the Bayesian filter works reasonably well, indicated by the accurate recovery at lead time-0.', '1501.06848-2-73-8': 'The forecast skills of SPEKF and diffusion forecast for the first mode are relatively similar and very skillful at lead times 4 and 8 with PC high scores, above 0.9.', '1501.06848-2-73-9': 'For the second mode, on the other hand, one can see the forecasts of both models become less accurate at these lead times with very low PC, less than 0.5.', '1501.06848-2-73-10': 'Here, notice that SPEKF forecasts are nearly zero.', '1501.06848-2-74-0': 'In Figures [REF], we show the RMSE and pattern correlation for the first two energetic modes in the ocean regime.', '1501.06848-2-74-1': 'Here, the first two energetic modes correspond to the two-dimensional horizontal wave numbers [MATH] and [MATH] with explained variance of about 50% and 97%, respectively, so the behavior is dominated by two competing modes, the Rossby mode [MATH] and the zonal jet mode [MATH] [CITATION].', '1501.06848-2-74-2': 'In this regime, the persistence model forecast has no skill at all on the first mode.', '1501.06848-2-74-3': 'On the second mode, however, the persistence forecast is quite reliable in the short term.', '1501.06848-2-74-4': 'The diffusion forecast is better than SPEKF on the first mode, but slightly worst than SPEKF on the second mode (indicated by worse PC score beyond lead time 2).', '1501.06848-2-74-5': 'In Figures [REF], we show the corresponding forecast time series of the real component of these two most energetic modes for lead times 0, 1, and 2.', '1501.06848-2-74-6': 'At these lead times, where the PC score is at least 0.6, we expect the forecasting skill to be comparable on the first mode while SPEKF should be slightly better than the diffusion forecast on the second mode.', '1501.06848-2-75-0': 'In Figure [REF], we report the RMSE and pattern correlation computed over the [MATH] observed grid points; obtained by inverse Fourier transforming 12 independent models for the corresponding modes.', '1501.06848-2-75-1': 'In terms of filtering, notice that the estimates from both the SPEKF and diffusion models are much more accurate compare to the estimates from the persistence model whose error is nothing but the observation noise standard deviation.', '1501.06848-2-75-2': 'In both regimes, the overall performances in terms of RMSE and PC scores from the best to the worst are the Diffusion model, SPEKF, and the persistence model.', '1501.06848-2-75-3': 'Notice that the forecasting skill of the diffusion model and SPEKF are competitive in the very short term forecast.', '1501.06848-2-75-4': 'In the long term, however, while the RMSE of the diffusion forecast always converges to the climatological standard deviation, SPEKF RMSE exceeds this climatological standard deviation, indicating a biased forecast.', '1501.06848-2-75-5': 'From the simulations in this section on a nontrivial prototypical example of midlatitude geophysical turbulent flows, we see that the proposed non-parametric diffusion models produce robust, skillful, and unbiased long term forecasts.', '1501.06848-2-76-0': '# Summary', '1501.06848-2-77-0': 'In this paper, we applied the nonparametric diffusion modeling approach proposed in [CITATION] to forecast Fourier modes of turbulent dynamical systems given a finite set of noisy observations of these modes.', '1501.06848-2-77-1': 'Motivated by the results in [CITATION], we built the nonparametric model in the delay embedding coordinates in order to account for the interaction between the resolved and unresolved modes as well as to smooth out the noisy observations.', '1501.06848-2-77-2': 'We empirically verify that larger [MATH] helps to reduce the influence of the observation noise, however our theoretical derivation suggests that taking [MATH] too large will project away less stable components of the dynamics.', '1501.06848-2-77-3': 'The choice of [MATH] should not be so large as to project away the unresolved modes, but large enough to smooth out the noise in the training data set, and currently this selection requires empirical tuning.', '1501.06848-2-77-4': 'To initiate each forecast, we introduced a simple Bayesian filtering method which accounts for the information of the noisy observations up to the current time.', '1501.06848-2-77-5': 'We compared the forecasting skills of the proposed nonparametric model with various standard modeling approaches when the underlying dynamics are unknown, including the persistence model, linear autoregressive models of order-1 (MSM) and order-3, the SPEKF model which involves combined additive and multiplicative noise terms.', '1501.06848-2-77-6': 'We also compared the diffusion forecast to the perfect model forecast in the L96 example above.', '1501.06848-2-77-7': 'In this example, we found that when the only available observations are the low-dimensional set of the modes that are being modeled, then the diffusion forecasting method is competitive with the perfect model.', '1501.06848-2-77-8': 'When full data (all modes) becomes available, the curse-of-dimensionality limits the ability of diffusion model to represent nontrivial interactions of the underlying dynamics that can only be achieved given the perfect model and accurate initial conditions.', '1501.06848-2-77-9': 'Overcoming this limitation of the diffusion forecast is a significant challenge which we hope to address in future work.', '1501.06848-2-77-10': 'From the QG experiments, we found that while the short-term forecast of the parametric model, SPEKF, is competitive with that of the diffusion model, the latter produces more robust and unbiased long term forecasts.', '1501.06848-2-78-0': 'One important thing to take away from this paper is that although the proposed nonparametric model seems to be competitive with the standard parametric models simulated here and even with the perfect model with partially observed data, we do not claim that this equation-free approach is the model to use for everything.', '1501.06848-2-78-1': 'If one knows the physics of the underlying dynamics (or the appropriate reduced parametric models), then one should use the physics-based model rather than the equation-free diffusion models.', '1501.06848-2-78-2': 'Indeed, it was shown rigorously in a simple setup that optimal filtering and accurate statistical prediction can be achieved with an appropriate stochastic parametric modeling of the unresolved scales [CITATION].', '1501.06848-2-78-3': 'However, there are at least two valuable aspects of the nonparametric model we develop here.', '1501.06848-2-78-4': 'First, one can use this model to diagnose whether his/her modeling approach is appropriate; we expect that appropriate physics-based models should beat this black box approach.', '1501.06848-2-78-5': 'Second, when an appropriate, physically motivated model is not available, then this approach will often outperform adhoc parametric models as demonstrated in this paper.', '1501.06848-2-78-6': 'Moreover, in practice it is difficult to guess the appropriate choice of parametric models even if some physics-based knowledge has been imposed in deriving a reduced model (e.g., in turbulent closure problems, one typically uses parameters to represent some small-scale processes or some physical processes which are not well understood).', '1501.06848-2-78-7': 'In this situation, we propose to extract the time series of these processes (if possible) and to subsequently build nonparametric models for such processes.', '1501.06848-2-78-8': 'This idea is what we refer to as the semi-parametric modeling approach and is introduced in the companion paper [CITATION].'}

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[]

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['1501.06848-1-17-0', '1501.06848-2-20-0', '1501.06848-2-40-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1501.06848

cond-mat-0510418

{'cond-mat-0510418-1-0-0': 'We present scaling estimates for characteristic times [MATH] and [MATH] of pulling ideal linear and randomly branched polymers of molecular weight [MATH] into a small hole by a force [MATH].', 'cond-mat-0510418-1-0-1': 'We show that the adsorption process develops as sequential straightening of folds of the initial polymer configuration.', 'cond-mat-0510418-1-0-2': 'On the basis of the forces balance analysis and estimating the typical size of the fold involved into the motion, we arrive at the following predictions: [MATH] and [MATH].', 'cond-mat-0510418-1-0-3': 'We also confirm our finding by the molecular dynamics experiment.', 'cond-mat-0510418-1-1-0': 'PACS: 05.40.-a; 05.70.-a; 87.15.', 'cond-mat-0510418-1-1-1': 'He', 'cond-mat-0510418-1-2-0': 'Adsorption of an ideal polymer chain at a potential well is one of the classical problems in the statistical physics of macromolecules used by the researchers for over a quarter of a century as a test ground for a variety of theories.', 'cond-mat-0510418-1-2-1': 'For a homopolymer, this is a clean simple yet interesting example of an ordinary second order phase transition [CITATION].', 'cond-mat-0510418-1-2-2': 'By contrast, for a heteropolymer, when chemical structure of a chain is quenched but not uniform, the problem of adsorption is nontrivial, its understanding is still incomplete - see, for example, [CITATION].', 'cond-mat-0510418-1-2-3': 'Furthermore, random heteropolymer adsorption at a point-like potential well is mathematically similar to the localization (pinning) transition in the solid-on-solid models with quenched impurities [CITATION].', 'cond-mat-0510418-1-3-0': 'Although equilibrium properties of adsorption of linear ideal polymers are addressed by many theorists (see [CITATION] and references therein), surprisingly the dynamics of the adsorption process is almost not discussed at all.', 'cond-mat-0510418-1-3-1': 'The related issue of the dynamics of coil-globule transition attracted steady interest over the years [CITATION], but the similarity between these two problems is limited, because globule formation after an abrupt solvent quench begins by a simultaneous chain condensation on many independent nucleation centers, while the adsorption process can be viewed as pulling (sacking) of a polymer with a constant force into a single hole.', 'cond-mat-0510418-1-3-2': 'Hence, only a single attractive center causes the chain condensation.', 'cond-mat-0510418-1-3-3': 'Similar problem arises when a force pulls polymer into a membrane channel, such as in DNA translocation [CITATION].', 'cond-mat-0510418-1-3-4': 'As we shall see, some scaling estimates obtained in [CITATION] could be directly transferred to the kinetics of polymer adsorption.', 'cond-mat-0510418-1-4-0': 'To describe in words the mechanism of polymer condensation into a hole, let us imagine a long rope randomly dropped at the desk top.', 'cond-mat-0510418-1-4-1': 'Pull now the rope by its end down from the desk edge.', 'cond-mat-0510418-1-4-2': 'It is obvious, and can be established through an experiment accessible even to theorists, that the rope does not move all at once.', 'cond-mat-0510418-1-4-3': 'What happens is only a part of the rope moves at the beginning, such that a fold closest to the end straightens out.', 'cond-mat-0510418-1-4-4': 'As soon as the first fold is straightened, it then transmits the force to the next fold, which starts moving, and so on.', 'cond-mat-0510418-1-4-5': 'The system sequentially straightens the folds of different scales, such that only parts of the rope about the size of a current fold are involved in the motion, while the rest of the rope remains immobile.', 'cond-mat-0510418-1-5-0': 'In the present letter we show how this mechanism applies to a homopolymer condensation dynamics and how it allows to shed light on the physics behind scaling estimates [CITATION] for the relaxation times for both linear, [MATH], and randomly branched, [MATH], polymers.', 'cond-mat-0510418-1-6-0': 'We address the problem on the simplest level of an ideal Rouse homopolymer of [MATH] monomers, each of size [MATH], embedded into an immobile solvent with the viscosity [MATH].', 'cond-mat-0510418-1-6-1': 'One end of the chain is put in the potential well (the "hole"), and the polymer is pulled down to the hole with constant force [MATH], acting locally on just one single monomer, the one currently entering the hole.', 'cond-mat-0510418-1-6-2': 'We will be interested in the relaxation time [MATH] of the complete pulling down of the entire polymer into the hole.', 'cond-mat-0510418-1-6-3': 'In particular we want to know how [MATH] depends on polymer length [MATH] and the pulling force [MATH].', 'cond-mat-0510418-1-7-0': 'Neglecting inertia, the balance of forces reads [MATH], where friction force [MATH] should be linear in velocity [MATH], with [MATH] being the number of monomers "swallowed by the hole" by the time [MATH].', 'cond-mat-0510418-1-7-1': 'More subtly, friction coefficient must be proportional to the number of monomers [MATH] currently involved in the forced motion (remember experiment with pulling rope from the desk): [MATH].', 'cond-mat-0510418-1-7-2': 'Our main task will be to estimate the length [MATH] involved in the motion for either linear or branched polymers.', 'cond-mat-0510418-1-8-0': 'In order to clarify our approach, let us compute the characteristic time [MATH] of pulling down of a linear polymer starting from a fully stretched conformation.', 'cond-mat-0510418-1-8-1': 'Force balance condition reads in this case [EQUATION] where we assumed that all [MATH] monomers not absorbed by the time [MATH] are involved in the motion, and, therefore, contribute to friction: [MATH].', 'cond-mat-0510418-1-8-2': 'Separating the variables and integrating ([REF]) in the intervals [MATH] and [MATH], we get [EQUATION]', 'cond-mat-0510418-1-8-3': 'In terms of dependence on [MATH], this result is similar to the Rouse relaxation time of a linear polymer, [MATH], where [MATH], and [MATH] is temperature.', 'cond-mat-0510418-1-8-4': 'Our estimate, ([REF]), remains valid as long as [MATH], because in this case the initially straight polymer has no time to relax and coil up on itself while being pulled up by the force [MATH].', 'cond-mat-0510418-1-8-5': 'Given the expression for [MATH], ([REF]), the condition [MATH] translates into [MATH], the latter making an obvious sense: pulling force should be strong enough to cause significant stretching of every bond, or, in other words, Pincus blobs [CITATION] must be as small as of the order of one monomer.', 'cond-mat-0510418-1-9-0': 'Now we turn to the dynamics of adsorption of a linear ideal polymer starting from a more natural initial configuration of a Gaussian coil, and look at [MATH].', 'cond-mat-0510418-1-9-1': 'To begin with, let us consider a simple scaling estimate [CITATION].', 'cond-mat-0510418-1-9-2': 'First of all, we note that the only time scale relevant for the problem is the above mentioned Rouse time [MATH], which is the relaxation time of the coil, or, in other words, it is the time during which the coil as a whole diffuses over the distance about its own size, [MATH].', 'cond-mat-0510418-1-9-3': 'With [MATH] being the solitary relevant time scale, the sacking time should be written in the scaling form [EQUATION] where the factor [MATH] depends on all other relevant parameters only in the dimensionless combination [MATH].', 'cond-mat-0510418-1-9-4': 'The second step follows from the fact that the speed of the process, [MATH], must be linear in the applied force, which means that the function [MATH] should be inversely proportional to its argument, [MATH].', 'cond-mat-0510418-1-9-5': 'This then yields the answer [EQUATION]', 'cond-mat-0510418-1-9-6': 'Scaling argument yields the answer, but does not give much insight.', 'cond-mat-0510418-1-9-7': 'To gain proper insight, let us show how the answer ([REF]) results from the process of sequential fold straightening.', 'cond-mat-0510418-1-9-8': 'The definition of "folds" and of their lengths [MATH] is illustrated in Fig. [REF].', 'cond-mat-0510418-1-9-9': 'Since the chain is being pulled all the time into an immobile point in the center, the definition of a fold arises from considering only the radial component of the random walk, [MATH], where [MATH] stands for the (natural) parametric representation of a path.', 'cond-mat-0510418-1-9-10': 'In 3D, for example, [MATH] represents the so-called "Bessel process" [CITATION].', 'cond-mat-0510418-1-9-11': 'To obtain lengths of folds, [MATH], one has to coarse grain the trajectory up to the one monomer scale [MATH], thus making a Wiener sausage [CITATION], and then find the local minima of [MATH] separated by the biggest maxima.', 'cond-mat-0510418-1-9-12': 'It is natural to expect that there are about [MATH] of such minima, separated by the intervals about [MATH] each.', 'cond-mat-0510418-1-9-13': 'The construction of folds described above is depicted in Fig. [REF].', 'cond-mat-0510418-1-9-14': 'The process of folds straightening in a random coil consists by replacing folds one by one with the straight segments.', 'cond-mat-0510418-1-10-0': 'When we pull down the Gaussian chain by its end with a constant force [MATH], only the current fold, of the length [MATH], is involved in the forced motion and experiences friction.', 'cond-mat-0510418-1-10-1': 'The equation ([REF]) is still approximately valid (since the velocities of all monomers involved into motion are of the same order of magnitude), but [MATH] should be integrated in the limits [MATH], yielding relaxation time for one fold [EQUATION]', 'cond-mat-0510418-1-10-2': 'All [MATH] folds relax sequentially, meaning that [MATH], and thus returning the scaling answer ([REF]).', 'cond-mat-0510418-1-11-0': 'We have tested our theoretical predictions by a molecular dynamics experiment.', 'cond-mat-0510418-1-11-1': 'We have compared the adsorption times of linear chains from initially straighten and from typically Gaussian conformations.', 'cond-mat-0510418-1-11-2': 'The corresponding results are shown in Fig. [REF] for ideal chains up to [MATH].', 'cond-mat-0510418-1-12-0': 'The numerical values of critical exponents for [MATH] and [MATH] are extracted from the slopes of log-log plots for sufficiently long chains ([MATH]).', 'cond-mat-0510418-1-12-1': 'While all the computational details will be published elsewhere, the figure shows that the data are in very satisfactory agreement with the theoretical predictions ([REF]) and ([REF]).', 'cond-mat-0510418-1-13-0': 'Notice that the adsorption time of a linear chain, [MATH], Eq. ([REF]), is much shorter than the typical Rouse time under the very weak condition [MATH], which means Pincus blob dictated by the force [MATH] should only be smaller than the entire coil.', 'cond-mat-0510418-1-13-1': 'Under this condition, during the adsorption process the configuration of the chain parts not involved into the straighten of a current fold, can be considered as quenched.', 'cond-mat-0510418-1-14-0': 'The same adsorption mechanism is valid for randomly branched chains schematically shown in Fig. [REF].', 'cond-mat-0510418-1-15-0': 'Scaling approach for the randomly branched polymer can be treated as follows.', 'cond-mat-0510418-1-15-1': 'Estimate a Rouse time as the time for a polymer to diffuse to its own size.', 'cond-mat-0510418-1-15-2': 'Since all monomers experience friction independently of each other in a Rouse model (there are neither solvent dynamics, nor hydrodynamic interactions), it follows that the diffusion coefficient of a Rouse coil, linear or branched, is given by [MATH].', 'cond-mat-0510418-1-15-3': 'However, the gyration radius of a branched polymer, unlike a linear one, is [MATH].', 'cond-mat-0510418-1-15-4': 'Therefore, the Rouse relaxation time for the branched polymer reads [MATH].', 'cond-mat-0510418-1-15-5': 'The rest of the argument repeats exactly what we have done for the linear chain, yielding the result [EQUATION]', 'cond-mat-0510418-1-15-6': 'Let us show now how one can re-derive ([REF]) generalizing the concept of Bessel process to randomly branched chains.', 'cond-mat-0510418-1-15-7': 'It is convenient to introduce a hierarchy of scales in the [MATH]-link ideal randomly branched chain.', 'cond-mat-0510418-1-15-8': 'Let us denote by [MATH] a number of monomers in a typical "primitive" (or "bare") path of a largest scale.', 'cond-mat-0510418-1-15-9': 'The size [MATH] of an ideal randomly branched polymer is [MATH].', 'cond-mat-0510418-1-15-10': 'The bare path of the maximal scale is a Gaussian random walk with [MATH].', 'cond-mat-0510418-1-15-11': 'Hence, [MATH].', 'cond-mat-0510418-1-15-12': 'The average number of monomers in all side branches [MATH], etc. attached to one bond of the bare path of the maximal scale (as shown in Fig. [REF]a), is of order of [MATH].', 'cond-mat-0510418-1-15-13': 'All these side branches form a randomly branched configuration of size [MATH].', 'cond-mat-0510418-1-15-14': 'Now, the number of monomers in the bare path of the second scale, [MATH], can be obtained repeating the above scaling arguments: [MATH], what gives [MATH], and so on: [MATH],- see Fig. [REF].', 'cond-mat-0510418-1-16-0': 'On all scales the bare paths [MATH], etc. form the Gaussian sub-coils, each of which can be divided in folds [MATH] in the same manner as it is done above for a linear chain.', 'cond-mat-0510418-1-16-1': 'For [MATH] we have the estimate [EQUATION]', 'cond-mat-0510418-1-16-2': 'If the randomly branched polymer is pulled by its end with a constant force, [MATH], then all Gaussian sub-coils on all scales straighten their folds simultaneously.', 'cond-mat-0510418-1-16-3': 'The total typical number of monomers [MATH] in all folds simultaneously involved into the motion on all scales can be estimated as [EQUATION]', 'cond-mat-0510418-1-16-4': 'We have seen on the example of the linear chain, that the characteristic time to straighten the fold of a typical length [MATH] is of order of [MATH] - see ([REF]).', 'cond-mat-0510418-1-16-5': 'So, as it follows from ([REF]), the characteristic time of straightening out all folds of length [MATH] is of the order of [EQUATION]', 'cond-mat-0510418-1-16-6': 'In a randomly branched polymer there are [MATH] independent parts which relax sequentially.', 'cond-mat-0510418-1-16-7': 'Hence, the total relaxation time of a randomly branched chain can be estimated as [MATH], which returns the result ([REF]).', 'cond-mat-0510418-1-17-0': 'To conclude, we have considered scaling estimates of the relaxation time associated with pulling linear or randomly branched polymer chain into a hole by a constant force [MATH].', 'cond-mat-0510418-1-17-1': 'We found that these estimates, [MATH] and [MATH] can be understood in terms of the scenario of sequential straightening of the polymer parts which we called folds and which were rigorously defined in terms of the radial component of the random walk representing the polymer chain.', 'cond-mat-0510418-1-17-2': 'Moreover, the theoretical prediction for typical time of pulling an ideal Rouse linear polymer to a small hole is consistent with the molecular dynamics simulation (see Fig. [REF]).', 'cond-mat-0510418-1-18-0': 'We think that the mechanism of sequential straightening of folds applies to a number of real physical situations, such as DNA translocation through the membrane pore, when the translocation is driven by the difference in chemical potentials of monomers on the two sides of the impermeable membrane.', 'cond-mat-0510418-1-19-0': 'This work is partially supported by the grant ACI-NIM-2004-243 "Nouvelles Interfaces des Mathematiques" (France).', 'cond-mat-0510418-1-19-1': 'A.Y.G. gratefully acknowledges warm hospitality he felt throughout his stay at the LPTMS where this work was done.'}

{'cond-mat-0510418-2-0-0': 'We present scaling estimates for characteristic times [MATH] and [MATH] of pulling ideal linear and randomly branched polymers of [MATH] monomers into a small hole by a force [MATH].', 'cond-mat-0510418-2-0-1': 'We show that the absorbtion process develops as sequential straightening of folds of the initial polymer configuration.', 'cond-mat-0510418-2-0-2': 'By estimating the typical size of the fold involved into the motion, we arrive at the following predictions: [MATH] and [MATH], and we also confirm them by the molecular dynamics experiment.', 'cond-mat-0510418-2-1-0': 'PACS: 05.40.-a; 05.70.-a; 87.15.', 'cond-mat-0510418-2-1-1': 'He', 'cond-mat-0510418-2-2-0': 'There are models in physics - most famously exemplified by an Ising model - whose importance is not due to their practical applicability to anything in particular, but because they provide the much needed training ground for our intuition and for the development of our theoretical methods.', 'cond-mat-0510418-2-2-1': 'In polymer physics, one of the very few such models is that of ideal polymer absorbtion into a point-like potential well.', 'cond-mat-0510418-2-2-2': 'For a homopolymer in equilibrium, this is a clean yet interesting example of a second order phase transition [CITATION].', 'cond-mat-0510418-2-2-3': 'For a heteropolymer, even in equilibrium, the problem is still not completely understood - see, for example, [CITATION]; mathematically it is similar to the localization (pinning) transition in the solid-on-solid models with quenched impurities [CITATION].', 'cond-mat-0510418-2-3-0': 'Surprisingly enough, the dynamics of a polymer chain pulled into a potential well is almost not discussed at all.', 'cond-mat-0510418-2-3-1': 'On the first glance, one might think that this dynamics should be related to the dynamics of coil-globule collapse [CITATION], but the similarity between these two problems is limited.', 'cond-mat-0510418-2-3-2': 'Indeed, globule formation after an abrupt solvent quench begins by a simultaneous chain condensation on many independent nucleation centers, while the absorbtion process can be viewed as pulling of a polymer with a constant force into a single hole.', 'cond-mat-0510418-2-3-3': 'Hence, only a single attractive center causes the chain condensation.', 'cond-mat-0510418-2-3-4': 'In this sense, dynamics of absorbtion into a potential well is more similar to a driven polymer translocation through a membrane channel [CITATION].', 'cond-mat-0510418-2-4-0': 'To describe in words the polymer pulled into a hole, let us imagine a long rope randomly dropped at the desk top.', 'cond-mat-0510418-2-4-1': 'Pull now the rope by its end down from the desk edge.', 'cond-mat-0510418-2-4-2': 'It is obvious, and can be established through an experiment accessible even to theorists, that the rope does not move all at once.', 'cond-mat-0510418-2-4-3': 'What happens is only a part of the rope moves at the beginning, such that a fold closest to the end straightens out.', 'cond-mat-0510418-2-4-4': 'As soon as the first fold is straightened, it then transmits the force to the next fold, which starts moving, and so on.', 'cond-mat-0510418-2-4-5': 'The rope sequentially straightens its folds, such that at any moment only part of the rope about the size of a current fold is involved in the motion, while the rest of the rope remains immobile.', 'cond-mat-0510418-2-5-0': 'In the present letter we show how this mechanism applies to an ideal homopolymer dynamics and how it sheds light on the physics behind scaling estimates [CITATION] for the relaxation times for both linear, [MATH], and randomly branched, [MATH], polymers.', 'cond-mat-0510418-2-6-0': 'We address the problem on the simplest level of an ideal Rouse homopolymer of [MATH] monomers, each of size [MATH], embedded into an immobile solvent with the viscosity [MATH].', 'cond-mat-0510418-2-6-1': 'One end of the chain is put in the potential well (the "hole"), and the polymer is pulled down to the hole with constant force [MATH], acting locally on just one single monomer, the one currently entering the hole.', 'cond-mat-0510418-2-6-2': 'We will be interested in the relaxation time [MATH] of the complete pulling down of the entire polymer into the hole, we want to know how [MATH] depends on polymer length [MATH] and the pulling force [MATH].', 'cond-mat-0510418-2-6-3': "Neglecting inertia, the balance of forces reads [MATH], where friction force [MATH] should be linear in velocity [MATH], with [MATH] being the number of monomers 'swallowed by the hole' by the time [MATH].", 'cond-mat-0510418-2-6-4': 'More subtly, friction coefficient must be proportional to the number of monomers [MATH] currently involved in the forced motion: [MATH].', 'cond-mat-0510418-2-6-5': 'Our main task will be to estimate the length [MATH] involved in the motion for either linear or branched polymers.', 'cond-mat-0510418-2-7-0': 'In order to clarify our approach, let us compute the characteristic time [MATH] of pulling down of an initially stretched linear polymer.', 'cond-mat-0510418-2-7-1': 'In this case, all monomers move except those already absorbed in the hole, so [MATH], and so the force balance condition reads [EQUATION]', 'cond-mat-0510418-2-7-2': 'Upon integration, this yields [EQUATION]', 'cond-mat-0510418-2-7-3': 'In terms of dependence on [MATH], this result is similar to the Rouse relaxation time of a linear polymer, [MATH], where [MATH], and [MATH] is temperature.', 'cond-mat-0510418-2-7-4': 'Our estimate, ([REF]), remains valid as long as [MATH], because in this case the initially straight polymer has no time to coil up on itself while being pulled up by the force [MATH].', 'cond-mat-0510418-2-7-5': 'Given the expression for [MATH], ([REF]), the condition [MATH] translates into [MATH], the latter making an obvious sense: pulling force should be strong enough to cause significant stretching of every bond, or, in other words, Pincus blobs must be as small as of the order of one monomer [CITATION].', 'cond-mat-0510418-2-8-0': 'Now we turn to more realistic dynamics starting from a Gaussian coil configuration, and look at [MATH].', 'cond-mat-0510418-2-8-1': 'To begin with, let us consider a simple scaling estimate [CITATION].', 'cond-mat-0510418-2-8-2': 'First of all, we note that the only time scale relevant for the problem is the above mentioned Rouse time [MATH], which is the time needed for the coil to diffuse over the distance about its own size, [MATH].', 'cond-mat-0510418-2-8-3': 'With [MATH] being the solitary relevant time scale, the absorbtion time should be written in the scaling form [EQUATION] where the factor [MATH] can depend on all other relevant parameters only in the dimensionless combination [MATH].', 'cond-mat-0510418-2-8-4': 'The second step follows from the fact that the speed of the process, [MATH], must be linear in the applied force, which means that the function [MATH] should be inversely proportional to its argument, [MATH].', 'cond-mat-0510418-2-8-5': 'This then yields [EQUATION]', 'cond-mat-0510418-2-8-6': 'Scaling argument gives the answer, but does not give much insight.', 'cond-mat-0510418-2-8-7': 'To gain proper insight, let us show how this same answer results from the process of sequential fold straightening.', 'cond-mat-0510418-2-8-8': 'The definition of "folds" and of their lengths [MATH] is illustrated in Fig. [REF].', 'cond-mat-0510418-2-8-9': 'Since the chain is being pulled all the time into an immobile point in the center, the definition of a fold arises from considering only the radial component of the random walk, [MATH], where [MATH] stands for the (natural) parametric representation of a path.', 'cond-mat-0510418-2-8-10': 'In [MATH], for example, [MATH] represents Bessel random process [CITATION].', 'cond-mat-0510418-2-8-11': 'To obtain lengths of folds, [MATH], one has to coarse grain the trajectory up to the one monomer scale [MATH], thus making a Wiener sausage, and then find the local minima of [MATH] separated by the biggest maxima.', 'cond-mat-0510418-2-8-12': 'It is obvious that there are about [MATH] of such minima, separated by the intervals about [MATH] each.', 'cond-mat-0510418-2-9-0': 'When we pull down the Gaussian chain by its end with a constant force [MATH], only the current fold, of the length [MATH], is involved in the forced motion and experiences friction.', 'cond-mat-0510418-2-9-1': 'The equation ([REF]) is still valid, but [MATH] should be replaced by [MATH] and [MATH] should be integrated in the limits [MATH], yielding relaxation time for one fold [EQUATION]', 'cond-mat-0510418-2-9-2': 'All [MATH] folds relax sequentially, meaning that [MATH], and thus returning the scaling answer ([REF]).', 'cond-mat-0510418-2-10-0': 'Notice that the absorbtion time of a linear chain, [MATH], Eq. ([REF]), is much shorter than the typical Rouse time under the very weak condition [MATH], which means Pincus blob dictated by the force [MATH] should only be smaller than the entire coil.', 'cond-mat-0510418-2-10-1': 'Under this condition, during the absorbtion process the configuration of the chain parts not involved into the straightening of a current fold, can be considered as quenched.', 'cond-mat-0510418-2-11-0': 'To understand the chain pulling dynamics even better, it is useful to consider another fundamental fractal model, that of randomly branched polymer, schematically shown in Fig. [REF].', 'cond-mat-0510418-2-11-1': 'To find scaling approach for the randomly branched polymer, we have to estimate the analog of Rouse time as the time for a polymer to diffuse to its own size.', 'cond-mat-0510418-2-11-2': 'Since all monomers experience friction independently of each other (there are neither solvent dynamics, nor hydrodynamic interactions), it follows that the diffusion coefficient of a Rouse coil, linear or branched, is given by [MATH].', 'cond-mat-0510418-2-11-3': 'However, the radius of a branched polymer, unlike a linear one, is only [MATH].', 'cond-mat-0510418-2-11-4': 'Therefore, the Rouse relaxation time for the branched polymer reads [MATH].', 'cond-mat-0510418-2-11-5': 'The rest of the argument repeats exactly what we have done for the linear chain, yielding the result [EQUATION]', 'cond-mat-0510418-2-11-6': 'Let us show now how one can re-derive ([REF]) generalizing the concept of Bessel process to randomly branched chains.', 'cond-mat-0510418-2-11-7': 'It is convenient to introduce a hierarchy of scales in the [MATH]-link ideal randomly branched chain.', 'cond-mat-0510418-2-11-8': 'Let us denote by [MATH] a number of monomers in a typical "primitive" (or "bare") path of a largest scale.', 'cond-mat-0510418-2-11-9': 'The size [MATH] of an ideal randomly branched polymer is [MATH].', 'cond-mat-0510418-2-11-10': 'The bare path of the maximal scale is a Gaussian random walk with [MATH].', 'cond-mat-0510418-2-11-11': 'Hence, [MATH].', 'cond-mat-0510418-2-11-12': 'The average number of monomers in all side branches [MATH], etc. attached to one bond of the bare path of the maximal scale (as shown in Fig. [REF]a), is of order of [MATH].', 'cond-mat-0510418-2-11-13': 'All these side branches form a randomly branched configuration of size [MATH].', 'cond-mat-0510418-2-11-14': 'Now, the number of monomers in the bare path of the second scale, [MATH], can be obtained repeating the above scaling arguments: [MATH], giving [MATH], and so on: [MATH],- see Fig. [REF].', 'cond-mat-0510418-2-12-0': 'On all scales the bare paths [MATH], etc. form the Gaussian sub-coils, each of which can be divided in folds [MATH] in the same manner as it is done above for a linear chain.', 'cond-mat-0510418-2-12-1': 'For [MATH] we have the estimate [EQUATION]', 'cond-mat-0510418-2-12-2': 'If the randomly branched polymer is pulled by its end with a constant force, [MATH], then all Gaussian sub-coils on all scales straighten their folds simultaneously.', 'cond-mat-0510418-2-12-3': 'The total typical number of monomers [MATH] in all folds simultaneously involved into the motion on all scales can be estimated as [EQUATION]', 'cond-mat-0510418-2-12-4': 'We have seen on the example of the linear chain, that the characteristic time to straighten the fold of a typical length [MATH] is of order of [MATH] - see ([REF]).', 'cond-mat-0510418-2-12-5': 'So, as it follows from ([REF]), the characteristic time of straightening out all folds of length [MATH] is of the order of [EQUATION]', 'cond-mat-0510418-2-12-6': 'In a randomly branched polymer there are [MATH] independent parts, which relax sequentially.', 'cond-mat-0510418-2-12-7': 'Hence, the total relaxation time can be estimated as [MATH], which returns the result ([REF]).', 'cond-mat-0510418-2-13-0': 'We have tested our results by a molecular dynamics experiment.', 'cond-mat-0510418-2-13-1': 'We used standard Rouse model with harmonic bonds between neighboring monomers along the chain and with the averaged bond length, [MATH].', 'cond-mat-0510418-2-13-2': 'The attracting force [MATH] acts at any moment on one monomer only and always points to the origin.', 'cond-mat-0510418-2-13-3': 'Once a given monomer [MATH] gets inside the absorbing hole ([MATH], where [MATH] is the position vector of the [MATH]th monomer) this monomer is considered absorbed, it does not move any more and does not exert any force on monomer [MATH], its neighbor along the chain - but instead neighbor becomes the subject of the steady pulling force [MATH].', 'cond-mat-0510418-2-13-4': 'The same mechanism was simulated for the randomly branched polymer, except different branches can be pulled in parallel.', 'cond-mat-0510418-2-13-5': 'The solution of dynamical equations is realized in the frameworks of the numerical velocity Verlet scheme [CITATION].', 'cond-mat-0510418-2-13-6': 'The initial state of the linear chain was prepared by placing the first monomer at [MATH].', 'cond-mat-0510418-2-13-7': 'The initial configuration of the randomly branched polymer was prepared as described in the work [CITATION], and then polymer was shifted as a whole to place one randomly chosen monomer on the absorbing surface [MATH].', 'cond-mat-0510418-2-13-8': 'By performing 100 runs, we have computed averaged pulling times of linear chains from initially elongated and initially Gaussian conformations, as well as for a randomly branched polymer.', 'cond-mat-0510418-2-13-9': 'The results are shown in the Fig. [REF] and demonstrate very satisfactory agreement with the theoretical predictions ([REF]) and ([REF]).', 'cond-mat-0510418-2-14-0': 'In order to check more directly our proposed mechanism of absorption, we have looked at how Gaussian polymer chain gets involved into the biased motion driven by the pulling force [MATH].', 'cond-mat-0510418-2-14-1': 'The result is shown in the inset of figure [REF].', 'cond-mat-0510418-2-14-2': 'In this inset, [MATH], marks the crossover between current fold and the rest of the chain.', 'cond-mat-0510418-2-14-3': 'The current fold is operationally defined to include all monomers, with numbers [MATH], whose motion is currently strongly affected by the pulling force [MATH].', 'cond-mat-0510418-2-14-4': 'Other monomers with the numbers [MATH] are not yet affected.', 'cond-mat-0510418-2-14-5': 'The simulation indicates that thus defined [MATH] follows quite accurately the power law [MATH] with the exponent independent of the applied force.', 'cond-mat-0510418-2-14-6': 'Inverting [MATH], we get [MATH], which suggests that the time of complete absorbtion of a [MATH]-length subchain, which is given by our theoretical prediction ([REF]), and the time of involving [MATH]-length subchain in the force-biased motion scale in the same way.', 'cond-mat-0510418-2-15-0': 'To conclude, we have considered scaling estimates of the relaxation time associated with pulling linear or randomly branched polymer chain into a hole by a constant force [MATH].', 'cond-mat-0510418-2-15-1': 'We found that these estimates, [MATH] and [MATH], are consistent with the molecular dynamics simulation and they can be understood in terms of the scenario of sequential straightening of the polymer parts which we called folds and which were defined in terms of the radial component of the random walk representing the polymer chain.', 'cond-mat-0510418-2-16-0': 'We have to emphasize that all consideration in this paper was restricted to polymer models without excluded volume in 3D.', 'cond-mat-0510418-2-16-1': 'There is no doubt that self-avoidance, as well as hydrodynamic interactions will significantly affect the specific scaling relations obtained here.', 'cond-mat-0510418-2-16-2': 'The account of excluded volume effects is, to some extent, simpler than that of hydrodynamic interactions.', 'cond-mat-0510418-2-16-3': 'On the basis of scaling approach we can predict the pulling time, [MATH], for the polymer characterized by the exponent [MATH] in the relation [MATH].', 'cond-mat-0510418-2-16-4': "Our consideration is also restricted in the sense that we did not consider the role of the polymer part already 'swallowed' by the potential well.", 'cond-mat-0510418-2-16-5': 'Although this might affect the result under some conditions, our aim was to elucidate the mechanism of sequential straightening of folds, for that purpose the absorbed tail is irrelevant.', 'cond-mat-0510418-2-16-6': 'Similarly, in reality polymer is usually pulled into a hole in a wall, such as membrane, and we have neglected the (presumably logarithmic) factors associated with the excluded half-space.', 'cond-mat-0510418-2-17-0': 'However, as we said in the beginning, the purpose of the model is to facilitate methods and intuition.', 'cond-mat-0510418-2-17-1': 'In this sense, we think that the consideration of ideal polymer in this letter was fruitful, because the mechanism of sequential straightening of folds obviously applies to a number of real physical situations, such as, e.g., DNA translocation through the membrane pore driven by the difference in chemical potentials.', 'cond-mat-0510418-2-18-0': 'AYG gratefully acknowledges warm hospitality he felt throughout his stay at the LPTMS where this work was done.', 'cond-mat-0510418-2-18-1': 'This work is partially supported by the grant ACI-NIM-2004-243 "Nouvelles Interfaces des Mathematiques" (France).'}

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[]

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[]

['cond-mat-0510418-1-1-0', 'cond-mat-0510418-1-1-1', 'cond-mat-0510418-1-15-11', 'cond-mat-0510418-2-1-0', 'cond-mat-0510418-2-1-1', 'cond-mat-0510418-2-11-11']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0510418

hep-ph-0506300

{'hep-ph-0506300-1-0-0': 'Probing nucleon strange asymmetry from charm production in neutrino deep inelastic scattering', 'hep-ph-0506300-1-1-0': 'We propose a means to detect the nucleon strange quark-antiquark asymmetry, which is predicted as a non-perturbative effect, but still unchecked directly by available experiments.', 'hep-ph-0506300-1-1-1': 'The difference for the [MATH] and [MATH] meson production cross sections in neutrino and antineutrino induced charged current deep inelastic scattering is illustrated to be sensitive to the nucleon strange asymmetry.', 'hep-ph-0506300-1-1-2': 'Prospect is given and the effect due to the light quark fragmentation is also discussed for the extraction of the strange asymmetry in future experiments.', 'hep-ph-0506300-1-2-0': 'Probing nucleon strange asymmetry from charm production in neutrino DIS', 'hep-ph-0506300-1-3-0': '# Introduction', 'hep-ph-0506300-1-4-0': 'Nucleon structure is a natural laboratory to understand QCD and is worth to study for its own sake.', 'hep-ph-0506300-1-4-1': 'The nucleon strange quark-antiquark asymmetry is an interesting feature predicted as a natural consequence of the non-perturbative aspect of the nucleon [CITATION].', 'hep-ph-0506300-1-4-2': 'Recently, the nucleon strange asymmetry has been suggested [CITATION] as a promising mechanism to explain the NuTeV anomaly [CITATION] within the framework of Standard Model.', 'hep-ph-0506300-1-5-0': 'While the experimental evidence for such an asymmetry is still inconclusive, there are some approaches such as the global analysis of deep inelastic scattering (DIS) data [CITATION], which show a favor for an asymmetric strange sea, in agreement qualitatively with the intrinsic sea theory.', 'hep-ph-0506300-1-5-1': 'On the other hand, the CCFR next-to-leading-order (NLO) analysis of the neutrino induced dimuon production result favors a symmetric strange sea [CITATION], which is also the case of a recent NuTeV analysis [CITATION].', 'hep-ph-0506300-1-5-2': 'It seems that more precise and dedicated research is needed to address the problem in a clear way.', 'hep-ph-0506300-1-6-0': 'The measurement of the strange quark distribution relies on charged current (CC) DIS processes.', 'hep-ph-0506300-1-6-1': 'One method is through parity violating structure functions for isoscalar target in CC DIS: [MATH], which gives the total distribution of the strange sea.', 'hep-ph-0506300-1-6-2': 'Another way is through the combination of CC parity conserving structure function [MATH] with the charged lepton DIS structure function [MATH], for isoscalar target, [MATH].', 'hep-ph-0506300-1-6-3': 'Such an idea has been applied using high statistic neutrino and charged lepton nucleon DIS data, and the result at low [MATH] shows a sizable disagreement with the direct measurement from the CCFR dimuon result [CITATION].', 'hep-ph-0506300-1-6-4': 'The extraction of a small quantity from the difference of two large quantities may suffer from systematic uncertainties, which also seems to be the case for the extraction of the strange asymmetry by CC parity conserving structure functions: [MATH].', 'hep-ph-0506300-1-7-0': 'A method free from the above drawback is through the charged current charm production process, which is the main idea of the CCFR and NuTeV dimuon experiments [CITATION], with its leading order (LO) subprocesses being [MATH] and [MATH].', 'hep-ph-0506300-1-7-1': 'The latter subprocess is Cabibbo suppressed, thus the charm production in [MATH]-induced process is most sensitive to the strange quark distribution in the nucleon.', 'hep-ph-0506300-1-7-2': 'Similarly, the anticharm production in [MATH]-induced process is sensitive to the antistrange quark distribution, as the corresponding partner subprocesses are [MATH] and [MATH], with the latter subprocess being Cabibbo suppressed.', 'hep-ph-0506300-1-8-0': 'The oppositely charged dimuon signature is easy to identify and measure in massive detectors, which allow for the collection of high statistics data samples, e.g., the CCFR experiment has a sample of data with 5030 [MATH] induced events and 1060 [MATH] induced events, and the NuTeV has 5102 [MATH] induced events and 1458 [MATH] induced events [CITATION].', 'hep-ph-0506300-1-8-1': 'However, these two experiments neither show strong support for an asymmetric strange sea, nor can they rule it out [CITATION].', 'hep-ph-0506300-1-8-2': 'There are uncertainties in the estimation of the semi-muonic decay of the charmed hadrons [CITATION], e.g., the average semi-leptonic branching ratio for [MATH] and [MATH] induced events was only constrained by [MATH] [CITATION].', 'hep-ph-0506300-1-8-3': 'Besides, the interplay of strange asymmetry and the light quark fragmentation (LQF) effect, as will be discussed in section 4, can only be drawn more clearly in inclusive measurement of charged and neutral charm productions.', 'hep-ph-0506300-1-8-4': 'Thus a direct measurement of charmed hadrons produced in [MATH] and [MATH] induced CC DIS will provide more valuable information to probe the s and [MATH] distributions of the nucleon.', 'hep-ph-0506300-1-8-5': 'It is the purpose of this work to show that inclusive charm productions in neutrino and antineutrino induced CC DIS processes will be a promising way to detect the strange quark-antiquark asymmetry.', 'hep-ph-0506300-1-9-0': '# Charged current charm production', 'hep-ph-0506300-1-10-0': 'The differential cross section for charmed hadron [MATH] production in neutrino induced CC DIS can be factorized as [EQUATION] where the function [MATH] describes the fragmentation of a quark q into the charmed hadron [MATH], with [MATH] being the momentum fraction of the quark [MATH] carried by the produced hadron [MATH].', 'hep-ph-0506300-1-10-1': 'For the purpose of this article, the charmed hadron [MATH] is taken to be [MATH] or [MATH] meson, with [MATH] denoting its antiparticle [MATH] or [MATH].', 'hep-ph-0506300-1-11-0': 'It is generally believed that the possibility for light quark fragmentation into charmed hadrons is very small.', 'hep-ph-0506300-1-11-1': 'For example, the Lund string model implemented in some popular Monte Carlo programs predicts a suppression proportional to [MATH] for [MATH] production in the process of hadronization [CITATION].', 'hep-ph-0506300-1-11-2': 'With a knowledge of the strange suppression [MATH] [CITATION], the suppression for charm will be lower than [MATH], which can be safely neglected.', 'hep-ph-0506300-1-12-0': 'In this case, at leading order, only the [MATH] and [MATH] subprocesses contribute to charmed hadron production.', 'hep-ph-0506300-1-12-1': 'For isoscalar target and neglecting target mass effects, the leading order differential cross section for charm production is given by [CITATION]: [EQUATION] where [MATH] is the momentum fraction of the struck quark in the infinite momentum frame.', 'hep-ph-0506300-1-12-2': 'It is introduced with the consideration of a non-negligible charm quark mass, and is related to Bjorken scaling variable [MATH] through (neglecting light quark mass): [MATH], referred to as slow-rescaling.', 'hep-ph-0506300-1-12-3': 'The term [MATH] in Eq. ([REF]) is introduced as an energy threshold for charm production and is supported by experiments [CITATION].', 'hep-ph-0506300-1-13-0': '# Probing the nucleon strange asymmetry', 'hep-ph-0506300-1-14-0': 'The differential cross sections for charmed hadrons, namely, [MATH] or [MATH]) and [MATH] or [MATH]), produced in [MATH] and [MATH] induced CC DIS respectively, are closely related to the [MATH] and [MATH] distributions of the nucleon, and their difference, as can be seen in the following, is quite sensitive to the nucleon strange asymmetry.', 'hep-ph-0506300-1-15-0': 'Neglecting the light quark fragmentation effect, and using Eq. ([REF]) and its corresponding partner process for [MATH] production [MATH], we can write the difference between [MATH] and [MATH] production cross sections in CC DIS: [EQUATION] where charge symmetry [MATH] for fragmentation process is assumed, and [MATH] and [MATH] are valence quark distributions of the proton.', 'hep-ph-0506300-1-16-0': 'From Eq. ([REF]), one sees that two terms, [MATH] and [MATH], contribute to the cross section difference [MATH], with [MATH] and [MATH] [CITATION] being their respective weights.', 'hep-ph-0506300-1-16-1': 'The strange asymmetric part of Eq. ([REF]) can be estimated from an integral on variable [MATH], to contribute a fraction [EQUATION] to the integral of the cross section difference [MATH].', 'hep-ph-0506300-1-16-2': 'Here, [MATH] and [MATH] are defined as [MATH] and [MATH].', 'hep-ph-0506300-1-17-0': 'In Table 1, results of the strange asymmetry from some models accounting for the NuTeV anomaly are listed, together with our estimations of the contributions due to strange asymmetry to the the cross section difference [MATH], namely, the [MATH] integrated fraction [MATH].', 'hep-ph-0506300-1-18-0': 'As shown in Table 1, from the model calculations [CITATION] that can explain the NuTeV anomaly, the strange asymmetry contributes a sizable proportion [MATH] to the cross section difference.', 'hep-ph-0506300-1-18-1': 'Note that the distribution functions [MATH] and [MATH] may evolve with [MATH], turning flatter and shifting towards smaller [MATH] region as [MATH] increases.', 'hep-ph-0506300-1-18-2': 'However, their relative feature will remain and the proportion of [MATH] to [MATH], will be of the same order in larger [MATH] and in [MATH].', 'hep-ph-0506300-1-18-3': 'Thus, as to their relative feature, it does not matter much whether the parton distributions are taken at [MATH] or at larger [MATH].', 'hep-ph-0506300-1-18-4': 'Since the peak of [MATH] is confined in narrower [MATH] region than [MATH], its contribution is expected to be more prominent than the integrated one in Table 1.', 'hep-ph-0506300-1-18-5': 'Thus it is promising to measure the strange quark-antiquark asymmetry from [MATH].', 'hep-ph-0506300-1-19-0': 'Compared to the sum of the cross sections [MATH], the cross section difference [MATH] is not a very small quantity, as can be seen from the ratio of their integrals, [EQUATION] where [MATH] and [MATH].', 'hep-ph-0506300-1-19-1': 'With a calculation of the [MATH], [MATH] and [MATH] from CTEQ5 parametrization at [MATH]GeV[MATH], together with [MATH] and [MATH], the ratio [MATH] is estimated to be about [MATH]) for [MATH] being 0.007 (0.022) from Table 1.', 'hep-ph-0506300-1-19-2': 'Thus the cross section difference [MATH] is a significant quantity that can be extracted from the semi-inclusive differential cross sections.', 'hep-ph-0506300-1-20-0': 'Neutrino experiment with emulsion target, like the CHORUS detector, is ideal for the study of charmed hadron production.', 'hep-ph-0506300-1-20-1': 'Compared to dimuon studies, it has a much lower level of background and is free from the uncertainties that exist in charm muonic weak decay processes [CITATION].', 'hep-ph-0506300-1-20-2': 'And for statistics, CHORUS reported in total about 94000 neutrino CC events located and fully reconstructed, in which about 2000 charm events were observed [CITATION].', 'hep-ph-0506300-1-20-3': 'This has been compatible with dimuon statistics.', 'hep-ph-0506300-1-20-4': 'If such (or higher) statistics can be achieved with both neutrino and antineutrino beams of high energies in future experiments, the question about strange asymmetry is promising to be settled.', 'hep-ph-0506300-1-21-0': '# Light quark fragmentation', 'hep-ph-0506300-1-22-0': 'The possibility that a light quark fragments into charmed hadrons (associated charm production) can be an interesting effect of non-perturbative QCD, and it has been explored [CITATION] to explain the unexpected high rate of like-sign dimuons production from many neutrino experiments [CITATION].', 'hep-ph-0506300-1-22-1': 'Although the field has been inactive for years, and in practice people generally assume the light quark fragmentation (LQF) to be negligible, the physical possibility of a small contribution is not ruled out.', 'hep-ph-0506300-1-22-2': 'In fact, as to our consideration, neutrino experiments can be slightly different from [MATH] experiments in this respect.', 'hep-ph-0506300-1-22-3': 'The scattered light quark with high momentum can pick up a charm quark or antiquark from nucleon sea to form a [MATH] meson, and the larger the energy of the light quark, the more the ability that it can pick up a charm from the sea.', 'hep-ph-0506300-1-22-4': 'This energy dependence is apparent in prompt like-sign dimuons production rates in many experiments [CITATION].', 'hep-ph-0506300-1-22-5': 'Since the scattered quark has most of the energy in the collision, it is much more promising to pick up the charm quark from nucleon sea than other produced quarks in fragmentation process.', 'hep-ph-0506300-1-22-6': 'Thus as to our consideration such fragmentation as [MATH], [MATH] can possibly be non-negligible in high energy neutrino experiments.', 'hep-ph-0506300-1-23-0': 'In case that light quark can fragment into charmed hadrons, the process can manifest itself in a number of observables, such as the prompt like-sign dilepton and trimuon productions in high energy neutrino experiments [CITATION], direct observations of two charmed hadrons in nuclear emulsion target [CITATION], and charm production in hadron collisions [CITATION].', 'hep-ph-0506300-1-23-1': 'Among these, the prompt like-sign dimuon productions have been the most seriously studied and we will use some of the data for a quantitative estimate of light quark fragmentation function and its influence on the extraction of strange asymmetry from CC charm production processes.', 'hep-ph-0506300-1-24-0': 'Prompt like-sigh dimuons ([MATH]) can be produced through the process: [MATH], with the scattered [MATH] to fragment into [MATH] or [MATH] meson.', 'hep-ph-0506300-1-24-1': 'Its differential cross section can be expressed as: [EQUATION] where [MATH] is the total fragmentation function for a light quark to fragment into charmed hadrons, defined as [MATH], with [MATH] and [MATH] simply assumed.', 'hep-ph-0506300-1-24-2': 'Note here that [MATH] is energy dependent in analogous to containing an energy suppression factor for charm production.', 'hep-ph-0506300-1-24-3': '[MATH] is the inclusive muonic decay ratio for [MATH] meson decay [MATH], which is the same for [MATH] meson, since all [MATH] will decay into [MATH] at first.', 'hep-ph-0506300-1-25-0': 'Similarly, the differential cross section for prompt [MATH] production in [MATH] induced DIS on isoscalar target is [EQUATION]', 'hep-ph-0506300-1-25-1': 'Many experimental groups have reported positive results on prompt like-sign dimoun production.', 'hep-ph-0506300-1-25-2': 'Among them the CDHSW [CITATION] and CCFR [CITATION] data have high precision and are somewhat show much consistency with each other.', 'hep-ph-0506300-1-25-3': 'Another high precision experiment CHARM [CITATION], which has reported a much higher [MATH] production rate, received doubts on their estimate of [MATH] decay background [CITATION].', 'hep-ph-0506300-1-25-4': 'Besides, their kinematic cut [MATH] GeV, which is lower than other experiments ([MATH] GeV), can permit more [MATH] events and thus can produce a higher rate.', 'hep-ph-0506300-1-25-5': 'Since kinematic cut on the second [MATH] reduces event number and thus the rate of like-sign dimuons in CC events: [MATH]/[MATH], it will probably underestimate the light quark fragmentation (LQF) effect to use the [MATH]/[MATH] data.', 'hep-ph-0506300-1-25-6': 'On the other hand, the ratio of prompt like-sign dimuons to opposite-sign dimuons [MATH]/[MATH] is expected to be less influenced by kinematic cut, as both second muons receive the same kinematic cut.', 'hep-ph-0506300-1-25-7': 'Thus, we consider it appropriate to use the [MATH]/[MATH] data other than [MATH]/[MATH] data to estimate the LQF effect.', 'hep-ph-0506300-1-26-0': 'The differential cross section for [MATH] production in [MATH] induced CC DIS on isoscalar target is given by [EQUATION] where [MATH] is the charm suppression factor [MATH], and [MATH] is the average muonic decay ratio for the charmed hadrons produced in CC DIS, [MATH], with H being [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0506300-1-26-1': 'The last term marked with LQF is the light quark fragmentation contribution to [MATH] production ([MATH]), [EQUATION] where [MATH] is the average muonic decay ratio of the [MATH] and [MATH] mesons produced from [MATH] quark fragmentation, [MATH].', 'hep-ph-0506300-1-26-2': 'Since [MATH] decays to [MATH] with branching ratio [MATH] [CITATION], and to [MATH] with ratio [MATH], we have [MATH], with [MATH].', 'hep-ph-0506300-1-27-0': 'Similarly, the differential cross section for [MATH] production in [MATH] induced CC DIS on isoscalar target reads [EQUATION] with [EQUATION]', 'hep-ph-0506300-1-27-1': 'CDHSW has reported prompt dimuon rates [MATH]/[MATH] and [MATH]/[MATH] in [MATH] and [MATH] induced DIS.', 'hep-ph-0506300-1-27-2': 'As mentioned previously, these data are less influenced by kinematic cut and thus are better suited for the extraction of the LQF effect.', 'hep-ph-0506300-1-27-3': 'The prompt dimoun rates from CDHSW with visible energy [MATH] in range [MATH] GeV are listed in Table 2.', 'hep-ph-0506300-1-28-0': 'As can be seen from Table 2, the prompt dimuons rate [MATH]/[MATH] still show a slight dependence on kinematic cut, though much smaller than the [MATH]/[MATH] data do.', 'hep-ph-0506300-1-28-1': 'Thus we can only estimate the order of magnitude for the LQF effect with the reported data.', 'hep-ph-0506300-1-29-0': 'The rate [MATH]/[MATH] (without any kinematic cut) can be deduced by Eq. ([REF]) and Eq. ([REF]) with an integral on kinematic variables to approximate [EQUATION] where [MATH], [MATH], and [MATH] denotes the average of energy suppression factor in Eq. ([REF]).', 'hep-ph-0506300-1-29-1': 'With the measured [MATH] and [MATH] [CITATION], together with [MATH] and [MATH] from CTEQ5 at [MATH] GeV[MATH], [MATH] is estimated to be: [EQUATION]', 'hep-ph-0506300-1-29-2': 'With the experimental data on [MATH] from Table 2, one can easily estimate [MATH] by Eq. ([REF]).', 'hep-ph-0506300-1-30-0': 'Since we are most interested in the strange quark-antiquark asymmetry here, we will directly address the influence of LQF effect on the extraction of strange asymmetry.', 'hep-ph-0506300-1-30-1': 'Because LQF effect contributes differently for [MATH] induced [MATH] production and for [MATH] induced [MATH] production, it will give different corrections to [MATH] and [MATH] distributions, and thus influence the measurement of strange asymmetry from opposite-sign dimuon method.', 'hep-ph-0506300-1-30-2': 'To illustrate this, we will compare the contribution of LQF effect with that of strange asymmetry on the difference between [MATH] and [MATH] induced opposite-sign dimuon production cross sections.', 'hep-ph-0506300-1-30-3': 'The latter (strange asymmetry contribution) can be drawn from model predictions in the last column of Table 1, when assuming the average muonic branding ratio of charmed hadrons to be the same for [MATH] and [MATH] induced CC DIS [MATH].', 'hep-ph-0506300-1-30-4': 'The former (LQF contribution) can be deduced from Eq. ([REF])-Eq. ([REF]) with an assumption [MATH], and be compared to the strange asymmetry part with an integral on kinematic variables.', 'hep-ph-0506300-1-30-5': 'The fraction of the LQF contribution is [EQUATION]', 'hep-ph-0506300-1-30-6': 'To assess [MATH], the value of [MATH] is needed.', 'hep-ph-0506300-1-30-7': 'Remember that [MATH], with [MATH].', 'hep-ph-0506300-1-30-8': 'The unknown [MATH] is the fraction of vector [MATH] meson in light quark fragmentation.', 'hep-ph-0506300-1-30-9': 'When we set [MATH] to be [MATH], and take [MATH], [MATH] [CITATION], we get [MATH].', 'hep-ph-0506300-1-30-10': 'Using Eq. ([REF]) and taking [MATH] from Table 1, we get [MATH].', 'hep-ph-0506300-1-30-11': 'Taking [MATH] from Table 2, we get [EQUATION]', 'hep-ph-0506300-1-30-12': 'Thus, we get an estimate of the LQF contribution to be a few percent compared to strange asymmetry contribution [MATH].', 'hep-ph-0506300-1-30-13': 'However, the constraint of [MATH] can also be done with [MATH] data, and the result is [MATH], which is very large compared to result from the [MATH] data.', 'hep-ph-0506300-1-30-14': 'This large discrepancy is difficult to explain at present, and may imply an uncertainty in the estimate of the LQF contribution in the opposite-sign dimuon measurements of strange asymmetry.', 'hep-ph-0506300-1-31-0': 'From the sign and size of [MATH], one sees that the LQF effect contributes oppositely to the predicted strange asymmetry contribution on the whole, with a rate that could be non-negligible in opposite-sign dimuon experiments.', 'hep-ph-0506300-1-32-0': 'The LQF effect also exists in the process of inclusive charm productions that we suggest.', 'hep-ph-0506300-1-32-1': 'For [MATH] production, the cross section difference, [MATH], for [MATH] and [MATH] induced CC DIS will include an additional term from light quark fragmentation: [EQUATION] where [MATH] is introduced with the consideration that part of [MATH] will decay into [MATH] and will not contribute to the cross sections.', 'hep-ph-0506300-1-33-0': 'For neutral charm production, LQF contributes to [MATH] production in [MATH] induced CC DIS ([MATH], [MATH]), and to [MATH] production in [MATH] induced CC DIS.', 'hep-ph-0506300-1-33-1': 'In case that [MATH] and [MATH] are not distinguished by emulsion target, the [MATH] production in [MATH]) induced CC DIS from LQF will be incorporated to [MATH] production in [MATH]) induced CC DIS.', 'hep-ph-0506300-1-33-2': 'Thus an additional term from LQF will contribute to [MATH]: [EQUATION] where [MATH], which is introduced from [MATH]) fragmentation into [MATH] mesons that then decay into [MATH] and contribute to cross section difference [MATH].', 'hep-ph-0506300-1-34-0': 'The proportion of LQF contribution to inclusive charm production cross section difference [MATH], namely [MATH], can be estimated similarly to that of dimuon productions.', 'hep-ph-0506300-1-34-1': 'With an integral on kinematic variables of Eq. ([REF]), Eq. ([REF]), Eq. ([REF]), and using charm production fractions [MATH] and [MATH] for [MATH]GeV [CITATION], [MATH] is estimated (in unite of [MATH]) to be: [MATH] for [MATH] meson productions, and [MATH] for [MATH], [MATH] meson productions.', 'hep-ph-0506300-1-35-0': 'If the LQF contribution [MATH] in opposite-sign dimuons measurement is in the order of a few percent percent and opposite to strange asymmetry contribution [MATH], just as we have estimated, the LQF will contribute to inclusive charm production with a larger proportion (in the order of about ten percent or even larger).', 'hep-ph-0506300-1-35-1': 'For inclusive [MATH] production, LQF contributes oppositely compared to strange asymmetry when [MATH].', 'hep-ph-0506300-1-35-2': 'On the other hand, for inclusive CC neutral charm [MATH] production, LQF contributes positively compared to strange asymmetry when [MATH].', 'hep-ph-0506300-1-35-3': 'A separation of the LQF effect and the strange asymmetry effect can be made from the distinct features of [MATH] and [MATH] measured by nuclear emulsion target.', 'hep-ph-0506300-1-35-4': 'Thus, the inclusive measurement of charged and neutral charm production in [MATH] and [MATH] induced CC DIS will shed light on both the strange asymmetry and the LQF effect.', 'hep-ph-0506300-1-36-0': 'Dedicated analysis of charm productions in neutrino experiments and in other processes will be helpful for a more precise estimate and constraint for the light quark fragmentation effect.', 'hep-ph-0506300-1-37-0': '# Conclusions', 'hep-ph-0506300-1-38-0': 'For probing the nucleon strange asymmetry, we analyzed the charged current charm production processes, in particular, the [MATH] induced [MATH] or [MATH]) production and the [MATH] induced [MATH] or [MATH]) production processes.', 'hep-ph-0506300-1-38-1': 'The strange asymmetry from various model calculations that can explain the NuTeV anomaly is shown in general to contribute a sizable proportion ([MATH]) to the [MATH] differential cross section difference [MATH].', 'hep-ph-0506300-1-38-2': 'Thus, measurement of these cross sections with high energy neutrino and antineutrino beams on nuclear emulsion target is very promising to detect the strange quark-antiquark asymmetry.', 'hep-ph-0506300-1-39-0': 'Meanwhile, we analyzed the possible light quark fragmentation (LQF) effect from prompt like-sign dimuon data and studied its influence for the measurement of strange asymmetry.', 'hep-ph-0506300-1-39-1': 'Our result is that the LQF may be an important source that reduces the effect of strange asymmetry from opposite sign dimuon studies.', 'hep-ph-0506300-1-39-2': 'And for inclusive charged current (CC) charm production with emulsion target, since the contributions of LQF are in opposite directions for [MATH] and for [MATH]) productions, a separation of the LQF effect from strange asymmetry effect can be made by the separate measurement of [MATH] and neutral charm differential cross sections in CC DIS.', 'hep-ph-0506300-1-39-3': 'Thus the inclusive measurement of charmed hadrons can shed light on both strange asymmetry and the LQF effect.', 'hep-ph-0506300-1-39-4': 'Further analysis and constraint for LQF effect from various experiments will also be helpful for the purpose of measuring the strange asymmetry more reliably.'}

{'hep-ph-0506300-2-0-0': 'Probing nucleon strange asymmetry from charm production in neutrino deep inelastic scattering', 'hep-ph-0506300-2-1-0': 'We propose a means to detect the nucleon strange quark-antiquark asymmetry, which is predicted as a non-perturbative effect, but still unchecked directly by available experiments.', 'hep-ph-0506300-2-1-1': 'The difference for the [MATH] and [MATH] meson production cross sections in neutrino and antineutrino induced charged current deep inelastic scattering is illustrated to be sensitive to the nucleon strange asymmetry.', 'hep-ph-0506300-2-1-2': 'Prospect is given and the effect due to the light quark fragmentation is also discussed for the extraction of the strange asymmetry in future experiments.', 'hep-ph-0506300-2-2-0': 'Probing nucleon strange asymmetry from charm production in neutrino DIS', 'hep-ph-0506300-2-3-0': '# Introduction', 'hep-ph-0506300-2-4-0': 'Nucleon structure is a natural laboratory to understand QCD and is worth to study for its own sake.', 'hep-ph-0506300-2-4-1': 'The nucleon strange quark-antiquark asymmetry is an interesting feature predicted as a natural consequence of the non-perturbative aspect of the nucleon [CITATION].', 'hep-ph-0506300-2-4-2': 'Recently, the nucleon strange asymmetry has been suggested [CITATION] as a promising mechanism to explain the NuTeV anomaly [CITATION] within the framework of Standard Model.', 'hep-ph-0506300-2-5-0': 'While the experimental evidence for such an asymmetry is still inconclusive, there are some approaches such as the global analysis of deep inelastic scattering (DIS) data [CITATION], which show a favor for an asymmetric strange sea, in agreement qualitatively with the intrinsic sea theory.', 'hep-ph-0506300-2-5-1': 'On the other hand, the CCFR next-to-leading-order (NLO) analysis of the neutrino induced dimuon production result favors a symmetric strange sea [CITATION], which is also the case of a recent NuTeV analysis [CITATION].', 'hep-ph-0506300-2-5-2': 'It seems that more precise and dedicated research is needed to address the problem in a clear way.', 'hep-ph-0506300-2-6-0': 'The measurement of the strange quark distribution relies on charged current (CC) DIS processes.', 'hep-ph-0506300-2-6-1': 'One method is through parity violating structure functions for isoscalar target in CC DIS: [MATH], which gives the total distribution of the strange sea.', 'hep-ph-0506300-2-6-2': 'Another way is through the combination of CC parity conserving structure function [MATH] with the charged lepton DIS structure function [MATH], for isoscalar target, [MATH].', 'hep-ph-0506300-2-6-3': 'Such an idea has been applied using high statistic neutrino and charged lepton nucleon DIS data, and the result at low [MATH] shows a sizable disagreement with the direct measurement from the CCFR dimuon result [CITATION].', 'hep-ph-0506300-2-6-4': 'The extraction of a small quantity from the difference of two large quantities may suffer from systematic uncertainties, which also seems to be the case for the extraction of the strange asymmetry by CC parity conserving structure functions: [MATH].', 'hep-ph-0506300-2-7-0': 'A method free from the above drawback is to use the charged current charm production process, which is the main idea of the CCFR and NuTeV dimuon experiments [CITATION], with its leading order (LO) subprocesses being [MATH] and [MATH].', 'hep-ph-0506300-2-7-1': 'The latter subprocess is Cabibbo suppressed, thus the charm production in [MATH]-induced process is most sensitive to the strange quark distribution in the nucleon.', 'hep-ph-0506300-2-7-2': 'Similarly, the anticharm production in [MATH]-induced process is sensitive to the antistrange quark distribution, as the corresponding partner subprocesses are [MATH] and [MATH], with the latter subprocess being Cabibbo suppressed.', 'hep-ph-0506300-2-8-0': 'The oppositely charged dimuon signature is easy to identify and measure in massive detectors, which allow for the collection of high statistics data samples, e.g., the CCFR experiment has a sample of data with 5030 [MATH] induced events and 1060 [MATH] induced events, and the NuTeV has 5102 [MATH] induced events and 1458 [MATH] induced events [CITATION].', 'hep-ph-0506300-2-8-1': 'However, these two experiments neither show strong support for an asymmetric strange sea, nor can they rule it out [CITATION].', 'hep-ph-0506300-2-8-2': 'There are uncertainties in the estimation of the semi-muonic decay of the charmed hadrons [CITATION], e.g., the average semi-leptonic branching ratio for [MATH] and [MATH] induced events was only constrained by [MATH] [CITATION].', 'hep-ph-0506300-2-8-3': 'Besides, the interplay of strange asymmetry and the light quark fragmentation (LQF) effect, as will be discussed in section 4, can only be drawn more clearly in inclusive measurement of charged and neutral charm productions.', 'hep-ph-0506300-2-8-4': 'Thus a direct measurement of charmed hadrons produced in [MATH] and [MATH] induced CC DIS will provide more valuable information to probe the s and [MATH] distributions of the nucleon.', 'hep-ph-0506300-2-8-5': 'It is the purpose of this work to show that inclusive charm productions in neutrino and antineutrino induced CC DIS processes will be a promising way to detect the strange quark-antiquark asymmetry.', 'hep-ph-0506300-2-9-0': '# Charged current charm production', 'hep-ph-0506300-2-10-0': 'The differential cross section for charmed hadron [MATH] production in neutrino induced CC DIS can be factorized as [EQUATION] where the function [MATH] describes the fragmentation of a quark q into the charmed hadron [MATH], with [MATH] being the momentum fraction of the quark [MATH] carried by the produced hadron [MATH].', 'hep-ph-0506300-2-10-1': 'For the purpose of this article, the charmed hadron [MATH] is taken to be [MATH] or [MATH] meson, with [MATH] denoting its antiparticle [MATH] or [MATH].', 'hep-ph-0506300-2-11-0': 'It is generally believed that the possibility for light quark fragmentation into charmed hadrons is very small.', 'hep-ph-0506300-2-11-1': 'For example, the Lund string model implemented in some popular Monte Carlo programs predicts a suppression proportional to [MATH] for [MATH] production in the process of hadronization [CITATION].', 'hep-ph-0506300-2-11-2': 'With a knowledge of the strange suppression [MATH] [CITATION], the suppression for charm will be lower than [MATH], which can be safely neglected.', 'hep-ph-0506300-2-12-0': 'In this case, at leading order, only the [MATH] and [MATH] subprocesses contribute to charmed hadron production.', 'hep-ph-0506300-2-12-1': 'For isoscalar target and neglecting target mass effects, the leading order differential cross section for charm production is given by [CITATION]: [EQUATION] where [MATH] is the momentum fraction of the struck quark in the infinite momentum frame.', 'hep-ph-0506300-2-12-2': 'It is introduced with the consideration of a non-negligible charm quark mass, and is related to Bjorken scaling variable [MATH] through (neglecting light quark mass): [MATH], referred to as slow-rescaling.', 'hep-ph-0506300-2-12-3': 'The term [MATH] in Eq. ([REF]) is introduced as an energy threshold for charm production and is supported by experiments [CITATION].', 'hep-ph-0506300-2-13-0': '# Probing the nucleon strange asymmetry', 'hep-ph-0506300-2-14-0': 'The differential cross sections for charmed hadrons, namely, [MATH] or [MATH]) and [MATH] or [MATH]), produced in [MATH] and [MATH] induced CC DIS respectively, are closely related to the [MATH] and [MATH] distributions of the nucleon, and their difference, as can be seen in the following, is quite sensitive to the nucleon strange asymmetry.', 'hep-ph-0506300-2-15-0': 'Neglecting the light quark fragmentation effect, and using Eq. ([REF]) and its corresponding partner process for [MATH] production [MATH], we can write the difference between [MATH] and [MATH] production cross sections in CC DIS: [EQUATION] where charge symmetry [MATH] for fragmentation process is assumed, and [MATH] and [MATH] are valence quark distributions of the proton.', 'hep-ph-0506300-2-16-0': 'From Eq. ([REF]), one sees that two terms, [MATH] and [MATH], contribute to the cross section difference [MATH], with [MATH] and [MATH] [CITATION] being their respective weights.', 'hep-ph-0506300-2-16-1': 'The strange asymmetric part of Eq. ([REF]) can be estimated from an integral on variable [MATH], to contribute a fraction [EQUATION] to the integral of the cross section difference [MATH].', 'hep-ph-0506300-2-16-2': 'Here, [MATH] and [MATH] are defined as [MATH] and [MATH].', 'hep-ph-0506300-2-17-0': 'In Table 1, results of the strange asymmetry from some models accounting for the NuTeV anomaly are listed, together with our estimations of the contributions due to strange asymmetry to the the cross section difference [MATH], namely, the [MATH] integrated fraction [MATH].', 'hep-ph-0506300-2-18-0': 'As shown in Table 1, from the model calculations [CITATION] that can explain the NuTeV anomaly, the strange asymmetry contributes a sizable proportion [MATH] to the cross section difference.', 'hep-ph-0506300-2-18-1': 'Note that the distribution functions [MATH] and [MATH] may evolve with [MATH], turning flatter and shifting towards smaller [MATH] region as [MATH] increases.', 'hep-ph-0506300-2-18-2': 'However, their relative feature will remain and the proportion of [MATH] to [MATH], will be of the same order in larger [MATH] and in [MATH].', 'hep-ph-0506300-2-18-3': 'Thus, as to their relative feature, it does not matter much whether the parton distributions are taken at [MATH] or at larger [MATH].', 'hep-ph-0506300-2-18-4': 'Since the peak of [MATH] is confined in narrower [MATH] region than [MATH], its contribution is expected to be more prominent than the integrated one in Table 1.', 'hep-ph-0506300-2-18-5': 'Thus it is promising to measure the strange quark-antiquark asymmetry from [MATH].', 'hep-ph-0506300-2-19-0': 'Compared to the sum of the cross sections [MATH], the cross section difference [MATH] is not a very small quantity, as can be seen from the ratio of their integrals, [EQUATION] where [MATH] and [MATH].', 'hep-ph-0506300-2-19-1': 'With a calculation of the [MATH], [MATH] and [MATH] from CTEQ5 parametrization at [MATH]GeV[MATH], together with [MATH] and [MATH], the ratio [MATH] is estimated to be about [MATH]) for [MATH] being 0.007 (0.022) from Table 1.', 'hep-ph-0506300-2-19-2': 'Thus the cross section difference [MATH] is a significant quantity that can be extracted from the semi-inclusive differential cross sections.', 'hep-ph-0506300-2-20-0': 'Neutrino experiment with emulsion target, like the CHORUS detector, is ideal for the study of charmed hadron production.', 'hep-ph-0506300-2-20-1': 'Compared to dimuon studies, it has a much lower level of background and is free from the uncertainties that exist in charm muonic weak decay processes [CITATION].', 'hep-ph-0506300-2-20-2': 'And for statistics, CHORUS reported in total about 94000 neutrino CC events located and fully reconstructed, in which about 2000 charm events were observed [CITATION].', 'hep-ph-0506300-2-20-3': 'This has been compatible with dimuon statistics.', 'hep-ph-0506300-2-20-4': 'If such (or higher) statistics can be achieved with both neutrino and antineutrino beams of high energies in future experiments, the question about strange asymmetry is promising to be settled.', 'hep-ph-0506300-2-21-0': '# Light quark fragmentation', 'hep-ph-0506300-2-22-0': 'The possibility that a light quark fragments into charmed hadrons (associated charm production) can be an interesting effect of non-perturbative QCD, and it has been explored [CITATION] to explain the unexpected high rate of like-sign dimuons production from many neutrino experiments [CITATION].', 'hep-ph-0506300-2-22-1': 'Although the field has been inactive for years, and in practice people generally assume the light quark fragmentation (LQF) to be negligible, the physical possibility of a small contribution is not ruled out.', 'hep-ph-0506300-2-22-2': 'In fact, as to our consideration, neutrino experiments can be slightly different from [MATH] experiments in this respect.', 'hep-ph-0506300-2-22-3': 'The scattered light quark with high momentum can pick up a charm quark or antiquark from nucleon sea to form a [MATH] meson, and the larger the energy of the light quark, the more the ability that it can pick up a charm from the sea.', 'hep-ph-0506300-2-22-4': 'This energy dependence is apparent in prompt like-sign dimuons production rates in many experiments [CITATION].', 'hep-ph-0506300-2-22-5': 'Since the scattered quark has most of the energy in the collision, it is much more promising to pick up the charm quark from nucleon sea than other produced quarks in fragmentation process.', 'hep-ph-0506300-2-22-6': 'Thus as to our consideration such fragmentation as [MATH], [MATH] can possibly be non-negligible in high energy neutrino experiments.', 'hep-ph-0506300-2-23-0': 'In case that light quark can fragment into charmed hadrons, the process can manifest itself in a number of observables, such as the prompt like-sign dilepton and trimuon productions in high energy neutrino experiments [CITATION], direct observations of two charmed hadrons in nuclear emulsion target [CITATION], and charm production in hadron collisions [CITATION].', 'hep-ph-0506300-2-23-1': 'Among these, the prompt like-sign dimuon productions have been the most seriously studied and we will use some of the data for a quantitative estimate of light quark fragmentation function and its influence on the extraction of strange asymmetry from CC charm production processes.', 'hep-ph-0506300-2-24-0': 'Prompt like-sigh dimuons ([MATH]) can be produced through the process: [MATH], with the scattered [MATH] to fragment into [MATH] or [MATH] meson.', 'hep-ph-0506300-2-24-1': 'Its differential cross section can be expressed as: [EQUATION] where [MATH] is the total fragmentation function for a light quark to fragment into charmed hadrons, defined as [MATH], with [MATH] and [MATH] simply assumed.', 'hep-ph-0506300-2-24-2': 'Note here that [MATH] is energy dependent in analogous to containing an energy suppression factor for charm production.', 'hep-ph-0506300-2-24-3': '[MATH] is the inclusive muonic decay ratio for [MATH] meson decay [MATH], which is the same for [MATH] meson, since all [MATH] will decay into [MATH] at first.', 'hep-ph-0506300-2-25-0': 'Similarly, the differential cross section for prompt [MATH] production in [MATH] induced DIS on isoscalar target is [EQUATION]', 'hep-ph-0506300-2-25-1': 'Many experimental groups have reported positive results on prompt like-sign dimoun production.', 'hep-ph-0506300-2-25-2': 'Among them the CDHSW [CITATION] and CCFR [CITATION] data have a high precision and show much consistency with each other.', 'hep-ph-0506300-2-25-3': 'Another high precision experiment CHARM [CITATION], which has reported a much higher [MATH] production rate, received doubts on their estimate of [MATH] decay background [CITATION].', 'hep-ph-0506300-2-25-4': 'Besides, their kinematic cut [MATH] GeV, which is lower than other experiments ([MATH] GeV), can permit more [MATH] events and thus can produce a higher rate.', 'hep-ph-0506300-2-25-5': 'Since kinematic cut on the second [MATH] reduces event number and thus the rate of like-sign dimuons in CC events: [MATH]/[MATH], it will probably underestimate the light quark fragmentation (LQF) effect to use the [MATH]/[MATH] data.', 'hep-ph-0506300-2-25-6': 'On the other hand, the ratio of prompt like-sign dimuons to opposite-sign dimuons [MATH]/[MATH] is expected to be less influenced by kinematic cut, as both second muons receive the same kinematic cut.', 'hep-ph-0506300-2-25-7': 'Thus, we consider it appropriate to use the [MATH]/[MATH] data other than [MATH]/[MATH] data to estimate the LQF effect.', 'hep-ph-0506300-2-26-0': 'The differential cross section for [MATH] production in [MATH] induced CC DIS on an isoscalar target is given by [EQUATION] where [MATH] is the charm suppression factor [MATH], and [MATH] is the average muonic decay ratio for the charmed hadrons produced in CC DIS, [MATH], with H being [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0506300-2-26-1': 'The last term marked with LQF is the light quark fragmentation contribution to [MATH] production ([MATH]), [EQUATION] where [MATH] is the average muonic decay ratio of the [MATH] and [MATH] mesons produced from [MATH] quark fragmentation, [MATH].', 'hep-ph-0506300-2-26-2': 'Since [MATH] decays to [MATH] with branching ratio [MATH] [CITATION], and to [MATH] with ratio [MATH], we have [MATH], with [MATH].', 'hep-ph-0506300-2-27-0': 'Similarly, the differential cross section for [MATH] production in [MATH] induced CC DIS on isoscalar target reads [EQUATION] with [EQUATION]', 'hep-ph-0506300-2-27-1': 'CDHSW has reported prompt dimuon rates [MATH]/[MATH] and [MATH]/[MATH] in [MATH] and [MATH] induced DIS.', 'hep-ph-0506300-2-27-2': 'As mentioned previously, these data are less influenced by kinematic cut and thus are better suited for the extraction of the LQF effect.', 'hep-ph-0506300-2-27-3': 'The prompt dimoun rates from CDHSW with visible energy [MATH] in range [MATH] GeV are listed in Table 2.', 'hep-ph-0506300-2-28-0': 'As can be seen from Table 2, the prompt dimuons rate [MATH]/[MATH] still show a slight dependence on kinematic cut, though much smaller than the [MATH]/[MATH] data do.', 'hep-ph-0506300-2-28-1': 'Thus we can only estimate the order of magnitude for the LQF effect with the reported data.', 'hep-ph-0506300-2-29-0': 'The rate [MATH]/[MATH] (without any kinematic cut) can be deduced by Eq. ([REF]) and Eq. ([REF]) with an integral on kinematic variables to approximate [EQUATION] where [MATH], [MATH], and [MATH] denotes the average of energy suppression factor in Eq. ([REF]).', 'hep-ph-0506300-2-29-1': 'With the measured [MATH] and [MATH] [CITATION], together with [MATH] and [MATH] from CTEQ5 at [MATH] GeV[MATH], [MATH] is estimated to be: [EQUATION]', 'hep-ph-0506300-2-29-2': 'With the experimental data on [MATH] from Table 2, one can easily estimate [MATH] by Eq. ([REF]).', 'hep-ph-0506300-2-30-0': 'Since we are most interested in the strange quark-antiquark asymmetry here, we will directly address the influence of LQF effect on the extraction of strange asymmetry.', 'hep-ph-0506300-2-30-1': 'Because LQF effect contributes differently for [MATH] induced [MATH] production and for [MATH] induced [MATH] production, it will give different corrections to [MATH] and [MATH] distributions, and thus influence the measurement of strange asymmetry from opposite-sign dimuon method.', 'hep-ph-0506300-2-30-2': 'To illustrate this, we will compare the contribution of LQF effect with that of strange asymmetry on the difference between [MATH] and [MATH] induced opposite-sign dimuon production cross sections.', 'hep-ph-0506300-2-30-3': 'The latter (strange asymmetry contribution) can be drawn from model predictions in the last column of Table 1, when assuming the average muonic branding ratio of charmed hadrons to be the same for [MATH] and [MATH] induced CC DIS [MATH].', 'hep-ph-0506300-2-30-4': 'The former (LQF contribution) can be deduced from Eq. ([REF])-Eq. ([REF]) with an assumption [MATH], and be compared to the strange asymmetry part with an integral on kinematic variables.', 'hep-ph-0506300-2-30-5': 'The fraction of the LQF contribution is [EQUATION]', 'hep-ph-0506300-2-30-6': 'To assess [MATH], the value of [MATH] is needed.', 'hep-ph-0506300-2-30-7': 'Remember that [MATH], with [MATH].', 'hep-ph-0506300-2-30-8': 'The unknown [MATH] is the fraction of vector [MATH] meson in light quark fragmentation.', 'hep-ph-0506300-2-30-9': 'When we set [MATH] to be [MATH], and take [MATH], [MATH] [CITATION], we get [MATH].', 'hep-ph-0506300-2-30-10': 'Using Eq. ([REF]) and taking [MATH] from Table 1, we get [MATH].', 'hep-ph-0506300-2-30-11': 'Taking [MATH] from Table 2, we get [EQUATION]', 'hep-ph-0506300-2-30-12': 'Thus, we get an estimate of the LQF contribution to be a few percent compared to strange asymmetry contribution [MATH].', 'hep-ph-0506300-2-30-13': 'However, the constraint of [MATH] can also be done with [MATH] data, and the result is [MATH], which is very large compared to result from the [MATH] data.', 'hep-ph-0506300-2-30-14': 'This large discrepancy is difficult to explain at present, and may imply an uncertainty in the estimate of the LQF contribution in the opposite-sign dimuon measurements of strange asymmetry.', 'hep-ph-0506300-2-31-0': 'From the sign and size of [MATH], one sees that the LQF effect contributes oppositely to the predicted strange asymmetry contribution on the whole, with a rate that could be non-negligible in opposite-sign dimuon experiments.', 'hep-ph-0506300-2-32-0': 'The LQF effect also exists in the process of inclusive charm productions that we suggest.', 'hep-ph-0506300-2-32-1': 'For [MATH] production, the cross section difference, [MATH], for [MATH] and [MATH] induced CC DIS will include an additional term from light quark fragmentation: [EQUATION] where [MATH] is introduced with the consideration that part of [MATH] will decay into [MATH] and will not contribute to the cross sections.', 'hep-ph-0506300-2-33-0': 'For neutral charm production, LQF contributes to [MATH] production in [MATH] induced CC DIS ([MATH], [MATH]), and to [MATH] production in [MATH] induced CC DIS.', 'hep-ph-0506300-2-33-1': 'In case that [MATH] and [MATH] are not distinguished by emulsion target, the [MATH] production in [MATH]) induced CC DIS from LQF will be incorporated to [MATH] production in [MATH]) induced CC DIS.', 'hep-ph-0506300-2-33-2': 'Thus an additional term from LQF will contribute to [MATH]: [EQUATION] where [MATH], which is introduced from [MATH]) fragmentation into [MATH] mesons that then decay into [MATH] and contribute to cross section difference [MATH].', 'hep-ph-0506300-2-34-0': 'The proportion of LQF contribution to inclusive charm production cross section difference [MATH], namely [MATH], can be estimated similarly to that of dimuon productions.', 'hep-ph-0506300-2-34-1': 'With an integral on kinematic variables of Eq. ([REF]), Eq. ([REF]), Eq. ([REF]), and using charm production fractions [MATH] and [MATH] for [MATH]GeV [CITATION], [MATH] is estimated (in unite of [MATH]) to be: [MATH] for [MATH] meson productions, and [MATH] for [MATH], [MATH] meson productions.', 'hep-ph-0506300-2-35-0': 'If the LQF contribution [MATH] in opposite-sign dimuons measurement is in the order of a few percent percent and opposite to strange asymmetry contribution [MATH], just as we have estimated, the LQF will contribute to inclusive charm production with a larger proportion (in the order of about ten percent or even larger).', 'hep-ph-0506300-2-35-1': 'For inclusive [MATH] production, LQF contributes oppositely compared to strange asymmetry when [MATH].', 'hep-ph-0506300-2-35-2': 'On the other hand, for inclusive CC neutral charm [MATH] production, LQF contributes positively compared to strange asymmetry when [MATH].', 'hep-ph-0506300-2-35-3': 'A separation of the LQF effect and the strange asymmetry effect can be made from the distinct features of [MATH] and [MATH] measured by nuclear emulsion target.', 'hep-ph-0506300-2-35-4': 'Thus, the inclusive measurement of charged and neutral charm production in [MATH] and [MATH] induced CC DIS will shed light on both the strange asymmetry and the LQF effect.', 'hep-ph-0506300-2-36-0': 'Dedicated analysis of charm productions in neutrino experiments and in other processes will be helpful for a more precise estimate and constraint for the light quark fragmentation effect.', 'hep-ph-0506300-2-37-0': '# Conclusions', 'hep-ph-0506300-2-38-0': 'For probing the nucleon strange asymmetry, we analyzed the charged current charm production processes, in particular, the [MATH] induced [MATH] or [MATH]) production and the [MATH] induced [MATH] or [MATH]) production processes.', 'hep-ph-0506300-2-38-1': 'The strange asymmetry from various model calculations that can explain the NuTeV anomaly is shown in general to contribute a sizable proportion ([MATH]) to the [MATH] differential cross section difference [MATH].', 'hep-ph-0506300-2-38-2': 'Thus, measurement of these cross sections with high energy neutrino and antineutrino beams on nuclear emulsion target is very promising to detect the strange quark-antiquark asymmetry.', 'hep-ph-0506300-2-39-0': 'Meanwhile, we analyzed the possible light quark fragmentation (LQF) effect from prompt like-sign dimuon data and studied its influence on the measurement of strange asymmetry.', 'hep-ph-0506300-2-39-1': 'Our result is that the LQF may be an important source that reduces the effect of strange asymmetry from opposite sign dimuon studies.', 'hep-ph-0506300-2-39-2': 'And for inclusive charged current (CC) charm production with emulsion target, since the contributions of LQF are in opposite directions for [MATH] and for [MATH]) productions, a separation of the LQF effect from strange asymmetry effect can be made by the separate measurement of [MATH] and neutral charm differential cross sections in CC DIS.', 'hep-ph-0506300-2-39-3': 'Thus the inclusive measurement of charmed hadrons can shed light on both strange asymmetry and the LQF effect.', 'hep-ph-0506300-2-39-4': 'Further analysis and constraint for LQF effect from various experiments will also be helpful for the purpose of measuring the strange asymmetry more reliably.'}

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['hep-ph-0506300-1-30-11', 'hep-ph-0506300-2-30-11'], ['hep-ph-0506300-1-30-12', 'hep-ph-0506300-2-30-12'], ['hep-ph-0506300-1-30-13', 'hep-ph-0506300-2-30-13'], ['hep-ph-0506300-1-30-14', 'hep-ph-0506300-2-30-14'], ['hep-ph-0506300-1-32-0', 'hep-ph-0506300-2-32-0'], ['hep-ph-0506300-1-32-1', 'hep-ph-0506300-2-32-1'], ['hep-ph-0506300-1-14-0', 'hep-ph-0506300-2-14-0'], ['hep-ph-0506300-1-22-0', 'hep-ph-0506300-2-22-0'], ['hep-ph-0506300-1-22-1', 'hep-ph-0506300-2-22-1'], ['hep-ph-0506300-1-22-2', 'hep-ph-0506300-2-22-2'], ['hep-ph-0506300-1-22-3', 'hep-ph-0506300-2-22-3'], ['hep-ph-0506300-1-22-4', 'hep-ph-0506300-2-22-4'], ['hep-ph-0506300-1-22-5', 'hep-ph-0506300-2-22-5'], ['hep-ph-0506300-1-22-6', 'hep-ph-0506300-2-22-6'], ['hep-ph-0506300-1-19-0', 'hep-ph-0506300-2-19-0'], ['hep-ph-0506300-1-19-1', 'hep-ph-0506300-2-19-1'], ['hep-ph-0506300-1-19-2', 'hep-ph-0506300-2-19-2'], ['hep-ph-0506300-1-39-1', 'hep-ph-0506300-2-39-1'], ['hep-ph-0506300-1-39-2', 'hep-ph-0506300-2-39-2'], 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[]

[]

[]

['hep-ph-0506300-1-2-0', 'hep-ph-0506300-2-2-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0506300

cond-mat-9603046

{'cond-mat-9603046-1-0-0': 'Recently proposed scenarios for the cuprates make extensive use of a "flat" quasiparticle (q.p.) dispersion and short-range hole-hole interactions in real-space, both caused by antiferromagnetic (AF) correlations.', 'cond-mat-9603046-1-0-1': 'The density of states (DOS) at half-filling has a robust peak which boosts the superconducting critical temperature [MATH] to large values as holes are introduced into the (rigid) q.p. band.', 'cond-mat-9603046-1-0-2': 'Here, the stability of such scenarios is studied after a [MATH] but small hole density is introduced.', 'cond-mat-9603046-1-0-3': 'The overall conclusion is that the main features of real-space AF-based approaches remain qualitatively similar, namely a large [MATH] is found and superconductivity (SC) appears in the [MATH] channel.', 'cond-mat-9603046-1-0-4': 'As the hole density grows the chemical potential [MATH] crosses a broad peak in the DOS.', 'cond-mat-9603046-1-0-5': 'We also observe that extended s-wave SC competes with d-wave in the overdoped regime.', 'cond-mat-9603046-1-1-0': 'The presence of "flat" regions in the normal-state q.p. dispersion is a remarkable feature of the phenomenology of hole-doped high temperature superconductors.[', 'cond-mat-9603046-1-1-1': '[CITATION] These flat bands are located around momenta [MATH] and [MATH], and at optimal doping they are [MATH] below the Fermi energy, according to angle-resolved photoemission (ARPES) studies.[', 'cond-mat-9603046-1-1-2': '[CITATION] Antiferromagnetic correlations may play an important role in the generation of these features, as has been suggested by studies of holes in the 2D t-J and Hubbard models at and away from half-filling.[', 'cond-mat-9603046-1-1-3': '[CITATION] In addition, the flat regions induce a large peak in the DOS which can be used to boost [MATH].', 'cond-mat-9603046-1-1-4': 'This leads to a natural explanation for the existence of an "optimal doping" which in this framework occurs when the flat regions are reached by [MATH].', 'cond-mat-9603046-1-1-5': 'We will refer to these ideas as the "Antiferromagnetic van Hove" (AFVH) scenario.[', 'cond-mat-9603046-1-1-6': '[CITATION] Superconductivity in the [MATH] channel is natural in the AFVH scenario due to the strong AF correlations.[', 'cond-mat-9603046-1-1-7': '[CITATION] The interaction of holes is better visualized in [MATH] i.e. with pairing occuring when dressed holes share a spin polaronic cloud, as in the spin-bag mechanism.[', 'cond-mat-9603046-1-1-8': '[CITATION] This real-space picture (see also Ref.[CITATION]) holds even for a small AF correlation length, [MATH], and in this approach there is no need to tune parameters to work very close to an AF instability.', 'cond-mat-9603046-1-1-9': 'Previous scenarios have also used van Hove (vH) singularities in the band structure to increase [MATH],[CITATION] but d-wave SC is not natural in this context unless AF correlations are included.', 'cond-mat-9603046-1-2-0': 'To obtain quantitative information from these intuitive ideas, holes moving with a dispersion calculated using one hole in an AF background, [MATH], and interacting through a nearest-neighbors (NN) attractive potential, that mimics the sharing of spin polarons, have been previously analyzed.', 'cond-mat-9603046-1-2-1': '[CITATION] Within a rigid band filling of [MATH] and using a BCS formalism, [MATH] superconductivity dominates with [MATH] caused by the large DOS implicit in the hole dispersion.', 'cond-mat-9603046-1-2-2': 'However, the accuracy of the rigid band approximation has not been studied before.', 'cond-mat-9603046-1-2-3': 'In particular, the following questions arise: (i) does the q.p. peak in the DOS found at [MATH] survive a finite hole density?;', 'cond-mat-9603046-1-2-4': '; (ii) how does the q.p. dispersion change with hole doping, and to what extend does this affect previous calculations in this framework?;', 'cond-mat-9603046-1-2-5': '; (iii) are the "shadow" regions generated by AF correlations[CITATION] important for real-space pairing approaches?', 'cond-mat-9603046-1-2-6': 'In this paper all these issues are discussed.', 'cond-mat-9603046-1-2-7': 'The overall conclusion is that as long as the hole density is such that the [MATH] is at least of a couple of lattice spacings, the predictions of the original AFVH scenario[CITATION] and other similar theories remain qualitatively the same.', 'cond-mat-9603046-1-3-0': 'To study the presence of a large DOS peak away from half-filling, [MATH] has been calculated numerically in systems with strong AF correlations.', 'cond-mat-9603046-1-3-1': 'While accurate extrapolations to the bulk limit are difficult, the simple qualitative picture emerging from these studies seems robust.', 'cond-mat-9603046-1-3-2': 'In Fig.1a, [MATH] for the 2D t-J model obtained with exact diagonalization (ED) techniques is shown at several densities.[', 'cond-mat-9603046-1-3-3': '[CITATION] At half-filling, a large DOS peak caused by flat regions in the q.p. dispersion appears at the top of the valence band, as discussed before.[', 'cond-mat-9603046-1-3-4': '[CITATION] Weight exists at energies far from [MATH] i.e. the large peak carries only part of the total weight.', 'cond-mat-9603046-1-3-5': 'As shown in Ref. [CITATION] the maximum in the DOS is [MATH] reached at the top of the valence band but at slightly smaller energies.', 'cond-mat-9603046-1-3-6': 'As [MATH] decreases, the peak is now much broader but it remains well-defined.', 'cond-mat-9603046-1-3-7': 'At [MATH], [MATH] is located close to the energy where [MATH] is maximized.', 'cond-mat-9603046-1-3-8': 'If a source of hole attraction exists in this system, a SC gap would open at [MATH], and the resulting [MATH] could be large.', 'cond-mat-9603046-1-3-9': 'However, even with [MATH] the SC gap would not alter much the already robust peak in the DOS shown in Fig.1a.', 'cond-mat-9603046-1-3-10': 'For [MATH], [MATH] moves to the left of the peak, where [MATH] should become smaller.', 'cond-mat-9603046-1-3-11': 'Then, as [MATH] is reduced away from half-filling in the 2D t-J model Fig.1a suggests that [MATH] travels across a broad DOS peak providing a natural definition for the "underdoped, optimal and overdoped" densities.', 'cond-mat-9603046-1-3-12': '[CITATION] Note that since the peak width increases substantially with hole density, strictly speaking the rigid band filling is invalid.', 'cond-mat-9603046-1-3-13': 'However, such an approximation may have captured the qualitative physics of the problem, as discussed in this paper, since the DOS peak is not washed out by a finite hole density.', 'cond-mat-9603046-1-4-0': 'Quantum Monte Carlo (QMC) data for the 2D one-band Hubbard model in strong coupling indicate a qualitative behavior similar to that of the t-J model, i.e. [MATH] also crosses a peak as [MATH] varies.', 'cond-mat-9603046-1-4-1': 'This effect was observed in previous simulations,[CITATION] and also in our own studies (Fig.1b).', 'cond-mat-9603046-1-4-2': '[CITATION] However, note that QMC simulations must be supplemented by a Maximum Entropy (ME) analysis to obtain [MATH].', 'cond-mat-9603046-1-4-3': 'ME predictions are more accurate close to [MATH] than at finite frequency, and thus we here do [MATH] interpret Fig.1b as suggesting the "generation" by doping of a Kondo-like peak at the top of the valence band which does not exist at [MATH].', 'cond-mat-9603046-1-4-4': 'We believe that the absence of a sharp peak at half-filling is caused by the systematic errors of the ME procedure and the finite temperature of the simulation.', 'cond-mat-9603046-1-4-5': 'Actually, a robust peak at [MATH] appears in other computational studies,[CITATION] and in the t-J model (Fig.1a).', 'cond-mat-9603046-1-4-6': 'Experimentally in the cuprates it has been already established[CITATION] that the states observed in PES upon doping are already present in the insulator and are [MATH] Kondo resonances.', 'cond-mat-9603046-1-4-7': 'In the present calculation, Fig.1a suggests that the broad peak observed at [MATH] in the t-J model is smoothly connected to the sharper peak found at half-filling.', 'cond-mat-9603046-1-5-0': 'Similar conclusions can be reached in geometries other than the 2D square lattice.', 'cond-mat-9603046-1-5-1': 'In particular, we studied the DOS of the t-J model on a [MATH], finding results very similar to those of Fig.1a, in agreement with previous calculations.[', 'cond-mat-9603046-1-5-2': '[CITATION] Note that on ladders the presence of pairing at finite hole density has been predicted.[', 'cond-mat-9603046-1-5-3': '[CITATION] Thus, a gap may open at [MATH] once couplings and lattice sizes are reached where pairing effects are important.', 'cond-mat-9603046-1-5-4': 'We have observed such effects in our studies, but we also found that at [MATH] the DOS has a sharp peak, as in Fig.1a.', 'cond-mat-9603046-1-5-5': 'Then, the ladder DOS peak is not exclusively caused by a BCS-induced redistribution of spectral weight upon doping.', 'cond-mat-9603046-1-6-0': 'In all cases discussed here, calculations of the spin-spin correlations show that [MATH] is approximately a couple of lattice spacings when [MATH] is located near the DOS peak, becoming negligible as the overdoped regime is reached.', 'cond-mat-9603046-1-6-1': 'Thus, a nonzero [MATH] and [MATH] near a large DOS peak are [MATH] features.', 'cond-mat-9603046-1-6-2': 'It is in this respect that scenarios where AF correlations produce a large DOS that enhances [MATH] differ from vH theories where divergences in the DOS are caused by band effects already present before interactions are switch on.', 'cond-mat-9603046-1-7-0': 'While the existence of a robust peak in the DOS is in good agreement with ARPES data,[CITATION] it is in apparent disagreement with specific heat studies for YBCO which have been interpreted as corresponding to a flat DOS.', 'cond-mat-9603046-1-7-1': '[CITATION] The lack of [MATH]-resolution in the specific heat measurements may solve this puzzle.', 'cond-mat-9603046-1-7-2': 'Actually, angle-[MATH] PES results for Bi2212 do not show the sharp flat features found in ARPES for the same material.', 'cond-mat-9603046-1-7-3': '[CITATION] Similar effects may affect the specific heat data.', 'cond-mat-9603046-1-8-0': 'Now let us discuss the [MATH]-dependence of the q.p. band obtained from [MATH].', 'cond-mat-9603046-1-8-1': 'Results are already available in the literature.[', 'cond-mat-9603046-1-8-2': '[CITATION] In Fig.2a-b, the q.p. dispersion is shown at [MATH], and at [MATH] for the t-J model.[', 'cond-mat-9603046-1-8-3': '[CITATION] Note that upon doping the small q.p. bandwidth and the flat regions remain well-defined, inducing a large DOS peak (Figs.1a).', 'cond-mat-9603046-1-8-4': 'However, the region around [MATH] has changed substantially, i.e. the AF shadow region clearly observed at [MATH] has reduced its intensity and considerable weight has been transferred to the inverse PES region (Fig.2b).', 'cond-mat-9603046-1-8-5': 'Actually, the q.p. dispersion at [MATH] can be fit by a tight-binding nearest-neighbors (NN) dispersion with a small effective hopping likely associated with [MATH].', 'cond-mat-9603046-1-8-6': 'This effect was also noticed in studies of the one band Hubbard model.', 'cond-mat-9603046-1-8-7': '[CITATION].', 'cond-mat-9603046-1-9-0': 'The changes in the q.p. dispersion with hole doping can be interpreted in two ways: First, note that [MATH] is influenced by matrix elements of [MATH] fermionic operators connecting states with [MATH] and [MATH] particles.', 'cond-mat-9603046-1-9-1': 'This is important when the q.p. weight is small i.e. when the state [MATH] does not have a large overlap with the ground state of the [MATH] particles subspace [MATH] being the ground state with [MATH] particles).', 'cond-mat-9603046-1-9-2': 'It has been suggested[CITATION] that if the hole excitation is created by a new operator [MATH] that incorporates the dressing of the hole by spin fluctuations, then [MATH] may now have a large overlap with [MATH].', 'cond-mat-9603046-1-9-3': 'In other words, if the dressed hole state resembles an extended spin polaron, then the physics deduced from PES studies, which rely on the sudden removal of a bare electron from the system, may be misleading.', 'cond-mat-9603046-1-9-4': 'To the extent that spin polarons remain well-defined at finite density, the use of [MATH] will induce spectral weight rearrangements between the PES and IPES regions, and Fig.2b could resemble Fig.2a after such spectral weight redistribution takes place, as some results already indicate.', 'cond-mat-9603046-1-9-5': '[CITATION] If this idea were correct, then the study of SC and transport, both regulated by [MATH] quasiparticles, could indeed be handled by filling a rigid band given by [MATH].', 'cond-mat-9603046-1-10-0': 'However, an alternative is that the difference between Figs.2a and 2b corresponds to an intrinsic change in the q.p. dispersion as [MATH] decreases.', 'cond-mat-9603046-1-10-1': 'This is the less favorable case for the AFVH approach, and thus we analyze this possibility in detail here.', 'cond-mat-9603046-1-10-2': "For this purpose, we have applied the standard BCS formalism to a model with a low density of q.p.'s having a dispersion [MATH], which roughly reproduces the dominant features of Fig.2b.", 'cond-mat-9603046-1-10-3': "As before, we include NN attraction between q.p.'s.", 'cond-mat-9603046-1-10-4': 'While naively it may seem dubious to use the same interaction both at and away from half-filling, the hole-hole potential should not be much affected at distances shorter than [MATH].', 'cond-mat-9603046-1-10-5': 'This can be illustrated by a real space analysis (Fig.2c) of a smeared [MATH]-function potential of AF origin [MATH], where the lattice spacing is set to 1, and [MATH].', 'cond-mat-9603046-1-10-6': 'Fig.2c shows that the NN potential ([MATH]) does not change noticeably as [MATH] is reduced, while [MATH] is rapidly suppressed.', 'cond-mat-9603046-1-10-7': 'Then, using the same NN form of the potential for many densities is justified.', 'cond-mat-9603046-1-10-8': 'Note also that the local character of the real-space potential does [MATH] imply small Cooper pairs.', 'cond-mat-9603046-1-10-9': 'Their size can be regulated using both its range [MATH] intensity.', 'cond-mat-9603046-1-11-0': 'Then, let us discuss the results obtained using the BCS formalism with an attractive NN potential [MATH],[CITATION] with [MATH], for both dispersions [MATH] and [MATH].', 'cond-mat-9603046-1-11-1': 'Solving numerically the gap equation, [MATH] is shown in Fig.3a.', 'cond-mat-9603046-1-11-2': 'As reported before,[CITATION] SC at [MATH] in the [MATH] channel appears naturally if the AF-dispersion is used.', 'cond-mat-9603046-1-11-3': 'If, instead, [MATH] is used, the flat bands present in this narrow dispersion produce [MATH] which is still large.', 'cond-mat-9603046-1-11-4': 'Even more remarkable is the fact that the [MATH] character of the SC state is maintained.', 'cond-mat-9603046-1-11-5': 'This result can be understood noticing that a combination of [MATH] with an attractive NN potential effectively locates us in the family of "t-U-V" models with U repulsive and V attractive, where it is known that for a "half-filled" electronic band the dominant SC state is [MATH]-wave.[', 'cond-mat-9603046-1-11-6': '[CITATION] In other words, when [MATH] is at the flat region in Fig.2b it approximately corresponds to a "half-filled" q.p. band, leading to a [MATH]-wave SC state (Fig.3a).[', 'cond-mat-9603046-1-11-7': '[CITATION] Then, one of our main results is that even if the q.p. dispersion changes substantially with doping near the [MATH] point, such an effect does [MATH] alter the main qualitative features found in previous studies.[', 'cond-mat-9603046-1-12-0': 'The present analysis also shows that the AF "shadow" regions of [MATH] are not crucial for the success of the real-space approach.', 'cond-mat-9603046-1-12-1': 'Actually, using [MATH], which does not contain weight in PES near [MATH], [MATH] is still robust and the d-wave state remains stable.', 'cond-mat-9603046-1-12-2': "To make this point more clear, we analyzed [MATH] using [MATH] but modulating the contribution of each momentum with a [MATH]-dependent weight [MATH] in the one particle Green's function.", 'cond-mat-9603046-1-12-3': 'We considered the special case where [MATH] is zero away from a window of total width [MATH] centered at the saddle point, which is located in the flat bands region.', 'cond-mat-9603046-1-12-4': 'Inside the window [MATH], [MATH].', 'cond-mat-9603046-1-12-5': 'Such a calculation also addresses indirectly possible concerns associated with widths [MATH] that q.p. peaks would acquire away from half-filling in standard Fermi liquids.', 'cond-mat-9603046-1-12-6': 'Results are shown in Fig.3b, for d-wave SC.', 'cond-mat-9603046-1-12-7': 'Note that even in the case where [MATH] is as small as just 5% of the total bandwidth (itself already small of order [MATH]), [MATH] remains robust and close to [MATH].', 'cond-mat-9603046-1-12-8': 'Then, it is clear that the dominant contribution to [MATH] comes from the flat regions and the shape of the q.p. dispersion away from them is not relevant for the sucess of the real-space approach.', 'cond-mat-9603046-1-13-0': 'We end this paper with novel predictions obtained from [MATH] and [MATH], when [MATH] reaches the bottom of these bands (which should [MATH] be confused with the bottom of the full hole spectrum since a large amount of weight lies at energies lower than those of the q.p. band).', 'cond-mat-9603046-1-13-1': 'In this regime, the BCS analysis applied to any of the two dispersions shows that extended [MATH]-wave SC dominates over [MATH]-wave SC in the "overdoped" regime (Fig.3a).', 'cond-mat-9603046-1-13-2': 'This occurs when the q.p. band is nearly empty which, according to Fig.1a, corresponds to an overall density of [MATH].', 'cond-mat-9603046-1-13-3': 'This change in the symmetry of the SC state can be understood recalling again that [MATH].', 'cond-mat-9603046-1-13-4': 'supplemented by a NN attraction formally corresponds to an effective "t-U-V" model.', 'cond-mat-9603046-1-13-5': 'It is well-known that in this model the SC state symmetry changes from d- to s-wave as the density is reduced away from half-filling to a nearly empty system.', 'cond-mat-9603046-1-13-6': '[CITATION] Actually the bound state of two particles with a NN tight binding dispersion and NN attraction is [MATH].', 'cond-mat-9603046-1-13-7': "To the extend that the AF- or NN-dispersions survive up to [MATH] hole doping, as suggested by numerical data,[CITATION] scenarios based on the real-space interaction of q.p.'s predict a competition between extended s-wave and [MATH]-wave SC in the [MATH] regime.", 'cond-mat-9603046-1-13-8': 'It is remarkable that calculations based on the analysis of the low electronic density [MATH] limit of the t-J model lead to analogous conclusions.', 'cond-mat-9603046-1-13-9': '[CITATION] Our approach is based on a very different formalism, but it arrives to similar results, and thus we believe that a crossover from d- to s-wave dominated SC in overdoped cuprates could occur.', 'cond-mat-9603046-1-13-10': 'Recent ARPES data for overdoped Bi2212 have indeed been interpreted as corresponding to a mixing of s- and d-wave states.', 'cond-mat-9603046-1-13-11': '[CITATION]', 'cond-mat-9603046-1-14-0': 'Summarizing, the predictions of real-space theories based on dispersions and interactions calculated at half-filling are qualitatively stable upon the introduction of a finite hole density.', 'cond-mat-9603046-1-14-1': 'These include a large [MATH], within the BCS formalism, and SC in the [MATH] channel.', 'cond-mat-9603046-1-14-2': 'In the overdoped regime a possible crossover from d- to extended s-wave SC was discussed.', 'cond-mat-9603046-1-15-0': 'We thank A. Fujimori, D. Dessau, Z.X. Shen, and D. M. Edwards for useful correspondence.', 'cond-mat-9603046-1-15-1': 'E. D. and A.M. are supported by grant NSF-DMR-9520776.', 'cond-mat-9603046-1-15-2': 'We thank the NHMFL and MARTECH for additional support.', 'cond-mat-9603046-1-16-0': 'bulut2 N. Bulut, D. J. Scalapino and S. R. White, Phys.', 'cond-mat-9603046-1-17-0': 'joynt Q. P. Li, B. E. C. Koltenbah, and R. J. Joynt, Phys.', 'cond-mat-9603046-1-18-0': 'prelovsek Recent specific heat calculations for the 2D t-J model are compatible with the presence of a maximum in [MATH] near half-filling (J. Jaklic and P. Prelovsek, preprint).', 'cond-mat-9603046-1-19-0': 'bulut1 N. Bulut, D. J. Scalapino and S. R. White, Phys.', 'cond-mat-9603046-1-20-0': 'comm2 Recently, Ulmke et al. (preprint) reported [MATH] using QMC on a [MATH] 3D cluster, with results similar to Fig.1b.', 'cond-mat-9603046-1-21-0': 'edwards For a different approach arriving to similar conclusions see J. Beenen and D. M. Edwards, J. of Low Temp.', 'cond-mat-9603046-1-22-0': 'kolte B. E. C. Koltenbah and R. Joynt, preprint, Jan. 1996.'}

{'cond-mat-9603046-2-0-0': 'Recently proposed scenarios for the cuprates make extensive use of a "flat" quasiparticle (q.p.) dispersion and short-range hole-hole interactions in real-space, both caused by antiferromagnetic (AF) correlations.', 'cond-mat-9603046-2-0-1': 'The density of states (DOS) at half-filling has a robust peak which boosts the superconducting critical temperature [MATH] to large values as holes are introduced into the (rigid) q.p. band.', 'cond-mat-9603046-2-0-2': 'Here, the stability of such scenarios is studied after a [MATH] but small hole density is introduced.', 'cond-mat-9603046-2-0-3': 'The overall conclusion is that the main features of real-space AF-based approaches remain qualitatively similar, namely a large [MATH] is found and superconductivity (SC) appears in the [MATH] channel.', 'cond-mat-9603046-2-0-4': 'As the hole density grows the chemical potential [MATH] crosses a broad peak in the DOS.', 'cond-mat-9603046-2-0-5': 'We also observe that extended s-wave SC competes with d-wave in the overdoped regime.', 'cond-mat-9603046-2-1-0': 'The presence of "flat" regions in the normal-state q.p. dispersion is a remarkable feature of the phenomenology of hole-doped high temperature superconductors.[', 'cond-mat-9603046-2-1-1': '[CITATION] These flat bands are located around momenta [MATH] and [MATH], and at optimal doping they are [MATH] below the Fermi energy, according to angle-resolved photoemission (ARPES) studies.[', 'cond-mat-9603046-2-1-2': '[CITATION] Antiferromagnetic correlations may play an important role in the generation of these features, as has been suggested by studies of holes in the 2D t-J and Hubbard models at and away from half-filling.[', 'cond-mat-9603046-2-1-3': '[CITATION] In addition, the flat regions induce a large peak in the DOS which can be used to boost [MATH].', 'cond-mat-9603046-2-1-4': 'This leads to a natural explanation for the existence of an "optimal doping" which in this framework occurs when the flat regions are reached by [MATH].', 'cond-mat-9603046-2-1-5': 'We will refer to these ideas as the "Antiferromagnetic van Hove" (AFVH) scenario.[', 'cond-mat-9603046-2-1-6': '[CITATION] Superconductivity in the [MATH] channel is natural in the AFVH scenario due to the strong AF correlations.[', 'cond-mat-9603046-2-1-7': '[CITATION] The interaction of holes is better visualized in [MATH] i.e. with pairing occuring when dressed holes share a spin polaronic cloud, as in the spin-bag mechanism.[', 'cond-mat-9603046-2-1-8': '[CITATION] This real-space picture (see also Ref.[CITATION]) holds even for a small AF correlation length, [MATH], and in this approach there is no need to tune parameters to work very close to an AF instability.', 'cond-mat-9603046-2-1-9': 'Previous scenarios have also used van Hove (vH) singularities in the band structure to increase [MATH],[CITATION] but d-wave SC is not natural in this context unless AF correlations are included.', 'cond-mat-9603046-2-2-0': 'To obtain quantitative information from these intuitive ideas, holes moving with a dispersion calculated using one hole in an AF background, [MATH], and interacting through a nearest-neighbors (NN) attractive potential, that mimics the sharing of spin polarons, have been previously analyzed.', 'cond-mat-9603046-2-2-1': '[CITATION] Within a rigid band filling of [MATH] and using a BCS formalism, [MATH] superconductivity dominates with [MATH] caused by the large DOS implicit in the hole dispersion.', 'cond-mat-9603046-2-2-2': 'However, the accuracy of the rigid band approximation has not been studied before.', 'cond-mat-9603046-2-2-3': 'In particular, the following questions arise: (i) does the q.p. peak in the DOS found at [MATH] survive a finite hole density?;', 'cond-mat-9603046-2-2-4': '; (ii) how does the q.p. dispersion change with hole doping, and to what extend does this affect previous calculations in this framework?;', 'cond-mat-9603046-2-2-5': '; (iii) are the "shadow" regions generated by AF correlations[CITATION] important for real-space pairing approaches?', 'cond-mat-9603046-2-2-6': 'In this paper all these issues are discussed.', 'cond-mat-9603046-2-2-7': 'The overall conclusion is that as long as the hole density is such that the [MATH] is at least of a couple of lattice spacings, the predictions of the original AFVH scenario[CITATION] and other similar theories remain qualitatively the same.', 'cond-mat-9603046-2-3-0': 'To study the presence of a large DOS peak away from half-filling, [MATH] has been calculated numerically in systems with strong AF correlations.', 'cond-mat-9603046-2-3-1': 'While accurate extrapolations to the bulk limit are difficult, the simple qualitative picture emerging from these studies seems robust.', 'cond-mat-9603046-2-3-2': 'In Fig.1a, [MATH] for the 2D t-J model obtained with exact diagonalization (ED) techniques is shown at several densities.[', 'cond-mat-9603046-2-3-3': '[CITATION] At half-filling, a large DOS peak caused by flat regions in the q.p. dispersion appears at the top of the valence band, as discussed before.[', 'cond-mat-9603046-2-3-4': '[CITATION] Weight exists at energies far from [MATH] i.e. the large peak carries only part of the total weight.', 'cond-mat-9603046-2-3-5': 'As shown in Ref. [CITATION] the maximum in the DOS is [MATH] reached at the top of the valence band but at slightly smaller energies.', 'cond-mat-9603046-2-3-6': 'As [MATH] decreases, the peak is now much broader but it remains well-defined.', 'cond-mat-9603046-2-3-7': 'At [MATH], [MATH] is located close to the energy where [MATH] is maximized.', 'cond-mat-9603046-2-3-8': 'If a source of hole attraction exists in this system, a SC gap would open at [MATH], and the resulting [MATH] could be large.', 'cond-mat-9603046-2-3-9': 'However, even with [MATH] the SC gap would not alter much the already robust peak in the DOS shown in Fig.1a.', 'cond-mat-9603046-2-3-10': 'For [MATH], [MATH] moves to the left of the peak, where [MATH] should become smaller.', 'cond-mat-9603046-2-3-11': 'Then, as [MATH] is reduced away from half-filling in the 2D t-J model Fig.1a suggests that [MATH] travels across a broad DOS peak providing a natural definition for the "underdoped, optimal and overdoped" densities.', 'cond-mat-9603046-2-3-12': '[CITATION] Note that since the peak width increases substantially with hole density, strictly speaking the rigid band filling is invalid.', 'cond-mat-9603046-2-3-13': 'However, such an approximation may have captured the qualitative physics of the problem, as discussed in this paper, since the DOS peak is not washed out by a finite hole density.', 'cond-mat-9603046-2-4-0': 'Quantum Monte Carlo (QMC) data for the 2D one-band Hubbard model in strong coupling indicate a qualitative behavior similar to that of the t-J model, i.e. [MATH] also crosses a peak as [MATH] varies.', 'cond-mat-9603046-2-4-1': 'This effect was observed in previous simulations,[CITATION] and also in our own studies (Fig.1b).', 'cond-mat-9603046-2-4-2': '[CITATION] However, note that QMC simulations must be supplemented by a Maximum Entropy (ME) analysis to obtain [MATH].', 'cond-mat-9603046-2-4-3': 'ME predictions are more accurate close to [MATH] than at finite frequency, and thus we here do [MATH] interpret Fig.1b as suggesting the "generation" by doping of a Kondo-like peak at the top of the valence band which does not exist at [MATH].', 'cond-mat-9603046-2-4-4': 'We believe that the absence of a sharp peak at half-filling is caused by the systematic errors of the ME procedure and the finite temperature of the simulation.', 'cond-mat-9603046-2-4-5': 'Actually, a robust peak at [MATH] appears in other computational studies,[CITATION] and in the t-J model (Fig.1a).', 'cond-mat-9603046-2-4-6': 'Experimentally in the cuprates it has been already established[CITATION] that the states observed in PES upon doping are already present in the insulator and are [MATH] Kondo resonances.', 'cond-mat-9603046-2-4-7': 'In the present calculation, Fig.1a suggests that the broad peak observed at [MATH] in the t-J model is smoothly connected to the sharper peak found at half-filling.', 'cond-mat-9603046-2-5-0': 'Similar conclusions can be reached in geometries other than the 2D square lattice.', 'cond-mat-9603046-2-5-1': 'In particular, we studied the DOS of the t-J model on a [MATH], finding results very similar to those of Fig.1a, in agreement with previous calculations.[', 'cond-mat-9603046-2-5-2': '[CITATION] Note that on ladders the presence of pairing at finite hole density has been predicted.[', 'cond-mat-9603046-2-5-3': '[CITATION] Thus, a gap may open at [MATH] once couplings and lattice sizes are reached where pairing effects are important.', 'cond-mat-9603046-2-5-4': 'We have observed such effects in our studies, but we also found that at [MATH] the DOS has a sharp peak, as in Fig.1a.', 'cond-mat-9603046-2-5-5': 'Then, the ladder DOS peak is not exclusively caused by a BCS-induced redistribution of spectral weight upon doping.', 'cond-mat-9603046-2-6-0': 'In all cases discussed here, calculations of the spin-spin correlations show that [MATH] is approximately a couple of lattice spacings when [MATH] is located near the DOS peak, becoming negligible as the overdoped regime is reached.', 'cond-mat-9603046-2-6-1': 'Thus, a nonzero [MATH] and [MATH] near a large DOS peak are [MATH] features.', 'cond-mat-9603046-2-6-2': 'It is in this respect that scenarios where AF correlations produce a large DOS that enhances [MATH] differ from vH theories where divergences in the DOS are caused by band effects already present before interactions are switch on.', 'cond-mat-9603046-2-7-0': 'While the existence of a robust peak in the DOS is in good agreement with ARPES data,[CITATION] it is in apparent disagreement with specific heat studies for YBCO which have been interpreted as corresponding to a flat DOS.', 'cond-mat-9603046-2-7-1': '[CITATION] The lack of [MATH]-resolution in the specific heat measurements may solve this puzzle.', 'cond-mat-9603046-2-7-2': 'Actually, angle-[MATH] PES results for Bi2212 do not show the sharp flat features found in ARPES for the same material.', 'cond-mat-9603046-2-7-3': '[CITATION] Similar effects may affect the specific heat data.', 'cond-mat-9603046-2-8-0': 'Now let us discuss the [MATH]-dependence of the q.p. band obtained from [MATH].', 'cond-mat-9603046-2-8-1': 'Results are already available in the literature.[', 'cond-mat-9603046-2-8-2': '[CITATION] In Fig.2a-b, the q.p. dispersion is shown at [MATH], and at [MATH] for the t-J model.[', 'cond-mat-9603046-2-8-3': '[CITATION] Note that upon doping the small q.p. bandwidth and the flat regions remain well-defined, inducing a large DOS peak (Figs.1a).', 'cond-mat-9603046-2-8-4': 'However, the region around [MATH] has changed substantially, i.e. the AF shadow region clearly observed at [MATH] has reduced its intensity and considerable weight has been transferred to the inverse PES region (Fig.2b).', 'cond-mat-9603046-2-8-5': 'Actually, the q.p. dispersion at [MATH] can be fit by a tight-binding nearest-neighbors (NN) dispersion with a small effective hopping likely associated with [MATH].', 'cond-mat-9603046-2-8-6': 'This effect was also noticed in studies of the Hubbard model.', 'cond-mat-9603046-2-8-7': '[CITATION].', 'cond-mat-9603046-2-9-0': 'The changes in the q.p. dispersion with hole doping can be interpreted in two ways: First, note that [MATH] is influenced by matrix elements of [MATH] fermionic operators connecting states with [MATH] and [MATH] particles.', 'cond-mat-9603046-2-9-1': 'This is important when the q.p. weight is small i.e. when the state [MATH] does not have a large overlap with the ground state of the [MATH] particles subspace [MATH] being the ground state with [MATH] particles).', 'cond-mat-9603046-2-9-2': 'It has been suggested[CITATION] that if the hole excitation is created by a new operator [MATH] that incorporates the dressing of the hole by spin fluctuations, then [MATH] may now have a large overlap with [MATH].', 'cond-mat-9603046-2-9-3': 'In other words, if the dressed hole state resembles an extended spin polaron, then the physics deduced from PES studies, which rely on the sudden removal of a bare electron from the system, may be misleading.', 'cond-mat-9603046-2-9-4': 'To the extent that spin polarons remain well-defined at finite density, the use of [MATH] will induce spectral weight rearrangements between the PES and IPES regions, and Fig.2b could resemble Fig.2a after such spectral weight redistribution takes place, as some results already indicate.', 'cond-mat-9603046-2-9-5': '[CITATION] If this idea were correct, then the study of SC and transport, both regulated by [MATH] quasiparticles, could indeed be handled by filling a rigid band given by [MATH].', 'cond-mat-9603046-2-10-0': 'However, an alternative is that the difference between Figs.2a and 2b corresponds to an intrinsic change in the q.p. dispersion as [MATH] decreases.', 'cond-mat-9603046-2-10-1': 'This is the less favorable case for the AFVH approach, and thus we analyze this possibility in detail here.', 'cond-mat-9603046-2-10-2': "For this purpose, we have applied the standard BCS formalism to a model with a low density of q.p.'s having a dispersion [MATH], which roughly reproduces the dominant features of Fig.2b.", 'cond-mat-9603046-2-10-3': "As before, we include NN attraction between q.p.'s.", 'cond-mat-9603046-2-10-4': 'While naively it may seem dubious to use the same interaction both at and away from half-filling, the hole-hole potential should not be much affected at distances shorter than [MATH].', 'cond-mat-9603046-2-10-5': 'This can be illustrated by a real space analysis (Fig.2c) of a smeared [MATH]-function potential of AF origin [MATH], where the lattice spacing is set to 1, and [MATH].', 'cond-mat-9603046-2-10-6': 'Fig.2c shows that the NN potential ([MATH]) does not change noticeably as [MATH] is reduced, while [MATH] is rapidly suppressed.', 'cond-mat-9603046-2-10-7': 'Then, using the same NN form of the potential for many densities is justified.', 'cond-mat-9603046-2-10-8': 'Note also that the local character of the real-space potential does [MATH] imply small Cooper pairs.', 'cond-mat-9603046-2-10-9': 'Their size can be regulated using both its range [MATH] intensity.', 'cond-mat-9603046-2-11-0': 'Then, let us discuss the results obtained using the BCS formalism with an attractive NN potential [MATH],[CITATION] with [MATH], for both dispersions [MATH] and [MATH].', 'cond-mat-9603046-2-11-1': 'Solving numerically the gap equation, [MATH] is shown in Fig.3a.', 'cond-mat-9603046-2-11-2': 'As reported before,[CITATION] SC at [MATH] in the [MATH] channel appears naturally if the AF-dispersion is used.', 'cond-mat-9603046-2-11-3': 'If, instead, [MATH] is used, the flat bands present in this narrow dispersion produce [MATH] which is still large.', 'cond-mat-9603046-2-11-4': 'Even more remarkable is the fact that the [MATH] character of the SC state is maintained.', 'cond-mat-9603046-2-11-5': 'This result can be understood noticing that a combination of [MATH] with an attractive NN potential effectively locates us in the family of "t-U-V" models with U repulsive and V attractive, where it is known that for a "half-filled" electronic band the dominant SC state is [MATH]-wave.[', 'cond-mat-9603046-2-11-6': '[CITATION] In other words, when [MATH] is at the flat region in Fig.2b it approximately corresponds to a "half-filled" q.p. band, leading to a [MATH]-wave SC state (Fig.3a).[', 'cond-mat-9603046-2-11-7': '[CITATION] Then, one of our main results is that even if the q.p. dispersion changes substantially with doping near the [MATH] point, such an effect does [MATH] alter the main qualitative features found in previous studies.[', 'cond-mat-9603046-2-12-0': 'The present analysis also shows that the AF "shadow" regions of [MATH] are not crucial for the success of the real-space approach.', 'cond-mat-9603046-2-12-1': 'Actually, using [MATH], which does not contain weight in PES near [MATH], [MATH] is still robust and the d-wave state remains stable.', 'cond-mat-9603046-2-12-2': "To make this point more clear, we analyzed [MATH] using [MATH] but modulating the contribution of each momentum with a [MATH]-dependent weight [MATH] in the one particle Green's function.", 'cond-mat-9603046-2-12-3': 'We considered the special case where [MATH] is zero away from a window of total width [MATH] centered at the saddle point, which is located in the flat bands region.', 'cond-mat-9603046-2-12-4': 'Inside the window [MATH], [MATH].', 'cond-mat-9603046-2-12-5': 'Such a calculation also addresses indirectly possible concerns associated with widths [MATH] that q.p. peaks would acquire away from half-filling in standard Fermi liquids.', 'cond-mat-9603046-2-12-6': 'Results are shown in Fig.3b, for d-wave SC.', 'cond-mat-9603046-2-12-7': 'Note that even in the case where [MATH] is as small as just 5% of the total bandwidth (itself already small of order [MATH]), [MATH] remains robust and close to [MATH].', 'cond-mat-9603046-2-12-8': 'Then, it is clear that the dominant contribution to [MATH] comes from the flat regions and the shape of the q.p. dispersion away from them is not relevant for the sucess of the real-space approach.', 'cond-mat-9603046-2-13-0': 'We end this paper with novel predictions obtained from [MATH] and [MATH], when [MATH] reaches the bottom of these bands (which should [MATH] be confused with the bottom of the full hole spectrum since a large amount of weight lies at energies lower than those of the q.p. band).', 'cond-mat-9603046-2-13-1': 'In this regime, the BCS analysis applied to any of the two dispersions shows that extended [MATH]-wave SC dominates over [MATH]-wave SC in the "overdoped" regime (Fig.3a).', 'cond-mat-9603046-2-13-2': 'This occurs when the q.p. band is nearly empty which, according to Fig.1a, corresponds to an overall density of [MATH].', 'cond-mat-9603046-2-13-3': 'This change in the symmetry of the SC state can be understood recalling again that [MATH].', 'cond-mat-9603046-2-13-4': 'supplemented by a NN attraction formally corresponds to an effective "t-U-V" model.', 'cond-mat-9603046-2-13-5': 'It is well-known that in this model the SC state symmetry changes from d- to s-wave as the density is reduced away from half-filling to a nearly empty system.', 'cond-mat-9603046-2-13-6': '[CITATION] Actually the bound state of two particles with a NN tight binding dispersion and NN attraction is [MATH].', 'cond-mat-9603046-2-13-7': "To the extend that the AF- or NN-dispersions survive up to [MATH] hole doping, as suggested by numerical data,[CITATION] scenarios based on the real-space interaction of q.p.'s predict a competition between extended s-wave and [MATH]-wave SC in the [MATH] regime.", 'cond-mat-9603046-2-13-8': 'It is remarkable that calculations based on the analysis of the low electronic density [MATH] limit of the t-J model lead to analogous conclusions.', 'cond-mat-9603046-2-13-9': '[CITATION] Our approach is based on a very different formalism, but it arrives to similar results, and thus we believe that a crossover from d- to s-wave dominated SC in overdoped cuprates could occur.', 'cond-mat-9603046-2-13-10': 'Recent ARPES data for overdoped Bi2212 have indeed been interpreted as corresponding to a mixing of s- and d-wave states.', 'cond-mat-9603046-2-13-11': '[CITATION]', 'cond-mat-9603046-2-14-0': 'Summarizing, the predictions of real-space theories based on dispersions and interactions calculated at half-filling are qualitatively stable upon the introduction of a finite hole density.', 'cond-mat-9603046-2-14-1': 'These include a large [MATH], within the BCS formalism, and SC in the [MATH] channel.', 'cond-mat-9603046-2-14-2': 'In the overdoped regime a possible crossover from d- to extended s-wave SC was discussed.', 'cond-mat-9603046-2-15-0': 'We thank A. Fujimori, D. Dessau, Z.X. Shen, and D. M. Edwards for useful correspondence.', 'cond-mat-9603046-2-15-1': 'E. D. and A.M. are supported by grant NSF-DMR-9520776.', 'cond-mat-9603046-2-15-2': 'We thank the NHMFL and MARTECH for additional support.', 'cond-mat-9603046-2-16-0': 'bulut2 N. Bulut, D. J. Scalapino and S. R. White, Phys.', 'cond-mat-9603046-2-17-0': 'hanke R. Preuss, W. Hanke, and W. von der Linden, Phys.', 'cond-mat-9603046-2-18-0': 'joynt Q. P. Li, B. E. C. Koltenbah, and R. J. Joynt, Phys.', 'cond-mat-9603046-2-19-0': 'prelovsek Recent specific heat calculations for the 2D t-J model are compatible with the presence of a maximum in [MATH] near half-filling (J. Jaklic and P. Prelovsek, preprint).', 'cond-mat-9603046-2-20-0': 'bulut1 N. Bulut, D. J. Scalapino and S. R. White, Phys.', 'cond-mat-9603046-2-21-0': 'comm2 Recently, Ulmke et al. (preprint) reported [MATH] using QMC on a [MATH] 3D cluster, with results similar to Fig.1b.', 'cond-mat-9603046-2-22-0': 'edwards For a different approach arriving to similar conclusions see J. Beenen and D. M. Edwards, J. of Low Temp.', 'cond-mat-9603046-2-23-0': 'kolte B. E. C. Koltenbah and R. Joynt, preprint, Jan. 1996.', 'cond-mat-9603046-2-23-1': 'See also D. van der Marel, Phys.'}

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[['cond-mat-9603046-1-8-6', 'cond-mat-9603046-2-8-6']]

[]

[]

[]

['cond-mat-9603046-1-8-7', 'cond-mat-9603046-1-13-11', 'cond-mat-9603046-1-15-1', 'cond-mat-9603046-2-8-7', 'cond-mat-9603046-2-13-11', 'cond-mat-9603046-2-15-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/9603046

1010.0748

{'1010.0748-1-0-0': 'We propose to make use of the off-shell recursive relations with the color-flow decomposition in the calculation of QCD amplitudes on MadGraph.', '1010.0748-1-0-1': 'We introduce colored quarks and their interactions with nine gluons in the color-flow basis plus an Abelian gluon on MadGraph, such that it generates helicity amplitudes in the color-flow basis with off-shell recursive formulae for multi-gluon sub-amplitudes.', '1010.0748-1-0-2': 'We demonstrate calculations of up to 5-jet processes such as [MATH], [MATH] and [MATH].', '1010.0748-1-0-3': 'Although our demonstration is limited, it paves the way to evaluate amplitudes with more quark lines and gluons with Madgraph.', '1010.0748-1-1-0': '# Introduction', '1010.0748-1-2-0': 'Large Hadron Collider (LHC) has been steadily running at 7 TeV since the end of March/2010.', '1010.0748-1-2-1': 'Since LHC is a hadron collider, understanding of QCD radiation processes is essential for the success of the experiments.', '1010.0748-1-2-2': 'To evaluate new physics signals and Standard Model (SM) backgrounds, we should resort to simulation programs which generate events with many hadrons and investigate obsevable distributions.', '1010.0748-1-2-3': 'In each simulation, one calculates matrix elements of hard processes with quarks and gluons, and this task is usually done by an automatic amplitude calculator.', '1010.0748-1-3-0': 'MadGraph[CITATION] is one of those programs.', '1010.0748-1-3-1': 'Although it is highly developed and has ample user-friendly utilities such as event generation with new physics signals matched to Parton Shower[CITATION], it cannot evaluate matrix elements with more than five jets in the final state[CITATION]', '1010.0748-1-4-0': 'This is a serious drawback since exact evaluation of multi-jet matrix elements is often required to study sub-structure of broad jets in identifying new physics signatures over QCD background.', '1010.0748-1-4-1': 'It is also disappointing because MadGraph generated HELAS amplitudes[CITATION] can be computed very fast on GPU (Graphic Processing Unit)[CITATION].', '1010.0748-1-5-0': 'Some of other simulation packages such as Alpgen[CITATION] employ QCD off-shell recursive relations to produce multi-parton scattering amplitudes efficiently.', '1010.0748-1-5-1': 'Successful computation of recursively evaluated QCD amplitudes on GPU has been reported[CITATION].', '1010.0748-1-5-2': 'It is therefore interesting to examine the possibility of implementing recursive relations for gluon currents in MadGraph without sacrificing its benefits.', '1010.0748-1-5-3': 'In this paper, we examine the use of the recursive amplitudes in the color-flow basis, since corresponding HELAS amplitude package has been developed in ref. [CITATION].', '1010.0748-1-5-4': 'The main purpose of this paper is to show that this approach can be accommodated in MadGraph with explicit examples.', '1010.0748-1-6-0': 'The outline of this paper is as follows.', '1010.0748-1-6-1': 'In sections 2, we briefly review the color-flow decomposition and off-shell recursive relations.', '1010.0748-1-6-2': 'We then discuss the implementation of those tequniques in MadGraph, using its "user mode" option[CITATION] in section 3.', '1010.0748-1-6-3': 'In section 4 and 5, we explain how we evaluate the color-summed amplitude squared and show the numerical results of [MATH]-jet production cross section.', '1010.0748-1-6-4': 'Section 6 is devoted to conclusions.', '1010.0748-1-7-0': '# The color-flow basis and off-shell recursive relations', '1010.0748-1-8-0': 'First, we briefly review the color-flow decomposition[CITATION].', '1010.0748-1-8-1': 'The Lagrangian of QCD can be expressed as [EQUATION] where [EQUATION]', '1010.0748-1-8-2': 'Upper and lower indeces are those of [MATH] and [MATH] representation, respectively.', '1010.0748-1-8-3': 'Note that the gluon fields, [MATH], are renormalized by a factor of [MATH], and hence the coupling [MATH] is divided by [MATH].', '1010.0748-1-8-4': 'At this stage, not all the nine gluon fields are independent because the traceslessness of the [MATH] generators gives [EQUATION]', '1010.0748-1-8-5': 'We then rewite eq. [REF] by adding and subtracting an additinoal [MATH] gauge boson (Abelian gluon), [EQUATION] such that the QCD Lagrangian is expressed as [EQUATION] with [EQUATION]', '1010.0748-1-8-6': 'Here all the nine gluons, [MATH] are now independent while the covariant tensors keep the same form [EQUATION]', '1010.0748-1-8-7': 'In this basis, nine gluons have one-to-one correspondence to a set of indeces of [MATH] and [MATH] representations of [MATH]: [MATH].', '1010.0748-1-8-8': 'Although the definition of [MATH] contains self-coupling terms for all the nine gluons, the Abelian gluon contribution drops out in the sum.', '1010.0748-1-8-9': 'Accordingly, the Feynman rules of the Lagrangian [REF] give directly the [MATH] (QCD) amplitudes for purely gluonic processes[CITATION].', '1010.0748-1-9-0': 'From the Lagrangian [REF], we derive Feynman rules and list them in Fig. [REF].', '1010.0748-1-10-0': 'It should be noted that the propagator of the Abelian gluon has an extra [MATH] factor.', '1010.0748-1-10-1': 'We can evaluate QCD amplitudes using these Feynman rules.', '1010.0748-1-10-2': 'For [MATH]-gluon scattering processes, it is written as [EQUATION] where [MATH] are gauge-invariant partial amplitudes, which depend only on particle momenta and helicities represented simply by the gluon indices, [MATH].', '1010.0748-1-10-3': 'The sumation is taken over all [MATH] non-cyclic permutaions, [MATH], of gluons.', '1010.0748-1-10-4': "A set of [MATH] Kronecker delta's in each term gives the color flow of the correponding partial amplitudes.", '1010.0748-1-10-5': "As an example, we list in Fig. [REF] all six color flows indicated by Kronecker delta's for the [MATH] process, [MATH].", '1010.0748-1-11-0': 'If a scattering process involves quarks, contributions from the Abelian gluon do not decouple.', '1010.0748-1-11-1': 'Therefore, we have to add all its contributions to the total amplitude.', '1010.0748-1-12-0': 'For processes with one quark line, such contributions come from Abelian gluon emitting diagrams.', '1010.0748-1-12-1': 'Therefore, the total amplitude for [MATH] process becomes [EQUATION] where [MATH] is the partial amplitude with [MATH] Abelian gluons, and [MATH] and [MATH] denote the momenta and the helicities of the quark and the anti-quark, respectively.', '1010.0748-1-12-2': 'The first term in the r.h.s. of eq. [REF] gives contributions from [MATH] gluons, [MATH], and the other terms give those from [MATH] gluons and [MATH] Abelian gluons, summed over [MATH] to [MATH].', '1010.0748-1-12-3': 'Note that we should multiply the [MATH] factor to partial amplitudes.', '1010.0748-1-12-4': 'This extra factor comes from the [MATH] factor in the Abelian gluon propagator.', '1010.0748-1-12-5': 'Corresponding color flows of those partial amplitudes are shown in Fig. [REF].', '1010.0748-1-13-0': 'For processes with two or more quark lines, the Abelian gluon can be exchanged between quark lines, which give rise to the amplitudes with [MATH] suppression from the propagating Abelian gluon.', '1010.0748-1-13-1': 'For processes such as [MATH] gluons, the total amplitude becomes [EQUATION] [MATH] (or [MATH]) denotes a partial amplitude with [MATH] gluons, where two sets of arguments divided by "[MATH]" belong to two different color-flow chains; one starts from [MATH] in the initial state) and ends with [MATH] in the final state), and the other starts from [MATH] and ends with [MATH], which are given explicitly by two sets of delta\'s before the amplitude.', '1010.0748-1-13-2': 'The difference between [MATH] and [MATH] is that the former consists of [MATH]-gluon-exchange diagrams while the latter consists of Abelian-gluon-exchange ones.', '1010.0748-1-13-3': 'Here we write down explicitly the partial amplitudes with external [MATH] gluons, [MATH] and [MATH], whereas those with external Abelian gluons should be added as in the previous one-quark-line case.', '1010.0748-1-13-4': 'For illustration, some typical diagrams for the [MATH] process are shown in Fig. [REF].', '1010.0748-1-14-0': '## Recursive relations of off-shell gluon currents', '1010.0748-1-15-0': 'Next, we discuss off-shell recursive relations in the color-flow basis.', '1010.0748-1-15-1': 'We define an [MATH]-point off-shell gluon current [MATH] recursively as [EQUATION] where the numbers in the argument of currents represent gluons, and their order respects the color flow as in the partial amplitudes, [MATH], in eq. [REF].', '1010.0748-1-15-2': 'As an initial value, the 2-point gluon current, [MATH], is defined as the polarization vector of the gluon [MATH], [MATH].', '1010.0748-1-15-3': '[MATH] is the partial sum of gluon momenta: [EQUATION] where [MATH] to [MATH] are defined as flowing-out momenta.', '1010.0748-1-15-4': '[MATH] and [MATH] denote the three- and four-gluon verteces from the Feynman rule of Fig.[REF]: [EQUATION]', '1010.0748-1-15-5': 'The summation over [MATH] and [MATH] in eq.[REF] are taken over the 3 and [MATH] indices, [MATH] and [MATH], of the off-shell gluons of off-shell currents, [MATH] and [MATH], respectively.', '1010.0748-1-15-6': 'For illustration, the 5-point off-shell current is shown explicitly in Fig.[REF].', '1010.0748-1-16-0': 'Helicity amplitudes of pure gluon processes in the color-flow basis are obtained from off-shell gluon currents as [EQUATION]', '1010.0748-1-16-1': 'Starting from the external wave functions [REF], the recursion relation computes the partial amplitudes very effectively by using the HELAS codes.', '1010.0748-1-16-2': 'When the off-shell line "[MATH]" in Fig. [REF] is set on-shell, the same set of diagrams give the complete helicity amplitudes in the color-flow basis.', '1010.0748-1-17-0': '# Implementation in MadGraph', '1010.0748-1-18-0': 'In this section, we discuss how we implement off-shell recursive relations in MadGraph in the color-flow basis and how we generate helicity amplitudes of QCD processes.', '1010.0748-1-19-0': '## Subroutines for off-shell recursive formulae', '1010.0748-1-20-0': 'First, we introduce new HELAS[CITATION] subroutines in MadGraph which make [MATH]-point gluon off-shell currents and [MATH]-gluon amplitudes in the color-flow basis, according to eq.[REF] and eq.[REF], respectively.', '1010.0748-1-20-1': 'Although the expression eq.[REF] gives the off-shell gluon current made from [MATH] on-shell gluons, in HELAS amplitudes any input on-shell gluon wave functions can be replaced by arbitrary off-shell gluon currents with the same color quantum numbers.', '1010.0748-1-20-2': 'Shown in Fig.[REF] is an example of such diagrams that are calculated by the HELAS subroutine for the 4-point off-shell gluon current.', '1010.0748-1-21-0': 'Thanks to this property of the HELAS amplitudes, we need to introduce only two types of subroutines: one which computes helicity amplitudes of [MATH]-gluon processes via eq.[REF] and the other which computes off-shell gluon currents from [MATH]-point gluon vertices via eq.[REF].', '1010.0748-1-21-1': 'We name the first type of subroutines as gluon and the second type as jgluo, where denotes the number of external on-shell and off-shell gluons.', '1010.0748-1-22-0': 'The number of new subroutines we should add to MadGraph depends on the number of external partons (quarks and gluons) in QCD processes.', '1010.0748-1-22-1': 'Processes with [MATH]-partons can be classified as those with [MATH] gluons, those with [MATH] gluons and one quark line, those with [MATH] gluons and two quark lines, and so on.', '1010.0748-1-23-0': 'In the color-flow basis, the first class of processes with [MATH] external gluons are calculated by just one amplitude subroutine, gluonn.', '1010.0748-1-24-0': 'For the second class of processes with [MATH] gluons and one quark line, we need up to [MATH]-point off-shell current subroutines, jgluo with [MATH] to [MATH].', '1010.0748-1-24-1': 'This is because the largest off-shell gluon current appears in the computation of diagrams where [MATH] on-shell gluons are connected to the quark line through one off-shell gluon, which can be computed by the [MATH]-point off-shell current and the [MATH] amplitude subroutine, iovxxx.', '1010.0748-1-24-2': 'Note that the same diagram can be computed by the off-shell gluon current made by the quark pair, jioxxx, and the [MATH]-point gluon amplitude subroutine, gluon(n-1).', '1010.0748-1-24-3': 'This type of redundancy in the computation of each Feynman diagram is inherent in the HELAS amplitudes, which is often useful in testing the codes.', '1010.0748-1-25-0': 'For [MATH]-parton processes with [MATH] gluons and two quark lines, we need up to [MATH]-point off-shell current subroutines.', '1010.0748-1-25-1': 'By also introducing multiple gluon amplitude subroutines up to [MATH]-point vertex, the maximum redundancy of HELAS amplitudes is achieved.', '1010.0748-1-25-2': 'Likewise, for [MATH]-parton processes with [MATH] quark lines, we need up to [MATH]-point off-shell current or amplitude subroutines.', '1010.0748-1-26-0': 'We list in Table [REF] the new HELAS subroutines we introduce in this study.', '1010.0748-1-27-0': 'The subroutine gluon evaluates a -gluon amplitude in the color-flow basis, and jgluo computes an off-shell current from ([MATH]) external gluons.', '1010.0748-1-27-1': 'Since we consider up to seven parton processes, [MATH], [MATH] and [MATH], in this study, we use gluon4 to gluon7 and jgluo4 to jgluo6.', '1010.0748-1-28-0': 'Beside these subroutines that computes amplitudes and currents recursively, we also introduce two subroutines: ggggcf and jgggcf.', '1010.0748-1-28-1': 'They evaluate an amplitude and an off-shell current from the contact 4-gluon vertex, following the Feynman rule of Fig. [REF].', '1010.0748-1-28-2': 'Although the amplitude subroutine and the off-shell current subroutine for the 4-gluon vertex already exist in MadGraph, ggggxx and jgggxx, we should introduce the new ones which evaluate the sum of s- and t-type or s- and u-type vertices for a given color flow, since the default subroutines compute only one type at a time.', '1010.0748-1-28-3': 'We show the codes of ggggcf, jgggcf, gluon5 and jgluo5 in Appendix.', '1010.0748-1-29-0': '## Introduction of a new Model: CFQCD', '1010.0748-1-30-0': 'Next, we define a new Model with which MadGraph generates HELAS amplitudes in the color-flow basis since the present MadGraph computes them in the color-ordered basis[CITATION].', '1010.0748-1-30-1': 'A Model is a set of definitions of particles, their interactions and couplings; there are about twenty preset Models in the present MadGraph package[CITATION] such as the Standard Model and the Minimal SUSY Standard Model[CITATION].', '1010.0748-1-30-2': 'There is also a template for further extension by users, User Mode (usrmod).', '1010.0748-1-30-3': 'Using this template, we can add new particles and their interactions to MadGraph.', '1010.0748-1-30-4': 'We make use of this utility and introduce a model which we call the CFQCD Model.', '1010.0748-1-31-0': 'In the CFQCD Model, we introduce gluons and quarks as new particles labeled by indices that dictate the color flow, such that the diagrams for partial amplitudes are generated according to the Feynman rules of Fig. [REF].', '1010.0748-1-31-1': 'We need one index for quarks and two indices for gluons, such as [MATH], [MATH] and [MATH].', '1010.0748-1-31-2': 'The Abelian gluon, [MATH], does not have an index.', '1010.0748-1-31-3': 'The index labels all possible color flows, and it runs from 1 to [MATH], where [EQUATION]', '1010.0748-1-31-4': "This number [MATH] is the number of Kronecker's delta symbols which dictate the color flow.", '1010.0748-1-31-5': "As an example, let us consider the purely gluonic process [EQUATION] for which we need seven delta's to specify the color flow: [EQUATION]", '1010.0748-1-31-6': "Here the numbers in parentheses label gluons whereas the numbers in the sub-indices of Kronecker's delta's count the color-flow lines, 1 to [MATH] as depicted in Fig. [REF].", '1010.0748-1-32-0': 'In CFQCD, we label gluons according to their flowing-out, 3, and flowing-in, [MATH], color-flow-line numbers, such that the partial amplitude with the color flow [REF] is generated as the amplitude for the process [EQUATION]', '1010.0748-1-32-1': 'This is the description of the process [REF] in our CFQCD Model, and we let MadGraph generate the corresponding partial amplitudes, such as [MATH] in eq. [REF], as the helicity amplitudes for the process [REF].', '1010.0748-1-32-2': 'This index number assignment has one-to-one correspondence with the color flow [REF].', '1010.0748-1-32-3': 'For instance, [MATH] in the process [REF] denotes the contribution of the [MATH] gluon [MATH] to the partial amplitude where its [MATH] index [MATH] terminates the color-flow line 2, and the 3 index [MATH] starts the new color-flow line 3.', '1010.0748-1-32-4': 'It should also be noted that we number the color-flow lines in the ascending order along the color flow, starting from the color-flow line 1 and ending with the color-flow line [MATH].', '1010.0748-1-32-5': 'This numbering scheme plays an important role in defining the interactions among CFQCD gluons and quarks.', '1010.0748-1-33-0': 'Let us now examine the case for one quark line.', '1010.0748-1-33-1': 'For the 5-jet production process [EQUATION] the color-flow index should run from 1 to 6 according to the rule [REF].', '1010.0748-1-33-2': 'Indeed the color flow [EQUATION] correponds to the process [EQUATION]', '1010.0748-1-33-3': 'We show in Fig. [REF] the color-flow diagram for this process.', '1010.0748-1-33-4': 'This is just that of the 6-gluon process cutted at the [MATH] gluon, where [MATH] gluon is replaced by the quark pair, [MATH] and [MATH].', '1010.0748-1-33-5': 'Shown in Fig. [REF] are a few representative Feynman diagrams contributing to the process [REF].', '1010.0748-1-34-0': 'Both the external and internal quarks and gluons in the CFQCD model are shown explicitly along the external and propagator lines.', '1010.0748-1-34-1': 'Following the MadGraph convention, we use flowing-out quantum numbers for gluons while the quantum numbers are along the fermion number flow for quarks.', '1010.0748-1-34-2': 'Gluon propagators attached to quark lines are named by their color-flow quantum numbers along arrows.', '1010.0748-1-34-3': 'The diagram (a) contains only [MATH] vertices, (b) contains an off-shell 4-point gluon current, jgluo4, or a 4-point gluon amplitude, gluon4, and (c) contains either an off-shell 6-point gluon current, jgluo6, or a 6-point gluon amplitude, gluon6.', '1010.0748-1-35-0': 'In CFQCD, not only [MATH] gluons but also the Abelian gluon contributes to the processes with quark lines.', '1010.0748-1-35-1': 'For the [MATH] process [REF], 1 to 5 gluons can be Abelian gluons, [MATH].', '1010.0748-1-35-2': 'If the number of Abelian gluons is [MATH], the color flow reads [EQUATION]', '1010.0748-1-35-3': 'When [MATH], all the five gluons are Abelian, and the first (the right-most) color flow should be [MATH], just as in [MATH] photons.', '1010.0748-1-35-4': 'In Fig. [REF], we show the color-flow diagram for the process [REF] with the color flow [REF] for [MATH] as an example.', '1010.0748-1-36-0': 'For the process with two quark lines, [EQUATION] the color-flow index should run from 1 to 5.', '1010.0748-1-36-1': 'All the possible color flows are obtained from the comb-like diagram for the one-quark-line process shown in Fig. [REF], by cutting one of the gluons into a quark pair, such as [MATH] to [MATH] and [MATH].', '1010.0748-1-36-2': 'Then the first color flow starting from [MATH] ends at [MATH], and the new color flow starts with [MATH] which ends at [MATH].', '1010.0748-1-36-3': 'An example of such color flow for [MATH] and [MATH] reads [EQUATION]', '1010.0748-1-36-4': 'The CFQCD model computes the partial amplitude for the above color flow as the helicity amplitude for the process [EQUATION]', '1010.0748-1-36-5': 'The other possible color-flow processes without Abelian gluons are [EQUATION] for [MATH], respectively.', '1010.0748-1-36-6': 'As in the single quark line case, all the external gluons can also be Abelian gluons, and we should sum over the contribution of one to three Abelian gluons.', '1010.0748-1-36-7': 'For instance, if the gluon [MATH] in the process [REF] is Abelian, the color flow becomes [EQUATION] and the corresponding partial amplitude is calculated for the process [EQUATION] in CFQCD.', '1010.0748-1-37-0': 'When there are more than one quark line, the Abelian gluon can be exchanged between two quark lines, and the color flow along each quark line is disconnected.', '1010.0748-1-37-1': 'For instance, the color flow [EQUATION] is obtained when the Abelian gluon is exchanged between the [MATH]-quark and [MATH]-quark lines.', '1010.0748-1-37-2': 'The corresponding partial amplitude is obtained for the CFQCD process [EQUATION]', '1010.0748-1-37-3': 'Note that in the process [REF] each flow starts from and ends with a quark pair which belongs to the same fermion line.', '1010.0748-1-38-0': 'We have shown so far that we can generate diagrams with definite color flow by assigning integer labels to quarks, [MATH], and gluons, [MATH], such that the labels [MATH] and [MATH] count the color-flow lines whose maximum number [MATH] is the sum of the number of external gluons and the numbers of quark lines (quark pairs), see eq. [REF].', '1010.0748-1-39-0': '## New particles and their interactions in the CFQCD Model', '1010.0748-1-40-0': 'In Table [REF], we list all the new particles in the CFQCD model for [MATH] in eq. [REF].', '1010.0748-1-41-0': 'The list is shown in the format of particles.dat in the usrmod of MadGraph[CITATION].', '1010.0748-1-41-1': 'As explained above, the [MATH] gluons have two color-flow indices, [MATH], while the Abelian gluon, [MATH], has no color-flow index.', '1010.0748-1-41-2': 'They are vector bosons (type[MATH]V), and we use curly lines (line [MATH] C) for [MATH] gluons while wavy lines (line [MATH] W) for the Abelian gluons.', '1010.0748-1-41-3': "The 'color' column of the list is used by MadGraph to perform color summation by using the color-ordered basis.", '1010.0748-1-41-4': "Since we sum over the color degrees of freedom by summing the 3 and [MATH] indices ([MATH]'s and [MATH]'s) over all possible color flows explicitly, we declare all our new particles as singlets (color [MATH] S).", '1010.0748-1-41-5': 'The last column gives the PDG code for the particles, and all our gluons, including the Abelian gluon, are given the number 21.', '1010.0748-1-42-0': 'All gluons, not only the Abelian gluon but also [MATH] gluons, are declared as Majorana particles (particle and anti-particle are the same) in CFQCD.', '1010.0748-1-42-1': 'We adopt this assignment in order to avoid generating gluon propagators between multi-gluon vertices.', '1010.0748-1-42-2': 'This goal is achieved because the Majorana [MATH] gluon propagators flip the color flow as shown in Fig. [REF], and hence CFQCD does not give diagrams with gluon propagator in color-flow conserving amplitudes.', '1010.0748-1-43-0': '[MATH] gluon propagators attached to quark lines are allowed by introducing the color-flow non-conserving [MATH] couplings that effectively recover a color-flow connection.', '1010.0748-1-43-1': 'See discussions on [MATH] vertices at the end of this section.', '1010.0748-1-44-0': 'In Table [REF], we list [MATH]-quarks, [MATH], and its anti-particles, [MATH], with the color-flow index [MATH] to [MATH].', '1010.0748-1-44-1': 'They are all femions (type [MATH] F) for which we use solid lines (line [MATH] S) in the diagram.', '1010.0748-1-44-2': 'Their colors are declared as singlets (color [MATH] S) as explained above.', '1010.0748-1-44-3': 'The list should be extended to the other five quarks, such as [MATH] and [MATH] for down and anti-down quarks.', '1010.0748-1-45-0': 'Before closing the explanation of new particles in CFQCD, let us note that the number of [MATH] gluons needed for [MATH]-gluon processes is [MATH].', '1010.0748-1-45-1': 'This follows from our color-flow-line numbering scheme, which counts the successive color-flow lines in the ascending order along the color flow as depicted in Fig. [REF].', '1010.0748-1-45-2': 'This is necessary to avoid double counting of the same color flow.', '1010.0748-1-45-3': 'According to this rule, the only gluonic vertex possible for [MATH] is the one among [MATH], [MATH] and [MATH].', '1010.0748-1-45-4': 'Although [MATH] can appear for processes with [MATH], [MATH] and [MATH] can never appear.', '1010.0748-1-45-5': "Generalizing this rule, we find that [MATH]'s with [MATH] mod([MATH]) as well as [MATH] cannot appear and hence we need only [MATH] gluons in CFQCD.", '1010.0748-1-46-0': 'In this paper, we report results up to 5-jet production processes.', '1010.0748-1-46-1': 'Purely gluonic [MATH] process has seven external gluons, and [MATH] is necessary only for this process.', '1010.0748-1-46-2': 'According to the above rule, there is only one interaction for this process, which is the one among [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1010.0748-1-46-3': 'Therefore, we should add [MATH] and [MATH] to Table [REF] in order to compute [MATH] amplitudes.', '1010.0748-1-47-0': 'Next, we list all the interactions among CFQCD particles.', '1010.0748-1-47-1': 'Once the interactions among particles of a user defined model are given, MadGraph generates Feynman diagrams for an arbitrary process and the corresponding HELAS amplitude code.', '1010.0748-1-47-2': 'In CFQCD, we introduce [MATH]-point gluon interactions and let MadGraph generate a code which calls just one HELAS subroutine (either gluon or jgluo with [MATH], which makes use of the recursion relations of eqs. [REF] or [REF], respectively) for each [MATH]-point vertex.', '1010.0748-1-47-3': 'In addition, we have quark-quark-gluon vertices.', '1010.0748-1-47-4': 'We list the interactions in the descending order of the number of participating particles in Table [REF] to [REF].', '1010.0748-1-48-0': 'First, we show the 7-point interaction in Table [REF] in the format of interactions.dat of usrmod in MadGraph[CITATION].', '1010.0748-1-48-1': 'As discussed above, it is needed only for generating [MATH] amplitudes.', '1010.0748-1-48-2': 'The vertex is proportional to [MATH], the fifth power of the strong coupling constant, and we give the five couplings, cpl1 to cpl5, as [EQUATION] according to the Feynman rules of Fig. [REF], whose types, type1 to type5, are all QCD.', '1010.0748-1-49-0': 'The 6-point interactions appear in [MATH] and also in [MATH] process in this study.', '1010.0748-1-49-1': 'Again, only one interaction is possible among the six gluons, [MATH], with [MATH] to [MATH] and [MATH], as shown in Table [REF].', '1010.0748-1-49-2': 'The coupling order is [MATH] and the four couplings, cpl1 to cpl4, are G2 all with the type QCD.', '1010.0748-1-50-0': 'The 5-point gluon vertices appear in [MATH], [MATH], [MATH] and [MATH] processes.', '1010.0748-1-50-1': 'The first two and the fourth processes have [MATH], for which the 5-point gluon vertex is unique as shown in the first row of Table [REF].', '1010.0748-1-50-2': 'The process [MATH] gives [MATH], and gluons with the sixth color-flow line can contribute to the 5-point gluon vertex.', '1010.0748-1-50-3': 'Because of the ascending color-flow numbering scheme, only one additional combination appears as shown in the second row of Table [REF].', '1010.0748-1-50-4': 'These vertices have [MATH] order and we have cpl[MATH] G2 and type[MATH] QCD for [MATH].', '1010.0748-1-51-0': 'The 4-point gluon vertices appear in [MATH]), [MATH] with [MATH] to [MATH], and in [MATH] with [MATH] and [MATH] in this study.', '1010.0748-1-51-1': 'As above, we have only one 4-point gluon vertex for processes with [MATH], which is shown in the first row of Table [REF].', '1010.0748-1-51-2': 'For the process [MATH]), one additional vertex appears as shown in the second row.', '1010.0748-1-51-3': 'In case of [MATH]), the ordering [MATH] also appears, and it is given in the third row.', '1010.0748-1-52-0': 'So far, we obtain multiple gluon vertices from the color-flow lines of consecutive numbers, corresponding to the color flows such as [EQUATION] for [MATH]-point gluon vertices.', '1010.0748-1-52-1': 'When there are two or more quark lines in the process, we can also have color flows which skips color-flow-line numbers, such as [EQUATION] for [MATH]-point gluon vertices, where [MATH] counts the number of skips.', '1010.0748-1-52-2': 'Because gluon propagators do not attach to gluon vertices in CFQCD, this skip can appear only when two or more gluons from the same vertex are connected to quark lines.', '1010.0748-1-52-3': 'For example, in the fourth row of Table [REF], the gluon [MATH] couples to the quark line, [MATH], and then the gluon [MATH] couples to the other quark line, [MATH], in the process [MATH].', '1010.0748-1-52-4': 'Likewise, [MATH] and [MATH] in the fifth and the sixth row, respectively, couple to the second quark line, [MATH], of the process.', '1010.0748-1-53-0': 'As for the 3-gluon vertices, ten vertices listed in Table [REF] appear in our study.', '1010.0748-1-53-1': 'The first four vertices with successive color-flow numbers appear in processes with one quark line, [MATH] with [MATH] to [MATH], and those with two quark lines, [MATH] with [MATH] to 5.', '1010.0748-1-53-2': 'The fifth and the sixth vertices starting with g14 appears for [MATH] with [MATH], for which only one unit of skip ([MATH]) appears, and also with [MATH].', '1010.0748-1-53-3': 'The last four vertices appear only in the [MATH] process with two quark lines, for which [MATH] is possible.', '1010.0748-1-53-4': 'In fact, the two vertices starting with g15 contain g52 or g41 with [MATH] skips in the color-flow number.', '1010.0748-1-53-5': 'This completes all the gluon self-interactions in CFQCD up to 5-jet production processes.', '1010.0748-1-54-0': 'There are two types of [MATH] vertices in CFQCD: the couplings of [MATH] gluons, [MATH], and those of the Abelian gluon, [MATH].', '1010.0748-1-54-1': 'All the [MATH] couplings for the [MATH]-quark, [MATH] to [MATH], are listed in Table [REF].', '1010.0748-1-54-2': 'In the HELAS convention, the fermion-fermion-boson couplings are two dimensional complex arrays, where the first and the second components are the couplings of the left- and the right-hand chirality of the flowing-in fermion.', '1010.0748-1-54-3': 'According to the Feynman rules of Fig. [REF], the couplings are [EQUATION] for [MATH] gluons and [EQUATION] for the Abelian gluon.', '1010.0748-1-55-0': '[MATH] gluons have interactions as [EQUATION] where the fermion number flows from [MATH] to [MATH] by emitting the out-going [MATH] gluon.', '1010.0748-1-55-1': 'All the diagrams with on-shell gluons attached to quark lines are obtained from the vertices [REF].', '1010.0748-1-55-2': 'Likewise, the abelian gluon couples to quarks as [EQUATION]', '1010.0748-1-55-3': 'In CFQCD, we generate an [MATH] gluon propagator between a quark line and a gluon vertex and between two quark lines by introducing color-flow flipping vertices [EQUATION]', '1010.0748-1-55-4': 'Here, color-flipped anti-[MATH]) is emitted from the quark line [MATH].', '1010.0748-1-55-5': 'These color-flip vertices can couple to the color-flip gluon propagators of CFQCD.', '1010.0748-1-55-6': '[MATH] gluons can propagate between a quark line and a gluonic vertex and also between two-quark lines when the other side of the [MATH] vertex is the color-flow conserving one, [REF].', '1010.0748-1-55-7': 'The condition, [MATH], for the color-flip vertices are needed to avoid double counting; If we allow color-flip vertices also for [MATH], the same color-flow amplitudes appear twice when [MATH] gluon is exchanged between two-quark lines.', '1010.0748-1-56-0': 'Now we exhaust all the interactions needed in the calculation of partial amplitudes and ready to generate diagrams and evaltate total amplitudes in the color-flow basis.', '1010.0748-1-57-0': '# Total amplitudes and the color summation', '1010.0748-1-58-0': 'In this section, we discuss how we evaluate the total amplitudes, eqs. [REF], [REF] and [REF], with the CFQCD model and how we perform the color summation of the total amplitude squared.', '1010.0748-1-59-0': '## [MATH] processes', '1010.0748-1-60-0': 'We consider pure gluon processes first.', '1010.0748-1-60-1': 'As discussed in Section [REF], the total amplitude of a [MATH]-gluon process is expressed as eq.[REF], and it is the sum of several partial amplitudes.', '1010.0748-1-60-2': "Because color factors, Kronecker's delta's, for partial amplitudes is either 1 or 0, the total amplitude for a given color configuration (a color assignment for each external gluons) consists of a subset of all the partial amplitudes in eq.[REF].", '1010.0748-1-60-3': 'Therefore, in order to evaluate the total amplitude for a given color configuration, we should find all possible color flows for the configuration, computate their partial amplitudes and simply sum them up.', '1010.0748-1-61-0': 'As an example, let us consider a [MATH] case, [EQUATION]', '1010.0748-1-61-1': 'The color configuration is expressed by sets of 3 and [MATH] indeces, [MATH] and [MATH], respectively, for each gluon; [EQUATION]', '1010.0748-1-61-2': 'Here the subscripts of each index pair label the gluon [MATH], [MATH] to 5, with the momentum [MATH] and the helicity [MATH].', '1010.0748-1-61-3': 'For instance, let us examine a color configuration [EQUATION]', '1010.0748-1-61-4': 'One of the possible color flows that gives the above color configuration is [EQUATION] whose associated amplitude can be calculated as the amplitude for the CFQCD process', '1010.0748-1-62-0': '[EQUATION]', '1010.0748-1-62-1': 'MadGraph generates the corresponding HELAS amplitude code, which calls the 5-gluon amplitude subroutine, gluon5.', '1010.0748-1-63-0': 'There is also another color flow [EQUATION] for the color configuration [REF].', '1010.0748-1-63-1': 'The corresponding partial amplitude can be obtained by one of the [MATH] permutations of [MATH] gluon momenta and helicities: [EQUATION] where [MATH] denotes the wave function of the external gluon [MATH].', '1010.0748-1-64-0': 'When all the partial amplitudes for each color flow are evaluated, we sum them up and obtain the color-fixed total amplitude, eq. [REF], for this color configuratoin.', '1010.0748-1-64-1': 'Therefore, for pure gluon process, we generate the HELAS amplitude once for all to evaluate the color-fixed total amplitude.', '1010.0748-1-64-2': 'The total amplitude is then squared and the summation over all color configurations is done by the Monte-Carlo method.', '1010.0748-1-65-0': '## [MATH] processes', '1010.0748-1-66-0': 'Next, we discuss the processes with one quark line, [MATH].', '1010.0748-1-66-1': 'The procedure to compute the color-summed amplitude squared is the same, but we should take into account the Abelian gluon contributions for processes with quarks.', '1010.0748-1-66-2': 'As shown in Fig. [REF], the Abelian gluons appear as an isolated color index pair which have the same number.', '1010.0748-1-66-3': 'Therefore, we can take into account its contribution by regarding the independent gluon index pair as the Abelian gluon.', '1010.0748-1-67-0': "As an example, let us consider the process [MATH] with the color configuration: [EQUATION] where the parenthesis with the subscript '[MATH]' gives the color charges of the [MATH]-quark pair; [MATH] denotes the annihilation of the [MATH]-quark with the 3 charge, [MATH], and the [MATH]-quark with the [MATH] charge, [MATH].", '1010.0748-1-67-1': 'Although this color configuration is essentially the same as [REF], and hence has the color flows like [REF] and [REF], we have an additional color flow for this process: [EQUATION]', '1010.0748-1-67-2': 'Here the index pair of the first gluon, [MATH] in [REF], forms an indipendent color flow, [MATH], which corresponds to the Abelian gluon, [MATH].', '1010.0748-1-67-3': 'The CFQCD process for this color flow reads', '1010.0748-1-68-0': '[MATH].', '1010.0748-1-69-0': 'Summing up, there are three color flows for the color configuration [REF]: [EQUATION]', '1010.0748-1-69-1': 'The partial amplitude [MATH] for the color flow [MATH] is obtained from that of the color flow (a), [MATH], by a permutation of gluon wave functions as in the case of [MATH] amplitudes for the color configuration [REF].', '1010.0748-1-69-2': 'The partial amplitude [MATH] for the color flow [MATH] is calculated with the Feynman rules of Fig. [REF], and the total amplitude [MATH] is obtained as [EQUATION] where the Abelian gluon factor, [MATH], is multiplied to [MATH] as shown in eq. [REF].', '1010.0748-1-70-0': 'When more than one gluon have the same color indices, [MATH] in [MATH], more Abelian gluons can contribute to the amplitude and the factor [MATH] should be multiplied for the partial amplitude with [MATH] Abelian gluons.', '1010.0748-1-71-0': '## [MATH] processes', '1010.0748-1-72-0': 'Finally, let us discuss the processes with two quark lines.', '1010.0748-1-72-1': 'As shown in Fig. [REF], there are two independent color flows because there are two sets of quark color indices, [MATH] and [MATH].', '1010.0748-1-72-2': 'Nevertheless, we can show color configurations and find possible color flows in the same way as in the [MATH] cases.', '1010.0748-1-72-3': 'Let us consider the process [EQUATION] with the color configuration [EQUATION] for illustration.', '1010.0748-1-72-4': 'Color charges of [MATH]-quarks are in the parenthesis [MATH], and those of [MATH]-quarks are in [MATH] as in the [MATH] processes.', '1010.0748-1-72-5': 'All the possible color flows are [EQUATION]', '1010.0748-1-72-6': 'We show in Fig. [REF] one representative Feynman diagram for each color flow, [MATH] to [MATH].', '1010.0748-1-73-0': 'The amplitudes for each color flow, [MATH] to [MATH], are calculated as amplitudes for the corresponding CFQCD processes: [EQUATION]', '1010.0748-1-73-1': 'For the CFQCD processes [MATH] and [MATH], the color flow lines starting with the [MATH]- and [MATH]-quarks terminate at the same quarks.', '1010.0748-1-73-2': 'Such amplitudes are generated by an exchange of the Abelian gluon, as shown by the representative diagrams in Fig.[REF].', '1010.0748-1-73-3': 'The total amplitude is now obtained as [EQUATION] according to eq.[REF].', '1010.0748-1-73-4': 'The color summation of the squared amplitudes [MATH] is performed by the MC method just as in the pure gluon case.', '1010.0748-1-74-0': '# Sample results', '1010.0748-1-75-0': 'In this section, we present numerical results for several multi-jet production processes as a demonstration of our CFQCD model on MadGraph.', '1010.0748-1-75-1': 'We compute [MATH]-jet production cross sections from [MATH], [MATH] and [MATH] subprocesses up to [MATH] in the [MATH] collision at [MATH] TeV.', '1010.0748-1-75-2': 'We define the final state cuts, the QCD coupling constant and the parton distribution function exactly as the same as those of ref. [CITATION], so that we can compare our results against those presented in ref. [CITATION], which have been tested by a few independent calculations.', '1010.0748-1-75-3': 'Specifically, we select jets (partons) that satisfy [EQUATION] where [MATH] and [MATH] are the pseudo-rapidity and the transverse momentum of the parton-[MATH], and [MATH] is the smaller of the relative transverse momentum between parton-[MATH] and parton-[MATH].', '1010.0748-1-75-4': 'We use CTEQ6L1 parton distribution functions[CITATION] at the factorization scale [MATH] GeV and the QCD coupling [MATH].', '1010.0748-1-75-5': 'Phase space intergration and summation over color and helicity are performed by an adaptive Monte Carlo (MC) integration program BASES[CITATION].', '1010.0748-1-76-0': 'Results are shown in Table [REF] and Fig. [REF].', '1010.0748-1-77-0': 'In Table [REF], the first row for each [MATH]-jet process gives the exact result for the [MATH]-jet production cross section, while the second row shows the cross section when we ignore all the Abelian gluon contributions.', '1010.0748-1-77-1': 'The third row shows the results where we include up to one Abelian gluon contributions: one Abelian gluon emission amplitudes without an Abelian gluon exchange for [MATH] and [MATH] processes, and Abelian gluon exchange amplitudes without an Abelian gluon emission for [MATH] processes.', '1010.0748-1-77-2': 'All the numerical results for the exact cross sections in Table [REF] agree with those presented in ref.[CITATION] within the accuracy of MC integration.', '1010.0748-1-77-3': 'In Fig.[REF], the multi-jet production cross sections are shown for [MATH] and 5 jets in units of fb.', '1010.0748-1-77-4': 'The upper line gives the results for the subprocess [MATH].', '1010.0748-1-77-5': 'The middle lines show those for the subprocess [MATH]; their amplitudes are obtained from those of the subprocess [MATH] outlined in this study, simply by anti-symmetrizing the amplitudes with respect to the two external quark wave functions.', '1010.0748-1-77-6': 'The solid line gives the exact results, while the dashed line gives the results when all the Abelian gluon contributions are ignored.', '1010.0748-1-77-7': 'The dotted line shows the results which include up to one Abelian gluon contributions, although it is hard to distinguish from the solid line in the figure.', '1010.0748-1-77-8': 'Despite their order [MATH] suppression, contributions of Abelian gluons can be significant; more than 30% for [MATH], about 13% for [MATH], while about 10% for [MATH]4 and 5.', '1010.0748-1-78-0': 'The bottom lines show results for the subprocess [MATH].', '1010.0748-1-78-1': 'As above, the solid line gives the exact cross sections, while the dashed line and the dotted line give the results when contributions from Abelian gluons are ignored and one Abelian gluon contributions are included, respectively.', '1010.0748-1-78-2': 'Unlike the case for [MATH] subprocesses, the Abelian gluon contributions remain at 30% level even for [MATH].', '1010.0748-1-79-0': 'Before closing this section, we would like to give two technical remarks on our implementation of CFQCD on MadGraph.', '1010.0748-1-79-1': 'First, since the present MadGraph[CITATION] does not allow vertices among more than 4 particles, we add HELAS codes for 5, 6, and 7 gluon vertices by hand to complete the MadGraph generated codes.', '1010.0748-1-79-2': 'This restriction will disappear once the new version of MadGraph, MG5, is available, since MG5 accepts vertices with arbitrary number of particles.', '1010.0748-1-79-3': 'Second, we do not expect difficulty in running CFQCD codes on GPU, since all the codes we developed (see Appendix) follow the standard HELAS surbourine rules.', '1010.0748-1-80-0': '# Conclusions', '1010.0748-1-81-0': 'In this paper, we have implemented off-shell recursive formulae for gluon currents in the color-flow basis in MadGraph and have shown that it is possible to generate QCD amplitudes in the color-flow basis by introducing a new model, CFQCD, in which quarks and gluons are labeled by color-flow numbers.', '1010.0748-1-81-1': 'We have', '1010.0748-1-82-0': '- introduced new subroutines for off-shell recursive formulae for gluon currents and the', '1010.0748-1-83-0': 'contact 3- and 4-point gluon verteces in the color flow basis,', '1010.0748-1-84-0': '- defined new MadGraph model: the CFQCD Model,', '1010.0748-1-85-0': '- generated HELAS amplitudes for given color flows and calculated', '1010.0748-1-86-0': 'the color-summed total amplitude squared, and', '1010.0748-1-87-0': '- showed the numerical results for [MATH]-jet production cross sections ([MATH]).', '1010.0748-1-88-0': 'Although we have studied only up to 5-jet production processes in this paper, it is straightforward to extend the method to higher [MATH]-jet production processes.'}

{'1010.0748-2-0-0': 'We propose to make use of the off-shell recursive relations with the color-flow decomposition in the calculation of QCD amplitudes on MadGraph.', '1010.0748-2-0-1': 'We introduce colored quarks and their interactions with nine gluons in the color-flow basis plus an Abelian gluon on MadGraph, such that it generates helicity amplitudes in the color-flow basis with off-shell recursive formulae for multi-gluon sub-amplitudes.', '1010.0748-2-0-2': 'We demonstrate calculations of up to 5-jet processes such as [MATH], [MATH] and [MATH].', '1010.0748-2-0-3': 'Although our demonstration is limited, it paves the way to evaluate amplitudes with more quark lines and gluons with Madgraph.', '1010.0748-2-1-0': '# Introduction', '1010.0748-2-2-0': 'The Large Hadron Collider (LHC) has been steadily running at 7 TeV since the end of March/2010.', '1010.0748-2-2-1': 'Since LHC is a hadron collider, understanding of QCD radiation processes is essential for the success of the experiments.', '1010.0748-2-2-2': 'To evaluate new physics signals and Standard Model (SM) backgrounds, we should resort to simulation programs which generate events with many hadrons and investigate observable distributions.', '1010.0748-2-2-3': 'In each simulation, one calculates matrix elements of hard processes with quarks and gluons, and this task is usually done by an automatic amplitude calculator.', '1010.0748-2-3-0': 'MadGraph[CITATION] is one of those programs.', '1010.0748-2-3-1': 'Although it is highly developed and has ample user-friendly utilities such as event generation with new physics signals matched to Parton Shower[CITATION], it cannot evaluate matrix elements with more than five jets in the final state[CITATION]', '1010.0748-2-4-0': 'This is a serious drawback since exact evaluation of multi-jet matrix elements is often required to study sub-structure of broad jets in identifying new physics signatures over QCD background.', '1010.0748-2-4-1': 'It is also disappointing because MadGraph generated HELAS amplitudes[CITATION] can be computed very fast on GPU (Graphic Processing Unit)[CITATION].', '1010.0748-2-5-0': 'Some of other simulation packages such as HELAC and Alpgen[CITATION] employ QCD off-shell recursive relations to produce multi-parton scattering amplitudes efficiently.', '1010.0748-2-5-1': 'Successful computation of recursively evaluated QCD amplitudes on GPU has been reported[CITATION].', '1010.0748-2-5-2': 'It is therefore interesting to examine the possibility of implementing recursive relations for gluon currents in MadGraph without sacrificing its benefits.', '1010.0748-2-5-3': 'In this paper, we examine the use of the recursive amplitudes in the color-flow basis, since corresponding HELAS amplitude package has been developed in ref. [CITATION].', '1010.0748-2-5-4': 'The main purpose of this paper is to show that this approach can be accommodated in MadGraph with explicit examples.', '1010.0748-2-6-0': 'The outline of this paper is as follows.', '1010.0748-2-6-1': 'In sections 2, we briefly review the color-flow decomposition and off-shell recursive relations.', '1010.0748-2-6-2': 'We then discuss the implementation of those techniques in MadGraph, using its "user mode" option[CITATION] in section 3.', '1010.0748-2-6-3': 'In section 4 and 5, we explain how we evaluate the color-summed amplitude squared and show the numerical results of [MATH]-jet production cross section.', '1010.0748-2-6-4': 'Section 6 is devoted to conclusions.', '1010.0748-2-7-0': '# The color-flow basis and off-shell recursive relations', '1010.0748-2-8-0': 'In this section we review the color-flow decomposition[CITATION] of QCD scattering amplitudes and the off-shell recursive relation[CITATION] of gluonic currents in the color flow basis.', '1010.0748-2-9-0': '## The color-flow decomposition', '1010.0748-2-10-0': 'First, we briefly review the color-flow decomposition.', '1010.0748-2-10-1': 'The Lagrangian of QCD can be expressed as [EQUATION] where [EQUATION]', '1010.0748-2-10-2': 'Upper and lower indices are those of [MATH] and [MATH] representations, respectively.', '1010.0748-2-10-3': 'Note that the gluon fields, [MATH], are renormalized by a factor of [MATH], and hence the coupling [MATH] is divided by [MATH].', '1010.0748-2-10-4': 'At this stage, not all the nine gluon fields are independent because of the traceslessness of the [MATH] generators, [EQUATION]', '1010.0748-2-10-5': 'Here we introduce the "Abelian gluon", [EQUATION]', '1010.0748-2-10-6': 'This is essentially the gauge boson , [MATH], of [MATH] subgroup of [MATH], combined with its generator, [MATH], which is normalized to 1/2.', '1010.0748-2-10-7': 'We then rewrite eq. [REF] by adding and subtracting the Abelian gluon, such that the QCD Lagrangian is expressed as [EQUATION] with [EQUATION]', '1010.0748-2-10-8': 'Here all the nine gluons, [MATH] are now independent while the covariant tensors keep the same form [EQUATION]', '1010.0748-2-10-9': 'In this basis, these nine gluons have one-to-one correspondence to a set of indices of [MATH] and [MATH] representations of [MATH], [MATH], and we address them and this basis as [MATH] gluons and the color-flow basis, respectively, in the following discussions.', '1010.0748-2-10-10': 'Although the definition of the covariant tensor [REF] contains self-coupling terms for [MATH] gluons, the Abelian gluon contribution actually drops out in the sum.', '1010.0748-2-10-11': 'Accordingly, the Feynman rules for [MATH] gluons give directly the [MATH] (QCD) amplitudes for purely gluonic processes[CITATION].', '1010.0748-2-11-0': 'We list the Feynman rules derived from the Lagrangian [REF] in Fig. [REF].', '1010.0748-2-12-0': 'It should be noted that both the propagator and the quark vertex of the Abelian gluon have an extra [MATH] factor because of the color-singlet projector, [MATH].', '1010.0748-2-12-1': 'The negative sign of this factor directly comes from the sign of the Abelian gluon kinetic term in the Lagrangian.', '1010.0748-2-12-2': 'We can evaluate QCD amplitudes by using these Feynman rules.', '1010.0748-2-13-0': 'For a [MATH]-gluon scattering process, it is written as [EQUATION] obtained from the corresponding [MATH] gluon scattering process.', '1010.0748-2-13-1': '[MATH] are gauge-invariant partial amplitudes, which depend only on particle momenta and helicities represented simply by the gluon indices, [MATH], and the summation is taken over all [MATH] non-cyclic permutations of gluons.', '1010.0748-2-13-2': 'A set of [MATH] Kronecker delta\'s in each term gives the "color flow" of the corresponding partial amplitudes.', '1010.0748-2-13-3': "As an example, we list in Fig. [REF] all six color flows indicated by Kronecker delta's for the [MATH] process, [MATH].", '1010.0748-2-14-0': 'On the other hand, if a scattering process involves quarks, contributions from the Abelian gluon do not decouple.', '1010.0748-2-14-1': 'Therefore, we have to add all its contributions to the total amplitude.', '1010.0748-2-14-2': 'For processes with one quark line, such contributions come from Abelian gluon emitting diagrams.', '1010.0748-2-14-3': 'The total amplitude for [MATH] process becomes [EQUATION] where [MATH] is the partial amplitude with [MATH] Abelian gluons, and [MATH] and [MATH] denote the momenta and the helicities of the quark and the anti-quark, respectively.', '1010.0748-2-14-4': 'The summation is taken over all the [MATH] permutations of [MATH] gluons.', '1010.0748-2-14-5': 'The first term in the r.h.s. of eq.[REF] gives contributions from [MATH] gluons, and the other terms give contributions from [MATH] gluons and [MATH] Abelian gluons, summed over [MATH] to [MATH], where [MATH] comes from the [MATH] factor in the Abelian gluon vertex as depicted in Fig. [REF].', '1010.0748-2-14-6': 'Corresponding color flows of those partial amplitudes are shown in Fig.[REF].', '1010.0748-2-15-0': 'For processes with two or more quark lines, the Abelian gluon can be exchanged between them, giving the amplitudes with at least [MATH] suppression.', '1010.0748-2-15-1': 'Therefore, the total amplitude of processes with two quark lines such as [MATH] gluons becomes [EQUATION] [MATH] (or [MATH]) denotes a partial amplitude with [MATH] gluons, where two sets of arguments divided by "[MATH]" belong to two different color-flow chains; one starts from [MATH] in the initial state) and ends with [MATH] in the final state), and the other starts from [MATH] and ends with [MATH], which are given explicitly by two sets of delta\'s before the partial amplitude.', '1010.0748-2-15-2': 'The difference between [MATH] and [MATH] is that the former consists of [MATH]-gluon-exchange diagrams while the latter consists of Abelian-gluon-exchange ones.', '1010.0748-2-15-3': 'Here we write down explicitly the partial amplitudes with external [MATH] gluons, [MATH] and [MATH], whereas those with external Abelian gluons should be added as in the previous one-quark-line case.', '1010.0748-2-15-4': 'For illustration, some typical diagrams for the [MATH] process are shown in Fig. [REF].', '1010.0748-2-16-0': '## Recursive relations of off-shell gluon currents', '1010.0748-2-17-0': 'Next, we mention the off-shell recursive relations in the color-flow basis.', '1010.0748-2-17-1': 'We define a [MATH]-point off-shell gluon current [MATH] recursively as [EQUATION] where the numbers in the argument of currents represent gluons, and their order respects the color flow as in the partial amplitudes, [MATH], in eq. [REF]; the argument after semicolon denotes the off-shell gluon.', '1010.0748-2-17-2': 'As an initial value, the 2-point gluon current, [MATH], is defined as the polarization vector of the gluon [MATH], [MATH].', '1010.0748-2-17-3': '[MATH] is the partial sum of gluon momenta: [EQUATION] where [MATH] to [MATH] are defined as flowing-out momenta.', '1010.0748-2-17-4': '[MATH] and [MATH] denote the three- and four-point gluon vertices from the Feynman rule of Fig.[REF]: [EQUATION]', '1010.0748-2-17-5': 'The summation over [MATH] and [MATH] in eq.[REF] are taken over the 3 and [MATH] indices, [MATH] and [MATH], of off-shell gluons in currents [MATH] and [MATH], respectively.', '1010.0748-2-17-6': 'For illustration, the 5-point off-shell current is shown explicitly in Fig.[REF].', '1010.0748-2-18-0': 'Helicity amplitudes of pure gluon processes in the color-flow basis are obtained from off-shell gluon currents as [EQUATION]', '1010.0748-2-18-1': 'Starting from the external wave functions [REF], the recursive relation computes partial amplitudes very effectively by using the HELAS code.', '1010.0748-2-18-2': 'When the off-shell line "[MATH]" in Fig. [REF] is set on-shell, the same set of diagrams give the complete helicity amplitudes in the color-flow basis.', '1010.0748-2-19-0': '# Implementation in MadGraph', '1010.0748-2-20-0': 'In this section, we discuss how we implement off-shell recursive relations in MadGraph in the color-flow basis and how we generate helicity amplitudes of QCD processes.', '1010.0748-2-21-0': '## Subroutines for off-shell recursive formulae', '1010.0748-2-22-0': 'First, we introduce new HELAS[CITATION] subroutines in MadGraph which make [MATH]-point gluon off-shell currents and [MATH]-gluon amplitudes in the color-flow basis, according to eq.[REF] and eq.[REF], respectively.', '1010.0748-2-22-1': 'Although the expression eq.[REF] gives the off-shell gluon current made from [MATH] on-shell gluons, in HELAS amplitudes any input on-shell gluon wave functions can be replaced by arbitrary off-shell gluon currents with the same color quantum numbers.', '1010.0748-2-22-2': 'Shown in Fig.[REF] is an example of such diagrams that are calculated by the HELAS subroutine for the 4-point off-shell gluon current.', '1010.0748-2-23-0': 'Thanks to this property of the HELAS amplitudes, we need to introduce only two types of subroutines: one which computes helicity amplitudes of [MATH]-gluon processes via eq.[REF] and the other which computes off-shell gluon currents from [MATH]-point gluon vertices via eq.[REF].', '1010.0748-2-23-1': 'We name the first type of subroutines as gluon and the second type as jgluo, where denotes the number of external on-shell and off-shell gluons.', '1010.0748-2-24-0': 'The number of new subroutines we should add to MadGraph depends on the number of external partons (quarks and gluons) in QCD processes.', '1010.0748-2-24-1': 'Processes with [MATH]-partons can be classified as those with [MATH] gluons, those with [MATH] gluons and one quark line, those with [MATH] gluons and two quark lines, and so on.', '1010.0748-2-25-0': 'In the color-flow basis, the first class of processes with [MATH] external gluons are calculated by just one amplitude subroutine, gluonn.', '1010.0748-2-26-0': 'For the second class of processes with [MATH] gluons and one quark line, we need up to [MATH]-point off-shell current subroutines, jgluo with [MATH] to [MATH].', '1010.0748-2-26-1': 'This is because the largest off-shell gluon current appears in the computation of diagrams where [MATH] on-shell gluons are connected to the quark line through one off-shell gluon, which can be computed by the [MATH]-point off-shell current and the [MATH] amplitude subroutine, iovxxx.', '1010.0748-2-26-2': 'Note that the same diagram can be computed by the off-shell gluon current subroutine made by the quark pair, jioxxx, and the [MATH]-point gluon amplitude subroutine, gluon(n-1).', '1010.0748-2-26-3': 'This type of redundancy in the computation of each Feynman diagram is inherent in the HELAS amplitudes, which is often useful in testing the codes.', '1010.0748-2-27-0': 'For [MATH]-parton processes with [MATH] gluons and two quark lines, we need up to [MATH]-point off-shell current subroutines.', '1010.0748-2-27-1': 'By also introducing multiple gluon amplitude subroutines up to [MATH]-point vertex, the maximum redundancy of HELAS amplitudes is achieved.', '1010.0748-2-27-2': 'Likewise, for [MATH]-parton processes with [MATH] quark lines, we need up to [MATH]-point off-shell current or amplitude subroutines.', '1010.0748-2-28-0': 'We list in Table [REF] the new HELAS subroutines we introduce in this study.', '1010.0748-2-29-0': 'The subroutine gluon evaluates a -gluon amplitude in the color-flow basis, and jgluo computes an off-shell current from ([MATH]) external gluons.', '1010.0748-2-29-1': 'Since we consider up to seven parton processes, [MATH], [MATH] and [MATH], in this study, we use gluon4 to gluon7 and jgluo4 to jgluo6.', '1010.0748-2-30-0': 'In addition to these subroutines that computes amplitudes and currents recursively, we also introduce three subroutines: ggggcf, jgggcf and jioaxx.', '1010.0748-2-30-1': 'Two of them, ggggcf and jgggcf, evaluate an amplitude and an off-shell current from the contact 4-gluon vertex, following the Feynman rule of Fig. [REF].', '1010.0748-2-30-2': 'Although the amplitude subroutine and the off-shell current subroutine for the 4-gluon vertex already exist in MadGraph, ggggxx and jgggxx, we should introduce the new ones which evaluate the sum of s- and t-type or s- and u-type vertices for a given color flow, since the default subroutines compute only one type at a time.', '1010.0748-2-30-3': 'jioaxx computes an off-shell Abelian gluon current made by a quark pair and is essentially the same as the off-shell gluon current subroutine, jioxxx, in the HELAS library[CITATION] except for an extra [MATH] factor: [EQUATION]', '1010.0748-2-30-4': 'Note that introducing this [MATH] factor is equivalent to summing up contributions from all Abelian gluon propagators, as we discussed in Section 2.', '1010.0748-2-30-5': 'We show the codes of ggggcf, jgggcf, gluon5 and jgluo5 in Appendix.', '1010.0748-2-31-0': '## Introduction of a new Model: CFQCD', '1010.0748-2-32-0': 'Next, we define a new Model with which MadGraph generates HELAS amplitudes in the color-flow basis since the present MadGraph computes them in the color-ordered basis[CITATION].', '1010.0748-2-32-1': 'A Model is a set of definitions of particles, their interactions and couplings; there are about twenty preset Models in the present MadGraph package[CITATION] such as the Standard Model and the Minimal SUSY Standard Model[CITATION].', '1010.0748-2-32-2': 'There is also a template for further extension by users, User Mode (usrmod).', '1010.0748-2-32-3': 'Using this template, we can add new particles and their interactions to MadGraph.', '1010.0748-2-32-4': 'We make use of this utility and introduce a model which we call the CFQCD Model.', '1010.0748-2-33-0': 'In the CFQCD Model, we introduce gluons and quarks as new particles labeled by indices that dictate the color flow, such that the diagrams for partial amplitudes are generated according to the Feynman rules of Fig. [REF].', '1010.0748-2-33-1': 'We need one index for quarks and two indices for gluons, such as [MATH], [MATH] and [MATH].', '1010.0748-2-33-2': 'The Abelian gluon, [MATH], does not have an index.', '1010.0748-2-33-3': 'The index labels all possible color flows, and it runs from 1 to [MATH], where [EQUATION]', '1010.0748-2-33-4': "This number [MATH] is the number of Kronecker's delta symbols which dictate the color flow.", '1010.0748-2-33-5': "As an example, let us consider the purely gluonic process [EQUATION] for which we need seven delta's to specify the color flow: [EQUATION]", '1010.0748-2-33-6': "Here the numbers in parentheses label gluons whereas the numbers in the sub-indices of Kronecker's delta's count the color-flow lines, 1 to [MATH] as depicted in Fig. [REF].", '1010.0748-2-34-0': 'In CFQCD, we label gluons according to their flowing-out, 3, and flowing-in, [MATH], color-flow-line numbers, such that the partial amplitude with the color flow [REF] is generated as the amplitude for the process [EQUATION]', '1010.0748-2-34-1': 'This is the description of the process [REF] in our CFQCD Model, and we let MadGraph generate the corresponding partial amplitudes, such as [MATH] in eq. [REF], as the helicity amplitudes for the process [REF].', '1010.0748-2-34-2': 'This index number assignment has one-to-one correspondence with the color flow [REF].', '1010.0748-2-34-3': 'For instance, [MATH] in the process [REF] denotes the contribution of the [MATH] gluon [MATH] to the partial amplitude where its [MATH] index [MATH] terminates the color-flow line 2, and the 3 index [MATH] starts the new color-flow line 3.', '1010.0748-2-34-4': 'It should also be noted that we number the color-flow lines in the ascending order along the color flow, starting from the color-flow line 1 and ending with the color-flow line [MATH].', '1010.0748-2-34-5': 'This numbering scheme plays an important role in defining the interactions among CFQCD gluons and quarks.', '1010.0748-2-35-0': 'Let us now examine the case for one quark line.', '1010.0748-2-35-1': 'For the 5-jet production process [EQUATION] the color-flow index should run from 1 to 6 according to the rule [REF].', '1010.0748-2-35-2': 'Indeed the color flow [EQUATION] corresponds to the process [EQUATION]', '1010.0748-2-35-3': 'We show in Fig. [REF] the color-flow diagram for this process.', '1010.0748-2-35-4': 'This is just that of the 6-gluon process cut at the [MATH] gluon, where [MATH] gluon is replaced by the quark pair, [MATH] and [MATH].', '1010.0748-2-35-5': 'Shown in Fig. [REF] are a few representative Feynman diagrams contributing to the process [REF].', '1010.0748-2-36-0': 'Both the external and internal quarks and gluons in the CFQCD model are shown explicitly along the external and propagator lines.', '1010.0748-2-36-1': 'Following the MadGraph convention, we use flowing-out quantum numbers for gluons while the quantum numbers are along the fermion number flow for quarks.', '1010.0748-2-36-2': 'Gluon propagators attached to quark lines are named by their color-flow quantum numbers along arrows.', '1010.0748-2-36-3': 'The diagram (a) contains only [MATH] vertices, (b) contains an off-shell 4-point gluon current, jgluo4, or a 4-point gluon amplitude, gluon4, and (c) contains either an off-shell 6-point gluon current, jgluo6, or a 6-point gluon amplitude, gluon6.', '1010.0748-2-37-0': 'In CFQCD, not only [MATH] gluons but also the Abelian gluon contributes to the processes with quark lines.', '1010.0748-2-37-1': 'For the [MATH] process [REF], 1 to 5 gluons can be Abelian gluons, [MATH].', '1010.0748-2-37-2': 'If the number of Abelian gluons is [MATH], the color flow reads [EQUATION]', '1010.0748-2-37-3': 'When [MATH], all the five gluons are Abelian, and the first (the right-most) color flow should be [MATH], just as in [MATH] photons.', '1010.0748-2-37-4': 'In Fig. [REF], we show the color-flow diagram for the process [REF] with the color flow [REF] for [MATH] as an example.', '1010.0748-2-38-0': 'For the process with two quark lines, [EQUATION] the color-flow index should run from 1 to 5.', '1010.0748-2-38-1': 'All the possible color flows are obtained from the comb-like diagram for the one-quark-line process shown in Fig. [REF], by cutting one of the gluons into a quark pair, such as [MATH] to [MATH] and [MATH].', '1010.0748-2-38-2': 'Then the first color flow starting from [MATH] ends at [MATH], and the new color flow starts with [MATH] which ends at [MATH].', '1010.0748-2-38-3': 'An example of such color flow for [MATH] and [MATH] reads [EQUATION]', '1010.0748-2-38-4': 'The CFQCD model computes the partial amplitude for the above color flow as the helicity amplitude for the process [EQUATION]', '1010.0748-2-38-5': 'The other possible color-flow processes without Abelian gluons are [EQUATION] for [MATH], respectively.', '1010.0748-2-38-6': 'As in the single quark line case, all the external gluons can also be Abelian gluons, and we should sum over contributions from external Abelian gluons.', '1010.0748-2-38-7': 'For instance, if the gluon [MATH] in the process [REF] is Abelian, the color flow becomes [EQUATION] and the corresponding partial amplitude is calculated for the process [EQUATION] in CFQCD.', '1010.0748-2-39-0': 'When there are more than one quark line, the Abelian gluon can be exchanged between two quark lines, and the color flow along each quark line is disconnected.', '1010.0748-2-39-1': 'For instance, the color flow [EQUATION] is obtained when the Abelian gluon is exchanged between the [MATH]-quark and [MATH]-quark lines.', '1010.0748-2-39-2': 'The corresponding partial amplitude is obtained for the CFQCD process [EQUATION]', '1010.0748-2-39-3': 'Note that in the process [REF] each flow starts from and ends with a quark pair which belongs to the same fermion line.', '1010.0748-2-40-0': 'We have shown so far that we can generate diagrams with definite color flow by assigning integer labels to quarks, [MATH], and gluons, [MATH], such that the labels [MATH] and [MATH] count the color-flow lines whose maximum number [MATH] is the sum of the number of external gluons and the numbers of quark lines (quark pairs), see eq. [REF].', '1010.0748-2-41-0': '## New particles and their interactions in the CFQCD Model', '1010.0748-2-42-0': 'In Table [REF], we list all the new particles in the CFQCD model for [MATH] in eq. [REF].', '1010.0748-2-43-0': 'The list is shown in the format of particles.dat in the usrmod of MadGraph[CITATION].', '1010.0748-2-43-1': 'As explained above, the [MATH] gluons have two color-flow indices, [MATH], while the Abelian gluon, [MATH], has no color-flow index.', '1010.0748-2-43-2': 'They are vector bosons (type[MATH]V), and we use curly lines (line [MATH] C) for [MATH] gluons while wavy lines (line [MATH] W) for the Abelian gluon.', '1010.0748-2-43-3': "The 'color' column of the list is used by MadGraph to perform color summation by using the color-ordered basis.", '1010.0748-2-43-4': "Since we sum over the color degrees of freedom by summing the 3 and [MATH] indices ([MATH]'s and [MATH]'s) over all possible color flows explicitly, we declare all our new particles as singlets (color [MATH] S).", '1010.0748-2-43-5': 'The last column gives the PDG code for the particles, and all our gluons, including the Abelian gluon, are given the number 21.', '1010.0748-2-44-0': 'All gluons, not only the Abelian gluon but also [MATH] gluons, are declared as Majorana particles (particle and anti-particle are the same) in CFQCD.', '1010.0748-2-44-1': 'We adopt this assignment in order to avoid generating gluon propagators between multi-gluon vertices.', '1010.0748-2-44-2': 'This goal is achieved because the Majorana [MATH] gluon propagators flip the color flow as shown in Fig. [REF], and hence CFQCD does not give diagrams with gluon propagator in color-flow conserving amplitudes.', '1010.0748-2-45-0': '[MATH] gluon propagators attached to quark lines are allowed by introducing the color-flow non-conserving [MATH] couplings that effectively recover a color-flow connection.', '1010.0748-2-45-1': 'See discussions on [MATH] vertices at the end of this section.', '1010.0748-2-46-0': 'In Table [REF], we list [MATH]-quarks, [MATH], and its anti-particles, [MATH], with the color-flow index [MATH] to [MATH].', '1010.0748-2-46-1': 'They are all femions (type [MATH] F) for which we use solid lines (line [MATH] S) in the diagram.', '1010.0748-2-46-2': 'Their colors are declared as singlets (color [MATH] S) as explained above.', '1010.0748-2-46-3': 'The list should be extended to the other five quarks, such as [MATH] and [MATH] for down and anti-down quarks.', '1010.0748-2-47-0': 'Before closing the explanation of new particles in CFQCD, let us note that the number of [MATH] gluons needed for [MATH]-gluon processes is [MATH].', '1010.0748-2-47-1': 'This follows from our color-flow-line numbering scheme, which counts the successive color-flow lines in the ascending order along the color flow as depicted in Fig. [REF].', '1010.0748-2-47-2': 'This is necessary to avoid double counting of the same color flow.', '1010.0748-2-47-3': 'According to this rule, the only gluonic vertex possible for [MATH] is the one among [MATH], [MATH] and [MATH].', '1010.0748-2-47-4': 'Although [MATH] can appear for processes with [MATH], [MATH] and [MATH] can never appear.', '1010.0748-2-47-5': "Generalizing this rule, we find that [MATH]'s with [MATH] mod([MATH]) as well as [MATH] cannot appear and hence we need only [MATH] gluons in CFQCD.", '1010.0748-2-48-0': 'In this paper, we report results up to 5-jet production processes.', '1010.0748-2-48-1': 'Purely gluonic [MATH] process has seven external gluons, and [MATH] is necessary only for this process.', '1010.0748-2-48-2': 'According to the above rule, there is only one interaction for this process, which is the one among [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1010.0748-2-48-3': 'Therefore, we should add [MATH] and [MATH] to Table [REF] in order to compute [MATH] amplitudes.', '1010.0748-2-49-0': 'Next, we list all the interactions among CFQCD particles.', '1010.0748-2-49-1': 'Once the interactions among particles of a user defined model are given, MadGraph generates Feynman diagrams for an arbitrary process and the corresponding HELAS amplitude code.', '1010.0748-2-49-2': 'In CFQCD, we introduce [MATH]-point gluon interactions and let MadGraph generate a code which calls just one HELAS subroutine (either gluon or jgluo with [MATH], which makes use of the recursion relations of eqs. [REF] or [REF], respectively) for each [MATH]-point vertex.', '1010.0748-2-49-3': 'In addition, we have quark-quark-gluon vertices.', '1010.0748-2-49-4': 'We list the interactions in the descending order of the number of participating particles in Table [REF] to [REF].', '1010.0748-2-50-0': 'First, we show the 7-point interaction in Table [REF] in the format of interactions.dat of usrmod in MadGraph[CITATION].', '1010.0748-2-50-1': 'As discussed above, it is needed only for generating [MATH] amplitudes.', '1010.0748-2-50-2': 'The vertex is proportional to [MATH], the fifth power of the strong coupling constant, and we give the five couplings, cpl1 to cpl5, as [EQUATION] according to the Feynman rules of Fig. [REF], whose types, type1 to type5, are all QCD.', '1010.0748-2-51-0': 'The 6-point interactions appear in [MATH] and also in [MATH] process in this study.', '1010.0748-2-51-1': 'Again, only one interaction is possible among the six gluons, [MATH], with [MATH] to [MATH] and [MATH], as shown in Table [REF].', '1010.0748-2-51-2': 'The coupling order is [MATH] and the four couplings, cpl1 to cpl4, are G2 all with the type QCD.', '1010.0748-2-52-0': 'The 5-point gluon vertices appear in [MATH], [MATH], [MATH] and [MATH] processes.', '1010.0748-2-52-1': 'The first two and the fourth processes have [MATH], for which the 5-point gluon vertex is unique as shown in the first row of Table [REF].', '1010.0748-2-52-2': 'The process [MATH] gives [MATH], and gluons with the sixth color-flow line can contribute to the 5-point gluon vertex.', '1010.0748-2-52-3': 'Because of the ascending color-flow numbering scheme, only one additional combination appears as shown in the second row of Table [REF].', '1010.0748-2-52-4': 'These vertices have [MATH] order and we have cpl[MATH] G2 and type[MATH] QCD for [MATH].', '1010.0748-2-53-0': 'The 4-point gluon vertices appear in [MATH]), [MATH] with [MATH] to [MATH], and in [MATH] with [MATH] and [MATH] in this study.', '1010.0748-2-53-1': 'As above, we have only one 4-point gluon vertex for processes with [MATH], which is shown in the first row of Table [REF].', '1010.0748-2-53-2': 'For the process [MATH]), one additional vertex appears as shown in the second row.', '1010.0748-2-53-3': 'In case of [MATH]), the ordering [MATH] also appears, and it is given in the third row.', '1010.0748-2-54-0': 'So far, we obtain multiple gluon vertices from the color-flow lines of consecutive numbers, corresponding to the color flows such as [EQUATION] for [MATH]-point gluon vertices.', '1010.0748-2-54-1': 'When there are two or more quark lines in the process, we can also have color flows which skips color-flow-line numbers, such as [EQUATION] for [MATH]-point gluon vertices, where [MATH] counts the number of skips.', '1010.0748-2-54-2': 'Because gluon propagators do not attach to gluon vertices in CFQCD, this skip can appear only when two or more gluons from the same vertex are connected to quark lines.', '1010.0748-2-54-3': 'For example, in the fourth row of Table [REF], the gluon [MATH] couples to the quark line, [MATH], and then the gluon [MATH] couples to the other quark line, [MATH], in the process [MATH].', '1010.0748-2-54-4': 'Likewise, [MATH] and [MATH] in the fifth and the sixth row, respectively, couple to the second quark line, [MATH], of the process.', '1010.0748-2-55-0': 'As for the 3-gluon vertices, ten vertices listed in Table [REF] appear in our study.', '1010.0748-2-55-1': 'The first four vertices with successive color-flow numbers appear in processes with one quark line, [MATH] with [MATH] to [MATH], and those with two quark lines, [MATH] with [MATH] to 5.', '1010.0748-2-55-2': 'The fifth and the sixth vertices starting with g14 appears for [MATH] with [MATH], for which only one unit of skip ([MATH]) appears, and also with [MATH].', '1010.0748-2-55-3': 'The last four vertices appear only in the [MATH] process with two quark lines, for which [MATH] is possible.', '1010.0748-2-55-4': 'In fact, the two vertices starting with g15 contain g52 or g41 with [MATH] skips in the color-flow number.', '1010.0748-2-55-5': 'This completes all the gluon self-interactions in CFQCD up to 5-jet production processes.', '1010.0748-2-56-0': 'There are two types of [MATH] vertices in CFQCD: the couplings of [MATH] gluons, [MATH], and those of the Abelian gluon, [MATH].', '1010.0748-2-56-1': 'All the [MATH] couplings for the [MATH]-quark, [MATH] to [MATH], are listed in Table [REF].', '1010.0748-2-56-2': 'In the HELAS convention, the fermion-fermion-boson couplings are two dimensional complex arrays, where the first and the second components are the couplings of the left- and the right-hand chirality of the flowing-in fermion.', '1010.0748-2-56-3': 'According to the Feynman rules of Fig. [REF], the couplings are [EQUATION] for [MATH] gluons and [EQUATION] for the Abelian gluon; Note the [MATH] factor for the Abelian gluon coupling.', '1010.0748-2-57-0': '[MATH] gluons have interactions as [EQUATION] where the fermion number flows from [MATH] to [MATH] by emitting the out-going [MATH] gluon.', '1010.0748-2-57-1': 'All the diagrams with on-shell [MATH] gluons attached to a quark line are obtained from the vertices [REF].', '1010.0748-2-57-2': 'Likewise, the Abelian gluon couples to quarks as [EQUATION]', '1010.0748-2-57-3': 'In CFQCD, we generate an [MATH] gluon propagator between a quark line and a gluon vertex and between two quark lines by introducing color-flow flipping vertices [EQUATION] as we discussed above.', '1010.0748-2-57-4': 'Here, the gluon [MATH] is emitted from the quark line [MATH], such that [MATH] gluons can propagate between a quark line and a gluonic vertex and also between two quark lines when the other side of the [MATH] vertex is the color-flow conserving one, [REF].', '1010.0748-2-57-5': 'In order to exchange gluons between an arbitrary gluonic vertex and a quark line, these color-flow-flipped [MATH] vertices should exist for all [MATH] gluons.', '1010.0748-2-57-6': 'However, since all [MATH] gluons have the color-flow conserving vertices [REF] as well, we find double counting of amplitudes where an [MATH] gluon is exchanged between two quark lines.', '1010.0748-2-57-7': 'This double counting can be avoided simply by discarding color-flow conserving [MATH] vertices [REF] for [MATH] as shown in Table 7.', '1010.0748-2-58-0': 'Now we exhaust all the interactions needed in the calculation of partial amplitudes and ready to generate diagrams and evaluate total amplitudes in the color-flow basis.', '1010.0748-2-59-0': '# Total amplitudes and the color summation', '1010.0748-2-60-0': 'In this section, we discuss how we evaluate the total amplitudes, eqs. [REF], [REF] and [REF], with the CFQCD model and how we perform the color summation of the total amplitude squared.', '1010.0748-2-61-0': '## [MATH] processes', '1010.0748-2-62-0': 'We consider pure gluon processes first.', '1010.0748-2-62-1': 'As discussed in Section [REF], the total amplitude of a [MATH]-gluon process is expressed as eq.[REF], and it is the sum of several partial amplitudes.', '1010.0748-2-62-2': "Because color factors, Kronecker's delta's, for partial amplitudes is either 1 or 0, the total amplitude for a given color configuration (a color assignment for each external gluons) consists of a subset of all the partial amplitudes in eq.[REF].", '1010.0748-2-62-3': 'Therefore, in order to evaluate the total amplitude for a given color configuration, we should find all possible color flows for the configuration, compute their partial amplitudes and simply sum them up.', '1010.0748-2-63-0': 'As an example, let us consider a [MATH] case, [EQUATION]', '1010.0748-2-63-1': 'The color configuration is expressed by sets of 3 and [MATH] indices, [MATH] and [MATH], respectively, for each gluon; [EQUATION]', '1010.0748-2-63-2': 'Here the subscripts of each index pair label the gluon [MATH], [MATH] to 5, with the momentum [MATH] and the helicity [MATH].', '1010.0748-2-63-3': 'For instance, let us examine a color configuration [EQUATION]', '1010.0748-2-63-4': 'One of the possible color flows that gives the above color configuration is [EQUATION] whose associated amplitude can be calculated as the amplitude for the CFQCD process', '1010.0748-2-64-0': '[EQUATION]', '1010.0748-2-64-1': 'MadGraph generates the corresponding HELAS amplitude code, which calls the 5-gluon amplitude subroutine, gluon5.', '1010.0748-2-65-0': 'There is also another color flow [EQUATION] for the color configuration [REF].', '1010.0748-2-65-1': 'The corresponding partial amplitude can be obtained by one of the [MATH] permutations of [MATH] gluon momenta and helicities: [EQUATION] where [MATH] denotes the wave function of the external gluon [MATH].', '1010.0748-2-66-0': 'When all the partial amplitudes for each color flow are evaluated, we sum them up and obtain the color-fixed total amplitude, eq. [REF], for this color configuration.', '1010.0748-2-66-1': 'Therefore, for pure gluon process, we generate the HELAS amplitude once for all to evaluate the color-fixed total amplitude.', '1010.0748-2-66-2': 'The total amplitude is then squared and the summation over all color configurations is done by the Monte-Carlo method.', '1010.0748-2-67-0': '## [MATH] processes', '1010.0748-2-68-0': 'Next, we discuss the processes with one quark line, [MATH].', '1010.0748-2-68-1': 'The procedure to compute the color-summed amplitude squared is the same, but we should take into account the Abelian gluon contributions for processes with quarks.', '1010.0748-2-68-2': 'As shown in Fig. [REF], the Abelian gluons appear as an isolated color index pair which have the same number.', '1010.0748-2-68-3': 'Therefore, we can take into account its contribution by regarding the independent gluon index pair as the Abelian gluon.', '1010.0748-2-69-0': "As an example, let us consider the process [MATH] with the color configuration: [EQUATION] where the parenthesis with the subscript '[MATH]' gives the color charges of the [MATH]-quark pair; [MATH] denotes the annihilation of the [MATH]-quark with the 3 charge, [MATH], and the [MATH]-quark with the [MATH] charge, [MATH].", '1010.0748-2-69-1': 'Although this color configuration is essentially the same as [REF], and hence has the color flows like [REF] and [REF], we have an additional color flow for this process: [EQUATION]', '1010.0748-2-69-2': 'Here the index pair of the first gluon, [MATH] in [REF], forms an independent color flow, [MATH], which corresponds to the Abelian gluon, [MATH].', '1010.0748-2-69-3': 'The CFQCD process for this color flow reads', '1010.0748-2-70-0': '[MATH].', '1010.0748-2-71-0': 'Summing up, there are three color flows for the color configuration [REF]: [EQUATION]', '1010.0748-2-71-1': 'The partial amplitude [MATH] for the color flow [MATH] is obtained from that of the color flow (a), [MATH], by a permutation of gluon wave functions as in the case of [MATH] amplitudes for the color configuration [REF].', '1010.0748-2-71-2': 'The partial amplitude for the color flow [MATH] is calculated with the Feynman rules of Fig. [REF], and the total amplitude [MATH] is obtained as [EQUATION] where the [MATH] factor comes from the Abelian gluon as shown in eq. ([REF]).', '1010.0748-2-71-3': 'When more than one gluon have the same color indices, [MATH] in [MATH], more Abelian gluons can contribute to the amplitude, and the factor [MATH] appears for the partial amplitude with [MATH] Abelian gluons.', '1010.0748-2-71-4': 'In the CFQCD model introduced in section 3, those factors are automatically taken into account in the HELAS code generated by MadGraph.', '1010.0748-2-72-0': '## [MATH] processes', '1010.0748-2-73-0': 'Finally, let us discuss the processes with two quark lines.', '1010.0748-2-73-1': 'As shown in Fig. [REF], there are two independent color flows because there are two sets of quark color indices, [MATH] and [MATH].', '1010.0748-2-73-2': 'Nevertheless, we can show color configurations and find possible color flows in the same way as in the [MATH] cases.', '1010.0748-2-73-3': 'Let us consider the process [EQUATION] with the color configuration [EQUATION] for illustration.', '1010.0748-2-73-4': 'Color charges of [MATH]-quarks are in the parenthesis [MATH], and those of [MATH]-quarks are in [MATH] as in the [MATH] processes.', '1010.0748-2-73-5': 'All the possible color flows are [EQUATION]', '1010.0748-2-73-6': 'We show in Fig. [REF] one representative Feynman diagram for each color flow, [MATH] to [MATH].', '1010.0748-2-74-0': 'The amplitudes for each color flow, [MATH] to [MATH], are calculated as amplitudes for the corresponding CFQCD processes: [EQUATION]', '1010.0748-2-74-1': 'For the CFQCD processes [MATH] and [MATH], the color flow lines starting with the [MATH]- and [MATH]-quarks terminate at the same quarks.', '1010.0748-2-74-2': 'Such amplitudes are generated by an exchange of the Abelian gluon, as shown by the representative diagrams in Fig.[REF].', '1010.0748-2-74-3': 'The total amplitude is now obtained as [EQUATION] according to eq.[REF].', '1010.0748-2-74-4': 'The color summation of the squared amplitudes [MATH] is performed by the MC method just as in the pure gluon case.', '1010.0748-2-75-0': '# Sample results', '1010.0748-2-76-0': 'In this section, we present numerical results for several multi-jet production processes as a demonstration of our CFQCD model on MadGraph.', '1010.0748-2-76-1': 'We compute [MATH]-jet production cross sections from [MATH], [MATH] and [MATH] subprocesses up to [MATH] in the [MATH] collision at [MATH] TeV.', '1010.0748-2-76-2': 'We define the final state cuts, the QCD coupling constant and the parton distribution function exactly as the same as those of ref. [CITATION], so that we can compare our results against those presented in ref. [CITATION], which have been tested by a few independent calculations.', '1010.0748-2-76-3': 'Specifically, we select jets (partons) that satisfy [EQUATION] where [MATH] and [MATH] are the pseudo-rapidity and the transverse momentum of the parton-[MATH], and [MATH] is the smaller of the relative transverse momentum between parton-[MATH] and parton-[MATH].', '1010.0748-2-76-4': 'We use CTEQ6L1 parton distribution functions[CITATION] at the factorization scale [MATH] GeV and the QCD coupling [MATH].', '1010.0748-2-76-5': 'Phase space integration and summation over color and helicity are performed by an adaptive Monte Carlo (MC) integration program BASES[CITATION].', '1010.0748-2-77-0': 'Results are shown in Table [REF] and Fig. [REF].', '1010.0748-2-78-0': 'In Table [REF], the first row for each [MATH]-jet process gives the exact result for the [MATH]-jet production cross section, while the second row shows the cross section when we ignore all the Abelian gluon contributions.', '1010.0748-2-78-1': 'The third row shows the results where we include up to one Abelian gluon contributions: one Abelian gluon emission amplitudes without an Abelian gluon exchange for [MATH] and [MATH] processes, and Abelian gluon exchange amplitudes without an Abelian gluon emission for [MATH] processes.', '1010.0748-2-78-2': 'All the numerical results for the exact cross sections in Table [REF] agree with those presented in ref.[CITATION] within the accuracy of MC integration.', '1010.0748-2-78-3': 'In Fig.[REF], the multi-jet production cross sections are shown for [MATH] and 5 jets in units of fb.', '1010.0748-2-78-4': 'The upper line gives the results for the subprocess [MATH].', '1010.0748-2-78-5': 'The middle lines show those for the subprocess [MATH]; their amplitudes are obtained from those of the subprocess [MATH] outlined in this study, simply by anti-symmetrizing the amplitudes with respect to the two external quark wave functions.', '1010.0748-2-78-6': 'The solid line gives the exact results, while the dashed line gives the results when all the Abelian gluon contributions are ignored.', '1010.0748-2-78-7': 'The dotted line shows the results which include up to one Abelian gluon contributions, although it is hard to distinguish from the solid line in the figure.', '1010.0748-2-78-8': 'Despite their order [MATH] suppression, contributions of Abelian gluons can be significant; more than 30% for [MATH], about 13% for [MATH], while about 10% for [MATH]4 and 5.', '1010.0748-2-79-0': 'The bottom lines show results for the subprocess [MATH].', '1010.0748-2-79-1': 'As above, the solid line gives the exact cross sections, while the dashed line and the dotted line give the results when contributions from Abelian gluons are ignored and one Abelian gluon contributions are included, respectively.', '1010.0748-2-79-2': 'Unlike the case for [MATH] subprocesses, the Abelian gluon contributions remain at 30% level even for [MATH].', '1010.0748-2-80-0': 'Before closing this section, we would like to give two technical remarks on our implementation of CFQCD on MadGraph.', '1010.0748-2-80-1': 'First, since the present MadGraph[CITATION] does not allow vertices among more than 4 particles, we add HELAS codes for 5, 6, and 7 gluon vertices by hand to complete the MadGraph generated codes.', '1010.0748-2-80-2': 'This restriction will disappear once the new version of MadGraph, MG5, is available, since MG5 accepts vertices with arbitrary number of particles.', '1010.0748-2-80-3': 'Second, we do not expect difficulty in running CFQCD codes on GPU, since all the codes we developed (see Appendix) follow the standard HELAS subroutine rules.', '1010.0748-2-81-0': '# Conclusions', '1010.0748-2-82-0': 'In this paper, we have implemented off-shell recursive formulae for gluon currents in the color-flow basis in MadGraph and have shown that it is possible to generate QCD amplitudes in the color-flow basis by introducing a new model, CFQCD, in which quarks and gluons are labeled by color-flow numbers.', '1010.0748-2-82-1': 'We have', '1010.0748-2-83-0': '- introduced new subroutines for off-shell recursive formulae for gluon currents, the', '1010.0748-2-84-0': 'contact 3- and 4-point gluon vertices in the color flow basis and the off-shell Abelian', '1010.0748-2-85-0': 'gluon current,', '1010.0748-2-86-0': '- defined new MadGraph model: the CFQCD Model,', '1010.0748-2-87-0': '- generated HELAS amplitudes for given color flows and calculated', '1010.0748-2-88-0': 'the color-summed total amplitude squared, and', '1010.0748-2-89-0': '- showed the numerical results for [MATH]-jet production cross sections ([MATH]).', '1010.0748-2-90-0': 'Although we have studied only up to 5-jet production processes in this paper, it is straightforward to extend the method to higher [MATH]-jet production processes.'}

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['1010.0748-2-78-0', '1010.0748-3-78-0'], ['1010.0748-2-78-1', '1010.0748-3-78-1'], ['1010.0748-2-78-2', '1010.0748-3-78-2'], ['1010.0748-2-78-3', '1010.0748-3-78-3'], ['1010.0748-2-78-4', '1010.0748-3-78-4'], ['1010.0748-2-78-5', '1010.0748-3-78-5'], ['1010.0748-2-78-6', '1010.0748-3-78-6'], ['1010.0748-2-78-7', '1010.0748-3-78-7'], ['1010.0748-2-78-8', '1010.0748-3-78-8'], ['1010.0748-2-2-0', '1010.0748-3-2-0'], ['1010.0748-2-2-1', '1010.0748-3-2-1'], ['1010.0748-2-2-2', '1010.0748-3-2-2'], ['1010.0748-2-2-3', '1010.0748-3-2-3'], ['1010.0748-2-27-0', '1010.0748-3-27-0'], ['1010.0748-2-27-1', '1010.0748-3-27-1'], ['1010.0748-2-27-2', '1010.0748-3-27-2'], ['1010.0748-2-24-0', '1010.0748-3-24-0'], ['1010.0748-2-24-1', '1010.0748-3-24-1'], ['1010.0748-2-5-0', '1010.0748-3-5-0'], ['1010.0748-2-5-1', '1010.0748-3-5-1'], ['1010.0748-2-5-2', '1010.0748-3-5-2'], ['1010.0748-2-5-3', '1010.0748-3-5-3'], ['1010.0748-2-5-4', '1010.0748-3-5-4'], ['1010.0748-1-6-0', '1010.0748-2-6-0'], ['1010.0748-1-6-1', '1010.0748-2-6-1'], ['1010.0748-1-6-3', '1010.0748-2-6-3'], ['1010.0748-1-6-4', '1010.0748-2-6-4'], ['1010.0748-1-10-5', '1010.0748-2-13-3'], ['1010.0748-1-11-1', '1010.0748-2-14-1'], ['1010.0748-1-12-0', '1010.0748-2-14-2'], ['1010.0748-1-69-0', '1010.0748-2-71-0'], ['1010.0748-1-69-1', '1010.0748-2-71-1']]

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[]

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[]

['1010.0748-1-62-0', '1010.0748-1-68-0', '1010.0748-1-76-0', '1010.0748-1-81-1', '1010.0748-1-83-0', '1010.0748-1-84-0', '1010.0748-1-85-0', '1010.0748-1-86-0', '1010.0748-2-64-0', '1010.0748-2-70-0', '1010.0748-2-77-0', '1010.0748-2-82-1', '1010.0748-2-85-0', '1010.0748-2-86-0', '1010.0748-2-87-0', '1010.0748-2-88-0', '1010.0748-3-64-0', '1010.0748-3-70-0', '1010.0748-3-77-0', '1010.0748-3-82-1', '1010.0748-3-83-0', '1010.0748-3-84-0', '1010.0748-3-88-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1010.0748

{'1010.0748-3-0-0': 'We propose to make use of the off-shell recursive relations with the color-flow decomposition in the calculation of QCD amplitudes on MadGraph.', '1010.0748-3-0-1': 'We introduce colored quarks and their interactions with nine gluons in the color-flow basis plus an Abelian gluon on MadGraph, such that it generates helicity amplitudes in the color-flow basis with off-shell recursive formulae for multi-gluon sub-amplitudes.', '1010.0748-3-0-2': 'We demonstrate calculations of up to 5-jet processes such as [MATH], [MATH] and [MATH].', '1010.0748-3-0-3': 'Although our demonstration is limited, it paves the way to evaluate amplitudes with more quark lines and gluons with Madgraph.', '1010.0748-3-1-0': '# Introduction', '1010.0748-3-2-0': 'The Large Hadron Collider (LHC) has been steadily running at 7 TeV since the end of March/2010.', '1010.0748-3-2-1': 'Since LHC is a hadron collider, understanding of QCD radiation processes is essential for the success of the experiments.', '1010.0748-3-2-2': 'To evaluate new physics signals and Standard Model (SM) backgrounds, we should resort to simulation programs which generate events with many hadrons and investigate observable distributions.', '1010.0748-3-2-3': 'In each simulation, one calculates matrix elements of hard processes with quarks and gluons, and this task is usually done by an automatic amplitude calculator.', '1010.0748-3-3-0': 'MadGraph[CITATION] is one of those programs.', '1010.0748-3-3-1': 'Although it is highly developed and has ample user-friendly utilities such as event generation with new physics signals matched to Parton Shower[CITATION], it cannot evaluate matrix elements with more than five jets in the final state[CITATION].', '1010.0748-3-4-0': 'This is a serious drawback since exact evaluation of multi-jet matrix elements is often required to study sub-structure of broad jets in identifying new physics signatures over QCD background.', '1010.0748-3-4-1': 'It is also disappointing because MadGraph generated HELAS amplitudes[CITATION] can be computed very fast on GPU (Graphic Processing Unit)[CITATION].', '1010.0748-3-5-0': 'Some of other simulation packages such as HELAC and Alpgen[CITATION] employ QCD off-shell recursive relations to produce multi-parton scattering amplitudes efficiently.', '1010.0748-3-5-1': 'Successful computation of recursively evaluated QCD amplitudes on GPU has been reported[CITATION].', '1010.0748-3-5-2': 'It is therefore interesting to examine the possibility of implementing recursive relations for gluon currents in MadGraph without sacrificing its benefits.', '1010.0748-3-5-3': 'In this paper, we examine the use of the recursive amplitudes in the color-flow basis, since corresponding HELAS amplitude package has been developed in ref. [CITATION].', '1010.0748-3-5-4': 'The main purpose of this paper is to show that this approach can be accommodated in MadGraph with explicit examples.', '1010.0748-3-6-0': 'The outline of this paper is as follows.', '1010.0748-3-6-1': 'In section [REF], we briefly review the color-flow decomposition and off-shell recursive relations.', '1010.0748-3-6-2': 'We then discuss the implementation of those techniques in MadGraph, using its "user mode" option[CITATION] in section [REF].', '1010.0748-3-6-3': 'In section [REF] and [REF], we explain how we evaluate the color-summed amplitude squared and show the numerical results of [MATH]-jet production cross section.', '1010.0748-3-6-4': 'Section [REF] is devoted to conclusions.', '1010.0748-3-7-0': '# The color-flow basis and off-shell recursive relations', '1010.0748-3-8-0': 'In this section we review the color-flow decomposition[CITATION] of QCD scattering amplitudes and the off-shell recursive relation[CITATION] of gluonic currents in the color flow basis.', '1010.0748-3-9-0': '## The color-flow decomposition', '1010.0748-3-10-0': 'First, we briefly review the color-flow decomposition.', '1010.0748-3-10-1': 'The Lagrangian of QCD can be expressed as [EQUATION] where [EQUATION]', '1010.0748-3-10-2': 'Upper and lower indices are those of [MATH] and [MATH] representations, respectively.', '1010.0748-3-10-3': 'Note that the gluon fields, [MATH], are renormalized by a factor of [MATH], and hence the coupling [MATH] is divided by [MATH].', '1010.0748-3-10-4': 'At this stage, not all the nine gluon fields are independent because of the traceslessness of the [MATH] generators, [EQUATION]', '1010.0748-3-10-5': 'Here we introduce the "Abelian gluon", [EQUATION]', '1010.0748-3-10-6': 'This is essentially the gauge boson , [MATH], of [MATH] subgroup of [MATH], combined with its generator, [MATH], which is normalized to 1/2.', '1010.0748-3-10-7': 'We then rewrite eq. ([REF]) by adding and subtracting the Abelian gluon, such that the QCD Lagrangian is expressed as [EQUATION] with [EQUATION]', '1010.0748-3-10-8': 'Here all the nine gluons, [MATH] are now independent while the covariant tensors keep the same form [EQUATION]', '1010.0748-3-10-9': 'In this basis, these nine gluons have one-to-one correspondence to a set of indices of [MATH] and [MATH] representations of [MATH], [MATH], and we address them and this basis as [MATH] gluons and the color-flow basis, respectively, in the following discussions.', '1010.0748-3-10-10': 'Although the definition of the covariant tensor ([REF]) contains self-coupling terms for [MATH] gluons, the Abelian gluon contribution actually drops out in the sum.', '1010.0748-3-10-11': 'Accordingly, the Feynman rules for [MATH] gluons give directly the [MATH] (QCD) amplitudes for purely gluonic processes[CITATION].', '1010.0748-3-11-0': 'We list the Feynman rules derived from the Lagrangian ([REF]) in Fig. [REF].', '1010.0748-3-12-0': 'It should be noted that both the propagator and the quark vertex of the Abelian gluon have an extra [MATH] factor because of the color-singlet projector, [MATH].', '1010.0748-3-12-1': 'The negative sign of this factor directly comes from the sign of the Abelian gluon kinetic term in the Lagrangian.', '1010.0748-3-12-2': 'We can evaluate QCD amplitudes by using these Feynman rules.', '1010.0748-3-13-0': 'For a [MATH]-gluon scattering process, it is written as [EQUATION] obtained from the corresponding [MATH] gluon scattering process.', '1010.0748-3-13-1': '[MATH] are gauge-invariant partial amplitudes, which depend only on particle momenta and helicities represented simply by the gluon indices, [MATH], and the summation is taken over all [MATH] non-cyclic permutations of gluons.', '1010.0748-3-13-2': 'A set of [MATH] Kronecker delta\'s in each term gives the "color flow" of the corresponding partial amplitudes.', '1010.0748-3-13-3': "As an example, we list in Fig. [REF] all six color flows indicated by Kronecker delta's for the [MATH] process, [MATH].", '1010.0748-3-14-0': 'On the other hand, if a scattering process involves quarks, contributions from the Abelian gluon do not decouple.', '1010.0748-3-14-1': 'Therefore, we have to add all its contributions to the total amplitude.', '1010.0748-3-14-2': 'For processes with one quark line, such contributions come from Abelian gluon emitting diagrams.', '1010.0748-3-14-3': 'The total amplitude for [MATH] process becomes [EQUATION] where [MATH] is the partial amplitude with [MATH] Abelian gluons, and [MATH] and [MATH] denote the momenta and the helicities of the quark and the anti-quark, respectively.', '1010.0748-3-14-4': 'The summation is taken over all the [MATH] permutations of [MATH] gluons.', '1010.0748-3-14-5': 'The first term in the r.h.s. of eq. ([REF]) gives contributions from [MATH] gluons, and the other terms give contributions from [MATH] gluons and [MATH] Abelian gluons, summed over [MATH] to [MATH], where [MATH] comes from the [MATH] factor in the Abelian gluon vertex as depicted in Fig. [REF].', '1010.0748-3-14-6': 'Corresponding color flows of those partial amplitudes are shown in Fig. [REF].', '1010.0748-3-15-0': 'For processes with two or more quark lines, the Abelian gluon can be exchanged between them, giving the amplitudes with at least [MATH] suppression.', '1010.0748-3-15-1': 'Therefore, the total amplitude of processes with two quark lines such as [MATH] gluons becomes [EQUATION] [MATH] (or [MATH]) denotes a partial amplitude with [MATH] gluons, where two sets of arguments divided by "[MATH]" belong to two different color-flow chains; one starts from [MATH] in the initial state) and ends with [MATH] in the final state), and the other starts from [MATH] and ends with [MATH], which are given explicitly by two sets of delta\'s before the partial amplitude.', '1010.0748-3-15-2': 'The difference between [MATH] and [MATH] is that the former consists of [MATH]-gluon-exchange diagrams while the latter consists of Abelian-gluon-exchange ones.', '1010.0748-3-15-3': 'Here we write down explicitly the partial amplitudes with external [MATH] gluons, [MATH] and [MATH], whereas those with external Abelian gluons should be added as in the previous one-quark-line case.', '1010.0748-3-15-4': 'For illustration, some typical diagrams for the [MATH] process are shown in Fig. [REF].', '1010.0748-3-16-0': '## Recursive relations of off-shell gluon currents', '1010.0748-3-17-0': 'Next, we mention the off-shell recursive relations in the color-flow basis.', '1010.0748-3-17-1': 'We define a [MATH]-point off-shell gluon current [MATH] recursively as [EQUATION] where the numbers in the argument of currents represent gluons, and their order respects the color flow as in the partial amplitudes, [MATH], in eq. ([REF]); the argument after semicolon denotes the off-shell gluon.', '1010.0748-3-17-2': 'As an initial value, the 2-point gluon current, [MATH], is defined as the polarization vector of the gluon [MATH], [MATH].', '1010.0748-3-17-3': '[MATH] is the partial sum of gluon momenta: [EQUATION] where [MATH] to [MATH] are defined as flowing-out momenta.', '1010.0748-3-17-4': '[MATH] and [MATH] denote the three- and four-point gluon vertices from the Feynman rule of Fig. [REF]: [EQUATION]', '1010.0748-3-17-5': 'The summation over [MATH] and [MATH] in eq. ([REF]) are taken over the 3 and [MATH] indices, [MATH] and [MATH], of off-shell gluons in currents [MATH] and [MATH], respectively.', '1010.0748-3-17-6': 'For illustration, the 5-point off-shell current is shown explicitly in Fig. [REF].', '1010.0748-3-18-0': 'Helicity amplitudes of pure gluon processes in the color-flow basis are obtained from off-shell gluon currents as [EQUATION]', '1010.0748-3-18-1': 'Starting from the external wave functions ([REF]), the recursive relation computes partial amplitudes very effectively by using the HELAS code.', '1010.0748-3-18-2': 'When the off-shell line "[MATH]" in Fig. [REF] is set on-shell, the same set of diagrams give the complete helicity amplitudes in the color-flow basis.', '1010.0748-3-19-0': '# Implementation in MadGraph', '1010.0748-3-20-0': 'In this section, we discuss how we implement off-shell recursive relations in MadGraph in the color-flow basis and how we generate helicity amplitudes of QCD processes.', '1010.0748-3-21-0': '## Subroutines for off-shell recursive formulae', '1010.0748-3-22-0': 'First, we introduce new HELAS[CITATION] subroutines in MadGraph which make [MATH]-point gluon off-shell currents and [MATH]-gluon amplitudes in the color-flow basis, according to eq. ([REF]) and eq. ([REF]), respectively.', '1010.0748-3-22-1': 'Although the expression eq. ([REF]) gives the off-shell gluon current made from [MATH] on-shell gluons, in HELAS amplitudes any input on-shell gluon wave functions can be replaced by arbitrary off-shell gluon currents with the same color quantum numbers.', '1010.0748-3-22-2': 'Shown in Fig. [REF] is an example of such diagrams that are calculated by the HELAS subroutine for the 4-point off-shell gluon current.', '1010.0748-3-23-0': 'Thanks to this property of the HELAS amplitudes, we need to introduce only two types of subroutines: one which computes helicity amplitudes of [MATH]-gluon processes via eq. ([REF]) and the other which computes off-shell gluon currents from [MATH]-point gluon vertices via eq. ([REF]).', '1010.0748-3-23-1': 'We name the first type of subroutines as gluon and the second type as jgluo, where denotes the number of external on-shell and off-shell gluons.', '1010.0748-3-24-0': 'The number of new subroutines we should add to MadGraph depends on the number of external partons (quarks and gluons) in QCD processes.', '1010.0748-3-24-1': 'Processes with [MATH]-partons can be classified as those with [MATH] gluons, those with [MATH] gluons and one quark line, those with [MATH] gluons and two quark lines, and so on.', '1010.0748-3-25-0': 'In the color-flow basis, the first class of processes with [MATH] external gluons are calculated by just one amplitude subroutine, gluonn.', '1010.0748-3-26-0': 'For the second class of processes with [MATH] gluons and one quark line, we need up to [MATH]-point off-shell current subroutines, jgluo with [MATH] to [MATH].', '1010.0748-3-26-1': 'This is because the largest off-shell gluon current appears in the computation of diagrams where [MATH] on-shell gluons are connected to the quark line through one off-shell gluon, which can be computed by the [MATH]-point off-shell current and the [MATH] amplitude subroutine, iovxxx.', '1010.0748-3-26-2': 'Note that the same diagram can be computed by the off-shell gluon current subroutine made by the quark pair, jioxxx, and the [MATH]-point gluon amplitude subroutine, gluon(n-1).', '1010.0748-3-26-3': 'This type of redundancy in the computation of each Feynman diagram is inherent in the HELAS amplitudes, which is often useful in testing the codes.', '1010.0748-3-27-0': 'For [MATH]-parton processes with [MATH] gluons and two quark lines, we need up to [MATH]-point off-shell current subroutines.', '1010.0748-3-27-1': 'By also introducing multiple gluon amplitude subroutines up to [MATH]-point vertex, the maximum redundancy of HELAS amplitudes is achieved.', '1010.0748-3-27-2': 'Likewise, for [MATH]-parton processes with [MATH] quark lines, we need up to [MATH]-point off-shell current or amplitude subroutines.', '1010.0748-3-28-0': 'We list in Table [REF] the new HELAS subroutines we introduce in this study.', '1010.0748-3-29-0': 'The subroutine gluon evaluates a -gluon amplitude in the color-flow basis, and jgluo computes an off-shell current from ([MATH]) external gluons.', '1010.0748-3-29-1': 'Since we consider up to seven parton processes, [MATH], [MATH] and [MATH], in this study, we use gluon4 to gluon7 and jgluo4 to jgluo6.', '1010.0748-3-30-0': 'In addition to these subroutines that computes amplitudes and currents recursively, we also introduce three subroutines: ggggcf, jgggcf and jioaxx.', '1010.0748-3-30-1': 'Two of them, ggggcf and jgggcf, evaluate an amplitude and an off-shell current from the contact 4-gluon vertex, following the Feynman rule of Fig. [REF].', '1010.0748-3-30-2': 'Although the amplitude subroutine and the off-shell current subroutine for the 4-gluon vertex already exist in MadGraph, ggggxx and jgggxx, we should introduce the new ones which evaluate the sum of s- and t-type or s- and u-type vertices for a given color flow, since the default subroutines compute only one type at a time.', '1010.0748-3-30-3': 'jioaxx computes an off-shell Abelian gluon current made by a quark pair and is essentially the same as the off-shell gluon current subroutine, jioxxx, in the HELAS library[CITATION] except for an extra [MATH] factor: [EQUATION]', '1010.0748-3-30-4': 'Note that introducing this [MATH] factor is equivalent to summing up contributions from all Abelian gluon propagators, as we discussed in section [REF].', '1010.0748-3-30-5': 'We show the codes of ggggcf, jgggcf, gluon5 and jgluo5 in Appendix.', '1010.0748-3-31-0': '## Introduction of a new Model: CFQCD', '1010.0748-3-32-0': 'Next, we define a new Model with which MadGraph generates HELAS amplitudes in the color-flow basis since the present MadGraph computes them in the color-ordered basis[CITATION].', '1010.0748-3-32-1': 'A Model is a set of definitions of particles, their interactions and couplings; there are about twenty preset Models in the present MadGraph package[CITATION] such as the Standard Model and the Minimal SUSY Standard Model[CITATION].', '1010.0748-3-32-2': 'There is also a template for further extension by users, User Mode (usrmod).', '1010.0748-3-32-3': 'Using this template, we can add new particles and their interactions to MadGraph.', '1010.0748-3-32-4': 'We make use of this utility and introduce a model which we call the CFQCD Model.', '1010.0748-3-33-0': 'In the CFQCD Model, we introduce gluons and quarks as new particles labeled by indices that dictate the color flow, such that the diagrams for partial amplitudes are generated according to the Feynman rules of Fig. [REF].', '1010.0748-3-33-1': 'We need one index for quarks and two indices for gluons, such as [MATH], [MATH] and [MATH].', '1010.0748-3-33-2': 'The Abelian gluon, [MATH], does not have an index.', '1010.0748-3-33-3': 'The index labels all possible color flows, and it runs from 1 to [MATH], where [EQUATION]', '1010.0748-3-33-4': "This number [MATH] is the number of Kronecker's delta symbols which dictate the color flow.", '1010.0748-3-33-5': "As an example, let us consider the purely gluonic process [EQUATION] for which we need seven delta's to specify the color flow: [EQUATION]", '1010.0748-3-33-6': "Here the numbers in parentheses label gluons whereas the numbers in the sub-indices of Kronecker's delta's count the color-flow lines, 1 to [MATH] as depicted in Fig. [REF].", '1010.0748-3-34-0': 'In CFQCD, we label gluons according to their flowing-out, 3, and flowing-in, [MATH], color-flow-line numbers, such that the partial amplitude with the color flow ([REF]) is generated as the amplitude for the process [EQUATION]', '1010.0748-3-34-1': 'This is the description of the process ([REF]) in our CFQCD Model, and we let MadGraph generate the corresponding partial amplitudes, such as [MATH] in eq. ([REF]), as the helicity amplitudes for the process ([REF]).', '1010.0748-3-34-2': 'This index number assignment has one-to-one correspondence with the color flow ([REF]).', '1010.0748-3-34-3': 'For instance, [MATH] in the process ([REF]) denotes the contribution of the [MATH] gluon [MATH] to the partial amplitude where its [MATH] index [MATH] terminates the color-flow line 2, and the 3 index [MATH] starts the new color-flow line 3.', '1010.0748-3-34-4': 'It should also be noted that we number the color-flow lines in the ascending order along the color flow, starting from the color-flow line 1 and ending with the color-flow line [MATH].', '1010.0748-3-34-5': 'This numbering scheme plays an important role in defining the interactions among CFQCD gluons and quarks.', '1010.0748-3-35-0': 'Let us now examine the case for one quark line.', '1010.0748-3-35-1': 'For the 5-jet production process [EQUATION] the color-flow index should run from 1 to 6 according to the rule ([REF]).', '1010.0748-3-35-2': 'Indeed the color flow [EQUATION] corresponds to the process [EQUATION]', '1010.0748-3-35-3': 'We show in Fig. [REF] the color-flow diagram for this process.', '1010.0748-3-35-4': 'This is just that of the 6-gluon process cut at the [MATH] gluon, where [MATH] gluon is replaced by the quark pair, [MATH] and [MATH].', '1010.0748-3-35-5': 'Shown in Fig. [REF] are a few representative Feynman diagrams contributing to the process ([REF]).', '1010.0748-3-36-0': 'Both the external and internal quarks and gluons in the CFQCD model are shown explicitly along the external and propagator lines.', '1010.0748-3-36-1': 'Following the MadGraph convention, we use flowing-out quantum numbers for gluons while the quantum numbers are along the fermion number flow for quarks.', '1010.0748-3-36-2': 'Gluon propagators attached to quark lines are named by their color-flow quantum numbers along arrows.', '1010.0748-3-36-3': 'The diagram (a) contains only [MATH] vertices, (b) contains an off-shell 4-point gluon current, jgluo4, or a 4-point gluon amplitude, gluon4, and (c) contains either an off-shell 6-point gluon current, jgluo6, or a 6-point gluon amplitude, gluon6.', '1010.0748-3-37-0': 'In CFQCD, not only [MATH] gluons but also the Abelian gluon contributes to the processes with quark lines.', '1010.0748-3-37-1': 'For the [MATH] process ([REF]), 1 to 5 gluons can be Abelian gluons, [MATH].', '1010.0748-3-37-2': 'If the number of Abelian gluons is [MATH], the color flow reads [EQUATION]', '1010.0748-3-37-3': 'When [MATH], all the five gluons are Abelian, and the first (the right-most) color flow should be [MATH], just as in [MATH] photons.', '1010.0748-3-37-4': 'In Fig. [REF], we show the color-flow diagram for the process ([REF]) with the color flow ([REF]) for [MATH] as an example.', '1010.0748-3-38-0': 'For the process with two quark lines, [EQUATION] the color-flow index should run from 1 to 5.', '1010.0748-3-38-1': 'All the possible color flows are obtained from the comb-like diagram for the one-quark-line process shown in Fig. [REF], by cutting one of the gluons into a quark pair, such as [MATH] to [MATH] and [MATH].', '1010.0748-3-38-2': 'Then the first color flow starting from [MATH] ends at [MATH], and the new color flow starts with [MATH] which ends at [MATH].', '1010.0748-3-38-3': 'An example of such color flow for [MATH] and [MATH] reads [EQUATION]', '1010.0748-3-38-4': 'The CFQCD model computes the partial amplitude for the above color flow as the helicity amplitude for the process [EQUATION]', '1010.0748-3-38-5': 'The other possible color-flow processes without Abelian gluons are [EQUATION] for [MATH], respectively.', '1010.0748-3-38-6': 'As in the single quark line case, all the external gluons can also be Abelian gluons, and we should sum over contributions from external Abelian gluons.', '1010.0748-3-38-7': 'For instance, if the gluon [MATH] in the process ([REF]) is Abelian, the color flow becomes [EQUATION] and the corresponding partial amplitude is calculated for the process [EQUATION] in CFQCD.', '1010.0748-3-39-0': 'When there are more than one quark line, the Abelian gluon can be exchanged between two quark lines, and the color flow along each quark line is disconnected.', '1010.0748-3-39-1': 'For instance, the color flow [EQUATION] is obtained when the Abelian gluon is exchanged between the [MATH]-quark and [MATH]-quark lines.', '1010.0748-3-39-2': 'The corresponding partial amplitude is obtained for the CFQCD process [EQUATION]', '1010.0748-3-39-3': 'Note that in the process ([REF]) each flow starts from and ends with a quark pair which belongs to the same fermion line.', '1010.0748-3-40-0': 'We have shown so far that we can generate diagrams with definite color flow by assigning integer labels to quarks, [MATH], and gluons, [MATH], such that the labels [MATH] and [MATH] count the color-flow lines whose maximum number [MATH] is the sum of the number of external gluons and the numbers of quark lines (quark pairs), see eq. ([REF]).', '1010.0748-3-41-0': '## New particles and their interactions in the CFQCD Model', '1010.0748-3-42-0': 'In Table [REF], we list all the new particles in the CFQCD model for [MATH] in eq. ([REF]).', '1010.0748-3-43-0': 'The list is shown in the format of particles.dat in the usrmod of MadGraph[CITATION].', '1010.0748-3-43-1': 'As explained above, the [MATH] gluons have two color-flow indices, [MATH], while the Abelian gluon, [MATH], has no color-flow index.', '1010.0748-3-43-2': 'They are vector bosons (type[MATH]V), and we use curly lines (line [MATH] C) for [MATH] gluons while wavy lines (line [MATH] W) for the Abelian gluon.', '1010.0748-3-43-3': "The 'color' column of the list is used by MadGraph to perform color summation by using the color-ordered basis.", '1010.0748-3-43-4': "Since we sum over the color degrees of freedom by summing the 3 and [MATH] indices ([MATH]'s and [MATH]'s) over all possible color flows explicitly, we declare all our new particles as singlets (color [MATH] S).", '1010.0748-3-43-5': 'The last column gives the PDG code for the particles, and all our gluons, including the Abelian gluon, are given the number 21.', '1010.0748-3-44-0': 'All gluons, not only the Abelian gluon but also [MATH] gluons, are declared as Majorana particles (particle and anti-particle are the same) in CFQCD.', '1010.0748-3-44-1': 'We adopt this assignment in order to avoid generating gluon propagators between multi-gluon vertices.', '1010.0748-3-44-2': 'Such a propagator is made from a particle coming from one of the vertices and its anti-particle from the other.', '1010.0748-3-44-3': 'Since we define the anti-particle of a [MATH] gluon as the gluon itself, the color flow of the propagator should be flipped as shown in Fig. [REF], according to the gluon naming scheme explained in the previous subsection.', '1010.0748-3-44-4': 'Therefore, CFQCD does not give diagrams with gluon propagator in "color-flow conserving" amplitudes.', '1010.0748-3-45-0': 'For example, to form a [MATH] gluon propagator between two multi-gluon vertices, we need a [MATH] gluon coming out from one of the vertices and a [MATH] gluon from the other, according to the gluon naming scheme explained in the previous subsection.', '1010.0748-3-45-1': 'However, because a [MATH] gluon is not an anti-particle of [MATH] but a different particle in our particle difinition, Table [REF] , [MATH] gluon propagator can not be formed.', '1010.0748-3-45-2': '[MATH] gluon propagators attached to quark lines are allowed by introducing the color-flow non-conserving [MATH] couplings that effectively recover a color-flow connection.', '1010.0748-3-45-3': 'See discussions on [MATH] vertices at the end of this section.', '1010.0748-3-46-0': 'In Table [REF], we list [MATH]-quarks, [MATH], and its anti-particles, [MATH], with the color-flow index [MATH] to [MATH].', '1010.0748-3-46-1': 'They are all femions (type [MATH] F) for which we use solid lines (line [MATH] S) in the diagram.', '1010.0748-3-46-2': 'Their colors are declared as singlets (color [MATH] S) as explained above.', '1010.0748-3-46-3': 'The list should be extended to the other five quarks, such as [MATH] and [MATH] for down and anti-down quarks.', '1010.0748-3-47-0': 'Before closing the explanation of new particles in CFQCD, let us note that the number of [MATH] gluons needed for [MATH]-gluon processes is [MATH].', '1010.0748-3-47-1': 'This follows from our color-flow-line numbering scheme, which counts the successive color-flow lines in the ascending order along the color flow as depicted in Fig. [REF].', '1010.0748-3-47-2': 'This is necessary to avoid double counting of the same color flow.', '1010.0748-3-47-3': 'According to this rule, the only gluonic vertex possible for [MATH] is the one among [MATH], [MATH] and [MATH].', '1010.0748-3-47-4': 'Although [MATH] can appear for processes with [MATH], [MATH] and [MATH] can never appear.', '1010.0748-3-47-5': "Generalizing this rule, we find that [MATH]'s with [MATH] mod([MATH]) as well as [MATH] cannot appear and hence we need only [MATH] gluons in CFQCD.", '1010.0748-3-48-0': 'In this paper, we report results up to 5-jet production processes.', '1010.0748-3-48-1': 'Purely gluonic [MATH] process has seven external gluons, and [MATH] is necessary only for this process.', '1010.0748-3-48-2': 'According to the above rule, there is only one interaction for this process, which is the one among [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1010.0748-3-48-3': 'Therefore, we should add [MATH] and [MATH] to Table [REF] in order to compute [MATH] amplitudes.', '1010.0748-3-49-0': 'Next, we list all the interactions among CFQCD particles.', '1010.0748-3-49-1': 'Once the interactions among particles of a user defined model are given, MadGraph generates Feynman diagrams for an arbitrary process and the corresponding HELAS amplitude code.', '1010.0748-3-49-2': 'In CFQCD, we introduce [MATH]-point gluon interactions and let MadGraph generate a code which calls just one HELAS subroutine (either gluon or jgluo with [MATH], which makes use of the recursion relations of eqs. ([REF]) or ([REF]), respectively) for each [MATH]-point vertex.', '1010.0748-3-49-3': 'In addition, we have quark-quark-gluon vertices.', '1010.0748-3-49-4': 'We list the interactions in the descending order of the number of participating particles in Table [REF] to [REF].', '1010.0748-3-50-0': 'First, we show the 7-point interaction in Table [REF] in the format of interactions.dat of usrmod in MadGraph[CITATION].', '1010.0748-3-50-1': 'As discussed above, it is needed only for generating [MATH] amplitudes.', '1010.0748-3-50-2': 'The vertex is proportional to [MATH], the fifth power of the strong coupling constant, and we give the five couplings, cpl1 to cpl5, as [EQUATION] according to the Feynman rules of Fig. [REF], whose types, type1 to type5, are all QCD.', '1010.0748-3-51-0': 'The 6-point interactions appear in [MATH] and also in [MATH] process in this study.', '1010.0748-3-51-1': 'Again, only one interaction is possible among the six gluons, [MATH], with [MATH] to [MATH] and [MATH], as shown in Table [REF].', '1010.0748-3-51-2': 'The coupling order is [MATH] and the four couplings, cpl1 to cpl4, are G2 all with the type QCD.', '1010.0748-3-52-0': 'The 5-point gluon vertices appear in [MATH], [MATH], [MATH] and [MATH] processes.', '1010.0748-3-52-1': 'The first two and the fourth processes have [MATH], for which the 5-point gluon vertex is unique as shown in the first row of Table [REF].', '1010.0748-3-52-2': 'The process [MATH] gives [MATH], and gluons with the sixth color-flow line can contribute to the 5-point gluon vertex.', '1010.0748-3-52-3': 'Because of the ascending color-flow numbering scheme, only one additional combination appears as shown in the second row of Table [REF].', '1010.0748-3-52-4': 'These vertices have [MATH] order and we have cpl[MATH] G2 and type[MATH] QCD for [MATH].', '1010.0748-3-53-0': 'The 4-point gluon vertices appear in [MATH]), [MATH] with [MATH] to [MATH], and in [MATH] with [MATH] and [MATH] in this study.', '1010.0748-3-53-1': 'As above, we have only one 4-point gluon vertex for processes with [MATH], which is shown in the first row of Table [REF].', '1010.0748-3-53-2': 'For the process [MATH]), one additional vertex appears as shown in the second row.', '1010.0748-3-53-3': 'In case of [MATH]), the ordering [MATH] also appears, and it is given in the third row.', '1010.0748-3-54-0': 'So far, we obtain multiple gluon vertices from the color-flow lines of consecutive numbers, corresponding to the color flows such as [EQUATION] for [MATH]-point gluon vertices.', '1010.0748-3-54-1': 'When there are two or more quark lines in the process, we can also have color flows which skips color-flow-line numbers, such as [EQUATION] for [MATH]-point gluon vertices, where [MATH] counts the number of skips.', '1010.0748-3-54-2': 'Because gluon propagators do not attach to gluon vertices in CFQCD, this skip can appear only when two or more gluons from the same vertex are connected to quark lines.', '1010.0748-3-54-3': 'For example, in the fourth row of Table [REF], the gluon [MATH] couples to the quark line, [MATH], and then the gluon [MATH] couples to the other quark line, [MATH], in the process [MATH].', '1010.0748-3-54-4': 'Likewise, [MATH] and [MATH] in the fifth and the sixth row, respectively, couple to the second quark line, [MATH], of the process.', '1010.0748-3-55-0': 'As for the 3-gluon vertices, ten vertices listed in Table [REF] appear in our study.', '1010.0748-3-55-1': 'The first four vertices with successive color-flow numbers appear in processes with one quark line, [MATH] with [MATH] to [MATH], and those with two quark lines, [MATH] with [MATH] to 5.', '1010.0748-3-55-2': 'The fifth and the sixth vertices starting with g14 appears for [MATH] with [MATH], for which only one unit of skip ([MATH]) appears, and also with [MATH].', '1010.0748-3-55-3': 'The last four vertices appear only in the [MATH] process with two quark lines, for which [MATH] is possible.', '1010.0748-3-55-4': 'In fact, the two vertices starting with g15 contain g52 or g41 with [MATH] skips in the color-flow number.', '1010.0748-3-55-5': 'This completes all the gluon self-interactions in CFQCD up to 5-jet production processes.', '1010.0748-3-56-0': 'There are two types of [MATH] vertices in CFQCD: the couplings of [MATH] gluons, [MATH], and those of the Abelian gluon, [MATH].', '1010.0748-3-56-1': 'All the [MATH] couplings for the [MATH]-quark, [MATH] to [MATH], are listed in Table [REF].', '1010.0748-3-56-2': 'In the HELAS convention, the fermion-fermion-boson couplings are two dimensional complex arrays, where the first and the second components are the couplings of the left- and the right-hand chirality of the flowing-in fermion.', '1010.0748-3-56-3': 'According to the Feynman rules of Fig. [REF], the couplings are [EQUATION] for [MATH] gluons and [EQUATION] for the Abelian gluon; Note the [MATH] factor for the Abelian gluon coupling.', '1010.0748-3-57-0': '[MATH] gluons have interactions as [EQUATION] where the fermion number flows from [MATH] to [MATH] by emitting the out-going [MATH] gluon.', '1010.0748-3-57-1': 'All the diagrams with on-shell [MATH] gluons attached to a quark line are obtained from the vertices ([REF]).', '1010.0748-3-57-2': 'Likewise, the Abelian gluon couples to quarks as [EQUATION]', '1010.0748-3-57-3': 'In CFQCD, we generate an [MATH] gluon propagator between a quark line and a gluon vertex and between two quark lines by introducing color-flow flipping vertices [EQUATION] as we discussed above.', '1010.0748-3-57-4': 'Here, the gluon [MATH] is emitted from the quark line [MATH], such that [MATH] gluons can propagate between a quark line and a gluonic vertex and also between two quark lines when the other side of the [MATH] vertex is the color-flow conserving one, ([REF]).', '1010.0748-3-57-5': 'In order to exchange gluons between an arbitrary gluonic vertex and a quark line, these color-flow-flipped [MATH] vertices should exist for all [MATH] gluons.', '1010.0748-3-57-6': 'However, since all [MATH] gluons have the color-flow conserving vertices ([REF]) as well, we find double counting of amplitudes where an [MATH] gluon is exchanged between two quark lines.', '1010.0748-3-57-7': 'This double counting can be avoided simply by discarding color-flow conserving [MATH] vertices ([REF]) for [MATH] as shown in Table 7.', '1010.0748-3-58-0': 'Now we exhaust all the interactions needed in the calculation of partial amplitudes and ready to generate diagrams and evaluate total amplitudes in the color-flow basis.', '1010.0748-3-59-0': '# Total amplitudes and the color summation', '1010.0748-3-60-0': 'In this section, we discuss how we evaluate the total amplitudes, eqs. ([REF]), ([REF]) and ([REF]), with the CFQCD model and how we perform the color summation of the total amplitude squared.', '1010.0748-3-61-0': '## [MATH] processes', '1010.0748-3-62-0': 'We consider pure gluon processes first.', '1010.0748-3-62-1': 'As discussed in section [REF], the total amplitude of a [MATH]-gluon process is expressed as eq. ([REF]), and it is the sum of several partial amplitudes.', '1010.0748-3-62-2': "Because color factors, Kronecker's delta's, for partial amplitudes is either 1 or 0, the total amplitude for a given color configuration (a color assignment for each external gluons) consists of a subset of all the partial amplitudes in eq. ([REF]).", '1010.0748-3-62-3': 'Therefore, in order to evaluate the total amplitude for a given color configuration, we should find all possible color flows for the configuration, compute their partial amplitudes and simply sum them up.', '1010.0748-3-63-0': 'As an example, let us consider a [MATH] case, [EQUATION]', '1010.0748-3-63-1': 'The color configuration is expressed by sets of 3 and [MATH] indices, [MATH] and [MATH], respectively, for each gluon; [EQUATION]', '1010.0748-3-63-2': 'Here the subscripts of each index pair label the gluon [MATH], [MATH] to 5, with the momentum [MATH] and the helicity [MATH].', '1010.0748-3-63-3': 'For instance, let us examine a color configuration [EQUATION]', '1010.0748-3-63-4': 'One of the possible color flows that gives the above color configuration is [EQUATION] whose associated amplitude can be calculated as the amplitude for the CFQCD process', '1010.0748-3-64-0': '[EQUATION]', '1010.0748-3-64-1': 'MadGraph generates the corresponding HELAS amplitude code, which calls the 5-gluon amplitude subroutine, gluon5.', '1010.0748-3-65-0': 'There is also another color flow [EQUATION] for the color configuration ([REF]).', '1010.0748-3-65-1': 'The corresponding partial amplitude can be obtained by one of the [MATH] permutations of [MATH] gluon momenta and helicities: [EQUATION] where [MATH] denotes the wave function of the external gluon [MATH].', '1010.0748-3-66-0': 'When all the partial amplitudes for each color flow are evaluated, we sum them up and obtain the color-fixed total amplitude, eq. ([REF]), for this color configuration.', '1010.0748-3-66-1': 'Therefore, for pure gluon process, we generate the HELAS amplitude once for all to evaluate the color-fixed total amplitude.', '1010.0748-3-66-2': 'The total amplitude is then squared and the summation over all color configurations is done by the Monte-Carlo method.', '1010.0748-3-67-0': '## [MATH] processes', '1010.0748-3-68-0': 'Next, we discuss the processes with one quark line, [MATH].', '1010.0748-3-68-1': 'The procedure to compute the color-summed amplitude squared is the same, but we should take into account the Abelian gluon contributions for processes with quarks.', '1010.0748-3-68-2': 'As shown in Fig. [REF], the Abelian gluons appear as an isolated color index pair which have the same number.', '1010.0748-3-68-3': 'Therefore, we can take into account its contribution by regarding the independent gluon index pair as the Abelian gluon.', '1010.0748-3-69-0': "As an example, let us consider the process [EQUATION] with the color configuration: [EQUATION] where the parenthesis with the subscript '[MATH]' gives the color charges of the [MATH]-quark pair; [MATH] denotes the annihilation of the [MATH]-quark with the 3 charge, [MATH], and the [MATH]-quark with the [MATH] charge, [MATH].", '1010.0748-3-69-1': 'Although this color configuration is essentially the same as ([REF]), and hence has the color flows like ([REF]) and ([REF]), we have an additional color flow for this process: [EQUATION]', '1010.0748-3-69-2': 'Here the index pair of the first gluon, [MATH] in ([REF]), forms an independent color flow, [MATH], which corresponds to the Abelian gluon, [MATH].', '1010.0748-3-69-3': 'The CFQCD process for this color flow reads', '1010.0748-3-70-0': '[MATH].', '1010.0748-3-71-0': 'Summing up, there are three color flows for the color configuration ([REF]): [EQUATION]', '1010.0748-3-71-1': 'The partial amplitude [MATH] for the color flow [MATH] is obtained from that of the color flow (a), [MATH], by a permutation of gluon wave functions as in the case of [MATH] amplitudes for the color configuration ([REF]).', '1010.0748-3-71-2': 'The partial amplitude for the color flow [MATH] is calculated with the Feynman rules of Fig. [REF], and the total amplitude [MATH] is obtained as [EQUATION] where the [MATH] factor comes from the Abelian gluon as shown in eq. ([REF]).', '1010.0748-3-71-3': 'When more than one gluon have the same color indices, [MATH] in [MATH], more Abelian gluons can contribute to the amplitude, and the factor [MATH] appears for the partial amplitude with [MATH] Abelian gluons.', '1010.0748-3-71-4': 'In the CFQCD model introduced in section [REF], those factors are automatically taken into account in the HELAS code generated by MadGraph.', '1010.0748-3-72-0': '## [MATH] processes', '1010.0748-3-73-0': 'Finally, let us discuss the processes with two quark lines.', '1010.0748-3-73-1': 'As shown in Fig. [REF], there are two independent color flows because there are two sets of quark color indices, [MATH] and [MATH].', '1010.0748-3-73-2': 'Nevertheless, we can show color configurations and find possible color flows in the same way as in the [MATH] cases.', '1010.0748-3-73-3': 'Let us consider the process [EQUATION] with the color configuration [EQUATION] for illustration.', '1010.0748-3-73-4': 'Color charges of [MATH]-quarks are in the parenthesis [MATH], and those of [MATH]-quarks are in [MATH] as in the [MATH] processes.', '1010.0748-3-73-5': 'All the possible color flows are [EQUATION]', '1010.0748-3-73-6': 'We show in Fig. [REF] one representative Feynman diagram for each color flow, [MATH] to [MATH].', '1010.0748-3-74-0': 'The amplitudes for each color flow, [MATH] to [MATH], are calculated as amplitudes for the corresponding CFQCD processes: [EQUATION]', '1010.0748-3-74-1': 'For the CFQCD processes [MATH] and [MATH], the color flow lines starting with the [MATH]- and [MATH]-quarks terminate at the same quarks.', '1010.0748-3-74-2': 'Such amplitudes are generated by an exchange of the Abelian gluon, as shown by the representative diagrams in Fig. [REF].', '1010.0748-3-74-3': 'The total amplitude is now obtained as [EQUATION] according to eq. ([REF]).', '1010.0748-3-74-4': 'The color summation of the squared amplitudes [MATH] is performed by the MC method just as in the pure gluon case.', '1010.0748-3-75-0': '# Sample results', '1010.0748-3-76-0': 'In this section, we present numerical results for several multi-jet production processes as a demonstration of our CFQCD model on MadGraph.', '1010.0748-3-76-1': 'We compute [MATH]-jet production cross sections from [MATH], [MATH] and [MATH] subprocesses up to [MATH] in the [MATH] collision at [MATH] TeV.', '1010.0748-3-76-2': 'We define the final state cuts, the QCD coupling constant and the parton distribution function exactly as the same as those of ref. [CITATION], so that we can compare our results against those presented in ref. [CITATION], which have been tested by a few independent calculations.', '1010.0748-3-76-3': 'Specifically, we select jets (partons) that satisfy [EQUATION] where [MATH] and [MATH] are the pseudo-rapidity and the transverse momentum of the parton-[MATH], and [MATH] is the smaller of the relative transverse momentum between parton-[MATH] and parton-[MATH].', '1010.0748-3-76-4': 'We use CTEQ6L1 parton distribution functions[CITATION] at the factorization scale [MATH] GeV and the QCD coupling [MATH].', '1010.0748-3-76-5': 'Phase space integration and summation over color and helicity are performed by an adaptive Monte Carlo (MC) integration program BASES[CITATION].', '1010.0748-3-77-0': 'Results are shown in Table [REF] and Fig. [REF].', '1010.0748-3-78-0': 'In Table [REF], the first row for each [MATH]-jet process gives the exact result for the [MATH]-jet production cross section, while the second row shows the cross section when we ignore all the Abelian gluon contributions.', '1010.0748-3-78-1': 'The third row shows the results where we include up to one Abelian gluon contributions: one Abelian gluon emission amplitudes without an Abelian gluon exchange for [MATH] and [MATH] processes, and Abelian gluon exchange amplitudes without an Abelian gluon emission for [MATH] processes.', '1010.0748-3-78-2': 'All the numerical results for the exact cross sections in Table [REF] agree with those presented in ref.[CITATION] within the accuracy of MC integration.', '1010.0748-3-78-3': 'In Fig.[REF], the multi-jet production cross sections are shown for [MATH] and 5 jets in units of fb.', '1010.0748-3-78-4': 'The upper line gives the results for the subprocess [MATH].', '1010.0748-3-78-5': 'The middle lines show those for the subprocess [MATH]; their amplitudes are obtained from those of the subprocess [MATH] outlined in this study, simply by anti-symmetrizing the amplitudes with respect to the two external quark wave functions.', '1010.0748-3-78-6': 'The solid line gives the exact results, while the dashed line gives the results when all the Abelian gluon contributions are ignored.', '1010.0748-3-78-7': 'The dotted line shows the results which include up to one Abelian gluon contributions, although it is hard to distinguish from the solid line in the figure.', '1010.0748-3-78-8': 'Despite their order [MATH] suppression, contributions of Abelian gluons can be significant; more than 30% for [MATH], about 13% for [MATH], while about 10% for [MATH]4 and 5.', '1010.0748-3-79-0': 'The bottom lines show results for the subprocess [MATH].', '1010.0748-3-79-1': 'As above, the solid line gives the exact cross sections, while the dashed line and the dotted line give the results when contributions from Abelian gluons are ignored and one Abelian gluon contributions are included, respectively.', '1010.0748-3-79-2': 'Unlike the case for [MATH] subprocesses, the Abelian gluon contributions remain at 30% level even for [MATH].', '1010.0748-3-80-0': 'Before closing this section, we would like to give two technical remarks on our implementation of CFQCD on MadGraph.', '1010.0748-3-80-1': 'First, since the present MadGraph[CITATION] does not allow vertices among more than 4 particles, we add HELAS codes for 5, 6, and 7 gluon vertices by hand to complete the MadGraph generated codes.', '1010.0748-3-80-2': 'This restriction will disappear once the new version of MadGraph, MG5, is available, since MG5 accepts vertices with arbitrary number of particles.', '1010.0748-3-80-3': 'Second, we do not expect difficulty in running CFQCD codes on GPU, since all the codes we developed (see Appendix) follow the standard HELAS subroutine rules.', '1010.0748-3-81-0': '# Conclusions', '1010.0748-3-82-0': 'In this paper, we have implemented off-shell recursive formulae for gluon currents in the color-flow basis in MadGraph and have shown that it is possible to generate QCD amplitudes in the color-flow basis by introducing a new model, CFQCD, in which quarks and gluons are labeled by color-flow numbers.', '1010.0748-3-82-1': 'We have', '1010.0748-3-83-0': '- introduced new subroutines for off-shell recursive formulae for gluon currents, the contact 3- and 4-point gluon vertices in the color flow basis and the off-shell Abelian gluon current,', '1010.0748-3-84-0': '- defined new MadGraph model: the CFQCD Model,', '1010.0748-3-85-0': '- generated HELAS amplitudes for given color flows and calculated the color-summed total amplitude squared, and', '1010.0748-3-86-0': '- showed the numerical results for [MATH]-jet production cross sections ([MATH]).', '1010.0748-3-87-0': 'Although we have studied only up to 5-jet production processes in this paper, it is straightforward to extend the method to higher [MATH]-jet production processes.', '1010.0748-3-88-0': '3em'}

0809.5067

{'0809.5067-1-0-0': 'Nonthermal radiation observed from astrophysical systems containing relativistic jets and shocks, e.g., gamma-ray bursts (GRBs), active galactic nuclei (AGNs), and microquasars commonly exhibit power-law emission spectra.', '0809.5067-1-0-1': 'Recent PIC simulations of relativistic electron-ion (or electron-positron) jets injected into a stationary medium show that particle acceleration occurs within the downstream jet.', '0809.5067-1-0-2': 'In collisionless, relativistic shocks, particle (electron, positron, and ion) acceleration is due to plasma waves and their associated instabilities (e.g., the Weibel (filamentation) instability) created in the shock region.', '0809.5067-1-0-3': 'The simulations show that the Weibel instability is responsible for generating and amplifying highly non-uniform, small-scale magnetic fields.', '0809.5067-1-0-4': "These fields contribute to the electron's transverse deflection behind the jet head.", '0809.5067-1-0-5': 'The resulting "jitter" radiation from deflected electrons has different properties compared to synchrotron radiation, which assumes a uniform magnetic field.', '0809.5067-1-0-6': 'Jitter radiation may be important for understanding the complex time evolution and/or spectra in gamma-ray bursts, relativistic jets in general, and supernova remnants.', '0809.5067-1-1-0': '# Introduction', '0809.5067-1-2-0': 'Shocks are believed to be responsible for prompt emission from gamma-ray bursts (GRBs) and their afterglows, for variable emission from blazars, and for particle acceleration processes in jets from active galactic nuclei (AGN) and supernova remnants (SNRs).', '0809.5067-1-2-1': 'The predominant contribution to the observed emission spectra is often assumed to be synchrotron- and inverse Compton radiation from these accelerated particles [CITATION].', '0809.5067-1-2-2': 'It is assumed that turbulent magnetic fields in the shock region lead to Fermi acceleration, producing higher energy particles [CITATION].', '0809.5067-1-2-3': 'To make progress in understanding emission from these object classes, it is essential to place modeling efforts on a firm physical basis.', '0809.5067-1-2-4': 'This requires studies of the microphysics of the shock process in a self-consistent manner [CITATION].', '0809.5067-1-3-0': '## New Numerical Method for Calculating Emission', '0809.5067-1-4-0': 'The retarded electric field from a charged particle moving with instantaneous velocity [MATH] under acceleration [MATH] is obtained [CITATION].', '0809.5067-1-4-1': 'After some calculation and simplifying assumptions the total energy [MATH] radiated per unit solid angle per unit frequency can be expressed as', '0809.5067-1-5-0': '[EQUATION]', '0809.5067-1-6-0': "Here, [MATH] is a unit vector that points from the particle's retarded position towards the observer.", '0809.5067-1-6-1': 'The first term on the right hand side, containing the velocity field, is the Coulomb field from a charge moving without influence from external forces in eq. 2.4 [CITATION].', '0809.5067-1-6-2': 'The second term is a correction term that arises when the charge is subject to acceleration.', '0809.5067-1-6-3': 'Since the velocity-dependent field falls off in distance as [MATH], while the acceleration-dependent field scales as [MATH], the latter becomes dominant when observing the charge at large distances ([MATH]).', '0809.5067-1-6-4': 'The choice of unit vector [MATH] along the direction of propagation of the jet (hereafter taken to be the [MATH]-axis) corresponds to head-on emission.', '0809.5067-1-6-5': 'For any other choice of [MATH] (e.g., [MATH]), off-axis emission is seen by the observer.', '0809.5067-1-6-6': "The observer's viewing angle is set by the choice of [MATH]).", '0809.5067-1-7-0': '# Radiation from two electrons', '0809.5067-1-8-0': 'In the previous section we discussed how to obtain the retarded electric field from relativistically moving particles (electrons) observed at large distance.', '0809.5067-1-8-1': 'Using eq. 1 we calculated the time evolution of the retarded electric field and the spectrum from a gyrating electron in a uniform magnetic field to verify the technique used in this calculation.', '0809.5067-1-8-2': 'We have calculated the radiation from two electrons gyrating in the [MATH] plane in the uniform magnetic field [MATH] with Lorentz factors ([MATH]) (Case P in Table 1) [CITATION].', '0809.5067-1-8-3': 'We have very good agreement between the spectrum obtained from the simulation and the theoretical synchrotron spectrum expectation from eq. 7.10 [CITATION].', '0809.5067-1-9-0': 'In order to calculate more realistic radiation from relativistic jets we included a parallel magnetic field ([MATH]).', '0809.5067-1-9-1': 'Relativistic jets are propagating along the [MATH] direction.', '0809.5067-1-9-2': 'Table 1 shows six cases including the previous case P (first row) [CITATION].', '0809.5067-1-9-3': 'The jet velocity is 0.99c (except Case B).', '0809.5067-1-9-4': 'Two different magnetic field strengths are used.', '0809.5067-1-9-5': 'Two electrons are injected with two different perpendicular velocities (Cases A - F).', '0809.5067-1-9-6': 'The maximum Lorenz factors, [MATH] are calculated for larger perpendicular velocity.', '0809.5067-1-9-7': 'The critical angles for the off-axis radiation is calculated with [MATH].', '0809.5067-1-10-0': 'Figure 1 shows the summary of the six cases.', '0809.5067-1-10-1': 'Trajectories of the two electrons are shown in the left column (red: larger perpendicular velocity, blue: smaller perpendicular velocity).', '0809.5067-1-10-2': 'The two electrons propagate from left to right with gyration in the [MATH] plane (not shown).', '0809.5067-1-10-3': 'The gyroradius is about [MATH]: the simulation grid length) for the electron with a larger perpendicular velocity (Case A).', '0809.5067-1-10-4': 'The radiation electric field from the two electrons is shown in the middle column.', '0809.5067-1-10-5': 'The spectra were calculated at the point [MATH] shown in he right column.', '0809.5067-1-10-6': 'The seven curves show the spectrum at the viewing angles 0[MATH] (red), 1[MATH] (orange), 2[MATH] (yellow), 3[MATH] (moss green), 4[MATH] (green), 5[MATH] (light blue), and 6[MATH] (blue) ([MATH]).', '0809.5067-1-10-7': 'The higher frequencies become stronger with the increasing viewing angle.', '0809.5067-1-10-8': 'For Case A the power spectrum is scaled as [MATH] as proposed for jitter radiation [CITATION].', '0809.5067-1-10-9': 'For all Cases the spectra are much steeper than the slope [MATH] for the synchrotron radiation.', '0809.5067-1-11-0': 'The second row in Fig. 1 shows Case B with a larger jet velocity [MATH] with the other parameters kept the same as Case A.', '0809.5067-1-11-1': 'The spectra with larger viewing angles are similar to those of Case A.', '0809.5067-1-11-2': 'The spectrum slope is smaller than that in Case A. However, due to the large jet velocity the higher frequencies at larger viewing angles (0[MATH], 1[MATH], 2[MATH]) become stronger.', '0809.5067-1-11-3': 'On the other hand, with the smaller perpendicular velocities (Cases C and D), the gyroradius becomes very small.', '0809.5067-1-11-4': 'The spectra become weaker than those in Case A.', '0809.5067-1-11-5': 'As shown in the third and fourth rows in Fig. 1, the viewing angle dependence becomes very small.', '0809.5067-1-11-6': 'It should be noted that the slope of spectra is very steep for Case C.', '0809.5067-1-12-0': 'Cases D - F have a weaker magnetic fields ([MATH]) than Cases A - C. Case D has a small perpendicular velocity.', '0809.5067-1-12-1': 'The trajectories are almost straight.', '0809.5067-1-12-2': 'The spectra look very similar to that for Bremsstrahlung [CITATION].', '0809.5067-1-12-3': 'The spectra become flat at lower frequencies.', '0809.5067-1-12-4': 'The peak spectral power is the weakest of all the cases.', '0809.5067-1-12-5': 'With larger perpendicular velocities the spectra become stronger than those with the smaller perpendicular velocities.', '0809.5067-1-12-6': 'Case F shows the case with a larger time step (5 times) with the same parameters as Case E.', '0809.5067-1-12-7': 'The spectrum slope is very steep.', '0809.5067-1-12-8': 'The spectra in this case show two differences with those in Case E. First there exist positive slopes in the lower frequency.', '0809.5067-1-12-9': 'Second, due to the gyro-motion the spectra split due to the viewing angles.', '0809.5067-1-12-10': 'In particular the spectrum with larger viewing angle becomes stronger at high frequencies.', '0809.5067-1-13-0': 'As shown in Table 1, the critical angles for the off-axis radiation [MATH] are different.', '0809.5067-1-13-1': 'In this study we have obtained the off-axis radiation for the angles 0[MATH], 1[MATH], 2[MATH], 3[MATH], 4[MATH], 5[MATH], and 6[MATH]).', '0809.5067-1-13-2': 'For cases D, E, and F the variation among different viewing angles is small since the angles are much smaller than the critical angle (25.3[MATH] and 13.35[MATH]).', '0809.5067-1-13-3': 'However, for case B ([MATH]) the radiation shows larger differences for different viewing angles due to the small critical angle.', '0809.5067-1-13-4': 'For Case F (a longer time ([MATH]) the spectra at high frequencies become stronger with larger viewing angles.', '0809.5067-1-14-0': 'These results validate the technique used in our code.', '0809.5067-1-14-1': 'It should be noted that the method based on the integration of the retarded electric fields calculated by tracing many electrons described in the previous section can provide a proper spectrum in turbulent electromagnetic fields.', '0809.5067-1-14-2': 'On the other hand, if the formula for the frequency spectrum of radiation emitted by a relativistic charged particle in instantaneous circular motion is used [CITATION], the complex particle accelerations and trajectories are not properly accounted for and the jitter radiation spectrum is not properly obtained.', '0809.5067-1-15-0': '# Discussion', '0809.5067-1-16-0': 'The procedure used to calculate jitter radiation using the technique described in the previous section has been implemented in our code.', '0809.5067-1-17-0': 'In order to obtain the spectrum of synchrotron (jitter) emission [CITATION], we consider an ensemble of electrons randomly selected in the region where the filamentation (Weibel) instability [CITATION] has fully developed, and electrons are accelerated in the generated magnetic fields.', '0809.5067-1-17-1': 'We calculate emission from about 20,000 electrons during the sampling time, [MATH] with Nyquist frequency [MATH] where [MATH] is the simulation time step and the frequency resolution [MATH].', '0809.5067-1-17-2': 'However, since the emission coordinate frame for each particle is different, we accumulate radiation at fixed angles in simulation system coordinates after transforming from the individual particle emission coordinate frame.', '0809.5067-1-17-3': 'This provides an intensity spectrum as a function of angle relative to the simulation frame [MATH]-axis (this can be any angle by changing the unit vector [MATH] in eq. (1)).', '0809.5067-1-17-4': 'A hypothetical observer in the ambient medium (viewing the external GRB shock) views emission along the system [MATH]-axis.', '0809.5067-1-17-5': 'This computation is carried out in the reference frame of the ambient medium in the numerical simulation.', '0809.5067-1-17-6': 'For an observer located outside the direction of bulk motion of the ambient medium, e.g., internal jet shocks in an ambient medium moving with respect to the observer, an additional Lorentz transformation would be needed along the line of sight to the observer.', '0809.5067-1-17-7': 'Spectra obtained from simulations can be rescaled to physical time scales.', '0809.5067-1-18-0': 'Emission obtained by the method described above is self-consistent, and automatically accounts for magnetic field structures on the small scales responsible for jitter emission.', '0809.5067-1-18-1': 'By performing such calculations for simulations with different parameters, we can then investigate and compare the quite contrasted regimes of jitter- and synchrotron-type emission [CITATION] for prompt and afterglow emission.', '0809.5067-1-18-2': 'The feasibility of this approach has been demonstrated and implemented [CITATION].', '0809.5067-1-18-3': 'Thus, we will be able to address the issue of low frequency GRB spectral index violation of the synchrotron line of death [CITATION].', '0809.5067-1-19-0': 'Simulations incorporating jitter radiation are in progress using an MPI code [CITATION] which speeds up considerably from the previous OpenMP code [CITATION].', '0809.5067-1-19-1': 'New results of jitter radiation will be presented separately.', '0809.5067-1-20-0': 'We have benefited from many useful discussions with A. J. van der Horst.', '0809.5067-1-20-1': 'This work is supported by AST-0506719, AST-0506666, NASA-NNG05GK73G and NNX07AJ88G.', '0809.5067-1-20-2': 'JN was supported by MNiSW research projects 1 P03D 003 29 and N N203 393034, and The Foundation for Polish Science through the HOMING program, which is supported through the EEA Financial Mechanism.Simulations were performed at the Columbia facility at the NASA Advanced Supercomputing (NAS).', '0809.5067-1-20-3': 'Part of this work was done while K.-I. N. was visiting the Niels Bohr Institute.', '0809.5067-1-20-4': 'He thanks the director of the institution for generous hospitality.'}

{'0809.5067-2-0-0': 'Nonthermal radiation observed from astrophysical systems containing relativistic jets and shocks, e.g., gamma-ray bursts (GRBs), active galactic nuclei (AGNs), and microquasars commonly exhibit power-law emission spectra.', '0809.5067-2-0-1': 'Recent PIC simulations of relativistic electron-ion (or electron-positron) jets injected into a stationary medium show that particle acceleration occurs within the downstream jet.', '0809.5067-2-0-2': 'In collisionless, relativistic shocks, particle (electron, positron, and ion) acceleration is due to plasma waves and their associated instabilities (e.g., the Weibel (filamentation) instability) created in the shock region.', '0809.5067-2-0-3': 'The simulations show that the Weibel instability is responsible for generating and amplifying highly non-uniform, small-scale magnetic fields.', '0809.5067-2-0-4': "These fields contribute to the electron's transverse deflection behind the jet head.", '0809.5067-2-0-5': 'The resulting "jitter" radiation from deflected electrons has different properties compared to synchrotron radiation, which assumes a uniform magnetic field.', '0809.5067-2-0-6': 'Jitter radiation may be important for understanding the complex time evolution and/or spectra in gamma-ray bursts, relativistic jets in general, and supernova remnants.', '0809.5067-2-1-0': '# Introduction', '0809.5067-2-2-0': 'Shocks are believed to be responsible for prompt emission from gamma-ray bursts (GRBs) and their afterglows, for variable emission from blazars, and for particle acceleration processes in jets from active galactic nuclei (AGN) and supernova remnants (SNRs).', '0809.5067-2-2-1': 'The predominant contribution to the observed emission spectra is often assumed to be synchrotron- and inverse Compton radiation from these accelerated particles [CITATION].', '0809.5067-2-2-2': 'It is assumed that turbulent magnetic fields in the shock region lead to Fermi acceleration, producing higher energy particles [CITATION].', '0809.5067-2-2-3': 'To make progress in understanding emission from these object classes, it is essential to place modeling efforts on a firm physical basis.', '0809.5067-2-2-4': 'This requires studies of the microphysics of the shock process in a self-consistent manner [CITATION].', '0809.5067-2-3-0': '## New Numerical Method for Calculating Emission', '0809.5067-2-4-0': 'The retarded electric field from a charged particle moving with instantaneous velocity [MATH] under acceleration [MATH] is obtained [CITATION].', '0809.5067-2-4-1': 'After some calculation and simplifying assumptions the total energy [MATH] radiated per unit solid angle per unit frequency can be expressed as', '0809.5067-2-5-0': '[EQUATION]', '0809.5067-2-6-0': "Here, [MATH] is a unit vector that points from the particle's retarded position towards the observer.", '0809.5067-2-6-1': 'The first term on the right hand side, containing the velocity field, is the Coulomb field from a charge moving without influence from external forces in eq. 2.4 [CITATION].', '0809.5067-2-6-2': 'The second term is a correction term that arises when the charge is subject to acceleration.', '0809.5067-2-6-3': 'Since the velocity-dependent field falls off in distance as [MATH], while the acceleration-dependent field scales as [MATH], the latter becomes dominant when observing the charge at large distances ([MATH]).', '0809.5067-2-6-4': 'The choice of unit vector [MATH] along the direction of propagation of the jet (hereafter taken to be the [MATH]-axis) corresponds to head-on emission.', '0809.5067-2-6-5': 'For any other choice of [MATH] (e.g., [MATH]), off-axis emission is seen by the observer.', '0809.5067-2-6-6': "The observer's viewing angle is set by the choice of [MATH]).", '0809.5067-2-7-0': '# Radiation from two electrons', '0809.5067-2-8-0': 'In the previous section we discussed how to obtain the retarded electric field from relativistically moving particles (electrons) observed at large distance.', '0809.5067-2-8-1': 'Using eq. 1 we calculated the time evolution of the retarded electric field and the spectrum from a gyrating electron in a uniform magnetic field to verify the technique used in this calculation.', '0809.5067-2-8-2': 'We have calculated the radiation from two electrons gyrating in the [MATH] plane in the uniform magnetic field [MATH] with Lorentz factors ([MATH]) (Case P in Table 1) [CITATION].', '0809.5067-2-8-3': 'We have very good agreement between the spectrum obtained from the simulation and the theoretical synchrotron spectrum expectation from eq. 7.10 [CITATION].', '0809.5067-2-9-0': 'In order to calculate more realistic radiation from relativistic jets we included a parallel magnetic field ([MATH]).', '0809.5067-2-9-1': 'Relativistic jets are propagating along the [MATH] direction.', '0809.5067-2-9-2': 'Table 1 shows six cases including the previous case P (first row) [CITATION].', '0809.5067-2-9-3': 'The jet velocity is 0.99c (except Case B).', '0809.5067-2-9-4': 'Two different magnetic field strengths are used.', '0809.5067-2-9-5': 'Two electrons are injected with two different perpendicular velocities (Cases A - F).', '0809.5067-2-9-6': 'The maximum Lorenz factors, [MATH] are calculated for larger perpendicular velocity.', '0809.5067-2-9-7': 'The critical angles for the off-axis radiation is calculated with [MATH].', '0809.5067-2-10-0': 'Figure 1 shows the summary of the six cases.', '0809.5067-2-10-1': 'Trajectories of the two electrons are shown in the left column (red: larger perpendicular velocity, blue: smaller perpendicular velocity).', '0809.5067-2-10-2': 'The two electrons propagate from left to right with gyration in the [MATH] plane (not shown).', '0809.5067-2-10-3': 'The gyroradius is about [MATH]: the simulation grid length) for the electron with a larger perpendicular velocity (Case A).', '0809.5067-2-10-4': 'The radiation electric field from the two electrons is shown in the middle column.', '0809.5067-2-10-5': 'The spectra were calculated at the point [MATH] shown in he right column.', '0809.5067-2-10-6': 'The seven curves show the spectrum at the viewing angles 0[MATH] (red), 1[MATH] (orange), 2[MATH] (yellow), 3[MATH] (moss green), 4[MATH] (green), 5[MATH] (light blue), and 6[MATH] (blue) ([MATH]).', '0809.5067-2-10-7': 'The higher frequencies become stronger with the increasing viewing angle.', '0809.5067-2-10-8': 'For Case A the power spectrum is scaled as [MATH] as proposed for jitter radiation [CITATION].', '0809.5067-2-10-9': 'For all Cases the spectra are much steeper than the slope [MATH] for the synchrotron radiation.', '0809.5067-2-11-0': 'The second row in Fig. 1 shows Case B with a larger jet velocity [MATH] with the other parameters kept the same as Case A.', '0809.5067-2-11-1': 'The spectra with larger viewing angles are similar to those of Case A.', '0809.5067-2-11-2': 'The spectrum slope is smaller than that in Case A. However, due to the large jet velocity the higher frequencies at larger viewing angles (0[MATH], 1[MATH], 2[MATH]) become stronger.', '0809.5067-2-11-3': 'On the other hand, with the smaller perpendicular velocities (Cases C and D), the gyroradius becomes very small.', '0809.5067-2-11-4': 'The spectra become weaker than those in Case A.', '0809.5067-2-11-5': 'As shown in the third and fourth rows in Fig. 1, the viewing angle dependence becomes very small.', '0809.5067-2-11-6': 'It should be noted that the slope of spectra is very steep for Case C. Spectral leakage is found.', '0809.5067-2-12-0': 'Cases D - F have a weaker magnetic fields ([MATH]) than Cases A - C. Case D has a small perpendicular velocity.', '0809.5067-2-12-1': 'The trajectories are almost straight.', '0809.5067-2-12-2': 'The spectra look very similar to that for Bremsstrahlung [CITATION].', '0809.5067-2-12-3': 'The spectra become flat at lower frequencies.', '0809.5067-2-12-4': 'The peak spectral power is the weakest of all the cases.', '0809.5067-2-12-5': 'With larger perpendicular velocities the spectra become stronger than those with the smaller perpendicular velocities.', '0809.5067-2-12-6': 'Case F shows the case with a larger time step (5 times) with the same parameters as Case E.', '0809.5067-2-12-7': 'The spectrum slope is very steep.', '0809.5067-2-12-8': 'The spectra in this case show two differences with those in Case E. First there exist positive slopes in the lower frequency.', '0809.5067-2-12-9': 'Second, due to the gyro-motion the spectra split due to the viewing angles.', '0809.5067-2-12-10': 'In particular the spectrum with larger viewing angle becomes stronger at high frequencies.', '0809.5067-2-13-0': 'As shown in Table 1, the critical angles for the off-axis radiation [MATH] are different.', '0809.5067-2-13-1': 'In this study we have obtained the off-axis radiation for the angles 0[MATH], 1[MATH], 2[MATH], 3[MATH], 4[MATH], 5[MATH], and 6[MATH]).', '0809.5067-2-13-2': 'For cases D, E, and F the variation among different viewing angles is small since the angles are much smaller than the critical angle (25.3[MATH] and 13.35[MATH]).', '0809.5067-2-13-3': 'However, for case B ([MATH]) the radiation shows larger differences for different viewing angles due to the small critical angle.', '0809.5067-2-13-4': 'For Case F (a longer time ([MATH]) the spectra at high frequencies become stronger with larger viewing angles.', '0809.5067-2-14-0': 'These results validate the technique used in our code.', '0809.5067-2-14-1': 'It should be noted that the method based on the integration of the retarded electric fields calculated by tracing many electrons described in the previous section can provide a proper spectrum in turbulent electromagnetic fields.', '0809.5067-2-14-2': 'On the other hand, if the formula for the frequency spectrum of radiation emitted by a relativistic charged particle in instantaneous circular motion is used [CITATION], the complex particle accelerations and trajectories are not properly accounted for and the jitter radiation spectrum is not properly obtained.', '0809.5067-2-15-0': '# Discussion', '0809.5067-2-16-0': 'The procedure used to calculate jitter radiation using the technique described in the previous section has been implemented in our code.', '0809.5067-2-17-0': 'In order to obtain the spectrum of synchrotron (jitter) emission [CITATION], we consider an ensemble of electrons randomly selected in the region where the filamentation (Weibel) instability [CITATION] has fully developed, and electrons are accelerated in the generated magnetic fields.', '0809.5067-2-17-1': 'We calculate emission from about 20,000 electrons during the sampling time, [MATH] with Nyquist frequency [MATH] where [MATH] is the simulation time step and the frequency resolution [MATH].', '0809.5067-2-17-2': 'However, since the emission coordinate frame for each particle is different, we accumulate radiation at fixed angles in simulation system coordinates after transforming from the individual particle emission coordinate frame.', '0809.5067-2-17-3': 'This provides an intensity spectrum as a function of angle relative to the simulation frame [MATH]-axis (this can be any angle by changing the unit vector [MATH] in eq. (1)).', '0809.5067-2-17-4': 'A hypothetical observer in the ambient medium (viewing the external GRB shock) views emission along the system [MATH]-axis.', '0809.5067-2-17-5': 'This computation is carried out in the reference frame of the ambient medium in the numerical simulation.', '0809.5067-2-17-6': 'For an observer located outside the direction of bulk motion of the ambient medium, e.g., internal jet shocks in an ambient medium moving with respect to the observer, an additional Lorentz transformation would be needed along the line of sight to the observer.', '0809.5067-2-17-7': 'Spectra obtained from simulations can be rescaled to physical time scales.', '0809.5067-2-18-0': 'Emission obtained by the method described above is self-consistent, and automatically accounts for magnetic field structures on the small scales responsible for jitter emission.', '0809.5067-2-18-1': 'By performing such calculations for simulations with different parameters, we can then investigate and compare the quite contrasted regimes of jitter- and synchrotron-type emission [CITATION] for prompt and afterglow emission.', '0809.5067-2-18-2': 'The feasibility of this approach has been demonstrated and implemented [CITATION].', '0809.5067-2-18-3': 'Thus, we will be able to address the issue of low frequency GRB spectral index violation of the synchrotron line of death [CITATION].', '0809.5067-2-19-0': 'Simulations incorporating jitter radiation are in progress using an MPI code [CITATION] which speeds up considerably from the previous OpenMP code [CITATION].', '0809.5067-2-19-1': 'New results of jitter radiation will be presented separately.', '0809.5067-2-20-0': 'We have benefited from many useful discussions with A. J. van der Horst.', '0809.5067-2-20-1': 'This work is supported by AST-0506719, AST-0506666, NASA-NNG05GK73G and NNX07AJ88G.', '0809.5067-2-20-2': 'JN was supported by MNiSW research projects 1 P03D 003 29 and N N203 393034, and The Foundation for Polish Science through the HOMING program, which is supported through the EEA Financial Mechanism.Simulations were performed at the Columbia facility at the NASA Advanced Supercomputing (NAS).', '0809.5067-2-20-3': 'Part of this work was done while K.-I. N. was visiting the Niels Bohr Institute.', '0809.5067-2-20-4': 'He thanks the director of the institution for generous hospitality.'}

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[['0809.5067-1-11-6', '0809.5067-2-11-6']]

[]

[]

[]

['0809.5067-1-5-0', '0809.5067-1-20-1', '0809.5067-1-20-3', '0809.5067-2-5-0', '0809.5067-2-20-1', '0809.5067-2-20-3']

{'1': 'http://creativecommons.org/licenses/publicdomain/', '2': 'http://creativecommons.org/licenses/publicdomain/'}

https://arxiv.org/abs/0809.5067

cond-mat-0603391

{'cond-mat-0603391-1-0-0': 'We describe in detail how to implement a coarse-grained hybrid Molecular Dynamics and Stochastic Rotation Dynamics simulation technique that captures the combined effects of Brownian and hydrodynamic forces in colloidal suspensions.', 'cond-mat-0603391-1-0-1': 'The importance of carefully tuning the simulation parameters to correctly resolve the multiple time and length-scales of this problem is emphasized.', 'cond-mat-0603391-1-0-2': 'We systematically analyze how our coarse-graining scheme resolves dimensionless hydrodynamic numbers such as the Reynolds number Re, which indicates the importance of inertial effects, the Schmidt number Sc, which indicates whether momentum transport is liquid-like or gas-like, the Mach number Ma, which measures compressibility effects, the Knudsen number Kn, which describes the importance of non-continuum molecular effects and the Peclet number Pe, which describes the relative effects of convective and diffusive transport.', 'cond-mat-0603391-1-0-3': 'With these dimensionless numbers in the correct regime the many Brownian and hydrodynamic time-scales can be telescoped together to maximize computational efficiency while still correctly resolving the physically relevant physical processes.', 'cond-mat-0603391-1-0-4': 'We also show how to control a number of numerical artifacts, such as finite size effects and solvent induced attractive depletion interactions.', 'cond-mat-0603391-1-0-5': 'When all these considerations are properly taken into account, the measured colloidal velocity auto-correlation functions and related self diffusion and friction coefficients compare quantitatively with theoretical calculations.', 'cond-mat-0603391-1-0-6': 'By contrast, these calculations demonstrate that, notwithstanding its seductive simplicity, the basic Langevin equation does a remarkably poor job of capturing the decay rate of the velocity auto-correlation function in the colloidal regime, strongly underestimating it at short times and strongly overestimating it at long times.', 'cond-mat-0603391-1-0-7': 'Finally, we discuss in detail how to map the parameters of our method onto physical systems, and from this extract more general lessons that may be relevant for other coarse-graining schemes such as Lattice Boltzmann or Dissipative Particle Dynamics.', 'cond-mat-0603391-1-1-0': '# Introduction', 'cond-mat-0603391-1-2-0': 'Calculating the non-equilibrium properties of colloidal suspensions is a highly non-trivial exercise because these depend both on the short-time thermal Brownian motion, and the long-time hydrodynamic behavior of the solvent [CITATION].', 'cond-mat-0603391-1-2-1': 'Moreover, the hydrodynamic interactions are of a many-body character, and cannot usually be decomposed into a pairwise sum of inter-colloid forces.', 'cond-mat-0603391-1-3-0': 'The fundamental difficulty of fully including the detailed solvent dynamics in computer simulations becomes apparent when considering the enormous time and length-scale differences between mesoscopic colloidal and microscopic solvent particles.', 'cond-mat-0603391-1-3-1': 'For example, a typical colloid of diameter [MATH] will displace on the order of [MATH] water molecules!', 'cond-mat-0603391-1-3-2': 'Furthermore, a molecular dynamics (MD) scheme for the solvent would need to resolve time-scales on the order of [MATH] s to describe the inter-molecular forces, while a colloid of diameter [MATH] in water diffuses over its own diameter in about [MATH]s.', 'cond-mat-0603391-1-4-0': 'Clearly, simulating even an extremely crude molecular model for the solvent on the time-scales of interest is completely out of the question: some form of coarse-graining is necessary.', 'cond-mat-0603391-1-4-1': 'The object of this paper is to describe in detail one such scheme.', 'cond-mat-0603391-1-4-2': 'But before we do so, we first briefly discuss a subset of the wide variety of different simulation techniques that have been devised to describe the dynamics of colloidal suspensions.', 'cond-mat-0603391-1-5-0': 'At the simplest level, the effects of the solvent can be taken into account through Brownian dynamics (BD) [CITATION], which assumes that collisions with the solvent molecules induce a random displacement of the colloidal particle positions, as well as a local friction proportional to the their velocity.', 'cond-mat-0603391-1-5-1': 'Although, due to its simplicity, Brownian dynamics is understandably very popular, it completely neglects momentum transport through the solvent - as described by the Navier Stokes equations - which leads to long-ranged hydrodynamic interactions (HI) between the suspended particles.', 'cond-mat-0603391-1-5-2': 'These HI may fall off as slowly as [MATH], and can qualitatively affect the dynamical behavior of the suspension [CITATION].', 'cond-mat-0603391-1-6-0': 'Beginning with the pioneering work of Ermak and McCammon [CITATION], who added a simple representation of the Oseen tensor [CITATION] to their implementation of BD, many authors have applied computational approaches that include the HI by an approximate analytical form.', 'cond-mat-0603391-1-6-1': 'The most successful of these methods is Stokesian Dynamics [CITATION], which can take into account higher order terms in a multipole expansion of the HI interactions.', 'cond-mat-0603391-1-6-2': 'Although some more sophisticated recent implementations of Stokesian Dynamics have achieved an [MATH] scaling with the number of colloids [CITATION], this method is still relatively slow, and becomes difficult to implement in complex boundary conditions.', 'cond-mat-0603391-1-7-0': 'Another way to solve for the HI is by direct numerical simulation (DNS) methods, where the solid particles are described by explicit boundary conditions for the Navier-Stokes equations [CITATION].', 'cond-mat-0603391-1-7-1': 'These methods are better adapted to non-Brownian particles, but can still be applied to understand the effects of HI on colloidal dynamics.', 'cond-mat-0603391-1-7-2': 'The fluid particle dynamics (FPD) method of Tanaka and Araki [CITATION], and a related method by Yamamoto et al. [CITATION], are two important recent approaches to solving the Navier Stokes equations that go a step beyond standard DNS, and simplify the problem of boundary conditions by using a smoothed step function at the colloid surfaces.', 'cond-mat-0603391-1-7-3': 'In principle Brownian motion can be added to these methods [CITATION].', 'cond-mat-0603391-1-8-0': 'The difficulty of including complex boundary conditions in DNS approaches has stimulated the use of lattice-gas based techniques to resolve the Navier Stokes equations.', 'cond-mat-0603391-1-8-1': 'These methods exploit the fact that only a few conditions, such as (local) energy and momentum conservation, need to be satisfied to allow the correct (thermo) hydrodynamics to emerge in the continuum limit.', 'cond-mat-0603391-1-8-2': 'Greatly simplified particle collision rules, easily amenable to efficient computer simulation, are therefore possible, and complex boundary conditions, such as those that characterize moving colloids, are easier to treat than in DNS methods.', 'cond-mat-0603391-1-8-3': 'The most popular of these techniques is Lattice Boltzmann (LB) where a linearized and pre-averaged Boltzmann equation is discretized and solved on a lattice [CITATION].', 'cond-mat-0603391-1-8-4': 'Ladd has pioneered the application of this method to solid-fluid suspensions [CITATION].', 'cond-mat-0603391-1-8-5': 'This is illustrated schematically in Fig. [REF].', 'cond-mat-0603391-1-8-6': 'The solid particles are modeled as hollow spheres, propagated with Newtonian dynamics, and coupled to the LB solvent through a bounce-back collision rule on the surface.', 'cond-mat-0603391-1-8-7': 'The fluctuations that lead to Brownian motion were modeled by adding a stochastic term to the stress tensor.', 'cond-mat-0603391-1-8-8': 'Recently this has been criticized and an improved method to include Brownian noise that does not suffer from some lattice induced artifacts was suggested [CITATION].', 'cond-mat-0603391-1-8-9': 'A number of other groups have recently derived alternative ways of coupling a colloidal particle to a LB fluid [CITATION], in part to simulate the dynamics of charged systems.', 'cond-mat-0603391-1-9-0': 'The discrete nature of lattice based methods can also bring disadvantages, particularly when treating fluids with more than one length-scale.', 'cond-mat-0603391-1-9-1': 'Dissipative Particle Dynamics (DPD) [CITATION] is a popular off-lattice alternative that includes hydrodynamics and Brownian fluctuations.', 'cond-mat-0603391-1-9-2': 'It can be viewed as an extension of standard Newtonian MD techniques, but with two important innovations: 1) soft potentials that allow large time-steps and rapid equilibration; 2) a Galilean invariant thermostat that locally conserves momentum and therefore generates the correct Navier Stokes hydrodynamics in the continuum limit.', 'cond-mat-0603391-1-10-0': 'In fact, these two methodological advances can be separated.', 'cond-mat-0603391-1-10-1': 'Soft potentials such as those postulated for innovation 1) may indeed arise from careful equilibrium coarse-graining of complex fluids [CITATION].', 'cond-mat-0603391-1-10-2': 'However, their proper statistical mechanical interpretation, even for equilibrium properties, is quite subtle.', 'cond-mat-0603391-1-10-3': 'For example, a potential that generates the correct pair structure will not normally reproduce the correct virial pressure due to so called representability problems [CITATION].', 'cond-mat-0603391-1-10-4': 'When used in dynamical simulations, the correct application of effective potentials is even more difficult to properly derive.', 'cond-mat-0603391-1-10-5': 'For polymer dynamics, for instance, uncrossability constraints must be re-introduced to prevent coarse-grained polymers from passing through one another [CITATION].', 'cond-mat-0603391-1-10-6': 'Depletion interactions in a multi component solution also depend on the relative rates of depletant diffusion coefficients and particle flow velocities [CITATION].', 'cond-mat-0603391-1-10-7': 'At present, therefore, the statistical mechanical origins of innovation 1) of DPD, the use of soft potentials out of equilibrium, are at best obscure.', 'cond-mat-0603391-1-10-8': 'We are convinced that a correct microscopic derivation of the coarse-grained DPD representation of the dynamics, if this can indeed be done, will show that the interpretation of such soft potentials depends on dynamic as well as static (phase-space) averages.', 'cond-mat-0603391-1-10-9': 'Viewing the DPD particles as static "clumps" of underlying fluid is almost certainly incorrect.', 'cond-mat-0603391-1-10-10': 'It may, in fact, be more fruitful to abandon simple analogies to the potential energy of a Hamiltonian system, and instead view the interactions as a kind of coarse-grained self-energy [CITATION].', 'cond-mat-0603391-1-11-0': 'Innovation 2), on the other hand, can be put on firmer statistical mechanical footing, see e.g. [CITATION], and can be usefully employed to study the dynamics of complex systems with other types of inter-particle interactions [CITATION].', 'cond-mat-0603391-1-11-1': 'The main advantage of the DPD thermostat is that, by preserving momentum conservation, the hydrodynamic interactions that intrinsically arise from microcanonical MD are preserved for calculations in the canonical ensemble.', 'cond-mat-0603391-1-11-2': 'Other thermostats typically screen the hydrodynamic interactions beyond a certain length-scale [CITATION].', 'cond-mat-0603391-1-11-3': 'For weak damping this may not be a problem, but for strong damping it could be.', 'cond-mat-0603391-1-12-0': 'Simulating only the colloids with DPD ignores the dominant solvent hydrodynamics.', 'cond-mat-0603391-1-12-1': 'While the solvent could be treated directly by DPD, as suggested in Fig. [REF], this method is quite computationally expensive because the "solvent" particles interact through pairwise potentials.', 'cond-mat-0603391-1-13-0': 'In an important paper [CITATION], Lowe took these ideas further and combined the local momentum conservation of the DPD thermostat with the stochastic nature of the Anderson thermostat to derive a coarse-grained scheme, now called the Lowe-Anderson thermostat, which is much more efficient than treating the solvent with full DPD.', 'cond-mat-0603391-1-13-1': 'It has recently been applied, for example, to polymer dynamics [CITATION].', 'cond-mat-0603391-1-14-0': 'Independently, Malevanets and Kapral [CITATION] derived a method now called stochastic rotation dynamics (SRD) or also multiple particle collision dynamics (we choose the former nomenclature here).', 'cond-mat-0603391-1-14-1': 'In many ways it resembles the Lowe-Anderson thermostat [CITATION], or the much older direct simulation Monte Carlo (DSMC) method of Bird [CITATION].', 'cond-mat-0603391-1-14-2': "For all three of these methods, the particles are ideal, and move in a continuous space, subject to Newton's laws of motion.", 'cond-mat-0603391-1-14-3': 'At discrete time-steps, a coarse-grained collision step allows particles to exchange momentum.', 'cond-mat-0603391-1-14-4': 'In SRD, space is partitioned into a rectangular grid, and at discrete time-steps the particles inside each cell exchange momentum by rotating their velocity vectors relative to the center of mass velocity of the cell (see Fig. [REF]).', 'cond-mat-0603391-1-14-5': "Similarly, in Lowe's method, a particle can, with a certain probability, exchange its relative velocity with another that lies within a certain radius.", 'cond-mat-0603391-1-14-6': 'One can imagine other collision rules as well.', 'cond-mat-0603391-1-14-7': 'As long as they locally conserve momentum and energy, just as the lattice gas methods do, they generate the correct Navier Stokes hydrodynamics, although for SRD it is necessary to include a grid-shift procedure to enforce Galilean invariance, something first pointed out by Ihle and Kroll [CITATION].', 'cond-mat-0603391-1-14-8': 'An important advantage of SRD is that its simplified dynamics have allowed the analytic calculation of several transport coefficients [CITATION], greatly facilitating its use.', 'cond-mat-0603391-1-14-9': 'In the rest of this paper we will concentrate on the SRD method, although many of our general conclusions and derivations should be easily extendible to the Lowe-Anderson thermostat and related methods summarized in Fig. [REF].', 'cond-mat-0603391-1-15-0': 'SRD can be applied to directly simulate flow, as done for example in refs [CITATION], but its stochastic nature means that a noise average must be performed to calculate flow lines, and this may make it less efficient than pre-averaged methods like LB.', 'cond-mat-0603391-1-15-1': 'Where SRD becomes more attractive is for the simulation of complex particles embedded in a solvent.', 'cond-mat-0603391-1-15-2': 'This is because in addition to long-ranged HI, it also naturally contains Brownian fluctuations, and both typically need to be resolved for a proper statistical mechanical treatment of the dynamics of mesoscopic suspended particles.', 'cond-mat-0603391-1-15-3': 'For example, SRD has been used to study polymers under flow [CITATION], the effect of hydrodynamics on protein folding and polymer collapse [CITATION], and the conformations of vesicles under flow [CITATION].', 'cond-mat-0603391-1-15-4': 'In each of the previously mentioned examples, the suspended particles are coupled to the solvent by participating in the collision step.', 'cond-mat-0603391-1-16-0': "A less coarse-grained coupling can be achieved by allowing direct collisions, obeying Newton's laws of motion, between the SRD solvent and the suspended particles.", 'cond-mat-0603391-1-16-1': 'Such an approach is important for systems, such as colloidal suspensions, where the solvent and colloid time and length-scales need to be clearly separated.', 'cond-mat-0603391-1-16-2': 'Malevanets and Kapral [CITATION] derived such a hybrid algorithm that combined a full MD scheme of the solute-solute and solute-solvent interactions, while treating the solvent-solvent interactions via SRD.', 'cond-mat-0603391-1-16-3': 'Early applications were to a two-dimensional many-particle system [CITATION], and to the aggregation of colloidal particles [CITATION].', 'cond-mat-0603391-1-17-0': 'We have recently extended this approach, and applied it to the sedimentation of up to 800 hard sphere (HS) like colloids as a function of volume fraction, for a number of different values of the Peclet number [CITATION].', 'cond-mat-0603391-1-17-1': 'To achieve these simulations we adapted the method of Malevanets and Kapral [CITATION] in a number of ways that were only briefly described in ref. [CITATION].', 'cond-mat-0603391-1-17-2': 'In the current paper we provide more in depth analysis of the simulation method we used, and describe some potential pitfalls.', 'cond-mat-0603391-1-17-3': 'In particular we focus on the different time and length-scales that arise from our coarse-graining approach, as well as the role of dimensionless hydrodynamic numbers that express the relative importance of competing physical phenomena.', 'cond-mat-0603391-1-17-4': 'Very recently, another similar study of colloidal sedimentation and aggregation has been carried out [CITATION].', 'cond-mat-0603391-1-17-5': 'Some of our results and analysis are similar, but some are different, and we will mention these where appropriate.', 'cond-mat-0603391-1-18-0': 'After the introductory overview above, we begin in section [REF] by carefully describing the properties of a pure SRD fluid, focusing on simple derivations that highlight the dominant physics involved for each transport coefficient.', 'cond-mat-0603391-1-18-1': 'Section [REF] explains how to implement the coupling of a colloidal particle to an SRD solvent bath, and shows how to avoid spurious depletion interactions and how to understand lubrication forces.', 'cond-mat-0603391-1-18-2': 'In Section [REF] we analyze how optimizing the efficiency of particle based coarse-graining schemes affects different dimensionless hydrodynamic numbers, such as the Schmidt number, the Mach number, the Reynolds number, the Knudsen number, and the Peclet number.', 'cond-mat-0603391-1-18-3': 'Section [REF] describes the hierarchy of time-scales that determine the physics of a colloidal suspension, and compares these to the compressed hierarchy in our coarse-grained method.', 'cond-mat-0603391-1-18-4': 'In section [REF] we tackle the question of how to map from a coarse-grained simulation, that has been optimized for computational efficiency, to a real colloidal system.', 'cond-mat-0603391-1-18-5': 'Finally after a Conclusions section, we include a few appendices that discuss amongst other things how to thermostat the SRD system (Appendix I); why the popular Langevin equation without memory effects does a remarkably poor job of capturing both the long and the short time dynamics of the colloidal velocity auto-correlation function (Appendix II); and, finally, how various physical properties and dimensionless numbers scale for an SRD simulation, and how these compare to a colloid of radius 10 nm or 1 [MATH]m in H[MATH]0 (Appendix III).', 'cond-mat-0603391-1-19-0': '# Properties of a pure SRD solvent', 'cond-mat-0603391-1-20-0': 'In SRD, the solvent is represented by a large number [MATH] of particles of mass [MATH].', 'cond-mat-0603391-1-20-1': 'Here and in the following, we will call these "fluid" particles, with the caveat that, however tempting, they should not be viewed as some kind of composite particles or clusters made up of the underlying (molecular) fluid.', 'cond-mat-0603391-1-20-2': 'Instead, SRD should be interpreted as a Navier Stokes solver that includes thermal noise.', 'cond-mat-0603391-1-20-3': 'The particles are merely a convenient computational device to facilitate the coarse-graining of the fluid properties.', 'cond-mat-0603391-1-21-0': '## Simulation method for a pure solvent', 'cond-mat-0603391-1-22-0': "In the first (propagation) step of the algorithm, the positions and velocities of the fluid particles are propagated for a time [MATH] (the time between collision steps) by accurately integrating Newton's equations of motion, [EQUATION] [MATH] and [MATH] are the position and velocity of fluid particle [MATH], respectively while [MATH] is the total (external) force on particle [MATH], which may come from an external field such as gravity, or fixed boundary conditions such as hard walls, or moving boundary conditions such as suspended colloids.", 'cond-mat-0603391-1-22-1': 'The direct forces between pairs of fluid particles are, however, neglected in the propagation step.', 'cond-mat-0603391-1-22-2': 'Herein lies the main advantage - the origin of the efficiency - of SRD.', 'cond-mat-0603391-1-22-3': 'Instead of directly treating the interactions between the fluid particles, a coarse-grained collision step is performed at each time-step [MATH]: First, space is partitioned into cubic cells of volume [MATH].', 'cond-mat-0603391-1-22-4': 'Next, for each cell, the particle velocities relative to the center of mass velocity [MATH] of the cell are rotated: [EQUATION] [MATH] is a rotation matrix which rotates velocities by a fixed angle [MATH] around a randomly oriented axis.', 'cond-mat-0603391-1-22-5': 'The aim of the collision step is to transfer momentum between the fluid particles while conserving the total momentum and energy of each cell.', 'cond-mat-0603391-1-22-6': 'Both the collision and the streaming step conserve phase-space volume, and it has been shown that the single particle velocity distribution evolves to a Maxwell-Boltzmann distribution [CITATION].', 'cond-mat-0603391-1-23-0': 'The rotation procedure can thus be viewed as a coarse-graining of particle collisions over space and time.', 'cond-mat-0603391-1-23-1': 'Because mass, momentum, and energy are conserved locally, the correct hydrodynamic (Navier Stokes) equations are captured in the continuum limit, including the effect of thermal noise [CITATION].', 'cond-mat-0603391-1-24-0': 'At low temperatures or small collision times [MATH], the transport coefficients of SRD, as originally formulated [CITATION], show anomalies caused by the fact that fluid particles in a given cell can remain in that cell and participate in several collision steps [CITATION].', 'cond-mat-0603391-1-24-1': 'Under these circumstances the assumption of molecular chaos and Galilean invariance are incorrect.', 'cond-mat-0603391-1-24-2': 'However, this anomaly can be cured by applying a random shift of the cell coordinates before the collision step [CITATION].', 'cond-mat-0603391-1-25-0': 'It should also be noted that the collision step in SRD does not locally conserve angular momentum.', 'cond-mat-0603391-1-25-1': 'As a consequence, the stress tensor [MATH] is not, in general, a symmetric function of the derivatives of the flow field (although it is still rotationally symmetric) [CITATION].', 'cond-mat-0603391-1-25-2': 'The asymmetric part can be interpreted as a viscous stress associated with the vorticity [MATH] of the velocity field [MATH].', 'cond-mat-0603391-1-25-3': 'The stress tensor will therefore depend on the amount of vorticity in the flow field.', 'cond-mat-0603391-1-25-4': 'However, the total force on a fluid element is determined not by the stress tensor itself, but by its divergence [MATH], which is what enters the Navier Stokes equations.', 'cond-mat-0603391-1-25-5': 'Taking the divergence causes the explicit vorticity dependence to drop out (the gradient of a curl is zero).', 'cond-mat-0603391-1-25-6': 'In principle, the stress tensor could be made symmetric by applying random rotations as well as random translations to the cell.', 'cond-mat-0603391-1-25-7': 'Or, alternatively, angular momentum can be explicitly conserved by dynamically adapting the collision rule, as done by Ryder et al [CITATION], who found no significant differences in fluid properties (although there is, of course, less flexibility in choosing simulation parameters).', 'cond-mat-0603391-1-25-8': 'This result has been confirmed by some recent theoretical calculations [CITATION], that demonstrate that the asymmetry of the stress tensor has only a few consequences, such as a correction to the sound wave attenuation associated with viscous dissipation of longitudinal density waves.', 'cond-mat-0603391-1-25-9': 'These are not important for the fluid properties we are trying to model, and so, on balance, we chose not to implement possible fixes to improve on angular momentum conservation.', 'cond-mat-0603391-1-26-0': '## Transport coefficients and the dimensionless mean-free path', 'cond-mat-0603391-1-27-0': 'The simplicity of SRD collisions has facilitated the analytical calculation of many transport coefficients [CITATION].', 'cond-mat-0603391-1-27-1': 'These analytical expressions are particularly useful because they enable us to efficiently tune the viscosity and other properties of the fluid, without the need for trial and error simulations.', 'cond-mat-0603391-1-27-2': 'In this section we will summarize a number of these transport coefficients, where possible giving a simple derivation of the dominant physics.', 'cond-mat-0603391-1-28-0': '### Units and the dimensionless mean-free path', 'cond-mat-0603391-1-29-0': 'In this paper we will use the following units: lengths will be in units of cell-size [MATH], energies in units of [MATH] and masses in units of [MATH] (This corresponds to setting [MATH], [MATH] and [MATH]).', 'cond-mat-0603391-1-29-1': 'Time, for example, is expressed in units of [MATH], the number density [MATH] and other derived units can be found in table [REF].', 'cond-mat-0603391-1-29-2': 'We find it instructive to express the transport coefficients and other parameters of the SRD fluid in terms of the dimensionless mean-free path [EQUATION] which provides a measure of the average fraction of a cell size that a fluid particle travels between collisions.', 'cond-mat-0603391-1-30-0': 'This particular choice of units helps highlight the basic physics of the coarse-graining method.', 'cond-mat-0603391-1-30-1': 'The (nontrivial) question of how to map them on to the units of real physical system will be discussed in section [REF].', 'cond-mat-0603391-1-31-0': '### Fluid self-diffusion constant', 'cond-mat-0603391-1-32-0': 'A simple back of the envelope estimate of the self-diffusion constant [MATH] of a fluid particle can be obtained from a random-walk picture.', 'cond-mat-0603391-1-32-1': 'In a unit of time [MATH], a particle will experience [MATH] collisions, in between which it moves an average distance [MATH].', 'cond-mat-0603391-1-32-2': 'Similarly, a heavier (tagged) particle of mass [MATH], which exchanges momentum with the fluid by participating in the coarse-grained collision step, will move an average distance [MATH] between collisions.', 'cond-mat-0603391-1-32-3': 'By viewing this motion as a random walk of step-size [MATH], the diffusion coefficient follows: [EQUATION] expressed in units of [MATH].', 'cond-mat-0603391-1-32-4': 'The diffusion coefficient [MATH] for a pure fluid particle of mass [MATH] is therefore given by [MATH].', 'cond-mat-0603391-1-33-0': 'A more systematic derivation of the diffusion coefficient of a fluid particle, but still within a random collision approximation, results in the following expression [CITATION]: [EQUATION]', 'cond-mat-0603391-1-33-1': 'The dependence on [MATH] is weak.', 'cond-mat-0603391-1-33-2': 'If, for example, we take [MATH], the value used in this paper, then [MATH], the same as Eq. ([REF]).', 'cond-mat-0603391-1-34-0': 'A similar expression can be derived for the self-diffusion coefficient of a heavier tagged particle of mass [MATH] [CITATION]: [EQUATION]', 'cond-mat-0603391-1-34-1': 'Note that this equation does not quite reduce to Eq. ([REF]) for [MATH] (because of slightly different approximations), but the relative discrepancy decreases with increasing [MATH].', 'cond-mat-0603391-1-35-0': 'While Eq. ([REF]) is accurate for larger mean-free paths, where the random collision approximation is expected to be valid, it begins to show deviations from simulations for [MATH] [CITATION], when longer-time kinetic correlations begin to develop.', 'cond-mat-0603391-1-35-1': 'Ripoll et al [CITATION] argue that these correlations induce interactions of a hydrodynamic nature that enhance the diffusion coefficient for a fluid particle.', 'cond-mat-0603391-1-35-2': 'For example, for [MATH] and [MATH], they measured a fluid self-diffusion constant [MATH] that is about [MATH] larger than the value found from Eq. ([REF]).', 'cond-mat-0603391-1-35-3': 'The enhancement is even more pronounced for heavier particles: for the same simulation parameters they found that [MATH] was enhanced by about [MATH] over the prediction of Eq. ([REF]) when [MATH].', 'cond-mat-0603391-1-36-0': 'We note that coupling a large particle to the solvent through participation in the coarse-grained collision step leads to a diffusion coefficient which scales with mass as [MATH], whereas if one couples a colloid of radius [MATH] to the solvent through direct MD collisions, one expects [MATH].', 'cond-mat-0603391-1-36-1': 'Moreover, it has been shown [CITATION] that the effective hydrodynamic particle radius is approximately given by [MATH].', 'cond-mat-0603391-1-36-2': 'For any reasonable average number of fluid particles per cell, the effective hydrodynamic radius [MATH] of the heavier particle is therefore much less than the collision cell size [MATH].', 'cond-mat-0603391-1-36-3': 'On the other hand, the hydrodynamic field is accurately resolved down to a scale comparable to [MATH] (vide infra).', 'cond-mat-0603391-1-36-4': 'What this implies is that this coupling method will yield correct hydrodynamic interactions only at large distances, when the colloids are more than several hydrodynamic radii apart.', 'cond-mat-0603391-1-36-5': 'All the above suggests that some care must be taken when interpreting the dynamics of heavier particles that couple through the coarse-grained collision step, especially when two or more heavy particles are in close proximity.', 'cond-mat-0603391-1-37-0': '### Kinematic viscosity', 'cond-mat-0603391-1-38-0': 'The spread of a velocity fluctuation [MATH] in a fluid can be described by a diffusion equation [CITATION]: [EQUATION] where [MATH] is the kinematic viscosity, which determines the rate at which momentum or vorticity "diffuses away".', 'cond-mat-0603391-1-38-1': 'The units of kinematic viscosity are [MATH] which are the same as those for particle self diffusion i.e. [MATH].', 'cond-mat-0603391-1-39-0': 'Momentum is transported through two mechanisms:', 'cond-mat-0603391-1-40-0': '1) By particles streaming between collision steps, leading to a "kinetic" contribution to the kinematic viscosity [MATH].', 'cond-mat-0603391-1-40-1': 'Since for this gas-like contribution the momentum is transported by particle motion, we expect [MATH] to scale like the particle self-diffusion coefficient [MATH], i.e. [MATH].', 'cond-mat-0603391-1-41-0': '2) By momentum being re-distributed among the particles of each cell during the collision step, resulting in a "collisional" contribution to the kinematic viscosity [MATH].', 'cond-mat-0603391-1-41-1': 'This mimics the way momentum is transferred due to inter-particle collisions, and would be the dominant contribution in a dense fluid such as water at standard temperature and pressure.', 'cond-mat-0603391-1-41-2': 'Again a simple random-walk argument explains the expected scaling in SRD: Each collision step distributes momentum among particles in a cell, making a step-size that scales like [MATH].', 'cond-mat-0603391-1-41-3': 'Since there are [MATH] collision steps per unit time [MATH], this suggests that the collisional contribution to the kinematic viscosity should scale as [MATH].', 'cond-mat-0603391-1-42-0': 'Accurate analytical expressions for the kinematic viscosity [MATH] of SRD have been derived [CITATION], and these can be rewritten in the following dimensionless form: [EQUATION] where the dependence on the collisional angle [MATH] and fluid number density [MATH] is subsumed in the following two factors: [EQUATION]', 'cond-mat-0603391-1-42-1': 'These factors only depend weakly on [MATH] for the typical parameters used in simulations.', 'cond-mat-0603391-1-42-2': 'For example, at [MATH] and [MATH], the angle and number density we use in this paper, they take the values [MATH] and [MATH].', 'cond-mat-0603391-1-42-3': 'For this choice of collision angle [MATH], they monotonically converge to [MATH] in the limit of large [MATH].', 'cond-mat-0603391-1-43-0': '### Shear viscosity', 'cond-mat-0603391-1-44-0': 'Suppose the fluid is sheared in the x direction, with the gradient of the average flow field in the y direction.', 'cond-mat-0603391-1-44-1': 'Two neighboring fluid elements with different y-coordinates will then experience a friction force in the x-direction, as expressed by the xy component of the stress tensor [MATH], which for a simple liquid is linearly proportional to the instantaneous flow field gradient, [EQUATION]', 'cond-mat-0603391-1-44-2': 'The coefficient of proportionality [MATH] is called the shear viscosity, and is it related to the kinematic viscosity by [MATH], where [MATH] is the fluid mass density.', 'cond-mat-0603391-1-44-3': 'From Eqs. [REF]-[REF] it follows that the two contributions to the shear viscosity can be written in dimensionless form as: [EQUATION] where [MATH] is the unit of shear viscosity.', 'cond-mat-0603391-1-45-0': 'In contrast to the expressions for the diffusion of a fluid particle or a tagged particle, Eqs. ([REF]) - ([REF]) compare quantitatively to simulations over a wide range of parameters [CITATION].', 'cond-mat-0603391-1-45-1': 'For the parameters we used in our simulations in [CITATION], i.e. [MATH], the collisional contribution to the viscosity dominates: [MATH] and [MATH].', 'cond-mat-0603391-1-45-2': 'This is typical for [MATH], where [MATH] and [MATH] can be taken to a good first approximation by the collisional contribution only.', 'cond-mat-0603391-1-45-3': 'In fact throughout this paper we will mainly focus on this small [MATH] limit.', 'cond-mat-0603391-1-46-0': "It is also instructive to compare the expressions derived in this section to what one would expect for simple gases, where, as famously first derived and demonstrated experimentally by Maxwell in the 1860's [CITATION], the shear viscosity is independent of density.", 'cond-mat-0603391-1-46-1': 'This result differs from the kinetic (gas-like) contribution to the viscosity in Eq. ([REF]), because in a real dilute gas the mean-free path scales as [MATH], canceling the dominant density dependence in [MATH].', 'cond-mat-0603391-1-46-2': 'The same argument explains why the self-diffusion and kinematic viscosity of the SRD fluid are, to first order, independent of [MATH], while in a gas they would scale as [MATH].', 'cond-mat-0603391-1-46-3': 'In SRD, the mean-free path [MATH] and the density [MATH] can be varied independently.', 'cond-mat-0603391-1-46-4': 'Moreover, the collisional contribution to the viscosity adds a new dimension, allowing a much wider range of physical fluids to be modeled than just simple gases.', 'cond-mat-0603391-1-47-0': '# Colloid simulation method', 'cond-mat-0603391-1-48-0': 'Malevanets and Kapral [CITATION] first showed how to implement a hybrid MD scheme that couples a set of colloids to a bath of SRD particles.', 'cond-mat-0603391-1-48-1': 'In this section, we expand on their method, describing in detail the implementation we used in ref. [CITATION].', 'cond-mat-0603391-1-48-2': 'We restrict ourselves to HS like colloids with steep interparticle repulsions, although attractions between colloids can easily be added on.', 'cond-mat-0603391-1-48-3': 'The colloid-colloid and colloid-fluid interactions, [MATH] and [MATH] respectively, are integrated via a normal MD procedure, while the fluid-fluid interactions are coarse-grained with SRD.', 'cond-mat-0603391-1-48-4': 'Because the number of fluid particles vastly outnumbers the number of HS colloids, treating their interactions approximately via SRD greatly speeds up the simulation.', 'cond-mat-0603391-1-49-0': '## Colloid-colloid and colloid-solvent interactions', 'cond-mat-0603391-1-50-0': 'Although it is possible to implement an event-driven dynamics of HS colloids in an SRD solvent [CITATION], here we approximate pure HS colloids by steep repulsive interactions of the WCA form [CITATION]: [EQUATION]', 'cond-mat-0603391-1-50-1': 'Similarly, the colloid-fluid interaction takes the WCA form: [EQUATION]', 'cond-mat-0603391-1-50-2': 'The mass [MATH] of a fluid particle is typically much smaller than the mass [MATH] of a colloid, so that the average thermal velocity of the fluid particles is larger than that of the colloid particles by a factor [MATH].', 'cond-mat-0603391-1-50-3': 'For this reason the time-step [MATH] is usually restricted by the fluid-colloid interaction ([REF]), allowing fairly large exponents [MATH] for the colloid-colloid interaction [MATH].', 'cond-mat-0603391-1-50-4': 'We choose [MATH], which makes the colloid-colloid potential steep, more like hard-spheres, while still soft enough to allow the time-step to be set by the colloid-solvent interaction.', 'cond-mat-0603391-1-51-0': 'The positions and velocities of the colloidal spheres are propagated through the Velocity Verlet algorithm [CITATION] with a time step [MATH]: [EQUATION] [MATH] and [MATH] are the position and velocity of colloid [MATH], respectively.', 'cond-mat-0603391-1-51-1': '[MATH] is the total force on that colloid, exerted by the fluid particles, an external field, such as gravity, external potentials such as repulsive walls, as well as other colloids within the range of the interaction potential ([REF]).', 'cond-mat-0603391-1-52-0': "The positions [MATH] and velocities [MATH] of SRD particles are updated by similarly solving Newton's Eqns. ([REF],[REF]) every time-step [MATH], and with the SRD procedure of Eq. ([REF]) every time-step [MATH].", 'cond-mat-0603391-1-53-0': 'Choosing both [MATH] and [MATH] as large as possible enhances the efficiency of a simulation.', 'cond-mat-0603391-1-53-1': 'To first order, each timestep is determined by different physics, [MATH] by the steepness of the potentials, and [MATH] by the desired fluid properties, and so there is some freedom in choosing their relative values.', 'cond-mat-0603391-1-53-2': 'We used [MATH] for our simulations of sedimentation [CITATION], but other authors have used ratios of [MATH] [CITATION] or even an order of magnitude larger than that [CITATION].', 'cond-mat-0603391-1-53-3': 'Later in the paper we will revisit this question, linking the time-steps to various dimensionless hydrodynamic numbers and Brownian time-scales.', 'cond-mat-0603391-1-54-0': '## Stick and slip boundary conditions', 'cond-mat-0603391-1-55-0': 'Because the surface of a colloid is never perfectly smooth, collisions with fluid particles transfer angular as well as linear momentum.', 'cond-mat-0603391-1-55-1': 'As demonstrated in Fig. [REF], the exact molecular details of the colloid-fluid interactions may be very complex, and mediated via co- and counter-ions, grafted polymer brushes etc.', 'cond-mat-0603391-1-55-2': 'However, on the time and length-scales over which our hybrid MD-SRD method coarse-grains the solvent, these interactions can be approximated by stick boundary conditions: the tangential velocity of the fluid, relative to the surface of the colloid, is zero at the surface of the colloid [CITATION].', 'cond-mat-0603391-1-55-3': 'For most situations, this boundary condition should be sufficient, although in some cases, such as a non-wetting surface, large slip-lengths may occur [CITATION].', 'cond-mat-0603391-1-56-0': 'In computer simulations, stick boundary conditions may be implemented by bounce-back rules, where both parallel and perpendicular components of the relative velocity are reversed upon a collision with a surface.', 'cond-mat-0603391-1-56-1': 'These have been applied by Lamura et al [CITATION], who needed to modify the bounce-back rules slightly to properly reproduce stick boundaries for a Poiseuille flow geometry.', 'cond-mat-0603391-1-57-0': 'Stick boundaries can also be modeled by a stochastic rule.', 'cond-mat-0603391-1-57-1': 'After a collision, the relative tangential velocity [MATH] and relative normal velocity [MATH] are taken from the distributions: [EQUATION] so that the colloid acts as an additional thermostat [CITATION].', 'cond-mat-0603391-1-57-2': 'Such stochastic boundary conditions have been used for colloidal particles by Inoue et al. [CITATION] and Hecht et al. [CITATION].', 'cond-mat-0603391-1-57-3': 'We have systematically studied several implementations of stick boundary conditions for spherical colloids [CITATION], and derived a version of the stochastic boundary conditions which reproduces linear and angular velocity correlation functions that agree with Enskog theory for short times, and hydrodynamic mode-coupling theory for long times.', 'cond-mat-0603391-1-57-4': 'We argue that the stochastic rule of Eq. ([REF]) is more like a real physical colloid - where fluid-surface interactions are mediated by steric stabilizing layers or local co- and counter-ion concentrations - than bounce-back rules are [CITATION].', 'cond-mat-0603391-1-58-0': 'Nevertheless, in this paper, many examples will be for radial interactions such as those described in Eq. ([REF]).', 'cond-mat-0603391-1-58-1': 'These do not transfer angular momentum to a spherical colloid, and so induce effective slip boundary conditions.', 'cond-mat-0603391-1-58-2': 'For many of the hydrodynamic effects we will discuss here the difference with stick boundary conditions is quantitative, not qualitative, and also well understood.', 'cond-mat-0603391-1-59-0': '## Depletion and lubrication forces', 'cond-mat-0603391-1-60-0': '### Spurious depletion forces induced by the fluid', 'cond-mat-0603391-1-61-0': 'We would like to issue a warning that the additional fluid degrees of freedom may inadvertently introduce depletion forces between the colloids.', 'cond-mat-0603391-1-61-1': 'Because the number density of SRD particles is much higher than that of the colloids, even a small overlap between two colloids can lead to enormous attractions.', 'cond-mat-0603391-1-62-0': 'For low colloid densities the equilibrium depletion interaction between any two colloids caused by the presence of the ideal fluid particles is given by [CITATION]: [EQUATION] where [MATH] is the number density of fluid particles and [MATH] is the (free) volume excluded to the fluid by the presence of two colloids separated by a distance [MATH].', 'cond-mat-0603391-1-62-1': 'The latter is given by [EQUATION] where [MATH].', 'cond-mat-0603391-1-62-2': 'An example is given in Fig. [REF] where we have plotted the resulting depletion potential for the colloid-solvent interaction ([REF]), with [MATH] as routinely used in our simulations, as well as the depletion interaction resulting from a truly HS colloid-solvent interaction.', 'cond-mat-0603391-1-62-3': 'The latter can easily be calculated analytically, with the result [EQUATION]', 'cond-mat-0603391-1-62-4': 'For the pure HS interactions, one could take [MATH] and the depletion forces would have no effect.', 'cond-mat-0603391-1-62-5': 'But for interactions such as those used in Eqs. ([REF]) and ([REF]), the softer repulsions mean that inter-colloid distances less than [MATH] are regularly sampled.', 'cond-mat-0603391-1-62-6': 'A more stringent criterion of [MATH] must therefore be used to avoid spurious depletion effects.', 'cond-mat-0603391-1-62-7': 'As an example, we re-analyze the simulations of ref. [CITATION], where a WCA form with [MATH] was used for the fluid-colloid interactions, and a normal Lennard Jones [MATH] potential with [MATH] was used for the colloid-colloid interactions.', 'cond-mat-0603391-1-62-8': 'The authors found differences between "vacuum" calculations without SRD particles, and a hybrid scheme coupling the colloids to an SRD solvent.', 'cond-mat-0603391-1-62-9': 'These were correctly attributed to "solvent induced pressure", which we quantify here as depletion interactions.', 'cond-mat-0603391-1-62-10': 'For one set of their parameters, [MATH], the effect is mainly to soften the repulsion, but for their other parameter set: [MATH], depletion attractions induce an effective attractive well-depth of over [MATH]!', 'cond-mat-0603391-1-63-0': 'Since the depletion potentials can be calculated analytically, one might try counteracting them by introducing a compensating repulsive potential of the form: [MATH] between the colloids.', 'cond-mat-0603391-1-63-1': 'However, there are three problems with this approach: Firstly, at higher colloid packing fractions, three and higher order interactions may also need to be added, and these are very difficult to calculate.', 'cond-mat-0603391-1-63-2': 'Secondly, the depletion interactions are not instantaneous, and in fact only converge to their equilibrium average algebraically in time [CITATION].', 'cond-mat-0603391-1-63-3': 'While this is not a problem for equilibrium properties, it will introduce errors for non-equilibrium properties.', 'cond-mat-0603391-1-63-4': 'Finally, when external fields drive the colloid, small but persistent anisotropies in the solvent density around a colloid may occur [CITATION].', 'cond-mat-0603391-1-63-5': 'Although these density variations are (and should be) small, the resulting variations in depletion interactions can be large.', 'cond-mat-0603391-1-64-0': 'To avoid these problems, we routinely choose the colloid-fluid interaction range [MATH] slightly below half the colloid diameter [MATH].', 'cond-mat-0603391-1-64-1': 'More precisely, we ensure that the colloid-colloid interaction equals [MATH] at a distance [MATH] where the depletion interactions have become zero, i.e., at a distance of twice the colloid-solvent interaction cut-off radius.', 'cond-mat-0603391-1-64-2': 'Smaller distances will consequently be rare, and adequately dealt with by the compensation potential.', 'cond-mat-0603391-1-64-3': 'This solution may be a more realistic representation anyhow, since in practice for charge and even for sterically stabilized colloids, the effective colloid-colloid diameter [MATH] is expected to be larger than twice the effective colloid-fluid diameter [MATH].', 'cond-mat-0603391-1-64-4': 'This is particularly so for charged colloids at large Debye screening lengths.', 'cond-mat-0603391-1-65-0': '### Lubrication forces depend on surface details', 'cond-mat-0603391-1-66-0': 'When two surfaces approach one another, they must displace the fluid between them, while if they move apart, fluid must flow into the space freed up between the surfaces.', 'cond-mat-0603391-1-66-1': 'At very short inter-surface distances, this results in so-called lubrication forces, which are repulsive for colloids approaching each other, and attractive for colloids moving apart [CITATION].', 'cond-mat-0603391-1-66-2': 'These forces are expected to be particularly important for driven dense colloidal suspensions; see e.g. [CITATION] for a recent review.', 'cond-mat-0603391-1-67-0': 'An additional advantage of our choice of diameters [MATH] above is that more fluid particles will fit in the space between two colloids, and consequently the lubrication forces will be more accurately represented.', 'cond-mat-0603391-1-67-1': 'It should be kept in mind that for colloids, the exact nature of the short-range lubrication forces will depend on physical details of the surface, such as its roughness, or presence of a grafted polymeric stabilizing layer [CITATION].', 'cond-mat-0603391-1-67-2': 'For perfectly smooth colloids, analytic limiting expressions for the lubrication forces can be derived [CITATION], showing a divergence at short distances.', 'cond-mat-0603391-1-67-3': 'We have confirmed that SRD resolves these lubrication forces down to surprisingly low interparticle distances.', 'cond-mat-0603391-1-67-4': 'But, at some point this will break down, depending of course on the choice of simulation parameters (such as [MATH], [MATH], and [MATH]), as well as the details of the particular type of colloidal particles that one wishes to model [CITATION].', 'cond-mat-0603391-1-67-5': 'An explicit analytic correction could be applied to properly resolve these forces for very small distances, as was recently implemented for Lattice Boltzmann [CITATION].', 'cond-mat-0603391-1-67-6': 'However, in this paper, we will assume that our choice of [MATH] is small enough for SRD to sufficiently resolve lubrication forces.', 'cond-mat-0603391-1-67-7': 'The lack of a complete divergence at very short distances may be a better model of what happens for real colloids anyway.', 'cond-mat-0603391-1-67-8': 'For dense suspensions under strong shear, explicit lubrication force corrections as well as other short-ranged colloid-colloid interactions arising from surface details such as polymer coats will almost certainly need to be put in by hand, see e.g. ref. [CITATION] for further discussion of this subtle problem.', 'cond-mat-0603391-1-68-0': '## Test of static properties', 'cond-mat-0603391-1-69-0': 'The equilibrium properties of a statistical mechanical system should be independent of the detailed dynamics by which it explores phase space.', 'cond-mat-0603391-1-69-1': 'Therefore one requisite condition imposed on our coarse-grained dynamics is that it reproduces the correct static properties of an ensemble of colloids.', 'cond-mat-0603391-1-70-0': 'An obvious property to measure is the radial distribution function [MATH] [CITATION].', 'cond-mat-0603391-1-70-1': 'In Fig. [REF] we depict some [MATH]s obtained from SRD simulations at a number of different densities, and compare these to similar simulations with standard Brownian dynamics.', 'cond-mat-0603391-1-70-2': 'The colloids interact via potentials of the form ([REF]) - ([REF]) with [MATH], [MATH], and [MATH].', 'cond-mat-0603391-1-70-3': 'For the SRD we used [MATH], [MATH] and [MATH], implying [MATH].', 'cond-mat-0603391-1-70-4': 'For the BD we used a background friction calculated from the effective hydrodynamic radius (vide infra) and a time-step equal to the [MATH] used above.', 'cond-mat-0603391-1-70-5': 'The colloids were placed in a box of dimensions [MATH], and the number of particles was varied from 64 to 540 in order to achieve different packing fractions [EQUATION] where the subscript in [MATH] refers to volume fraction based on the colloid-colloid interaction, to distinguished it from the volume fraction [MATH] based on the colloid hydrodynamic radius [CITATION].', 'cond-mat-0603391-1-70-6': 'From Fig. [REF] is clear that the two simulations are indistinguishable within statistical errors, as required.', 'cond-mat-0603391-1-70-7': 'Even though [MATH], we still found it necessary to include an explicit compensating potential without which the [MATH] generated by SRD is increased noticeably at contact when compared to BD simulations.', 'cond-mat-0603391-1-71-0': '# Dimensionless numbers', 'cond-mat-0603391-1-72-0': 'Different regimes of hydrodynamic behavior can be characterized by a series of dimensionless numbers that indicate the relative strengths of competing physical processes [CITATION].', 'cond-mat-0603391-1-72-1': 'If two distinct physical systems can be described by the same set of hydrodynamic numbers, then their flow behavior will be governed by the same physics, even if their time and length scales differ by orders of magnitude.', 'cond-mat-0603391-1-72-2': 'In other words, a macroscopic boulder sedimenting in viscous molten magma and a mesoscopic colloid sedimenting in water may show similar behavior if they share key hydrodynamic dimensionless numbers.', 'cond-mat-0603391-1-72-3': 'It is therefore very instructive to analyze where a system described by SRD sits in "hydrodynamic parameter space".', 'cond-mat-0603391-1-72-4': 'In this section we review this parameter space, one "dimension" at a time, by exploring the hydrodynamic numbers listed in Table [REF].', 'cond-mat-0603391-1-73-0': '## Schmidt number', 'cond-mat-0603391-1-74-0': 'The Schmidt number [EQUATION] is important for characterizing a pure fluid.', 'cond-mat-0603391-1-74-1': 'It expresses the rate of momentum transfer, measured by the kinematic viscosity [MATH], relative to the rate of mass transfer, measured by the fluid particle diffusion coefficient [MATH].', 'cond-mat-0603391-1-74-2': 'For a gas, momentum transport is dominated by mass diffusion, so that Sc [MATH], whereas for a liquid, momentum transport is dominated by inter-particle collisions, and Sc [MATH].', 'cond-mat-0603391-1-74-3': 'For low Schmidt numbers, the dynamics of SRD is indeed "gas-like", while for larger Sc, the dynamics shows collective behavior reminiscent of the hydrodynamics of a liquid [CITATION].', 'cond-mat-0603391-1-74-4': 'This distinction will be particularly important for simulating the dynamics of embedded particles that couple to the solvent through participation in the collision step.', 'cond-mat-0603391-1-74-5': 'For particles that couple directly, either through a potential, or through bounce-back or stochastic boundary conditions, this distinction is less important because the fluid-particle diffusion coefficient [MATH] does not directly enter the Navier Stokes equations.', 'cond-mat-0603391-1-74-6': 'Of course other physical properties can be affected.', 'cond-mat-0603391-1-74-7': 'For example, we expect that the self diffusion coefficient of a colloid [MATH] should be much smaller than the fluid diffusion coefficient, i.e. [MATH] to achieve the correct time-scale separation, as we will see in Section [REF].', 'cond-mat-0603391-1-75-0': '### Dependence of Schmidt number on simulation parameters', 'cond-mat-0603391-1-76-0': 'From Eqs. ([REF]) - ([REF]) it follows that the Schmidt number can be rewritten as [EQUATION] where we have ignored the dependence on [MATH] and [MATH] since the dominant scaling is with the dimensionless mean-free path [MATH].', 'cond-mat-0603391-1-76-1': 'For larger [MATH], where the kinetic contributions to the viscosity dominate, the Sc number is small, and the dynamics is gas-like.', 'cond-mat-0603391-1-76-2': 'Because both the fluid diffusion coefficient ([REF]) and the kinetic contribution to the viscosity ([REF]) scale linearly with [MATH], the only way to obtain a large Sc number is in the limit [MATH] where the collisional contribution to [MATH] dominates.', 'cond-mat-0603391-1-77-0': 'Low Sc numbers may be a more general characteristic of particle-based coarse-graining methods.', 'cond-mat-0603391-1-77-1': 'For computational reasons, the number of particles is normally greatly reduced compared to the solvent one is modeling.', 'cond-mat-0603391-1-77-2': 'Thus the average inter-particle separation and mean-free path are substantially increased, typically leading to a smaller kinematic viscosity and a larger fluid diffusion coefficient [MATH].', 'cond-mat-0603391-1-77-3': 'With DPD, for example, one typically finds [MATH] [CITATION].', 'cond-mat-0603391-1-77-4': 'It is therefore difficult to achieve large Sc numbers, as in a real liquid, without sacrificing computational efficiency.', 'cond-mat-0603391-1-77-5': 'But this may not always be necessary.', 'cond-mat-0603391-1-77-6': 'As long as momentum transport is clearly faster than mass transport, the solvent should behave in a liquid-like fashion.', 'cond-mat-0603391-1-77-7': 'With this argument in mind, and also because the Sc number does not directly enter the Navier Stokes equations we want to solve, we use [MATH] in this paper and in ref. [CITATION], which leads to a Sc [MATH] for [MATH] and [MATH].', 'cond-mat-0603391-1-77-8': 'This should be sufficient for the problems we study.', 'cond-mat-0603391-1-77-9': 'For other systems, such as those where the suspended particles are coupled to the solvent through the collision step, more care may be needed to ensure that the Sc number is indeed large enough [CITATION].', 'cond-mat-0603391-1-78-0': '## Mach number', 'cond-mat-0603391-1-79-0': 'The Mach number measures the ratio [EQUATION] between [MATH], the speed of solvent or colloid flow, and [MATH], the speed of sound.', 'cond-mat-0603391-1-79-1': 'In contrast to the Schmidt number, which is an intrinsic property of the solvent, it depends directly on flow velocity.', 'cond-mat-0603391-1-79-2': 'The Ma number measures compressibility effects [CITATION] since sound speed is related to the compressibility of a liquid.', 'cond-mat-0603391-1-79-3': 'Because [MATH] in many liquids is of order [MATH] m/s, the Ma numbers for physical colloidal systems are extremely small under normally achievable flow conditions.', 'cond-mat-0603391-1-79-4': 'Just as for the Sc number, however, particle based coarse-graining schemes drastically lower the Ma number.', 'cond-mat-0603391-1-79-5': 'The particle mass [MATH] is typically much greater than the mass of a molecule of the underlying fluid, resulting in lower velocities, and moreover, due to the lower density, collisions also occur less frequently.', 'cond-mat-0603391-1-79-6': 'These effects mean that the speed of sound is much lower in a coarse-grained system than it is in the underlying physical fluid.', 'cond-mat-0603391-1-79-7': 'Or, in other words, particle based coarse-graining systems are typically much more compressible than the solvents they model.', 'cond-mat-0603391-1-80-0': 'Ma number effects typically scale with [MATH] [CITATION], and so the Ma number does not need to be nearly as small as for a realistic fluid to still be in the correct regime of hydrodynamic parameter space.', 'cond-mat-0603391-1-80-1': 'This is convenient, because to lower the Ma number, one would need to integrate over longer fluid particle trajectories to allow, for example, a colloidal particle to flow over a given distance, making the simulation computationally more expensive.', 'cond-mat-0603391-1-80-2': 'So there is a compromise between small Ma numbers and computational efficiency.', 'cond-mat-0603391-1-80-3': 'We limit our Ma numbers to values such that [MATH], but it might be possible, in some situations, to double or triple that limit without causing undue error.', 'cond-mat-0603391-1-80-4': 'For example, incompressible hydrodynamics is used for aerodynamic flows up to such Ma numbers since the errors are expected to scale as [MATH] [CITATION].', 'cond-mat-0603391-1-80-5': 'When working in units of [MATH] and [MATH], the only way to keep the Ma number below our upper limit is to restrict the maximum flow velocity to [MATH].', 'cond-mat-0603391-1-80-6': 'The flow velocity itself is, of course, determined by the external fields, but also by other parameters of the system.', 'cond-mat-0603391-1-81-0': '## Reynolds number', 'cond-mat-0603391-1-82-0': 'The Reynolds number is one of the most important dimensionless numbers characterizing hydrodynamic flows.', 'cond-mat-0603391-1-82-1': 'Mathematically, it measures the relative importance of the non-linear terms in the Navier-Stokes equations [CITATION].', 'cond-mat-0603391-1-82-2': 'Physically, it determines the relative importance of inertial over viscous forces, and can be expressed as: [EQUATION] where [MATH] is a length-scale, in our case, the hydrodynamic radius of a colloid, i.e. [MATH], and [MATH] is a flow velocity.', 'cond-mat-0603391-1-83-0': 'For a spherical particle in a flow, the following heuristic argument helps clarify the physics behind the Reynolds number: If the Stokes time [EQUATION] it takes a particle to move over its own radius is about the same as the kinematic time [EQUATION] it takes momentum to diffuse over that distance, i.e. Re=[MATH], then the particle will feel vorticity effects from its own motion a distance [MATH] away, leading to non-linear inertial effects.', 'cond-mat-0603391-1-83-1': 'Since hydrodynamic interactions can decay as slowly as [MATH], their effects can be non-negligible.', 'cond-mat-0603391-1-83-2': 'If, on the other hand, Re [MATH], then vorticity will have diffused away and the particle will only feel very weak hydrodynamic effects from its own motion.', 'cond-mat-0603391-1-84-0': 'Exactly when inertial finite Re effects become significant depends on the physical system under investigation.', 'cond-mat-0603391-1-84-1': 'For example, in a pipe, where the length-scale [MATH] in Eq. ([REF]) is its diameter, the transition from simpler laminar to more complex turbulent flow is at a pipe Reynolds number of Re [MATH] [CITATION].', 'cond-mat-0603391-1-84-2': 'On the other hand, for a single spherical particle, a non-linear dependence of the friction [MATH] on the velocity [MATH], induced by inertial effects, starts to become noticeable for a particle Reynolds number of Re [MATH] [CITATION], while deviations in the symmetry of the streamlines around a rotating sphere have been observed in calculations for Re [MATH] [CITATION].', 'cond-mat-0603391-1-85-0': 'For typical colloidal suspensions, where the particle diameter is on the order of a few [MATH]m down to a few nm, the particle Reynolds number is rarely more than [MATH].', 'cond-mat-0603391-1-85-1': 'In this so-called Stokes regime viscous forces dominate and inertial effects can be completely ignored.', 'cond-mat-0603391-1-85-2': 'The Navier-Stokes equations can be replaced by the linear Stokes equations [CITATION] so that analytic solutions are easier to obtain [CITATION].', 'cond-mat-0603391-1-85-3': 'However, some of the resulting behavior is non-intuitive for those used to hydrodynamic effects on a macroscopic scale.', 'cond-mat-0603391-1-85-4': 'For example, as famously explained by Purcell in a talk entitled "Life at low Reynolds numbers" [CITATION], many simple processes in biology occur on small length scales, well into the Stokes regime.', 'cond-mat-0603391-1-85-5': 'The conditions that bacteria, typically a few [MATH]m long, experience in water are more akin to those humans would experience in extremely thick molasses.', 'cond-mat-0603391-1-85-6': 'Similarly, colloids, polymers, vesicles and other small suspended objects are all subject to the same physics, their motion dominated by viscous forces.', 'cond-mat-0603391-1-85-7': 'For example, if for a colloid sedimenting at 1 [MATH]m/s, gravity were instantaneously turned off, then the Stokes equations suggest that it would come to a complete halt in a distance significantly less than one , reflecting the irrelevance of inertial forces in this low Re number regime.', 'cond-mat-0603391-1-85-8': 'It should be kept in mind that when the Stokes regime is reached because of small length scales (as opposed to very large viscosities such as those found for volcanic lava flows), then thermal fluctuations are also important.', 'cond-mat-0603391-1-85-9': 'These will drive diffusive behavior [CITATION].', 'cond-mat-0603391-1-85-10': 'In many ways SRD is ideally suited for this regime of small particles because the thermal fluctuations are naturally included [CITATION].', 'cond-mat-0603391-1-86-0': '### Dependence of the Reynolds number on simulation parameters', 'cond-mat-0603391-1-87-0': 'From Eqs. ([REF]) and ([REF]), it follows that the Reynolds number for a colloid of hydrodynamic radius [MATH] can be written as: [EQUATION]', 'cond-mat-0603391-1-87-1': 'Equating the hydrodynamic radius [MATH] to [MATH] from the fluid-colloid WCA interaction of Eq. ([REF]) is not quite correct, as we will see in section [REF], but is a good enough approximation.', 'cond-mat-0603391-1-88-0': 'In order to keep the Reynolds number low, one must either use small particles, or very viscous fluids, or low velocities [MATH].', 'cond-mat-0603391-1-88-1': 'The latter condition is commensurate with a low Ma number, which is also desirable.', 'cond-mat-0603391-1-88-2': 'For small enough [MATH] (and ignoring for simplicty the factor [MATH]) in Eq. ([REF]), the Re number then scales as [EQUATION]', 'cond-mat-0603391-1-88-3': 'Again, we see that a smaller mean-free path [MATH], which enhances the collisional viscosity, also helps bring the Re number down.', 'cond-mat-0603391-1-88-4': 'This parameter choice is also consistent with a larger Sc number.', 'cond-mat-0603391-1-88-5': 'Thus larger Sc and smaller Ma numbers both help keep the Re number low for SRD.', 'cond-mat-0603391-1-88-6': 'In principle a large viscosity can also be obtained for large [MATH], which enhances the kinetic viscosity, but this choice also lowers the Sc number and raises the Knudsen number, which, as we will see in the next sub-section, is not desirable.', 'cond-mat-0603391-1-89-0': 'Just as was found for the Ma number, it is relatively speaking more expensive computationally to simulate for low Re numbers because the flow velocity must be kept low, which means longer simulation times are necessary to reach time-scales where the suspended particles or fluid flows have moved a significant distance.', 'cond-mat-0603391-1-89-1': 'We therefore compromise and normally keep Re [MATH], which is similar to the choice made for LB simulations [CITATION].', 'cond-mat-0603391-1-89-2': 'For many situations related to the flow of colloids, this should be sufficiently stringent.', 'cond-mat-0603391-1-90-0': '## Knudsen number', 'cond-mat-0603391-1-91-0': 'The Knudsen number Kn measures rarefaction effects, and can be written as [EQUATION] where [MATH] is a characteristic length-scale of the fluid flow, and [MATH] is the mean-free path of the fluid or gas.', 'cond-mat-0603391-1-91-1': 'For a colloid in SRD one could take [MATH] and [MATH].', 'cond-mat-0603391-1-91-2': 'For large Knudsen numbers [MATH] continuum Navier-Stokes equations completely break down, but even for much smaller Knudsen numbers, significant rarefaction effects are seen.', 'cond-mat-0603391-1-91-3': 'For example, for flow in a pipe, where [MATH] in Eq. ([REF]) is taken to be the pipe radius, important non-continuum effects are seen when [MATH].', 'cond-mat-0603391-1-91-4': 'In fact it is exactly for these conditions of modest Kn numbers, when corrections to the Navier Stokes become noticeable, that DSMC approaches are often used [CITATION].', 'cond-mat-0603391-1-91-5': 'SRD, which is related to DSMC, may also be expected to work well for such flows.', 'cond-mat-0603391-1-91-6': 'It could find important applications in microfluidics and other micromechanical devices where Kn effects are expected to play a role [CITATION].', 'cond-mat-0603391-1-92-0': 'Colloidal dispersions normally have very small Kn numbers because the mean-free path of most liquid solvents is very small.', 'cond-mat-0603391-1-92-1': 'For water at standard temperature and pressure [MATH] .', 'cond-mat-0603391-1-92-2': 'Just as found for the other dimensionless numbers, coarse-graining typically leads to larger Kn numbers because of the increase of the mean-free path.', 'cond-mat-0603391-1-92-3': 'Making the Kn number smaller also typically increases the computational cost because a larger number of collisions need to be calculated.', 'cond-mat-0603391-1-92-4': 'In our simulations, we keep Kn[MATH] for spheres.', 'cond-mat-0603391-1-92-5': 'This rough criterion is based on the observation that for small Kn numbers the friction coefficient on a sphere is expected to be decreased by a factor [MATH] Kn, where [MATH] is a material dependent constant of order [MATH] [CITATION], so that we expect Kn number effects to be of the same order as other coarse-graining errors.', 'cond-mat-0603391-1-92-6': 'There are two ways to achieve small Kn numbers: one is by increasing [MATH], the other is by decreasing [MATH].', 'cond-mat-0603391-1-92-7': 'The second condition is commensurate with a large Sc number or a small Re number.', 'cond-mat-0603391-1-93-0': 'In ref. [CITATION], the Kn numbers for colloids were [MATH] and [MATH], depending on their colloid-solvent coupling.', 'cond-mat-0603391-1-93-1': 'Such large Kn numbers should have a significant effect on particle friction coefficients, and this may explain why these authors find that volume fraction [MATH] has less of an effect on the sedimentation velocity than we do [CITATION].', 'cond-mat-0603391-1-94-0': '## Peclet number', 'cond-mat-0603391-1-95-0': 'The Peclet number Pe measures the relative strength of convective transport to diffusive transport.', 'cond-mat-0603391-1-95-1': 'For example, for a colloid of radius [MATH], traveling at an average velocity [MATH], the Pe number is defined as: [EQUATION] where [MATH] is the colloid diffusion coefficient.', 'cond-mat-0603391-1-95-2': 'Just as for the Re number, the Pe number can be interpreted as a ratio of a diffusive to a convective time-scale, but now the former time-scale is not for the diffusion of momentum but rather it is given by the colloid diffusion time [EQUATION] which measures how long it takes for a colloid to diffuse over a distance [MATH].', 'cond-mat-0603391-1-95-3': 'We again take [MATH] so that, using Eq. ([REF]), the Pe number can be written as: [EQUATION]', 'cond-mat-0603391-1-95-4': 'If Pe [MATH] then the colloid moves convectively over a distance much larger than its radius [MATH] in the time [MATH] that it diffuses over that same distance.', 'cond-mat-0603391-1-95-5': 'Brownian fluctuations are expected to be less important in this regime.', 'cond-mat-0603391-1-95-6': 'For Pe [MATH], on the other hand, the opposite is the case, and the main transport mechanism is diffusive (note that on long enough time-scales ([MATH]) convection will always eventually "outrun" diffusion [CITATION]).', 'cond-mat-0603391-1-95-7': 'It is sometimes thought that for low Pe numbers hydrodynamic effects can be safely ignored, but this is not always true.', 'cond-mat-0603391-1-95-8': 'For example, we found that the reduction of average sedimentation velocity with particle volume fraction, famously first explained by Batchelor [CITATION], is independent of Pe number down to Pe [MATH] at least [CITATION].', 'cond-mat-0603391-1-96-0': '### Dependence of Peclet number on simulation parameters', 'cond-mat-0603391-1-97-0': 'The highest Pe number achievable in simulation is limited by the constraints on the Ma and Re numbers.', 'cond-mat-0603391-1-97-1': 'For example the Ma number sets an upper limit on the maximum Pe number by limiting [MATH].', 'cond-mat-0603391-1-97-2': 'From Eqs. ([REF]) and ([REF]), it follows that the Peclet number can be re-written in terms of the Reynolds number as [EQUATION] where we have approximated [MATH].', 'cond-mat-0603391-1-97-3': 'This shows that for a given constraint on the Re number, increasing [MATH] or [MATH] increases the range of accessible Pe numbers.', 'cond-mat-0603391-1-97-4': 'Similarly, when the kinematic viscosity is dominated by the collisional contribution, decreasing the dimensionless mean-free path [MATH] will also increase the maximum Pe number allowed since [MATH].', 'cond-mat-0603391-1-97-5': 'However, these changes increase the computational cost of the simulation.', 'cond-mat-0603391-1-98-0': '# finite-size, discretization, and inertial effects', 'cond-mat-0603391-1-99-0': 'The cost of an SRD simulation scales almost linearly with the number of fluid particles [MATH] in the system, and this contribution is usually much larger than the cost of including the colloidal degrees of freedom.', 'cond-mat-0603391-1-99-1': 'To optimize the efficiency of a simulation, one would therefore like to keep [MATH] as small as possible.', 'cond-mat-0603391-1-99-2': 'This objective can be achieved by keeping [MATH] and the box-size [MATH] small.', 'cond-mat-0603391-1-99-3': 'Unfortunately both these choices are constrained by the errors they introduce.', 'cond-mat-0603391-1-99-4': 'Reducing [MATH] means that short-ranged hydrodynamic fields become less accurately resolved due to Kn number and discretization effects.', 'cond-mat-0603391-1-99-5': 'Decreasing the box-size [MATH] falls foul of the long-ranged nature of the HI, and therefore, just as for Coulomb interactions [CITATION], finite size effects such as those induced by periodic images must be treated with special care.', 'cond-mat-0603391-1-100-0': 'Increasing the flow velocity may also be desirable since more Stokes times [MATH] can be achieved for the same number of SRD collision steps, thus increasing computational efficiency.', 'cond-mat-0603391-1-100-1': 'The Ma number gives one constraint on [MATH], but usually the more stringent constraint comes from keeping the Re number low to prevent unwanted inertial effects.', 'cond-mat-0603391-1-101-0': '## Finite-size effects', 'cond-mat-0603391-1-102-0': '### Finite-size correction to the friction', 'cond-mat-0603391-1-103-0': 'The friction coefficient [MATH] can be extracted from the Stokes drag [MATH] on a fixed colloid in fluid flow: [EQUATION] where [MATH] is the flow field at large distances.', 'cond-mat-0603391-1-103-1': 'The pre-factor 4 comes from using slip boundary conditions; it would be 6 for stick-boundary conditions, as verified in [CITATION].', 'cond-mat-0603391-1-103-2': 'In principle, since [MATH] is accurately known from theory for SRD, this expression can be used to extract the hydrodynamic radius [MATH], which is not necessarily the same as [MATH], from a simulation, as we did in [CITATION].', 'cond-mat-0603391-1-104-0': 'The hydrodynamic radius [MATH] can also be directly calculated from theory.', 'cond-mat-0603391-1-104-1': 'To derive this, it is important to recognize that there are two sources of friction [CITATION].', 'cond-mat-0603391-1-104-2': 'The first comes from the local Brownian collisions with the small particles, and can be calculated by a simplified Enskog/Boltzmann type kinetic theory [CITATION]: [EQUATION] which is here adapted for slip boundary conditions.', 'cond-mat-0603391-1-104-3': 'A related expression for stick boundary conditions, including that for rotational frictions, is described in [CITATION], where it was shown that the short-time exponential decays of the linear and angular velocity auto-correlation functions are quantitatively described by Enskog theory.', 'cond-mat-0603391-1-105-0': 'The second contribution to the friction, [MATH], comes from integrating the Stokes solution to the hydrodynamic field over the surface of the particle, defined here as [MATH].', 'cond-mat-0603391-1-105-1': 'These two contributions to the friction should be added in parallel to obtain the total friction [CITATION]: [EQUATION]', 'cond-mat-0603391-1-105-2': 'In contrast to the Enskog friction, which is local, we expect substantial box-size effects on the Stokes friction [MATH] since it depends on long-ranged hydrodynamic effects.', 'cond-mat-0603391-1-105-3': 'These can be expressed in terms of a correction factor [MATH] that should go to 1 for very large systems : [EQUATION]', 'cond-mat-0603391-1-105-4': 'To measure this correction factor we plot, in Fig. [REF], the form [MATH] for various system box sizes for which we have measured [MATH] from the simulation, and estimated [MATH] from Eq. ([REF]).', 'cond-mat-0603391-1-105-5': 'As expected, the correction factor tends to 1 for smaller [MATH].', 'cond-mat-0603391-1-105-6': 'More detailed calculations [CITATION], taking into account the effect of periodic boundaries, suggests that to lowest order in [MATH] the correction factor should scale as [EQUATION]', 'cond-mat-0603391-1-105-7': 'Indeed, a least squares fit of the data to this form gives a slope of [MATH], close to the theoretical value and in agreement with similar Lattice Boltzmann simulations of a single colloidal sphere [CITATION].', 'cond-mat-0603391-1-106-0': 'With these ingredients in hand, we can calculate the theoretical expected friction from Eqns. ([REF]) - ([REF]).', 'cond-mat-0603391-1-106-1': 'We know from previous work that the Enskog contribution at short times and the hydrodynamic contribution at long times quantitatively reproduce the rotational and translational velocity auto-correlation functions (VACF) for [MATH] [CITATION], and so we expect that the friction coefficients should also be accurately described by these theories.', 'cond-mat-0603391-1-106-2': 'We test this further in Fig. [REF] for a number of different values of [MATH], and find excellent agreement with theory for [MATH] and [MATH].', 'cond-mat-0603391-1-106-3': 'For [MATH] and below, on the other hand, we find deviations from the theory.', 'cond-mat-0603391-1-106-4': 'These are most likely due to Kn number and discretization effects, to be discussed in the next sub-section.', 'cond-mat-0603391-1-107-0': 'For the smallest spheres the mass ratio [MATH] may also change the measured friction if instead of fixing the sphere we were to let it move freely.', 'cond-mat-0603391-1-107-1': 'For [MATH], [MATH], which is small enough to have a significant effect [CITATION].', 'cond-mat-0603391-1-107-2': 'For larger spheres, this is not expected to be a problem.', 'cond-mat-0603391-1-107-3': 'For example [MATH] for [MATH] and [MATH] for [MATH].', 'cond-mat-0603391-1-108-0': '### Effective hydrodynamic radius', 'cond-mat-0603391-1-109-0': 'From the calculated or measured friction we can also obtain the effective hydrodynamic radius [MATH], which is continuum concept, from a comparison of the microscopic frictions from Eq. ([REF]) with Eq. ([REF]): [EQUATION] (for stick-boundaries the [MATH] should be replaced by [MATH]).', 'cond-mat-0603391-1-109-1': 'We find [MATH], [MATH], and [MATH].', 'cond-mat-0603391-1-109-2': 'The effective hydrodynamic radius is increased by the finite-size effects, but lowered by the Enskog contribution which is added in parallel.', 'cond-mat-0603391-1-109-3': 'Because this latter contribution is relatively more important for small colloids, [MATH] for smaller [MATH].', 'cond-mat-0603391-1-109-4': 'For [MATH] the Enskog contribution is smaller than the finite-size effects, and so the effective hydrodynamic radius is larger than [MATH].', 'cond-mat-0603391-1-109-5': 'Obviously this latter effect depends on the box-size.', 'cond-mat-0603391-1-109-6': 'In an infinite box [MATH] for all values of [MATH], due to the Enskog contribution.', 'cond-mat-0603391-1-109-7': 'Note that it is the effective hydrodynamic radius [MATH] which sets the long-ranged hydrodynamic fields, as we will see below.', 'cond-mat-0603391-1-110-0': '### Turning off long-ranged hydrodynamics', 'cond-mat-0603391-1-111-0': 'Hydrodynamic forces can be turned off in SRD by regularly randomizing the absolute fluid particle velocities (with for example a naive Langevin thermostat).', 'cond-mat-0603391-1-111-1': 'For particles embedded in an SRD solvent through participation in the collision step, this trick can be used to compare the effects of hydrodynamics to a purely Brownian simulation[CITATION].', 'cond-mat-0603391-1-111-2': 'The Yeomans group has successfully applied this idea in a study of polymer collapse and protein folding [CITATION].', 'cond-mat-0603391-1-112-0': 'However for the case of the colloids embedded through direct solvent collisions, turning off the hydrodynamic forces by randomizing the velocities greatly enhances the friction because, as is clear from Eqns. ([REF]) - ([REF]), the two contributions add in parallel.', 'cond-mat-0603391-1-112-1': 'Without long-ranged hydrodynamics, the friction would be entirely dominated by the Enskog contribution ([REF]) which scales with [MATH], and can be much larger than the hydrodynamic contribution which scales as [MATH].', 'cond-mat-0603391-1-112-2': 'Another way of stating this would be: By locally conserving momentum, SRD allows the development of long-ranged hydrodynamic fluid velocity correlations that greatly reduce the friction felt by a larger colloidal particle compared to the friction it would feel from a purely random Brownian heat-bath at the same temperature and number density [MATH].', 'cond-mat-0603391-1-113-0': '## Discretization effects on flow-field and friction', 'cond-mat-0603391-1-114-0': 'For pure Stokes flow (Re = 0), the velocity around a fixed slip-boundary sphere of (effective) hydrodynamic radius [MATH] can be exactly calculated: [EQUATION] where [MATH] is the velocity field far away from the sphere, and [MATH] the vector pointing from the center of the sphere to a position inside the fluid, with corresponding unit vector [MATH].', 'cond-mat-0603391-1-115-0': 'In Fig. [REF] we plot the difference between the measured field and the theoretical expected field ([REF]) for four different colloid radius to cell size ratios [MATH], using the values of the effective hydrodynamic radius [MATH] calculated from combining Eq. ([REF]) and Eq. [REF].', 'cond-mat-0603391-1-115-1': 'As expected, the field is more accurately reproduced as the colloid radius becomes larger with respect to the cell size [MATH].', 'cond-mat-0603391-1-115-2': 'This improvement arises because SRD discretization and hydrodynamic Knudsen number effects become smaller.', 'cond-mat-0603391-1-115-3': 'In Fig. [REF] we observe quite large deviations in the hydrodynamic field for [MATH].', 'cond-mat-0603391-1-115-4': 'These discretization effects may explain why the measured frictions in Fig. [REF] do not agree with theory for these smallest sphere sizes.', 'cond-mat-0603391-1-115-5': 'We also note that using [MATH] instead of the more accurate value of the effective hydrodynamic radius [MATH] calculated from theory results in significantly larger deviations between measured and theoretical flow-fields.', 'cond-mat-0603391-1-115-6': 'This independently confirms the values of the effective hydrodynamic radius.', 'cond-mat-0603391-1-116-0': 'The increased accuracy from using larger [MATH] comes at a sharp increase in computational cost.', 'cond-mat-0603391-1-116-1': 'To make sure that the finite size effects in each simulation of Fig. [REF] are approximately the same, the box-size was scaled as [MATH].', 'cond-mat-0603391-1-116-2': 'This means that doubling the colloid size leads to an eightfold increase in the number of fluid particles.', 'cond-mat-0603391-1-116-3': 'Moreover, the maximum velocity of the fluid must go down linearly in colloid size to keep the Re number, defined in Eq. ([REF]), constant.', 'cond-mat-0603391-1-116-4': 'If in addition we keep the number of Stokes times fixed, meaning that the fluid flows a certain multiple of the colloid radius or the box size, then a larger particle also means the fluid needs to flow over a proportionally longer total distance.', 'cond-mat-0603391-1-116-5': 'The overall computational costs for this calculation then scales at least as [MATH], which is quite steep.', 'cond-mat-0603391-1-116-6': 'For that reason, we advocate using smaller colloids wherever possible.', 'cond-mat-0603391-1-117-0': 'In most of our simulations we choose [MATH], which leads to a small relative error in the full velocity field and for which we can fully explain the observed friction (as seen in the previous subsection).', 'cond-mat-0603391-1-117-1': 'This size is similar to what is commonly used in LB [CITATION].', 'cond-mat-0603391-1-118-0': '## Inertial effects on flow-field and friction', 'cond-mat-0603391-1-119-0': 'One of the challenges in SRD is to keep the Re number down.', 'cond-mat-0603391-1-119-1': 'It is virtually impossible to reach the extremely low Re number (Stokes) regime of realistic colloids, but that is not necessary either.', 'cond-mat-0603391-1-119-2': 'In Fig. [REF] we see that the friction only begins to noticeably vary from the Stokes limit at Re [MATH] (at least on a logarithmic scale).', 'cond-mat-0603391-1-119-3': 'We expect the flow-field itself to be more sensitive to finite Re number effects.', 'cond-mat-0603391-1-119-4': 'To study this directly, we examine, in Fig. [REF], the flow-field around a fixed colloid of size [MATH] in a box of [MATH] for for different values of Re.', 'cond-mat-0603391-1-119-5': 'The differences with the Stokes flow field of Eq. ([REF]) are shown in Fig. [REF](b)-(d) for Re [MATH], [MATH], and [MATH] respectively.', 'cond-mat-0603391-1-119-6': 'In all cases we used a hydrodynamic radius of [MATH] for the theoretical comparison, as explained in the previous sub-section.', 'cond-mat-0603391-1-119-7': 'The lengths of the vectors in Figs. (b)-(d) are multiplied by 10 for clarity.', 'cond-mat-0603391-1-119-8': 'We observe that the relative errors increase with Re.', 'cond-mat-0603391-1-119-9': 'They are on the order of [MATH] for Re=0.08, and increase to something on the order of [MATH] for Re=2.', 'cond-mat-0603391-1-119-10': 'This is exactly what is expected, of course, as we are moving away from the Stokes regime with which these flow lines are being compared.', 'cond-mat-0603391-1-119-11': 'Since we also expect effects on the order of a few % from Kn number, Ma number and various finite size effects, we argue that keeping Re[MATH] should be good enough for most of the applications we have in mind.', 'cond-mat-0603391-1-120-0': '# The hierarchy of time scales', 'cond-mat-0603391-1-121-0': '## Colloidal time-scales', 'cond-mat-0603391-1-122-0': 'Many different time-scales govern the physics of a colloid of mass [MATH] embedded in a solvent [CITATION].', 'cond-mat-0603391-1-122-1': 'The most important ones are summarized in table [REF], and discussed in more detail below.', 'cond-mat-0603391-1-123-0': '### Fluid time-scales', 'cond-mat-0603391-1-124-0': 'The shortest of these is the solvent collision time [MATH] over which fluid molecules interact with each other.', 'cond-mat-0603391-1-124-1': 'For a typical molecular fluid, [MATH] is on the order of a few tens of fs, and any MD scheme for a molecular fluid must use a discretization time-step [MATH] to properly integrate the equations of motion.', 'cond-mat-0603391-1-125-0': 'The next time-scale up is the solvent relaxation time [MATH], which measures how fast the solvent VACF decays.', 'cond-mat-0603391-1-125-1': 'For a typical molecular liquid, [MATH] s. (For water, at room temperature, for example, it is about 50 femtoseconds)', 'cond-mat-0603391-1-126-0': '### Hydrodynamic time-scales for colloids', 'cond-mat-0603391-1-127-0': 'Hydrodynamic interactions propagate by momentum (vorticity) diffusion, and also by sound.', 'cond-mat-0603391-1-127-1': 'The sonic time [EQUATION] it takes a sound wave to travel the radius of a colloid is typically very small, on the order of 1 ns for a colloid of radius [MATH]m. For that reason, sound effects are often ignored for colloidal suspensions under the assumption that they will have dissipated so quickly that they have no noticeably influence on the dynamics.', 'cond-mat-0603391-1-127-2': 'However, some experiments [CITATION] and theory [CITATION] do find effects from sound waves on colloidal hydrodynamics, so that this issue is not completely settled yet.', 'cond-mat-0603391-1-128-0': 'The kinematic time, [MATH], defined in Eq. ([REF]) (and Table [REF]) as the time for momentum (vorticity) to diffuse over the radius of a colloid, is particularly important for hydrodynamics.', 'cond-mat-0603391-1-128-1': 'It sets the time-scale over which hydrodynamic interactions develop.', 'cond-mat-0603391-1-128-2': 'For a colloid of radius [MATH]m, [MATH] s, which is much faster the colloidal diffusion time [MATH].', 'cond-mat-0603391-1-129-0': 'When studying problems with a finite flow velocity, another hydrodynamic time-scale emerges.', 'cond-mat-0603391-1-129-1': 'The Stokes time [MATH], defined in Eq. ([REF]) (and Table [REF]) as the time for a colloid to advect over its own radius, can be related to the kinematic time by the relation [MATH].', 'cond-mat-0603391-1-129-2': 'Because colloidal particles are in the Stokes regime where Re [MATH], we find [MATH].', 'cond-mat-0603391-1-130-0': 'Simulation and analytical methods based on the Oseen tensor and its generalizations, e.g. [CITATION], implicitly assume that the hydrodynamic interactions develop instantaneously, i.e. that [MATH].', 'cond-mat-0603391-1-130-1': 'For the low Re number regime of colloidal dispersions this is indeed a good approximation.', 'cond-mat-0603391-1-130-2': 'Of course it must be kept in mind that [MATH] is a diffusive time-scale, so that the distance over which it propagates grows as [MATH].', 'cond-mat-0603391-1-130-3': 'Thus the approximation of instantaneous hydrodynamics must be interpreted with some care for effects on larger length-scales.', 'cond-mat-0603391-1-131-0': '### Brownian time-scales for colloids', 'cond-mat-0603391-1-132-0': 'The fastest time-scale relevant to the colloidal Brownian motion is the Fokker-Planck time [MATH] defined in [CITATION] as the time over which the force-force correlation function decays.', 'cond-mat-0603391-1-132-1': 'It is related to [MATH], the time over which the fluid looses memory of its velocity because the forces on the colloid are caused by velocity differences between the colloid and fluid particles.', 'cond-mat-0603391-1-132-2': 'These velocity differences de-correlate on a time scale of order [MATH].', 'cond-mat-0603391-1-133-0': 'The next time-scale in this series is the Brownian time: [EQUATION] where [MATH] is the Stokes friction for stick boundary conditions.', 'cond-mat-0603391-1-133-1': 'It is often claimed that this measures the time for a colloid to lose memory of its velocity.', 'cond-mat-0603391-1-133-2': 'However, as discussed in Appendix [REF], this picture, based on the Langevin equation, is in fact incorrect for colloids.', 'cond-mat-0603391-1-133-3': 'Nevertheless, [MATH] has the advantage that it can easily be calculated and so we will use it as a simple upper bound on the colloid velocity de-correlation time.', 'cond-mat-0603391-1-134-0': 'Perhaps the most important time-scale relevant for Brownian motion is the diffusion time [MATH], described in Eq. ([REF]) as the time for a colloidal particle to diffuse over its radius.', 'cond-mat-0603391-1-134-1': 'For a colloid of radius [MATH]m, [MATH] s, and even though [MATH], it remains much larger than most microscopic time-scales in the mesoscopic colloidal regime.', 'cond-mat-0603391-1-135-0': '## Time-scales for coarse-grained simulation', 'cond-mat-0603391-1-136-0': 'In the previous subsection we saw that the relevant time-scales for a single colloid in a solvent can span as many as 15 orders of magnitude.', 'cond-mat-0603391-1-136-1': 'Clearly it would be impossible to bridge all the time-scales of a physical colloidal system - from the molecular [MATH] to the mesoscopic [MATH] - in a single simulation.', 'cond-mat-0603391-1-136-2': 'Thankfully, it is not necessary to exactly reproduce each of the different time-scales in order to achieve a correct coarse-graining of colloidal dynamics.', 'cond-mat-0603391-1-136-3': 'As long as they are clearly separated, the correct physics should still emerge.', 'cond-mat-0603391-1-137-0': 'Since we take the view that the "solvent" particles are really a Navier Stokes solver with noise, what is needed to reproduce Brownian behavior is first of all that the colloid experiences random kicks from the solvent on a short enough time-scale.', 'cond-mat-0603391-1-137-1': 'By dramatically reducing the number of "molecules" in the solvent, the number of kicks per unit of time is similarly reduced.', 'cond-mat-0603391-1-137-2': 'Here we identify the Fokker-Planck time [MATH] as the time-scale on which the colloid experiences random Brownian motion.', 'cond-mat-0603391-1-137-3': "For Brownian motion it doesn't really matter how the kicks are produced, they could be completely uncorrelated, but since we also require that the solvent transports momentum, solvent particles must have some kind of correlation, which in our case is represented by the time [MATH].", 'cond-mat-0603391-1-137-4': 'Other colloid relaxation times should be much longer than this.', 'cond-mat-0603391-1-137-5': 'We therefore require first of all that [MATH] and [MATH].', 'cond-mat-0603391-1-138-0': "Secondly, the colloid's VACF should decay to zero well before it has diffused or convected over its own radius.", 'cond-mat-0603391-1-138-1': 'A proper separation of time-scales in a coarse-graining scheme then requires that [MATH] as well as [MATH] for systems where convection is important.', 'cond-mat-0603391-1-139-0': 'Finally, for the correct hydrodynamics to emerge we require that [MATH].', 'cond-mat-0603391-1-139-1': 'But this separation no longer needs to be over many orders of magnitude.', 'cond-mat-0603391-1-139-2': 'As argued in [CITATION], one order of magnitude separation between time-scales should be sufficient for most applications.', 'cond-mat-0603391-1-140-0': 'In the next few subsections we discuss in more quantitative detail how our coarse-graining strategy telescopes down the hierarchy of time-scales in order to maximize the efficiency of a simulation while still retaining key physical features.', 'cond-mat-0603391-1-141-0': '### SRD fluid time-scales', 'cond-mat-0603391-1-142-0': 'In SRD the physical time-scale [MATH] is coarse-grained out, and the effect of the collisions calculated in an average way every time-step [MATH].', 'cond-mat-0603391-1-142-1': 'The time-scale [MATH] on which the velocity correlations decay can be quite easily calculated from a random-collision approximation.', 'cond-mat-0603391-1-142-2': 'Following [CITATION]: [MATH].', 'cond-mat-0603391-1-142-3': 'For our parameters, [MATH], we find [MATH].', 'cond-mat-0603391-1-142-4': 'However, it should be kept in mind that for small [MATH] the exponential decay with [MATH] turns over to a slower algebraic decay at larger [MATH] [CITATION].', 'cond-mat-0603391-1-142-5': 'The amplitude of this "tail" is, however, quite small.', 'cond-mat-0603391-1-143-0': '### Hydrodynamic time-scales for simulation', 'cond-mat-0603391-1-144-0': 'For our choice of units [MATH], so that the sonic time of Eq. ([REF]) reduces to [EQUATION] which is independent of [MATH] or [MATH].', 'cond-mat-0603391-1-145-0': 'In the limit of small [MATH], the ratio of the kinematic time [MATH] to [MATH] can be simplified to the following form: [EQUATION] so that the condition [MATH] is very easy to fulfill.', 'cond-mat-0603391-1-145-1': 'Furthermore, under the same approximations, the ratio [MATH] so that for [MATH] too small the kinematic time becomes faster than the sonic time.', 'cond-mat-0603391-1-145-2': 'For the simulation parameters used in [CITATION] (and in Fig. [REF]), we find [MATH].', 'cond-mat-0603391-1-146-0': '### Brownian time-scales for simulation', 'cond-mat-0603391-1-147-0': 'We expect the Fokker Planck time [MATH] to scale as [MATH], since this is roughly equivalent to the time [MATH] over which the fluid velocities will have randomized.', 'cond-mat-0603391-1-147-1': 'In Fig. [REF] we plot the force-force correlation function [EQUATION]', 'cond-mat-0603391-1-147-2': 'Its short time behavior is dominated by the random forces, and the initial decay time gives a good estimate of [MATH].', 'cond-mat-0603391-1-147-3': 'As can be seen in the inset of Fig. [REF], [MATH], which is indeed on the order of [MATH] or [MATH].', 'cond-mat-0603391-1-148-0': 'If we make the reasonable approximation that [MATH], then the ratio of the Brownian time [MATH] to the kinematic time [MATH] simplifies to: [EQUATION] so that the ratio of [MATH] to [MATH] is the same as for a real physical system.', 'cond-mat-0603391-1-148-1': 'Since buoyancy requirements mean that normally [MATH], the time-scales are ordered as [MATH].', 'cond-mat-0603391-1-148-2': 'Moreover, since the ratio [MATH], independently of [MATH] or even [MATH] (as long as [MATH]), the condition that [MATH] is also not hard to fulfill in an SRD simulation.', 'cond-mat-0603391-1-149-0': 'These time-scales are illustrated in Fig. [REF] where we plot the normalized time correlation function: [EQUATION] with [MATH] the Cartesian velocity coordinate of a single colloid.', 'cond-mat-0603391-1-149-1': 'After the initial non-Markovian quadratic decay, the subsequent short-time exponential decay is not given by [MATH] but instead by the Enskog time [EQUATION] where [MATH] is the Enskog friction which can be calculated from kinetic theory [CITATION], see Eq. ([REF]), and generally [MATH].', 'cond-mat-0603391-1-149-2': 'The physical origins of this behavior are described in more detail in Appendix [REF].', 'cond-mat-0603391-1-150-0': 'The colloid diffusion coefficient is directly related to the friction by the Stokes-Einstein relation: [EQUATION]', 'cond-mat-0603391-1-150-1': 'If we assume that [MATH] and, for simplicity, that [MATH], i.e. we ignore the Enskog contribution, then the diffusion time scales as: [EQUATION] so that [MATH] is not hard to fulfill.', 'cond-mat-0603391-1-151-0': 'It is also instructive to examine the ratio of the diffusion time [MATH] to the kinematic time [MATH]: [EQUATION]', 'cond-mat-0603391-1-151-1': 'In general we advocate keeping [MATH] small to increase the Sc number [MATH], and since another obvious constraint is [MATH], there is not too much difficulty achieving the desired separation of time-scales [MATH].', 'cond-mat-0603391-1-152-0': 'As a concrete example of how these time-scales are separated in a simulation, consider the parameters used in Fig. [REF] for which we find [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0603391-1-152-1': 'But more generally, what the analysis of this section shows is that obtaining the correct hierarchy of time-scales: [EQUATION] is virtually guaranteed once the conditions on dimensionless numbers, detailed in Section [REF], are fulfilled.', 'cond-mat-0603391-1-153-0': '## Measurements of diffusion', 'cond-mat-0603391-1-154-0': 'To further test the hybrid MD-SRD coarse-graining scheme, we plot in Fig. [REF] the time-dependent diffusion coefficient [MATH], defined as: [EQUATION] for a number of different volume fractions [MATH].', 'cond-mat-0603391-1-154-1': 'Note that the convergence slows with increasing volume fraction [MATH].', 'cond-mat-0603391-1-154-2': 'The infinite time integral gives the diffusion coefficient, i.e. [MATH].', 'cond-mat-0603391-1-154-3': 'In the limit of low densities the diffusion coefficient has been predicted to take the form [MATH] with 1.1795 for slip boundary spheres [CITATION], and this provides a good fit at the lower volume fractions.', 'cond-mat-0603391-1-155-0': 'In Figure [REF] we plot the mean-square displacement of a Cartesian component of the colloidal particle position.', 'cond-mat-0603391-1-155-1': 'As highlighted in the inset, at short times there is an initial ballistic regime due to motion at an average mean square velocity [MATH], resulting in a mean-square displacement [MATH].', 'cond-mat-0603391-1-155-2': 'At times [MATH], the motion is clearly diffusive, as expected, with a linear dependence on [MATH] and a slope of [MATH], where [MATH] is given by the infinite time limit of Eq. ([REF]).', 'cond-mat-0603391-1-155-3': 'On the scale of the plot the asymptotic regime is reached well before the time [MATH] over which the particle has diffused, on average, over its radius.', 'cond-mat-0603391-1-155-4': 'In Appendix [REF] the behavior of [MATH] and the related mean-square displacement are discussed in more detail for time-scales [MATH].', 'cond-mat-0603391-1-156-0': '# Mapping between a physical and coarse-grained systems', 'cond-mat-0603391-1-157-0': 'The ultimate goal of any coarse-graining procedure is to correctly describe physical phenomena in the natural world.', 'cond-mat-0603391-1-157-1': 'As we have argued in this paper, it is impossible to bridge, in a single simulation, all the time and length-scales relevant to a colloid suspended in a solvent.', 'cond-mat-0603391-1-157-2': 'Compromises must be made.', 'cond-mat-0603391-1-157-3': 'Nevertheless, a fruitful mapping from a coarse-grained simulation to a physical system is possible, and greatly facilitated by expressing the physical properties of interest in dimensionless terms.', 'cond-mat-0603391-1-157-4': 'The best way to illustrate this is with some examples.', 'cond-mat-0603391-1-158-0': '## Example 1: Mapping to diffusive and Stokes time-scales', 'cond-mat-0603391-1-159-0': 'For many dynamic phenomena in colloidal dispersions, the most important time-scale is the diffusion time [MATH].', 'cond-mat-0603391-1-159-1': 'To map a coarse-grained simulation onto a real physical system one could therefore equate the diffusion time and the colloid radius [MATH] of the simulation to that of the real physical system.', 'cond-mat-0603391-1-159-2': 'For example, take the simulation parameters from table [REF] in Appendix [REF] for [MATH], and compare these to the properties of colloids of radius [MATH]m and [MATH] nm from table [REF].', 'cond-mat-0603391-1-159-3': 'For the larger colloids, [MATH]s, and equating this to the simulation [MATH] means that [MATH]m and [MATH] s. Performing the same exercise for the smaller colloid, where [MATH] s, results in [MATH] nm and [MATH] s.', 'cond-mat-0603391-1-159-4': 'The first thing that becomes obvious from this exercise is that "physical" time is set here not by the coarse-grained simulation, but by the particular system that one is comparing to.', 'cond-mat-0603391-1-160-0': 'If the system is also subjected to flow, then a second time-scale emerges, the Stokes time [MATH].', 'cond-mat-0603391-1-160-1': 'As long as the physical Pe number is achievable without compromising the simulation quality, then this time-scale can simultaneously be set equal to that of the physical system.', 'cond-mat-0603391-1-160-2': 'Behavior that depends primarily on these two time-scales should then be correctly rendered by the SRD hybrid MD scheme described in this paper.', 'cond-mat-0603391-1-161-0': '## Example 2: Mapping to kinematic time-scales', 'cond-mat-0603391-1-162-0': 'By fixing [MATH] or [MATH], as in example 1, other time-scales are not correctly reproduced.', 'cond-mat-0603391-1-162-1': 'For the large and small colloid systems described above, we would find [MATH] s and [MATH] s respectively, which contrasts with the physical values of [MATH] s and [MATH] s shown in Table [REF].', 'cond-mat-0603391-1-162-2': 'In other words, fixing [MATH] results in values of [MATH] that are too large by orders of magnitude.', 'cond-mat-0603391-1-162-3': 'But of course these much larger values of [MATH] are by design, because the SRD simulation is optimized by compacting the very large physical hierarchy of time-scales.', 'cond-mat-0603391-1-162-4': "If we are interested in processes dominated by [MATH], having different [MATH] doesn't matter, so long as [MATH].", 'cond-mat-0603391-1-163-0': 'If, on the other hand, we want to describe processes that occur on much smaller time-scales, for example long-time tails in colloidal VACFs, then simulation times [MATH] could be mapped onto the physical [MATH] instead.', 'cond-mat-0603391-1-163-1': "For the same [MATH] simulation parameters used above, the [MATH]m system now maps onto [MATH]m, and [MATH] s with of course a strongly underestimated diffusion time: [MATH] s instead of [MATH] s. Similarly for the [MATH] nm system, the mapping is to [MATH] nm, [MATH] s and [MATH] s instead of [MATH] s. For many processes on time-scales [MATH], this underestimate of [MATH] doesn't really matter.", 'cond-mat-0603391-1-163-2': 'It is only when time-scales are mixed that extra care must be employed in interpreting the simulation results.', 'cond-mat-0603391-1-163-3': 'And, if the processes can be described in terms of different dimensionless times, then it should normally still be possible to disentangle the simulation results and correctly map onto a real physical system.', 'cond-mat-0603391-1-164-0': 'Another lesson from these examples of mapping to different time-scales comes from comparing the kinematic viscosity of a physical system, which say takes the value [MATH]m[MATH]/s for water, to its value in SRD simulations.', 'cond-mat-0603391-1-164-1': 'If for the [MATH] SRD system described above we map time onto [MATH], as in Example 1, then for the [MATH]m system [MATH]m[MATH]/s, while for the [MATH] nm system [MATH]m[MATH]/s.', 'cond-mat-0603391-1-164-2': 'As expected, both are much smaller than the true value in water.', 'cond-mat-0603391-1-164-3': 'If we instead map the times to [MATH], then of course the kinematic viscosity takes the same value as for the physical system (since we also set the length-scales equal to the physical length-scales).', 'cond-mat-0603391-1-164-4': 'This apparent ambiguity in defining a property like the kinematic viscosity is inherent in many coarse-graining schemes.', 'cond-mat-0603391-1-164-5': 'We take the view that the precise value of [MATH] is not important, the only thing that matters is that it is large enough to ensure a proper separation of times-scales and the correct regime of hydrodynamic numbers.', 'cond-mat-0603391-1-165-0': '## Example 3: Mapping mass and temperature', 'cond-mat-0603391-1-166-0': 'In the examples above, we have only discussed setting lengths and times.', 'cond-mat-0603391-1-166-1': 'This leaves either the mass [MATH] or temperature (thermal energy) [MATH] still to be set.', 'cond-mat-0603391-1-166-2': 'Because for many dynamic properties the absolute mass is effectively an irrelevant variable (although relative masses may not be) there is some freedom to choose.', 'cond-mat-0603391-1-166-3': 'One possibility would be to set [MATH], the mass of the colloid, equal to the physical mass.', 'cond-mat-0603391-1-166-4': 'For an [MATH]m neutrally buoyant colloid this sets [MATH] g, (equal to about [MATH] water molecules) while for an [MATH] nm colloid we find [MATH] g (equal to about [MATH] water molecules).', 'cond-mat-0603391-1-166-5': 'By construction this procedure produces the correct physical [MATH].', 'cond-mat-0603391-1-166-6': 'If we fix time with [MATH], this will therefore also generate the correct shear viscosity [MATH].', 'cond-mat-0603391-1-166-7': 'If instead we fix [MATH], then [MATH] will be much smaller than the physical value.', 'cond-mat-0603391-1-166-8': 'For either choice of time-scales, other properties, such as the fluid self-diffusion constant or the number density, will have different values from the underlying physical system.', 'cond-mat-0603391-1-167-0': 'Setting the mass in this way means that the unit of thermal energy [MATH] would also be very large, leading to the appearance of very low temperatures.', 'cond-mat-0603391-1-167-1': 'However, because temperature only determines behavior through the way it scales potentials, it is quite easy to simply re-normalize the effective energy scales and obtain the same behavior as for the physical system.', 'cond-mat-0603391-1-167-2': 'Alternatively one could set the temperature equal to that of a physical system, but that would mean that the masses would again differ significantly from the real physical system.', 'cond-mat-0603391-1-167-3': 'At any rate, for the same simulation (say of sedimentation at a given value of Pe) the values of mass or temperature depend on the physical system one compares to.', 'cond-mat-0603391-1-167-4': 'This apparent ambiguity (or flexibility) simply reflects the fact that the dominant effects of temperature and mass come into play as relative ratios and not as absolute values.', 'cond-mat-0603391-1-168-0': '## Example 4: Mapping to attractive potentials: problems with length-scales?', 'cond-mat-0603391-1-169-0': 'So far we have only treated one length-scale in the problem, the radius [MATH].', 'cond-mat-0603391-1-169-1': 'Implicitly we have therefore assumed that the colloid-colloid interaction does not have another intrinsic length-scale of its own.', 'cond-mat-0603391-1-169-2': 'For hard-sphere colloids this is strictly true, and for steep-repulsions such as the WCA form of Eq. ([REF]) it is also a fairly good assumption that the exact choice of the exponent [MATH] in this equation does not significantly affect the dynamical properties [CITATION].', 'cond-mat-0603391-1-169-3': 'Similarly, there is some freedom to set the repulsive fluid-colloid potential of Eq. ([REF]) in such a way as to optimize the simulation parameters, in particular [MATH].', 'cond-mat-0603391-1-169-4': 'The physical liquid-colloid interaction would obviously have a much more complex form, but SRD coarse-grains out such details.', 'cond-mat-0603391-1-169-5': 'A similar argument can be made for colloid and fluid interactions with hard walls, needed, for example, to study problems with confinement, for which, inter alia, SRD is particularly well suited.', 'cond-mat-0603391-1-170-0': 'Things become more complicated for colloids with an explicit attractive interaction, such as DLVO or a depletion potential [CITATION].', 'cond-mat-0603391-1-170-1': 'These potentials introduce a new length-scale, the range of the attraction.', 'cond-mat-0603391-1-170-2': 'The ratio of this length to the hard-core diameter helps determine the equilibrium properties of the fluid [CITATION].', 'cond-mat-0603391-1-170-3': 'In a physical system this ratio may be quite small.', 'cond-mat-0603391-1-170-4': 'Keeping the ratio the same in the simulations then leads to potentials that are very steep on the scale of [MATH].', 'cond-mat-0603391-1-170-5': 'The MD time-step [MATH] may need to be very small in order to properly integrate the MD equations of motion.', 'cond-mat-0603391-1-170-6': 'Such a small [MATH] can make a simulation very inefficient, as was found in [CITATION] who followed this strategy and were forced to use [MATH]!', 'cond-mat-0603391-1-170-7': '(This is mainly caused by their explicit lubrication force) In general, we advocate adapting the potential to maximize simulation efficiency, while preserving key physical properties such as the topology of the phase diagram.', 'cond-mat-0603391-1-170-8': 'For example, the attractive energy scale can be set by the dimensionless reduced second virial coefficient, which gives an excellent approximation for how far one is from the liquid-liquid critical point [CITATION].', 'cond-mat-0603391-1-170-9': 'When the attractive interaction range is less than about [MATH] of the colloid hard-core diameter, the colloidal "liquid" phase becomes metastable with respect to the fluid-solid phase-transition.', 'cond-mat-0603391-1-170-10': 'At low colloid packing fraction [MATH] this leads to so-called "energetic fluid" behavior [CITATION].', 'cond-mat-0603391-1-170-11': 'To simulate a colloidal suspension in this "energetic fluid" regime, having a range of less than [MATH] of [MATH] should suffice.', 'cond-mat-0603391-1-170-12': 'In this way adding attractions does not have to make the simulation much less efficient.', 'cond-mat-0603391-1-170-13': 'Moreover the effects of these constraints, placed by the colloid-colloid interaction on [MATH], are to some degree mitigated by the fact that it is usually the colloid-fluid interaction which sets this time-scale.', 'cond-mat-0603391-1-171-0': 'There are still a few final subtleties to mention.', 'cond-mat-0603391-1-171-1': 'First of all, using small [MATH] in the simulations may greatly increase efficiency, but it also leads to a relatively larger contribution of the Enskog friction to the total friction in Eq. ([REF]).', 'cond-mat-0603391-1-171-2': 'For physical colloids in a molecular solvent the exact magnitude of the Enskog friction is still an open question, but undoubtedly for larger colloids it is almost negligible.', 'cond-mat-0603391-1-171-3': 'Some of the Enskog effects can simply be absorbed into an effective hydrodynamic radius [MATH], but others must be interpreted with some care.', 'cond-mat-0603391-1-172-0': 'Another regime where we might expect to see deviations beyond a simple re-scaling of time-scales would be at very short times.', 'cond-mat-0603391-1-172-1': 'For example, when colloids form a crystal, their oscillations can be de-composed into lattice phonons.', 'cond-mat-0603391-1-172-2': 'All but the longest wave-length longitudinal oscillations are overdamped due to the viscous drag and backflow effects of the solvent [CITATION].', 'cond-mat-0603391-1-172-3': 'In SRD, however, some of the very short wavelength oscillations might be preserved due to the fact that the motion on those time-scales is still ballistic.', 'cond-mat-0603391-1-172-4': 'These may have an impact on the interpretation of SRD simulations of microrheology.', 'cond-mat-0603391-1-173-0': 'In summary, these examples demonstrate that as long as the coarse-grained simulation produces the correct hydrodynamic and Brownian behavior then there is considerable freedom in assigning real physical values to the parameters in the simulation.', 'cond-mat-0603391-1-173-1': 'Attractive interactions may introduce a new length-scale, but a judicious choice of a model potential should still reproduce the correct physical behavior.', 'cond-mat-0603391-1-173-2': 'The same simulation may therefore be mapped onto multiple different physical systems.', 'cond-mat-0603391-1-173-3': 'Conversely, for the same physical system, different choices can be made for the time-scales that are mapped to physical values.', 'cond-mat-0603391-1-174-0': '# Conclusion', 'cond-mat-0603391-1-175-0': 'Correctly rendering the principal Brownian and hydrodynamic properties of colloids in solution is a challenging task for computer simulation.', 'cond-mat-0603391-1-175-1': 'In this article we have explored how treating the solvent with SRD, which coarse-grains the collisions between solvent particles over both time and space, leads to an efficient solution of the thermo-hydrodynamic equations of the solvent, in the external field provided by the colloids.', 'cond-mat-0603391-1-175-2': 'Although it is impossible to simulate the entire range of time-scales encountered in real colloidal suspensions, which can span as much as 15 orders of magnitude, we argue that this is also not necessary.', 'cond-mat-0603391-1-175-3': 'Instead, by recognizing firstly that hydrodynamic effects are governed by a set of dimensionless numbers, and secondly that the full hierarchy of time-scales can be telescoped down to a much more manageable range while still being properly physically separated, we demonstrate that the key Brownian and hydrodynamic properties of a colloidal suspension can indeed be captured by our SRD based coarse-graining scheme.', 'cond-mat-0603391-1-176-0': 'In particular, to simulate in the colloidal regime, the modeler should ensure that the dimensionless hydrodynamic numbers such as the Mach, Reynolds and Knudsen numbers are small enough ([MATH]), and the Schmidt number is large enough ([MATH]).', 'cond-mat-0603391-1-176-1': 'While these numbers are constrained by approximate limits, the Peclet number, which measures the relative strength of the convective transport to the diffusive transport of the colloids, can be either larger or smaller than 1, depending on physical properties such as the colloidal size and the strength of the external field that one wants to study.', 'cond-mat-0603391-1-176-2': 'It turns out that if the hydrodynamic numbers above are chosen correctly, it is usually not so difficult to also satisfy the proper separation of the time-scales.', 'cond-mat-0603391-1-177-0': 'We explored how to reach the desired regime of hydrodynamic numbers and time-scales by tuning the simulation parameters.', 'cond-mat-0603391-1-177-1': 'The two most important parameters are the dimensionless mean-free path [MATH] which measures what fraction of a cell size [MATH] an SRD particle travels between collisions performed at every time-step [MATH], and the ratio of the colloid radius [MATH] to the SRD cell size [MATH].', 'cond-mat-0603391-1-177-2': 'The general picture that emerges is that the collision time [MATH] must be chosen so that [MATH] is small since this helps keep the Schmidt number high, and both the Knudsen and Reynolds numbers low.', 'cond-mat-0603391-1-177-3': 'Of course for computational efficiency, the collision time should not be too small either.', 'cond-mat-0603391-1-177-4': 'Similarly although a larger colloid radius means that the hydrodynamic fields are more accurately rendered, this should be tempered by the fact that the simulation efficiency drops off rapidly with increasing colloid radius.', 'cond-mat-0603391-1-177-5': 'We find that choosing [MATH], which may seem small, already leads to accurate hydrodynamic properties.', 'cond-mat-0603391-1-178-0': 'A number of subtleties occur at the fluid-colloid interface, which may induce spurious depletion interactions between the colloids, but these can be avoided by careful choice of potential parameters for slip boundary conditions.', 'cond-mat-0603391-1-178-1': 'Stick boundary conditions can also be implemented, and we advocate using stochastic bounce-back rules to achieve this coarse-graining over the fluid-colloid interactions.', 'cond-mat-0603391-1-179-0': 'The simplicty of the SRD solvent facilitates the calculation of the the short-time behavior of the VACF from kinetic theory.', 'cond-mat-0603391-1-179-1': 'We argue that for times shorter than the sonic time [MATH], where hydrodynamic collective modes will not yet have fully developed, the decay of the VACF is typically much more rapid than the prediction of the simple Langevin equation.', 'cond-mat-0603391-1-179-2': 'The ensuing Enskog contribution to the total friction should be added in parallel to the microscopic hydrodynamic friction, and from the sum an analytic expression for the effective hydrodynamic radius [MATH] can be derived which is consistent with independent measurements of the long-ranged hydrodynamic fields.', 'cond-mat-0603391-1-180-0': 'Although we have shown how to correctly render the principal Brownian and hydrodynamic behavior of colloids in solution there is, as always, no such thing as a free lunch.', 'cond-mat-0603391-1-180-1': 'The price to be paid is an inevitable consequence of compressing together the hierarchy of time and length-scales: not all the simulation parameters can be simultaneously mapped onto a physical system of interest.', 'cond-mat-0603391-1-180-2': 'However the cost of the "lunch" can be haggled down by recognizing that only a few physical parameters are usually important.', 'cond-mat-0603391-1-180-3': 'For example, one could map the simulation onto real physical time, say through [MATH] and [MATH], or alternatively through [MATH]; there is some freedom (or ambiguity) in the choice of which time scale to use.', 'cond-mat-0603391-1-180-4': 'The correspondence with physical reality becomes more complex when different physical time-scales mix, but in practice they can often be disentangled when both the coarse-grained simulation and the physical system are expressed in terms of dimensionless numbers and ratios.', 'cond-mat-0603391-1-180-5': 'In other words, there is no substitute for careful physical insight which is, as always, priceless.', 'cond-mat-0603391-1-181-0': 'A number of lessons can be drawn from this exercise that may be relevant for other coarse-graining schemes.', 'cond-mat-0603391-1-181-1': 'SRD is closely related to Lattice-Boltzmann and to the Lowe-Anderson thermostat, and many of the conclusions above should transfer in an obvious way.', 'cond-mat-0603391-1-181-2': 'Similarly, a number of lessons can be drawn for Dissipative Particle Dynamics.', 'cond-mat-0603391-1-181-3': 'For example, we argue that the physical interpretation is facilitated by recognizing that, just as for SRD, the DPD particles should not be viewed as "clumps of fluid" but rather as a convenient computational tool to solve the underlying thermo-hydrodynamic equations.', 'cond-mat-0603391-1-181-4': 'And finally, our measurements of the VACF show explicitly that Brownian Dynamics does not correctly capture either the long or the short-time decay of colloidal VACFs.', 'cond-mat-0603391-1-182-0': 'A major advantage of SRD is the relative ease with which solutes and external boundary conditions (walls) are introduced.', 'cond-mat-0603391-1-182-1': 'Boundaries may be hard or soft, with either stick or slip conditions.', 'cond-mat-0603391-1-182-2': 'This method is therefore particularly promising for simulations in the fields of bio- and nanofluidics, where small meso particles flow in constrained geometries, possibly confined by soft fluctuating walls.', 'cond-mat-0603391-1-182-3': 'We are currently exploring these possibilities.', 'cond-mat-0603391-1-183-0': 'JTP thanks the EPSRC and IMPACT Faraday, and AAL thanks the Royal Society (London) for financial support.', 'cond-mat-0603391-1-183-1': 'We thank W. Briels, H. Lowen, J. Lopez Lopez, J. Sane, A. Mayhew Seers, I. Pagonabarraga, A. Wilber, and A. Wysocki for very helpful conversations, and W. den Otter for a careful reading of the manuscript.', 'cond-mat-0603391-1-184-0': '# Thermostating of SRD under flow', 'cond-mat-0603391-1-185-0': 'When an external field is applied to the fluid (or to objects embedded in the fluid), energy is pumped into the system and, as a consequence, the average temperature will rise.', 'cond-mat-0603391-1-185-1': 'To prevent this from happening, the system must be coupled to a thermostat.', 'cond-mat-0603391-1-185-2': 'In order not to influence the average flow, thermostating requires a local and Galilean invariant definition of temperature.', 'cond-mat-0603391-1-186-0': 'We achieve this by relating the instantaneous local temperature in a cell to the mean square deviation of the fluid particle velocities from the center of mass velocity of that cell.', 'cond-mat-0603391-1-186-1': 'To minimize interference of the thermostat with the dynamics, we choose to measure the overall temperature as: [EQUATION]', 'cond-mat-0603391-1-186-2': 'In Eq. ([REF]), [MATH] is the instantaneous number of fluid particles within cell [MATH].', 'cond-mat-0603391-1-186-3': 'Three degrees of freedom must be subtracted for fixing the center of mass velocity of each cell (note that this implies that the local temperature is not defined in cells containing 0 or 1 particle).', 'cond-mat-0603391-1-186-4': 'During the simulations, the temperature [MATH] is measured every [MATH] (this can be done very efficiently because the relative velocities are readily available in the collision step routine).', 'cond-mat-0603391-1-186-5': 'The thermostat then acts by rescaling all relative velocities [MATH] by a factor [MATH].', 'cond-mat-0603391-1-186-6': 'This strict enforcement of the overall temperature may be relaxed by allowing appropriate fluctuations, but in view of the very large number of fluid particles (typically [MATH]) this will not have a measurable effect on the dynamics of the fluid particles.', 'cond-mat-0603391-1-187-0': '# The Langevin equation and memory effects', 'cond-mat-0603391-1-188-0': 'Within the Brownian approximation each Cartesian component of the colloid velocity [MATH] is described by a simple Langevin equation of the form: [EQUATION] without any memory effect in the friction coefficient [MATH], or in the random force [MATH].', 'cond-mat-0603391-1-188-1': 'This Langevin equation fundamentally arises from the assumption that the force-kicks are Markovian: each step is independent of any previous behavior.', 'cond-mat-0603391-1-188-2': 'Of course on a very short time-scale this is not true, but since Brownian motion is self-similar, the Markovian assumptions underlying Eq. ([REF]) might be expected to be accurate on longer time-scales.', 'cond-mat-0603391-1-188-3': 'Inspection of Fig. [REF] shows that the force-force correlation function ([REF]) does indeed decay on a time-scale [MATH] which is much faster than other Brownian time-scales in the system.', 'cond-mat-0603391-1-189-0': 'From the Langevin Eq. ([REF]) it follows that the velocity auto-correlation function (VACF) takes the form [EQUATION] where [MATH].', 'cond-mat-0603391-1-189-1': 'The VACF is directly related to the diffusion constant through the Green-Kubo relation: [EQUATION] and Eq. ([REF]) has the attractive property that its integral gives the correct Stokes-Einstein form for [MATH], and that its [MATH] limit gives the expected value from equipartition.', 'cond-mat-0603391-1-189-2': 'It is therefore a popular pedagogical device for introducing Brownian motion.', 'cond-mat-0603391-1-189-3': 'Notwithstanding its seductive simplicity, we will argue that this Langevin approach strongly underestimates the initial decay rate of the VACF, and badly overestimates its long-time decay rate.', 'cond-mat-0603391-1-190-0': '## VACF at long times', 'cond-mat-0603391-1-191-0': 'At longer times, as first shown in MD simulations by Alder and Wainwright [CITATION], the VACF shows an algebraic decay that is much slower than the exponential decay of Eq. ([REF]).', 'cond-mat-0603391-1-191-1': 'The difference is due to the breakdown of the Markovian assumption for times [MATH].', 'cond-mat-0603391-1-191-2': 'Memory effects arise because the momentum that a colloidal particle transfers to the fluid is conserved, with dynamics described by the Navier Stokes equations, resulting in long-ranged flow patterns that affect a colloid on much longer time-scales than Eq. ([REF]) suggests.', 'cond-mat-0603391-1-191-3': 'Calculations taking into account the full time-dependent hydrodynamics were performed by Hauge [CITATION] and Hinch [CITATION] (and others apparently much earlier [CITATION]).', 'cond-mat-0603391-1-191-4': 'For [MATH], i.e. a neutrally buoyant particle, these expressions can be written as [EQUATION] where we have chosen to use an integral form as in [CITATION].', 'cond-mat-0603391-1-191-5': 'Under the assumption that the sound effects have dissipated away, the only hydrodynamic time-scale is the kinematic time [MATH] defined in Eq. ([REF]).', 'cond-mat-0603391-1-191-6': 'The VACF must therefore scale with [MATH], independently of colloid size.', 'cond-mat-0603391-1-191-7': 'Eq. ([REF]) is written in a way to emphasize this scaling.', 'cond-mat-0603391-1-192-0': 'By expanding the denominator it is not hard to see that at longer times the well-known algebraic tail results: [EQUATION] with the same pre-factor that was found from mode-coupling theory [CITATION].', 'cond-mat-0603391-1-193-0': 'The integral of Eq. ([REF]) is [EQUATION] which means that all the hydrodynamic contributions to the friction [MATH] come from [MATH].', 'cond-mat-0603391-1-193-1': 'The integral converges slowly, as [MATH], because of the long-time tail.', 'cond-mat-0603391-1-193-2': 'The VACF can easily be related to the mean-square displacement, and deviations from purely diffusive motion are still observable for [MATH], and have been measured in experiments [CITATION].', 'cond-mat-0603391-1-194-0': 'Although the last equality in Eq. ([REF]) shows that the VACF tail amplitude is independent of colloid size, it should be kept in mind that the algebraic decay does not set in until [MATH].', 'cond-mat-0603391-1-194-1': 'Thus it is the relative (not absolute) effect of the tail that is the same for all colloids.', 'cond-mat-0603391-1-194-2': 'For example, at [MATH], the VACF will have decayed to [MATH] of its initial value of [MATH], independently of colloid mass [MATH].', 'cond-mat-0603391-1-194-3': 'Even though this tail amplitude may seem small, about half the weight of the integral ([REF]) that determines the friction is from [MATH].', 'cond-mat-0603391-1-194-4': 'The dominance of the non-Markovian hydrodynamic behavior in determining the friction for colloids does not appear to hold for microscopic fluids, where, for example for LJ liquids near the triple point, it is thought that the long-time tail only adds a few [MATH] to the friction coefficient [CITATION].', 'cond-mat-0603391-1-194-5': 'However, this is not due to the simple Langevin equation ([REF]) being more accurate, but rather to the modified short-time behavior of the VACF, caused by fluid correlations (a different non-Markovian effect).', 'cond-mat-0603391-1-195-0': '## VACF at short times', 'cond-mat-0603391-1-196-0': 'At short times the hydrodynamic contribution to the VACF ([REF]) reduces to [EQUATION] because, within a continuum description, [MATH] of the energy is dissipated as a sound wave at velocity [MATH], [EQUATION] where the sonic time [MATH] is defined in Table [REF].', 'cond-mat-0603391-1-196-1': "The sound wave doesn't contribute to the friction or diffusion since [EQUATION]", 'cond-mat-0603391-1-196-2': 'Because [MATH], the sound-wave contribution to the VACF decays much faster than the hydrodynamic contribution [MATH] or even than the Langevin approximation [MATH] (recall that [MATH] for neutrally buoyant colloids.)', 'cond-mat-0603391-1-196-3': 'Therefore, in the continuum picture, the short-time decay of the VACF is much faster than that suggested by the simple Langevin equation.', 'cond-mat-0603391-1-196-4': 'And even after the sound wave has decayed, [MATH] decays much more rapidly (faster than simple exponential) than Eq. ([REF]) for times [MATH].', 'cond-mat-0603391-1-196-5': 'SRD simulations confirm this more rapid short time decay.', 'cond-mat-0603391-1-196-6': 'For example, in Fig [REF] we show a direct comparison of the measured VACF to the continuum approach, where [MATH].', 'cond-mat-0603391-1-196-7': 'For short times the decay predicted from sound agrees reasonable well, and for long times the full hydrodynamic theory quantitatively fits the simulation data.', 'cond-mat-0603391-1-197-0': 'A closer look at the short-time behavior in our SRD simulations, both for slip boundaries as in Fig. [REF] and for stick-boundaries as in Fig [REF] or in [CITATION], shows that a better fit to the short-time exponential decay of the VACF is given by [EQUATION] where the Enskog time ([REF]) scales as: [EQUATION] for heavy (stick boundary) colloids in an SRD fluid.', 'cond-mat-0603391-1-197-1': 'Note that the Enskog decay time is very close to the decay time [MATH] of the sound wave ([REF]).', 'cond-mat-0603391-1-197-2': 'That the SRD Enskog time should scale as [MATH] is perhaps not too surprising, because that is the natural time-scale of the SRD fluid.', 'cond-mat-0603391-1-197-3': 'However, the fact that the pre-factors in the exponential of Eqs ([REF]) and ([REF]) are virtually the same may be accidental.', 'cond-mat-0603391-1-197-4': "Although in a real liquid the increased compressibility factor means [MATH] and [MATH] are lower than their values for an ideal gas, the two times change at slightly different rates so that we don't expect the same equivalence in physical systems.", 'cond-mat-0603391-1-198-0': 'In contrast to the sound wave, the integral over the Enskog VACF leads to a finite value: [EQUATION] which quantitatively explains the friction we measured in our simulations, as discussed in section [REF].', 'cond-mat-0603391-1-198-1': 'In contrast to the VACF from sound, the Enskog result cannot simply be added to the VACF from Eq. ([REF]) because this would violate the [MATH] limit of the VACF, set by equipartition.', 'cond-mat-0603391-1-198-2': 'However, because the Enskog contribution occurs on times [MATH], where sound and other collective modes of (compressible) hydrodynamics are not yet fully developed, it may be that Eqs. ([REF]) and ([REF]), derived from continuum theory, should be modified on these short time-scales.', 'cond-mat-0603391-1-198-3': 'Since the dominant contribution to the integral in Eq. ([REF]) is for longer times, the exact form of the hydrodynamic contribution ([REF]) on these short times ([MATH]) does not have much influence on the hydrodynamic part of the friction.', 'cond-mat-0603391-1-198-4': 'Thus the approximation of the short-time VACF by the Enskog result of Eq. ([REF]), and the longer-time ([MATH], but still [MATH]) VACF by Eq. ([REF]) may be a fairly reasonable empirical approach.', 'cond-mat-0603391-1-198-5': 'This approximation, together with the Green-Kubo relation ([REF]), then justifies, a-posteori, the parallel addition of frictions in Eq. ([REF]).', 'cond-mat-0603391-1-199-0': 'Just as was found for the long-time behavior, the short-time behavior of the VACF displays important deviations from the simple Langevin picture.', 'cond-mat-0603391-1-199-1': 'We find that the VACF decays much faster, on the order of the sonic time [MATH] rather than the Brownian time [MATH].', 'cond-mat-0603391-1-199-2': 'From the tables [REF] and [REF] one can see how large these differences can be.', 'cond-mat-0603391-1-199-3': 'For example, for colloids with [MATH], the short-time Enskog or sonic decay times are of order [MATH] s, while [MATH] is of order [MATH] s, so that the decay of the VACF will be dominated by [MATH], with its non-exponential behavior, over almost the entire time regime.', 'cond-mat-0603391-1-200-0': 'An example of the effect of the Enskog contribution on the friction or the mean-square displacement is given in Fig. [REF].', 'cond-mat-0603391-1-200-1': 'The integral of the VACF, [MATH] from Eq. ([REF]), which is directly related to the mean-square displacement [CITATION], is extracted from the simulations described in Fig [REF], and compared to [EQUATION] calculated using Eqs. ([REF]) and ([REF]).', 'cond-mat-0603391-1-200-2': 'Both show exactly the same [MATH] scaling for [MATH], but the simulated [MATH] is larger because the initial Enskog contribution means that [MATH], for [MATH], with [MATH] defined in Eq. ([REF]).', 'cond-mat-0603391-1-201-0': 'The slow algebraic convergence to the final diffusion coefficient, or related friction, also helps explain the scaling of the finite size effects of Eq. ([REF]).', 'cond-mat-0603391-1-201-1': 'A finite box of length [MATH] means that the measured [MATH] is cut off roughly at [MATH], and because [MATH] for [MATH] the effect on the diffusion constant can be written as [MATH] which explains the scaling form of Eq. ([REF]).', 'cond-mat-0603391-1-201-2': 'Fig. [REF] also demonstrates how much more rapidly the [MATH] from the Langevin Eq. ([REF]) converges to its final result [MATH] when compared to the simulations and hydrodynamic theories.', 'cond-mat-0603391-1-202-0': 'In summary, even though the simple Langevin equation ([REF]), without memory, may have pedagogical merit, it does a poor job of describing the VACF for colloids in solution.', 'cond-mat-0603391-1-203-0': '# Comparison of SRD to physical parameters', 'cond-mat-0603391-1-204-0': 'At the end of the day one would like to compare a coarse-grained simulation to a real physical colloidal dispersion.', 'cond-mat-0603391-1-204-1': 'To help facilitate such comparisons, we have summarized a number of physical parameters and dimensionless numbers for colloids in water, and also colloids in an SRD fluid.', 'cond-mat-0603391-1-205-0': 'Table [REF] compares the properties of a pure SRD fluid, in the limit of small mean-free path [MATH], to the properties of water.', 'cond-mat-0603391-1-206-0': 'Table [REF] summarizes the main scaling and values for a number of different properties of two different size neutrally buoyant colloids in water.', 'cond-mat-0603391-1-206-1': 'To approximate the Enskog friction we used a simplified form valid for stick boundary conditions in an ideal fluid, as in [CITATION], which ignores the compressibility and other expected complexities of real water.', 'cond-mat-0603391-1-206-2': 'These predictions should therefore be taken with a grain of salt.', 'cond-mat-0603391-1-206-3': 'For simplicity, we have ignored any potential Enskog effects on the diffusion constant or time-scales.', 'cond-mat-0603391-1-207-0': 'Table [REF] shows a very similar comparison for two different size colloids in an SRD fluid.', 'cond-mat-0603391-1-207-1': 'For simplicity we assume that there are no finite-size effects, which explains small differences when compared to similar parameters used in the text of the main article.', 'cond-mat-0603391-1-207-2': 'However, in contrast to Table [REF], we do include the Enskog contribution when calculating the total friction and related properties.'}

{'cond-mat-0603391-2-0-0': 'We describe in detail how to implement a coarse-grained hybrid Molecular Dynamics and Stochastic Rotation Dynamics simulation technique that captures the combined effects of Brownian and hydrodynamic forces in colloidal suspensions.', 'cond-mat-0603391-2-0-1': 'The importance of carefully tuning the simulation parameters to correctly resolve the multiple time and length-scales of this problem is emphasized.', 'cond-mat-0603391-2-0-2': 'We systematically analyze how our coarse-graining scheme resolves dimensionless hydrodynamic numbers such as the Reynolds number Re, which indicates the importance of inertial effects, the Schmidt number Sc, which indicates whether momentum transport is liquid-like or gas-like, the Mach number Ma, which measures compressibility effects, the Knudsen number Kn, which describes the importance of non-continuum molecular effects and the Peclet number Pe, which describes the relative effects of convective and diffusive transport.', 'cond-mat-0603391-2-0-3': 'With these dimensionless numbers in the correct regime the many Brownian and hydrodynamic time-scales can be telescoped together to maximize computational efficiency while still correctly resolving the physically relevant physical processes.', 'cond-mat-0603391-2-0-4': 'We also show how to control a number of numerical artifacts, such as finite size effects and solvent induced attractive depletion interactions.', 'cond-mat-0603391-2-0-5': 'When all these considerations are properly taken into account, the measured colloidal velocity auto-correlation functions and related self diffusion and friction coefficients compare quantitatively with theoretical calculations.', 'cond-mat-0603391-2-0-6': 'By contrast, these calculations demonstrate that, notwithstanding its seductive simplicity, the basic Langevin equation does a remarkably poor job of capturing the decay rate of the velocity auto-correlation function in the colloidal regime, strongly underestimating it at short times and strongly overestimating it at long times.', 'cond-mat-0603391-2-0-7': 'Finally, we discuss in detail how to map the parameters of our method onto physical systems, and from this extract more general lessons - keeping in mind that there is no such thing as a free lunch - that may be relevant for other coarse-graining schemes such as Lattice Boltzmann or Dissipative Particle Dynamics.', 'cond-mat-0603391-2-1-0': '# Introduction', 'cond-mat-0603391-2-2-0': 'Calculating the non-equilibrium properties of colloidal suspensions is a highly non-trivial exercise because these depend both on the short-time thermal Brownian motion, and the long-time hydrodynamic behavior of the solvent [CITATION].', 'cond-mat-0603391-2-2-1': 'Moreover, the hydrodynamic interactions are of a many-body character, and cannot usually be decomposed into a pairwise sum of inter-colloid forces.', 'cond-mat-0603391-2-3-0': 'The fundamental difficulty of fully including the detailed solvent dynamics in computer simulations becomes apparent when considering the enormous time and length-scale differences between mesoscopic colloidal and microscopic solvent particles.', 'cond-mat-0603391-2-3-1': 'For example, a typical colloid of diameter [MATH] will displace on the order of [MATH] water molecules!', 'cond-mat-0603391-2-3-2': 'Furthermore, a molecular dynamics (MD) scheme for the solvent would need to resolve time-scales on the order of [MATH] s to describe the inter-molecular forces, while a colloid of diameter [MATH] in water diffuses over its own diameter in about [MATH]s.', 'cond-mat-0603391-2-4-0': 'Clearly, simulating even an extremely crude molecular model for the solvent on the time-scales of interest is completely out of the question: some form of coarse-graining is necessary.', 'cond-mat-0603391-2-4-1': 'The object of this paper is to describe in detail one such scheme.', 'cond-mat-0603391-2-4-2': 'But before we do so, we first briefly discuss a subset of the wide variety of different simulation techniques that have been devised to describe the dynamics of colloidal suspensions.', 'cond-mat-0603391-2-5-0': 'At the simplest level, the effects of the solvent can be taken into account through Brownian dynamics (BD) [CITATION], which assumes that collisions with the solvent molecules induce a random displacement of the colloidal particle positions, as well as a local friction proportional to the their velocity.', 'cond-mat-0603391-2-5-1': 'Although, due to its simplicity, Brownian dynamics is understandably very popular, it completely neglects momentum transport through the solvent - as described by the Navier Stokes equations - which leads to long-ranged hydrodynamic interactions (HI) between the suspended particles.', 'cond-mat-0603391-2-5-2': 'These HI may fall off as slowly as [MATH], and can qualitatively affect the dynamical behavior of the suspension [CITATION].', 'cond-mat-0603391-2-6-0': 'Beginning with the pioneering work of Ermak and McCammon [CITATION], who added a simple representation of the Oseen tensor [CITATION] to their implementation of BD, many authors have applied computational approaches that include the HI by an approximate analytical form.', 'cond-mat-0603391-2-6-1': 'The most successful of these methods is Stokesian Dynamics [CITATION], which can take into account higher order terms in a multipole expansion of the HI interactions.', 'cond-mat-0603391-2-6-2': 'Although some more sophisticated recent implementations of Stokesian Dynamics have achieved an [MATH] scaling with the number of colloids [CITATION], this method is still relatively slow, and becomes difficult to implement in complex boundary conditions.', 'cond-mat-0603391-2-7-0': 'Another way to solve for the HI is by direct numerical simulation (DNS) methods, where the solid particles are described by explicit boundary conditions for the Navier-Stokes equations [CITATION].', 'cond-mat-0603391-2-7-1': 'These methods are better adapted to non-Brownian particles, but can still be applied to understand the effects of HI on colloidal dynamics.', 'cond-mat-0603391-2-7-2': 'The fluid particle dynamics (FPD) method of Tanaka and Araki [CITATION], and a related method by Yamamoto et al. [CITATION], are two important recent approaches to solving the Navier Stokes equations that go a step beyond standard DNS, and simplify the problem of boundary conditions by using a smoothed step function at the colloid surfaces.', 'cond-mat-0603391-2-7-3': 'In principle Brownian motion can be added to these methods [CITATION].', 'cond-mat-0603391-2-8-0': 'The difficulty of including complex boundary conditions in DNS approaches has stimulated the use of lattice-gas based techniques to resolve the Navier Stokes equations.', 'cond-mat-0603391-2-8-1': 'These methods exploit the fact that only a few conditions, such as (local) energy and momentum conservation, need to be satisfied to allow the correct (thermo) hydrodynamics to emerge in the continuum limit.', 'cond-mat-0603391-2-8-2': 'Greatly simplified particle collision rules, easily amenable to efficient computer simulation, are therefore possible, and complex boundary conditions, such as those that characterize moving colloids, are easier to treat than in DNS methods.', 'cond-mat-0603391-2-8-3': 'The most popular of these techniques is Lattice Boltzmann (LB) where a linearized and pre-averaged Boltzmann equation is discretized and solved on a lattice [CITATION].', 'cond-mat-0603391-2-8-4': 'Ladd has pioneered the application of this method to solid-fluid suspensions [CITATION].', 'cond-mat-0603391-2-8-5': 'This is illustrated schematically in Fig. [REF].', 'cond-mat-0603391-2-8-6': 'The solid particles are modeled as hollow spheres, propagated with Newtonian dynamics, and coupled to the LB solvent through a bounce-back collision rule on the surface.', 'cond-mat-0603391-2-8-7': 'The fluctuations that lead to Brownian motion were modeled by adding a stochastic term to the stress tensor.', 'cond-mat-0603391-2-8-8': 'Recently this has been criticized and an improved method to include Brownian noise that does not suffer from some lattice induced artifacts was suggested [CITATION].', 'cond-mat-0603391-2-8-9': 'A number of other groups have recently derived alternative ways of coupling a colloidal particle to a LB fluid [CITATION], in part to simulate the dynamics of charged systems.', 'cond-mat-0603391-2-9-0': 'The discrete nature of lattice based methods can also bring disadvantages, particularly when treating fluids with more than one length-scale.', 'cond-mat-0603391-2-9-1': 'Dissipative Particle Dynamics (DPD) [CITATION] is a popular off-lattice alternative that includes hydrodynamics and Brownian fluctuations.', 'cond-mat-0603391-2-9-2': 'It can be viewed as an extension of standard Newtonian MD techniques, but with two important innovations: 1) soft potentials that allow large time-steps and rapid equilibration; 2) a Galilean invariant thermostat that locally conserves momentum and therefore generates the correct Navier Stokes hydrodynamics in the continuum limit.', 'cond-mat-0603391-2-10-0': 'In fact, these two methodological advances can be separated.', 'cond-mat-0603391-2-10-1': 'Soft potentials such as those postulated for innovation 1) may indeed arise from careful equilibrium coarse-graining of complex fluids [CITATION].', 'cond-mat-0603391-2-10-2': 'However, their proper statistical mechanical interpretation, even for equilibrium properties, is quite subtle.', 'cond-mat-0603391-2-10-3': 'For example, a potential that generates the correct pair structure will not normally reproduce the correct virial pressure due to so called representability problems [CITATION].', 'cond-mat-0603391-2-10-4': 'When used in dynamical simulations, the correct application of effective potentials is even more difficult to properly derive.', 'cond-mat-0603391-2-10-5': 'For polymer dynamics, for instance, uncrossability constraints must be re-introduced to prevent coarse-grained polymers from passing through one another [CITATION].', 'cond-mat-0603391-2-10-6': 'Depletion interactions in a multi component solution also depend on the relative rates of depletant diffusion coefficients and particle flow velocities [CITATION].', 'cond-mat-0603391-2-10-7': 'At present, therefore, the statistical mechanical origins of innovation 1) of DPD, the use of soft potentials out of equilibrium, are at best obscure.', 'cond-mat-0603391-2-10-8': 'We are convinced that a correct microscopic derivation of the coarse-grained DPD representation of the dynamics, if this can indeed be done, will show that the interpretation of such soft potentials depends on dynamic as well as static (phase-space) averages.', 'cond-mat-0603391-2-10-9': 'Viewing the DPD particles as static "clumps" of underlying fluid is almost certainly incorrect.', 'cond-mat-0603391-2-10-10': 'It may, in fact, be more fruitful to abandon simple analogies to the potential energy of a Hamiltonian system, and instead view the interactions as a kind of coarse-grained self-energy [CITATION].', 'cond-mat-0603391-2-11-0': 'Innovation 2), on the other hand, can be put on firmer statistical mechanical footing, see e.g. [CITATION], and can be usefully employed to study the dynamics of complex systems with other types of inter-particle interactions [CITATION].', 'cond-mat-0603391-2-11-1': 'The main advantage of the DPD thermostat is that, by preserving momentum conservation, the hydrodynamic interactions that intrinsically arise from microcanonical MD are preserved for calculations in the canonical ensemble.', 'cond-mat-0603391-2-11-2': 'Other thermostats typically screen the hydrodynamic interactions beyond a certain length-scale [CITATION].', 'cond-mat-0603391-2-11-3': 'For weak damping this may not be a problem, but for strong damping it could be.', 'cond-mat-0603391-2-12-0': 'Simulating only the colloids with DPD ignores the dominant solvent hydrodynamics.', 'cond-mat-0603391-2-12-1': 'While the solvent could be treated directly by DPD, as suggested in Fig. [REF] (see also [CITATION]), this method is quite computationally expensive because the "solvent" particles interact through pairwise potentials.', 'cond-mat-0603391-2-13-0': 'In an important paper [CITATION], Lowe took these ideas further and combined the local momentum conservation of the DPD thermostat with the stochastic nature of the Anderson thermostat to derive a coarse-grained scheme, now called the Lowe-Anderson thermostat, which is much more efficient than treating the solvent with full DPD.', 'cond-mat-0603391-2-13-1': 'It has recently been applied, for example, to polymer dynamics [CITATION].', 'cond-mat-0603391-2-14-0': 'Independently, Malevanets and Kapral [CITATION] derived a method now called stochastic rotation dynamics (SRD) or also multiple particle collision dynamics (we choose the former nomenclature here).', 'cond-mat-0603391-2-14-1': 'In many ways it resembles the Lowe-Anderson thermostat [CITATION], or the much older direct simulation Monte Carlo (DSMC) method of Bird [CITATION].', 'cond-mat-0603391-2-14-2': "For all three of these methods, the particles are ideal, and move in a continuous space, subject to Newton's laws of motion.", 'cond-mat-0603391-2-14-3': 'At discrete time-steps, a coarse-grained collision step allows particles to exchange momentum.', 'cond-mat-0603391-2-14-4': 'In SRD, space is partitioned into a rectangular grid, and at discrete time-steps the particles inside each cell exchange momentum by rotating their velocity vectors relative to the center of mass velocity of the cell (see Fig. [REF]).', 'cond-mat-0603391-2-14-5': "Similarly, in Lowe's method, a particle can, with a certain probability, exchange its relative velocity with another that lies within a certain radius.", 'cond-mat-0603391-2-14-6': 'One can imagine other collision rules as well.', 'cond-mat-0603391-2-14-7': 'As long as they locally conserve momentum and energy, just as the lattice gas methods do, they generate the correct Navier Stokes hydrodynamics, although for SRD it is necessary to include a grid-shift procedure to enforce Galilean invariance, something first pointed out by Ihle and Kroll [CITATION].', 'cond-mat-0603391-2-14-8': 'An important advantage of SRD is that its simplified dynamics have allowed the analytic calculation of several transport coefficients [CITATION], greatly facilitating its use.', 'cond-mat-0603391-2-14-9': 'In the rest of this paper we will concentrate on the SRD method, although many of our general conclusions and derivations should be easily extendible to the Lowe-Anderson thermostat and related methods summarized in Fig. [REF].', 'cond-mat-0603391-2-15-0': 'SRD can be applied to directly simulate flow, as done for example in refs [CITATION], but its stochastic nature means that a noise average must be performed to calculate flow lines, and this may make it less efficient than pre-averaged methods like LB.', 'cond-mat-0603391-2-15-1': 'Where SRD becomes more attractive is for the simulation of complex particles embedded in a solvent.', 'cond-mat-0603391-2-15-2': 'This is because in addition to long-ranged HI, it also naturally contains Brownian fluctuations, and both typically need to be resolved for a proper statistical mechanical treatment of the dynamics of mesoscopic suspended particles.', 'cond-mat-0603391-2-15-3': 'For example, SRD has been used to study polymers under flow [CITATION], the effect of hydrodynamics on protein folding and polymer collapse [CITATION], and the conformations of vesicles under flow [CITATION].', 'cond-mat-0603391-2-15-4': 'In each of the previously mentioned examples, the suspended particles are coupled to the solvent by participating in the collision step.', 'cond-mat-0603391-2-16-0': "A less coarse-grained coupling can be achieved by allowing direct collisions, obeying Newton's laws of motion, between the SRD solvent and the suspended particles.", 'cond-mat-0603391-2-16-1': 'Such an approach is important for systems, such as colloidal suspensions, where the solvent and colloid time and length-scales need to be clearly separated.', 'cond-mat-0603391-2-16-2': 'Malevanets and Kapral [CITATION] derived such a hybrid algorithm that combined a full MD scheme of the solute-solute and solute-solvent interactions, while treating the solvent-solvent interactions via SRD.', 'cond-mat-0603391-2-16-3': 'Early applications were to a two-dimensional many-particle system [CITATION], and to the aggregation of colloidal particles [CITATION].', 'cond-mat-0603391-2-17-0': 'We have recently extended this approach, and applied it to the sedimentation of up to 800 hard sphere (HS) like colloids as a function of volume fraction, for a number of different values of the Peclet number [CITATION].', 'cond-mat-0603391-2-17-1': 'To achieve these simulations we adapted the method of Malevanets and Kapral [CITATION] in a number of ways that were only briefly described in ref. [CITATION].', 'cond-mat-0603391-2-17-2': 'In the current paper we provide more in depth analysis of the simulation method we used, and describe some potential pitfalls.', 'cond-mat-0603391-2-17-3': 'In particular we focus on the different time and length-scales that arise from our coarse-graining approach, as well as the role of dimensionless hydrodynamic numbers that express the relative importance of competing physical phenomena.', 'cond-mat-0603391-2-17-4': 'Very recently, another similar study of colloidal sedimentation and aggregation has been carried out [CITATION].', 'cond-mat-0603391-2-17-5': 'Some of our results and analysis are similar, but some are different, and we will mention these where appropriate.', 'cond-mat-0603391-2-18-0': 'After the introductory overview above, we begin in section [REF] by carefully describing the properties of a pure SRD fluid, focusing on simple derivations that highlight the dominant physics involved for each transport coefficient.', 'cond-mat-0603391-2-18-1': 'Section [REF] explains how to implement the coupling of a colloidal particle to an SRD solvent bath, and shows how to avoid spurious depletion interactions and how to understand lubrication forces.', 'cond-mat-0603391-2-18-2': 'In Section [REF] we analyze how optimizing the efficiency of particle based coarse-graining schemes affects different dimensionless hydrodynamic numbers, such as the Schmidt number, the Mach number, the Reynolds number, the Knudsen number, and the Peclet number.', 'cond-mat-0603391-2-18-3': 'Section [REF] describes the hierarchy of time-scales that determine the physics of a colloidal suspension, and compares these to the compressed hierarchy in our coarse-grained method.', 'cond-mat-0603391-2-18-4': 'In section [REF] we tackle the question of how to map from a coarse-grained simulation, optimized for computational efficiency, to a real colloidal system.', 'cond-mat-0603391-2-18-5': 'Finally after a Conclusions section, we include a few appendices that discuss amongst other things how to thermostat the SRD system (Appendix I); why the popular Langevin equation without memory effects does a remarkably poor job of capturing both the long and the short time dynamics of the colloidal velocity auto-correlation function (Appendix II); and, finally, how various physical properties and dimensionless numbers scale for an SRD simulation, and how these compare to a colloid of radius 10 nm or 1 [MATH]m in H[MATH]0 (Appendix III).', 'cond-mat-0603391-2-19-0': '# Properties of a pure SRD solvent', 'cond-mat-0603391-2-20-0': 'In SRD, the solvent is represented by a large number [MATH] of particles of mass [MATH].', 'cond-mat-0603391-2-20-1': 'Here and in the following, we will call these "fluid" particles, with the caveat that, however tempting, they should not be viewed as some kind of composite particles or clusters made up of the underlying (molecular) fluid.', 'cond-mat-0603391-2-20-2': 'Instead, SRD should be interpreted as a Navier Stokes solver that includes thermal noise.', 'cond-mat-0603391-2-20-3': 'The particles are merely a convenient computational device to facilitate the coarse-graining of the fluid properties.', 'cond-mat-0603391-2-21-0': '## Simulation method for a pure solvent', 'cond-mat-0603391-2-22-0': "In the first (propagation) step of the algorithm, the positions and velocities of the fluid particles are propagated for a time [MATH] (the time between collision steps) by accurately integrating Newton's equations of motion, [EQUATION] [MATH] and [MATH] are the position and velocity of fluid particle [MATH], respectively while [MATH] is the total (external) force on particle [MATH], which may come from an external field such as gravity, or fixed boundary conditions such as hard walls, or moving boundary conditions such as suspended colloids.", 'cond-mat-0603391-2-22-1': 'The direct forces between pairs of fluid particles are, however, neglected in the propagation step.', 'cond-mat-0603391-2-22-2': 'Herein lies the main advantage - the origin of the efficiency - of SRD.', 'cond-mat-0603391-2-22-3': 'Instead of directly treating the interactions between the fluid particles, a coarse-grained collision step is performed at each time-step [MATH]: First, space is partitioned into cubic cells of volume [MATH].', 'cond-mat-0603391-2-22-4': 'Next, for each cell, the particle velocities relative to the center of mass velocity [MATH] of the cell are rotated: [EQUATION] [MATH] is a rotation matrix which rotates velocities by a fixed angle [MATH] around a randomly oriented axis.', 'cond-mat-0603391-2-22-5': 'The aim of the collision step is to transfer momentum between the fluid particles while conserving the total momentum and energy of each cell.', 'cond-mat-0603391-2-22-6': 'Both the collision and the streaming step conserve phase-space volume, and it has been shown that the single particle velocity distribution evolves to a Maxwell-Boltzmann distribution [CITATION].', 'cond-mat-0603391-2-23-0': 'The rotation procedure can thus be viewed as a coarse-graining of particle collisions over space and time.', 'cond-mat-0603391-2-23-1': 'Because mass, momentum, and energy are conserved locally, the correct hydrodynamic (Navier Stokes) equations are captured in the continuum limit, including the effect of thermal noise [CITATION].', 'cond-mat-0603391-2-24-0': 'At low temperatures or small collision times [MATH], the transport coefficients of SRD, as originally formulated [CITATION], show anomalies caused by the fact that fluid particles in a given cell can remain in that cell and participate in several collision steps [CITATION].', 'cond-mat-0603391-2-24-1': 'Under these circumstances the assumption of molecular chaos and Galilean invariance are incorrect.', 'cond-mat-0603391-2-24-2': 'However, this anomaly can be cured by applying a random shift of the cell coordinates before the collision step [CITATION].', 'cond-mat-0603391-2-25-0': 'It should also be noted that the collision step in SRD does not locally conserve angular momentum.', 'cond-mat-0603391-2-25-1': 'As a consequence, the stress tensor [MATH] is not, in general, a symmetric function of the derivatives of the flow field (although it is still rotationally symmetric) [CITATION].', 'cond-mat-0603391-2-25-2': 'The asymmetric part can be interpreted as a viscous stress associated with the vorticity [MATH] of the velocity field [MATH].', 'cond-mat-0603391-2-25-3': 'The stress tensor will therefore depend on the amount of vorticity in the flow field.', 'cond-mat-0603391-2-25-4': 'However, the total force on a fluid element is determined not by the stress tensor itself, but by its divergence [MATH], which is what enters the Navier Stokes equations.', 'cond-mat-0603391-2-25-5': 'Taking the divergence causes the explicit vorticity dependence to drop out (the gradient of a curl is zero).', 'cond-mat-0603391-2-25-6': 'In principle, the stress tensor could be made symmetric by applying random rotations as well as random translations to the cell.', 'cond-mat-0603391-2-25-7': 'Or, alternatively, angular momentum can be explicitly conserved by dynamically adapting the collision rule, as done by Ryder et al [CITATION], who found no significant differences in fluid properties (although there is, of course, less flexibility in choosing simulation parameters).', 'cond-mat-0603391-2-25-8': 'This result has been confirmed by some recent theoretical calculations [CITATION], that demonstrate that the asymmetry of the stress tensor has only a few consequences, such as a correction to the sound wave attenuation associated with viscous dissipation of longitudinal density waves.', 'cond-mat-0603391-2-25-9': 'These are not important for the fluid properties we are trying to model, and so, on balance, we chose not to implement possible fixes to improve on angular momentum conservation.', 'cond-mat-0603391-2-26-0': '## Transport coefficients and the dimensionless mean-free path', 'cond-mat-0603391-2-27-0': 'The simplicity of SRD collisions has facilitated the analytical calculation of many transport coefficients [CITATION].', 'cond-mat-0603391-2-27-1': 'These analytical expressions are particularly useful because they enable us to efficiently tune the viscosity and other properties of the fluid, without the need for trial and error simulations.', 'cond-mat-0603391-2-27-2': 'In this section we will summarize a number of these transport coefficients, where possible giving a simple derivation of the dominant physics.', 'cond-mat-0603391-2-28-0': '### Units and the dimensionless mean-free path', 'cond-mat-0603391-2-29-0': 'In this paper we will use the following units: lengths will be in units of cell-size [MATH], energies in units of [MATH] and masses in units of [MATH] (This corresponds to setting [MATH], [MATH] and [MATH]).', 'cond-mat-0603391-2-29-1': 'Time, for example, is expressed in units of [MATH], the number density [MATH] and other derived units can be found in table [REF].', 'cond-mat-0603391-2-29-2': 'We find it instructive to express the transport coefficients and other parameters of the SRD fluid in terms of the dimensionless mean-free path [EQUATION] which provides a measure of the average fraction of a cell size that a fluid particle travels between collisions.', 'cond-mat-0603391-2-30-0': 'This particular choice of units helps highlight the basic physics of the coarse-graining method.', 'cond-mat-0603391-2-30-1': 'The (nontrivial) question of how to map them on to the units of real physical system will be discussed in section [REF].', 'cond-mat-0603391-2-31-0': '### Fluid self-diffusion constant', 'cond-mat-0603391-2-32-0': 'A simple back of the envelope estimate of the self-diffusion constant [MATH] of a fluid particle can be obtained from a random-walk picture.', 'cond-mat-0603391-2-32-1': 'In a unit of time [MATH], a particle will experience [MATH] collisions, in between which it moves an average distance [MATH].', 'cond-mat-0603391-2-32-2': 'Similarly, a heavier (tagged) particle of mass [MATH], which exchanges momentum with the fluid by participating in the coarse-grained collision step, will move an average distance [MATH] between collisions.', 'cond-mat-0603391-2-32-3': 'By viewing this motion as a random walk of step-size [MATH], the diffusion coefficient follows: [EQUATION] expressed in units of [MATH].', 'cond-mat-0603391-2-32-4': 'The diffusion coefficient [MATH] for a pure fluid particle of mass [MATH] is therefore given by [MATH].', 'cond-mat-0603391-2-33-0': 'A more systematic derivation of the diffusion coefficient of a fluid particle, but still within a random collision approximation, results in the following expression [CITATION]: [EQUATION]', 'cond-mat-0603391-2-33-1': 'The dependence on [MATH] is weak.', 'cond-mat-0603391-2-33-2': 'If, for example, we take [MATH], the value used in this paper, then [MATH], the same as Eq. ([REF]).', 'cond-mat-0603391-2-34-0': 'A similar expression can be derived for the self-diffusion coefficient of a heavier tagged particle of mass [MATH] [CITATION]: [EQUATION]', 'cond-mat-0603391-2-34-1': 'Note that this equation does not quite reduce to Eq. ([REF]) for [MATH] (because of slightly different approximations), but the relative discrepancy decreases with increasing [MATH].', 'cond-mat-0603391-2-35-0': 'While Eq. ([REF]) is accurate for larger mean-free paths, where the random collision approximation is expected to be valid, it begins to show deviations from simulations for [MATH] [CITATION], when longer-time kinetic correlations begin to develop.', 'cond-mat-0603391-2-35-1': 'Ripoll et al [CITATION] argue that these correlations induce interactions of a hydrodynamic nature that enhance the diffusion coefficient for a fluid particle.', 'cond-mat-0603391-2-35-2': 'For example, for [MATH] and [MATH], they measured a fluid self-diffusion constant [MATH] that is about [MATH] larger than the value found from Eq. ([REF]).', 'cond-mat-0603391-2-35-3': 'The enhancement is even more pronounced for heavier particles: for the same simulation parameters they found that [MATH] was enhanced by about [MATH] over the prediction of Eq. ([REF]) when [MATH].', 'cond-mat-0603391-2-36-0': 'We note that coupling a large particle to the solvent through participation in the coarse-grained collision step leads to a diffusion coefficient which scales with mass as [MATH], whereas if one couples a colloid of radius [MATH] to the solvent through direct MD collisions, one expects [MATH].', 'cond-mat-0603391-2-36-1': 'Moreover, it has been shown [CITATION] that the effective hydrodynamic particle radius is approximately given by [MATH].', 'cond-mat-0603391-2-36-2': 'For any reasonable average number of fluid particles per cell, the effective hydrodynamic radius [MATH] of the heavier particle is therefore much less than the collision cell size [MATH].', 'cond-mat-0603391-2-36-3': 'On the other hand, the hydrodynamic field is accurately resolved down to a scale comparable to [MATH] (vide infra).', 'cond-mat-0603391-2-36-4': 'What this implies is that this coupling method will yield correct hydrodynamic interactions only at large distances, when the colloids are more than several hydrodynamic radii apart.', 'cond-mat-0603391-2-36-5': 'All the above suggests that some care must be taken when interpreting the dynamics of heavier particles that couple through the coarse-grained collision step, especially when two or more heavy particles are in close proximity.', 'cond-mat-0603391-2-37-0': '### Kinematic viscosity', 'cond-mat-0603391-2-38-0': 'The spread of a velocity fluctuation [MATH] in a fluid can be described by a diffusion equation [CITATION]: [EQUATION] where [MATH] is the kinematic viscosity, which determines the rate at which momentum or vorticity "diffuses away".', 'cond-mat-0603391-2-38-1': 'The units of kinematic viscosity are [MATH] which are the same as those for particle self diffusion i.e. [MATH].', 'cond-mat-0603391-2-39-0': 'Momentum is transported through two mechanisms:', 'cond-mat-0603391-2-40-0': '1) By particles streaming between collision steps, leading to a "kinetic" contribution to the kinematic viscosity [MATH].', 'cond-mat-0603391-2-40-1': 'Since for this gas-like contribution the momentum is transported by particle motion, we expect [MATH] to scale like the particle self-diffusion coefficient [MATH], i.e. [MATH].', 'cond-mat-0603391-2-41-0': '2) By momentum being re-distributed among the particles of each cell during the collision step, resulting in a "collisional" contribution to the kinematic viscosity [MATH].', 'cond-mat-0603391-2-41-1': 'This mimics the way momentum is transferred due to inter-particle collisions, and would be the dominant contribution in a dense fluid such as water at standard temperature and pressure.', 'cond-mat-0603391-2-41-2': 'Again a simple random-walk argument explains the expected scaling in SRD: Each collision step distributes momentum among particles in a cell, making a step-size that scales like [MATH].', 'cond-mat-0603391-2-41-3': 'Since there are [MATH] collision steps per unit time [MATH], this suggests that the collisional contribution to the kinematic viscosity should scale as [MATH].', 'cond-mat-0603391-2-42-0': 'Accurate analytical expressions for the kinematic viscosity [MATH] of SRD have been derived [CITATION], and these can be rewritten in the following dimensionless form: [EQUATION] where the dependence on the collisional angle [MATH] and fluid number density [MATH] is subsumed in the following two factors: [EQUATION]', 'cond-mat-0603391-2-42-1': 'These factors only depend weakly on [MATH] for the typical parameters used in simulations.', 'cond-mat-0603391-2-42-2': 'For example, at [MATH] and [MATH], the angle and number density we use in this paper, they take the values [MATH] and [MATH].', 'cond-mat-0603391-2-42-3': 'For this choice of collision angle [MATH], they monotonically converge to [MATH] in the limit of large [MATH].', 'cond-mat-0603391-2-43-0': '### Shear viscosity', 'cond-mat-0603391-2-44-0': 'Suppose the fluid is sheared in the x direction, with the gradient of the average flow field in the y direction.', 'cond-mat-0603391-2-44-1': 'Two neighboring fluid elements with different y-coordinates will then experience a friction force in the x-direction, as expressed by the xy component of the stress tensor [MATH], which for a simple liquid is linearly proportional to the instantaneous flow field gradient, [EQUATION]', 'cond-mat-0603391-2-44-2': 'The coefficient of proportionality [MATH] is called the shear viscosity, and is it related to the kinematic viscosity by [MATH], where [MATH] is the fluid mass density.', 'cond-mat-0603391-2-44-3': 'From Eqs. [REF]-[REF] it follows that the two contributions to the shear viscosity can be written in dimensionless form as: [EQUATION] where [MATH] is the unit of shear viscosity.', 'cond-mat-0603391-2-45-0': 'In contrast to the expressions for the diffusion of a fluid particle or a tagged particle, Eqs. ([REF]) - ([REF]) compare quantitatively to simulations over a wide range of parameters [CITATION].', 'cond-mat-0603391-2-45-1': 'For the parameters we used in our simulations in [CITATION], i.e. [MATH], the collisional contribution to the viscosity dominates: [MATH] and [MATH].', 'cond-mat-0603391-2-45-2': 'This is typical for [MATH], where [MATH] and [MATH] can be taken to a good first approximation by the collisional contribution only.', 'cond-mat-0603391-2-45-3': 'In fact throughout this paper we will mainly focus on this small [MATH] limit.', 'cond-mat-0603391-2-46-0': "It is also instructive to compare the expressions derived in this section to what one would expect for simple gases, where, as famously first derived and demonstrated experimentally by Maxwell in the 1860's [CITATION], the shear viscosity is independent of density.", 'cond-mat-0603391-2-46-1': 'This result differs from the kinetic (gas-like) contribution to the viscosity in Eq. ([REF]), because in a real dilute gas the mean-free path scales as [MATH], canceling the dominant density dependence in [MATH].', 'cond-mat-0603391-2-46-2': 'The same argument explains why the self-diffusion and kinematic viscosity of the SRD fluid are, to first order, independent of [MATH], while in a gas they would scale as [MATH].', 'cond-mat-0603391-2-46-3': 'In SRD, the mean-free path [MATH] and the density [MATH] can be varied independently.', 'cond-mat-0603391-2-46-4': 'Moreover, the collisional contribution to the viscosity adds a new dimension, allowing a much wider range of physical fluids to be modeled than just simple gases.', 'cond-mat-0603391-2-47-0': '# Colloid simulation method', 'cond-mat-0603391-2-48-0': 'Malevanets and Kapral [CITATION] first showed how to implement a hybrid MD scheme that couples a set of colloids to a bath of SRD particles.', 'cond-mat-0603391-2-48-1': 'In this section, we expand on their method, describing in detail the implementation we used in ref. [CITATION].', 'cond-mat-0603391-2-48-2': 'We restrict ourselves to HS like colloids with steep interparticle repulsions, although attractions between colloids can easily be added on.', 'cond-mat-0603391-2-48-3': 'The colloid-colloid and colloid-fluid interactions, [MATH] and [MATH] respectively, are integrated via a normal MD procedure, while the fluid-fluid interactions are coarse-grained with SRD.', 'cond-mat-0603391-2-48-4': 'Because the number of fluid particles vastly outnumbers the number of HS colloids, treating their interactions approximately via SRD greatly speeds up the simulation.', 'cond-mat-0603391-2-49-0': '## Colloid-colloid and colloid-solvent interactions', 'cond-mat-0603391-2-50-0': 'Although it is possible to implement an event-driven dynamics of HS colloids in an SRD solvent [CITATION], here we approximate pure HS colloids by steep repulsive interactions of the WCA form [CITATION]: [EQUATION]', 'cond-mat-0603391-2-50-1': 'Similarly, the colloid-fluid interaction takes the WCA form: [EQUATION]', 'cond-mat-0603391-2-50-2': 'The mass [MATH] of a fluid particle is typically much smaller than the mass [MATH] of a colloid, so that the average thermal velocity of the fluid particles is larger than that of the colloid particles by a factor [MATH].', 'cond-mat-0603391-2-50-3': 'For this reason the time-step [MATH] is usually restricted by the fluid-colloid interaction ([REF]), allowing fairly large exponents [MATH] for the colloid-colloid interaction [MATH].', 'cond-mat-0603391-2-50-4': 'We choose [MATH], which makes the colloid-colloid potential steep, more like hard-spheres, while still soft enough to allow the time-step to be set by the colloid-solvent interaction.', 'cond-mat-0603391-2-51-0': 'The positions and velocities of the colloidal spheres are propagated through the Velocity Verlet algorithm [CITATION] with a time step [MATH]: [EQUATION] [MATH] and [MATH] are the position and velocity of colloid [MATH], respectively.', 'cond-mat-0603391-2-51-1': '[MATH] is the total force on that colloid, exerted by the fluid particles, an external field, such as gravity, external potentials such as repulsive walls, as well as other colloids within the range of the interaction potential ([REF]).', 'cond-mat-0603391-2-52-0': "The positions [MATH] and velocities [MATH] of SRD particles are updated by similarly solving Newton's Eqns. ([REF],[REF]) every time-step [MATH], and with the SRD procedure of Eq. ([REF]) every time-step [MATH].", 'cond-mat-0603391-2-53-0': 'Choosing both [MATH] and [MATH] as large as possible enhances the efficiency of a simulation.', 'cond-mat-0603391-2-53-1': 'To first order, each timestep is determined by different physics [MATH] by the steepness of the potentials, and [MATH] by the desired fluid properties, and so there is some freedom in choosing their relative values.', 'cond-mat-0603391-2-53-2': 'We used [MATH] for our simulations of sedimentation [CITATION], but other authors have used ratios of [MATH] [CITATION] or even an order of magnitude larger than that [CITATION].', 'cond-mat-0603391-2-53-3': 'Later in the paper we will revisit this question, linking the time-steps to various dimensionless hydrodynamic numbers and Brownian time-scales.', 'cond-mat-0603391-2-54-0': '## Stick and slip boundary conditions', 'cond-mat-0603391-2-55-0': 'Because the surface of a colloid is never perfectly smooth, collisions with fluid particles transfer angular as well as linear momentum.', 'cond-mat-0603391-2-55-1': 'As demonstrated in Fig. [REF], the exact molecular details of the colloid-fluid interactions may be very complex, and mediated via co- and counter-ions, grafted polymer brushes etc.', 'cond-mat-0603391-2-55-2': 'However, on the time and length-scales over which our hybrid MD-SRD method coarse-grains the solvent, these interactions can be approximated by stick boundary conditions: the tangential velocity of the fluid, relative to the surface of the colloid, is zero at the surface of the colloid [CITATION].', 'cond-mat-0603391-2-55-3': 'For most situations, this boundary condition should be sufficient, although in some cases, such as a non-wetting surface, large slip-lengths may occur [CITATION].', 'cond-mat-0603391-2-56-0': 'In computer simulations, stick boundary conditions may be implemented by bounce-back rules, where both parallel and perpendicular components of the relative velocity are reversed upon a collision with a surface.', 'cond-mat-0603391-2-56-1': 'These have been applied by Lamura et al [CITATION], who needed to modify the bounce-back rules slightly to properly reproduce stick boundaries for a Poiseuille flow geometry.', 'cond-mat-0603391-2-57-0': 'Stick boundaries can also be modeled by a stochastic rule.', 'cond-mat-0603391-2-57-1': 'After a collision, the relative tangential velocity [MATH] and relative normal velocity [MATH] are taken from the distributions: [EQUATION] so that the colloid acts as an additional thermostat [CITATION].', 'cond-mat-0603391-2-57-2': 'Such stochastic boundary conditions have been used for colloidal particles by Inoue et al. [CITATION] and Hecht et al. [CITATION].', 'cond-mat-0603391-2-57-3': 'We have systematically studied several implementations of stick boundary conditions for spherical colloids [CITATION], and derived a version of the stochastic boundary conditions which reproduces linear and angular velocity correlation functions that agree with Enskog theory for short times, and hydrodynamic mode-coupling theory for long times.', 'cond-mat-0603391-2-57-4': 'We argue that the stochastic rule of Eq. ([REF]) is more like a real physical colloid - where fluid-surface interactions are mediated by steric stabilizing layers or local co- and counter-ion concentrations - than bounce-back rules are [CITATION].', 'cond-mat-0603391-2-58-0': 'Nevertheless, in this paper, many examples will be for radial interactions such as those described in Eq. ([REF]).', 'cond-mat-0603391-2-58-1': 'These do not transfer angular momentum to a spherical colloid, and so induce effective slip boundary conditions.', 'cond-mat-0603391-2-58-2': 'For many of the hydrodynamic effects we will discuss here the difference with stick boundary conditions is quantitative, not qualitative, and also well understood.', 'cond-mat-0603391-2-59-0': '## Depletion and lubrication forces', 'cond-mat-0603391-2-60-0': '### Spurious depletion forces induced by the fluid', 'cond-mat-0603391-2-61-0': 'We would like to issue a warning that the additional fluid degrees of freedom may inadvertently introduce depletion forces between the colloids.', 'cond-mat-0603391-2-61-1': 'Because the number density of SRD particles is much higher than that of the colloids, even a small overlap between two colloids can lead to enormous attractions.', 'cond-mat-0603391-2-62-0': 'For low colloid densities the equilibrium depletion interaction between any two colloids caused by the presence of the ideal fluid particles is given by [CITATION]: [EQUATION] where [MATH] is the number density of fluid particles and [MATH] is the (free) volume excluded to the fluid by the presence of two colloids separated by a distance [MATH].', 'cond-mat-0603391-2-62-1': 'The latter is given by [EQUATION] where [MATH].', 'cond-mat-0603391-2-62-2': 'An example is given in Fig. [REF] where we have plotted the resulting depletion potential for the colloid-solvent interaction ([REF]), with [MATH] as routinely used in our simulations, as well as the depletion interaction resulting from a truly HS colloid-solvent interaction.', 'cond-mat-0603391-2-62-3': 'The latter can easily be calculated analytically, with the result [EQUATION]', 'cond-mat-0603391-2-62-4': 'For the pure HS interactions, one could take [MATH] and the depletion forces would have no effect.', 'cond-mat-0603391-2-62-5': 'But for interactions such as those used in Eqs. ([REF]) and ([REF]), the softer repulsions mean that inter-colloid distances less than [MATH] are regularly sampled.', 'cond-mat-0603391-2-62-6': 'A more stringent criterion of [MATH] must therefore be used to avoid spurious depletion effects.', 'cond-mat-0603391-2-62-7': 'As an example, we re-analyze the simulations of ref. [CITATION], where a WCA form with [MATH] was used for the fluid-colloid interactions, and a normal Lennard Jones [MATH] potential with [MATH] was used for the colloid-colloid interactions.', 'cond-mat-0603391-2-62-8': 'The authors found differences between "vacuum" calculations without SRD particles, and a hybrid scheme coupling the colloids to an SRD solvent.', 'cond-mat-0603391-2-62-9': 'These were correctly attributed to "solvent induced pressure", which we quantify here as depletion interactions.', 'cond-mat-0603391-2-62-10': 'For one set of their parameters, [MATH], the effect is mainly to soften the repulsion, but for their other parameter set: [MATH], depletion attractions induce an effective attractive well-depth of over [MATH]!', 'cond-mat-0603391-2-63-0': 'Since the depletion potentials can be calculated analytically, one might try counteracting them by introducing a compensating repulsive potential of the form: [MATH] between the colloids.', 'cond-mat-0603391-2-63-1': 'However, there are three problems with this approach: Firstly, at higher colloid packing fractions, three and higher order interactions may also need to be added, and these are very difficult to calculate.', 'cond-mat-0603391-2-63-2': 'Secondly, the depletion interactions are not instantaneous, and in fact only converge to their equilibrium average algebraically in time [CITATION].', 'cond-mat-0603391-2-63-3': 'While this is not a problem for equilibrium properties, it will introduce errors for non-equilibrium properties.', 'cond-mat-0603391-2-63-4': 'Finally, when external fields drive the colloid, small but persistent anisotropies in the solvent density around a colloid may occur [CITATION].', 'cond-mat-0603391-2-63-5': 'Although these density variations are (and should be) small, the resulting variations in depletion interactions can be large.', 'cond-mat-0603391-2-64-0': 'To avoid these problems, we routinely choose the colloid-fluid interaction range [MATH] slightly below half the colloid diameter [MATH].', 'cond-mat-0603391-2-64-1': 'More precisely, we ensure that the colloid-colloid interaction equals [MATH] at a distance [MATH] where the depletion interactions have become zero, i.e., at a distance of twice the colloid-solvent interaction cut-off radius.', 'cond-mat-0603391-2-64-2': 'Smaller distances will consequently be rare, and adequately dealt with by the compensation potential.', 'cond-mat-0603391-2-64-3': 'This solution may be a more realistic representation anyhow, since in practice for charge and even for sterically stabilized colloids, the effective colloid-colloid diameter [MATH] is expected to be larger than twice the effective colloid-fluid diameter [MATH].', 'cond-mat-0603391-2-64-4': 'This is particularly so for charged colloids at large Debye screening lengths.', 'cond-mat-0603391-2-65-0': '### Lubrication forces depend on surface details', 'cond-mat-0603391-2-66-0': 'When two surfaces approach one another, they must displace the fluid between them, while if they move apart, fluid must flow into the space freed up between the surfaces.', 'cond-mat-0603391-2-66-1': 'At very short inter-surface distances, this results in so-called lubrication forces, which are repulsive for colloids approaching each other, and attractive for colloids moving apart [CITATION].', 'cond-mat-0603391-2-66-2': 'These forces are expected to be particularly important for driven dense colloidal suspensions; see e.g. [CITATION] for a recent review.', 'cond-mat-0603391-2-67-0': 'An additional advantage of our choice of diameters [MATH] above is that more fluid particles will fit in the space between two colloids, and consequently the lubrication forces will be more accurately represented.', 'cond-mat-0603391-2-67-1': 'It should be kept in mind that for colloids, the exact nature of the short-range lubrication forces will depend on physical details of the surface, such as its roughness, or presence of a grafted polymeric stabilizing layer [CITATION].', 'cond-mat-0603391-2-67-2': 'For perfectly smooth colloids, analytic limiting expressions for the lubrication forces can be derived [CITATION], showing a divergence at short distances.', 'cond-mat-0603391-2-67-3': 'We have confirmed that SRD resolves these lubrication forces down to surprisingly low interparticle distances.', 'cond-mat-0603391-2-67-4': 'But, at some point, this will break down, depending of course on the choice of simulation parameters (such as [MATH], [MATH], and [MATH]), as well as the details of the particular type of colloidal particles that one wishes to model [CITATION].', 'cond-mat-0603391-2-67-5': 'An explicit analytic correction could be applied to properly resolve these forces for very small distances, as was recently implemented for Lattice Boltzmann [CITATION].', 'cond-mat-0603391-2-67-6': 'However, in this paper, we will assume that our choice of [MATH] is small enough for SRD to sufficiently resolve lubrication forces.', 'cond-mat-0603391-2-67-7': 'The lack of a complete divergence at very short distances may be a better model of what happens for real colloids anyway.', 'cond-mat-0603391-2-67-8': 'For dense suspensions under strong shear, explicit lubrication force corrections as well as other short-ranged colloid-colloid interactions arising from surface details such as polymer coats will almost certainly need to be put in by hand, see e.g. ref. [CITATION] for further discussion of this subtle problem.', 'cond-mat-0603391-2-68-0': '## Test of static properties', 'cond-mat-0603391-2-69-0': 'The equilibrium properties of a statistical mechanical system should be independent of the detailed dynamics by which it explores phase space.', 'cond-mat-0603391-2-69-1': 'Therefore one requisite condition imposed on our coarse-grained dynamics is that it reproduces the correct static properties of an ensemble of colloids.', 'cond-mat-0603391-2-70-0': 'An obvious property to measure is the radial distribution function [MATH] [CITATION].', 'cond-mat-0603391-2-70-1': 'In Fig. [REF] we depict some [MATH]s obtained from SRD simulations at a number of different densities, and compare these to similar simulations with standard Brownian dynamics.', 'cond-mat-0603391-2-70-2': 'The colloids interact via potentials of the form ([REF]) - ([REF]) with [MATH], [MATH], and [MATH].', 'cond-mat-0603391-2-70-3': 'For the SRD we used [MATH], [MATH] and [MATH], implying [MATH].', 'cond-mat-0603391-2-70-4': 'For the BD we used a background friction calculated from the effective hydrodynamic radius (vide infra) and a time-step equal to the [MATH] used above.', 'cond-mat-0603391-2-70-5': 'The colloids were placed in a box of dimensions [MATH], and the number of particles was varied from 64 to 540 in order to achieve different packing fractions [EQUATION] where the subscript in [MATH] refers to volume fraction based on the colloid-colloid interaction, to distinguished it from the volume fraction [MATH] based on the colloid hydrodynamic radius [CITATION].', 'cond-mat-0603391-2-70-6': 'From Fig. [REF] is clear that the two simulations are indistinguishable within statistical errors, as required.', 'cond-mat-0603391-2-70-7': 'Even though [MATH], we still found it necessary to include an explicit compensating potential without which the [MATH] generated by SRD is increased noticeably at contact when compared to BD simulations.', 'cond-mat-0603391-2-71-0': 'Finally, we note that any simulation will show finite size effects.', 'cond-mat-0603391-2-71-1': 'These may be thermodynamic as well as dynamic.', 'cond-mat-0603391-2-71-2': 'If there are thermodynamic finite-size effects (for example due to a long correlation length caused by proximity to a critical point), then a correct simulation technique will show the same behaviour regardless of the underlying dynamics.', 'cond-mat-0603391-2-72-0': '# Dimensionless numbers', 'cond-mat-0603391-2-73-0': 'Different regimes of hydrodynamic behavior can be characterized by a series of dimensionless numbers that indicate the relative strengths of competing physical processes [CITATION].', 'cond-mat-0603391-2-73-1': 'If two distinct physical systems can be described by the same set of hydrodynamic numbers, then their flow behavior will be governed by the same physics, even if their time and length scales differ by orders of magnitude.', 'cond-mat-0603391-2-73-2': 'In other words, a macroscopic boulder sedimenting in viscous molten magma and a mesoscopic colloid sedimenting in water may show similar behavior if they share key hydrodynamic dimensionless numbers.', 'cond-mat-0603391-2-73-3': 'It is therefore very instructive to analyze where a system described by SRD sits in "hydrodynamic parameter space".', 'cond-mat-0603391-2-73-4': 'In this section we review this parameter space, one "dimension" at a time, by exploring the hydrodynamic numbers listed in Table [REF].', 'cond-mat-0603391-2-74-0': '## Schmidt number', 'cond-mat-0603391-2-75-0': 'The Schmidt number [EQUATION] is important for characterizing a pure fluid.', 'cond-mat-0603391-2-75-1': 'It expresses the rate of diffusive momentum transfer, measured by the kinematic viscosity [MATH], relative to the rate of diffusive mass transfer, measured by the fluid particle self-diffusion coefficient [MATH].', 'cond-mat-0603391-2-75-2': 'For a gas, momentum transport is dominated by mass diffusion, so that Sc [MATH], whereas for a liquid, momentum transport is dominated by inter-particle collisions, and Sc [MATH].', 'cond-mat-0603391-2-75-3': 'For low Schmidt numbers, the dynamics of SRD is indeed "gas-like", while for larger Sc, the dynamics shows collective behavior reminiscent of the hydrodynamics of a liquid [CITATION].', 'cond-mat-0603391-2-75-4': 'This distinction will be particularly important for simulating the dynamics of embedded particles that couple to the solvent through participation in the collision step.', 'cond-mat-0603391-2-75-5': 'For particles that couple directly, either through a potential, or through bounce-back or stochastic boundary conditions, this distinction is less important because the fluid-particle diffusion coefficient [MATH] does not directly enter the Navier Stokes equations.', 'cond-mat-0603391-2-75-6': 'Of course other physical properties can be affected.', 'cond-mat-0603391-2-75-7': 'For example, we expect that the self diffusion coefficient of a colloid [MATH] should be much smaller than the fluid diffusion coefficient, i.e. [MATH] to achieve the correct time-scale separation, as we will see in Section [REF].', 'cond-mat-0603391-2-76-0': '### Dependence of Schmidt number on simulation parameters', 'cond-mat-0603391-2-77-0': 'From Eqs. ([REF]) - ([REF]) it follows that the Schmidt number can be rewritten as [EQUATION] where we have ignored the dependence on [MATH] and [MATH] since the dominant scaling is with the dimensionless mean-free path [MATH].', 'cond-mat-0603391-2-77-1': 'For larger [MATH], where the kinetic contributions to the viscosity dominate, the Sc number is small, and the dynamics is gas-like.', 'cond-mat-0603391-2-77-2': 'Because both the fluid diffusion coefficient ([REF]) and the kinetic contribution to the viscosity ([REF]) scale linearly with [MATH], the only way to obtain a large Sc number is in the limit [MATH] where the collisional contribution to [MATH] dominates.', 'cond-mat-0603391-2-78-0': 'Low Sc numbers may be a more general characteristic of particle-based coarse-graining methods.', 'cond-mat-0603391-2-78-1': 'For computational reasons, the number of particles is normally greatly reduced compared to the solvent one is modeling.', 'cond-mat-0603391-2-78-2': 'Thus the average inter-particle separation and mean-free path are substantially increased, typically leading to a smaller kinematic viscosity and a larger fluid diffusion coefficient [MATH].', 'cond-mat-0603391-2-78-3': 'With DPD, for example, one typically finds [MATH] [CITATION].', 'cond-mat-0603391-2-78-4': 'It is therefore difficult to achieve large Sc numbers, as in a real liquid, without sacrificing computational efficiency.', 'cond-mat-0603391-2-78-5': 'But this may not always be necessary.', 'cond-mat-0603391-2-78-6': 'As long as momentum transport is clearly faster than mass transport, the solvent should behave in a liquid-like fashion.', 'cond-mat-0603391-2-78-7': 'With this argument in mind, and also because the Sc number does not directly enter the Navier Stokes equations we want to solve, we use [MATH] in this paper and in ref. [CITATION], which leads to a Sc [MATH] for [MATH] and [MATH].', 'cond-mat-0603391-2-78-8': 'This should be sufficient for the problems we study.', 'cond-mat-0603391-2-78-9': 'For other systems, such as those where the suspended particles are coupled to the solvent through the collision step, more care may be needed to ensure that the Sc number is indeed large enough [CITATION].', 'cond-mat-0603391-2-79-0': '## Mach number', 'cond-mat-0603391-2-80-0': 'The Mach number measures the ratio [EQUATION] between [MATH], the speed of solvent or colloid flow, and [MATH], the speed of sound.', 'cond-mat-0603391-2-80-1': 'In contrast to the Schmidt number, which is an intrinsic property of the solvent, it depends directly on flow velocity.', 'cond-mat-0603391-2-80-2': 'The Ma number measures compressibility effects [CITATION] since sound speed is related to the compressibility of a liquid.', 'cond-mat-0603391-2-80-3': 'Because [MATH] in many liquids is of order [MATH] m/s, the Ma numbers for physical colloidal systems are extremely small under normally achievable flow conditions.', 'cond-mat-0603391-2-80-4': 'Just as for the Sc number, however, particle based coarse-graining schemes drastically lower the Ma number.', 'cond-mat-0603391-2-80-5': 'The particle mass [MATH] is typically much greater than the mass of a molecule of the underlying fluid, resulting in lower velocities, and moreover, due to the lower density, collisions also occur less frequently.', 'cond-mat-0603391-2-80-6': 'These effects mean that the speed of sound is much lower in a coarse-grained system than it is in the underlying physical fluid.', 'cond-mat-0603391-2-80-7': 'Or, in other words, particle based coarse-graining systems are typically much more compressible than the solvents they model.', 'cond-mat-0603391-2-81-0': 'Ma number effects typically scale with [MATH] [CITATION], and so the Ma number does not need to be nearly as small as for a realistic fluid to still be in the correct regime of hydrodynamic parameter space.', 'cond-mat-0603391-2-81-1': 'This is convenient, because to lower the Ma number, one would need to integrate over longer fluid particle trajectories to allow, for example, a colloidal particle to flow over a given distance, making the simulation computationally more expensive.', 'cond-mat-0603391-2-81-2': 'So there is a compromise between small Ma numbers and computational efficiency.', 'cond-mat-0603391-2-81-3': 'We limit our Ma numbers to values such that [MATH], but it might be possible, in some situations, to double or triple that limit without causing undue error.', 'cond-mat-0603391-2-81-4': 'For example, incompressible hydrodynamics is used for aerodynamic flows up to such Ma numbers since the errors are expected to scale as [MATH] [CITATION].', 'cond-mat-0603391-2-81-5': 'When working in units of [MATH] and [MATH], the only way to keep the Ma number below our upper limit is to restrict the maximum flow velocity to [MATH].', 'cond-mat-0603391-2-81-6': 'The flow velocity itself is, of course, determined by the external fields, but also by other parameters of the system.', 'cond-mat-0603391-2-82-0': '## Reynolds number', 'cond-mat-0603391-2-83-0': 'The Reynolds number is one of the most important dimensionless numbers characterizing hydrodynamic flows.', 'cond-mat-0603391-2-83-1': 'Mathematically, it measures the relative importance of the non-linear terms in the Navier-Stokes equations [CITATION].', 'cond-mat-0603391-2-83-2': 'Physically, it determines the relative importance of inertial over viscous forces, and can be expressed as: [EQUATION] where [MATH] is a length-scale, in our case, the hydrodynamic radius of a colloid, i.e. [MATH], and [MATH] is a flow velocity.', 'cond-mat-0603391-2-84-0': 'For a spherical particle in a flow, the following heuristic argument helps clarify the physics behind the Reynolds number: If the Stokes time [EQUATION] it takes a particle to advect over its own radius is about the same as the kinematic time [EQUATION] it takes momentum to diffuse over that distance, i.e. Re=[MATH], then the particle will feel vorticity effects from its own motion a distance [MATH] away, leading to non-linear inertial contributions to its motion.', 'cond-mat-0603391-2-84-1': 'Since hydrodynamic interactions can decay as slowly as [MATH], their influence can be non-negligible.', 'cond-mat-0603391-2-84-2': 'If, on the other hand, Re [MATH], then vorticity will have diffused away and the particle will only feel very weak hydrodynamic effects from its own motion.', 'cond-mat-0603391-2-85-0': 'Exactly when inertial finite Re effects become significant depends on the physical system under investigation.', 'cond-mat-0603391-2-85-1': 'For example, in a pipe, where the length-scale [MATH] in Eq. ([REF]) is its diameter, the transition from simpler laminar to more complex turbulent flow is at a pipe Reynolds number of Re [MATH] [CITATION].', 'cond-mat-0603391-2-85-2': 'On the other hand, for a single spherical particle, a non-linear dependence of the friction [MATH] on the velocity [MATH], induced by inertial effects, starts to become noticeable for a particle Reynolds number of Re [MATH] [CITATION], while deviations in the symmetry of the streamlines around a rotating sphere have been observed in calculations for Re [MATH] [CITATION].', 'cond-mat-0603391-2-86-0': 'For typical colloidal suspensions, where the particle diameter is on the order of a few [MATH]m down to a few nm, the particle Reynolds number is rarely more than [MATH].', 'cond-mat-0603391-2-86-1': 'In this so-called Stokes regime viscous forces dominate and inertial effects can be completely ignored.', 'cond-mat-0603391-2-86-2': 'The Navier-Stokes equations can be replaced by the linear Stokes equations [CITATION] so that analytic solutions are easier to obtain [CITATION].', 'cond-mat-0603391-2-86-3': 'However, some of the resulting behavior is non-intuitive for those used to hydrodynamic effects on a macroscopic scale.', 'cond-mat-0603391-2-86-4': 'For example, as famously explained by Purcell in a talk entitled "Life at low Reynolds numbers" [CITATION], many simple processes in biology occur on small length scales, well into the Stokes regime.', 'cond-mat-0603391-2-86-5': 'The conditions that bacteria, typically a few [MATH]m long, experience in water are more akin to those humans would experience in extremely thick molasses.', 'cond-mat-0603391-2-86-6': 'Similarly, colloids, polymers, vesicles and other small suspended objects are all subject to the same physics, their motion dominated by viscous forces.', 'cond-mat-0603391-2-86-7': 'For example, if for a colloid sedimenting at 1 [MATH]m/s, gravity were instantaneously turned off, then the Stokes equations suggest that it would come to a complete halt in a distance significantly less than one , reflecting the irrelevance of inertial forces in this low Re number regime.', 'cond-mat-0603391-2-86-8': 'It should be kept in mind that when the Stokes regime is reached because of small length scales (as opposed to very large viscosities such as those found for volcanic lava flows), then thermal fluctuations are also important.', 'cond-mat-0603391-2-86-9': 'These will drive diffusive behavior [CITATION].', 'cond-mat-0603391-2-86-10': 'In many ways SRD is ideally suited for this regime of small particles because the thermal fluctuations are naturally included [CITATION].', 'cond-mat-0603391-2-87-0': '### Dependence of the Reynolds number on simulation parameters', 'cond-mat-0603391-2-88-0': 'From Eqs. ([REF]) and ([REF]), it follows that the Reynolds number for a colloid of hydrodynamic radius [MATH] can be written as: [EQUATION]', 'cond-mat-0603391-2-88-1': 'Equating the hydrodynamic radius [MATH] to [MATH] from the fluid-colloid WCA interaction of Eq. ([REF]) is not quite correct, as we will see in section [REF], but is a good enough approximation in light of the fact that these hydrodynamic numbers are qualitative rather than quantitative indicators.', 'cond-mat-0603391-2-89-0': 'In order to keep the Reynolds number low, one must either use small particles, or very viscous fluids, or low velocities [MATH].', 'cond-mat-0603391-2-89-1': 'The latter condition is commensurate with a low Ma number, which is also desirable.', 'cond-mat-0603391-2-89-2': 'For small enough [MATH] (and ignoring for simplicty the factor [MATH] in Eq. ([REF])) the Re number then scales as [EQUATION]', 'cond-mat-0603391-2-89-3': 'Again, we see that a smaller mean-free path [MATH], which enhances the collisional viscosity, also helps bring the Re number down.', 'cond-mat-0603391-2-89-4': 'This parameter choice is also consistent with a larger Sc number.', 'cond-mat-0603391-2-89-5': 'Thus larger Sc and smaller Ma numbers both help keep the Re number low for SRD.', 'cond-mat-0603391-2-89-6': 'In principle a large viscosity can also be obtained for large [MATH], which enhances the kinetic viscosity, but this choice also lowers the Sc number and raises the Knudsen number, which, as we will see in the next sub-section, is not desirable.', 'cond-mat-0603391-2-90-0': 'Just as was found for the Ma number, it is relatively speaking more expensive computationally to simulate for low Re numbers because the flow velocity must be kept low, which means longer simulation times are necessary to reach time-scales where the suspended particles or fluid flows have moved a significant distance.', 'cond-mat-0603391-2-90-1': 'We therefore compromise and normally keep Re [MATH], which is similar to the choice made for LB simulations [CITATION].', 'cond-mat-0603391-2-90-2': 'For many situations related to the flow of colloids, this should be sufficiently stringent.', 'cond-mat-0603391-2-91-0': '## Knudsen number', 'cond-mat-0603391-2-92-0': 'The Knudsen number Kn measures rarefaction effects, and can be written as [EQUATION] where [MATH] is a characteristic length-scale of the fluid flow, and [MATH] is the mean-free path of the fluid or gas.', 'cond-mat-0603391-2-92-1': 'For a colloid in SRD one could take [MATH] and [MATH].', 'cond-mat-0603391-2-92-2': 'For large Knudsen numbers [MATH] continuum Navier-Stokes equations completely break down, but even for much smaller Knudsen numbers, significant rarefaction effects are seen.', 'cond-mat-0603391-2-92-3': 'For example, for flow in a pipe, where [MATH] in Eq. ([REF]) is taken to be the pipe radius, important non-continuum effects are seen when [MATH].', 'cond-mat-0603391-2-92-4': 'In fact it is exactly for these conditions of modest Kn numbers, when corrections to the Navier Stokes become noticeable, that DSMC approaches are often used [CITATION].', 'cond-mat-0603391-2-92-5': 'SRD, which is related to DSMC, may also be expected to work well for such flows.', 'cond-mat-0603391-2-92-6': 'It could find important applications in microfluidics and other micromechanical devices where Kn effects are expected to play a role [CITATION].', 'cond-mat-0603391-2-93-0': 'Colloidal dispersions normally have very small Kn numbers because the mean-free path of most liquid solvents is very small.', 'cond-mat-0603391-2-93-1': 'For water at standard temperature and pressure [MATH] .', 'cond-mat-0603391-2-93-2': 'Just as found for the other dimensionless numbers, coarse-graining typically leads to larger Kn numbers because of the increase of the mean-free path.', 'cond-mat-0603391-2-93-3': 'Making the Kn number smaller also typically increases the computational cost because a larger number of collisions need to be calculated.', 'cond-mat-0603391-2-93-4': 'In our simulations, we keep Kn[MATH] for spheres.', 'cond-mat-0603391-2-93-5': 'This rough criterion is based on the observation that for small Kn numbers the friction coefficient on a sphere is expected to be decreased by a factor [MATH] Kn, where [MATH] is a material dependent constant of order [MATH] [CITATION], so that we expect Kn number effects to be of the same order as other coarse-graining errors.', 'cond-mat-0603391-2-93-6': 'There are two ways to achieve small Kn numbers: one is by increasing [MATH], the other is by decreasing [MATH].', 'cond-mat-0603391-2-93-7': 'The second condition is commensurate with a large Sc number or a small Re number.', 'cond-mat-0603391-2-94-0': 'In ref. [CITATION], the Kn numbers for colloids were [MATH] and [MATH], depending on their colloid-solvent coupling.', 'cond-mat-0603391-2-94-1': 'Such large Kn numbers should have a significant effect on particle friction coefficients, and this may explain why these authors find that volume fraction [MATH] has less of an effect on the sedimentation velocity than we do [CITATION].', 'cond-mat-0603391-2-95-0': '## Peclet number', 'cond-mat-0603391-2-96-0': 'The Peclet number Pe measures the relative strength of convective transport to diffusive transport.', 'cond-mat-0603391-2-96-1': 'For example, for a colloid of radius [MATH], travelling at an average velocity [MATH], the Pe number is defined as: [EQUATION] where [MATH] is the colloid diffusion coefficient.', 'cond-mat-0603391-2-96-2': 'Just as for the Re number, the Pe number can be interpreted as a ratio of a diffusive to a convective time-scale, but now the former time-scale is not for the diffusion of momentum but rather it is given by the colloid diffusion time [EQUATION] which measures how long it takes for a colloid to diffuse over a distance [MATH].', 'cond-mat-0603391-2-96-3': 'We again take [MATH] so that, using Eq. ([REF]), the Pe number can be written as: [EQUATION]', 'cond-mat-0603391-2-96-4': 'If Pe [MATH] then the colloid moves convectively over a distance much larger than its radius [MATH] in the time [MATH] that it diffuses over that same distance.', 'cond-mat-0603391-2-96-5': 'Brownian fluctuations are expected to be less important in this regime.', 'cond-mat-0603391-2-96-6': 'For Pe [MATH], on the other hand, the opposite is the case, and the main transport mechanism is diffusive (note that on long enough time-scales ([MATH]) convection will always eventually "outrun" diffusion [CITATION]).', 'cond-mat-0603391-2-96-7': 'It is sometimes thought that for low Pe numbers hydrodynamic effects can be safely ignored, but this is not always true.', 'cond-mat-0603391-2-96-8': 'For example, we found that the reduction of average sedimentation velocity with particle volume fraction, famously first explained by Batchelor [CITATION], is independent of Pe number down to Pe [MATH] at least [CITATION].', 'cond-mat-0603391-2-97-0': '### Dependence of Peclet number on simulation parameters', 'cond-mat-0603391-2-98-0': 'The highest Pe number achievable in simulation is limited by the constraints on the Ma and Re numbers.', 'cond-mat-0603391-2-98-1': 'For example the Ma number sets an upper limit on the maximum Pe number by limiting [MATH].', 'cond-mat-0603391-2-98-2': 'From Eqs. ([REF]) and ([REF]), it follows that the Peclet number can be re-written in terms of the Reynolds number as [EQUATION] where we have approximated [MATH].', 'cond-mat-0603391-2-98-3': 'This shows that for a given constraint on the Re number, increasing [MATH] or [MATH] increases the range of accessible Pe numbers.', 'cond-mat-0603391-2-98-4': 'Similarly, when the kinematic viscosity is dominated by the collisional contribution, decreasing the dimensionless mean-free path [MATH] will also increase the maximum Pe number allowed since [MATH].', 'cond-mat-0603391-2-98-5': 'However, these changes increase the computational cost of the simulation.', 'cond-mat-0603391-2-99-0': '# finite-size, discretization, and inertial effects', 'cond-mat-0603391-2-100-0': 'The cost of an SRD simulation scales almost linearly with the number of fluid particles [MATH] in the system, and this contribution is usually much larger than the cost of including the colloidal degrees of freedom.', 'cond-mat-0603391-2-100-1': 'To optimize the efficiency of a simulation, one would therefore like to keep [MATH] as small as possible.', 'cond-mat-0603391-2-100-2': 'This objective can be achieved by keeping [MATH] and the box-size [MATH] small.', 'cond-mat-0603391-2-100-3': 'Unfortunately both these choices are constrained by the errors they introduce.', 'cond-mat-0603391-2-100-4': 'Reducing [MATH] means that short-ranged hydrodynamic fields become less accurately resolved due to Kn number and discretization effects.', 'cond-mat-0603391-2-100-5': 'Decreasing the box-size [MATH] falls foul of the long-ranged nature of the HI, and therefore, just as for Coulomb interactions [CITATION], finite size effects such as those induced by periodic images must be treated with special care.', 'cond-mat-0603391-2-101-0': 'Increasing the flow velocity may also be desirable since more Stokes times [MATH] can be achieved for the same number of SRD collision steps, thus increasing computational efficiency.', 'cond-mat-0603391-2-101-1': 'The Ma number gives one constraint on [MATH], but usually the more stringent constraint comes from keeping the Re number low to prevent unwanted inertial effects.', 'cond-mat-0603391-2-102-0': '## Finite-size effects', 'cond-mat-0603391-2-103-0': '### Finite-size correction to the friction', 'cond-mat-0603391-2-104-0': 'The friction coefficient [MATH] can be extracted from the Stokes drag [MATH] on a fixed colloid in fluid flow: [EQUATION] where [MATH] is the flow field at large distances.', 'cond-mat-0603391-2-104-1': 'The pre-factor 4 comes from using slip boundary conditions; it would be 6 for stick-boundary conditions, as verified in [CITATION].', 'cond-mat-0603391-2-104-2': 'In principle, since [MATH] is accurately known from theory for SRD, this expression can be used to extract the hydrodynamic radius [MATH], which is not necessarily the same as [MATH], from a simulation, as we did in [CITATION].', 'cond-mat-0603391-2-105-0': 'The hydrodynamic radius [MATH] can also be directly calculated from theory.', 'cond-mat-0603391-2-105-1': 'To derive this, it is important to recognize that there are two sources of friction [CITATION].', 'cond-mat-0603391-2-105-2': 'The first comes from the local Brownian collisions with the small particles, and can be calculated by a simplified Enskog/Boltzmann type kinetic theory [CITATION]: [EQUATION] which is here adapted for slip boundary conditions.', 'cond-mat-0603391-2-105-3': 'A related expression for stick boundary conditions, including that for rotational frictions, is described in [CITATION], where it was shown that the short-time exponential decays of the linear and angular velocity auto-correlation functions are quantitatively described by Enskog theory.', 'cond-mat-0603391-2-106-0': 'The second contribution to the friction, [MATH], comes from integrating the Stokes solution to the hydrodynamic field over the surface of the particle, defined here as [MATH].', 'cond-mat-0603391-2-106-1': 'These two contributions to the friction should be added in parallel to obtain the total friction [CITATION] (see also Appendix B): [EQUATION]', 'cond-mat-0603391-2-106-2': 'In contrast to the Enskog friction, which is local, we expect substantial box-size effects on the Stokes friction [MATH] since it depends on long-ranged hydrodynamic effects.', 'cond-mat-0603391-2-106-3': 'These can be expressed in terms of a correction factor [MATH] that should go to 1 for very large systems : [EQUATION]', 'cond-mat-0603391-2-106-4': 'To measure this correction factor we plot, in Fig. [REF], the form [MATH] for various system box sizes for which we have measured [MATH] from the simulation, and estimated [MATH] from Eq. ([REF]).', 'cond-mat-0603391-2-106-5': 'As expected, the correction factor tends to 1 for smaller [MATH].', 'cond-mat-0603391-2-106-6': 'More detailed calculations [CITATION], taking into account the effect of periodic boundaries, suggests that to lowest order in [MATH] the correction factor should scale as [EQUATION]', 'cond-mat-0603391-2-106-7': 'Indeed, a least squares fit of the data to this form gives a slope of [MATH], close to the theoretical value and in agreement with similar Lattice Boltzmann simulations of a single colloidal sphere [CITATION].', 'cond-mat-0603391-2-107-0': 'With these ingredients in hand, we can calculate the theoretical expected friction from Eqns. ([REF]) - ([REF]).', 'cond-mat-0603391-2-107-1': 'We know from previous work that the Enskog contribution at short times and the hydrodynamic contribution at long times quantitatively reproduce the rotational and translational velocity auto-correlation functions (VACF) for [MATH] [CITATION], and so we expect that the friction coefficients should also be accurately described by these theories.', 'cond-mat-0603391-2-107-2': 'We test this further in Fig. [REF] for a number of different values of [MATH], and find excellent agreement with theory for [MATH] and [MATH].', 'cond-mat-0603391-2-107-3': 'For [MATH] and below, on the other hand, we find deviations from the theory.', 'cond-mat-0603391-2-107-4': 'These are most likely due to Kn number and discretization effects, to be discussed in the next sub-section.', 'cond-mat-0603391-2-108-0': 'For the smallest spheres the mass ratio [MATH] may also change the measured friction if instead of fixing the sphere we were to let it move freely.', 'cond-mat-0603391-2-108-1': 'For [MATH], [MATH], which is small enough to have a significant effect [CITATION].', 'cond-mat-0603391-2-108-2': 'For larger spheres, this is not expected to be a problem.', 'cond-mat-0603391-2-108-3': 'For example [MATH] for [MATH] and [MATH] for [MATH].', 'cond-mat-0603391-2-109-0': '### Effective hydrodynamic radius', 'cond-mat-0603391-2-110-0': 'From the calculated or measured friction we can also obtain the effective hydrodynamic radius [MATH], which is continuum concept, from a comparison of the microscopic frictions from Eq. ([REF]) with Eq. ([REF]): [EQUATION] (for stick-boundaries the [MATH] should be replaced by [MATH]).', 'cond-mat-0603391-2-110-1': 'We find [MATH], [MATH], and [MATH].', 'cond-mat-0603391-2-110-2': 'The effective hydrodynamic radius is increased by the finite-size effects, but lowered by the Enskog contribution which is added in parallel.', 'cond-mat-0603391-2-110-3': 'Because this latter contribution is relatively more important for small colloids, [MATH] for smaller [MATH].', 'cond-mat-0603391-2-110-4': 'For [MATH] the Enskog contribution is smaller than the finite-size effects, and so the effective hydrodynamic radius is larger than [MATH].', 'cond-mat-0603391-2-110-5': 'Obviously this latter effect depends on the box-size.', 'cond-mat-0603391-2-110-6': 'In an infinite box [MATH] for all values of [MATH], due to the Enskog contribution.', 'cond-mat-0603391-2-110-7': 'Note that it is the effective hydrodynamic radius [MATH] which sets the long-ranged hydrodynamic fields, as we will see below.', 'cond-mat-0603391-2-111-0': '### Turning off long-ranged hydrodynamics', 'cond-mat-0603391-2-112-0': 'Hydrodynamic forces can be turned off in SRD by regularly randomizing the absolute fluid particle velocities (with for example a naive Langevin thermostat).', 'cond-mat-0603391-2-112-1': 'For particles embedded in an SRD solvent through participation in the collision step, this trick can be used to compare the effects of hydrodynamics to a purely Brownian simulation[CITATION].', 'cond-mat-0603391-2-112-2': 'The Yeomans group has successfully applied this idea in a study of polymer collapse and protein folding [CITATION].', 'cond-mat-0603391-2-113-0': 'However for the case of the colloids embedded through direct solvent collisions, turning off the hydrodynamic forces by randomizing the velocities greatly enhances the friction because, as is clear from Eqns. ([REF]) - ([REF]), the two contributions add in parallel.', 'cond-mat-0603391-2-113-1': 'Without long-ranged hydrodynamics, the friction would be entirely dominated by the Enskog contribution ([REF]) which scales with [MATH], and can be much larger than the hydrodynamic contribution which scales as [MATH].', 'cond-mat-0603391-2-113-2': 'Another way of stating this would be: By locally conserving momentum, SRD allows the development of long-ranged hydrodynamic fluid velocity correlations that greatly reduce the friction felt by a larger colloidal particle compared to the friction it would feel from a purely random Brownian heat-bath at the same temperature and number density [MATH].', 'cond-mat-0603391-2-114-0': '## Discretization effects on flow-field and friction', 'cond-mat-0603391-2-115-0': 'For pure Stokes flow (Re = 0), the velocity around a fixed slip-boundary sphere of (effective) hydrodynamic radius [MATH] can be exactly calculated: [EQUATION] where [MATH] is the velocity field far away from the sphere, and [MATH] the vector pointing from the center of the sphere to a position inside the fluid, with corresponding unit vector [MATH].', 'cond-mat-0603391-2-116-0': 'In Fig. [REF] we plot the difference between the measured field and the theoretical expected field ([REF]) for four different colloid radius to cell size ratios [MATH], using the values of the effective hydrodynamic radius [MATH] calculated from combining Eq. ([REF]) and Eq. [REF].', 'cond-mat-0603391-2-116-1': 'As expected, the field is more accurately reproduced as the colloid radius becomes larger with respect to the cell size [MATH].', 'cond-mat-0603391-2-116-2': 'This improvement arises because SRD discretization and hydrodynamic Knudsen number effects become smaller.', 'cond-mat-0603391-2-116-3': 'In Fig. [REF] we observe quite large deviations in the hydrodynamic field for [MATH].', 'cond-mat-0603391-2-116-4': 'These discretization effects may explain why the measured frictions in Fig. [REF] do not agree with theory for these smallest sphere sizes.', 'cond-mat-0603391-2-116-5': 'We also note that using [MATH] instead of the more accurate value of the effective hydrodynamic radius [MATH] calculated from theory results in significantly larger deviations between measured and theoretical flow-fields.', 'cond-mat-0603391-2-116-6': 'This independently confirms the values of the effective hydrodynamic radius.', 'cond-mat-0603391-2-117-0': 'The increased accuracy from using larger [MATH] comes at a sharp increase in computational cost.', 'cond-mat-0603391-2-117-1': 'To make sure that the finite size effects in each simulation of Fig. [REF] are approximately the same, the box-size was scaled as [MATH].', 'cond-mat-0603391-2-117-2': 'This means that doubling the colloid size leads to an eightfold increase in the number of fluid particles.', 'cond-mat-0603391-2-117-3': 'Moreover, the maximum velocity of the fluid must go down linearly in colloid size to keep the Re number, defined in Eq. ([REF]), constant.', 'cond-mat-0603391-2-117-4': 'If in addition we keep the number of Stokes times fixed, meaning that the fluid flows a certain multiple of the colloid radius or the box size, then a larger particle also means the fluid needs to flow over a proportionally longer total distance.', 'cond-mat-0603391-2-117-5': 'The overall computational costs for this calculation then scales at least as [MATH], which is quite steep.', 'cond-mat-0603391-2-117-6': 'For that reason, we advocate using smaller colloids wherever possible.', 'cond-mat-0603391-2-118-0': 'In most of our simulations we choose [MATH], which leads to a small relative error in the full velocity field and for which we can fully explain the observed friction (as seen in the previous subsection).', 'cond-mat-0603391-2-118-1': 'This size is similar to what is commonly used in LB [CITATION].', 'cond-mat-0603391-2-119-0': '## Inertial effects on flow-field and friction', 'cond-mat-0603391-2-120-0': 'One of the challenges in SRD is to keep the Re number down.', 'cond-mat-0603391-2-120-1': 'It is virtually impossible to reach the extremely low Re number (Stokes) regime of realistic colloids, but that is not necessary either.', 'cond-mat-0603391-2-120-2': 'In Fig. [REF] we see that the friction only begins to noticeably vary from the Stokes limit at Re [MATH] (at least on a logarithmic scale).', 'cond-mat-0603391-2-120-3': 'We expect the flow-field itself to be more sensitive to finite Re number effects.', 'cond-mat-0603391-2-120-4': 'To study this directly, we examine, in Fig. [REF], the flow-field around a fixed colloid of size [MATH] in a box of [MATH] for for different values of Re.', 'cond-mat-0603391-2-120-5': 'The differences with the Stokes flow field of Eq. ([REF]) are shown in Fig. [REF](b)-(d) for Re [MATH], [MATH], and [MATH] respectively.', 'cond-mat-0603391-2-120-6': 'In all cases we used a hydrodynamic radius of [MATH] for the theoretical comparison, as explained in the previous sub-section.', 'cond-mat-0603391-2-120-7': 'The lengths of the vectors in Figs. (b)-(d) are multiplied by 10 for clarity.', 'cond-mat-0603391-2-120-8': 'We observe that the relative errors increase with Re.', 'cond-mat-0603391-2-120-9': 'They are on the order of [MATH] for Re=0.08, and increase to something on the order of [MATH] for Re=2.', 'cond-mat-0603391-2-120-10': 'This is exactly what is expected, of course, as we are moving away from the Stokes regime with which these flow lines are being compared.', 'cond-mat-0603391-2-120-11': 'Since we also expect effects on the order of a few % from Kn number, Ma number and various finite size effects, we argue that keeping Re[MATH] should be good enough for most of the applications we have in mind.', 'cond-mat-0603391-2-121-0': '# The hierarchy of time scales', 'cond-mat-0603391-2-122-0': '## Colloidal time-scales', 'cond-mat-0603391-2-123-0': 'Many different time-scales govern the physics of a colloid of mass [MATH] embedded in a solvent [CITATION].', 'cond-mat-0603391-2-123-1': 'The most important ones are summarized in table [REF], and discussed in more detail below.', 'cond-mat-0603391-2-124-0': '### Fluid time-scales', 'cond-mat-0603391-2-125-0': 'The shortest of these is the solvent collision time [MATH] over which fluid molecules interact with each other.', 'cond-mat-0603391-2-125-1': 'For a typical molecular fluid, [MATH] is on the order of a few tens of fs, and any MD scheme for a molecular fluid must use a discretization time-step [MATH] to properly integrate the equations of motion.', 'cond-mat-0603391-2-126-0': 'The next time-scale up is the solvent relaxation time [MATH], which measures how fast the solvent VACF decays.', 'cond-mat-0603391-2-126-1': 'For a typical molecular liquid, [MATH] s. (For water, at room temperature, for example, it is on the order of 50 femtoseconds)', 'cond-mat-0603391-2-127-0': '### Hydrodynamic time-scales for colloids', 'cond-mat-0603391-2-128-0': 'Hydrodynamic interactions propagate by momentum (vorticity) diffusion, and also by sound.', 'cond-mat-0603391-2-128-1': 'The sonic time [EQUATION] it takes a sound wave to travel the radius of a colloid is typically very small, on the order of 1 ns for a colloid of radius [MATH]m. For that reason, sound effects are often ignored for colloidal suspensions under the assumption that they will have dissipated so quickly that they have no noticeably influence on the dynamics.', 'cond-mat-0603391-2-128-2': 'However, some experiments [CITATION] and theory [CITATION] do find effects from sound waves on colloidal hydrodynamics, so that this issue is not completely settled yet.', 'cond-mat-0603391-2-129-0': 'The kinematic time, [MATH], defined in Eq. ([REF]) (and Table [REF]) as the time for momentum (vorticity) to diffuse over the radius of a colloid, is particularly important for hydrodynamics.', 'cond-mat-0603391-2-129-1': 'It sets the time-scale over which hydrodynamic interactions develop.', 'cond-mat-0603391-2-129-2': 'For a colloid of radius [MATH]m, [MATH] s, which is much faster the colloidal diffusion time [MATH].', 'cond-mat-0603391-2-130-0': 'When studying problems with a finite flow velocity, another hydrodynamic time-scale emerges.', 'cond-mat-0603391-2-130-1': 'The Stokes time [MATH], defined in Eq. ([REF]) (and Table [REF]) as the time for a colloid to advect over its own radius, can be related to the kinematic time by the relation [MATH].', 'cond-mat-0603391-2-130-2': 'Because colloidal particles are in the Stokes regime where Re [MATH], we find [MATH].', 'cond-mat-0603391-2-131-0': 'Simulation and analytical methods based on the Oseen tensor and its generalizations, e.g. [CITATION], implicitly assume that the hydrodynamic interactions develop instantaneously, i.e. that [MATH].', 'cond-mat-0603391-2-131-1': 'For the low Re number regime of colloidal dispersions this is indeed a good approximation.', 'cond-mat-0603391-2-131-2': 'Of course it must be kept in mind that [MATH] is a diffusive time-scale, so that the distance over which it propagates grows as [MATH].', 'cond-mat-0603391-2-131-3': 'Thus the approximation of instantaneous hydrodynamics must be interpreted with some care for effects on larger length-scales.', 'cond-mat-0603391-2-132-0': '### Brownian time-scales for colloids', 'cond-mat-0603391-2-133-0': 'The fastest time-scale relevant to the colloidal Brownian motion is the Fokker-Planck time [MATH] defined in [CITATION] as the time over which the force-force correlation function decays.', 'cond-mat-0603391-2-133-1': 'It is related to [MATH], the time over which the fluid looses memory of its velocity because the forces on the colloid are caused by collisions with the fluid particles, velocity differences de-correlate on a time scale of order [MATH].', 'cond-mat-0603391-2-134-0': 'The next time-scale in this series is the Brownian time: [EQUATION] where [MATH] is the Stokes friction for stick boundary conditions.', 'cond-mat-0603391-2-134-1': 'It is often claimed that this measures the time for a colloid to lose memory of its velocity.', 'cond-mat-0603391-2-134-2': 'However, as discussed in Appendix [REF], this picture, based on the Langevin equation, is in fact incorrect for colloids.', 'cond-mat-0603391-2-134-3': 'Nevertheless, [MATH] has the advantage that it can easily be calculated and so we will use it as a crude upper bound on the colloid velocity de-correlation time.', 'cond-mat-0603391-2-135-0': 'Perhaps the most important time-scale relevant for Brownian motion is the diffusion time [MATH], described in Eq. ([REF]) as the time for a colloidal particle to diffuse over its radius.', 'cond-mat-0603391-2-135-1': 'For a colloid of radius [MATH]m, [MATH] s, and even though [MATH], it remains much larger than most microscopic time-scales in the mesoscopic colloidal regime.', 'cond-mat-0603391-2-136-0': '## Time-scales for coarse-grained simulation', 'cond-mat-0603391-2-137-0': 'In the previous subsection we saw that the relevant time-scales for a single colloid in a solvent can span as many as 15 orders of magnitude.', 'cond-mat-0603391-2-137-1': 'Clearly it would be impossible to bridge all the time-scales of a physical colloidal system - from the molecular [MATH] to the mesoscopic [MATH] - in a single simulation.', 'cond-mat-0603391-2-137-2': 'Thankfully, it is not necessary to exactly reproduce each of the different time-scales in order to achieve a correct coarse-graining of colloidal dynamics.', 'cond-mat-0603391-2-137-3': 'As long as they are clearly separated, the correct physics should still emerge.', 'cond-mat-0603391-2-138-0': 'Since we take the view that the "solvent" particles are really a Navier Stokes solver with noise, what is needed to reproduce Brownian behavior is first of all that the colloid experiences random kicks from the solvent on a short enough time-scale.', 'cond-mat-0603391-2-138-1': 'By dramatically reducing the number of "molecules" in the solvent, the number of kicks per unit of time is similarly reduced.', 'cond-mat-0603391-2-138-2': 'Here we identify the Fokker-Planck time [MATH] as the time-scale on which the colloid experiences random Brownian motion.', 'cond-mat-0603391-2-138-3': "For Brownian motion it doesn't really matter how the kicks are produced, they could be completely uncorrelated, but since we also require that the solvent transports momentum, solvent particles must have some kind of correlation, which in our case is represented by the time [MATH].", 'cond-mat-0603391-2-138-4': 'Other colloid relaxation times should be much longer than this.', 'cond-mat-0603391-2-138-5': 'We therefore require first of all that [MATH] and [MATH].', 'cond-mat-0603391-2-139-0': "Secondly, the colloid's VACF should decay to zero well before it has diffused or convected over its own radius.", 'cond-mat-0603391-2-139-1': 'A proper separation of time-scales in a coarse-graining scheme then requires that [MATH] as well as [MATH] for systems where convection is important.', 'cond-mat-0603391-2-140-0': 'Finally, for the correct hydrodynamics to emerge we require that [MATH].', 'cond-mat-0603391-2-140-1': 'But this separation no longer needs to be over many orders of magnitude.', 'cond-mat-0603391-2-140-2': 'As argued in [CITATION], one order of magnitude separation between time-scales should be sufficient for most applications.', 'cond-mat-0603391-2-140-3': 'We illustrate this approach in Fig. [REF].', 'cond-mat-0603391-2-141-0': 'In the next few subsections we discuss in more quantitative detail how our coarse-graining strategy telescopes down the hierarchy of time-scales in order to maximize the efficiency of a simulation while still retaining key physical features.', 'cond-mat-0603391-2-142-0': '### SRD fluid time-scales', 'cond-mat-0603391-2-143-0': 'In SRD the physical time-scale [MATH] is coarse-grained out, and the effect of the collisions calculated in an average way every time-step [MATH].', 'cond-mat-0603391-2-143-1': 'The time-scale [MATH] on which the velocity correlations decay can be quite easily calculated from a random-collision approximation.', 'cond-mat-0603391-2-143-2': 'Following [CITATION]: [MATH].', 'cond-mat-0603391-2-143-3': 'For our parameters, [MATH], we find [MATH].', 'cond-mat-0603391-2-143-4': 'However, it should be kept in mind that for small [MATH] the exponential decay with [MATH] turns over to a slower algebraic decay at larger [MATH] [CITATION].', 'cond-mat-0603391-2-143-5': 'The amplitude of this "tail" is, however, quite small.', 'cond-mat-0603391-2-144-0': '### Hydrodynamic time-scales for simulation', 'cond-mat-0603391-2-145-0': 'For our choice of units [MATH], so that the sonic time of Eq. ([REF]) reduces to [EQUATION] which is independent of [MATH] or [MATH].', 'cond-mat-0603391-2-146-0': 'In the limit of small [MATH], the ratio of the kinematic time [MATH] to [MATH] can be simplified to the following form: [EQUATION] so that the condition [MATH] is very easy to fulfil.', 'cond-mat-0603391-2-146-1': 'Furthermore, under the same approximations, the ratio [MATH] so that for [MATH] too small the kinematic time becomes faster than the sonic time.', 'cond-mat-0603391-2-146-2': 'For the simulation parameters used in [CITATION] (and in Fig. [REF]), we find [MATH].', 'cond-mat-0603391-2-147-0': '### Brownian time-scales for simulation', 'cond-mat-0603391-2-148-0': 'We expect the Fokker Planck time [MATH] to scale as [MATH], since this is roughly equivalent to the time [MATH] over which the fluid velocities will have randomized.', 'cond-mat-0603391-2-148-1': 'In Fig. [REF] we plot the force-force correlation function [EQUATION]', 'cond-mat-0603391-2-148-2': 'Its short time behavior is dominated by the random forces, and the initial decay time gives a good estimate of [MATH].', 'cond-mat-0603391-2-148-3': 'As can be seen in the inset of Fig. [REF], [MATH], which is indeed on the order of [MATH] or [MATH].', 'cond-mat-0603391-2-149-0': 'If we make the reasonable approximation that [MATH], then the ratio of the Brownian time [MATH] to the kinematic time [MATH] simplifies to: [EQUATION] so that the ratio of [MATH] to [MATH] is the same as for a real physical system.', 'cond-mat-0603391-2-149-1': 'Since buoyancy requirements mean that normally [MATH], the time-scales are ordered as [MATH].', 'cond-mat-0603391-2-149-2': 'Moreover, since the ratio [MATH], independently of [MATH] or even [MATH] (as long as [MATH]), the condition that [MATH] is also not hard to fulfil in an SRD simulation.', 'cond-mat-0603391-2-150-0': 'These time-scales are illustrated in Fig. [REF] where we plot the normalized time correlation function: [EQUATION] with [MATH] the Cartesian velocity coordinate of a single colloid.', 'cond-mat-0603391-2-150-1': 'After the initial non-Markovian quadratic decay, the subsequent short-time exponential decay is not given by [MATH] but instead by the Enskog time [EQUATION] where [MATH] is the Enskog friction which can be calculated from kinetic theory [CITATION], see Eq. ([REF]), and generally [MATH].', 'cond-mat-0603391-2-150-2': 'The physical origins of this behavior are described in more detail in Appendix [REF].', 'cond-mat-0603391-2-151-0': 'The colloid diffusion coefficient is directly related to the friction by the Stokes-Einstein relation: [EQUATION]', 'cond-mat-0603391-2-151-1': 'If we assume that [MATH] and, for simplicity, that [MATH], i.e. we ignore the Enskog contribution, then the diffusion time scales as: [EQUATION] so that [MATH] is not hard to fulfil.', 'cond-mat-0603391-2-152-0': 'It is also instructive to examine the ratio of the diffusion time [MATH] to the kinematic time [MATH]: [EQUATION]', 'cond-mat-0603391-2-152-1': 'In general we advocate keeping [MATH] small to increase the Sc number [MATH], and since another obvious constraint is [MATH], there is not too much difficulty achieving the desired separation of time-scales [MATH].', 'cond-mat-0603391-2-153-0': 'As a concrete example of how these time-scales are separated in a simulation, consider the parameters used in Fig. [REF] for which we find [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0603391-2-153-1': 'But more generally, what the analysis of this section shows is that obtaining the correct hierarchy of time-scales: [EQUATION] is virtually guaranteed once the conditions on dimensionless numbers, detailed in Section [REF], are fulfilled.', 'cond-mat-0603391-2-154-0': '## Measurements of diffusion', 'cond-mat-0603391-2-155-0': 'To further test the hybrid MD-SRD coarse-graining scheme, we plot in Fig. [REF] the time-dependent diffusion coefficient [MATH], defined as: [EQUATION] for a number of different volume fractions [MATH].', 'cond-mat-0603391-2-155-1': 'Note that the convergence slows with increasing volume fraction [MATH].', 'cond-mat-0603391-2-155-2': 'The infinite time integral gives the diffusion coefficient, i.e. [MATH].', 'cond-mat-0603391-2-155-3': 'In the limit of low densities the diffusion coefficient has been predicted to take the form [MATH] with 1.1795 for slip boundary spheres [CITATION], and this provides a good fit at the lower volume fractions.', 'cond-mat-0603391-2-156-0': 'In Figure [REF] we plot the mean-square displacement of a Cartesian component of the colloidal particle position.', 'cond-mat-0603391-2-156-1': 'As highlighted in the inset, at short times there is an initial ballistic regime due to motion at an average mean square velocity [MATH], resulting in a mean-square displacement [MATH].', 'cond-mat-0603391-2-156-2': 'At times [MATH], the motion is clearly diffusive, as expected, with a linear dependence on [MATH] and a slope of [MATH], where [MATH] is given by the infinite time limit of Eq. ([REF]).', 'cond-mat-0603391-2-156-3': 'On the scale of the plot the asymptotic regime is reached well before the time [MATH] over which the particle has diffused, on average, over its radius.', 'cond-mat-0603391-2-156-4': 'In Appendix [REF] the behavior of [MATH] and the related mean-square displacement are discussed in more detail for time-scales [MATH].', 'cond-mat-0603391-2-157-0': '# Mapping between a physical and coarse-grained systems: No free lunch?', 'cond-mat-0603391-2-158-0': 'The ultimate goal of any coarse-graining procedure is to correctly describe physical phenomena in the natural world.', 'cond-mat-0603391-2-158-1': 'As we have argued in this paper, it is impossible to bridge, in a single simulation, all the time and length-scales relevant to a colloid suspended in a solvent.', 'cond-mat-0603391-2-158-2': 'Compromises must be made.', 'cond-mat-0603391-2-158-3': 'Nevertheless, a fruitful mapping from a coarse-grained simulation to a physical system is possible, and greatly facilitated by expressing the physical properties of interest in dimensionless terms.', 'cond-mat-0603391-2-158-4': 'The best way to illustrate this is with some examples.', 'cond-mat-0603391-2-159-0': '## Example 1: Mapping to diffusive and Stokes time-scales', 'cond-mat-0603391-2-160-0': 'For many dynamic phenomena in colloidal dispersions, the most important time-scale is the diffusion time [MATH].', 'cond-mat-0603391-2-160-1': 'To map a coarse-grained simulation onto a real physical system one could therefore equate the diffusion time and the colloid radius [MATH] of the simulation to that of the real physical system.', 'cond-mat-0603391-2-160-2': 'For example, take the simulation parameters from table [REF] in Appendix [REF] for [MATH], and compare these to the properties of colloids of radius [MATH]m and [MATH] nm from table [REF].', 'cond-mat-0603391-2-160-3': 'For the larger colloids, [MATH]s, and equating this to the simulation [MATH] means that [MATH]m and [MATH] s. Performing the same exercise for the smaller colloid, where [MATH] s, results in [MATH] nm and [MATH] s.', 'cond-mat-0603391-2-160-4': 'The first thing that becomes obvious from this exercise is that "physical" time is set here not by the coarse-grained simulation, but by the particular system that one is comparing to.', 'cond-mat-0603391-2-160-5': 'In other words, many different physical systems could be mapped to the same simulation.', 'cond-mat-0603391-2-161-0': 'If the system is also subjected to flow, then a second time-scale emerges, the Stokes time [MATH].', 'cond-mat-0603391-2-161-1': 'As long as the physical Pe number is achievable without compromising the simulation quality, then this time-scale can simultaneously be set equal to that of the physical system.', 'cond-mat-0603391-2-161-2': 'Behavior that depends primarily on these two time-scales should then be correctly rendered by the SRD hybrid MD scheme described in this paper.', 'cond-mat-0603391-2-162-0': '## Example 2: Mapping to kinematic time-scales', 'cond-mat-0603391-2-163-0': 'By fixing [MATH] or [MATH], as in example 1, other time-scales are not correctly reproduced.', 'cond-mat-0603391-2-163-1': 'For the large and small colloid systems described above, we would find [MATH] s and [MATH] s respectively, which contrasts with the physical values of [MATH] s and [MATH] s shown in Table [REF].', 'cond-mat-0603391-2-163-2': 'In other words, fixing [MATH] results in values of [MATH] that are too large by orders of magnitude.', 'cond-mat-0603391-2-163-3': 'But of course these much larger values of [MATH] are by design, because the SRD simulation is optimized by compacting the very large physical hierarchy of time-scales.', 'cond-mat-0603391-2-163-4': "If we are interested in processes dominated by [MATH], having different [MATH] doesn't matter, so long as [MATH].", 'cond-mat-0603391-2-164-0': 'On the other hand, if we want to describe processes that occur on much smaller time-scales, for example long-time tails in colloidal VACFs, then simulation times [MATH] could be mapped onto the physical [MATH] instead.', 'cond-mat-0603391-2-164-1': "For the same [MATH] simulation parameters used above, the [MATH]m system now maps onto [MATH]m, and [MATH] s with of course a strongly underestimated diffusion time: [MATH] s instead of [MATH] s. Similarly for the [MATH] nm system, the mapping is to [MATH] nm, [MATH] s and [MATH] s instead of [MATH] s. For many processes on time-scales [MATH], this underestimate of [MATH] doesn't really matter.", 'cond-mat-0603391-2-164-2': 'It is only when time-scales are mixed that extra care must be employed in interpreting the simulation results.', 'cond-mat-0603391-2-164-3': 'And, if the processes can be described in terms of different dimensionless times, then it should normally still be possible to disentangle the simulation results and correctly map onto a real physical system.', 'cond-mat-0603391-2-165-0': 'Another lesson from these examples of mapping to different time-scales comes from comparing the kinematic viscosity of a physical system, which say takes the value [MATH]m[MATH]/s for water, to its value in SRD simulations.', 'cond-mat-0603391-2-165-1': 'If for the [MATH] SRD system described above we map time onto [MATH], as in Example 1, then for the [MATH]m system [MATH]m[MATH]/s, while for the [MATH] nm system [MATH]m[MATH]/s.', 'cond-mat-0603391-2-165-2': 'As expected, both are much smaller than the true value in water.', 'cond-mat-0603391-2-165-3': 'If we instead map the times to [MATH], then of course the kinematic viscosity takes the same value as for the physical system (since we also set the length-scales equal to the physical length-scales).', 'cond-mat-0603391-2-165-4': 'This apparent ambiguity in defining a property like the kinematic viscosity is inherent in many coarse-graining schemes.', 'cond-mat-0603391-2-165-5': 'We take the view that the precise value of [MATH] is not important, the only thing that matters is that it is large enough to ensure a proper separation of times-scales and the correct regime of hydrodynamic numbers.', 'cond-mat-0603391-2-166-0': '## Example 3: Mapping mass and temperature', 'cond-mat-0603391-2-167-0': 'In the examples above, we have only discussed setting lengths and times.', 'cond-mat-0603391-2-167-1': 'This leaves either the mass [MATH] or temperature (thermal energy) [MATH] still to be set.', 'cond-mat-0603391-2-167-2': 'Because for many dynamic properties the absolute mass is effectively an irrelevant variable (although relative masses may not be) there is some freedom to choose.', 'cond-mat-0603391-2-167-3': 'One possibility would be to set [MATH], the mass of the colloid, equal to the physical mass.', 'cond-mat-0603391-2-167-4': 'For an [MATH]m neutrally buoyant colloid this sets [MATH] g, (equal to about [MATH] water molecules) while for an [MATH] nm colloid we find [MATH] g (equal to about [MATH] water molecules).', 'cond-mat-0603391-2-167-5': 'By construction this procedure produces the correct physical [MATH].', 'cond-mat-0603391-2-167-6': 'If we fix time with [MATH], this will therefore also generate the correct shear viscosity [MATH].', 'cond-mat-0603391-2-167-7': 'If instead we fix [MATH], then [MATH] will be much smaller than the physical value.', 'cond-mat-0603391-2-167-8': 'For either choice of time-scales, other properties, such as the fluid self-diffusion constant or the number density, will have different values from the underlying physical system.', 'cond-mat-0603391-2-168-0': 'Setting the mass in this way means that the unit of thermal energy [MATH] would also be very large, leading to the appearance of very low temperatures.', 'cond-mat-0603391-2-168-1': 'However, because temperature only determines behavior through the way it scales potentials, it is quite easy to simply re-normalize the effective energy scales and obtain the same behavior as for the physical system.', 'cond-mat-0603391-2-168-2': 'Alternatively one could set the temperature equal to that of a physical system, but that would mean that the masses would again differ significantly from the real physical system.', 'cond-mat-0603391-2-168-3': 'At any rate, for the same simulation (say of sedimentation at a given value of Pe) the values of mass or temperature depend on the physical system one compares to.', 'cond-mat-0603391-2-168-4': 'This apparent ambiguity (or flexibility) simply reflects the fact that the dominant effects of temperature and mass come into play as relative ratios and not as absolute values.', 'cond-mat-0603391-2-169-0': '## Example 4: Mapping to attractive potentials: problems with length-scales?', 'cond-mat-0603391-2-170-0': 'So far we have only treated one length-scale in the problem, the radius [MATH].', 'cond-mat-0603391-2-170-1': 'Implicitly we have therefore assumed that the colloid-colloid interaction does not have another intrinsic length-scale of its own.', 'cond-mat-0603391-2-170-2': 'For hard-sphere colloids this is strictly true, and for steep-repulsions such as the WCA form of Eq. ([REF]) it is also a fairly good assumption that the exact choice of the exponent [MATH] in this equation does not significantly affect the dynamical properties [CITATION].', 'cond-mat-0603391-2-170-3': 'Similarly, there is some freedom to set the repulsive fluid-colloid potential of Eq. ([REF]) in such a way as to optimize the simulation parameters, in particular [MATH].', 'cond-mat-0603391-2-170-4': 'The physical liquid-colloid interaction would obviously have a much more complex form, but SRD coarse-grains out such details.', 'cond-mat-0603391-2-170-5': 'A similar argument can be made for colloid and fluid interactions with hard walls, needed, for example, to study problems with confinement, for which, inter alia, SRD is particularly well suited.', 'cond-mat-0603391-2-171-0': 'Things become more complicated for colloids with an explicit attractive interaction, such as DLVO or a depletion potential [CITATION].', 'cond-mat-0603391-2-171-1': 'These potentials introduce a new length-scale, the range of the attraction.', 'cond-mat-0603391-2-171-2': 'The ratio of this length to the hard-core diameter helps determine the equilibrium properties of the fluid [CITATION].', 'cond-mat-0603391-2-171-3': 'In a physical system this ratio may be quite small.', 'cond-mat-0603391-2-171-4': 'Keeping the ratio the same in the simulations then leads to potentials that are very steep on the scale of [MATH].', 'cond-mat-0603391-2-171-5': 'The MD time-step [MATH] may need to be very small in order to properly integrate the MD equations of motion.', 'cond-mat-0603391-2-171-6': 'Such a small [MATH] can make a simulation very inefficient, as was found in [CITATION] who followed this strategy and were forced to use [MATH].', 'cond-mat-0603391-2-171-7': 'However, in that case the DLVO potential dominates the behaviour, so that SRD was only included to roughly resolve the long range hydrodynamics[CITATION].', 'cond-mat-0603391-2-171-8': 'In general, we advocate adapting the potential to maximize simulation efficiency, while preserving key physical properties such as the topology of the phase diagram.', 'cond-mat-0603391-2-171-9': 'For example, the attractive energy scale can be set by the dimensionless reduced second virial coefficient, which gives an excellent approximation for how far one is from the liquid-liquid critical point [CITATION].', 'cond-mat-0603391-2-171-10': 'When the attractive interaction range is less than about [MATH] of the colloid hard-core diameter, the colloidal "liquid" phase becomes metastable with respect to the fluid-solid phase-transition.', 'cond-mat-0603391-2-171-11': 'At low colloid packing fraction [MATH] this leads to so-called "energetic fluid" behavior [CITATION].', 'cond-mat-0603391-2-171-12': 'To simulate a colloidal suspension in this "energetic fluid" regime, having a range of less than [MATH] of [MATH] should suffice.', 'cond-mat-0603391-2-171-13': 'In this way adding attractions does not have to make the simulation much less efficient.', 'cond-mat-0603391-2-171-14': 'Moreover the effects of these constraints, placed by the colloid-colloid interaction on [MATH], are to some degree mitigated by the fact that it is usually the colloid-fluid interaction which sets this time-scale.', 'cond-mat-0603391-2-172-0': 'There are still a few final subtleties to mention.', 'cond-mat-0603391-2-172-1': 'First of all, using small [MATH] in the simulations may greatly increase efficiency, but it also leads to a relatively larger contribution of the Enskog friction to the total friction in Eq. ([REF]).', 'cond-mat-0603391-2-172-2': 'For physical colloids in a molecular solvent the exact magnitude of the Enskog friction is still an open question, but undoubtedly for larger colloids it is almost negligible.', 'cond-mat-0603391-2-172-3': 'Some of the Enskog effects can simply be absorbed into an effective hydrodynamic radius [MATH], but others must be interpreted with some care.', 'cond-mat-0603391-2-173-0': 'Another regime where we might expect to see deviations beyond a simple re-scaling of time-scales would be at very short times.', 'cond-mat-0603391-2-173-1': 'For example, when colloids form a crystal, their oscillations can be de-composed into lattice phonons.', 'cond-mat-0603391-2-173-2': 'All but the longest wave-length longitudinal oscillations are overdamped due to the viscous drag and backflow effects of the solvent [CITATION].', 'cond-mat-0603391-2-173-3': 'In SRD, however, some of the very short wavelength oscillations might be preserved due to the fact that the motion on those time-scales is still ballistic.', 'cond-mat-0603391-2-173-4': 'These may have an impact on the interpretation of SRD simulations of microrheology.', 'cond-mat-0603391-2-174-0': 'In summary, these examples demonstrate that as long as the coarse-grained simulation produces the correct hydrodynamic and Brownian behavior then there is considerable freedom in assigning real physical values to the parameters in the simulation.', 'cond-mat-0603391-2-174-1': 'Attractive interactions may introduce a new length-scale, but a judicious choice of a model potential should still reproduce the correct physical behavior.', 'cond-mat-0603391-2-174-2': 'The same simulation may therefore be mapped onto multiple different physical systems.', 'cond-mat-0603391-2-174-3': 'Conversely, for the same physical system, different choices can be made for the time-scales that are mapped to physical values, but by design, not all time-scales can be simultaneously resolved.', 'cond-mat-0603391-2-175-0': '# Conclusion', 'cond-mat-0603391-2-176-0': 'Correctly rendering the principal Brownian and hydrodynamic properties of colloids in solution is a challenging task for computer simulation.', 'cond-mat-0603391-2-176-1': 'In this article we have explored how treating the solvent with SRD, which coarse-grains the collisions between solvent particles over both time and space, leads to an efficient solution of the thermo-hydrodynamic equations of the solvent, in the external field provided by the colloids.', 'cond-mat-0603391-2-176-2': 'Although it is impossible to simulate the entire range of time-scales encountered in real colloidal suspensions, which can span as much as 15 orders of magnitude, we argue that this is also not necessary.', 'cond-mat-0603391-2-176-3': 'Instead, by recognizing firstly that hydrodynamic effects are governed by a set of dimensionless numbers, and secondly that the full hierarchy of time-scales can be telescoped down to a much more manageable range while still being properly physically separated, we demonstrate that the key Brownian and hydrodynamic properties of a colloidal suspension can indeed be captured by our SRD based coarse-graining scheme.', 'cond-mat-0603391-2-177-0': 'In particular, to simulate in the colloidal regime, the modeler should ensure that the dimensionless hydrodynamic numbers such as the Mach, Reynolds and Knudsen numbers are small enough ([MATH]), and the Schmidt number is large enough ([MATH]).', 'cond-mat-0603391-2-177-1': 'While these numbers are constrained by approximate limits, the Peclet number, which measures the relative strength of the convective transport to the diffusive transport of the colloids, can be either larger or smaller than 1, depending on physical properties such as the colloidal size and the strength of the external field that one wants to study.', 'cond-mat-0603391-2-177-2': 'It turns out that if the hydrodynamic numbers above are chosen correctly, it is usually not so difficult to also satisfy the proper separation of the time-scales.', 'cond-mat-0603391-2-178-0': 'We explored how to reach the desired regime of hydrodynamic numbers and time-scales by tuning the simulation parameters.', 'cond-mat-0603391-2-178-1': 'The two most important parameters are the dimensionless mean-free path [MATH] which measures what fraction of a cell size [MATH] an SRD particle travels between collisions performed at every time-step [MATH], and the ratio of the colloid radius [MATH] to the SRD cell size [MATH].', 'cond-mat-0603391-2-178-2': 'The general picture that emerges is that the collision time [MATH] must be chosen so that [MATH] is small since this helps keep the Schmidt number high, and both the Knudsen and Reynolds numbers low.', 'cond-mat-0603391-2-178-3': 'Of course for computational efficiency, the collision time should not be too small either.', 'cond-mat-0603391-2-178-4': 'Similarly although a larger colloid radius means that the hydrodynamic fields are more accurately rendered, this should be tempered by the fact that the simulation efficiency drops off rapidly with increasing colloid radius.', 'cond-mat-0603391-2-178-5': 'We find that choosing [MATH], which may seem small, already leads to accurate hydrodynamic properties.', 'cond-mat-0603391-2-179-0': 'A number of subtleties occur at the fluid-colloid interface, which may induce spurious depletion interactions between the colloids, but these can be avoided by careful choice of potential parameters for slip boundary conditions.', 'cond-mat-0603391-2-179-1': 'Stick boundary conditions can also be implemented, and we advocate using stochastic bounce-back rules to achieve this coarse-graining over the fluid-colloid interactions.', 'cond-mat-0603391-2-180-0': 'The simplicity of the SRD solvent facilitates the calculation of the the short-time behavior of the VACF from kinetic theory.', 'cond-mat-0603391-2-180-1': 'We argue that for times shorter than the sonic time [MATH], where hydrodynamic collective modes will not yet have fully developed, the decay of the VACF is typically much more rapid than the prediction of the simple Langevin equation.', 'cond-mat-0603391-2-180-2': 'The ensuing Enskog contribution to the total friction should be added in parallel to the microscopic hydrodynamic friction, and from the sum an analytic expression for the effective hydrodynamic radius [MATH] can be derived which is consistent with independent measurements of the long-ranged hydrodynamic fields.', 'cond-mat-0603391-2-181-0': 'Although we have shown how to correctly render the principal Brownian and hydrodynamic behavior of colloids in solution there is, as always, no such thing as a free lunch.', 'cond-mat-0603391-2-181-1': 'The price to be paid is an inevitable consequence of compressing together the hierarchy of time and length-scales: not all the simulation parameters can be simultaneously mapped onto a physical system of interest.', 'cond-mat-0603391-2-181-2': 'However the cost of the "lunch" can be haggled down by recognizing that only a few physical parameters are usually important.', 'cond-mat-0603391-2-181-3': 'For example, one could map the simulation onto real physical time, say through [MATH] and [MATH], or alternatively through [MATH]; there is some freedom (or ambiguity) in the choice of which time scale to use.', 'cond-mat-0603391-2-181-4': 'The correspondence with physical reality becomes more complex when different physical time-scales mix, but in practice they can often be disentangled when both the coarse-grained simulation and the physical system are expressed in terms of dimensionless numbers and ratios.', 'cond-mat-0603391-2-181-5': 'In other words, there is no substitute for careful physical insight which is, as always, priceless.', 'cond-mat-0603391-2-182-0': 'A number of lessons can be drawn from this exercise that may be relevant for other coarse-graining schemes.', 'cond-mat-0603391-2-182-1': 'SRD is closely related to Lattice-Boltzmann and to the Lowe-Anderson thermostat, and many of the conclusions above should transfer in an obvious way.', 'cond-mat-0603391-2-182-2': 'For dissipative particle dynamics, we argue that the physical interpretation is facilitated by recognizing that, just as for SRD, the DPD particles should not be viewed as "clumps of fluid" but rather as a convenient computational tool to solve the underlying thermo-hydrodynamic equations.', 'cond-mat-0603391-2-182-3': 'All these methods must correctly resolve the time-scale hierarchy, and satisfy the relevant hydrodynamic numbers, in order to reproduce the right underlying physics.', 'cond-mat-0603391-2-183-0': 'Our measurements of the VACF and the discussion in Appendix B show explicitly that Brownian Dynamics simulations, which are based on the simple Langevin equation, do not correctly capture either the long or the short-time decay of colloidal VACFs.', 'cond-mat-0603391-2-184-0': 'A major advantage of SRD is the relative ease with which solutes and external boundary conditions (walls) are introduced.', 'cond-mat-0603391-2-184-1': 'Boundaries may be hard or soft, with either stick or slip conditions.', 'cond-mat-0603391-2-184-2': 'This method is therefore particularly promising for simulations in the fields of bio- and nanofluidics, where small meso particles flow in constrained geometries, possibly confined by soft fluctuating walls.', 'cond-mat-0603391-2-184-3': 'We are currently exploring these possibilities.', 'cond-mat-0603391-2-185-0': 'JTP thanks the EPSRC and IMPACT Faraday, and AAL thanks the Royal Society (London) for financial support.', 'cond-mat-0603391-2-185-1': 'We thank W. Briels, H. Lowen, J. Lopez Lopez, J. Sane, A. Mayhew Seers, I. Pagonabarraga, A. Wilber, and A. Wysocki for very helpful conversations, and W. den Otter for a careful reading of the manuscript.', 'cond-mat-0603391-2-186-0': '# Thermostating of SRD under flow', 'cond-mat-0603391-2-187-0': 'When an external field is applied to the fluid (or to objects embedded in the fluid), energy is pumped into the system and, as a consequence, the average temperature will rise.', 'cond-mat-0603391-2-187-1': 'To prevent this from happening, the system must be coupled to a thermostat.', 'cond-mat-0603391-2-187-2': 'In order not to influence the average flow, thermostating requires a local and Galilean invariant definition of temperature.', 'cond-mat-0603391-2-188-0': 'We achieve this by relating the instantaneous local temperature in a cell to the mean square deviation of the fluid particle velocities from the center of mass velocity of that cell.', 'cond-mat-0603391-2-188-1': 'To minimize interference of the thermostat with the dynamics, we choose to measure the overall temperature as: [EQUATION]', 'cond-mat-0603391-2-188-2': 'In Eq. ([REF]), [MATH] is the instantaneous number of fluid particles within cell [MATH].', 'cond-mat-0603391-2-188-3': 'Three degrees of freedom must be subtracted for fixing the center of mass velocity of each cell (note that this implies that the local temperature is not defined in cells containing 0 or 1 particle).', 'cond-mat-0603391-2-188-4': 'During the simulations, the temperature [MATH] is measured every [MATH] (this can be done very efficiently because the relative velocities are readily available in the collision step routine).', 'cond-mat-0603391-2-188-5': 'The thermostat then acts by rescaling all relative velocities [MATH] by a factor [MATH].', 'cond-mat-0603391-2-188-6': 'This strict enforcement of the overall temperature may be relaxed by allowing appropriate fluctuations, but in view of the very large number of fluid particles (typically [MATH]) this will not have a measurable effect on the dynamics of the fluid particles.', 'cond-mat-0603391-2-189-0': '# The Langevin equation and memory effects', 'cond-mat-0603391-2-190-0': 'Within the Brownian approximation each Cartesian component of the colloid velocity [MATH] is described by a simple Langevin equation of the form: [EQUATION] without any memory effect in the friction coefficient [MATH], or in the random force [MATH].', 'cond-mat-0603391-2-190-1': 'This Langevin equation fundamentally arises from the assumption that the force-kicks are Markovian: each step is independent of any previous behavior.', 'cond-mat-0603391-2-190-2': 'Of course on a very short time-scale this is not true, but since Brownian motion is self-similar, the Markovian assumptions underlying Eq. ([REF]) might be expected to be accurate on longer time-scales.', 'cond-mat-0603391-2-190-3': 'Inspection of Fig. [REF] shows that the force-force correlation function ([REF]) does indeed decay on a time-scale [MATH] which is much faster than other Brownian time-scales in the system.', 'cond-mat-0603391-2-191-0': 'From the Langevin Eq. ([REF]) it follows that the velocity auto-correlation function (VACF) takes the form [EQUATION] where [MATH].', 'cond-mat-0603391-2-191-1': 'The VACF is directly related to the diffusion constant through the Green-Kubo relation: [EQUATION] and Eq. ([REF]) has the attractive property that its integral gives the correct Stokes-Einstein form for [MATH], and that its [MATH] limit gives the expected value from equipartition.', 'cond-mat-0603391-2-191-2': 'It is therefore a popular pedagogical device for introducing Brownian motion.', 'cond-mat-0603391-2-191-3': 'Notwithstanding its seductive simplicity, we will argue that this Langevin approach strongly underestimates the initial decay rate of the VACF, and badly overestimates its long-time decay rate.', 'cond-mat-0603391-2-192-0': '## VACF at long times', 'cond-mat-0603391-2-193-0': 'At longer times, as first shown in MD simulations by Alder and Wainwright [CITATION], the VACF shows an algebraic decay that is much slower than the exponential decay of Eq. ([REF]).', 'cond-mat-0603391-2-193-1': 'The difference is due to the breakdown of the Markovian assumption for times [MATH].', 'cond-mat-0603391-2-193-2': 'Memory effects arise because the momentum that a colloidal particle transfers to the fluid is conserved, with dynamics described by the Navier Stokes equations, resulting in long-ranged flow patterns that affect a colloid on much longer time-scales than Eq. ([REF]) suggests.', 'cond-mat-0603391-2-193-3': 'Calculations taking into account the full time-dependent hydrodynamics were performed by Hauge [CITATION] and Hinch [CITATION] (and others apparently much earlier [CITATION]).', 'cond-mat-0603391-2-193-4': 'For [MATH], i.e. a neutrally buoyant particle, these expressions can be written as [EQUATION] where we have chosen to use an integral form as in [CITATION].', 'cond-mat-0603391-2-193-5': 'Under the assumption that the sound effects have dissipated away, the only hydrodynamic time-scale is the kinematic time [MATH] defined in Eq. ([REF]).', 'cond-mat-0603391-2-193-6': 'The VACF must therefore scale with [MATH], independently of colloid size.', 'cond-mat-0603391-2-193-7': 'Eq. ([REF]) is written in a way to emphasize this scaling.', 'cond-mat-0603391-2-194-0': 'By expanding the denominator it is not hard to see that at longer times the well-known algebraic tail results: [EQUATION] with the same pre-factor that was found from mode-coupling theory [CITATION].', 'cond-mat-0603391-2-195-0': 'The integral of Eq. ([REF]) is [EQUATION] which means that all the hydrodynamic contributions to the friction [MATH] come from [MATH].', 'cond-mat-0603391-2-195-1': 'The integral converges slowly, as [MATH], because of the long-time tail.', 'cond-mat-0603391-2-195-2': 'The VACF can easily be related to the mean-square displacement, and deviations from purely diffusive motion are still observable for [MATH], and have been measured in experiments [CITATION].', 'cond-mat-0603391-2-196-0': 'Although the last equality in Eq. ([REF]) shows that the VACF tail amplitude is independent of colloid size, it should be kept in mind that the algebraic decay does not set in until [MATH].', 'cond-mat-0603391-2-196-1': 'Thus it is the relative (not absolute) effect of the tail that is the same for all colloids.', 'cond-mat-0603391-2-196-2': 'For example, at [MATH], the VACF will have decayed to [MATH] of its initial value of [MATH], independently of colloid mass [MATH].', 'cond-mat-0603391-2-196-3': 'Even though this tail amplitude may seem small, about half the weight of the integral ([REF]) that determines the friction is from [MATH].', 'cond-mat-0603391-2-196-4': 'The dominance of the non-Markovian hydrodynamic behavior in determining the friction for colloids does not appear to hold for microscopic fluids, where, for example for LJ liquids near the triple point, it is thought that the long-time tail only adds a few [MATH] to the friction coefficient [CITATION].', 'cond-mat-0603391-2-196-5': 'However, this is not due to the simple Langevin equation ([REF]) being more accurate, but rather to the modified short-time behavior of the VACF, caused by fluid correlations (a different non-Markovian effect).', 'cond-mat-0603391-2-197-0': '## VACF at short times', 'cond-mat-0603391-2-198-0': 'At short times the hydrodynamic contribution to the VACF ([REF]) reduces to [EQUATION] because, within a continuum description, [MATH] of the energy is dissipated as a sound wave at velocity [MATH], [EQUATION] where the sonic time [MATH] is defined in Table [REF].', 'cond-mat-0603391-2-198-1': "The sound wave doesn't contribute to the friction or diffusion since [EQUATION]", 'cond-mat-0603391-2-198-2': 'For a more extended discussion of the role of sound on hydrodynamics see e.g. ref. [CITATION].', 'cond-mat-0603391-2-199-0': 'Because [MATH], the sound-wave contribution to the VACF decays much faster than the hydrodynamic contribution [MATH] or even than the Langevin approximation [MATH] (recall that [MATH] for neutrally buoyant colloids.)', 'cond-mat-0603391-2-199-1': 'Therefore, in the continuum picture, the short-time decay of the VACF is much faster than that suggested by the simple Langevin equation.', 'cond-mat-0603391-2-199-2': 'And even after the sound wave has decayed, [MATH] decays much more rapidly (faster than simple exponential) than Eq. ([REF]) for times [MATH].', 'cond-mat-0603391-2-199-3': 'SRD simulations confirm this more rapid short time decay.', 'cond-mat-0603391-2-199-4': 'For example, in Fig [REF] we show a direct comparison of the measured VACF to the continuum approach, where [MATH].', 'cond-mat-0603391-2-199-5': 'For short times the decay predicted from sound agrees reasonable well, and for long times the full hydrodynamic theory quantitatively fits the simulation data.', 'cond-mat-0603391-2-200-0': 'A closer look at the short-time behavior in our SRD simulations, both for slip boundaries as in Fig. [REF] and for stick-boundaries as in Fig [REF] or in [CITATION], shows that a better fit to the short-time exponential decay of the VACF is given by [EQUATION] where the Enskog time ([REF]) scales as: [EQUATION] for heavy (stick boundary) colloids in an SRD fluid.', 'cond-mat-0603391-2-200-1': 'Note that the Enskog decay time is very close to the decay time [MATH] of the sound wave ([REF]).', 'cond-mat-0603391-2-200-2': 'That the SRD Enskog time should scale as [MATH] is perhaps not too surprising, because that is the natural time-scale of the SRD fluid.', 'cond-mat-0603391-2-200-3': 'However, the fact that the pre-factors in the exponential of Eqs ([REF]) and ([REF]) are virtually the same may be accidental.', 'cond-mat-0603391-2-200-4': "Although in a real liquid the increased compressibility factor means [MATH] and [MATH] are lower than their values for an ideal gas, the two times change at slightly different rates so that we don't expect the same equivalence in physical systems.", 'cond-mat-0603391-2-201-0': 'In contrast to the sound wave, the integral over the Enskog VACF leads to a finite value: [EQUATION] which quantitatively explains the friction we measured in our simulations, as discussed in section [REF].', 'cond-mat-0603391-2-201-1': 'In contrast to the VACF from sound, the Enskog result cannot simply be added to the VACF from Eq. ([REF]) because this would violate the [MATH] limit of the VACF, set by equipartition.', 'cond-mat-0603391-2-201-2': 'However, because the Enskog contribution occurs on times [MATH], where sound and other collective modes of (compressible) hydrodynamics are not yet fully developed, it may be that Eqs. ([REF]) and ([REF]), derived from continuum theory, should be modified on these short time-scales.', 'cond-mat-0603391-2-201-3': 'Since the dominant contribution to the integral in Eq. ([REF]) is for longer times, the exact form of the hydrodynamic contribution ([REF]) on these short times ([MATH]) does not have much influence on the hydrodynamic part of the friction.', 'cond-mat-0603391-2-201-4': 'Thus the approximation of the short-time VACF by the Enskog result of Eq. ([REF]), and the longer-time ([MATH], but still [MATH]) VACF by Eq. ([REF]) may be a fairly reasonable empirical approach.', 'cond-mat-0603391-2-201-5': 'This approximation, together with the Green-Kubo relation ([REF]), then justifies, a-posteori, the parallel addition of frictions in Eq. ([REF]).', 'cond-mat-0603391-2-202-0': 'Just as was found for the long-time behavior, the short-time behavior of the VACF displays important deviations from the simple Langevin picture.', 'cond-mat-0603391-2-202-1': 'We find that the VACF decays much faster, on the order of the sonic time [MATH] rather than the Brownian time [MATH].', 'cond-mat-0603391-2-202-2': 'From the tables [REF] and [REF] one can see how large these differences can be.', 'cond-mat-0603391-2-202-3': 'For example, for colloids with [MATH], the short-time Enskog or sonic decay times are of order [MATH] s, while [MATH] is of order [MATH] s, so that the decay of the VACF will be dominated by [MATH], with its non-exponential behavior, over almost the entire time regime.', 'cond-mat-0603391-2-203-0': 'An example of the effect of the Enskog contribution on the friction or the mean-square displacement is given in Fig. [REF].', 'cond-mat-0603391-2-203-1': 'The integral of the VACF, [MATH] from Eq. ([REF]), which is directly related to the mean-square displacement [CITATION], is extracted from the simulations described in Fig [REF], and compared to [EQUATION] calculated using Eqs. ([REF]) and ([REF]).', 'cond-mat-0603391-2-203-2': 'Both show exactly the same [MATH] scaling for [MATH], but the simulated [MATH] is larger because the initial Enskog contribution means that [MATH], for [MATH], with [MATH] defined in Eq. ([REF]).', 'cond-mat-0603391-2-204-0': 'The slow algebraic convergence to the final diffusion coefficient, or related friction, also helps explain the scaling of the finite size effects of Eq. ([REF]).', 'cond-mat-0603391-2-204-1': 'A finite box of length [MATH] means that the measured [MATH] is cut off roughly at [MATH], and because [MATH] for [MATH] the effect on the diffusion constant can be written as [MATH] which explains the scaling form of Eq. ([REF]).', 'cond-mat-0603391-2-204-2': 'Fig. [REF] also demonstrates how much more rapidly the [MATH] from the Langevin Eq. ([REF]) converges to its final result [MATH] when compared to the simulations and hydrodynamic theories.', 'cond-mat-0603391-2-205-0': 'In summary, even though the simple Langevin equation ([REF]), without memory, may have pedagogical merit, it does a poor job of describing the VACF for colloids in solution.', 'cond-mat-0603391-2-206-0': '# Comparison of SRD to physical parameters', 'cond-mat-0603391-2-207-0': 'At the end of the day one would like to compare a coarse-grained simulation to a real physical colloidal dispersion.', 'cond-mat-0603391-2-207-1': 'To help facilitate such comparisons, we have summarized a number of physical parameters and dimensionless numbers for colloids in water, and also colloids in an SRD fluid.', 'cond-mat-0603391-2-208-0': 'Table [REF] compares the properties of a pure SRD fluid, in the limit of small mean-free path [MATH], to the properties of water.', 'cond-mat-0603391-2-209-0': 'Table [REF] summarizes the main scaling and values for a number of different properties of two different size neutrally buoyant colloids in water.', 'cond-mat-0603391-2-209-1': 'To approximate the Enskog friction we used a simplified form valid for stick boundary conditions in an ideal fluid, as in [CITATION], which ignores the compressibility and other expected complexities of real water.', 'cond-mat-0603391-2-209-2': 'These predictions should therefore be taken with a grain of salt.', 'cond-mat-0603391-2-209-3': 'For simplicity, we have ignored any potential Enskog effects on the diffusion constant or time-scales.', 'cond-mat-0603391-2-210-0': 'Table [REF] shows a very similar comparison for two different size colloids in an SRD fluid.', 'cond-mat-0603391-2-210-1': 'For simplicity we assume that there are no finite-size effects, which explains small differences when compared to similar parameters used in the text of the main article.', 'cond-mat-0603391-2-210-2': 'However, in contrast to Table [REF], we do include the Enskog contribution when calculating the total friction and related properties.'}

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[]

[['cond-mat-0603391-1-87-1', 'cond-mat-0603391-2-88-1'], ['cond-mat-0603391-1-105-1', 'cond-mat-0603391-2-106-1'], ['cond-mat-0603391-1-132-1', 'cond-mat-0603391-2-133-1'], ['cond-mat-0603391-1-132-2', 'cond-mat-0603391-2-133-1'], ['cond-mat-0603391-1-173-3', 'cond-mat-0603391-2-174-3'], ['cond-mat-0603391-1-181-4', 'cond-mat-0603391-2-183-0']]

[]

['cond-mat-0603391-1-39-0', 'cond-mat-0603391-1-109-1', 'cond-mat-0603391-1-142-2', 'cond-mat-0603391-2-39-0', 'cond-mat-0603391-2-110-1', 'cond-mat-0603391-2-143-2', 'cond-mat-0603391-3-39-0', 'cond-mat-0603391-3-110-1', 'cond-mat-0603391-3-143-2']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0603391

{'cond-mat-0603391-3-0-0': 'We describe in detail how to implement a coarse-grained hybrid Molecular Dynamics and Stochastic Rotation Dynamics simulation technique that captures the combined effects of Brownian and hydrodynamic forces in colloidal suspensions.', 'cond-mat-0603391-3-0-1': 'The importance of carefully tuning the simulation parameters to correctly resolve the multiple time and length-scales of this problem is emphasized.', 'cond-mat-0603391-3-0-2': 'We systematically analyze how our coarse-graining scheme resolves dimensionless hydrodynamic numbers such as the Reynolds number Re, which indicates the importance of inertial effects, the Schmidt number Sc, which indicates whether momentum transport is liquid-like or gas-like, the Mach number Ma, which measures compressibility effects, the Knudsen number Kn, which describes the importance of non-continuum molecular effects and the Peclet number Pe, which describes the relative effects of convective and diffusive transport.', 'cond-mat-0603391-3-0-3': 'With these dimensionless numbers in the correct regime the many Brownian and hydrodynamic time-scales can be telescoped together to maximize computational efficiency while still correctly resolving the physically relevant physical processes.', 'cond-mat-0603391-3-0-4': 'We also show how to control a number of numerical artifacts, such as finite size effects and solvent induced attractive depletion interactions.', 'cond-mat-0603391-3-0-5': 'When all these considerations are properly taken into account, the measured colloidal velocity auto-correlation functions and related self diffusion and friction coefficients compare quantitatively with theoretical calculations.', 'cond-mat-0603391-3-0-6': 'By contrast, these calculations demonstrate that, notwithstanding its seductive simplicity, the basic Langevin equation does a remarkably poor job of capturing the decay rate of the velocity auto-correlation function in the colloidal regime, strongly underestimating it at short times and strongly overestimating it at long times.', 'cond-mat-0603391-3-0-7': 'Finally, we discuss in detail how to map the parameters of our method onto physical systems, and from this extract more general lessons - keeping in mind that there is no such thing as a free lunch - that may be relevant for other coarse-graining schemes such as Lattice Boltzmann or Dissipative Particle Dynamics.', 'cond-mat-0603391-3-1-0': '# Introduction', 'cond-mat-0603391-3-2-0': 'Calculating the non-equilibrium properties of colloidal suspensions is a highly non-trivial exercise because these depend both on the short-time thermal Brownian motion, and the long-time hydrodynamic behavior of the solvent [CITATION].', 'cond-mat-0603391-3-2-1': 'Moreover, the hydrodynamic interactions are of a many-body character, and cannot usually be decomposed into a pairwise sum of inter-colloid forces.', 'cond-mat-0603391-3-3-0': 'The fundamental difficulty of fully including the detailed solvent dynamics in computer simulations becomes apparent when considering the enormous time and length-scale differences between mesoscopic colloidal and microscopic solvent particles.', 'cond-mat-0603391-3-3-1': 'For example, a typical colloid of diameter [MATH] will displace on the order of [MATH] water molecules!', 'cond-mat-0603391-3-3-2': 'Furthermore, a molecular dynamics (MD) scheme for the solvent would need to resolve time-scales on the order of [MATH] s to describe the inter-molecular forces, while a colloid of diameter [MATH] in water diffuses over its own diameter in about [MATH]s.', 'cond-mat-0603391-3-4-0': 'Clearly, simulating even an extremely crude molecular model for the solvent on the time-scales of interest is completely out of the question: some form of coarse-graining is necessary.', 'cond-mat-0603391-3-4-1': 'The object of this paper is to describe in detail one such scheme.', 'cond-mat-0603391-3-4-2': 'But before we do so, we first briefly discuss a subset of the wide variety of different simulation techniques that have been devised to describe the dynamics of colloidal suspensions.', 'cond-mat-0603391-3-5-0': 'At the simplest level, the effects of the solvent can be taken into account through Brownian dynamics (BD) [CITATION], which assumes that collisions with the solvent molecules induce a random displacement of the colloidal particle positions, as well as a local friction proportional to the their velocity.', 'cond-mat-0603391-3-5-1': 'Although, due to its simplicity, Brownian dynamics is understandably very popular, it completely neglects momentum transport through the solvent - as described by the Navier Stokes equations - which leads to long-ranged hydrodynamic interactions (HI) between the suspended particles.', 'cond-mat-0603391-3-5-2': 'These HI may fall off as slowly as [MATH], and can qualitatively affect the dynamical behavior of the suspension [CITATION].', 'cond-mat-0603391-3-6-0': 'Beginning with the pioneering work of Ermak and McCammon [CITATION], who added a simple representation of the Oseen tensor [CITATION] to their implementation of BD, many authors have applied computational approaches that include the HI by an approximate analytical form.', 'cond-mat-0603391-3-6-1': 'The most successful of these methods is Stokesian Dynamics [CITATION], which can take into account higher order terms in a multipole expansion of the HI interactions.', 'cond-mat-0603391-3-6-2': 'Although some more sophisticated recent implementations of Stokesian Dynamics have achieved an [MATH] scaling with the number of colloids [CITATION], this method is still relatively slow, and becomes difficult to implement in complex boundary conditions.', 'cond-mat-0603391-3-7-0': 'Another way to solve for the HI is by direct numerical simulation (DNS) methods, where the solid particles are described by explicit boundary conditions for the Navier-Stokes equations [CITATION].', 'cond-mat-0603391-3-7-1': 'These methods are better adapted to non-Brownian particles, but can still be applied to understand the effects of HI on colloidal dynamics.', 'cond-mat-0603391-3-7-2': 'The fluid particle dynamics (FPD) method of Tanaka and Araki [CITATION], and a related method by Yamamoto et al. [CITATION], are two important recent approaches to solving the Navier Stokes equations that go a step beyond standard DNS, and simplify the problem of boundary conditions by using a smoothed step function at the colloid surfaces.', 'cond-mat-0603391-3-7-3': 'In principle Brownian motion can be added to these methods [CITATION].', 'cond-mat-0603391-3-8-0': 'The difficulty of including complex boundary conditions in DNS approaches has stimulated the use of lattice-gas based techniques to resolve the Navier Stokes equations.', 'cond-mat-0603391-3-8-1': 'These methods exploit the fact that only a few conditions, such as (local) energy and momentum conservation, need to be satisfied to allow the correct (thermo) hydrodynamics to emerge in the continuum limit.', 'cond-mat-0603391-3-8-2': 'Greatly simplified particle collision rules, easily amenable to efficient computer simulation, are therefore possible, and complex boundary conditions, such as those that characterize moving colloids, are easier to treat than in DNS methods.', 'cond-mat-0603391-3-8-3': 'The most popular of these techniques is Lattice Boltzmann (LB) where a linearized and pre-averaged Boltzmann equation is discretized and solved on a lattice [CITATION].', 'cond-mat-0603391-3-8-4': 'Ladd has pioneered the application of this method to solid-fluid suspensions [CITATION].', 'cond-mat-0603391-3-8-5': 'This is illustrated schematically in Fig. [REF].', 'cond-mat-0603391-3-8-6': 'The solid particles are modeled as hollow spheres, propagated with Newtonian dynamics, and coupled to the LB solvent through a bounce-back collision rule on the surface.', 'cond-mat-0603391-3-8-7': 'The fluctuations that lead to Brownian motion were modeled by adding a stochastic term to the stress tensor.', 'cond-mat-0603391-3-8-8': 'Recently this has been criticized and an improved method to include Brownian noise that does not suffer from some lattice induced artifacts was suggested [CITATION].', 'cond-mat-0603391-3-8-9': 'A number of other groups have recently derived alternative ways of coupling a colloidal particle to a LB fluid [CITATION], in part to simulate the dynamics of charged systems.', 'cond-mat-0603391-3-9-0': 'The discrete nature of lattice based methods can also bring disadvantages, particularly when treating fluids with more than one length-scale.', 'cond-mat-0603391-3-9-1': 'Dissipative Particle Dynamics (DPD) [CITATION] is a popular off-lattice alternative that includes hydrodynamics and Brownian fluctuations.', 'cond-mat-0603391-3-9-2': 'It can be viewed as an extension of standard Newtonian MD techniques, but with two important innovations: 1) soft potentials that allow large time-steps and rapid equilibration; 2) a Galilean invariant thermostat that locally conserves momentum and therefore generates the correct Navier Stokes hydrodynamics in the continuum limit.', 'cond-mat-0603391-3-10-0': 'In fact, these two methodological advances can be separated.', 'cond-mat-0603391-3-10-1': 'Soft potentials such as those postulated for innovation 1) may indeed arise from careful equilibrium coarse-graining of complex fluids [CITATION].', 'cond-mat-0603391-3-10-2': 'However, their proper statistical mechanical interpretation, even for equilibrium properties, is quite subtle.', 'cond-mat-0603391-3-10-3': 'For example, a potential that generates the correct pair structure will not normally reproduce the correct virial pressure due to so called representability problems [CITATION].', 'cond-mat-0603391-3-10-4': 'When used in dynamical simulations, the correct application of effective potentials is even more difficult to properly derive.', 'cond-mat-0603391-3-10-5': 'For polymer dynamics, for instance, uncrossability constraints must be re-introduced to prevent coarse-grained polymers from passing through one another [CITATION].', 'cond-mat-0603391-3-10-6': 'Depletion interactions in a multi component solution also depend on the relative rates of depletant diffusion coefficients and particle flow velocities [CITATION].', 'cond-mat-0603391-3-10-7': 'At present, therefore, the statistical mechanical origins of innovation 1) of DPD, the use of soft potentials out of equilibrium, are at best obscure.', 'cond-mat-0603391-3-10-8': 'We are convinced that a correct microscopic derivation of the coarse-grained DPD representation of the dynamics, if this can indeed be done, will show that the interpretation of such soft potentials depends on dynamic as well as static (phase-space) averages.', 'cond-mat-0603391-3-10-9': 'Viewing the DPD particles as static "clumps" of underlying fluid is almost certainly incorrect.', 'cond-mat-0603391-3-10-10': 'It may, in fact, be more fruitful to abandon simple analogies to the potential energy of a Hamiltonian system, and instead view the interactions as a kind of coarse-grained self-energy [CITATION].', 'cond-mat-0603391-3-11-0': 'Innovation 2), on the other hand, can be put on firmer statistical mechanical footing, see e.g. [CITATION], and can be usefully employed to study the dynamics of complex systems with other types of inter-particle interactions [CITATION].', 'cond-mat-0603391-3-11-1': 'The main advantage of the DPD thermostat is that, by preserving momentum conservation, the hydrodynamic interactions that intrinsically arise from microcanonical MD are preserved for calculations in the canonical ensemble.', 'cond-mat-0603391-3-11-2': 'Other thermostats typically screen the hydrodynamic interactions beyond a certain length-scale [CITATION].', 'cond-mat-0603391-3-11-3': 'For weak damping this may not be a problem, but for strong damping it could be.', 'cond-mat-0603391-3-12-0': 'Simulating only the colloids with DPD ignores the dominant solvent hydrodynamics.', 'cond-mat-0603391-3-12-1': 'While the solvent could be treated directly by DPD, as suggested in Fig. [REF] (see also [CITATION]), this method is quite computationally expensive because the "solvent" particles interact through pairwise potentials.', 'cond-mat-0603391-3-13-0': 'In an important paper [CITATION], Lowe took these ideas further and combined the local momentum conservation of the DPD thermostat with the stochastic nature of the Anderson thermostat to derive a coarse-grained scheme, now called the Lowe-Anderson thermostat, which is much more efficient than treating the solvent with full DPD.', 'cond-mat-0603391-3-13-1': 'It has recently been applied, for example, to polymer dynamics [CITATION].', 'cond-mat-0603391-3-14-0': 'Independently, Malevanets and Kapral [CITATION] derived a method now called stochastic rotation dynamics (SRD) or also multiple particle collision dynamics (we choose the former nomenclature here).', 'cond-mat-0603391-3-14-1': 'In many ways it resembles the Lowe-Anderson thermostat [CITATION], or the much older direct simulation Monte Carlo (DSMC) method of Bird [CITATION].', 'cond-mat-0603391-3-14-2': "For all three of these methods, the particles are ideal, and move in a continuous space, subject to Newton's laws of motion.", 'cond-mat-0603391-3-14-3': 'At discrete time-steps, a coarse-grained collision step allows particles to exchange momentum.', 'cond-mat-0603391-3-14-4': 'In SRD, space is partitioned into a rectangular grid, and at discrete time-steps the particles inside each cell exchange momentum by rotating their velocity vectors relative to the center of mass velocity of the cell (see Fig. [REF]).', 'cond-mat-0603391-3-14-5': "Similarly, in Lowe's method, a particle can, with a certain probability, exchange its relative velocity with another that lies within a certain radius.", 'cond-mat-0603391-3-14-6': 'One can imagine other collision rules as well.', 'cond-mat-0603391-3-14-7': 'As long as they locally conserve momentum and energy, just as the lattice gas methods do, they generate the correct Navier Stokes hydrodynamics, although for SRD it is necessary to include a grid-shift procedure to enforce Galilean invariance, something first pointed out by Ihle and Kroll [CITATION].', 'cond-mat-0603391-3-14-8': 'An important advantage of SRD is that its simplified dynamics have allowed the analytic calculation of several transport coefficients [CITATION], greatly facilitating its use.', 'cond-mat-0603391-3-14-9': 'In the rest of this paper we will concentrate on the SRD method, although many of our general conclusions and derivations should be easily extendible to the Lowe-Anderson thermostat and related methods summarized in Fig. [REF].', 'cond-mat-0603391-3-15-0': 'SRD can be applied to directly simulate flow, as done for example in refs [CITATION], but its stochastic nature means that a noise average must be performed to calculate flow lines, and this may make it less efficient than pre-averaged methods like LB.', 'cond-mat-0603391-3-15-1': 'Where SRD becomes more attractive is for the simulation of complex particles embedded in a solvent.', 'cond-mat-0603391-3-15-2': 'This is because in addition to long-ranged HI, it also naturally contains Brownian fluctuations, and both typically need to be resolved for a proper statistical mechanical treatment of the dynamics of mesoscopic suspended particles.', 'cond-mat-0603391-3-15-3': 'For example, SRD has been used to study polymers under flow [CITATION], the effect of hydrodynamics on protein folding and polymer collapse [CITATION], and the conformations of vesicles under flow [CITATION].', 'cond-mat-0603391-3-15-4': 'In each of the previously mentioned examples, the suspended particles are coupled to the solvent by participating in the collision step.', 'cond-mat-0603391-3-16-0': "A less coarse-grained coupling can be achieved by allowing direct collisions, obeying Newton's laws of motion, between the SRD solvent and the suspended particles.", 'cond-mat-0603391-3-16-1': 'Such an approach is important for systems, such as colloidal suspensions, where the solvent and colloid time and length-scales need to be clearly separated.', 'cond-mat-0603391-3-16-2': 'Malevanets and Kapral [CITATION] derived such a hybrid algorithm that combined a full MD scheme of the solute-solute and solute-solvent interactions, while treating the solvent-solvent interactions via SRD.', 'cond-mat-0603391-3-16-3': 'Early applications were to a two-dimensional many-particle system [CITATION], and to the aggregation of colloidal particles [CITATION].', 'cond-mat-0603391-3-17-0': 'We have recently extended this approach, and applied it to the sedimentation of up to 800 hard sphere (HS) like colloids as a function of volume fraction, for a number of different values of the Peclet number [CITATION].', 'cond-mat-0603391-3-17-1': 'To achieve these simulations we adapted the method of Malevanets and Kapral [CITATION] in a number of ways that were only briefly described in ref. [CITATION].', 'cond-mat-0603391-3-17-2': 'In the current paper we provide more in depth analysis of the simulation method we used, and describe some potential pitfalls.', 'cond-mat-0603391-3-17-3': 'In particular we focus on the different time and length-scales that arise from our coarse-graining approach, as well as the role of dimensionless hydrodynamic numbers that express the relative importance of competing physical phenomena.', 'cond-mat-0603391-3-17-4': 'Very recently, another similar study of colloidal sedimentation and aggregation has been carried out [CITATION].', 'cond-mat-0603391-3-17-5': 'Some of our results and analysis are similar, but some are different, and we will mention these where appropriate.', 'cond-mat-0603391-3-18-0': 'After the introductory overview above, we begin in section [REF] by carefully describing the properties of a pure SRD fluid, focusing on simple derivations that highlight the dominant physics involved for each transport coefficient.', 'cond-mat-0603391-3-18-1': 'Section [REF] explains how to implement the coupling of a colloidal particle to an SRD solvent bath, and shows how to avoid spurious depletion interactions and how to understand lubrication forces.', 'cond-mat-0603391-3-18-2': 'In Section [REF] we analyze how optimizing the efficiency of particle based coarse-graining schemes affects different dimensionless hydrodynamic numbers, such as the Schmidt number, the Mach number, the Reynolds number, the Knudsen number, and the Peclet number.', 'cond-mat-0603391-3-18-3': 'Section [REF] describes the hierarchy of time-scales that determine the physics of a colloidal suspension, and compares these to the compressed hierarchy in our coarse-grained method.', 'cond-mat-0603391-3-18-4': 'In section [REF] we tackle the question of how to map from a coarse-grained simulation, optimized for computational efficiency, to a real colloidal system.', 'cond-mat-0603391-3-18-5': 'Finally after a Conclusions section, we include a few appendices that discuss amongst other things how to thermostat the SRD system (Appendix I); why the popular Langevin equation without memory effects does a remarkably poor job of capturing both the long and the short time dynamics of the colloidal velocity auto-correlation function (Appendix II); and, finally, how various physical properties and dimensionless numbers scale for an SRD simulation, and how these compare to a colloid of radius 10 nm or 1 [MATH]m in H[MATH]0 (Appendix III).', 'cond-mat-0603391-3-19-0': '# Properties of a pure SRD solvent', 'cond-mat-0603391-3-20-0': 'In SRD, the solvent is represented by a large number [MATH] of particles of mass [MATH].', 'cond-mat-0603391-3-20-1': 'Here and in the following, we will call these "fluid" particles, with the caveat that, however tempting, they should not be viewed as some kind of composite particles or clusters made up of the underlying (molecular) fluid.', 'cond-mat-0603391-3-20-2': 'Instead, SRD should be interpreted as a Navier Stokes solver that includes thermal noise.', 'cond-mat-0603391-3-20-3': 'The particles are merely a convenient computational device to facilitate the coarse-graining of the fluid properties.', 'cond-mat-0603391-3-21-0': '## Simulation method for a pure solvent', 'cond-mat-0603391-3-22-0': "In the first (propagation) step of the algorithm, the positions and velocities of the fluid particles are propagated for a time [MATH] (the time between collision steps) by accurately integrating Newton's equations of motion, [EQUATION] [MATH] and [MATH] are the position and velocity of fluid particle [MATH], respectively while [MATH] is the total (external) force on particle [MATH], which may come from an external field such as gravity, or fixed boundary conditions such as hard walls, or moving boundary conditions such as suspended colloids.", 'cond-mat-0603391-3-22-1': 'The direct forces between pairs of fluid particles are, however, neglected in the propagation step.', 'cond-mat-0603391-3-22-2': 'Herein lies the main advantage - the origin of the efficiency - of SRD.', 'cond-mat-0603391-3-22-3': 'Instead of directly treating the interactions between the fluid particles, a coarse-grained collision step is performed at each time-step [MATH]: First, space is partitioned into cubic cells of volume [MATH].', 'cond-mat-0603391-3-22-4': 'Next, for each cell, the particle velocities relative to the center of mass velocity [MATH] of the cell are rotated: [EQUATION] [MATH] is a rotation matrix which rotates velocities by a fixed angle [MATH] around a randomly oriented axis.', 'cond-mat-0603391-3-22-5': 'The aim of the collision step is to transfer momentum between the fluid particles while conserving the total momentum and energy of each cell.', 'cond-mat-0603391-3-22-6': 'Both the collision and the streaming step conserve phase-space volume, and it has been shown that the single particle velocity distribution evolves to a Maxwell-Boltzmann distribution [CITATION].', 'cond-mat-0603391-3-23-0': 'The rotation procedure can thus be viewed as a coarse-graining of particle collisions over space and time.', 'cond-mat-0603391-3-23-1': 'Because mass, momentum, and energy are conserved locally, the correct hydrodynamic (Navier Stokes) equations are captured in the continuum limit, including the effect of thermal noise [CITATION].', 'cond-mat-0603391-3-24-0': 'At low temperatures or small collision times [MATH], the transport coefficients of SRD, as originally formulated [CITATION], show anomalies caused by the fact that fluid particles in a given cell can remain in that cell and participate in several collision steps [CITATION].', 'cond-mat-0603391-3-24-1': 'Under these circumstances the assumption of molecular chaos and Galilean invariance are incorrect.', 'cond-mat-0603391-3-24-2': 'However, this anomaly can be cured by applying a random shift of the cell coordinates before the collision step [CITATION].', 'cond-mat-0603391-3-25-0': 'It should also be noted that the collision step in SRD does not locally conserve angular momentum.', 'cond-mat-0603391-3-25-1': 'As a consequence, the stress tensor [MATH] is not, in general, a symmetric function of the derivatives of the flow field (although it is still rotationally symmetric) [CITATION].', 'cond-mat-0603391-3-25-2': 'The asymmetric part can be interpreted as a viscous stress associated with the vorticity [MATH] of the velocity field [MATH].', 'cond-mat-0603391-3-25-3': 'The stress tensor will therefore depend on the amount of vorticity in the flow field.', 'cond-mat-0603391-3-25-4': 'However, the total force on a fluid element is determined not by the stress tensor itself, but by its divergence [MATH], which is what enters the Navier Stokes equations.', 'cond-mat-0603391-3-25-5': 'Taking the divergence causes the explicit vorticity dependence to drop out (the gradient of a curl is zero).', 'cond-mat-0603391-3-25-6': 'In principle, the stress tensor could be made symmetric by applying random rotations as well as random translations to the cell.', 'cond-mat-0603391-3-25-7': 'Or, alternatively, angular momentum can be explicitly conserved by dynamically adapting the collision rule, as done by Ryder et al [CITATION], who found no significant differences in fluid properties (although there is, of course, less flexibility in choosing simulation parameters).', 'cond-mat-0603391-3-25-8': 'This result has been confirmed by some recent theoretical calculations [CITATION], that demonstrate that the asymmetry of the stress tensor has only a few consequences, such as a correction to the sound wave attenuation associated with viscous dissipation of longitudinal density waves.', 'cond-mat-0603391-3-25-9': 'These are not important for the fluid properties we are trying to model, and so, on balance, we chose not to implement possible fixes to improve on angular momentum conservation.', 'cond-mat-0603391-3-26-0': '## Transport coefficients and the dimensionless mean-free path', 'cond-mat-0603391-3-27-0': 'The simplicity of SRD collisions has facilitated the analytical calculation of many transport coefficients [CITATION].', 'cond-mat-0603391-3-27-1': 'These analytical expressions are particularly useful because they enable us to efficiently tune the viscosity and other properties of the fluid, without the need for trial and error simulations.', 'cond-mat-0603391-3-27-2': 'In this section we will summarize a number of these transport coefficients, where possible giving a simple derivation of the dominant physics.', 'cond-mat-0603391-3-28-0': '### Units and the dimensionless mean-free path', 'cond-mat-0603391-3-29-0': 'In this paper we will use the following units: lengths will be in units of cell-size [MATH], energies in units of [MATH] and masses in units of [MATH] (This corresponds to setting [MATH], [MATH] and [MATH]).', 'cond-mat-0603391-3-29-1': 'Time, for example, is expressed in units of [MATH], the number density [MATH] and other derived units can be found in table [REF].', 'cond-mat-0603391-3-29-2': 'We find it instructive to express the transport coefficients and other parameters of the SRD fluid in terms of the dimensionless mean-free path [EQUATION] which provides a measure of the average fraction of a cell size that a fluid particle travels between collisions.', 'cond-mat-0603391-3-30-0': 'This particular choice of units helps highlight the basic physics of the coarse-graining method.', 'cond-mat-0603391-3-30-1': 'The (nontrivial) question of how to map them on to the units of real physical system will be discussed in section [REF].', 'cond-mat-0603391-3-31-0': '### Fluid self-diffusion constant', 'cond-mat-0603391-3-32-0': 'A simple back of the envelope estimate of the self-diffusion constant [MATH] of a fluid particle can be obtained from a random-walk picture.', 'cond-mat-0603391-3-32-1': 'In a unit of time [MATH], a particle will experience [MATH] collisions, in between which it moves an average distance [MATH].', 'cond-mat-0603391-3-32-2': 'Similarly, a heavier (tagged) particle of mass [MATH], which exchanges momentum with the fluid by participating in the coarse-grained collision step, will move an average distance [MATH] between collisions.', 'cond-mat-0603391-3-32-3': 'By viewing this motion as a random walk of step-size [MATH], the diffusion coefficient follows: [EQUATION] expressed in units of [MATH].', 'cond-mat-0603391-3-32-4': 'The diffusion coefficient [MATH] for a pure fluid particle of mass [MATH] is therefore given by [MATH].', 'cond-mat-0603391-3-33-0': 'A more systematic derivation of the diffusion coefficient of a fluid particle, but still within a random collision approximation, results in the following expression [CITATION]: [EQUATION]', 'cond-mat-0603391-3-33-1': 'The dependence on [MATH] is weak.', 'cond-mat-0603391-3-33-2': 'If, for example, we take [MATH], the value used in this paper, then [MATH], the same as Eq. ([REF]).', 'cond-mat-0603391-3-34-0': 'A similar expression can be derived for the self-diffusion coefficient of a heavier tagged particle of mass [MATH] [CITATION]: [EQUATION]', 'cond-mat-0603391-3-34-1': 'Note that this equation does not quite reduce to Eq. ([REF]) for [MATH] (because of slightly different approximations), but the relative discrepancy decreases with increasing [MATH].', 'cond-mat-0603391-3-35-0': 'While Eq. ([REF]) is accurate for larger mean-free paths, where the random collision approximation is expected to be valid, it begins to show deviations from simulations for [MATH] [CITATION], when longer-time kinetic correlations begin to develop.', 'cond-mat-0603391-3-35-1': 'Ripoll et al [CITATION] argue that these correlations induce interactions of a hydrodynamic nature that enhance the diffusion coefficient for a fluid particle.', 'cond-mat-0603391-3-35-2': 'For example, for [MATH] and [MATH], they measured a fluid self-diffusion constant [MATH] that is about [MATH] larger than the value found from Eq. ([REF]).', 'cond-mat-0603391-3-35-3': 'The enhancement is even more pronounced for heavier particles: for the same simulation parameters they found that [MATH] was enhanced by about [MATH] over the prediction of Eq. ([REF]) when [MATH].', 'cond-mat-0603391-3-36-0': 'We note that coupling a large particle to the solvent through participation in the coarse-grained collision step leads to a diffusion coefficient which scales with mass as [MATH], whereas if one couples a colloid of radius [MATH] to the solvent through direct MD collisions, one expects [MATH].', 'cond-mat-0603391-3-36-1': 'Moreover, it has been shown [CITATION] that the effective hydrodynamic particle radius is approximately given by [MATH].', 'cond-mat-0603391-3-36-2': 'For any reasonable average number of fluid particles per cell, the effective hydrodynamic radius [MATH] of the heavier particle is therefore much less than the collision cell size [MATH].', 'cond-mat-0603391-3-36-3': 'On the other hand, the hydrodynamic field is accurately resolved down to a scale comparable to [MATH] (vide infra).', 'cond-mat-0603391-3-36-4': 'What this implies is that this coupling method will yield correct hydrodynamic interactions only at large distances, when the colloids are more than several hydrodynamic radii apart.', 'cond-mat-0603391-3-36-5': 'All the above suggests that some care must be taken when interpreting the dynamics of heavier particles that couple through the coarse-grained collision step, especially when two or more heavy particles are in close proximity.', 'cond-mat-0603391-3-37-0': '### Kinematic viscosity', 'cond-mat-0603391-3-38-0': 'The spread of a velocity fluctuation [MATH] in a fluid can be described by a diffusion equation [CITATION]: [EQUATION] where [MATH] is the kinematic viscosity, which determines the rate at which momentum or vorticity "diffuses away".', 'cond-mat-0603391-3-38-1': 'The units of kinematic viscosity are [MATH] which are the same as those for particle self diffusion i.e. [MATH].', 'cond-mat-0603391-3-39-0': 'Momentum is transported through two mechanisms:', 'cond-mat-0603391-3-40-0': '1) By particles streaming between collision steps, leading to a "kinetic" contribution to the kinematic viscosity [MATH].', 'cond-mat-0603391-3-40-1': 'Since for this gas-like contribution the momentum is transported by particle motion, we expect [MATH] to scale like the particle self-diffusion coefficient [MATH], i.e. [MATH].', 'cond-mat-0603391-3-41-0': '2) By momentum being re-distributed among the particles of each cell during the collision step, resulting in a "collisional" contribution to the kinematic viscosity [MATH].', 'cond-mat-0603391-3-41-1': 'This mimics the way momentum is transferred due to inter-particle collisions, and would be the dominant contribution in a dense fluid such as water at standard temperature and pressure.', 'cond-mat-0603391-3-41-2': 'Again a simple random-walk argument explains the expected scaling in SRD: Each collision step distributes momentum among particles in a cell, making a step-size that scales like [MATH].', 'cond-mat-0603391-3-41-3': 'Since there are [MATH] collision steps per unit time [MATH], this suggests that the collisional contribution to the kinematic viscosity should scale as [MATH].', 'cond-mat-0603391-3-42-0': 'Accurate analytical expressions for the kinematic viscosity [MATH] of SRD have been derived [CITATION], and these can be rewritten in the following dimensionless form: [EQUATION] where the dependence on the collisional angle [MATH] and fluid number density [MATH] is subsumed in the following two factors: [EQUATION]', 'cond-mat-0603391-3-42-1': 'These factors only depend weakly on [MATH] for the typical parameters used in simulations.', 'cond-mat-0603391-3-42-2': 'For example, at [MATH] and [MATH], the angle and number density we use in this paper, they take the values [MATH] and [MATH].', 'cond-mat-0603391-3-42-3': 'For this choice of collision angle [MATH], they monotonically converge to [MATH] in the limit of large [MATH].', 'cond-mat-0603391-3-43-0': '### Shear viscosity', 'cond-mat-0603391-3-44-0': 'Suppose the fluid is sheared in the x direction, with the gradient of the average flow field in the y direction.', 'cond-mat-0603391-3-44-1': 'Two neighboring fluid elements with different y-coordinates will then experience a friction force in the x-direction, as expressed by the xy component of the stress tensor [MATH], which for a simple liquid is linearly proportional to the instantaneous flow field gradient, [EQUATION]', 'cond-mat-0603391-3-44-2': 'The coefficient of proportionality [MATH] is called the shear viscosity, and is it related to the kinematic viscosity by [MATH], where [MATH] is the fluid mass density.', 'cond-mat-0603391-3-44-3': 'From Eqs. [REF]-[REF] it follows that the two contributions to the shear viscosity can be written in dimensionless form as: [EQUATION] where [MATH] is the unit of shear viscosity.', 'cond-mat-0603391-3-45-0': 'In contrast to the expressions for the diffusion of a fluid particle or a tagged particle, Eqs. ([REF]) - ([REF]) compare quantitatively to simulations over a wide range of parameters [CITATION].', 'cond-mat-0603391-3-45-1': 'For the parameters we used in our simulations in [CITATION], i.e. [MATH], the collisional contribution to the viscosity dominates: [MATH] and [MATH].', 'cond-mat-0603391-3-45-2': 'This is typical for [MATH], where [MATH] and [MATH] can be taken to a good first approximation by the collisional contribution only.', 'cond-mat-0603391-3-45-3': 'In fact throughout this paper we will mainly focus on this small [MATH] limit.', 'cond-mat-0603391-3-46-0': "It is also instructive to compare the expressions derived in this section to what one would expect for simple gases, where, as famously first derived and demonstrated experimentally by Maxwell in the 1860's [CITATION], the shear viscosity is independent of density.", 'cond-mat-0603391-3-46-1': 'This result differs from the kinetic (gas-like) contribution to the viscosity in Eq. ([REF]), because in a real dilute gas the mean-free path scales as [MATH], canceling the dominant density dependence in [MATH].', 'cond-mat-0603391-3-46-2': 'The same argument explains why the self-diffusion and kinematic viscosity of the SRD fluid are, to first order, independent of [MATH], while in a gas they would scale as [MATH].', 'cond-mat-0603391-3-46-3': 'In SRD, the mean-free path [MATH] and the density [MATH] can be varied independently.', 'cond-mat-0603391-3-46-4': 'Moreover, the collisional contribution to the viscosity adds a new dimension, allowing a much wider range of physical fluids to be modeled than just simple gases.', 'cond-mat-0603391-3-47-0': '# Colloid simulation method', 'cond-mat-0603391-3-48-0': 'Malevanets and Kapral [CITATION] first showed how to implement a hybrid MD scheme that couples a set of colloids to a bath of SRD particles.', 'cond-mat-0603391-3-48-1': 'In this section, we expand on their method, describing in detail the implementation we used in ref. [CITATION].', 'cond-mat-0603391-3-48-2': 'We restrict ourselves to HS like colloids with steep interparticle repulsions, although attractions between colloids can easily be added on.', 'cond-mat-0603391-3-48-3': 'The colloid-colloid and colloid-fluid interactions, [MATH] and [MATH] respectively, are integrated via a normal MD procedure, while the fluid-fluid interactions are coarse-grained with SRD.', 'cond-mat-0603391-3-48-4': 'Because the number of fluid particles vastly outnumbers the number of HS colloids, treating their interactions approximately via SRD greatly speeds up the simulation.', 'cond-mat-0603391-3-49-0': '## Colloid-colloid and colloid-solvent interactions', 'cond-mat-0603391-3-50-0': 'Although it is possible to implement an event-driven dynamics of HS colloids in an SRD solvent [CITATION], here we approximate pure HS colloids by steep repulsive interactions of the WCA form [CITATION]: [EQUATION]', 'cond-mat-0603391-3-50-1': 'Similarly, the colloid-fluid interaction takes the WCA form: [EQUATION]', 'cond-mat-0603391-3-50-2': 'The mass [MATH] of a fluid particle is typically much smaller than the mass [MATH] of a colloid, so that the average thermal velocity of the fluid particles is larger than that of the colloid particles by a factor [MATH].', 'cond-mat-0603391-3-50-3': 'For this reason the time-step [MATH] is usually restricted by the fluid-colloid interaction ([REF]), allowing fairly large exponents [MATH] for the colloid-colloid interaction [MATH].', 'cond-mat-0603391-3-50-4': 'We choose [MATH], which makes the colloid-colloid potential steep, more like hard-spheres, while still soft enough to allow the time-step to be set by the colloid-solvent interaction.', 'cond-mat-0603391-3-51-0': 'The positions and velocities of the colloidal spheres are propagated through the Velocity Verlet algorithm [CITATION] with a time step [MATH]: [EQUATION] [MATH] and [MATH] are the position and velocity of colloid [MATH], respectively.', 'cond-mat-0603391-3-51-1': '[MATH] is the total force on that colloid, exerted by the fluid particles, an external field, such as gravity, external potentials such as repulsive walls, as well as other colloids within the range of the interaction potential ([REF]).', 'cond-mat-0603391-3-52-0': "The positions [MATH] and velocities [MATH] of SRD particles are updated by similarly solving Newton's Eqns. ([REF],[REF]) every time-step [MATH], and with the SRD procedure of Eq. ([REF]) every time-step [MATH].", 'cond-mat-0603391-3-53-0': 'Choosing both [MATH] and [MATH] as large as possible enhances the efficiency of a simulation.', 'cond-mat-0603391-3-53-1': 'To first order, each timestep is determined by different physics [MATH] by the steepness of the potentials, and [MATH] by the desired fluid properties, and so there is some freedom in choosing their relative values.', 'cond-mat-0603391-3-53-2': 'We used [MATH] for our simulations of sedimentation [CITATION], but other authors have used ratios of [MATH] [CITATION] or even an order of magnitude larger than that [CITATION].', 'cond-mat-0603391-3-53-3': 'Later in the paper we will revisit this question, linking the time-steps to various dimensionless hydrodynamic numbers and Brownian time-scales.', 'cond-mat-0603391-3-54-0': '## Stick and slip boundary conditions', 'cond-mat-0603391-3-55-0': 'Because the surface of a colloid is never perfectly smooth, collisions with fluid particles transfer angular as well as linear momentum.', 'cond-mat-0603391-3-55-1': 'As demonstrated in Fig. [REF], the exact molecular details of the colloid-fluid interactions may be very complex, and mediated via co- and counter-ions, grafted polymer brushes etc.', 'cond-mat-0603391-3-55-2': 'However, on the time and length-scales over which our hybrid MD-SRD method coarse-grains the solvent, these interactions can be approximated by stick boundary conditions: the tangential velocity of the fluid, relative to the surface of the colloid, is zero at the surface of the colloid [CITATION].', 'cond-mat-0603391-3-55-3': 'For most situations, this boundary condition should be sufficient, although in some cases, such as a non-wetting surface, large slip-lengths may occur [CITATION].', 'cond-mat-0603391-3-56-0': 'In computer simulations, stick boundary conditions may be implemented by bounce-back rules, where both parallel and perpendicular components of the relative velocity are reversed upon a collision with a surface.', 'cond-mat-0603391-3-56-1': 'These have been applied by Lamura et al [CITATION], who needed to modify the bounce-back rules slightly to properly reproduce stick boundaries for a Poiseuille flow geometry.', 'cond-mat-0603391-3-57-0': 'Stick boundaries can also be modeled by a stochastic rule.', 'cond-mat-0603391-3-57-1': 'After a collision, the relative tangential velocity [MATH] and relative normal velocity [MATH] are taken from the distributions: [EQUATION] so that the colloid acts as an additional thermostat [CITATION].', 'cond-mat-0603391-3-57-2': 'Such stochastic boundary conditions have been used for colloidal particles by Inoue et al. [CITATION] and Hecht et al. [CITATION].', 'cond-mat-0603391-3-57-3': 'We have systematically studied several implementations of stick boundary conditions for spherical colloids [CITATION], and derived a version of the stochastic boundary conditions which reproduces linear and angular velocity correlation functions that agree with Enskog theory for short times, and hydrodynamic mode-coupling theory for long times.', 'cond-mat-0603391-3-57-4': 'We argue that the stochastic rule of Eq. ([REF]) is more like a real physical colloid - where fluid-surface interactions are mediated by steric stabilizing layers or local co- and counter-ion concentrations - than bounce-back rules are [CITATION].', 'cond-mat-0603391-3-58-0': 'Nevertheless, in this paper, many examples will be for radial interactions such as those described in Eq. ([REF]).', 'cond-mat-0603391-3-58-1': 'These do not transfer angular momentum to a spherical colloid, and so induce effective slip boundary conditions.', 'cond-mat-0603391-3-58-2': 'For many of the hydrodynamic effects we will discuss here the difference with stick boundary conditions is quantitative, not qualitative, and also well understood.', 'cond-mat-0603391-3-59-0': '## Depletion and lubrication forces', 'cond-mat-0603391-3-60-0': '### Spurious depletion forces induced by the fluid', 'cond-mat-0603391-3-61-0': 'We would like to issue a warning that the additional fluid degrees of freedom may inadvertently introduce depletion forces between the colloids.', 'cond-mat-0603391-3-61-1': 'Because the number density of SRD particles is much higher than that of the colloids, even a small overlap between two colloids can lead to enormous attractions.', 'cond-mat-0603391-3-62-0': 'For low colloid densities the equilibrium depletion interaction between any two colloids caused by the presence of the ideal fluid particles is given by [CITATION]: [EQUATION] where [MATH] is the number density of fluid particles and [MATH] is the (free) volume excluded to the fluid by the presence of two colloids separated by a distance [MATH].', 'cond-mat-0603391-3-62-1': 'The latter is given by [EQUATION] where [MATH].', 'cond-mat-0603391-3-62-2': 'An example is given in Fig. [REF] where we have plotted the resulting depletion potential for the colloid-solvent interaction ([REF]), with [MATH] as routinely used in our simulations, as well as the depletion interaction resulting from a truly HS colloid-solvent interaction.', 'cond-mat-0603391-3-62-3': 'The latter can easily be calculated analytically, with the result [EQUATION]', 'cond-mat-0603391-3-62-4': 'For the pure HS interactions, one could take [MATH] and the depletion forces would have no effect.', 'cond-mat-0603391-3-62-5': 'But for interactions such as those used in Eqs. ([REF]) and ([REF]), the softer repulsions mean that inter-colloid distances less than [MATH] are regularly sampled.', 'cond-mat-0603391-3-62-6': 'A more stringent criterion of [MATH] must therefore be used to avoid spurious depletion effects.', 'cond-mat-0603391-3-62-7': 'As an example, we re-analyze the simulations of ref. [CITATION], where a WCA form with [MATH] was used for the fluid-colloid interactions, and a normal Lennard Jones [MATH] potential with [MATH] was used for the colloid-colloid interactions.', 'cond-mat-0603391-3-62-8': 'The authors found differences between "vacuum" calculations without SRD particles, and a hybrid scheme coupling the colloids to an SRD solvent.', 'cond-mat-0603391-3-62-9': 'These were correctly attributed to "solvent induced pressure", which we quantify here as depletion interactions.', 'cond-mat-0603391-3-62-10': 'For one set of their parameters, [MATH], the effect is mainly to soften the repulsion, but for their other parameter set: [MATH], depletion attractions induce an effective attractive well-depth of over [MATH]!', 'cond-mat-0603391-3-63-0': 'Since the depletion potentials can be calculated analytically, one might try counteracting them by introducing a compensating repulsive potential of the form: [MATH] between the colloids.', 'cond-mat-0603391-3-63-1': 'However, there are three problems with this approach: Firstly, at higher colloid packing fractions, three and higher order interactions may also need to be added, and these are very difficult to calculate.', 'cond-mat-0603391-3-63-2': 'Secondly, the depletion interactions are not instantaneous, and in fact only converge to their equilibrium average algebraically in time [CITATION].', 'cond-mat-0603391-3-63-3': 'While this is not a problem for equilibrium properties, it will introduce errors for non-equilibrium properties.', 'cond-mat-0603391-3-63-4': 'Finally, when external fields drive the colloid, small but persistent anisotropies in the solvent density around a colloid may occur [CITATION].', 'cond-mat-0603391-3-63-5': 'Although these density variations are (and should be) small, the resulting variations in depletion interactions can be large.', 'cond-mat-0603391-3-64-0': 'To avoid these problems, we routinely choose the colloid-fluid interaction range [MATH] slightly below half the colloid diameter [MATH].', 'cond-mat-0603391-3-64-1': 'More precisely, we ensure that the colloid-colloid interaction equals [MATH] at a distance [MATH] where the depletion interactions have become zero, i.e., at a distance of twice the colloid-solvent interaction cut-off radius.', 'cond-mat-0603391-3-64-2': 'Smaller distances will consequently be rare, and adequately dealt with by the compensation potential.', 'cond-mat-0603391-3-64-3': 'This solution may be a more realistic representation anyhow, since in practice for charge and even for sterically stabilized colloids, the effective colloid-colloid diameter [MATH] is expected to be larger than twice the effective colloid-fluid diameter [MATH].', 'cond-mat-0603391-3-64-4': 'This is particularly so for charged colloids at large Debye screening lengths.', 'cond-mat-0603391-3-65-0': '### Lubrication forces depend on surface details', 'cond-mat-0603391-3-66-0': 'When two surfaces approach one another, they must displace the fluid between them, while if they move apart, fluid must flow into the space freed up between the surfaces.', 'cond-mat-0603391-3-66-1': 'At very short inter-surface distances, this results in so-called lubrication forces, which are repulsive for colloids approaching each other, and attractive for colloids moving apart [CITATION].', 'cond-mat-0603391-3-66-2': 'These forces are expected to be particularly important for driven dense colloidal suspensions; see e.g. [CITATION] for a recent review.', 'cond-mat-0603391-3-67-0': 'An additional advantage of our choice of diameters [MATH] above is that more fluid particles will fit in the space between two colloids, and consequently the lubrication forces will be more accurately represented.', 'cond-mat-0603391-3-67-1': 'It should be kept in mind that for colloids, the exact nature of the short-range lubrication forces will depend on physical details of the surface, such as its roughness, or presence of a grafted polymeric stabilizing layer [CITATION].', 'cond-mat-0603391-3-67-2': 'For perfectly smooth colloids, analytic limiting expressions for the lubrication forces can be derived [CITATION], showing a divergence at short distances.', 'cond-mat-0603391-3-67-3': 'We have confirmed that SRD resolves these lubrication forces down to surprisingly low interparticle distances.', 'cond-mat-0603391-3-67-4': 'But, at some point, this will break down, depending of course on the choice of simulation parameters (such as [MATH], [MATH], and [MATH]), as well as the details of the particular type of colloidal particles that one wishes to model [CITATION].', 'cond-mat-0603391-3-67-5': 'An explicit analytic correction could be applied to properly resolve these forces for very small distances, as was recently implemented for Lattice Boltzmann [CITATION].', 'cond-mat-0603391-3-67-6': 'However, in this paper, we will assume that our choice of [MATH] is small enough for SRD to sufficiently resolve lubrication forces.', 'cond-mat-0603391-3-67-7': 'The lack of a complete divergence at very short distances may be a better model of what happens for real colloids anyway.', 'cond-mat-0603391-3-67-8': 'For dense suspensions under strong shear, explicit lubrication force corrections as well as other short-ranged colloid-colloid interactions arising from surface details such as polymer coats will almost certainly need to be put in by hand, see e.g. ref. [CITATION] for further discussion of this subtle problem.', 'cond-mat-0603391-3-68-0': '## Test of static properties', 'cond-mat-0603391-3-69-0': 'The equilibrium properties of a statistical mechanical system should be independent of the detailed dynamics by which it explores phase space.', 'cond-mat-0603391-3-69-1': 'Therefore one requisite condition imposed on our coarse-grained dynamics is that it reproduces the correct static properties of an ensemble of colloids.', 'cond-mat-0603391-3-70-0': 'An obvious property to measure is the radial distribution function [MATH] [CITATION].', 'cond-mat-0603391-3-70-1': 'In Fig. [REF] we depict some [MATH]s obtained from SRD simulations at a number of different densities, and compare these to similar simulations with standard Brownian dynamics.', 'cond-mat-0603391-3-70-2': 'The colloids interact via potentials of the form ([REF]) - ([REF]) with [MATH], [MATH], and [MATH].', 'cond-mat-0603391-3-70-3': 'For the SRD we used [MATH], [MATH] and [MATH], implying [MATH].', 'cond-mat-0603391-3-70-4': 'For the BD we used a background friction calculated from the effective hydrodynamic radius (vide infra) and a time-step equal to the [MATH] used above.', 'cond-mat-0603391-3-70-5': 'The colloids were placed in a box of dimensions [MATH], and the number of particles was varied from 64 to 540 in order to achieve different packing fractions [EQUATION] where the subscript in [MATH] refers to volume fraction based on the colloid-colloid interaction, to distinguished it from the volume fraction [MATH] based on the colloid hydrodynamic radius [CITATION].', 'cond-mat-0603391-3-70-6': 'From Fig. [REF] is clear that the two simulations are indistinguishable within statistical errors, as required.', 'cond-mat-0603391-3-70-7': 'Even though [MATH], we still found it necessary to include an explicit compensating potential without which the [MATH] generated by SRD is increased noticeably at contact when compared to BD simulations.', 'cond-mat-0603391-3-71-0': 'Finally, we note that any simulation will show finite size effects.', 'cond-mat-0603391-3-71-1': 'These may be thermodynamic as well as dynamic.', 'cond-mat-0603391-3-71-2': 'If there are thermodynamic finite-size effects (for example due to a long correlation length caused by proximity to a critical point), then a correct simulation technique will show the same behaviour regardless of the underlying dynamics.', 'cond-mat-0603391-3-72-0': '# Dimensionless numbers', 'cond-mat-0603391-3-73-0': 'Different regimes of hydrodynamic behavior can be characterized by a series of dimensionless numbers that indicate the relative strengths of competing physical processes [CITATION].', 'cond-mat-0603391-3-73-1': 'If two distinct physical systems can be described by the same set of hydrodynamic numbers, then their flow behavior will be governed by the same physics, even if their time and length scales differ by orders of magnitude.', 'cond-mat-0603391-3-73-2': 'In other words, a macroscopic boulder sedimenting in viscous molten magma and a mesoscopic colloid sedimenting in water may show similar behavior if they share key hydrodynamic dimensionless numbers.', 'cond-mat-0603391-3-73-3': 'It is therefore very instructive to analyze where a system described by SRD sits in "hydrodynamic parameter space".', 'cond-mat-0603391-3-73-4': 'In this section we review this parameter space, one "dimension" at a time, by exploring the hydrodynamic numbers listed in Table [REF].', 'cond-mat-0603391-3-74-0': '## Schmidt number', 'cond-mat-0603391-3-75-0': 'The Schmidt number [EQUATION] is important for characterizing a pure fluid.', 'cond-mat-0603391-3-75-1': 'It expresses the rate of diffusive momentum transfer, measured by the kinematic viscosity [MATH], relative to the rate of diffusive mass transfer, measured by the fluid particle self-diffusion coefficient [MATH].', 'cond-mat-0603391-3-75-2': 'For a gas, momentum transport is dominated by mass diffusion, so that Sc [MATH], whereas for a liquid, momentum transport is dominated by inter-particle collisions, and Sc [MATH].', 'cond-mat-0603391-3-75-3': 'For low Schmidt numbers, the dynamics of SRD is indeed "gas-like", while for larger Sc, the dynamics shows collective behavior reminiscent of the hydrodynamics of a liquid [CITATION].', 'cond-mat-0603391-3-75-4': 'This distinction will be particularly important for simulating the dynamics of embedded particles that couple to the solvent through participation in the collision step.', 'cond-mat-0603391-3-75-5': 'For particles that couple directly, either through a potential, or through bounce-back or stochastic boundary conditions, this distinction is less important because the fluid-particle diffusion coefficient [MATH] does not directly enter the Navier Stokes equations.', 'cond-mat-0603391-3-75-6': 'Of course other physical properties can be affected.', 'cond-mat-0603391-3-75-7': 'For example, we expect that the self diffusion coefficient of a colloid [MATH] should be much smaller than the fluid diffusion coefficient, i.e. [MATH] to achieve the correct time-scale separation, as we will see in Section [REF].', 'cond-mat-0603391-3-76-0': '### Dependence of Schmidt number on simulation parameters', 'cond-mat-0603391-3-77-0': 'From Eqs. ([REF]) - ([REF]) it follows that the Schmidt number can be rewritten as [EQUATION] where we have ignored the dependence on [MATH] and [MATH] since the dominant scaling is with the dimensionless mean-free path [MATH].', 'cond-mat-0603391-3-77-1': 'For larger [MATH], where the kinetic contributions to the viscosity dominate, the Sc number is small, and the dynamics is gas-like.', 'cond-mat-0603391-3-77-2': 'Because both the fluid diffusion coefficient ([REF]) and the kinetic contribution to the viscosity ([REF]) scale linearly with [MATH], the only way to obtain a large Sc number is in the limit [MATH] where the collisional contribution to [MATH] dominates.', 'cond-mat-0603391-3-78-0': 'Low Sc numbers may be a more general characteristic of particle-based coarse-graining methods.', 'cond-mat-0603391-3-78-1': 'For computational reasons, the number of particles is normally greatly reduced compared to the solvent one is modeling.', 'cond-mat-0603391-3-78-2': 'Thus the average inter-particle separation and mean-free path are substantially increased, typically leading to a smaller kinematic viscosity and a larger fluid diffusion coefficient [MATH].', 'cond-mat-0603391-3-78-3': 'With DPD, for example, one typically finds [MATH] [CITATION].', 'cond-mat-0603391-3-78-4': 'It is therefore difficult to achieve large Sc numbers, as in a real liquid, without sacrificing computational efficiency.', 'cond-mat-0603391-3-78-5': 'But this may not always be necessary.', 'cond-mat-0603391-3-78-6': 'As long as momentum transport is clearly faster than mass transport, the solvent should behave in a liquid-like fashion.', 'cond-mat-0603391-3-78-7': 'With this argument in mind, and also because the Sc number does not directly enter the Navier Stokes equations we want to solve, we use [MATH] in this paper and in ref. [CITATION], which leads to a Sc [MATH] for [MATH] and [MATH].', 'cond-mat-0603391-3-78-8': 'This should be sufficient for the problems we study.', 'cond-mat-0603391-3-78-9': 'For other systems, such as those where the suspended particles are coupled to the solvent through the collision step, more care may be needed to ensure that the Sc number is indeed large enough [CITATION].', 'cond-mat-0603391-3-79-0': '## Mach number', 'cond-mat-0603391-3-80-0': 'The Mach number measures the ratio [EQUATION] between [MATH], the speed of solvent or colloid flow, and [MATH], the speed of sound.', 'cond-mat-0603391-3-80-1': 'In contrast to the Schmidt number, which is an intrinsic property of the solvent, it depends directly on flow velocity.', 'cond-mat-0603391-3-80-2': 'The Ma number measures compressibility effects [CITATION] since sound speed is related to the compressibility of a liquid.', 'cond-mat-0603391-3-80-3': 'Because [MATH] in many liquids is of order [MATH] m/s, the Ma numbers for physical colloidal systems are extremely small under normally achievable flow conditions.', 'cond-mat-0603391-3-80-4': 'Just as for the Sc number, however, particle based coarse-graining schemes drastically lower the Ma number.', 'cond-mat-0603391-3-80-5': 'The particle mass [MATH] is typically much greater than the mass of a molecule of the underlying fluid, resulting in lower velocities, and moreover, due to the lower density, collisions also occur less frequently.', 'cond-mat-0603391-3-80-6': 'These effects mean that the speed of sound is much lower in a coarse-grained system than it is in the underlying physical fluid.', 'cond-mat-0603391-3-80-7': 'Or, in other words, particle based coarse-graining systems are typically much more compressible than the solvents they model.', 'cond-mat-0603391-3-81-0': 'Ma number effects typically scale with [MATH] [CITATION], and so the Ma number does not need to be nearly as small as for a realistic fluid to still be in the correct regime of hydrodynamic parameter space.', 'cond-mat-0603391-3-81-1': 'This is convenient, because to lower the Ma number, one would need to integrate over longer fluid particle trajectories to allow, for example, a colloidal particle to flow over a given distance, making the simulation computationally more expensive.', 'cond-mat-0603391-3-81-2': 'So there is a compromise between small Ma numbers and computational efficiency.', 'cond-mat-0603391-3-81-3': 'We limit our Ma numbers to values such that [MATH], but it might be possible, in some situations, to double or triple that limit without causing undue error.', 'cond-mat-0603391-3-81-4': 'For example, incompressible hydrodynamics is used for aerodynamic flows up to such Ma numbers since the errors are expected to scale as [MATH] [CITATION].', 'cond-mat-0603391-3-81-5': 'When working in units of [MATH] and [MATH], the only way to keep the Ma number below our upper limit is to restrict the maximum flow velocity to [MATH].', 'cond-mat-0603391-3-81-6': 'The flow velocity itself is, of course, determined by the external fields, but also by other parameters of the system.', 'cond-mat-0603391-3-82-0': '## Reynolds number', 'cond-mat-0603391-3-83-0': 'The Reynolds number is one of the most important dimensionless numbers characterizing hydrodynamic flows.', 'cond-mat-0603391-3-83-1': 'Mathematically, it measures the relative importance of the non-linear terms in the Navier-Stokes equations [CITATION].', 'cond-mat-0603391-3-83-2': 'Physically, it determines the relative importance of inertial over viscous forces, and can be expressed as: [EQUATION] where [MATH] is a length-scale, in our case, the hydrodynamic radius of a colloid, i.e. [MATH], and [MATH] is a flow velocity.', 'cond-mat-0603391-3-84-0': 'For a spherical particle in a flow, the following heuristic argument helps clarify the physics behind the Reynolds number: If the Stokes time [EQUATION] it takes a particle to advect over its own radius is about the same as the kinematic time [EQUATION] it takes momentum to diffuse over that distance, i.e. Re=[MATH], then the particle will feel vorticity effects from its own motion a distance [MATH] away, leading to non-linear inertial contributions to its motion.', 'cond-mat-0603391-3-84-1': 'Since hydrodynamic interactions can decay as slowly as [MATH], their influence can be non-negligible.', 'cond-mat-0603391-3-84-2': 'If, on the other hand, Re [MATH], then vorticity will have diffused away and the particle will only feel very weak hydrodynamic effects from its own motion.', 'cond-mat-0603391-3-85-0': 'Exactly when inertial finite Re effects become significant depends on the physical system under investigation.', 'cond-mat-0603391-3-85-1': 'For example, in a pipe, where the length-scale [MATH] in Eq. ([REF]) is its diameter, the transition from simpler laminar to more complex turbulent flow is at a pipe Reynolds number of Re [MATH] [CITATION].', 'cond-mat-0603391-3-85-2': 'On the other hand, for a single spherical particle, a non-linear dependence of the friction [MATH] on the velocity [MATH], induced by inertial effects, starts to become noticeable for a particle Reynolds number of Re [MATH] [CITATION], while deviations in the symmetry of the streamlines around a rotating sphere have been observed in calculations for Re [MATH] [CITATION].', 'cond-mat-0603391-3-86-0': 'For typical colloidal suspensions, where the particle diameter is on the order of a few [MATH]m down to a few nm, the particle Reynolds number is rarely more than [MATH].', 'cond-mat-0603391-3-86-1': 'In this so-called Stokes regime viscous forces dominate and inertial effects can be completely ignored.', 'cond-mat-0603391-3-86-2': 'The Navier-Stokes equations can be replaced by the linear Stokes equations [CITATION] so that analytic solutions are easier to obtain [CITATION].', 'cond-mat-0603391-3-86-3': 'However, some of the resulting behavior is non-intuitive for those used to hydrodynamic effects on a macroscopic scale.', 'cond-mat-0603391-3-86-4': 'For example, as famously explained by Purcell in a talk entitled "Life at low Reynolds numbers" [CITATION], many simple processes in biology occur on small length scales, well into the Stokes regime.', 'cond-mat-0603391-3-86-5': 'The conditions that bacteria, typically a few [MATH]m long, experience in water are more akin to those humans would experience in extremely thick molasses.', 'cond-mat-0603391-3-86-6': 'Similarly, colloids, polymers, vesicles and other small suspended objects are all subject to the same physics, their motion dominated by viscous forces.', 'cond-mat-0603391-3-86-7': 'For example, if for a colloid sedimenting at 1 [MATH]m/s, gravity were instantaneously turned off, then the Stokes equations suggest that it would come to a complete halt in a distance significantly less than one , reflecting the irrelevance of inertial forces in this low Re number regime.', 'cond-mat-0603391-3-86-8': 'It should be kept in mind that when the Stokes regime is reached because of small length scales (as opposed to very large viscosities such as those found for volcanic lava flows), then thermal fluctuations are also important.', 'cond-mat-0603391-3-86-9': 'These will drive diffusive behavior [CITATION].', 'cond-mat-0603391-3-86-10': 'In many ways SRD is ideally suited for this regime of small particles because the thermal fluctuations are naturally included [CITATION].', 'cond-mat-0603391-3-87-0': '### Dependence of the Reynolds number on simulation parameters', 'cond-mat-0603391-3-88-0': 'From Eqs. ([REF]) and ([REF]), it follows that the Reynolds number for a colloid of hydrodynamic radius [MATH] can be written as: [EQUATION]', 'cond-mat-0603391-3-88-1': 'Equating the hydrodynamic radius [MATH] to [MATH] from the fluid-colloid WCA interaction of Eq. ([REF]) is not quite correct, as we will see in section [REF], but is a good enough approximation in light of the fact that these hydrodynamic numbers are qualitative rather than quantitative indicators.', 'cond-mat-0603391-3-89-0': 'In order to keep the Reynolds number low, one must either use small particles, or very viscous fluids, or low velocities [MATH].', 'cond-mat-0603391-3-89-1': 'The latter condition is commensurate with a low Ma number, which is also desirable.', 'cond-mat-0603391-3-89-2': 'For small enough [MATH] (and ignoring for simplicty the factor [MATH] in Eq. ([REF])) the Re number then scales as [EQUATION]', 'cond-mat-0603391-3-89-3': 'Again, we see that a smaller mean-free path [MATH], which enhances the collisional viscosity, also helps bring the Re number down.', 'cond-mat-0603391-3-89-4': 'This parameter choice is also consistent with a larger Sc number.', 'cond-mat-0603391-3-89-5': 'Thus larger Sc and smaller Ma numbers both help keep the Re number low for SRD.', 'cond-mat-0603391-3-89-6': 'In principle a large viscosity can also be obtained for large [MATH], which enhances the kinetic viscosity, but this choice also lowers the Sc number and raises the Knudsen number, which, as we will see in the next sub-section, is not desirable.', 'cond-mat-0603391-3-90-0': 'Just as was found for the Ma number, it is relatively speaking more expensive computationally to simulate for low Re numbers because the flow velocity must be kept low, which means longer simulation times are necessary to reach time-scales where the suspended particles or fluid flows have moved a significant distance.', 'cond-mat-0603391-3-90-1': 'We therefore compromise and normally keep Re [MATH], which is similar to the choice made for LB simulations [CITATION].', 'cond-mat-0603391-3-90-2': 'For many situations related to the flow of colloids, this should be sufficiently stringent.', 'cond-mat-0603391-3-91-0': '## Knudsen number', 'cond-mat-0603391-3-92-0': 'The Knudsen number Kn measures rarefaction effects, and can be written as [EQUATION] where [MATH] is a characteristic length-scale of the fluid flow, and [MATH] is the mean-free path of the fluid or gas.', 'cond-mat-0603391-3-92-1': 'For a colloid in SRD one could take [MATH] and [MATH].', 'cond-mat-0603391-3-92-2': 'For large Knudsen numbers [MATH] continuum Navier-Stokes equations completely break down, but even for much smaller Knudsen numbers, significant rarefaction effects are seen.', 'cond-mat-0603391-3-92-3': 'For example, for flow in a pipe, where [MATH] in Eq. ([REF]) is taken to be the pipe radius, important non-continuum effects are seen when [MATH].', 'cond-mat-0603391-3-92-4': 'In fact it is exactly for these conditions of modest Kn numbers, when corrections to the Navier Stokes become noticeable, that DSMC approaches are often used [CITATION].', 'cond-mat-0603391-3-92-5': 'SRD, which is related to DSMC, may also be expected to work well for such flows.', 'cond-mat-0603391-3-92-6': 'It could find important applications in microfluidics and other micromechanical devices where Kn effects are expected to play a role [CITATION].', 'cond-mat-0603391-3-93-0': 'Colloidal dispersions normally have very small Kn numbers because the mean-free path of most liquid solvents is very small.', 'cond-mat-0603391-3-93-1': 'For water at standard temperature and pressure [MATH] .', 'cond-mat-0603391-3-93-2': 'Just as found for the other dimensionless numbers, coarse-graining typically leads to larger Kn numbers because of the increase of the mean-free path.', 'cond-mat-0603391-3-93-3': 'Making the Kn number smaller also typically increases the computational cost because a larger number of collisions need to be calculated.', 'cond-mat-0603391-3-93-4': 'In our simulations, we keep Kn[MATH] for spheres.', 'cond-mat-0603391-3-93-5': 'This rough criterion is based on the observation that for small Kn numbers the friction coefficient on a sphere is expected to be decreased by a factor [MATH] Kn, where [MATH] is a material dependent constant of order [MATH] [CITATION], so that we expect Kn number effects to be of the same order as other coarse-graining errors.', 'cond-mat-0603391-3-93-6': 'There are two ways to achieve small Kn numbers: one is by increasing [MATH], the other is by decreasing [MATH].', 'cond-mat-0603391-3-93-7': 'The second condition is commensurate with a large Sc number or a small Re number.', 'cond-mat-0603391-3-94-0': 'In ref. [CITATION], the Kn numbers for colloids were [MATH] and [MATH], depending on their colloid-solvent coupling.', 'cond-mat-0603391-3-94-1': 'Such large Kn numbers should have a significant effect on particle friction coefficients, and this may explain why these authors find that volume fraction [MATH] has less of an effect on the sedimentation velocity than we do [CITATION].', 'cond-mat-0603391-3-95-0': '## Peclet number', 'cond-mat-0603391-3-96-0': 'The Peclet number Pe measures the relative strength of convective transport to diffusive transport.', 'cond-mat-0603391-3-96-1': 'For example, for a colloid of radius [MATH], travelling at an average velocity [MATH], the Pe number is defined as: [EQUATION] where [MATH] is the colloid diffusion coefficient.', 'cond-mat-0603391-3-96-2': 'Just as for the Re number, the Pe number can be interpreted as a ratio of a diffusive to a convective time-scale, but now the former time-scale is not for the diffusion of momentum but rather it is given by the colloid diffusion time [EQUATION] which measures how long it takes for a colloid to diffuse over a distance [MATH].', 'cond-mat-0603391-3-96-3': 'We again take [MATH] so that, using Eq. ([REF]), the Pe number can be written as: [EQUATION]', 'cond-mat-0603391-3-96-4': 'If Pe [MATH] then the colloid moves convectively over a distance much larger than its radius [MATH] in the time [MATH] that it diffuses over that same distance.', 'cond-mat-0603391-3-96-5': 'Brownian fluctuations are expected to be less important in this regime.', 'cond-mat-0603391-3-96-6': 'For Pe [MATH], on the other hand, the opposite is the case, and the main transport mechanism is diffusive (note that on long enough time-scales ([MATH]) convection will always eventually "outrun" diffusion [CITATION]).', 'cond-mat-0603391-3-96-7': 'It is sometimes thought that for low Pe numbers hydrodynamic effects can be safely ignored, but this is not always true.', 'cond-mat-0603391-3-96-8': 'For example, we found that the reduction of average sedimentation velocity with particle volume fraction, famously first explained by Batchelor [CITATION], is independent of Pe number down to Pe [MATH] at least [CITATION].', 'cond-mat-0603391-3-97-0': '### Dependence of Peclet number on simulation parameters', 'cond-mat-0603391-3-98-0': 'The highest Pe number achievable in simulation is limited by the constraints on the Ma and Re numbers.', 'cond-mat-0603391-3-98-1': 'For example the Ma number sets an upper limit on the maximum Pe number by limiting [MATH].', 'cond-mat-0603391-3-98-2': 'From Eqs. ([REF]) and ([REF]), it follows that the Peclet number can be re-written in terms of the Reynolds number as [EQUATION] where we have approximated [MATH].', 'cond-mat-0603391-3-98-3': 'This shows that for a given constraint on the Re number, increasing [MATH] or [MATH] increases the range of accessible Pe numbers.', 'cond-mat-0603391-3-98-4': 'Similarly, when the kinematic viscosity is dominated by the collisional contribution, decreasing the dimensionless mean-free path [MATH] will also increase the maximum Pe number allowed since [MATH].', 'cond-mat-0603391-3-98-5': 'However, these changes increase the computational cost of the simulation.', 'cond-mat-0603391-3-99-0': '# finite-size, discretization, and inertial effects', 'cond-mat-0603391-3-100-0': 'The cost of an SRD simulation scales almost linearly with the number of fluid particles [MATH] in the system, and this contribution is usually much larger than the cost of including the colloidal degrees of freedom.', 'cond-mat-0603391-3-100-1': 'To optimize the efficiency of a simulation, one would therefore like to keep [MATH] as small as possible.', 'cond-mat-0603391-3-100-2': 'This objective can be achieved by keeping [MATH] and the box-size [MATH] small.', 'cond-mat-0603391-3-100-3': 'Unfortunately both these choices are constrained by the errors they introduce.', 'cond-mat-0603391-3-100-4': 'Reducing [MATH] means that short-ranged hydrodynamic fields become less accurately resolved due to Kn number and discretization effects.', 'cond-mat-0603391-3-100-5': 'Decreasing the box-size [MATH] falls foul of the long-ranged nature of the HI, and therefore, just as for Coulomb interactions [CITATION], finite size effects such as those induced by periodic images must be treated with special care.', 'cond-mat-0603391-3-101-0': 'Increasing the flow velocity may also be desirable since more Stokes times [MATH] can be achieved for the same number of SRD collision steps, thus increasing computational efficiency.', 'cond-mat-0603391-3-101-1': 'The Ma number gives one constraint on [MATH], but usually the more stringent constraint comes from keeping the Re number low to prevent unwanted inertial effects.', 'cond-mat-0603391-3-102-0': '## Finite-size effects', 'cond-mat-0603391-3-103-0': '### Finite-size correction to the friction', 'cond-mat-0603391-3-104-0': 'The friction coefficient [MATH] can be extracted from the Stokes drag [MATH] on a fixed colloid in fluid flow: [EQUATION] where [MATH] is the flow field at large distances.', 'cond-mat-0603391-3-104-1': 'The pre-factor 4 comes from using slip boundary conditions; it would be 6 for stick-boundary conditions, as verified in [CITATION].', 'cond-mat-0603391-3-104-2': 'In principle, since [MATH] is accurately known from theory for SRD, this expression can be used to extract the hydrodynamic radius [MATH], which is not necessarily the same as [MATH], from a simulation, as we did in [CITATION].', 'cond-mat-0603391-3-105-0': 'The hydrodynamic radius [MATH] can also be directly calculated from theory.', 'cond-mat-0603391-3-105-1': 'To derive this, it is important to recognize that there are two sources of friction [CITATION].', 'cond-mat-0603391-3-105-2': 'The first comes from the local Brownian collisions with the small particles, and can be calculated by a simplified Enskog/Boltzmann type kinetic theory [CITATION]: [EQUATION] which is here adapted for slip boundary conditions.', 'cond-mat-0603391-3-105-3': 'A related expression for stick boundary conditions, including that for rotational frictions, is described in [CITATION], where it was shown that the short-time exponential decays of the linear and angular velocity auto-correlation functions are quantitatively described by Enskog theory.', 'cond-mat-0603391-3-106-0': 'The second contribution to the friction, [MATH], comes from integrating the Stokes solution to the hydrodynamic field over the surface of the particle, defined here as [MATH].', 'cond-mat-0603391-3-106-1': 'These two contributions to the friction should be added in parallel to obtain the total friction [CITATION] (see also Appendix B): [EQUATION]', 'cond-mat-0603391-3-106-2': 'In contrast to the Enskog friction, which is local, we expect substantial box-size effects on the Stokes friction [MATH] since it depends on long-ranged hydrodynamic effects.', 'cond-mat-0603391-3-106-3': 'These can be expressed in terms of a correction factor [MATH] that should go to 1 for very large systems : [EQUATION]', 'cond-mat-0603391-3-106-4': 'To measure this correction factor we plot, in Fig. [REF], the form [MATH] for various system box sizes for which we have measured [MATH] from the simulation, and estimated [MATH] from Eq. ([REF]).', 'cond-mat-0603391-3-106-5': 'As expected, the correction factor tends to 1 for smaller [MATH].', 'cond-mat-0603391-3-106-6': 'More detailed calculations [CITATION], taking into account the effect of periodic boundaries, suggests that to lowest order in [MATH] the correction factor should scale as [EQUATION]', 'cond-mat-0603391-3-106-7': 'Indeed, a least squares fit of the data to this form gives a slope of [MATH], close to the theoretical value and in agreement with similar Lattice Boltzmann simulations of a single colloidal sphere [CITATION].', 'cond-mat-0603391-3-107-0': 'With these ingredients in hand, we can calculate the theoretical expected friction from Eqns. ([REF]) - ([REF]).', 'cond-mat-0603391-3-107-1': 'We know from previous work that the Enskog contribution at short times and the hydrodynamic contribution at long times quantitatively reproduce the rotational and translational velocity auto-correlation functions (VACF) for [MATH] [CITATION], and so we expect that the friction coefficients should also be accurately described by these theories.', 'cond-mat-0603391-3-107-2': 'We test this further in Fig. [REF] for a number of different values of [MATH], and find excellent agreement with theory for [MATH] and [MATH].', 'cond-mat-0603391-3-107-3': 'For [MATH] and below, on the other hand, we find deviations from the theory.', 'cond-mat-0603391-3-107-4': 'These are most likely due to Kn number and discretization effects, to be discussed in the next sub-section.', 'cond-mat-0603391-3-108-0': 'For the smallest spheres the mass ratio [MATH] may also change the measured friction if instead of fixing the sphere we were to let it move freely.', 'cond-mat-0603391-3-108-1': 'For [MATH], [MATH], which is small enough to have a significant effect [CITATION].', 'cond-mat-0603391-3-108-2': 'For larger spheres, this is not expected to be a problem.', 'cond-mat-0603391-3-108-3': 'For example [MATH] for [MATH] and [MATH] for [MATH].', 'cond-mat-0603391-3-109-0': '### Effective hydrodynamic radius', 'cond-mat-0603391-3-110-0': 'From the calculated or measured friction we can also obtain the effective hydrodynamic radius [MATH], which is continuum concept, from a comparison of the microscopic frictions from Eq. ([REF]) with Eq. ([REF]): [EQUATION] (for stick-boundaries the [MATH] should be replaced by [MATH]).', 'cond-mat-0603391-3-110-1': 'We find [MATH], [MATH], and [MATH].', 'cond-mat-0603391-3-110-2': 'The effective hydrodynamic radius is increased by the finite-size effects, but lowered by the Enskog contribution which is added in parallel.', 'cond-mat-0603391-3-110-3': 'Because this latter contribution is relatively more important for small colloids, [MATH] for smaller [MATH].', 'cond-mat-0603391-3-110-4': 'For [MATH] the Enskog contribution is smaller than the finite-size effects, and so the effective hydrodynamic radius is larger than [MATH].', 'cond-mat-0603391-3-110-5': 'Obviously this latter effect depends on the box-size.', 'cond-mat-0603391-3-110-6': 'In an infinite box [MATH] for all values of [MATH], due to the Enskog contribution.', 'cond-mat-0603391-3-110-7': 'Note that it is the effective hydrodynamic radius [MATH] which sets the long-ranged hydrodynamic fields, as we will see below.', 'cond-mat-0603391-3-111-0': '### Turning off long-ranged hydrodynamics', 'cond-mat-0603391-3-112-0': 'Hydrodynamic forces can be turned off in SRD by regularly randomizing the absolute fluid particle velocities (with for example a naive Langevin thermostat).', 'cond-mat-0603391-3-112-1': 'For particles embedded in an SRD solvent through participation in the collision step, this trick can be used to compare the effects of hydrodynamics to a purely Brownian simulation[CITATION].', 'cond-mat-0603391-3-112-2': 'The Yeomans group has successfully applied this idea in a study of polymer collapse and protein folding [CITATION].', 'cond-mat-0603391-3-113-0': 'However for the case of the colloids embedded through direct solvent collisions, turning off the hydrodynamic forces by randomizing the velocities greatly enhances the friction because, as is clear from Eqns. ([REF]) - ([REF]), the two contributions add in parallel.', 'cond-mat-0603391-3-113-1': 'Without long-ranged hydrodynamics, the friction would be entirely dominated by the Enskog contribution ([REF]) which scales with [MATH], and can be much larger than the hydrodynamic contribution which scales as [MATH].', 'cond-mat-0603391-3-113-2': 'Another way of stating this would be: By locally conserving momentum, SRD allows the development of long-ranged hydrodynamic fluid velocity correlations that greatly reduce the friction felt by a larger colloidal particle compared to the friction it would feel from a purely random Brownian heat-bath at the same temperature and number density [MATH].', 'cond-mat-0603391-3-114-0': '## Discretization effects on flow-field and friction', 'cond-mat-0603391-3-115-0': 'For pure Stokes flow (Re = 0), the velocity around a fixed slip-boundary sphere of (effective) hydrodynamic radius [MATH] can be exactly calculated: [EQUATION] where [MATH] is the velocity field far away from the sphere, and [MATH] the vector pointing from the center of the sphere to a position inside the fluid, with corresponding unit vector [MATH].', 'cond-mat-0603391-3-116-0': 'In Fig. [REF] we plot the difference between the measured field and the theoretical expected field ([REF]) for four different colloid radius to cell size ratios [MATH], using the values of the effective hydrodynamic radius [MATH] calculated from combining Eq. ([REF]) and Eq. [REF].', 'cond-mat-0603391-3-116-1': 'As expected, the field is more accurately reproduced as the colloid radius becomes larger with respect to the cell size [MATH].', 'cond-mat-0603391-3-116-2': 'This improvement arises because SRD discretization and hydrodynamic Knudsen number effects become smaller.', 'cond-mat-0603391-3-116-3': 'In Fig. [REF] we observe quite large deviations in the hydrodynamic field for [MATH].', 'cond-mat-0603391-3-116-4': 'These discretization effects may explain why the measured frictions in Fig. [REF] do not agree with theory for these smallest sphere sizes.', 'cond-mat-0603391-3-116-5': 'We also note that using [MATH] instead of the more accurate value of the effective hydrodynamic radius [MATH] calculated from theory results in significantly larger deviations between measured and theoretical flow-fields.', 'cond-mat-0603391-3-116-6': 'This independently confirms the values of the effective hydrodynamic radius.', 'cond-mat-0603391-3-117-0': 'The increased accuracy from using larger [MATH] comes at a sharp increase in computational cost.', 'cond-mat-0603391-3-117-1': 'To make sure that the finite size effects in each simulation of Fig. [REF] are approximately the same, the box-size was scaled as [MATH].', 'cond-mat-0603391-3-117-2': 'This means that doubling the colloid size leads to an eightfold increase in the number of fluid particles.', 'cond-mat-0603391-3-117-3': 'Moreover, the maximum velocity of the fluid must go down linearly in colloid size to keep the Re number, defined in Eq. ([REF]), constant.', 'cond-mat-0603391-3-117-4': 'If in addition we keep the number of Stokes times fixed, meaning that the fluid flows a certain multiple of the colloid radius or the box size, then a larger particle also means the fluid needs to flow over a proportionally longer total distance.', 'cond-mat-0603391-3-117-5': 'The overall computational costs for this calculation then scales at least as [MATH], which is quite steep.', 'cond-mat-0603391-3-117-6': 'For that reason, we advocate using smaller colloids wherever possible.', 'cond-mat-0603391-3-118-0': 'In most of our simulations we choose [MATH], which leads to a small relative error in the full velocity field and for which we can fully explain the observed friction (as seen in the previous subsection).', 'cond-mat-0603391-3-118-1': 'This size is similar to what is commonly used in LB [CITATION].', 'cond-mat-0603391-3-119-0': '## Inertial effects on flow-field and friction', 'cond-mat-0603391-3-120-0': 'One of the challenges in SRD is to keep the Re number down.', 'cond-mat-0603391-3-120-1': 'It is virtually impossible to reach the extremely low Re number (Stokes) regime of realistic colloids, but that is not necessary either.', 'cond-mat-0603391-3-120-2': 'In Fig. [REF] we see that the friction only begins to noticeably vary from the Stokes limit at Re [MATH] (at least on a logarithmic scale).', 'cond-mat-0603391-3-120-3': 'We expect the flow-field itself to be more sensitive to finite Re number effects.', 'cond-mat-0603391-3-120-4': 'To study this directly, we examine, in Fig. [REF], the flow-field around a fixed colloid of size [MATH] in a box of [MATH] for for different values of Re.', 'cond-mat-0603391-3-120-5': 'The differences with the Stokes flow field of Eq. ([REF]) are shown in Fig. [REF](b)-(d) for Re [MATH], [MATH], and [MATH] respectively.', 'cond-mat-0603391-3-120-6': 'In all cases we used a hydrodynamic radius of [MATH] for the theoretical comparison, as explained in the previous sub-section.', 'cond-mat-0603391-3-120-7': 'The lengths of the vectors in Figs. (b)-(d) are multiplied by 10 for clarity.', 'cond-mat-0603391-3-120-8': 'We observe that the relative errors increase with Re.', 'cond-mat-0603391-3-120-9': 'They are on the order of [MATH] for Re=0.08, and increase to something on the order of [MATH] for Re=2.', 'cond-mat-0603391-3-120-10': 'This is exactly what is expected, of course, as we are moving away from the Stokes regime with which these flow lines are being compared.', 'cond-mat-0603391-3-120-11': 'Since we also expect effects on the order of a few % from Kn number, Ma number and various finite size effects, we argue that keeping Re[MATH] should be good enough for most of the applications we have in mind.', 'cond-mat-0603391-3-121-0': '# The hierarchy of time scales', 'cond-mat-0603391-3-122-0': '## Colloidal time-scales', 'cond-mat-0603391-3-123-0': 'Many different time-scales govern the physics of a colloid of mass [MATH] embedded in a solvent [CITATION].', 'cond-mat-0603391-3-123-1': 'The most important ones are summarized in table [REF], and discussed in more detail below.', 'cond-mat-0603391-3-124-0': '### Fluid time-scales', 'cond-mat-0603391-3-125-0': 'The shortest of these is the solvent collision time [MATH] over which fluid molecules interact with each other.', 'cond-mat-0603391-3-125-1': 'For a typical molecular fluid, [MATH] is on the order of a few tens of fs, and any MD scheme for a molecular fluid must use a discretization time-step [MATH] to properly integrate the equations of motion.', 'cond-mat-0603391-3-126-0': 'The next time-scale up is the solvent relaxation time [MATH], which measures how fast the solvent VACF decays.', 'cond-mat-0603391-3-126-1': 'For a typical molecular liquid, [MATH] s. (For water, at room temperature, for example, it is on the order of 50 femtoseconds)', 'cond-mat-0603391-3-127-0': '### Hydrodynamic time-scales for colloids', 'cond-mat-0603391-3-128-0': 'Hydrodynamic interactions propagate by momentum (vorticity) diffusion, and also by sound.', 'cond-mat-0603391-3-128-1': 'The sonic time [EQUATION] it takes a sound wave to travel the radius of a colloid is typically very small, on the order of 1 ns for a colloid of radius [MATH]m. For that reason, sound effects are often ignored for colloidal suspensions under the assumption that they will have dissipated so quickly that they have no noticeably influence on the dynamics.', 'cond-mat-0603391-3-128-2': 'However, some experiments [CITATION] and theory [CITATION] do find effects from sound waves on colloidal hydrodynamics, so that this issue is not completely settled yet.', 'cond-mat-0603391-3-129-0': 'The kinematic time, [MATH], defined in Eq. ([REF]) (and Table [REF]) as the time for momentum (vorticity) to diffuse over the radius of a colloid, is particularly important for hydrodynamics.', 'cond-mat-0603391-3-129-1': 'It sets the time-scale over which hydrodynamic interactions develop.', 'cond-mat-0603391-3-129-2': 'For a colloid of radius [MATH]m, [MATH] s, which is much faster the colloidal diffusion time [MATH].', 'cond-mat-0603391-3-130-0': 'When studying problems with a finite flow velocity, another hydrodynamic time-scale emerges.', 'cond-mat-0603391-3-130-1': 'The Stokes time [MATH], defined in Eq. ([REF]) (and Table [REF]) as the time for a colloid to advect over its own radius, can be related to the kinematic time by the relation [MATH].', 'cond-mat-0603391-3-130-2': 'Because colloidal particles are in the Stokes regime where Re [MATH], we find [MATH].', 'cond-mat-0603391-3-131-0': 'Simulation and analytical methods based on the Oseen tensor and its generalizations, e.g. [CITATION], implicitly assume that the hydrodynamic interactions develop instantaneously, i.e. that [MATH].', 'cond-mat-0603391-3-131-1': 'For the low Re number regime of colloidal dispersions this is indeed a good approximation.', 'cond-mat-0603391-3-131-2': 'Of course it must be kept in mind that [MATH] is a diffusive time-scale, so that the distance over which it propagates grows as [MATH].', 'cond-mat-0603391-3-131-3': 'Thus the approximation of instantaneous hydrodynamics must be interpreted with some care for effects on larger length-scales.', 'cond-mat-0603391-3-132-0': '### Brownian time-scales for colloids', 'cond-mat-0603391-3-133-0': 'The fastest time-scale relevant to the colloidal Brownian motion is the Fokker-Planck time [MATH] defined in [CITATION] as the time over which the force-force correlation function decays.', 'cond-mat-0603391-3-133-1': 'It is related to [MATH], the time over which the fluid looses memory of its velocity because the forces on the colloid are caused by collisions with the fluid particles, velocity differences de-correlate on a time scale of order [MATH].', 'cond-mat-0603391-3-134-0': 'The next time-scale in this series is the Brownian time: [EQUATION] where [MATH] is the Stokes friction for stick boundary conditions.', 'cond-mat-0603391-3-134-1': 'It is often claimed that this measures the time for a colloid to lose memory of its velocity.', 'cond-mat-0603391-3-134-2': 'However, as discussed in Appendix [REF], this picture, based on the Langevin equation, is in fact incorrect for colloids.', 'cond-mat-0603391-3-134-3': 'Nevertheless, [MATH] has the advantage that it can easily be calculated and so we will use it as a crude upper bound on the colloid velocity de-correlation time.', 'cond-mat-0603391-3-135-0': 'Perhaps the most important time-scale relevant for Brownian motion is the diffusion time [MATH], described in Eq. ([REF]) as the time for a colloidal particle to diffuse over its radius.', 'cond-mat-0603391-3-135-1': 'For a colloid of radius [MATH]m, [MATH] s, and even though [MATH], it remains much larger than most microscopic time-scales in the mesoscopic colloidal regime.', 'cond-mat-0603391-3-136-0': '## Time-scales for coarse-grained simulation', 'cond-mat-0603391-3-137-0': 'In the previous subsection we saw that the relevant time-scales for a single colloid in a solvent can span as many as 15 orders of magnitude.', 'cond-mat-0603391-3-137-1': 'Clearly it would be impossible to bridge all the time-scales of a physical colloidal system - from the molecular [MATH] to the mesoscopic [MATH] - in a single simulation.', 'cond-mat-0603391-3-137-2': 'Thankfully, it is not necessary to exactly reproduce each of the different time-scales in order to achieve a correct coarse-graining of colloidal dynamics.', 'cond-mat-0603391-3-137-3': 'As long as they are clearly separated, the correct physics should still emerge.', 'cond-mat-0603391-3-138-0': 'Since we take the view that the "solvent" particles are really a Navier Stokes solver with noise, what is needed to reproduce Brownian behavior is first of all that the colloid experiences random kicks from the solvent on a short enough time-scale.', 'cond-mat-0603391-3-138-1': 'By dramatically reducing the number of "molecules" in the solvent, the number of kicks per unit of time is similarly reduced.', 'cond-mat-0603391-3-138-2': 'Here we identify the Fokker-Planck time [MATH] as the time-scale on which the colloid experiences random Brownian motion.', 'cond-mat-0603391-3-138-3': "For Brownian motion it doesn't really matter how the kicks are produced, they could be completely uncorrelated, but since we also require that the solvent transports momentum, solvent particles must have some kind of correlation, which in our case is represented by the time [MATH].", 'cond-mat-0603391-3-138-4': 'Other colloid relaxation times should be much longer than this.', 'cond-mat-0603391-3-138-5': 'We therefore require first of all that [MATH] and [MATH].', 'cond-mat-0603391-3-139-0': "Secondly, the colloid's VACF should decay to zero well before it has diffused or convected over its own radius.", 'cond-mat-0603391-3-139-1': 'A proper separation of time-scales in a coarse-graining scheme then requires that [MATH] as well as [MATH] for systems where convection is important.', 'cond-mat-0603391-3-140-0': 'Finally, for the correct hydrodynamics to emerge we require that [MATH].', 'cond-mat-0603391-3-140-1': 'But this separation no longer needs to be over many orders of magnitude.', 'cond-mat-0603391-3-140-2': 'As argued in [CITATION], one order of magnitude separation between time-scales should be sufficient for most applications.', 'cond-mat-0603391-3-140-3': 'We illustrate this approach in Fig. [REF].', 'cond-mat-0603391-3-141-0': 'In the next few subsections we discuss in more quantitative detail how our coarse-graining strategy telescopes down the hierarchy of time-scales in order to maximize the efficiency of a simulation while still retaining key physical features.', 'cond-mat-0603391-3-142-0': '### SRD fluid time-scales', 'cond-mat-0603391-3-143-0': 'In SRD the physical time-scale [MATH] is coarse-grained out, and the effect of the collisions calculated in an average way every time-step [MATH].', 'cond-mat-0603391-3-143-1': 'The time-scale [MATH] on which the velocity correlations decay can be quite easily calculated from a random-collision approximation.', 'cond-mat-0603391-3-143-2': 'Following [CITATION]: [MATH].', 'cond-mat-0603391-3-143-3': 'For our parameters, [MATH], we find [MATH].', 'cond-mat-0603391-3-143-4': 'However, it should be kept in mind that for small [MATH] the exponential decay with [MATH] turns over to a slower algebraic decay at larger [MATH] [CITATION].', 'cond-mat-0603391-3-143-5': 'The amplitude of this "tail" is, however, quite small.', 'cond-mat-0603391-3-144-0': '### Hydrodynamic time-scales for simulation', 'cond-mat-0603391-3-145-0': 'For our choice of units [MATH], so that the sonic time of Eq. ([REF]) reduces to [EQUATION] which is independent of [MATH] or [MATH].', 'cond-mat-0603391-3-146-0': 'In the limit of small [MATH], the ratio of the kinematic time [MATH] to [MATH] can be simplified to the following form: [EQUATION] so that the condition [MATH] is very easy to fulfil.', 'cond-mat-0603391-3-146-1': 'Furthermore, under the same approximations, the ratio [MATH] so that for [MATH] too small the kinematic time becomes faster than the sonic time.', 'cond-mat-0603391-3-146-2': 'For the simulation parameters used in [CITATION] (and in Fig. [REF]), we find [MATH].', 'cond-mat-0603391-3-147-0': '### Brownian time-scales for simulation', 'cond-mat-0603391-3-148-0': 'We expect the Fokker Planck time [MATH] to scale as [MATH], since this is roughly equivalent to the time [MATH] over which the fluid velocities will have randomized.', 'cond-mat-0603391-3-148-1': 'In Fig. [REF] we plot the force-force correlation function [EQUATION]', 'cond-mat-0603391-3-148-2': 'Its short time behavior is dominated by the random forces, and the initial decay time gives a good estimate of [MATH].', 'cond-mat-0603391-3-148-3': 'As can be seen in the inset of Fig. [REF], [MATH], which is indeed on the order of [MATH] or [MATH].', 'cond-mat-0603391-3-149-0': 'If we make the reasonable approximation that [MATH], then the ratio of the Brownian time [MATH] to the kinematic time [MATH] simplifies to: [EQUATION] so that the ratio of [MATH] to [MATH] is the same as for a real physical system.', 'cond-mat-0603391-3-149-1': 'Since buoyancy requirements mean that normally [MATH], the time-scales are ordered as [MATH].', 'cond-mat-0603391-3-149-2': 'Moreover, since the ratio [MATH], independently of [MATH] or even [MATH] (as long as [MATH]), the condition that [MATH] is also not hard to fulfil in an SRD simulation.', 'cond-mat-0603391-3-150-0': 'These time-scales are illustrated in Fig. [REF] where we plot the normalized time correlation function: [EQUATION] with [MATH] the Cartesian velocity coordinate of a single colloid.', 'cond-mat-0603391-3-150-1': 'After the initial non-Markovian quadratic decay, the subsequent short-time exponential decay is not given by [MATH] but instead by the Enskog time [EQUATION] where [MATH] is the Enskog friction which can be calculated from kinetic theory [CITATION], see Eq. ([REF]), and generally [MATH].', 'cond-mat-0603391-3-150-2': 'The physical origins of this behavior are described in more detail in Appendix [REF].', 'cond-mat-0603391-3-151-0': 'The colloid diffusion coefficient is directly related to the friction by the Stokes-Einstein relation: [EQUATION]', 'cond-mat-0603391-3-151-1': 'If we assume that [MATH] and, for simplicity, that [MATH], i.e. we ignore the Enskog contribution, then the diffusion time scales as: [EQUATION] so that [MATH] is not hard to fulfil.', 'cond-mat-0603391-3-152-0': 'It is also instructive to examine the ratio of the diffusion time [MATH] to the kinematic time [MATH]: [EQUATION]', 'cond-mat-0603391-3-152-1': 'In general we advocate keeping [MATH] small to increase the Sc number [MATH], and since another obvious constraint is [MATH], there is not too much difficulty achieving the desired separation of time-scales [MATH].', 'cond-mat-0603391-3-153-0': 'As a concrete example of how these time-scales are separated in a simulation, consider the parameters used in Fig. [REF] for which we find [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0603391-3-153-1': 'But more generally, what the analysis of this section shows is that obtaining the correct hierarchy of time-scales: [EQUATION] is virtually guaranteed once the conditions on dimensionless numbers, detailed in Section [REF], are fulfilled.', 'cond-mat-0603391-3-154-0': '## Measurements of diffusion', 'cond-mat-0603391-3-155-0': 'To further test the hybrid MD-SRD coarse-graining scheme, we plot in Fig. [REF] the time-dependent diffusion coefficient [MATH], defined as: [EQUATION] for a number of different volume fractions [MATH].', 'cond-mat-0603391-3-155-1': 'Note that the convergence slows with increasing volume fraction [MATH].', 'cond-mat-0603391-3-155-2': 'The infinite time integral gives the diffusion coefficient, i.e. [MATH].', 'cond-mat-0603391-3-155-3': 'In the limit of low densities the diffusion coefficient has been predicted to take the form [MATH] with 1.1795 for slip boundary spheres [CITATION], and this provides a good fit at the lower volume fractions.', 'cond-mat-0603391-3-156-0': 'In Figure [REF] we plot the mean-square displacement of a Cartesian component of the colloidal particle position.', 'cond-mat-0603391-3-156-1': 'As highlighted in the inset, at short times there is an initial ballistic regime due to motion at an average mean square velocity [MATH], resulting in a mean-square displacement [MATH].', 'cond-mat-0603391-3-156-2': 'At times [MATH], the motion is clearly diffusive, as expected, with a linear dependence on [MATH] and a slope of [MATH], where [MATH] is given by the infinite time limit of Eq. ([REF]).', 'cond-mat-0603391-3-156-3': 'On the scale of the plot the asymptotic regime is reached well before the time [MATH] over which the particle has diffused, on average, over its radius.', 'cond-mat-0603391-3-156-4': 'In Appendix [REF] the behavior of [MATH] and the related mean-square displacement are discussed in more detail for time-scales [MATH].', 'cond-mat-0603391-3-157-0': '# Mapping between a physical and coarse-grained systems: No free lunch?', 'cond-mat-0603391-3-158-0': 'The ultimate goal of any coarse-graining procedure is to correctly describe physical phenomena in the natural world.', 'cond-mat-0603391-3-158-1': 'As we have argued in this paper, it is impossible to bridge, in a single simulation, all the time and length-scales relevant to a colloid suspended in a solvent.', 'cond-mat-0603391-3-158-2': 'Compromises must be made.', 'cond-mat-0603391-3-158-3': 'Nevertheless, a fruitful mapping from a coarse-grained simulation to a physical system is possible, and greatly facilitated by expressing the physical properties of interest in dimensionless terms.', 'cond-mat-0603391-3-158-4': 'The best way to illustrate this is with some examples.', 'cond-mat-0603391-3-159-0': '## Example 1: Mapping to diffusive and Stokes time-scales', 'cond-mat-0603391-3-160-0': 'For many dynamic phenomena in colloidal dispersions, the most important time-scale is the diffusion time [MATH].', 'cond-mat-0603391-3-160-1': 'To map a coarse-grained simulation onto a real physical system one could therefore equate the diffusion time and the colloid radius [MATH] of the simulation to that of the real physical system.', 'cond-mat-0603391-3-160-2': 'For example, take the simulation parameters from table [REF] in Appendix [REF] for [MATH], and compare these to the properties of colloids of radius [MATH]m and [MATH] nm from table [REF].', 'cond-mat-0603391-3-160-3': 'For the larger colloids, [MATH]s, and equating this to the simulation [MATH] means that [MATH]m and [MATH] s. Performing the same exercise for the smaller colloid, where [MATH] s, results in [MATH] nm and [MATH] s.', 'cond-mat-0603391-3-160-4': 'The first thing that becomes obvious from this exercise is that "physical" time is set here not by the coarse-grained simulation, but by the particular system that one is comparing to.', 'cond-mat-0603391-3-160-5': 'In other words, many different physical systems could be mapped to the same simulation.', 'cond-mat-0603391-3-161-0': 'If the system is also subjected to flow, then a second time-scale emerges, the Stokes time [MATH].', 'cond-mat-0603391-3-161-1': 'As long as the physical Pe number is achievable without compromising the simulation quality, then this time-scale can simultaneously be set equal to that of the physical system.', 'cond-mat-0603391-3-161-2': 'Behavior that depends primarily on these two time-scales should then be correctly rendered by the SRD hybrid MD scheme described in this paper.', 'cond-mat-0603391-3-162-0': '## Example 2: Mapping to kinematic time-scales', 'cond-mat-0603391-3-163-0': 'By fixing [MATH] or [MATH], as in example 1, other time-scales are not correctly reproduced.', 'cond-mat-0603391-3-163-1': 'For the large and small colloid systems described above, we would find [MATH] s and [MATH] s respectively, which contrasts with the physical values of [MATH] s and [MATH] s shown in Table [REF].', 'cond-mat-0603391-3-163-2': 'In other words, fixing [MATH] results in values of [MATH] that are too large by orders of magnitude.', 'cond-mat-0603391-3-163-3': 'But of course these much larger values of [MATH] are by design, because the SRD simulation is optimized by compacting the very large physical hierarchy of time-scales.', 'cond-mat-0603391-3-163-4': "If we are interested in processes dominated by [MATH], having different [MATH] doesn't matter, so long as [MATH].", 'cond-mat-0603391-3-164-0': 'On the other hand, if we want to describe processes that occur on much smaller time-scales, for example long-time tails in colloidal VACFs, then simulation times [MATH] could be mapped onto the physical [MATH] instead.', 'cond-mat-0603391-3-164-1': "For the same [MATH] simulation parameters used above, the [MATH]m system now maps onto [MATH]m, and [MATH] s with of course a strongly underestimated diffusion time: [MATH] s instead of [MATH] s. Similarly for the [MATH] nm system, the mapping is to [MATH] nm, [MATH] s and [MATH] s instead of [MATH] s. For many processes on time-scales [MATH], this underestimate of [MATH] doesn't really matter.", 'cond-mat-0603391-3-164-2': 'It is only when time-scales are mixed that extra care must be employed in interpreting the simulation results.', 'cond-mat-0603391-3-164-3': 'And, if the processes can be described in terms of different dimensionless times, then it should normally still be possible to disentangle the simulation results and correctly map onto a real physical system.', 'cond-mat-0603391-3-165-0': 'Another lesson from these examples of mapping to different time-scales comes from comparing the kinematic viscosity of a physical system, which say takes the value [MATH]m[MATH]/s for water, to its value in SRD simulations.', 'cond-mat-0603391-3-165-1': 'If for the [MATH] SRD system described above we map time onto [MATH], as in Example 1, then for the [MATH]m system [MATH]m[MATH]/s, while for the [MATH] nm system [MATH]m[MATH]/s.', 'cond-mat-0603391-3-165-2': 'As expected, both are much smaller than the true value in water.', 'cond-mat-0603391-3-165-3': 'If we instead map the times to [MATH], then of course the kinematic viscosity takes the same value as for the physical system (since we also set the length-scales equal to the physical length-scales).', 'cond-mat-0603391-3-165-4': 'This apparent ambiguity in defining a property like the kinematic viscosity is inherent in many coarse-graining schemes.', 'cond-mat-0603391-3-165-5': 'We take the view that the precise value of [MATH] is not important, the only thing that matters is that it is large enough to ensure a proper separation of times-scales and the correct regime of hydrodynamic numbers.', 'cond-mat-0603391-3-166-0': '## Example 3: Mapping mass and temperature', 'cond-mat-0603391-3-167-0': 'In the examples above, we have only discussed setting lengths and times.', 'cond-mat-0603391-3-167-1': 'This leaves either the mass [MATH] or temperature (thermal energy) [MATH] still to be set.', 'cond-mat-0603391-3-167-2': 'Because for many dynamic properties the absolute mass is effectively an irrelevant variable (although relative masses may not be) there is some freedom to choose.', 'cond-mat-0603391-3-167-3': 'One possibility would be to set [MATH], the mass of the colloid, equal to the physical mass.', 'cond-mat-0603391-3-167-4': 'For an [MATH]m neutrally buoyant colloid this sets [MATH] g, (equal to about [MATH] water molecules) while for an [MATH] nm colloid we find [MATH] g (equal to about [MATH] water molecules).', 'cond-mat-0603391-3-167-5': 'By construction this procedure produces the correct physical [MATH].', 'cond-mat-0603391-3-167-6': 'If we fix time with [MATH], this will therefore also generate the correct shear viscosity [MATH].', 'cond-mat-0603391-3-167-7': 'If instead we fix [MATH], then [MATH] will be much smaller than the physical value.', 'cond-mat-0603391-3-167-8': 'For either choice of time-scales, other properties, such as the fluid self-diffusion constant or the number density, will have different values from the underlying physical system.', 'cond-mat-0603391-3-168-0': 'Setting the mass in this way means that the unit of thermal energy [MATH] would also be very large, leading to the appearance of very low temperatures.', 'cond-mat-0603391-3-168-1': 'However, because temperature only determines behavior through the way it scales potentials, it is quite easy to simply re-normalize the effective energy scales and obtain the same behavior as for the physical system.', 'cond-mat-0603391-3-168-2': 'Alternatively one could set the temperature equal to that of a physical system, but that would mean that the masses would again differ significantly from the real physical system.', 'cond-mat-0603391-3-168-3': 'At any rate, for the same simulation (say of sedimentation at a given value of Pe) the values of mass or temperature depend on the physical system one compares to.', 'cond-mat-0603391-3-168-4': 'This apparent ambiguity (or flexibility) simply reflects the fact that the dominant effects of temperature and mass come into play as relative ratios and not as absolute values.', 'cond-mat-0603391-3-169-0': '## Example 4: Mapping to attractive potentials: problems with length-scales?', 'cond-mat-0603391-3-170-0': 'So far we have only treated one length-scale in the problem, the radius [MATH].', 'cond-mat-0603391-3-170-1': 'Implicitly we have therefore assumed that the colloid-colloid interaction does not have another intrinsic length-scale of its own.', 'cond-mat-0603391-3-170-2': 'For hard-sphere colloids this is strictly true, and for steep-repulsions such as the WCA form of Eq. ([REF]) it is also a fairly good assumption that the exact choice of the exponent [MATH] in this equation does not significantly affect the dynamical properties [CITATION].', 'cond-mat-0603391-3-170-3': 'Similarly, there is some freedom to set the repulsive fluid-colloid potential of Eq. ([REF]) in such a way as to optimize the simulation parameters, in particular [MATH].', 'cond-mat-0603391-3-170-4': 'The physical liquid-colloid interaction would obviously have a much more complex form, but SRD coarse-grains out such details.', 'cond-mat-0603391-3-170-5': 'A similar argument can be made for colloid and fluid interactions with hard walls, needed, for example, to study problems with confinement, for which, inter alia, SRD is particularly well suited.', 'cond-mat-0603391-3-171-0': 'Things become more complicated for colloids with an explicit attractive interaction, such as DLVO or a depletion potential [CITATION].', 'cond-mat-0603391-3-171-1': 'These potentials introduce a new length-scale, the range of the attraction.', 'cond-mat-0603391-3-171-2': 'The ratio of this length to the hard-core diameter helps determine the equilibrium properties of the fluid [CITATION].', 'cond-mat-0603391-3-171-3': 'In a physical system this ratio may be quite small.', 'cond-mat-0603391-3-171-4': 'Keeping the ratio the same in the simulations then leads to potentials that are very steep on the scale of [MATH].', 'cond-mat-0603391-3-171-5': 'The MD time-step [MATH] may need to be very small in order to properly integrate the MD equations of motion.', 'cond-mat-0603391-3-171-6': 'Such a small [MATH] can make a simulation very inefficient, as was found in [CITATION] who followed this strategy and were forced to use [MATH].', 'cond-mat-0603391-3-171-7': 'However, in that case the DLVO potential dominates the behaviour, so that SRD was only included to roughly resolve the long range hydrodynamics[CITATION].', 'cond-mat-0603391-3-171-8': 'In general, we advocate adapting the potential to maximize simulation efficiency, while preserving key physical properties such as the topology of the phase diagram.', 'cond-mat-0603391-3-171-9': 'For example, the attractive energy scale can be set by the dimensionless reduced second virial coefficient, which gives an excellent approximation for how far one is from the liquid-liquid critical point [CITATION].', 'cond-mat-0603391-3-171-10': 'When the attractive interaction range is less than about [MATH] of the colloid hard-core diameter, the colloidal "liquid" phase becomes metastable with respect to the fluid-solid phase-transition.', 'cond-mat-0603391-3-171-11': 'At low colloid packing fraction [MATH] this leads to so-called "energetic fluid" behavior [CITATION].', 'cond-mat-0603391-3-171-12': 'To simulate a colloidal suspension in this "energetic fluid" regime, having a range of less than [MATH] of [MATH] should suffice.', 'cond-mat-0603391-3-171-13': 'In this way adding attractions does not have to make the simulation much less efficient.', 'cond-mat-0603391-3-171-14': 'Moreover the effects of these constraints, placed by the colloid-colloid interaction on [MATH], are to some degree mitigated by the fact that it is usually the colloid-fluid interaction which sets this time-scale.', 'cond-mat-0603391-3-172-0': 'There are still a few final subtleties to mention.', 'cond-mat-0603391-3-172-1': 'First of all, using small [MATH] in the simulations may greatly increase efficiency, but it also leads to a relatively larger contribution of the Enskog friction to the total friction in Eq. ([REF]).', 'cond-mat-0603391-3-172-2': 'For physical colloids in a molecular solvent the exact magnitude of the Enskog friction is still an open question, but undoubtedly for larger colloids it is almost negligible.', 'cond-mat-0603391-3-172-3': 'Some of the Enskog effects can simply be absorbed into an effective hydrodynamic radius [MATH], but others must be interpreted with some care.', 'cond-mat-0603391-3-173-0': 'Another regime where we might expect to see deviations beyond a simple re-scaling of time-scales would be at very short times.', 'cond-mat-0603391-3-173-1': 'For example, when colloids form a crystal, their oscillations can be de-composed into lattice phonons.', 'cond-mat-0603391-3-173-2': 'All but the longest wave-length longitudinal oscillations are overdamped due to the viscous drag and backflow effects of the solvent [CITATION].', 'cond-mat-0603391-3-173-3': 'In SRD, however, some of the very short wavelength oscillations might be preserved due to the fact that the motion on those time-scales is still ballistic.', 'cond-mat-0603391-3-173-4': 'These may have an impact on the interpretation of SRD simulations of microrheology.', 'cond-mat-0603391-3-174-0': 'In summary, these examples demonstrate that as long as the coarse-grained simulation produces the correct hydrodynamic and Brownian behavior then there is considerable freedom in assigning real physical values to the parameters in the simulation.', 'cond-mat-0603391-3-174-1': 'Attractive interactions may introduce a new length-scale, but a judicious choice of a model potential should still reproduce the correct physical behavior.', 'cond-mat-0603391-3-174-2': 'The same simulation may therefore be mapped onto multiple different physical systems.', 'cond-mat-0603391-3-174-3': 'Conversely, for the same physical system, different choices can be made for the time-scales that are mapped to physical values, but by design, not all time-scales can be simultaneously resolved.', 'cond-mat-0603391-3-175-0': '# Conclusion', 'cond-mat-0603391-3-176-0': 'Correctly rendering the principal Brownian and hydrodynamic properties of colloids in solution is a challenging task for computer simulation.', 'cond-mat-0603391-3-176-1': 'In this article we have explored how treating the solvent with SRD, which coarse-grains the collisions between solvent particles over both time and space, leads to an efficient solution of the thermo-hydrodynamic equations of the solvent, in the external field provided by the colloids.', 'cond-mat-0603391-3-176-2': 'Although it is impossible to simulate the entire range of time-scales encountered in real colloidal suspensions, which can span as much as 15 orders of magnitude, we argue that this is also not necessary.', 'cond-mat-0603391-3-176-3': 'Instead, by recognizing firstly that hydrodynamic effects are governed by a set of dimensionless numbers, and secondly that the full hierarchy of time-scales can be telescoped down to a much more manageable range while still being properly physically separated, we demonstrate that the key Brownian and hydrodynamic properties of a colloidal suspension can indeed be captured by our SRD based coarse-graining scheme.', 'cond-mat-0603391-3-177-0': 'In particular, to simulate in the colloidal regime, the modeler should ensure that the dimensionless hydrodynamic numbers such as the Mach, Reynolds and Knudsen numbers are small enough ([MATH]), and the Schmidt number is large enough ([MATH]).', 'cond-mat-0603391-3-177-1': 'While these numbers are constrained by approximate limits, the Peclet number, which measures the relative strength of the convective transport to the diffusive transport of the colloids, can be either larger or smaller than 1, depending on physical properties such as the colloidal size and the strength of the external field that one wants to study.', 'cond-mat-0603391-3-177-2': 'It turns out that if the hydrodynamic numbers above are chosen correctly, it is usually not so difficult to also satisfy the proper separation of the time-scales.', 'cond-mat-0603391-3-178-0': 'We explored how to reach the desired regime of hydrodynamic numbers and time-scales by tuning the simulation parameters.', 'cond-mat-0603391-3-178-1': 'The two most important parameters are the dimensionless mean-free path [MATH] which measures what fraction of a cell size [MATH] an SRD particle travels between collisions performed at every time-step [MATH], and the ratio of the colloid radius [MATH] to the SRD cell size [MATH].', 'cond-mat-0603391-3-178-2': 'The general picture that emerges is that the collision time [MATH] must be chosen so that [MATH] is small since this helps keep the Schmidt number high, and both the Knudsen and Reynolds numbers low.', 'cond-mat-0603391-3-178-3': 'Of course for computational efficiency, the collision time should not be too small either.', 'cond-mat-0603391-3-178-4': 'Similarly although a larger colloid radius means that the hydrodynamic fields are more accurately rendered, this should be tempered by the fact that the simulation efficiency drops off rapidly with increasing colloid radius.', 'cond-mat-0603391-3-178-5': 'We find that choosing [MATH], which may seem small, already leads to accurate hydrodynamic properties.', 'cond-mat-0603391-3-179-0': 'A number of subtleties occur at the fluid-colloid interface, which may induce spurious depletion interactions between the colloids, but these can be avoided by careful choice of potential parameters for slip boundary conditions.', 'cond-mat-0603391-3-179-1': 'Stick boundary conditions can also be implemented, and we advocate using stochastic bounce-back rules to achieve this coarse-graining over the fluid-colloid interactions.', 'cond-mat-0603391-3-180-0': 'The simplicity of the SRD solvent facilitates the calculation of the the short-time behavior of the VACF from kinetic theory.', 'cond-mat-0603391-3-180-1': 'We argue that for times shorter than the sonic time [MATH], where hydrodynamic collective modes will not yet have fully developed, the decay of the VACF is typically much more rapid than the prediction of the simple Langevin equation.', 'cond-mat-0603391-3-180-2': 'The ensuing Enskog contribution to the total friction should be added in parallel to the microscopic hydrodynamic friction, and from the sum an analytic expression for the effective hydrodynamic radius [MATH] can be derived which is consistent with independent measurements of the long-ranged hydrodynamic fields.', 'cond-mat-0603391-3-181-0': 'Although we have shown how to correctly render the principal Brownian and hydrodynamic behavior of colloids in solution there is, as always, no such thing as a free lunch.', 'cond-mat-0603391-3-181-1': 'The price to be paid is an inevitable consequence of compressing together the hierarchy of time and length-scales: not all the simulation parameters can be simultaneously mapped onto a physical system of interest.', 'cond-mat-0603391-3-181-2': 'However the cost of the "lunch" can be haggled down by recognizing that only a few physical parameters are usually important.', 'cond-mat-0603391-3-181-3': 'For example, one could map the simulation onto real physical time, say through [MATH] and [MATH], or alternatively through [MATH]; there is some freedom (or ambiguity) in the choice of which time scale to use.', 'cond-mat-0603391-3-181-4': 'The correspondence with physical reality becomes more complex when different physical time-scales mix, but in practice they can often be disentangled when both the coarse-grained simulation and the physical system are expressed in terms of dimensionless numbers and ratios.', 'cond-mat-0603391-3-181-5': 'In other words, there is no substitute for careful physical insight which is, as always, priceless.', 'cond-mat-0603391-3-182-0': 'A number of lessons can be drawn from this exercise that may be relevant for other coarse-graining schemes.', 'cond-mat-0603391-3-182-1': 'SRD is closely related to Lattice-Boltzmann and to the Lowe-Anderson thermostat, and many of the conclusions above should transfer in an obvious way.', 'cond-mat-0603391-3-182-2': 'For dissipative particle dynamics, we argue that the physical interpretation is facilitated by recognizing that, just as for SRD, the DPD particles should not be viewed as "clumps of fluid" but rather as a convenient computational tool to solve the underlying thermo-hydrodynamic equations.', 'cond-mat-0603391-3-182-3': 'All these methods must correctly resolve the time-scale hierarchy, and satisfy the relevant hydrodynamic numbers, in order to reproduce the right underlying physics.', 'cond-mat-0603391-3-183-0': 'Our measurements of the VACF and the discussion in Appendix B show explicitly that Brownian Dynamics simulations, which are based on the simple Langevin equation, do not correctly capture either the long or the short-time decay of colloidal VACFs.', 'cond-mat-0603391-3-184-0': 'A major advantage of SRD is the relative ease with which solutes and external boundary conditions (walls) are introduced.', 'cond-mat-0603391-3-184-1': 'Boundaries may be hard or soft, with either stick or slip conditions.', 'cond-mat-0603391-3-184-2': 'This method is therefore particularly promising for simulations in the fields of bio- and nanofluidics, where small meso particles flow in constrained geometries, possibly confined by soft fluctuating walls.', 'cond-mat-0603391-3-184-3': 'We are currently exploring these possibilities.', 'cond-mat-0603391-3-185-0': 'JTP thanks the EPSRC and IMPACT Faraday, and AAL thanks the Royal Society (London) for financial support.', 'cond-mat-0603391-3-185-1': 'We thank W. Briels, H. Lowen, J. Lopez Lopez, J. Sane, A. Mayhew Seers, I. Pagonabarraga, A. Wilber, and A. Wysocki for very helpful conversations, and W. den Otter for a careful reading of the manuscript.', 'cond-mat-0603391-3-186-0': '# Thermostating of SRD under flow', 'cond-mat-0603391-3-187-0': 'When an external field is applied to the fluid (or to objects embedded in the fluid), energy is pumped into the system and, as a consequence, the average temperature will rise.', 'cond-mat-0603391-3-187-1': 'To prevent this from happening, the system must be coupled to a thermostat.', 'cond-mat-0603391-3-187-2': 'In order not to influence the average flow, thermostating requires a local and Galilean invariant definition of temperature.', 'cond-mat-0603391-3-188-0': 'We achieve this by relating the instantaneous local temperature in a cell to the mean square deviation of the fluid particle velocities from the center of mass velocity of that cell.', 'cond-mat-0603391-3-188-1': 'To minimize interference of the thermostat with the dynamics, we choose to measure the overall temperature as: [EQUATION]', 'cond-mat-0603391-3-188-2': 'In Eq. ([REF]), [MATH] is the instantaneous number of fluid particles within cell [MATH].', 'cond-mat-0603391-3-188-3': 'Three degrees of freedom must be subtracted for fixing the center of mass velocity of each cell (note that this implies that the local temperature is not defined in cells containing 0 or 1 particle).', 'cond-mat-0603391-3-188-4': 'During the simulations, the temperature [MATH] is measured every [MATH] (this can be done very efficiently because the relative velocities are readily available in the collision step routine).', 'cond-mat-0603391-3-188-5': 'The thermostat then acts by rescaling all relative velocities [MATH] by a factor [MATH].', 'cond-mat-0603391-3-188-6': 'This strict enforcement of the overall temperature may be relaxed by allowing appropriate fluctuations, but in view of the very large number of fluid particles (typically [MATH]) this will not have a measurable effect on the dynamics of the fluid particles.', 'cond-mat-0603391-3-189-0': '# The Langevin equation and memory effects', 'cond-mat-0603391-3-190-0': 'Within the Brownian approximation each Cartesian component of the colloid velocity [MATH] is described by a simple Langevin equation of the form: [EQUATION] without any memory effect in the friction coefficient [MATH], or in the random force [MATH].', 'cond-mat-0603391-3-190-1': 'This Langevin equation fundamentally arises from the assumption that the force-kicks are Markovian: each step is independent of any previous behavior.', 'cond-mat-0603391-3-190-2': 'Of course on a very short time-scale this is not true, but since Brownian motion is self-similar, the Markovian assumptions underlying Eq. ([REF]) might be expected to be accurate on longer time-scales.', 'cond-mat-0603391-3-190-3': 'Inspection of Fig. [REF] shows that the force-force correlation function ([REF]) does indeed decay on a time-scale [MATH] which is much faster than other Brownian time-scales in the system.', 'cond-mat-0603391-3-191-0': 'From the Langevin Eq. ([REF]) it follows that the velocity auto-correlation function (VACF) takes the form [EQUATION] where [MATH].', 'cond-mat-0603391-3-191-1': 'The VACF is directly related to the diffusion constant through the Green-Kubo relation: [EQUATION] and Eq. ([REF]) has the attractive property that its integral gives the correct Stokes-Einstein form for [MATH], and that its [MATH] limit gives the expected value from equipartition.', 'cond-mat-0603391-3-191-2': 'It is therefore a popular pedagogical device for introducing Brownian motion.', 'cond-mat-0603391-3-191-3': 'Notwithstanding its seductive simplicity, we will argue that this Langevin approach strongly underestimates the initial decay rate of the VACF, and badly overestimates its long-time decay rate.', 'cond-mat-0603391-3-192-0': '## VACF at long times', 'cond-mat-0603391-3-193-0': 'At longer times, as first shown in MD simulations by Alder and Wainwright [CITATION], the VACF shows an algebraic decay that is much slower than the exponential decay of Eq. ([REF]).', 'cond-mat-0603391-3-193-1': 'The difference is due to the breakdown of the Markovian assumption for times [MATH].', 'cond-mat-0603391-3-193-2': 'Memory effects arise because the momentum that a colloidal particle transfers to the fluid is conserved, with dynamics described by the Navier Stokes equations, resulting in long-ranged flow patterns that affect a colloid on much longer time-scales than Eq. ([REF]) suggests.', 'cond-mat-0603391-3-193-3': 'Calculations taking into account the full time-dependent hydrodynamics were performed by Hauge [CITATION] and Hinch [CITATION] (and others apparently much earlier [CITATION]).', 'cond-mat-0603391-3-193-4': 'For [MATH], i.e. a neutrally buoyant particle, these expressions can be written as [EQUATION] where we have chosen to use an integral form as in [CITATION].', 'cond-mat-0603391-3-193-5': 'Under the assumption that the sound effects have dissipated away, the only hydrodynamic time-scale is the kinematic time [MATH] defined in Eq. ([REF]).', 'cond-mat-0603391-3-193-6': 'The VACF must therefore scale with [MATH], independently of colloid size.', 'cond-mat-0603391-3-193-7': 'Eq. ([REF]) is written in a way to emphasize this scaling.', 'cond-mat-0603391-3-194-0': 'By expanding the denominator it is not hard to see that at longer times the well-known algebraic tail results: [EQUATION] with the same pre-factor that was found from mode-coupling theory [CITATION].', 'cond-mat-0603391-3-195-0': 'The integral of Eq. ([REF]) is [EQUATION] which means that all the hydrodynamic contributions to the friction [MATH] come from [MATH].', 'cond-mat-0603391-3-195-1': 'The integral converges slowly, as [MATH], because of the long-time tail.', 'cond-mat-0603391-3-195-2': 'The VACF can easily be related to the mean-square displacement, and deviations from purely diffusive motion are still observable for [MATH], and have been measured in experiments [CITATION].', 'cond-mat-0603391-3-196-0': 'Although the last equality in Eq. ([REF]) shows that the VACF tail amplitude is independent of colloid size, it should be kept in mind that the algebraic decay does not set in until [MATH].', 'cond-mat-0603391-3-196-1': 'Thus it is the relative (not absolute) effect of the tail that is the same for all colloids.', 'cond-mat-0603391-3-196-2': 'For example, at [MATH], the VACF will have decayed to [MATH] of its initial value of [MATH], independently of colloid mass [MATH].', 'cond-mat-0603391-3-196-3': 'Even though this tail amplitude may seem small, about half the weight of the integral ([REF]) that determines the friction is from [MATH].', 'cond-mat-0603391-3-196-4': 'The dominance of the non-Markovian hydrodynamic behavior in determining the friction for colloids does not appear to hold for microscopic fluids, where, for example for LJ liquids near the triple point, it is thought that the long-time tail only adds a few [MATH] to the friction coefficient [CITATION].', 'cond-mat-0603391-3-196-5': 'However, this is not due to the simple Langevin equation ([REF]) being more accurate, but rather to the modified short-time behavior of the VACF, caused by fluid correlations (a different non-Markovian effect).', 'cond-mat-0603391-3-197-0': '## VACF at short times', 'cond-mat-0603391-3-198-0': 'At short times the hydrodynamic contribution to the VACF ([REF]) reduces to [EQUATION] because, within a continuum description, [MATH] of the energy is dissipated as a sound wave at velocity [MATH], [EQUATION] where the sonic time [MATH] is defined in Table [REF].', 'cond-mat-0603391-3-198-1': "The sound wave doesn't contribute to the friction or diffusion since [EQUATION]", 'cond-mat-0603391-3-198-2': 'For a more extended discussion of the role of sound on hydrodynamics see e.g. ref. [CITATION].', 'cond-mat-0603391-3-199-0': 'Because [MATH], the sound-wave contribution to the VACF decays much faster than the hydrodynamic contribution [MATH] or even than the Langevin approximation [MATH] (recall that [MATH] for neutrally buoyant colloids.)', 'cond-mat-0603391-3-199-1': 'Therefore, in the continuum picture, the short-time decay of the VACF is much faster than that suggested by the simple Langevin equation.', 'cond-mat-0603391-3-199-2': 'And even after the sound wave has decayed, [MATH] decays much more rapidly (faster than simple exponential) than Eq. ([REF]) for times [MATH].', 'cond-mat-0603391-3-199-3': 'SRD simulations confirm this more rapid short time decay.', 'cond-mat-0603391-3-199-4': 'For example, in Fig [REF] we show a direct comparison of the measured VACF to the continuum approach, where [MATH].', 'cond-mat-0603391-3-199-5': 'For short times the decay predicted from sound agrees reasonable well, and for long times the full hydrodynamic theory quantitatively fits the simulation data.', 'cond-mat-0603391-3-200-0': 'A closer look at the short-time behavior in our SRD simulations, both for slip boundaries as in Fig. [REF] and for stick-boundaries as in Fig [REF] or in [CITATION], shows that a better fit to the short-time exponential decay of the VACF is given by [EQUATION] where the Enskog time ([REF]) scales as: [EQUATION] for heavy (stick boundary) colloids in an SRD fluid.', 'cond-mat-0603391-3-200-1': 'Note that the Enskog decay time is very close to the decay time [MATH] of the sound wave ([REF]).', 'cond-mat-0603391-3-200-2': 'That the SRD Enskog time should scale as [MATH] is perhaps not too surprising, because that is the natural time-scale of the SRD fluid.', 'cond-mat-0603391-3-200-3': 'However, the fact that the pre-factors in the exponential of Eqs ([REF]) and ([REF]) are virtually the same may be accidental.', 'cond-mat-0603391-3-200-4': "Although in a real liquid the increased compressibility factor means [MATH] and [MATH] are lower than their values for an ideal gas, the two times change at slightly different rates so that we don't expect the same equivalence in physical systems.", 'cond-mat-0603391-3-201-0': 'In contrast to the sound wave, the integral over the Enskog VACF leads to a finite value: [EQUATION] which quantitatively explains the friction we measured in our simulations, as discussed in section [REF].', 'cond-mat-0603391-3-201-1': 'In contrast to the VACF from sound, the Enskog result cannot simply be added to the VACF from Eq. ([REF]) because this would violate the [MATH] limit of the VACF, set by equipartition.', 'cond-mat-0603391-3-201-2': 'However, because the Enskog contribution occurs on times [MATH], where sound and other collective modes of (compressible) hydrodynamics are not yet fully developed, it may be that Eqs. ([REF]) and ([REF]), derived from continuum theory, should be modified on these short time-scales.', 'cond-mat-0603391-3-201-3': 'Since the dominant contribution to the integral in Eq. ([REF]) is for longer times, the exact form of the hydrodynamic contribution ([REF]) on these short times ([MATH]) does not have much influence on the hydrodynamic part of the friction.', 'cond-mat-0603391-3-201-4': 'Thus the approximation of the short-time VACF by the Enskog result of Eq. ([REF]), and the longer-time ([MATH], but still [MATH]) VACF by Eq. ([REF]) may be a fairly reasonable empirical approach.', 'cond-mat-0603391-3-201-5': 'This approximation, together with the Green-Kubo relation ([REF]), then justifies, a-posteori, the parallel addition of frictions in Eq. ([REF]).', 'cond-mat-0603391-3-202-0': 'Just as was found for the long-time behavior, the short-time behavior of the VACF displays important deviations from the simple Langevin picture.', 'cond-mat-0603391-3-202-1': 'We find that the VACF decays much faster, on the order of the sonic time [MATH] rather than the Brownian time [MATH].', 'cond-mat-0603391-3-202-2': 'From the tables [REF] and [REF] one can see how large these differences can be.', 'cond-mat-0603391-3-202-3': 'For example, for colloids with [MATH], the short-time Enskog or sonic decay times are of order [MATH] s, while [MATH] is of order [MATH] s, so that the decay of the VACF will be dominated by [MATH], with its non-exponential behavior, over almost the entire time regime.', 'cond-mat-0603391-3-203-0': 'An example of the effect of the Enskog contribution on the friction or the mean-square displacement is given in Fig. [REF].', 'cond-mat-0603391-3-203-1': 'The integral of the VACF, [MATH] from Eq. ([REF]), which is directly related to the mean-square displacement [CITATION], is extracted from the simulations described in Fig [REF], and compared to [EQUATION] calculated using Eqs. ([REF]) and ([REF]).', 'cond-mat-0603391-3-203-2': 'Both show exactly the same [MATH] scaling for [MATH], but the simulated [MATH] is larger because the initial Enskog contribution means that [MATH], for [MATH], with [MATH] defined in Eq. ([REF]).', 'cond-mat-0603391-3-204-0': 'The slow algebraic convergence to the final diffusion coefficient, or related friction, also helps explain the scaling of the finite size effects of Eq. ([REF]).', 'cond-mat-0603391-3-204-1': 'A finite box of length [MATH] means that the measured [MATH] is cut off roughly at [MATH], and because [MATH] for [MATH] the effect on the diffusion constant can be written as [MATH] which explains the scaling form of Eq. ([REF]).', 'cond-mat-0603391-3-204-2': 'Fig. [REF] also demonstrates how much more rapidly the [MATH] from the Langevin Eq. ([REF]) converges to its final result [MATH] when compared to the simulations and hydrodynamic theories.', 'cond-mat-0603391-3-205-0': 'In summary, even though the simple Langevin equation ([REF]), without memory, may have pedagogical merit, it does a poor job of describing the VACF for colloids in solution.', 'cond-mat-0603391-3-206-0': '# Comparison of SRD to physical parameters', 'cond-mat-0603391-3-207-0': 'At the end of the day one would like to compare a coarse-grained simulation to a real physical colloidal dispersion.', 'cond-mat-0603391-3-207-1': 'To help facilitate such comparisons, we have summarized a number of physical parameters and dimensionless numbers for colloids in water, and also colloids in an SRD fluid.', 'cond-mat-0603391-3-208-0': 'Table [REF] compares the properties of a pure SRD fluid, in the limit of small mean-free path [MATH], to the properties of water.', 'cond-mat-0603391-3-209-0': 'Table [REF] summarizes the main scaling and values for a number of different properties of two different size neutrally buoyant colloids in water.', 'cond-mat-0603391-3-209-1': 'To approximate the Enskog friction we used a simplified form valid for stick boundary conditions in an ideal fluid, as in [CITATION], which ignores the compressibility and other expected complexities of real water.', 'cond-mat-0603391-3-209-2': 'These predictions should therefore be taken with a grain of salt.', 'cond-mat-0603391-3-209-3': 'For simplicity, we have ignored any potential Enskog effects on the diffusion constant or time-scales.', 'cond-mat-0603391-3-210-0': 'Table [REF] shows a very similar comparison for two different size colloids in an SRD fluid.', 'cond-mat-0603391-3-210-1': 'For simplicity we assume that there are no finite-size effects, which explains small differences when compared to similar parameters used in the text of the main article.', 'cond-mat-0603391-3-210-2': 'However, in contrast to Table [REF], we do include the Enskog contribution when calculating the total friction and related properties.'}

1610.04648

{'1610.04648-1-0-0': 'Let [MATH] be an arc belonging to a Burau pair [MATH] subject to the reductions of Section [REF].', '1610.04648-1-0-1': 'The boundary of a regular neighborhood of [MATH] is a loop [MATH] supported on [MATH], and thus we can describe [MATH] using the set of weights for [MATH].', '1610.04648-1-0-2': 'In this section we go the opposite direction, finding necessary and sufficient conditions that a set of weights produces a multicurve [MATH], one component of which is a regular neighborhood of an arc [MATH] belonging to a Burau pair [MATH] subject to the reductions of Section [REF].', '1610.04648-1-0-3': 'Not being able to give conditions on the weights to prevent proper multicurves from arising, we instead detect proper multicurves by performing a preliminary computation on a given weight set.', '1610.04648-1-0-4': 'Details on this as well as our implementation are given at the end of this section.', '1610.04648-1-1-0': 'Let [MATH], where [MATH] is the weight of the [MATH]th rail of [MATH], as labeled in Figure [REF].', '1610.04648-1-1-1': 'The switch conditions are expressed by the following equations.', '1610.04648-1-1-2': 'As every weight is freely determined by [MATH], [MATH], [MATH], [MATH], and [MATH], we focus our attention on the tuple [MATH].', '1610.04648-1-2-0': 'Divisibility: The arc [MATH] starts on rail [MATH] and ends on rail [MATH], so weights [MATH] and [MATH] are even and weights [MATH] and [MATH] are odd.', '1610.04648-1-2-1': 'If [MATH], then [MATH], guaranteeing the weights will give rise to a proper multicurve.', '1610.04648-1-3-0': 'In the next two groups of conditions, we "steer" the initial and terminal segments as dictated by the reductions in Section [REF].', '1610.04648-1-3-1': 'Consider the track segment seen in Figure [REF].', '1610.04648-1-3-2': 'Which rail the arc follows is determined by the number of lines [MATH] that are to the left of the arc as it approaches a switch.', '1610.04648-1-3-3': 'Beginning at the switch of [MATH] and [MATH] traveling left, [MATH].', '1610.04648-1-3-4': 'The arc is steered left if and only if [MATH].', '1610.04648-1-3-5': 'Assuming this is the case, after passing through the switch of [MATH] and [MATH] we still have [MATH].', '1610.04648-1-3-6': 'On the other hand if the arc is steered right, then after passing through the switch of [MATH] and [MATH], we now have [MATH].', '1610.04648-1-3-7': 'As the notions "initial" and "terminal" are merely a choice of orientation, these remarks equally apply to the terminal segment of the arc.', '1610.04648-1-4-0': 'Initial segment: The arc [MATH] starts on rail [MATH] and then travels on rail [MATH], meaning [MATH].', '1610.04648-1-4-1': 'After this, [MATH] intersects the arc [MATH] from the top along rail [MATH] or [MATH], or from the bottom along rail [MATH], meaning [MATH] or [MATH].', '1610.04648-1-4-2': 'In the case where [MATH] intersects [MATH] from the bottom while traveling along rail [MATH], we must enforce that [MATH] does not then undo this intersection by traveling along rail [MATH] and intersecting [MATH] from the top, as seen in figure [REF].', '1610.04648-1-5-0': 'Terminal segment: The terminal segment of [MATH] is seen in Figure [REF].', '1610.04648-1-6-0': 'This translates into the conditions [MATH], [MATH], and [MATH].', '1610.04648-1-6-1': 'Immediately prior to taking this path, the arc cannot travel on rail [MATH] intersecting [MATH] from below.', '1610.04648-1-6-2': 'This gives the fourth condition [EQUATION].', '1610.04648-1-6-3': 'Again using the switch conditions, we express these in terms of [MATH], [MATH], [MATH], [MATH], and [MATH].', '1610.04648-1-7-0': 'In order to prove that the conditions we have given are sufficient, we will use the following lemma.', '1610.04648-1-8-0': 'Two transverse simple closed curves [MATH] and [MATH] in a surface [MATH] cannot be homotoped to have fewer intersections if and only if they do not form a "bigon", that is, an embedded disk in [MATH] whose boundary is the union of an arc of [MATH] and an arc of [MATH].', '1610.04648-1-9-0': 'The conditions D1-D4, N1-N6, I1-I3, and T1-T4 are necessary and sufficient for the weights [MATH] to give rise to a multicurve [MATH], one component of which is a regular neighborhood of an arc [MATH] belonging to a Burau pair [MATH] subject to the reductions of Section [REF].', '1610.04648-1-10-0': 'The necessity is clear.', '1610.04648-1-10-1': 'To show the sufficiency of the conditions, we must show that the initial and terminal intersections are essential.', '1610.04648-1-10-2': 'We prove a stronger statement, that the only nonessential intersection between [MATH] and [MATH] supported on [MATH] is of the form seen in Figure 7.', '1610.04648-1-10-3': 'Let [MATH] and [MATH] intersect at [MATH].', '1610.04648-1-10-4': 'Continuing to follow [MATH], assuming it is not of the form in Figure 7, it must wind around a puncture [MATH] before intersecting [MATH] again.', '1610.04648-1-10-5': 'Denote the next intersection of [MATH] with [MATH] by [MATH], and consider the loop [MATH] formed by [MATH] from [MATH] to [MATH] and [MATH] from [MATH] to [MATH].', '1610.04648-1-10-6': 'As [MATH] is supported on the track, there is a corresponding set of weights [MATH].', '1610.04648-1-10-7': 'The puncture [MATH] will be interior to [MATH] if and only if the corresponding weight [MATH] is odd.', '1610.04648-1-10-8': 'If [MATH], [MATH], [MATH], and [MATH] are even, then by the switch conditions all the weights are even, implying that [MATH] is a proper multicurve.', '1610.04648-1-10-9': 'Therefore one weight must be odd, so some [MATH] is interior to [MATH], and the result now follows from Lemma [REF].', '1610.04648-1-11-0': 'Let [MATH] be a set of weights as in Proposition [REF] giving rise to a loop (and not a proper multicurve).', '1610.04648-1-11-1': 'Denote by [MATH] the resulting arc.', '1610.04648-1-11-2': 'Then the number of essential intersections of [MATH] with [MATH] is [EQUATION].', '1610.04648-1-12-0': 'There are two ways we can prevent certain proper multicurves from arising.', '1610.04648-1-12-1': 'The first is to insist that [MATH], as otherwise all weights will share a non-trivial common factor and give rise to multiple copies of a single, simple closed curve.', '1610.04648-1-12-2': 'The second is to insist that some weight is nonzero, as otherwise the resulting multicurve will have a loop encompassing the entire train track and thus be a proper multicurve.', '1610.04648-1-12-3': 'The paths of the initial and terminal segments of the corresponding arc imply that all weights are nonzero except for [MATH], [MATH], [MATH], and [MATH].', '1610.04648-1-12-4': 'Since [MATH] being zero implies that [MATH] and [MATH] are zero, we have three cases.', '1610.04648-1-13-0': 'All three cases can occur simultaneously.', '1610.04648-1-13-1': 'In each case, many of the above conditions become satisfied or simplified.', '1610.04648-1-13-2': 'The conditions D1-D4 remain in all cases, and the other conditions become the following.', '1610.04648-1-14-0': 'When traveling on rail [MATH] either moving left or right, the conditions on the initial and terminal segments imply that there are only a finite number of possible paths one may take before returning to rail [MATH].', '1610.04648-1-14-1': 'These paths are enumerated in Figure [REF] and Figure [REF].', '1610.04648-1-15-0': 'Let [MATH] denote the number of lines to the left of the arc as it travels on rail 14.', '1610.04648-1-15-1': 'When traveling to the left, the path and direction are determined by the position of the first number that [MATH] is smaller than in the following list.', '1610.04648-1-15-2': 'Likewise when traveling to the right, the path and direction are determined by the position of the first number that [MATH] is smaller than in the following list.', '1610.04648-1-15-3': 'Note that for a given set of weights, it may be that either of the lists are not ascending.', '1610.04648-1-15-4': 'This can occur when an arc never takes one of the paths enumerated in Figure [REF] and Figure [REF].', '1610.04648-1-16-0': 'To compute the resulting Burau polynomial, our algorithm begins with [MATH] and the Burau polynomial being [MATH].', '1610.04648-1-16-1': 'We then alternate traveling left and right on rail [MATH].', '1610.04648-1-16-2': 'Each time a path is taken, we update [MATH] and the Burau polynomial accordingly.', '1610.04648-1-16-3': 'This process terminates when traveling right on rail 14, we have either [MATH] and [MATH], or [MATH] is greater than all the numbers in the list above and [MATH].', '1610.04648-1-16-4': 'Along with being computationally efficient, this allows us to concisely describe the path of an arc.', '1610.04648-1-16-5': 'Each time we return to rail 14, using the appropriate list we record the position of the first number that [MATH] is smaller than, [MATH] and [MATH] respectively.', '1610.04648-1-16-6': 'We call the resulting list the record of the arc [MATH].', '1610.04648-1-17-0': 'We achieve a significant speedup by doing a preliminary run on each arc, recording the number of positive and negative crossings.', '1610.04648-1-17-1': 'We discard the weight set if a proper multicurve is ever detected, which by Corollary [REF] is the case if and only if the number of crossings is less than [MATH].', '1610.04648-1-17-2': 'During the preliminary run, we also use a static array of size [MATH] to compute the Burau polynomial where the powers of [MATH] are treated modulo [MATH].', '1610.04648-1-17-3': 'This is similar to a reduction made by Bigelow while calculating Burau polynomials [CITATION].', '1610.04648-1-17-4': 'On our system, [MATH] appears to be optimal.', '1610.04648-1-17-5': 'When we are solely searching for the zero polynomial, we short circuit the computation if the array is not all zero.'}

{'1610.04648-2-0-0': 'Let [MATH] be an arc belonging to a Burau pair [MATH] subject to the reductions of Section [REF].', '1610.04648-2-0-1': 'The boundary of a regular neighborhood of [MATH] is a loop [MATH] supported on [MATH], and thus we can get a set of weights for [MATH].', '1610.04648-2-0-2': 'In this section we go the opposite direction, finding necessary and sufficient conditions that a set of weights produces a multicurve [MATH], one component of which is a regular neighborhood of an arc [MATH] belonging to a Burau pair [MATH] subject to the reductions of Section [REF].', '1610.04648-2-0-3': 'Not being able to give conditions on the weights to prevent proper multicurves from arising, we instead detect proper multicurves by performing a preliminary computation on a given weight set.', '1610.04648-2-0-4': 'Details on this as well as our algorithm are given at the end of this section.', '1610.04648-2-1-0': 'With [MATH] the nonnegative integers, let [MATH] where [MATH] is the weight of the [MATH]th rail of [MATH] as labeled in Figure [REF].', '1610.04648-2-1-1': 'The switch conditions are expressed by the following equations.', '1610.04648-2-1-2': 'As every weight is freely determined by [MATH], [MATH], [MATH], [MATH], and [MATH], we focus our attention on the tuple [MATH].', '1610.04648-2-2-0': 'Divisibility: The arc [MATH] starts on rail [MATH] and ends on rail [MATH], so weights [MATH] and [MATH] are even and weights [MATH] and [MATH] are odd.', '1610.04648-2-3-0': 'In the next two groups of conditions, we "steer" the initial and terminal segments as dictated by the reductions in Section [REF].', '1610.04648-2-3-1': 'Consider the track segment seen in Figure [REF].', '1610.04648-2-3-2': 'Which rail the arc follows is determined by the number of lines [MATH] that are to the left of the arc as it approaches a switch.', '1610.04648-2-3-3': 'Beginning at the switch of [MATH] and [MATH] traveling left, [MATH].', '1610.04648-2-3-4': 'The arc is steered left if and only if [MATH].', '1610.04648-2-3-5': 'Assuming this is the case, after passing through the switch of [MATH] and [MATH] we still have [MATH].', '1610.04648-2-3-6': 'On the other hand if the arc is steered right, then after passing through the switch of [MATH] and [MATH] we now have [MATH].', '1610.04648-2-3-7': 'As the notions "initial" and "terminal" are merely a choice of orientation, these remarks equally apply to the terminal segment of the arc.', '1610.04648-2-4-0': 'Initial segment: The arc [MATH] starts on rail [MATH] and then travels on rail [MATH], meaning [MATH].', '1610.04648-2-4-1': 'After this, [MATH] intersects the arc [MATH] from the top along rail [MATH] or [MATH], or from the bottom along rail [MATH], meaning [MATH] or [MATH].', '1610.04648-2-4-2': 'In the case where [MATH] intersects [MATH] from the bottom while traveling along rail [MATH], we must enforce that [MATH] does not then undo this intersection by traveling along rail [MATH] and intersecting [MATH] from the top, as seen in Figure [REF].', '1610.04648-2-5-0': 'Terminal segment: The terminal segment of [MATH] is seen in Figure [REF].', '1610.04648-2-6-0': 'This translates into the conditions [MATH], [MATH], and [MATH].', '1610.04648-2-6-1': 'Immediately prior to taking this path, the arc cannot travel on rail [MATH] intersecting [MATH] from below.', '1610.04648-2-6-2': 'This gives the fourth condition [EQUATION].', '1610.04648-2-6-3': 'Again using the switch conditions, we express these in terms of [MATH], [MATH], [MATH], [MATH], and [MATH].', '1610.04648-2-7-0': 'In order to prove that the conditions we have given are sufficient, we will use the following lemma.', '1610.04648-2-8-0': 'Two transverse simple closed curves [MATH] and [MATH] in a surface [MATH] cannot be homotoped to have fewer intersections if and only if they do not form a "bigon", that is, an embedded disk in [MATH] whose boundary is the union of an arc of [MATH] and an arc of [MATH].', '1610.04648-2-9-0': 'The conditions D1-D4, N1-N6, I1-I3, and T1-T4 are necessary and sufficient for the weights [MATH] to give rise to a multicurve [MATH], one component of which is a regular neighborhood of an arc [MATH] belonging to a Burau pair [MATH] subject to the reductions of Section [REF].', '1610.04648-2-10-0': 'The necessity is clear.', '1610.04648-2-10-1': 'To show the sufficiency of the conditions, we must show that the initial and terminal intersections are essential.', '1610.04648-2-10-2': 'We prove a stronger statement, that the only nonessential intersection between [MATH] and [MATH] supported on [MATH] is of the form seen in Figure [REF].', '1610.04648-2-10-3': 'Let [MATH] and [MATH] intersect at [MATH].', '1610.04648-2-10-4': 'Continuing to follow [MATH], assuming it is not of the form in Figure [REF], it must wind around a puncture [MATH] before intersecting [MATH] again.', '1610.04648-2-10-5': 'Denote the next intersection of [MATH] with [MATH] by [MATH], and consider the loop [MATH] formed by [MATH] from [MATH] to [MATH] and [MATH] from [MATH] to [MATH].', '1610.04648-2-10-6': 'As [MATH] is supported on the track, there is a corresponding set of weights [MATH].', '1610.04648-2-10-7': 'The puncture [MATH] will be interior to [MATH] if and only if the corresponding weight [MATH] is odd.', '1610.04648-2-10-8': 'If [MATH], [MATH], [MATH], and [MATH] are even, then by the switch conditions all the weights are even, implying that [MATH] is a proper multicurve.', '1610.04648-2-10-9': 'Therefore one weight must be odd, so some [MATH] is interior to [MATH], and the result now follows from Lemma [REF].', '1610.04648-2-11-0': 'Let [MATH] be a set of weights as in Proposition [REF] giving rise to a loop (and not a proper multicurve).', '1610.04648-2-11-1': 'Denote by [MATH] the resulting arc.', '1610.04648-2-11-2': 'Then the number of essential intersections of [MATH] with [MATH] is [EQUATION].', '1610.04648-2-12-0': 'There are two ways we can prevent certain proper multicurves from arising.', '1610.04648-2-12-1': 'The first is to insist that [MATH], as otherwise all weights will share a non-trivial common factor and give rise to multiple copies of a single, simple closed curve.', '1610.04648-2-12-2': 'The second is to insist that some weight is zero, as otherwise the resulting multicurve will have a loop encompassing the entire train track and thus be a proper multicurve.', '1610.04648-2-12-3': 'The paths of the initial and terminal segments of the corresponding arc imply that all weights except for [MATH], [MATH], [MATH], and [MATH] are non-zero.', '1610.04648-2-12-4': 'Since [MATH] being zero implies that [MATH] and [MATH] are zero, we have three cases.', '1610.04648-2-13-0': 'All three cases can occur simultaneously.', '1610.04648-2-13-1': 'In each case, many of the above conditions become satisfied or simplified.', '1610.04648-2-13-2': 'The conditions D1-D4 remain in all cases, and the other conditions become the following.', '1610.04648-2-14-0': 'When traveling on rail [MATH] either moving left or right, the conditions on the initial and terminal segments imply that there are only a finite number of possible paths one may take before returning to rail [MATH].', '1610.04648-2-14-1': 'These paths are enumerated in Figure [REF].', '1610.04648-2-15-0': 'Let [MATH] denote the number of lines to the left of the arc as it travels on rail 14.', '1610.04648-2-15-1': 'Which of the enumerated paths is taken, along with the direction this path is traversed, is determined by comparing [MATH] to the train track weights.', '1610.04648-2-15-2': 'When traveling to the left, the path and direction are determined by the position of the first number that [MATH] is smaller than in the following list.', '1610.04648-2-15-3': 'Likewise when traveling to the right, the path and direction are determined by the position of the first number that [MATH] is smaller than in the following list.', '1610.04648-2-15-4': 'Note that for a given set of weights, it may be that either of the lists are not ascending.', '1610.04648-2-15-5': 'This can occur when an arc never takes one of the paths enumerated in Figure [REF] and Figure [REF].', '1610.04648-2-16-0': 'To compute the resulting Burau polynomial, our algorithm begins with [MATH] and the Burau polynomial being [MATH].', '1610.04648-2-16-1': 'We then alternate traveling left and right on rail [MATH].', '1610.04648-2-16-2': 'Each time a path is taken, we update [MATH] and the Burau polynomial accordingly.', '1610.04648-2-16-3': 'We end the computation when the arc reaches rail [MATH] with [MATH].', '1610.04648-2-16-4': 'While traveling right on rail 14, this is precisely when we have either [MATH] and [MATH], or [MATH] is greater than all the numbers in the list above and [MATH].', '1610.04648-2-16-5': 'Along with being computationally efficient, this allows us to concisely describe the path of an arc.', '1610.04648-2-16-6': 'Each time we return to rail 14, using the appropriate list we record the position of the first number that [MATH] is smaller than, [MATH] and [MATH] respectively.', '1610.04648-2-16-7': 'We call the resulting list the record of the arc [MATH].', '1610.04648-2-17-0': 'We achieve a significant speedup by doing a preliminary run on each arc, recording the number of positive and negative crossings.', '1610.04648-2-17-1': 'We discard the weight set if a proper multicurve is ever detected, which by Corollary [REF] is the case if and only if the number of crossings is less than [MATH].', '1610.04648-2-17-2': 'During the preliminary run, we also use a static array of size [MATH] to compute the Burau polynomial where the powers of [MATH] are treated modulo [MATH].', '1610.04648-2-17-3': 'On our system, [MATH] appears to be optimal.', '1610.04648-2-17-4': 'When we are solely searching for the zero polynomial, we short-circuit the computation if the array is not all zero.'}

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[['1610.04648-1-4-0', '1610.04648-2-4-0'], ['1610.04648-1-4-1', '1610.04648-2-4-1'], ['1610.04648-1-5-0', '1610.04648-2-5-0'], ['1610.04648-1-16-0', '1610.04648-2-16-0'], ['1610.04648-1-16-1', '1610.04648-2-16-1'], ['1610.04648-1-16-2', '1610.04648-2-16-2'], ['1610.04648-1-16-4', '1610.04648-2-16-5'], ['1610.04648-1-16-5', '1610.04648-2-16-6'], ['1610.04648-1-16-6', '1610.04648-2-16-7'], ['1610.04648-1-3-0', '1610.04648-2-3-0'], ['1610.04648-1-3-1', '1610.04648-2-3-1'], ['1610.04648-1-3-2', '1610.04648-2-3-2'], ['1610.04648-1-3-3', '1610.04648-2-3-3'], ['1610.04648-1-3-4', '1610.04648-2-3-4'], ['1610.04648-1-3-5', '1610.04648-2-3-5'], ['1610.04648-1-3-7', '1610.04648-2-3-7'], ['1610.04648-1-7-0', '1610.04648-2-7-0'], ['1610.04648-1-15-0', '1610.04648-2-15-0'], ['1610.04648-1-15-1', '1610.04648-2-15-2'], ['1610.04648-1-15-2', '1610.04648-2-15-3'], ['1610.04648-1-15-3', '1610.04648-2-15-4'], ['1610.04648-1-15-4', '1610.04648-2-15-5'], ['1610.04648-1-12-0', '1610.04648-2-12-0'], ['1610.04648-1-12-1', '1610.04648-2-12-1'], ['1610.04648-1-12-4', '1610.04648-2-12-4'], ['1610.04648-1-14-0', '1610.04648-2-14-0'], ['1610.04648-1-11-0', '1610.04648-2-11-0'], ['1610.04648-1-11-1', '1610.04648-2-11-1'], ['1610.04648-1-11-2', '1610.04648-2-11-2'], ['1610.04648-1-17-0', '1610.04648-2-17-0'], ['1610.04648-1-17-1', '1610.04648-2-17-1'], ['1610.04648-1-17-2', '1610.04648-2-17-2'], ['1610.04648-1-17-4', '1610.04648-2-17-3'], ['1610.04648-1-1-1', '1610.04648-2-1-1'], ['1610.04648-1-1-2', '1610.04648-2-1-2'], ['1610.04648-1-2-0', '1610.04648-2-2-0'], ['1610.04648-1-6-0', '1610.04648-2-6-0'], ['1610.04648-1-6-1', '1610.04648-2-6-1'], ['1610.04648-1-6-2', '1610.04648-2-6-2'], ['1610.04648-1-6-3', '1610.04648-2-6-3'], ['1610.04648-1-8-0', '1610.04648-2-8-0'], ['1610.04648-1-9-0', '1610.04648-2-9-0'], ['1610.04648-1-13-0', '1610.04648-2-13-0'], ['1610.04648-1-13-1', '1610.04648-2-13-1'], ['1610.04648-1-13-2', '1610.04648-2-13-2'], ['1610.04648-1-0-0', '1610.04648-2-0-0'], ['1610.04648-1-0-2', '1610.04648-2-0-2'], ['1610.04648-1-0-3', '1610.04648-2-0-3'], ['1610.04648-1-10-0', '1610.04648-2-10-0'], ['1610.04648-1-10-1', '1610.04648-2-10-1'], ['1610.04648-1-10-3', '1610.04648-2-10-3'], ['1610.04648-1-10-5', '1610.04648-2-10-5'], ['1610.04648-1-10-6', '1610.04648-2-10-6'], ['1610.04648-1-10-7', '1610.04648-2-10-7'], ['1610.04648-1-10-8', '1610.04648-2-10-8'], ['1610.04648-1-10-9', '1610.04648-2-10-9']]

[['1610.04648-1-4-2', '1610.04648-2-4-2'], ['1610.04648-1-16-3', '1610.04648-2-16-4'], ['1610.04648-1-3-6', '1610.04648-2-3-6'], ['1610.04648-1-12-2', '1610.04648-2-12-2'], ['1610.04648-1-12-3', '1610.04648-2-12-3'], ['1610.04648-1-14-1', '1610.04648-2-14-1'], ['1610.04648-1-17-5', '1610.04648-2-17-4'], ['1610.04648-1-1-0', '1610.04648-2-1-0'], ['1610.04648-1-0-1', '1610.04648-2-0-1'], ['1610.04648-1-0-4', '1610.04648-2-0-4'], ['1610.04648-1-10-2', '1610.04648-2-10-2'], ['1610.04648-1-10-4', '1610.04648-2-10-4']]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1610.04648

1504.02295

{'1504.02295-1-0-0': 'In this work we have analysed some cosmological bounds concerning an open FLRW solution of massive gravity.', '1504.02295-1-0-1': 'The constraints with recent observational [MATH] data were found and the best fit values for the cosmological parameters are in agreement with the [MATH]CDM model, and also point to a nearly open spatial curvature, as expected from the model.', '1504.02295-1-0-2': 'The graviton mass dependence with the constant parameters [MATH] and [MATH], related to the additional lagrangians terms of the model, are also analysed, and we have obtained a strong dependence with such parameters, although the condition [MATH] seems dominant for a long range of the parameters [MATH] and [MATH].', '1504.02295-1-0-3': 'The limit [MATH] forbid one of the branches of accelerated solution, which indicates the necessity of the corresponding lagrangian term.', '1504.02295-1-1-0': '# Introduction', '1504.02295-1-2-0': 'Current observations of supernovae type Ia [CITATION], cosmic microwave background radiation [CITATION] and Hubble parameter data [CITATION] indicate an accelerated expansion of the universe, being the [MATH]CDM model the best model to fit the observational data.', '1504.02295-1-2-1': 'The [MATH] term corresponds to a cosmological constant energy density which is plagued with several fundamental issues [CITATION], which has motivated the search for alternatives models of gravity that could explain the observations.', '1504.02295-1-3-0': 'Massive gravity theories [CITATION] are good and old candidates to explain the accelerated expansion of the universe, since that the graviton mass could perfectly induce and mimic a cosmological constant term.', '1504.02295-1-3-1': 'However, such kinds of theories were considered for long time as being unsuitable due to the appearing of Boulware-Deser (BD) ghosts [CITATION].', '1504.02295-1-3-2': 'Recently it was discovered a nonlinear massive gravity theory that was shown to be BD ghost free [CITATION], called dRGT (de Rham-Gabadadze-Tolley) model (see [CITATION] for a review), bringing back the cosmological interest for such theories [CITATION].', '1504.02295-1-3-3': 'Self-accelerating cosmologies with ghost-free massive gravitons have been studied thereafter [CITATION], although there is no full concordance about the stability of the solutions concerning the flat, open and closed Friedmann-Lemaitre-Robertson-Walker (FLRW) isotropic and homogeneous solutions of the metric.', '1504.02295-1-3-4': 'While some authors defend the existence of isotropic FLRW solutions [CITATION], others indicate the appearance of some kinds of nonlinear ghosts instabilities, thus just anisotropic FLRW solutions are ghost-free [CITATION].', '1504.02295-1-3-5': 'Finally, in [CITATION], there is a good explanation on these questions, showing that zero-curvature solutions are in one to one correspondence with exact open FLRW solutions.', '1504.02295-1-4-0': 'We assume the existence of stable solutions in open isotropic FLRW cosmologies and study some bounds on the free parameters present in the massive gravity theory compared to recent observational data.', '1504.02295-1-4-1': 'In Section II we present the general massive gravity theory and the cosmological equations in Section III.', '1504.02295-1-4-2': 'In Section IV we present the constraints from [MATH] data and in Section V some bounds on the graviton mass are presented.', '1504.02295-1-4-3': 'We conclude in Section VI.', '1504.02295-1-5-0': '# Massive gravity theory', '1504.02295-1-6-0': 'Our starting point for the cosmological analysis is the massive theory for gravity proposed in [CITATION].', '1504.02295-1-6-1': 'The nonlinear action besides a functional of the physical metric [MATH] include four spurious scalar fields [MATH] with [MATH] called the Stuckelberg fields.', '1504.02295-1-6-2': 'They are introduced in order to make the action manifestly invariant under diffeomorphism, see for example [CITATION].', '1504.02295-1-6-3': 'Let us start by observing that these scalar fields are related to the physical metric and enter into the action as follows: [EQUATION] where it is defined the fiducial metric [MATH] which is written in terms of the Stuckelberg fields: [EQUATION]', '1504.02295-1-6-4': 'Usually [MATH] is called the reference metric and for the purpose of this paper, one can use [MATH] once the scheme proposed by dRGT respects Poincare symmetry.', '1504.02295-1-6-5': 'Hence the fiducial metric in ([REF]) is nothing but the Minkowski metric in the coordinate system defined by the Stuckelberg fields.', '1504.02295-1-6-6': 'So we have automatically defined the covariant tensor [MATH] which propagates on Minkowski space and the action is then a functional of the fiducial metric and the physical metric [MATH].', '1504.02295-1-7-0': 'The covariant action for massive general relativity that we are going to work with, can be written as: [EQUATION] where [MATH] is a potential without derivatives in the interaction terms between [MATH] and [MATH] that gives mass to the spin-2 mode described by the Einstein-Hilbert term.', '1504.02295-1-7-1': 'As observed in [CITATION] a necessary condition for the theory ([REF]) to be free of the Bouware-Deser ghost in the decoupling limit is that [MATH] be a total derivative.', '1504.02295-1-7-2': 'The most general covariant mass term which respects this condition is composed by: [EQUATION] where [MATH] are constants and the three lagrangians in ([REF]) are written as: [EQUATION] where one has defined the tensor [MATH].', '1504.02295-1-7-3': 'In general there are other polynomial terms in [MATH] and the procedure to generate it can be found in [CITATION], however it has been shown that all terms after the quartic order vanishes (see an explanation for this in [CITATION]).', '1504.02295-1-7-4': 'Notice also that, contrary to most papers in this subject, we have maintained the constant term [MATH] in order to better study its consequences on the evolution equations.', '1504.02295-1-8-0': '# Cosmology of massive gravity', '1504.02295-1-9-0': 'Let us begin by considering an open [MATH], homogeneous and isotropic FRW universe for the physical metric: [EQUATION] where, [MATH] and [MATH], with [MATH], [MATH], [MATH], [MATH].', '1504.02295-1-9-1': 'Adopting the same ansatz for the Stuckelberg fields used in [CITATION], i.e: [EQUATION] after plugging back the metric and ([REF]) in ([REF]), one obtains the following Lagrangian [CITATION] for [MATH] and [MATH], where overdot will denote the time derivative: [EQUATION] where [EQUATION]', '1504.02295-1-9-2': 'The matter content is assumed to be of the form [MATH].', '1504.02295-1-9-3': 'Taking the Euler-Lagrange equation of [MATH] with respect to [MATH] leads to [EQUATION] where [MATH].', '1504.02295-1-9-4': 'The two interesting solutions are: [EQUATION] where [MATH].', '1504.02295-1-9-5': 'We take Euler-Lagrange equation of ([REF]) with respect to [MATH] and use ([REF]) to obtain the Friedmann equation: [EQUATION] where [MATH] and [EQUATION] is a dimensionless parameter depending only on the constants [MATH], [MATH] and [MATH].', '1504.02295-1-9-6': 'In this equation one can recognize [MATH] as being the energy density of massive gravity ([MATH]).', '1504.02295-1-9-7': 'Equally, it is expected to find the pressure term ([MATH]) in the second Friedmann equation.', '1504.02295-1-10-0': 'Such equation can be obtained by combining Eq. ([REF]) with the variation of Eq. ([REF]) with respect to [MATH].', '1504.02295-1-10-1': 'Namely, we subtract Eq. ([REF]) from the Euler-Lagrange equation of [MATH] in order to get [EQUATION] where [MATH].', '1504.02295-1-10-2': 'Notice that the r.h.s of the above equation contains only the contribution from the matter part, which indicates that the graviton mass contribution satisfies an equation of state of the form [MATH], exactly as a vacuum behaviour.', '1504.02295-1-10-3': 'This same result was observed in [CITATION].', '1504.02295-1-11-0': 'Another combination is possible in order to get a direct relation among [MATH], pressure and energy density of matter and graviton.', '1504.02295-1-11-1': 'At this time we eliminate [MATH] of the expression from variation of Eq. ([REF]) with respect to [MATH].', '1504.02295-1-11-2': 'Thus, it is possible to get [EQUATION]', '1504.02295-1-11-3': 'With such equation it is much easier to analyse the universe acceleration.', '1504.02295-1-11-4': 'We conclude that an accelerated expansion occurs when [MATH].', '1504.02295-1-12-0': 'It is also easy to see that [MATH] acts exactly like an effective cosmological constant in Eq. ([REF]).', '1504.02295-1-12-1': 'In both Friedmann equations, ([REF]), ([REF]) and also ([REF]), there should be set [MATH] in order to reproduce a cosmological scenario.', '1504.02295-1-12-2': 'In order to reproduce a positive cosmological constant (which leads to an accelerating universe), we must have [MATH], which implies [MATH].', '1504.02295-1-12-3': 'We have kept the [MATH] parameter in order to analyse its consequences on the evolution equations.', '1504.02295-1-13-0': 'Taking the limit [MATH] into ([REF]) leads to an interesting relation between the parameters [MATH] and [MATH] in order to have a positive effective cosmological constant, which is a minimal condition to have an accelerated expansion.', '1504.02295-1-13-1': 'We must have [MATH], and the only possibility is [MATH], so that [MATH].', '1504.02295-1-13-2': 'It is interesting to notice that the case [MATH] is forbidden.', '1504.02295-1-13-3': 'The conditions are represented in the following table:', '1504.02295-1-14-0': 'Having analysed the effect of the [MATH] parameter in the evolution equation, from now on we will assume [MATH] according to the original dRGT theory [CITATION].', '1504.02295-1-14-1': 'The Friedmann equation ([REF]) can be rewritten in terms of the present critical energy density [MATH], [EQUATION] where [MATH] is the Hubble parameter and [MATH] km/s/Mpc is its present day value.', '1504.02295-1-14-2': 'Writing [MATH], where [MATH] is the present day value for the matter energy density, and introducing the density parameters [EQUATION] the Friedmann equation ([REF]) can be expressed as [EQUATION] or in terms of the redshift parameter, defined by [MATH], [EQUATION] where we have used the Friedmann constraint [EQUATION] that follows from ([REF]) and ([REF]).', '1504.02295-1-14-3': 'Observational data can be used to constrain the values of such parameters and this will be done in the next section.', '1504.02295-1-15-0': '# Constraints from Observational [MATH] Data', '1504.02295-1-16-0': 'Observational [MATH] data provide one of the most straightforward and model independent tests of cosmological models, as [MATH] data estimation relies on astrophysical rather than cosmological assumptions.', '1504.02295-1-16-1': 'In this work, we use the data compilation of [MATH] from Sharov and Vorontsova [CITATION], which is, currently, the most complete compilation, with 34 measurements.', '1504.02295-1-17-0': 'From these data, we perform a [MATH]-statistics, generating the [MATH] function of free parameters: [EQUATION] where [MATH] and [MATH] is obtained by Eq. ([REF]).', '1504.02295-1-18-0': 'As the function to be fitted, [MATH], is linear on the Hubble constant, [MATH], we may analytically project over [MATH], yielding [MATH]: [EQUATION] where [MATH], [MATH], [MATH] and [MATH].', '1504.02295-1-19-0': 'The result of such analysis can be seen on Figure [REF].', '1504.02295-1-19-1': 'As can be seen, the results from [MATH] data alone yield nice constraints on the plane [MATH] - [MATH].', '1504.02295-1-19-2': 'The flatness limit, which corresponds to [MATH], can be seen as an straight line on this plane (dashed line on Fig. [REF]).', '1504.02295-1-19-3': 'Points on and above this line were not considered, as they correspond to non-open models.', '1504.02295-1-19-4': 'One may see that the best fit relies right below this line, indicating that [MATH] data alone favour a slightly open Universe.', '1504.02295-1-20-0': 'Furthermore, we have considered the prior [MATH], with the baryon density parameter, [MATH], estimated by Planck and WMAP: [MATH] [CITATION], a value which is in agreement with Big Bang Nucleosynthesis (BBN), as shown on Ref. [CITATION].', '1504.02295-1-20-1': 'As a result of this prior, the 3[MATH] c.l. contour alone is cut for low matter density parameter, as we may see on Fig. [REF].', '1504.02295-1-21-0': 'The minimum [MATH] was [MATH], yielding a [MATH] per degree of freedom [MATH].', '1504.02295-1-21-1': 'The best fit parameters were [MATH], [MATH], for 68.3%, 95.4% and 99.7% c.l., respectively, in the joint analysis.', '1504.02295-1-22-0': 'As expected, this result is in agreement with [MATH]CDM constraints, as this model mimics the concordance model.', '1504.02295-1-22-1': 'Moreover, the best fit values of [MATH] and [MATH] leads to [MATH], which corresponds to a negative value of [MATH], as expected for this model.', '1504.02295-1-22-2': 'Sharov and Vorontsova [CITATION] have found, for [MATH]CDM: [MATH], [MATH], for 1[MATH] c.l., where they have combined [MATH] with SN Ia and BAO data.', '1504.02295-1-22-3': 'Given the uncertainties on massive gravity parameters above, the results are in good agreement, even considering the open Universe restriction for massive gravity, while [MATH]CDM has no restriction on curvature.', '1504.02295-1-23-0': '# Bounds on the graviton mass', '1504.02295-1-24-0': 'Having obtained the best fit values for the parameters, we show in Fig. [REF] the plot of Massive Gravity theory (red line) with [MATH] limit (red dotted line).', '1504.02295-1-24-1': 'The [MATH]CDM according to best fit data of Sharov and Vorontsova [CITATION] are also represented (black line).', '1504.02295-1-25-0': 'This model also gives an expression to the graviton mass depending on the [MATH] and [MATH] parameters through [MATH]: [EQUATION]', '1504.02295-1-25-1': 'If we fix some of the parameters, we can see how the mass depends on the others.', '1504.02295-1-25-2': 'In Figure [REF] we show a typical mass dependence with [MATH] when we set the [MATH] as a constant and we choose to work with [MATH].', '1504.02295-1-25-3': 'Such behaviour is also observed for others positive and negative values of [MATH].', '1504.02295-1-25-4': 'It is easy to see that the mass increases and diverges for some specific value of [MATH], corresponding to the limit [MATH].', '1504.02295-1-25-5': 'In some cases the mass can also abruptly decrease to zero, as shown in the cases [MATH], [MATH] and [MATH].', '1504.02295-1-25-6': "This shows that the graviton mass is strongly dependent on the [MATH]'s parameters, although in the limit of very negative [MATH] values the graviton mass goes to [MATH].", '1504.02295-1-26-0': '# Conclusion', '1504.02295-1-27-0': 'In this work we have analysed an open FLRW solution of massive gravity which admits accelerated expansion of the universe, in full concordance to observations.', '1504.02295-1-27-1': 'The constraints with recent observational [MATH] data were found and the best fit values obtained for the cosmological parameters are [MATH], in accordance with the [MATH]CDM model, and also point to a nearly open curvature [MATH], for which the model is valid.', '1504.02295-1-27-2': 'The graviton mass dependence with the constant parameters [MATH] and [MATH] of the model were also analysed, and we have verified a strong dependence with such parameters.', '1504.02295-1-27-3': 'We have also obtained that the condition [MATH] seems dominant for a long range of the parameters [MATH] and [MATH], although cosmological observations cannot be used to determine such parameters.', '1504.02295-1-27-4': 'The study of the positivity of the cosmological constant like term, [MATH], when the constant [MATH] indicates a complete supression of solutions for the case [MATH] and admits solutions only if [MATH] for the case [MATH].', '1504.02295-1-27-5': 'This indicates the necessity of a [MATH] term in the construction of the model, at least for the [MATH] branch, in order to get a positive cosmological constant like term.', '1504.02295-1-28-0': 'SHP is grateful to CNPq, process number 477872/2010-7.', '1504.02295-1-28-1': 'E.L.M thanks CNPq 449806/2014-6.', '1504.02295-1-28-2': 'APSS thanks CAPES - Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior, for financial support.', '1504.02295-1-28-3': 'JFJ is grateful to Unesp - Campus de Guaratingueta for hospitality and facilities.'}

{'1504.02295-2-0-0': 'In this work we have analysed some cosmological bounds concerning an open FLRW solution of massive gravity.', '1504.02295-2-0-1': 'The constraints with recent observational [MATH] data were found and the best fit values for the cosmological parameters are in agreement with the [MATH]CDM model, and also point to a nearly open spatial curvature, as expected from the model.', '1504.02295-2-0-2': 'The graviton mass dependence with the constant parameters [MATH] and [MATH], related to the additional lagrangians terms of the model, are also analysed, and we have obtained a strong dependence with such parameters, although the condition [MATH] seems dominant for a long range of the parameters [MATH] and [MATH].', '1504.02295-2-1-0': '# Introduction', '1504.02295-2-2-0': 'Current observations of Supernovae type Ia (SNIa) [CITATION], Cosmic Microwave Background (CMB) radiation [CITATION] and Hubble parameter data [CITATION] indicate an accelerated expansion of the universe, being the [MATH]CDM model the best model to fit the observational data.', '1504.02295-2-2-1': 'The [MATH] term corresponds to a cosmological constant energy density which is plagued with several fundamental issues [CITATION], which has motivated the search for alternatives models of gravity that could explain the observations.', '1504.02295-2-3-0': 'Massive gravity theories [CITATION] are old candidates to explain the accelerated expansion of the universe, since that the graviton mass could perfectly induce and mimic a cosmological constant term.', '1504.02295-2-3-1': 'However, such kinds of theories were considered for long time as being unsuitable due to the appearing of Boulware-Deser (BD) ghosts [CITATION].', '1504.02295-2-3-2': 'Recently it was discovered a nonlinear massive gravity theory that was first shown to be BD ghost free by Hassan et al. [CITATION], and also in the Stuckelberg formulation in [CITATION].', '1504.02295-2-3-3': 'Then such theory was also developed by de Rham et al.[CITATION], sometimes called dRGT (de Rham-Gabadadze-Tolley) model (see [CITATION] for a review), bringing back the cosmological interest for such theories [CITATION].', '1504.02295-2-3-4': 'Self-accelerating cosmologies with ghost-free massive gravitons have been studied thereafter [CITATION].', '1504.02295-2-3-5': 'Nowadays there is a general agreement that in dRGT models which are defined with a flat reference metric, isotropic flat and closed FLRW cosmologies do not exist, even at the background level.', '1504.02295-2-3-6': 'Nevertheless isotropic open cosmologies exist as classical solutions but have unstable perturbations.', '1504.02295-2-3-7': 'In the case of a non-flat reference metric a ghost free theory exist for all types of background cosmologies [CITATION], though the perturbations are still unstable in isotropic cases [CITATION].', '1504.02295-2-4-0': 'Although the theoretical aspects of massive gravity has been severely studied in the last years, the cosmological constraints with observational data does not.', '1504.02295-2-4-1': 'In [CITATION] has been investigated the cosmological behavior in the quasi-Dilaton nonlinear massive gravity, and the parameters of the theory has been constrained with observational data from SNIa, Baryon Acoustic Oscillations (BAO) and CMB.', '1504.02295-2-5-0': 'In order to study some cosmological bounds concerning the free parameters of the theory, we must have a massive gravity theory that has a FLRW limit well established for the reference metric.', '1504.02295-2-5-1': 'As showed in [CITATION] the perturbations in unisotropic FLRW are stable, thus if we suppose that the unisotropies that cure the instability of the model do not greatly affect the cosmic evolution, we can use the results involving isotropic solutions as a first approximation to the stable unisotropic case.', '1504.02295-2-5-2': 'For this case the results of [CITATION] are a good starting point to study some bounds in the massive gravity theory compared to recent observational data.', '1504.02295-2-5-3': 'In Section II we present the general massive gravity theory and the cosmological equations in Section III.', '1504.02295-2-5-4': 'In Section IV we present the constraints from [MATH] data and in Section V some bounds on the graviton mass are presented.', '1504.02295-2-5-5': 'We conclude in Section VI.', '1504.02295-2-6-0': '# Massive gravity theory', '1504.02295-2-7-0': 'Our starting point for the cosmological analysis is the massive theory for gravity proposed in [CITATION].', '1504.02295-2-7-1': 'The nonlinear action besides a functional of the physical metric [MATH] include four spurious scalar fields [MATH] with [MATH] called the Stuckelberg fields.', '1504.02295-2-7-2': 'They are introduced in order to make the action manifestly invariant under diffeomorphism, see for example [CITATION].', '1504.02295-2-7-3': 'Let us start by observing that these scalar fields are related to the physical metric and enter into the action as follows: [EQUATION] where it is defined the fiducial metric [MATH] which is written in terms of the Stuckelberg fields: [EQUATION]', '1504.02295-2-7-4': 'Usually [MATH] is called the reference metric and for the purpose of this paper, one can use [MATH] once the scheme proposed by dRGT respects Poincare symmetry.', '1504.02295-2-7-5': 'Hence the fiducial metric in ([REF]) is nothing but the Minkowski metric in the coordinate system defined by the Stuckelberg fields.', '1504.02295-2-7-6': 'So we have automatically defined the covariant tensor [MATH] which propagates on Minkowski space and the action is then a functional of the fiducial metric and the physical metric [MATH].', '1504.02295-2-8-0': 'The covariant action for massive general relativity that we are going to work with, can be written as: [EQUATION] where [MATH] is a potential without derivatives in the interaction terms between [MATH] and [MATH] that gives mass to the spin-2 mode described by the Einstein-Hilbert term.', '1504.02295-2-8-1': 'As observed in [CITATION] a necessary condition for the theory ([REF]) to be free of the Bouware-Deser ghost in the decoupling limit is that [MATH] be a total derivative.', '1504.02295-2-8-2': 'The most general covariant mass term which respects this condition is composed by: [EQUATION] where [MATH] are constants and the three lagrangians in ([REF]) are written as: [EQUATION] where one has defined the tensor [MATH].', '1504.02295-2-8-3': 'In general there are other polynomial terms in [MATH] and the procedure to generate it can be found in [CITATION], however it has been shown that all terms after the quartic order vanishes (see also [CITATION]).', '1504.02295-2-8-4': 'Notice also that we have maintained the constant term [MATH] in order to see its consequences on the evolution equations.', '1504.02295-2-8-5': 'The case [MATH] is pathological since in this case the linearised theory and the non-linear one have different number of propagating modes.', '1504.02295-2-9-0': '# Cosmology of massive gravity', '1504.02295-2-10-0': 'Let us begin by considering an open [MATH], homogeneous and isotropic FRW universe for the physical metric: [EQUATION] where, [MATH] and [MATH], with [MATH], [MATH], [MATH], [MATH].', '1504.02295-2-10-1': 'Adopting the same ansatz for the Stuckelberg fields used in [CITATION], i.e: [EQUATION] after plugging back the metric and ([REF]) in ([REF]), one obtains the following Lagrangian [CITATION] for [MATH] and [MATH], where overdot will denote the time derivative: [EQUATION] where [EQUATION]', '1504.02295-2-10-2': 'The matter content is assumed to be of the form [MATH].', '1504.02295-2-10-3': 'Taking the Euler-Lagrange equation of [MATH] with respect to [MATH] leads to [EQUATION] where [MATH].', '1504.02295-2-10-4': 'The two interesting solutions are: [EQUATION] where [MATH].', '1504.02295-2-10-5': 'We take Euler-Lagrange equation of ([REF]) with respect to [MATH] and use ([REF]) to obtain the Friedmann equation: [EQUATION] where [MATH] and [EQUATION] is a dimensionless parameter depending only on the constants [MATH], [MATH] and [MATH].', '1504.02295-2-10-6': 'In this equation one can recognize [MATH] as being the energy density of massive gravity ([MATH]).', '1504.02295-2-10-7': 'Equally, it is expected to find the pressure term ([MATH]) in the second Friedmann equation.', '1504.02295-2-11-0': 'Such equation can be obtained by combining Eq. ([REF]) with the variation of Eq. ([REF]) with respect to [MATH].', '1504.02295-2-11-1': 'Namely, we subtract Eq. ([REF]) from the Euler-Lagrange equation of [MATH] in order to get [EQUATION] where [MATH].', '1504.02295-2-11-2': 'Notice that the r.h.s of the above equation contains only the contribution from the matter part, which indicates that the graviton mass contribution satisfies an equation of state of the form [MATH], exactly as a vacuum behaviour.', '1504.02295-2-11-3': 'This same result was observed in [CITATION].', '1504.02295-2-12-0': 'Another combination is possible in order to get a direct relation among [MATH], pressure and energy density of matter and graviton.', '1504.02295-2-12-1': 'At this time we eliminate [MATH] of the expression from variation of Eq. ([REF]) with respect to [MATH].', '1504.02295-2-12-2': 'Thus, it is possible to get [EQUATION]', '1504.02295-2-12-3': 'With such equation it is much easier to analyse the universe acceleration.', '1504.02295-2-12-4': 'We conclude that an accelerated expansion occurs when [MATH].', '1504.02295-2-13-0': 'It is also easy to see that [MATH] acts exactly like an effective cosmological constant in Eq. ([REF]).', '1504.02295-2-13-1': 'In both Friedmann equations, ([REF]), ([REF]) and also ([REF]), there should be set [MATH] in order to reproduce a cosmological scenario.', '1504.02295-2-13-2': 'In order to reproduce a positive cosmological constant (which leads to an accelerating universe), we must have [MATH], which implies [MATH].', '1504.02295-2-14-0': 'From now on we will assume [MATH] according to the original dRGT theory [CITATION].', '1504.02295-2-14-1': 'The Friedmann equation ([REF]) can be rewritten in terms of the present critical energy density [MATH], [EQUATION] where [MATH] is the Hubble parameter and [MATH] km/s/Mpc is its present day value.', '1504.02295-2-14-2': 'Writing [MATH], where [MATH] is the present day value for the matter energy density, and introducing the density parameters [EQUATION] the Friedmann equation ([REF]) can be expressed as [EQUATION] or in terms of the redshift parameter, defined by [MATH], [EQUATION] where we have used the Friedmann constraint [EQUATION] that follows from ([REF]) and ([REF]).', '1504.02295-2-14-3': 'Observational data can be used to constrain the values of such parameters and this will be done in the next section.', '1504.02295-2-15-0': '# Constraints from Observational [MATH] Data', '1504.02295-2-16-0': 'Observational [MATH] data provide one of the most straightforward and model independent tests of cosmological models, as [MATH] data estimation relies on astrophysical rather than cosmological assumptions.', '1504.02295-2-16-1': 'In this work, we use the data compilation of [MATH] from Sharov and Vorontsova [CITATION], which is, currently, the most complete compilation, with 34 measurements.', '1504.02295-2-17-0': 'From these data, we perform a [MATH]-statistics, generating the [MATH] function of free parameters: [EQUATION] where [MATH] and [MATH] is obtained by Eq. ([REF]).', '1504.02295-2-18-0': 'As the function to be fitted, [MATH], is linear on the Hubble constant, [MATH], we may analytically project over [MATH], yielding [MATH]: [EQUATION] where [MATH], [MATH], [MATH] and [MATH].', '1504.02295-2-19-0': 'The result of such analysis can be seen on Figure [REF].', '1504.02295-2-19-1': 'As can be seen, the results from [MATH] data alone yield nice constraints on the plane [MATH] - [MATH].', '1504.02295-2-19-2': 'The flatness limit, which corresponds to [MATH], can be seen as an straight line on this plane (dashed line on Fig. [REF]).', '1504.02295-2-19-3': 'Points on and above this line were not considered, as they correspond to non-open models.', '1504.02295-2-19-4': 'One may see that the best fit relies right below this line, indicating that [MATH] data alone favour a slightly open Universe.', '1504.02295-2-20-0': 'Furthermore, we have considered the prior [MATH], with the baryon density parameter, [MATH], estimated by Planck and WMAP: [MATH] [CITATION], a value which is in agreement with Big Bang Nucleosynthesis (BBN), as shown on Ref. [CITATION].', '1504.02295-2-20-1': 'As a result of this prior, the 3[MATH] c.l. contour alone is cut for low matter density parameter, as we may see on Fig. [REF].', '1504.02295-2-21-0': 'The minimum [MATH] was [MATH], yielding a [MATH] per degree of freedom [MATH].', '1504.02295-2-21-1': 'The best fit parameters were [MATH], [MATH], for 68.3%, 95.4% and 99.7% c.l., respectively, in the joint analysis.', '1504.02295-2-22-0': 'As expected, this result is in agreement with [MATH]CDM constraints, as this model mimics the concordance model.', '1504.02295-2-22-1': 'Moreover, the best fit values of [MATH] and [MATH] leads to [MATH], which corresponds to a negative value of [MATH], as expected for this model.', '1504.02295-2-22-2': 'Sharov and Vorontsova [CITATION] have found, for [MATH]CDM: [MATH], [MATH], for 1[MATH] c.l., where they have combined [MATH] with SN Ia and BAO data.', '1504.02295-2-22-3': 'Given the uncertainties on massive gravity parameters above, the results are in good agreement, even considering the open Universe restriction for massive gravity, while [MATH]CDM has no restriction on curvature.', '1504.02295-2-23-0': '# Bounds on the graviton mass', '1504.02295-2-24-0': 'Having obtained the best fit values for the parameters, we show in Fig. [REF] the plot of Massive Gravity theory (red line) with [MATH] limit (red dotted line).', '1504.02295-2-24-1': 'The [MATH]CDM according to best fit data of Sharov and Vorontsova [CITATION] are also represented (black line).', '1504.02295-2-25-0': 'This model also gives an expression to the graviton mass depending on the [MATH] and [MATH] parameters through [MATH]: [EQUATION]', '1504.02295-2-25-1': 'If we fix some of the parameters, we can see how the mass depends on the others.', '1504.02295-2-25-2': 'In Figure [REF] we show a typical mass dependence with [MATH] when we set the [MATH] as a constant and we choose to work with [MATH].', '1504.02295-2-25-3': 'Such behaviour is also observed for others positive and negative values of [MATH].', '1504.02295-2-25-4': 'It is easy to see that the mass increases and diverges for some specific value of [MATH], corresponding to the limit [MATH].', '1504.02295-2-25-5': 'In some cases the mass can also abruptly decrease to zero, as shown in the cases [MATH], [MATH] and [MATH].', '1504.02295-2-25-6': "This shows that the graviton mass is strongly dependent on the [MATH]'s parameters, although in the limit of very negative [MATH] values the graviton mass goes to [MATH].", '1504.02295-2-26-0': '# Conclusion', '1504.02295-2-27-0': 'In this work we have analysed an open FLRW solution of massive gravity which admits accelerated expansion of the universe, in full concordance to observations.', '1504.02295-2-27-1': 'The constraints with recent observational [MATH] data were found and the best fit values obtained for the cosmological parameters are [MATH], in accordance with the [MATH]CDM model, and also point to a nearly open curvature [MATH], for which the model is valid.', '1504.02295-2-27-2': 'The graviton mass dependence with the constant parameters [MATH] and [MATH] of the model were also analysed, and we have verified a strong dependence with such parameters.', '1504.02295-2-27-3': 'We have also obtained that the condition [MATH] seems dominant for a long range of the parameters [MATH] and [MATH], although cosmological observations cannot be used to determine such parameters.', '1504.02295-2-28-0': 'SHP is grateful to CNPq, process number 477872/2010-7.', '1504.02295-2-28-1': 'E.L.M thanks CNPq 449806/2014-6.', '1504.02295-2-28-2': 'APSS thanks CAPES - Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior, for financial support.', '1504.02295-2-28-3': 'JFJ is grateful to Unesp - Campus de Guaratingueta for hospitality and facilities.'}

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[]

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[]

['1504.02295-1-13-3', '1504.02295-1-18-0', '1504.02295-1-28-0', '1504.02295-1-28-1', '1504.02295-2-18-0', '1504.02295-2-28-0', '1504.02295-2-28-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1504.02295

1406.3521

{'1406.3521-1-0-0': 'Linear mixed-effect models with two variance components are often used when variability comes from two sources.', '1406.3521-1-0-1': 'In genetics applications, variation in observed traits can be attributed to biological and environmental effects, and the heritability coefficient is a fundamental quantity that measures the proportion of total variability due to the biological effect.', '1406.3521-1-0-2': 'We propose an inferential model approach which yields exact prior-free probabilistic inference on the heritability coefficient.', '1406.3521-1-0-3': "In particular we construct exact confidence intervals and demonstrate numerically our method's efficiency compared to that of existing methods.", '1406.3521-1-1-0': 'Keywords and phrases: Differential equation; inferential model; plausibility; unbalanced design; variance components.', '1406.3521-1-2-0': '# Introduction', '1406.3521-1-3-0': 'Normal linear mixed effects models are useful in a variety of biological, physical, and social scientific applications with variability coming from multiple sources.', '1406.3521-1-3-1': '[CITATION] provide a survey on linear mixed effects models, and [CITATION] is a classic book on the subject.', '1406.3521-1-3-2': 'In this paper, we focus on the case of two variance components, and the general model can be written as [EQUATION] where [MATH] is a [MATH]-vector of response variables, [MATH] and [MATH] are design matrices for the fixed and random effects, of dimension [MATH] and [MATH], respectively, [MATH] is a [MATH]-vector of unknown parameters, [MATH] is a normal random [MATH]-vector with mean 0 and covariance matrix [MATH], and [MATH] is a normal random [MATH]-vector with mean 0 and covariance matrix [MATH].', '1406.3521-1-3-3': 'Here, [MATH] is the pair of variance components.', '1406.3521-1-3-4': 'The matrix [MATH] is known, and [MATH] is of rank [MATH].', '1406.3521-1-3-5': 'The unknown parameters in this model are the [MATH] fixed-effect coefficients [MATH] and the two variance components [MATH], so the parameter space is [MATH]-dimensional.', '1406.3521-1-4-0': 'In biological applications, the quantities [MATH] and [MATH] in [REF] denote the genetic and environmental effects, respectively.', '1406.3521-1-4-1': 'Given that "a central question in biology is whether observed variation in a particular trait is due to environmental or biological factors" , the heritability coefficient, [MATH], which represents the proportion of phenotypic variance attributed to variation in genotypic values, is a fundamentally important quantity.', '1406.3521-1-4-2': 'Indeed, mixed-effect models and inference on the heritability coefficient has been applied recently in genome-wide association studies ; see Section [REF] for more on these applications.', '1406.3521-1-5-0': 'Given the importance of the heritability coefficient, there are a number of methods available to construct confidence intervals for [MATH] or, equivalently, for the variance ratio [MATH].', '1406.3521-1-5-1': 'When the design is balanced, [CITATION] and [CITATION] give a confidence intervals for [MATH] and other quantities.', '1406.3521-1-5-2': 'When the design is possibly unbalanced, as we assume here, the problem is more challenging; in particular, exact ANOVA-based confidence intervals often are not available.', '1406.3521-1-5-3': '[CITATION] gave intervals for [MATH] in the unbalanced case, and subsequent contributions include [CITATION], [CITATION], [CITATION], and [CITATION].', '1406.3521-1-5-4': 'Bayesian and fiducial methods are also available.', '1406.3521-1-6-0': 'The focus in this paper is exact prior-free probabilistic inference on the heritability coefficient based on the recently proposed inferential model (IM) framework.', '1406.3521-1-6-1': '[CITATION] give a detailed account of the general framework, along with comparisons to other related approaches, including fiducial .', '1406.3521-1-6-2': 'The IM approach is driven by the specification of an association between unobservable auxiliary variables, the observable data, and the unknown parameters.', '1406.3521-1-6-3': 'This association is followed up by using properly calibrated random sets to predict these auxiliary variables.', '1406.3521-1-6-4': '[CITATION] argue that it is precisely this prediction of the auxiliary variables that sets the IM approach apart from fiducial, Dempster-Shafer , and structural inference .', '1406.3521-1-6-5': 'The IM output is a plausibility function that can be used for inference on the parameter of interest.', '1406.3521-1-6-6': 'In particular, plausibility function often yield confidence intervals with exact coverage .', '1406.3521-1-6-7': 'A brief overview of the general IM framework is given in Section [REF].', '1406.3521-1-6-8': "A key feature of the IM approach, and the motivation for this work, is that, besides being exact, the IM approach's careful handling of uncertainty generally leads to more efficient inference.", '1406.3521-1-7-0': 'A key to the construction of an efficient IM is to reduce the dimension of the auxiliary variable as much as possible and, for the heritability coefficient, there are several dimension reduction steps.', '1406.3521-1-7-1': 'A first dimension reduction eliminates the nuisance parameter [MATH].', '1406.3521-1-7-2': 'These well-known calculations are reviewed in Section [REF].', '1406.3521-1-7-3': 'The main technical contribution here is in Section [REF], where we employ a differential equation-driven technique to reduce the auxiliary variable dimension, leading to a conditional IM for the heritability coefficient.', '1406.3521-1-7-4': 'A predictive random set is introduced in Section [REF], and in Section [REF] we show that the corresponding plausibility function is properly calibrated and, therefore, the plausibility function-based confidence intervals are provably exact.', '1406.3521-1-7-5': 'Sections [REF] and [REF] illustrate the proposed method in simulated and real data examples.', '1406.3521-1-7-6': 'The general message is that our proposed confidence intervals for the heritability coefficient are exact and tend to be more efficient compared to existing methods.', '1406.3521-1-8-0': '# Preliminaries', '1406.3521-1-9-0': '## Overview of inferential models', '1406.3521-1-10-0': 'The goal of the IM framework is to get valid prior-free probabilistic inference.', '1406.3521-1-10-1': 'This is facilitated by first associating the observable data and unknown parameters to a set of unobservable auxiliary variables.', '1406.3521-1-10-2': 'For example, the marginal distribution of [MATH] from [REF] is [EQUATION] which can be written in association form: [EQUATION]', '1406.3521-1-10-3': 'That is, observable data [MATH] and unknown parameters [MATH] are associated with unobservable auxiliary variables [MATH], in this case, a [MATH]-vector of independent standard normal random variables.', '1406.3521-1-10-4': 'Given this association, the basic IM approach is to introduce a predictive random set for [MATH] and combine with the observed value of [MATH] to get a plausibility function.', '1406.3521-1-10-5': 'Roughly, this plausibility function assigns, to assertions about the parameter of interest, values in [MATH] measuring the plausibility that the assertion is true.', '1406.3521-1-10-6': 'This plausibility function can be used to design exact frequentist tests or confidence regions.', '1406.3521-1-11-0': '[CITATION] give a general description of this three-step IM construction and its properties; in Section [REF] below we will flesh out these three steps, including choice of a predictive random set, for the variance components problem at hand.', '1406.3521-1-11-1': 'The construction of exact plausibility intervals for the heritability coefficient is presented in Section [REF], along with a theoretical justification of the claimed exactness.', '1406.3521-1-12-0': 'Notice, in the association [REF], that the auxiliary variable [MATH] is, in general, of higher dimension than that of the parameter [MATH].', '1406.3521-1-12-1': 'There are two reasons for trying to reduce the auxiliary variable dimension.', '1406.3521-1-12-2': 'First, it is much easier to specify good predictive random sets when the auxiliary variable is of low dimension.', '1406.3521-1-12-3': 'Second, inference is more efficient when only a relatively small number of auxiliary variables require prediction.', '1406.3521-1-12-4': 'This dimension reduction is via a conditioning operation, and [CITATION] give a general explanation of the gains as well as a novel technique for carrying out this reduction.', '1406.3521-1-12-5': 'We shall employ these techniques in Section [REF] to get a dimension-reduced association for the heritability coefficient in the linear mixed model.', '1406.3521-1-13-0': 'Another important challenge is when nuisance parameters are present.', '1406.3521-1-13-1': 'Our interest here is in the heritability coefficient, a function of the full parameter [MATH], so we need to marginalize over the nuisance parameters.', '1406.3521-1-13-2': 'As [CITATION] demonstrate, marginalization over nuisance parameters can often be carried out simply by deleting auxiliary variables.', '1406.3521-1-13-3': 'The next subsection gives a first marginalization over [MATH] using these techniques.', '1406.3521-1-13-4': 'Further marginalization will be needed in Section [REF].', '1406.3521-1-14-0': '## Marginalizing out the fixed effect', '1406.3521-1-15-0': 'Start with the linear mixed model [REF].', '1406.3521-1-15-1': 'Following the setup in [CITATION], let [MATH] be a [MATH] matrix such that [MATH] and [MATH].', '1406.3521-1-15-2': 'Next, let [MATH].', '1406.3521-1-15-3': 'Then [MATH] is a one-to-one mapping.', '1406.3521-1-15-4': 'Moreover, the distribution of [MATH] depends on [MATH] only, and the distribution of [MATH] depends on [MATH], with [MATH] a location parameter.', '1406.3521-1-15-5': 'In particular, from [REF], we get [EQUATION] where [MATH] and [MATH] is a [MATH] covariance matrix of a known form that depends on [MATH]; its precise form is not important.', '1406.3521-1-15-6': 'From the distributional statement in the previous display, we can extract an association: [EQUATION].', '1406.3521-1-15-7': 'For any [MATH], there exists a [MATH] such that the above equation holds; so, the model is "regular" in the sense of [CITATION].', '1406.3521-1-15-8': 'According to their theory, regularity allows us to marginalize over [MATH] by deleting the [MATH] component in the transformation [MATH].', '1406.3521-1-15-9': 'Therefore, a marginal association for [MATH], free of the nuisance parameter [MATH], is given by [EQUATION].', '1406.3521-1-15-10': 'This marginalization reduces the auxiliary variable dimension from [MATH] to [MATH].', '1406.3521-1-16-0': 'In the marginal association for [MATH] above, there are [MATH] auxiliary variables but only two parameters.', '1406.3521-1-16-1': 'Since the goal is to reduce the dimension of the auxiliary variable as much as possible, we should keep pushing.', '1406.3521-1-16-2': 'Classical results on sufficient statistics in mixed effects model that will facilitate further dimension reduction.', '1406.3521-1-16-3': 'For the matrix [MATH] defined above, let [MATH] denote the (distinct) ordered eigenvalues with multiplicities [MATH], respectively.', '1406.3521-1-16-4': 'Let [MATH] be a [MATH] orthogonal matrix such that [MATH] is diagonal with eigenvalues [MATH], in their multiplicities, on the diagonal.', '1406.3521-1-16-5': 'For [MATH], a [MATH] matrix, define [EQUATION] [CITATION] showed that [MATH] is a minimal sufficient statistic for [MATH].', '1406.3521-1-16-6': 'Moreover, the distribution of [MATH] is determined by [EQUATION]', '1406.3521-1-16-7': 'This changes the auxiliary variable dimension from [MATH] to [MATH].', '1406.3521-1-16-8': 'We take [REF] as our "baseline association."', '1406.3521-1-16-9': 'In this baseline association, however, there are [MATH] auxiliary variables but only two parameters; this means there is room for even further dimension reduction.', '1406.3521-1-16-10': 'In particular, our interest is in the scalar heritability coefficient, and the next section shows how to reduce to a scalar auxiliary variable.', '1406.3521-1-17-0': '# Inferential model for heritability', '1406.3521-1-18-0': '## Association', '1406.3521-1-19-0': 'For the moment, it will be convenient to work with the variance ratio, [MATH].', '1406.3521-1-19-1': 'Since [MATH] is a one-to-one function of [MATH], the two parametrizations are equivalent.', '1406.3521-1-19-2': 'Rewrite the baseline association [REF] as [EQUATION]', '1406.3521-1-19-3': 'If we make the following transformations, [EQUATION] then the association [REF] becomes [EQUATION].', '1406.3521-1-19-4': 'Since for every [MATH], there exists a [MATH] that solves the right-most equation, it follows from the general theory in [CITATION] that a marginal association for [MATH] is obtained by deleting the component above involving [MATH].', '1406.3521-1-19-5': 'In particular, a marginal association for [MATH] is [EQUATION].', '1406.3521-1-19-6': 'If we write [EQUATION] then we get a marginal association for [MATH] of the form [EQUATION]', '1406.3521-1-19-7': 'Marginalization reduces the auxiliary variable dimension by 1.', '1406.3521-1-19-8': 'Further dimension reduction will be considered next.', '1406.3521-1-19-9': 'Note that the new auxiliary variable [MATH] is a multivariate F-distributed random vector .', '1406.3521-1-20-0': 'Here we construct a local conditional IM for [MATH] as described in [CITATION].', '1406.3521-1-20-1': 'Select a fixed value [MATH]; more on this choice in Section [REF].', '1406.3521-1-20-2': 'To reduce the dimension of the auxiliary variable [MATH], in [REF], from [MATH] to [MATH], we construct two pairs of functions-one pair, [MATH], on the sample space, and one pair, [MATH], on the auxiliary variable space.', '1406.3521-1-20-3': 'We insist that [MATH] and [MATH] are both one-to-one, and [MATH] and [MATH] are allowed to depend on the selected [MATH].', '1406.3521-1-20-4': 'The goal is to find [MATH] which is, in a certain sense, not sensitive to changes in the auxiliary variable.', '1406.3521-1-21-0': 'Write the association [REF] so that [MATH] is a function of [MATH] and [MATH], i.e., [EQUATION].', '1406.3521-1-21-1': 'We want to choose the function [MATH] such that the partial derivative of [MATH] with respect to [MATH] vanishes at [MATH].', '1406.3521-1-21-2': 'By the chain rule, we have [EQUATION] so our goal is to find [MATH] to solve the following partial differential equation: [EQUATION] here [MATH] is a [MATH] vector and [MATH] is a [MATH] matrix of rank [MATH].', '1406.3521-1-21-3': 'Toward solving the partial differential equation, first we get that the partial derivative of [MATH] with respect to [MATH] satisfies [EQUATION] where [MATH].', '1406.3521-1-21-4': 'This simplifies the relevant partial differential equation to the following: [EQUATION] where [MATH] is a diagonal matrix constructed from a vector [MATH].', '1406.3521-1-21-5': 'The method of characteristics for solving partial differential equations identifies a logarithmic function of the form [EQUATION] where [MATH] is a [MATH] matrix with rows orthogonal to [MATH] at [MATH].', '1406.3521-1-21-6': 'For example, since the matrix that projects to the orthogonal complement to the column space of [MATH] has rank [MATH], we can take [MATH] to be a matrix whose [MATH] rows form a basis for that space.', '1406.3521-1-21-7': 'For [MATH] defined in this way, it is easy to check that [MATH] in [REF] is indeed a solution to the partial differential equation [REF].', '1406.3521-1-22-0': 'We have completed specification of the [MATH] function; it remains to specify [MATH] and [MATH].', '1406.3521-1-22-1': 'The easiest to specify next is [MATH], the value of [MATH], as a function of [MATH]: [EQUATION].', '1406.3521-1-22-2': 'As we describe below, the goal is to condition on the observed value of [MATH].', '1406.3521-1-23-0': 'Next, we define [MATH] and [MATH] to supplement [MATH] and [MATH], respectively.', '1406.3521-1-23-1': 'In particular, take a [MATH]-vector [MATH] which is not orthogonal to [MATH] at [MATH].', '1406.3521-1-23-2': 'It is easy to check that the entries in [MATH] are strictly positive for all [MATH].', '1406.3521-1-23-3': 'Therefore, we can take [MATH] independent of [MATH], e.g., [MATH], is not orthogonal to [MATH].', '1406.3521-1-23-4': 'Now set [EQUATION]', '1406.3521-1-23-5': 'This is a log-linear transformation, and the linear part is non-singular, so this is a one-to-one mapping.', '1406.3521-1-23-6': 'Finally, we take [MATH] as [EQUATION].', '1406.3521-1-23-7': 'Since [MATH] is log-linear, just like [MATH], it is also one-to-one.', '1406.3521-1-24-0': 'We are now prepared to write the conditional association for [MATH].', '1406.3521-1-24-1': 'For the given [MATH], the mapping [MATH] describes a split of our previous association [REF] into two pieces: [EQUATION].', '1406.3521-1-24-2': 'The first piece carries direct information about [MATH].', '1406.3521-1-24-3': 'The second piece plays a conditioning role, correcting for the fact that some information was lost in reducing the [MATH]-dimensional [MATH] to a one-dimensional [MATH].', '1406.3521-1-24-4': 'To complete the specification of the conditional association, write [MATH] and [MATH].', '1406.3521-1-24-5': 'Then we have [EQUATION] where [MATH] is the conditional distribution of [MATH], given that [MATH] equals to the observed value [MATH] of [MATH].', '1406.3521-1-24-6': 'To summarize, [REF] completes the association step that describes the connection between observable data, unknown parameter of interest, and unobservable auxiliary variables.', '1406.3521-1-24-7': 'Of particular interest is that this association involves only a one-dimensional auxiliary variable.', '1406.3521-1-24-8': 'This dimension reduction will come in handy for the choice of predictive random set in the following section.', '1406.3521-1-24-9': 'The price we paid for this dimension reduction was the choice of a particular localization point [MATH].', '1406.3521-1-24-10': 'In Section [REF] we employ a trick to side-step this issue when the goal is, as in this paper, to construct confidence intervals.', '1406.3521-1-25-0': '## Predictive random sets', '1406.3521-1-26-0': 'Having reduced the auxiliary variable to a scalar in [REF], the choice of an efficient predictive random set is now relatively simple.', '1406.3521-1-26-1': 'Though there is an available theory of optimal predictive random sets , here we opt for simplicity; in particular, we propose a default predictive random set that is theoretically sound and computational and intuitively simple.', '1406.3521-1-27-0': 'Consider the following version of what [CITATION] call the "default" predictive random set: [EQUATION]', '1406.3521-1-27-1': 'This [MATH], with distribution [MATH], is a random interval, centered at the mean [MATH] of the conditional distribution [MATH].', '1406.3521-1-27-2': 'One key feature of this predictive random set is that it is nested, i.e., for any two distinct realizations of [MATH], one is a subset of the other.', '1406.3521-1-27-3': 'The second key feature is a calibration of the predictive random set distribution with the underlying distribution of the auxiliary variable.', '1406.3521-1-27-4': 'Following [CITATION], define the contour function [EQUATION] which represents the probability that the predictive random set contains the value [MATH] of the auxiliary variable.', '1406.3521-1-27-5': 'We shall require that [EQUATION]', '1406.3521-1-27-6': 'For the default predictive random set in [REF], it is easy to check that [EQUATION] where [MATH] is the distribution function of [MATH] for [MATH].', '1406.3521-1-27-7': 'From the construction above, it is clear that it is a continuous distribution.', '1406.3521-1-27-8': 'Then, [MATH] is a continuous random variable, so [MATH] is uniformly distributed on [MATH].', '1406.3521-1-27-9': 'Therefore, [REF] holds for the default predictive random set [MATH].', '1406.3521-1-27-10': 'Results on optimal predictive random sets are available , but here, again, our focus is on simplicity.', '1406.3521-1-27-11': 'See Section [REF].', '1406.3521-1-28-0': '## Plausibility intervals', '1406.3521-1-29-0': 'Here we combine the association in Section [REF] with the predictive random set described above to produce a plausibility function for inference about [MATH].', '1406.3521-1-29-1': 'In general, a plausibility function is a data-dependent mapping that assigns, to each assertion about the parameter, a value in [MATH], with the interpretation that small values suggest the assertion is not likely to be true, given the data; see [CITATION].', '1406.3521-1-29-2': 'For simplicity, we focus only on a plausibility function defined over the collection of singleton assertions, i.e., [MATH] for [MATH].', '1406.3521-1-29-3': 'These are also the relevant assertions for constructing interval estimates based on the IM output.', '1406.3521-1-30-0': 'Let [MATH] be the observations in [REF].', '1406.3521-1-30-1': 'The association step in Section [REF] yields a data-dependent collection of sets indexed by the auxiliary variable.', '1406.3521-1-30-2': 'In particular, write [MATH], a set-valued function of [MATH].', '1406.3521-1-30-3': 'These sets are combined with the predictive random set in Section [REF] to get an enlarged [MATH]-dependent random set: [EQUATION]', '1406.3521-1-30-4': 'Now, for a given assertion [MATH], we compute the plausibility function, [EQUATION] the probability that the random set [MATH] contains the asserted value [MATH] of [MATH].', '1406.3521-1-30-5': 'A simple calculation shows that, in this case with singleton assertions, we have [EQUATION] where [MATH] is defined in Section [REF].', '1406.3521-1-30-6': 'The above display shows that the plausibility function can be expressed directly in terms of the distribution of the predictive random set, without needing to go through the construction of [MATH] as in [REF].', '1406.3521-1-31-0': 'We pause here to answer a question that was left open from Section [REF], namely, how to choose the localization point [MATH].', '1406.3521-1-31-1': 'Following [CITATION], we propose here to choose [MATH] to match the value of [MATH] specified by the singleton assertion.', '1406.3521-1-31-2': 'That is, we propose to let the localization point depend on the assertion.', '1406.3521-1-31-3': 'All the elements in the plausibility function above with a 0 subscript, denoting dependence on [MATH], are changed in an obvious way to get a new plausibility function [EQUATION]', '1406.3521-1-31-4': 'We treat this as a function of [MATH] to be used for inference.', '1406.3521-1-31-5': 'In particular, we can construct a [MATH]% plausibility interval for [MATH] as follows: [EQUATION]', '1406.3521-1-31-6': 'The plausibility function, and the corresponding plausibility region, are easy to compute, as we describe in Section [REF].', '1406.3521-1-31-7': 'Moreover, the calibration [REF] of the predictive random set leads to exact plausibility function-based confidence intervals, as we now show.', '1406.3521-1-32-0': 'We need some notation for the sampling distribution of [MATH], given all the relevant parameters.', '1406.3521-1-32-1': 'Recall that the distribution of [MATH] actually depends on [MATH] or, equivalently, [MATH].', '1406.3521-1-32-2': 'The error variance [MATH] is a nuisance parameter, but [MATH] still appears in the sampling model for [MATH].', '1406.3521-1-32-3': 'We write this sampling distribution as [MATH].', '1406.3521-1-33-0': 'Take the association [REF] and the default predictive random set [MATH] in [REF].', '1406.3521-1-33-1': 'Then for any [MATH], any value [MATH] of [MATH], and any [MATH], the plausibility function satisfies [EQUATION]', '1406.3521-1-33-2': 'For given [MATH], if [MATH] is the value of [MATH], then it follows from the conditional distribution construction that the plausibility function in [REF], as a function of [MATH] with [MATH], is uniformly distributed on [MATH].', '1406.3521-1-33-3': 'Then the equality in [REF] follows immediately.', '1406.3521-1-34-0': 'Averaging the left-hand side of [REF] over [MATH], with respect to the distribution of [MATH], and using iterated expectation gives the following unconditional version of Theorem [REF].', '1406.3521-1-35-0': 'Under the conditions of Theorem [REF], for any [MATH], [EQUATION].', '1406.3521-1-36-0': 'Since we have proper calibration of the plausibility function, both conditionally and unconditionally, coverage probability results for the plausibility interval [REF] are also available.', '1406.3521-1-36-1': 'This justifies our choice to call [MATH] a [MATH]% plausibility interval, i.e., the frequentist coverage probability of [MATH] is exactly [MATH].', '1406.3521-1-37-0': 'The coverage probability of [MATH] in [REF] is exactly [MATH].', '1406.3521-1-38-0': '# Numerical results', '1406.3521-1-39-0': '## Implementation', '1406.3521-1-40-0': 'Evaluation of the plausibility function in [REF] requires computation of the distribution function [MATH] of [MATH] corresponding to the conditional distribution [MATH] of [MATH], given [MATH].', '1406.3521-1-40-1': 'This conditional distribution is not of a convenient form, so Monte Carlo methods are needed.', '1406.3521-1-40-2': 'For [MATH] fixed, since the transformation [REF] from [MATH] to [MATH] is of a log-linear form, and the density function of [MATH] can be written in closed-form, we can evaluate the joint density for [MATH] and, hence, the conditional density of [MATH].', '1406.3521-1-40-3': 'We then employ a Metropolis-Hastings algorithm with a random walk proposal distribution to generate a Markov chain whose stationary distribution is [MATH].', '1406.3521-1-40-4': 'R code is available at www.math.uic.edu/ rgmartin.', '1406.3521-1-40-5': 'In our experience, this Metropolis-Hastings chain is fast mixing and converges quickly to the stationary distribution.', '1406.3521-1-40-6': 'Once this sample is available, a Monte Carlo approximation for [MATH] for fixed [MATH] is readily obtained.', '1406.3521-1-41-0': 'To compute the plausibility function over a range of [MATH] values, different Monte Carlo samples are needed for each [MATH] value.', '1406.3521-1-41-1': 'One must be careful to use a fixed random seed for each [MATH]-specific Monte Carlo simulation so that the effect of [MATH] is not confounded with simulation error; this care also gives faster and more stable computations.', '1406.3521-1-41-2': 'To compute the [MATH]% plausibility interval, one solves the equation [MATH], which, again, will require several [MATH]-specific Monte Carlo simulations; the available R code contains the details.', '1406.3521-1-42-0': '## Simulation results', '1406.3521-1-43-0': 'In this section, we consider a standard one-way random effects model, i.e., [EQUATION] where [MATH] are independent with common distribution [MATH], and the [MATH]s are independent with common distribution [MATH]; the [MATH]s and [MATH]s are also mutually independent.', '1406.3521-1-43-1': 'Our goal is to compare the proposed IM-based plausibility intervals for [MATH] with the confidence intervals based on several competing methods.', '1406.3521-1-43-2': 'Of course, the properties of the various intervals depend on the design, in this case, the within-group sample sizes [MATH], and the values of [MATH].', '1406.3521-1-43-3': 'Following [CITATION], we consider designs with a varying range of imbalance, parametrized by the [MATH] index of [CITATION]; the measure [MATH] is defined as [MATH], where [MATH] is the total sample size, and [MATH] is the harmonic mean of [MATH].', '1406.3521-1-43-4': 'We consider seven different design patterns, specified in Table [REF], and nine [MATH] pairs: [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH].', '1406.3521-1-43-5': 'Without loss of generality, we set [MATH].', '1406.3521-1-44-0': 'For each pattern and pair of [MATH], 1000 independent data sets were generated and 95% two-sided interval estimates for [MATH] were computed based on the exact method of [CITATION], the fiducial method of [CITATION], and the proposed IM method.', '1406.3521-1-44-1': 'Empirical coverage probabilities and average length of the confidence interval under each setting were compared to investigate the performance of each method.', '1406.3521-1-44-2': 'A Bayesian method was also implemented, but the three aforementioned methods all gave intervals with better frequentist properties, so these results are not reported.', '1406.3521-1-45-0': 'A summary of the simulation results is displayed in Figure [REF].', '1406.3521-1-45-1': 'Panel (a) displays the coverage probabilities, and Panel (b) displays the relative length difference, which is defined as the length of the particular interval minus the length of the IM interval, scaled by the length of the IM interval.', '1406.3521-1-45-2': 'The simulations are summarized over three different groupings and displayed with three colors.', '1406.3521-1-45-3': 'The white boxes correspond to all simulation settings; the dark gray boxes correspond to settings with [MATH] and [MATH]; and the light gray boxes correspond to the settings under Pattern 6 in Table [REF].', '1406.3521-1-45-4': 'As we expect from Corollary [REF], the IM plausibility intervals have coverage at, or slightly above, the nominal 95% level in all cases.', '1406.3521-1-45-5': 'However, we also see that, overall, the IM plausibility intervals are shorter than the fiducial and Burch-Iyer confidence intervals.', '1406.3521-1-45-6': 'In particular, under the setting [MATH], the average length of IM is 10-20% smaller than that resulting from other competing methods.', '1406.3521-1-45-7': 'Similar conclusions can be reached based on the Pattern 6 setting.', '1406.3521-1-45-8': 'The fiducial intervals have coverage probability exceeding the nominal 95% level, but this comes at the expense of longer intervals on average.', '1406.3521-1-45-9': 'Overall, the proposed IM-based method performs quite well compared to these existing methods.', '1406.3521-1-46-0': '## Real-data analysis', '1406.3521-1-47-0': 'Equal numbers of subjects are tested under each standard and test preparations and a blank dose under a ([MATH])-point symmetrical slope-ratio assay.', '1406.3521-1-47-1': 'The response, on logarithmic scale, is assumed to depend linearly on the dose level.', '1406.3521-1-47-2': 'A modified balanced incomplete block design with [MATH] block size is introduced by [CITATION].', '1406.3521-1-47-3': 'The [MATH]th dose levels for standard and test preparations are represented by [MATH] and [MATH], [MATH].', '1406.3521-1-47-4': 'Under this design, the dose will be equally spaced and listed in ascending order.', '1406.3521-1-47-5': 'A balanced incomplete block design with [MATH] doses of the standard preparation inside [MATH] blocks is constructed and used as the basic design.', '1406.3521-1-47-6': 'Then a modified design is constructed by adding a blank dose and [MATH] doses of the test preparation into every block, under the rule that dose [MATH] should accompany [MATH] in every blocks.', '1406.3521-1-47-7': 'The model developed by Das and Kulkarni can be written as [EQUATION] where [MATH], [MATH], [MATH] represent observation response in [MATH]th block for [MATH]th dose of standard preparation, test preparation and blank dose; [MATH] and [MATH] represent [MATH]th dose level for standard and test preparation; [MATH] is zero by default, [MATH] denotes [MATH]th block effect; and [MATH] denotes independent random errors with common distribution [MATH].', '1406.3521-1-47-8': 'We consider random block effects and assume that [MATH] are independent with common distribution [MATH].', '1406.3521-1-47-9': 'Independence of [MATH] and [MATH] is also assumed.', '1406.3521-1-48-0': 'We analyze data coming from a nine-point slope-ratio assay on riboflavin content of yeast, with two replications in each dose; see Table 2 in [CITATION] for the design and data.', '1406.3521-1-48-1': 'For this design, we have [MATH] distinct eigenvalues, namely, 4.55, 1, and 0, with multiplicities 1, 1, and 10, respectively.', '1406.3521-1-48-2': 'A plot of the plausibility function for [MATH] is shown in Figure [REF](a).', '1406.3521-1-48-3': 'The function exceeds 0.2 at [MATH], which implies that 90 and 95 plausibility intervals include zero.', '1406.3521-1-48-4': 'The left panel of Table [REF] shows the 90 and 95 interval estimates for [MATH] based on the Burch-Iyer, fiducial, and IM methods.', '1406.3521-1-48-5': 'In this case, the IM intervals are provably exact and shorter.', '1406.3521-1-49-0': '[CITATION] analyzed data on birth weights of lambs.', '1406.3521-1-49-1': 'These data consist of the weights information at the birth of 62 single-birth male lambs, and were collected from three selection lines and two control lines.', '1406.3521-1-49-2': 'Each lamb was the offspring of one of the 23 rams and each lamb had a distinct dam.', '1406.3521-1-49-3': 'Age of the dam was also recorded and separated into three categories, numbered 1 (1-2 years), 2 (2-3 years), and 3 (over 3 years).', '1406.3521-1-49-4': 'A linear mixed model for these data is [EQUATION] where [MATH] represents the weight of the [MATH]th offspring of the [MATH]th sire in the [MATH]th population lines and of a dam in the [MATH]th age category; [MATH] represents [MATH]th level age effect; [MATH] represents the [MATH]th line effects; [MATH] denotes random sire effects and are assumed to be independently distributed as [MATH]; and random errors denoted by [MATH] is supposed to be independently distributed as [MATH].', '1406.3521-1-49-5': 'Furthermore, the [MATH]s and [MATH]s are assumed to be independent.', '1406.3521-1-49-6': 'In this case, [MATH], [MATH], [MATH] and [MATH]; all non-zero eigenvalues have multiplicity 1.', '1406.3521-1-49-7': 'A plot of the plausibility function for [MATH] is shown in Figure [REF](b).', '1406.3521-1-49-8': 'As in the previous example, the plausibility function is positive at [MATH], which means that plausibility intervals with any reasonable level will contain [MATH].', '1406.3521-1-49-9': 'We also used each of the three methods considered above to compute 90% and 95% interval estimates for [MATH].', '1406.3521-1-49-10': 'The results are shown in the right panel of Table [REF].', '1406.3521-1-49-11': 'In this case, IM gives a shorter interval compared to Burch-Iyer.', '1406.3521-1-49-12': 'The fiducial interval, however, is shorter than both exact intervals.', '1406.3521-1-49-13': 'We expect that the IM interval should be most efficient, so we explore the relative performance a bit further by simulating 1000 independent data sets from the fitted model in this case, i.e., with [MATH] and [MATH] as the true values.', '1406.3521-1-49-14': 'In these simulations, the fiducial and IM coverage probabilities were 0.944 and 0.954, respectively, both within an acceptable range of the nominal level, but the average lengths of the intervals are 0.488 and 0.456.', '1406.3521-1-49-15': 'That is, the IM intervals tend to be shorter than the fiducial intervals in this example, as we would expect.', '1406.3521-1-50-0': '# Concluding remarks', '1406.3521-1-51-0': 'The IM method proposed here gives exact confidence intervals for the heritability coefficient [MATH], as well as the variance ratio [MATH], and numerical results suggest increased efficiency compared to existing methods.', '1406.3521-1-51-1': 'A question is if these same techniques can be employed for exact prior-free probabilistic inference on other quantities related to the variance components [MATH].', '1406.3521-1-51-2': 'It is well-known that, for the unbalanced design case, exact marginalization is challenging.', '1406.3521-1-51-3': 'In the IM context, this means that the association is not "regular" in the sense of [CITATION].', '1406.3521-1-51-4': 'Therefore, some more sophisticated tools are needed for exact inference on, say, the individual variance components [MATH] and [MATH].', '1406.3521-1-51-5': 'This application provides clear motivation for our ongoing investigations into more general IM-based marginalization strategies.', '1406.3521-1-52-0': 'Another important question is if the techniques presented herein can be applied in more complex and high-dimensional mixed-effect models.', '1406.3521-1-52-1': 'In genome-wide association studies, for example, the dimensions of the problem are extremely large.', '1406.3521-1-52-2': 'We expect that, conceptually, the techniques described here will carry over to the more complex scenario.', '1406.3521-1-52-3': 'However, there will be computational challenges to overcome, as with all approaches .', '1406.3521-1-52-4': 'This, along with the incorporation of optimal predictive random sets is a focus of future research.', '1406.3521-1-53-0': 'Finally, recall that our plausibility interval construction requires several Monte Carlo runs at different [MATH] values.', '1406.3521-1-53-1': 'This is more expensive compared to fiducial or Bayesian methods that require only one Monte Carlo run.', '1406.3521-1-53-2': 'We believe that this is the price that one must pay for exact and efficient inference, though running the IM Monte Carlo runs in parallel for different [MATH] values would help to reduce the price.', '1406.3521-1-53-3': 'However, one can consider some modifications that will speed up the computations.', '1406.3521-1-53-4': 'One possible speed-up strategy is to choose an initial localization point, say [MATH], and produce a single Monte Carlo sample corresponding to [MATH].', '1406.3521-1-53-5': 'One might take this [MATH] to be the maximum likelihood estimator.', '1406.3521-1-53-6': 'This strategy reduces the overall computation time, but the theoretical properties of the genuine IM approach are lost, though they would hold asymptotically.'}

{'1406.3521-2-0-0': 'Linear mixed-effect models with two variance components are often used when variability comes from two sources.', '1406.3521-2-0-1': 'In genetics applications, variation in observed traits can be attributed to biological and environmental effects, and the heritability coefficient is a fundamental quantity that measures the proportion of total variability due to the biological effect.', '1406.3521-2-0-2': 'We propose a new inferential model approach which yields exact prior-free probabilistic inference on the heritability coefficient.', '1406.3521-2-0-3': "In particular we construct exact confidence intervals and demonstrate numerically our method's efficiency compared to that of existing methods.", '1406.3521-2-1-0': 'Keywords and phrases: Differential equation; inferential model; plausibility; unbalanced design; variance components.', '1406.3521-2-2-0': '# Introduction', '1406.3521-2-3-0': 'Normal linear mixed effects models are useful in a variety of biological, physical, and social scientific applications with variability coming from multiple sources; see [CITATION] and [CITATION].', '1406.3521-2-3-1': 'In this paper, we focus on the case of two variance components, and the general model can be written as [EQUATION] where [MATH] is a [MATH]-vector of response variables, [MATH] and [MATH] are design matrices for the fixed and random effects, of dimension [MATH] and [MATH], respectively, [MATH] is a [MATH]-vector of unknown parameters, [MATH] is a normal random [MATH]-vector with mean 0 and covariance matrix [MATH], and [MATH] is a normal random [MATH]-vector with mean 0 and covariance matrix [MATH].', '1406.3521-2-3-2': 'Here, [MATH] is the pair of variance components.', '1406.3521-2-3-3': 'Assume [MATH] is known and [MATH] is of rank [MATH].', '1406.3521-2-3-4': 'The unknown parameters in this model are the [MATH] fixed-effect coefficients [MATH] and the two variance components [MATH], so the parameter space is [MATH]-dimensional.', '1406.3521-2-4-0': 'In biological applications, the quantities [MATH] and [MATH] in [REF] denote the genetic and environmental effects, respectively.', '1406.3521-2-4-1': 'Given that "a central question in biology is whether observed variation in a particular trait is due to environmental or biological factors" , the heritability coefficient, [MATH], which represents the proportion of phenotypic variance attributed to variation in genotypic values, is a fundamentally important quantity.', '1406.3521-2-4-2': 'Indeed, mixed-effect models and inference on the heritability coefficient has been applied recently in genome-wide association studies ; see Section [REF] for more on these applications.', '1406.3521-2-5-0': 'Given the importance of the heritability coefficient, there are a number of methods available to construct confidence intervals for [MATH] or, equivalently, for the variance ratio [MATH].', '1406.3521-2-5-1': 'When the design is balanced, [CITATION] and [CITATION] give a confidence intervals for [MATH] and other quantities.', '1406.3521-2-5-2': 'When the design is possibly unbalanced, as we assume here, the problem is more challenging; in particular, exact ANOVA-based confidence intervals generally are not available.', '1406.3521-2-5-3': '[CITATION] gave intervals for [MATH] in the unbalanced case, and subsequent contributions include [CITATION], [CITATION], [CITATION], and [CITATION].', '1406.3521-2-5-4': 'Bayesian and fiducial methods are also available.', '1406.3521-2-6-0': 'The focus in this paper is exact prior-free probabilistic inference on the heritability coefficient based on the recently proposed inferential model (IM) framework.', '1406.3521-2-6-1': '[CITATION] give a detailed account of the general framework, along with comparisons to other related approaches, including fiducial .', '1406.3521-2-6-2': 'The IM approach is driven by the specification of an association between unobservable auxiliary variables, the observable data, and the unknown parameters.', '1406.3521-2-6-3': 'This association is followed up by using properly calibrated random sets to predict these auxiliary variables.', '1406.3521-2-6-4': '[CITATION] argue that it is precisely this prediction of the auxiliary variables that sets the IM approach apart from fiducial, Dempster-Shafer , and structural inference .', '1406.3521-2-6-5': 'The IM output is a plausibility function that can be used for inference on the parameter of interest.', '1406.3521-2-6-6': 'In particular, the plausibility function yields confidence intervals with exact coverage .', '1406.3521-2-6-7': 'A brief overview of the general IM framework is given in Section [REF].', '1406.3521-2-6-8': "A key feature of the IM approach, and the motivation for this work, is that, besides being exact, the IM approach's careful handling of uncertainty often leads to more efficient inference.", '1406.3521-2-7-0': 'A key to the construction of an efficient IM is to reduce the dimension of the auxiliary variable as much as possible and, for the heritability coefficient, there are several dimension reduction steps.', '1406.3521-2-7-1': 'A first dimension reduction eliminates the nuisance parameter [MATH].', '1406.3521-2-7-2': 'These well-known calculations are reviewed in Section [REF].', '1406.3521-2-7-3': 'The main technical contribution here is in Section [REF], where we employ a differential equation-driven technique to reduce the auxiliary variable dimension, leading to a conditional IM for the heritability coefficient.', '1406.3521-2-7-4': 'In particular, this step allows us to reduce the dimension beyond that allowed by sufficiency.', '1406.3521-2-7-5': 'A predictive random set is introduced in Section [REF], and in Section [REF] we show that the corresponding plausibility function is properly calibrated and, therefore, the plausibility function-based confidence intervals are provably exact.', '1406.3521-2-7-6': 'Sections [REF] and [REF] illustrate the proposed method in simulated and real data examples.', '1406.3521-2-7-7': 'The general message is that our proposed confidence intervals for the heritability coefficient are exact and tend to be more efficient compared to existing methods.', '1406.3521-2-8-0': '# Preliminaries', '1406.3521-2-9-0': '## Overview of inferential models', '1406.3521-2-10-0': 'The goal of the IM framework is to get valid prior-free probabilistic inference.', '1406.3521-2-10-1': 'This is facilitated by first associating the observable data and unknown parameters to a set of unobservable auxiliary variables.', '1406.3521-2-10-2': 'For example, the marginal distribution of [MATH] from [REF] is [EQUATION] which can be written in association form: [EQUATION]', '1406.3521-2-10-3': 'That is, observable data [MATH] and unknown parameters [MATH] are associated with unobservable auxiliary variables [MATH], in this case, a [MATH]-vector of independent standard normal random variables.', '1406.3521-2-10-4': 'Given this association, the basic IM approach is to introduce a predictive random set for [MATH] and combine with the observed value of [MATH] to get a plausibility function.', '1406.3521-2-10-5': 'Roughly, this plausibility function assigns, to assertions about the parameter of interest, values in [MATH] measuring the plausibility that the assertion is true.', '1406.3521-2-10-6': 'This plausibility function can be used to design exact frequentist tests or confidence regions.', '1406.3521-2-11-0': '[CITATION] give a general description of this three-step IM construction and its properties; in Section [REF] below we will flesh out these three steps, including choice of a predictive random set, for the variance components problem at hand.', '1406.3521-2-11-1': 'The construction of exact plausibility intervals for the heritability coefficient is presented in Section [REF], along with a theoretical justification of the claimed exactness.', '1406.3521-2-12-0': 'Notice, in the association [REF], that the auxiliary variable [MATH] is, in general, of higher dimension than that of the parameter [MATH].', '1406.3521-2-12-1': 'There are two reasons for trying to reduce the auxiliary variable dimension.', '1406.3521-2-12-2': 'First, it is much easier to specify good predictive random sets when the auxiliary variable is of low dimension.', '1406.3521-2-12-3': 'Second, inference is more efficient when only a relatively small number of auxiliary variables require prediction.', '1406.3521-2-12-4': 'This dimension reduction is via a conditioning operation, and [CITATION] give a general explanation of the gains as well as a novel technique for carrying out this reduction.', '1406.3521-2-12-5': 'We employ these techniques in Section [REF] to get a dimension-reduced association for the heritability coefficient in the linear mixed model.', '1406.3521-2-13-0': 'Another important challenge is when nuisance parameters are present.', '1406.3521-2-13-1': 'Our interest is in the heritability coefficient, a function of the full parameter [MATH], so we need to marginalize over the nuisance parameters.', '1406.3521-2-13-2': 'The next section marginalizes over [MATH] using standard techniques; further marginalization will be carried out in Section [REF].', '1406.3521-2-14-0': '## Marginalizing out the fixed effect', '1406.3521-2-15-0': 'Start with the linear mixed model [REF].', '1406.3521-2-15-1': 'Following the setup in [CITATION], let [MATH] be a [MATH] matrix such that [MATH] and [MATH].', '1406.3521-2-15-2': 'Next, let [MATH].', '1406.3521-2-15-3': 'Then [MATH] is a one-to-one mapping.', '1406.3521-2-15-4': 'Moreover, the distribution of [MATH] depends on [MATH] only, and the distribution of [MATH] depends on [MATH], with [MATH] a location parameter.', '1406.3521-2-15-5': 'In particular, from [REF], we get [EQUATION] where [MATH] and [MATH] is a [MATH] covariance matrix of a known form that depends on [MATH]; its precise form is not important.', '1406.3521-2-15-6': 'From this point, the general theory in [CITATION] allows us to marginalize over [MATH] by simply deleting the [MATH] component.', '1406.3521-2-15-7': 'Therefore, a marginal association for [MATH] is [EQUATION].', '1406.3521-2-15-8': 'This marginalization reduces the auxiliary variable dimension from [MATH] to [MATH].', '1406.3521-2-16-0': 'In the marginal association for [MATH] above, there are [MATH] auxiliary variables but only two parameters.', '1406.3521-2-16-1': 'Classical results on sufficient statistics in mixed effects model that will facilitate further dimension reduction.', '1406.3521-2-16-2': 'For the matrix [MATH] defined above, let [MATH] denote the (distinct) ordered eigenvalues with multiplicities [MATH], respectively.', '1406.3521-2-16-3': 'Let [MATH] be a [MATH] orthogonal matrix such that [MATH] is diagonal with eigenvalues [MATH], in their multiplicities, on the diagonal.', '1406.3521-2-16-4': 'For [MATH], a [MATH] matrix, define [EQUATION] [CITATION] showed that [MATH] is a minimal sufficient statistic for [MATH].', '1406.3521-2-16-5': 'Moreover, the distribution of [MATH] is determined by [EQUATION]', '1406.3521-2-16-6': 'This reduces the auxiliary variable dimension from [MATH] to [MATH].', '1406.3521-2-16-7': 'We take [REF] as our "baseline association."', '1406.3521-2-16-8': 'Even in this reduced baseline association, there are [MATH] auxiliary variables but only two parameters, which means there is room for even further dimension reduction.', '1406.3521-2-16-9': 'The next section shows how to reduce to a scalar auxiliary variable when the parameter of interest is the scalar heritability coefficient.', '1406.3521-2-17-0': '# Inferential model for heritability', '1406.3521-2-18-0': '## Association', '1406.3521-2-19-0': 'For the moment, it will be convenient to work with the variance ratio, [MATH].', '1406.3521-2-19-1': 'Since [MATH] is a one-to-one function of [MATH], the two parametrizations are equivalent.', '1406.3521-2-19-2': 'Rewrite the baseline association [REF] as [EQUATION]', '1406.3521-2-19-3': 'If we make the following transformations, [EQUATION] then the association [REF] becomes [EQUATION].', '1406.3521-2-19-4': 'Since for every [MATH], there exists a [MATH] that solves the right-most equation, it follows from the general theory in [CITATION] that a marginal association for [MATH] is obtained by deleting the component above involving [MATH].', '1406.3521-2-19-5': 'In particular, a marginal association for [MATH] is [EQUATION].', '1406.3521-2-19-6': 'If we write [EQUATION] then we get a marginal association for [MATH] of the form [EQUATION]', '1406.3521-2-19-7': 'Marginalization reduces the auxiliary variable dimension by 1.', '1406.3521-2-19-8': 'Further dimension reduction will be considered next.', '1406.3521-2-19-9': 'Note that the new auxiliary variable [MATH] is a multivariate F-distributed random vector .', '1406.3521-2-20-0': 'Here we construct a local conditional IM for [MATH] as described in [CITATION].', '1406.3521-2-20-1': 'Select a fixed value [MATH]; more on this choice in Section [REF].', '1406.3521-2-20-2': 'To reduce the dimension of the auxiliary variable [MATH], in [REF], from [MATH] to [MATH], we construct two pairs of functions-one pair, [MATH], on the sample space, and one pair, [MATH], on the auxiliary variable space.', '1406.3521-2-20-3': 'We insist that [MATH] and [MATH] are both one-to-one, and [MATH] and [MATH] are allowed to depend on the selected [MATH].', '1406.3521-2-20-4': 'The goal is to find [MATH] which is, in a certain sense, not sensitive to changes in the auxiliary variable.', '1406.3521-2-21-0': 'Write the association [REF] so that [MATH] is a function of [MATH] and [MATH], i.e., [EQUATION].', '1406.3521-2-21-1': 'We want to choose the function [MATH] such that the partial derivative of [MATH] with respect to [MATH] vanishes at [MATH].', '1406.3521-2-21-2': 'By the chain rule, we have [EQUATION] so our goal is to find [MATH] to solve the following partial differential equation: [EQUATION] here [MATH] is a [MATH] vector and [MATH] is a [MATH] matrix of rank [MATH].', '1406.3521-2-21-3': 'Toward solving the partial differential equation, first we get that the partial derivative of [MATH] with respect to [MATH] satisfies [EQUATION] where [MATH].', '1406.3521-2-21-4': 'This simplifies the relevant partial differential equation to the following: [EQUATION] where [MATH] is a diagonal matrix constructed from a vector [MATH].', '1406.3521-2-21-5': 'The method of characteristics for solving partial differential equations identifies a logarithmic function of the form [EQUATION] where [MATH] is a [MATH] matrix with rows orthogonal to [MATH] at [MATH].', '1406.3521-2-21-6': 'For example, since the matrix that projects to the orthogonal complement to the column space of [MATH] has rank [MATH], we can take [MATH] to be a matrix whose [MATH] rows form a basis for that space.', '1406.3521-2-21-7': 'For [MATH] defined in this way, it is easy to check that [MATH] in [REF] is indeed a solution to the partial differential equation [REF].', '1406.3521-2-22-0': 'We have completed specification of the [MATH] function; it remains to specify [MATH] and [MATH].', '1406.3521-2-22-1': 'The easiest to specify next is [MATH], the value of [MATH], as a function of [MATH]: [EQUATION].', '1406.3521-2-22-2': 'As we describe below, the goal is to condition on the observed value of [MATH].', '1406.3521-2-23-0': 'Next, we define [MATH] and [MATH] to supplement [MATH] and [MATH], respectively.', '1406.3521-2-23-1': 'In particular, take a [MATH]-vector [MATH] which is not orthogonal to [MATH] at [MATH].', '1406.3521-2-23-2': 'It is easy to check that the entries in [MATH] are strictly positive for all [MATH].', '1406.3521-2-23-3': 'Therefore, we can take [MATH] independent of [MATH], e.g., [MATH], is not orthogonal to [MATH].', '1406.3521-2-23-4': 'Now set [EQUATION]', '1406.3521-2-23-5': 'This is a log-linear transformation, and the linear part is non-singular, so this is a one-to-one mapping.', '1406.3521-2-23-6': 'Finally, we take [MATH] as [EQUATION].', '1406.3521-2-23-7': 'Since [MATH] is log-linear, just like [MATH], it is also one-to-one.', '1406.3521-2-24-0': 'We can now write the conditional association for [MATH].', '1406.3521-2-24-1': 'For the given [MATH], the mapping [MATH] describes a split of our previous association [REF] into two pieces: [EQUATION].', '1406.3521-2-24-2': 'The first piece carries direct information about [MATH].', '1406.3521-2-24-3': 'The second piece plays a conditioning role, correcting for the fact that some information was lost in reducing the [MATH]-dimensional [MATH] to a one-dimensional [MATH].', '1406.3521-2-24-4': 'To complete the specification of the conditional association, write [MATH] and [MATH].', '1406.3521-2-24-5': 'Then we have [EQUATION] where [MATH] is the conditional distribution of [MATH], given that [MATH] equals to the observed value [MATH] of [MATH].', '1406.3521-2-24-6': 'To summarize, [REF] completes the association step that describes the connection between observable data, unknown parameter of interest, and unobservable auxiliary variables.', '1406.3521-2-24-7': 'Of particular interest is that this association involves only a one-dimensional auxiliary variable compared to the association [REF] obtained from the minimal sufficient statistics that involves an [MATH]-dimensional auxiliary variable.', '1406.3521-2-24-8': 'This dimension reduction will come in handy for the choice of predictive random set in the following section.', '1406.3521-2-24-9': 'The price we paid for this dimension reduction was the choice of a particular localization point [MATH].', '1406.3521-2-24-10': 'In Section [REF] we employ a trick to side-step this issue when the goal is, as in this paper, to construct confidence intervals.', '1406.3521-2-25-0': '## Predictive random sets', '1406.3521-2-26-0': 'Having reduced the auxiliary variable to a scalar in [REF], the choice of an efficient predictive random set is now relatively simple.', '1406.3521-2-26-1': 'Though there is an available theory of optimal predictive random sets , here we opt for simplicity; in particular, we propose a default predictive random set that is theoretically sound and computational and intuitively simple.', '1406.3521-2-27-0': 'Consider the following version of what [CITATION] call the "default" predictive random set: [EQUATION]', '1406.3521-2-27-1': 'This [MATH], with distribution [MATH], is a random interval, centered at the mean [MATH] of the conditional distribution [MATH].', '1406.3521-2-27-2': 'One key feature of this predictive random set is that it is nested, i.e., for any two distinct realizations of [MATH], one is a subset of the other.', '1406.3521-2-27-3': 'The second key feature is a calibration of the predictive random set distribution with the underlying distribution of the auxiliary variable.', '1406.3521-2-27-4': 'Following [CITATION], define the contour function [EQUATION] which represents the probability that the predictive random set contains the value [MATH] of the auxiliary variable.', '1406.3521-2-27-5': 'We shall require that [EQUATION]', '1406.3521-2-27-6': 'For the default predictive random set in [REF], it is easy to check that [EQUATION] where [MATH] is the distribution function of [MATH] for [MATH].', '1406.3521-2-27-7': 'From the construction above, it is clear that it is a continuous distribution.', '1406.3521-2-27-8': 'Then, [MATH] is a continuous random variable, so [MATH] is uniformly distributed on [MATH].', '1406.3521-2-27-9': 'Therefore, [REF] holds for the default predictive random set [MATH].', '1406.3521-2-27-10': 'Results on optimal predictive random sets are available , but here, again, our focus is on simplicity.', '1406.3521-2-27-11': 'See Section [REF].', '1406.3521-2-28-0': '## Plausibility intervals', '1406.3521-2-29-0': 'Here we combine the association in Section [REF] with the predictive random set described above to produce a plausibility function for inference about [MATH].', '1406.3521-2-29-1': 'In general, a plausibility function is a data-dependent mapping that assigns, to each assertion about the parameter, a value in [MATH], with the interpretation that small values suggest the assertion is not likely to be true, given the data; see [CITATION].', '1406.3521-2-29-2': 'For simplicity, we focus only on the collection of singleton assertions, i.e., [MATH] for [MATH].', '1406.3521-2-29-3': 'These are also the relevant assertions for constructing interval estimates based on the IM output.', '1406.3521-2-30-0': 'Let [MATH] be the observations in [REF].', '1406.3521-2-30-1': 'The association step in Section [REF] yields a data-dependent collection of sets indexed by the auxiliary variable.', '1406.3521-2-30-2': 'In particular, write [MATH], a set-valued function of [MATH].', '1406.3521-2-30-3': 'These sets are combined with the predictive random set in Section [REF] to get an enlarged [MATH]-dependent random set: [EQUATION]', '1406.3521-2-30-4': 'Now, for a given assertion [MATH], we compute the plausibility function, [EQUATION] the probability that the random set [MATH] contains the asserted value [MATH] of [MATH].', '1406.3521-2-30-5': 'A simple calculation shows that, in this case with singleton assertions, we have [EQUATION] where [MATH] is defined in Section [REF].', '1406.3521-2-30-6': 'The above display shows that the plausibility function can be expressed directly in terms of the distribution of the predictive random set, without needing to go through the construction of [MATH] as in [REF].', '1406.3521-2-31-0': 'We pause here to answer a question that was left open from Section [REF], namely, how to choose the localization point [MATH].', '1406.3521-2-31-1': 'Following [CITATION], we propose here to choose [MATH] to match the value of [MATH] specified by the singleton assertion.', '1406.3521-2-31-2': 'That is, we propose to let the localization point depend on the assertion.', '1406.3521-2-31-3': 'All the elements in the plausibility function above with a 0 subscript, denoting dependence on [MATH], are changed in an obvious way to get a new plausibility function [EQUATION]', '1406.3521-2-31-4': 'We treat this as a function of [MATH] to be used for inference.', '1406.3521-2-31-5': 'In particular, we can construct a [MATH]% plausibility interval for [MATH] as follows: [EQUATION]', '1406.3521-2-31-6': 'The plausibility function, and the corresponding plausibility region, are easy to compute, as we describe in Section [REF].', '1406.3521-2-31-7': 'Moreover, the calibration [REF] of the predictive random set leads to exact plausibility function-based confidence intervals, as we now show.', '1406.3521-2-32-0': 'We need some notation for the sampling distribution of [MATH], given all the relevant parameters.', '1406.3521-2-32-1': 'Recall that the distribution of [MATH] actually depends on [MATH] or, equivalently, [MATH].', '1406.3521-2-32-2': 'The error variance [MATH] is a nuisance parameter, but [MATH] still appears in the sampling model for [MATH].', '1406.3521-2-32-3': 'We write this sampling distribution as [MATH].', '1406.3521-2-33-0': 'Take the association [REF] and the default predictive random set [MATH] in [REF].', '1406.3521-2-33-1': 'Then for any [MATH], any value [MATH] of [MATH], and any [MATH], the plausibility function satisfies [EQUATION]', '1406.3521-2-33-2': 'For given [MATH], if [MATH] is the value of [MATH], then it follows from the conditional distribution construction that the plausibility function in [REF], as a function of [MATH] with [MATH], is [MATH].', '1406.3521-2-33-3': 'Then the equality in [REF] follows immediately.', '1406.3521-2-34-0': 'Averaging the left-hand side of [REF] over [MATH], with respect to the distribution of [MATH], and using iterated expectation gives the following unconditional version of Theorem [REF].', '1406.3521-2-35-0': 'Under the conditions of Theorem [REF], for any [MATH], [EQUATION].', '1406.3521-2-36-0': 'Since we have proper calibration of the plausibility function, both conditionally and unconditionally, coverage probability results for the plausibility interval [REF] are also available.', '1406.3521-2-36-1': 'This justifies our choice to call [MATH] a [MATH]% plausibility interval, i.e., the frequentist coverage probability of [MATH] is exactly [MATH].', '1406.3521-2-37-0': 'The coverage probability of [MATH] in [REF] is exactly [MATH].', '1406.3521-2-38-0': '# Numerical results', '1406.3521-2-39-0': '## Implementation', '1406.3521-2-40-0': 'Evaluation of the plausibility function in [REF] requires the distribution function [MATH] of [MATH] corresponding to the conditional distribution [MATH] of [MATH], given [MATH].', '1406.3521-2-40-1': 'This conditional distribution is not of a convenient form, so numerical methods are needed.', '1406.3521-2-40-2': 'For [MATH] fixed, since the transformation [REF] from [MATH] to [MATH] is of a log-linear form, and the density function of [MATH] can be written in closed-form, we can evaluate the joint density for [MATH] and, hence, the conditional density of [MATH].', '1406.3521-2-40-3': 'Numerical integration is used to evaluate the normalizing constant, the mean [MATH], and the distribution function [MATH].', '1406.3521-2-40-4': 'R code is available at www.math.uic.edu/ rgmartin.', '1406.3521-2-41-0': '## Simulation results', '1406.3521-2-42-0': 'In this section, we consider a standard one-way random effects model, i.e., [EQUATION] where [MATH] are independent with common distribution [MATH], and the [MATH]s are independent with common distribution [MATH]; the [MATH]s and [MATH]s are also mutually independent.', '1406.3521-2-42-1': 'Our goal is to compare the proposed IM-based plausibility intervals for [MATH] with the confidence intervals based on several competing methods.', '1406.3521-2-42-2': 'Of course, the properties of the various intervals depend on the design, in this case, the within-group sample sizes [MATH], and the values of [MATH].', '1406.3521-2-42-3': 'Our focus here is on cases with small sample sizes, namely, where the total sample size [MATH] is fixed at 15.', '1406.3521-2-42-4': 'The three design patterns [MATH] considered are: [MATH], [MATH], and [MATH].', '1406.3521-2-42-5': 'The nine [MATH] pairs considered are: [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH].', '1406.3521-2-42-6': 'Without loss of generality, we set [MATH].', '1406.3521-2-43-0': 'For each design pattern and pair of [MATH], 1000 independent data sets were generated and 95% two-sided interval estimates for [MATH] were computed based on the exact method of [CITATION], the fiducial method of [CITATION], and the proposed IM method.', '1406.3521-2-43-1': 'Empirical coverage probabilities and average length of the confidence interval under each setting were compared to investigate the performance of each method.', '1406.3521-2-43-2': 'Besides these three methods, we also implemented Bayesian and profile likelihood approaches.', '1406.3521-2-43-3': 'The three aforementioned methods all gave intervals with better coverage and length properties than the Bayesian method, and the profile likelihood method was very unstable with small sample sizes, often having very high coverage with very wide intervals or very low coverage with very narrow intervals.', '1406.3521-2-43-4': 'So, these results are not reported.', '1406.3521-2-44-0': 'A summary of the simulation results is displayed in Figure [REF].', '1406.3521-2-44-1': 'Panel (a) displays the coverage probabilities, and Panel (b) displays the relative length difference, which is defined as the length of the particular interval minus the length of the IM interval, scaled by the length of the IM interval.', '1406.3521-2-44-2': 'As we expect from Corollary [REF], the IM plausibility intervals have coverage at the nominal 95% level.', '1406.3521-2-44-3': 'We also see that the IM intervals tend to be shorter than the fiducial and Burch-Iyer confidence intervals.', '1406.3521-2-44-4': 'The fiducial intervals have coverage probability exceeding the nominal 95% level, but this comes at the expense of longer intervals on average.', '1406.3521-2-44-5': 'Overall, the proposed IM-based method performs quite well compared to these existing methods.', '1406.3521-2-44-6': 'We also replicated the simulation study in [CITATION], which involves larger sample sizes and a broader range of imbalance, and the relative comparisons between these three methods are the same as here.', '1406.3521-2-45-0': '## Real-data analysis', '1406.3521-2-46-0': 'Equal numbers of subjects are tested under each standard and test preparations and a blank dose under a ([MATH])-point symmetrical slope-ratio assay.', '1406.3521-2-46-1': 'The response, on logarithmic scale, is assumed to depend linearly on the dose level.', '1406.3521-2-46-2': 'A modified balanced incomplete block design with [MATH] block size is introduced by [CITATION].', '1406.3521-2-46-3': 'The [MATH]th dose levels for standard and test preparations are represented by [MATH] and [MATH], [MATH].', '1406.3521-2-46-4': 'Under this design, the dose will be equally spaced and listed in ascending order.', '1406.3521-2-46-5': 'A balanced incomplete block design with [MATH] doses of the standard preparation inside [MATH] blocks is constructed and used as the basic design.', '1406.3521-2-46-6': 'Then a modified design is constructed by adding a blank dose and [MATH] doses of the test preparation into every block, under the rule that dose [MATH] should accompany [MATH] in every blocks.', '1406.3521-2-46-7': 'The model developed by Das and Kulkarni can be written as [EQUATION] where [MATH], [MATH], [MATH] represent observation response in [MATH]th block for [MATH]th dose of standard preparation, test preparation and blank dose; [MATH] and [MATH] represent [MATH]th dose level for standard and test preparation; [MATH] is zero by default, [MATH] denotes [MATH]th block effect; and [MATH] denotes independent random errors with common distribution [MATH].', '1406.3521-2-46-8': 'We consider random block effects and assume that [MATH] are independent with common distribution [MATH].', '1406.3521-2-46-9': 'Independence of [MATH] and [MATH] is also assumed.', '1406.3521-2-47-0': 'We analyze data coming from a nine-point slope-ratio assay on riboflavin content of yeast, with two replications in each dose; see Table 2 in [CITATION] for the design and data.', '1406.3521-2-47-1': 'For this design, we have [MATH] distinct eigenvalues, namely, 4.55, 1, and 0, with multiplicities 1, 1, and 10, respectively.', '1406.3521-2-47-2': 'A plot of the plausibility function for [MATH] is shown in Figure [REF](a).', '1406.3521-2-47-3': 'The function exceeds 0.2 at [MATH], which implies that 90 and 95 plausibility intervals include zero.', '1406.3521-2-47-4': 'The left panel of Table [REF] shows the 90 and 95 interval estimates for [MATH] based on the Burch-Iyer, fiducial, and IM methods.', '1406.3521-2-47-5': 'In this case, the IM intervals are provably exact and shorter.', '1406.3521-2-48-0': '1=1', '1406.3521-2-49-0': '# Results for new IM with numerical integration:', '1406.3521-2-50-0': '> o <- assay.example(alpha=0.05, plot.pl=FALSE) [1] 0.0000000 0.9264546 > o <- assay.example(alpha=0.1, plot.pl=FALSE) [1] 0.0000000 0.8816783 > o <- lamb.example(alpha=0.05, plot.pl=FALSE) [1] 0.0000000 0.6071329 > o <- lamb.example(alpha=0.1, plot.pl=FALSE) [1] 0.0000000 0.5539082', '1406.3521-2-51-0': '# Results for REML (using observed information for variance):', '1406.3521-2-52-0': '> o <- assay.example(alpha=0.05, plot.pl=FALSE) [1] 0.0000000 0.9264546 > o <- assay.example(alpha=0.1, plot.pl=FALSE) [1] 0.0000000 0.8816783 > o <- lamb.example(alpha=0.05, plot.pl=FALSE) [1] -0.1336466 0.5226253 > o <- lamb.example(alpha=0.1, plot.pl=FALSE) [1] -0.0808910 0.4698697', '1406.3521-2-53-0': '[CITATION] analyzed data on birth weights of lambs.', '1406.3521-2-53-1': 'These data consist of the weights information at the birth of 62 single-birth male lambs, and were collected from three selection lines and two control lines.', '1406.3521-2-53-2': 'Each lamb was the offspring of one of the 23 rams and each lamb had a distinct dam.', '1406.3521-2-53-3': 'Age of the dam was also recorded and separated into three categories, numbered 1 (1-2 years), 2 (2-3 years), and 3 (over 3 years).', '1406.3521-2-53-4': 'A linear mixed model for these data is [EQUATION] where [MATH] represents the weight of the [MATH]th offspring of the [MATH]th sire in the [MATH]th population lines and of a dam in the [MATH]th age category; [MATH] represents [MATH]th level age effect; [MATH] represents the [MATH]th line effects; [MATH] denotes random sire effects and are assumed to be independently distributed as [MATH]; and random errors denoted by [MATH] is supposed to be independently distributed as [MATH].', '1406.3521-2-53-5': 'Furthermore, the [MATH]s and [MATH]s are assumed to be independent.', '1406.3521-2-53-6': 'In this case, [MATH], [MATH], [MATH] and [MATH]; all non-zero eigenvalues have multiplicity 1 except [MATH] with multiplicity 2.', '1406.3521-2-53-7': 'A plot of the plausibility function for [MATH] is shown in Figure [REF](b).', '1406.3521-2-53-8': 'As in the previous example, the plausibility function is positive at [MATH], which means that plausibility intervals with any reasonable level will contain [MATH].', '1406.3521-2-53-9': 'We also used each of the three methods considered above to compute 90% and 95% interval estimates for [MATH].', '1406.3521-2-53-10': 'The results are shown in the right panel of Table [REF].', '1406.3521-2-53-11': 'In this case, IM gives a shorter interval compared to Burch-Iyer.', '1406.3521-2-53-12': 'The fiducial interval, however, is shorter than both exact intervals.', '1406.3521-2-53-13': 'We expect the IM interval to be most efficient, so we explore the relative performance a bit further by simulating 1000 independent data sets from the fitted model in this case, i.e., with [MATH] and [MATH] as the true values.', '1406.3521-2-53-14': 'In these simulations, the fiducial and IM coverage probabilities were 0.944 and 0.954, respectively, both within an acceptable range of the nominal level, but the average lengths of the intervals are 0.488 and 0.456.', '1406.3521-2-53-15': 'That is, the IM intervals tend to be shorter than the fiducial intervals in problems similar to this example, as we would expect.', '1406.3521-2-54-0': '1=1 Some results from "lamb weight data"', '1406.3521-2-55-0': '1) results of the real data 90 BI 0.567 0.615 FI 0.466 0.530 IM 0.554 0.597 LI 0.505 0.573', '1406.3521-2-56-0': '2) results of simulations based on "lamb weight data"', '1406.3521-2-57-0': 'Coverage Probability Length of C.I (95 FI 94.4 0.488 IM 95.4 0.456 LI 97.5 0.584', '1406.3521-2-58-0': '# Concluding remarks', '1406.3521-2-59-0': 'The IM method proposed here gives exact confidence intervals for the heritability coefficient [MATH], as well as the variance ratio [MATH], and numerical results suggest increased efficiency compared to existing methods.', '1406.3521-2-59-1': 'A question is if these same techniques can be employed for exact prior-free probabilistic inference on other quantities related to the variance components [MATH].', '1406.3521-2-59-2': 'It is well-known that, for the unbalanced design case, exact marginalization is challenging.', '1406.3521-2-59-3': 'In the IM context, this means that the association is not "regular" in the sense of [CITATION].', '1406.3521-2-59-4': 'Therefore, some more sophisticated tools are needed for exact inference on, say, the individual variance components [MATH] and [MATH].', '1406.3521-2-59-5': 'This application provides clear motivation for our ongoing investigations into more general IM-based marginalization strategies.', '1406.3521-2-60-0': 'Another important question is if the techniques presented herein can be applied in more complex and high-dimensional mixed-effect models.', '1406.3521-2-60-1': 'In genome-wide association studies, for example, the dimensions of the problem are extremely large.', '1406.3521-2-60-2': 'We expect that, conceptually, the techniques described here will carry over to the more complex scenario.', '1406.3521-2-60-3': 'However, there will be computational challenges to overcome, as with all approaches .', '1406.3521-2-60-4': 'This, along with the incorporation of optimal predictive random sets is a focus of ongoing research.'}

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['1406.3521-1-47-3', '1406.3521-2-46-3'], ['1406.3521-1-47-4', '1406.3521-2-46-4'], ['1406.3521-1-47-5', '1406.3521-2-46-5'], ['1406.3521-1-47-6', '1406.3521-2-46-6'], ['1406.3521-1-47-7', '1406.3521-2-46-7'], ['1406.3521-1-47-8', '1406.3521-2-46-8'], ['1406.3521-1-47-9', '1406.3521-2-46-9'], ['1406.3521-1-21-0', '1406.3521-2-21-0'], ['1406.3521-1-21-1', '1406.3521-2-21-1'], ['1406.3521-1-21-2', '1406.3521-2-21-2'], ['1406.3521-1-21-3', '1406.3521-2-21-3'], ['1406.3521-1-21-4', '1406.3521-2-21-4'], ['1406.3521-1-21-5', '1406.3521-2-21-5'], ['1406.3521-1-21-6', '1406.3521-2-21-6'], ['1406.3521-1-21-7', '1406.3521-2-21-7'], ['1406.3521-1-6-0', '1406.3521-2-6-0'], ['1406.3521-1-6-1', '1406.3521-2-6-1'], ['1406.3521-1-6-2', '1406.3521-2-6-2'], ['1406.3521-1-6-3', '1406.3521-2-6-3'], ['1406.3521-1-6-4', '1406.3521-2-6-4'], ['1406.3521-1-6-5', '1406.3521-2-6-5'], ['1406.3521-1-6-7', '1406.3521-2-6-7'], ['1406.3521-1-33-0', '1406.3521-2-33-0'], ['1406.3521-1-33-1', '1406.3521-2-33-1'], 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'1406.3521-2-23-0'], ['1406.3521-1-23-1', '1406.3521-2-23-1'], ['1406.3521-1-23-2', '1406.3521-2-23-2'], ['1406.3521-1-23-3', '1406.3521-2-23-3'], ['1406.3521-1-23-5', '1406.3521-2-23-5'], ['1406.3521-1-23-6', '1406.3521-2-23-6'], ['1406.3521-1-23-7', '1406.3521-2-23-7'], ['1406.3521-1-31-0', '1406.3521-2-31-0'], ['1406.3521-1-31-1', '1406.3521-2-31-1'], ['1406.3521-1-31-2', '1406.3521-2-31-2'], ['1406.3521-1-31-3', '1406.3521-2-31-3'], ['1406.3521-1-31-4', '1406.3521-2-31-4'], ['1406.3521-1-31-5', '1406.3521-2-31-5'], ['1406.3521-1-31-6', '1406.3521-2-31-6'], ['1406.3521-1-31-7', '1406.3521-2-31-7'], ['1406.3521-1-44-1', '1406.3521-2-43-1'], ['1406.3521-1-48-0', '1406.3521-2-47-0'], ['1406.3521-1-48-1', '1406.3521-2-47-1'], ['1406.3521-1-48-2', '1406.3521-2-47-2'], ['1406.3521-1-48-3', '1406.3521-2-47-3'], ['1406.3521-1-48-4', '1406.3521-2-47-4'], ['1406.3521-1-48-5', '1406.3521-2-47-5'], ['1406.3521-1-1-0', '1406.3521-2-1-0'], ['1406.3521-1-29-0', '1406.3521-2-29-0'], ['1406.3521-1-29-1', '1406.3521-2-29-1'], ['1406.3521-1-29-3', '1406.3521-2-29-3'], ['1406.3521-1-13-0', '1406.3521-2-13-0'], ['1406.3521-1-0-0', '1406.3521-2-0-0'], ['1406.3521-1-0-1', '1406.3521-2-0-1'], ['1406.3521-1-0-3', '1406.3521-2-0-3'], ['1406.3521-1-16-0', '1406.3521-2-16-0'], ['1406.3521-1-16-2', '1406.3521-2-16-1'], ['1406.3521-1-16-3', '1406.3521-2-16-2'], ['1406.3521-1-16-4', '1406.3521-2-16-3'], ['1406.3521-1-16-5', '1406.3521-2-16-4'], ['1406.3521-1-16-6', '1406.3521-2-16-5'], ['1406.3521-1-16-8', '1406.3521-2-16-7'], ['1406.3521-1-51-0', '1406.3521-2-59-0'], ['1406.3521-1-51-1', '1406.3521-2-59-1'], ['1406.3521-1-51-2', '1406.3521-2-59-2'], ['1406.3521-1-51-3', '1406.3521-2-59-3'], ['1406.3521-1-51-4', '1406.3521-2-59-4'], ['1406.3521-1-51-5', '1406.3521-2-59-5'], ['1406.3521-1-15-0', '1406.3521-2-15-0'], ['1406.3521-1-15-1', '1406.3521-2-15-1'], ['1406.3521-1-15-3', '1406.3521-2-15-3'], ['1406.3521-1-15-4', '1406.3521-2-15-4'], ['1406.3521-1-15-5', '1406.3521-2-15-5'], ['1406.3521-1-15-10', '1406.3521-2-15-8'], ['1406.3521-1-5-0', '1406.3521-2-5-0'], ['1406.3521-1-5-1', '1406.3521-2-5-1'], ['1406.3521-1-5-3', '1406.3521-2-5-3'], ['1406.3521-1-5-4', '1406.3521-2-5-4'], ['1406.3521-1-3-2', '1406.3521-2-3-1'], ['1406.3521-1-3-3', '1406.3521-2-3-2'], ['1406.3521-1-3-5', '1406.3521-2-3-4'], ['1406.3521-1-45-0', '1406.3521-2-44-0'], ['1406.3521-1-45-1', '1406.3521-2-44-1'], ['1406.3521-1-45-8', '1406.3521-2-44-4'], ['1406.3521-1-45-9', '1406.3521-2-44-5'], ['1406.3521-1-10-0', '1406.3521-2-10-0'], ['1406.3521-1-10-1', '1406.3521-2-10-1'], ['1406.3521-1-10-2', '1406.3521-2-10-2'], ['1406.3521-1-10-3', '1406.3521-2-10-3'], ['1406.3521-1-10-4', '1406.3521-2-10-4'], ['1406.3521-1-10-5', '1406.3521-2-10-5'], ['1406.3521-1-10-6', '1406.3521-2-10-6'], ['1406.3521-1-32-0', '1406.3521-2-32-0'], ['1406.3521-1-32-1', '1406.3521-2-32-1'], ['1406.3521-1-32-2', '1406.3521-2-32-2'], ['1406.3521-1-32-3', '1406.3521-2-32-3']]

[['1406.3521-1-12-5', '1406.3521-2-12-5'], ['1406.3521-1-6-6', '1406.3521-2-6-6'], ['1406.3521-1-6-8', '1406.3521-2-6-8'], ['1406.3521-1-33-2', '1406.3521-2-33-2'], ['1406.3521-1-49-6', '1406.3521-2-53-6'], ['1406.3521-1-49-13', '1406.3521-2-53-13'], ['1406.3521-1-49-15', '1406.3521-2-53-15'], ['1406.3521-1-52-4', '1406.3521-2-60-4'], ['1406.3521-1-40-0', '1406.3521-2-40-0'], ['1406.3521-1-40-1', '1406.3521-2-40-1'], ['1406.3521-1-44-0', '1406.3521-2-43-0'], ['1406.3521-1-29-2', '1406.3521-2-29-2'], ['1406.3521-1-13-1', '1406.3521-2-13-1'], ['1406.3521-1-0-2', '1406.3521-2-0-2'], ['1406.3521-1-16-7', '1406.3521-2-16-6'], ['1406.3521-1-16-9', '1406.3521-2-16-8'], ['1406.3521-1-5-2', '1406.3521-2-5-2'], ['1406.3521-1-3-0', '1406.3521-2-3-0']]

[]

[['1406.3521-1-24-0', '1406.3521-2-24-0'], ['1406.3521-1-24-7', '1406.3521-2-24-7'], ['1406.3521-1-44-2', '1406.3521-2-43-4'], ['1406.3521-1-13-3', '1406.3521-2-13-2'], ['1406.3521-1-15-8', '1406.3521-2-15-6'], ['1406.3521-1-15-9', '1406.3521-2-15-7'], ['1406.3521-1-3-4', '1406.3521-2-3-3'], ['1406.3521-1-45-4', '1406.3521-2-44-2'], ['1406.3521-1-45-5', '1406.3521-2-44-3']]

[]

['1406.3521-1-15-2', '1406.3521-1-23-4', '1406.3521-1-27-11', '1406.3521-1-35-0', '1406.3521-1-37-0', '1406.3521-2-15-2', '1406.3521-2-23-4', '1406.3521-2-27-11', '1406.3521-2-35-0', '1406.3521-2-37-0', '1406.3521-2-42-5', '1406.3521-2-48-0', '1406.3521-2-50-0', '1406.3521-2-52-0', '1406.3521-2-54-0', '1406.3521-2-55-0', '1406.3521-2-56-0', '1406.3521-2-57-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1406.3521

1112.4147

{'1112.4147-1-0-0': 'In this article, we consider Leray solutions of the Navier-Stokes equations in the exterior of one obstacle in 3D and we study the asymptotic behavior of these solutions when the obstacle shrinks to a curve or to a surface.', '1112.4147-1-0-1': 'In particular, we will prove that a curve has no effect on the motion of a viscous fluid, so it is an erasable singularity for these equations.', '1112.4147-1-1-0': '# Introduction', '1112.4147-1-2-0': 'The present article studies the influence of a thin obstacle on the motion of a three-dimensional incompressible viscous flow.', '1112.4147-1-3-0': 'Such a subject present a large literature on experiments and simulations (see the references in [CITATION]), but concerning a PDE approach, we have waited 2006 with the first work of Iftimie, Lopes and Lopes in [CITATION].', '1112.4147-1-3-1': 'Therein, the authors treated the case of an obstacle which shrinks to a point in a 2D viscous fluid.', '1112.4147-1-3-2': 'They proved that we do not see the effect of the point on the motion of the flow, except if we assume a non-zero circulation for the initial velocity.', '1112.4147-1-3-3': 'Later, the present author proved in [CITATION] that a curve in 2D is not erasable, as in the previous case.', '1112.4147-1-3-4': 'Finally, Iftimie and Kelliher showed in [CITATION] that a point in 3D has no influence on a viscous fluid.', '1112.4147-1-4-0': 'By our physical common sense, all these results are easy to interpret, and we could guess to prove that a surface in 3D has a non-negligible effect on a viscous fluid as the curve in 2D, and that a curve in 3D is an erasable singularity as the point in 2D for the Navier-Stokes equations.', '1112.4147-1-4-1': 'The goal of this article is exactly to show these two points.', '1112.4147-1-4-2': 'We will note that the first point can be obtained by a similar proof than [CITATION], but the case of the curve requires a different way to cut a divergence free function.', '1112.4147-1-5-0': 'More precisely, the motion of an incompressible viscous flow in a domain [MATH] is governed by the Navier-Stokes equations: [EQUATION] where [MATH] denotes the velocity, [MATH] the pressure and [MATH] the viscosity.', '1112.4147-1-5-1': 'The incompressibility condition reads [EQUATION]', '1112.4147-1-5-2': 'The natural boundary condition is the no-slip boundary condition: [EQUATION]', '1112.4147-1-5-3': 'As we will study the asymptotic behavior when an obstacle [MATH] shrinks to a point, a curve or a surface, we need to give an initial data which allows us to compare solutions defined on different domains.', '1112.4147-1-5-4': 'For this reason, we use an important quantity for the 2D ideal flows (ideal means that [MATH]), which is the vorticity: [EQUATION]', '1112.4147-1-5-5': 'As in all the previous works [CITATION], we fix a smooth initial vorticity [MATH], which is divergence free and compactly supported in [MATH].', '1112.4147-1-5-6': 'Let [MATH] be a family of smooth simply connected compact set of [MATH], then we claim that there exists a unique vector field [MATH] on [MATH] such that: [EQUATION]', '1112.4147-1-5-7': 'For such an initial velocity, it is well known that there exists a global weak solution [MATH] of the Navier-Stokes equations [REF]-[REF] on [MATH] in the Leray sense (see Definition [REF]), and the main goal is to study the asymptotic behavior of [MATH] when [MATH] shrinks to a point, a curve or a surface.', '1112.4147-1-6-0': 'In dimension two, the exterior of one obstacle is not simply connected, and the problem [REF] have an infinite number of solution in [MATH].', '1112.4147-1-6-1': 'Indeed, we have to consider the harmonic vector field which behaves like [MATH] at infinity.', '1112.4147-1-6-2': 'Then, to uniquely determine [MATH], the authors in [CITATION] have to fix the initial circulation: [EQUATION].', '1112.4147-1-7-0': 'In the case of one obstacle [MATH] (containing the origin) which shrinks homothetically to [MATH], the main result of [CITATION] states that [MATH] converges to [MATH] in [MATH], where [MATH] is the solution of the Navier-Stokes equations in [MATH] with initial velocity [MATH] verifying: [EQUATION] where [MATH] is the Dirac at the origin.', '1112.4147-1-7-1': 'Therefore, we do not feel the presence of the point, except in the initial data.', '1112.4147-1-7-2': 'In dimension three, the exterior of one obstacle is simply connected, and the circulation has no sense, so the reader should keep in mind the previous result in the case [MATH], i.e. a point in 2D is erasable for the Navier-Stokes equations.', '1112.4147-1-8-0': 'When the obstacle shrinks to a Jordan curve [MATH], [CITATION] shows that the solutions [MATH] converges to the unique solution in the exterior of the curve, where the initial velocity [MATH] verifies [EQUATION] with [MATH] the Dirac on the curve [MATH] and [MATH] a density which can be expressed in terms of [MATH], [MATH] and the shape of the curve.', '1112.4147-1-8-1': 'This function [MATH] is continuous on [MATH] and blows up near the end-points of the curve.', '1112.4147-1-8-2': 'Actually, we can show that [MATH] is a vector field continuous up to the curve, with different values on each sides, except near the end-points where it blows up like the inverse of the square root of the distance.', '1112.4147-1-8-3': 'Moreover, [MATH] is tangent to the curve, and [MATH] is equal to the jump of the velocity through the curve.', '1112.4147-1-8-4': 'Then we see an important difference with the case of the point: even if initially [MATH], the function [MATH] which implies a non-negligible effect of the curve.', '1112.4147-1-9-0': 'A key point in dimension two is the explicit formula for the Biot-Savart law, law giving [MATH] in terms of [MATH], [MATH], and the Riemann mapping [MATH] which sends [MATH] to the exterior of the unit disk.', '1112.4147-1-9-1': 'In [CITATION], the authors chose an homothetic convergence in order to write: [MATH], and putting it in the Biot-Savart formula they obtained some good estimates.', '1112.4147-1-9-2': 'In [CITATION], we assumed the convergence of [MATH] and [MATH], and we exhibited an example of [MATH] where such a convergence holds.', '1112.4147-1-9-3': 'In that article, we were not be able to show that the convergence of the Riemann mappings holds for some"good geometrical convergence".', '1112.4147-1-9-4': 'The convergence of the biholomorphisms when the domains convergence is now established in [CITATION].', '1112.4147-1-10-0': 'To give the result in dimension three, we introduce the Biot-Savart law in [MATH]: [EQUATION] with [MATH] denotes the standart cross product of vectors in [MATH], then [MATH] is the unique vector field in [MATH] verifying: [EQUATION]', '1112.4147-1-10-1': 'With this definition, we will prove that the solution [MATH] of [REF] appears to be the Leray projection (see later for the precise definition) of [MATH]: [EQUATION]', '1112.4147-1-10-2': 'When one obstacle [MATH] shrinks to a point in dimension three, the authors in [CITATION] shows that [MATH] converges to a solution of the Navier-Stokes equations in [MATH] with initial velocity [MATH].', '1112.4147-1-10-3': 'Therefore, a 3D viscous fluid does not feel a material point.', '1112.4147-1-11-0': 'The goal of this article is to treat the case of a curve and a surface in dimension three.', '1112.4147-1-12-0': 'To give a sense to a convergence of vector fields defined on different domains, we set the notation of extensions.', '1112.4147-1-12-1': 'For any function [MATH] defined on [MATH], we denote by [MATH] the extension of [MATH] on [MATH], vanishing on [MATH].', '1112.4147-1-12-2': 'If [MATH] is regular enough and vanishes on [MATH], then [MATH].', '1112.4147-1-12-3': 'Similarly, if [MATH] is a vector field regular enough and tangent to the boundary, then [MATH].', '1112.4147-1-13-0': 'The goal of Section [REF] is to show a similar result to the case of the curve in dimension two.', '1112.4147-1-14-0': 'Let [MATH] be a simply connected compact set of [MATH], and [MATH] be a divergence free function in [MATH].', '1112.4147-1-14-1': 'We consider [MATH] a family of smooth simply connected compact sets converging to [MATH] (in the Hausdorff sense), such that [MATH].', '1112.4147-1-14-2': 'Let [MATH] be a global weak solution to the Navier-Stokes equations on [MATH] (in the sense of Definition [REF]) with initial velocity [MATH] ([REF] the solution of [REF]), then we can extract a subsequence such that [MATH] converges to [MATH] in [MATH], where [MATH] is a global weak solution of the Navier-Stokes equations on [MATH] with initial velocity [MATH].', '1112.4147-1-15-0': 'In this theorem, we do not need any assumption on the regularity of [MATH].', '1112.4147-1-15-1': 'We recall that the existence of a weak solution in non smooth domain is already known in literature (see e.g. [CITATION]).', '1112.4147-1-16-0': 'In the case where [MATH] is a smooth surface, we discuss in Subsection [REF] about some properties of the initial datum, but it is clear from the no-slip condition that the surface is a non-negligible singularity.', '1112.4147-1-17-0': 'Section [REF] is devoted to prove that the curve is an erasable singularity for the 3D Navier-Stokes equations.', '1112.4147-1-18-0': 'Let [MATH] be a [MATH] injective compact curve of [MATH], and [MATH] be a divergence free function in [MATH].', '1112.4147-1-18-1': 'We consider [MATH] a family of smooth simply connected compact sets converging to [MATH] (in the Hausdorff sense), such that [MATH].', '1112.4147-1-18-2': 'Let [MATH] be a global weak solution to the Navier-Stokes equations on [MATH] with initial velocity [MATH] (see [REF]), then we can extract a subsequence such that [MATH] converges to [MATH] in [MATH], where [MATH] is a global weak solution of the Navier-Stokes equations on [MATH] with initial velocity [MATH].', '1112.4147-1-19-0': 'Therefore, we obtain exactly the same theorem than [CITATION], but in the case of a curve, which is the similar result to the point in 2D (see [CITATION]).', '1112.4147-1-19-1': 'If such a theorem seems natural, the proof requires a new way to cut divergence free test function.', '1112.4147-1-19-2': 'The method is based on the Bogovskii results.', '1112.4147-1-20-0': '## Notation and recall', '1112.4147-1-21-0': 'We start by introducing the classical vector fields spaces for the study of incompressible flows.', '1112.4147-1-22-0': 'Let [MATH] an open subset of [MATH].', '1112.4147-1-22-1': 'We denote by', '1112.4147-1-23-0': 'Now, we give some results on these solenoidal spaces.', '1112.4147-1-23-1': 'The following theorem concerns the Leray projection [CITATION].', '1112.4147-1-24-0': 'Let [MATH] be an arbitrary domain in [MATH].', '1112.4147-1-24-1': 'Then [MATH] and [MATH] are orthogonal subspaces in [MATH].', '1112.4147-1-24-2': 'Moreover [EQUATION].', '1112.4147-1-25-0': 'Let us note that the previous theorem does not require any assumption of [MATH].', '1112.4147-1-25-1': 'The validity of such a decomposition in [MATH] requires some restrictions, whereas for [MATH], the Hilbert structure allows us to stay general.', '1112.4147-1-25-2': 'Moreover, we deduce from this theorem the existence of a unique projection operator (called Leray projection): [EQUATION] and by orthogonality we have: [EQUATION].', '1112.4147-1-26-0': 'Let us also mention that Theorem [REF] implies the existence-uniqueness of a solution of [REF], and that this solution is [REF]:', '1112.4147-1-27-0': 'Let [MATH] be a simply connected compact set, and [MATH].', '1112.4147-1-27-1': 'If [MATH], then there exists a unique solution of [EQUATION]', '1112.4147-1-27-2': 'Moreover, [MATH], where [MATH] is given in terms of [MATH] by [REF].', '1112.4147-1-28-0': 'By Theorem [REF], we know that [MATH] is a gradient and then [MATH].', '1112.4147-1-28-1': 'Then [MATH] is a solution, which prove the existence part.', '1112.4147-1-29-0': 'Concerning the uniqueness, let us assume that [MATH] and [MATH] are two solutions.', '1112.4147-1-29-1': 'As [MATH] is simply connected in [MATH], [MATH] on [MATH] implies that there exists [MATH] such that [MATH].', '1112.4147-1-29-2': 'Then [MATH] belongs to [MATH] which is possible only for [MATH] (see Theorem [REF]).', '1112.4147-1-29-3': 'Its ends the proof.', '1112.4147-1-30-0': 'Actually, to infer that the previous corollary implies the existence-uniqueness of [REF], we have to note that some spaces are similar.', '1112.4147-1-30-1': 'If [MATH] and [MATH] seems to be the same (as [MATH] compared to [MATH]), we aware that there exist some counter examples, even in the case of smooth domains (see e.g. [CITATION]).', '1112.4147-1-30-2': 'However, it will not be the case for the domains considered here.', '1112.4147-1-30-3': 'Indeed, in this article we always consider exterior domains [MATH] where [MATH] is a simply connected compact set.', '1112.4147-1-30-4': 'Then, [CITATION] implies that [EQUATION]', '1112.4147-1-30-5': 'The reader can look for the precise definitions in [CITATION], but we state that the exterior of a smooth compact or of a thin surface (as in Subsection [REF]) verifies the cone condition.', '1112.4147-1-31-0': 'Then, Theorems III.2.3 and III.4.2 from [CITATION] can be applied to get the following.', '1112.4147-1-32-0': 'Let [MATH] be the complementary of one simply connected compact set.', '1112.4147-1-32-1': 'If [MATH] verifies the cone condition, then [EQUATION].', '1112.4147-1-33-0': 'The exterior of the curve verifies also the cone condition, but [MATH] has no sense in this case.', '1112.4147-1-34-0': 'Putting together Corollary [REF] and Theorem [REF], we finally get, in the case of smooth simply connected compact obstacle [MATH], the existence-uniqueness of a solution of [REF], and that this solution is [REF].', '1112.4147-1-35-0': 'Now we give precise definition of a global weak solution of the Navier-Stokes equations in the Leray sense.', '1112.4147-1-36-0': 'Let [MATH].', '1112.4147-1-36-1': 'We say that [MATH] is a global weak solution of the Navier-Stokes equations on [MATH] with initial velocity [MATH] iff', '1112.4147-1-37-0': 'Without any assumption about the regularity of [MATH], the Leray theorem states that there exists a global weak solution of the Navier-Stokes equations in the sense of the previous definition (see e.g. [CITATION] and [CITATION]).', '1112.4147-1-37-1': 'So the goal here is to choose one solution [MATH] on [MATH] and to prove that the limit of [MATH] is a weak solution of the Navier-Stokes equations on [MATH] or on [MATH].', '1112.4147-1-38-0': '# Convergence of viscous flows when the obstacles converge to a singular compact', '1112.4147-1-39-0': '## Convergence of the initial velocity', '1112.4147-1-40-0': 'Let [MATH] be a simply connected compact set.', '1112.4147-1-40-1': 'We consider [MATH] a family of compact, simply connected, smooth set of [MATH] such that [MATH] and [MATH] converges to [MATH] in the Hausdorff topology when [MATH].', '1112.4147-1-41-0': 'The only property needed here concerning Hausdorff topology is the following.', '1112.4147-1-42-0': 'If [MATH] converges in the Hausdorff sense to [MATH] and [MATH] is a compact subset of [MATH], then there exists [MATH] such that [MATH] for all [MATH].', '1112.4147-1-43-0': 'As mentioned in the introduction, we fix [MATH] and we define [MATH] as in [REF], which is the unique vector field verifying [REF].', '1112.4147-1-43-1': 'Let us show that [MATH] converges strongly to [MATH].', '1112.4147-1-43-2': 'We recall that [MATH] is the extension by zero in [MATH].', '1112.4147-1-44-0': 'With the above notations, we have that [EQUATION].', '1112.4147-1-45-0': 'As the Leray projection is orthogonal in [MATH], we get from [REF] that [EQUATION].', '1112.4147-1-45-1': "By the Banach-Alaoglu's theorem, we infer that there exists [MATH] and a subsequence [MATH], such that [EQUATION].", '1112.4147-1-45-2': 'This weak convergence implies in particular that [EQUATION] and [EQUATION].', '1112.4147-1-45-3': 'In the proof of Corollary [REF] we prove that [MATH].', '1112.4147-1-45-4': 'Then, Corollary [REF] implies that [MATH] and then [EQUATION].', '1112.4147-1-45-5': 'Putting together the two last inequalities, we get [MATH].', '1112.4147-1-45-6': 'Using the weak convergence in [MATH] of [MATH] to [MATH] we obtain the strong convergence in [MATH].', '1112.4147-1-46-0': 'As [MATH] belongs to [MATH], we deduce directly from the [MATH] strong convergence that [MATH] belongs to [MATH].', '1112.4147-1-46-1': 'Then we have two functions in [MATH] having the same vorticity, which implies that [MATH] (see Corollary [REF]).', '1112.4147-1-47-0': 'We have obtained in the previous proof that [EQUATION] but we do not have [EQUATION].', '1112.4147-1-47-1': 'Even in the case where [MATH] is a thin surface, we can just pretend that [MATH] belongs to [MATH] and supported on the surface.', '1112.4147-1-47-2': 'Actually, we will prove in Subsection [REF] that [EQUATION] where [MATH] is the jump of the tangential component of [MATH].', '1112.4147-1-48-0': 'In the case of the curve, we will show in Subsection [REF] that there is no function belonging in [MATH] compactly supported on a curve, and we will obtain that [MATH].', '1112.4147-1-49-0': '## Time evolution', '1112.4147-1-50-0': 'For all [MATH], we denote by [MATH] a global weak solution of the Navier-Stokes equations (in the sense of Definition [REF]) on [MATH] with initial data [MATH].', '1112.4147-1-51-0': 'By Proposition [REF], we already know that [MATH] in [MATH].', '1112.4147-1-51-1': 'Moreover, thanks to the energy inequality [REF], we state that [EQUATION]', '1112.4147-1-51-2': 'Now, we need to establish a temporal estimates.', '1112.4147-1-51-3': 'First, we use that [MATH] verifies the Dirichlet boundary condition for [MATH] in order to write the following Sobolev inequalities: [EQUATION] where [MATH] is independent of [MATH].', '1112.4147-1-51-4': 'We consider in a first step [MATH] an open bounded set relatively compact in [MATH].', '1112.4147-1-51-5': 'By Proposition [REF], there exists [MATH] such that [MATH] for all [MATH], then we have for any [MATH] and [MATH]: [EQUATION] where we have used [REF].', '1112.4147-1-51-6': 'This inequality implies that [MATH] is equicontinous as a family of functions from [MATH] to [MATH].', '1112.4147-1-51-7': 'Using that this family is bounded in [MATH], Ascoli theorem gives that [MATH] is precompact in [MATH].', '1112.4147-1-51-8': 'Moreover, [MATH] is also bounded in [MATH], then we get by interpolation that this family is precompact in [MATH].', '1112.4147-1-51-9': 'Finally, by a diagonal extraction on the compact sets of [MATH], we conclude that [EQUATION].', '1112.4147-1-52-0': 'Therefore, there exists a subsequence [MATH] such that [MATH] strongly in [MATH].', '1112.4147-1-52-1': 'Moreover, extracting a subsequence if necessary, we know from [REF] that [EQUATION]', '1112.4147-1-52-2': 'For any test function [MATH], there exist [MATH], [MATH] such that [EQUATION].', '1112.4147-1-52-3': 'For [MATH] fixed, there exists [MATH] such that [MATH], then [REF] reads [EQUATION].', '1112.4147-1-52-4': 'Thanks to the strong convergence of [MATH] in [MATH] and the weak of [MATH] in [MATH] then we can pass to the limit [MATH] to get: [EQUATION].', '1112.4147-1-52-5': 'In particular, this equality putting together with [REF] and [REF] gives that [EQUATION].', '1112.4147-1-52-6': 'Then we can pass to the limit in [EQUATION] as [MATH] to get [EQUATION].', '1112.4147-1-52-7': 'This equality implies that [EQUATION] in the sense of distribution in [MATH].', '1112.4147-1-52-8': 'Since the right hand side term belongs to [MATH], the equality holds in [MATH].', '1112.4147-1-53-0': 'Now, we check that [MATH] belongs in the good functional space.', '1112.4147-1-53-1': 'Thanks to the previous equality, and [REF], then we can easily prove that [EQUATION].', '1112.4147-1-53-2': 'This argument can be found in [CITATION]: as [MATH] belongs to [MATH], then its implies that [MATH] is almost everywhere equal to a function continuous from [MATH] into [MATH].', '1112.4147-1-53-3': 'Moreover, using the fact that [MATH], then [CITATION] states that the continuity in [MATH] implies the weak continuity in time with values in [MATH].', '1112.4147-1-54-0': 'Moreover, thanks to the continuity in [MATH], we infer that the equality [REF] in the sense of [MATH] implies that the integral equality [REF] holds for all [MATH].', '1112.4147-1-54-1': 'Indeed, for the initial data we know from the uniform convergence in [MATH] that [MATH] in [MATH].', '1112.4147-1-54-2': 'However, we proved in Proposition [REF] that [MATH] in [MATH], which allows us to state by the uniqueness of the limit in [MATH] that the initial velocity is [MATH].', '1112.4147-1-55-0': 'To finish the proof of Theorem [REF], we have to prove the energy inequality.', '1112.4147-1-55-1': 'We take the liminf of [REF]: [EQUATION] and we note that the weak limit in [MATH] of [MATH] to [MATH] and the weak limit in [MATH] of [MATH] to [MATH] imply that [EQUATION].', '1112.4147-1-55-2': 'It gives the last point required in Definition [REF], which ends the proof of Theorem [REF].', '1112.4147-1-56-0': 'Therefore, we have shown that the Navier-Stokes solutions converge when domains convergence to the exterior of a simply connected compact set [MATH].', '1112.4147-1-56-1': 'We note here that we do not assume any assumption on the regularity of [MATH].', '1112.4147-1-56-2': 'In particular, this result holds if [MATH] is a surface [MATH].', '1112.4147-1-56-3': 'As [MATH] for a.e. time, it is clear that the surface has a non-negligible effect on the motion of 3D viscous flow: [MATH] verifies for almost every time the no slip boundary condition.', '1112.4147-1-56-4': 'In the following subsection, we discuss about the initial velocity properties in the particular case of the surface, and the goal is to get some similarities to the curve in 2D.', '1112.4147-1-57-0': '## Remark on the behavior of the initial velocity near a smooth surface', '1112.4147-1-58-0': 'In the two dimensional case, we obtain in [CITATION] an explicit formula of the initial velocity in terms of the biholomorphisms.', '1112.4147-1-58-1': 'This formula allows us to state that [MATH] is continuous up to the curve with different values on each side, except near the end-points where it behaves as the inverse of the square root of the distance.', '1112.4147-1-58-2': 'In our case, we do not have this formula, and we use in this subsection classical elliptic theory in order to get similar results in the case where [MATH] is a bounded orientable surface of codimension 1 in [MATH].', '1112.4147-1-59-0': 'As mentioned in Remark [REF], [MATH] is curl free in [MATH] and as [MATH] is simply connected we infer that there exists [MATH] such that [MATH].', '1112.4147-1-59-1': 'We know that [MATH] is continuous up to the boundary, then the goal is to determine the behavior of [MATH] near [MATH] where [MATH] verifies the following elliptic problem [EQUATION] where [MATH] is regular on [MATH].', '1112.4147-1-60-0': 'This subsection is independent of the convergence theory, and the goal here is just to give an example of behavior of [MATH].', '1112.4147-1-60-1': 'Therefore, we add here some conditions on [MATH] in order to apply classical elliptic results.', '1112.4147-1-61-0': 'We assume that [MATH] is a [MATH] manifold and its boundary [MATH] is a [MATH] closed curve.', '1112.4147-1-62-0': 'The study of elliptic equations in the exterior of a surface with the Neumann condition is standard for the crack problem in 3D linear elasticity.', '1112.4147-1-62-1': 'Actually, to get exactly the Neumann boundary condition, we add a regular function [MATH] such that [MATH] verifies [EQUATION].', '1112.4147-1-62-2': 'For [MATH] regular enough, we rich the necessary regularity for [MATH] in order to state that [MATH] have an expansion near [MATH] on the form [EQUATION] in local polar coordinates [MATH].', '1112.4147-1-62-3': 'Such a result is proved in [CITATION] (see also the references therein).', '1112.4147-1-62-4': 'In particular, it implies that [MATH] is continuous up to [MATH], with possibly different values on each side, except near the boundary [MATH] where [MATH] behaves like the inverse of the square root of the distance.', '1112.4147-1-62-5': 'Therefore, we obtain exactly the same behavior in 3D in the exterior of a surface than in 2D in the exterior of a curve.', '1112.4147-1-63-0': 'Thanks to the continuity up to the surface, it is easy to see that the tangent condition implies that [EQUATION] in [MATH], where [MATH] is the jump of the tangential component of [MATH] through the surface.', '1112.4147-1-64-0': '# Viscous flow around a curve', '1112.4147-1-65-0': 'As in the previous section and as in [CITATION], we consider [MATH] a family of compact, simply connected, smooth set of [MATH] such that [MATH] converges to [MATH] in the Hausdorff topology when [MATH].', '1112.4147-1-65-1': 'Here, [MATH] is assumed to be a compact injective [MATH] curve, i.e. of dimension 1 in [MATH].', '1112.4147-1-65-2': 'The goal of this section is to prove that the curve is a negligible singularity for the Naviers-Stokes solutions in [MATH].', '1112.4147-1-66-0': '## Convergence of the initial velocity', '1112.4147-1-67-0': 'In the following subsections, we will need to use a suitable cutoff function of a small neighborhood of [MATH].', '1112.4147-1-67-1': 'Let [MATH] be a function verifying [EQUATION] then we define [EQUATION]', '1112.4147-1-67-2': 'Although it is obvious that [MATH] vanishes in a small neighborhood of [MATH], we need the following estimates of [MATH].', '1112.4147-1-68-0': 'Let [MATH] be a [MATH] compact injective curve.', '1112.4147-1-68-1': 'There exist [MATH] and [MATH] such that for all [MATH], [EQUATION] where mes is the Lebesgue measure.', '1112.4147-1-69-0': 'Even if such a proposition seems standard, we write the details in Appendix [REF] for a sake of completeness.', '1112.4147-1-69-1': 'Thanks to this proposition, we can prove that there is no function supported on [MATH] which belongs to [MATH].', '1112.4147-1-70-0': 'Let [MATH] a function belonging in [MATH].', '1112.4147-1-70-1': 'If [EQUATION] then [MATH] in [MATH].', '1112.4147-1-71-0': 'We fix [MATH], and we introduce [EQUATION] where [MATH] is the cutoff function defined in [REF].', '1112.4147-1-71-1': 'As we proved in Appendix [REF] that [MATH] is [MATH] supported in the exterior of a small neighborhood of [MATH], then by assumption (and density of [MATH] in [MATH] for [MATH] norm), we have that [EQUATION].', '1112.4147-1-71-2': 'Thanks to Proposition [REF], we get easily that [EQUATION].', '1112.4147-1-71-3': 'By Banach-Alaoglu theorem, we can extract a subsequence such that [MATH] weakly in [MATH].', '1112.4147-1-71-4': 'In particular, it implies that [EQUATION] as [MATH].', '1112.4147-1-71-5': 'Then, we have proved that for all [MATH], we have [MATH].', '1112.4147-1-72-0': 'Now, we can come back to our main problem.', '1112.4147-1-72-1': 'Let [MATH] be an injective compact smooth curve in [MATH].', '1112.4147-1-72-2': 'We consider [MATH] a family of compact, simply connected, smooth set of [MATH] such that [MATH] converges to [MATH] in the Hausdorff topology when [MATH].', '1112.4147-1-72-3': 'Let [MATH] an initial vorticity which is bounded and compactly supported in [MATH].', '1112.4147-1-72-4': 'As in the previous section we set [EQUATION] then we define by [MATH] the unique vector field in [MATH] solving [REF].', '1112.4147-1-72-5': 'We prove in the following proposition that we do not feel the presence of the curve for the limit initial velocity.', '1112.4147-1-73-0': 'With the above notation, we have [EQUATION] where [MATH] is the velocity field without obstacle [REF].', '1112.4147-1-74-0': 'As [MATH] is continuous and behaves like [MATH] at infinity, we obtain directly that [MATH] belongs to [MATH] for any [MATH].', '1112.4147-1-75-0': 'Firstly, we introduce the stream function corresponding to [MATH]: [EQUATION] which verifies [MATH], [MATH] and [MATH] (see e.g. [CITATION]), so [MATH] is bounded.', '1112.4147-1-76-0': 'Secondly, we define [EQUATION] where [MATH] is the cutoff function [REF].', '1112.4147-1-76-1': 'By construction, we deduce that [MATH] belongs to [MATH].', '1112.4147-1-77-0': 'Finally, we use that [MATH] is the [MATH] projection of [MATH] on [MATH], to compute [EQUATION] where we have used Proposition [REF].', '1112.4147-1-77-1': 'This inequality allows us to pretend that there exist [MATH] and a subsequence such that [EQUATION].', '1112.4147-1-77-2': 'By passing to the limit, we obtain that [MATH] for all test function in [MATH].', '1112.4147-1-77-3': 'Moreover, [MATH] and [MATH] belongs to [MATH] and we apply the previous lemma to state that [MATH] on [MATH], and then [MATH].', '1112.4147-1-78-0': 'Moreover, reasoning as in the proof of Proposition [REF], we pass from the weak convergence to the strong convergence of [MATH] to [MATH], which ends this proof.', '1112.4147-1-79-0': '## Proof of Theorem [REF]', '1112.4147-1-80-0': 'The begin of the proof follows the same idea of the proof of Theorem [REF]: we prove that we have a strong limit of [MATH] to [MATH] a solution of the Navier-Stokes equations in "the exterior of the curve".', '1112.4147-1-80-1': 'The second step is to show that [MATH] is actually a solution in [MATH].', '1112.4147-1-81-0': 'For all [MATH], we consider [MATH] a global weak solution, in the sense of Definition [REF], of [REF]-[REF] in [MATH] with initial datum [MATH], which is defined in the previous subsection.', '1112.4147-1-82-0': 'By Proposition [REF], we already know that [MATH] in [MATH].', '1112.4147-1-82-1': 'Next, we do exactly the same thing than in Subsection [REF]:', '1112.4147-1-83-0': 'Now, we need to prove that the momentum equation is verified for all [MATH].', '1112.4147-1-83-1': 'Then, to finish the proof of Theorem [REF], we have to establish [REF] for test functions which meet the curve.', '1112.4147-1-83-2': 'The following lemma will be the key of this extension.', '1112.4147-1-84-0': 'For all [MATH] such that [MATH], then there exists a sequence [MATH] such that [EQUATION].', '1112.4147-1-85-0': 'All the difficulty comes from the condition [MATH].', '1112.4147-1-85-1': 'Indeed, without this condition, it is sufficient to multiply by the cutoff function.', '1112.4147-1-85-2': 'In [CITATION], the standard way to construct divergence free functions compactly supported outside the obstacle is to multiply the stream function by the cutoff function.', '1112.4147-1-85-3': 'However, we see in the proof of Proposition [REF] that the computation of the [MATH] norm makes appear [MATH] which blows up strongly in our case.', '1112.4147-1-86-0': 'So, we present here a new way to approximate divergence free function, which is not explicit as in the standard way.', '1112.4147-1-86-1': 'The following method comes from [CITATION] and is based on the Bogovskii operator.', '1112.4147-1-86-2': 'We set [EQUATION] which belongs to [MATH] but which is not divergence free.', '1112.4147-1-86-3': 'Its divergence [EQUATION] verifies the following estimates for all [MATH]: [EQUATION] where [MATH] depends only on [MATH].', '1112.4147-1-86-4': 'To correct this divergence, we use the result of Bogovskii.', '1112.4147-1-86-5': 'Let [MATH] be a ball big enough containing the support of [MATH].', '1112.4147-1-86-6': 'As the set [MATH] verifies the cone condition, and as [MATH] with [MATH], then Theorem 3.1 in [CITATION] states that there exists at least one solution [MATH] of the following problem: [EQUATION].', '1112.4147-1-86-7': 'Moreover, the constant [MATH] depends only on [MATH] and [MATH], and as [MATH] as a compact support in [MATH], so is [MATH].', '1112.4147-1-86-8': 'Then, we can define for all [MATH], [MATH], and we have that [MATH] strongly in [MATH] for all [MATH] and uniformly bounded in [MATH].', '1112.4147-1-86-9': 'So we can extract a subsequence such that [MATH] converges weak-[MATH] in [MATH] and by uniqueness of the limit in [MATH] it converges to [MATH].', '1112.4147-1-87-0': 'From the definition we get that [MATH] converges to [MATH] strongly in [MATH] for any [MATH] and weak-[MATH] in [MATH].', '1112.4147-1-88-0': 'In conclusion, [MATH] is divergence free, compactly supported in [MATH] and converges to [MATH] weak-[MATH] in [MATH].', '1112.4147-1-88-1': 'Smoothing [MATH] by some mollifiers, we obtain the result.', '1112.4147-1-89-0': 'In the previous proof, we see that the approximation constructed verifies also [EQUATION].', '1112.4147-1-89-1': 'Therefore, we apply the previous lemma in order to pass to the limit in [REF], which implies that this equality is verified for all test function in [MATH].', '1112.4147-1-89-2': 'Now, we finish as we did in Subsection [REF]:', '1112.4147-1-90-0': 'Its ends to prove that [MATH] is a global weak solution of the Navier-Stokes solutions in [MATH], in the sense of Definition [REF].', '1112.4147-1-91-0': 'We finish the proof of Theorem [REF] by noting that the uniform estimates [REF] and the convergence in [MATH] implies that [EQUATION].', '1112.4147-1-91-1': 'Indeed, for any [MATH] and [MATH] a compact of [MATH], we decompose the [MATH] norm thanks to [REF]: [EQUATION] where [MATH] is a time bigger than all [MATH].', '1112.4147-1-91-2': 'Using the uniform estimates, we deduce from Proposition [REF] that the second right hand side term tends to zero as [MATH], uniformly in [MATH].', '1112.4147-1-91-3': 'So there exists [MATH] such that this term is less than [MATH].', '1112.4147-1-91-4': 'For this [MATH] fixed, the first right hand side term tends to zero as [MATH] because the compact does not meet [MATH].', '1112.4147-1-91-5': 'Then, there exists [MATH] such that this term is less than [MATH] for any [MATH], which ends the proof.', '1112.4147-1-92-0': '# Cutoff functions', '1112.4147-1-93-0': 'In order to differentiate the function [MATH], we have to check that the minimal distance [MATH] is reached for a unique [MATH], at least for [MATH] closed enough to [MATH].', '1112.4147-1-93-1': 'We need to assume that the curve is at least [MATH], because such a property is false near a corner.', '1112.4147-1-93-2': 'For example, if [MATH] is the curve [MATH] then for all [MATH] on the half line [MATH], we have two minimums on [MATH]: [MATH] and [MATH] and [MATH].', '1112.4147-1-94-0': 'This example shows that we have to assume some regularity for the curve.', '1112.4147-1-95-0': 'Let [MATH] be a [MATH] injective curve, then there exists [MATH] such that [EQUATION].', '1112.4147-1-96-0': 'Let [MATH] be the arclength parametrization.', '1112.4147-1-96-1': 'Then, we recall the standard definition:', '1112.4147-1-97-0': 'As [MATH] is assumed to be [MATH], there exists [MATH] such that [MATH] on [MATH].', '1112.4147-1-98-0': 'The existence of [MATH] such that [MATH] is obvious because the map [MATH] is continuous on the compact [MATH].', '1112.4147-1-98-1': 'Let us assume that the conclusion of the lemma is false, i.e. that the infimum is reached twice: [EQUATION] such that [EQUATION].', '1112.4147-1-98-2': 'Extracting a subsequence, we have by compactness that [MATH].', '1112.4147-1-98-3': 'As the curve is assumed to be injective (so without cross-point), we infer that [MATH].', '1112.4147-1-98-4': 'So, by continuity of [MATH] in [MATH], we note easily that there exists [MATH] such that [EQUATION]', '1112.4147-1-98-5': 'Next, we introduce the following function [EQUATION] and we differentiate twice to get [EQUATION].', '1112.4147-1-98-6': 'The previous computation holds even if [MATH].', '1112.4147-1-99-0': 'By assumption, [MATH] is minimal for [MATH] and [MATH], so [EQUATION] but [REF] implies that [EQUATION] which is impossible.', '1112.4147-1-99-1': 'This contradiction allows us to end this proof.', '1112.4147-1-100-0': 'In the previous proof, we infer from [MATH] that [MATH] when [MATH] is the closest point of [MATH] to [MATH].', '1112.4147-1-101-0': 'Now, we link the differentiability of [MATH] and the previous lemma.', '1112.4147-1-102-0': 'Let [MATH] be a [MATH] injective curve and [MATH] from Lemma [REF].', '1112.4147-1-102-1': 'Then for all [MATH] such that [MATH], [MATH] is differentiable in [MATH] and [EQUATION] where [MATH] is the unique point of [MATH] such that [MATH].', '1112.4147-1-103-0': 'This proof can be found in the exercise book [CITATION] and we copy it for a sake of completeness.', '1112.4147-1-103-1': 'We fix [MATH] such that [MATH], then Lemma [REF] states that there exists a unique [MATH] verifying [MATH].', '1112.4147-1-104-0': 'The first step consists to show the following property: for all [MATH] small enough, we set [MATH] by the unique point of [MATH] such that [MATH], then [EQUATION]', '1112.4147-1-104-1': 'Let us assume that [REF] is false, then there exist a sequence [MATH] and [MATH] such that [MATH] for all [MATH].', '1112.4147-1-104-2': 'We introduce the following compact set [EQUATION] and the function [EQUATION].', '1112.4147-1-104-3': 'This function is continuous and reaches his minimum.', '1112.4147-1-104-4': 'As [MATH], the uniqueness part of Lemma [REF] allows us to check that this minimum is strictly greater than [MATH].', '1112.4147-1-104-5': 'Hence, we have [EQUATION].', '1112.4147-1-104-6': 'However, we use the fact that [MATH] belongs to [MATH] to write [EQUATION] implying that [EQUATION] which is impossible by the continuity of [MATH].', '1112.4147-1-104-7': 'The property [REF] follows from this contradiction.', '1112.4147-1-105-0': 'Let us prove in the second step that [REF] implies that [MATH] is differentiable.', '1112.4147-1-105-1': 'On the one hand, we have that [EQUATION]', '1112.4147-1-105-2': 'On the other hand, we compute [EQUATION]', '1112.4147-1-105-3': 'Hence, [REF] implies that [EQUATION]', '1112.4147-1-105-4': 'Putting together [REF]-[REF], we conclude that [MATH] is differentiable and that [EQUATION] which ends the proof.', '1112.4147-1-106-0': 'From the value of [MATH], we see that the uniqueness part of Lemma [REF] is mandatory to obtain the differentiability of [MATH] (or of [MATH]).', '1112.4147-1-107-0': 'Finally, we can establish the estimates of the cutoff function by proving Proposition [REF].', '1112.4147-1-108-0': 'Thanks to the definition and Lemma [REF], we compute for [MATH] [EQUATION] where [MATH] is the unique point of [MATH] such that [MATH].', '1112.4147-1-108-1': 'Indeed, [MATH] iff [MATH], so [MATH] and [MATH] which allows us to apply the previous lemmas.', '1112.4147-1-109-0': 'Hence, we obtain directly the first point: [EQUATION]', '1112.4147-1-110-0': 'Concerning the support, we use Remark [REF] to note that for all [MATH], then the unique [MATH] such that [MATH] verifies [MATH].', '1112.4147-1-110-1': 'If there is an interval where [MATH], then it implies that [MATH] on this part is a segment.', '1112.4147-1-110-2': 'Then, it is obvious that the volume such that [MATH] is [MATH] on this section.', '1112.4147-1-110-3': 'So, without loss of generality, we assume that [MATH] on [MATH], and the previous remark implies that [EQUATION] and the application [MATH]: [MATH] is a diffeomorphism invertible where [MATH] gives the parameter of [MATH] where [MATH] is reached, the distance [MATH], and the angle between [MATH] and [MATH].', '1112.4147-1-110-4': 'In the expression of [MATH], [MATH] denotes the half ball on each endpoints, which obviously verify [MATH].', '1112.4147-1-111-0': 'Using the general relation [EQUATION] where [MATH] is a smooth function called the torsion, then we can compute [MATH] in the orthonormal base [MATH]: [EQUATION] which implies that [MATH] if [MATH] is chosen small enough (i.e. [MATH]).', '1112.4147-1-111-1': 'Finally, we can compute for [MATH] big enough: [EQUATION] which ends the proof of Proposition [REF].'}

{'1112.4147-2-0-0': 'In this article, we consider Leray solutions of the Navier-Stokes equations in the exterior of one obstacle in 3D and we study the asymptotic behavior of these solutions when the obstacle shrinks to a curve or to a surface.', '1112.4147-2-0-1': 'In particular, we will prove that a solid curve has no effect on the motion of a viscous fluid, so it is a removable singularity for these equations.', '1112.4147-2-1-0': '# Introduction', '1112.4147-2-2-0': 'The present article is devoted to the stability of the Navier-Stokes equations when one obstacle shrinks to a curve or a surface, and to determine the influence of a thin obstacle on the motion of a three-dimensional incompressible viscous flow.', '1112.4147-2-2-1': 'More precisely, for any obstacle [MATH], i.e. verifying [EQUATION] we consider the 3D-Navier-Stokes equations on [MATH]: [EQUATION] where [MATH] denotes the velocity, [MATH] the pressure and [MATH] the viscosity.', '1112.4147-2-2-2': 'The incompressibility and the no-slip boundary conditions reads [EQUATION]', '1112.4147-2-2-3': 'A natural quantity for incompressible flows is the vorticity: [EQUATION]', '1112.4147-2-2-4': 'As the domains [MATH] depend on [MATH], it is standard to give an initial condition in terms of a vorticity independent of [MATH] (see [CITATION]): for any smooth initial vorticity [MATH] which is divergence free and compactly supported in [MATH], Lemma [REF] states that there exists a unique vector field [MATH] on [MATH] such that: [EQUATION]', '1112.4147-2-2-5': 'For such an initial velocity, it is well known that there exists a global weak solution [MATH] of the Navier-Stokes equations [REF]-[REF] on [MATH] in the sense of Leray (see Definition [REF]).', '1112.4147-2-3-0': 'The purpose of this paper is to study the asymptotic behavior of [MATH] when [MATH] shrinks to a curve or a surface.', '1112.4147-2-4-0': '## Leray solutions', '1112.4147-2-5-0': 'In order to give precisely the main theorems, we recall here the notion of weak solution in the sense of Leray.', '1112.4147-2-6-0': 'We begin by introducing the classical solenoidal vector fields spaces.', '1112.4147-2-7-0': 'Let [MATH] an open subset of [MATH].', '1112.4147-2-7-1': 'We denote by', '1112.4147-2-8-0': 'For any arbitrary domain [MATH] in [MATH], we know that [MATH] and [MATH] are orthogonal subspaces in [MATH] (see e.g. [CITATION]).', '1112.4147-2-8-1': 'Moreover [EQUATION] which implies the existence of a unique projection operator (called Leray projection): [EQUATION] and by orthogonality we have: [EQUATION].', '1112.4147-2-9-0': 'Let us also mention that [REF] implies the existence and uniqueness of a solution of [REF]:', '1112.4147-2-10-0': 'Let [MATH] be a smooth obstacle (in the sense of [REF]) and [MATH].', '1112.4147-2-10-1': 'If [MATH], then there exists a unique solution of [REF].', '1112.4147-2-10-2': 'Moreover, [EQUATION] where [MATH] is the Biot-Savart law in [MATH]: [EQUATION]', '1112.4147-2-10-3': 'In [REF], [MATH] denotes the standard cross product of vectors in [MATH].', '1112.4147-2-10-4': 'We note that [MATH] is the unique vector field in [MATH] verifying: [EQUATION]', '1112.4147-2-10-5': 'For sake of completeness, this lemma is proved in Appendix [REF].', '1112.4147-2-11-0': 'Now we can give the definition of a global weak solution of the Navier-Stokes equations in the Leray sense.', '1112.4147-2-12-0': 'Let [MATH].', '1112.4147-2-12-1': 'We say that [MATH] is a global weak solution of the Navier-Stokes equations on [MATH] with initial velocity [MATH] iff', '1112.4147-2-13-0': 'Without any assumption about the regularity of [MATH], the Leray theorem states that there exists a global weak solution of the Navier-Stokes equations in the sense of the previous definition (see e.g. [CITATION] and [CITATION]).', '1112.4147-2-14-0': '## Main results', '1112.4147-2-15-0': 'In Section [REF], we establish that the Navier-Stokes equations is structurally stable under Hausdorff approximations of the fluid domains:', '1112.4147-2-16-0': 'Let [MATH] be an obstacle of [MATH] (in the sense of [REF]) which is a limit (in the Hausdorff sense) of a sequence of smooth obstacles [MATH].', '1112.4147-2-16-1': 'Let [MATH] and [MATH] be a global weak solution to the Navier-Stokes equations on [MATH] (in the sense of Definition [REF]) with initial velocity [MATH] (given by [REF], which is the solution of [REF]).', '1112.4147-2-16-2': 'Then we can extract a subsequence such that [MATH] converges weakly-[MATH] to [MATH] in [MATH], where [MATH] is a global weak solution of the Navier-Stokes equations on [MATH] with initial velocity [MATH].', '1112.4147-2-17-0': 'Here, [MATH] is the extension of [MATH] on [MATH], vanishing on [MATH], and [MATH] converges to [MATH] in the Hausdorff sense if and only if the Hausdorff distance between [MATH] and [MATH] converges to zero.', '1112.4147-2-17-1': 'See for example [CITATION] for more details about the Hausdorff topology, in particular the Hausdorff convergence implies the following proposition: [EQUATION]', '1112.4147-2-17-2': 'Actually, we will also show that for any sequence [MATH] which verifies [MATH] (for the [MATH] norm), then we can extract a subsequence such that [MATH] converges weakly-[MATH] to [MATH] in [MATH], where [MATH] is a global weak solution of the Navier-Stokes equations on [MATH] with initial velocity [MATH].', '1112.4147-2-17-3': 'Of course, to pass to the limit in the non-linear term, we will need a strong compactness argument in [MATH].', '1112.4147-2-17-4': 'However, the precise statement is not very convenient to give here.', '1112.4147-2-17-5': 'Indeed, we will decompose the velocity in two parts (depending on the compact subset of [MATH]) and we will prove the strong compactness only of one part of [MATH] (see Subsection [REF] for more details).', '1112.4147-2-18-0': 'More importantly, we wonder for which condition [MATH] is removable, i.e. [MATH] is the solution of the Navier-Stokes on the full space [MATH].', '1112.4147-2-18-1': 'Such an issue presents a large literature on experiments and simulations (see e.g. [CITATION] and references therein).', '1112.4147-2-18-2': 'As the solution [MATH] belongs to [MATH] for a.e. time, the natural notion is the Sobolev [MATH] capacity of the obstacle, which is defined by [EQUATION].', '1112.4147-2-18-3': 'The capacity is not a measure, but has similar good properties.', '1112.4147-2-18-4': 'For nice sets [MATH] in [MATH], the capacity of [MATH] can be thought very roughly as some [MATH] dimensional Hausdorff measure of its boundary.', '1112.4147-2-18-5': 'More precisely:', '1112.4147-2-19-0': 'For all compact set [MATH] included in a bounded open set [MATH],', '1112.4147-2-20-0': '[MATH].', '1112.4147-2-20-1': 'If [MATH] is contained in a manifold of dimension [MATH], then [MATH].', '1112.4147-2-20-2': 'If [MATH] contains a piece of some smooth hypersurface (manifold of dimension N-1), then [MATH].', '1112.4147-2-20-3': 'Let [MATH] and [MATH] be open sets such that [MATH].', '1112.4147-2-20-4': 'Then [EQUATION] which means that [MATH] except on a set with zero capacity.', '1112.4147-2-21-0': 'We refer to [CITATION] for all details on the Sobolev capacity (see [CITATION] for a short summary).', '1112.4147-2-22-0': 'If [MATH] is a compact subset of [MATH] which contains a piece of smooth hypersurface, then we infer from (3) and (4) that for a.e. [MATH], [MATH] vanishes quasi everywhere in [MATH], where [MATH].', '1112.4147-2-22-1': 'Therefore, a surface is not removable for the 3D viscous fluid, as a curve for the 2D viscous fluid (case treated in [CITATION]).', '1112.4147-2-23-0': 'Then, we turn to the obstacles with zero capacity.', '1112.4147-2-23-1': 'Iftimie, Lopes Filho and Nussenzveig Lopes have considered in [CITATION] the 2D case where one obstacle shrinks homotetically to a point, whereas Iftimie and Kelliher show in [CITATION] that a point in 3D has no influence on a viscous fluid.', '1112.4147-2-24-0': 'There is a small restriction in dimension two, due to the fact that the exterior of one obstacle is not simply connected.', '1112.4147-2-24-1': 'In this case the vorticity is not sufficient to determine uniquely the velocity, and we have also to prescribe the initial circulation of [MATH] around [MATH].', '1112.4147-2-24-2': 'Nevertheless, if this initial circulation is assumed to be zero, [CITATION] exactly states that a material point has no effect on a 2D fluid (see [CITATION] for more details).', '1112.4147-2-25-0': 'Section [REF] is devoted to prove that a curve is a removable singularity for the 3D Navier-Stokes equations.', '1112.4147-2-26-0': 'Let [MATH] be a [MATH] injective compact curve of [MATH], and [MATH].', '1112.4147-2-26-1': 'Let [MATH] a family of smooth obstacles (in the sense of [REF]) converging to [MATH] (in the Hausdorff sense), such that [MATH].', '1112.4147-2-26-2': 'Let [MATH] be a global weak solution to the Navier-Stokes equations on [MATH] with initial velocity [MATH] (given by [REF]), then we can extract a subsequence such that [MATH] converges weakly-[MATH] to [MATH] in [MATH], where [MATH] is a global weak solution of the Navier-Stokes equations on [MATH] with initial velocity [MATH].', '1112.4147-2-27-0': 'Even if this theorem is the natural extension of [CITATION], the proof requires a new way to cut-off divergence free test functions.', '1112.4147-2-27-1': 'In the two previous articles, the classical cutting-off of stream function was sufficient to get the stability result.', '1112.4147-2-27-2': 'Such a method does not hold in our case, and we will introduce a non-explicit approximation, based on Bogovskii results.', '1112.4147-2-28-0': 'Notation: for any function [MATH] defined on [MATH], we denote by [MATH] the extension of [MATH] on [MATH], vanishing on [MATH].', '1112.4147-2-28-1': 'If [MATH] is regular enough and vanishes on [MATH], then [MATH].', '1112.4147-2-28-2': 'Similarly, if [MATH] is a vector field regular enough and tangent to the boundary, then [MATH].', '1112.4147-2-29-0': '# Stability under Hausdorff approximations', '1112.4147-2-30-0': '## Convergence of the initial velocity', '1112.4147-2-31-0': 'Let [MATH] be an obstacle of [MATH] (in the sense of [REF]) and [MATH] be a family of smooth obstacles converging to [MATH] in the Hausdorff topology when [MATH] and such that [MATH].', '1112.4147-2-31-1': 'As mentioned in the introduction, we fix [MATH] and we define [MATH] as in [REF], which is the unique vector field verifying [REF].', '1112.4147-2-31-2': 'Let us show that [MATH] converges strongly to [MATH].', '1112.4147-2-32-0': 'With the above notations, we have that [EQUATION].', '1112.4147-2-33-0': 'As the Leray projection is orthogonal in [MATH], we get from [REF] that [EQUATION].', '1112.4147-2-33-1': "By the Banach-Alaoglu's theorem, we infer that there exists [MATH] and a subsequence [MATH], such that [EQUATION].", '1112.4147-2-33-2': 'This weak convergence implies in particular that [EQUATION] and [EQUATION].', '1112.4147-2-33-3': 'In the proof of Lemma [REF] (see Appendix [REF]), we have proved that [MATH].', '1112.4147-2-33-4': 'The uniqueness part of Lemma [REF] implies that [MATH] and then [EQUATION].', '1112.4147-2-33-5': 'Putting together the two last inequalities, we get [MATH].', '1112.4147-2-33-6': 'Using the weak convergence in [MATH] of [MATH] to [MATH], we obtain the strong convergence in [MATH].', '1112.4147-2-34-0': 'As [MATH] belongs to [MATH], we deduce directly from the [MATH] strong convergence that [MATH] belongs to [MATH].', '1112.4147-2-34-1': 'Then we have two functions in [MATH] having the same vorticity, which implies that [MATH], without assuming nothing about the regularity of [MATH] (see Appendix [REF]).', '1112.4147-2-34-2': 'The uniqueness also implies that the limit holds without extracting a subsequence.', '1112.4147-2-35-0': 'We have obtained in the previous proof that [EQUATION] but we do not have [EQUATION].', '1112.4147-2-35-1': 'Even in the case where [MATH] is a surface, we can just pretend that [MATH] belongs to [MATH] and is supported on the surface.', '1112.4147-2-35-2': 'Actually, we will prove in Subsection [REF] that [EQUATION] where [MATH] is the jump of the tangential component of [MATH] and [MATH] is the Dirac measure on [MATH].', '1112.4147-2-36-0': 'In the case of the curve, we will show in Subsection [REF] that there is no function belonging in [MATH] compactly supported on a curve, and we will obtain that [MATH].', '1112.4147-2-37-0': '## Time evolution', '1112.4147-2-38-0': 'For all [MATH], we denote by [MATH] a global weak solution of the Navier-Stokes equations (in the sense of Definition [REF]) on [MATH] with initial data [MATH].', '1112.4147-2-39-0': 'By Proposition [REF], we already know that [MATH] in [MATH].', '1112.4147-2-39-1': 'Moreover, thanks to the energy inequality [REF], we state that [EQUATION]', '1112.4147-2-39-2': 'Now, we need to establish a temporal estimate.', '1112.4147-2-39-3': 'First, we use that [MATH] verifies the Dirichlet boundary condition for a.e. [MATH] in order to write the following Sobolev inequality: [EQUATION] where [MATH] is independent of [MATH].', '1112.4147-2-39-4': 'Let us consider [MATH] and [MATH] an open smooth bounded set relatively compact in [MATH].', '1112.4147-2-39-5': 'By [REF], there exists [MATH] such that [MATH] for all [MATH].', '1112.4147-2-39-6': 'Even if it is possible to show that [MATH] is equicontinuous in [MATH], we cannot deduce precompactness in [MATH]: indeed there is no injection from [MATH] to [MATH] (the gradients of harmonic function are divergence free, but their [MATH] norm are equal to zero).', '1112.4147-2-39-7': 'To use interpolation, we have to consider [MATH] which embeds in [MATH]: the boundary condition forbids the gradients of harmonic function.', '1112.4147-2-39-8': 'Then, we write in [MATH] [EQUATION] with [EQUATION] which imply that [MATH] converges to [MATH] weak-[MATH] in [MATH] and in [MATH].', '1112.4147-2-39-9': 'For [MATH], we perform a strong compactness argument as follows: for any [MATH] and [MATH]: [EQUATION] where we have used [REF].', '1112.4147-2-39-10': 'This inequality implies that [MATH] is equicontinous as a family of functions from [MATH] to [MATH].', '1112.4147-2-39-11': 'Using that this family is bounded in [MATH] and the compact embedding in [MATH], Ascoli theorem gives that [MATH] is precompact in [MATH].', '1112.4147-2-39-12': 'Moreover, [MATH] is also bounded in [MATH], then we get by interpolation that this family is precompact in [MATH].', '1112.4147-2-39-13': 'By a diagonal extraction on the compact sets of [MATH] and [MATH], we find a subsequence [MATH] such that we have the following property: for any compact set [MATH], then there exists [MATH] a relatively compact set of [MATH] belonging in the sequence where the diagonal extraction was considered, such that [MATH] and [MATH] is precompact in [MATH].', '1112.4147-2-40-0': 'Moreover, extracting again a subsequence if necessary, we know from [REF] that the limit verifies [EQUATION]', '1112.4147-2-40-1': 'For any test function [MATH], there exist [MATH], [MATH] such that [EQUATION].', '1112.4147-2-40-2': 'For [MATH] fixed, there exists [MATH] a relatively compact set of [MATH], belonging in the sequence where the diagonal extraction was considered, where [MATH].', '1112.4147-2-40-3': 'As there exists [MATH] such that [MATH], then [REF] reads [EQUATION]', '1112.4147-2-40-4': 'Actually, the last term is equal to zero: indeed we can check that [MATH] and as it is a gradient, the last part vanishes by the divergence free condition of [MATH].', '1112.4147-2-40-5': 'Thanks to the strong convergence of [MATH] in [MATH] and the weak of [MATH], [MATH], [MATH], we can pass to the limit [MATH] to get: [EQUATION].', '1112.4147-2-40-6': 'In particular, this equality putting together with [REF] and [REF] gives that [EQUATION].', '1112.4147-2-40-7': 'Then we can pass to the limit in [EQUATION] as [MATH] to get [EQUATION].', '1112.4147-2-40-8': 'This equality implies that [EQUATION] in the sense of distribution in [MATH].', '1112.4147-2-40-9': 'Since the right hand side term belongs to [MATH], the equality holds in [MATH].', '1112.4147-2-41-0': 'Now, we check that [MATH] belongs in the good functional space.', '1112.4147-2-41-1': 'Thanks to the previous equality, and [REF], then we can easily prove that [EQUATION].', '1112.4147-2-41-2': 'This argument can be found in [CITATION]: as [MATH] belongs to [MATH], then its implies that [MATH] is almost everywhere equal to a function continuous from [MATH] into [MATH].', '1112.4147-2-41-3': 'Moreover, using the fact that [MATH], then [CITATION] states that the continuity in [MATH] implies the weak continuity in time with values in [MATH].', '1112.4147-2-42-0': 'Moreover, thanks to the continuity in [MATH], we infer that the equality [REF] in the sense of [MATH] implies that the integral equality [REF] holds for all [MATH].', '1112.4147-2-42-1': 'Indeed, for the initial data we know from the uniform convergence in [MATH] that [MATH] in [MATH].', '1112.4147-2-42-2': 'However, we proved in Proposition [REF] that [MATH] in [MATH], which allows us to state by the uniqueness of the limit in [MATH] that the initial velocity is [MATH].', '1112.4147-2-43-0': 'To finish the proof of Theorem [REF], we have to prove the energy inequality.', '1112.4147-2-43-1': 'We take the liminf of [REF]: [EQUATION] and we note that the weak limit in [MATH] of [MATH] to [MATH] and the weak limit in [MATH] of [MATH] to [MATH] imply that [EQUATION].', '1112.4147-2-43-2': 'It gives the last point required in Definition [REF], which ends the proof of Theorem [REF].', '1112.4147-2-44-0': 'Therefore, we have shown that the Navier-Stokes solutions converge when the smooth obstacles convergence to a obstacle [MATH] verifying [REF].', '1112.4147-2-44-1': 'We note here that we do not assume any assumption on the regularity of [MATH].', '1112.4147-2-44-2': 'In particular, this result holds if [MATH] is a surface.', '1112.4147-2-44-3': 'As [MATH] for a.e. time, it is clear that the surface has a non-negligible effect on the motion of 3D viscous flow: [MATH] verifies for almost every time the no slip boundary condition.', '1112.4147-2-44-4': 'In the following subsection, we discuss about the initial velocity properties in the particular case of the surface, and the goal is to get some similarities to the curve in 2D.', '1112.4147-2-45-0': '## Remark on the behavior of the initial velocity near a smooth surface', '1112.4147-2-46-0': 'In the two dimensional case, we obtain in [CITATION] an explicit formula of the initial velocity in terms of Riemann maps (identifying [MATH] and [MATH]).', '1112.4147-2-46-1': 'This formula allows us to state that [MATH] is continuous up to the curve with different values on each side, except near the end-points where it behaves as the inverse of the square root of the distance.', '1112.4147-2-46-2': 'In our case, we do not have this formula, and we use in this subsection classical elliptic theory in order to get similar results in the case where [MATH] is a bounded orientable surface of codimension 1 in [MATH].', '1112.4147-2-47-0': 'As mentioned in Remark [REF], [MATH] is curl free in [MATH] and as [MATH] is simply connected we infer that there exists [MATH] such that [MATH].', '1112.4147-2-47-1': 'We know that [MATH] is continuous up to the boundary, then the goal is to determine the behavior of [MATH] near [MATH] where [MATH] verifies the following elliptic problem [EQUATION] where [MATH] is regular on [MATH].', '1112.4147-2-48-0': 'This subsection is independent of the convergence theory, and the goal here is to give an example of behavior of [MATH].', '1112.4147-2-48-1': 'Therefore, we add here some assumptions on [MATH] in order to apply classical elliptic results.', '1112.4147-2-49-0': 'We assume that [MATH] is a [MATH] manifold and its boundary [MATH] is a [MATH] closed curve.', '1112.4147-2-49-1': 'For example, if [MATH] we denote by [MATH].', '1112.4147-2-50-0': 'The study of elliptic equations in the exterior of a surface with the Neumann condition is standard for the crack problem in 3D linear elasticity.', '1112.4147-2-50-1': 'Actually, to get exactly the Neumann boundary condition, we add a regular function [MATH] such that [MATH] verifies [EQUATION].', '1112.4147-2-50-2': 'For [MATH] regular enough, we rich the necessary regularity for [MATH] in order that [MATH] has an expansion near [MATH] on the form [EQUATION] in local polar coordinates [MATH].', '1112.4147-2-50-3': 'Such a result is proved in [CITATION] (see also the references therein).', '1112.4147-2-50-4': 'In particular, it implies that [MATH] is continuous up to [MATH], with possibly different values on each side, except near the boundary [MATH] where [MATH] behaves like the inverse of the square root of the distance.', '1112.4147-2-50-5': 'Therefore, we obtain exactly the same behavior in 3D in the exterior of a surface than in 2D in the exterior of a curve.', '1112.4147-2-51-0': 'Thanks to the continuity up to the surface, it is easy to see that the tangent condition implies that [EQUATION]', '1112.4147-2-51-1': 'C^(R), (s) 0 on (-,1), (s) 1 on (2,+), [EQUATION] _n _L^(R^3)Cn and meas( supp(1- _n)) C/n^2, [EQUATION] f, _H^-1,H^1 = 0, C^_c(R^3 C) [EQUATION] _n(x):= _n(x) (x) [EQUATION] f, _n _H^-1,H^1 = 0, n. [EQUATION] _n-_L^2C() / n 0 and _n-_L^2C().', '1112.4147-2-51-2': '[EQUATION] f,= f,_n +f,f,0 [EQUATION] _0^n:= _0__n with _n= R^3C_n [EQUATION]', '1112.4147-2-51-3': 'Eu_0^nv_0 strongly in L^2(R^3), [EQUATION] (x)=-_R^3x-y4x-y^3v_0(y) dy , [EQUATION] w^n : = curl (_n ) = _n v_0 + _n , [EQUATION]', '1112.4147-2-51-4': 'E u_0^n - v_0 v weakly in L^2.', '1112.4147-2-51-5': '[EQUATION] div _n =0 and _n weak-[MATH] in L^(R^+; H^1(R^3)).', '1112.4147-2-51-6': '[EQUATION] g_n(t,x):= div f_n(t,x)= (t,x)_n(x) [EQUATION] _R^3 g_n(t,)=0 and g_n(t, )_L^p(R^3) C n^1-2p, [EQUATION] div h_n = -g_n,h_nW^1,p_0(), h_n _W^1,p() c_p g_n _L^p().', '1112.4147-2-51-7': '[EQUATION] _t _n_t weak-[MATH] in L^(R^+; H^1(R^3)).', '1112.4147-2-51-8': '[EQUATION] x d(x,C) <1/n_0, !', '1112.4147-2-51-9': 'y C d(x,y)=d(x,C).', '1112.4147-2-51-10': '[EQUATION] nN, x_n d(x_n,C) <1/n, y_n=C(s_1,n), z_n=C(s_2,n) [EQUATION] s_1,n<s_2,n and d(x_n,y_n)=d(x_n,C)= d(x_n,z_n).', '1112.4147-2-51-11': '[EQUATION] f: sd(x_N, C(s))^2= x_N M(s) ^2, [EQUATION] f\'(s)= 2 x_N M(s) (s),f"(s)=2(1+(s) x_N M(s) (s)).', '1112.4147-2-51-12': '[EQUATION] f\'(s_1,N)=0=f\'(s_2,N), [EQUATION] f"(s) 1, s [s_1,N, s_2,N] [EQUATION] g(x) = x-yx-y, [EQUATION]', '1112.4147-2-51-13': 'G:= zC z-y [EQUATION] h : zG x-z^2.', '1112.4147-2-51-14': '[EQUATION] d(x,G)> d(x,C).', '1112.4147-2-51-15': '[EQUATION] g(x+h_n)= d(x+h_n,C)=x + h_n - y_h_n=d(x+h_n,G) [EQUATION] g(x+h_n) d(x,G)> g(x) [EQUATION] (g^2(x))=2g(x)g(x) = 2 (x-y) [EQUATION] _n(x)= n ( n d(x,C) ) d(x,C)= n ( n d(x,C) ) x-yx-y, [EQUATION] _n _L^(R^3)n _L^(R).', '1112.4147-2-51-16': '[EQUATION] supp(1-_n) = C(s) + r(s)+r(s) (s,r,)[0,L][0,2/n][0,2) _i=1,2 O_i [EQUATION] d d s=(s) (s), d d s=-(s) (s)-(s) (s), [EQUATION]', '1112.4147-2-51-17': 'D(r,s,f) = 1- r &0&0 r & & -r -r & & r [MATH]D(r,s,f) r/2[MATH]r[MATH]r1/(2 _M)[MATH]n[MATH]'}

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[]

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['1112.4147-1-22-0', '1112.4147-1-22-1', '1112.4147-1-24-2', '1112.4147-1-26-0', '1112.4147-1-36-0', '1112.4147-1-44-0', '1112.4147-1-70-0', '1112.4147-1-70-1', '1112.4147-1-75-0', '1112.4147-1-82-0', '1112.4147-1-82-1', '1112.4147-1-89-2', '1112.4147-1-96-0', '1112.4147-1-96-1', '1112.4147-1-97-0', '1112.4147-1-102-0', '1112.4147-1-102-1', '1112.4147-1-109-0', '1112.4147-2-6-0', '1112.4147-2-7-0', '1112.4147-2-7-1', '1112.4147-2-9-0', '1112.4147-2-12-0', '1112.4147-2-15-0', '1112.4147-2-18-5', '1112.4147-2-19-0', '1112.4147-2-20-0', '1112.4147-2-32-0', '1112.4147-2-51-1', '1112.4147-2-51-2', '1112.4147-2-51-3', '1112.4147-2-51-4', '1112.4147-2-51-5', '1112.4147-2-51-6', '1112.4147-2-51-7', '1112.4147-2-51-8', '1112.4147-2-51-9', '1112.4147-2-51-10', '1112.4147-2-51-11', '1112.4147-2-51-12', '1112.4147-2-51-13', '1112.4147-2-51-14', '1112.4147-2-51-15', '1112.4147-2-51-16', '1112.4147-2-51-17']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1112.4147

quant-ph-0602207

{'quant-ph-0602207-1-0-0': 'We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases.', 'quant-ph-0602207-1-0-1': 'The "self-orthogonality" phenomenon is clarified in terms of a correct spectral decomposition and it is shown that "self-orthogonal" states never jeopardize decomposition of unity and thereby quantum averages of observables.', 'quant-ph-0602207-1-0-2': 'The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated.', 'quant-ph-0602207-1-0-3': 'Some puzzles with zero-binorm bound states in continuous spectrum are unraveled with the help of a correct decomposition of unity.', 'quant-ph-0602207-1-1-0': 'Journal of Physics A: Math.', 'quant-ph-0602207-1-1-1': 'Gen.', 'quant-ph-0602207-1-2-0': '# Introduction', 'quant-ph-0602207-1-3-0': 'The variety of complex potentials in Quantum Physics is associated typically with open systems when a control of information is partially lost and thereby the unitarity of observable evolution is broken.', 'quant-ph-0602207-1-3-1': 'For this class of quantum systems the energy eigenvalues may have an imaginary part which signals the opening of new channels not directly measured in a given experiment.', 'quant-ph-0602207-1-3-2': 'In this context non-Hermitian interactions have been used in Field Theory and Statistical Mechanics for many years with applications to Condensed Matter, Quantum Optics and Hadronic and Nuclear Physics [CITATION] - [CITATION].', 'quant-ph-0602207-1-3-3': 'The subject of non-self-adjoint operators has been also under intensive mathematical investigations [CITATION].', 'quant-ph-0602207-1-4-0': 'An important class of complex Hamiltonians deals with a real spectrum [CITATION], in particular, in the PT-symmetric Quantum Mechanics [CITATION]-[CITATION] and its pseudo-Hermitian generalization [CITATION].', 'quant-ph-0602207-1-4-1': 'Scattering problems for such Hamiltonians have been investigated in [CITATION] .', 'quant-ph-0602207-1-5-0': 'For complex, non-Hermitian potentials the natural spectral decomposition exploits the sets of biorthogonal states [CITATION], and within this framework one can discover new features that never happen for closed systems with Hermitian Hamiltonians : namely, certain Hamiltonians may not be diagonalizable [CITATION] and can be reduced only to a quasi-diagonal form with a number of Jordan cells [CITATION].', 'quant-ph-0602207-1-5-1': 'This feature can be realized by level crossing which, in fact, occurs under certain circumstances in atomic and molecular spectra [CITATION] and Optics [CITATION] (see more examples in [CITATION]) as well as in PT-symmetric quantum systems [CITATION].', 'quant-ph-0602207-1-5-2': 'In this case some of eigenstates seem to be "self-orthogonal" in respect to a binorm [CITATION] .', 'quant-ph-0602207-1-5-3': 'This quite intriguing phenomenon has been interpreted as a sort of phase transition [CITATION] .', 'quant-ph-0602207-1-6-0': 'The main purpose of the present work is to clarify the "self-orthogonality" in terms of a correct spectral decomposition both for discrete and for continuous spectra and to show that, at least, in one-dimensional Quantum Mechanics such states never jeopardize decomposition of unity and thereby don\'t affect quantum averages of observables.', 'quant-ph-0602207-1-7-0': 'We start introducing the notion of biorthogonal basis and, correspondingly, the decomposition of unity for a non-Hermitian diagonalizable Hamiltonian.', 'quant-ph-0602207-1-7-1': 'In Sec. 2 the appearance of associated functions is discussed and in Sec. 3 non-diagonalizable (but quasi-diagonalizable) Hamiltonians with finite-size Jordan cells are analyzed.', 'quant-ph-0602207-1-7-2': 'Special attention is paid to the definition of a biorthogonal diagonal basis and the meaning of zero-binorm states is clarified.', 'quant-ph-0602207-1-7-3': 'Namely, it is shown that the apparent self-orthogonality of eigenfunctions and associated functions is misleading as they never replicate as relative pairs in decomposition of unity.', 'quant-ph-0602207-1-7-4': 'Instead, the "self-orthogonality" concerns the different elements in the basis thereby being related to a conventional orthogonality.', 'quant-ph-0602207-1-7-5': 'The existence of biorthogonal bases with pairs of mutually complex-conjugated base functions is proved .', 'quant-ph-0602207-1-7-6': 'In Sec. 4 another, manifestly symmetric representation of non-diagonalizable Hamiltonians is given, compatible with a diagonal decomposition of unity.', 'quant-ph-0602207-1-7-7': 'In Sec. 5 the example of a (transparent) complex potential leading to the non-diagonalizable Hamiltonian with one Jordan [MATH] cell is constructed and its origin from level coalescence is illustrated.', 'quant-ph-0602207-1-8-0': 'Some puzzles with zero-binorm bound states in continuous spectrum are unraveled in Sec.6 with the help of a correct decomposition of unity.', 'quant-ph-0602207-1-8-1': 'In Sec. 7 we complete our analysis with discussion of resolvents and scattering characteristics for previous examples focusing on the possibility of spectral singularities for non-Hermitian Hamiltonians.', 'quant-ph-0602207-1-8-2': 'We conclude with some proposals for probabilistic interpretation of wave functions defined in respect to a biorthogonal basis which does not allow any negative or zero-norm states.', 'quant-ph-0602207-1-9-0': 'In our paper we deal with complex one-dimensional potentials [MATH] and respectively with non-Hermitian Hamiltonians [MATH] of Schrodinger type, defined on the real axis, [EQUATION] which are assumed to be symmetric or self-transposed under the [MATH] - transposition operation, [MATH].', 'quant-ph-0602207-1-9-1': 'Only scalar local potentials will be analyzed which are obviously symmetric under transposition (for some matrix non-diagonalizable problems, see [CITATION]).', 'quant-ph-0602207-1-10-0': 'Let us first consider a one-dimensional non-Hermitian diagonalizable Hamiltonian [MATH] with discrete spectrum.', 'quant-ph-0602207-1-10-1': 'For this Hamiltonian it is assumed that a biorthogonal system [MATH] exists, [EQUATION] such that the complete decomposition of unity and the spectral decomposition of the Hamiltonian hold, [EQUATION]', 'quant-ph-0602207-1-10-2': 'In the coordinate representation, [EQUATION] the decomposition of unity has the form, [EQUATION]', 'quant-ph-0602207-1-10-3': 'The differential equations, [EQUATION] and the fact that there is only one normalizable eigenfunction of [MATH] for the eigenvalue [MATH] (up to a constant factor), allow one to conclude that [EQUATION]', 'quant-ph-0602207-1-10-4': 'Hence the system [MATH] can be redefined [EQUATION] so that [EQUATION]', 'quant-ph-0602207-1-10-5': 'Indeed if some of the states in Eq. ([REF]) were "self-orthogonal" (as it has been accepted in [CITATION]) , i.e. had zero binorms in ([REF]), the would-be unity in ([REF]) would annihilate such states thereby signalling the incompleteness.', 'quant-ph-0602207-1-11-0': '# Non-diagonalizable Hamiltonians and zero-binorm states', 'quant-ph-0602207-1-12-0': 'For complex Hamiltonians one can formulate the extended eigenvalue problem, searching not only for normalizable eigenfunctions but also for normalizable associated functions for discrete part of the energy spectrum.', 'quant-ph-0602207-1-12-1': 'Some related problems have been known for a long time in mathematics of linear differential equations (see for instance, [CITATION]) .', 'quant-ph-0602207-1-13-0': 'Let us make the formal definition.', 'quant-ph-0602207-1-14-0': "Definition The function [MATH] is called a formal associated function of [MATH]-th order of the Hamiltonian [MATH] for a spectral value [MATH], if [EQUATION] where 'formal' emphasizes that a related function is not necessarily normalizable.", 'quant-ph-0602207-1-15-0': 'In particular, the associated function of zero order [MATH] is a formal eigenfunction of [MATH] (a solution of the homogeneous Schrodinger equation, not necessarily normalizable).', 'quant-ph-0602207-1-16-0': 'Let us single out normalizable associated functions and the case when [MATH] maps them into normalizable functions.', 'quant-ph-0602207-1-16-1': 'Evidently this may occur only for non-Hermitian Hamiltonians.', 'quant-ph-0602207-1-16-2': 'Then for any normalizable associated functions [MATH] and [MATH] the transposition symmetry holds [EQUATION]', 'quant-ph-0602207-1-16-3': 'Furthermore one can prove the following relations: [EQUATION] where [MATH] is scalar product.', 'quant-ph-0602207-1-17-0': 'As well, if there are normalizable associated functions [MATH] and [MATH] such that [EQUATION] then [EQUATION]', 'quant-ph-0602207-1-17-1': 'In particular, for some normalizable associated function [MATH], the "self-orthogonality" [CITATION] is realized , [EQUATION]', 'quant-ph-0602207-1-17-2': 'Thus, when assigning [CITATION] the probabilistic meaning for the binorm [MATH] one comes to conclusion that a sort of intriguing phase transition occurs in such a system, signalled by the puzzling divergence of some averages of observables, [EQUATION]', 'quant-ph-0602207-1-17-3': 'All the above relations are derived from the symmetry of a Hamiltonian under transposition and the very definition of associated functions and therefore the existence of self-orthogonal states seems to be inherent for any non-diagonalizable Hamiltonians with normalizable associated functions.', 'quant-ph-0602207-1-18-0': '# Towards resolution of puzzle with self-orthogonal states for Hamiltonians with finite-size Jordan cells', 'quant-ph-0602207-1-19-0': 'Let us show that the puzzle with self-orthogonal states may appear, in fact, due to misinterpretation of what are the pairs of orthogonal states in a true biorthogonal basis.', 'quant-ph-0602207-1-19-1': 'We proceed to the special class of Hamiltonians for which the spectrum is discrete and there is a complete biorthogonal system [MATH] such that, [EQUATION] where [MATH] is an index of an [MATH] eigenvalue [MATH], , [MATH] is an index of a Jordan cell (block) for the given eigenvalue, [MATH]; [MATH] is a number of Jordan cells for [MATH];', 'quant-ph-0602207-1-20-0': '[MATH], , [MATH] is an index of associated function in the Jordan cell with indexes [MATH]', 'quant-ph-0602207-1-21-0': 'and [MATH] is a dimension of this Jordan cell.', 'quant-ph-0602207-1-21-1': 'We have taken a general framework which is applicable also for matrix and/or multidimensional Hamiltonians.', 'quant-ph-0602207-1-21-2': 'But the main results of this and the next sections are guaranteed only for scalar Hamiltonians.', 'quant-ph-0602207-1-22-0': 'We remark that the number [MATH] is called a geometric multiplicity of the eigenvalue [MATH] .', 'quant-ph-0602207-1-22-1': 'For a scalar one-dimensional Schrodinger equation it cannot exceed 2.', 'quant-ph-0602207-1-22-2': 'In turn, the sum [MATH] is called an algebraic multiplicity of the eigenvalue [MATH].', 'quant-ph-0602207-1-23-0': 'The completeness implies the biorthogonality relations [EQUATION] and the unity decomposition [EQUATION]', 'quant-ph-0602207-1-23-1': 'The spectral decomposition for the Hamiltonian can be constructed as well, [EQUATION]', 'quant-ph-0602207-1-23-2': 'It represents the analog of the block-diagonal Jordan form for arbitrary non-Hermitian matrices [CITATION].', 'quant-ph-0602207-1-24-0': 'If existing such biorthogonal systems are not unique.', 'quant-ph-0602207-1-24-1': 'Indeed the relations ([REF]) remain invariant under the group of triangle transformations, [EQUATION] where the matrix elements must obey the following equations, [EQUATION]', 'quant-ph-0602207-1-24-2': 'The biorthogonality ([REF]) restricts the choice of pairs of matrices [MATH] and [MATH] in ([REF]) to be, [EQUATION]', 'quant-ph-0602207-1-24-3': 'This freedom in the redefinition of the biorthogonal basis is similar to Eq. ([REF]) and it can be exploited to define the pairs of biorthogonal functions [MATH] and [MATH] in accordance with ([REF]).', 'quant-ph-0602207-1-24-4': 'However one has to take into account our enumeration of associated functions [MATH] vs. their conjugated ones [MATH] as it is introduced in Eqs. ([REF]) [EQUATION]', 'quant-ph-0602207-1-24-5': 'Then the analog of Eq. ([REF]) reads, [EQUATION]', 'quant-ph-0602207-1-24-6': 'We stress that this kind of biorthogonal systems is determined uniquely up to an overall sign.', 'quant-ph-0602207-1-25-0': 'In these terms it becomes clear that the relations ([REF]) have the meaning of orthogonality of some off-diagonal pairs in the biorthogonal system [MATH] as [EQUATION]', 'quant-ph-0602207-1-25-1': 'When comparing with specification of indices in Eq. ([REF]) one identifies [MATH].', 'quant-ph-0602207-1-25-2': 'In both cases [MATH] .', 'quant-ph-0602207-1-25-3': 'Then the inequality ([REF]) singles out off-diagonal binorms, [MATH].', 'quant-ph-0602207-1-25-4': 'One can show easily that in order to have all diagonal binorms non-vanishing it is sufficient to prove that at least one of them is not zero which equivalent to the requirement [EQUATION] thereby providing the completeness of the basis.', 'quant-ph-0602207-1-26-0': 'Going back to the definition of quantum-state averages of certain observables we realize that the matrix element used in ([REF]) is not diagonal and therefore this relation cannot be interpreted as an average (compare with [CITATION]) of a putative order-parameter like operator.', 'quant-ph-0602207-1-27-0': '# Symmetric representation of non-diagonalizable Hamiltonians', 'quant-ph-0602207-1-28-0': 'We still notice that the biorthogonal basis ([REF]) does not provide a manifestly symmetric representation of the Hamiltonian (which is a symmetric operator).', 'quant-ph-0602207-1-28-1': 'One can obtain another biorthogonal basis using the canonical set of (normalizable) associated functions given by Eq. ([REF]) and their complex conjugates in an analogy to ([REF]) .', 'quant-ph-0602207-1-28-2': 'It can be achieved by means of renumbering of conjugated elements of the biorthogonal system ([REF]) , [EQUATION]', 'quant-ph-0602207-1-28-3': 'Eventually one arrives to the symmetric spectral decomposition for [MATH]: [EQUATION] which looks like a Jordan decomposition along the secondary diagonal.', 'quant-ph-0602207-1-28-4': 'Evidently in the coordinate representation the Hamiltonian operator is manifestly symmetric when the special biorthogonal basis ([REF]), [EQUATION] is chosen.', 'quant-ph-0602207-1-29-0': 'But the unity decomposition in this case is not diagonal, [EQUATION] although symmetric.', 'quant-ph-0602207-1-29-1': 'One can diagonalize this unity decomposition by a non-degenerate orthogonal transformation [MATH] of sub-bases in each non-diagonal sub-block, [EQUATION] retaining the type of the basis ([REF]) .', 'quant-ph-0602207-1-29-2': 'Then one finds a number eigenvalues [MATH].', 'quant-ph-0602207-1-29-3': 'In order to come to the canonical form of a basis ([REF]) one has to rotate by the complex unit [MATH] the pairs in the basis ([REF]) normalized on [MATH] .', 'quant-ph-0602207-1-29-4': 'Evidently the combination of the transformation [MATH] and such a rotation contains complex elements and is not orthogonal.', 'quant-ph-0602207-1-30-0': 'The remaining freedom of basis redefinition with the help of orthogonal rotations cannot provide the consequent diagonalization of the symmetric Hamiltonian matrix in each non-diagonal block.', 'quant-ph-0602207-1-30-1': 'The reason is that some of eigenvectors of the Hamiltonian sub-matrices have zero binorms, in particular, those ones which are related to the true Hamiltonian eigenfunctions.', 'quant-ph-0602207-1-30-2': 'Thus while being a symmetric operator with symmetric matrix representation, the Hamiltonian remains essentially non-diagonalizable.', 'quant-ph-0602207-1-31-0': 'We remark that in the general case the existence and the completeness of a biorthogonal system is not obvious (especially if the continuous spectrum is present) and needs a careful examination.', 'quant-ph-0602207-1-31-1': 'In particular, at the border between discrete and continuous spectra and in the continuous spectrum itself one can anticipate to have puzzling states with non-trivial role in the spectral decomposition.', 'quant-ph-0602207-1-31-2': 'This peculiarities will be discussed in the next Sections.', 'quant-ph-0602207-1-32-0': '# Example of a Non-diagonalizable Hamiltonian and its origin from level coalescence', 'quant-ph-0602207-1-33-0': 'In this Section we build a model with non-Hermitian Hamiltonian which possesses a Jordan cell spanned on the bound state and a normalizable associated state and further on demonstrate how this kind of degeneracy arises from coalescence of a pair of non-degenerate levels.', 'quant-ph-0602207-1-34-0': '## Jordan cell for bound state', 'quant-ph-0602207-1-35-0': 'The model Hamiltonian contains the potential with coordinates selectively shifted into complex plane, [EQUATION]', 'quant-ph-0602207-1-35-1': 'This Hamiltonian is not PT-symmetric unless [MATH] .', 'quant-ph-0602207-1-35-2': 'It has the Jordan cell, spanned by the normalizable eigenfunction [MATH] and associated function [MATH] on the level [MATH], [EQUATION]', 'quant-ph-0602207-1-35-3': 'In turn, the eigenfunctions of [MATH] for continuous spectrum read, [EQUATION]', 'quant-ph-0602207-1-35-4': 'For this model, unity decomposition built of eigenfunctions and associated functions of [MATH] can be obtained by conventional Green function methods, [EQUATION] this decomposition can be represented in the operator form, [EQUATION]', 'quant-ph-0602207-1-35-5': 'Evidently the basis [MATH] corresponds to the basis [MATH] of Sec. 4 and therefore gives the symmetric spectral decomposition for the Hamiltonian, [EQUATION]', 'quant-ph-0602207-1-35-6': 'The diagonalization of the decomposition of unity ([REF]) may be arranged in two different ways.', 'quant-ph-0602207-1-35-7': 'First one can exploit the scheme of Sec. 3 performing re-numeration of certain elements of conjugated basis, [EQUATION] namely, [EQUATION]', 'quant-ph-0602207-1-35-8': 'With this notation the decomposition of unity reads, [EQUATION]', 'quant-ph-0602207-1-35-9': 'We stress that [MATH] are related to the basis [MATH] in Sec. 3.', 'quant-ph-0602207-1-35-10': 'The relevant spectral decomposition of the Hamiltonian takes the quasi-diagonal form with one Jordan cell, [EQUATION]', 'quant-ph-0602207-1-35-11': 'On the other hand, the decomposition of unity ([REF]) can be diagonalized by complex non-degenerate rotations , i.e. by using the construction of Sec. 4 .', 'quant-ph-0602207-1-35-12': 'The relevant basis is given by, [EQUATION] and [MATH] is an arbitrary constant with dimension of a length.', 'quant-ph-0602207-1-35-13': 'The decomposition of unity becomes diagonal , [EQUATION] or in the operator form, [EQUATION] where again the Dirac notations have been used, [EQUATION]', 'quant-ph-0602207-1-35-14': 'Accordingly, the manifestly symmetric spectral decomposition of [MATH] can be easily obtained, [EQUATION]', 'quant-ph-0602207-1-35-15': 'Notice that it cannot be diagonalized further, since the symmetric [MATH] matrix in ([REF]), [EQUATION] has one degenerate eigenvalue [MATH] and possesses only one eigenvector [MATH] with zero scalar product [MATH].', 'quant-ph-0602207-1-35-16': 'Its existence means that the orthogonal non-degenerate matrix required for diagonalization cannot be built conventionally from a set of eigenvectors.', 'quant-ph-0602207-1-35-17': 'Evidently, this vector maps the pair of basis functions [MATH] into the eigenstate of the Hamiltonian [MATH] whose partner in the biorthogonal basis is [MATH] .', 'quant-ph-0602207-1-35-18': 'Thus the existence of the zero norm vector [MATH] does not entail the self-orthogonality of basis elements [MATH] and [MATH].', 'quant-ph-0602207-1-36-0': '## Level coalescence for complex coordinates', 'quant-ph-0602207-1-37-0': 'The Hamiltonian [MATH] with a Jordan cell for bound state ([REF]) can be obtained as a limiting case, of the Hamiltonian [MATH] with two non-degenerate bound states (of algebraic multiplicity 1), corresponding [MATH], [EQUATION].', 'quant-ph-0602207-1-37-1': 'For this Hamiltonian [MATH] there are two normalized eigenfunctions for bound states, [EQUATION] with eigenvalues, [EQUATION]', 'quant-ph-0602207-1-37-2': 'Eigenfunctions of [MATH] for continuous spectrum take the form [EQUATION] i.e. the unity decomposition ([REF]) is reproduced.', 'quant-ph-0602207-1-38-0': '# Puzzles with zero-binorm bound states in the continuum', 'quant-ph-0602207-1-39-0': 'In what follows we develop another type of models in which the continuous spectrum is essentially involved in non-diagonal part of a Hamiltonian and elaborate the decomposition of unity.', 'quant-ph-0602207-1-39-1': 'The standard treatment of continuous spectrum physics deals with reflection and transmission coefficients whose definition implies the existence of two linearly independent scattering solutions for a given spectral parameter.', 'quant-ph-0602207-1-39-2': 'Here we provide an example when this is not realized for a non-Hermitian Hamiltonian defined on the whole axis.', 'quant-ph-0602207-1-39-3': 'This point can be thought of as a spectral singularity for the continuous spectrum.', 'quant-ph-0602207-1-40-0': '## Non-Hermitian Hamiltonian with normalizable bound state at the continuum threshold', 'quant-ph-0602207-1-41-0': 'Let us now consider the Hamiltonian [EQUATION] raising up the puzzle of "self-orthogonality" [CITATION].', 'quant-ph-0602207-1-42-0': 'In order to unravel this puzzle we examine the unity decomposition made of eigenfunctions of [MATH], [EQUATION]', 'quant-ph-0602207-1-42-1': 'Hence it is the 3rd term in the right side of ([REF]) that provides the opportunity to reproduce [MATH] and thereby to complete the unity decomposition.', 'quant-ph-0602207-1-42-2': 'Thus one concludes that the state [MATH] is inseparable from the bottom of continuous spectrum and the decomposition of unity in this sense is not diagonal.', 'quant-ph-0602207-1-43-0': 'We notice that the Hamiltonian ([REF]) is PT-symmetric and can be derived from the Hamiltonian ([REF]) in the limit [MATH] but the parameter [MATH] must be taken as a half of [MATH] from ([REF]).', 'quant-ph-0602207-1-44-0': 'We also remark that the Hamiltonian ([REF]) makes sense also for arbitrary coupling constants of "centrifugal" potential, and for the following set, [EQUATION] with positive [MATH], the Jordan cell, spanned by [MATH] normalizable eigenfunction and associated functions, appears at the threshold of continuous spectrum, [EQUATION]', 'quant-ph-0602207-1-44-1': 'All these zero-energy bound and associated states have zero binorms and are biorthogonal to each other (the multiple puzzle of "self-orthogonality").', 'quant-ph-0602207-1-44-2': 'Unity decomposition in such cases can be derived in a similar way although its form will be more cumbersome.', 'quant-ph-0602207-1-45-0': '## Hamiltonian with spectral singularity', 'quant-ph-0602207-1-46-0': 'Let us force the bound state energy [MATH] in the Hamiltonian ([REF]) to move towards the continuous spectrum, [MATH] .', 'quant-ph-0602207-1-46-1': 'Then for the Hamiltonian, [EQUATION] on the level [MATH] in the continuous spectrum, one finds the Jordan cell, spanned by the normalizable eigenfunction [MATH] and the associated function [MATH], whose asymptotics for [MATH] correspond to superposition of incoming and outgoing waves (standing wave), [EQUATION]', 'quant-ph-0602207-1-46-2': 'The asymptotics of this standing wave is, [EQUATION] therefore it does not belong to the scattering spectrum.', 'quant-ph-0602207-1-47-0': 'In turn the eigenfunctions of [MATH] for the scattering spectrum read, [EQUATION]', 'quant-ph-0602207-1-47-1': 'For this model, unity decomposition made of eigenfunctions and associated functions of [MATH] can be obtained by conventional Green function methods, [EQUATION] where [MATH] is an integration path in the complex momentum plane, obtained from real axis by its simultaneous displacement near the points [MATH] up or down.', 'quant-ph-0602207-1-48-0': 'For test functions from [MATH], [MATH] this decomposition can be presented in the form, [EQUATION] where the solutions [MATH] are made by analytical continuation of [MATH] in [MATH] into complex plane, and the branch of [MATH] is uniquely defined by the condition [MATH] in the plane with the cut on positive part of real axis.', 'quant-ph-0602207-1-48-1': 'In virtue of ([REF]) the Green function for the diagonalizable Hamiltonian of Subsec. 5.2.', 'quant-ph-0602207-1-48-2': 'has two poles of the first order (if [MATH]) at the points [MATH] where [MATH] can be either real or imaginary.', 'quant-ph-0602207-1-48-3': 'If [MATH] and level confluence emerges, two poles coalesce into one pole of the second order in both cases ([REF]) and ([REF]) when [MATH] .', 'quant-ph-0602207-1-48-4': 'However the examples of Subsec. 5.1 and 6.2 have different meaning: in the first case the double pole does not appear on the physical cut [MATH] and its order enumerates the rank of the Jordan cell.', 'quant-ph-0602207-1-48-5': 'On the contrary, in the second case the double pole is placed exactly on the cut [MATH] and strictly speaking signifies the spectral singularity as it generates only one eigenstate with the eigenvalue [MATH] in unity decomposition.', 'quant-ph-0602207-1-48-6': 'The second state - the associated function, represents a standing wave and does not influence the spectral decomposition.', 'quant-ph-0602207-1-48-7': 'In the example of Subsec. 6.1.', 'quant-ph-0602207-1-48-8': 'the Green function has only a branch point at [MATH] of the following type [MATH].', 'quant-ph-0602207-1-48-9': 'This branch point can be thought of as a confluence of the double pole in the variable [MATH] and the branch point of order [MATH]', 'quant-ph-0602207-1-49-0': 'From the explicit form of wave functions one can see that all potentials in Sec.5 and 6 are transparent as the reflection coefficient is zero.', 'quant-ph-0602207-1-49-1': 'The transmission coefficient in the non-degenerate case of Subsec. 5.2.', 'quant-ph-0602207-1-49-2': 'takes the form, [EQUATION]', 'quant-ph-0602207-1-49-3': 'In different limits one can derive the transmission coefficients for three other Hamiltonians.', 'quant-ph-0602207-1-49-4': 'Namely, when [MATH] and [MATH] (the case 5.1.)', 'quant-ph-0602207-1-49-5': 'the scattering is described by [EQUATION] and for [MATH] and [MATH] or imaginary (the cases 6.1.', 'quant-ph-0602207-1-49-6': 'and 6.2.)', 'quant-ph-0602207-1-49-7': 'the scattering is absent, [EQUATION]', 'quant-ph-0602207-1-49-8': 'Thus the colliding particle in such cases is not "influenced" by the bound states in the continuum.', 'quant-ph-0602207-1-50-0': '# Conclusions', 'quant-ph-0602207-1-51-0': 'In this paper we have presented a thorough analysis of the phenomenon of apparent self-orthogonality of some eigenstates for non-Hermitian Hamiltonians.', 'quant-ph-0602207-1-51-1': 'For the discrete part of energy spectrum, it has been shown that such a phenomenon should take place only for non-diagonalizable Hamiltonians the spectrum of which consists not only of eigenfunctions and but also of associated functions.', 'quant-ph-0602207-1-51-2': 'However the genuine diagonal biorthogonal basis related to the spectral decomposition of such Hamiltonians normally does not contain pairs made of the same eigenfunctions or the associated functions of the same order.', 'quant-ph-0602207-1-51-3': 'Rather they are complementary: for instance, eigenfunctions in the direct basis are paired to those associated functions in the conjugated basis, which have the maximal order in the same Jordan cell.', 'quant-ph-0602207-1-51-4': 'One possible exception exists for Jordan cells of odd order where one basis pair consists of the same function which is not self-orthogonal.', 'quant-ph-0602207-1-52-0': 'The situation in the continuous spectrum is more subtle: namely, the spectral decomposition does not include any obvious Jordan cells and associated functions.', 'quant-ph-0602207-1-52-1': 'However we have established that when a zero-binorm normalizable state arises it remains inseparable from the nearest scattering states of the continuum and eventually the existence of this state does not destroy the completeness of decomposition of unity.', 'quant-ph-0602207-1-53-0': 'Finally, let us outline the measurability of quantum observables and, for this purpose, prepare a wave packet, [EQUATION] where the possibility to have continuous spectrum is made explicit in the notation and, for brevity, all indices enumerating eigenvalues, Jordan cells and their elements are encoded in the index [MATH].', 'quant-ph-0602207-1-54-0': 'In order to perform the quantum averaging of an operator of observable [MATH] one can use the conventional Hilbert space scalar product and the complex conjugated wave function, [MATH] .', 'quant-ph-0602207-1-54-1': 'In this way one defines the wave packet of the conjugated state and, respectively, average values of the operator [MATH] , [EQUATION]', 'quant-ph-0602207-1-54-2': 'For such a definition the averages of an operator of observable, [MATH] cannot be infinite and the phenomena of (pseudo) phase transitions at the level crossing [CITATION] cannot appear.', 'quant-ph-0602207-1-54-3': 'In this relation the basis from Sec. 3 with [MATH] may be used equally well.', 'quant-ph-0602207-1-55-0': 'However one has to keep in mind that such a definition of probabilities, to some extent, depends on a particular set of biorthogonal bases.', 'quant-ph-0602207-1-55-1': 'We hope to examine this approach and its applications elsewhere.', 'quant-ph-0602207-1-56-0': 'The work of A.A. and A.S. was supported by Grant RFBR 06-01 and by the Program "Development of scientific potential of Higher Education in Russia" (2.2.1.)', 'quant-ph-0602207-1-56-1': 'A.Sokolov was partially supported by Grant INFN-IS/BO31.', 'quant-ph-0602207-1-56-2': 'We are grateful to the organizers of 3rd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics (June 2005, Koc, Turkey), especially to Prof. A. Mostafazadeh and Prof. M. Znojil, for financial support and opportunity to present the preliminary results of this paper.', 'quant-ph-0602207-1-57-0': '# References'}

{'quant-ph-0602207-2-0-0': 'We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases.', 'quant-ph-0602207-2-0-1': 'The "self-orthogonality" phenomenon is clarified in terms of a correct spectral decomposition and it is shown that "self-orthogonal" states never jeopardize decomposition of unity and thereby quantum averages of observables.', 'quant-ph-0602207-2-0-2': 'The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated.', 'quant-ph-0602207-2-0-3': 'Some puzzles with zero-binorm bound states in continuous spectrum are unraveled with the help of a correct decomposition of unity.', 'quant-ph-0602207-2-1-0': 'Journal of Physics A: Math.', 'quant-ph-0602207-2-1-1': 'Gen.', 'quant-ph-0602207-2-2-0': '# Introduction', 'quant-ph-0602207-2-3-0': 'The variety of complex potentials in Quantum Physics is associated typically with open systems when a control of information is partially lost and thereby the unitarity of observable evolution is broken.', 'quant-ph-0602207-2-3-1': 'For this class of quantum systems the energy eigenvalues may have an imaginary part which signals the opening of new channels not directly measured in a given experiment.', 'quant-ph-0602207-2-3-2': 'In this context non-Hermitian interactions have been used in Field Theory and Statistical Mechanics for many years with applications to Condensed Matter, Quantum Optics and Hadronic and Nuclear Physics [CITATION] - [CITATION].', 'quant-ph-0602207-2-3-3': 'The subject of non-self-adjoint operators has been also under intensive mathematical investigations [CITATION].', 'quant-ph-0602207-2-4-0': 'An important class of complex Hamiltonians deals with a real spectrum [CITATION], in particular, in the PT-symmetric Quantum Mechanics [CITATION]-[CITATION] and its pseudo-Hermitian generalization [CITATION].', 'quant-ph-0602207-2-4-1': 'Scattering problems for such Hamiltonians have been investigated in [CITATION] .', 'quant-ph-0602207-2-5-0': 'For complex, non-Hermitian potentials the natural spectral decomposition exploits the sets of biorthogonal states [CITATION], and within this framework one can discover new features that never happen for closed systems with Hermitian Hamiltonians : namely, certain Hamiltonians may not be diagonalizable [CITATION] and can be reduced only to a quasi-diagonal form with a number of Jordan cells [CITATION].', 'quant-ph-0602207-2-5-1': 'This feature can be realized by level crossing which, in fact, occurs under certain circumstances in atomic and molecular spectra [CITATION] and Optics [CITATION] (see more examples in [CITATION]) as well as in PT-symmetric quantum systems [CITATION].', 'quant-ph-0602207-2-5-2': 'In this case some of eigenstates seem to be "self-orthogonal" in respect to a binorm [CITATION] .', 'quant-ph-0602207-2-5-3': 'This quite intriguing phenomenon has been interpreted as a sort of phase transition [CITATION] .', 'quant-ph-0602207-2-6-0': 'The main purpose of the present work is to clarify the "self-orthogonality" in terms of a correct spectral decomposition both for discrete and for continuous spectra and to show that, at least, in one-dimensional Quantum Mechanics such states never jeopardize decomposition of unity and thereby don\'t affect quantum averages of observables.', 'quant-ph-0602207-2-7-0': 'We start introducing the notion of biorthogonal basis and, correspondingly, the decomposition of unity for a non-Hermitian diagonalizable Hamiltonian.', 'quant-ph-0602207-2-7-1': 'In Sec. 2 the appearance of associated functions is discussed and in Sec. 3 non-diagonalizable (but quasi-diagonalizable) Hamiltonians with finite-size Jordan cells are analyzed.', 'quant-ph-0602207-2-7-2': 'Special attention is paid to the definition of a biorthogonal diagonal basis and the meaning of zero-binorm states is clarified.', 'quant-ph-0602207-2-7-3': 'Namely, it is shown that the apparent self-orthogonality of eigenfunctions and associated functions is misleading as they never replicate as relative pairs in decomposition of unity.', 'quant-ph-0602207-2-7-4': 'Instead, the "self-orthogonality" concerns the different elements in the basis thereby being related to a conventional orthogonality.', 'quant-ph-0602207-2-7-5': 'The existence of biorthogonal bases with pairs of mutually complex-conjugated base functions is proved .', 'quant-ph-0602207-2-7-6': 'In Sec. 4 another, manifestly symmetric representation of non-diagonalizable Hamiltonians is given, compatible with a diagonal decomposition of unity.', 'quant-ph-0602207-2-7-7': 'In Sec. 5 the example of a (transparent) complex potential leading to the non-diagonalizable Hamiltonian with one Jordan [MATH] cell is constructed and its origin from level coalescence is illustrated.', 'quant-ph-0602207-2-8-0': 'Some puzzles with zero-binorm bound states in continuous spectrum are unraveled in Sec.6 with the help of a correct decomposition of unity.', 'quant-ph-0602207-2-8-1': 'In Sec. 7 we complete our analysis with discussion of resolvents and scattering characteristics for previous examples focusing on the possibility of spectral singularities for non-Hermitian Hamiltonians.', 'quant-ph-0602207-2-8-2': 'We conclude with some proposals for probabilistic interpretation of wave functions defined in respect to a biorthogonal basis which does not allow any negative or zero-norm states.', 'quant-ph-0602207-2-9-0': 'In our paper we deal with complex one-dimensional potentials [MATH] and respectively with non-Hermitian Hamiltonians [MATH] of Schrodinger type, defined on the real axis, [EQUATION] which are assumed to be symmetric or self-transposed under the [MATH] - transposition operation, [MATH].', 'quant-ph-0602207-2-9-1': 'Only scalar local potentials will be analyzed which are obviously symmetric under transposition (for some matrix non-diagonalizable problems, see [CITATION]).', 'quant-ph-0602207-2-10-0': 'Let us first consider a one-dimensional non-Hermitian diagonalizable Hamiltonian [MATH] with discrete spectrum.', 'quant-ph-0602207-2-10-1': 'For this Hamiltonian it is assumed that a biorthogonal system [MATH] exists, [EQUATION] such that the complete decomposition of unity and the spectral decomposition of the Hamiltonian hold, [EQUATION]', 'quant-ph-0602207-2-10-2': 'In the coordinate representation, [EQUATION] the decomposition of unity has the form, [EQUATION]', 'quant-ph-0602207-2-10-3': 'The differential equations, [EQUATION] and the fact that there is only one normalizable eigenfunction of [MATH] for the eigenvalue [MATH] (up to a constant factor), allow one to conclude that [EQUATION]', 'quant-ph-0602207-2-10-4': 'Hence the system [MATH] can be redefined [EQUATION] so that [EQUATION]', 'quant-ph-0602207-2-10-5': 'Indeed if some of the states in Eq. ([REF]) were "self-orthogonal" (as it has been accepted in [CITATION]) , i.e. had zero binorms in ([REF]), the would-be unity in ([REF]) would annihilate such states thereby signalling the incompleteness.', 'quant-ph-0602207-2-11-0': '# Non-diagonalizable Hamiltonians and zero-binorm states', 'quant-ph-0602207-2-12-0': 'For complex Hamiltonians one can formulate the extended eigenvalue problem, searching not only for normalizable eigenfunctions but also for normalizable associated functions for discrete part of the energy spectrum.', 'quant-ph-0602207-2-12-1': 'Some related problems have been known for a long time in mathematics of linear differential equations (see for instance, [CITATION]) .', 'quant-ph-0602207-2-13-0': 'Let us make the formal definition.', 'quant-ph-0602207-2-14-0': "Definition The function [MATH] is called a formal associated function of [MATH]-th order of the Hamiltonian [MATH] for a spectral value [MATH], if [EQUATION] where 'formal' emphasizes that a related function is not necessarily normalizable.", 'quant-ph-0602207-2-15-0': 'In particular, the associated function of zero order [MATH] is a formal eigenfunction of [MATH] (a solution of the homogeneous Schrodinger equation, not necessarily normalizable).', 'quant-ph-0602207-2-16-0': 'Let us single out normalizable associated functions and the case when [MATH] maps them into normalizable functions.', 'quant-ph-0602207-2-16-1': 'Evidently this may occur only for non-Hermitian Hamiltonians.', 'quant-ph-0602207-2-16-2': 'Then for any normalizable associated functions [MATH] and [MATH] the transposition symmetry holds [EQUATION]', 'quant-ph-0602207-2-16-3': 'Furthermore one can prove the following relations: [EQUATION] where [MATH] is scalar product.', 'quant-ph-0602207-2-17-0': 'As well, if there are normalizable associated functions [MATH] and [MATH] such that [EQUATION] then [EQUATION]', 'quant-ph-0602207-2-17-1': 'In particular, for some normalizable associated function [MATH], the "self-orthogonality" [CITATION] is realized , [EQUATION]', 'quant-ph-0602207-2-17-2': 'Thus, when assigning [CITATION] the probabilistic meaning for the binorm [MATH] one comes to conclusion that a sort of intriguing phase transition occurs in such a system, signalled by the puzzling divergence of some averages of observables, [EQUATION]', 'quant-ph-0602207-2-17-3': 'All the above relations are derived from the symmetry of a Hamiltonian under transposition and the very definition of associated functions and therefore the existence of self-orthogonal states seems to be inherent for any non-diagonalizable Hamiltonians with normalizable associated functions.', 'quant-ph-0602207-2-18-0': '# Towards resolution of puzzle with self-orthogonal states for Hamiltonians with finite-size Jordan cells', 'quant-ph-0602207-2-19-0': 'Let us show that the puzzle with self-orthogonal states may appear, in fact, due to misinterpretation of what are the pairs of orthogonal states in a true biorthogonal basis.', 'quant-ph-0602207-2-19-1': 'We proceed to the special class of Hamiltonians for which the spectrum is discrete and there is a complete biorthogonal system [MATH] such that, [EQUATION] where [MATH] is an index of an [MATH] eigenvalue [MATH], , [MATH] is an index of a Jordan cell (block) for the given eigenvalue, [MATH]; [MATH] is a number of Jordan cells for [MATH];', 'quant-ph-0602207-2-20-0': '[MATH], , [MATH] is an index of associated function in the Jordan cell with indexes [MATH]', 'quant-ph-0602207-2-21-0': 'and [MATH] is a dimension of this Jordan cell.', 'quant-ph-0602207-2-21-1': 'We have taken a general framework which is applicable also for matrix and/or multidimensional Hamiltonians.', 'quant-ph-0602207-2-21-2': 'But the main results of this and the next sections are guaranteed only for scalar Hamiltonians.', 'quant-ph-0602207-2-22-0': 'We remark that the number [MATH] is called a geometric multiplicity of the eigenvalue [MATH] .', 'quant-ph-0602207-2-22-1': 'For a scalar one-dimensional Schrodinger equation it cannot exceed 2.', 'quant-ph-0602207-2-22-2': 'In turn, the sum [MATH] is called an algebraic multiplicity of the eigenvalue [MATH].', 'quant-ph-0602207-2-23-0': 'The completeness implies the biorthogonality relations [EQUATION] and the unity decomposition [EQUATION]', 'quant-ph-0602207-2-23-1': 'The spectral decomposition for the Hamiltonian can be constructed as well, [EQUATION]', 'quant-ph-0602207-2-23-2': 'It represents the analog of the block-diagonal Jordan form for arbitrary non-Hermitian matrices [CITATION].', 'quant-ph-0602207-2-24-0': 'If existing such biorthogonal systems are not unique.', 'quant-ph-0602207-2-24-1': 'Indeed the relations ([REF]) remain invariant under the group of triangle transformations, [EQUATION] where the matrix elements must obey the following equations, [EQUATION]', 'quant-ph-0602207-2-24-2': 'The biorthogonality ([REF]) restricts the choice of pairs of matrices [MATH] and [MATH] in ([REF]) to be, [EQUATION]', 'quant-ph-0602207-2-24-3': 'This freedom in the redefinition of the biorthogonal basis is similar to Eq. ([REF]) and it can be exploited to define the pairs of biorthogonal functions [MATH] and [MATH] in accordance with ([REF]).', 'quant-ph-0602207-2-24-4': 'However one has to take into account our enumeration of associated functions [MATH] vs. their conjugated ones [MATH] as it is introduced in Eqs. ([REF]) [EQUATION]', 'quant-ph-0602207-2-24-5': 'Then the analog of Eq. ([REF]) reads, [EQUATION]', 'quant-ph-0602207-2-24-6': 'We stress that this kind of biorthogonal systems is determined uniquely up to an overall sign.', 'quant-ph-0602207-2-25-0': 'In these terms it becomes clear that the relations ([REF]) have the meaning of orthogonality of some off-diagonal pairs in the biorthogonal system [MATH] as [EQUATION]', 'quant-ph-0602207-2-25-1': 'When comparing with specification of indices in Eq. ([REF]) one identifies [MATH].', 'quant-ph-0602207-2-25-2': 'In both cases [MATH] .', 'quant-ph-0602207-2-25-3': 'Then the inequality ([REF]) singles out off-diagonal binorms, [MATH].', 'quant-ph-0602207-2-25-4': 'One can show easily that in order to have all diagonal binorms non-vanishing it is sufficient to prove that at least one of them is not zero which equivalent to the requirement [EQUATION] thereby providing the completeness of the basis.', 'quant-ph-0602207-2-26-0': 'Going back to the definition of quantum-state averages of certain observables we realize that the matrix element used in ([REF]) is not diagonal and therefore this relation cannot be interpreted as an average (compare with [CITATION]) of a putative order-parameter like operator.', 'quant-ph-0602207-2-27-0': '# Symmetric representation of non-diagonalizable Hamiltonians', 'quant-ph-0602207-2-28-0': 'We still notice that the biorthogonal basis ([REF]) does not provide a manifestly symmetric representation of the Hamiltonian (which is a symmetric operator).', 'quant-ph-0602207-2-28-1': 'One can obtain another biorthogonal basis using the canonical set of (normalizable) associated functions given by Eq. ([REF]) and their complex conjugates in an analogy to ([REF]) .', 'quant-ph-0602207-2-28-2': 'It can be achieved by means of renumbering of conjugated elements of the biorthogonal system ([REF]) , [EQUATION]', 'quant-ph-0602207-2-28-3': 'Eventually one arrives to the symmetric spectral decomposition for [MATH]: [EQUATION] which looks like a Jordan decomposition along the secondary diagonal.', 'quant-ph-0602207-2-28-4': 'Evidently in the coordinate representation the Hamiltonian operator is manifestly symmetric when the special biorthogonal basis ([REF]), [EQUATION] is chosen.', 'quant-ph-0602207-2-29-0': 'But the unity decomposition in this case is not diagonal, [EQUATION] although symmetric.', 'quant-ph-0602207-2-29-1': 'One can diagonalize this unity decomposition by a non-degenerate orthogonal transformation [MATH] of sub-bases in each non-diagonal sub-block, [EQUATION] retaining the type of the basis ([REF]) .', 'quant-ph-0602207-2-29-2': 'Then one finds a number eigenvalues [MATH].', 'quant-ph-0602207-2-29-3': 'In order to come to the canonical form of a basis ([REF]) one has to rotate by the complex unit [MATH] the pairs in the basis ([REF]) normalized on [MATH] .', 'quant-ph-0602207-2-29-4': 'Evidently the combination of the transformation [MATH] and such a rotation contains complex elements and is not orthogonal.', 'quant-ph-0602207-2-30-0': 'The remaining freedom of basis redefinition with the help of orthogonal rotations cannot provide the consequent diagonalization of the symmetric Hamiltonian matrix in each non-diagonal block.', 'quant-ph-0602207-2-30-1': 'The reason is that some of eigenvectors of the Hamiltonian sub-matrices have zero binorms, in particular, those ones which are related to the true Hamiltonian eigenfunctions.', 'quant-ph-0602207-2-30-2': 'Thus while being a symmetric operator with symmetric matrix representation, the Hamiltonian remains essentially non-diagonalizable.', 'quant-ph-0602207-2-31-0': 'We remark that in the general case the existence and the completeness of a biorthogonal system is not obvious (especially if the continuous spectrum is present) and needs a careful examination.', 'quant-ph-0602207-2-31-1': 'In particular, at the border between discrete and continuous spectra and in the continuous spectrum itself one can anticipate to have puzzling states with non-trivial role in the spectral decomposition.', 'quant-ph-0602207-2-31-2': 'This peculiarities will be discussed in the next Sections.', 'quant-ph-0602207-2-32-0': '# Example of a Non-diagonalizable Hamiltonian and its origin from level coalescence', 'quant-ph-0602207-2-33-0': 'In this Section we build a model with non-Hermitian Hamiltonian which possesses a Jordan cell spanned on the bound state and a normalizable associated state and further on demonstrate how this kind of degeneracy arises from coalescence of a pair of non-degenerate levels.', 'quant-ph-0602207-2-34-0': '## Jordan cell for bound state', 'quant-ph-0602207-2-35-0': 'The model Hamiltonian contains the potential with coordinates selectively shifted into complex plane, [EQUATION]', 'quant-ph-0602207-2-35-1': 'This Hamiltonian is not PT-symmetric unless [MATH] .', 'quant-ph-0602207-2-35-2': 'It has the Jordan cell, spanned by the normalizable eigenfunction [MATH] and associated function [MATH] on the level [MATH], [EQUATION]', 'quant-ph-0602207-2-35-3': 'In turn, the eigenfunctions of [MATH] for continuous spectrum read, [EQUATION]', 'quant-ph-0602207-2-35-4': 'For this model, unity decomposition built of eigenfunctions and associated functions of [MATH] can be obtained by conventional Green function methods, [EQUATION] this decomposition can be represented in the operator form, [EQUATION]', 'quant-ph-0602207-2-35-5': 'Evidently the basis [MATH] corresponds to the basis [MATH] of Sec. 4 and therefore gives the symmetric spectral decomposition for the Hamiltonian, [EQUATION]', 'quant-ph-0602207-2-35-6': 'The diagonalization of the decomposition of unity ([REF]) may be arranged in two different ways.', 'quant-ph-0602207-2-35-7': 'First one can exploit the scheme of Sec. 3 performing re-numeration of certain elements of conjugated basis, [EQUATION] namely, [EQUATION]', 'quant-ph-0602207-2-35-8': 'With this notation the decomposition of unity reads, [EQUATION]', 'quant-ph-0602207-2-35-9': 'We stress that [MATH] are related to the basis [MATH] in Sec. 3.', 'quant-ph-0602207-2-35-10': 'The relevant spectral decomposition of the Hamiltonian takes the quasi-diagonal form with one Jordan cell, [EQUATION]', 'quant-ph-0602207-2-35-11': 'On the other hand, the decomposition of unity ([REF]) can be diagonalized by complex non-degenerate rotations , i.e. by using the construction of Sec. 4 .', 'quant-ph-0602207-2-35-12': 'The relevant basis is given by, [EQUATION] and [MATH] is an arbitrary constant with dimension of a length.', 'quant-ph-0602207-2-35-13': 'The decomposition of unity becomes diagonal , [EQUATION] or in the operator form, [EQUATION] where again the Dirac notations have been used, [EQUATION]', 'quant-ph-0602207-2-35-14': 'Accordingly, the manifestly symmetric spectral decomposition of [MATH] can be easily obtained, [EQUATION]', 'quant-ph-0602207-2-35-15': 'Notice that it cannot be diagonalized further, since the symmetric [MATH] matrix in ([REF]), [EQUATION] has one degenerate eigenvalue [MATH] and possesses only one eigenvector [MATH] with zero scalar product [MATH].', 'quant-ph-0602207-2-35-16': 'Its existence means that the orthogonal non-degenerate matrix required for diagonalization cannot be built conventionally from a set of eigenvectors.', 'quant-ph-0602207-2-35-17': 'Evidently, this vector maps the pair of basis functions [MATH] into the eigenstate of the Hamiltonian [MATH] whose partner in the biorthogonal basis is [MATH] .', 'quant-ph-0602207-2-35-18': 'Thus the existence of the zero norm vector [MATH] does not entail the self-orthogonality of basis elements [MATH] and [MATH].', 'quant-ph-0602207-2-36-0': '## Level coalescence for complex coordinates', 'quant-ph-0602207-2-37-0': 'The Hamiltonian [MATH] with a Jordan cell for bound state ([REF]) can be obtained as a limiting case, of the Hamiltonian [MATH] with two non-degenerate bound states (of algebraic multiplicity 1), corresponding [MATH], [EQUATION].', 'quant-ph-0602207-2-37-1': 'For this Hamiltonian [MATH] there are two normalized eigenfunctions for bound states, [EQUATION] with eigenvalues, [EQUATION]', 'quant-ph-0602207-2-37-2': 'Eigenfunctions of [MATH] for continuous spectrum take the form [EQUATION] i.e. the unity decomposition ([REF]) is reproduced.', 'quant-ph-0602207-2-38-0': '# Puzzles with zero-binorm bound states in the continuum', 'quant-ph-0602207-2-39-0': 'In what follows we develop another type of models in which the continuous spectrum is essentially involved in non-diagonal part of a Hamiltonian and elaborate the decomposition of unity.', 'quant-ph-0602207-2-39-1': 'The standard treatment of continuous spectrum physics deals with reflection and transmission coefficients whose definition implies the existence of two linearly independent scattering solutions for a given spectral parameter.', 'quant-ph-0602207-2-39-2': 'Here we provide an example when this is not realized for a non-Hermitian Hamiltonian defined on the whole axis.', 'quant-ph-0602207-2-39-3': 'This point can be thought of as a spectral singularity for the continuous spectrum.', 'quant-ph-0602207-2-40-0': '## Non-Hermitian Hamiltonian with normalizable bound state at the continuum threshold', 'quant-ph-0602207-2-41-0': 'Let us now consider the Hamiltonian [EQUATION] raising up the puzzle of "self-orthogonality" [CITATION].', 'quant-ph-0602207-2-42-0': 'In order to unravel this puzzle we examine the unity decomposition made of eigenfunctions of [MATH], [EQUATION]', 'quant-ph-0602207-2-42-1': 'Hence it is the 3rd term in the right side of ([REF]) that provides the opportunity to reproduce [MATH] and thereby to complete the unity decomposition.', 'quant-ph-0602207-2-42-2': 'Thus one concludes that the state [MATH] is inseparable from the bottom of continuous spectrum and the decomposition of unity in this sense is not diagonal.', 'quant-ph-0602207-2-43-0': 'We notice that the Hamiltonian ([REF]) is PT-symmetric and can be derived from the Hamiltonian ([REF]) in the limit [MATH] but the parameter [MATH] must be taken as a half of [MATH] from ([REF]).', 'quant-ph-0602207-2-44-0': 'We also remark that the Hamiltonian ([REF]) makes sense also for arbitrary coupling constants of "centrifugal" potential, and for the following set, [EQUATION] with positive [MATH], the Jordan cell, spanned by [MATH] normalizable eigenfunction and associated functions, appears at the threshold of continuous spectrum, [EQUATION]', 'quant-ph-0602207-2-44-1': 'All these zero-energy bound and associated states have zero binorms and are biorthogonal to each other (the multiple puzzle of "self-orthogonality").', 'quant-ph-0602207-2-44-2': 'Unity decomposition in such cases can be derived in a similar way although its form will be more cumbersome.', 'quant-ph-0602207-2-45-0': '## Hamiltonian with spectral singularity', 'quant-ph-0602207-2-46-0': 'Let us force the bound state energy [MATH] in the Hamiltonian ([REF]) to move towards the continuous spectrum, [MATH] .', 'quant-ph-0602207-2-46-1': 'Then for the Hamiltonian, [EQUATION] on the level [MATH] in the continuous spectrum, one finds the Jordan cell, spanned by the normalizable eigenfunction [MATH] and the associated function [MATH], whose asymptotics for [MATH] correspond to superposition of incoming and outgoing waves (standing wave), [EQUATION]', 'quant-ph-0602207-2-46-2': 'The asymptotics of this standing wave is, [EQUATION] therefore it does not belong to the scattering spectrum.', 'quant-ph-0602207-2-47-0': 'In turn the eigenfunctions of [MATH] for the scattering spectrum read, [EQUATION]', 'quant-ph-0602207-2-47-1': 'For this model, unity decomposition made of eigenfunctions and associated functions of [MATH] can be obtained by conventional Green function methods, [EQUATION] where [MATH] is an integration path in the complex momentum plane, obtained from real axis by its simultaneous displacement near the points [MATH] up or down.', 'quant-ph-0602207-2-48-0': 'For test functions from [MATH], [MATH] this decomposition can be presented in the form, [EQUATION] where the solutions [MATH] are made by analytical continuation of [MATH] in [MATH] into complex plane, and the branch of [MATH] is uniquely defined by the condition [MATH] in the plane with the cut on positive part of real axis.', 'quant-ph-0602207-2-48-1': 'In virtue of ([REF]) the Green function for the diagonalizable Hamiltonian of Subsec. 5.2.', 'quant-ph-0602207-2-48-2': 'has two poles of the first order (if [MATH]) at the points [MATH] where [MATH] can be either real or imaginary.', 'quant-ph-0602207-2-48-3': 'If [MATH] and level confluence emerges, two poles coalesce into one pole of the second order in both cases ([REF]) and ([REF]) when [MATH] .', 'quant-ph-0602207-2-48-4': 'However the examples of Subsec. 5.1 and 6.2 have different meaning: in the first case the double pole does not appear on the physical cut [MATH] and its order enumerates the rank of the Jordan cell.', 'quant-ph-0602207-2-48-5': 'On the contrary, in the second case the double pole is placed exactly on the cut [MATH] and strictly speaking signifies the spectral singularity as it generates only one eigenstate with the eigenvalue [MATH] in unity decomposition.', 'quant-ph-0602207-2-48-6': 'The second state - the associated function, represents a standing wave and does not influence the spectral decomposition.', 'quant-ph-0602207-2-48-7': 'In the example of Subsec. 6.1.', 'quant-ph-0602207-2-48-8': 'the Green function has only a branch point at [MATH] of the following type [MATH].', 'quant-ph-0602207-2-48-9': 'This branch point can be thought of as a confluence of the double pole in the variable [MATH] and the branch point of order [MATH]', 'quant-ph-0602207-2-49-0': 'From the explicit form of wave functions one can see that all potentials in Sec.5 and 6 are transparent as the reflection coefficient is zero.', 'quant-ph-0602207-2-49-1': 'The transmission coefficient in the non-degenerate case of Subsec. 5.2.', 'quant-ph-0602207-2-49-2': 'takes the form, [EQUATION]', 'quant-ph-0602207-2-49-3': 'In different limits one can derive the transmission coefficients for three other Hamiltonians.', 'quant-ph-0602207-2-49-4': 'Namely, when [MATH] and [MATH] (the case 5.1.)', 'quant-ph-0602207-2-49-5': 'the scattering is described by [EQUATION] and for [MATH] and [MATH] or imaginary (the cases 6.1.', 'quant-ph-0602207-2-49-6': 'and 6.2.)', 'quant-ph-0602207-2-49-7': 'the scattering is absent, [EQUATION]', 'quant-ph-0602207-2-49-8': 'Thus the colliding particle in such cases is not "influenced" by the bound states in the continuum.', 'quant-ph-0602207-2-50-0': '# Conclusions', 'quant-ph-0602207-2-51-0': 'In this paper we have presented a thorough analysis of the phenomenon of apparent self-orthogonality of some eigenstates for non-Hermitian Hamiltonians.', 'quant-ph-0602207-2-51-1': 'For the discrete part of energy spectrum, it has been shown that such a phenomenon should take place only for non-diagonalizable Hamiltonians the spectrum of which consists not only of eigenfunctions and but also of associated functions.', 'quant-ph-0602207-2-51-2': 'However the genuine diagonal biorthogonal basis related to the spectral decomposition of such Hamiltonians normally does not contain pairs made of the same eigenfunctions or the associated functions of the same order.', 'quant-ph-0602207-2-51-3': 'Rather they are complementary: for instance, eigenfunctions in the direct basis are paired to those associated functions in the conjugated basis, which have the maximal order in the same Jordan cell.', 'quant-ph-0602207-2-51-4': 'One possible exception exists for Jordan cells of odd order where one basis pair consists of the same function which is not self-orthogonal.', 'quant-ph-0602207-2-52-0': 'The situation in the continuous spectrum is more subtle: namely, the spectral decomposition does not include any obvious Jordan cells and associated functions.', 'quant-ph-0602207-2-52-1': 'However we have established that when a zero-binorm normalizable state arises it remains inseparable from the nearest scattering states of the continuum and eventually the existence of this state does not destroy the completeness of decomposition of unity.', 'quant-ph-0602207-2-53-0': 'Finally, let us outline the measurability of quantum observables and, for this purpose, prepare a wave packet, [EQUATION] where the possibility to have continuous spectrum is made explicit in the notation and, for brevity, all indices enumerating eigenvalues, Jordan cells and their elements are encoded in the index [MATH].', 'quant-ph-0602207-2-54-0': 'In order to perform the quantum averaging of an operator of observable [MATH] one can use the conventional Hilbert space scalar product and the complex conjugated wave function, [MATH] .', 'quant-ph-0602207-2-54-1': 'In this way one defines the wave packet of the conjugated state and, respectively, average values of the operator [MATH] , [EQUATION]', 'quant-ph-0602207-2-54-2': 'For such a definition the averages of an operator of observable, [MATH] cannot be infinite and the phenomena of (pseudo) phase transitions at the level crossing [CITATION] cannot appear.', 'quant-ph-0602207-2-54-3': 'In this relation the basis from Sec. 3 with [MATH] may be used equally well.', 'quant-ph-0602207-2-55-0': 'However one has to keep in mind that such a definition of probabilities, to some extent, depends on a particular set of biorthogonal bases.', 'quant-ph-0602207-2-55-1': 'We hope to examine this approach and its applications elsewhere.', 'quant-ph-0602207-2-56-0': 'The work of A.A. and A.S. was supported by Grant RFBR 06-01-00186-a and by the Program "Development of scientific potential of Higher Education in Russia" RNP 2.1.1.1112.', 'quant-ph-0602207-2-56-1': 'A.Sokolov was partially supported by Grant INFN-IS/BO31.', 'quant-ph-0602207-2-56-2': 'We are grateful to the organizers of 3rd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics (June 2005, Istanbul, Turkey), especially to Prof. A. Mostafazadeh and Prof. M. Znojil, for financial support and opportunity to present the preliminary results of this paper.', 'quant-ph-0602207-2-57-0': '# References'}

[['quant-ph-0602207-1-7-0', 'quant-ph-0602207-2-7-0'], ['quant-ph-0602207-1-7-1', 'quant-ph-0602207-2-7-1'], ['quant-ph-0602207-1-7-2', 'quant-ph-0602207-2-7-2'], ['quant-ph-0602207-1-7-3', 'quant-ph-0602207-2-7-3'], ['quant-ph-0602207-1-7-4', 'quant-ph-0602207-2-7-4'], ['quant-ph-0602207-1-7-5', 'quant-ph-0602207-2-7-5'], ['quant-ph-0602207-1-7-6', 'quant-ph-0602207-2-7-6'], ['quant-ph-0602207-1-7-7', 'quant-ph-0602207-2-7-7'], ['quant-ph-0602207-1-17-0', 'quant-ph-0602207-2-17-0'], ['quant-ph-0602207-1-17-1', 'quant-ph-0602207-2-17-1'], ['quant-ph-0602207-1-17-2', 'quant-ph-0602207-2-17-2'], ['quant-ph-0602207-1-17-3', 'quant-ph-0602207-2-17-3'], ['quant-ph-0602207-1-54-0', 'quant-ph-0602207-2-54-0'], ['quant-ph-0602207-1-54-1', 'quant-ph-0602207-2-54-1'], ['quant-ph-0602207-1-54-2', 'quant-ph-0602207-2-54-2'], ['quant-ph-0602207-1-54-3', 'quant-ph-0602207-2-54-3'], ['quant-ph-0602207-1-8-0', 'quant-ph-0602207-2-8-0'], ['quant-ph-0602207-1-8-1', 'quant-ph-0602207-2-8-1'], ['quant-ph-0602207-1-8-2', 'quant-ph-0602207-2-8-2'], ['quant-ph-0602207-1-31-0', 'quant-ph-0602207-2-31-0'], ['quant-ph-0602207-1-31-1', 'quant-ph-0602207-2-31-1'], ['quant-ph-0602207-1-31-2', 'quant-ph-0602207-2-31-2'], ['quant-ph-0602207-1-23-0', 'quant-ph-0602207-2-23-0'], ['quant-ph-0602207-1-23-1', 'quant-ph-0602207-2-23-1'], ['quant-ph-0602207-1-23-2', 'quant-ph-0602207-2-23-2'], ['quant-ph-0602207-1-42-0', 'quant-ph-0602207-2-42-0'], ['quant-ph-0602207-1-42-1', 'quant-ph-0602207-2-42-1'], ['quant-ph-0602207-1-42-2', 'quant-ph-0602207-2-42-2'], ['quant-ph-0602207-1-52-0', 'quant-ph-0602207-2-52-0'], ['quant-ph-0602207-1-52-1', 'quant-ph-0602207-2-52-1'], ['quant-ph-0602207-1-56-1', 'quant-ph-0602207-2-56-1'], ['quant-ph-0602207-1-39-0', 'quant-ph-0602207-2-39-0'], ['quant-ph-0602207-1-39-1', 'quant-ph-0602207-2-39-1'], ['quant-ph-0602207-1-39-2', 'quant-ph-0602207-2-39-2'], ['quant-ph-0602207-1-39-3', 'quant-ph-0602207-2-39-3'], ['quant-ph-0602207-1-44-0', 'quant-ph-0602207-2-44-0'], ['quant-ph-0602207-1-44-1', 'quant-ph-0602207-2-44-1'], ['quant-ph-0602207-1-44-2', 'quant-ph-0602207-2-44-2'], ['quant-ph-0602207-1-49-0', 'quant-ph-0602207-2-49-0'], ['quant-ph-0602207-1-49-1', 'quant-ph-0602207-2-49-1'], ['quant-ph-0602207-1-49-2', 'quant-ph-0602207-2-49-2'], ['quant-ph-0602207-1-49-3', 'quant-ph-0602207-2-49-3'], ['quant-ph-0602207-1-49-5', 'quant-ph-0602207-2-49-5'], ['quant-ph-0602207-1-49-7', 'quant-ph-0602207-2-49-7'], ['quant-ph-0602207-1-49-8', 'quant-ph-0602207-2-49-8'], ['quant-ph-0602207-1-5-0', 'quant-ph-0602207-2-5-0'], ['quant-ph-0602207-1-5-1', 'quant-ph-0602207-2-5-1'], ['quant-ph-0602207-1-5-2', 'quant-ph-0602207-2-5-2'], ['quant-ph-0602207-1-5-3', 'quant-ph-0602207-2-5-3'], ['quant-ph-0602207-1-3-0', 'quant-ph-0602207-2-3-0'], ['quant-ph-0602207-1-3-1', 'quant-ph-0602207-2-3-1'], ['quant-ph-0602207-1-3-2', 'quant-ph-0602207-2-3-2'], ['quant-ph-0602207-1-3-3', 'quant-ph-0602207-2-3-3'], ['quant-ph-0602207-1-20-0', 'quant-ph-0602207-2-20-0'], ['quant-ph-0602207-1-51-0', 'quant-ph-0602207-2-51-0'], ['quant-ph-0602207-1-51-1', 'quant-ph-0602207-2-51-1'], ['quant-ph-0602207-1-51-2', 'quant-ph-0602207-2-51-2'], ['quant-ph-0602207-1-51-3', 'quant-ph-0602207-2-51-3'], ['quant-ph-0602207-1-51-4', 'quant-ph-0602207-2-51-4'], ['quant-ph-0602207-1-47-0', 'quant-ph-0602207-2-47-0'], ['quant-ph-0602207-1-47-1', 'quant-ph-0602207-2-47-1'], ['quant-ph-0602207-1-4-0', 'quant-ph-0602207-2-4-0'], ['quant-ph-0602207-1-4-1', 'quant-ph-0602207-2-4-1'], ['quant-ph-0602207-1-9-0', 'quant-ph-0602207-2-9-0'], ['quant-ph-0602207-1-9-1', 'quant-ph-0602207-2-9-1'], ['quant-ph-0602207-1-19-0', 'quant-ph-0602207-2-19-0'], ['quant-ph-0602207-1-19-1', 'quant-ph-0602207-2-19-1'], ['quant-ph-0602207-1-55-0', 'quant-ph-0602207-2-55-0'], ['quant-ph-0602207-1-55-1', 'quant-ph-0602207-2-55-1'], ['quant-ph-0602207-1-15-0', 'quant-ph-0602207-2-15-0'], ['quant-ph-0602207-1-24-0', 'quant-ph-0602207-2-24-0'], ['quant-ph-0602207-1-24-1', 'quant-ph-0602207-2-24-1'], ['quant-ph-0602207-1-24-2', 'quant-ph-0602207-2-24-2'], ['quant-ph-0602207-1-24-3', 'quant-ph-0602207-2-24-3'], ['quant-ph-0602207-1-24-4', 'quant-ph-0602207-2-24-4'], ['quant-ph-0602207-1-24-5', 'quant-ph-0602207-2-24-5'], ['quant-ph-0602207-1-24-6', 'quant-ph-0602207-2-24-6'], ['quant-ph-0602207-1-30-0', 'quant-ph-0602207-2-30-0'], ['quant-ph-0602207-1-30-1', 'quant-ph-0602207-2-30-1'], ['quant-ph-0602207-1-30-2', 'quant-ph-0602207-2-30-2'], ['quant-ph-0602207-1-22-0', 'quant-ph-0602207-2-22-0'], ['quant-ph-0602207-1-22-1', 'quant-ph-0602207-2-22-1'], ['quant-ph-0602207-1-22-2', 'quant-ph-0602207-2-22-2'], ['quant-ph-0602207-1-28-0', 'quant-ph-0602207-2-28-0'], ['quant-ph-0602207-1-28-1', 'quant-ph-0602207-2-28-1'], ['quant-ph-0602207-1-28-2', 'quant-ph-0602207-2-28-2'], ['quant-ph-0602207-1-28-3', 'quant-ph-0602207-2-28-3'], ['quant-ph-0602207-1-28-4', 'quant-ph-0602207-2-28-4'], ['quant-ph-0602207-1-16-0', 'quant-ph-0602207-2-16-0'], ['quant-ph-0602207-1-16-1', 'quant-ph-0602207-2-16-1'], ['quant-ph-0602207-1-16-2', 'quant-ph-0602207-2-16-2'], ['quant-ph-0602207-1-16-3', 'quant-ph-0602207-2-16-3'], ['quant-ph-0602207-1-48-0', 'quant-ph-0602207-2-48-0'], ['quant-ph-0602207-1-48-1', 'quant-ph-0602207-2-48-1'], ['quant-ph-0602207-1-48-2', 'quant-ph-0602207-2-48-2'], ['quant-ph-0602207-1-48-3', 'quant-ph-0602207-2-48-3'], ['quant-ph-0602207-1-48-4', 'quant-ph-0602207-2-48-4'], ['quant-ph-0602207-1-48-5', 'quant-ph-0602207-2-48-5'], ['quant-ph-0602207-1-48-6', 'quant-ph-0602207-2-48-6'], ['quant-ph-0602207-1-48-7', 'quant-ph-0602207-2-48-7'], ['quant-ph-0602207-1-48-8', 'quant-ph-0602207-2-48-8'], ['quant-ph-0602207-1-48-9', 'quant-ph-0602207-2-48-9'], ['quant-ph-0602207-1-26-0', 'quant-ph-0602207-2-26-0'], ['quant-ph-0602207-1-10-0', 'quant-ph-0602207-2-10-0'], ['quant-ph-0602207-1-10-1', 'quant-ph-0602207-2-10-1'], ['quant-ph-0602207-1-10-2', 'quant-ph-0602207-2-10-2'], ['quant-ph-0602207-1-10-3', 'quant-ph-0602207-2-10-3'], ['quant-ph-0602207-1-10-4', 'quant-ph-0602207-2-10-4'], ['quant-ph-0602207-1-10-5', 'quant-ph-0602207-2-10-5'], ['quant-ph-0602207-1-25-0', 'quant-ph-0602207-2-25-0'], ['quant-ph-0602207-1-25-1', 'quant-ph-0602207-2-25-1'], ['quant-ph-0602207-1-25-2', 'quant-ph-0602207-2-25-2'], ['quant-ph-0602207-1-25-3', 'quant-ph-0602207-2-25-3'], ['quant-ph-0602207-1-25-4', 'quant-ph-0602207-2-25-4'], ['quant-ph-0602207-1-29-0', 'quant-ph-0602207-2-29-0'], ['quant-ph-0602207-1-29-1', 'quant-ph-0602207-2-29-1'], ['quant-ph-0602207-1-29-2', 'quant-ph-0602207-2-29-2'], ['quant-ph-0602207-1-29-3', 'quant-ph-0602207-2-29-3'], ['quant-ph-0602207-1-29-4', 'quant-ph-0602207-2-29-4'], ['quant-ph-0602207-1-12-0', 'quant-ph-0602207-2-12-0'], ['quant-ph-0602207-1-12-1', 'quant-ph-0602207-2-12-1'], ['quant-ph-0602207-1-41-0', 'quant-ph-0602207-2-41-0'], ['quant-ph-0602207-1-0-0', 'quant-ph-0602207-2-0-0'], ['quant-ph-0602207-1-0-1', 'quant-ph-0602207-2-0-1'], ['quant-ph-0602207-1-0-2', 'quant-ph-0602207-2-0-2'], ['quant-ph-0602207-1-0-3', 'quant-ph-0602207-2-0-3'], ['quant-ph-0602207-1-37-0', 'quant-ph-0602207-2-37-0'], ['quant-ph-0602207-1-37-1', 'quant-ph-0602207-2-37-1'], ['quant-ph-0602207-1-37-2', 'quant-ph-0602207-2-37-2'], ['quant-ph-0602207-1-33-0', 'quant-ph-0602207-2-33-0'], ['quant-ph-0602207-1-14-0', 'quant-ph-0602207-2-14-0'], ['quant-ph-0602207-1-53-0', 'quant-ph-0602207-2-53-0'], ['quant-ph-0602207-1-6-0', 'quant-ph-0602207-2-6-0'], ['quant-ph-0602207-1-46-0', 'quant-ph-0602207-2-46-0'], ['quant-ph-0602207-1-46-1', 'quant-ph-0602207-2-46-1'], ['quant-ph-0602207-1-46-2', 'quant-ph-0602207-2-46-2'], ['quant-ph-0602207-1-21-0', 'quant-ph-0602207-2-21-0'], ['quant-ph-0602207-1-21-1', 'quant-ph-0602207-2-21-1'], ['quant-ph-0602207-1-21-2', 'quant-ph-0602207-2-21-2'], ['quant-ph-0602207-1-35-0', 'quant-ph-0602207-2-35-0'], ['quant-ph-0602207-1-35-1', 'quant-ph-0602207-2-35-1'], ['quant-ph-0602207-1-35-2', 'quant-ph-0602207-2-35-2'], ['quant-ph-0602207-1-35-3', 'quant-ph-0602207-2-35-3'], ['quant-ph-0602207-1-35-4', 'quant-ph-0602207-2-35-4'], ['quant-ph-0602207-1-35-5', 'quant-ph-0602207-2-35-5'], ['quant-ph-0602207-1-35-6', 'quant-ph-0602207-2-35-6'], ['quant-ph-0602207-1-35-7', 'quant-ph-0602207-2-35-7'], ['quant-ph-0602207-1-35-8', 'quant-ph-0602207-2-35-8'], ['quant-ph-0602207-1-35-9', 'quant-ph-0602207-2-35-9'], ['quant-ph-0602207-1-35-10', 'quant-ph-0602207-2-35-10'], ['quant-ph-0602207-1-35-11', 'quant-ph-0602207-2-35-11'], ['quant-ph-0602207-1-35-12', 'quant-ph-0602207-2-35-12'], ['quant-ph-0602207-1-35-13', 'quant-ph-0602207-2-35-13'], ['quant-ph-0602207-1-35-14', 'quant-ph-0602207-2-35-14'], ['quant-ph-0602207-1-35-15', 'quant-ph-0602207-2-35-15'], ['quant-ph-0602207-1-35-16', 'quant-ph-0602207-2-35-16'], ['quant-ph-0602207-1-35-17', 'quant-ph-0602207-2-35-17'], ['quant-ph-0602207-1-35-18', 'quant-ph-0602207-2-35-18'], ['quant-ph-0602207-1-43-0', 'quant-ph-0602207-2-43-0'], ['quant-ph-0602207-2-9-0', 'quant-ph-0602207-3-9-0'], ['quant-ph-0602207-2-9-1', 'quant-ph-0602207-3-9-1'], ['quant-ph-0602207-2-30-0', 'quant-ph-0602207-3-30-0'], ['quant-ph-0602207-2-30-1', 'quant-ph-0602207-3-30-1'], ['quant-ph-0602207-2-30-2', 'quant-ph-0602207-3-30-2'], ['quant-ph-0602207-2-31-0', 'quant-ph-0602207-3-31-0'], ['quant-ph-0602207-2-31-1', 'quant-ph-0602207-3-31-1'], ['quant-ph-0602207-2-31-2', 'quant-ph-0602207-3-31-2'], ['quant-ph-0602207-2-43-0', 'quant-ph-0602207-3-43-0'], ['quant-ph-0602207-2-42-0', 'quant-ph-0602207-3-42-0'], ['quant-ph-0602207-2-42-1', 'quant-ph-0602207-3-42-1'], ['quant-ph-0602207-2-10-0', 'quant-ph-0602207-3-10-0'], ['quant-ph-0602207-2-10-3', 'quant-ph-0602207-3-10-3'], ['quant-ph-0602207-2-10-4', 'quant-ph-0602207-3-10-4'], ['quant-ph-0602207-2-10-5', 'quant-ph-0602207-3-10-5'], ['quant-ph-0602207-2-35-0', 'quant-ph-0602207-3-35-0'], ['quant-ph-0602207-2-35-1', 'quant-ph-0602207-3-35-1'], ['quant-ph-0602207-2-35-2', 'quant-ph-0602207-3-35-2'], ['quant-ph-0602207-2-35-3', 'quant-ph-0602207-3-35-3'], ['quant-ph-0602207-2-35-4', 'quant-ph-0602207-3-35-4'], ['quant-ph-0602207-2-35-5', 'quant-ph-0602207-3-35-5'], ['quant-ph-0602207-2-35-7', 'quant-ph-0602207-3-35-7'], ['quant-ph-0602207-2-35-9', 'quant-ph-0602207-3-35-9'], ['quant-ph-0602207-2-35-10', 'quant-ph-0602207-3-35-10'], ['quant-ph-0602207-2-35-12', 'quant-ph-0602207-3-35-12'], ['quant-ph-0602207-2-35-13', 'quant-ph-0602207-3-35-13'], ['quant-ph-0602207-2-35-14', 'quant-ph-0602207-3-35-14'], ['quant-ph-0602207-2-35-15', 'quant-ph-0602207-3-35-15'], ['quant-ph-0602207-2-35-16', 'quant-ph-0602207-3-35-16'], ['quant-ph-0602207-2-35-17', 'quant-ph-0602207-3-35-17'], ['quant-ph-0602207-2-35-18', 'quant-ph-0602207-3-35-18'], ['quant-ph-0602207-2-51-0', 'quant-ph-0602207-3-51-0'], ['quant-ph-0602207-2-51-1', 'quant-ph-0602207-3-51-1'], ['quant-ph-0602207-2-51-2', 'quant-ph-0602207-3-51-2'], ['quant-ph-0602207-2-51-3', 'quant-ph-0602207-3-51-3'], 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['quant-ph-0602207-3-35-11', 'quant-ph-0602207-4-37-11'], ['quant-ph-0602207-3-35-16', 'quant-ph-0602207-4-37-16'], ['quant-ph-0602207-3-54-0', 'quant-ph-0602207-4-60-0'], ['quant-ph-0602207-3-54-1', 'quant-ph-0602207-4-60-1'], ['quant-ph-0602207-3-54-2', 'quant-ph-0602207-4-60-2'], ['quant-ph-0602207-3-26-0', 'quant-ph-0602207-4-28-0'], ['quant-ph-0602207-3-8-2', 'quant-ph-0602207-4-8-3'], ['quant-ph-0602207-3-19-0', 'quant-ph-0602207-4-21-0'], ['quant-ph-0602207-3-19-1', 'quant-ph-0602207-4-21-1'], ['quant-ph-0602207-3-43-0', 'quant-ph-0602207-4-46-0'], ['quant-ph-0602207-3-39-0', 'quant-ph-0602207-4-41-0']]

[['quant-ph-0602207-1-56-0', 'quant-ph-0602207-2-56-0'], ['quant-ph-0602207-1-56-2', 'quant-ph-0602207-2-56-2'], ['quant-ph-0602207-2-42-2', 'quant-ph-0602207-3-42-2'], ['quant-ph-0602207-2-10-1', 'quant-ph-0602207-3-10-1'], ['quant-ph-0602207-2-35-6', 'quant-ph-0602207-3-35-6'], ['quant-ph-0602207-2-35-11', 'quant-ph-0602207-3-35-11'], ['quant-ph-0602207-2-52-1', 'quant-ph-0602207-3-52-1'], ['quant-ph-0602207-2-0-1', 'quant-ph-0602207-3-0-1'], ['quant-ph-0602207-2-0-3', 'quant-ph-0602207-3-0-3'], ['quant-ph-0602207-2-6-0', 'quant-ph-0602207-3-6-0'], ['quant-ph-0602207-2-39-0', 'quant-ph-0602207-3-39-0'], ['quant-ph-0602207-2-7-0', 'quant-ph-0602207-3-7-0'], ['quant-ph-0602207-2-7-3', 'quant-ph-0602207-3-7-3'], ['quant-ph-0602207-2-7-6', 'quant-ph-0602207-3-7-6'], ['quant-ph-0602207-2-8-0', 'quant-ph-0602207-3-8-0'], ['quant-ph-0602207-3-9-0', 'quant-ph-0602207-4-10-0'], ['quant-ph-0602207-3-37-2', 'quant-ph-0602207-4-39-2'], ['quant-ph-0602207-3-5-0', 'quant-ph-0602207-4-5-0'], ['quant-ph-0602207-3-5-1', 'quant-ph-0602207-4-5-1'], ['quant-ph-0602207-3-5-2', 'quant-ph-0602207-4-5-2'], ['quant-ph-0602207-3-5-3', 'quant-ph-0602207-4-5-3'], ['quant-ph-0602207-3-44-2', 'quant-ph-0602207-4-47-2'], ['quant-ph-0602207-3-29-1', 'quant-ph-0602207-4-31-1'], ['quant-ph-0602207-3-42-0', 'quant-ph-0602207-4-45-0'], ['quant-ph-0602207-3-42-1', 'quant-ph-0602207-4-45-1'], ['quant-ph-0602207-3-7-3', 'quant-ph-0602207-4-7-3'], ['quant-ph-0602207-3-7-4', 'quant-ph-0602207-4-7-4'], ['quant-ph-0602207-3-7-5', 'quant-ph-0602207-4-7-5'], ['quant-ph-0602207-3-7-6', 'quant-ph-0602207-4-7-6'], ['quant-ph-0602207-3-14-0', 'quant-ph-0602207-4-16-1'], ['quant-ph-0602207-3-28-0', 'quant-ph-0602207-4-30-0'], ['quant-ph-0602207-3-28-3', 'quant-ph-0602207-4-30-3'], ['quant-ph-0602207-3-28-4', 'quant-ph-0602207-4-30-4'], ['quant-ph-0602207-3-48-0', 'quant-ph-0602207-4-54-0'], ['quant-ph-0602207-3-48-5', 'quant-ph-0602207-4-54-5'], ['quant-ph-0602207-3-48-9', 'quant-ph-0602207-4-54-9'], ['quant-ph-0602207-3-6-0', 'quant-ph-0602207-4-6-0'], ['quant-ph-0602207-3-30-2', 'quant-ph-0602207-4-32-2'], ['quant-ph-0602207-3-47-1', 'quant-ph-0602207-4-53-1'], ['quant-ph-0602207-3-35-4', 'quant-ph-0602207-4-37-4'], ['quant-ph-0602207-3-35-5', 'quant-ph-0602207-4-37-5'], ['quant-ph-0602207-3-35-13', 'quant-ph-0602207-4-37-13'], ['quant-ph-0602207-3-35-14', 'quant-ph-0602207-4-37-14'], ['quant-ph-0602207-3-35-15', 'quant-ph-0602207-4-37-15'], ['quant-ph-0602207-3-54-3', 'quant-ph-0602207-4-60-3'], ['quant-ph-0602207-3-8-0', 'quant-ph-0602207-4-8-0'], ['quant-ph-0602207-3-39-1', 'quant-ph-0602207-4-42-0'], ['quant-ph-0602207-3-39-2', 'quant-ph-0602207-4-42-1']]

[]

[['quant-ph-0602207-2-10-2', 'quant-ph-0602207-3-10-2'], ['quant-ph-0602207-2-35-8', 'quant-ph-0602207-3-35-8'], ['quant-ph-0602207-2-3-3', 'quant-ph-0602207-3-3-3'], ['quant-ph-0602207-3-25-4', 'quant-ph-0602207-4-27-4'], ['quant-ph-0602207-3-17-0', 'quant-ph-0602207-4-19-0'], ['quant-ph-0602207-3-21-2', 'quant-ph-0602207-4-23-2'], ['quant-ph-0602207-3-29-0', 'quant-ph-0602207-4-31-0'], ['quant-ph-0602207-3-46-2', 'quant-ph-0602207-4-52-2'], ['quant-ph-0602207-3-22-1', 'quant-ph-0602207-4-24-1'], ['quant-ph-0602207-3-10-1', 'quant-ph-0602207-4-12-0'], ['quant-ph-0602207-3-23-0', 'quant-ph-0602207-4-25-0'], ['quant-ph-0602207-3-33-0', 'quant-ph-0602207-4-35-0'], ['quant-ph-0602207-3-33-0', 'quant-ph-0602207-4-35-2'], ['quant-ph-0602207-3-35-12', 'quant-ph-0602207-4-37-12'], ['quant-ph-0602207-3-35-17', 'quant-ph-0602207-4-37-17'], ['quant-ph-0602207-3-35-18', 'quant-ph-0602207-4-37-19'], ['quant-ph-0602207-3-8-1', 'quant-ph-0602207-4-8-2']]

[]

['quant-ph-0602207-1-1-0', 'quant-ph-0602207-1-1-1', 'quant-ph-0602207-1-13-0', 'quant-ph-0602207-1-49-4', 'quant-ph-0602207-1-49-6', 'quant-ph-0602207-2-1-0', 'quant-ph-0602207-2-1-1', 'quant-ph-0602207-2-13-0', 'quant-ph-0602207-2-49-4', 'quant-ph-0602207-2-49-6', 'quant-ph-0602207-3-1-0', 'quant-ph-0602207-3-1-1', 'quant-ph-0602207-3-13-0', 'quant-ph-0602207-3-49-4', 'quant-ph-0602207-3-49-6', 'quant-ph-0602207-4-1-0', 'quant-ph-0602207-4-1-1', 'quant-ph-0602207-4-11-0', 'quant-ph-0602207-4-15-0', 'quant-ph-0602207-4-16-0', 'quant-ph-0602207-4-19-1', 'quant-ph-0602207-4-55-4', 'quant-ph-0602207-4-55-6', 'quant-ph-0602207-4-62-0', 'quant-ph-0602207-4-66-0', 'quant-ph-0602207-4-66-3', 'quant-ph-0602207-4-66-5', 'quant-ph-0602207-4-67-0', 'quant-ph-0602207-4-67-3', 'quant-ph-0602207-4-68-0', 'quant-ph-0602207-4-68-3', 'quant-ph-0602207-4-69-0', 'quant-ph-0602207-4-69-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/', '4': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/quant-ph/0602207

{'quant-ph-0602207-3-0-0': 'We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases.', 'quant-ph-0602207-3-0-1': 'The "self-orthogonality" phenomenon is clarified in terms of a correct spectral decomposition and it is shown that "self-orthogonal" states never jeopardize resolution of identity and thereby quantum averages of observables.', 'quant-ph-0602207-3-0-2': 'The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated.', 'quant-ph-0602207-3-0-3': 'Some puzzles with zero-binorm bound states in continuous spectrum are unraveled with the help of a correct resolution of identity.', 'quant-ph-0602207-3-1-0': 'Journal of Physics A: Math.', 'quant-ph-0602207-3-1-1': 'Gen.', 'quant-ph-0602207-3-2-0': '# Introduction', 'quant-ph-0602207-3-3-0': 'The variety of complex potentials in Quantum Physics is associated typically with open systems when a control of information is partially lost and thereby the unitarity of observable evolution is broken.', 'quant-ph-0602207-3-3-1': 'For this class of quantum systems the energy eigenvalues may have an imaginary part which signals the opening of new channels not directly measured in a given experiment.', 'quant-ph-0602207-3-3-2': 'In this context non-Hermitian interactions have been used in Field Theory and Statistical Mechanics for many years with applications to Condensed Matter, Quantum Optics and Hadronic and Nuclear Physics [CITATION] - [CITATION].', 'quant-ph-0602207-3-3-3': 'The subject of non-self-adjoint operators has been also under intensive mathematical investigations [CITATION], in particular, interesting examples of non-Hermitian effective Hamiltonian operators have been found for the Faddeev equations [CITATION].', 'quant-ph-0602207-3-4-0': 'An important class of complex Hamiltonians deals with a real spectrum [CITATION], in particular, in the PT-symmetric Quantum Mechanics [CITATION]-[CITATION] and its pseudo-Hermitian generalization [CITATION].', 'quant-ph-0602207-3-4-1': 'Scattering problems for such Hamiltonians have been investigated in [CITATION] .', 'quant-ph-0602207-3-5-0': 'For complex, non-Hermitian potentials the natural spectral decomposition exploits the sets of biorthogonal states [CITATION], and within this framework one can discover new features that never happen for closed systems with Hermitian Hamiltonians : namely, certain Hamiltonians may not be diagonalizable [CITATION] and can be reduced only to a quasi-diagonal form with a number of Jordan cells [CITATION].', 'quant-ph-0602207-3-5-1': 'This feature can be realized by level crossing which, in fact, occurs under certain circumstances in atomic and molecular spectra [CITATION] and Optics [CITATION] (see more examples in [CITATION]) as well as in PT-symmetric quantum systems [CITATION].', 'quant-ph-0602207-3-5-2': 'In this case some of eigenstates seem to be "self-orthogonal" in respect to a binorm [CITATION] .', 'quant-ph-0602207-3-5-3': 'This quite intriguing phenomenon has been interpreted as a sort of phase transition [CITATION] .', 'quant-ph-0602207-3-6-0': 'The main purpose of the present work is to clarify the "self-orthogonality" in terms of a correct spectral decomposition both for discrete and for continuous spectra and to show that, at least, in one-dimensional Quantum Mechanics such states never jeopardize resolution of identity and thereby don\'t affect quantum averages of observables.', 'quant-ph-0602207-3-7-0': 'We start introducing the notion of biorthogonal basis and, correspondingly, the resolution of identity for a non-Hermitian diagonalizable Hamiltonian.', 'quant-ph-0602207-3-7-1': 'In Sec. 2 the appearance of associated functions is discussed and in Sec. 3 non-diagonalizable (but quasi-diagonalizable) Hamiltonians with finite-size Jordan cells are analyzed.', 'quant-ph-0602207-3-7-2': 'Special attention is paid to the definition of a biorthogonal diagonal basis and the meaning of zero-binorm states is clarified.', 'quant-ph-0602207-3-7-3': 'Namely, it is shown that the apparent self-orthogonality of eigenfunctions and associated functions is misleading as they never replicate as relative pairs in resolution of identity.', 'quant-ph-0602207-3-7-4': 'Instead, the "self-orthogonality" concerns the different elements in the basis thereby being related to a conventional orthogonality.', 'quant-ph-0602207-3-7-5': 'The existence of biorthogonal bases with pairs of mutually complex-conjugated base functions is proved .', 'quant-ph-0602207-3-7-6': 'In Sec. 4 another, manifestly symmetric representation of non-diagonalizable Hamiltonians is given, compatible with a diagonal resolution of identity.', 'quant-ph-0602207-3-7-7': 'In Sec. 5 the example of a (transparent) complex potential leading to the non-diagonalizable Hamiltonian with one Jordan [MATH] cell is constructed and its origin from level coalescence is illustrated.', 'quant-ph-0602207-3-8-0': 'Some puzzles with zero-binorm bound states in continuous spectrum are unraveled in Sec.6 with the help of a correct resolution of identity.', 'quant-ph-0602207-3-8-1': 'In Sec. 7 we complete our analysis with discussion of resolvents and scattering characteristics for previous examples focusing on the possibility of spectral singularities for non-Hermitian Hamiltonians.', 'quant-ph-0602207-3-8-2': 'We conclude with some proposals for probabilistic interpretation of wave functions defined in respect to a biorthogonal basis which does not allow any negative or zero-norm states.', 'quant-ph-0602207-3-9-0': 'In our paper we deal with complex one-dimensional potentials [MATH] and respectively with non-Hermitian Hamiltonians [MATH] of Schrodinger type, defined on the real axis, [EQUATION] which are assumed to be symmetric or self-transposed under the [MATH] - transposition operation, [MATH].', 'quant-ph-0602207-3-9-1': 'Only scalar local potentials will be analyzed which are obviously symmetric under transposition (for some matrix non-diagonalizable problems, see [CITATION]).', 'quant-ph-0602207-3-10-0': 'Let us first consider a one-dimensional non-Hermitian diagonalizable Hamiltonian [MATH] with discrete spectrum.', 'quant-ph-0602207-3-10-1': 'For this Hamiltonian it is assumed that a biorthogonal system [MATH] exists, [EQUATION] such that the complete resolution of identity and the spectral decomposition of the Hamiltonian hold, [EQUATION]', 'quant-ph-0602207-3-10-2': 'In the coordinate representation, [EQUATION] the resolution of identity has the form, [EQUATION]', 'quant-ph-0602207-3-10-3': 'The differential equations, [EQUATION] and the fact that there is only one normalizable eigenfunction of [MATH] for the eigenvalue [MATH] (up to a constant factor), allow one to conclude that [EQUATION]', 'quant-ph-0602207-3-10-4': 'Hence the system [MATH] can be redefined [EQUATION] so that [EQUATION]', 'quant-ph-0602207-3-10-5': 'Indeed if some of the states in Eq. ([REF]) were "self-orthogonal" (as it has been accepted in [CITATION]) , i.e. had zero binorms in ([REF]), the would-be unity in ([REF]) would annihilate such states thereby signalling the incompleteness.', 'quant-ph-0602207-3-11-0': '# Non-diagonalizable Hamiltonians and zero-binorm states', 'quant-ph-0602207-3-12-0': 'For complex Hamiltonians one can formulate the extended eigenvalue problem, searching not only for normalizable eigenfunctions but also for normalizable associated functions for discrete part of the energy spectrum.', 'quant-ph-0602207-3-12-1': 'Some related problems have been known for a long time in mathematics of linear differential equations (see for instance, [CITATION]) .', 'quant-ph-0602207-3-13-0': 'Let us make the formal definition.', 'quant-ph-0602207-3-14-0': "Definition The function [MATH] is called a formal associated function of [MATH]-th order of the Hamiltonian [MATH] for a spectral value [MATH], if [EQUATION] where 'formal' emphasizes that a related function is not necessarily normalizable.", 'quant-ph-0602207-3-15-0': 'In particular, the associated function of zero order [MATH] is a formal eigenfunction of [MATH] (a solution of the homogeneous Schrodinger equation, not necessarily normalizable).', 'quant-ph-0602207-3-16-0': 'Let us single out normalizable associated functions and the case when [MATH] maps them into normalizable functions.', 'quant-ph-0602207-3-16-1': 'Evidently this may occur only for non-Hermitian Hamiltonians.', 'quant-ph-0602207-3-16-2': 'Then for any normalizable associated functions [MATH] and [MATH] the transposition symmetry holds [EQUATION]', 'quant-ph-0602207-3-16-3': 'Furthermore one can prove the following relations: [EQUATION] where [MATH] is scalar product.', 'quant-ph-0602207-3-17-0': 'As well, if there are normalizable associated functions [MATH] and [MATH] such that [EQUATION] then [EQUATION]', 'quant-ph-0602207-3-17-1': 'In particular, for some normalizable associated function [MATH], the "self-orthogonality" [CITATION] is realized , [EQUATION]', 'quant-ph-0602207-3-17-2': 'Thus, when assigning [CITATION] the probabilistic meaning for the binorm [MATH] one comes to conclusion that a sort of intriguing phase transition occurs in such a system, signalled by the puzzling divergence of some averages of observables, [EQUATION]', 'quant-ph-0602207-3-17-3': 'All the above relations are derived from the symmetry of a Hamiltonian under transposition and the very definition of associated functions and therefore the existence of self-orthogonal states seems to be inherent for any non-diagonalizable Hamiltonians with normalizable associated functions.', 'quant-ph-0602207-3-18-0': '# Towards resolution of puzzle with self-orthogonal states for Hamiltonians with finite-size Jordan cells', 'quant-ph-0602207-3-19-0': 'Let us show that the puzzle with self-orthogonal states may appear, in fact, due to misinterpretation of what are the pairs of orthogonal states in a true biorthogonal basis.', 'quant-ph-0602207-3-19-1': 'We proceed to the special class of Hamiltonians for which the spectrum is discrete and there is a complete biorthogonal system [MATH] such that, [EQUATION] where [MATH] is an index of an [MATH] eigenvalue [MATH], , [MATH] is an index of a Jordan cell (block) for the given eigenvalue, [MATH]; [MATH] is a number of Jordan cells for [MATH];', 'quant-ph-0602207-3-20-0': '[MATH], , [MATH] is an index of associated function in the Jordan cell with indexes [MATH]', 'quant-ph-0602207-3-21-0': 'and [MATH] is a dimension of this Jordan cell.', 'quant-ph-0602207-3-21-1': 'We have taken a general framework which is applicable also for matrix and/or multidimensional Hamiltonians.', 'quant-ph-0602207-3-21-2': 'But the main results of this and the next sections are guaranteed only for scalar Hamiltonians.', 'quant-ph-0602207-3-22-0': 'We remark that the number [MATH] is called a geometric multiplicity of the eigenvalue [MATH] .', 'quant-ph-0602207-3-22-1': 'For a scalar one-dimensional Schrodinger equation it cannot exceed 2.', 'quant-ph-0602207-3-22-2': 'In turn, the sum [MATH] is called an algebraic multiplicity of the eigenvalue [MATH].', 'quant-ph-0602207-3-23-0': 'The completeness implies the biorthogonality relations [EQUATION] and the unity decomposition [EQUATION]', 'quant-ph-0602207-3-23-1': 'The spectral decomposition for the Hamiltonian can be constructed as well, [EQUATION]', 'quant-ph-0602207-3-23-2': 'It represents the analog of the block-diagonal Jordan form for arbitrary non-Hermitian matrices [CITATION].', 'quant-ph-0602207-3-24-0': 'If existing such biorthogonal systems are not unique.', 'quant-ph-0602207-3-24-1': 'Indeed the relations ([REF]) remain invariant under the group of triangle transformations, [EQUATION] where the matrix elements must obey the following equations, [EQUATION]', 'quant-ph-0602207-3-24-2': 'The biorthogonality ([REF]) restricts the choice of pairs of matrices [MATH] and [MATH] in ([REF]) to be, [EQUATION]', 'quant-ph-0602207-3-24-3': 'This freedom in the redefinition of the biorthogonal basis is similar to Eq. ([REF]) and it can be exploited to define the pairs of biorthogonal functions [MATH] and [MATH] in accordance with ([REF]).', 'quant-ph-0602207-3-24-4': 'However one has to take into account our enumeration of associated functions [MATH] vs. their conjugated ones [MATH] as it is introduced in Eqs. ([REF]) [EQUATION]', 'quant-ph-0602207-3-24-5': 'Then the analog of Eq. ([REF]) reads, [EQUATION]', 'quant-ph-0602207-3-24-6': 'We stress that this kind of biorthogonal systems is determined uniquely up to an overall sign.', 'quant-ph-0602207-3-25-0': 'In these terms it becomes clear that the relations ([REF]) have the meaning of orthogonality of some off-diagonal pairs in the biorthogonal system [MATH] as [EQUATION]', 'quant-ph-0602207-3-25-1': 'When comparing with specification of indices in Eq. ([REF]) one identifies [MATH].', 'quant-ph-0602207-3-25-2': 'In both cases [MATH] .', 'quant-ph-0602207-3-25-3': 'Then the inequality ([REF]) singles out off-diagonal binorms, [MATH].', 'quant-ph-0602207-3-25-4': 'One can show easily that in order to have all diagonal binorms non-vanishing it is sufficient to prove that at least one of them is not zero which equivalent to the requirement [EQUATION] thereby providing the completeness of the basis.', 'quant-ph-0602207-3-26-0': 'Going back to the definition of quantum-state averages of certain observables we realize that the matrix element used in ([REF]) is not diagonal and therefore this relation cannot be interpreted as an average (compare with [CITATION]) of a putative order-parameter like operator.', 'quant-ph-0602207-3-27-0': '# Symmetric representation of non-diagonalizable Hamiltonians', 'quant-ph-0602207-3-28-0': 'We still notice that the biorthogonal basis ([REF]) does not provide a manifestly symmetric representation of the Hamiltonian (which is a symmetric operator).', 'quant-ph-0602207-3-28-1': 'One can obtain another biorthogonal basis using the canonical set of (normalizable) associated functions given by Eq. ([REF]) and their complex conjugates in an analogy to ([REF]) .', 'quant-ph-0602207-3-28-2': 'It can be achieved by means of renumbering of conjugated elements of the biorthogonal system ([REF]) , [EQUATION]', 'quant-ph-0602207-3-28-3': 'Eventually one arrives to the symmetric spectral decomposition for [MATH]: [EQUATION] which looks like a Jordan decomposition along the secondary diagonal.', 'quant-ph-0602207-3-28-4': 'Evidently in the coordinate representation the Hamiltonian operator is manifestly symmetric when the special biorthogonal basis ([REF]), [EQUATION] is chosen.', 'quant-ph-0602207-3-29-0': 'But the unity decomposition in this case is not diagonal, [EQUATION] although symmetric.', 'quant-ph-0602207-3-29-1': 'One can diagonalize this unity decomposition by a non-degenerate orthogonal transformation [MATH] of sub-bases in each non-diagonal sub-block, [EQUATION] retaining the type of the basis ([REF]) .', 'quant-ph-0602207-3-29-2': 'Then one finds a number eigenvalues [MATH].', 'quant-ph-0602207-3-29-3': 'In order to come to the canonical form of a basis ([REF]) one has to rotate by the complex unit [MATH] the pairs in the basis ([REF]) normalized on [MATH] .', 'quant-ph-0602207-3-29-4': 'Evidently the combination of the transformation [MATH] and such a rotation contains complex elements and is not orthogonal.', 'quant-ph-0602207-3-30-0': 'The remaining freedom of basis redefinition with the help of orthogonal rotations cannot provide the consequent diagonalization of the symmetric Hamiltonian matrix in each non-diagonal block.', 'quant-ph-0602207-3-30-1': 'The reason is that some of eigenvectors of the Hamiltonian sub-matrices have zero binorms, in particular, those ones which are related to the true Hamiltonian eigenfunctions.', 'quant-ph-0602207-3-30-2': 'Thus while being a symmetric operator with symmetric matrix representation, the Hamiltonian remains essentially non-diagonalizable.', 'quant-ph-0602207-3-31-0': 'We remark that in the general case the existence and the completeness of a biorthogonal system is not obvious (especially if the continuous spectrum is present) and needs a careful examination.', 'quant-ph-0602207-3-31-1': 'In particular, at the border between discrete and continuous spectra and in the continuous spectrum itself one can anticipate to have puzzling states with non-trivial role in the spectral decomposition.', 'quant-ph-0602207-3-31-2': 'This peculiarities will be discussed in the next Sections.', 'quant-ph-0602207-3-32-0': '# Example of a Non-diagonalizable Hamiltonian and its origin from level coalescence', 'quant-ph-0602207-3-33-0': 'In this Section we build a model with non-Hermitian Hamiltonian which possesses a Jordan cell spanned on the bound state and a normalizable associated state and further on demonstrate how this kind of degeneracy arises from coalescence of a pair of non-degenerate levels.', 'quant-ph-0602207-3-34-0': '## Jordan cell for bound state', 'quant-ph-0602207-3-35-0': 'The model Hamiltonian contains the potential with coordinates selectively shifted into complex plane, [EQUATION]', 'quant-ph-0602207-3-35-1': 'This Hamiltonian is not PT-symmetric unless [MATH] .', 'quant-ph-0602207-3-35-2': 'It has the Jordan cell, spanned by the normalizable eigenfunction [MATH] and associated function [MATH] on the level [MATH], [EQUATION]', 'quant-ph-0602207-3-35-3': 'In turn, the eigenfunctions of [MATH] for continuous spectrum read, [EQUATION]', 'quant-ph-0602207-3-35-4': 'For this model, unity decomposition built of eigenfunctions and associated functions of [MATH] can be obtained by conventional Green function methods, [EQUATION] this decomposition can be represented in the operator form, [EQUATION]', 'quant-ph-0602207-3-35-5': 'Evidently the basis [MATH] corresponds to the basis [MATH] of Sec. 4 and therefore gives the symmetric spectral decomposition for the Hamiltonian, [EQUATION]', 'quant-ph-0602207-3-35-6': 'The diagonalization of the resolution of identity ([REF]) may be arranged in two different ways.', 'quant-ph-0602207-3-35-7': 'First one can exploit the scheme of Sec. 3 performing re-numeration of certain elements of conjugated basis, [EQUATION] namely, [EQUATION]', 'quant-ph-0602207-3-35-8': 'With this notation the resolution of identity reads, [EQUATION]', 'quant-ph-0602207-3-35-9': 'We stress that [MATH] are related to the basis [MATH] in Sec. 3.', 'quant-ph-0602207-3-35-10': 'The relevant spectral decomposition of the Hamiltonian takes the quasi-diagonal form with one Jordan cell, [EQUATION]', 'quant-ph-0602207-3-35-11': 'On the other hand, the resolution of identity ([REF]) can be diagonalized by complex non-degenerate rotations , i.e. by using the construction of Sec. 4 .', 'quant-ph-0602207-3-35-12': 'The relevant basis is given by, [EQUATION] and [MATH] is an arbitrary constant with dimension of a length.', 'quant-ph-0602207-3-35-13': 'The decomposition of unity becomes diagonal , [EQUATION] or in the operator form, [EQUATION] where again the Dirac notations have been used, [EQUATION]', 'quant-ph-0602207-3-35-14': 'Accordingly, the manifestly symmetric spectral decomposition of [MATH] can be easily obtained, [EQUATION]', 'quant-ph-0602207-3-35-15': 'Notice that it cannot be diagonalized further, since the symmetric [MATH] matrix in ([REF]), [EQUATION] has one degenerate eigenvalue [MATH] and possesses only one eigenvector [MATH] with zero scalar product [MATH].', 'quant-ph-0602207-3-35-16': 'Its existence means that the orthogonal non-degenerate matrix required for diagonalization cannot be built conventionally from a set of eigenvectors.', 'quant-ph-0602207-3-35-17': 'Evidently, this vector maps the pair of basis functions [MATH] into the eigenstate of the Hamiltonian [MATH] whose partner in the biorthogonal basis is [MATH] .', 'quant-ph-0602207-3-35-18': 'Thus the existence of the zero norm vector [MATH] does not entail the self-orthogonality of basis elements [MATH] and [MATH].', 'quant-ph-0602207-3-36-0': '## Level coalescence for complex coordinates', 'quant-ph-0602207-3-37-0': 'The Hamiltonian [MATH] with a Jordan cell for bound state ([REF]) can be obtained as a limiting case, of the Hamiltonian [MATH] with two non-degenerate bound states (of algebraic multiplicity 1), corresponding [MATH], [EQUATION].', 'quant-ph-0602207-3-37-1': 'For this Hamiltonian [MATH] there are two normalized eigenfunctions for bound states, [EQUATION] with eigenvalues, [EQUATION]', 'quant-ph-0602207-3-37-2': 'Eigenfunctions of [MATH] for continuous spectrum take the form [EQUATION] i.e. the unity decomposition ([REF]) is reproduced.', 'quant-ph-0602207-3-38-0': '# Puzzles with zero-binorm bound states in the continuum', 'quant-ph-0602207-3-39-0': 'In what follows we develop another type of models in which the continuous spectrum is essentially involved in non-diagonal part of a Hamiltonian and elaborate the resolution of identity.', 'quant-ph-0602207-3-39-1': 'The standard treatment of continuous spectrum physics deals with reflection and transmission coefficients whose definition implies the existence of two linearly independent scattering solutions for a given spectral parameter.', 'quant-ph-0602207-3-39-2': 'Here we provide an example when this is not realized for a non-Hermitian Hamiltonian defined on the whole axis.', 'quant-ph-0602207-3-39-3': 'This point can be thought of as a spectral singularity for the continuous spectrum.', 'quant-ph-0602207-3-40-0': '## Non-Hermitian Hamiltonian with normalizable bound state at the continuum threshold', 'quant-ph-0602207-3-41-0': 'Let us now consider the Hamiltonian [EQUATION] raising up the puzzle of "self-orthogonality" [CITATION].', 'quant-ph-0602207-3-42-0': 'In order to unravel this puzzle we examine the unity decomposition made of eigenfunctions of [MATH], [EQUATION]', 'quant-ph-0602207-3-42-1': 'Hence it is the 3rd term in the right side of ([REF]) that provides the opportunity to reproduce [MATH] and thereby to complete the unity decomposition.', 'quant-ph-0602207-3-42-2': 'Thus one concludes that the state [MATH] is inseparable from the bottom of continuous spectrum and the resolution of identity in this sense is not diagonal.', 'quant-ph-0602207-3-43-0': 'We notice that the Hamiltonian ([REF]) is PT-symmetric and can be derived from the Hamiltonian ([REF]) in the limit [MATH] but the parameter [MATH] must be taken as a half of [MATH] from ([REF]).', 'quant-ph-0602207-3-44-0': 'We also remark that the Hamiltonian ([REF]) makes sense also for arbitrary coupling constants of "centrifugal" potential, and for the following set, [EQUATION] with positive [MATH], the Jordan cell, spanned by [MATH] normalizable eigenfunction and associated functions, appears at the threshold of continuous spectrum, [EQUATION]', 'quant-ph-0602207-3-44-1': 'All these zero-energy bound and associated states have zero binorms and are biorthogonal to each other (the multiple puzzle of "self-orthogonality").', 'quant-ph-0602207-3-44-2': 'Unity decomposition in such cases can be derived in a similar way although its form will be more cumbersome.', 'quant-ph-0602207-3-45-0': '## Hamiltonian with spectral singularity', 'quant-ph-0602207-3-46-0': 'Let us force the bound state energy [MATH] in the Hamiltonian ([REF]) to move towards the continuous spectrum, [MATH] .', 'quant-ph-0602207-3-46-1': 'Then for the Hamiltonian, [EQUATION] on the level [MATH] in the continuous spectrum, one finds the Jordan cell, spanned by the normalizable eigenfunction [MATH] and the associated function [MATH], whose asymptotics for [MATH] correspond to superposition of incoming and outgoing waves (standing wave), [EQUATION]', 'quant-ph-0602207-3-46-2': 'The asymptotics of this standing wave is, [EQUATION] therefore it does not belong to the scattering spectrum.', 'quant-ph-0602207-3-47-0': 'In turn the eigenfunctions of [MATH] for the scattering spectrum read, [EQUATION]', 'quant-ph-0602207-3-47-1': 'For this model, unity decomposition made of eigenfunctions and associated functions of [MATH] can be obtained by conventional Green function methods, [EQUATION] where [MATH] is an integration path in the complex momentum plane, obtained from real axis by its simultaneous displacement near the points [MATH] up or down.', 'quant-ph-0602207-3-48-0': 'For test functions from [MATH], [MATH] this decomposition can be presented in the form, [EQUATION] where the solutions [MATH] are made by analytical continuation of [MATH] in [MATH] into complex plane, and the branch of [MATH] is uniquely defined by the condition [MATH] in the plane with the cut on positive part of real axis.', 'quant-ph-0602207-3-48-1': 'In virtue of ([REF]) the Green function for the diagonalizable Hamiltonian of Subsec. 5.2.', 'quant-ph-0602207-3-48-2': 'has two poles of the first order (if [MATH]) at the points [MATH] where [MATH] can be either real or imaginary.', 'quant-ph-0602207-3-48-3': 'If [MATH] and level confluence emerges, two poles coalesce into one pole of the second order in both cases ([REF]) and ([REF]) when [MATH] .', 'quant-ph-0602207-3-48-4': 'However the examples of Subsec. 5.1 and 6.2 have different meaning: in the first case the double pole does not appear on the physical cut [MATH] and its order enumerates the rank of the Jordan cell.', 'quant-ph-0602207-3-48-5': 'On the contrary, in the second case the double pole is placed exactly on the cut [MATH] and strictly speaking signifies the spectral singularity as it generates only one eigenstate with the eigenvalue [MATH] in unity decomposition.', 'quant-ph-0602207-3-48-6': 'The second state - the associated function, represents a standing wave and does not influence the spectral decomposition.', 'quant-ph-0602207-3-48-7': 'In the example of Subsec. 6.1.', 'quant-ph-0602207-3-48-8': 'the Green function has only a branch point at [MATH] of the following type [MATH].', 'quant-ph-0602207-3-48-9': 'This branch point can be thought of as a confluence of the double pole in the variable [MATH] and the branch point of order [MATH]', 'quant-ph-0602207-3-49-0': 'From the explicit form of wave functions one can see that all potentials in Sec.5 and 6 are transparent as the reflection coefficient is zero.', 'quant-ph-0602207-3-49-1': 'The transmission coefficient in the non-degenerate case of Subsec. 5.2.', 'quant-ph-0602207-3-49-2': 'takes the form, [EQUATION]', 'quant-ph-0602207-3-49-3': 'In different limits one can derive the transmission coefficients for three other Hamiltonians.', 'quant-ph-0602207-3-49-4': 'Namely, when [MATH] and [MATH] (the case 5.1.)', 'quant-ph-0602207-3-49-5': 'the scattering is described by [EQUATION] and for [MATH] and [MATH] or imaginary (the cases 6.1.', 'quant-ph-0602207-3-49-6': 'and 6.2.)', 'quant-ph-0602207-3-49-7': 'the scattering is absent, [EQUATION]', 'quant-ph-0602207-3-49-8': 'Thus the colliding particle in such cases is not "influenced" by the bound states in the continuum.', 'quant-ph-0602207-3-50-0': '# Conclusions', 'quant-ph-0602207-3-51-0': 'In this paper we have presented a thorough analysis of the phenomenon of apparent self-orthogonality of some eigenstates for non-Hermitian Hamiltonians.', 'quant-ph-0602207-3-51-1': 'For the discrete part of energy spectrum, it has been shown that such a phenomenon should take place only for non-diagonalizable Hamiltonians the spectrum of which consists not only of eigenfunctions and but also of associated functions.', 'quant-ph-0602207-3-51-2': 'However the genuine diagonal biorthogonal basis related to the spectral decomposition of such Hamiltonians normally does not contain pairs made of the same eigenfunctions or the associated functions of the same order.', 'quant-ph-0602207-3-51-3': 'Rather they are complementary: for instance, eigenfunctions in the direct basis are paired to those associated functions in the conjugated basis, which have the maximal order in the same Jordan cell.', 'quant-ph-0602207-3-51-4': 'One possible exception exists for Jordan cells of odd order where one basis pair consists of the same function which is not self-orthogonal.', 'quant-ph-0602207-3-52-0': 'The situation in the continuous spectrum is more subtle: namely, the spectral decomposition does not include any obvious Jordan cells and associated functions.', 'quant-ph-0602207-3-52-1': 'However we have established that when a zero-binorm normalizable state arises it remains inseparable from the nearest scattering states of the continuum and eventually the existence of this state does not destroy the completeness of resolution of identity.', 'quant-ph-0602207-3-53-0': 'Finally, let us outline the measurability of quantum observables and, for this purpose, prepare a wave packet, [EQUATION] where the possibility to have continuous spectrum is made explicit in the notation and, for brevity, all indices enumerating eigenvalues, Jordan cells and their elements are encoded in the index [MATH].', 'quant-ph-0602207-3-54-0': 'In order to perform the quantum averaging of an operator of observable [MATH] one can use the conventional Hilbert space scalar product and the complex conjugated wave function, [MATH] .', 'quant-ph-0602207-3-54-1': 'In this way one defines the wave packet of the conjugated state and, respectively, average values of the operator [MATH] , [EQUATION]', 'quant-ph-0602207-3-54-2': 'For such a definition the averages of an operator of observable, [MATH] cannot be infinite and the phenomena of (pseudo) phase transitions at the level crossing [CITATION] cannot appear.', 'quant-ph-0602207-3-54-3': 'In this relation the basis from Sec. 3 with [MATH] may be used equally well.', 'quant-ph-0602207-3-55-0': 'However one has to keep in mind that such a definition of probabilities, to some extent, depends on a particular set of biorthogonal bases.', 'quant-ph-0602207-3-55-1': 'We hope to examine this approach and its applications elsewhere.', 'quant-ph-0602207-3-56-0': 'The work of A.A. and A.S. was supported by Grant RFBR 06-01-00186-a and by the Program "Development of scientific potential of Higher Education in Russia" RNP 2.1.1.1112.', 'quant-ph-0602207-3-56-1': 'A.Sokolov was partially supported by Grant INFN-IS/BO31.', 'quant-ph-0602207-3-56-2': 'We acknowledge Prof. S.L. Yakovlev for useful remarks and relevant references.', 'quant-ph-0602207-3-56-3': 'We are grateful to the organizers of 3rd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics (June 2005, Istanbul, Turkey), especially to Prof. A. Mostafazadeh and Prof. M. Znojil, for financial support and opportunity to present the preliminary results of this paper.', 'quant-ph-0602207-3-57-0': '# References'}

{'quant-ph-0602207-4-0-0': 'We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases.', 'quant-ph-0602207-4-0-1': 'The "self-orthogonality" phenomenon is clarified in terms of a correct spectral decomposition and it is shown that "self-orthogonal" states never jeopardize resolution of identity and thereby quantum averages of observables.', 'quant-ph-0602207-4-0-2': 'The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated.', 'quant-ph-0602207-4-0-3': 'Some puzzles with zero-binorm bound states in continuous spectrum are unraveled with the help of a correct resolution of identity.', 'quant-ph-0602207-4-1-0': 'Journal of Physics A: Math.', 'quant-ph-0602207-4-1-1': 'Gen.', 'quant-ph-0602207-4-2-0': '# Introduction', 'quant-ph-0602207-4-3-0': 'The variety of complex potentials in Quantum Physics is associated typically with open systems when a control of information is partially lost and thereby the unitarity of observable evolution is broken.', 'quant-ph-0602207-4-3-1': 'For this class of quantum systems the energy eigenvalues may have an imaginary part which signals the opening of new channels not directly measured in a given experiment.', 'quant-ph-0602207-4-3-2': 'In this context non-Hermitian interactions have been used in Field Theory and Statistical Mechanics for many years with applications to Condensed Matter, Quantum Optics and Hadronic and Nuclear Physics [CITATION] - [CITATION].', 'quant-ph-0602207-4-3-3': 'The subject of non-self-adjoint operators has been also under intensive mathematical investigations [CITATION], in particular, interesting examples of non-Hermitian effective Hamiltonian operators have been found for the Faddeev equations [CITATION].', 'quant-ph-0602207-4-4-0': 'An important class of complex Hamiltonians deals with a real spectrum [CITATION], in particular, in the PT-symmetric Quantum Mechanics [CITATION]-[CITATION] and its pseudo-Hermitian generalization [CITATION].', 'quant-ph-0602207-4-4-1': 'Scattering problems for such Hamiltonians have been investigated in [CITATION] .', 'quant-ph-0602207-4-5-0': 'For complex, non-Hermitian potentials the natural spectral decomposition exploits the sets of biorthogonal states [CITATION] , and within this framework one can discover new features that never happen for closed systems with Hermitian Hamiltonians possessing real spectrum : namely, certain Hamiltonians may not be diagonalizable [CITATION] with a help of biorthogonal bases and can be reduced only to a quasi-diagonal form with a number of Jordan cells [CITATION].', 'quant-ph-0602207-4-5-1': 'Such a feature can be realized by level crossing which, in fact, occurs (after some kind of complexification) in atomic and molecular spectra [CITATION] and Optics [CITATION] (see more examples in [CITATION]) as well as in PT-symmetric quantum systems [CITATION].', 'quant-ph-0602207-4-5-2': 'In this case some eigenstates seem to be "self-orthogonal" in respect to a binorm [CITATION] .', 'quant-ph-0602207-4-5-3': 'The latter quite intriguing phenomenon has been interpreted as a sort of phase transition [CITATION] .', 'quant-ph-0602207-4-6-0': 'The main purpose of the present work is to clarify the "self-orthogonality" in terms of a correct spectral decomposition both for discrete and for continuous spectra and to show that, at least, in one-dimensional Quantum Mechanics such states never jeopardize resolution of identity for the discrete or bound state spectrum and thereby don\'t affect quantum averages of observables.', 'quant-ph-0602207-4-7-0': 'We start introducing the notion of biorthogonal basis and, correspondingly, the resolution of identity for a non-Hermitian diagonalizable Hamiltonian.', 'quant-ph-0602207-4-7-1': 'In Sec. 2 the appearance of associated functions is discussed and in Sec. 3 non-diagonalizable (but quasi-diagonalizable) Hamiltonians with finite-size Jordan cells are analyzed.', 'quant-ph-0602207-4-7-2': 'Special attention is paid to the definition of a biorthogonal diagonal basis and the meaning of zero-binorm states is clarified.', 'quant-ph-0602207-4-7-3': 'Namely, it is shown that the apparent self-orthogonality of eigenfunctions and associated functions is misleading as they never replicate themselves as relative pairs in diagonal resolution of identity.', 'quant-ph-0602207-4-7-4': 'Instead, the "self-orthogonality" involves the different elements in the related basis thereby being addressed to a conventional orthogonality.', 'quant-ph-0602207-4-7-5': 'The construction of such biorthogonal bases with pairs of mutually complex-conjugated base functions is described .', 'quant-ph-0602207-4-7-6': 'In Sec. 4 another representation of non-diagonalizable Hamiltonians , manifestly symmetric under transposition is given, compatible with a diagonal resolution of identity.', 'quant-ph-0602207-4-7-7': 'In Sec. 5 the example of a (transparent) complex potential leading to the non-diagonalizable Hamiltonian with one Jordan [MATH] cell is constructed and its origin from level coalescence is illustrated.', 'quant-ph-0602207-4-8-0': 'On the other hand some puzzles with zero-binorm bound states arise in continuous spectrum and they are unraveled in Sec.6 with the help of a correct resolution of identity.', 'quant-ph-0602207-4-8-1': 'Its proof is relegated to the Appendix.', 'quant-ph-0602207-4-8-2': 'In Sec. 7 we complete our analysis with discussion of singularities in the spectral parameter for resolvents and of scattering characteristics for previous examples.', 'quant-ph-0602207-4-8-3': 'We conclude with some proposals for probabilistic interpretation of wave functions defined in respect to a biorthogonal basis which does not allow any negative or zero-norm states.', 'quant-ph-0602207-4-9-0': 'There are certain links of our approach to the works [CITATION] on Jordan cells associated with the occurrence of non-Hermitian degeneracies for essentially Hermitian Hamiltonians where the description has been developed for complex eigenvalue Gamow states (resonances) unbounded in their asymptotics and, in general, not belonging to the Hilbert space.', 'quant-ph-0602207-4-9-1': 'On the contrary, we instead examine Nonhermitian Hamiltonians with normalizable bound and associated states.', 'quant-ph-0602207-4-10-0': 'In our paper we deal with complex one-dimensional potentials [MATH] and respectively with non-Hermitian Hamiltonians [MATH] of Schrodinger type, defined on the real axis, [EQUATION] which are assumed to be [MATH]-symmetric or self-transposed under the [MATH] - transposition operation, [MATH].', 'quant-ph-0602207-4-10-1': 'Only scalar local potentials will be analyzed which are obviously symmetric under transposition (for some matrix non-diagonalizable problems, see [CITATION]).', 'quant-ph-0602207-4-10-2': 'Throughout this work the units will be used with [MATH] which leads to dimensionless energies .', 'quant-ph-0602207-4-11-0': 'Let us first define a class of one-dimensional non-Hermitian diagonalizable Hamiltonian [MATH] with discrete spectrum such that:', 'quant-ph-0602207-4-12-0': 'a) a biorthogonal system [MATH] exists, [EQUATION] b) the complete resolution of identity in terms of these bases and the spectral decomposition of the Hamiltonian hold, [EQUATION]', 'quant-ph-0602207-4-12-1': 'In the coordinate representation, [EQUATION] the resolution of identity has the form, [EQUATION]', 'quant-ph-0602207-4-12-2': 'The differential equations, [EQUATION] and the fact that there is only one normalizable eigenfunction of [MATH] for the eigenvalue [MATH] (up to a constant factor), allow one to conclude that [EQUATION]', 'quant-ph-0602207-4-12-3': 'Hence the system [MATH] can be redefined [EQUATION] so that [EQUATION]', 'quant-ph-0602207-4-12-4': 'Indeed if some of the states in Eq. ([REF]) were "self-orthogonal" (as it has been accepted in [CITATION]) , i.e. had zero binorms in ([REF]), the would-be unity in ([REF]) would annihilate such states thereby signalling the incompleteness.', 'quant-ph-0602207-4-13-0': '# Non-diagonalizable Hamiltonians and zero-binorm states', 'quant-ph-0602207-4-14-0': 'For complex Hamiltonians one can formulate the extended eigenvalue problem, searching not only for normalizable eigenfunctions but also for normalizable associated functions for discrete part of the energy spectrum.', 'quant-ph-0602207-4-14-1': 'Some related problems have been known for a long time in mathematics of linear differential equations (see for instance, [CITATION]) .', 'quant-ph-0602207-4-15-0': 'Let us give the formal definition.', 'quant-ph-0602207-4-16-0': 'Definition.', 'quant-ph-0602207-4-16-1': "The function [MATH] is called a formal associated function of [MATH]-th order of the Hamiltonian [MATH] for a spectral value [MATH], if [EQUATION] where 'formal' emphasizes that a related function is not necessarily normalizable.", 'quant-ph-0602207-4-17-0': 'In particular, the associated function of zero order [MATH] is a formal eigenfunction of [MATH] (a solution of the homogeneous Schrodinger equation, not necessarily normalizable).', 'quant-ph-0602207-4-18-0': 'Let us single out normalizable associated functions and the case when [MATH] maps them into normalizable functions.', 'quant-ph-0602207-4-18-1': 'Evidently this may occur only for non-Hermitian Hamiltonians.', 'quant-ph-0602207-4-18-2': 'Then for any normalizable associated functions [MATH] and [MATH] the transposition symmetry holds [EQUATION]', 'quant-ph-0602207-4-18-3': 'Furthermore one can prove the following relations: [EQUATION] where [MATH] is scalar product.', 'quant-ph-0602207-4-19-0': "As well, let's take two normalizable associated functions [MATH] and [MATH] so that, in general, [MATH] and there are two different sequences of associated functions for [MATH] and [MATH] [EQUATION]", 'quant-ph-0602207-4-19-1': 'Then [EQUATION]', 'quant-ph-0602207-4-19-2': 'In particular, for some normalizable associated function [MATH], the "self-orthogonality" [CITATION] is realized , [EQUATION]', 'quant-ph-0602207-4-19-3': 'Thus, when assigning [CITATION] the probabilistic meaning for the binorm [MATH] one comes to conclusion that a sort of intriguing phase transition occurs in such a system, signalled by the puzzling divergence of some averages of observables, [EQUATION]', 'quant-ph-0602207-4-19-4': 'All the above relations are derived from the symmetry of a Hamiltonian under transposition and the very definition of associated functions and therefore the existence of self-orthogonal states seems to be inherent for any non-diagonalizable Hamiltonians with normalizable associated functions.', 'quant-ph-0602207-4-20-0': '# Towards resolution of puzzle with self-orthogonal states for Hamiltonians with finite-size Jordan cells', 'quant-ph-0602207-4-21-0': 'Let us show that the puzzle with self-orthogonal states may appear, in fact, due to misinterpretation of what are the pairs of orthogonal states in a true biorthogonal basis.', 'quant-ph-0602207-4-21-1': 'We proceed to the special class of Hamiltonians for which the spectrum is discrete and there is a complete biorthogonal system [MATH] such that, [EQUATION] where [MATH] is an index of an [MATH] eigenvalue [MATH], , [MATH] is an index of a Jordan cell (block) for the given eigenvalue, [MATH]; [MATH] is a number of Jordan cells for [MATH];', 'quant-ph-0602207-4-22-0': '[MATH], , [MATH] is an index of associated function in the Jordan cell with indexes [MATH]', 'quant-ph-0602207-4-23-0': 'and [MATH] is a dimension of this Jordan cell.', 'quant-ph-0602207-4-23-1': 'We have taken a general framework which is applicable also for matrix and/or multidimensional Hamiltonians.', 'quant-ph-0602207-4-23-2': 'But the main results of this and the next sections are guaranteed only for scalar one-dimensional Hamiltonians with local potentials.', 'quant-ph-0602207-4-24-0': 'We remark that the number [MATH] is called a geometric multiplicity of the eigenvalue [MATH] .', 'quant-ph-0602207-4-24-1': 'For a scalar one-dimensional Schrodinger equation it cannot normally exceed 1 (but may reach 2 in specific cases of periodic potentials and of potentials unbounded from below).', 'quant-ph-0602207-4-24-2': 'In turn, the sum [MATH] is called an algebraic multiplicity of the eigenvalue [MATH].', 'quant-ph-0602207-4-25-0': 'The completeness implies the biorthogonality relations (in line with the enumeration of states [MATH] given in Eq. ([REF])) [EQUATION] and the resolution of identity [EQUATION]', 'quant-ph-0602207-4-25-1': 'The spectral decomposition for the Hamiltonian can be constructed as well, [EQUATION]', 'quant-ph-0602207-4-25-2': 'It represents the analog of the block-diagonal Jordan form for arbitrary non-Hermitian matrices [CITATION].', 'quant-ph-0602207-4-26-0': 'If existing such biorthogonal systems are not unique.', 'quant-ph-0602207-4-26-1': 'Indeed the relations ([REF]) remain invariant under the group of triangle transformations, [EQUATION] where the matrix elements must obey the following equations, [EQUATION]', 'quant-ph-0602207-4-26-2': 'The biorthogonality ([REF]) restricts the choice of pairs of matrices [MATH] and [MATH] in ([REF]) to be, [EQUATION]', 'quant-ph-0602207-4-26-3': 'This freedom in the redefinition of the biorthogonal basis is similar to Eq. ([REF]) and it can be exploited to define the pairs of biorthogonal functions [MATH] and [MATH] in accordance with ([REF]).', 'quant-ph-0602207-4-26-4': 'However one has to take into account our enumeration of associated functions [MATH] vs. their conjugated ones [MATH] as it is introduced in Eqs. ([REF]) [EQUATION]', 'quant-ph-0602207-4-26-5': 'Then the analog of Eq. ([REF]) reads, [EQUATION]', 'quant-ph-0602207-4-26-6': 'We stress that this kind of biorthogonal systems is determined uniquely up to an overall sign.', 'quant-ph-0602207-4-27-0': 'In these terms it becomes clear that the relations ([REF]) have the meaning of orthogonality of some off-diagonal pairs in the biorthogonal system [MATH] as [EQUATION]', 'quant-ph-0602207-4-27-1': 'When comparing with specification of indices in Eq. ([REF]) one identifies [MATH].', 'quant-ph-0602207-4-27-2': 'In both cases [MATH] .', 'quant-ph-0602207-4-27-3': 'Then the inequality ([REF]) singles out off-diagonal binorms, [MATH].', 'quant-ph-0602207-4-27-4': 'From Eq. ([REF]) it follows that in order to have all diagonal binorms non-vanishing it is sufficient to prove that at least one of them is not zero because [EQUATION]', 'quant-ph-0602207-4-27-5': 'The latter is necessary for the completeness of the basis ( because of the absence of self-orthogonal pairs of basis elements made of bound and associated functions when resolution of identity is diagonal).', 'quant-ph-0602207-4-28-0': 'Going back to the definition of quantum-state averages of certain observables we realize that the matrix element used in ([REF]) is not diagonal and therefore this relation cannot be interpreted as an average (compare with [CITATION]) of a putative order-parameter like operator.', 'quant-ph-0602207-4-29-0': '# [MATH]-symmetric representation of non-diagonalizable Hamiltonians', 'quant-ph-0602207-4-30-0': 'We still notice that the biorthogonal basis ([REF]) does not provide a manifestly [MATH]-symmetric representation of the Hamiltonian (which is symmetric under transposition [MATH] in the coordinate representation as a finite-order differential operator).', 'quant-ph-0602207-4-30-1': 'One can obtain another biorthogonal basis using the canonical set of (normalizable) associated functions given by Eq. ([REF]) and their complex conjugates in an analogy to ([REF]) .', 'quant-ph-0602207-4-30-2': 'It can be achieved by means of renumbering of conjugated elements of the biorthogonal system ([REF]) , [EQUATION]', 'quant-ph-0602207-4-30-3': 'Eventually one arrives to the [MATH]-symmetric spectral decomposition for [MATH]: [EQUATION] which looks like a Jordan decomposition along the secondary diagonal.', 'quant-ph-0602207-4-30-4': 'Evidently in the coordinate representation the Hamiltonian operator is manifestly [MATH]-symmetric when the special biorthogonal basis ([REF]), [EQUATION] is chosen.', 'quant-ph-0602207-4-31-0': 'But the resolution of identity in this case is not diagonal, [EQUATION] although [MATH]-symmetric.', 'quant-ph-0602207-4-31-1': 'One can diagonalize this resolution of identity by a non-degenerate orthogonal transformation [MATH] of sub-bases in each non-diagonal sub-block, [EQUATION] retaining the type of the basis ([REF]) .', 'quant-ph-0602207-4-31-2': 'Then one finds a number eigenvalues [MATH].', 'quant-ph-0602207-4-31-3': 'In order to come to the canonical form of a basis ([REF]) one has to rotate by the complex unit [MATH] the pairs in the basis ([REF]) normalized on [MATH] .', 'quant-ph-0602207-4-31-4': 'Evidently the combination of the transformation [MATH] and such a rotation contains complex elements and is not orthogonal.', 'quant-ph-0602207-4-32-0': 'The remaining freedom of basis redefinition with the help of orthogonal rotations cannot provide the consequent diagonalization of the symmetric Hamiltonian matrix in each non-diagonal block.', 'quant-ph-0602207-4-32-1': 'The reason is that some of eigenvectors of the Hamiltonian sub-matrices have zero binorms, in particular, those ones which are related to the true Hamiltonian eigenfunctions.', 'quant-ph-0602207-4-32-2': 'Thus while being a [MATH]-symmetric operator with symmetric matrix representation, the Hamiltonian remains essentially non-diagonalizable .', 'quant-ph-0602207-4-33-0': 'We remark that in the general case the existence and the completeness of a biorthogonal system is not obvious (especially if the continuous spectrum is present) and needs a careful examination.', 'quant-ph-0602207-4-33-1': 'In particular, at the border between discrete and continuous spectra and in the continuous spectrum itself one can anticipate to have puzzling states with non-trivial role in the spectral decomposition.', 'quant-ph-0602207-4-33-2': 'This peculiarities will be discussed in the next Sections.', 'quant-ph-0602207-4-34-0': '# A model with Non-diagonalizable Hamiltonian and its origin from level coalescence', 'quant-ph-0602207-4-35-0': 'In this Section we build a model with non-Hermitian Hamiltonian which has a real continuous spectrum and, in addition, possesses a Jordan cell spanned on the bound state and a normalizable associated state.', 'quant-ph-0602207-4-35-1': 'This model does not belong to the class of Hamiltonians with purely discrete spectrum considered in the preceding sections but being block-diagonal it inherits some of their properties in the bound state sector.', 'quant-ph-0602207-4-35-2': 'Further on we demonstrate how this kind of degeneracy arises from coalescence of a pair of non-degenerate levels.', 'quant-ph-0602207-4-36-0': '## Jordan cell for bound state', 'quant-ph-0602207-4-37-0': 'The model Hamiltonian contains the potential with coordinates selectively shifted into complex plane, [EQUATION]', 'quant-ph-0602207-4-37-1': 'This Hamiltonian is not PT-symmetric unless [MATH] .', 'quant-ph-0602207-4-37-2': 'It has the Jordan cell, spanned by the normalizable eigenfunction [MATH] and associated function [MATH] on the level [MATH], [EQUATION]', 'quant-ph-0602207-4-37-3': 'In turn, the eigenfunctions of [MATH] for continuous spectrum read, [EQUATION]', 'quant-ph-0602207-4-37-4': 'For this model, the resolution of identity built of eigenfunctions and associated functions of [MATH] can be obtained by conventional Green function methods, [EQUATION] this resolution of identity can be represented in the operator form, [EQUATION]', 'quant-ph-0602207-4-37-5': 'Evidently the basis [MATH] corresponds to the basis [MATH] of Sec. 4 and therefore gives the [MATH]-symmetric spectral decomposition for the Hamiltonian, [EQUATION]', 'quant-ph-0602207-4-37-6': 'The diagonalization of the resolution of identity ([REF]) may be arranged in two different ways.', 'quant-ph-0602207-4-37-7': 'First one can exploit the scheme of Sec. 3 performing re-numeration of certain elements of conjugated basis, [EQUATION] namely, [EQUATION]', 'quant-ph-0602207-4-37-8': 'With this notation the resolution of identity reads, [EQUATION]', 'quant-ph-0602207-4-37-9': 'We stress that [MATH] are related to the basis [MATH] in Sec. 3.', 'quant-ph-0602207-4-37-10': 'The relevant spectral decomposition of the Hamiltonian takes the quasi-diagonal form with one Jordan cell, [EQUATION]', 'quant-ph-0602207-4-37-11': 'On the other hand, the resolution of identity ([REF]) can be diagonalized by complex non-degenerate rotations , i.e. by using the construction of Sec. 4 .', 'quant-ph-0602207-4-37-12': 'The relevant basis is given by, [EQUATION] where [MATH] is an arbitrary constant .', 'quant-ph-0602207-4-37-13': 'The resolution of identity becomes diagonal , [EQUATION] or in the operator form, [EQUATION] where again the Dirac notations have been used, [EQUATION]', 'quant-ph-0602207-4-37-14': 'Accordingly, the manifestly [MATH]-symmetric spectral decomposition of [MATH] can be easily obtained, [EQUATION]', 'quant-ph-0602207-4-37-15': 'Notice that it cannot be diagonalized further, since the symmetric [MATH] matrix in ([REF]), [EQUATION] has one degenerate eigenvalue [MATH] and possesses only one eigenvector [MATH] with zero norm [MATH].', 'quant-ph-0602207-4-37-16': 'Its existence means that the orthogonal non-degenerate matrix required for diagonalization cannot be built conventionally from a set of eigenvectors.', 'quant-ph-0602207-4-37-17': 'Evidently, this vector [MATH] maps the pair of basis functions [MATH] into the self-biorthogonal eigenstate of the Hamiltonian [MATH].', 'quant-ph-0602207-4-37-18': 'However its partner in the biorthogonal basis is [MATH] with [MATH].', 'quant-ph-0602207-4-37-19': 'Thus the existence of the zero norm vector [MATH] does not entail the breakdown of resolution of identity.', 'quant-ph-0602207-4-38-0': '## Level coalescence for complex coordinates', 'quant-ph-0602207-4-39-0': 'The Hamiltonian [MATH] with a Jordan cell for bound state ([REF]) can be obtained as a limiting case, of the Hamiltonian [MATH] with two non-degenerate bound states (of algebraic multiplicity 1), corresponding [MATH], [EQUATION].', 'quant-ph-0602207-4-39-1': 'For this Hamiltonian [MATH] there are two normalized eigenfunctions for bound states, [EQUATION] with eigenvalues, [EQUATION]', 'quant-ph-0602207-4-39-2': 'Eigenfunctions of [MATH] for continuous spectrum take the form [EQUATION] i.e. the resolution of identity ([REF]) is reproduced.', 'quant-ph-0602207-4-40-0': '# Puzzles with zero-binorm bound states in the continuum', 'quant-ph-0602207-4-41-0': 'In what follows we develop another type of models in which the continuous spectrum is essentially involved in non-diagonal part of a Hamiltonian and elaborate the resolution of identity.', 'quant-ph-0602207-4-41-1': 'First we built the model with self-orthogonal bound state which however is essentially entangled with the lower end of continuous spectrum.', 'quant-ph-0602207-4-41-2': 'As a consequence, the self-orthogonality does not lead to infinite average values of observables like kinetic or potential energies if these averages are treated with the help of wave packet regularization.', 'quant-ph-0602207-4-42-0': 'Conventionally the continuous spectrum physics deals with reflection and transmission coefficients whose definition implies the existence of two linearly independent scattering solutions for a given spectral parameter.', 'quant-ph-0602207-4-42-1': 'The second model provides an example when this is not realized for a non-Hermitian Hamiltonian defined on the whole axis.', 'quant-ph-0602207-4-43-0': '## Non-Hermitian Hamiltonian with normalizable bound state at the continuum threshold', 'quant-ph-0602207-4-44-0': 'Let us now consider the Hamiltonian [EQUATION] raising up the puzzle of "self-orthogonality" [CITATION].', 'quant-ph-0602207-4-45-0': 'In order to unravel this puzzle we examine the resolution of identity made of eigenfunctions of [MATH], [EQUATION]', 'quant-ph-0602207-4-45-1': 'Hence it is the 3rd term in the right side of ([REF]) that provides the opportunity to reproduce [MATH] and thereby to complete the resolution of identity.', 'quant-ph-0602207-4-45-2': 'Thus one concludes that the state [MATH] is inseparable from the bottom of continuous spectrum and the resolution of identity in this sense is not diagonal.', 'quant-ph-0602207-4-46-0': 'We notice that the Hamiltonian ([REF]) is PT-symmetric and can be derived from the Hamiltonian ([REF]) in the limit [MATH] but the parameter [MATH] must be taken as a half of [MATH] from ([REF]).', 'quant-ph-0602207-4-47-0': 'We also remark that the Hamiltonian ([REF]) makes sense also for arbitrary coupling constants of "centrifugal" potential, and for the following set, [EQUATION] with positive [MATH], the Jordan cell, spanned by [MATH] normalizable eigenfunction and associated functions, appears at the threshold of continuous spectrum, [EQUATION]', 'quant-ph-0602207-4-47-1': 'All these zero-energy bound and associated states have zero binorms and are biorthogonal to each other (the multiple puzzle of "self-orthogonality").', 'quant-ph-0602207-4-47-2': 'Resolution of identity in such cases can be derived in a similar way although its form will be more cumbersome.', 'quant-ph-0602207-4-48-0': '## Expectation values (e.v.) of kinetic and potential energies in the vicinity of zero-energy bound state', 'quant-ph-0602207-4-49-0': 'As the binorm of the bound state ([REF]) vanishes it seems that the quantum averages of basic observables like the kinetic [MATH] or potential [MATH] energy in this system described by the Hamiltonian ([REF]) [MATH] tend to diverge.', 'quant-ph-0602207-4-49-1': 'But it is, in fact, not the case.', 'quant-ph-0602207-4-49-2': "Indeed, the e.v.'s of these observables vanish as well, [EQUATION]", 'quant-ph-0602207-4-49-3': 'Thus one comes to the classical uncertainty of [MATH] type.', 'quant-ph-0602207-4-49-4': 'In order to unravel it one has to built a wave packet which reproduces the function [MATH] in the limit of its form parameters.', 'quant-ph-0602207-4-49-5': 'We choose the Gaussian wave packet, [EQUATION] which evidently approaches uniformly [MATH] when [MATH] .', 'quant-ph-0602207-4-49-6': 'The binorm of this wave packet , [EQUATION] rapidly vanishes when [MATH] .', 'quant-ph-0602207-4-50-0': 'In turn the e.v. of the total energy, [EQUATION] decreases with [MATH] faster than the normalization ([REF]) .', 'quant-ph-0602207-4-50-1': 'The e.v. of the potential energy, [EQUATION] behaves as the total energy and therefore the e.v. for kinetic energy decreases also as [MATH], much faster than the binorm ([REF]) .', 'quant-ph-0602207-4-50-2': 'Thus one concludes that their ratios, i.e. the quantum averages of kinetic and potential energies vanish for the self-orthogonal bound state in contrast to the superficial divergence .', 'quant-ph-0602207-4-50-3': 'Therefore the puzzle with self-orthogonality is resolved.', 'quant-ph-0602207-4-51-0': '## Hamiltonian with bound state in continuum', 'quant-ph-0602207-4-52-0': 'Let us force the bound state energy [MATH] in the Hamiltonian ([REF]) to move towards the continuous spectrum, [MATH] .', 'quant-ph-0602207-4-52-1': 'Then for the Hamiltonian, [EQUATION] on the level [MATH] in the continuous spectrum, one finds the Jordan cell, spanned by the normalizable eigenfunction [MATH] and the associated function [MATH], whose asymptotics for [MATH] correspond to superposition of incoming and outgoing waves (standing wave), [EQUATION]', 'quant-ph-0602207-4-52-2': 'The asymptotics of the associated state is given by the standing wave, [EQUATION] but it does not appear in the resolution of identity (see below).', 'quant-ph-0602207-4-52-3': 'Thereby this associated state does not belong to the physical state space.', 'quant-ph-0602207-4-53-0': 'In turn the eigenfunctions of [MATH] for the scattering spectrum read, [EQUATION]', 'quant-ph-0602207-4-53-1': 'For this model, resolution of identity made of eigenfunctions and associated functions of [MATH] can be obtained by conventional Green function methods, [EQUATION] where [MATH] is an integration path in the complex momentum plane, obtained from real axis by its simultaneous displacement near the points [MATH] up or down.', 'quant-ph-0602207-4-54-0': 'For test functions from [MATH], [MATH] this resolution of identity can be presented in the form, [EQUATION] where the solutions [MATH] are made by analytical continuation of [MATH] in [MATH] into complex plane, and the branch of [MATH] is uniquely defined by the condition [MATH] in the plane with the cut on positive part of real axis.', 'quant-ph-0602207-4-54-1': 'In virtue of ([REF]) the Green function for the diagonalizable Hamiltonian of Subsec. 5.2.', 'quant-ph-0602207-4-54-2': 'has two poles of the first order (if [MATH]) at the points [MATH] where [MATH] can be either real or imaginary.', 'quant-ph-0602207-4-54-3': 'If [MATH] and level confluence emerges, two poles coalesce into one pole of the second order in both cases ([REF]) and ([REF]) when [MATH] .', 'quant-ph-0602207-4-54-4': 'However the examples of Subsec. 5.1 and 6.2 have different meaning: in the first case the double pole does not appear on the physical cut [MATH] and its order enumerates the rank of the Jordan cell.', 'quant-ph-0602207-4-54-5': 'On the contrary, in the second case the double pole is placed exactly on the cut [MATH] and strictly speaking signifies the spectral pathology as it generates only one eigenstate with the eigenvalue [MATH] in resolution of identity.', 'quant-ph-0602207-4-54-6': 'The second state - the associated function, represents a standing wave and does not influence the spectral decomposition.', 'quant-ph-0602207-4-54-7': 'In the example of Subsec. 6.1.', 'quant-ph-0602207-4-54-8': 'the Green function has only a branch point at [MATH] of the following type [MATH].', 'quant-ph-0602207-4-54-9': 'This branch point can be thought of as a confluence of the double pole in the variable [MATH] and the branch point of order [MATH] .', 'quant-ph-0602207-4-55-0': 'From the explicit form of wave functions one can see that all potentials in Sec.5 and 6 are transparent as the reflection coefficient is zero.', 'quant-ph-0602207-4-55-1': 'The transmission coefficient in the non-degenerate case of Subsec. 5.2.', 'quant-ph-0602207-4-55-2': 'takes the form, [EQUATION]', 'quant-ph-0602207-4-55-3': 'In different limits one can derive the transmission coefficients for three other Hamiltonians.', 'quant-ph-0602207-4-55-4': 'Namely, when [MATH] and [MATH] (the case 5.1.)', 'quant-ph-0602207-4-55-5': 'the scattering is described by [EQUATION] and for [MATH] and [MATH] or imaginary (the cases 6.1.', 'quant-ph-0602207-4-55-6': 'and 6.2.)', 'quant-ph-0602207-4-55-7': 'the scattering is absent, [EQUATION]', 'quant-ph-0602207-4-55-8': 'Thus the colliding particle in such cases is not "influenced" by the bound states in the continuum.', 'quant-ph-0602207-4-56-0': '# Conclusions', 'quant-ph-0602207-4-57-0': 'In this paper we have presented a thorough analysis of the phenomenon of apparent self-orthogonality of some eigenstates for non-Hermitian Hamiltonians.', 'quant-ph-0602207-4-57-1': 'For the discrete part of energy spectrum, it has been shown that such a phenomenon should take place only for non-diagonalizable Hamiltonians the spectrum of which consists not only of eigenfunctions and but also of associated functions.', 'quant-ph-0602207-4-57-2': 'However the genuine diagonal biorthogonal basis related to the spectral decomposition of such Hamiltonians normally does not contain pairs made of the same eigenfunctions or the associated functions of the same order.', 'quant-ph-0602207-4-57-3': 'Rather they are complementary: for instance, eigenfunctions in the direct basis are paired to those associated functions in the conjugated basis, which have the maximal order in the same Jordan cell.', 'quant-ph-0602207-4-57-4': 'One possible exception exists for Jordan cells of odd order where one basis pair consists of the same function which is not self-orthogonal.', 'quant-ph-0602207-4-58-0': 'The situation in the continuous spectrum is more subtle: namely, the spectral decomposition does not include any obvious Jordan cells and associated functions.', 'quant-ph-0602207-4-58-1': 'However we have established that when a zero-binorm normalizable state arises it remains inseparable from the nearest scattering states of the continuum and eventually the existence of this state does not destroy the completeness of resolution of identity.', 'quant-ph-0602207-4-59-0': 'Finally, let us outline the measurability of quantum observables and, for this purpose, prepare a wave packet, [EQUATION] where the possibility to have continuous spectrum is made explicit in the notation and, for brevity, all indices enumerating eigenvalues, Jordan cells and their elements are encoded in the index [MATH].', 'quant-ph-0602207-4-60-0': 'In order to perform the quantum averaging of an operator of observable [MATH] one can use the conventional Hilbert space scalar product and the complex conjugated wave function, [MATH] .', 'quant-ph-0602207-4-60-1': 'In this way one defines the wave packet of the conjugated state and, respectively, average values of the operator [MATH] , [EQUATION]', 'quant-ph-0602207-4-60-2': 'For such a definition the averages of an operator of observable, [MATH] cannot be infinite and the phenomena of (pseudo) phase transitions at the level crossing [CITATION] cannot appear.', 'quant-ph-0602207-4-60-3': 'In this relation the basis from Sec. 4 with [MATH] may be used equally well.', 'quant-ph-0602207-4-61-0': 'However one has to keep in mind that such a definition of probabilities, to some extent, depends on a particular set of biorthogonal bases.', 'quant-ph-0602207-4-61-1': 'We hope to examine this approach and its applications elsewhere.', 'quant-ph-0602207-4-62-0': 'The work of A.A. and A.S. was supported by Grant RFBR 06-01-00186-a and by the Programs RNP 2.1.1.1112 and RSS-5538.2006.2.', 'quant-ph-0602207-4-62-1': 'A.Sokolov was partially supported by Grant INFN-IS/BO31.', 'quant-ph-0602207-4-62-2': 'We acknowledge Prof. S.L. Yakovlev for useful remarks and relevant references.', 'quant-ph-0602207-4-62-3': 'We are grateful to the organizers of 3rd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics (June 2005, Istanbul, Turkey), especially to Prof. A. Mostafazadeh and Prof. M. Znojil, for financial support and opportunity to present the preliminary results of this paper.', 'quant-ph-0602207-4-63-0': '# Appendix', 'quant-ph-0602207-4-64-0': 'Let [MATH], [MATH] be the space of test functions.', 'quant-ph-0602207-4-64-1': 'The sequence [MATH] is called convergent in [MATH] to [MATH], [EQUATION] if [EQUATION] and for any [MATH], [MATH], [EQUATION]', 'quant-ph-0602207-4-64-2': 'We shall denote the value of a functional [MATH] on [MATH] conventionally as [MATH].', 'quant-ph-0602207-4-64-3': 'A functional [MATH] is called continuous, if for any sequence [MATH] convergent in [MATH] to zero the equality , [EQUATION] is valid.', 'quant-ph-0602207-4-64-4': 'The space of distributions over [MATH], i.e. of linear continuous functionals over [MATH] is denoted as [MATH].', 'quant-ph-0602207-4-64-5': 'The sequence [MATH] is called convergent in [MATH] to [MATH], [EQUATION] if for any [MATH] the relation takes place [EQUATION]', 'quant-ph-0602207-4-64-6': 'A functional [MATH] is called regular, if there is [MATH] such that for any [MATH] the equality [EQUATION] holds.', 'quant-ph-0602207-4-64-7': 'In this case we shall identify the distribution [MATH] with the function [MATH].', 'quant-ph-0602207-4-64-8': 'In virtue of the Bunyakovskii inequality, [EQUATION] it is evident that [MATH].', 'quant-ph-0602207-4-65-0': 'Let us notice also that the Dirac delta-function [MATH] belongs to [MATH], [MATH].', 'quant-ph-0602207-4-66-0': 'Lemma 1.', 'quant-ph-0602207-4-66-1': 'Suppose that: 1) [EQUATION] 2) [MATH] is a path in the complex plane of [MATH], made of the segment [MATH] of real axis by deformation of its central part up or down of zero and the positive direction of [MATH] is specified from [MATH] to [MATH]; 3) [MATH], [MATH], [MATH].', 'quant-ph-0602207-4-66-2': 'Then the following relations hold, [EQUATION] wherefrom it follows that [EQUATION]', 'quant-ph-0602207-4-66-3': 'Hence, [EQUATION] and in view of ([REF]) Eqs. ([REF]) and ([REF]) hold.', 'quant-ph-0602207-4-66-4': 'Eq. ([REF]) follows from Eq. ([REF]) and from additivity of classical path integrals.', 'quant-ph-0602207-4-66-5': 'Lemma 1 is proved.', 'quant-ph-0602207-4-67-0': 'Lemma 2.', 'quant-ph-0602207-4-67-1': 'For any [MATH] and [MATH] the relation, [EQUATION] is valid.', 'quant-ph-0602207-4-67-2': 'But its validity follows from the Bunyakovskii inequality [EQUATION]', 'quant-ph-0602207-4-67-3': 'Lemma 2 is proved.', 'quant-ph-0602207-4-68-0': 'Lemma 3.', 'quant-ph-0602207-4-68-1': 'For any [MATH], [MATH], [MATH] and [MATH] the relation holds, [EQUATION] holds.', 'quant-ph-0602207-4-68-2': 'This equality can be obtained from the chain of inequalities [EQUATION] derived with the help of Bunyakovskii inequality and trivial inequalities [MATH], [MATH], [MATH].', 'quant-ph-0602207-4-68-3': 'Lemma 3 is proved.', 'quant-ph-0602207-4-69-0': 'Corollary 1.', 'quant-ph-0602207-4-69-1': "Let's define [EQUATION]"}

1307.8312

{'1307.8312-1-0-0': 'We investigate the linearly (quadratically) coupled cubic Galileon models which may explain the late-time acceleration.', '1307.8312-1-0-1': 'In these cases, we need two equations of state parameter named the native and effective equations of state to test whether the universe is accelerating or not because there is coupling between the cold dark matter and Galileon.', '1307.8312-1-0-2': 'It turns out that there is no transition from accelerating phase to phantom phase in the future.', '1307.8312-1-1-0': '# Introduction', '1307.8312-1-2-0': 'Observational data indicate that our universe undergoes accelerating phase since the recent past [CITATION].', '1307.8312-1-2-1': 'The cosmological constant could be considered as the dark energy to explain the observational result in the [MATH]CDM model.', '1307.8312-1-2-2': 'However, this model has problems of fine tuning and coincidence and thus, an alternative candidate is a dynamical dark energy based on a scalar field dubbed the quintessence [CITATION] which has a canonical kinetic term.', '1307.8312-1-2-3': 'Here, a scalar is minimally coupled to gravity.', '1307.8312-1-2-4': 'A cosmological constant has a constant equation of state [MATH], while the scalar model has a time-varying equation of state with [MATH].', '1307.8312-1-3-0': 'Recently, Planck observation [CITATION] has shown four combined data that i) [MATH]; Panck+WMAP+BAO) which is in good agreement with a cosmological constant, ii) [MATH]; Panck+WMAP+Union2.1) which is more consistent with a cosmological constant, iii) [MATH]; Panck+WMAP+SNLS) which favors the phantom phase ([MATH]) at [MATH] level, and iv) [MATH]; Panck+WMAP+[MATH]) which is in tension with a cosmological constant at more than [MATH] level.', '1307.8312-1-3-1': 'The last two might draw the universe into the phantom phase at about 2[MATH] level.', '1307.8312-1-3-2': 'However, if one uses the BAO data in addition to the CMB, one finds no strong evidence for the phantom phase which is incompatible with a cosmological constant.', '1307.8312-1-4-0': 'On the other hand, one of modified gravities named the Galileon gravity was considered another model of the dynamical dark energy [CITATION].', '1307.8312-1-4-1': 'This model is composed of a scalar field theory which contains nonlinear-derivative self couplings.', '1307.8312-1-4-2': 'In the absence of gravity, the Galileon action is invariant under Galilean symmetry of [MATH].', '1307.8312-1-4-3': 'The field equations contain at most second derivatives of [MATH], implying that it is free from Ostrogradsky ghosts.', '1307.8312-1-4-4': 'Turning on gravity breaks the symmetry.', '1307.8312-1-4-5': 'Thus, a covariant Galilean action has been constructed [CITATION] where the Galilean symmetry is softly broken, but it preserves the shift symmetry of [MATH].', '1307.8312-1-4-6': 'Its cosmological implications have been investigated to explain the late-time acceleration in [CITATION].', '1307.8312-1-5-0': 'In this work, we investigate mainly the late-time acceleration induced by the linearly coupled cubic Galileon model [CITATION] with a linear potential [MATH] which is similar to the DGP model [CITATION].', '1307.8312-1-5-1': 'In this case, we need two equations of state parameter named the native and effective equations of state to test whether the universe is accelerating or not because there is coupling between the cold dark matter and Galileon.', '1307.8312-1-5-2': 'In the uncoupled case of [MATH], the model corresponds to the cubic Galileon gravity.', '1307.8312-1-5-3': 'This was studied in [CITATION] which shows that the data of SN+BAO+[MATH] prefers the Galileon gravity over the quintessence.', '1307.8312-1-5-4': 'The authors [CITATION] have shown a phantom phase, but it arose from choosing negative [MATH] and [MATH].', '1307.8312-1-5-5': 'More recently, it turned out that the equation of state for the cubic Galileon model is indistinguishable from [MATH] [CITATION].', '1307.8312-1-5-6': 'For [MATH], the phantom phase appeared for [MATH] case [CITATION] and [MATH] [CITATION].', '1307.8312-1-5-7': 'For a quadratic coupling with [MATH], its late-time evolution was investigated in [CITATION] where shows crossing of the phantom divide line.', '1307.8312-1-5-8': 'However, these phantom phases arose from choosing negative [MATH].', '1307.8312-1-6-0': 'If one suppresses a term of [MATH] indicating a decisive feature of the Galileon gravity, this is similar to the Brans-Dicke cosmology with the power-law potential [MATH] where one could observe a future crossing of the phantom divide line only for [MATH] [CITATION].', '1307.8312-1-6-1': 'Thus, it conjectures that the linear potential [MATH] seems not to provide the phantom phase.', '1307.8312-1-6-2': 'Very recently, the model of Slotheon gravity with [MATH] and [MATH] provides the cosmic acceleration at late time [CITATION].', '1307.8312-1-6-3': 'Hence, we will focus on observing whether the phantom phase appears in the future within the linearly (quadratically) coupled cubic Galileon models.', '1307.8312-1-7-0': '# Evolution equations', '1307.8312-1-8-0': "The covariant Galileon action can be written as [CITATION] [EQUATION] where [MATH] are arbitrary dimensionless constants, and [MATH] is the reduced Planck mass, [MATH] to make the [MATH]'s dimensionless, and [MATH] denotes the Lagrangian for the cold dark matter.", '1307.8312-1-8-1': 'Here, [MATH] and its trace [MATH] represent the energy-momentum tensor.', '1307.8312-1-8-2': 'The action is classified into three classes: the uncoupled Galileon with [MATH]; the linearly coupled Galileon with [MATH] and [MATH]; the derivative coupled Galileon with [MATH] and [MATH].', '1307.8312-1-9-0': 'We are working in the Jordan frame where the explicit coupling between [MATH] and [MATH] is removed when using a metric redefinition [CITATION].', '1307.8312-1-9-1': 'Among three classes of model, we focus on the linearly coupled cubic Galileon (lcG) model as [CITATION] [EQUATION] where the linear potential of [MATH] is introduced for our late-time evolution.', '1307.8312-1-9-2': 'Here we demand that [MATH] are positive to avoid the phantom scalar field.', '1307.8312-1-9-3': 'The case of [MATH] corresponds to the cubic Galileon gravity [CITATION].', '1307.8312-1-9-4': 'This model with [MATH] was extensively studied in [CITATION] and the uncoupled case of [MATH], [MATH] and [MATH] was investigated in [CITATION].', '1307.8312-1-9-5': 'In addition to [MATH], we use [MATH] instead of [MATH] which shows a decisive feature of Galileon gravity when comparing it with the quintessence and the Brans-Dicke cosmology.', '1307.8312-1-10-0': 'The Einstein equation is derived from ([REF]) as [EQUATION] where [MATH] is the Einstein tensor and the energy-momentum tensors are defined to be [CITATION] [EQUATION]', '1307.8312-1-10-1': 'The pressureless matter [MATH] from [MATH] is given by [EQUATION] with [MATH] the four velocity.', '1307.8312-1-11-0': 'The Galileon equation takes the form [EQUATION] where [MATH] denotes [MATH].', '1307.8312-1-11-1': "In order to study its cosmological implications, it would be better to convert ([REF]) into the standard form of the Einstein equation [EQUATION] where [MATH] denotes the energy-momentum tensor from all [MATH]'s contributions [MATH].", '1307.8312-1-12-0': 'Importantly, the Bianchi identity obtained by acting [MATH] on ([REF]) leads to the total conservation-law as [CITATION] [EQUATION] when using the conservation-law for the cold dark matter [EQUATION]', '1307.8312-1-12-1': 'At this stage, we consider the flat Friedmann-Robertson-Walker (FRW) metric as [EQUATION] where [MATH] is the scale factor.', '1307.8312-1-12-2': 'Then, we write the Einstein equation ([REF]) and the Galileon equation ([REF]) as [EQUATION]', '1307.8312-1-12-3': 'In the case of [MATH], the above all equations reduce to (2)-(4) of Ref.[CITATION].', '1307.8312-1-12-4': 'We can rewrite Eqs.([REF]) and ([REF]) as the standard forms of Friedmann equations [EQUATION] which can read off from ([REF]) directly.', '1307.8312-1-12-5': 'Since Eqs. ([REF]) and ([REF]) are the first and second Friedmann equations, respectively, we can read off the energy density and pressure for the Galileon as [EQUATION]', '1307.8312-1-12-6': 'We express the total conservation-law ([REF]) in terms of density and pressure for the Galileon and matter density as [EQUATION] which is very similar to the Brans-Dicke cosmology [CITATION].', '1307.8312-1-12-7': 'We observe that the left-hand side of ([REF]) implies the energy transfer between Galileon and cold dark matter.', '1307.8312-1-13-0': '# Background Evolution', '1307.8312-1-14-0': 'In order to solve three coupled equations ([REF])-([REF]), we introduce the new variables [EQUATION] where [MATH] is introduced instead of the scale factor [MATH].', '1307.8312-1-14-1': '([REF])-([REF]) are transformed into the first-order coupled equations [EQUATION] where [MATH], prime([MATH]) denotes the differentiation with respect to [MATH].', '1307.8312-1-14-2': 'Here, [MATH] and [MATH] are given as [EQUATION]', '1307.8312-1-14-3': 'Importantly, the total conservation-law ([REF]) can be rewritten as the new variables as [EQUATION] where the native equation of state is defined to be [EQUATION]', '1307.8312-1-14-4': 'One may interpret Eq. ([REF]) as [EQUATION]', '1307.8312-1-14-5': 'From this equation, one can define the effective equations of state as [EQUATION] where the density parameter [MATH] is given by [EQUATION]', '1307.8312-1-14-6': 'Here [MATH] is the critical energy density and we used the relation [EQUATION]', '1307.8312-1-14-7': 'In the case of [MATH], we have [MATH], which implies that [MATH] is not necessary for describing the uncoupled Galileon.', '1307.8312-1-14-8': 'Since for [MATH], [MATH] and [MATH] blow up at [MATH], it requires [MATH].', '1307.8312-1-15-0': 'We rewrite the first Friedmann equation ([REF]) in terms of the new variables as [EQUATION] which will be used to find the initial values of the evolving variables.', '1307.8312-1-15-1': 'According to the Planck mission [CITATION], the current dark matter content is [MATH].', '1307.8312-1-15-2': 'We take this value as an initial condition for the numerical evolution.', '1307.8312-1-15-3': 'After solving the first-order coupled equations, one finds the background evolution for the uncoupled case ([MATH]).', '1307.8312-1-15-4': 'As is depicted in Fig. [REF], we observe the accelerating universe of [MATH] in the future.', '1307.8312-1-15-5': 'Furthermore, Fig. [REF] shows a typical background evolution for the linearly coupled case ([MATH]).', '1307.8312-1-15-6': 'In this case, we find the accelerating universe confirmed by [MATH] as well as [MATH].', '1307.8312-1-15-7': 'Definitely, no signal to give a phantom phase with [MATH] in the future.', '1307.8312-1-16-0': '# Quadratic Coupling', '1307.8312-1-17-0': 'Recently, there was a work for the quadratic coupling which shows crossing of the phantom divide [CITATION].', '1307.8312-1-17-1': 'In order to see whether this happens or not, we consider the action without [MATH] and [MATH] for simplicity [EQUATION]', '1307.8312-1-17-2': 'Following the computation steps of the previous section, two equations that are different from the linearly coupled case are [EQUATION] where [MATH] and [MATH] are given as [EQUATION]', '1307.8312-1-17-3': 'Also, the native and effective equations of state are given as [EQUATION]', '1307.8312-1-17-4': 'We note that the first Friedmann equation for the initial condition is slightly modified to be [EQUATION]', '1307.8312-1-17-5': 'We observe from Fig. [REF] that there is no crossing of the phantom divide in the future.', '1307.8312-1-17-6': 'No essential difference is found between the linearly coupled and quadratically coupled Galileon models.', '1307.8312-1-18-0': '# Discussions', '1307.8312-1-19-0': 'First of all, we observe that the native and effective equations of state do not cross the phantom divide of [MATH] in the linearly (quadratically) coupled cubic Galileon models.', '1307.8312-1-19-1': 'This means that there is no difference between the Brans-Dicke cosmology and the cubic Galileon models.', '1307.8312-1-19-2': 'In other words, the role of [MATH] showing a feature of the Galileon gravity is not significant for deriving the late-time evolution.', '1307.8312-1-20-0': 'At this stage, one may ask which one is an observable quantity between [MATH] and [MATH] in the Jordan frame [CITATION].', '1307.8312-1-20-1': 'It is well known that the Jordan frame is a physical frame because of a minimal coupling to matter.', '1307.8312-1-20-2': 'However, this frame gives rises to the non-conservation of continuity equation ([REF]) which shows that [MATH] plays the role of a source to generate a new dark fluid.', '1307.8312-1-20-3': 'Even though [MATH] indicates a genuine equation of state for a Galileon-fluid, it cannot satisfy the continuity equation.', '1307.8312-1-20-4': 'On the other hand, although [MATH] is not a genuine equation of state for a Galileon-fluid (because it contains cold dark matter), it satisfies the continuity equation.', '1307.8312-1-20-5': 'This implies that two are not a perfect observable for the linearly (quadratically) coupled Galileon models.', '1307.8312-1-20-6': 'Therefore, we have to use both [MATH] and [MATH] to show the presence of crossing of the phantom divide.', '1307.8312-1-20-7': 'If the phantom phase is observed from both, one believes that it really happens in the evolution of the linearly (quadratically) coupled cubic Galileon gravity models.', '1307.8312-1-20-8': 'Otherwise, one could not confirm the appearance of the phantom phase in the Jordan frame.', '1307.8312-1-20-9': 'Surely, our analysis shows the disappearance of any phantom phase in the coupled cubic Galileon models.', '1307.8312-1-21-0': 'Finally, there is the total equation of state [CITATION] [EQUATION] which leads to [EQUATION]', '1307.8312-1-21-1': 'The evolution of [MATH] is given in Figs. [REF] and [REF] for linearly coupled and quadratically coupled cases, respectively.', '1307.8312-1-21-2': 'These show clearly that there is no phantom phase in the future.', '1307.8312-1-22-0': 'This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (NO. 2012R1A1A3012776).'}

{'1307.8312-2-0-0': 'We investigate the linearly and quadratically coupled cubic Galileon models that include linear potentials.', '1307.8312-2-0-1': 'These models may explain the late-time acceleration.', '1307.8312-2-0-2': 'In these cases, we need two equations of state parameter named the native and effective equations of state to test whether the universe is accelerating or not because there is coupling between the cold dark matter and Galileon.', '1307.8312-2-0-3': 'It turns out that there is no transition from accelerating phase to phantom phase in the future.', '1307.8312-2-1-0': '# Introduction', '1307.8312-2-2-0': 'Observational data indicate that our universe undergoes an accelerating phase since the recent past [CITATION].', '1307.8312-2-2-1': 'The cosmological constant could be considered as a candidate for the dark energy to explain the observational result in the [MATH]CDM model.', '1307.8312-2-2-2': 'However, this model has two problems of the fine tuning and the coincidence and thus, an alternative candidate was needed.', '1307.8312-2-2-3': 'One promising candidate is a dynamical dark energy model based on scalar field theory which is dubbed the quintessence [CITATION].', '1307.8312-2-2-4': 'This model has a canonical kinetic term and thus, a scalar is minimally coupled to gravity.', '1307.8312-2-2-5': 'We wish to point out that the cosmological constant has a constant equation of state [MATH], while the scalar field model has a time-dependent equation of state with [MATH].', '1307.8312-2-3-0': 'Recently, Planck observation [CITATION] has shown four combined data on equation of state: i) [MATH]; Panck+WMAP+BAO) which is in good agreement with a cosmological constant, ii) [MATH]; Panck+WMAP+Union2.1) that is more consistent with a cosmological constant, iii) [MATH]; Panck+WMAP+SNLS) which favors the phantom phase ([MATH]) at the [MATH] level, and iv) [MATH]; Panck+WMAP+[MATH]) which is in tension with a cosmological constant at more than the [MATH] level.', '1307.8312-2-3-1': 'The last two might draw the universe into a phantom phase about at the 2[MATH] level.', '1307.8312-2-3-2': 'However, if one uses the BAO data in addition to the CMB, there is no strong evidence for the phantom phase that is incompatible with a cosmological constant.', '1307.8312-2-4-0': 'On the other hand, one modified gravity named the Galileon gravity was considered another model of the dynamical dark energy [CITATION].', '1307.8312-2-4-1': 'This model is also described by a scalar field theory which contains terms of nonlinear-derivative self couplings.', '1307.8312-2-4-2': 'Turning off gravity, the Galileon action is invariant under the Galilean transformation of [MATH].', '1307.8312-2-4-3': 'The field equations contain at most second derivatives of [MATH], implying that it is surely free from the Ostrogradsky ghosts.', '1307.8312-2-4-4': 'Turning on gravity, however, breaks the symmetry.', '1307.8312-2-4-5': 'Therefore, a covariant Galilean action has been constructed [CITATION], where the Galilean symmetry is softly broken but it preserves the shift symmetry of [MATH].', '1307.8312-2-4-6': 'Its cosmological implications have been extensively investigated to explain the late-time acceleration in [CITATION].', '1307.8312-2-5-0': 'In this work, we wish to investigate the late-time acceleration by using the linearly coupled cubic Galileon model [CITATION] together with a linear potential [MATH].', '1307.8312-2-5-1': 'This model is similar to the DGP model [CITATION].', '1307.8312-2-5-2': 'We note that adding the potential breaks the shift symmetry, which might make the nonlinear-derivative self coupling term trivial.', '1307.8312-2-5-3': 'In this case, one needs two equations of state parameter named the native and effective equations of state to say whether the universe is accelerating or not.', '1307.8312-2-5-4': 'This is because there is coupling between the cold dark matter and Galileon.', '1307.8312-2-5-5': 'In the uncoupled case of [MATH], the model under consideration corresponds to the cubic Galileon gravity.', '1307.8312-2-5-6': 'This has been studied in [CITATION], which shows that the data of SN+BAO+[MATH] prefers the Galileon gravity over the quintessence.', '1307.8312-2-5-7': 'The authors in [CITATION] have shown an appearance of the phantom phase, but it arose from choosing negative [MATH] and [MATH].', '1307.8312-2-5-8': 'More recently, it turned out that the equation of state obtained from the cubic Galileon model is indistinguishable from [MATH] of the cosmological constant [CITATION].', '1307.8312-2-5-9': 'For [MATH] and [MATH], a phantom phase appeared for both [MATH] case [CITATION] and [MATH] [CITATION].', '1307.8312-2-5-10': 'For a quadratic coupling with [MATH] and [MATH], its late-time evolution was investigated in [CITATION] which indicates a crossing of the phantom divide line.', '1307.8312-2-5-11': 'However, we mention that these phantom phases arose from when one chooses negative coefficients [MATH].', '1307.8312-2-6-0': 'Previous works on the Galilean models have shown phantom phases, depending on the choice of coefficients.', '1307.8312-2-6-1': 'Hence, it is very curious to check if a phantom phase happens really in the Galilean models because phantom behavior is quite interesting and it is possibly bad for the future evolution of the universe.', '1307.8312-2-6-2': 'Hence, we hope to find a phantom phase in the linearly and quadratically coupled cubic Galileon model that include linear potential.', '1307.8312-2-6-3': 'However, we have found no phantom phase when we did make a complete computation.', '1307.8312-2-6-4': 'We wish to understand why there is no phantom by comparing it with the Brans-Dicke theory with a linear potential.', '1307.8312-2-6-5': 'If one suppresses a nonlinear-derivative self coupling term of [MATH] capturing a decisive feature of the Galileon gravity, the linearly coupled cubic Galileon model with the potential is similar to the Brans-Dicke cosmology with the power-law potential [MATH] where one could observe a future crossing of the phantom divide line only for [MATH] [CITATION].', '1307.8312-2-6-6': 'Thus, it seems that choosing the linear potential [MATH] does not provide the phantom phase because this potential breaks the shift symmetry.', '1307.8312-2-6-7': 'Explicitly, the addition of the linear potential did not make the nonlinear-derivative self coupling term of [MATH] matter and thus, it did not play a role in the late-time evolution.', '1307.8312-2-6-8': 'Very recently, a model of Slotheon gravity with [MATH] and [MATH] has provided the cosmic acceleration but not the phantom phase in the late-time evolution [CITATION].', '1307.8312-2-7-0': 'Inspired by the above motivation, we will focus on observing whether the phantom phase appears in the future when we use the linearly and quadratically coupled cubic Galileon models that include linear potentials.', '1307.8312-2-8-0': '# Evolution equations', '1307.8312-2-9-0': "The covariant Galileon action can be written as [CITATION] [EQUATION] where [MATH] are arbitrary dimensionless constants, and [MATH] is the reduced Planck mass, and [MATH] to make the [MATH]'s dimensionless.", '1307.8312-2-9-1': '[MATH] denotes the Lagrangian for the cold dark matter and [MATH] (its trace [MATH]) represent the energy-momentum tensor.', '1307.8312-2-9-2': 'The Galileon action is usually classified into three classes: the uncoupled Galileon with [MATH]; the linearly coupled Galileon with [MATH] and [MATH]; the derivative coupled Galileon with [MATH] and [MATH].', '1307.8312-2-10-0': 'We are working in the Jordan frame where the explicit coupling between [MATH] and [MATH] is removed when we use a metric redefinition [CITATION].', '1307.8312-2-10-1': 'Among three classes of the Galileon model, we focus on the linearly coupled cubic Galileon (lcG) model as [CITATION] [EQUATION] where the linear potential of [MATH] is introduced for our late-time evolution.', '1307.8312-2-10-2': 'Here we demand that [MATH] are positive to avoid the phantom scalar field.', '1307.8312-2-10-3': 'The case of [MATH] corresponds to the cubic Galileon gravity [CITATION].', '1307.8312-2-10-4': 'This model with [MATH] was extensively studied in [CITATION] and the uncoupled case of [MATH], [MATH] and [MATH] was investigated in [CITATION].', '1307.8312-2-10-5': 'In addition to the choice of [MATH], we use [MATH] instead of [MATH] which shows a decisive feature of the Galileon gravity when we compare it with the quintessence and the Brans-Dicke cosmology.', '1307.8312-2-11-0': 'The Einstein equation is derived from ([REF]) as [EQUATION] where [MATH] is the Einstein tensor and the energy-momentum tensors are defined to be [CITATION] [EQUATION]', '1307.8312-2-11-1': 'Here the energy-momentum tensor [MATH] for a pressureless matter of [MATH] is given by [EQUATION] with [MATH] the four velocity.', '1307.8312-2-12-0': 'The Galileon equation takes the form [EQUATION] where [MATH] denotes the derivative with respect to [MATH].', '1307.8312-2-12-1': "To study its cosmological implications, it would be better to convert Eq. ([REF]) into a standard form of the Einstein equation [EQUATION] where [MATH] denotes the energy-momentum tensor from all [MATH]'s contributions[MATH].", '1307.8312-2-13-0': 'Importantly, the Bianchi identity obtained by acting [MATH] on Eq. ([REF]) leads to the total conservation-law as [CITATION] [EQUATION] when one uses the conservation-law for the cold dark matter [EQUATION]', '1307.8312-2-13-1': 'At this stage, we introduce the flat Friedmann-Robertson-Walker (FRW) metric as [EQUATION] where [MATH] is the scale factor.', '1307.8312-2-13-2': 'Then, we write the Einstein equation ([REF]) and the Galileon equation ([REF]) as [EQUATION]', '1307.8312-2-13-3': 'In the case of [MATH], the above all equations reduce to Eqs. (2)-(4) of Ref. [CITATION].', '1307.8312-2-13-4': 'We can rewrite Eqs. ([REF]) and ([REF]) as the standard forms of the Friedmann equations [EQUATION] which can also found from ([REF]) directly.', '1307.8312-2-13-5': 'Since Eqs. ([REF]) and ([REF]) are the first and second Friedmann equations, respectively, we can read off the energy density and pressure for the Galileon as [EQUATION]', '1307.8312-2-13-6': 'Now we express the total conservation-law ([REF]) in terms of density and pressure for the Galileon and matter density as [EQUATION] which is very similar to the Brans-Dicke cosmology [CITATION].', '1307.8312-2-13-7': 'We observe that the left-hand side of Eq. ([REF]) implies an energy (matter) transfer between Galileon and cold dark matter.', '1307.8312-2-14-0': '# Background Evolution', '1307.8312-2-15-0': 'To solve three coupled equations ([REF])-([REF]), we introduce the new variables [EQUATION] where [MATH] is introduced instead of the scale factor [MATH].', '1307.8312-2-15-1': 'Equations ([REF])-([REF]) are transformed into the first-order coupled equations [EQUATION] where [MATH] and prime ([MATH]) denotes the differentiation with respect to [MATH].', '1307.8312-2-15-2': 'Here, [MATH] and [MATH] are given as [EQUATION]', '1307.8312-2-15-3': 'Importantly, the total conservation-law ([REF]) can be rewritten as the new variables as [EQUATION] where the native equation of state is defined to be [EQUATION]', '1307.8312-2-15-4': 'One may interpret Eq. ([REF]) as [EQUATION]', '1307.8312-2-15-5': 'From this equation, one can obtain the effective equations of state as [EQUATION] where the two density parameter [MATH] are given by [EQUATION]', '1307.8312-2-15-6': 'Here [MATH] is the critical energy density and we used the relation [EQUATION]', '1307.8312-2-15-7': 'In case of [MATH], we have [MATH], which implies that [MATH] is not necessary to describe the uncoupled Galileon model.', '1307.8312-2-15-8': 'Since [MATH] and [MATH] blow up at [MATH] for [MATH], we require [MATH].', '1307.8312-2-16-0': 'We rewrite the first Friedmann equation ([REF]) in terms of the new variables as [EQUATION] which was used to find the initial values of the evolving variables.', '1307.8312-2-16-1': 'According to the Planck mission [CITATION], the current dark matter content is [MATH].', '1307.8312-2-16-2': 'We take this value as an initial condition for the numerical evolution.', '1307.8312-2-16-3': 'After solving the first-order coupled equations, one finds the background evolution for the uncoupled case ([MATH]).', '1307.8312-2-16-4': 'As is depicted in Fig. [REF], we observe the accelerating universe of [MATH] in the future.', '1307.8312-2-16-5': 'Furthermore, Fig. [REF] shows a typical background evolution for the linearly coupled case ([MATH]).', '1307.8312-2-16-6': 'In this case, we find the accelerating universe which is shown by [MATH] as well as [MATH].', '1307.8312-2-16-7': 'Definitely, there is no signal to give a phantom phase with [MATH] in the future.', '1307.8312-2-17-0': '# Quadratic Coupling', '1307.8312-2-18-0': 'Recently, there was a work for the quadratic coupling which shows a crossing of the phantom divide [CITATION].', '1307.8312-2-18-1': 'To see whether this happens really or not, we consider the action without [MATH] and [MATH] for simplicity [EQUATION]', '1307.8312-2-18-2': 'Following the computation steps of the previous section, two equations that are different from the linearly coupled case are [EQUATION] where [MATH] and [MATH] are given as [EQUATION]', '1307.8312-2-18-3': 'Also, the native and effective equations of state are given as [EQUATION]', '1307.8312-2-18-4': 'We note that the first Friedmann equation for the initial condition is slightly modified to be [EQUATION]', '1307.8312-2-18-5': 'We observe from Fig. [REF] that there is no crossing of the phantom divide in the future.', '1307.8312-2-18-6': 'There is no essential difference between the linearly coupled and quadratically coupled Galileon models.', '1307.8312-2-19-0': '# Discussions', '1307.8312-2-20-0': 'First of all, we observe that the native and effective equations of state do not cross the phantom divide of [MATH] in the linearly (quadratically) coupled cubic Galileon models.', '1307.8312-2-20-1': 'This means that there is no essential difference between the Brans-Dicke cosmology and the cubic Galileon models.', '1307.8312-2-20-2': 'In other words, the term of [MATH] showing a feature of the Galileon gravity did not contribute significantly to deriving the late-time acceleration.', '1307.8312-2-20-3': 'To explain it, let us compare our result with Ref. [CITATION] where acceleration was found, even though the potential [MATH] was not introduced.', '1307.8312-2-20-4': 'The reason seems to be clear because this potential breaks the shift symmetry of [MATH].', '1307.8312-2-20-5': 'The addition of the potential did not make the nonlinear-derivative self coupling term of [MATH] matter and thus, it did not play a role in the late-time evolution.', '1307.8312-2-21-0': 'At this stage, one may ask which one is an observable quantity between [MATH] and [MATH] in the Jordan frame [CITATION].', '1307.8312-2-21-1': 'It is well known that the Jordan frame is a physical frame because of a minimal coupling to matter.', '1307.8312-2-21-2': 'However, this frame gives rises to the non-conservation of continuity equation ([REF]) which shows that [MATH] plays the role of a source to generate a new dark fluid.', '1307.8312-2-21-3': 'Even though [MATH] indicates a genuine equation of state for a Galileon-fluid, it cannot satisfy the continuity equation.', '1307.8312-2-21-4': 'On the other hand, although [MATH] is not a genuine equation of state for a Galileon-fluid (because it contains cold dark matter), it satisfies the continuity equation.', '1307.8312-2-21-5': 'This implies that each of them is not a perfect observable for the linearly (quadratically) coupled Galileon models.', '1307.8312-2-21-6': 'Therefore, we have to use both [MATH] and [MATH] to show the presence of a crossing of the phantom divide.', '1307.8312-2-21-7': 'If the phantom phase is observed from both, one believes that it really happens in the evolution of the linearly (quadratically) coupled cubic Galileon gravity models.', '1307.8312-2-21-8': 'Otherwise, one is hard to confirm the appearance of the phantom phase in the Jordan frame.', '1307.8312-2-21-9': 'Surely, our analysis shows the disappearance of any phantom phase in the coupled cubic Galileon models.', '1307.8312-2-22-0': 'Finally, we mention that there is the equation [CITATION] [EQUATION] which defines the total equation of state [EQUATION]', '1307.8312-2-22-1': 'The evolution of [MATH] is given in Figs. [REF] and [REF] for linearly coupled and quadratically coupled cases, respectively.', '1307.8312-2-22-2': 'These also show that there is no phantom phase in the future.', '1307.8312-2-23-0': 'This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NO. 2012R1A1A3012776).'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1307.8312

1412.5031

{'1412.5031-1-0-0': 'Cluster dynamical mean field methods are used to calculate the normal and anomalous components of the electron self energy of the two dimensional Hubbard model.', '1412.5031-1-0-1': 'From these the evolution of the superconducting gap and the momentum dependent photoemission and inverse photoemission spectra across the phase diagram are determined.', '1412.5031-1-0-2': "In the pseudogap regime, decreasing the temperature into the superconducting state leads to a decrease in the energy gap and the formation of a 'peak-dip-hump' structure in the electronic density of states.", '1412.5031-1-0-3': 'The peak feature disperses very weakly.', '1412.5031-1-0-4': 'The calculated spectral functions are in good qualitative agreement with published data.', '1412.5031-1-0-5': 'The mathematical origin of the behavior is found to be the effect of the superconductivity on the pole structure giving rise to the normal state pseudogap.', '1412.5031-1-0-6': 'In particular the "hump" feature is found to arise from a zero crossing of the real part of the electron self energy rather than from an onset of scattering.', '1412.5031-1-0-7': 'The effect of superconductivity on the zone diagonal spectra is presented.', '1412.5031-1-1-0': '# Overview', '1412.5031-1-2-0': 'After more than a quarter century of research, our theoretical understanding of the unusual electronic properties of the high transition temperature superconductivity observed [CITATION] in layered copper oxide materials remains incomplete.', '1412.5031-1-2-1': "A plethora of remarkable dynamical phenomena have been reported, including a correlation-driven insulating phase occurring when the conduction band is half filled,[CITATION] non-Fermi-liquid transport properties,[CITATION] a 'pseudogap' in the electronic spectrum [CITATION] as well as ordered phases including 'stripe' states with spin and charge order,[CITATION] charge order apparently unconnected with spin order,[CITATION] 'nematic' (rotational symmetry breaking) states,[CITATION] and time reversal symmetry breaking states also apparently unaccompanied by conventional spin order.[", '1412.5031-1-2-2': '[CITATION] Angle-resolved photoemission studies report unusual quasiparticle properties at essentially all carrier concentrations.[', '1412.5031-1-3-0': 'The diversity of reported phenomena has led to debate about what is the essential physics to include in a minimal theoretical model while the apparent strong coupling nature of the problem suggests that whatever model is adopted, a nonperturbative treatment is required.', '1412.5031-1-3-1': "Important open questions include the mechanism for superconductivity, the nature of the minimal low energy model describing the high-[MATH] phenomenon,[CITATION] and the physics of the 'pseudogap' and its interplay with the superconducting gap.", '1412.5031-1-3-2': '[CITATION]', '1412.5031-1-4-0': 'This paper presents a theoretical study of the electron excitation spectrum of the superconducting and normal states of the two dimensional Hubbard model.', '1412.5031-1-4-1': 'This model is the minimal model of the physics of strongly correlated electrons on a lattice.', '1412.5031-1-4-2': 'Although it is not yet known if this model contains the full panoply of phenomena observed in the high-[MATH] materials, it clearly contains some important aspects of the physics and is accepted as one of the candidate models [CITATION] for describing the low energy (energies of order [MATH] or less) physics of the copper-oxide superconductors.', '1412.5031-1-4-3': 'Determining the properties of this model to the level at which a clear comparison to experiment can be made is an important goal of theory.', '1412.5031-1-5-0': "We address this problem using the 'dynamical cluster approximation' (DCA) version [CITATION] of the cluster dynamical mean field method.[", '1412.5031-1-5-1': '[CITATION] The method is based on approximating the full spatial dependence of the electron self energy in terms of a finite number [MATH] of functions of frequency,[CITATION] with the exact properties recovered in the [MATH] limit.[', '1412.5031-1-5-2': '[CITATION] Within this approximation the method provides an unbiased (in the sense of not pre-selecting a particular interaction channel or class of diagrams) numerical approach to the correlated electron problem and allows comprehensive investigation of the frequency dependence, and some aspects of the momentum dependence, of the electronic properties.', '1412.5031-1-6-0': 'Our ability to solve the equations of dynamical mean field theory[CITATION] has reached the point where approximation sizes [MATH] large enough [CITATION] to be representative of the [MATH] limit can be studied at the low temperatures required to stabilize superconductivity.', '1412.5031-1-6-1': '[CITATION] The superconducting state has been constructed [CITATION] and physical properties including the superconducting condensation energy,[CITATION] the c-axis and Raman response [CITATION] and the structure of the gap function and pairing potential [CITATION] have been computed and found to be in remarkable agreement with experiment.', '1412.5031-1-6-2': 'Here we use the new methodologies to study the electronic excitation spectrum of the normal and superconducting states in detail.', '1412.5031-1-6-3': 'While some of these issues have been previously studied in a cluster size [MATH] approximation, [CITATION] our results, obtained on larger [MATH] clusters, provide a better representation of the physics including a clear separation of nodal and antinodal behavior.', '1412.5031-1-7-0': 'The DCA equations are solved in imaginary time, and an analytical continuation[CITATION] procedure is needed to obtain the real frequency information needed for the electronic excitation spectrum.', '1412.5031-1-7-1': 'Especially in the pseudogap regime of the model, the novel physics we find presents challenges for the analytical continuation of our computed results.', '1412.5031-1-7-2': 'These are discussed at length below.', '1412.5031-1-8-0': 'The rest of the paper is organized as follows.', '1412.5031-1-8-1': 'In Section [REF] we define the quantities of interest and present the specifics of the dynamical mean field method.', '1412.5031-1-8-2': 'Section [REF] discusses issues related to the analytic continuation.', '1412.5031-1-8-3': 'Section [REF] displays the normal and anomalous components of the self energy, drawing attention to an unusual pole structure related to sector-selective Mott nature of the pseudogap phase.', '1412.5031-1-8-4': '[CITATION] Section [REF] displays the gap function constructed from the ratio of anomalous and normal components of the self energy.', '1412.5031-1-8-5': 'Sections [REF] and [REF] display the photoemission and inverse photoemission spectra predicted by the model while section [REF] presents our findings on superconductivity-induced changes to the electron scattering and mass renormalization for states near the zone diagonal where the superconducting order parameter vanishes.', '1412.5031-1-8-6': 'Section [REF] presents a summary and conclusions.', '1412.5031-1-9-0': '# Formalism', '1412.5031-1-10-0': '## Model', '1412.5031-1-11-0': 'We study the two dimensional Hubbard model of electrons hopping on a square lattice and subject to a local interaction [MATH] which we take to be repulsive.', '1412.5031-1-11-1': 'The model may be written in a mixed momentum ([MATH])/position ([MATH]) representation as [EQUATION]', '1412.5031-1-11-2': 'In the first term we have represented the electronic degrees of freedom by the Nambu spinor defined in terms of [MATH], the Fourier transform to momentum space of the operator [MATH] which creates an electron of spin [MATH] on lattice site [MATH] as [EQUATION]', '1412.5031-1-11-3': 'We set the lattice constant to unity.', '1412.5031-1-11-4': 'The momentum index [MATH] runs over the Brillouin zone of the two dimensional square lattice [MATH].', '1412.5031-1-11-5': 'The trace is over the Nambu indices and [MATH] denote the Pauli matrices operating in Nambu space.', '1412.5031-1-11-6': 'The chemical potential is [MATH], [MATH] is the energy dispersion and [MATH] is the operator measuring the density of spin [MATH] electrons on site [MATH].', '1412.5031-1-11-7': 'In the computations presented below we take [MATH] and present our results in units of [MATH].', '1412.5031-1-11-8': 'A reasonable estimate of the energy scales pertaining to the physical copper oxide materials is [MATH].', '1412.5031-1-11-9': 'At carrier density [MATH] per site this version of the Hubbard model is particle-hole symmetric ([MATH]), but particle-hole symmetry is broken at carrier concentrations [MATH].', '1412.5031-1-12-0': '## Green function and self energy', '1412.5031-1-13-0': 'Our analysis proceeds from the components of the matrix Nambu Green function defined for imaginary time [MATH] (T is the temperature) as [EQUATION]', '1412.5031-1-13-1': 'It is useful to Fourier transform [MATH] to the Matsubara frequency axis [EQUATION] with [MATH].', '1412.5031-1-13-2': 'The self energy is defined as [EQUATION] and may be written explicitly in terms of normal (N) and anomalous (A) components as (note we have chosen the phase of the superconducting order parameter to be real) [EQUATION]', '1412.5031-1-13-3': 'Under spatial (k) transformations [MATH] has the full symmetry of the lattice, while depending on the superconducting state [MATH] may have lower symmetry.', '1412.5031-1-13-4': 'Our investigations,[CITATION] consistent with a large body of previous work, [CITATION] indicate that for the interesting carrier concentrations [MATH] and for temperatures higher than [MATH] the only stable superconducting state of this model is of [MATH] symmetry, so that on the square lattice studied here [MATH] changes sign if [MATH] is rotated by [MATH] or reflected through one of the axes [MATH] that lie at [MATH] to the lattice vectors, but remains invariant under reflections through the bond axes [MATH] or [MATH] and under rotations by [MATH].', '1412.5031-1-14-0': 'We now discuss the frequency dependence at fixed [MATH].', '1412.5031-1-14-1': 'In this discussion because the wave vector is fixed we do not explicitly denote it.', '1412.5031-1-14-2': 'The anomalous self energy is an even function of Matsubara frequency: [MATH].', '1412.5031-1-14-3': 'However the normal component at positive Matsubara frequency [MATH] is not simply related to the normal component at negative Matsubara frequency [MATH] except in the special case of particle-hole symmetry where [MATH].', '1412.5031-1-14-4': 'For later purposes it is convenient to distinguish components of [MATH] even (e) and odd (o) in Matsubara frequency[CITATION] as [EQUATION] so that in Nambu notation we may write [EQUATION]', '1412.5031-1-14-5': 'It is convenient also to define the gap function [MATH] as[CITATION] [EQUATION]', '1412.5031-1-14-6': 'This definition is motivated by the observation that at the low frequencies of interest here, the even (under sign change of Matsubara frequency) component of [MATH] can be absorbed into a shift of chemical potential.', '1412.5031-1-15-0': '## Analytic Structure', '1412.5031-1-16-0': '[MATH] and [MATH] are analytic functions of complex frequency argument [MATH] except for a branch cut along the real axis [MATH].', '1412.5031-1-16-1': 'Apart from a Hartree term in the normal component of [MATH] they decay as [MATH].', '1412.5031-1-16-2': 'They thus obey a Kramers-Kronig relation (written here for [MATH], with [MATH] one half of the branch-cut discontinuity) [EQUATION] [MATH] is a matrix with eigenvalues that are non-negative and related by time reversal, thus: [EQUATION] with [MATH].', '1412.5031-1-16-3': 'Here [EQUATION] is a rotation matrix in Nambu space parametrized by an angle [MATH] which in general is frequency dependent and lies in the range [MATH].', '1412.5031-1-16-4': 'That only [MATH] appears in Eq. [REF] follows from our phase convention for the superconducting state, Eq. [REF].', '1412.5031-1-17-0': 'Evaluating Eq. [REF] for [MATH] approaching the real axis gives the real-time Green functions.', '1412.5031-1-17-1': 'Of particular interest are the retarded functions obtained by continuation from below to the real frequency axis: [MATH] with [MATH] real and [MATH] (typically not explicitly written) a positive infinitesimal.', '1412.5031-1-17-2': 'The retarded real axis axis functions have real and imaginary parts [MATH] and [MATH], respectively.', '1412.5031-1-17-3': 'Our choice of superconducting phase convention (upper right and lower left entries in self energy matrix identical) implies that [MATH] is purely real so that [MATH] is identical to the branch cut discontinuity [MATH] introduced in Eq. [REF].', '1412.5031-1-18-0': 'The Kramers-Kronig relation along with the positivity of [MATH] implies that [MATH] and [MATH] are a real functions of [MATH] while [MATH] is a purely imaginary function of [MATH].', '1412.5031-1-18-1': 'The symmetry properties of [MATH] are those of [MATH].', '1412.5031-1-19-0': 'The non-negativity of the two eigenvalues of the spectral function implies that the diagonal components of the imaginary parts of [MATH] and [MATH] are non-negative so that on the real frequency axis [MATH] and [MATH].', '1412.5031-1-20-0': '## Method', '1412.5031-1-21-0': 'To solve the Hubbard model we employ the dynamical mean field approximation [CITATION] in its DCA cluster[CITATION] form.', '1412.5031-1-21-1': 'In this approximation the Brillouin zone is partitioned into [MATH] equal area tiles labeled by central momentum [MATH] and the self energy is approximated as a piecewise continuous function [EQUATION] with [MATH] if [MATH] is in the tile centered on [MATH] and 0 otherwise.', '1412.5031-1-21-2': 'The functions [MATH] are Nambu matrices with normal (N) and anomalous (A) components which are determined from the solution of an auxiliary quantum impurity model as described in detail in Refs. Lichtenstein00,Maier06,Gull11,Gull13.', '1412.5031-1-22-0': 'The quantum impurity model is solved by the continuous-time auxiliary field quantum Monte-Carlo method introduced in Ref. Gull08 and discussed in detail in Ref. Gull11.', '1412.5031-1-22-1': 'Fast update techniques [CITATION] are crucial for accessing the range of interaction strengths and temperatures needed to study superconductivity and its interplay with the pseudogap.', '1412.5031-1-22-2': 'The quantum Monte-Carlo calculations are performed on the Matsubara axis and maximum-entropy analytical continuation techniques[CITATION] are employed to obtain real-frequency results.', '1412.5031-1-23-0': 'The expense of the computation increases rapidly as the interaction strength [MATH] or number of approximants [MATH] increases, or as the temperature [MATH] decreases.', '1412.5031-1-23-1': 'We present results for [MATH] using the momentum-space tiling shown in the left panel of Fig. [REF].', '1412.5031-1-23-2': 'Previous work [CITATION] has shown that this cluster is in many aspects representative of the [MATH] limit; in particular it is large enough to enable a clear distinction between zone-diagonal and zone-face electronic properties, yet small enough to permit calculations in the superconducting phase of the precision needed for analytical continuation.', '1412.5031-1-24-0': 'The d-wave symmetry of the superconducting state of the two dimensional Hubbard model means that in the coarse-grained momentum resolution available in the [MATH] DCA approximation we have [EQUATION] [MATH] has the full point group symmetry of the lattice and thus is one function of frequency in the [MATH] and [MATH] sectors, a different function of frequency in the [MATH] sectors and yet a different function of frequency in the [MATH] and in the [MATH] sectors.', '1412.5031-1-24-1': 'The main focus of attention in this paper will be on the [MATH] momentum sector but some results will be presented on superconductivity-induced changes in the zone-diagonal [MATH] momentum sectors.', '1412.5031-1-24-2': 'Because in the 8-site DCA approximation the anomalous self energy is non-zero only in the [MATH] sectors we will typically omit the momentum argument in our discussions of [MATH] and [MATH].', '1412.5031-1-25-0': 'The right panel of Fig. [REF] shows the phase diagram in the plane of interaction strength and carrier concentration obtained [CITATION] from the solution of the DCA equations at temperature [MATH], about half of the maximal computed superconducting transition temperature [MATH].', '1412.5031-1-25-1': 'The temperature [MATH] corresponds in physical units to [MATH].', '1412.5031-1-25-2': 'A Mott insulating phase (heavy line, green online) is found at density [MATH] and interaction strength [MATH].', '1412.5031-1-25-3': 'A non-superconducting, pseudo gapped phase (diamonds and darker shading, blue online) separates the Mott insulator from a superconducting phase of [MATH] symmetry (circles, intermediate shading, pink online).', '1412.5031-1-25-4': 'At larger doping or weaker correlation strength a non-superconducting (at the temperatures accessible to us) Fermi liquid phase is found (squares, lightest shading, yellow online).', '1412.5031-1-25-5': 'The onset of the normal state pseudogap is indicated by a light dotted line (purple online) running through the superconducting phase.', '1412.5031-1-25-6': 'In the pseudogapped regime the electronic spectrum is gapped for momenta near the zone face but is gapless for momenta near the zone diagonal.', '1412.5031-1-25-7': '[CITATION]', '1412.5031-1-26-0': 'We study the electronic properties as a function of doping at [MATH] and as a function of interaction strength at carrier concentration [MATH].', '1412.5031-1-26-1': '[MATH] is the largest interaction strength for which high precision data could be obtained with the resources available to us for temperatures substantially below [MATH] and general dopings.', '1412.5031-1-26-2': 'Simulations at larger interaction strengths are severely hampered by the fermionic sign problem.', '1412.5031-1-26-3': 'As can be seen from the phase diagram, this interaction strength is such that the model is conducting (although pseudo gapped) at [MATH].', '1412.5031-1-26-4': 'Thus, we believe that [MATH] is slightly lower than the [MATH] which is relevant to the real materials, so the quantitative values for example of the carrier concentrations at which different behaviors occur will be somewhat lower than is realistic.', '1412.5031-1-26-5': 'The information available to us [CITATION] indicates that all of the qualitative features of the doping dependence are well reproduced by the [MATH] computations.', '1412.5031-1-26-6': 'As we will see, considerable insight can be obtained from examination of the [MATH]-dependence of the superconducting self energy in the special case [MATH].', '1412.5031-1-26-7': 'The actual model also has an antiferromagnetic state, which competes with the superconducting state but is not studied here.', '1412.5031-1-27-0': '# Analytical Continuation of Normal and Anomalous Self Energies', '1412.5031-1-28-0': '## Formalism', '1412.5031-1-29-0': 'The main objects of interest in this paper are the real-frequency normal (N) and anomalous (A) components of the Nambu matrix electron self energy [MATH] as well as the gap function [MATH] defined from Eq. [REF].', '1412.5031-1-29-1': 'These are functions of a complex frequency argument [MATH].', '1412.5031-1-29-2': 'Our results for [MATH] and thus [MATH] are obtained on the imaginary frequency (Matsubara) points [MATH].', '1412.5031-1-29-3': 'Theorems from complex variable theory guarantee that knowledge of the function on the Matsubara points fully determines the function at all [MATH], but direct inversion of Eq. [REF] to obtain [MATH] from measurements of [MATH] is a mathematically ill-posed problem.', '1412.5031-1-29-4': 'In this paper we invert Eq. [REF] using the maximum entropy method [CITATION] and verify the results with the Pade method.[', '1412.5031-1-30-0': 'To continue the diagonal component [MATH] we follow the methods employed for self energy continuation in Ref. Wang09a.', '1412.5031-1-30-1': 'While there are no rigorous methods to control errors in this procedure, our experience is that given sufficiently accurate input data (relative statistical errors smaller than [MATH] on each Matsubara point) this procedure produces reasonably reliable results for the lowest frequency features in the real-axis spectral function at low temperature and qualitatively reasonable results (estimates of characteristic energy scales and integrated areas) for the higher frequency features.', '1412.5031-1-30-2': 'The relative statistical errors in our calculations are typically smaller than [MATH] so we have reasonable confidence in the qualitative features of the continuations.', '1412.5031-1-31-0': 'A variation of the procedure outlined in Wang09a is needed to obtain the off-diagonal component of the self energy [MATH] because [MATH] so that the spectral function [MATH] is an odd function of frequency and is thus not non-negative.', '1412.5031-1-31-1': 'We rewrite Eq. [REF] as [EQUATION] and continue [MATH] by standard methods.', '1412.5031-1-31-2': '[CITATION] The normalization of [MATH] is fixed from [MATH]: We obtain [MATH] by fitting [MATH] at the three lowest positive Matsubara frequencies to a parabola.', '1412.5031-1-31-3': 'A similar procedure is used to continue [MATH].', '1412.5031-1-32-0': '## Non-negativity of [MATH]', '1412.5031-1-33-0': 'While the usual maximum entropy analytic continuation formalism requires a non-negative spectral function, there is no guarantee that [MATH] is of definite sign.', '1412.5031-1-33-1': 'For example, in Migdal-Eliashberg theory the Coulomb pseudopotential leads to a negative contribution to [MATH] at frequencies of the order of the plasma frequency,[CITATION] while a recent solution of the Eliashberg equations for a model involving two competing spin fluctuations also displayed a sign change in the gap function as frequency was increased above the lower of the two characteristic frequencies.', '1412.5031-1-33-2': '[CITATION]', '1412.5031-1-34-0': 'For the two dimensional Hubbard model, our data are consistent with a positive definite [MATH] but we do not have a rigorous proof that this is always the case.', '1412.5031-1-34-1': 'Evidence for a positive definite [MATH] may be found from consideration of the particle-hole symmetric ([MATH]) situation.', '1412.5031-1-34-2': "In this case in the [MATH] sector the normal components of the hybridization function and self energy are also particle-hole symmetric and are odd functions of Matsubara frequency, so that [MATH] at all frequencies and there is a frequency-independent basis choice (the 'Majorana combination' [MATH]) that diagonalizes the self energy matrix in the [MATH] sector.", '1412.5031-1-34-3': "The corresponding 'Majorana' self energy [MATH] obeys the Kramers-Kronig relation [EQUATION]", '1412.5031-1-34-4': 'Thus analytical continuation of the Majorana combination of self energies gives direct access to [MATH] from which the normal and anomalous components of the self energy can easily be reconstructed from Eqs. [REF],[REF] as [EQUATION]', '1412.5031-1-34-5': 'Fig. [REF] presents a comparison of [MATH] obtained by direction continuation of the normal and anomalous parts of the Matsubara Green function and reconstructed using Eqs. [REF] and [REF] from continuations obtained from the [MATH] component of [MATH] in the Majorana basis (continuation of the [MATH] component yields values differing by [MATH]).', '1412.5031-1-34-6': 'A reconstruction of [MATH] from the continued [MATH] and [MATH] using Eq. [REF] is also shown.', '1412.5031-1-35-0': 'We see that the methods yield very similar results with some differences in the height and width of the low frequency peak (the integrated peak areas, not shown, are the same).', '1412.5031-1-35-1': 'The differences between the curves are an indication of the continuation errors.', '1412.5031-1-35-2': 'In the [MATH] obtained from [MATH] a relatively small amplitude oscillation is present, which leads to a small negative value in some regions where the directly continued [MATH] along with an overshoot (relative to the directly continued [MATH]) at slightly higher frequencies.', '1412.5031-1-35-3': 'Our experience is that these oscillations do not vary systematically with input data or details of the maximum entropy procedure, and we believe they are artifacts of the continuation procedure related to the requirement that norm of the function is conserved in the entropy minimization.', '1412.5031-1-35-4': 'We thus believe that [MATH] is generically non-negative for the [MATH] superconducting state of the Hubbard model.', '1412.5031-1-35-5': 'This conclusion further supported by continuations (not shown) that we have performed using the Pade method at both [MATH] and [MATH].', '1412.5031-1-35-6': 'The Pade method makes no assumptions as to the sign of the spectral function, and in the cases we have studied leads always to a non-negative [MATH].', '1412.5031-1-35-7': 'A non-negative [MATH] was also found by Civelli09b in [MATH] calculations using an exact diagonalization solver.', '1412.5031-1-36-0': '## Pole structure and continuation of gap function', '1412.5031-1-37-0': 'Fig. [REF] reveals that in certain parameter regimes the normal and anomalous self energies exhibit strong peaks at relatively low frequencies.', '1412.5031-1-37-1': 'As will be discussed at length below, these peaks do not correspond to physical excitations of the system; rather they are the expression in the superconducting state of the physics of the normal-state pseudogap, which in the DCA approximation is associated with the formation of a low frequency pole in the self energy of the [MATH] momentum sector.', '1412.5031-1-37-2': '[CITATION] It is useful to discuss the pole structure in terms of the representation in Eq. [REF].', '1412.5031-1-37-3': 'All of our data are consistent with the statement that [MATH] for the [MATH] sector is the sum of a pole and a regular part: [EQUATION] with [MATH] a smooth function of [MATH].', '1412.5031-1-37-4': 'The pole structure was also noted in a very recent paper by Sakai and collaborators.', '1412.5031-1-37-5': '[CITATION]', '1412.5031-1-38-0': 'Retaining for the moment only the pole term and explicitly evaluating Eq. [REF] and [REF] using the Nambu angle [MATH] corresponding to [MATH] gives [EQUATION]', '1412.5031-1-38-1': 'Thus in general we expect the normal and anomalous components of the self energy to have poles at exactly the same frequencies.', '1412.5031-1-38-2': 'In the particle-hole symmetric case, where [MATH], the poles in the normal and anomalous parts of the self energy have exactly the same amplitude.', '1412.5031-1-38-3': 'This is demonstrated in Fig. [REF], which shows the imaginary parts of the continuations of the normal and anomalous components of the self energy, along with the continuations of [MATH] obtained from continuation of the [MATH] component of the Majorana-basis representation of [MATH] and of [MATH] obtained from continuation of the [MATH] component of the Majorana-basis representation of [MATH].', '1412.5031-1-38-4': 'That [MATH] for [MATH] in the vicinity of the pole is strong evidence of the cancellation.', '1412.5031-1-38-5': 'In the general case both normal and anomalous components of the self energy have poles at [MATH] but because [MATH] the average of the strengths of the poles in [MATH] will in general be greater than the strengths of the poles in [MATH].', '1412.5031-1-38-6': 'However, as will be seen in our detailed examination of the self energy below, the background contributions are such that unambiguously identifying the pole strengths is not possible.', '1412.5031-1-39-0': 'It also follows from Eqs. [REF], [REF] and [REF] that the pole contribution to the gap function [MATH] is [EQUATION] so that [MATH] does not have poles at [MATH].', '1412.5031-1-39-1': 'All of our continuations are consistent with this result.', '1412.5031-1-40-0': 'The structure defined by Eq. [REF] creates challenges for analytical continuation.', '1412.5031-1-40-1': 'Intrinsic errors in the analytical continuation process mean that independent continuations the different component of [MATH] may lead to slightly different estimates of the pole positions, amplitudes and widths.', '1412.5031-1-40-2': 'This can be seen for example by comparison of the upper and lower panels of Fig. [REF] or by comparing [MATH] and [MATH] in Fig. [REF].', '1412.5031-1-40-3': 'The difficulties are exacerbated if particle-hole symmetry is broken because the continuation process may not place the poles in [MATH] at exactly opposite frequencies.', '1412.5031-1-40-4': 'We do not know how to control the continuation process so as to force the precise alignment of the poles in the different components of [MATH].', '1412.5031-1-40-5': 'For studies of the particle-hole symmetric situation, we work with [MATH] defined from continuation of the [MATH] component of [MATH] in the Majorana basis.', '1412.5031-1-40-6': 'In the doped case, we focus on the gap function.', '1412.5031-1-41-0': '# The normal and anomalous self energies, [MATH] sector', '1412.5031-1-42-0': 'It has been accepted for many years that the two dimensional Hubbard model exhibits a Mott insulating phase at carrier concentration [MATH] and large interaction, and a Fermi liquid phase at small interaction or carrier concentration sufficiently different from [MATH].', '1412.5031-1-42-1': 'Work [CITATION] has established that if superconductivity is neglected then the Mott and Fermi liquid phases are separated by a pseudogap regime, in which regions of momentum space near the zone face are gapped while regions of momentum space near the zone diagonal are not.', '1412.5031-1-42-2': 'Mathematically, within DCA the gapping is a consequence of the appearance of a pole in the electron self energy pertaining to the [MATH] sector.[', '1412.5031-1-42-3': '[CITATION] If the Mott insulator is approached by varying interaction strength in the particle-hole symmetric case ([MATH], [MATH]) the pole occurs at [MATH];[CITATION] in a non-particle-hole symmetric situation (for example if the Mott insulator is approached by varying doping) the pole appears at a frequency close to, but slightly different, from zero.[', '1412.5031-1-42-4': '[CITATION] In this section we show how the onset of superconductivity affects this pole structure.', '1412.5031-1-43-0': 'Fig. [REF] presents the imaginary parts of the analytically continued self energies and the gap function for the three [MATH]-values at [MATH] presented in Fig. [REF].', '1412.5031-1-43-1': 'The upper panel shows results obtained in the normal state at temperature [MATH] (at [MATH] is slightly greater than [MATH] but superconductivity has been suppressed here for clarity of presentation).', '1412.5031-1-43-2': 'At [MATH] the imaginary part has the Fermi liquid form, with [MATH] exhibiting an approximately quadratic minimum at zero.', '1412.5031-1-43-3': 'As [MATH] is increased a thermally broadened pole appears at [MATH].', '1412.5031-1-43-4': 'The pole increases rapidly in strength as [MATH] is increased.', '1412.5031-1-43-5': 'These results are consistent with our previous analysis of the pseudogap.', '1412.5031-1-43-6': '[CITATION] The middle two panels present the imaginary parts of the normal and anomalous components of the self energy at a temperature [MATH].', '1412.5031-1-43-7': 'One sees that the pole centered at [MATH] has split into two and appears with comparable strength in both the normal and anomalous components.', '1412.5031-1-43-8': 'The pole frequency weakly decreases as [MATH] is increased.', '1412.5031-1-43-9': 'The larger width seen in the [MATH] calculation is likely to be an artifact of the analytical continuation.', '1412.5031-1-43-10': 'The lowest panel shows the imaginary part of the gap function.', '1412.5031-1-43-11': 'We see that the gap function is much smaller in magnitude than the self energy, that the structure in the gap function occurs at a higher frequency than the structure in the self energy (as also noted by Sakai et al [CITATION]), and that the gap function varies less dramatically with interaction strength than does the self energy.', '1412.5031-1-44-0': 'Fig. [REF] shows the analogous plots as a function of doping at [MATH].', '1412.5031-1-44-1': 'The upper panel (normal state) shows again a pole that appears and grows in strength as doping is decreased.', '1412.5031-1-44-2': 'The particle-hole symmetry breaking provided by the doping means that the pole is slightly displaced from the origin.', '1412.5031-1-44-3': 'In the superconducting state (middle panels) the pole splits; in [MATH] the pole strength is different for the positive frequency than for the negative frequency pole.', '1412.5031-1-44-4': 'The strength of the pole in [MATH] is of the order of that in [MATH].', '1412.5031-1-44-5': 'The detailed line shapes of the poles depend to on the details of the continuation process.', '1412.5031-1-44-6': 'The difference in line shapes and small differences in pole position can lead to unphysical structures in calculated spectra.', '1412.5031-1-44-7': 'As in the interaction-driven case, there is no structure in [MATH] at the pole frequencies of [MATH].', '1412.5031-1-45-0': 'Poles in the anomalous component of the self energy were reported by Civelli [CITATION] and the alignment of poles in [MATH] and [MATH] and the cancellation of the [MATH] poles in [MATH] were very recently discussed in the context of 4-site CDMFT calculations [CITATION].', '1412.5031-1-46-0': '# The Superconducting Gap and the pseudogap', '1412.5031-1-47-0': 'In this section we present results for the energy gap in the superconducting and non-superconducting states.', '1412.5031-1-47-1': 'The gap may be estimated from an analytical continuation of the normal component of the Green function, but the inevitable broadening associated with continuations means that it is not clear a priori how to estimate a gap.', '1412.5031-1-47-2': 'As discussed in Ref. [CITATION] for Mott insulators, an alternative approach provides a more accurate estimate.', '1412.5031-1-47-3': 'This approach is based on the argument that for frequencies less than the gap, all imaginary parts are zero except for the poles in [MATH], which do not correspond to physical excitations.', '1412.5031-1-47-4': 'Thus one may determine the gap energy [MATH] from the vanishing of the real part of the denominator of the Green function.', '1412.5031-1-47-5': 'For the normal state, this criterion involves consideration of [EQUATION]', '1412.5031-1-47-6': 'For a given [MATH] this equation will have two solutions, [MATH].', '1412.5031-1-47-7': 'The pseudogap is typically indirect (the [MATH] that maximizes [MATH] is different from the [MATH] that minimizes [MATH]).', '1412.5031-1-47-8': 'The direct gap is determined as [EQUATION]', '1412.5031-1-47-9': 'The points of minimum direct gap are found to be the renormalized Fermi surface points [MATH] satisfying [MATH] and that at these points [MATH].', '1412.5031-1-48-0': 'In the superconducting state, these ideas lead to the consideration of [MATH].', '1412.5031-1-48-1': 'Rearranging the expression for [MATH] gives [EQUATION] with [MATH] defined in Eq. [REF] and [EQUATION]', '1412.5031-1-48-2': 'We then define the gap frequency [MATH] in the superconducting state as the frequency at which the real part of [MATH] vanishes, i.e. as [EQUATION]', '1412.5031-1-48-3': 'We define the Fermi surface as the locus of [MATH]-points for which [MATH].', '1412.5031-1-48-4': 'We will find that the the [MATH] dependence of [MATH] is modest so that we may identify the gap [MATH] in the superconducting state as the value of [MATH] which solves [MATH].', '1412.5031-1-49-0': 'In the particle-hole symmetric case where [MATH] we may alternatively write the gap equation as the solution of [EQUATION]', '1412.5031-1-49-1': 'Fig. [REF] demonstrates this procedure.', '1412.5031-1-49-2': 'The upper panel shows results obtained in the superconducting state at [MATH] (the normal state is not shown because at this [MATH] there is no normal state pseudogap).', '1412.5031-1-49-3': 'The figure plots the spectral function (imaginary part of continued Green function), the real part of the Majorana combination of the self energy and the real part of the gap function as a function of frequency.', '1412.5031-1-49-4': 'We see by comparing the continued spectral function to the self energy and gap curves that the criteria [MATH] or [MATH] identifies a point close to that at which the spectral function is maximal.', '1412.5031-1-49-5': 'We believe that the appearance of weight at lower frequencies in the spectral function comes from artificial broadening induced by the continuation process.', '1412.5031-1-50-0': 'The middle panel shows the same analysis in the normal state at [MATH].', '1412.5031-1-50-1': 'The normal state pseudogap is evident, and again we see that the quasiparticle equation picks out as the gap the point at which the spectral function is maximal.', '1412.5031-1-50-2': 'The lower panel shows the superconducting state, also at [MATH].', '1412.5031-1-50-3': 'We also see that at this [MATH] value the line [MATH] is tangent to (in fact very slightly below) the [MATH] curve at the point that one would naturally identify as the gap.', '1412.5031-1-50-4': 'That [MATH] intersects [MATH] while [MATH] is peaked at the point of near tangency suggests that the absence of an exact intersection is a continuation artifact and that if all quantities were exactly and consistently continued the line [MATH] would intersect the [MATH] line.', '1412.5031-1-51-0': 'Comparison of the middle and lower panels of Fig. [REF] shows that the superconducting gap is unambiguously smaller than the normal state pseudogap, consistent with results presented in Ref. [CITATION].', '1412.5031-1-51-1': 'Also, comparison of the lower panel of Fig. [REF] to Fig. [REF] shows that for this value of [MATH] the pole in the self energy lies inside the excitation gap of the superconductor, further confirming that the self energy pole does not represent a physical excitation of the system.', '1412.5031-1-52-0': 'The two panels of Fig. [REF] show the dependence of the low-T ([MATH]) gap on interaction strength (at [MATH], computed from Eq. [REF]) and carrier concentration (at [MATH], computed from Eq. [REF]).', '1412.5031-1-52-1': 'In the weak interaction or large doping limits, there is no superconductivity.', '1412.5031-1-52-2': 'As the doping decreases or interaction strength increases the gap increases smoothly down to the low doping/high interaction end of the phase diagram.', '1412.5031-1-52-3': 'Also shown on the plots is the normal state pseudogap, computed at the same low temperature [MATH] by suppressing superconductivity in the calculation.', '1412.5031-1-52-4': 'One sees that in the low doping/strong interaction regime, turning on superconductivity leads to a decrease in the energy of the lowest-lying excitation.', '1412.5031-1-53-0': 'Finally, Fig. [REF] shows the temperature dependence of the gap function at the three [MATH] values considered above.', '1412.5031-1-53-1': 'We see that in the moderate coupling regime, there is no normal state gap and the superconducting gap increases from zero as the temperature is decreased through the transition temperature.', '1412.5031-1-53-2': 'At intermediate coupling a small but non-zero gap is already present in the normal state and the gap increases with the onset of superconductivity, while at stronger coupling the gap actually decreases as the temperature is decreased into the superconducting state.', '1412.5031-1-54-0': '# Fermi surface spectra', '1412.5031-1-55-0': 'In this section we present spectra computed at the Fermi surface using continued self energies.', '1412.5031-1-55-1': 'The issues discussed above relating to the difficulties of analytical continuation in non-particle-hole symmetric case mean that we limit ourselves here to study of the interaction-driven transition at density [MATH] but all of the information available to us suggests that the behavior in the doped case shares the essential features found at half-filling.', '1412.5031-1-56-0': 'Fig. [REF] shows the temperature evolution of the electron spectral function computed for a momentum on the Fermi surface at the antinode ([MATH]) at carrier density [MATH] and different interaction strengths.', '1412.5031-1-56-1': 'The upper panel shows results obtained for a moderate interaction [MATH].', '1412.5031-1-56-2': 'We see that the normal state spectral function is peaked at the chemical potential [MATH].', '1412.5031-1-56-3': 'The onset of the superconductivity ([MATH] is between [MATH] and [MATH]) induces a suppression of the low frequency density of states.', '1412.5031-1-56-4': 'The gap grows and sharpens as temperature is decreased.', '1412.5031-1-56-5': 'For the lower temperatures one sees that the spectral function has a sharp quasiparticle peak, a weak minimum at slightly higher frequencies, followed by a weak maximum at yet higher frequencies.', '1412.5031-1-56-6': 'This structure in the spectral function is frequently observed in photoemission [CITATION] and scanning tunneling microscopy [CITATION] experiments on copper-oxide high-[MATH] materials and is referred to as a "peak-dip-hump" feature.', '1412.5031-1-56-7': 'The higher frequency "hump" is typically interpreted as arising from the interaction of electrons with some kind of bosonic excitation, with the hump frequency determined by the boson energy and the minimum energy to create an electron-hole pair.', '1412.5031-1-57-0': 'The middle panel of Fig. [REF] shows shows results obtained for an intermediate interaction [MATH].', '1412.5031-1-57-1': "We see that a weak minimum is evident in the density of states even temperature [MATH]; this weak suppression of the density of states marks the onset of the 'pseudogap'.", '1412.5031-1-57-2': 'As the temperature is decreased below the transition temperature the low frequency intensity drops very rapidly.', '1412.5031-1-57-3': 'The gap (peak in the spectral function) increase and saturates at a value rather greater than that found in the moderate interaction case.', '1412.5031-1-57-4': 'The peak-dip-hump structure is more evident.', '1412.5031-1-57-5': 'In this case the "hump" energy increases as T decreases except for the lowest temperature.', '1412.5031-1-58-0': 'The lower panel of Fig. [REF] shows results obtained for a strong interaction ([MATH]).', '1412.5031-1-58-1': 'In this case the normal state pseudogap is well established even at [MATH].', '1412.5031-1-58-2': 'The transition to superconductivity is associated with the formation of a quasiparticle peak which lies inside the pseudogap and with a shift outwards of the second peak structure.', '1412.5031-1-58-3': 'For both [MATH] and [MATH] the energy of the second peak increases as temperature decreases, with the exception of [MATH], [MATH].', '1412.5031-1-58-4': 'We believe that at this particular [MATH] and [MATH] the continuations are of lower quality than at other parameter values.', '1412.5031-1-58-5': 'For example the errors associated with back continuation are slightly larger.', '1412.5031-1-59-0': 'The question of the origin of the "peak-dip-hump" structure has been discussed extensively in the literature on angle-resolved photoemission in the cuprates.', '1412.5031-1-59-1': '[CITATION] The most widely accepted explanation is that the higher energy "hump" is evidence for a "shakeoff" process in which an electron emits a bosonic excitation such as a spin fluctuation [CITATION] or a phonon,[CITATION] with the hump energy determined as the sum of the energy of the boson and the superconducting gap energy, although structure in the bare dispersion associated with interbilayer hopping may play a role in certain materials.', '1412.5031-1-59-2': '[CITATION] Such a shakeoff process would appear in the imaginary part of the self energy as an upward step, corresponding to the opening of a scattering channel.', '1412.5031-1-59-3': 'However, a quantitative and generally accepted identification of the boson responsible for the hump energy is lacking.', '1412.5031-1-60-0': 'Mathematically, structures in the spectral function may be understood from the fundamental expression [EQUATION] where for clarity we have not explicitly written the change of basis matrices [MATH] (Eq. [REF]).', '1412.5031-1-60-1': 'From Eq. [REF] we see that structure can arise from resonance ([MATH]) or, off resonance, from an increase in [MATH].', '1412.5031-1-60-2': 'The former case produces the quasiparticle peak of Fermi liquid theory; the latter is the origin of the "shakeoff" explanation of the peak-dip-hump structure.', '1412.5031-1-60-3': '[CITATION] Fig. [REF] investigates the origin of the "hump" structure in the present calculation by comparing the computed spectral function (lower panel, evaluated at the Fermi surface [MATH]) to the electron self energy (heavy lines, upper panel) in the Majorana basis that diagonalizes the Nambu Greens function at all [MATH] for [MATH].', '1412.5031-1-60-4': 'The real part is presented in the combination [MATH].', '1412.5031-1-60-5': 'We see that the location of the "hump" feature in fact does not correspond to any significant feature in the imaginary part of the self energy (top panel), so that it cannot be interpreted as an onset of scattering.', '1412.5031-1-60-6': 'Instead, inspection of [MATH] energy (middle panel) shows that for the two stronger couplings the frequency of the hump corresponds to the frequency at which the real part of the inverse Greens function vanishes, while for the weaker coupling the hump frequency corresponds to a point where the real part is minimized.', '1412.5031-1-60-7': 'In other words, the "hump" is a resonance phenomenon, not a scattering phenomenon, and its frequency does not correspond directly to the energy of any excitation of the system.', '1412.5031-1-61-0': '# Momentum-dependent Spectra', '1412.5031-1-62-0': 'In this section we present and discuss the momentum dependence of spectral functions calculated from the normal and anomalous components of the self energy.', '1412.5031-1-62-1': 'The left panel of Fig. [REF] shows a sequence of energy distribution curves (EDC) calculated in the normal state for a sequence of momenta cutting across the Fermi surface in the [MATH] sector for the moderate interaction strength [MATH].', '1412.5031-1-62-2': 'A quasiparticle peak is visible, which disperses through the Fermi surface.', '1412.5031-1-62-3': 'The right panel shows EDC at the same momenta, this time in the superconducting state.', '1412.5031-1-62-4': 'Comparison of the two panels reveals the behavior expected of a moderate-coupling BCS-like superconductor, with the gap having the greatest effect on the Fermi surface trace and the particle-hole symmetry breaking producing a peak dispersing away from zero as momentum is increased above the Fermi level.', '1412.5031-1-63-0': 'Fig. [REF] shows the EDC curves obtained for a strong interaction [MATH].', '1412.5031-1-63-1': 'The weakly dispersing and rather broadened normal state pseudogap is seen in the left panel.', '1412.5031-1-63-2': 'The right panel shows that the onset of superconductivity produces very weakly dispersing peak at an energy well inside of the pseudogap.', '1412.5031-1-63-3': 'This essentially non-dispersing zone-edge feature is characteristically observed in photoemission experiments on high [MATH] copper oxide superconductors.', '1412.5031-1-63-4': '[CITATION]', '1412.5031-1-64-0': '# Superconducting effects on zone diagonal spectra', '1412.5031-1-65-0': 'In the cuprates and in the Hubbard model the dramatic effects of superconductivity are visible in the electronic states near the zone face ([MATH]) point, because it is at this point that the superconducting gap and the pseudogap are maximal.', '1412.5031-1-65-1': 'But in addition to opening, or changing the value of, a gap, the onset of superconductivity may affect other aspects of the physics, for example by changing the density of final states that enter into scattering processes [CITATION] or, more profoundly, by changing the electronic state itself and thus the nature of the scattering mechanisms.', '1412.5031-1-65-2': 'Some understanding of these effects may be gleaned from consideration of electronic states with momentum along the zone diagonal.', '1412.5031-1-65-3': 'The superconducting gap vanishes for these momenta and thus any effects on electron propagation must arise from changes in scattering and electronic state.', '1412.5031-1-66-0': 'We find that at all dopings and interaction strengths we have studied the electronic self energy in the zone diagonal momentum sector has approximately the Fermi liquid form.', '1412.5031-1-66-1': 'In particular the self energy does not have a low frequency pole.', '1412.5031-1-66-2': 'Its imaginary part is minimal at zero frequency and the value at zero frequency, [MATH], decreases as temperature [MATH].', '1412.5031-1-66-3': 'The real part is linear in frequency at low frequency, defining a zone diagonal mass enhancement [MATH].', '1412.5031-1-66-4': 'We extract estimates of the Fermi level scattering rate [MATH] and mass enhancement [MATH] by fitting the lowest four Matsubara frequencies to the cubic form [EQUATION]', '1412.5031-1-66-5': 'Representative results are shown in Fig. [REF].', '1412.5031-1-66-6': 'We see that the mass enhancement has a modest doping dependence, and decreases markedly as temperature is decreased.', '1412.5031-1-66-7': 'Superconductivity has only a small effect on the mass enhancement.', '1412.5031-1-66-8': 'At low doping the onset of superconductivity leads to a very small decrease in the mass enhancement; at higher doping the effect is of opposite sign and is slightly larger.', '1412.5031-1-66-9': 'It is interesting to note that the change in sign of the superconducting contribution to the mass enhancement occurs at a lower doping than the change from potential energy-driven to kinetic energy driven pairing discussed in Ref. [CITATION].', '1412.5031-1-66-10': 'Thus the change in electronic state associated with the onset of superconductivity has a weaker effect on the nodal quasiparticles than it does on other properties.', '1412.5031-1-67-0': 'The normal state scattering rate also has a dramatic doping dependence, especially at the higher temperature, and at all dopings exhibits a marked temperature dependence.', '1412.5031-1-67-1': 'The effect of superconductivity on the scattering rate is much more noticeable than the effect on the mass enhancement: the onset of superconductivity leads to an almost factor of two drop in the Fermi surface scattering rate, except at the very lowest doping which is right on the boundary of superconducting phase.', '1412.5031-1-67-2': 'These finding are consistent with angle-resolved photoemission measurements on Bi[MATH]Sr[MATH]CaCu[MATH]O[MATH].[', '1412.5031-1-67-3': '[CITATION] Fig. 4c of Ref. Valla06 reveals that the onset of superconductivity leads to an approximately factor of two decrease in the momentum distribution width (a good proxy for scattering rate) below the extrapolation of the normal state rate to low temperature, while Fig. 3 reveals a much smaller change in the mass enhancement (albeit of opposite sign to that predicted here).', '1412.5031-1-67-4': 'The more coherent nature of the zone diagonal quasiparticles is also qualitatively consistent with conclusions drawn from pump-probe experiments,[CITATION] but because our calculations are restricted to equilibrium a direct comparison cannot be made.', '1412.5031-1-68-0': '# Conclusions', '1412.5031-1-69-0': "Two characteristic features of the high transition temperature copper-oxide superconductors are superconductivity with [MATH] symmetry and a 'pseudogap'[CITATION], a suppression of electronic density of states for momenta near the Brillouin zone face ([MATH] point).", '1412.5031-1-69-1': 'The interplay between these two phenomena has been the focus of considerable attention in the literature.', '1412.5031-1-69-2': 'In this paper we present cluster dynamical mean field calculations of the electronic self energy and spectral function of the normal and superconducting phase of the two dimensional Hubbard model that shed new light on the subject.', '1412.5031-1-70-0': 'The correspondence of the results to the essential features of superconductivity in the cuprates is striking.', '1412.5031-1-70-1': 'We find, consistent with a large body of experimental literature, that when superconductivity emerges from the pseudogap regime the onset of superconductivity is associated with the appearance of very weakly dispersing states inside the pseudogap and with the appearance of a peak-dip-hump structure in the spectral function.', '1412.5031-1-70-2': 'Superconductivity affects the zone-diagonal states via a significant ([MATH] factor of 2) decrease in the scattering rate.', '1412.5031-1-70-3': 'We have also determined the superconducting and pseudogaps accurately and have shown that the superconducting gap systematically increases as doping is decreased or interaction strength is increased, until it abruptly drops to zero at the low-doping/high interaction boundary of the superconducting phase.', '1412.5031-1-70-4': 'These results support the notion that the pseudogap and superconductivity are competing phenomena and strengthen the case made in prior papers [CITATION] that as P. W. Anderson predicted in 1987 [CITATION] the Hubbard model contains the essential physics of the high-[MATH] observed in layered copper-oxide materials.', '1412.5031-1-71-0': 'Mathematically, we find (in agreement with previous work [CITATION]) that the pseudogap is a Mott-transition-like phenomenon associated with the appearance of a low frequency pole in the self energy and that the interplay between superconductivity and the pseudogap is controlled by the superconductivity-induced changes in the pole structure.', '1412.5031-1-71-1': "In particular, in our calculation the 'hump' feature in the spectral function arises mathematically from a zero-crossing in the real part of the self energy and not from the onset of a scattering process, in other words, not from a bosonic excitation at all.", '1412.5031-1-71-2': 'On the technical side we have clarified the mathematical structure of the pseudogap-induced poles associated with the normal and anomalous self energies, shown how the inevitable analytical continuation errors make it difficult to construct real-frequency spectra in situations where the pseudogap poles are important.', '1412.5031-1-71-3': 'The pole structure has also been discussed by Sakai14 who present an interesting alternative interpretation.', '1412.5031-1-72-0': 'It is important to consider the limitations of the methods used here.', '1412.5031-1-72-1': 'The essential technical step is the use of continuous-time auxiliary field methods [CITATION] and submatrix updates.', '1412.5031-1-72-2': '[CITATION] These methods enable highly accurate simulations at temperatures as low (in physical units) as [MATH], far below the basic scales of the model and, crucially, well below the superconducting transition temperature.', '1412.5031-1-72-3': 'However, even with these improvements, obtaining results of the needed precision at the required low temperatures requires approximations.', '1412.5031-1-73-0': 'The key approximation used in this paper is the cluster dynamical mean field approximation.', '1412.5031-1-73-1': "[CITATION] In the 'DCA' form used here [CITATION] this amounts to approximating the normal and anomalous components of the electron self energy as piecewise constant functions of momentum, taking different values in each of [MATH] momentum sectors that tile the Brillouin zone.", '1412.5031-1-73-2': 'We have adopted the [MATH] approximation.', '1412.5031-1-73-3': 'This is the smallest momentum decomposition that permits a clear separation between the zone face and zone diagonal regions of momentum space.', '1412.5031-1-73-4': 'Previous work [CITATION] indicates that although we do not have quantitative convergence to the [MATH] limit, the [MATH] approximation correctly captures the physics of the normal state pseudogap.', '1412.5031-1-74-0': 'A second approximation is the use of an interaction [MATH] which is likely to be slightly weaker than needed to quantitatively capture the physics of the cuprates.', '1412.5031-1-74-1': 'This approximation is needed because computation time increases rapidly as [MATH] increases (as [MATH], with additional complications from the sign problem) and we needed to undertake a broad survey of parameter space.', '1412.5031-1-74-2': 'For similar computational reasons we restricted attention to the particle-hole symmetric version of the model (second neighbor hopping [MATH] and further neighbor hoppings set to zero).', '1412.5031-1-75-0': 'Finally, because the basic computations use imaginary time methods, obtaining spectra of interest requires an analytical continuation method.', '1412.5031-1-75-1': 'Analytical continuation is an ill-posed problem.', '1412.5031-1-75-2': 'We used maximum entropy analytical continuation methods (which are widely employed but essentially uncontrolled) to extract real frequency information.', '1412.5031-1-75-3': 'This requires extremely high quality Monte Carlo data, further constraining the parameter ranges that could be examined.', '1412.5031-1-75-4': 'In this context the properties of the anomalous self energy are of particular importance.', '1412.5031-1-75-5': 'With the exception of the Pade method, all methods known to us require a positive definite spectral function.', '1412.5031-1-75-6': 'While we have presented evidence that in at least some cases the spectral function associated with the anomalous self energy has an appropriate positivity property, the possibility of a sign change at high frequency cannot be ruled out.', '1412.5031-1-75-7': 'At high frequencies the anomalous part is very small and the intrinsic limitations of the continuation process mean that small systematic errors could induce (or mask) a sign change.', '1412.5031-1-76-0': 'A key physical limitation of our work is that we have not considered other ordered states (for example Neel antiferromagnetic or striped order) that might preempt the phases considered here.', '1412.5031-1-76-1': 'The dynamical mean field method captures (within the rather coarse momentum resolution of the cluster dynamical mean field method) fluctuations associated with these states, but because we have symmetrized over spin degrees of freedom, long range ordered antiferromagnetic states are excluded.', '1412.5031-1-76-2': 'Also our calculation lacks the momentum resolution needed to provide a clear account of striped states.', '1412.5031-1-76-3': 'Thus we view the results as providing a reasonable qualitative account of the properties of the superconducting phase and pseudogap regime of the two dimensional Hubbard model, but not as a quantitatively accurate account of the properties of the Hubbard model.', '1412.5031-1-77-0': 'For these reasons, extensions of the work presented here would be desirable.', '1412.5031-1-77-1': 'Pushing the calculation to somewhat larger [MATH] so that the [MATH] endpoint is well within the Mott insulating phase should become feasible as computer power improves.', '1412.5031-1-77-2': 'Study of larger [MATH] would provide more insight into the interplay of superconductivity and Mott physics.', '1412.5031-1-77-3': 'Extending the calculations to [MATH] site approximation would similarly be useful.', '1412.5031-1-77-4': 'In particular, examination of differences between gap values and spectra calculated with [MATH] and [MATH] will provide insight into the quantitative aspects of the results.', '1412.5031-1-77-5': 'If feasible, real-time calculations and comprehensive Pade continuations, that could investigate the possibility of sign changes in the anomalous component of the self energy would be reassuring.', '1412.5031-1-77-6': 'On the conceptual side, a deeper understanding of the pseudogap state and of the meaning of the self energy poles would be desirable.', '1412.5031-1-77-7': 'Most importantly, investigation of competing states such as Neel antiferromagnet and stripe orders is needed.'}

{'1412.5031-2-0-0': 'Cluster dynamical mean field methods are used to calculate the normal and anomalous components of the electron self energy of the two dimensional Hubbard model.', '1412.5031-2-0-1': 'Issues associated with the analytical continuation of the normal and anomalous parts of the gap function are discussed.', '1412.5031-2-0-2': 'Methods of minimizing the uncertainties associated with the pseudogap-related pole in the self energy are discussed.', '1412.5031-2-0-3': 'From these the evolution of the superconducting gap and the momentum dependent electron spectral function across the phase diagram are determined.', '1412.5031-2-0-4': "In the pseudogap regime, decreasing the temperature into the superconducting state leads to a decrease in the energy gap and the formation of a 'peak-dip-hump' structure in the electronic density of states.", '1412.5031-2-0-5': 'The peak feature disperses very weakly.', '1412.5031-2-0-6': 'The calculated spectral functions are in good qualitative agreement with published data.', '1412.5031-2-0-7': 'The mathematical origin of the behavior is found to be the effect of the superconductivity on the pole structure giving rise to the normal state pseudogap.', '1412.5031-2-0-8': 'In particular the "hump" feature is found to arise from a zero crossing of the real part of the electron self energy rather than from an onset of scattering.', '1412.5031-2-0-9': 'The effect of superconductivity on the zone diagonal spectra is presented.', '1412.5031-2-1-0': '# Overview', '1412.5031-2-2-0': 'After more than a quarter century of research, our theoretical understanding of the unusual electronic properties of the high transition temperature superconductivity observed [CITATION] in layered copper oxide materials remains incomplete.', '1412.5031-2-2-1': "A plethora of remarkable dynamical phenomena have been reported, including a correlation-driven insulating phase occurring when the conduction band is half filled,[CITATION] non-Fermi-liquid transport properties,[CITATION] a 'pseudogap' in the electronic spectrum [CITATION] as well as ordered phases including 'stripe' states with spin and charge order,[CITATION] charge order apparently unconnected with spin order,[CITATION] 'nematic' (rotational symmetry breaking) states,[CITATION] and time reversal symmetry breaking states also apparently unaccompanied by conventional spin order.[", '1412.5031-2-2-2': '[CITATION] Angle-resolved photoemission studies report unusual quasiparticle properties at essentially all carrier concentrations.[', '1412.5031-2-3-0': 'The diversity of reported phenomena has led to debate about what is the essential physics to include in a minimal theoretical model while the apparent strong coupling nature of the problem suggests that whatever model is adopted, a nonperturbative treatment is required.', '1412.5031-2-3-1': "Important open questions include the mechanism for superconductivity, the nature of the minimal low energy model describing the high-[MATH] phenomenon,[CITATION] and the physics of the 'pseudogap' and its interplay with the superconducting gap.", '1412.5031-2-3-2': '[CITATION]', '1412.5031-2-4-0': 'This paper presents a theoretical study of the electron excitation spectrum of the superconducting and normal states of the two dimensional Hubbard model.', '1412.5031-2-4-1': 'This model is the minimal model of the physics of strongly correlated electrons on a lattice.', '1412.5031-2-4-2': 'Although it is not yet known if this model contains the full panoply of phenomena observed in the high-[MATH] materials, it clearly contains some important aspects of the physics and is accepted as one of the candidate models [CITATION] for describing the low energy (energies of order [MATH] or less) physics of the copper-oxide superconductors.', '1412.5031-2-4-3': 'Determining the properties of this model to the level at which a clear comparison to experiment can be made is an important goal of theory.', '1412.5031-2-5-0': "We address this problem using the 'dynamical cluster approximation' (DCA) version [CITATION] of the cluster dynamical mean field method.[", '1412.5031-2-5-1': '[CITATION] The method is based on approximating the full spatial dependence of the electron self energy in terms of a finite number [MATH] of functions of frequency,[CITATION] with the exact properties recovered in the [MATH] limit.[', '1412.5031-2-5-2': '[CITATION] Within this approximation the method provides an unbiased (in the sense of not pre-selecting a particular interaction channel or class of diagrams) numerical approach to the correlated electron problem and allows comprehensive investigation of the frequency dependence, and some aspects of the momentum dependence, of the electronic properties.', '1412.5031-2-6-0': 'Our ability to solve the equations of dynamical mean field theory[CITATION] has reached the point where approximation sizes [MATH] that in many aspects are representative [CITATION] of the [MATH] limit can be studied at the low temperatures required to stabilize superconductivity.', '1412.5031-2-6-1': '[CITATION] The superconducting state has been constructed [CITATION] and physical properties including the superconducting condensation energy,[CITATION] the c-axis and Raman response [CITATION] and the structure of the gap function and pairing potential [CITATION] have been computed and found to be in remarkable agreement with experiment.', '1412.5031-2-6-2': 'Here we use the new methodologies to study the electronic excitation spectrum of the normal and superconducting states in detail.', '1412.5031-2-6-3': 'While some of these issues have been previously studied in a cluster size [MATH] approximation, [CITATION] our results, obtained on larger [MATH] clusters, provide a better representation of the physics including a clear separation of nodal and antinodal behavior.', '1412.5031-2-7-0': 'The DCA equations are solved in imaginary time, and an analytical continuation[CITATION] procedure is needed to obtain the real frequency information needed for the electronic excitation spectrum.', '1412.5031-2-7-1': 'Especially in the pseudogap regime of the model, the novel physics we find presents challenges for the analytical continuation of our computed results.', '1412.5031-2-7-2': 'These are discussed at length below.', '1412.5031-2-8-0': 'The rest of the paper is organized as follows.', '1412.5031-2-8-1': 'In Section [REF] we define the quantities of interest and present the specifics of the dynamical mean field method.', '1412.5031-2-8-2': 'Section [REF] discusses issues related to the analytic continuation.', '1412.5031-2-8-3': 'Section [REF] displays the normal and anomalous components of the self energy, drawing attention to an unusual pole structure related to sector-selective Mott nature of the pseudogap phase.', '1412.5031-2-8-4': '[CITATION] Section [REF] displays the gap function constructed from the ratio of anomalous and normal components of the self energy.', '1412.5031-2-8-5': 'Sections [REF] and [REF] display the photoemission and inverse photoemission spectra predicted by the model while section [REF] presents our findings on superconductivity-induced changes to the electron scattering and mass renormalization for states near the zone diagonal where the superconducting order parameter vanishes.', '1412.5031-2-8-6': 'Section [REF] presents a summary and conclusions.', '1412.5031-2-9-0': '# Formalism', '1412.5031-2-10-0': '## Model', '1412.5031-2-11-0': 'We study the two dimensional Hubbard model of electrons hopping on a square lattice and subject to a local interaction [MATH] which we take to be repulsive.', '1412.5031-2-11-1': 'The model may be written in a mixed momentum ([MATH])/position ([MATH]) representation as [EQUATION]', '1412.5031-2-11-2': 'In the first term we have represented the electronic degrees of freedom by the Nambu spinor defined in terms of [MATH], the Fourier transform to momentum space of the operator [MATH] which creates an electron of spin [MATH] on lattice site [MATH] as [EQUATION]', '1412.5031-2-11-3': 'We set the lattice constant to unity.', '1412.5031-2-11-4': 'The momentum index [MATH] runs over the Brillouin zone of the two dimensional square lattice [MATH].', '1412.5031-2-11-5': 'The trace is over the Nambu indices and [MATH] denote the Pauli matrices operating in Nambu space.', '1412.5031-2-11-6': 'The chemical potential is [MATH], [MATH] is the energy dispersion and [MATH] is the operator measuring the density of spin [MATH] electrons on site [MATH].', '1412.5031-2-11-7': 'In the computations presented below we take [MATH] and present our results in units of [MATH].', '1412.5031-2-11-8': 'A reasonable estimate of the energy scales pertaining to the physical copper oxide materials is [MATH].', '1412.5031-2-11-9': 'At carrier density [MATH] per site this version of the Hubbard model is particle-hole symmetric ([MATH]), but particle-hole symmetry is broken at carrier concentrations [MATH].', '1412.5031-2-12-0': '## Green function and self energy', '1412.5031-2-13-0': 'Our analysis proceeds from the components of the matrix Nambu Green function defined for imaginary time [MATH] (T is the temperature) as [EQUATION]', '1412.5031-2-13-1': 'It is useful to Fourier transform [MATH] to the Matsubara frequency axis [EQUATION] with [MATH].', '1412.5031-2-13-2': 'The self energy is defined as [EQUATION] and may be written explicitly in terms of normal (N) and anomalous (A) components as (note we have chosen the phase of the superconducting order parameter to be real) [EQUATION]', '1412.5031-2-13-3': 'Under spatial (k) transformations [MATH] has the full symmetry of the lattice, while depending on the superconducting state [MATH] may have lower symmetry.', '1412.5031-2-13-4': 'Our investigations,[CITATION] consistent with a large body of previous work, [CITATION] indicate that for the interesting carrier concentrations [MATH] and for temperatures higher than [MATH] the only stable superconducting state of this model is of [MATH] symmetry, so that on the square lattice studied here [MATH] changes sign if [MATH] is rotated by [MATH] or reflected through one of the axes [MATH] that lie at [MATH] to the lattice vectors, but remains invariant under reflections through the bond axes [MATH] or [MATH] and under rotations by [MATH].', '1412.5031-2-14-0': 'We now discuss the frequency dependence at fixed [MATH].', '1412.5031-2-14-1': 'In this discussion because the wave vector is fixed we do not explicitly denote it.', '1412.5031-2-14-2': 'The anomalous self energy is an even function of Matsubara frequency: [MATH].', '1412.5031-2-14-3': 'However the normal component at positive Matsubara frequency [MATH] is not simply related to the normal component at negative Matsubara frequency [MATH] except in the special case of particle-hole symmetry where [MATH].', '1412.5031-2-14-4': 'For later purposes it is convenient to distinguish components of [MATH] even (e) and odd (o) in Matsubara frequency[CITATION] as [EQUATION] so that in Nambu notation we may write [EQUATION]', '1412.5031-2-14-5': 'It is convenient also to define the gap function [MATH] as[CITATION] [EQUATION]', '1412.5031-2-14-6': 'This definition is motivated by the observation that at the low frequencies of interest here, the even (under sign change of Matsubara frequency) component of [MATH] can be absorbed into a shift of chemical potential.', '1412.5031-2-15-0': '## Analytic Structure', '1412.5031-2-16-0': '[MATH] and [MATH] are analytic functions of complex frequency argument [MATH] except for a branch cut along the real axis [MATH].', '1412.5031-2-16-1': 'Apart from a Hartree term in the normal component of [MATH] they decay as [MATH].', '1412.5031-2-16-2': 'They thus obey a Kramers-Kronig relation (written here for [MATH], with [MATH] one half of the branch-cut discontinuity) [EQUATION] [MATH] is a matrix with eigenvalues that are non-negative and related by time reversal, thus: [EQUATION] with [MATH].', '1412.5031-2-16-3': 'Here [EQUATION] is a rotation matrix in Nambu space parametrized by an angle [MATH] which in general is frequency dependent and lies in the range [MATH].', '1412.5031-2-16-4': 'That only [MATH] appears in Eq. [REF] follows from our phase convention for the superconducting state, Eq. [REF].', '1412.5031-2-17-0': 'Evaluating Eq. [REF] for [MATH] approaching the real axis gives the real-time Green functions.', '1412.5031-2-17-1': 'Of particular interest are the retarded functions obtained by continuation from below to the real frequency axis: [MATH] with [MATH] real and [MATH] (typically not explicitly written) a positive infinitesimal.', '1412.5031-2-17-2': 'The retarded real axis axis functions have real and imaginary parts [MATH] and [MATH], respectively.', '1412.5031-2-17-3': 'Our choice of superconducting phase convention (upper right and lower left entries in self energy matrix identical) implies that [MATH] is purely real so that [MATH] is identical to the branch cut discontinuity [MATH] introduced in Eq. [REF].', '1412.5031-2-18-0': 'The Kramers-Kronig relation along with the positivity of [MATH] implies that [MATH] and [MATH] are a real functions of [MATH] while [MATH] is a purely imaginary function of [MATH].', '1412.5031-2-18-1': 'The symmetry properties of [MATH] are those of [MATH].', '1412.5031-2-19-0': 'The non-negativity of the two eigenvalues of the spectral function implies that the diagonal components of the imaginary parts of [MATH] and [MATH] are non-negative so that on the real frequency axis [MATH] and [MATH].', '1412.5031-2-20-0': '## Method', '1412.5031-2-21-0': 'To solve the Hubbard model we employ the dynamical mean field approximation [CITATION] in its DCA cluster[CITATION] form.', '1412.5031-2-21-1': 'In this approximation the Brillouin zone is partitioned into [MATH] equal area tiles labeled by central momentum [MATH] and the self energy is approximated as a piecewise continuous function [EQUATION] with [MATH] if [MATH] is in the tile centered on [MATH] and 0 otherwise.', '1412.5031-2-21-2': 'The functions [MATH] are Nambu matrices with normal (N) and anomalous (A) components which are determined from the solution of an auxiliary quantum impurity model as described in detail in Refs. Lichtenstein00,Maier06,Gull11,Gull13.', '1412.5031-2-22-0': 'The quantum impurity model is solved by the continuous-time auxiliary field quantum Monte-Carlo method introduced in Ref. Gull08 and discussed in detail in Ref. Gull11.', '1412.5031-2-22-1': 'Fast update techniques [CITATION] are crucial for accessing the range of interaction strengths and temperatures needed to study superconductivity and its interplay with the pseudogap.', '1412.5031-2-22-2': 'The quantum Monte-Carlo calculations are performed on the Matsubara axis and maximum-entropy analytical continuation techniques[CITATION] are employed to obtain real-frequency results.', '1412.5031-2-23-0': 'The expense of the computation increases rapidly as the interaction strength [MATH] or number of approximants [MATH] increases, or as the temperature [MATH] decreases.', '1412.5031-2-23-1': 'We present results for [MATH] using the momentum-space tiling shown in the left panel of Fig. [REF].', '1412.5031-2-23-2': 'Previous work [CITATION] has shown that this cluster is in many aspects representative of the [MATH] limit; in particular it is large enough to enable a clear distinction between zone-diagonal and zone-face electronic properties, yet small enough to permit calculations in the superconducting phase of the precision needed for analytical continuation.', '1412.5031-2-24-0': 'The d-wave symmetry of the superconducting state of the two dimensional Hubbard model means that in the coarse-grained momentum resolution available in the [MATH] DCA we have [EQUATION] [MATH] has the full point group symmetry of the lattice and thus is one function of frequency in the [MATH] and [MATH] sectors, a different function of frequency in the [MATH] sectors and yet a different function of frequency in the [MATH] and in the [MATH] sectors.', '1412.5031-2-24-1': 'The main focus of attention in this paper will be on the [MATH] momentum sector but some results will be presented on superconductivity-induced changes in the zone-diagonal [MATH] momentum sectors.', '1412.5031-2-24-2': 'Because in the 8-site DCA the anomalous self energy is non-zero only in the [MATH] sectors we will typically omit the momentum argument in our discussions of [MATH] and [MATH].', '1412.5031-2-25-0': 'The right panel of Fig. [REF] shows the phase diagram in the plane of interaction strength and carrier concentration obtained [CITATION] from the solution of the DCA equations at temperature [MATH], about half of the maximal computed superconducting transition temperature [MATH].', '1412.5031-2-25-1': 'The temperature [MATH] corresponds in physical units to [MATH].', '1412.5031-2-25-2': '[CITATION]', '1412.5031-2-26-0': 'We study the electronic properties as a function of doping at [MATH] and as a function of interaction strength at carrier concentration [MATH].', '1412.5031-2-26-1': '[MATH] is the largest interaction strength for which high precision data could be obtained with the resources available to us for temperatures substantially below [MATH] and general dopings.', '1412.5031-2-26-2': 'Simulations at larger interaction strengths are severely hampered by the fermionic sign problem.', '1412.5031-2-26-3': 'As can be seen from the phase diagram, this interaction strength is such that the model is conducting (although pseudo gapped) at [MATH].', '1412.5031-2-26-4': 'Thus, we believe that [MATH] is slightly lower than the [MATH] which is relevant to the real materials, so the quantitative values for example of the carrier concentrations at which different behaviors occur will be somewhat lower than is realistic.', '1412.5031-2-26-5': 'The information available to us [CITATION] indicates that all of the qualitative features of the doping dependence are well reproduced by the [MATH] computations.', '1412.5031-2-26-6': 'As we will see, considerable insight can be obtained from examination of the [MATH]-dependence of the superconducting self energy in the special case [MATH].', '1412.5031-2-26-7': 'The actual model also has an antiferromagnetic state, which competes with the superconducting state but is not studied here.', '1412.5031-2-27-0': '# Analytical Continuation of Normal and Anomalous Self Energies', '1412.5031-2-28-0': '## Formalism', '1412.5031-2-29-0': 'The main objects of interest in this paper are the real-frequency normal (N) and anomalous (A) components of the Nambu matrix electron self energy [MATH] as well as the gap function [MATH] defined from Eq. [REF].', '1412.5031-2-29-1': 'These are functions of a complex frequency argument [MATH].', '1412.5031-2-29-2': 'Our results for [MATH] and thus [MATH] are obtained on the imaginary frequency (Matsubara) points [MATH].', '1412.5031-2-29-3': 'Theorems from complex variable theory guarantee that knowledge of the function on the Matsubara points fully determines the function at all [MATH], but direct inversion of Eq. [REF] to obtain [MATH] from measurements of [MATH] is a mathematically ill-posed problem.', '1412.5031-2-29-4': 'In this paper we invert Eq. [REF] using the maximum entropy method [CITATION] and verify the results with the Pade method.[', '1412.5031-2-30-0': 'To continue the diagonal component [MATH] we follow the methods employed for self energy continuation in Ref. Wang09a.', '1412.5031-2-30-1': 'While there are no rigorous methods to control errors in this procedure, our experience is that given sufficiently accurate input data (relative statistical errors smaller than [MATH] on each Matsubara point) this procedure produces reasonably reliable results for the lowest frequency features in the real-axis spectral function at low temperature and qualitatively reasonable results (estimates of characteristic energy scales and integrated areas) for the higher frequency features.', '1412.5031-2-30-2': 'The relative statistical errors in our calculations are typically smaller than [MATH] so we have reasonable confidence in the qualitative features of the continuations.', '1412.5031-2-31-0': 'A variation of the procedure outlined in Wang09a is needed to obtain the off-diagonal component of the self energy [MATH] because [MATH] so that the spectral function [MATH] is an odd function of frequency and is thus not non-negative.', '1412.5031-2-31-1': 'We rewrite Eq. [REF] as [EQUATION] and continue [MATH] by standard methods.', '1412.5031-2-31-2': '[CITATION] The normalization of [MATH] is fixed from [MATH]: We obtain [MATH] by fitting [MATH] at the three lowest positive Matsubara frequencies to a parabola.', '1412.5031-2-31-3': 'A similar procedure is used to continue [MATH].', '1412.5031-2-32-0': '## Non-negativity of [MATH]', '1412.5031-2-33-0': 'While the usual maximum entropy analytic continuation formalism requires a non-negative spectral function, there is no guarantee that [MATH] is of definite sign.', '1412.5031-2-33-1': 'For example, in Migdal-Eliashberg theory the Coulomb pseudopotential leads to a negative contribution to [MATH] at frequencies of the order of the plasma frequency,[CITATION] while a recent solution of the Eliashberg equations for a model involving two competing spin fluctuations also displayed a sign change in the gap function as frequency was increased above the lower of the two characteristic frequencies.', '1412.5031-2-33-2': '[CITATION]', '1412.5031-2-34-0': 'For the two dimensional Hubbard model, our data are consistent with a positive definite [MATH] but we do not have a rigorous proof that this is always the case.', '1412.5031-2-34-1': 'Evidence for a positive definite [MATH] may be found from consideration of the particle-hole symmetric ([MATH]) situation.', '1412.5031-2-34-2': "In this case in the [MATH] sector the normal components of the hybridization function and self energy are also particle-hole symmetric and are odd functions of Matsubara frequency, so that [MATH] at all frequencies and there is a frequency-independent basis choice (the 'Majorana combination' [MATH]) that diagonalizes the self energy matrix in the [MATH] sector.", '1412.5031-2-34-3': "The corresponding 'Majorana' self energy [MATH] obeys the Kramers-Kronig relation [EQUATION]", '1412.5031-2-34-4': 'Thus analytical continuation of the Majorana combination of self energies gives direct access to [MATH] from which the normal and anomalous components of the self energy can easily be reconstructed from Eqs. [REF],[REF] as [EQUATION]', '1412.5031-2-34-5': 'Fig. [REF] presents a comparison of [MATH] obtained by direction continuation of the normal and anomalous parts of the Matsubara Green function and reconstructed using Eqs. [REF] and [REF] from continuations obtained from the [MATH] component of [MATH] in the Majorana basis (continuation of the [MATH] component yields values differing by [MATH]).', '1412.5031-2-34-6': 'A reconstruction of [MATH] from the continued [MATH] and [MATH] using Eq. [REF] is also shown.', '1412.5031-2-35-0': 'We see that the methods yield very similar results with some differences in the height and width of the low frequency peak (the integrated peak areas, not shown, are the same).', '1412.5031-2-35-1': 'The differences between the curves are an indication of the continuation errors.', '1412.5031-2-35-2': 'In the [MATH] obtained from [MATH] a relatively small amplitude oscillation is present, which leads to a small negative value in some regions where the directly continued [MATH] along with an overshoot (relative to the directly continued [MATH]) at slightly higher frequencies.', '1412.5031-2-35-3': 'Our experience is that these oscillations do not vary systematically with input data or details of the maximum entropy procedure, and we believe they are artifacts of the continuation procedure related to the requirement that norm of the function is conserved in the entropy minimization.', '1412.5031-2-35-4': 'We thus believe that [MATH] is generically non-negative for the [MATH] superconducting state of the Hubbard model.', '1412.5031-2-35-5': 'This conclusion further supported by continuations (not shown) that we have performed using the Pade method at both [MATH] and [MATH].', '1412.5031-2-35-6': 'The Pade method makes no assumptions as to the sign of the spectral function, and in the cases we have studied leads always to a non-negative [MATH].', '1412.5031-2-35-7': 'A [MATH] which was non-negative was also found by Civelli09b in [MATH] calculations using an exact diagonalization (ED) solver (we believe that the small negative excursions are artifacts of the ED method but the issue deserves further investigation).', '1412.5031-2-36-0': '## Pole structure and continuation of gap function', '1412.5031-2-37-0': 'Fig. [REF] reveals that in certain parameter regimes the normal and anomalous self energies exhibit strong peaks at relatively low frequencies.', '1412.5031-2-37-1': 'As will be discussed at length below, these peaks do not correspond to physical excitations of the system; rather they are the expression in the superconducting state of the physics of the normal-state pseudogap, which in the DCA is associated with the formation of a low frequency pole in the self energy of the [MATH] momentum sector.', '1412.5031-2-37-2': '[CITATION] It is useful to discuss the pole structure in terms of the representation in Eq. [REF].', '1412.5031-2-37-3': 'All of our data are consistent with the statement that [MATH] for the [MATH] sector is the sum of a pole and a regular part: [EQUATION] with [MATH] a smooth function of [MATH].', '1412.5031-2-37-4': 'The pole structure was also noted in a very recent paper by Sakai and collaborators.', '1412.5031-2-37-5': '[CITATION]', '1412.5031-2-38-0': 'Retaining for the moment only the pole term and explicitly evaluating Eq. [REF] and [REF] using the Nambu angle [MATH] corresponding to [MATH] gives [EQUATION]', '1412.5031-2-38-1': 'Thus in general we expect the normal and anomalous components of the self energy to have poles at exactly the same frequencies.', '1412.5031-2-38-2': 'In the particle-hole symmetric case, where [MATH], we expect the poles in the normal and anomalous parts of the self energy to have exactly the same amplitude.', '1412.5031-2-38-3': 'This is demonstrated in Fig. [REF], which shows the imaginary parts of the continuations of the normal and anomalous components of the self energy, along with the continuations of [MATH] obtained from continuation of the [MATH] component of the Majorana-basis representation of [MATH] and of [MATH] obtained from continuation of the [MATH] component of the Majorana-basis representation of [MATH].', '1412.5031-2-38-4': 'That [MATH] for [MATH] in the vicinity of the pole is strong evidence of the cancellation.', '1412.5031-2-38-5': 'In the general case both normal and anomalous components of the self energy have poles at [MATH] but because [MATH] the average of the strengths of the poles in [MATH] will in general be greater than the strengths of the poles in [MATH].', '1412.5031-2-38-6': 'However, as will be seen in our detailed examination of the self energy below, the background contributions are such that unambiguously identifying the pole strengths is not possible.', '1412.5031-2-39-0': 'It also follows from Eqs. [REF], [REF] and [REF] that the pole contribution to the gap function [MATH] is [EQUATION] so that [MATH] does not have poles at [MATH].', '1412.5031-2-39-1': 'All of our continuations are consistent with this result.', '1412.5031-2-40-0': 'The structure defined by Eq. [REF] creates challenges for analytical continuation.', '1412.5031-2-40-1': 'Intrinsic errors in the analytical continuation process mean that independent continuations of the different components of [MATH] may lead to slightly different estimates of the pole positions, amplitudes and widths.', '1412.5031-2-40-2': 'This can be seen for example by comparison of the upper and lower panels of Fig. [REF] or by comparing [MATH] and [MATH] in Fig. [REF].', '1412.5031-2-40-3': 'The difficulties are exacerbated if particle-hole symmetry is broken because the continuation process may not place the poles in [MATH] at exactly opposite frequencies.', '1412.5031-2-40-4': 'We do not know how to control the continuation process so as to force the precise alignment of the poles in the different components of [MATH].', '1412.5031-2-40-5': 'For studies of the particle-hole symmetric situation, we work with [MATH] defined from continuation of the [MATH] component of [MATH] in the Majorana basis.', '1412.5031-2-40-6': 'In the doped case, we focus on the gap function.', '1412.5031-2-41-0': '# The normal and anomalous self energies, [MATH] sector', '1412.5031-2-42-0': 'It has been accepted for many years that the two dimensional Hubbard model exhibits a Mott insulating phase at carrier concentration [MATH] and large interaction, and a Fermi liquid phase at small interaction or carrier concentration sufficiently different from [MATH].', '1412.5031-2-42-1': 'Work [CITATION] has established that if superconductivity is neglected then the Mott and Fermi liquid phases are separated by a pseudogap regime, in which regions of momentum space near the zone face are gapped while regions of momentum space near the zone diagonal are not.', '1412.5031-2-42-2': 'Mathematically, within DCA the gapping is a consequence of the appearance of a pole in the electron self energy pertaining to the [MATH] sector.[', '1412.5031-2-42-3': '[CITATION] If the Mott insulator is approached by varying interaction strength in the particle-hole symmetric case ([MATH], [MATH]) the pole occurs at [MATH];[CITATION] in a non-particle-hole symmetric situation (for example if the Mott insulator is approached by varying doping) the pole appears at a frequency close to, but slightly different, from zero.[', '1412.5031-2-42-4': '[CITATION] In this section we show how the onset of superconductivity affects this pole structure.', '1412.5031-2-43-0': 'Fig. [REF] presents the imaginary parts of the analytically continued self energies and the gap function for the three [MATH]-values at [MATH] presented in Fig. [REF].', '1412.5031-2-43-1': 'The upper panel shows results obtained in the normal state at temperature [MATH] (at [MATH] is slightly greater than [MATH] but superconductivity has been suppressed here for clarity of presentation, i.e. [MATH] set to zero).', '1412.5031-2-43-2': 'At [MATH] the imaginary part has the Fermi liquid form, with [MATH] exhibiting an approximately quadratic minimum at zero.', '1412.5031-2-43-3': 'As [MATH] is increased a thermally broadened pole appears at [MATH].', '1412.5031-2-43-4': 'The pole increases rapidly in strength as [MATH] is increased.', '1412.5031-2-43-5': 'These results are consistent with our previous analysis of the pseudogap.', '1412.5031-2-43-6': '[CITATION] The middle two panels present the imaginary parts of the normal and anomalous components of the self energy at a temperature [MATH].', '1412.5031-2-43-7': 'One sees that the pole centered at [MATH] has split into two and appears with comparable strength in both the normal and anomalous components.', '1412.5031-2-43-8': 'The pole frequency weakly decreases as [MATH] is increased.', '1412.5031-2-43-9': 'The larger width seen in the [MATH] calculation is likely to be an artifact of the analytical continuation.', '1412.5031-2-43-10': 'The lowest panel shows the imaginary part of the gap function.', '1412.5031-2-43-11': 'We see that the gap function is much smaller in magnitude than the self energy, that the structure in the gap function occurs at a higher frequency than the structure in the self energy (as also noted by Sakai et al [CITATION]), and that the gap function varies less dramatically with interaction strength than does the self energy.', '1412.5031-2-44-0': 'Fig. [REF] shows the analogous plots as a function of doping at [MATH].', '1412.5031-2-44-1': 'The upper panel (normal state) shows again a pole that appears and grows in strength as doping is decreased.', '1412.5031-2-44-2': 'The particle-hole symmetry breaking provided by the doping means that the pole is slightly displaced from the origin.', '1412.5031-2-44-3': 'In the superconducting state (middle panels) the pole splits; in [MATH] the pole strength is different for the positive frequency than for the negative frequency pole.', '1412.5031-2-44-4': 'The strength of the pole in [MATH] is of the order of that in [MATH].', '1412.5031-2-44-5': 'The detailed line shapes of the poles depend to on the details of the continuation process.', '1412.5031-2-44-6': 'The difference in line shapes and small differences in pole position can lead to unphysical structures in calculated spectra.', '1412.5031-2-44-7': 'As in the interaction-driven case, there is no structure in [MATH] at the pole frequencies of [MATH].', '1412.5031-2-45-0': 'Poles in the anomalous component of the self energy were reported by Civelli [CITATION] and the alignment of poles in [MATH] and [MATH] and the cancellation of the [MATH] poles in [MATH] were very recently discussed in the context of 4-site CDMFT calculations [CITATION].', '1412.5031-2-46-0': '# The Superconducting Gap and the pseudogap', '1412.5031-2-47-0': 'In this section we present results for the energy gap in the superconducting and non-superconducting states.', '1412.5031-2-47-1': 'The gap may be estimated from an analytical continuation of the normal component of the Green function, but the inevitable broadening associated with continuations means that it is not clear a priori how to estimate a gap.', '1412.5031-2-47-2': 'As discussed in Ref. [CITATION] for Mott insulators, an alternative approach provides a more accurate estimate.', '1412.5031-2-47-3': 'This approach is based on the argument that for frequencies less than the gap, all imaginary parts are zero except for the poles in [MATH], which do not correspond to physical excitations.', '1412.5031-2-47-4': 'Thus one may determine the gap energy [MATH] from the vanishing of the real part of the denominator of the Green function.', '1412.5031-2-47-5': 'For the normal state, this criterion involves consideration of [EQUATION]', '1412.5031-2-47-6': 'For a given [MATH] this equation will have two solutions, [MATH].', '1412.5031-2-47-7': 'The pseudogap is typically indirect (the [MATH] that maximizes [MATH] is different from the [MATH] that minimizes [MATH]).', '1412.5031-2-47-8': 'The direct gap is determined as [EQUATION]', '1412.5031-2-47-9': 'The points of minimum direct gap are found to be the renormalized Fermi surface points [MATH] satisfying [MATH] and that at these points [MATH].', '1412.5031-2-48-0': 'In the superconducting state, these ideas lead to the consideration of [MATH].', '1412.5031-2-48-1': 'Rearranging the expression for [MATH] gives [EQUATION] with [MATH] defined in Eq. [REF] and [EQUATION]', '1412.5031-2-48-2': 'We then define the gap frequency [MATH] in the superconducting state as the frequency at which the real part of [MATH] vanishes, i.e. as [EQUATION]', '1412.5031-2-48-3': 'We define the Fermi surface as the locus of [MATH]-points for which [MATH].', '1412.5031-2-48-4': 'We will find that the the [MATH] dependence of [MATH] is modest so that we may identify the gap [MATH] in the superconducting state as the value of [MATH] which solves [MATH].', '1412.5031-2-49-0': 'In the particle-hole symmetric case where [MATH] we may alternatively write the gap equation as the solution of [EQUATION]', '1412.5031-2-49-1': 'Fig. [REF] demonstrates this procedure.', '1412.5031-2-49-2': 'The upper panel shows results obtained in the superconducting state at [MATH] (the normal state is not shown because at this [MATH] there is no normal state pseudogap).', '1412.5031-2-49-3': 'The figure plots the spectral function (imaginary part of continued Green function), the real part of the Majorana combination of the self energy and the real part of the gap function as a function of frequency.', '1412.5031-2-49-4': 'We see by comparing the continued spectral function to the self energy and gap curves that the criteria [MATH] or [MATH] identifies a point close to that at which the spectral function is maximal.', '1412.5031-2-49-5': 'We believe that the appearance of weight at lower frequencies in the spectral function comes from artificial broadening induced by the continuation process.', '1412.5031-2-50-0': 'The middle panel shows the same analysis in the normal state at [MATH].', '1412.5031-2-50-1': 'The normal state pseudogap is evident, and again we see that the quasiparticle equation picks out as the gap the point at which the spectral function is maximal.', '1412.5031-2-50-2': 'The lower panel shows the superconducting state, also at [MATH].', '1412.5031-2-50-3': 'We also see that at this [MATH] value the line [MATH] is tangent to (in fact very slightly below) the [MATH] curve at the point that one would naturally identify as the gap.', '1412.5031-2-50-4': 'That [MATH] intersects [MATH] while [MATH] is peaked at the point of near tangency suggests that the absence of an exact intersection is a continuation artifact and that if all quantities were exactly and consistently continued the line [MATH] would intersect the [MATH] line.', '1412.5031-2-51-0': 'Comparison of the middle and lower panels of Fig. [REF] shows that the superconducting gap is unambiguously smaller than the normal state pseudogap, consistent with results presented in Ref. [CITATION].', '1412.5031-2-51-1': 'Also, comparison of the lower panel of Fig. [REF] to Fig. [REF] shows that for this value of [MATH] the pole in the self energy lies inside the excitation gap of the superconductor, further confirming that the self energy pole does not represent a physical excitation of the system.', '1412.5031-2-52-0': 'The two panels of Fig. [REF] show the dependence of the low-T ([MATH]) gap on interaction strength at [MATH], computed from Eq. [REF] and the dependence of the low-T gap carrier concentration at [MATH], computed from Eq. [REF].', '1412.5031-2-52-1': 'In the weak interaction or large doping limits, there is no superconductivity.', '1412.5031-2-52-2': 'As the doping decreases or interaction strength increases the gap increases smoothly down to the low doping/high interaction end of the phase diagram (note that at the two endpoints of the superconducting phase, [MATH] and [MATH] the superconducting state is not fully formed so the gap values are not meaningful).', '1412.5031-2-52-3': 'Also shown on the plots is the normal state pseudogap, computed at the same low temperature [MATH] by suppressing superconductivity in the calculation.', '1412.5031-2-52-4': 'One sees that in the low doping/strong interaction regime, turning on superconductivity leads to a decrease in the energy of the lowest-lying excitation.', '1412.5031-2-53-0': 'Finally, Fig. [REF] shows the temperature dependence of the energy gap at the three [MATH] values considered above.', '1412.5031-2-53-1': 'We see that in the moderate coupling regime, there is no normal state gap and the superconducting gap increases from zero as the temperature is decreased through the transition temperature.', '1412.5031-2-53-2': 'At intermediate coupling a small but non-zero gap is already present in the normal state and the gap increases with the onset of superconductivity, while at stronger coupling the gap actually decreases as the temperature is decreased into the superconducting state.', '1412.5031-2-54-0': '# Fermi surface spectra', '1412.5031-2-55-0': 'In this section we present spectra computed at the Fermi surface using continued self energies.', '1412.5031-2-55-1': 'The issues discussed above relating to the difficulties of analytical continuation in non-particle-hole symmetric case mean that we limit ourselves here to study of the interaction-driven transition at density [MATH] but all of the information available to us suggests that the behavior in the doped case shares the essential features found at half-filling.', '1412.5031-2-56-0': 'Fig. [REF] shows the temperature evolution of the electron spectral function computed for a momentum on the Fermi surface at the antinode ([MATH]) at carrier density [MATH] and different interaction strengths.', '1412.5031-2-56-1': 'The upper panel shows results obtained for a moderate interaction [MATH].', '1412.5031-2-56-2': 'We see that the normal state spectral function is peaked at the chemical potential [MATH].', '1412.5031-2-56-3': 'The onset of the superconductivity ([MATH] is between [MATH] and [MATH]) induces a suppression of the low frequency density of states.', '1412.5031-2-56-4': 'The gap grows and sharpens as temperature is decreased.', '1412.5031-2-56-5': 'For the lower temperatures one sees that the spectral function has a sharp quasiparticle peak, a weak minimum at slightly higher frequencies, followed by a weak maximum at yet higher frequencies.', '1412.5031-2-56-6': 'This structure in the spectral function is frequently observed in photoemission [CITATION] and scanning tunneling microscopy [CITATION] experiments on copper-oxide high-[MATH] materials and is referred to as a "peak-dip-hump" feature.', '1412.5031-2-56-7': 'The higher frequency "hump" is typically interpreted as arising from the interaction of electrons with some kind of bosonic excitation, with the hump frequency determined by the boson energy and the minimum energy to create an electron-hole pair.', '1412.5031-2-57-0': 'The middle panel of Fig. [REF] shows shows results obtained for an intermediate interaction [MATH].', '1412.5031-2-57-1': "We see that a weak minimum is evident in the density of states even temperature [MATH]; this weak suppression of the density of states marks the onset of the 'pseudogap'.", '1412.5031-2-57-2': 'As the temperature is decreased below the transition temperature the low frequency intensity drops very rapidly.', '1412.5031-2-57-3': 'The gap (peak in the spectral function) increase and saturates at a value rather greater than that found in the moderate interaction case.', '1412.5031-2-57-4': 'The peak-dip-hump structure is more evident.', '1412.5031-2-57-5': 'In this case the "hump" energy increases as T decreases except for the lowest temperature.', '1412.5031-2-58-0': 'The lower panel of Fig. [REF] shows results obtained for the strongest interaction ([MATH]).', '1412.5031-2-58-1': 'In this case the normal state pseudogap is well established even at [MATH].', '1412.5031-2-58-2': 'The transition to superconductivity is associated with the formation of a quasiparticle peak which lies inside the pseudogap and with a shift outwards of the second peak structure.', '1412.5031-2-58-3': 'For both [MATH] and [MATH] the energy of the second peak increases as temperature decreases, with the exception of [MATH], [MATH].', '1412.5031-2-58-4': 'We believe that at this particular [MATH] and [MATH] the continuations are of lower quality than at other parameter values.', '1412.5031-2-58-5': 'For example the errors associated with back continuation are slightly larger.', '1412.5031-2-59-0': 'The question of the origin of the "peak-dip-hump" structure has been discussed extensively in the literature on angle-resolved photoemission in the cuprates.', '1412.5031-2-59-1': '[CITATION] The most widely accepted explanation is that the higher energy "hump" is evidence for a "shakeoff" process in which an electron emits a bosonic excitation such as a spin fluctuation [CITATION] or a phonon,[CITATION] while the peak feature is the quasiparticle excitation above the gap.', '1412.5031-2-59-2': 'In this picture the onset of the hump feature determined as the sum of the energy of the boson and the superconducting gap energy, (structure in the bare dispersion associated with interbilayer hopping may also play a role in certain materials).', '1412.5031-2-59-3': '[CITATION] Such a shakeoff process would appear in the imaginary part of the self energy as an upward step, corresponding to the opening of a scattering channel.', '1412.5031-2-59-4': 'However, a quantitative and generally accepted identification of the boson responsible for the hump energy is lacking.', '1412.5031-2-60-0': 'Mathematically, structures in the spectral function may be understood from the fundamental expression [EQUATION] where for clarity we have not explicitly written the change of basis matrices [MATH] (Eq. [REF]).', '1412.5031-2-60-1': 'From Eq. [REF] we see that structure can arise from resonance ([MATH]) or, off resonance, from an increase in [MATH].', '1412.5031-2-60-2': 'The former case produces the quasiparticle peak of Fermi liquid theory; the latter is the origin of the "shakeoff" explanation of the peak-dip-hump structure.', '1412.5031-2-60-3': '[CITATION] Fig. [REF] investigates the origin of the "hump" structure in the present calculation by comparing the computed spectral function (lower panel, evaluated at the Fermi surface [MATH]) to the electron self energy (heavy lines, upper panel) in the Majorana basis that diagonalizes the Nambu Greens function at all [MATH] for [MATH].', '1412.5031-2-60-4': 'The real part is presented in the combination [MATH].', '1412.5031-2-60-5': 'We see that the location of the "hump" feature in fact does not correspond to any significant feature in the imaginary part of the self energy (top panel), so that it cannot be interpreted as an onset of scattering.', '1412.5031-2-60-6': 'Instead, inspection of [MATH] energy (middle panel) shows that for the two stronger couplings the frequency of the hump corresponds to the frequency at which the real part of the inverse Greens function vanishes, while for the weaker coupling the hump frequency corresponds to a point where the absolute value of [MATH] is minimized.', '1412.5031-2-60-7': 'In other words, the "hump" is a resonance phenomenon, not a scattering phenomenon, and its frequency does not correspond directly to the energy of any excitation of the system.', '1412.5031-2-61-0': '# Momentum-dependent Spectra', '1412.5031-2-62-0': 'In this section we present and discuss the momentum dependence of spectral functions calculated from the normal and anomalous components of the self energy.', '1412.5031-2-62-1': 'The left panel of Fig. [REF] shows a sequence of energy distribution curves (EDC) calculated in the normal state for a sequence of momenta cutting across the Fermi surface in the [MATH] sector for the moderate interaction strength [MATH].', '1412.5031-2-62-2': 'A quasiparticle peak is visible, which disperses through the Fermi surface.', '1412.5031-2-62-3': 'The right panel shows EDC at the same momenta, this time in the superconducting state.', '1412.5031-2-62-4': 'Comparison of the two panels reveals the behavior expected of a moderate-coupling BCS-like superconductor, with the gap having the greatest effect on the Fermi surface trace and the superconductivity-induced particle-hole mixing producing a peak dispersing away from zero as momentum is increased above the Fermi level.', '1412.5031-2-63-0': 'Fig. [REF] shows the EDC curves obtained for a strong interaction [MATH].', '1412.5031-2-63-1': 'The weakly dispersing and rather broadened normal state pseudogap is seen in the left panel.', '1412.5031-2-63-2': 'The right panel shows that the onset of superconductivity produces a very weakly dispersing peak at an energy well inside of the pseudogap.', '1412.5031-2-63-3': 'This essentially non-dispersing zone-edge feature is characteristically observed in photoemission experiments on high [MATH] copper oxide superconductors.', '1412.5031-2-63-4': '[CITATION] In the present calculation the weak dispersion arises mathematically from the very strong frequency dependence of the self energy caused by the proximity of the pseudogap pole (cf Eq. [REF]).', '1412.5031-2-64-0': '# Superconducting effects on zone diagonal spectra', '1412.5031-2-65-0': 'In the cuprates and in the Hubbard model the dramatic effects of superconductivity are visible in the electronic states near the zone face ([MATH]) point, because it is at this point that the superconducting gap and the pseudogap are maximal.', '1412.5031-2-65-1': 'But in addition to opening, or changing the value of, a gap, the onset of superconductivity may affect other aspects of the physics, for example by changing the density of final states that enter into scattering processes [CITATION] or, more profoundly, by changing the electronic state itself and thus the nature of the scattering mechanisms.', '1412.5031-2-65-2': 'Some understanding of these effects may be gleaned from consideration of electronic states with momentum along the zone diagonal.', '1412.5031-2-65-3': 'The superconducting gap vanishes for these momenta and thus any effects on electron propagation must arise from changes in scattering and electronic state.', '1412.5031-2-66-0': 'We find that at all dopings and interaction strengths we have studied the electronic self energy in the zone diagonal momentum sector has approximately the Fermi liquid form.', '1412.5031-2-66-1': 'In particular the self energy does not have a low frequency pole.', '1412.5031-2-66-2': 'Its imaginary part is minimal at zero frequency and the value at zero frequency, [MATH], decreases as temperature [MATH].', '1412.5031-2-66-3': 'The real part is linear in frequency over a reasonable range of low frequencies, allowing us to define a zone diagonal mass enhancement [MATH] from the Matsurbara data (note also that in DCA, the self-energy is k-independent except at the sector boundaries, so the k-derivative contribution to the disperson renormalization is not relevant).', '1412.5031-2-66-4': 'We extract estimates of the Fermi level scattering rate [MATH] and mass enhancement [MATH] by fitting the lowest four Matsubara frequencies to the cubic form [EQUATION]', '1412.5031-2-66-5': 'Representative results are shown in Fig. [REF].', '1412.5031-2-66-6': 'We see that the mass enhancement has a modest doping dependence, and decreases markedly as temperature is decreased.', '1412.5031-2-66-7': 'Superconductivity has only a small effect on the mass enhancement.', '1412.5031-2-66-8': 'At low doping the onset of superconductivity leads to a very small decrease in the mass enhancement; at higher doping the effect is of opposite sign and is slightly larger.', '1412.5031-2-66-9': 'It is interesting to note that the change in sign of the superconducting contribution to the mass enhancement occurs at a lower doping than the change from potential energy-driven to kinetic energy driven pairing discussed in Ref. [CITATION].', '1412.5031-2-66-10': 'Thus the change in electronic state associated with the onset of superconductivity has a weaker effect on the nodal quasiparticles than it does on other properties.', '1412.5031-2-67-0': 'The normal state scattering rate also has a dramatic doping dependence, especially at the higher temperature, and at all dopings exhibits a marked temperature dependence.', '1412.5031-2-67-1': 'The effect of superconductivity on the scattering rate is much more noticeable than the effect on the mass enhancement: the onset of superconductivity leads to an almost factor of two drop in the Fermi surface scattering rate, except at the very lowest doping which is right on the boundary of superconducting phase.', '1412.5031-2-67-2': 'These finding are consistent with angle-resolved photoemission measurements on Bi[MATH]Sr[MATH]CaCu[MATH]O[MATH].[', '1412.5031-2-67-3': '[CITATION] Fig. 4c of Ref. Valla06 reveals that the onset of superconductivity leads to an approximately factor of two decrease in the MDC width (a good proxy for scattering rate) below the extrapolation of the normal state rate to low temperature, while Fig. 3 of Ref. Valla06 reveals a much smaller change in the mass enhancement (albeit of opposite sign to that predicted here).', '1412.5031-2-67-4': 'The more coherent nature of the zone diagonal quasiparticles is also qualitatively consistent with conclusions drawn from pump-probe experiments,[CITATION] but because our calculations are restricted to equilibrium a direct comparison cannot be made.', '1412.5031-2-68-0': '# Conclusions', '1412.5031-2-69-0': "Two characteristic features of the high transition temperature copper-oxide superconductors are superconductivity with [MATH] symmetry and a 'pseudogap'[CITATION], a suppression of electronic density of states for momenta near the Brillouin zone face ([MATH] point).", '1412.5031-2-69-1': 'The interplay between these two phenomena has been the focus of considerable attention in the literature.', '1412.5031-2-69-2': 'In this paper we present cluster dynamical mean field calculations of the electronic self energy and spectral function of the normal and superconducting phase of the two dimensional Hubbard model that shed new light on the subject.', '1412.5031-2-70-0': 'The correspondence of the results to the essential features of superconductivity in the cuprates is striking.', '1412.5031-2-70-1': 'We find, consistent with a large body of experimental literature, that when superconductivity emerges from the pseudogap regime the onset of superconductivity is associated with the appearance of very weakly dispersing states inside the pseudogap and with the appearance of a peak-dip-hump structure in the spectral function.', '1412.5031-2-70-2': 'Superconductivity affects the zone-diagonal states via a significant ([MATH] factor of 2) decrease in the scattering rate.', '1412.5031-2-70-3': 'We have also determined the superconducting and pseudogaps accurately and have shown that the superconducting gap systematically increases as doping is decreased or interaction strength is increased, until it abruptly drops to zero at the low-doping/high interaction boundary of the superconducting phase.', '1412.5031-2-70-4': 'These results support the notion that the pseudogap and superconductivity are competing phenomena and strengthen the case made in prior papers [CITATION] that as P. W. Anderson predicted in 1987 [CITATION] the Hubbard model contains the essential physics of the high-[MATH] observed in layered copper-oxide materials.', '1412.5031-2-71-0': 'Mathematically, we find (in agreement with previous work [CITATION]) that the pseudogap is a Mott-transition-like phenomenon ("sector-selective Mott transition") associated with the appearance of a low frequency pole in the self energy in the zone-face self energy.', '1412.5031-2-71-1': 'The interplay between superconductivity and the pseudogap is controlled by the superconductivity-induced changes in the pole structure.', '1412.5031-2-71-2': "In particular, in our calculation the 'hump' feature in the spectral function arises mathematically from a zero-crossing in the real part of the self energy and not from the onset of a scattering process, in other words, not from a bosonic excitation at all.", '1412.5031-2-71-3': 'On the technical side we have clarified the mathematical structure of the pseudogap-induced poles associated with the normal and anomalous self energies, shown how the inevitable analytical continuation errors make it difficult to construct real-frequency spectra in situations where the pseudogap poles are important.', '1412.5031-2-71-4': 'The pole structure has also been discussed by Sakai14 who present an interesting alternative interpretation.', '1412.5031-2-72-0': 'It is important to consider the limitations of the methods used here.', '1412.5031-2-72-1': 'The essential technical step is the use of continuous-time auxiliary field methods [CITATION] and submatrix updates.', '1412.5031-2-72-2': '[CITATION] These methods enable highly accurate simulations at temperatures as low (in physical units) as [MATH], far below the basic scales of the model and, crucially, well below the superconducting transition temperature.', '1412.5031-2-72-3': 'However, even with these improvements, obtaining results of the needed precision at the required low temperatures requires approximations.', '1412.5031-2-73-0': 'The key approximation used in this paper is the cluster dynamical mean field approximation.', '1412.5031-2-73-1': "[CITATION] In the 'DCA' form used here [CITATION] this amounts to approximating the normal and anomalous components of the electron self energy as piecewise constant functions of momentum, taking different values in each of [MATH] momentum sectors that tile the Brillouin zone.", '1412.5031-2-73-2': 'We have adopted the [MATH] approximation.', '1412.5031-2-73-3': 'This is the smallest momentum decomposition that permits a clear separation between the zone face and zone diagonal regions of momentum space.', '1412.5031-2-73-4': 'Previous work [CITATION] indicates that although we do not have quantitative convergence to the [MATH] limit, the [MATH] approximation correctly captures the physics of the normal state pseudogap.', '1412.5031-2-74-0': 'A second approximation is the use of an interaction [MATH] which is likely to be slightly weaker than needed to quantitatively capture the physics of the cuprates.', '1412.5031-2-74-1': 'This approximation is needed because computation time increases rapidly as [MATH] increases (as [MATH], with additional complications from the sign problem) and we needed to undertake a broad survey of parameter space.', '1412.5031-2-74-2': 'For similar computational reasons we restricted attention to the particle-hole symmetric version of the model (second neighbor hopping [MATH] and further neighbor hoppings set to zero).', '1412.5031-2-75-0': 'Finally, because the basic computations use imaginary time methods, obtaining spectra of interest requires an analytical continuation method.', '1412.5031-2-75-1': 'Analytical continuation is an ill-posed problem.', '1412.5031-2-75-2': 'We used maximum entropy analytical continuation methods (which are widely employed but essentially uncontrolled) to extract real frequency information.', '1412.5031-2-75-3': 'This requires extremely high quality Monte Carlo data, further constraining the parameter ranges that could be examined.', '1412.5031-2-75-4': 'In this context the properties of the anomalous self energy are of particular importance.', '1412.5031-2-75-5': 'With the exception of the Pade method, all methods known to us require a positive definite spectral function.', '1412.5031-2-75-6': 'While we have presented evidence that in at least some cases the spectral function associated with the anomalous self energy has an appropriate positivity property, the possibility of a sign change at high frequency cannot be ruled out.', '1412.5031-2-75-7': 'At high frequencies the anomalous part is very small and the intrinsic limitations of the continuation process mean that small systematic errors could induce (or mask) a sign change.', '1412.5031-2-76-0': 'A key physical limitation of our work is that we have not considered other ordered states (for example Neel antiferromagnetic or striped order) that might preempt the phases considered here.', '1412.5031-2-76-1': 'The dynamical mean field method captures (within the rather coarse momentum resolution of the cluster dynamical mean field method) fluctuations associated with these states, but because we have symmetrized over spin degrees of freedom, long range ordered antiferromagnetic states are excluded.', '1412.5031-2-76-2': 'Also our calculation lacks the momentum resolution needed to provide a clear account of striped states.', '1412.5031-2-76-3': 'Thus we view the results as providing a reasonable qualitative account of the properties of the superconducting phase and pseudogap regime of the two dimensional Hubbard model, but not as a quantitatively accurate account of the properties of the Hubbard model.', '1412.5031-2-77-0': 'For these reasons, extensions of the work presented here would be desirable.', '1412.5031-2-77-1': 'Pushing the calculation on large clusters to somewhat larger [MATH] so that the [MATH] endpoint is well within the Mott insulating phase should become feasible as computer power improves.', '1412.5031-2-77-2': 'Study of larger [MATH] would provide more insight into the interplay of superconductivity and Mott physics.', '1412.5031-2-77-3': 'Extending the calculations to [MATH] site approximation would similarly be useful.', '1412.5031-2-77-4': 'In particular, examination of differences between gap values and spectra calculated with [MATH] and [MATH] will provide insight into the quantitative aspects of the results.', '1412.5031-2-77-5': 'If feasible, real-time calculations and comprehensive Pade continuations, that could investigate the possibility of sign changes in the anomalous component of the self energy would be reassuring.', '1412.5031-2-77-6': 'On the conceptual side, a deeper understanding of the pseudogap state and of the meaning of the self energy poles would be desirable.', '1412.5031-2-77-7': 'Most importantly, investigation of competing states such as Neel antiferromagnet and stripe orders is needed.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1412.5031

1712.05206

{'1712.05206-1-0-0': 'We propose a way to define uniformly accelerated fields.', '1712.05206-1-0-1': 'It is introduced by the operator version of the relativistic kinematics relation [MATH] as the required field equation, and following the canonical procedure we perform the field quantization in the momentum space, claiming that the resulting excitations can be interpreted as uniformly accelerated particles.', '1712.05206-1-0-2': 'The behavior of accelerated particle detectors with definite mass is investigated, and it is shown that a massive detector experiences a temperature inversely proportional to its mass, and consequently finds the vacuum state full of accelerated particles.', '1712.05206-1-1-0': '# Introduction', '1712.05206-1-2-0': 'The original reasoning that led Schrödinger to write down the first relativistic wave equation [CITATION], was based on the energy-momentum relation for a massive particle, [MATH], and the assumption that the particle would be described by a wave function satisfying the equation obtained by considering the same identifications from which the de Broglie’s relations came out for a free particle described by a plane wave.', '1712.05206-1-3-0': 'In accordance with special relativity, a particle moving at a uniform acceleration [MATH] undergoes a hyperbolic motion defined by the relation [MATH], asymptotic to the null paths [MATH] in the past and [MATH] in the future.', '1712.05206-1-3-1': 'In this work, we take the view that the fundamental entities are the field and not the particles.', '1712.05206-1-3-2': 'Therefore, it makes sense to transfer the distinction between massive and accelerated particles (i.e. systems satisfying different relations) directly to the fields.', '1712.05206-1-3-3': 'Our definition of accelerated fields rests on this observation.', '1712.05206-1-3-4': 'One of the main results of this paper is the argument that one can define accelerated fields by canonically quantizing the momentum space.', '1712.05206-1-3-5': 'Therefore, we need to proceed beyond the standard canonical field quantization in Minkowski spacetime, with the establishment of the field equation and the commutation relations in momentum space and then with the construction of the associated framework of observables and quantum states.', '1712.05206-1-3-6': 'Our procedure is in close analogy to the standard quantization of a free scalar field in flat spacetime and can be shortly described as follows.', '1712.05206-1-3-7': "First, following Schrodinger's reasoning, we transform the relativistic relation [MATH] into a wave equation in momentum space.", '1712.05206-1-3-8': 'Subsequently, after promoting the wave function to a field operator, we solve the field equation and quantize the resulting normal mode structure.', '1712.05206-1-3-9': 'It is then shown that the emerging notions of particles and vacuum states are different from those of the standard Fock representation.', '1712.05206-1-3-10': 'In the standard field quantization, excitations are interpreted as particles with specific mass, while in the case we study here, excitations are regarded as particles with constant acceleration.', '1712.05206-1-4-0': "Some time after Hawking's suggestion that black holes evaporate into thermal radiation [CITATION], Unruh, in an attempt to investigate how physically reasonable such a process is, concluded that an uniformly accelerating observer in the ordinary Minkowski vacuum will observe a thermal spectrum of Rindler particles [CITATION].", '1712.05206-1-4-1': 'In the present work we change the point of view and investigate the behavior of an accelerated particle detector with finite rest mass [MATH], finding that, when in the vacuum state, it behaves like an infinitely massive one in a thermal bath at temperature [MATH].', '1712.05206-1-5-0': 'Regarding Unruh effect, what we do is to perform the field quantization on a flat spacetime background in two different coordinate systems, Rindler for the accelerated observers and Minkowski for inertial ones.', '1712.05206-1-5-1': "The idea behind the definition of accelerated fields proposed in this work is to encode observer's acceleration directly to the field, offering a new way to address not only the Unruh effect, but also other properties of quantum field theories in non-inertial frames, while working in flat spacetime.", '1712.05206-1-5-2': 'A deeper understanding of Unruh effect is of crucial importance not only for fundamental questions, but also to engineer a practical method to experimentally detect such an effect.', '1712.05206-1-5-3': 'In the field of relativistic quantum information, the question of understanding entanglement in noninertial frames has became crucial.', '1712.05206-1-5-4': 'Previous analyses show that entanglement between modes of bosonic or fermionic fields is degraded from the perspective of observers moving in uniform acceleration [CITATION].', '1712.05206-1-5-5': 'Accelerated fields could then be a fruitful approach for addressing these and related questions.', '1712.05206-1-6-0': "Einstein's equivalence principle states that we cannot convince, by local effects, an accelerated observer that what he feels is not gravity.", '1712.05206-1-6-1': 'Although the main goal of the present work is not related with quantum gravity theories, we note that the quantization procedure presented here for accelerated field applies equally well, locally, to gravitational field.', '1712.05206-1-6-2': 'Therefore, keeping in mind that we have used only special relativity, we believe that quantized accelerated fields can be useful to shed new light in the search for a theory of quantum gravity.', '1712.05206-1-7-0': 'We proceed as follows.', '1712.05206-1-7-1': 'Section [REF] gives some background on the standard canonical field quantization from the point of view of inertial and accelerated observers, presenting some mathematical definitions that will be useful for later sections.', '1712.05206-1-7-2': 'Our main results are presented in section [REF], followed by the final discussions in Sec. [REF].', '1712.05206-1-7-3': 'In order to avoid unnecessary complications, we consider a quantum field theory in [MATH] dimension and adopt the metric signature [MATH].', '1712.05206-1-7-4': 'Furthermore, we shall use units such that [MATH] throughout the paper, unless specified otherwise.', '1712.05206-1-7-5': 'Quantities defined in or refereeing to the momentum space will be denoted with a tilde.', '1712.05206-1-8-0': '# Backgroung', '1712.05206-1-9-0': 'Specific elements in the definition of accelerating fields, as it will become clear in the next section, distinguish them from the massive fields.', '1712.05206-1-9-1': 'However, the canonical procedure of quantization we know for scalar fields, appropriately modified, can be applied also to accelerated fields.', '1712.05206-1-9-2': 'In this view, we present here the well known canonical field quantization on a flat spacetime background in order to set the stage for the quantization in momentum space, making the procedure comprehensible and less demanding.', '1712.05206-1-10-0': 'The simplest example of canonical field quantization in Minkowski space is a real scalar field described by the Klein-Gordon equation [EQUATION] whose Lagrangian density reads [EQUATION]', '1712.05206-1-10-1': 'Following the canonical procedure [CITATION], we define the conjugate momentum [EQUATION] and the Hamiltonian [EQUATION]', '1712.05206-1-10-2': 'The quantization of the field is performed by defining the field operators [MATH] and [MATH] and imposing the canonical equal-time commutation relations [EQUATION]', '1712.05206-1-10-3': 'The solutions to the Klein-Gordon equation ([REF]) can be given in terms of the set of plane waves [MATH] as [EQUATION] where [MATH] is a normalization factor while [MATH] is the annihilation operator for the field.', '1712.05206-1-10-4': 'By substituting this expression into Eq. ([REF]), we obtain the equation of motion for [MATH] in the following form [EQUATION] whose general solution is given by [EQUATION] with [MATH] and [MATH] being time independent and the frequency is defined by the relativistic dispersion relation [EQUATION]', '1712.05206-1-10-5': 'Since we have chosen to work with real-valued field, the field operator should be hermitian, [MATH].', '1712.05206-1-10-6': 'This constraint allows us to express one of [MATH] in terms of its counterpart and finally write the field operator as [EQUATION] where we have defined [EQUATION] with [MATH], as an orthonormal set of mode solutions of the Klein-Gordon equation.', '1712.05206-1-10-7': 'These modes form a complete set with respect to the Klein-Gordon inner product [EQUATION] where [MATH] is a future pointing normal vector to the spacelike hypersurface [MATH] and [MATH] is the volume element on [MATH] [CITATION].', '1712.05206-1-10-8': 'A feature distinguishing relativistic field theories from the non-relativistic ones is that the Klein-Gordon inner product is not positive definite [EQUATION]', '1712.05206-1-10-9': "One chooses the positive-frequency modes as [MATH]'s and, consequently, the negative-frequency modes as [MATH]'s, with temporal behavior [MATH] and [MATH], respectively.", '1712.05206-1-11-0': 'Equation ([REF]) defines the annihilation and creation operators, [MATH] and [MATH] satisfying the usual algebra [EQUATION]', '1712.05206-1-11-1': 'In this way, the positive-frequency (negative-frequency) modes are the coefficients of annihilation (creation) operators.', '1712.05206-1-11-2': 'This idea will also be applied to momentum space in the following section.', '1712.05206-1-12-0': 'Once we have these operators, the vacuum state [MATH] is defined by demanding that it is annihilated by each [MATH], [EQUATION]', '1712.05206-1-12-1': 'A state with [MATH] excitations of momenta [MATH] is obtained by the action of the creation operator on this vacuum state [EQUATION] while the number operator for each momentum [MATH], [MATH], obeys [EQUATION]', '1712.05206-1-12-2': 'In terms of the creation and annihilation operators, the Hamiltonian becomes [EQUATION] where we have used the dispersion relation [MATH].', '1712.05206-1-12-3': 'By similar logic, we construct an operator corresponding to the spacial components of the total momentum [EQUATION]', '1712.05206-1-12-4': 'Since the field excitations carry momentum and energy (and they are counted by the number operator), they are interpreted as particles.', '1712.05206-1-13-0': 'Turning now to the coordinate space, let us compute [EQUATION] which corresponds to the superposition of a single particle momentum eigenstate and thus represents a particle at position [MATH].', '1712.05206-1-13-1': 'Since we are considering a real scalar field, [MATH] corresponds to a particle at the same position [MATH] as well.', '1712.05206-1-14-0': 'Let us now ask how a uniformly accelerated observer describes the Minkowski vacuum.', '1712.05206-1-14-1': "We suppose the observer follows a [MATH] trajectory in Rindler coordinates [MATH], which are related to the coordinates [MATH] by [EQUATION] with [MATH] being the observer's proper acceleration and [MATH] the proper time along the trajectory.", '1712.05206-1-14-2': 'The Rindler coordinates cover only two portions of spacetime, namely the right Rindler wedge, [MATH], and the left Rindler wedge, [MATH].', '1712.05206-1-14-3': 'As pointed out in Ref. [CITATION], each of these Rindler wedges is actually a spacetime on its own.', '1712.05206-1-14-4': 'Hence, one can apply the canonical quantization procedure to each wedge taking a quantization of Minkowski space inequivalent to the normal Minkowski quantization [CITATION].', '1712.05206-1-14-5': 'If someone wants to determine explicitly the relation between these two kinds of quantization, he needs to compare the corresponding quantization modes.', '1712.05206-1-14-6': 'The comparison gives a relation between the Minkowski vacuum state and the product of the left and right wedge Rindler vacuums, symbolically expressed as [EQUATION] where the operator [MATH] is characterized by the parameter [MATH] [CITATION].', '1712.05206-1-15-0': 'A partial trace operation over the Rindler modes with respect to one of the wedges, gives us a thermal state characterized by the temperature [EQUATION]', '1712.05206-1-15-1': 'An alternative approach to the derivation of Unruh effect was presented in Ref. [CITATION].', '1712.05206-1-15-2': 'There, the author obtained Eq. ([REF]) by comparing the field correlation functions of a field in equilibrium at temperature [MATH] and a scalar field in the ground state as measured by an observer with uniform acceleration [MATH].', '1712.05206-1-16-0': '# Canonical field quantization in momentum space', '1712.05206-1-17-0': 'In this section we will extend the canonical quantization formulation we shortly described in Sec. [REF] to momentum space.', '1712.05206-1-17-1': 'In order to simplify notation, we will omit the hat over the operators since it will be clear from the context which quantities are operators and which are not.', '1712.05206-1-17-2': 'The physical motivation for this is the relativistic invariant relation which describes accelerated particles [EQUATION]', '1712.05206-1-17-3': 'Inserting the operators for position and time, [EQUATION] into Eq. ([REF]) we obtain the differential equation [EQUATION]', '1712.05206-1-17-4': 'This is a wave equation with an extra term associated with the proper acceleration.', '1712.05206-1-17-5': 'As a wave equation it bears some resemblance to Klein-Gordon equation.', '1712.05206-1-17-6': 'The difference arises from the fact that, contrary to Klein-Gordon equation, Eq. ([REF]) is associated with the spacelike equation shown in ([REF]).', '1712.05206-1-17-7': "We know that each solution of the d'Alembert wave equation [MATH] is a function in spatial part, [MATH], and [MATH]-dependent on the temporal part, [MATH], as a parameter.", '1712.05206-1-17-8': 'Of course, [MATH] can also be regarded as a function in [MATH] that is [MATH]-dependent on [MATH] as a parameter (see [CITATION]).', '1712.05206-1-17-9': 'While the set of solutions to the Klein-Gordon equation are parameterized according to the first case, this form of parameterization for the function [MATH] leads the obtained frequency to acquire an imaginary part, resulting in an unstable (exponential) growth.', '1712.05206-1-17-10': 'Thus, we choose to associate with [MATH] the family of functions [EQUATION] which are [MATH]-dependent on [MATH] and satisfy the differential equation [EQUATION]', '1712.05206-1-17-11': 'The Lagrange density of this equation can be constructed by inverting the Euler-Lagrange equation and is given by [EQUATION]', '1712.05206-1-17-12': 'Due to the chosen parameterization of [MATH], its uniqueness as solution of ([REF]) is satisfied when the initial data [MATH] are selected on a hypersurface of constant [MATH], or a timelike Cauchy surface.', '1712.05206-1-17-13': 'So far the field configuration is based only on the relativistic kinematic equation ([REF]), the energy-momentum relation has not been encompassed in any way, thus the shape of the fields is supposed not to change under translation in momentum space.', '1712.05206-1-17-14': "Therefore, from the homogeneity of the momentum space, Noether's theorem provides us with a conserved current given by [EQUATION]", '1712.05206-1-17-15': 'Particularly in our case, for the Lagrangian density ([REF]), the quantities that are "momentum" independent are [EQUATION]', '1712.05206-1-17-16': 'Translating this to the Hamiltonian description is a straightforward procedure, we just need to be careful with the peculiarity of our case.', '1712.05206-1-17-17': 'The conjugate momentum for [MATH] is defined as [EQUATION] thus leading to the Hamiltonian [EQUATION] in agreement with [MATH].', '1712.05206-1-18-0': 'Our next task is to find the spectrum of this Hamiltonian.', '1712.05206-1-18-1': 'We do this by finding the solutions of Eq. ([REF]).', '1712.05206-1-18-2': 'Our guide on this will be the procedure we followed to solve the Klein-Gordon equation in the preceding section.', '1712.05206-1-18-3': 'Using the set of plane waves [MATH], we expand [MATH] as [EQUATION] with [MATH] a normalization factor which will be fixed later.', '1712.05206-1-18-4': 'Equation ([REF]) then becomes [EQUATION] whose general solution can be written as [EQUATION] where [MATH] and [MATH] being momentum-independent and the frequency is now defined by the relation [EQUATION]', '1712.05206-1-18-5': 'Since we continue working with real fields, we must have [MATH].', '1712.05206-1-18-6': 'This constraint allows us to write the field operator as [EQUATION] where [EQUATION] with [MATH], defines the set of orthonormal-mode solutions of ([REF]).', '1712.05206-1-19-0': 'For this to make sense, we need to define an inner product on the space of solutions.', '1712.05206-1-19-1': 'The appropriate one is expressed as an integral over a constant-momentum hypersurface [MATH] [EQUATION] with [MATH] a normal vector to the time-like hypersurface [MATH].', '1712.05206-1-19-2': 'We can verify that the plane waves ([REF]), for [EQUATION] with respect to this product, form an orthonormal set [EQUATION]', '1712.05206-1-19-3': 'Note that, like the Klein-Gordon inner product, the inner product ([REF]) is not positive definite.', '1712.05206-1-19-4': "From this we can choose the positive-frequency modes as [MATH]'s and, consequently, the negative-frequency modes as [MATH]'s.", '1712.05206-1-19-5': 'It is important to emphasize that in this construction, unlike the Klein-Gordon case, the division of the modes into positive- and negative-frequency occurs with the spatial dependence of the type [MATH] and [MATH], respectively (see Fig. [REF]).', '1712.05206-1-20-0': 'If we define the annihilation and creation operators associated with [MATH] and [MATH] by the relations [EQUATION] we can write [MATH] and [MATH] as a linear combination of an infinite number of creation and annihilation operators [MATH] and [MATH], indexed by the time [MATH] as [EQUATION] and [EQUATION]', '1712.05206-1-20-1': 'The operators [MATH] and [MATH] should fulfill the typical algebra for creation and annihilation operators, i.e. [EQUATION]', '1712.05206-1-20-2': 'Due to Eqs. ([REF]) and ([REF]), the commutation relations ([REF]) in coordinate space are equivalent to the equal-momentum canonical commutation relations [EQUATION] in momentum space.', '1712.05206-1-20-3': 'This is the first important result of our formulation.', '1712.05206-1-20-4': 'It might seem disturbing at first sight, but it is consistent with the nature of the wave equation we employed to quantize the field in the momentum space.', '1712.05206-1-21-0': 'After we have defined the creation and annihilation operators, we can then express the Hamiltonian in terms of those operators [EQUATION]', '1712.05206-1-21-1': 'The second term is proportional to [MATH], an infinite c-number.', '1712.05206-1-21-2': 'It is the sum over all modes of the zero-point "positions" [MATH].', '1712.05206-1-21-3': 'We cannot avoid the existence of this term, since our treatment resembles that of harmonic oscillator and the infinite c-number term is the field analogue of the harmonic oscillator zero-point energy.', '1712.05206-1-21-4': 'Considerations coming from the standard field theory suggests discarding this term.', '1712.05206-1-22-0': 'Using Eq. ([REF]) for the Hamiltonian, it is straightforward to evaluate the commutators [EQUATION]', '1712.05206-1-22-1': 'We can then write down the spectrum of the theory.', '1712.05206-1-22-2': 'There will be a single vacuum state [MATH], characterized by the fact that it is annihilated by all [MATH], [EQUATION]', '1712.05206-1-22-3': 'All other eigenstates can be built by letting [MATH] acting on the vacuum.', '1712.05206-1-22-4': 'Let [EQUATION]', '1712.05206-1-22-5': 'Having found the spectrum of the Hamiltonian, let us try to interpret its eigenstates.', '1712.05206-1-22-6': 'By similar logic applied to the Hamiltonian [MATH], we can construct an operator corresponding to the one showed in Eq. ([REF]) [EQUATION]', '1712.05206-1-22-7': 'So the operator [MATH] creates excitations at time [MATH] and position [MATH].', '1712.05206-1-22-8': 'These excitations are discrete entities that have the proper relativistic kinematics relation which describes uniform accelerated particles.', '1712.05206-1-22-9': 'We claim that we can call these excitations accelerating particles and the associated fields as accelerated fields.', '1712.05206-1-22-10': 'By a particle here we mean something that is localized in space, since [MATH] creates space eigenstates.', '1712.05206-1-22-11': 'But this operator says nothing regarding the momentum of the particle.', '1712.05206-1-22-12': 'This information can be derived considering the interpretation of the state [MATH].', '1712.05206-1-22-13': 'From the expansion ([REF]) we see that [EQUATION] is a linear superposition of single particle states that have well-defined time-position, hence we will claim that the operator [MATH], acting on the vacuum, creates a particle with momentum [MATH] and energy [MATH].', '1712.05206-1-22-14': 'Note that, since we considered an hermitian field operator, [MATH] creates particles with the same momentum and energy as [MATH] does.', '1712.05206-1-22-15': 'Therefore, we quantized the accelerated field in the Schrödinger picture and interpreted the resulting theory in terms of accelerated relativistic particles.', '1712.05206-1-22-16': 'In the Heisenberg picture, the operator [MATH] allows us to relate [MATH] to [MATH] by means of the relation [EQUATION]', '1712.05206-1-22-17': 'For one particle state of well-defined time, [MATH] is the eigenvalue of [MATH].', '1712.05206-1-22-18': 'Equation ([REF]) makes explicit the dual particle and wave interpretations of the quantum field [MATH].', '1712.05206-1-22-19': 'On the one hand, [MATH] is written as a Hilbert space operator, which creates particles that are the quanta of the field.', '1712.05206-1-22-20': 'On the other hand, [MATH] is written as a linear combination of the solutions [MATH] of the wave equation ([REF]).', '1712.05206-1-22-21': 'If these were single-particle wavefunctions, they would correspond to states of positive and negative position; We have chosen to refer to them more generally as positive- and negative-frequency modes, keeping in mind the difference with the corresponding modes coming from Klein-Gordon equation.', '1712.05206-1-22-22': 'The connection between the particle creation operators and the waveforms has already been mentioned above.', '1712.05206-1-22-23': 'A positive-frequency solution of the field equation has as its coefficients the operator that destroys a particle in that particular mode.', '1712.05206-1-22-24': 'While a negative-frequency solution, being the Hermitian conjugate of a positive-frequency solution, has as its coefficient the operator that creates a particle in that positive-position single-particle wavefunction.', '1712.05206-1-23-0': 'To illustrate the difference between the Klein-Gordon field and the field theory we developed from the kinematic wave equation ([REF]) let us compute the quantities [MATH] and [MATH], which can be interpret as the position representation of the single-particle wavefunction of the state [MATH] and the momentum representation of the state [MATH], respectively.', '1712.05206-1-23-1': 'We find [EQUATION]', '1712.05206-1-23-2': 'In the first case we recognize the relativistic dispersion relation for a particle with mass [MATH], while in the second one we identify the relativistic kinematic relation for uniformly accelerated particle.', '1712.05206-1-24-0': '## Causality', '1712.05206-1-25-0': 'After the introduction of Heisenberg picture, we can proceed to the calculation of the amplitude for a particle to propagate from [MATH] to [MATH], which in our present formalism is [MATH].', '1712.05206-1-25-1': 'We will call this quantity [MATH] and we easily check that it is given by the integral [EQUATION]', '1712.05206-1-25-2': 'Let us evaluate this integral for some particular values of [MATH].', '1712.05206-1-26-0': 'First consider the case where the difference is: [MATH], [MATH].', '1712.05206-1-26-1': 'Then we have [EQUATION] where [MATH] is the modified Bessel function for [MATH].', '1712.05206-1-27-0': 'Next consider the case where [MATH], [MATH].', '1712.05206-1-27-1': 'The amplitude is [EQUATION]', '1712.05206-1-27-2': 'We find that the propagation amplitudes are not zero for particles propagating over all intervals, timelike and spacelike.', '1712.05206-1-28-0': 'A measurement performed at one point can affect a measurement at another point if the associated commutator does not vanish.', '1712.05206-1-28-1': 'Thus, a theory, in order to be causal, should require all the operators defined outside the light-cone to commute.', '1712.05206-1-28-2': 'Let us compute the commutator [MATH] whose general form is given by [EQUATION]', '1712.05206-1-28-3': 'As shown in Fig. [REF], when [MATH], being outside the light-cone, an appropriate continuous Lorentz transformation can take us from [MATH] to [MATH].', '1712.05206-1-28-4': 'If we perform this transformation on the second term in ([REF]), the two terms inside parenthesis are therefore equal and cancel.', '1712.05206-1-28-5': 'In case [MATH], there is no continuous Lorentz transformation that can take [MATH] to [MATH], thus the amplitude is nonzero.', '1712.05206-1-28-6': 'We then conclude that no measurement in our theory of accelerated fields can affect another measurement outside the light-cone.', '1712.05206-1-28-7': 'Hence, causality is maintained.', '1712.05206-1-28-8': "We intentionally didn't discuss causality by recruiting spacelike/timelike intervals.", '1712.05206-1-28-9': 'Contrary to Klein-Gordon theory, our field commutator vanishes for timelike separations while survives for spacelike separations.', '1712.05206-1-28-10': 'This outcome is consistent with our choice the role of "time" to be played by the spacelike coordinate [MATH], resulting in a light-cone with different direction (see Fig. [REF]) and interchanged notions of timelike and spacelike intervals.', '1712.05206-1-29-0': 'It is crucial, in order to satisfy the causality requirement, that we quantize the accelerated field using commutators.', '1712.05206-1-29-1': 'Had we used anticommutators instead of commutators, we would get a nonvanishing result for timelike intervals.', '1712.05206-1-29-2': 'Thus, this is an indication that the "spin-statistics theorem" also holds for relativistic particles.', '1712.05206-1-29-3': 'Another crucial element in our construction is essential for causality: The negative-space eigenfunctions in the field operators.', '1712.05206-1-29-4': 'Again, without negative-space contributions Eq. ([REF]) does not vanish for spacelike separations.', '1712.05206-1-30-0': '## How a massive observer describes the vacuum', '1712.05206-1-31-0': 'We have seen that in case the field is defined by the kinematics wave equation ([REF]), the corresponding annihilation and creation operators [MATH] and [MATH], annihilate and create accelerated particles of magnitude [MATH] at time [MATH] and position [MATH], while the field operator [MATH], acting on [MATH], corresponds to a particle with four-momentum [MATH].', '1712.05206-1-31-1': 'Note that, since the energy momentum relation has not been considered in any step of the procedure, the state [MATH], defined on a specific point of momentum space is not characterized by any value of mass.', '1712.05206-1-32-0': 'Let us consider now a relativistic observer with specific rest mass [MATH].', '1712.05206-1-32-1': 'The relativistic invariant energy-momentum relation [MATH], that the observer should satisfy, in momentum space turns into an equation of motion.', '1712.05206-1-32-2': 'Thus, for each fixed [MATH], this equation represents an observer with rest-mass [MATH] following the hyperbolic trajectory [MATH] given by [EQUATION] with [MATH] being a parameter.', '1712.05206-1-32-3': 'As [MATH] the paths fit ever more snugly into the limiting photon "paths" [MATH].', '1712.05206-1-33-0': 'We consider the field correlation function [MATH] at a point in momentum space for a field in equilibrium at temperature [MATH].', '1712.05206-1-33-1': 'In order to compute this, we impose [EQUATION] which simply implies that different modes of a thermal field are uncorrelated and that a mode of frequency [MATH] has an average number of quanta [MATH].', '1712.05206-1-33-2': 'Thus we take [EQUATION]', '1712.05206-1-33-3': 'For the field [MATH] satisfying the wave equation ([REF]) we consider the correlation function [MATH], computed in the vacuum state.', '1712.05206-1-33-4': 'In this case it holds [MATH] and [MATH].', '1712.05206-1-33-5': 'Therefore [EQUATION]', '1712.05206-1-33-6': 'The vacuum correlation function [MATH] measured by our massive observer is given by ([REF]) with [MATH] and [MATH].', '1712.05206-1-33-7': 'Since [EQUATION] it follows from Eq. ([REF]) that [EQUATION]', '1712.05206-1-33-8': 'Comparing the thermal-field correlation function ([REF]) with the correlation function ([REF]) we find that they are equivalent for the temperature [EQUATION]', '1712.05206-1-33-9': 'The meaning of this result is that a detector with finite rest mass [MATH] in the vacuum responds as an infinitely massive one in a thermal bath at temperature [MATH].', '1712.05206-1-33-10': 'The argument given for Hawking and Unruh effects for the structure of the vacuum near a black hole and acceleration horizon, respectively, applies equally well here.', '1712.05206-1-33-11': 'Let us compare Eq. ([REF]) with ([REF]).', '1712.05206-1-33-12': 'As [MATH] the temperature is red-shifted to zero, so an observer at infinity sees only the zero temperature vacuum.', '1712.05206-1-33-13': 'As the mass horizon, [MATH], is approached the observer sees a diverging temperature.', '1712.05206-1-34-0': '# Discussion', '1712.05206-1-35-0': 'In the framework of quantum field theory both photons and particles, placed on equal footing, are considered as derived concepts, arising from fields after quantization.', '1712.05206-1-35-1': 'Photons arise from the quantization of the electromagnetic field, while massive, charged particles with specific spin, arise from the quantization of matter fields.', '1712.05206-1-35-2': 'Extending this reasoning, in this work we showed how one can define accelerated scalar fields in the sense that their excitations will be uniformly accelerated scalar particles.', '1712.05206-1-36-0': 'The ground state of a quantum field is a highly structured state.', '1712.05206-1-36-1': 'From a global point of view, is the state identified by the zero eigenvalue for particle number operators.', '1712.05206-1-36-2': 'But locally it is seething with activity.', '1712.05206-1-36-3': "Local observables not only exhibit fluctuations, for example, fluctuations of charge densities produce very accurately predicted contributions to the magnetic moment of the electron, but also exhibit correlations as was first exhibited in Unruh's work [CITATION].", '1712.05206-1-36-4': 'The modern view of the vacuum is closely related to zero-point energy.', '1712.05206-1-36-5': 'All quantum fields have zero-point energies and vacuum fluctuations [CITATION].', '1712.05206-1-36-6': 'Considering a wave equation in momentum space, entails the fields in coordinate space to experience zero-point quantum correlations as well.', '1712.05206-1-36-7': 'Thus, according to this approach, vacuum fluctuations and vacuum correlations can be treated on an equal footing.', '1712.05206-1-37-0': 'Several directions follow from this work.', '1712.05206-1-37-1': 'The question of understanding entanglement in non-inertial frames has been central to the development of the emerging field of relativistic quantum information [CITATION].', '1712.05206-1-37-2': 'The approach presented in this work offers a new ground to address this fundamental question from a completely different point of view.', '1712.05206-1-38-0': 'A natural next step is to extend this analysis to fermionic fields, defining accelerated fermions.', '1712.05206-1-38-1': 'The second quantization of Eq. ([REF]) has brought out an important point.', '1712.05206-1-38-2': 'Causality and Lorentz invariance, given the "spin-statistics theorem", dictates the existence of two different kinematics field equations, one for non-spinning accelerated particles, which was studied in this work, and one for spinning accelerated particles, which will be studied in a future work.', '1712.05206-1-38-3': 'In addition, while the construction of the spin integrals can be considered as a solved problem for free particles, this is not true for the case for particles in external fields.', '1712.05206-1-38-4': 'Our approach may be used as a tool to address this important open problem.', '1712.05206-1-39-0': 'Finally, the full description of an accelerated, massive relativistic particle requires both the relativistic invariant energy-momentum and kinematics relations.', '1712.05206-1-39-1': 'A detailed description of this case will be given in a future publication.', '1712.05206-1-40-0': 'We would like to thanks George E. A. Matsas, Daniel Terno and Daniel A. T. Vanzella for discussions.', '1712.05206-1-40-1': 'We acknowledge financial support from the Brazilian funding agencies CNPq (Grants No. 401230/2014-7, 305086/2013-8 and 445516/2014-) and CAPES (Grant No. 6531/2014-08), the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ).'}

{'1712.05206-2-0-0': 'We propose a way to define uniformly accelerated fields.', '1712.05206-2-0-1': 'It is introduced by the operator version of the relativistic kinematics relation [MATH] being the acceleration) as the required field equation.', '1712.05206-2-0-2': 'We then apply the canonical procedure in order to perform the field quantization in the momentum space, claiming that the resulting excitations can be interpreted as uniformly accelerated particles.', '1712.05206-2-0-3': 'This leads to the quantization of the spacetime in the same sense that energy and momentum are quantized in the usual quantum field theory.', '1712.05206-2-0-4': 'The behavior of accelerated particle detectors with definite mass is investigated, and it is shown that a massive detector experiences a temperature inversely proportional to its mass and, consequently, finds the vacuum state full of accelerated particles.', '1712.05206-2-1-0': '# Introduction', '1712.05206-2-2-0': 'Quantum field theory is the best physical theory that we have in order to describe our world.', '1712.05206-2-2-1': 'In this view, quantum fields are the essential elements and everything else can by derived as a consequence of the dynamics of such fields.', '1712.05206-2-2-2': 'The only exception is the gravitational field.', '1712.05206-2-2-3': "This happens because, according to Einstein's general relativity, gravity is a property of the spacetime, the stage on what every other field acts.", '1712.05206-2-3-0': 'Einstein’s equivalence principle states that we cannot convince, by local effects, an accelerated observer that what he feels is not gravity.', '1712.05206-2-3-1': 'It helps in analyzing gravitation to consider a situation where gravity is mocked up by acceleration.', '1712.05206-2-3-2': 'In this perspective, Unruh, in an attempt to investigate how physically reasonable is Hawking radiation [CITATION], concluded that an uniformly accelerated observer in the ordinary Minkowski vacuum will observe a thermal spectrum of Rindler particles [CITATION].', '1712.05206-2-4-0': 'Accelerated motion and accelerated observers can be analyzed using special relativity.', '1712.05206-2-4-1': 'Since quantum field theories are subject to the rules of special relativity and quantum mechanics, we can ask if there is any quantum field theory capable of embracing acceleration as a fundamental characteristic of the fields.', '1712.05206-2-4-2': 'In this paper we answer this question in the affirmative by defining the accelerated fields.', '1712.05206-2-4-3': 'In this way, we change the point of view, from the observer to the field itself.', '1712.05206-2-5-0': 'Schrodinger wrote the first relativistic wave equation [CITATION] based on two assumptions: ([MATH]) the energy-momentum relation for a massive particle is given by [MATH] and ([MATH]) the assumption that the particle dynamics would be described by a wave function.', '1712.05206-2-5-1': "Then, by following the same reasoning that lead to de Broglie's theory, he was able to find the dynamical equation for this wave function.", '1712.05206-2-6-0': 'Now, in accordance with special relativity, a particle moving at a uniform acceleration [MATH] undergoes a hyperbolic motion defined by the relation [MATH], asymptotic to the null paths [MATH] in the past and [MATH] in the future.', '1712.05206-2-6-1': 'Here, by taking the view that the fundamental entities are the fields and not the particles, we transfer the kinematical description of the particles directly to the fields.', '1712.05206-2-7-0': 'Our definition of accelerated fields rests on this observation.', '1712.05206-2-7-1': 'One of the main results of this paper is the argument that one can define accelerated fields by canonically quantizing the momentum space.', '1712.05206-2-7-2': 'Therefore, we need to proceed beyond the standard canonical field quantization in Minkowski spacetime, with the establishment of the field equation and the commutation relations in momentum space and then with the construction of the associated framework of observables and quantum states.', '1712.05206-2-7-3': 'Our procedure is in close analogy to the standard quantization of a free scalar field in flat spacetime and can be shortly described as follows.', '1712.05206-2-7-4': 'First, following Schrodinger’s reasoning, we transform the relativistic relation [MATH] into a wave equation in momentum space.', '1712.05206-2-7-5': 'Subsequently, after promoting the wave function to a field operator, we solve the field equation and quantize the resulting normal mode structure.', '1712.05206-2-8-0': 'It is then shown that the emerging notions of particles and vacuum states are different from those of the standard Fock representation.', '1712.05206-2-8-1': 'In the theory of accelerated fields, the wave-particle duality appears in the association of the angular frequency and wave vector with space and time, respectively.', '1712.05206-2-8-2': 'Here, space and time are quantized, analogously to the way quantities like energy and momentum are quantized in usual quantum field theories.', '1712.05206-2-9-0': 'By construction, the theory of accelerated fields does not convey any information regarding mass.', '1712.05206-2-9-1': 'Since accelerated fields have been established in momentum space, the consideration of the energy momentum relation modifies the metric of the space (since for any point it holds [MATH]).', '1712.05206-2-9-2': 'As a result we find an effect similar to Unruh effect for usual quantum field theories, but now the key-role is on the mass and not on acceleration.', '1712.05206-2-10-0': 'By considering a quantum field theory in [MATH] dimension and adopting the metric signature [MATH], our main results are presented in section [REF].', '1712.05206-2-10-1': 'We dedicate Sec. [REF] for our final discussions, including a possible extension of the theory to [MATH] dimensions.', '1712.05206-2-10-2': 'Furthermore, we shall use units such that [MATH] throughout the paper.', '1712.05206-2-10-3': 'Quantities defined in or refereeing to the momentum space will be denoted with a tilde.', '1712.05206-2-11-0': '# Canonical field quantization in momentum space', '1712.05206-2-12-0': 'Specific elements in the definition of accelerating fields distinguish them from the massive fields.', '1712.05206-2-12-1': 'However, the canonical procedure of quantization we know for scalar fields [CITATION], appropriately modified, can be applied also to accelerated fields.', '1712.05206-2-12-2': 'The physical motivation for this is the relativistic invariant relation which describes accelerated particles [EQUATION]', '1712.05206-2-12-3': 'Inserting the operators for position and time, [EQUATION] into Eq. ([REF]) we obtain the differential equation [EQUATION]', '1712.05206-2-12-4': 'This is a wave equation with an extra term associated with the proper acceleration.', '1712.05206-2-12-5': 'As a wave equation it bears some resemblance to Klein-Gordon equation.', '1712.05206-2-12-6': 'The difference arises from the fact that, contrary to Klein-Gordon equation, Eq. ([REF]) is associated with the spacelike equation shown in ([REF]).', '1712.05206-2-12-7': "We know that each solution of the d'Alembert wave equation [MATH] is a function in spatial part, [MATH], and [MATH]-dependent on the temporal part, [MATH], as a parameter.", '1712.05206-2-12-8': 'Of course, [MATH] can also be regarded as a function in [MATH] that is [MATH]-dependent on [MATH] as a parameter (see [CITATION]).', '1712.05206-2-12-9': 'While the set of solutions to the Klein-Gordon equation are parameterized according to the first case, this form of parameterization for the function [MATH] leads the obtained frequency to acquire an imaginary part, resulting in an unstable (exponential) growth.', '1712.05206-2-12-10': 'Thus, we choose to associate with [MATH] the family of functions [EQUATION] which are [MATH]-dependent on [MATH] and satisfy the differential equation [EQUATION]', '1712.05206-2-12-11': 'The Lagrange density of this equation can be constructed by inverting the Euler-Lagrange equation and is given by [EQUATION]', '1712.05206-2-12-12': 'Due to the chosen parameterization of [MATH], its uniqueness as solution of ([REF]) is satisfied when the initial data [MATH] are selected on a hypersurface of constant [MATH], or a timelike Cauchy surface.', '1712.05206-2-12-13': 'So far the field configuration is based only on the relativistic kinematic equation ([REF]), the energy-momentum relation has not been encompassed in any way, thus the shape of the fields is supposed not to change under translation in momentum space.', '1712.05206-2-12-14': "Therefore, from the homogeneity of the momentum space, Noether's theorem provides us with a conserved current given by [EQUATION]", '1712.05206-2-12-15': 'Particularly in our case, for the Lagrangian density ([REF]), the quantities that are "momentum" independent are [EQUATION]', '1712.05206-2-12-16': 'Translating this to the Hamiltonian description is a straightforward procedure, we just need to be careful with the peculiarity of our case.', '1712.05206-2-12-17': 'The conjugate momentum for [MATH] is defined as [EQUATION] thus leading to the Hamiltonian [EQUATION] in agreement with [MATH].', '1712.05206-2-13-0': 'Our next task is to find the spectrum of this Hamiltonian.', '1712.05206-2-13-1': 'We do this by finding the solutions of Eq. ([REF]).', '1712.05206-2-13-2': 'Our guide on this will be the procedure it is followed to solve the Klein-Gordon equation.', '1712.05206-2-13-3': 'Using the set of plane waves [MATH], we expand [MATH] as [EQUATION] with [MATH] a normalization factor which will be fixed later.', '1712.05206-2-13-4': 'Equation ([REF]) then becomes [EQUATION] whose general solution can be written as [EQUATION] where [MATH] and [MATH] being momentum-independent and the frequency is now defined by the relation [EQUATION]', '1712.05206-2-13-5': 'Since we work with real fields, we must have [MATH].', '1712.05206-2-13-6': 'This constraint allows us to write the field operator as [EQUATION] where [EQUATION] with [MATH], defines the set of orthonormal-mode solutions of ([REF]).', '1712.05206-2-14-0': 'For this to make sense, we need to define an inner product on the space of solutions.', '1712.05206-2-14-1': 'The appropriate one is expressed as an integral over a constant-momentum hypersurface [MATH] [EQUATION] with [MATH] a normal vector to the time-like hypersurface [MATH].', '1712.05206-2-14-2': 'We can verify that the plane waves ([REF]), for [EQUATION] with respect to this product, form an orthonormal set [EQUATION]', '1712.05206-2-14-3': 'Note that, like the Klein-Gordon inner product, the inner product ([REF]) is not positive definite.', '1712.05206-2-14-4': "From this we can choose the positive-frequency modes as [MATH]'s and, consequently, the negative-frequency modes as [MATH]'s.", '1712.05206-2-14-5': 'It is important to emphasize that in this construction, unlike the Klein-Gordon case, the division of the modes into positive- and negative-frequency occurs with the spatial dependence of the type [MATH] and [MATH], respectively (see Fig. [REF]).', '1712.05206-2-15-0': 'If we define the annihilation and creation operators associated with [MATH] and [MATH] by the relations [EQUATION] we can write [MATH] and [MATH] as a linear combination of an infinite number of creation and annihilation operators [MATH] and [MATH], indexed by the time [MATH] as [EQUATION] and [EQUATION]', '1712.05206-2-15-1': 'The operators [MATH] and [MATH] should fulfill the typical algebra for creation and annihilation operators, i.e. [EQUATION]', '1712.05206-2-15-2': 'Due to Eqs. ([REF]) and ([REF]), the commutation relations ([REF]) in coordinate space are equivalent to the equal-momentum canonical commutation relations [EQUATION] in momentum space.', '1712.05206-2-15-3': 'This is the first important result of our formulation.', '1712.05206-2-15-4': 'It might seem disturbing at first sight, but it is consistent with the nature of the wave equation we employed to quantize the field in the momentum space.', '1712.05206-2-16-0': 'After we have defined the creation and annihilation operators, we can then express the Hamiltonian in terms of those operators [EQUATION]', '1712.05206-2-16-1': 'The second term is proportional to [MATH], an infinite c-number.', '1712.05206-2-16-2': 'It is the sum over all modes of the zero-point "positions" [MATH].', '1712.05206-2-16-3': 'We cannot avoid the existence of this term, since our treatment resembles that of harmonic oscillator and the infinite c-number term is the field analogue of the harmonic oscillator zero-point energy.', '1712.05206-2-16-4': 'Considerations coming from the standard field theory suggests discarding this term.', '1712.05206-2-17-0': 'Using Eq. ([REF]) for the Hamiltonian, it is straightforward to evaluate the commutators [EQUATION]', '1712.05206-2-17-1': 'We can then write down the spectrum of the theory.', '1712.05206-2-17-2': 'There will be a single vacuum state [MATH], characterized by the fact that it is annihilated by all [MATH], [EQUATION]', '1712.05206-2-17-3': 'All other eigenstates can be built by letting [MATH] acting on the vacuum.', '1712.05206-2-17-4': 'Let [EQUATION]', '1712.05206-2-17-5': 'Having found the spectrum of the Hamiltonian, let us try to interpret its eigenstates.', '1712.05206-2-17-6': 'By similar logic applied to the Hamiltonian [MATH], we can construct an operator corresponding to the one showed in Eq. ([REF]) [EQUATION]', '1712.05206-2-17-7': 'So the operator [MATH] creates excitations at time [MATH] and position [MATH].', '1712.05206-2-17-8': 'These excitations are discrete entities that have the proper relativistic kinematics relation which describes uniform accelerated particles.', '1712.05206-2-17-9': 'We claim that we can call these excitations accelerating particles and the associated fields as accelerated fields.', '1712.05206-2-17-10': 'By a particle here we mean something that is localized in space, since [MATH] creates space eigenstates.', '1712.05206-2-17-11': 'But this operator says nothing regarding the momentum of the particle.', '1712.05206-2-17-12': 'This information can be derived considering the interpretation of the state [MATH].', '1712.05206-2-17-13': 'From the expansion ([REF]) we see that [EQUATION] is a linear superposition of single particle states that have well-defined time-position, hence we will claim that the operator [MATH], acting on the vacuum, creates a particle with momentum [MATH] and energy [MATH].', '1712.05206-2-17-14': 'Note that, since we considered an hermitian field operator, [MATH] creates particles with the same momentum and energy as [MATH] does.', '1712.05206-2-17-15': 'Therefore, we quantized the accelerated field in the Schrödinger picture and interpreted the resulting theory in terms of accelerated relativistic particles.', '1712.05206-2-17-16': 'In the Heisenberg picture, the operator [MATH] allows us to relate [MATH] to [MATH] by means of the relation [EQUATION]', '1712.05206-2-17-17': 'For one particle state of well-defined time, [MATH] is the eigenvalue of [MATH].', '1712.05206-2-17-18': 'Equation ([REF]) makes explicit the dual particle and wave interpretations of the quantum field [MATH].', '1712.05206-2-17-19': 'On the one hand, [MATH] is written as a Hilbert space operator, which creates particles that are the quanta of the field.', '1712.05206-2-17-20': 'On the other hand, [MATH] is written as a linear combination of the solutions [MATH] of the wave equation ([REF]).', '1712.05206-2-17-21': 'If these were single-particle wavefunctions, they would correspond to states of positive and negative position; We have chosen to refer to them more generally as positive- and negative-frequency modes, keeping in mind the difference with the corresponding modes coming from Klein-Gordon equation.', '1712.05206-2-17-22': 'The connection between the particle creation operators and the waveforms has already been mentioned above.', '1712.05206-2-17-23': 'A positive-frequency solution of the field equation has as its coefficients the operator that destroys a particle in that particular mode.', '1712.05206-2-17-24': 'While a negative-frequency solution, being the Hermitian conjugate of a positive-frequency solution, has as its coefficient the operator that creates a particle in that positive-position single-particle wavefunction.', '1712.05206-2-18-0': 'Let us consider a real scalar field described by the Klein-Gordon equation [MATH].', '1712.05206-2-18-1': 'To illustrate the difference between the Klein-Gordon field and the field theory we developed from the kinematic wave equation ([REF]) let us compute the quantities [MATH] and [MATH], which can be interpret as the position representation of the single-particle wavefunction of the state [MATH] and the momentum representation of the state [MATH], respectively.', '1712.05206-2-18-2': 'We find [EQUATION]', '1712.05206-2-18-3': 'In the first case we recognize the relativistic dispersion relation for a particle with mass [MATH], while in the second one we identify the relativistic kinematic relation for uniformly accelerated particle.', '1712.05206-2-19-0': '## Causality', '1712.05206-2-20-0': 'After the introduction of Heisenberg picture, we can proceed to the calculation of the amplitude for a particle to propagate from [MATH] to [MATH], which in our present formalism is [MATH].', '1712.05206-2-20-1': 'We will call this quantity [MATH] and we easily check that it is given by the integral [EQUATION]', '1712.05206-2-20-2': 'Let us evaluate this integral for some particular values of [MATH].', '1712.05206-2-21-0': 'First consider the case where the difference is: [MATH], [MATH].', '1712.05206-2-21-1': 'Then we have [EQUATION] where [MATH] is the modified Bessel function for [MATH].', '1712.05206-2-22-0': 'Next consider the case where [MATH], [MATH].', '1712.05206-2-22-1': 'The amplitude is [EQUATION]', '1712.05206-2-22-2': 'We find that the propagation amplitudes are not zero for particles propagating over all intervals, timelike and spacelike.', '1712.05206-2-23-0': 'A measurement performed at one point can affect a measurement at another point if the associated commutator does not vanish.', '1712.05206-2-23-1': 'Thus, a theory, in order to be causal, should require all the operators defined outside the light-cone to commute.', '1712.05206-2-23-2': 'Let us compute the commutator [MATH] whose general form is given by [EQUATION]', '1712.05206-2-23-3': 'As shown in Fig. [REF], when [MATH], being outside the light-cone, an appropriate continuous Lorentz transformation can take us from [MATH] to [MATH].', '1712.05206-2-23-4': 'If we perform this transformation on the second term in ([REF]), the two terms inside parenthesis are therefore equal and cancel.', '1712.05206-2-23-5': 'In case [MATH], there is no continuous Lorentz transformation that can take [MATH] to [MATH], thus the amplitude is nonzero.', '1712.05206-2-23-6': 'We then conclude that no measurement in our theory of accelerated fields can affect another measurement outside the light-cone.', '1712.05206-2-23-7': 'Hence, causality is maintained.', '1712.05206-2-23-8': "We intentionally didn't discuss causality by recruiting spacelike/timelike intervals.", '1712.05206-2-23-9': 'Contrary to Klein-Gordon theory, our field commutator vanishes for timelike separations while survives for spacelike separations.', '1712.05206-2-23-10': 'This outcome is consistent with our choice the role of "time" to be played by the spacelike coordinate [MATH], resulting in a light-cone with different direction (see Fig. [REF]) and interchanged notions of timelike and spacelike intervals.', '1712.05206-2-24-0': 'It is crucial, in order to satisfy the causality requirement, that we quantize the accelerated field using commutators.', '1712.05206-2-24-1': 'Had we used anticommutators instead of commutators, we would get a nonvanishing result for timelike intervals.', '1712.05206-2-24-2': 'Thus, this is an indication that the "spin-statistics theorem" also holds for relativistic particles.', '1712.05206-2-24-3': 'Another crucial element in our construction is essential for causality: The negative-space eigenfunctions in the field operators.', '1712.05206-2-24-4': 'Again, without negative-space contributions Eq. ([REF]) does not vanish for spacelike separations.', '1712.05206-2-25-0': '## How a massive observer describes the vacuum', '1712.05206-2-26-0': 'We have seen that in case the field is defined by the kinematics wave equation ([REF]), the corresponding annihilation and creation operators [MATH] and [MATH], annihilate and create accelerated particles of magnitude [MATH] at time [MATH] and position [MATH], while the field operator [MATH], acting on [MATH], corresponds to a particle with four-momentum [MATH].', '1712.05206-2-26-1': 'Note that, since the energy momentum relation has not been considered in any step of the procedure, the state [MATH], defined on a specific point of momentum space is not characterized by any value of mass.', '1712.05206-2-27-0': 'Let us consider now a relativistic observer with specific rest mass [MATH].', '1712.05206-2-27-1': 'The relativistic invariant energy-momentum relation [MATH], that the observer should satisfy, in momentum space turns into an equation of motion.', '1712.05206-2-27-2': 'Thus, for each fixed [MATH], this equation represents an observer with rest-mass [MATH] following the hyperbolic trajectory [MATH] given by [EQUATION] with [MATH] being a parameter.', '1712.05206-2-27-3': 'As [MATH] the paths fit ever more snugly into the limiting photon "paths" [MATH].', '1712.05206-2-28-0': 'We consider the field correlation function [MATH] at a point in momentum space for a field in equilibrium at temperature [MATH].', '1712.05206-2-28-1': 'In order to compute this, we impose [EQUATION] which simply implies that different modes of a thermal field are uncorrelated and that a mode of frequency [MATH] has an average number of quanta [MATH].', '1712.05206-2-28-2': 'Thus we take [EQUATION]', '1712.05206-2-28-3': 'For the field [MATH] satisfying the wave equation ([REF]) we consider the correlation function [MATH], computed in the vacuum state.', '1712.05206-2-28-4': 'In this case it holds [MATH] and [MATH].', '1712.05206-2-28-5': 'Therefore [EQUATION]', '1712.05206-2-28-6': 'The vacuum correlation function [MATH] measured by our massive observer is given by ([REF]) with [MATH] and [MATH].', '1712.05206-2-28-7': 'Since [EQUATION] it follows from Eq. ([REF]) that [EQUATION]', '1712.05206-2-28-8': 'Comparing the thermal-field correlation function ([REF]) with the correlation function ([REF]) we find that they are equivalent for the temperature [EQUATION]', '1712.05206-2-28-9': 'The meaning of this result is that a detector with finite rest mass [MATH] in the vacuum responds as an infinitely massive one in a thermal bath at temperature [MATH].', '1712.05206-2-28-10': 'The argument given for Hawking and Unruh effects for the structure of the vacuum near a black hole and acceleration horizon, respectively, applies equally well here.', '1712.05206-2-28-11': 'Let us compare Eq. ([REF]) with ([REF]).', '1712.05206-2-28-12': 'As [MATH] the temperature is red-shifted to zero, so an observer at infinity sees only the zero temperature vacuum.', '1712.05206-2-28-13': 'As the mass horizon, [MATH], is approached the observer sees a diverging temperature.', '1712.05206-2-29-0': '# Discussion', '1712.05206-2-30-0': 'Although we have considered field theories in [MATH] dimensions, our results can be extended to physical dimensions.', '1712.05206-2-30-1': 'In terms of a global inertial coordinate system [MATH], let us consider the killing field which generates a boost about the origin in the [MATH] direction.', '1712.05206-2-30-2': 'In this case, the hyperbolic surface [MATH] is invariant under translation in [MATH] and [MATH] direction.', '1712.05206-2-30-3': 'Given the fact that the definition of accelerated fields is based on this hyperbolic cylinder surface, in our theory, accelerated fields become invariant under translations in [MATH] and [MATH] direction.', '1712.05206-2-30-4': 'This clearly reproduces all the results presented above.', '1712.05206-2-30-5': 'In the general case, we will have three dimensional surface that will be projected into the planes ([MATH]), ([MATH]) and ([MATH]), thus defining three independent accelerated fields.', '1712.05206-2-30-6': 'This reflects the fact that, in our theory, the geometry is translated into fields.', '1712.05206-2-31-0': 'Even though gravity, as described in general relativity, appears not to be compatible with the quantum field theories, this is not the case for the accelerated fields.', '1712.05206-2-31-1': 'In this work we propose a new quantum field theory, that encodes acceleration directly into the fields and provides a mathematical consistent way to locally quantize spacetime.', '1712.05206-2-31-2': 'This is a strong evidence that not only gravity but also acceleration can reshape the notions of space and time.', '1712.05206-2-32-0': 'The problem of finding a quantum theory of gravity is the main problem in modern theoretical physics.', '1712.05206-2-32-1': 'Today, the two main approaches are string theory and loop quantum gravity.', '1712.05206-2-32-2': 'These theories addresses the problem of quantum gravity in totally different ways.', '1712.05206-2-32-3': 'String theory is based on the idea of a fundamental description of gravity in terms of physical excitations of a string that lives in a background metric space.', '1712.05206-2-32-4': 'Loop quantum gravity on the other hand, in an attempt to comprehend what quantum spacetime is at the fundamental level, was formulated without a background spacetime.', '1712.05206-2-33-0': 'The theory of accelerated fields shares elements from both of these theories.', '1712.05206-2-33-1': 'Even though the description of acceleration is in terms of excitations over a background metric space, the fact that this space is the momentum space allows the quantization of spacetime, the background of all the others fields.', '1712.05206-2-33-2': 'From this view one can define accelerated Klein-Gordon fields establishing a notion of scalar fields being located with respect to the accelerated field.', '1712.05206-2-34-0': 'Several directions follow from this work.', '1712.05206-2-34-1': 'For instance, the question of understanding entanglement in non-inertial frames has been central to the development of the emerging field of relativistic quantum information [CITATION].', '1712.05206-2-34-2': 'The approach presented in this work offers a new ground to address this fundamental question from a completely different point of view.', '1712.05206-2-35-0': 'We would like to thanks George E. A. Matsas, Daniel Terno and Daniel A. T. Vanzella for discussions.', '1712.05206-2-35-1': 'We acknowledge financial support from the Brazilian funding agencies CNPq (Grants No. 401230/2014-7, 305086/2013-8 and 445516/2014-) and CAPES (Grant No. 6531/2014-08), the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ).'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1712.05206

{'1712.05206-3-0-0': 'Fictitious forces arise in general relativity where they were used by Einstein in the geometrization of gravity.', '1712.05206-3-0-1': 'We believe that fictitious forces could play a key role also in quantization of gravity.', '1712.05206-3-0-2': 'We propose a way to encode acceleration directly into field, thus establishing a new kind of fields, the accelerated fields.', '1712.05206-3-0-3': 'Accelerated fields have some very interesting properties.', '1712.05206-3-0-4': 'They can be considered as the quantum field analog of fictitious forces since they are absent in an inertial frame.', '1712.05206-3-0-5': 'They provides a mathematical consistent way to quantize spacetime in the same sense that energy and momentum are quantized in the usual quantum field theory.', '1712.05206-3-0-6': 'Mass never appears in the equations of motion of the accelerated field, something that is true also for gravitational field.', '1712.05206-3-0-7': 'The consideration of the energy momentum relation results in an effect similar to Unruh effect for usual quantum field theories, but the key-role is on the mass and not on acceleration.', '1712.05206-3-1-0': '# Introduction', '1712.05206-3-2-0': 'The description of a classical system from a non-inertial reference frame exhibits fictitious forces.', '1712.05206-3-2-1': 'These forces do not stem from any physical interaction, being solely a consequence of the non-inertial feature of the reference frame itself.', '1712.05206-3-2-2': 'The notion of fictitious forces also emerge in general theory of relativity.', '1712.05206-3-2-3': 'Einstein used accelerated systems and their accompanying fictitious forces, as a guide in order to comprehend the nature of gravity in classical physics.', '1712.05206-3-2-4': "Einstein's motivation was the observation that mass never appears in the equations of motion under fictitious forces, a feature which is also encountered with gravity, already known from Galileo's famous experiments.", '1712.05206-3-3-0': 'Quantum field theory is the best physical model we have in order to describe our world.', '1712.05206-3-3-1': 'All the fundamental forces of physics are described within this framework, except one, gravity.', '1712.05206-3-3-2': 'Several difficulties arise from the application of the usual prescription of quantum field theory to the quantization of the gravitational field as described by general relativity.', '1712.05206-3-3-3': 'This fact led theorists to consider more radical approaches to the problem of quantum gravity.', '1712.05206-3-3-4': 'Among them, loop quantum gravity which provides a genuine quantum theory of geometry [CITATION].', '1712.05206-3-4-0': 'Einstein concluded that spacetime and the gravitational field are the same physical quantity after recognizing the equivalence between fictitious forces and gravitational forces.', '1712.05206-3-4-1': 'Due to the importance of fictitious forces in the geometrization of gravity we believe that fictitious forces could play a key role also in quantization of gravity.', '1712.05206-3-4-2': "A natural question that comes up is the role of these forces in quantum field theory and whether Einstein's idea can also be applied in quantum theory.", '1712.05206-3-4-3': 'The investigation of these issues is the main purpose of this paper.', '1712.05206-3-5-0': 'As is well known, accelerated systems can be analyzed using special relativity.', '1712.05206-3-5-1': 'Since quantum field theories are subject to the rules of special relativity and quantum mechanics, we can ask if there is any quantum field theory capable of embracing acceleration as a fundamental characteristic of the fields.', '1712.05206-3-5-2': 'We answer this question in the affirmative by defining the accelerated quantum fields.', '1712.05206-3-5-3': 'The structure of accelerated quantum fields permits us to interpret them as fictitious in the sense that they disappear in an inertial reference frame.', '1712.05206-3-6-0': 'Schrodinger wrote the first relativistic wave equation [CITATION] based on two assumptions: ([MATH]) the energy-momentum relation for a massive particle is given by [MATH] and ([MATH]) the particle dynamics should be described by a wave function.', '1712.05206-3-6-1': 'Our definition of accelerated fields rests on the kinematical description of the particles and the assumption that their kinematics are described by a different wave function.', '1712.05206-3-6-2': 'Our procedure is close to the standard quantization of a free scalar field in flat spacetime and can be shortly described as follows.', '1712.05206-3-6-3': 'First, following Schrodinger’s reasoning, we transform the relativistic relation [MATH] into a wave equation in momentum space.', '1712.05206-3-6-4': 'Subsequently, after promoting the wave function to a field operator, we solve the field equation and quantize the resulting normal mode structure.', '1712.05206-3-7-0': 'One of the main results of this paper is the argument that one can define accelerated fields by canonically quantizing the momentum space.', '1712.05206-3-7-1': 'Therefore, we need to proceed beyond the standard canonical field quantization in Minkowski spacetime, with the establishment of the field equation and the commutation relations in momentum space and, then, with the construction of the associated framework of observables and quantum states.', '1712.05206-3-7-2': 'It is then shown that the emerging notions of particles and vacuum states are different from those of the standard Fock representation.', '1712.05206-3-7-3': 'In the theory of accelerated fields, the wave-particle duality appears in the association of the angular frequency and wave vector with space and time, respectively.', '1712.05206-3-7-4': 'Here, space and time are quantized, analogously to the way quantities like energy and momentum are quantized in usual quantum field theories.', '1712.05206-3-8-0': 'In a different approach, Sanchez investigated the spacetime structure with quantum theory by promoting the length [MATH] and the time [MATH] to quantum non-commutative coordinates [CITATION].', '1712.05206-3-8-1': 'In that work, she found the quantum light cone which is generated by the hyperbolae [MATH] and replaces the classical light cone.', '1712.05206-3-8-2': 'Our conclusions regarding accelerated field support such result.', '1712.05206-3-9-0': 'The idea of quantized spacetime dates back to the early days of quantum theory as a way to eliminate infinities from quantum field theories [CITATION] or just to survey the consequences of this assumption [CITATION].', '1712.05206-3-9-1': 'Later it became obvious that quantum spacetime is closely related to proposed theories of quantum gravity.', '1712.05206-3-9-2': 'While it is generally accepted that spacetime is quantized, there is disagreement as to how quantization manifests itself [CITATION].', '1712.05206-3-9-3': 'In this work we advance arguments that inertial acceleration encoded as quantum field provides a mathematical consistent way to quantize spacetime.', '1712.05206-3-10-0': 'It is important to notice that the mass never appears in the equations of motion of the accelerated field.', '1712.05206-3-10-1': 'Something that is true also for gravitational fields.', '1712.05206-3-10-2': 'If this is the only criterion for the identification of a field as gravitational, the presence of accelerated fields can never be distinguished from the presence of certain gravitational field.', '1712.05206-3-10-3': 'Based on this we reveal a feature that may be attributed to quantum gravity.', '1712.05206-3-10-4': 'By construction, the theory of accelerated fields does not convey any information regarding mass.', '1712.05206-3-10-5': 'Since accelerated fields have been established in momentum space, the consideration of the energy momentum relation modifies the metric of the space (since for any point it holds [MATH]).', '1712.05206-3-10-6': 'As a result we find an effect similar to Unruh effect for usual quantum field theories, but now the key-role is on the mass and not on acceleration.', '1712.05206-3-10-7': 'This result is necessary to keep the consistency of accelerated field theory from the point of view of a massive observer.', '1712.05206-3-11-0': 'By considering a quantum field theory in [MATH] dimension and adopting the metric signature [MATH], our main results are presented in section [REF].', '1712.05206-3-11-1': 'We dedicate Sec. [REF] for our final discussions, including a possible extension of the theory to [MATH] dimensions.', '1712.05206-3-11-2': 'Furthermore, we shall use units such that [MATH] throughout the paper.', '1712.05206-3-11-3': 'Quantities defined in or refereeing to the momentum space will be denoted with a tilde.', '1712.05206-3-12-0': '# Canonical field quantization in momentum space', '1712.05206-3-13-0': 'Specific elements in the definition of accelerating fields distinguish them from the massive fields.', '1712.05206-3-13-1': 'However, the canonical procedure of quantization we know for scalar fields [CITATION], appropriately modified, can be applied also to accelerated fields.', '1712.05206-3-13-2': 'The physical motivation for this is the relativistic invariant relation which describes accelerated particles [EQUATION]', '1712.05206-3-13-3': 'Inserting the operators for position and time, [EQUATION] into Eq. ([REF]) we obtain the differential equation [EQUATION]', '1712.05206-3-13-4': 'This is a wave equation with an extra term associated with the proper acceleration.', '1712.05206-3-13-5': 'As a wave equation it bears some resemblance to Klein-Gordon equation.', '1712.05206-3-13-6': 'The difference arises from the fact that, contrary to Klein-Gordon equation, Eq. ([REF]) is associated with the spacelike equation shown in ([REF]).', '1712.05206-3-13-7': "We know that each solution of the d'Alembert wave equation [MATH] is a function in spatial part, [MATH], and [MATH]-dependent on the temporal part, [MATH], as a parameter.", '1712.05206-3-13-8': 'Of course, [MATH] can also be regarded as a function in [MATH] that is [MATH]-dependent on [MATH] as a parameter (see [CITATION]).', '1712.05206-3-13-9': 'While the set of solutions to the Klein-Gordon equation are parameterized according to the first case, this form of parameterization for the function [MATH] leads the obtained frequency to acquire an imaginary part, resulting in an unstable (exponential) growth.', '1712.05206-3-13-10': 'Thus, we choose to associate with [MATH] the family of functions [EQUATION] which are [MATH]-dependent on [MATH] and satisfy the differential equation [EQUATION]', '1712.05206-3-13-11': 'The Lagrange density of this equation can be constructed by inverting the Euler-Lagrange equation and is given by [EQUATION]', '1712.05206-3-13-12': 'Due to the chosen parameterization of [MATH], its uniqueness as solution of ([REF]) is satisfied when the initial data [MATH] are selected on a hypersurface of constant [MATH], or a timelike Cauchy surface.', '1712.05206-3-13-13': 'So far the field configuration is based only on the relativistic kinematic equation ([REF]), the energy-momentum relation has not been encompassed in any way, thus the shape of the fields is supposed not to change under translation in momentum space.', '1712.05206-3-13-14': "Therefore, from the homogeneity of the momentum space, Noether's theorem provides us with a conserved current given by [EQUATION]", '1712.05206-3-13-15': 'Particularly in our case, for the Lagrangian density ([REF]), the quantities that are "momentum" independent are [EQUATION]', '1712.05206-3-13-16': 'Translating this to the Hamiltonian description is a straightforward procedure, we just need to be careful with the peculiarity of our case.', '1712.05206-3-13-17': 'The conjugate momentum for [MATH] is defined as [EQUATION] thus leading to the Hamiltonian [EQUATION] in agreement with [MATH].', '1712.05206-3-14-0': 'Our next task is to find the spectrum of this Hamiltonian.', '1712.05206-3-14-1': 'We do this by finding the solutions of Eq. ([REF]).', '1712.05206-3-14-2': 'Our guide on this will be the procedure it is followed to solve the Klein-Gordon equation.', '1712.05206-3-14-3': 'Using the set of plane waves [MATH], we expand [MATH] as [EQUATION] with [MATH] a normalization factor which will be fixed later.', '1712.05206-3-14-4': 'Equation ([REF]) then becomes [EQUATION] whose general solution can be written as [EQUATION] where [MATH] and [MATH] being momentum-independent and the frequency is now defined by the relation [EQUATION]', '1712.05206-3-14-5': 'Since we work with real fields, we must have [MATH].', '1712.05206-3-14-6': 'This constraint allows us to write the field operator as [EQUATION] where [EQUATION] with [MATH], defines the set of orthonormal-mode solutions of ([REF]).', '1712.05206-3-15-0': 'For this to make sense, we need to define an inner product on the space of solutions.', '1712.05206-3-15-1': 'The appropriate one is expressed as an integral over a constant-momentum hypersurface [MATH] [EQUATION] with [MATH] a normal vector to the time-like hypersurface [MATH].', '1712.05206-3-15-2': 'We can verify that the plane waves ([REF]), for [EQUATION] with respect to this product, form an orthonormal set [EQUATION]', '1712.05206-3-15-3': 'Note that, like the Klein-Gordon inner product, the inner product ([REF]) is not positive definite.', '1712.05206-3-15-4': "From this we can choose the positive-frequency modes as [MATH]'s and, consequently, the negative-frequency modes as [MATH]'s.", '1712.05206-3-15-5': 'It is important to emphasize that in this construction, unlike the Klein-Gordon case, the division of the modes into positive- and negative-frequency occurs with the spatial dependence of the type [MATH] and [MATH], respectively (see Fig. [REF]).', '1712.05206-3-16-0': 'If we define the annihilation and creation operators associated with [MATH] and [MATH] by the relations [EQUATION] we can write [MATH] and [MATH] as a linear combination of an infinite number of creation and annihilation operators [MATH] and [MATH], indexed by the time [MATH] as [EQUATION] and [EQUATION]', '1712.05206-3-16-1': 'The operators [MATH] and [MATH] should fulfill the typical algebra for creation and annihilation operators, i.e. [EQUATION]', '1712.05206-3-16-2': 'Due to Eqs. ([REF]) and ([REF]), the commutation relations ([REF]) in coordinate space are equivalent to the equal-momentum canonical commutation relations [EQUATION] in momentum space.', '1712.05206-3-16-3': 'This is the first important result of our formulation.', '1712.05206-3-16-4': 'It might seem disturbing at first sight, but it is consistent with the nature of the wave equation we employed to quantize the field in the momentum space.', '1712.05206-3-17-0': 'After we have defined the creation and annihilation operators, we can then express the Hamiltonian in terms of those operators [EQUATION]', '1712.05206-3-17-1': 'The second term is proportional to [MATH], an infinite c-number.', '1712.05206-3-17-2': 'It is the sum over all modes of the zero-point "positions" [MATH].', '1712.05206-3-17-3': 'We cannot avoid the existence of this term, since our treatment resembles that of harmonic oscillator and the infinite c-number term is the field analogue of the harmonic oscillator zero-point energy.', '1712.05206-3-17-4': 'Considerations coming from the standard field theory suggests discarding this term.', '1712.05206-3-18-0': 'Using Eq. ([REF]) for the Hamiltonian, it is straightforward to evaluate the commutators [EQUATION]', '1712.05206-3-18-1': 'We can then write down the spectrum of the theory.', '1712.05206-3-18-2': 'There will be a single vacuum state [MATH], characterized by the fact that it is annihilated by all [MATH], [EQUATION]', '1712.05206-3-18-3': 'All other eigenstates can be built by letting [MATH] acting on the vacuum.', '1712.05206-3-18-4': 'Let [EQUATION]', '1712.05206-3-18-5': 'Having found the spectrum of the Hamiltonian, let us try to interpret its eigenstates.', '1712.05206-3-18-6': 'By similar logic applied to the Hamiltonian [MATH], we can construct an operator corresponding to the one showed in Eq. ([REF]) [EQUATION]', '1712.05206-3-18-7': 'So the operator [MATH] creates excitations at time [MATH] and position [MATH].', '1712.05206-3-18-8': 'These excitations are discrete entities that have the proper relativistic kinematics relation which describes uniform accelerated particles.', '1712.05206-3-18-9': 'We claim that we can call these excitations accelerating particles and the associated fields as accelerated fields.', '1712.05206-3-18-10': 'By a particle here we mean something that is localized in space, since [MATH] creates space eigenstates.', '1712.05206-3-18-11': 'But this operator says nothing regarding the momentum of the particle.', '1712.05206-3-18-12': 'This information can be derived considering the interpretation of the state [MATH].', '1712.05206-3-18-13': 'From the expansion ([REF]) we see that [EQUATION] is a linear superposition of single particle states that have well-defined time-position, hence we will claim that the operator [MATH], acting on the vacuum, creates a particle with momentum [MATH] and energy [MATH].', '1712.05206-3-18-14': 'Note that, since we considered an hermitian field operator, [MATH] creates particles with the same momentum and energy as [MATH] does.', '1712.05206-3-18-15': 'Therefore, we quantized the accelerated field in the Schrödinger picture and interpreted the resulting theory in terms of accelerated relativistic particles.', '1712.05206-3-18-16': 'In the Heisenberg picture, the operator [MATH] allows us to relate [MATH] to [MATH] by means of the relation [EQUATION]', '1712.05206-3-18-17': 'For one particle state of well-defined time, [MATH] is the eigenvalue of [MATH].', '1712.05206-3-18-18': 'Equation ([REF]) makes explicit the dual particle and wave interpretations of the quantum field [MATH].', '1712.05206-3-18-19': 'On the one hand, [MATH] is written as a Hilbert space operator, which creates particles that are the quanta of the field.', '1712.05206-3-18-20': 'On the other hand, [MATH] is written as a linear combination of the solutions [MATH] of the wave equation ([REF]).', '1712.05206-3-18-21': 'If these were single-particle wavefunctions, they would correspond to states of positive and negative position; We have chosen to refer to them more generally as positive- and negative-frequency modes, keeping in mind the difference with the corresponding modes coming from Klein-Gordon equation.', '1712.05206-3-18-22': 'The connection between the particle creation operators and the waveforms has already been mentioned above.', '1712.05206-3-18-23': 'A positive-frequency solution of the field equation has as its coefficients the operator that destroys a particle in that particular mode.', '1712.05206-3-18-24': 'While a negative-frequency solution, being the Hermitian conjugate of a positive-frequency solution, has as its coefficient the operator that creates a particle in that positive-position single-particle wavefunction.', '1712.05206-3-19-0': 'Let us consider a real scalar field described by the Klein-Gordon equation [MATH].', '1712.05206-3-19-1': 'To illustrate the difference between the Klein-Gordon field and the field theory we developed from the kinematic wave equation ([REF]) let us compute the quantities [MATH] and [MATH], which can be interpret as the position representation of the single-particle wavefunction of the state [MATH] and the momentum representation of the state [MATH], respectively.', '1712.05206-3-19-2': 'We find [EQUATION]', '1712.05206-3-19-3': 'In the first case we recognize the relativistic dispersion relation for a particle with mass [MATH], while in the second one we identify the relativistic kinematic relation for uniformly accelerated particle.', '1712.05206-3-20-0': '## Causality', '1712.05206-3-21-0': 'After the introduction of Heisenberg picture, we can proceed to the calculation of the amplitude for a particle to propagate from [MATH] to [MATH], which in our present formalism is [MATH].', '1712.05206-3-21-1': 'We will call this quantity [MATH] and we easily check that it is given by the integral [EQUATION]', '1712.05206-3-21-2': 'Let us evaluate this integral for some particular values of [MATH].', '1712.05206-3-22-0': 'First consider the case where the difference is: [MATH], [MATH].', '1712.05206-3-22-1': 'Then we have [EQUATION] where [MATH] is the modified Bessel function for [MATH].', '1712.05206-3-23-0': 'Next consider the case where [MATH], [MATH].', '1712.05206-3-23-1': 'The amplitude is [EQUATION]', '1712.05206-3-23-2': 'We find that the propagation amplitudes are not zero for particles propagating over all intervals, timelike and spacelike.', '1712.05206-3-24-0': 'A measurement performed at one point can affect a measurement at another point if the associated commutator does not vanish.', '1712.05206-3-24-1': 'Thus, a theory, in order to be causal, should require all the operators defined outside the light-cone to commute.', '1712.05206-3-24-2': 'Let us compute the commutator [MATH] whose general form is given by [EQUATION]', '1712.05206-3-24-3': 'As shown in Fig. [REF], when [MATH], being outside the light-cone, an appropriate continuous Lorentz transformation can take us from [MATH] to [MATH].', '1712.05206-3-24-4': 'If we perform this transformation on the second term in ([REF]), the two terms inside parenthesis are therefore equal and cancel.', '1712.05206-3-24-5': 'In case [MATH], there is no continuous Lorentz transformation that can take [MATH] to [MATH], thus the amplitude is nonzero.', '1712.05206-3-24-6': 'We then conclude that no measurement in our theory of accelerated fields can affect another measurement outside the light-cone.', '1712.05206-3-24-7': 'Hence, causality is maintained.', '1712.05206-3-24-8': "We intentionally didn't discuss causality by recruiting spacelike/timelike intervals.", '1712.05206-3-24-9': 'Contrary to Klein-Gordon theory, our field commutator vanishes for timelike separations while survives for spacelike separations.', '1712.05206-3-24-10': 'This outcome is consistent with our choice the role of "time" to be played by the spacelike coordinate [MATH], resulting in a light-cone with different direction (see Fig. [REF]) and interchanged notions of timelike and spacelike intervals.', '1712.05206-3-25-0': 'It is crucial, in order to satisfy the causality requirement, that we quantize the accelerated field using commutators.', '1712.05206-3-25-1': 'Had we used anticommutators instead of commutators, we would get a nonvanishing result for timelike intervals.', '1712.05206-3-25-2': 'Thus, this is an indication that the "spin-statistics theorem" also holds for relativistic particles.', '1712.05206-3-25-3': 'Another crucial element in our construction is essential for causality: The negative-space eigenfunctions in the field operators.', '1712.05206-3-25-4': 'Again, without negative-space contributions Eq. ([REF]) does not vanish for spacelike separations.', '1712.05206-3-26-0': '## How a massive observer describes the vacuum', '1712.05206-3-27-0': 'We have seen that in case the field is defined by the kinematics wave equation ([REF]), the corresponding annihilation and creation operators [MATH] and [MATH], annihilate and create accelerated particles of magnitude [MATH] at time [MATH] and position [MATH], while the field operator [MATH], acting on [MATH], corresponds to a particle with four-momentum [MATH].', '1712.05206-3-27-1': 'Note that, since the energy momentum relation has not been considered in any step of the procedure, the state [MATH], defined on a specific point of momentum space is not characterized by any value of mass.', '1712.05206-3-28-0': 'Let us consider now a relativistic observer with specific rest mass [MATH].', '1712.05206-3-28-1': 'The relativistic invariant energy-momentum relation [MATH], that the observer should satisfy, in momentum space turns into an equation of motion.', '1712.05206-3-28-2': 'Thus, for each fixed [MATH], this equation represents an observer with rest-mass [MATH] following the hyperbolic trajectory [MATH] given by [EQUATION] with [MATH] being a parameter.', '1712.05206-3-28-3': 'As [MATH] the paths fit ever more snugly into the limiting photon "paths" [MATH].', '1712.05206-3-29-0': 'We consider the field correlation function [MATH] at a point in momentum space for a field in equilibrium at temperature [MATH].', '1712.05206-3-29-1': 'In order to compute this, we impose [EQUATION] which simply implies that different modes of a thermal field are uncorrelated and that a mode of frequency [MATH] has an average number of quanta [MATH].', '1712.05206-3-29-2': 'Thus we take [EQUATION]', '1712.05206-3-29-3': 'For the field [MATH] satisfying the wave equation ([REF]) we consider the correlation function [MATH], computed in the vacuum state.', '1712.05206-3-29-4': 'In this case it holds [MATH] and [MATH].', '1712.05206-3-29-5': 'Therefore [EQUATION]', '1712.05206-3-29-6': 'The vacuum correlation function [MATH] measured by our massive observer is given by ([REF]) with [MATH] and [MATH].', '1712.05206-3-29-7': 'Since [EQUATION] it follows from Eq. ([REF]) that [EQUATION]', '1712.05206-3-29-8': 'Comparing the thermal-field correlation function ([REF]) with the correlation function ([REF]) we find that they are equivalent for the temperature [EQUATION]', '1712.05206-3-29-9': 'The meaning of this result is that a detector with finite rest mass [MATH] in the vacuum responds as an infinitely massive one in a thermal bath at temperature [MATH].', '1712.05206-3-29-10': 'The argument given for Hawking and Unruh effects for the structure of the vacuum near a black hole and acceleration horizon, respectively, applies equally well here.', '1712.05206-3-29-11': 'Let us compare Eq. ([REF]) with ([REF]).', '1712.05206-3-29-12': 'As [MATH] the temperature is red-shifted to zero, so an observer at infinity sees only the zero temperature vacuum.', '1712.05206-3-29-13': 'As the mass horizon, [MATH], is approached the observer sees a diverging temperature.', '1712.05206-3-30-0': '# Discussion', '1712.05206-3-31-0': 'Although we have considered field theories in [MATH] dimensions, our results can be extended to physical dimensions.', '1712.05206-3-31-1': 'In terms of a global inertial coordinate system [MATH], let us consider the killing field which generates a boost about the origin in the [MATH] direction.', '1712.05206-3-31-2': 'In this case, the hyperbolic surface [MATH] is invariant under translation in [MATH] and [MATH] direction.', '1712.05206-3-31-3': 'Given the fact that the definition of accelerated fields is based on this hyperbolic cylinder surface, in our theory, accelerated fields become invariant under translations in [MATH] and [MATH] direction.', '1712.05206-3-31-4': 'This clearly reproduces all the results presented above.', '1712.05206-3-31-5': 'In the general case, we will have three dimensional surface that will be projected into the planes ([MATH]), ([MATH]) and ([MATH]), thus defining three independent accelerated fields.', '1712.05206-3-31-6': 'This reflects the fact that, in our theory, geometry is translated into fields.', '1712.05206-3-32-0': 'Even though gravity, as described in general relativity, appears not to be compatible with the quantum field theories, this is not the case for the accelerated fields.', '1712.05206-3-32-1': 'In this work we propose a new quantum field theory, that encodes acceleration directly into the fields and provides a mathematical consistent way to locally quantize spacetime.', '1712.05206-3-32-2': 'This is a strong evidence that not only gravity but also acceleration can reshape the notions of space and time.', '1712.05206-3-33-0': 'The problem of finding a quantum theory of gravity is the main problem in modern theoretical physics.', '1712.05206-3-33-1': 'Today, the two main approaches are string theory and loop quantum gravity.', '1712.05206-3-33-2': 'These theories address the problem of quantum gravity in totally different ways.', '1712.05206-3-33-3': 'String theory is based on the idea of a fundamental description of gravity in terms of physical excitations of a string that lives in a background metric space.', '1712.05206-3-33-4': 'Loop quantum gravity, on the other hand, in an attempt to comprehend what quantum spacetime is at the fundamental level, was formulated without a background spacetime.', '1712.05206-3-34-0': 'The theory of accelerated fields shares elements from both of these theories.', '1712.05206-3-34-1': 'Even though the description of acceleration takes place in terms of excitations over a background metric space, the fact that this space is the momentum space allows the quantization of spacetime, the background of all the others fields.', '1712.05206-3-34-2': 'From this view one can define accelerated Klein-Gordon fields establishing a notion of scalar fields being located with respect to the accelerated field.', '1712.05206-3-35-0': 'The theory of accelerated fields, as presented here, is an effort to quantize spacetime, while it is kept separate from the matter fields.', '1712.05206-3-35-1': 'In a subsequent study, we intend to investigate the effects of accelerated fields on the Klein-Gordon massive field, by defining accelerated Klein-Gordon massive fields and comparing them with the inertial ones.', '1712.05206-3-36-0': 'Several other directions follow from this work.', '1712.05206-3-36-1': 'For instance, the question of understanding entanglement in non-inertial frames has been central to the development of the emerging field of relativistic quantum information [CITATION].', '1712.05206-3-36-2': 'The approach presented in this work offers a new ground to address this fundamental question from a completely different point of view.', '1712.05206-3-37-0': 'We would like to thanks George E. A. Matsas, Daniel Terno and Daniel A. T. Vanzella for discussions.', '1712.05206-3-37-1': 'We acknowledge financial support from the Brazilian funding agencies CNPq (Grants No. 401230/2014-7, 305086/2013-8 and 445516/2014-) and CAPES (Grant No. 6531/2014-08), the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ).'}

{'1712.05206-4-0-0': 'We propose a way to encode acceleration directly into quantum fields, establishing a new class of fields.', '1712.05206-4-0-1': 'Accelerated quantum fields, as we have named them, have some very interesting properties.', '1712.05206-4-0-2': 'The most important is that they provide a mathematically consistent way to quantize space-time in the same way that energy and momentum are quantized in standard quantum field theories.', '1712.05206-4-1-0': '# Introduction', '1712.05206-4-2-0': 'The description of a classical system from a non-inertial reference frame exhibits fictitious forces.', '1712.05206-4-2-1': 'These forces do not stem from any physical interaction, being solely a consequence of the non-inertial feature of the reference frame itself.', '1712.05206-4-2-2': 'The notion of fictitious forces also emerge in general theory of relativity.', '1712.05206-4-2-3': 'Einstein used accelerated systems and their accompanying fictitious forces, as a guide in order to comprehend the nature of gravity in classical physics.', '1712.05206-4-2-4': "Einstein's motivation was the observation that mass never appears in the equations of motion under fictitious forces, a feature which is also encountered with gravity, already known from Galileo's famous experiments.", '1712.05206-4-3-0': 'Quantum field theory is the best physical model we have in order to describe our world.', '1712.05206-4-3-1': 'All the fundamental forces of physics are described within this framework, except one, gravity.', '1712.05206-4-3-2': 'Several difficulties arise from the application of the usual prescription of quantum field theory to the quantization of the gravitational field as described by general relativity.', '1712.05206-4-3-3': 'This fact led theorists to consider more radical approaches to the problem of quantum gravity.', '1712.05206-4-3-4': 'Among them, loop quantum gravity which provides a genuine quantum theory of geometry [CITATION].', '1712.05206-4-4-0': 'Einstein concluded that spacetime and the gravitational field are the same physical quantity after recognizing the equivalence between fictitious forces and gravitational forces.', '1712.05206-4-4-1': 'Due to the importance of fictitious forces in the geometrization of gravity we believe that fictitious forces could play a key role also in quantization of gravity.', '1712.05206-4-4-2': "A natural question that comes up is the role of these forces in quantum field theory and whether Einstein's idea can also be applied in quantum theory.", '1712.05206-4-4-3': 'The investigation of these issues is the main purpose of this paper.', '1712.05206-4-5-0': 'As is well known, accelerated systems can be analyzed using special relativity.', '1712.05206-4-5-1': 'Since quantum field theories are subject to the rules of special relativity and quantum mechanics, we can ask if there is any quantum field theory capable of embracing acceleration as a fundamental characteristic of the fields.', '1712.05206-4-5-2': 'We answer this question in the affirmative by defining the accelerated quantum fields.', '1712.05206-4-5-3': 'The structure of accelerated quantum fields permits us to interpret them as fictitious in the sense that they disappear in an inertial reference frame.', '1712.05206-4-6-0': 'Schrodinger wrote the first relativistic wave equation [CITATION] based on two assumptions: ([MATH]) the energy-momentum relation for a massive particle is given by [MATH] and ([MATH]) the particle dynamics should be described by a wave function.', '1712.05206-4-6-1': 'Our definition of accelerated fields rests on the kinematical description of the particles and the assumption that their kinematics are described by a different wave function.', '1712.05206-4-6-2': 'Our procedure is close to the standard quantization of a free scalar field in flat spacetime and can be shortly described as follows.', '1712.05206-4-6-3': 'First, following Schrodinger’s reasoning, we transform the relativistic relation [MATH] into a wave equation in momentum space.', '1712.05206-4-6-4': 'Subsequently, after promoting the wave function to a field operator, we solve the field equation and quantize the resulting normal mode structure.', '1712.05206-4-7-0': 'One of the main results of this paper is the argument that one can define accelerated fields by canonically quantizing the momentum space.', '1712.05206-4-7-1': 'Therefore, we need to proceed beyond the standard canonical field quantization in Minkowski spacetime, with the establishment of the field equation and the commutation relations in momentum space and, then, with the construction of the associated framework of observables and quantum states.', '1712.05206-4-7-2': 'It is then shown that the emerging notions of particles and vacuum states are different from those of the standard Fock representation.', '1712.05206-4-7-3': 'In the theory of accelerated fields, the wave-particle duality appears in the association of the angular frequency and wave vector with space and time, respectively.', '1712.05206-4-7-4': 'Here, space and time are quantized, analogously to the way quantities like energy and momentum are quantized in usual quantum field theories.', '1712.05206-4-8-0': 'In a different approach, Sanchez investigated the spacetime structure with quantum theory by promoting the length [MATH] and the time [MATH] to quantum non-commutative coordinates [CITATION].', '1712.05206-4-8-1': 'In that work, she found the quantum light cone which is generated by the hyperbolae [MATH] and replaces the classical light cone.', '1712.05206-4-8-2': 'Our conclusions regarding accelerated field support such result.', '1712.05206-4-8-3': 'Furthermore, keeping in mind that the mathematical language it is used is that of quantum field theory and not that of string theory, our formulation integrates some of the ideas developed in metastring theory [CITATION].', '1712.05206-4-9-0': 'The idea of quantized spacetime dates back to the early days of quantum theory as a way to eliminate infinities from quantum field theories [CITATION] or just to survey the consequences of this assumption [CITATION].', '1712.05206-4-9-1': 'Later it became obvious that quantum spacetime is closely related to proposed theories of quantum gravity.', '1712.05206-4-9-2': 'While it is generally accepted that spacetime is quantized, there is disagreement as to how quantization manifests itself [CITATION].', '1712.05206-4-9-3': 'In this work we advance arguments that inertial acceleration encoded as quantum field provides a mathematical consistent way to quantize spacetime.', '1712.05206-4-10-0': 'By considering a quantum field theory in [MATH] dimension and adopting the metric signature [MATH], our main results are presented in section [REF].', '1712.05206-4-10-1': 'We dedicate Sec. [REF] for our final discussions, including a possible extension of the theory to [MATH] dimensions.', '1712.05206-4-10-2': 'Furthermore, we shall use units such that [MATH] throughout the paper.', '1712.05206-4-10-3': 'Quantities defined in or refereeing to the momentum space will be denoted with a tilde.', '1712.05206-4-11-0': '# Canonical field quantization in momentum space', '1712.05206-4-12-0': 'Specific elements in the definition of accelerating fields distinguish them from the massive fields.', '1712.05206-4-12-1': 'However, the canonical procedure of quantization we know for scalar fields [CITATION], appropriately modified, can be applied also to accelerated fields.', '1712.05206-4-12-2': 'The physical motivation for this is the relativistic invariant relation which describes accelerated particles [EQUATION]', '1712.05206-4-12-3': 'Inserting the operators for position and time, [EQUATION] into Eq. ([REF]) we obtain the differential equation [EQUATION]', '1712.05206-4-12-4': 'This is a wave equation with an extra term associated with the proper acceleration.', '1712.05206-4-12-5': 'As a wave equation it bears some resemblance to Klein-Gordon equation.', '1712.05206-4-12-6': 'The difference arises from the fact that, contrary to Klein-Gordon equation, Eq. ([REF]) is associated with the spacelike equation shown in ([REF]).', '1712.05206-4-12-7': "We know that each solution of the d'Alembert wave equation [MATH] is a function in spatial part, [MATH], and [MATH]-dependent on the temporal part, [MATH], as a parameter.", '1712.05206-4-12-8': 'Of course, [MATH] can also be regarded as a function in [MATH] that is [MATH]-dependent on [MATH] as a parameter (see [CITATION]).', '1712.05206-4-12-9': 'While the set of solutions to the Klein-Gordon equation are parameterized according to the first case, this form of parameterization for the function [MATH] leads the obtained frequency to acquire an imaginary part, resulting in an unstable (exponential) growth.', '1712.05206-4-12-10': 'Thus, we choose to associate with [MATH] the family of functions [EQUATION] which are [MATH]-dependent on [MATH] and satisfy the differential equation [EQUATION]', '1712.05206-4-12-11': 'The Lagrange density of this equation can be constructed by inverting the Euler-Lagrange equation and is given by [EQUATION]', '1712.05206-4-12-12': 'Due to the chosen parameterization of [MATH], its uniqueness as solution of ([REF]) is satisfied when the initial data [MATH] are selected on a hypersurface of constant [MATH], or a timelike Cauchy surface.', '1712.05206-4-12-13': 'So far the field configuration is based only on the relativistic kinematic equation ([REF]), the energy-momentum relation has not been encompassed in any way, thus the shape of the fields is supposed not to change under translation in momentum space.', '1712.05206-4-12-14': "Therefore, from the homogeneity of the momentum space, Noether's theorem provides us with a conserved current given by [EQUATION]", '1712.05206-4-12-15': 'Particularly in our case, for the Lagrangian density ([REF]), the quantities that are "momentum" independent are [EQUATION]', '1712.05206-4-12-16': 'Translating this to the Hamiltonian description is a straightforward procedure, we just need to be careful with the peculiarity of our case.', '1712.05206-4-12-17': 'The conjugate momentum for [MATH] is defined as [EQUATION] thus leading to the Hamiltonian [EQUATION] in agreement with [MATH].', '1712.05206-4-13-0': 'Our next task is to find the spectrum of this Hamiltonian.', '1712.05206-4-13-1': 'We do this by finding the solutions of Eq. ([REF]).', '1712.05206-4-13-2': 'Our guide on this will be the procedure it is followed to solve the Klein-Gordon equation.', '1712.05206-4-13-3': 'Using the set of plane waves [MATH], we expand [MATH] as [EQUATION] with [MATH] a normalization factor which will be fixed later.', '1712.05206-4-13-4': 'Equation ([REF]) then becomes [EQUATION] whose general solution can be written as [EQUATION] where [MATH] and [MATH] being momentum-independent and the frequency is now defined by the relation [EQUATION]', '1712.05206-4-13-5': 'Since we work with real fields, we must have [MATH].', '1712.05206-4-13-6': 'This constraint allows us to write the field operator as [EQUATION] where [EQUATION] with [MATH], defines the set of orthonormal-mode solutions of ([REF]).', '1712.05206-4-14-0': 'For this to make sense, we need to define an inner product on the space of solutions.', '1712.05206-4-14-1': 'The appropriate one is expressed as an integral over a constant-momentum hypersurface [MATH] [EQUATION] with [MATH] a normal vector to the time-like hypersurface [MATH].', '1712.05206-4-14-2': 'We can verify that the plane waves ([REF]), for [EQUATION] with respect to this product, form an orthonormal set [EQUATION]', '1712.05206-4-14-3': 'Note that, like the Klein-Gordon inner product, the inner product ([REF]) is not positive definite.', '1712.05206-4-14-4': "From this we can choose the positive-frequency modes as [MATH]'s and, consequently, the negative-frequency modes as [MATH]'s.", '1712.05206-4-14-5': 'It is important to emphasize that in this construction, unlike the Klein-Gordon case, the division of the modes into positive- and negative-frequency occurs with the spatial dependence of the type [MATH] and [MATH], respectively (see Fig. [REF]).', '1712.05206-4-15-0': 'If we define the annihilation and creation operators associated with [MATH] and [MATH] by the relations [EQUATION] we can write [MATH] and [MATH] as a linear combination of an infinite number of creation and annihilation operators [MATH] and [MATH], indexed by the time [MATH] as [EQUATION] and [EQUATION]', '1712.05206-4-15-1': 'The operators [MATH] and [MATH] should fulfill the typical algebra for creation and annihilation operators, i.e. [EQUATION]', '1712.05206-4-15-2': 'Due to Eqs. ([REF]) and ([REF]), the commutation relations ([REF]) in coordinate space are equivalent to the equal-momentum canonical commutation relations [EQUATION] in momentum space.', '1712.05206-4-15-3': 'This is the first important result of our formulation.', '1712.05206-4-15-4': 'It might seem disturbing at first sight, but it is consistent with the nature of the wave equation we employed to quantize the field in the momentum space.', '1712.05206-4-16-0': 'After we have defined the creation and annihilation operators, we can then express the Hamiltonian in terms of those operators [EQUATION]', '1712.05206-4-16-1': 'The second term is proportional to [MATH], an infinite c-number.', '1712.05206-4-16-2': 'It is the sum over all modes of the zero-point "positions" [MATH].', '1712.05206-4-16-3': 'We cannot avoid the existence of this term, since our treatment resembles that of harmonic oscillator and the infinite c-number term is the field analogue of the harmonic oscillator zero-point energy.', '1712.05206-4-16-4': 'Considerations coming from the standard field theory suggests discarding this term.', '1712.05206-4-17-0': 'Using Eq. ([REF]) for the Hamiltonian, it is straightforward to evaluate the commutators [EQUATION]', '1712.05206-4-17-1': 'We can then write down the spectrum of the theory.', '1712.05206-4-17-2': 'There will be a single vacuum state [MATH], characterized by the fact that it is annihilated by all [MATH], [EQUATION]', '1712.05206-4-17-3': 'All other eigenstates can be built by letting [MATH] acting on the vacuum.', '1712.05206-4-17-4': 'Let [EQUATION]', '1712.05206-4-17-5': 'Having found the spectrum of the Hamiltonian, let us try to interpret its eigenstates.', '1712.05206-4-17-6': 'By similar logic applied to the Hamiltonian [MATH], we can construct an operator corresponding to the one showed in Eq. ([REF]) [EQUATION]', '1712.05206-4-17-7': 'So the operator [MATH] creates excitations at time [MATH] and position [MATH].', '1712.05206-4-17-8': 'These excitations are discrete entities that have the proper relativistic kinematics relation which describes uniform accelerated particles.', '1712.05206-4-17-9': 'We claim that we can call these excitations accelerating particles and the associated fields as accelerated fields.', '1712.05206-4-17-10': 'By a particle here we mean something that is localized in space, since [MATH] creates space eigenstates.', '1712.05206-4-17-11': 'But this operator says nothing regarding the momentum of the particle.', '1712.05206-4-17-12': 'This information can be derived considering the interpretation of the state [MATH].', '1712.05206-4-17-13': 'From the expansion ([REF]) we see that [EQUATION] is a linear superposition of single particle states that have well-defined time-position, hence we will claim that the operator [MATH], acting on the vacuum, creates a particle with momentum [MATH] and energy [MATH].', '1712.05206-4-17-14': 'Note that, since we considered an hermitian field operator, [MATH] creates particles with the same momentum and energy as [MATH] does.', '1712.05206-4-17-15': 'Therefore, we quantized the accelerated field in the Schrödinger picture and interpreted the resulting theory in terms of accelerated relativistic particles.', '1712.05206-4-17-16': 'In the Heisenberg picture, the operator [MATH] allows us to relate [MATH] to [MATH] by means of the relation [EQUATION]', '1712.05206-4-17-17': 'For one particle state of well-defined time, [MATH] is the eigenvalue of [MATH].', '1712.05206-4-17-18': 'Equation ([REF]) makes explicit the dual particle and wave interpretations of the quantum field [MATH].', '1712.05206-4-17-19': 'On the one hand, [MATH] is written as a Hilbert space operator, which creates particles that are the quanta of the field.', '1712.05206-4-17-20': 'On the other hand, [MATH] is written as a linear combination of the solutions [MATH] of the wave equation ([REF]).', '1712.05206-4-17-21': 'If these were single-particle wavefunctions, they would correspond to states of positive and negative position; We have chosen to refer to them more generally as positive- and negative-frequency modes, keeping in mind the difference with the corresponding modes coming from Klein-Gordon equation.', '1712.05206-4-17-22': 'The connection between the particle creation operators and the waveforms has already been mentioned above.', '1712.05206-4-17-23': 'A positive-frequency solution of the field equation has as its coefficients the operator that destroys a particle in that particular mode.', '1712.05206-4-17-24': 'While a negative-frequency solution, being the Hermitian conjugate of a positive-frequency solution, has as its coefficient the operator that creates a particle in that positive-position single-particle wavefunction.', '1712.05206-4-18-0': 'Let us consider a real scalar field described by the Klein-Gordon equation [MATH].', '1712.05206-4-18-1': 'To illustrate the difference between the Klein-Gordon field and the field theory we developed from the kinematic wave equation ([REF]) let us compute the quantities [MATH] and [MATH], which can be interpret as the position representation of the single-particle wavefunction of the state [MATH] and the momentum representation of the state [MATH], respectively.', '1712.05206-4-18-2': 'We find [EQUATION]', '1712.05206-4-18-3': 'In the first case we recognize the relativistic dispersion relation for a particle with mass [MATH], while in the second one we identify the relativistic kinematic relation for uniformly accelerated particle.', '1712.05206-4-19-0': 'After the introduction of Heisenberg picture, we can proceed to the calculation of the amplitude for a particle to propagate from [MATH] to [MATH], which in our present formalism is [MATH].', '1712.05206-4-19-1': 'We will call this quantity [MATH] and we easily check that it is given by the integral [EQUATION]', '1712.05206-4-19-2': 'Let us evaluate this integral for some particular values of [MATH].', '1712.05206-4-20-0': 'First consider the case where the difference is: [MATH], [MATH].', '1712.05206-4-20-1': 'Then we have [EQUATION] where [MATH] is the modified Bessel function for [MATH].', '1712.05206-4-21-0': 'Next consider the case where [MATH], [MATH].', '1712.05206-4-21-1': 'The amplitude is [EQUATION]', '1712.05206-4-21-2': 'We find that the propagation amplitudes are not zero for particles propagating over all intervals, timelike and spacelike.', '1712.05206-4-22-0': 'A measurement performed at one point can affect a measurement at another point if the associated commutator does not vanish.', '1712.05206-4-22-1': 'Thus, a theory, in order to be causal, should require all the operators defined outside the light-cone to commute.', '1712.05206-4-22-2': 'Let us compute the commutator [MATH] whose general form is given by [EQUATION]', '1712.05206-4-22-3': 'As shown in Fig. [REF], when [MATH], being outside the light-cone, an appropriate continuous Lorentz transformation can take us from [MATH] to [MATH].', '1712.05206-4-22-4': 'If we perform this transformation on the second term in ([REF]), the two terms inside parenthesis are therefore equal and cancel.', '1712.05206-4-22-5': 'In case [MATH], there is no continuous Lorentz transformation that can take [MATH] to [MATH], thus the amplitude is nonzero.', '1712.05206-4-22-6': 'We then conclude that no measurement in our theory of accelerated fields can affect another measurement outside the light-cone.', '1712.05206-4-22-7': 'Hence, causality is maintained.', '1712.05206-4-22-8': "We intentionally didn't discuss causality by recruiting spacelike/timelike intervals.", '1712.05206-4-22-9': 'Contrary to Klein-Gordon theory, our field commutator vanishes for timelike separations while survives for spacelike separations.', '1712.05206-4-22-10': 'This outcome is consistent with our choice the role of "time" to be played by the spacelike coordinate [MATH], resulting in a light-cone with different direction (see Fig. [REF]) and interchanged notions of timelike and spacelike intervals.', '1712.05206-4-23-0': 'It is crucial, in order to satisfy the causality requirement, that we quantize the accelerated field using commutators.', '1712.05206-4-23-1': 'Had we used anticommutators instead of commutators, we would get a nonvanishing result for timelike intervals.', '1712.05206-4-23-2': 'Thus, this is an indication that the "spin-statistics theorem" also holds for relativistic particles.', '1712.05206-4-23-3': 'Another crucial element in our construction is essential for causality: The negative-space eigenfunctions in the field operators.', '1712.05206-4-23-4': 'Again, without negative-space contributions Eq. ([REF]) does not vanish for spacelike separations.', '1712.05206-4-24-0': '# Discussion', '1712.05206-4-25-0': 'Although we have considered field theories in [MATH] dimensions, our results can be extended to physical dimensions.', '1712.05206-4-25-1': 'In terms of a global inertial coordinate system [MATH], let us consider the killing field which generates a boost about the origin in the [MATH] direction.', '1712.05206-4-25-2': 'In this case, the hyperbolic surface [MATH] is invariant under translation in [MATH] and [MATH] direction.', '1712.05206-4-25-3': 'Given the fact that the definition of accelerated fields is based on this hyperbolic cylinder surface, in our theory, accelerated fields become invariant under translations in [MATH] and [MATH] direction.', '1712.05206-4-25-4': 'This clearly reproduces all the results presented above.', '1712.05206-4-25-5': 'In the general case, we will have three dimensional surface that will be projected into the planes ([MATH]), ([MATH]) and ([MATH]), thus defining three independent accelerated fields.', '1712.05206-4-25-6': 'This reflects the fact that, in our theory, geometry is translated into fields.', '1712.05206-4-26-0': 'Even though gravity, as described in general relativity, appears not to be compatible with the quantum field theories, this is not the case for the accelerated fields.', '1712.05206-4-26-1': 'In this work we propose a new quantum field theory, that encodes acceleration directly into the fields and provides a mathematical consistent way to locally quantize spacetime.', '1712.05206-4-26-2': 'This is a strong evidence that not only gravity but also acceleration can reshape the notions of space and time.', '1712.05206-4-27-0': 'The problem of finding a quantum theory of gravity is the main problem in modern theoretical physics.', '1712.05206-4-27-1': 'Today, the two main approaches are string theory and loop quantum gravity.', '1712.05206-4-27-2': 'These theories address the problem of quantum gravity in totally different ways.', '1712.05206-4-27-3': 'String theory is based on the idea of a fundamental description of gravity in terms of physical excitations of a string that lives in a background metric space.', '1712.05206-4-27-4': 'Loop quantum gravity, on the other hand, in an attempt to comprehend what quantum spacetime is at the fundamental level, was formulated without a background spacetime.', '1712.05206-4-28-0': 'The theory of accelerated fields shares elements from both of these theories.', '1712.05206-4-28-1': 'Even though the description of acceleration takes place in terms of excitations over a background metric space, the fact that this space is the momentum space allows the quantization of spacetime, the background of all the others fields.', '1712.05206-4-28-2': 'From this view one can define accelerated Klein-Gordon fields establishing a notion of scalar fields being located with respect to the accelerated field.', '1712.05206-4-29-0': 'The theory of accelerated fields, as presented here, is an effort to quantize spacetime, while it is kept separate from the matter fields.', '1712.05206-4-29-1': 'In a subsequent study, we intend to investigate the effects of accelerated fields on the Klein-Gordon massive field, by defining accelerated Klein-Gordon massive fields and comparing them with the inertial ones.', '1712.05206-4-30-0': 'Several other directions follow from this work.', '1712.05206-4-30-1': 'For instance, the question of understanding entanglement in non-inertial frames has been central to the development of the emerging field of relativistic quantum information [CITATION].', '1712.05206-4-30-2': 'The approach presented in this work offers a new ground to address this fundamental question from a completely different point of view.', '1712.05206-4-31-0': 'We would like to thanks George E. A. Matsas, Daniel Terno and Daniel A. T. Vanzella for discussions.', '1712.05206-4-31-1': 'We acknowledge financial support from the Brazilian funding agencies CNPq (Grants No. 401230/2014-7, 305086/2013-8 and 445516/2014-) and CAPES (Grant No. 6531/2014-08), the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ).'}

cond-mat-0602398

{'cond-mat-0602398-1-0-0': 'We discuss localization properties of the Dirac-like electronic states in monolayers of graphite.', 'cond-mat-0602398-1-0-1': 'In the framework of a general disorder model, we identify the conditions under which such standard localization effects as logarithmic temperature-dependent conductivity corrections appear to be strongly suppressed, as compared to the case of a two-dimensional electron gas with parabolic dispersion, in agreement with recent experimental observations.', 'cond-mat-0602398-1-1-0': 'After several decades of predominantly application-driven studies, graphene has finally been recognized as a unique example of the system of two-dimensional fermions with a linear dispersion and pseudo-relativistic kinematical properties.', 'cond-mat-0602398-1-1-1': 'The recent advances in microfabrication of graphitic samples that are only a few carbon layers thick [CITATION] have made it possible to test the early theoretical predictions of the anomalous properties of this system [CITATION].', 'cond-mat-0602398-1-2-0': 'The most striking experimental observation up to date was that of an anomalous quantization of the Hall conductivity [CITATION] which is characteristic of the pseudo-relativistic nature of the quasiparticle excitations in this system.', 'cond-mat-0602398-1-2-1': 'The other properties that manifest this peculiar single-particle kinematics have been revealed by magnetotransport measurements, including the [MATH] dependence of the energies of the (non-equidistant) Landau levels and the intrinsic [MATH]-shift of the phase of the Shubnikov-de Haas oscillations [CITATION].', 'cond-mat-0602398-1-3-0': 'In a (nearly) degenerate semimetal such as graphite, the Coulomb interactions are expected to play an important role, largely due to their poor screening [CITATION].', 'cond-mat-0602398-1-3-1': 'Besides, any interplay between the Coulomb interactions and disorder is likely to further modify the behavior of the idealized (clean and non-interacting) Dirac fermion system.', 'cond-mat-0602398-1-3-2': 'Thus far, however, experiment has not yet provided a direct evidence of any interaction-induced phenomena, some of which were predicted in the recent years [CITATION].', 'cond-mat-0602398-1-4-0': 'Nonetheless, contrary to the situation in the conventional 2DEG, the data of Refs. [CITATION] do not seem to manifest any pronounced weak-localization effects.', 'cond-mat-0602398-1-4-1': 'Motivated by these puzzling reports, in the present work we set out to study electron localization in graphene.', 'cond-mat-0602398-1-4-2': 'Although this topic has already attracted some recent attention [CITATION], we shall demonstrate that the analysis of Ref. [CITATION] is neither complete, nor (suriously enough, considering that many assertions made in Ref. [CITATION] are basically correct) do the arguments presented in that work appear to be technically sound.', 'cond-mat-0602398-1-5-0': 'The electronic band structure of graphene is characterized by the presence of a pair of inequivalent nodal points at the wave vectors [MATH] where [MATH] is the lattice spacing.', 'cond-mat-0602398-1-5-1': 'At these two and the four other points in the Brillouin zone obtainable from [MATH] with a shift by one of the reciprocal lattice vectors [MATH], the valence and conduction bands touch upon each other as a pair of opposing cones with the opening angle given by the Fermi velocity [MATH].', 'cond-mat-0602398-1-6-0': 'In the leading approximation, quasiparticle excitations in the vicinity of the nodal points (hereafter referred to as valleys) can be described by the Dirac Hamiltonian [CITATION] [EQUATION] where [MATH] is the triplet of the Pauli matrices acting in the space of spinors [MATH] composed of the values of the electron wave function on the [MATH] and [MATH] sublattices of the hexagonal lattice of graphene, whereas the triplet [MATH] acts in the valley subspace.', 'cond-mat-0602398-1-6-1': 'In the absence of magnetic field, the Hamiltonian (1) remains a unity matrix in the physical spin subspace, though.', 'cond-mat-0602398-1-7-0': 'In the presence of disorder, the quasiparticles experience both intra- and inter-valley scattering.', 'cond-mat-0602398-1-7-1': 'Upon averaging over disorder, the most general form of the elastic four-fermion vertex induced by disorder can be described as follows [EQUATION] where we use the customary "g-oloqical" notations to denote the processes of forward scattering between fermions from the same ([MATH]) and different ([MATH]) valleys, as well as those of "backward" scattering ([MATH]) with the momentum transfer close to [MATH].', 'cond-mat-0602398-1-7-2': 'Besides, we have included the possibility of "umklapp" scattering ([MATH]), in which case the total momentum of two fermions changes by [MATH].', 'cond-mat-0602398-1-7-3': 'A justification for this (not immediately obvious, considering that the total momentum changes by only a fraction of the reciprocal latice vector [MATH]) extension of the disorder model will be discussed below.', 'cond-mat-0602398-1-8-0': 'In the framework of the self-consistent Born approximation (SCBA), the effect of disorder on the quasiparticle spectrum is described by a self-energy which obeys the equation [EQUATION] where the retarded Green function is given by the expression [EQUATION] that includes a chemical potential [MATH] corresponding to a variable electron density.', 'cond-mat-0602398-1-8-1': 'Notably, Eq.(3) for the elastic self-energy allows only for solutions that are diagonal in the valley subspace.', 'cond-mat-0602398-1-9-0': 'By solving Eq.(3), we obtain [EQUATION] where [MATH] is the density of states at the Fermi energy.', 'cond-mat-0602398-1-9-1': 'In the low-doping limit ([MATH]), a finite value of [MATH] is dominated by the effect of disorder.', 'cond-mat-0602398-1-9-2': 'This phenomenon has been extensively studied in the context of normal quasiparticle transport in dirty [MATH]-wave superconductors where [MATH] and the spectrum possesses an exact particle-hole symmetry [CITATION].', 'cond-mat-0602398-1-9-3': 'Likewise, in the case of our interest one obtains [MATH].', 'cond-mat-0602398-1-10-0': 'By contrast, most of the data of Refs. [CITATION] pertain to the regime of relatively high doping where [MATH], as indicated by, e.g., the measured mean free path which was found to be of order [MATH] at electron densities [MATH].', 'cond-mat-0602398-1-10-1': 'In this regime, the value of [MATH] is controlled by the finite radius of the Fermi surface [MATH] and is only weakly affected by disorder.', 'cond-mat-0602398-1-11-0': 'In this case, the double-pole Green function (4) given by a four-by-four matrix can be well approximated by a pair of two-by-two matrix-valued single-pole expressions [EQUATION] where [MATH] and [MATH].', 'cond-mat-0602398-1-12-0': 'Even more importantly, the presence of a large parameter [MATH] facilitates a systematic account of quantum interference corrections to the SCBA results.', 'cond-mat-0602398-1-12-1': 'We mention, in passing, that in the low-doping ([MATH]) limit the only parameter that can (at least, in principle) be used for the analysis of such corrections is the (inverse) number of valleys, whose actual value is, of course, [MATH].', 'cond-mat-0602398-1-12-2': 'The same caveat plagues the analysis of the dirty [MATH]-wave superconductors where the strength of the conductivity corrections is governed by the number of pairs of opposite nodes of the order parameter, whose physical value is again equal two.', 'cond-mat-0602398-1-13-0': 'Luckily, in the high-density regime that, so far, has been probed in graphene, the leading quantum correction to the zeroth-order (Drude) conductivity stem from the standard single-Cooperon (fan-shaped) diagram.', 'cond-mat-0602398-1-13-1': 'From the technical standpoint, however, the calculation of the corresponding correction appears to be somewhat more involved due to the matrix nature of the Green function (6) and the vertex (2).', 'cond-mat-0602398-1-14-0': 'The Drude conductivity itself is given by the standard expression [MATH] where the renormalization factor [MATH] accounts for a ladder series of vertex corrections associated with one of the two current operators [MATH] inserted into the fermion loop in the diagrammatic representation of the Kubo formula.', 'cond-mat-0602398-1-14-1': 'Being given by a non-singular (angular momentum [MATH]) diffusion mode in the expansion over the angular harmonics [MATH], [MATH] is a function of the parameters [MATH].', 'cond-mat-0602398-1-15-0': 'Expanding the expression for the Cooperon in the same basis as that used in Eq. (2), we obtain equations for the corresponding amplitudes [MATH], each of which is a matrix in the direct product of two [MATH] subspaces [EQUATION] where [MATH] denotes a convolution of a pair of Green functions.', 'cond-mat-0602398-1-16-0': 'Computing the latter at [MATH], we obtain [EQUATION]', 'cond-mat-0602398-1-16-1': 'It can be readily seen that Eqs. (7) split onto two pairs which only couple [MATH] and [MATH], respectively.', 'cond-mat-0602398-1-16-2': 'Their solutions read [EQUATION] where [MATH], [MATH], [MATH], and [MATH].', 'cond-mat-0602398-1-17-0': 'The quantum conductivity correction [EQUATION] involves the [MATH] and [MATH] components of the Cooperon, whose contributions turn out to be negative and positive, respectively.', 'cond-mat-0602398-1-17-1': 'The logarithmic temperature-dependent part of Eq.(10) can be cast in the form [EQUATION] where we introduced [MATH] proportional to the inelastic phase relaxation rate [MATH] which provides a cutoff in the momentum integration and diverges at [MATH].', 'cond-mat-0602398-1-18-0': 'The analysis of Eq.(11) reveals that, in the absence of a fine tuning between the amplitudes of backward scattering and umklapp processes ([MATH]) the [MATH] Cooperon always acquires a gap [MATH].', 'cond-mat-0602398-1-18-1': 'On the other hand, the [MATH] mode remains gapless, provided that all the forward scattering processes are controlled by the same amplitude (i.e., [MATH]).', 'cond-mat-0602398-1-18-2': 'Under these general conditions the overall logarithmic correction (11) becomes negative, suggesting the onset of rather conventional weak localization.', 'cond-mat-0602398-1-19-0': 'Conversely, making the [MATH] mode gapful and inverting the sign of the conductivity correction would only be possible under the condition [MATH] which is unlikely to be satisfied for any realistic impurity potential that yields equal amplitudes of the processes of intra- and inter-valley forward scattering.', 'cond-mat-0602398-1-19-1': 'It can also be readily seen that the antilocalizing behavior predicted in Ref. [CITATION] occurs in the absence of any scattering at momenta close to [MATH]).', 'cond-mat-0602398-1-20-0': 'The above conclusions were drawn for the general disorder model (2).', 'cond-mat-0602398-1-20-1': 'However, in the situation where disorder is realized as a random distribution of impurities with a concentration [MATH] and a (short-range) potential [MATH], it suffices to introduce only two independent parameters [EQUATION]', 'cond-mat-0602398-1-20-2': 'The above expressions represent a [MATH]-matrix computed for an arbitrary strength of disorder, the customary Born and unitarity (where the scattering phase approaches [MATH]) limits corresponding to [MATH] and [MATH], respectively (in the case of a genuine long-range impurity potential, one should keep the dependence [MATH] on the transferred momentum, though).', 'cond-mat-0602398-1-21-0': 'Provided that the relations (12) between the parameters [MATH] hold, one obtains [MATH], and the logarithmic term in (11) vanishes as a result of the exact cancellation between the contributions of the localizing ([MATH]) and antilocalizing ([MATH]) Cooperon modes.', 'cond-mat-0602398-1-21-1': 'It has to be stressed, however, that such a strong suppression of the (anti)localization effects could only be possible due to an opening of the umklapp channel.', 'cond-mat-0602398-1-21-2': 'In turn, the latter requires an emergence of a crystal superstructure with the wave vector [MATH] (or equivalent).', 'cond-mat-0602398-1-22-0': 'While an isolated sheet of weakly-interacting graphene apparently would lack such a superstructure, it is conceivable that the latter might emerge, if a commensurate substrate were used during the process of microfabfication.', 'cond-mat-0602398-1-22-1': 'Besides, a commensurate corrugation could occur [CITATION] due to the Coulomb correlations that can induce spatially periodic patterns of the electron density itself.', 'cond-mat-0602398-1-22-2': 'The possibility of a spontaneous formation of such charge density wave states has long been discussed in the general context of degenerate semimetals and, specifically, in graphene [CITATION].', 'cond-mat-0602398-1-23-0': 'Next, we comment on the technical details presented in Ref.[CITATION] where the first prediction of the antilocalization behavior for [MATH] and [MATH] was made.', 'cond-mat-0602398-1-23-1': 'In essence, instead of working directly with the [MATH]-matrices in the sublattice subspace, the authors of Ref.[CITATION] used single-component Green functions [MATH] complemented by matrix elements of a (purely forward-scattering) impurity potential between different Bloch wave functions.', 'cond-mat-0602398-1-24-0': 'To that end, in Ref.[CITATION] the latter were chosen in a particular (asymmetrical) gauge [EQUATION]', 'cond-mat-0602398-1-24-1': 'The aforementioned matrix element of the impurity potential is then given by a complex-valued expression [MATH].', 'cond-mat-0602398-1-25-0': 'Next, the authors of Ref.[CITATION] asserted that the equations for the Cooperon mode involve the disorder-induced vertex [EQUATION] which describes scattering from the Bloch states with momenta [MATH] and [MATH] into [MATH] and [MATH] in the limit [MATH].', 'cond-mat-0602398-1-26-0': 'It was then argued in Ref.[CITATION] that the full Cooperon amplitude inherits the phase factor from Eq.(14) and, therefore, becomes negative for [MATH] where [MATH], which configuration of the momenta provides a dominant contribution to the weak localization correction.', 'cond-mat-0602398-1-26-1': 'Thus obtained, the negative sign of the Cooperon amplitude was claimed to be instrumental for the onset of antilocalizing behavior.', 'cond-mat-0602398-1-27-0': 'The validity of the above argument disputed by observing that choosing a different (symmetrical) gauge for the Bloch wave functions [EQUATION] would result in a purely real matrix element [MATH], the use of which yields Eq. (14) without the said phase factor and, hence the (deemed to be crucially important) sign change of the Cooperon amplitude does not seem to occur.', 'cond-mat-0602398-1-28-0': 'Although the observed gauge dependence of the Cooperon is perfectly consistent with its being a gauge non-invariant two-particle amplitude, the conductivity correction, of course, is supposed to be gauge invariant.', 'cond-mat-0602398-1-28-1': 'A resolution of such apparent contradiction goes as following.', 'cond-mat-0602398-1-29-0': 'In fact, the vertex that ought to be used in the construction of the Cooperon is the particle-hole (exchange) amplitude [MATH] which is manifestly gauge invariant and given by Eq. (14) without the phase factor in question.', 'cond-mat-0602398-1-29-1': 'In this regard, it is important to realize that the amplitudes [MATH] and [MATH] represent two non-commuting limits of the general vertex (2).', 'cond-mat-0602398-1-30-0': 'On a side note, it might also be tempting to try to explain the experimentally found suppression of localization by the presence of the factor [MATH] vanishing for [MATH] in Eq. (14).', 'cond-mat-0602398-1-30-1': 'Notice, however, that the Cooperon built out of such a vertex lacks this factor, since the gapless pole develops for only [MATH] (namely, [MATH]) angular harmonic, whereas the other two harmonics that constitute the above angular-dependent factor ([MATH]) receive only a finite enhancement by a factor of two.', 'cond-mat-0602398-1-31-0': 'Thus, an obvious suppression of the backward scattering off of an individual impurity (which fact has often been mentioned in the context of electron transport in 1D carbon nanotubes) does not necessarily imply the absence of (anti)localization in the deep diffusive regime in 2D graphene.', 'cond-mat-0602398-1-32-0': 'The true origin of the antilocalizing behavior exhibited by Eq. (11) for [MATH], can be traced back to the negative signature of the expansion of the Cooperon mode [MATH] over the product basis [MATH], as opposed to that of [MATH] (see Eqs. (9)).', 'cond-mat-0602398-1-32-1': 'Interpreting the sublattice index as a fictitious spin one-half ([MATH]), one can associate the [MATH] component with the singlet mode [MATH]) which, in the spin-orbit context, is known to be a common source of antilocalization.', 'cond-mat-0602398-1-33-0': 'In that regard, this anilocalizing behavior is similar to that predicted in the case of a 2DEG with a strong spin-orbit coupling of either Rashba or Dresselhaus kind [CITATION] [EQUATION]', 'cond-mat-0602398-1-33-1': 'The details of the calculations presented above differ from those pertaining to the case of the spin-orbit coupling (for one, except for the splitting, the SO-coupling does not alter the fermion dispersion).', 'cond-mat-0602398-1-33-2': 'In particular, for [MATH] the coefficient in front of the logarithmic term in Eq.(11) appears to be twice larger than that in the case of a 2DEG with a strong SO-coupling and [MATH].', 'cond-mat-0602398-1-34-0': 'Yet another argument invoked by the authors of Ref.[CITATION] to support their prediction of the antilocalizing behavior at intermediate temperatures [MATH] exploits the notion of the Berry phase associated with the electron wavefunctions (13).', 'cond-mat-0602398-1-35-0': 'Indeed, the first Chern number ("vorticity") [EQUATION] where the momentum space integral is taken along any contour enclosing one of the Dirac points [MATH], yields [MATH] for both wavefunctions (13).', 'cond-mat-0602398-1-35-1': 'In Ref. [CITATION] (see also [CITATION]) this observation was used to assert that the interference between any pair of counter-propagating electron waves becomes destructive, thereby giving way to the antilocalizing behavior.', 'cond-mat-0602398-1-36-0': 'However, the Berry phase (17) would be identically zero, if the wave functions (14) were used instead of (13), which suggests that its relationship with the Cooperon is somewhat more subtle.', 'cond-mat-0602398-1-36-1': 'To that end, it is worth mentioning that the two Dirac species emerging in the continuous (low-energy) description of the original lattice system must have opposite chiralities [CITATION], as required for compliance with the ubiquitous Nielsen-Ninomiya ("fermion doubling") theorem [CITATION].', 'cond-mat-0602398-1-36-2': 'In turn, this observation seems to suggest that, even if the Berry phases of the individual species happen to be finite ([MATH]), they should still add up to zero.', 'cond-mat-0602398-1-37-0': 'A cancellation of the overall Berry phase would also be consistent with the vanishing of the quantum conductivity correction in the presence of equally strong intervalley backward and umklapp scattering, which, in effect, make the two Dirac points indistinguishable (the shortest reciprocal lattice vector becomes equivalent to [MATH]).', 'cond-mat-0602398-1-38-0': 'Recently, the role of the Berry phase has also been brought up in the context of the spin Hall effect generated by a spin-orbit coupling.', 'cond-mat-0602398-1-38-1': 'Interstingly enough, the dependence of the spin Hall conductivity upon the coefficients [MATH] and [MATH] from Eq. (16) (and, especially, its exact vanishing at [MATH]) can be established with the use of the Bloch wave functions in either asymmetrical (similar to Eq. (13)) or symmetrical (similar to Eq. (14)) gauge, where the corresponding Berry phase is non-zero and zero, respectively (cf., e.g., the two references in [CITATION]).', 'cond-mat-0602398-1-38-2': 'This observation, too, indicates that the relationship between the Berry phase and transport properties might be rather intricate.', 'cond-mat-0602398-1-39-0': 'Before concluding, a few more comments are in order.', 'cond-mat-0602398-1-40-0': 'In reality, the weak (anti)localization corrections are accompanied by the Aronov-Altshuler ones, which stem from the interference between disorder and Coulomb interactions.', 'cond-mat-0602398-1-40-1': 'Despite a similar temperature dependence ([MATH]), the former, but not the latter, gets quenched by an applied magnetic field, thus allowing one to discriminate between the two.', 'cond-mat-0602398-1-41-0': 'Also, considering the potential attainability of the very low-density regime, it would be of interest to extend the analysis of the localization effects to the case [MATH].', 'cond-mat-0602398-1-41-1': 'However, as we have already pointed out, in this regime a systematic expansion of the conductivity corrections would only be possible for an (unphysically) large number of valleys.', 'cond-mat-0602398-1-41-2': 'Moreover, even if this number were indeed large, the calculation itself would still pose a significant challenge.', 'cond-mat-0602398-1-42-0': 'The main technical difficulty here is due to a double-pole structure of the Green functions (4), which results in the emergence of additional gapless Cooperon and diffusion modes in the RR (AA) channel, alongside the conventional (RA) one.', 'cond-mat-0602398-1-42-1': 'A similar situation has been encountered in the context of dirty [MATH]-wave superconductors [CITATION] where it has been argued that the quasiparticle conductivity receives aditional logarithmic contributions from the processes involving one RA and one RR (AA) Cooperon [CITATION].', 'cond-mat-0602398-1-43-0': 'Furthermore, the authors of Ref. [CITATION] found the overall logarithmic correction to the spin (thermal) conductivity of normal [MATH]-wave quasiparticles to be negative, contrary to (in this case, ill-defined due to a non-conservation of charge) electrical one.', 'cond-mat-0602398-1-43-1': 'In view of the formal differences between the Dirac-like descriptions of planar [MATH]-wave superconductors and graphene, it is conceivable that the correction to the electrical conductivity of graphene could change sign in the low-density regime, too.', 'cond-mat-0602398-1-44-0': 'Lastly, we comment on the possiblity (as well as the need) of including other types of randomness.', 'cond-mat-0602398-1-44-1': 'While the vertex (2) is devised as a general model of short-range (screened) impurities generating a random scalar potential, it obviously misses out on those types of disorder that can be best represented by either a random vector potential or a random mass of the Dirac fermions.', 'cond-mat-0602398-1-45-0': 'The latter might be relevant in the situation where a spatially inhomogeneous charge or spin density wave ordering sets in.', 'cond-mat-0602398-1-45-1': 'In turn, the former can be utilized to describe extended lattice defects (dislocations, disclinations, and cracks) in terms of a random magnetic flux [CITATION].', 'cond-mat-0602398-1-45-2': 'In Ref. [CITATION], it was assumed that the corresponding random vector potential (rather then the flux) is [MATH]-correlated in space ([MATH]).', 'cond-mat-0602398-1-45-3': 'Although this model possesses a conformal invariance and, therefore, can be analyzed in quite some detail [CITATION], a more adequate to the task would obviously be the model of a long-range correlated random vector potential ([MATH]).', 'cond-mat-0602398-1-45-4': 'The latter is known to show a rather different behavior even in the high-energy (ballistic) regime, as demonstrated in Ref. [CITATION].', 'cond-mat-0602398-1-46-0': 'A further investigation into these and related issues will be presented elsewhere.', 'cond-mat-0602398-1-47-0': 'In summary, we carried out a comprehensive analysis of the first-order quantum correction to the conductivity of doped graphene.', 'cond-mat-0602398-1-47-1': 'We identified the conditions under which the conductivity correction becomes positive, negative, or zero.', 'cond-mat-0602398-1-47-2': 'The earlier analysis of Ref. [CITATION] has been critically assessed, and the open problems have been outlined.', 'cond-mat-0602398-1-48-0': 'This research was supported by NSF under Grant DMR-0349881.', 'cond-mat-0602398-1-48-1': 'The author acknowledges valuable communications with E. Abrahams, A. Geim, and H. Suzuura.'}

{'cond-mat-0602398-2-0-0': 'We discuss localization properties of the Dirac-like electronic states in monolayers of graphite.', 'cond-mat-0602398-2-0-1': 'In the framework of a general disorder model, we identify the conditions under which such standard localization effects as, e.g., the logarithmic temperature-dependent conductivity correction appear to be strongly suppressed, as compared to the case of a two-dimensional electron gas with parabolic dispersion, in agreement with recent experimental observations.', 'cond-mat-0602398-2-0-2': 'After several decades of predominantly application-driven studies, graphene has finally been recognized as a unique example of the system of two-dimensional fermions with a linear dispersion and pseudo-relativistic kinematical properties.', 'cond-mat-0602398-2-0-3': 'The recent advances in microfabrication of graphitic samples that are only a few carbon layers thick [CITATION] have made it possible to test the early theoretical predictions of the anomalous properties of this system [CITATION].', 'cond-mat-0602398-2-0-4': 'The most striking experimental observation up to date was that of an anomalous quantization of the Hall conductivity [CITATION] which is characteristic of the pseudo-relativistic nature of the quasiparticle excitations in this system.', 'cond-mat-0602398-2-0-5': 'The other properties manifesting this peculiar single-particle kinematics have been revealed by magnetotransport measurements, including the [MATH] dependence of the energies of the (non-equidistant) Landau levels and the intrinsic [MATH]-shift of the phase of the Shubnikov-de Haas oscillations [CITATION].', 'cond-mat-0602398-2-0-6': 'In a (nearly) degenerate semimetal such as graphite, the Coulomb interactions are expected to play an important role, largely due to their poor screening [CITATION].', 'cond-mat-0602398-2-0-7': 'Besides, any interplay between the Coulomb interactions and disorder is likely to further modify the behavior of the idealized (clean and non-interacting) Dirac fermion system.', 'cond-mat-0602398-2-0-8': 'Thus far, however, experiment has not yet provided a direct evidence of any interaction-induced phenomena, some of which were predicted in the recent years [CITATION].', 'cond-mat-0602398-2-0-9': 'Also, contrary to the situation in the conventional 2DEG, the data of Refs. [CITATION] do not seem to manifest any pronounced weak-localization effects, either.', 'cond-mat-0602398-2-0-10': 'Motivated by these observations, in the present work we set out to study electron localization in graphene.', 'cond-mat-0602398-2-0-11': 'Although this topic has already attracted some recent attention [CITATION], we shall demonstrate that the analysis of Ref. [CITATION] is neither complete, nor (curiously enough, considering that the assertions made in Ref. [CITATION] are qualitatively correct) do the arguments presented in that work appear to be technically sound.', 'cond-mat-0602398-2-0-12': 'The electronic band structure of graphene is characterized by the presence of a pair of inequivalent nodal points at the wave vectors [MATH] where [MATH] is the lattice spacing.', 'cond-mat-0602398-2-0-13': 'At these two and the four other points in the Brillouin zone obtainable from [MATH] with a shift by one of the reciprocal lattice vectors [MATH], the valence and conduction bands touch upon each other as a pair of opposing cones with the opening angle given by the Fermi velocity [MATH].', 'cond-mat-0602398-2-0-14': 'In the leading approximation, quasiparticle excitations in the vicinity of the nodal points (hereafter referred to as valleys) can be described by the Dirac Hamiltonian [CITATION] [EQUATION] where [MATH] is the triplet of the Pauli matrices acting in the space of spinors [MATH] composed of the values of the electron wave function on the [MATH] and [MATH] sublattices of the hexagonal lattice of graphene, whereas the triplet [MATH] acts in the valley subspace.', 'cond-mat-0602398-2-0-15': 'In the absence of magnetic field, the Hamiltonian (1) remains a unity matrix in the physical spin subspace.', 'cond-mat-0602398-2-0-16': 'In the presence of disorder, the quasiparticles experience both intra- and inter-valley scattering.', 'cond-mat-0602398-2-0-17': 'Upon averaging over disorder, the most general form of the elastic four-fermion vertex induced by disorder can be described by the expression [EQUATION] where we used the customary "g-oloqical" notations to denote the processes of forward scattering between fermions from the same ([MATH]) and different ([MATH]) valleys, as well as those of "backward" scattering ([MATH]) with the momentum transfer close to [MATH].', 'cond-mat-0602398-2-0-18': 'Besides, we have included the possibility of "umklapp" scattering ([MATH]), in which case the total momentum of two fermions changes by [MATH].', 'cond-mat-0602398-2-0-19': 'A justification for this (not immediately obvious, considering that the total momentum changes by only a fraction of the reciprocal latice vector) extension of the disorder model will be discussed below.', 'cond-mat-0602398-2-0-20': 'In the framework of the self-consistent Born approximation (SCBA), the effect of disorder on the quasiparticle spectrum is described by a self-energy which obeys the equation [EQUATION] where the retarded Green function is given by the expression [EQUATION] that includes a chemical potential [MATH] which allows one to account for a variable electron density.', 'cond-mat-0602398-2-1-0': 'Notably, the solution to Eq. (3) turns out to be diagonal in the valley subspace, [MATH], and proportional to the inverse elastic lifetime [EQUATION] where [MATH] is the density of states at the Fermi energy.', 'cond-mat-0602398-2-1-1': 'In the low-doping limit ([MATH]), a finite value of [MATH] is dominated by disorder.', 'cond-mat-0602398-2-1-2': 'A similar phenomenon has been extensively studied in the context of normal quasiparticle transport in dirty [MATH]-wave superconductors where [MATH] and the spectrum possesses an exact particle-hole symmetry [CITATION].', 'cond-mat-0602398-2-1-3': 'In both cases one obtains the self-consistent DOS as [MATH].', 'cond-mat-0602398-2-1-4': 'However, most of the data of Refs. [CITATION] pertain to the regime of relatively high dopings ([MATH]), as indicated by, e.g., the measured mean free path, which was found to be of order [MATH] at electron densities [MATH].', 'cond-mat-0602398-2-1-5': 'In this regime, the value of [MATH] is controlled by the finite radius of the Fermi surface [MATH] and is only weakly affected by disorder.', 'cond-mat-0602398-2-1-6': 'Therefore, the double-pole Green function (4) given by a four-by-four matrix can be well approximated by a pair of two-by-two matrices containing single-poles [EQUATION] where [MATH] and [MATH].', 'cond-mat-0602398-2-1-7': 'Even more importantly, the presence of a large parameter [MATH] facilitates a systematic account of quantum interference corrections to the SCBA results.', 'cond-mat-0602398-2-1-8': 'We mention, in passing, that in the low-doping ([MATH]) limit the only parameter that can (at least, in principle) be used for the analysis of such corrections is the (inverse) number of valleys, whose actual value is, of course, [MATH].', 'cond-mat-0602398-2-1-9': 'The same caveat plagues the analysis of dirty [MATH]-wave superconductors where the strength of the conductivity corrections is governed by the number of pairs of opposite nodes of the order parameter, whose physical value is again equal two.', 'cond-mat-0602398-2-1-10': 'Luckily, in the high-density regime that, so far, has been probed in graphene, the leading quantum correction to the zeroth-order (Drude) conductivity stem from the standard single-Cooperon (fan-shaped) diagram.', 'cond-mat-0602398-2-1-11': 'From the technical standpoint, however, the calculation of the corresponding correction appears to be somewhat more involved due to the matrix structure of the Green function (6) and the vertex (2).', 'cond-mat-0602398-2-1-12': 'The Drude conductivity itself is given by the standard expression [MATH] where the renormalization factor [MATH] accounts for a ladder series of vertex corrections associated with one of the two current operators [MATH] inserted into the fermion loop in the diagrammatic representation of the Kubo formula.', 'cond-mat-0602398-2-1-13': 'Being given by a non-singular (angular momentum [MATH]) diffusion mode in the expansion over the angular harmonics [MATH], [MATH] is a function of the parameters [MATH].', 'cond-mat-0602398-2-1-14': 'Expanding the expression for the Cooperon in the same basis as that used in Eq. (2), we obtain equations for the corresponding amplitudes [MATH], each of which is a matrix in the direct product of two [MATH] subspaces [EQUATION] where [MATH] denotes a convolution of a pair of Green functions.', 'cond-mat-0602398-2-1-15': 'Computing the latter at [MATH], we obtain [EQUATION]', 'cond-mat-0602398-2-1-16': 'It can be readily seen that Eqs. (7) split onto two pairs which only couple [MATH] and [MATH], respectively.', 'cond-mat-0602398-2-1-17': 'Their solutions read [EQUATION] where [MATH], [MATH], [MATH], and [MATH].', 'cond-mat-0602398-2-1-18': 'The quantum conductivity correction [EQUATION] involves the [MATH] and [MATH] components of the Cooperon, whose contributions turn out to be negative and positive, respectively.', 'cond-mat-0602398-2-1-19': 'The logarithmic temperature-dependent part of Eq.(10) can be cast in the form [EQUATION] where we introduced an inelastic phase relaxation rate [MATH] which provides a cutoff in the momentum integration and diverges at [MATH].', 'cond-mat-0602398-2-1-20': 'The analysis of Eq.(11) reveals that, in the absence of a fine tuning between the amplitudes of backward scattering and umklapp processes ([MATH]) the [MATH] Cooperon always acquires a gap [MATH].', 'cond-mat-0602398-2-1-21': 'On the other hand, the [MATH] mode remains gapless, provided that all the forward scattering processes are controlled by the same amplitude (i.e., [MATH]).', 'cond-mat-0602398-2-1-22': 'Conversely, making the [MATH] mode gapful and inverting the sign of the conductivity correction would only be possible under the condition [MATH] which is unlikely to be satisfied for any realistic impurity potential that yields equal amplitudes of the processes of intra- and inter-valley forward scattering.', 'cond-mat-0602398-2-1-23': 'The antilocalizing behavior predicted in Ref. [CITATION] for [MATH] and [MATH]) can only occur at intermediate temperatures (namely, at [MATH]), whereas at still lower temperatures the overall sign of (11) reverts to negative, thereby suggesting the onset of rather conventional weak localization.', 'cond-mat-0602398-2-1-24': 'The above conclusions apply to the general disorder model (2).', 'cond-mat-0602398-2-1-25': 'However, in the situation where disorder is realized as a random distribution of impurities with a concentration [MATH] and a (short-range) potential [MATH], it suffices to introduce only two independent parameters [EQUATION]', 'cond-mat-0602398-2-1-26': 'The above expressions represent a [MATH]-matrix computed for an arbitrary strength of disorder, the customary Born and unitarity (where the scattering phase approaches [MATH]) limits corresponding to [MATH] and [MATH], respectively.', 'cond-mat-0602398-2-1-27': '[In the case of a genuine long-range (unscreened) impurity potential, the dependence [MATH] on the transferred momentum makes the explicit formulas for [MATH] more involved, though.]', 'cond-mat-0602398-2-1-28': 'Provided that the relations (12) between the parameters [MATH] hold, one obtains [MATH], and the logarithmic term in (11) vanishes as a result of the exact cancellation between the contributions of the localizing ([MATH]) and antilocalizing ([MATH]) Cooperon modes.', 'cond-mat-0602398-2-1-29': 'It has to be stressed, however, that such a strong suppression of (anti)localization would only be possible due to an opening of the umklapp channel.', 'cond-mat-0602398-2-1-30': 'In turn, the latter requires an emergence of a crystal superstructure with the wave vector [MATH] (or equivalent).', 'cond-mat-0602398-2-1-31': 'While an isolated sheet of weakly-interacting graphene would apparently lack such a superstructure, it is conceivable that the latter might emerge if a commensurate substrate were used during the process of microfabfication.', 'cond-mat-0602398-2-1-32': 'Besides, a commensurate corrugation could occur [CITATION] due to the Coulomb correlations that can induce spatially periodic patterns of the electron density itself.', 'cond-mat-0602398-2-1-33': 'The possibility of a spontaneous formation of such charge density wave states has long been discussed in the general context of degenerate semimetals and, specifically, in graphene [CITATION].', 'cond-mat-0602398-2-1-34': 'Next, we comment on the technical details presented in Ref.[CITATION] where the first prediction of the antilocalization behavior for [MATH] and [MATH] was made.', 'cond-mat-0602398-2-1-35': 'In essence, instead of working directly with the matrices in the sublattice subspace, the authors of Ref.[CITATION] used single-component (projected) Green functions [MATH] complemented by matrix elements of a (purely forward-scattering) impurity potential between different Bloch wave functions.', 'cond-mat-0602398-2-1-36': 'To that end, in Ref.[CITATION] the latter were chosen in a particular (asymmetrical) gauge [EQUATION]', 'cond-mat-0602398-2-1-37': 'The aforementioned matrix element of the impurity potential is then given by a complex-valued expression [MATH].', 'cond-mat-0602398-2-1-38': 'Next, the authors of Ref.[CITATION] asserted that the equations for the Cooperon mode involve the disorder-induced vertex [EQUATION] which describes scattering from the Bloch states with momenta [MATH] and [MATH] into [MATH] and [MATH] in the limit [MATH].', 'cond-mat-0602398-2-1-39': 'It was then argued in Ref.[CITATION] that the full Cooperon amplitude inherits the phase factor from Eq.(14) and, therefore, becomes negative for [MATH] where [MATH], which configuration of the momenta provides a dominant contribution to the weak localization correction.', 'cond-mat-0602398-2-1-40': 'Thus obtained, the negative sign of the Cooperon amplitude was claimed to be instrumental for the onset of antilocalizing behavior.', 'cond-mat-0602398-2-1-41': 'The validity of the above argument can be disputed by observing that the choice of a different (symmetrical) gauge for the Bloch wave functions [EQUATION] would result in a purely real matrix element [MATH], the use of which yields Eq. (14) without the said phase factor, hence the (deemed to be crucially important) sign change of the Cooperon amplitude does not seem to occur.', 'cond-mat-0602398-2-1-42': 'Although the observed gauge dependence of the Cooperon is perfectly consistent with its being a gauge non-invariant two-particle amplitude, the conductivity correction is, of course, supposed to be gauge invariant.', 'cond-mat-0602398-2-1-43': 'A resolution of such apparent contradiction goes as follows.', 'cond-mat-0602398-2-1-44': 'In fact, the vertex that ought to be used in the construction of the Cooperon is the particle-hole (exchange) amplitude [MATH] which is manifestly gauge invariant and given by Eq. (14) without the phase factor in question.', 'cond-mat-0602398-2-1-45': 'In this regard, it is important to realize that the amplitudes [MATH] and [MATH] represent two non-commuting (and, in this particular case, unequal) limits of the general vertex (2).', 'cond-mat-0602398-2-1-46': 'On a side note, it might also be tempting to try to explain the experimentally found suppression of localization by the presence of the factor [MATH] vanishing for [MATH] in Eq. (14).', 'cond-mat-0602398-2-1-47': 'Notice, however, that the Cooperon built out of such a vertex lacks this factor, since the gapless pole develops for only one (namely, [MATH]) angular harmonic, whereas the other two harmonics that constitute the above angular-dependent factor ([MATH]) receive only a finite enhancement by a factor of two.', 'cond-mat-0602398-2-1-48': 'Thus, an obvious suppression of the backward scattering off of an individual impurity (which fact has often been mentioned in the context of electron transport in 1D carbon nanotubes) does not necessarily imply the absence of (anti)localization in the deep diffusive regime in 2D graphene.', 'cond-mat-0602398-2-1-49': 'The true origin of the antilocalizing behavior exhibited by Eq. (11) for [MATH], can be traced back to the negative signature of the expansion of the Cooperon mode [MATH] over the product basis [MATH], as opposed to that of [MATH] (see Eqs. (9)).', 'cond-mat-0602398-2-1-50': 'Interpreting the sublattice index as a fictitious spin one-half ([MATH]), one can associate the [MATH] component with the singlet mode [MATH]) which, in the spin-orbit context, is known to be a common source of antilocalization.', 'cond-mat-0602398-2-1-51': 'In this regard, the present anilocalizing behavior is similar to that predicted in the case of 2DEG with a strong spin-orbit coupling of either Rashba or Dresselhaus kind [CITATION] [EQUATION]', 'cond-mat-0602398-2-1-52': 'The details of the calculations presented above differ from those pertaining to the case of the spin-orbit coupling (for one, except for the splitting, the SO-coupling does not alter the fermion dispersion).', 'cond-mat-0602398-2-1-53': 'In particular, for [MATH] the coefficient in front of the logarithmic term in Eq.(11) appears to be twice larger than that in the case of 2DEG with a strong SO-coupling and [MATH].', 'cond-mat-0602398-2-1-54': 'Yet another argument invoked by the authors of Ref.[CITATION] to support their prediction of the antilocalizing behavior at intermediate temperatures exploits the notion of the Berry phase associated with the electron wavefunctions (13).', 'cond-mat-0602398-2-1-55': 'Indeed, the first Chern number ("vorticity") [EQUATION] where the momentum space integral is taken along any contour enclosing one of the Dirac points [MATH], yields [MATH] for both wavefunctions (13).', 'cond-mat-0602398-2-1-56': 'In Ref. [CITATION] (see also [CITATION]) this observation was used to assert that the interference between any pair of counter-propagating electron waves becomes destructive, thereby giving way to the antilocalizing behavior.', 'cond-mat-0602398-2-1-57': 'However, the Berry phase (17) would be identically zero, if the wave functions (14) were used instead of (13), which suggests that its relationship with the Cooperon is somewhat more subtle.', 'cond-mat-0602398-2-1-58': 'To that end, it is worth mentioning that the two Dirac species emerging in the continuous (low-energy) description of the original lattice system must have opposite chiralities [CITATION], as required for compliance with the ubiquitous Nielsen-Ninomiya ("fermion doubling") theorem [CITATION].', 'cond-mat-0602398-2-1-59': 'In turn, this observation seems to suggest that, even if the Berry phases of the individual species happen to be finite ([MATH]), they should still add up to zero.', 'cond-mat-0602398-2-1-60': 'A cancellation of the overall Berry phase would also be consistent with the vanishing of the quantum conductivity correction in the presence of equally strong intervalley backward and umklapp scattering, which, in effect, make the two Dirac points indistinguishable (the shortest reciprocal lattice vector becomes equivalent to [MATH]).', 'cond-mat-0602398-2-1-61': 'Recently, the role of the Berry phase has also been brought up in the context of the spin Hall effect generated by a spin-orbit coupling.', 'cond-mat-0602398-2-1-62': 'Interestingly enough, the dependence of the spin Hall conductivity upon the coefficients [MATH] and [MATH] from Eq. (16) (and, especially, its exact vanishing at [MATH]) can be established with the use of the Bloch wave functions in either asymmetrical (similar to Eq. (13)) or symmetrical (similar to Eq. (14)) gauge, where the corresponding Berry phase is non-zero and zero, respectively (cf., e.g., the two references in [CITATION]).', 'cond-mat-0602398-2-1-63': 'This observation, too, indicates that the relationship between the Berry phase and transport properties might be rather intricate.', 'cond-mat-0602398-2-1-64': 'Before concluding, a few more comments are in order.', 'cond-mat-0602398-2-1-65': 'In reality, the weak (anti)localization corrections are accompanied by the Aronov-Altshuler ones, which stem from the interference between disorder and Coulomb interactions.', 'cond-mat-0602398-2-1-66': 'Despite a similar temperature dependence ([MATH]), the former, but not the latter, gets quenched by an applied magnetic field, thus allowing one to discriminate between the two.', 'cond-mat-0602398-2-1-67': 'Also, considering the potential attainability of the very low-density regime, it would be of interest to extend the analysis of the localization effects to the case [MATH].', 'cond-mat-0602398-2-1-68': 'However, as we have already pointed out, in this regime a systematic expansion of the conductivity corrections would only be possible for an (unphysically) large number of valleys.', 'cond-mat-0602398-2-1-69': 'Moreover, even if this number were indeed large, the calculation itself would still pose a significant challenge.', 'cond-mat-0602398-2-1-70': 'The main technical difficulty here is due to the double-pole structure of the Green functions (4), which results in the emergence of additional gapless Cooperon and diffusion modes in the RR (AA) channel, alongside the conventional (RA) one.', 'cond-mat-0602398-2-1-71': 'A similar situation has been encountered in the context of dirty [MATH]-wave superconductors [CITATION] where it has been argued that the quasiparticle conductivity receives aditional logarithmic contributions from the processes involving RR (AA) Cooperons [CITATION].', 'cond-mat-0602398-2-1-72': 'Furthermore, the authors of Ref. [CITATION] found the overall logarithmic correction to the spin (thermal) conductivity of normal [MATH]-wave quasiparticles to be negative, contrary to (in this case, ill-defined due to a non-conservation of charge) electrical one.', 'cond-mat-0602398-2-1-73': 'In view of the formal differences between the Dirac-like descriptions of planar [MATH]-wave superconductors and graphene, it is, at least, conceivable that the correction to the electrical conductivity of graphene could change sign in the low-density regime, too.', 'cond-mat-0602398-2-1-74': 'Lastly, we comment on the possiblity (as well as the need) of including other types of randomness.', 'cond-mat-0602398-2-1-75': 'While the vertex (2) is devised as a general model of short-range (screened) impurities generating a random scalar potential, it obviously misses out on those types of disorder that can be best represented by either a random vector potential or a random mass of the Dirac fermions.', 'cond-mat-0602398-2-1-76': 'The latter might be relevant in the situation where a spatially inhomogeneous charge or spin density wave ordering sets in.', 'cond-mat-0602398-2-1-77': 'In turn, the former can be utilized to describe extended lattice defects (dislocations, disclinations, and cracks) in terms of a random magnetic flux [CITATION].', 'cond-mat-0602398-2-1-78': 'In Ref. [CITATION], it was assumed that the corresponding random vector potential (rather then the flux) is [MATH]-correlated in space ([MATH]).', 'cond-mat-0602398-2-1-79': 'Although this model possesses a conformal invariance and, therefore, can be analyzed in quite some detail [CITATION], a more adequate to the task would obviously be the model of a long-range correlated random vector potential ([MATH]).', 'cond-mat-0602398-2-1-80': 'The latter is known to show a rather different behavior even in the high-energy (ballistic) regime, as demonstrated in Ref. [CITATION].', 'cond-mat-0602398-2-1-81': 'A further investigation into these and related issues will be presented elsewhere.', 'cond-mat-0602398-2-1-82': 'In summary, we carried out a comprehensive analysis of the first-order quantum correction to the conductivity of doped graphene.', 'cond-mat-0602398-2-1-83': 'We identified the conditions under which the conductivity correction becomes positive, negative, or zero.', 'cond-mat-0602398-2-1-84': 'The earlier analysis of Ref. [CITATION] has been critically assessed, and the open problems have been outlined.', 'cond-mat-0602398-2-1-85': 'This research was supported by NSF under Grant DMR-0349881.', 'cond-mat-0602398-2-1-86': 'The author acknowledges valuable communications with E. Abrahams, A. Geim, and H. Suzuura.'}

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[]

[['cond-mat-0602398-1-20-0', 'cond-mat-0602398-2-1-24'], ['cond-mat-0602398-1-20-2', 'cond-mat-0602398-2-1-26'], ['cond-mat-0602398-1-20-2', 'cond-mat-0602398-2-1-27']]

[]

['cond-mat-0602398-1-39-0', 'cond-mat-0602398-3-39-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0602398

{'cond-mat-0602398-3-0-0': 'We discuss localization properties of the Dirac-like electronic states in monolayers of graphite.', 'cond-mat-0602398-3-0-1': 'In the framework of a general disorder model, we identify the conditions under which such standard localization effects as, e.g., the logarithmic temperature-dependent conductivity correction appear to be strongly suppressed, as compared to the case of a two-dimensional electron gas with parabolic dispersion, in agreement with recent experimental observations.', 'cond-mat-0602398-3-1-0': 'After several decades of predominantly application-driven studies, graphene has finally been recognized as a unique example of the system of two-dimensional fermions with a linear dispersion and pseudo-relativistic kinematical properties.', 'cond-mat-0602398-3-1-1': 'The recent advances in microfabrication of graphitic samples that are only a few carbon layers thick [CITATION] have made it possible to test the early theoretical predictions of the anomalous properties of this system [CITATION].', 'cond-mat-0602398-3-2-0': 'The most striking experimental observation up to date was that of an anomalous quantization of the Hall conductivity [CITATION] which is characteristic of the pseudo-relativistic nature of the quasiparticle excitations in this system.', 'cond-mat-0602398-3-2-1': 'The other properties manifesting this peculiar single-particle kinematics have been revealed by magnetotransport measurements, including the [MATH] dependence of the energies of the (non-equidistant) Landau levels and the intrinsic [MATH]-shift of the phase of the Shubnikov-de Haas oscillations [CITATION].', 'cond-mat-0602398-3-3-0': 'In a (nearly) degenerate semimetal such as graphite, the Coulomb interactions are expected to play an important role, largely due to their poor screening [CITATION].', 'cond-mat-0602398-3-3-1': 'Besides, any interplay between the Coulomb interactions and disorder is likely to further modify the behavior of the idealized (clean and non-interacting) Dirac fermion system.', 'cond-mat-0602398-3-3-2': 'Thus far, however, experiment has not yet provided a direct evidence of any interaction-induced phenomena, some of which were predicted in the recent years [CITATION].', 'cond-mat-0602398-3-4-0': 'Also, contrary to the situation in the conventional 2DEG, the data of Refs. [CITATION] do not seem to manifest any pronounced weak-localization effects, either.', 'cond-mat-0602398-3-4-1': 'Motivated by these observations, in the present work we set out to study electron localization in graphene.', 'cond-mat-0602398-3-4-2': 'Although this topic has already attracted some recent attention [CITATION], we shall demonstrate that the analysis of Ref. [CITATION] is neither complete, nor (curiously enough, considering that the assertions made in Ref. [CITATION] are mostly correct) do the arguments presented in that work appear to be technically sound.', 'cond-mat-0602398-3-5-0': 'The electronic band structure of graphene is characterized by the presence of a pair of inequivalent nodal points at the wave vectors [MATH] where [MATH] is the lattice spacing.', 'cond-mat-0602398-3-5-1': 'At these two and the four other points in the Brillouin zone obtainable from [MATH] with a shift by one of the reciprocal lattice vectors [MATH], the valence and conduction bands touch upon each other as a pair of opposing cones with the opening angle given by the Fermi velocity [MATH].', 'cond-mat-0602398-3-6-0': 'In the leading approximation, quasiparticle excitations in the vicinity of the nodal points (hereafter referred to as valleys) can be described by the Dirac Hamiltonian [CITATION] [EQUATION] where [MATH] is the triplet of the Pauli matrices acting in the space of spinors [MATH] composed of the values of the electron wave function on the [MATH] and [MATH] sublattices of the hexagonal lattice of graphene, whereas the triplet [MATH] acts in the valley subspace.', 'cond-mat-0602398-3-6-1': 'In the absence of magnetic field, the Hamiltonian (1) remains a unity matrix in the physical spin subspace.', 'cond-mat-0602398-3-7-0': 'In the presence of disorder, the quasiparticles experience both intra- and inter-valley scattering.', 'cond-mat-0602398-3-7-1': 'Upon averaging over disorder, the most general form of the elastic four-fermion vertex induced by disorder can be described by the expression [EQUATION] where we used the customary "g-oloqical" notations to denote the processes of forward scattering between fermions from the same ([MATH]) and different ([MATH]) valleys, as well as those of "backward" scattering ([MATH]) with the momentum transfer close to [MATH].', 'cond-mat-0602398-3-7-2': 'Besides, we have included the possibility of "umklapp" scattering ([MATH]), in which case the total momentum of two fermions changes by [MATH].', 'cond-mat-0602398-3-7-3': 'A justification for this (not immediately obvious, considering that the total momentum changes by only a fraction of the reciprocal latice vector) extension of the disorder model will be discussed below.', 'cond-mat-0602398-3-8-0': 'In the framework of the self-consistent Born approximation (SCBA), the effect of disorder on the quasiparticle spectrum is described by a self-energy which obeys the equation [EQUATION] where the retarded Green function is given by the expression [EQUATION] that includes a chemical potential [MATH] which allows one to account for a variable electron density.', 'cond-mat-0602398-3-9-0': 'Notably, the solution to Eq. (3) turns out to be diagonal in the valley subspace, [MATH], and proportional to the inverse elastic lifetime [EQUATION] where [MATH] is the density of states at the Fermi energy.', 'cond-mat-0602398-3-9-1': 'In the low-doping limit ([MATH]), a finite value of [MATH] is dominated by disorder.', 'cond-mat-0602398-3-9-2': 'A similar phenomenon has been extensively studied in the context of normal quasiparticle transport in dirty [MATH]-wave superconductors where [MATH] and the spectrum possesses an exact particle-hole symmetry [CITATION].', 'cond-mat-0602398-3-9-3': 'In both cases one obtains the self-consistent DOS as [MATH].', 'cond-mat-0602398-3-10-0': 'However, most of the data of Refs. [CITATION] pertain to the regime of relatively high dopings ([MATH]), as indicated by, e.g., the measured mean free path, which was found to be of order [MATH] at electron densities [MATH].', 'cond-mat-0602398-3-10-1': 'In this regime, the value of [MATH] is controlled by the finite radius of the Fermi surface [MATH] and is only weakly affected by disorder.', 'cond-mat-0602398-3-11-0': 'Therefore, the double-pole Green function (4) given by a four-by-four matrix can be well approximated by a pair of two-by-two matrices containing single-poles [EQUATION] where [MATH] and [MATH].', 'cond-mat-0602398-3-12-0': 'Even more importantly, the presence of a large parameter [MATH] facilitates a systematic account of quantum interference corrections to the SCBA results.', 'cond-mat-0602398-3-12-1': 'We mention, in passing, that in the low-doping ([MATH]) limit the only parameter that can (at least, in principle) be used for the analysis of such corrections is the (inverse) number of valleys, whose actual value is, of course, [MATH].', 'cond-mat-0602398-3-12-2': 'The same caveat plagues the analysis of dirty [MATH]-wave superconductors where the strength of the conductivity corrections is governed by the number of pairs of opposite nodes of the order parameter, whose physical value is again equal two.', 'cond-mat-0602398-3-13-0': 'Luckily, in the high-density regime that, so far, has been probed in graphene, the leading quantum correction to the zeroth-order (Drude) conductivity stem from the standard single-Cooperon (fan-shaped) diagram.', 'cond-mat-0602398-3-13-1': 'From the technical standpoint, however, the calculation of the corresponding correction appears to be somewhat more involved due to the matrix structure of the Green function (6) and the vertex (2).', 'cond-mat-0602398-3-14-0': 'The Drude conductivity itself is given by the standard expression [MATH] where the renormalization factor [MATH] accounts for a ladder series of vertex corrections associated with one of the two current operators [MATH] inserted into the fermion loop in the diagrammatic representation of the Kubo formula.', 'cond-mat-0602398-3-14-1': 'Being given by a non-singular (angular momentum [MATH]) diffusion mode in the expansion over the angular harmonics [MATH], [MATH] is a function of the parameters [MATH].', 'cond-mat-0602398-3-15-0': 'Expanding the expression for the Cooperon in the same basis as that used in Eq. (2), we obtain equations for the corresponding amplitudes [MATH], each of which is a matrix in the direct product of two [MATH] subspaces [EQUATION] where [MATH] denotes a convolution of a pair of Green functions.', 'cond-mat-0602398-3-15-1': 'Computing the latter at [MATH], we obtain [EQUATION]', 'cond-mat-0602398-3-15-2': 'It can be readily seen that Eqs. (7) split onto two pairs which only couple [MATH] and [MATH], respectively.', 'cond-mat-0602398-3-15-3': 'Their solutions read [EQUATION] where [MATH], [MATH], [MATH], and [MATH].', 'cond-mat-0602398-3-16-0': 'The quantum conductivity correction (including spin) [EQUATION] involves the [MATH] and [MATH] components of the Cooperon, whose contributions turn out to be negative and positive, respectively.', 'cond-mat-0602398-3-16-1': 'The logarithmic temperature-dependent part of Eq.(10) can be cast in the form [EQUATION] where we introduced an inelastic phase relaxation rate [MATH] which provides a cutoff in the momentum integration and diverges at [MATH].', 'cond-mat-0602398-3-17-0': 'The analysis of Eq.(11) reveals that, in the absence of a fine tuning between the amplitudes of backward scattering and umklapp processes ([MATH]) the [MATH] Cooperon always acquires a gap [MATH].', 'cond-mat-0602398-3-17-1': 'On the other hand, the [MATH] mode remains gapless, provided that all the forward scattering processes are controlled by the same amplitude (i.e., [MATH]).', 'cond-mat-0602398-3-18-0': 'Conversely, making the [MATH] mode gapful and inverting the sign of the conductivity correction would only be possible under the condition [MATH] which is unlikely to be satisfied for any realistic impurity potential that yields equal amplitudes of the processes of intra- and inter-valley forward scattering.', 'cond-mat-0602398-3-19-0': 'The antilocalizing behavior predicted in Ref. [CITATION] for [MATH] and [MATH]) can only occur at intermediate temperatures (namely, at [MATH]), whereas at still lower temperatures the overall sign of (11) reverts to negative, thereby suggesting the onset of rather conventional weak localization.', 'cond-mat-0602398-3-20-0': 'The above conclusions apply to the general disorder model (2).', 'cond-mat-0602398-3-20-1': 'However, in the situation where disorder is realized as a random distribution of impurities with a concentration [MATH] and a (short-range) potential [MATH], it suffices to introduce only two independent parameters [EQUATION]', 'cond-mat-0602398-3-20-2': 'The above expressions represent a [MATH]-matrix computed for an arbitrary strength of disorder, the customary Born and unitarity (where the scattering phase approaches [MATH]) limits corresponding to [MATH] and [MATH], respectively.', 'cond-mat-0602398-3-20-3': '[In the case of a genuine long-range (unscreened) impurity potential, the dependence [MATH] on the transferred momentum makes the explicit formulas for [MATH] more involved, though.]', 'cond-mat-0602398-3-21-0': 'Provided that the relations (12) between the parameters [MATH] hold, one obtains [MATH], and the logarithmic term in (11) vanishes as a result of the exact cancellation between the contributions of the localizing ([MATH]) and antilocalizing ([MATH]) Cooperon modes.', 'cond-mat-0602398-3-21-1': 'It has to be stressed, however, that such a strong suppression of (anti)localization would only be possible due to an opening of the umklapp channel.', 'cond-mat-0602398-3-21-2': 'In turn, the latter requires an emergence of a crystal superstructure with the wave vector [MATH] (or equivalent).', 'cond-mat-0602398-3-22-0': 'While an isolated sheet of weakly-interacting graphene would apparently lack such a superstructure, it is conceivable that the latter might emerge if a commensurate substrate were used during the process of microfabfication.', 'cond-mat-0602398-3-22-1': 'Besides, a commensurate corrugation could occur [CITATION] due to the Coulomb correlations that can induce spatially periodic patterns of the electron density itself.', 'cond-mat-0602398-3-22-2': 'The possibility of a spontaneous formation of such charge density wave states has long been discussed in the general context of degenerate semimetals and, specifically, in graphene [CITATION].', 'cond-mat-0602398-3-23-0': 'Next, we comment on the technical details presented in Ref.[CITATION] where the first prediction of the antilocalization behavior for [MATH] and [MATH] was made.', 'cond-mat-0602398-3-23-1': 'In essence, instead of working directly with the matrices in the sublattice subspace, the authors of Ref.[CITATION] used single-component (projected) Green functions [MATH] complemented by matrix elements of a (purely forward-scattering) impurity potential between different Bloch wave functions.', 'cond-mat-0602398-3-24-0': 'To that end, in Ref.[CITATION] the latter were chosen in a particular (asymmetrical) gauge [EQUATION]', 'cond-mat-0602398-3-24-1': 'The aforementioned matrix element of the impurity potential is then given by a complex-valued expression [MATH].', 'cond-mat-0602398-3-25-0': 'Next, the authors of Ref.[CITATION] asserted that the equations for the Cooperon mode involve the disorder-induced vertex [EQUATION] which describes scattering from the Bloch states with momenta [MATH] and [MATH] into [MATH] and [MATH] in the limit [MATH].', 'cond-mat-0602398-3-26-0': 'It was then argued in Ref.[CITATION] that the full Cooperon amplitude inherits the phase factor from Eq.(14) and, therefore, becomes negative for [MATH] where [MATH], which configuration of the momenta provides a dominant contribution to the weak localization correction.', 'cond-mat-0602398-3-26-1': 'Thus obtained, the negative sign of the Cooperon amplitude was claimed to be instrumental for the onset of antilocalizing behavior.', 'cond-mat-0602398-3-27-0': 'The validity of the above argument can be disputed by observing that the choice of a different (symmetrical) gauge for the Bloch wave functions [EQUATION] would result in a purely real matrix element [MATH], the use of which yields Eq. (14) without the said phase factor, hence the (deemed to be crucially important) sign change of the Cooperon amplitude does not seem to occur.', 'cond-mat-0602398-3-28-0': 'Although the observed gauge dependence of the Cooperon is perfectly consistent with its being a gauge non-invariant two-particle amplitude, the conductivity correction is, of course, supposed to be gauge invariant.', 'cond-mat-0602398-3-28-1': 'A resolution of such apparent contradiction goes as follows.', 'cond-mat-0602398-3-29-0': 'In fact, the vertex that ought to be used in the construction of the Cooperon is the particle-hole (exchange) amplitude [MATH] which is manifestly gauge invariant and given by Eq. (14) without the phase factor in question.', 'cond-mat-0602398-3-29-1': 'In this regard, it is important to realize that the amplitudes [MATH] and [MATH] represent two non-commuting (and, in this particular case, unequal) limits of the general vertex (2).', 'cond-mat-0602398-3-30-0': 'On a side note, it might also be tempting to try to explain the experimentally found suppression of localization by the presence of the factor [MATH] vanishing for [MATH] in Eq. (14).', 'cond-mat-0602398-3-30-1': 'Notice, however, that the Cooperon built out of such a vertex lacks this factor, since the gapless pole develops for only one (namely, [MATH]) angular harmonic, whereas the other two harmonics that constitute the above angular-dependent factor ([MATH]) receive only a finite enhancement by a factor of two.', 'cond-mat-0602398-3-31-0': 'Thus, an obvious suppression of the backward scattering off of an individual impurity (which fact has often been mentioned in the context of electron transport in 1D carbon nanotubes) does not necessarily imply the absence of (anti)localization in the deep diffusive regime in 2D graphene.', 'cond-mat-0602398-3-32-0': 'The true origin of the antilocalizing behavior exhibited by Eq. (11) for [MATH], can be traced back to the negative signature of the expansion of the Cooperon mode [MATH] over the product basis [MATH], as opposed to that of [MATH] (see Eqs. (9)).', 'cond-mat-0602398-3-32-1': 'Interpreting the sublattice index as a fictitious spin one-half ([MATH]), one can associate the [MATH] component with the singlet mode [MATH]) which, in the spin-orbit context, is known to be a common source of antilocalization.', 'cond-mat-0602398-3-33-0': 'In this regard, the present anilocalizing behavior is similar to that predicted in the case of 2DEG with a strong spin-orbit coupling of either Rashba or Dresselhaus kind [CITATION] [EQUATION]', 'cond-mat-0602398-3-33-1': 'The details of the calculations presented above differ from those pertaining to the case of the spin-orbit coupling (for one thing, except for a splitting, the linear in momentum SO-coupling does not alter the fermion dispersion, whereas in the case of graphene it is the linear term which is solely responsible for the dispersion).', 'cond-mat-0602398-3-34-0': 'Yet another argument invoked by the authors of Ref. [CITATION] to support their prediction of the antilocalizing behavior intermediate temperatures exploits the notion of the Berry phase associated with the electron wavefunctions (13).', 'cond-mat-0602398-3-35-0': 'Indeed, the first Chern number ("vorticity") [EQUATION] where the momentum space integral is taken along any contour enclosing one of the Dirac points [MATH], yields [MATH] for both wavefunctions (13).', 'cond-mat-0602398-3-35-1': 'In Ref. [CITATION] (see also [CITATION]) this observation was used to assert that the interference between any pair of counter-propagating electron waves becomes destructive, thereby giving way to the antilocalizing behavior.', 'cond-mat-0602398-3-36-0': 'However, the Berry phase (17) would be identically zero, if the wave functions (14) were used instead of (13), which suggests that its relationship with the Cooperon is somewhat more subtle.', 'cond-mat-0602398-3-36-1': 'To that end, it is worth mentioning that the two Dirac species emerging in the continuous (low-energy) description of the original lattice system must have opposite chiralities [CITATION], as required for compliance with the ubiquitous Nielsen-Ninomiya ("fermion doubling") theorem [CITATION].', 'cond-mat-0602398-3-36-2': 'In turn, this observation seems to suggest that, even if the Berry phases of the individual species happen to be finite ([MATH]), they should still add up to zero.', 'cond-mat-0602398-3-37-0': 'A cancellation of the overall Berry phase would also be consistent with the vanishing of the quantum conductivity correction in the presence of equally strong intervalley backward and umklapp scattering, which, in effect, make the two Dirac points indistinguishable (the shortest reciprocal lattice vector becomes equivalent to [MATH]).', 'cond-mat-0602398-3-38-0': 'Recently, the role of the Berry phase has also been brought up in the context of the spin Hall effect generated by a spin-orbit coupling.', 'cond-mat-0602398-3-38-1': 'Interestingly enough, the dependence of the spin Hall conductivity upon the coefficients [MATH] and [MATH] from Eq. (16) (and, especially, its exact vanishing at [MATH]) can be established with the use of the Bloch wave functions in either asymmetrical (similar to Eq. (13)) or symmetrical (similar to Eq. (14)) gauge, where the corresponding Berry phase is non-zero and zero, respectively (cf., e.g., the two references in [CITATION]).', 'cond-mat-0602398-3-38-2': 'This observation, too, indicates that the relationship between the Berry phase and transport properties might be rather intricate.', 'cond-mat-0602398-3-39-0': 'Before concluding, a few more comments are in order.', 'cond-mat-0602398-3-40-0': 'In reality, the weak (anti)localization corrections are accompanied by the Aronov-Altshuler ones, which stem from the interference between disorder and Coulomb interactions.', 'cond-mat-0602398-3-40-1': 'Despite a similar temperature dependence ([MATH]), the former, but not the latter, gets quenched by an applied magnetic field, thus allowing one to discriminate between the two.', 'cond-mat-0602398-3-41-0': 'Also, considering the potential attainability of the very low-density regime, it would be of interest to extend the analysis of the localization effects to the case [MATH].', 'cond-mat-0602398-3-41-1': 'However, as we have already pointed out, in this regime a systematic expansion of the conductivity corrections would only be possible for an (unphysically) large number of valleys.', 'cond-mat-0602398-3-42-0': 'Moreover, even if this number were indeed large, the calculation itself would still pose a significant challenge.', 'cond-mat-0602398-3-42-1': 'The main technical difficulty here is due to the double-pole structure of the Green functions (4), which results in the emergence of additional gapless Cooperon and diffusion modes in the RR (AA) channel, alongside the conventional (RA) one.', 'cond-mat-0602398-3-42-2': 'A similar situation has been encountered in the context of dirty [MATH]-wave superconductors [CITATION] where it has been argued that the quasiparticle conductivity receives aditional logarithmic contributions from the processes involving RR (AA) Cooperons [CITATION].', 'cond-mat-0602398-3-43-0': 'Furthermore, the authors of Ref. [CITATION] found the overall logarithmic correction to the spin (thermal) conductivity of normal [MATH]-wave quasiparticles to be negative, contrary to (in this case, ill-defined due to a non-conservation of charge) electrical one.', 'cond-mat-0602398-3-43-1': 'In view of the formal differences between the Dirac-like descriptions of planar [MATH]-wave superconductors and graphene, it is, at least, conceivable that the correction to the electrical conductivity of graphene could change sign in the low-density regime, too.', 'cond-mat-0602398-3-44-0': 'Lastly, we comment on the possiblity (as well as the need) of including other types of randomness.', 'cond-mat-0602398-3-44-1': 'While the vertex (2) is devised as a general model of short-range (screened) impurities generating a random scalar potential, it obviously misses out on those types of disorder that can be best represented by either a random vector potential or a random mass of the Dirac fermions.', 'cond-mat-0602398-3-45-0': 'The latter might be relevant in the situation where a spatially inhomogeneous charge or spin density wave ordering sets in.', 'cond-mat-0602398-3-45-1': 'In turn, the former can be utilized to describe extended lattice defects (dislocations, disclinations, and cracks) in terms of a random magnetic flux [CITATION].', 'cond-mat-0602398-3-45-2': 'In Ref. [CITATION], it was assumed that the corresponding random vector potential (rather then the flux) is [MATH]-correlated in space ([MATH]).', 'cond-mat-0602398-3-45-3': 'Although this model possesses a conformal invariance and, therefore, can be analyzed in quite some detail [CITATION], a more adequate to the task would obviously be the model of a long-range correlated random vector potential ([MATH]).', 'cond-mat-0602398-3-45-4': 'The latter is known to show a rather different behavior even in the high-energy (ballistic) regime, as demonstrated in Ref. [CITATION].', 'cond-mat-0602398-3-46-0': 'A further investigation into these and related issues will be presented elsewhere.', 'cond-mat-0602398-3-47-0': 'In summary, we carried out a comprehensive analysis of the first-order quantum correction to the conductivity of doped graphene.', 'cond-mat-0602398-3-47-1': 'We identified the conditions under which the conductivity correction becomes positive, negative, or zero.', 'cond-mat-0602398-3-47-2': 'The earlier analysis of Ref. [CITATION] has been critically assessed, and the open problems have been outlined.', 'cond-mat-0602398-3-48-0': 'This research was supported by NSF under Grant DMR-0349881.', 'cond-mat-0602398-3-48-1': 'The author acknowledges valuable communications with E. Abrahams, A. Geim, V. Gusynin, and H. Suzuura.'}

1007.1811

{'1007.1811-1-0-0': 'We study a special class of the cognitive radio channel in which the receiver of the cognitive pair does not suffer interference from the primary user.', '1007.1811-1-0-1': 'This class of cognitive Z-interference channels has not been investigated before.', '1007.1811-1-0-2': 'Previously developed general encoding schemes are complex as they attempt to cope with arbitrary channel conditions and thus typically result in rate regions that are difficult to evaluate.', '1007.1811-1-0-3': 'Instead, in this paper we derive simple rate regions that are easily computable.', '1007.1811-1-0-4': 'We first present several explicit achievable regions for the general discrete memoryless case.', '1007.1811-1-0-5': 'We then extend these regions to Gaussian channels.', '1007.1811-1-0-6': 'With a simple outer bound we establish a new capacity region in the strong-interference regime.', '1007.1811-1-0-7': 'Lastly, we provide numerical comparisons between the derived achievable rate regions and the outer bound.', '1007.1811-1-1-0': '# Introduction', '1007.1811-1-2-0': 'Cognitive radio techniques hold the potential to significantly improve the efficiency of spectrum usage.', '1007.1811-1-2-1': 'As a result, the capacity gains associated with cognitive radios has received significant attention [CITATION].', '1007.1811-1-2-2': "In these works, several achievable rate regions have been obtained by developing coding schemes based on rate splitting, Gel'fand-Pinsker coding (dirty paper coding), and superposition coding.", '1007.1811-1-2-3': 'The capacity region for the general case has not been determined except for the case of the weak interference [CITATION], and the case of very strong interference [CITATION].', '1007.1811-1-2-4': 'For a special case of the cognitive radio channel referred to as the cognitive Z-interference channel, the capacity region has been established under the assumption of a noiseless link from the primary user to its receiver in [CITATION].', '1007.1811-1-2-5': 'A more generalized model of the cognitive Z-interference channel has also been considered in [CITATION].', '1007.1811-1-2-6': "Both of these works assume that the primary user's receiver does not suffer from interference generated by the cognitive user.", '1007.1811-1-2-7': 'Complementing the existing works, in this paper we consider another type of cognitive Z-interference channel, in which the receiver of the cognitive user does not suffer from interference generated by the primary user.', '1007.1811-1-2-8': 'This channel models the scenario where the link from the primary user to the receiver of the cognitive user suffers from strong shadowing, e.g., the cognitive pair and a TV receiver are in a room which is far from the primary user (the TV broadcasting station).', '1007.1811-1-3-0': 'For this new type of cognitive Z-interference channel, we may obtain quite a few achievable rate regions by specializing some known achievable rate regions developed for the cognitive radio channel.', '1007.1811-1-3-1': 'As some of the coding schemes developed for the general case are intended to cope with arbitrary channel conditions, the associated achievable rate regions are very complex and hard to compute.', '1007.1811-1-3-2': 'If we directly apply the general formulas to the Z-interference channel, the rate regions do not simplify despite the missing interference link.', '1007.1811-1-3-3': 'In the Z-interference channel under consideration, it appears that not all components of the general coding schemes are needed to achieve tight upper and lower bounds, which is the main focus of this paper.', '1007.1811-1-4-0': 'It has been shown that simple coding schemes are capacity achieving when the cognitive radio channel is in a certain regimes.', '1007.1811-1-4-1': 'For example, the capacity region for the cognitive radio channel in the weak-interference regime is achieved with a very simple coding scheme, and the capacity consists of only two inequalities.', '1007.1811-1-4-2': 'Similar phenomenon can be found for the interference channel and the broadcast channel as well.', '1007.1811-1-4-3': "In particular, the best achievable rate region for the broadcast channel is Marton's region [CITATION], while when the channel is Gaussian, the capacity region can be achieved by a simple superposition coding or a particular order of dirty paper coding.", '1007.1811-1-4-4': 'For the interference channel, the best general coding scheme is the rate splitting scheme and the associated Han-Kobayashi rate region is quite complicated [CITATION].', '1007.1811-1-4-5': 'However, when the interference is very weak, the capacity region is achieved by a naive coding scheme that simply ignores the interference; when the interference is strong, the capacity region is achieve by a simple scheme requiring the two receivers to decode both messages.', '1007.1811-1-5-0': 'Therefore, in this paper, we look into simple coding schemes and simple achievable rate regions for the cognitive Z-interference channel.', '1007.1811-1-5-1': 'We focus on the high-interference regime, since the capacity region for the weak-interference regime is implied by that of the cognitive radio channel.', '1007.1811-1-5-2': 'We wish to examine how these simple coding schemes perform by comparing their achievable rate regions with the capacity outer bound.', '1007.1811-1-5-3': 'The simple coding schemes are in fact not new, but rather are extracted from the coding schemes in existing works.', '1007.1811-1-5-4': 'Specifically, we first derive a few different simple achievable rate regions extended from the achievable rate region for the cognitive radio channel developed in [CITATION], and also obtain a simple rate region with a coding scheme which has been applied as a component coding scheme in [CITATION].', '1007.1811-1-5-5': 'We then extended their rate regions to the Gaussian case.', '1007.1811-1-5-6': 'We find that the last simple achievable rate region is the capacity region for the cognitive Z-interference channel, when the interference link is not too strong in the high-interference regime.', '1007.1811-1-5-7': 'Through numerical comparisons, we also find another simple achievable rate region that dominates all other simple achievable rate regions when the interference link is very strong.', '1007.1811-1-6-0': 'The paper is organized as follows.', '1007.1811-1-6-1': 'In Section II, we present the channel model and the related definitions.', '1007.1811-1-6-2': 'In Section III, we derive the aforementioned simple achievable rate regions for the channel in the discrete memoryless case.', '1007.1811-1-6-3': 'In Section IV, we present the achievable rate regions for the Gaussian case and our new special case capacity result.', '1007.1811-1-6-4': 'We also present the numerical results.', '1007.1811-1-6-5': 'The paper is concluded in Section V.', '1007.1811-1-7-0': '# Channel Model', '1007.1811-1-8-0': 'We consider the following channel that involves one primary user and one cognitive user.', '1007.1811-1-8-1': "As shown in in Fig. [REF], both users need to send a message to their individual receivers, while the cognitive user is assumed to have non-causal knowledge of the primary user's message.", '1007.1811-1-8-2': 'The channel is defined by [MATH], where [MATH] and [MATH], [MATH], denote the channel input and output alphabets, and [MATH] denotes the collection of channel transition probabilities.', '1007.1811-1-8-3': "Furthermore, we assume [MATH] can be factored as [MATH], i.e., the cognitive user's receiver output is not affected by the primary user's channel input.", '1007.1811-1-8-4': 'The source messages [MATH], [MATH], are assumed to be uniformly generated over the respective ranges: [MATH], [MATH].', '1007.1811-1-8-5': 'We call this channel the cognitive Z-interference channel, and denote it as [MATH].', '1007.1811-1-9-0': 'An [MATH] code for [MATH] consists of an encoding function at the cognitive user [MATH], an encoding function at the primary user [MATH], and one decoding function at each receiver [MATH], [MATH], with the probability of decoding errors defined as [MATH], where the individual error probability at each receiver is computed as [MATH].', '1007.1811-1-10-0': 'A non-negative rate pair [MATH] is achievable for [MATH] if there exists a sequence of codes [MATH] for the channel such that [MATH] approaches 0 as [MATH].', '1007.1811-1-11-0': 'The capacity region of [MATH] is the closure of the set of all achievable rate pairs.', '1007.1811-1-11-1': 'Any subset of the capacity region is an achievable rate region.', '1007.1811-1-12-0': '# Achievable Rate Regions', '1007.1811-1-13-0': 'In this section, we derive a few simple achievable rate regions with explicit descriptions based on the achievable rate region developed in [CITATION], and a simple achievable rate region obtained using a component of the coding schemes developed in [CITATION].', '1007.1811-1-14-0': 'Let [MATH], [MATH], [MATH], and [MATH] be arbitrary auxiliary random variables defined over finite alphabets: [MATH], [MATH], [MATH], and [MATH], respectively.', '1007.1811-1-14-1': 'Let [MATH] denote the set of all joint distributions [MATH] that factor in the following form [EQUATION]', '1007.1811-1-14-2': 'For any joint distribution [MATH], define [MATH] as the set of non-negative rate pairs [MATH] such that [EQUATION]', '1007.1811-1-14-3': 'Define [MATH], and we have the following result.', '1007.1811-1-15-0': 'The rate region [MATH] is an achievable rate region for the discrete memoryless cognitive Z-interference channel [MATH].', '1007.1811-1-16-0': 'The proof is based on [CITATION].', '1007.1811-1-16-1': 'First swap the indices, e.g., change [MATH] to [MATH], change [MATH] to [MATH], etc. due to the positions of the cognitive user and the primary user being switched in the current model.', '1007.1811-1-16-2': 'Next, drop the common information of the primary user, i.e., set [MATH] and [MATH] to be a constant to obtain the rate region in the implicit form.', '1007.1811-1-16-3': 'Lastly, perform Fourier-Motzkin elimination on the implicit rate region, and to obtain the set of inequalities defining [MATH].', '1007.1811-1-16-4': 'The proposition follows.', '1007.1811-1-17-0': "In this coding scheme, rate splitting is only applied on the cognitive user's message, i.e., [MATH].", '1007.1811-1-17-1': 'More specifically, the common message [MATH] is encoded with codewords generated using [MATH], and the private message [MATH] is encoded with codewords generated using [MATH].', '1007.1811-1-17-2': "The primary user's message [MATH] is encoded with codewords generated using both [MATH] and [MATH].", '1007.1811-1-17-3': "The cognitive user's receiver decodes its own messages [MATH] only, while the primary user's receiver jointly decodes its own message [MATH] and the common message from the cognitive user [MATH].", '1007.1811-1-18-0': 'Next, we specialize the rate region [MATH] into two simple achievable rate regions.', '1007.1811-1-19-0': "First, we completely remove the 'artificially' created common information, i.e., no rate splitting is applied to the cognitive user's message.", '1007.1811-1-19-1': 'The following rate region can be obtained by setting [MATH] as a constant.', '1007.1811-1-20-0': 'Any non-negative rate pair [MATH] satisfying [EQUATION] for any joint distribution [EQUATION] is achievable for the channel [MATH].', '1007.1811-1-21-0': 'Denote rate region [REF]-[REF] as [MATH].', '1007.1811-1-21-1': "Note that [MATH] and [MATH] are employed to perform Marton's binning, therefore when it is extended to the Gaussian case, it becomes two different rate regions resulted from dirty paper coding in different orders.", '1007.1811-1-22-0': 'Instead of removing the common information of the cognitive user, we now remove private information.', '1007.1811-1-22-1': 'Specifically, set [MATH] as a constant, and merge [MATH] and [MATH] into one random variable [MATH], i.e., set [MATH].', '1007.1811-1-22-2': 'We have the following simple achievable rate region.', '1007.1811-1-23-0': 'Any non-negative rate pair [MATH] satisfying [EQUATION] for any joint distribution [EQUATION] is achievable for the channel [MATH].', '1007.1811-1-24-0': 'Denote the rate region [REF]-[REF] as [MATH].', '1007.1811-1-24-1': 'This rate region is achieved by using only superposition coding.', '1007.1811-1-24-2': "Nevertheless, it turns out to be the largest amongst all these simple rate regions when the interference link is 'substantially' strong, which is numerically shown in the next section.", '1007.1811-1-25-0': "We present another simple achievable rate region, based on the idea that the artificial common message of the cognitive user can be dirty-paper coded against the primary user's private message.", '1007.1811-1-25-1': "The purpose of dirty paper coding in this coding scheme is to protect the message of the cognitive user from the interference created by the primary user's message.", '1007.1811-1-26-0': 'Any non-negative rate pair [MATH] satisfying [EQUATION] for any joint distribution [EQUATION] is achievable for the channel [MATH].', '1007.1811-1-27-0': 'Let us denote this rate region by [MATH].', '1007.1811-1-27-1': 'The proof can be easily obtained by following the argument in [CITATION].', '1007.1811-1-27-2': "The intention of the dirty paper coding applied in this coding scheme is to let the encoding of the cognitive user's common message take advantage of its non-causal knowledge about the primary user's private message.", '1007.1811-1-27-3': 'As a component of the respective coding schemes, this technique has been employed in several existing works [CITATION].', '1007.1811-1-27-4': 'Nevertheless, there has been no evidence showing that this feature is indeed helpful in terms of increasing achievable rates.', '1007.1811-1-27-5': 'In the next section, we will show that this coding scheme is in fact capacity achieving, when the channel is in the high-interference regime, but the interference is not too strong.', '1007.1811-1-28-0': '# Gaussian Cognitive Z-interference Channel', '1007.1811-1-29-0': 'In this section, we first define the channel model.', '1007.1811-1-29-1': 'We then extend the achievable rate regions derived in the previous section to the Gaussian case, and present a new capacity result for the Cognitive Z-interference Channel in the high-interference regime.', '1007.1811-1-30-0': 'The channel input-output relationship is given by [EQUATION] where [MATH] is the interference link gain and is assumed to be greater than [MATH]; [MATH] and [MATH] are additive white Gaussian noise of zero mean and unit variance.', '1007.1811-1-30-1': 'The power constraints are given by [EQUATION]', '1007.1811-1-30-2': 'Note that when [MATH], the channel belongs to the weak-interference regime, for which the capacity region has been established for the general cognitive radio channel [CITATION].', '1007.1811-1-30-3': 'The capacity achieving scheme is rather simple: the cognitive user spares a portion of its own power to cooperate with the primary user for the transmission of message [MATH], while it performs dirty paper coding on its own message [MATH] by treating the signals carrying [MATH] as the non-causally known interference.', '1007.1811-1-30-4': 'No rate splitting is applied, and each receiver only decodes its own intended message.', '1007.1811-1-30-5': 'We focus on the high-interference regime, which is in general open.', '1007.1811-1-30-6': 'In what follows, we present the Gaussian counterparts of the achievable regions presented in the previous section.', '1007.1811-1-31-0': '## Rate Regions for the Gaussian Cognitive Z-interference Channel', '1007.1811-1-32-0': 'We first extend the rate region [MATH] to the Gaussian case.', '1007.1811-1-32-1': "When Marton's coding scheme is extended to the Gaussian case, the double binning first extends to Gel'fand-Pinsker coding in two different orders, and subsequently dirty-paper coding in two different orders.", '1007.1811-1-32-2': "Similarly, in order to extend [MATH] to the Gaussian case, we also extend the double binning to Gel'fand-Pinsker coding in two different orders, which results in two achievable rate regions described by the follow two groups of rate constraints: [EQUATION] and [EQUATION] respectively.", '1007.1811-1-32-3': "The first rate region denoted by [MATH] corresponds to the case of applying Gel'fand Pinsker coding on [MATH] by treating [MATH] as the known state information, and simultaneously applying Gel'fand Pinsker coding on [MATH] by treating [MATH] as the known state information.", '1007.1811-1-32-4': "The second one denoted by [MATH] corresponds to the case of applying Gel'fand Pinsker coding on [MATH] by treating both [MATH] and [MATH] as the known state information, while [MATH] and [MATH] are merged into only one random variable [MATH].", '1007.1811-1-32-5': 'To further extend [MATH] into the Gaussian case, we apply the following mapping of random variables: [EQUATION] where [MATH], [MATH], [MATH], and [MATH] are Gaussian random variables with zero mean and unit variance; [MATH], [MATH], and [MATH]; and [MATH].', '1007.1811-1-32-6': 'By choosing the optimal values for [MATH] and [MATH], and evaluating the mutual information terms with respect to the mapping and channel input-output relationship, we have the rate region [MATH] defined by the following constraints: [EQUATION]', '1007.1811-1-32-7': 'Similarly, to extend [MATH] to the Gaussian counterpart, we apply the following mapping [EQUATION] and we have the Gaussian achievable rate region [MATH] defined by the following rate constraints [EQUATION]', '1007.1811-1-32-8': 'Now define [MATH].', '1007.1811-1-33-0': 'Any rate pair [MATH] is achievable for the Gaussian Z-interference channel.', '1007.1811-1-34-0': 'Next, we extend the rate region [MATH] to the Gaussian case.', '1007.1811-1-34-1': 'Note that [MATH] is extended from [MATH] by setting [MATH] as a constant.', '1007.1811-1-34-2': 'Hence, it is straightforward to obtain the Gaussian counterpart of [MATH] by setting [MATH] which corresponds to [MATH].', '1007.1811-1-34-3': 'This Gaussian rate region can be expressed as [MATH].', '1007.1811-1-35-0': 'We also extend [MATH] to the its Gaussian counterpart [MATH] as follows.', '1007.1811-1-35-1': 'Let [EQUATION] and we easily find [MATH] defined by the following rate constraints [EQUATION]', '1007.1811-1-35-2': 'Lastly, we extend [MATH] to obtain the Gaussian rate region [MATH].', '1007.1811-1-35-3': 'The following mapping of random variables is applied [EQUATION]', '1007.1811-1-35-4': 'Evaluating the rate constraints defining [MATH] with respect to the above mapping, we obtain the rate region [MATH] defined by the following rate constraints [EQUATION]', '1007.1811-1-35-5': 'By choosing [MATH], which is the dirty-paper coding coefficient maximizing the righthand side of [REF], we have another achievable rate region [MATH] defined by the following rate constraints [EQUATION]', '1007.1811-1-35-6': 'It is easy to see that [MATH].', '1007.1811-1-35-7': 'However, the rate region [MATH] cannot be easily computed, due to the fact that the choice of [MATH] is unbounded, i.e., [MATH], while [MATH] can be easily computed.', '1007.1811-1-35-8': 'Furthermore, the rate region [MATH] even turns out to be the capacity region when interference link gain falls into a certain range in the high-interference regime.', '1007.1811-1-36-0': '## An Outer Bound', '1007.1811-1-37-0': 'Due to the assumption on the cross link gain [MATH] that it is greater than 1, the channel satisfies the conditions of the capacity outer bound derived in [CITATION].', '1007.1811-1-37-1': 'Hence, this outer bound directly applies to the cognitive Z-interference channel in the high-interference regime.', '1007.1811-1-38-0': 'Any achievable rate pair [MATH] for the Gaussian cognitive Z-interference channel defined by [REF]-[REF] with [MATH], satisfies the following constraints [EQUATION] for any [MATH].', '1007.1811-1-39-0': '## Capacity in the High-interference Regime', '1007.1811-1-40-0': 'We observe that the bounds [REF] and [REF] are equal to [REF] and [REF], when we let [MATH].', '1007.1811-1-40-1': 'Hence, the rate region [MATH] meets the outer bound if the rate constraint [REF] is loose.', '1007.1811-1-41-0': 'For the Gaussian cognitive Z-interference channel defined by [REF]-[REF], the capacity region is defined by [EQUATION] with [MATH], when the interference link gain [EQUATION].', '1007.1811-1-42-0': 'The proof is omitted, as it is a cumbersome but simple algebra.', '1007.1811-1-42-1': 'For the rate constraint [REF] to be slack, the righthand side of [REF] has to be greater than or equal to the difference between the terms on the righthand side of [REF] and [REF].', '1007.1811-1-42-2': 'Through the algebra, the condition can be found as [EQUATION].', '1007.1811-1-42-3': 'For this condition to be satisfied for all choices of [MATH], we need [EQUATION]', '1007.1811-1-43-0': '## Numerical Results', '1007.1811-1-44-0': 'In this subsection, we numerically compare the rate regions and the outer bound.', '1007.1811-1-44-1': 'For all the comparisons, we let [MATH], leaving the interference link gain [MATH] to be variable.', '1007.1811-1-45-0': 'First, in Fig. [REF], the rate region [MATH] is compared against the upper bound.', '1007.1811-1-45-1': 'We can observe from the figure that, the achievable rate region meets the outer bound for both [MATH] and [MATH], while the rate region becomes strictly smaller than the outer bound for [MATH].', '1007.1811-1-45-2': 'When the interference link becomes very strong, the coding scheme to achieve [MATH] turns out to be suboptimal, which is shown in the last comparison.', '1007.1811-1-46-0': 'Next, in Fig. [REF], we compare the rate regions [MATH], [MATH], [MATH], and the outer bound in the very high-interference regime, i.e., [MATH].', '1007.1811-1-46-1': 'It can be observed that [MATH] is strictly smaller than [MATH] and [MATH], while [MATH] and [MATH] always coincide with each other.', '1007.1811-1-46-2': 'Note that [MATH] is in fact a subset of [MATH], as the coding scheme to achieve [MATH] generalizes the one for [MATH].', '1007.1811-1-46-3': 'However, the numerical results indicate that the coding scheme for [MATH] dominates other coding components in the coding scheme to achieve [MATH].', '1007.1811-1-46-4': 'This potentially imply that other coding features, especially the one to achieve [MATH], are redundant, which needs further investigation.', '1007.1811-1-47-0': 'Lastly, we compare the rate regions [MATH], [MATH], [MATH], and the outer bound in the very high-interference regime in Fig. [REF].', '1007.1811-1-47-1': 'At the boundary, [MATH], the rate region [MATH] meets the outer bound, and is strictly larger than both [MATH] and [MATH].', '1007.1811-1-47-2': 'However, when [MATH] is large enough, i.e., [MATH], the rate region [MATH] becomes strictly larger than [MATH].', '1007.1811-1-47-3': 'It is yet a question to find the critical value of [MATH], at which [MATH] becomes larger than [MATH].', '1007.1811-1-48-0': '# Concluding Remarks', '1007.1811-1-49-0': 'In this paper, we have investigated the cognitive Z-interference channel in which the primary user suffers no interference.', '1007.1811-1-49-1': 'Our results demonstrate that simple coding schemes perform well in the strong-interference regime.', '1007.1811-1-49-2': 'In particular, we have shown that the capacity region is achieved with an encoding scheme that combines superposition coding and dirty paper coding (corresponding to [MATH])) for the certain regime of interference, i.e. [MATH], Furthermore, our numerical results suggest that the superposition coding scheme (corresponding to [MATH]) strictly dominates other schemes when interference is very strong.', '1007.1811-1-49-3': 'However, there is a gap between the achievable rate region and the considered outer bound.', '1007.1811-1-49-4': 'We suspect that this gap is due to the looseness of the outer bound.', '1007.1811-1-49-5': 'Tightening of the outer bound is a possible direction of the future work.'}

{'1007.1811-2-0-0': 'We study a special class of the cognitive radio channel in which the receiver of the cognitive pair does not suffer interference from the primary user.', '1007.1811-2-0-1': 'Previously developed general encoding schemes for this channel are complex as they attempt to cope with arbitrary channel conditions, which leads to rate regions that are difficult to evaluate.', '1007.1811-2-0-2': 'The focus of our work is to derive simple rate regions that are easily computable, thereby providing more insights into achievable rates and good coding strategies under different channel conditions.', '1007.1811-2-0-3': 'We first present several explicit achievable regions for the general discrete memoryless case.', '1007.1811-2-0-4': 'We also present an improved outer bound on the capacity region for the case of high interference.', '1007.1811-2-0-5': 'We then extend these regions to Gaussian channels.', '1007.1811-2-0-6': 'With a simple outer bound we establish a new capacity region in the high-interference regime.', '1007.1811-2-0-7': 'Lastly, we provide numerical comparisons between the derived achievable rate regions and the outer bounds.', '1007.1811-2-1-0': '# Introduction', '1007.1811-2-2-0': 'Cognitive radio techniques bear the potential to significantly improve the efficiency of spectrum usage.', '1007.1811-2-2-1': 'As a result, the information-theoretic capacity gains associated with cognitive radios have been the subject of much investigation [CITATION].', '1007.1811-2-2-2': "In these works, several achievable rate regions have been obtained by developing coding schemes based on rate splitting, Gel'fand-Pinsker coding (dirty paper coding), and superposition coding.", '1007.1811-2-2-3': 'The capacity regions for the general cases have not been determined except for the case of weak interference [CITATION], and the case of very strong interference [CITATION].', '1007.1811-2-2-4': 'For a special case of the cognitive radio channel referred to as the cognitive Z-interference channel, the capacity region has been established under the assumption of a noiseless link from the primary transmitter to its receiver in [CITATION].', '1007.1811-2-2-5': 'A more generalized model of the cognitive Z-interference channel has also been considered in [CITATION].', '1007.1811-2-2-6': "Both of these works assume that the primary user's receiver does not suffer from interference generated by the cognitive user.", '1007.1811-2-2-7': 'Complementing the existing works, in this paper we consider another type of cognitive Z-interference channel, in which the receiver of the cognitive user does not suffer from interference generated by the primary user.', '1007.1811-2-2-8': 'This channel models the scenario where the link from the primary transmitter to the receiver of the cognitive user suffers from strong shadowing or other channel losses.', '1007.1811-2-3-0': 'For this new type of cognitive Z-interference channel, achievable rate regions may be obtained by specializing some known achievable rate regions developed for the cognitive radio channel.', '1007.1811-2-3-1': 'As some of the coding schemes developed for the general case are intended to cope with arbitrary channel conditions, the associated achievable rate regions are very complex and hard to compute.', '1007.1811-2-3-2': 'As such, if we directly apply the general formulas to the considered Z-interference channel, the rate regions do not simplify despite the missing interference link.', '1007.1811-2-3-3': 'Instead, we will show in this paper that not all components of the general coding schemes are needed to achieve tight upper and lower bounds.', '1007.1811-2-4-0': 'It has been shown that simple coding schemes are capacity achieving when the cognitive radio channel is in certain regimes.', '1007.1811-2-4-1': 'For example, the capacity region for the cognitive radio channel in the weak-interference regime is achievable with a very simple coding scheme [CITATION], and the capacity is defined by two inequalities.', '1007.1811-2-4-2': 'Similar phenomena can be found for the interference channel and the broadcast channel as well.', '1007.1811-2-4-3': "In particular, the best achievable rate region for the general broadcast channel is Marton's region [CITATION], while when the channel is Gaussian, the capacity region can be achievable by a simple superposition coding or a particular order of dirty paper coding.", '1007.1811-2-4-4': 'This is because an arbitrary Gaussian broadcast channel always falls into one of the two regimes, degraded or reversely degraded.', '1007.1811-2-4-5': 'For the interference channel, the best general coding scheme is the rate splitting scheme and the associated Han-Kobayashi rate region is quite complicated [CITATION].', '1007.1811-2-4-6': 'However, when the interference is very weak, the capacity region is achievable by a naive coding scheme that simply ignores the interference; when the interference is strong, the capacity region is achievable by a simple scheme requiring the two receivers to decode both messages.', '1007.1811-2-5-0': 'Therefore, in this paper, we look into simple coding schemes with simple achievable rate regions for the cognitive Z-interference channel.', '1007.1811-2-5-1': 'When the interference is weak, the capacity region can be obtained by directly extending the existing capacity result the cognitive radio channel in the weak-interference regime.', '1007.1811-2-5-2': 'Hence, we focus on the high-interference regime.', '1007.1811-2-5-3': 'We wish to examine the achievable rates associated with these simple coding schemes by comparing their achievable rate regions with the capacity region outer bound.', '1007.1811-2-5-4': 'Some of the simple coding schemes are in fact not new, but rather are extracted from the coding schemes previously proposed for various cognitive and interference channel models.', '1007.1811-2-5-5': 'Specifically, we first derive a few different simple achievable rate regions extended from the achievable rate region for the cognitive radio channel developed in [CITATION] and from a component code proposed in [CITATION].', '1007.1811-2-5-6': 'We then extend these rate regions to the Gaussian case, and derive new outer bounds on the capacity region.', '1007.1811-2-5-7': 'We find that the latter simple achievable rate region is the capacity region, when the interference is in the lower range of the high-interference regime.', '1007.1811-2-5-8': 'Note that a similar capacity result and proof techniques were reported in an independent study on the Gaussian cognitive channel [CITATION].', '1007.1811-2-5-9': 'In addition, our new outer bounds can be applied on the general cognitive channel in the high-interference regime.', '1007.1811-2-6-0': 'The remainder of the paper is organized as follows.', '1007.1811-2-6-1': 'In Section II, we present the channel model and the related definitions.', '1007.1811-2-6-2': 'In Section III, we derive the aforementioned simple achievable rate regions and a new capacity region outer bound for the channel in the discrete memoryless case.', '1007.1811-2-6-3': 'In Section IV, we present the achievable rate regions and outer bounds for the Gaussian case, and a new capacity result in the high-interference regime, along with numerical comparisons between the achievable rate regions and the outer bounds.', '1007.1811-2-6-4': 'The paper is concluded in Section V.', '1007.1811-2-7-0': '# Channel Model', '1007.1811-2-8-0': 'We consider the following channel model that involves one primary user and one cognitive user.', '1007.1811-2-8-1': "As shown in Fig. [REF], each user needs to send a message to its corresponding receiver, while the cognitive user is assumed to have non-causal knowledge of the primary user's message.", '1007.1811-2-8-2': 'The channel is defined by [MATH], where [MATH] and [MATH], [MATH], denote the channel input and output alphabets, and [MATH] denotes the collection of channel transition probabilities.', '1007.1811-2-8-3': "Furthermore, we assume [MATH] can be factored as [MATH], i.e., the cognitive user's receiver output is not affected by the primary user's channel input.", '1007.1811-2-8-4': 'The source messages [MATH], [MATH], are assumed to be uniformly generated over the respective ranges: [MATH], [MATH].', '1007.1811-2-8-5': 'We call this channel the cognitive Z-interference channel, and denote it as [MATH].', '1007.1811-2-9-0': 'An [MATH] code for [MATH] consists of an encoding function at the cognitive user [MATH], an encoding function at the primary user [MATH], and one decoding function at each receiver [MATH], [MATH], with the probability of decoding errors defined as [MATH], where the individual error probability at each receiver is computed as [MATH].', '1007.1811-2-10-0': 'A non-negative rate pair [MATH] is achievable for [MATH] if there exists a sequence of codes [MATH] for the channel such that [MATH] approaches 0 as [MATH].', '1007.1811-2-10-1': 'The capacity region of [MATH] is the closure over the set of all achievable rate pairs.', '1007.1811-2-10-2': 'Any subset of the capacity region is an achievable rate region.', '1007.1811-2-11-0': '# Discrete Memoryless Channels', '1007.1811-2-12-0': 'In this section, we first derive a few simple achievable rate regions with explicit descriptions based on the rate region in [CITATION], and with a component code applied in [CITATION].', '1007.1811-2-12-1': 'We also derive a new outer bound on the capacity region of the channel in the high-interference regime.', '1007.1811-2-13-0': '## Achievable Rate Regions', '1007.1811-2-14-0': 'Let [MATH], [MATH], [MATH], and [MATH] be arbitrary auxiliary random variables defined over finite alphabets: [MATH], [MATH], [MATH], and [MATH], respectively.', '1007.1811-2-14-1': 'Let [MATH] denote the set of all joint distributions [MATH] that factor in the following form [EQUATION]', '1007.1811-2-14-2': 'For any joint distribution [MATH], define [MATH] as the set of non-negative rate pairs [MATH] such that [EQUATION] and define [MATH].', '1007.1811-2-15-0': 'The rate region [MATH] is an achievable rate region for the cognitive Z-interference channel [MATH].', '1007.1811-2-16-0': 'Proof: The details of the proof is omitted here due to space limit.', '1007.1811-2-16-1': 'The proof is based on [CITATION].', '1007.1811-2-16-2': 'First, note that the positions of the primary user and cognitive user are switched relative to the channel model in [CITATION], and thus we switch the indices in the region description.', '1007.1811-2-16-3': 'Next, we drop the common information of primary user by setting the corresponding rate to be 0.', '1007.1811-2-16-4': 'Lastly, perform Fourier-Motzkin elimination on the implicit rate region, which obtains the set of inequalities defining [MATH].', '1007.1811-2-16-5': '[MATH]', '1007.1811-2-17-0': "In this coding scheme, rate splitting is only applied on the cognitive user's message, i.e., [MATH].", '1007.1811-2-17-1': 'More specifically, the common message [MATH] is encoded with codewords generated with [MATH], and the private message [MATH] is encoded with codewords generated with [MATH].', '1007.1811-2-17-2': "The primary user's message [MATH] is encoded with codewords generated with both [MATH] and [MATH].", '1007.1811-2-17-3': "The cognitive user's receiver decodes its own messages [MATH] only, while the primary user's receiver jointly decodes its own message [MATH] and the common message from the cognitive user [MATH].", '1007.1811-2-17-4': 'Next, we specialize the rate region [MATH] into two more simple achievable rate regions.', '1007.1811-2-18-0': "First, we completely remove the 'artificially' created common information, i.e., no rate splitting is applied to the cognitive user's message.", '1007.1811-2-18-1': 'The following rate region can be obtained by setting [MATH] as a constant.', '1007.1811-2-19-0': 'Any rate pair [MATH] satisfying [EQUATION] for any joint distribution [EQUATION] is achievable for the channel [MATH].', '1007.1811-2-20-0': 'Denote the rate region [REF]-[REF] as [MATH].', '1007.1811-2-20-1': "Note that [MATH] and [MATH] are used to perform Marton's binning.", '1007.1811-2-20-2': 'Therefore, when it is extended to the Gaussian case, it becomes two different rate regions as a result of dirty paper coding in different orders.', '1007.1811-2-21-0': 'For the second one, instead of removing the common information of the cognitive user, we remove the private information.', '1007.1811-2-21-1': 'Specifically, set [MATH] as a constant, and merge [MATH] and [MATH] into one random variable [MATH], i.e., set [MATH].', '1007.1811-2-21-2': 'We have the following simple achievable rate region.', '1007.1811-2-22-0': 'Any rate pair [MATH] satisfying [EQUATION] for any joint distribution [EQUATION] is achievable for the channel [MATH].', '1007.1811-2-23-0': 'Denote the rate region [REF]-[REF] as [MATH].', '1007.1811-2-23-1': 'This rate region is achieved by using only superposition coding.', '1007.1811-2-23-2': "Nevertheless, it turns out to be the largest amongst all these simple rate regions when the interference link is 'substantially' strong, which is numerically shown in the next section.", '1007.1811-2-24-0': "We now present another simple achievable rate region, based on the idea that the artificial common message of the cognitive user can be dirty-paper coded against the primary user's private message.", '1007.1811-2-24-1': "The purpose of dirty paper coding in this coding scheme is to protect the message of the cognitive user from the interference created by the primary user's message during the cognitive user's encoding process.", '1007.1811-2-25-0': 'Any rate pair [MATH] satisfying [EQUATION] for any joint distribution [EQUATION] is achievable for the channel [MATH].', '1007.1811-2-26-0': 'Let us denote this rate region by [MATH].', '1007.1811-2-26-1': 'The proof can be easily obtained by following the argument in [CITATION].', '1007.1811-2-26-2': "The intention of the dirty paper coding applied in this coding scheme is to let the encoding of the cognitive user's common message take advantage of its non-causal knowledge about the primary user's private message.", '1007.1811-2-26-3': 'As a component of the respective coding schemes, this technique has been employed in several existing works [CITATION].', '1007.1811-2-26-4': 'Nevertheless, there has been no evidence showing that this feature is indeed helpful in terms of increasing achievable rates.', '1007.1811-2-26-5': 'In the next section, we will show that this coding scheme is in fact capacity-achieving, when the channel is in the high-interference regime, but the interference belongs to the lower range.', '1007.1811-2-27-0': '## A New Outer Bound', '1007.1811-2-28-0': 'In [CITATION], an outer bound on the capacity region of the strong cognitive interference channel was derived.', '1007.1811-2-28-1': 'This outer bound also directly applies to the cognitive Z-interference channel in the high-interference regime.', '1007.1811-2-28-2': "It was also observed in [CITATION] that, under the strong/high interference assumption, the primary user can decode the cognitive user's message without any rate penalty.", '1007.1811-2-28-3': 'In fact, an equivalence between the capacity regions of the strong cognitive interference channel and its variant with degraded message sets can be established.', '1007.1811-2-28-4': 'The capacity region of strong cognitive interference channel can now be outer-bounded by the capacity region of the broadcast channel with degraded message sets [CITATION], where we let the primary user and the cognitive user fully cooperate.', '1007.1811-2-28-5': 'By combining this with the outer bound derived in [CITATION], we obtain a new outer bound for the cognitive interference channel in the high-interference regime.', '1007.1811-2-28-6': 'Denote the outer bound given in [CITATION] as [MATH].', '1007.1811-2-29-0': 'Define [MATH] as the set of all rate pairs satisfying [EQUATION] for all joint input distributions [MATH].', '1007.1811-2-30-0': 'For a cognitive Z-interference channel satisfying [MATH] for all input distributions [MATH], [MATH] is an outer bound on the capacity region.', '1007.1811-2-31-0': 'In general, the new outer bound [MATH] can be interpreted as an intersection between [MATH] and capacity region of the associated broadcast channel with degraded message sets, and thus it is a subset of the outer bound [MATH].', '1007.1811-2-31-1': 'This outer bound is strictly smaller in some cases, which is demonstrated using a numerical example in the next section.', '1007.1811-2-32-0': '# Gaussian Channels', '1007.1811-2-33-0': 'In this section, we first define the Gaussian channel model.', '1007.1811-2-33-1': 'We then extend the achievable rate regions derived in the previous section to the Gaussian case, and present a new capacity result in the high-interference regime.', '1007.1811-2-34-0': 'The channel input-output relationship is given by [EQUATION] where [MATH] is the interference link gain and is assumed to be greater than [MATH]; [MATH] and [MATH] are additive white Gaussian noises of zero mean and unit variance.', '1007.1811-2-34-1': 'The power constraints are given by [MATH].', '1007.1811-2-35-0': 'Note that when [MATH], the channel belongs to the weak-interference regime, for which the capacity region has been established for the general cognitive radio channel [CITATION].', '1007.1811-2-35-1': 'The capacity achieving scheme is rather simple: the cognitive user spares a portion of its own power to cooperate with the primary user for the transmission of message [MATH], while it performs dirty paper coding on its own message [MATH] by treating the signals carrying [MATH] as the non-causally known interference.', '1007.1811-2-35-2': 'No rate splitting is applied, and each receiver only decodes its own intended message.', '1007.1811-2-35-3': 'Here, we focus on the high-interference regime, which is in general open.', '1007.1811-2-35-4': 'In what follows, we present the Gaussian counterparts of the achievable regions presented in the previous section.', '1007.1811-2-36-0': '## Achievable Rate Regions', '1007.1811-2-37-0': 'We first extend the rate region [MATH] to the Gaussian case.', '1007.1811-2-37-1': "When Marton's coding scheme is extended to the Gaussian case, the double binning first extends to Gel'fand-Pinsker coding in two different orders, and subsequently dirty-paper coding in two different orders.", '1007.1811-2-37-2': 'The rate region [MATH] can thus be represented as a union of two rate regions corresponding to the respective dirty-paper coding orders.', '1007.1811-2-37-3': 'Define [MATH] as the set of rate pairs satisfying: [EQUATION] and [MATH] as the set of rate pairs satisfying: [EQUATION] where [MATH], [MATH], and [MATH]; and [MATH].', '1007.1811-2-37-4': 'Next, define [MATH], where [MATH] stands for the convex hull operation.', '1007.1811-2-38-0': 'Any rate pair [MATH] is achievable for the Gaussian Z-interference channel.', '1007.1811-2-39-0': 'Next, we extend the rate region [MATH] to the Gaussian case.', '1007.1811-2-39-1': 'Note that [MATH] is extended from [MATH] by setting [MATH] as a constant.', '1007.1811-2-39-2': 'Hence, it is straightforward to obtain the Gaussian counterpart of [MATH] by setting [MATH] which corresponds to a constant [MATH].', '1007.1811-2-39-3': 'This Gaussian rate region can be expressed as [MATH].', '1007.1811-2-40-0': 'We also extend [MATH] to the its Gaussian counterpart [MATH], which is defined as the set of rate pairs satisfying: [EQUATION]', '1007.1811-2-40-1': 'Lastly, we extend [MATH] to obtain the Gaussian rate region [MATH], and we have [EQUATION]', '1007.1811-2-40-2': 'By choosing [MATH], which is the dirty-paper coding coefficient maximizing the righthand side of [REF], we have another achievable rate region [MATH] defined by the following rate constraints [EQUATION]', '1007.1811-2-40-3': 'It is easy to see that [MATH].', '1007.1811-2-40-4': 'However, the rate region [MATH] cannot be easily computed, due to the fact that the choice of [MATH] is unbounded, i.e., [MATH], while [MATH] can be easily computed.', '1007.1811-2-40-5': 'Furthermore, the rate region [MATH] even turns out to be the capacity region when interference link gain falls into a certain range in the high-interference regime, as shown in Section IV.C.', '1007.1811-2-41-0': '## Outer Bounds', '1007.1811-2-42-0': 'As we assume that the cross link gain [MATH] is greater than 1, the channel satisfies the conditions for the capacity outer bound derived in [CITATION].', '1007.1811-2-42-1': 'Hence, this outer bound directly applies to the cognitive Z-interference channel in the high-interference regime.', '1007.1811-2-43-0': 'Any achievable rate pair [MATH] for the Gaussian cognitive Z-interference channel defined by [REF]-[REF] with [MATH], satisfies the following constraints [EQUATION] for any [MATH].', '1007.1811-2-44-0': 'Our improved outer bound [MATH] for the Gaussian case can be obtained by intersecting the outer bound in Proposition [REF] with the capacity region of the associated Gaussian MISO broadcast channel with degraded message sets [CITATION].', '1007.1811-2-44-1': 'Note that the computation of the region [MATH] requires optimization with respect to the individual antenna power constraints.', '1007.1811-2-45-0': '## Capacity in the High-interference Regime', '1007.1811-2-46-0': 'We observe that the rate constraints [REF] and [REF] are equivalent to [REF] and [REF], when we let [MATH].', '1007.1811-2-46-1': 'Hence, the rate region [MATH] meets the outer bound if the rate constraint [REF] is loose.', '1007.1811-2-47-0': 'For the Gaussian cognitive Z-interference channel defined by [REF]-[REF], the capacity region is defined by [EQUATION] with [MATH], when the interference link gain satisfies [EQUATION].', '1007.1811-2-48-0': 'The above proposition is proved by showing every rate pair on the boundary of the outer bound is achievable.', '1007.1811-2-48-1': 'For the rate constraint [REF] to be loose, the righthand side of [REF] has to be greater than or equal to the difference between the righthand sides of [REF] and [REF].', '1007.1811-2-48-2': 'Through cumbersome but simple algebra, the condition can be found as [EQUATION].', '1007.1811-2-48-3': 'For this condition to be satisfied for all the choices of [MATH], we need [EQUATION]', '1007.1811-2-49-0': '## Numerical Results', '1007.1811-2-50-0': 'In this subsection, we numerically compare the rate regions and the outer bound.', '1007.1811-2-50-1': 'For all the comparisons, we let [MATH], leaving the interference link gain [MATH] to be variable.', '1007.1811-2-51-0': 'First, in Fig. [REF], the rate region [MATH] is compared against the outer bound [MATH].', '1007.1811-2-51-1': 'We observe from the figure that the achievable rate region meets the outer bound for both [MATH] and [MATH], while the rate region becomes strictly smaller than the outer bound for [MATH].', '1007.1811-2-51-2': 'When the interference link becomes very strong, the coding scheme to achieve [MATH] turns out to be suboptimal, which is shown in the later comparisons.', '1007.1811-2-52-0': 'Next, in Fig. [REF], we compare the rate regions [MATH], [MATH], [MATH], and the outer bound [MATH] in the very-high interference regime, i.e., [MATH].', '1007.1811-2-52-1': 'It can be observed that [MATH] is strictly smaller than [MATH] and [MATH], while [MATH] and [MATH] always coincide with each other.', '1007.1811-2-52-2': 'Note that [MATH] is in fact a subset of [MATH], as the coding scheme to achieve [MATH] generalizes the one for [MATH].', '1007.1811-2-52-3': 'However, the numerical results indicate that the coding scheme for [MATH] dominates other coding components in the coding scheme to achieve [MATH].', '1007.1811-2-52-4': 'This may imply that other coding features, especially the one to achieve [MATH], are redundant; this conjecture is the subject of further investigation.', '1007.1811-2-53-0': 'We also compare the rate regions [MATH], [MATH], [MATH], and the outer bound [MATH] in the very high-interference regime in Fig. [REF].', '1007.1811-2-53-1': 'At the boundary, when [MATH], the rate region [MATH] meets the outer bound, and is strictly larger than both [MATH] and [MATH].', '1007.1811-2-53-2': 'However, when [MATH] is large enough, i.e., when [MATH], the rate region [MATH] becomes strictly larger than [MATH].', '1007.1811-2-53-3': 'It is a current topic of investigation to find the critical value of [MATH], at which [MATH] becomes larger than [MATH].', '1007.1811-2-54-0': 'Lastly, we compare the new outer bound [MATH] with [MATH] under an extreme case setting: [MATH], [MATH], and [MATH].', '1007.1811-2-54-1': 'In Fig. [REF], it can be easily observed that [MATH] is strictly smaller than [MATH].', '1007.1811-2-55-0': '# Conclusions', '1007.1811-2-56-0': "We have investigated the cognitive Z-interference channel where the cognitive user's receiver suffers no interference from the primary user.", '1007.1811-2-56-1': 'Our results demonstrate that simple coding schemes perform well in the strong-interference regime.', '1007.1811-2-56-2': 'In particular, we have shown that the capacity region is achieved with an encoding scheme that combines superposition coding and dirty paper coding (corresponding to [MATH])) over certain regime of interference, i.e. when [MATH], Furthermore, our numerical results suggest that the superposition coding scheme (corresponding to [MATH]) strictly dominates other schemes when interference is very strong.', '1007.1811-2-56-3': 'However, there is a gap between the achievable rate region and the considered outer bound.', '1007.1811-2-56-4': 'We also cannot show that the improved outer bound is tight.', '1007.1811-2-56-5': 'Therefore, further tightening of the outer bounds or deriving a new one is a possible direction of the future work.'}

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[]

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[]

['1007.1811-1-16-4', '1007.1811-1-32-8', '1007.1811-2-16-5']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1007.1811

0706.3147

{'0706.3147-1-0-0': 'We present a systematic classification of half-supersymmetric solutions of gauged [MATH], [MATH] supergravity coupled to an arbitrary number of abelian vector multiplets for which at least one of the Killing spinors generate a time-like Killing vector.', '0706.3147-1-1-0': '# Introduction', '0706.3147-1-2-0': 'There has been a considerable research effort in recent years devoted to the analysis and study of solutions of supergravity theories in various dimensions and in particular those obtained as low energy limits of superstring and M-theory.', '0706.3147-1-2-1': 'This effort is motivated by the important role that black holes and domain walls have played in some of the recent developments that took place in superstring theory.', '0706.3147-1-2-2': 'These include the conjectured equivalence between string theory on anti-de Sitter (AdS) spaces and certain superconformal gauge theories living on the boundary [CITATION] known as the AdS/CFT correspondence, the understanding of the microscopic analysis of black hole entropy [CITATION] and the understanding of various duality symmetries relating string theories to each other and to M-theory.', '0706.3147-1-2-3': 'An interesting possibility that arises from the conjectured AdS/CFT correspondence is the ability to obtain information of the nonperturbative structure of field theories by studying dual classical gravitational configurations.', '0706.3147-1-2-4': 'A notable example in this direction is the Hawking-Page phase transition [CITATION] which was interpreted in [CITATION] as a thermal phase transition from a confining to a deconfining phase in the dual [MATH], [MATH] super Yang-Mills theory.', '0706.3147-1-2-5': 'Various interesting results using Anti-de Sitter black holes and their CFT duals have been obtained in recent years (see for example [CITATION]).', '0706.3147-1-3-0': 'In this paper we will focus on the study of supersymmetric solutions in five-dimensional [MATH] gauged supergravity coupled to abelian vector multiplets [CITATION].', '0706.3147-1-3-1': 'These solutions are relevant for the holographic descriptions of four dimensional field theories with less than maximal supersymmetry.', '0706.3147-1-3-2': 'Explicit supersymmetric black holes for these theories were constructed in [CITATION].', '0706.3147-1-3-3': 'However, these solutions have naked singularities or naked closed time-like curves.', '0706.3147-1-3-4': 'Domain walls and magnetic strings were also constructed in [CITATION].', '0706.3147-1-3-5': 'Obviously one would like to study the general structure of supersymmetric solutions in five dimensions rather than some specific solutions based on a certain ansatz.', '0706.3147-1-3-6': 'The main purpose of this paper is to construct a systematic classification of half-supersymmetric solutions.', '0706.3147-1-4-0': 'The first systematic classification of supersymmetric solutions, following the results of [CITATION], was performed in [CITATION] for minimal [MATH] supergravity in [MATH].', '0706.3147-1-4-1': 'In [CITATION] it was shown that supersymmetric solutions fall into two classes which depend on whether the Killing vector obtained from the Killing spinor is time-like or null.', '0706.3147-1-4-2': 'For the time-like case, one obtains the Israel-Wilson-Perjes class of solutions and the null solutions are pp-waves.', '0706.3147-1-4-3': 'Further generalizations were presented in [CITATION].', '0706.3147-1-4-4': 'More recently and motivated by the results of Tod, purely bosonic supersymmetric solutions of minimal [MATH] were classified in [CITATION].', '0706.3147-1-4-5': 'The basic idea in this analysis is to assume the existence of a Killing spinor, (i.e., to assume that the solution preserves at least one supersymmetry) and construct differential forms as bilinears in the Killing spinor.', '0706.3147-1-4-6': 'Then Fierz identities and the vanishing of the supersymmetry transformation of the fermionic fields in a bosonic background provide a set of algebraic and differential equations for the spinor bilinear differential forms which can be used to deduce the form of the metric and gauge fields.', '0706.3147-1-4-7': 'Such a general framework provides a powerful method for obtaining many new solutions, in contrast to the earlier methods that start with an ansatz for the metric and assume certain symmetries for the solution from the outset.', '0706.3147-1-4-8': 'The strategy of [CITATION] was used later to perform similar classifications of supersymmetric solutions in various supergravity theories.', '0706.3147-1-4-9': 'In particular, in [CITATION] the classification of [MATH] supersymmetric solutions of the minimal gauged [MATH] supergravity was performed.', '0706.3147-1-5-0': 'Explicit supersymmetric asymptotically anti-de Sitter black hole solutions with no closed time loops or naked singularities were constructed for the minimal supergravity theory in [CITATION].', '0706.3147-1-5-1': 'The results of [CITATION] for the time-like solutions were generalized in [CITATION] to the non-minimal case where the scalar fields live on symmetric spaces and explicit solutions for the [MATH] theory (with three [MATH] -charges) were also constructed.', '0706.3147-1-5-2': 'The constraint of symmetric spaces was relaxed in [CITATION] , where solutions with a null Killing vector in both gauged and ungauged theories were also obtained.', '0706.3147-1-6-0': 'In this paper we focus on the classification of half supersymmetric solutions in gauged [MATH] supergravity with vector multiplets.', '0706.3147-1-6-1': 'Half supersymmetric solutions have two Killing spinors from which one can construct two Killing vectors as bilinears in the Killing spinors.', '0706.3147-1-6-2': 'These vectors could be either time-like or null.', '0706.3147-1-6-3': 'Therefore one has to consider three cases depending on the nature of the Killing spinors and vectors considered.', '0706.3147-1-6-4': 'In our present work we will focus on the cases where the solutions contain at least one Killing spinor with an associated time-like Killing vector.', '0706.3147-1-6-5': 'In order to investigate supersymmetric solutions with more than one Killing spinor, it is very useful to express the Killing spinors in terms of differential forms [CITATION], [CITATION], [CITATION].', '0706.3147-1-6-6': 'Such a method, known now by the spinorial geometry method, has been very efficient in classifying solutions of supergravity theories in ten and eleven dimensions [CITATION], [CITATION], [CITATION] [CITATION].', '0706.3147-1-7-0': 'We organize our work as follows.', '0706.3147-1-7-1': 'In section two, we present the basic structure of the theory of [MATH], [MATH] gauged supergravity coupled to abelian vector multiplets and the equations of motion.', '0706.3147-1-7-2': 'In section three we express spinors in five dimensions as differential forms on [MATH].', '0706.3147-1-7-3': 'We start with the generic form of the spinor and then use the gauge symmetries ([MATH] and [MATH] preserving the symplectic Majorana condition to write down two canonical forms for a single symplectic Majorana spinor corresponding to time-like and null Killing vectors.', '0706.3147-1-7-4': 'In section four, we derive the conditions for quarter supersymmetric solutions with time-like Killing vector.', '0706.3147-1-7-5': 'In section five, the [MATH] Killing constraints, i. e., the conditions for a time-like quarter supersymmetric solution, are then substituted into the generic Killing spinor equations and the resulting equations are rewritten in the form of constraints on the Kahler base.', '0706.3147-1-7-6': 'Section six contains a detailed classification of half-supersymmetric solutions.', '0706.3147-1-7-7': 'Our paper ends with two appendices.', '0706.3147-1-7-8': 'Appendix A deals with the determination of the linear system obtained from the Killing spinor equations.', '0706.3147-1-7-9': 'Appendix B discusses the integrability conditions of the Killing spinor equations.', '0706.3147-1-7-10': 'There it is demonstrated that for a given background preserving at least half of the supersymmetry, where at least one of the Killing spinors generates a time-like Killing vector, all of the Einstein, gauge and scalar field equations of motion hold automatically provided that the Bianchi identity is satisfied.', '0706.3147-1-8-0': '# [MATH] supergravity', '0706.3147-1-9-0': 'In this section, we review briefly some aspects of the [MATH], [MATH] gauged supergravity coupled to abelian vector multiplets is [CITATION].', '0706.3147-1-9-1': 'The bosonic action of the theory is [EQUATION] where [MATH] take values [MATH] and [MATH] are the two-forms representing gauge field strengths (one of the gauge fields corresponds to the graviphoton).', '0706.3147-1-9-2': 'The metric has mostly negative signature.', '0706.3147-1-9-3': 'The constants [MATH] are symmetric in [MATH] and are not assumed to satisfy the non-linear adjoint identity which arises when the scalars lie in a symmetric space [CITATION]; though we will assume that [MATH] is invertible, with inverse [MATH].', '0706.3147-1-9-4': 'The [MATH] are scalar fields subject to the constraint [EQUATION]', '0706.3147-1-9-5': 'The fields [MATH] can thus be regarded as being functions of [MATH] unconstrained scalars [MATH].', '0706.3147-1-9-6': 'It is convenient to define [EQUATION] so that the condition ([REF]) becomes [EQUATION]', '0706.3147-1-9-7': 'In addition, the coupling [MATH] depends on the scalars via [EQUATION] so in particular', '0706.3147-1-10-0': '[EQUATION] where [MATH] denotes differentiation with respect to [MATH].', '0706.3147-1-10-1': 'The scalar potential can be written as [EQUATION] where [MATH] are constants.', '0706.3147-1-11-0': 'Bosonic backgrounds are said to be supersymmetric if there exists a spinor [MATH] for which the supersymmetry variations of the gravitino and dilatino vanish in the given background.', '0706.3147-1-11-1': 'For the gravitino this requires [EQUATION] and for the dilatino it requires [EQUATION]', '0706.3147-1-11-2': 'The Einstein equation derived from ([REF]) is given by [EQUATION]', '0706.3147-1-11-3': 'The Maxwell equations (varying [MATH]) are [EQUATION]', '0706.3147-1-11-4': 'The scalar equations (varying [MATH]) are [EQUATION]', '0706.3147-1-11-5': 'If a quantity [MATH] satisfies [MATH] then there must be a function [MATH] such that [MATH].', '0706.3147-1-11-6': 'This implies that the dilatino equation ([REF]) can be simplified to [EQUATION] and the scalar equation can be written as [EQUATION]', '0706.3147-1-12-0': '# Spinors in Five Dimensions', '0706.3147-1-13-0': 'Following [CITATION] , we write spinors in five dimensions as forms on [MATH].', '0706.3147-1-13-1': 'We represent a generic spinor [MATH] in the form [EQUATION] where [MATH], [MATH] are 1-forms on [MATH], [MATH] and [MATH] and [MATH] are complex functions.', '0706.3147-1-14-0': 'The action of [MATH]-matrices on these forms is given by [EQUATION]', '0706.3147-1-14-1': 'We define [MATH] by [MATH].', '0706.3147-1-14-2': 'This satisfies [EQUATION]', '0706.3147-1-14-3': 'The charge conjugation operator [MATH] is defined by [EQUATION] where [MATH] is antisymmetric with [MATH].', '0706.3147-1-15-0': 'The Killing spinors [MATH] of the theory satisfy a symplectic Majorana constraint which is [EQUATION] so if one writes [EQUATION] then [MATH] is fixed via [EQUATION]', '0706.3147-1-15-1': 'We note the useful identity [EQUATION]', '0706.3147-1-15-2': 'It will be particularly useful in our work to complexify the gamma operators.', '0706.3147-1-15-3': 'Therefore we write [EQUATION]', '0706.3147-1-16-0': '## Gauge transformations and [MATH] spinors', '0706.3147-1-17-0': 'There are two types of gauge transformation that preserve the symplectic Majorana condition ([REF]).', '0706.3147-1-17-1': 'First, we have the [MATH] gauge transformations described by [EQUATION] and there are also [MATH] gauge transformations of the form [EQUATION] for real functions [MATH].', '0706.3147-1-18-0': 'Note in particular that [MATH], [MATH] and [MATH] generate a [MATH] which leaves [MATH] and [MATH] invariant and acts on [MATH] , [MATH]; whereas [MATH], [MATH] and [MATH] generate another [MATH] which leaves the [MATH] invariant but acts on [MATH] and [MATH].', '0706.3147-1-18-1': 'In addition, [MATH] generates a [MATH] which acts (simultaneously) on [MATH] and [MATH] , whereas [MATH] generates another [MATH] which acts (simultaneously) on [MATH] and [MATH].', '0706.3147-1-19-0': 'Therefore, for a single symplectic Majorana spinor, one can always use [MATH] gauge transformations to write [EQUATION] or [EQUATION] or [EQUATION] for some real function [MATH].', '0706.3147-1-19-1': 'However, under the transformation [EQUATION] the spinor in ([REF]) transforms as [EQUATION] and [EQUATION] and [MATH] is unchanged.', '0706.3147-1-19-2': 'This transformation corresponds to reflections in the [MATH] directions.', '0706.3147-1-19-3': 'Moreover, the spinor in ([REF]) is equivalent to that in ([REF]) under a [MATH] gauge transformation.', '0706.3147-1-19-4': 'The spinors corresponding to ([REF]) and ([REF]) are therefore equivalent under these transformations.', '0706.3147-1-19-5': 'Hence, for a single spinor, one need only consider the cases ([REF]) and ([REF]).', '0706.3147-1-20-0': '## Differential Forms from Spinors', '0706.3147-1-21-0': 'In order to define differential forms, we first define a Hermitian inner product on [MATH] by [EQUATION]', '0706.3147-1-21-1': 'Then [MATH] gauge-invariant [MATH]-forms are obtained from spinors [MATH], [MATH] via [EQUATION]', '0706.3147-1-21-2': 'In particular, for the generic Majorana spinor given in ([REF]) and ( [REF]) one finds [EQUATION]', '0706.3147-1-21-3': 'The scalars are then given by [EQUATION]', '0706.3147-1-21-4': 'Hence, by comparing with [CITATION], it is clear that the spinor given in ( [REF]) corresponds to the time-like class of solutions, whereas that in ([REF]) is in the null class of solutions.', '0706.3147-1-21-5': 'With a slight abuse of notation, we shall refer to the corresponding Killing spinors as being either time-like or null.', '0706.3147-1-22-0': '## Canonical [MATH] spinors', '0706.3147-1-23-0': 'We will now assume that there are two linearly independent symplectic Majorana Killing spinors [MATH], [MATH], where [MATH] is time-like.', '0706.3147-1-23-1': 'In appendix B it is demonstrated that the existence of such spinors is sufficient to ensure that the scalar, gauge and Einstein equations of motion hold automatically from the integrability conditions, provided one assumes that the Bianchi identities are satisfied.', '0706.3147-1-23-2': 'So the only equations which must be solved are the Killing spinor equations together with the Bianchi identity.', '0706.3147-1-24-0': 'From the previous reasoning, we can take [MATH] to have the canonical form.', '0706.3147-1-24-1': '[EQUATION] for [MATH].', '0706.3147-1-24-2': 'Next consider [MATH] given by [EQUATION] for complex [MATH].', '0706.3147-1-24-3': 'It is possible to simplify [MATH] a little using gauge transformations which leave [MATH] invariant.', '0706.3147-1-24-4': 'In particular, by using an appropriate [MATH] transformation, one could for example set [MATH] with [MATH].', '0706.3147-1-24-5': 'However, we will not make this gauge choice.', '0706.3147-1-25-0': '## The 1/4 Supersymmetric time-like Solution', '0706.3147-1-26-0': 'In this section we obtain the time-like solutions preserving a quarter of the supersymmetry using the spinorial geometry method.', '0706.3147-1-26-1': 'These solutions were derived in [CITATION].', '0706.3147-1-26-2': 'In order to obtain 1/4 supersymmetric solutions with time-like Killing spinor, it suffices to consider the equations ([REF])-([REF]) and set [MATH] and [MATH].', '0706.3147-1-26-3': 'Then from the dilatino equation, we find [EQUATION] whereas from the gravitino equation we find [EQUATION]', '0706.3147-1-26-4': 'To analyze this linear system, we will first consider the gravitino equations.', '0706.3147-1-26-5': 'Note that ([REF]) implies that [EQUATION] and [EQUATION]', '0706.3147-1-26-6': 'Next, consider ([REF]), ([REF]) and ([REF]).', '0706.3147-1-26-7': 'These imply [EQUATION]', '0706.3147-1-26-8': 'From ([REF]) and ([REF]) we find [EQUATION] and', '0706.3147-1-27-0': '[EQUATION]', '0706.3147-1-27-1': 'From ([REF]) we find [EQUATION] and from ([REF]) and ([REF]) we obtain [EQUATION] and [EQUATION]', '0706.3147-1-27-2': 'Hence, to summarize, we obtain the following purely geometric constraints [EQUATION] together with [EQUATION]', '0706.3147-1-27-3': 'It is straightforward to show that the constraints ([REF]) are the necessary and sufficient conditions for the 1-form [EQUATION] to define a Killing vector [MATH].', '0706.3147-1-27-4': 'This form is, as expected, the 1-form spinor bilinear which is obtained from [MATH].', '0706.3147-1-27-5': 'Note that this Killing vector satisfies [EQUATION]', '0706.3147-1-27-6': 'In fact, one can choose a gauge in which the Lie derivative of the vielbein with respect to [MATH] vanishes.', '0706.3147-1-27-7': 'To see this, note that [EQUATION]', '0706.3147-1-27-8': 'From ([REF]) we note that [MATH].', '0706.3147-1-27-9': 'Without loss of generality we can make a [MATH] gauge transformation in order to set [EQUATION]', '0706.3147-1-27-10': 'This gauge transformation alters the form of the Killing spinors via the transformation given in ([REF]).', '0706.3147-1-27-11': 'However, the Killing spinors can be restored to their original form by making a [MATH] gauge transformation generated by [MATH].', '0706.3147-1-27-12': 'Working in this gauge, ([REF]) implies that [MATH] and hence [EQUATION] where the constraints in ([REF]), ([REF]) and ([REF]) imply that [MATH].', '0706.3147-1-27-13': 'By making a further [MATH] gauge transformation generated by [MATH] which leaves [MATH] invariant and maps [MATH] for [MATH], we can without loss of generality take [MATH].', '0706.3147-1-27-14': 'In this basis the vielbein is time-independent.', '0706.3147-1-28-0': 'Equation ([REF]) also has a simple geometric interpretation.', '0706.3147-1-28-1': 'First note that the only [MATH] gauge-invariant 2-form which can be obtained from [MATH] is the real part of the 2-form [EQUATION]', '0706.3147-1-28-2': 'The real part of this form (denoted by [MATH]) is then given by [EQUATION]', '0706.3147-1-28-3': 'It is convenient to make a conformal rescaling of the complexified basis and define [EQUATION]', '0706.3147-1-28-4': 'We shall refer to the 4-manifold with metric [EQUATION] as the base space [MATH].', '0706.3147-1-28-5': 'Then it is clear that [MATH] defines an almost complex structure on this 4-manifold.', '0706.3147-1-28-6': 'In fact, ([REF]) implies that [MATH] is covariantly constant with respect to the Levi-civita connection of the base manifold, and hence [MATH] is a Kahler manifold (as expected) with Ka hler form [MATH].', '0706.3147-1-29-0': 'Also, from ([REF]) we have [EQUATION]', '0706.3147-1-29-1': 'We remark that ([REF]) implies that [EQUATION] where [MATH] is (locally) the potential for the Ricci form of the K ahler base [MATH] .', '0706.3147-1-30-0': 'The remaining constraints on the [MATH]-flux are then [EQUATION]', '0706.3147-1-30-1': 'Finally, we substitute these constraints into the dilatino equations.', '0706.3147-1-30-2': 'From ( [REF])-([REF]), we find [EQUATION]', '0706.3147-1-31-0': '# Killing spinor in [MATH] background', '0706.3147-1-32-0': 'In this section, we substitute the constraints obtained in the previous section back into the generic Killing spinor equation ([REF])-([REF]) and simplify as much as possible.', '0706.3147-1-32-1': 'We find from the dilatino equation: [EQUATION]', '0706.3147-1-32-2': 'And from the gravitino equations we find [EQUATION] and [EQUATION]', '0706.3147-1-32-3': 'Note that these equations admit a solution of the form [MATH] , [MATH], [MATH] with [MATH], [MATH] real constants, and no additional constraints on the fluxes or geometry.', '0706.3147-1-32-4': 'Hence we observe that the generic time-like solution preserves [MATH] supersymmetry.', '0706.3147-1-32-5': 'More generally, if [MATH], [MATH] are symplectic Majorana Killing spinors, then so are [EQUATION] which is just a special case of ([REF]) with [MATH].', '0706.3147-1-32-6': 'In particular, the equations computed above are invariant under the transformations [EQUATION] therefore it is clear that the Killing spinors arise in pairs.', '0706.3147-1-33-0': '## Solutions with [MATH]', '0706.3147-1-34-0': 'Suppose we consider the case when [MATH].', '0706.3147-1-34-1': 'Then, assuming that [MATH], we find from ([REF]) and ([REF]) that [EQUATION]', '0706.3147-1-34-2': 'If, however, [MATH] then ([REF]) and ([REF]) again imply ([REF]), as we assume that not all of the [MATH] vanish.', '0706.3147-1-35-0': 'Hence, from ([REF]), ([REF]), ([REF]) and ([REF]) it follows directly that [MATH], [MATH], [MATH] with [MATH], [MATH] real constants.', '0706.3147-1-35-1': 'The solution is therefore only 1/4-supersymmetric.', '0706.3147-1-35-2': 'Thus, to find new solutions with enhanced supersymmetry, one must take [MATH]; henceforth we shall assume that [MATH].', '0706.3147-1-36-0': '## Constraints on the base space', '0706.3147-1-37-0': 'It will be particularly useful to rewrite the equations ([REF])-( [REF]) in terms of constraints on the Kahler base.', '0706.3147-1-37-1': 'Throughout this section, unless stated otherwise, tensor indices are evaluated with respect to the 4-dimensional complex basis [MATH], [MATH]; so we shall drop the [MATH] from all expressions.', '0706.3147-1-37-2': 'It is convenient to define a real vector field [MATH] on the Kahler base as follows [EQUATION]', '0706.3147-1-37-3': 'In order to rewrite the constraints, we define a time co-ordinate [MATH] so that the Killing vector field associated with the Killing spinor [MATH] is [EQUATION] and set [EQUATION] where [MATH] is a 1-form defined on the Kahler base.', '0706.3147-1-38-0': 'Then ([REF]) is equivalent to [EQUATION] where here [MATH] denotes the Levi-civita connection of the Kahler base metric given in ([REF]).', '0706.3147-1-38-1': 'Also, ([REF]) can be rewritten as [EQUATION]', '0706.3147-1-38-2': 'It is also useful to define [EQUATION]', '0706.3147-1-38-3': 'It is then straightforward to show that [EQUATION]', '0706.3147-1-38-4': 'The commutator is given by [EQUATION]', '0706.3147-1-38-5': 'Next, ([REF]) and ([REF]) are equivalent to [EQUATION]', '0706.3147-1-38-6': 'These equations simply imply that [EQUATION]', '0706.3147-1-38-7': 'In addition, ([REF]) and ([REF]) can be rewritten as [EQUATION] and [EQUATION]', '0706.3147-1-38-8': 'In order to simplify the remainder of the equations, observe that for indices [MATH], [EQUATION] and [EQUATION] where on the LHS of ([REF]) and ([REF]), spatial indices are taken with respect to the original five-dimensional basis, whereas on the RHS, they are taken with respect to the conformally rescaled K ahler basis; and [MATH] denotes the spin connection of the K ahler base space.', '0706.3147-1-38-9': 'From henceforth, the hat will be dropped, and we will work solely on the Kahler base space.', '0706.3147-1-39-0': 'Then ([REF]) is equivalent to [EQUATION] and using this, ([REF]) can be rewritten as [EQUATION]', '0706.3147-1-39-1': 'It is also useful to rewrite ([REF]) as [EQUATION]', '0706.3147-1-39-2': 'In addition, ([REF]), ([REF]), ([REF]) and ([REF]) can be rewritten as [EQUATION]', '0706.3147-1-40-0': '# Half Supersymmetric Solutions', '0706.3147-1-41-0': 'Suppose that the solution preserves exactly four of the supersymmetries.', '0706.3147-1-41-1': 'Then the four linearly independent Killing spinors are [MATH], [MATH] and [EQUATION]', '0706.3147-1-41-2': 'As all of the scalars, gauge field strengths and components of the spin connection are [MATH]-independent, it follows that [MATH] is also a Killing spinor.', '0706.3147-1-41-3': 'As the solution is exactly half-supersymmetric, it follows that there must be real constants [MATH] such that [EQUATION] or equivalently [EQUATION]', '0706.3147-1-41-4': 'On substituting these constraints into ([REF]) we find that [EQUATION]', '0706.3147-1-41-5': 'Contracting this expression with [MATH] we find that [EQUATION]', '0706.3147-1-41-6': 'The real part of this expression implies that [MATH].', '0706.3147-1-42-0': 'Suppose now that [MATH].', '0706.3147-1-42-1': 'From ([REF]) we find that [EQUATION] and hence [EQUATION] where [MATH].', '0706.3147-1-42-2': 'Note that by making a redefinition of the type [EQUATION] we can without loss of generality set [MATH] and drop the primes on [MATH] and [MATH].', '0706.3147-1-43-0': 'It will be convenient to split the solutions into three classes.', '0706.3147-1-43-1': 'For the first class [MATH] and [MATH], for the second [MATH] but [MATH] and for the third [MATH].', '0706.3147-1-44-0': '## Solutions with [MATH], [MATH]', '0706.3147-1-45-0': 'For this class of solutions we have [EQUATION] where [MATH].', '0706.3147-1-45-1': 'Here we have the conditions [MATH] and [MATH].', '0706.3147-1-46-0': 'To proceed with the analysis for these solutions, we define the 1-forms [MATH], [MATH] and [MATH] on the Kahler base via [EQUATION]', '0706.3147-1-46-1': 'The components of these 1-forms can be easily shown to be [MATH]-independent [EQUATION]', '0706.3147-1-46-2': 'For convenience we set [MATH] and [MATH].', '0706.3147-1-47-0': 'In order to evaluate various integrability constraints, it is useful to compute the components of the covariant derivatives: [EQUATION] and [EQUATION] and [EQUATION]', '0706.3147-1-47-1': 'It immediately follows that [MATH] and [MATH] defines a Killing vector on the Kahler base space.', '0706.3147-1-47-2': 'In particular, setting [MATH], [MATH] is exact and satisfies [EQUATION]', '0706.3147-1-47-3': 'The first integrability condition we shall examine is obtained by considering the constraints ([REF]), ([REF]), ([REF]) and ([REF]).', '0706.3147-1-47-4': 'These are equivalent to [EQUATION]', '0706.3147-1-47-5': 'Taking the exterior derivative of this equation, we obtain the constraint [EQUATION]', '0706.3147-1-47-6': 'Hence, using ([REF]) and ([REF]) we obtain the constraints [EQUATION] and [EQUATION]', '0706.3147-1-47-7': 'Observe also that ([REF]) is equivalent to [EQUATION]', '0706.3147-1-47-8': 'Using ([REF]) and ([REF]), we compute [EQUATION]', '0706.3147-1-47-9': 'To proceed, we impose the integrability condition, [MATH].', '0706.3147-1-47-10': 'We note the following useful identities: [EQUATION] and [EQUATION] from whence we obtain [EQUATION]', '0706.3147-1-47-11': 'Using this expression, the constraint [MATH] implies that [EQUATION] for real constant [MATH].', '0706.3147-1-48-0': 'Next we consider the integrability condition [MATH].', '0706.3147-1-48-1': 'It is straightforward to show that [MATH] satisfies [EQUATION]', '0706.3147-1-48-2': 'Hence the integrability condition [MATH] implies that [EQUATION]', '0706.3147-1-48-3': 'We observe that [EQUATION]', '0706.3147-1-48-4': 'It is then straightforward but tedious to show that ([REF]) implies [EQUATION]', '0706.3147-1-48-5': 'Using this condition the above relations simplify and we obtain [EQUATION]', '0706.3147-1-49-0': 'The components of [MATH] are therefore given by [EQUATION]', '0706.3147-1-49-1': 'Using the expressions for [MATH] which we have obtained, we next examine ([REF]) and ([REF]).', '0706.3147-1-49-2': 'These may be rewritten as [EQUATION] and [EQUATION]', '0706.3147-1-49-3': 'We note the useful identities: [EQUATION] where we have set [MATH].', '0706.3147-1-49-4': 'Then from the integrability condition [MATH] we find the constraint [EQUATION]', '0706.3147-1-49-5': 'Note that [EQUATION] then upon differentiating ([REF]) with respect to [MATH] gives [EQUATION]', '0706.3147-1-49-6': 'It turns out that the constraints ([REF]) and ([REF]) are also sufficient to ensure that [EQUATION]', '0706.3147-1-49-7': 'Next, note that the constraints ([REF]) and ([REF]) can be used to write the gauge field strengths [MATH] as [EQUATION] and note that as [MATH] it follows that [EQUATION]', '0706.3147-1-49-8': 'It is then straightforward to show that the Bianchi identity [MATH] follows automatically from the constraints we have obtained.', '0706.3147-1-49-9': 'To proceed further, it is useful to consider the cases for which [MATH] and [MATH] separately.', '0706.3147-1-49-10': 'Observe that [MATH] implies that [MATH] and [MATH] are either both real or both imaginary.', '0706.3147-1-50-0': '### Solutions with [MATH]', '0706.3147-1-51-0': 'In order to introduce a local co-ordinate system for solutions with [MATH], recall that [MATH] is a Killing vector on the Kahler base space.', '0706.3147-1-51-1': 'Furthermore, as [MATH], the closure of [MATH] implies that [MATH] preserves the Kahler form; [EQUATION]', '0706.3147-1-51-2': 'It is also straightforward to show that [EQUATION]', '0706.3147-1-51-3': 'Hence it follows that [MATH] defines a symmetry of the full five dimensional solution.', '0706.3147-1-51-4': 'As [MATH], ([REF]) and ([REF]) can be inverted to obtain [EQUATION]', '0706.3147-1-51-5': 'It is convenient to define the real 1-forms [MATH] and [MATH] by [EQUATION]', '0706.3147-1-51-6': 'It is then straightforward, but tedious, to show that [EQUATION] and also that [EQUATION]', '0706.3147-1-51-7': 'In addition, we find that [EQUATION]', '0706.3147-1-51-8': 'Hence we define the following orthonormal basis on the Kahler base space [EQUATION] where [EQUATION]', '0706.3147-1-51-9': 'As [MATH], [MATH] are commuting vector fields, we can choose co-ordinates [MATH], [MATH] such that [EQUATION]', '0706.3147-1-51-10': 'Then, defining [EQUATION] we have [EQUATION] and from ([REF]) we see that there must be a function [MATH] such that [EQUATION]', '0706.3147-1-51-11': 'As [MATH], [MATH], [MATH], [MATH] are orthogonal, it follows that [MATH] form a local co-ordinate system on the base space.', '0706.3147-1-51-12': 'One can then write [EQUATION]', '0706.3147-1-51-13': 'As [MATH] is a Killing vector, the functions [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] do not depend on [MATH] (or [MATH]).', '0706.3147-1-51-14': 'Furthermore, [EQUATION]', '0706.3147-1-51-15': 'Therefore [MATH], [MATH] and [MATH] are functions of [MATH] and [MATH] only.', '0706.3147-1-51-16': 'However, there is a non-trivial [MATH] -dependence in [MATH]; [MATH] is not a Killing vector.', '0706.3147-1-52-0': 'It is also useful to observe that the identity [MATH] implies [EQUATION] and hence we shall set [EQUATION]', '0706.3147-1-52-1': 'Here [MATH] is a real function which satisfies [EQUATION] which implies [MATH].', '0706.3147-1-52-2': 'With these conventions, it is straightforward to compute [EQUATION] and [EQUATION]', '0706.3147-1-52-3': 'Also note that ([REF]) can be rewritten as [EQUATION]', '0706.3147-1-52-4': 'If [MATH] then this constraint can be integrated up to give [EQUATION] for constant [MATH].', '0706.3147-1-53-0': 'To proceed, consider the equation ([REF]), which can be rewritten as [EQUATION]', '0706.3147-1-53-1': 'Taking the self-dual projection of ([REF]) yields the constraints [EQUATION] together with [EQUATION]', '0706.3147-1-53-2': 'Using these constraints, the anti-self-dual projection of ([REF] ) fixes [MATH] to be given by [EQUATION]', '0706.3147-1-53-3': 'Imposing the covariant constancy condition [MATH] imposes two additional constraints: [EQUATION] and [EQUATION]', '0706.3147-1-53-4': 'Finally, we compare the spin connection components [MATH] computed in this basis with the expression given in ([REF]), noting that [MATH].', '0706.3147-1-54-0': 'This implies that [EQUATION]', '0706.3147-1-54-1': 'Then from ([REF]) and ([REF]) we obtain [EQUATION]', '0706.3147-1-54-2': 'Then ([REF]) and ([REF]) can be integrated up to give [EQUATION] and ([REF]) and ([REF]) then imply [EQUATION]', '0706.3147-1-54-3': 'Hence, in these co-ordinates, the orthonormal basis of the Kahler base space is [EQUATION] and if [MATH], then [MATH] can be written as [EQUATION]', '0706.3147-1-54-4': 'By considering ([REF]), [MATH] is fixed (up to a total derivative) by [EQUATION] and by considering ([REF]) we find the gauge field strengths are given by [EQUATION]', '0706.3147-1-55-0': '### Solutions with [MATH], [MATH]', '0706.3147-1-56-0': 'If [MATH] and [MATH] are imaginary but non-vanishing, then from ([REF]) we obtain [MATH], and the constraints on [MATH] and [MATH] as given in ([REF]) and ([REF]) imply that [EQUATION]', '0706.3147-1-56-1': 'Moreover, ([REF]) can be integrated up to give [EQUATION] for constants [MATH].', '0706.3147-1-56-2': 'The Kahler form can be expressed by [EQUATION]', '0706.3147-1-56-3': 'It can also be demonstrated that [MATH] is given in these cases by [EQUATION] for some real function [MATH] with [EQUATION] and hence [EQUATION] for some real function [MATH] where [MATH] satisfy [EQUATION]', '0706.3147-1-56-4': 'Note that [MATH] with [MATH].', '0706.3147-1-56-5': 'Without loss of generality, we can work in a gauge for which [MATH] and [MATH].', '0706.3147-1-57-0': 'It is then straightforward to prove the following identities: [EQUATION]', '0706.3147-1-57-1': 'There are then three cases to consider.', '0706.3147-1-57-2': '[EQUATION]', '0706.3147-1-57-3': 'It is then straightforward to show that [EQUATION] where', '0706.3147-1-58-0': '[EQUATION] so that [EQUATION]', '0706.3147-1-58-1': 'Hence the 3-manifold with metric [MATH] is either [MATH], the Nil-manifold or [MATH] according as to whether [MATH], [MATH] or [MATH] respectively.', '0706.3147-1-59-0': '### Solutions with [MATH], [MATH]', '0706.3147-1-60-0': 'If [MATH] and [MATH] are real but non-vanishing, then [MATH], and from ([REF]) and ([REF]) we obtain [EQUATION]', '0706.3147-1-60-1': 'Moreover, from ([REF]) we obtain [EQUATION]', '0706.3147-1-60-2': 'This implies that [MATH], and hence without loss of generality we can set [MATH] and the scalars are therefore constants.', '0706.3147-1-60-3': 'Furthermore, we also find that [EQUATION] and [EQUATION] with [EQUATION] and hence we can work in a gauge for which [EQUATION]', '0706.3147-1-60-4': 'It is then straightforward to prove the following identities: [EQUATION]', '0706.3147-1-60-5': 'There are then three cases to consider.', '0706.3147-1-61-0': 'It is then straightforward to show that [EQUATION] where [EQUATION] so that [EQUATION]', '0706.3147-1-61-1': 'Hence the 3-manifold with metric [MATH] is either [MATH], the Nil-manifold or [MATH] according as to whether [MATH], [MATH] or [MATH] respectively.', '0706.3147-1-62-0': '### Solutions with [MATH]', '0706.3147-1-63-0': 'If [MATH], then [MATH] and the scalars [MATH] are constant and [MATH] is constant.', '0706.3147-1-63-1': 'Without loss of generality, we set [MATH].', '0706.3147-1-63-2': 'The following constraints also hold: [EQUATION] so that [MATH] for these solutions.', '0706.3147-1-63-3': 'Furthermore, we also find [EQUATION] and the gauge field strengths are given by [EQUATION]', '0706.3147-1-63-4': 'Note also that [MATH] and [MATH] satisfy [EQUATION] where here [MATH] denotes the covariant derivative restricted to the base space, and [MATH], [MATH] are base space indices.', '0706.3147-1-64-0': 'It is convenient to define [EQUATION]', '0706.3147-1-64-1': 'Then [EQUATION] and [EQUATION] where [MATH] is defined by [EQUATION]', '0706.3147-1-64-2': 'Note that ([REF]) implies that [EQUATION] and hence [EQUATION] where [MATH] is fixed by comparing the integrability condition associated with ([REF]) with the expression for the gauge field strengths ([REF]).', '0706.3147-1-64-3': 'We find [EQUATION]', '0706.3147-1-64-4': 'Next we define [EQUATION] so that [EQUATION]', '0706.3147-1-64-5': 'We also define [EQUATION]', '0706.3147-1-64-6': 'Then [MATH] form an orthonormal time-independent basis for the Kahler base space, such that [EQUATION] and [EQUATION]', '0706.3147-1-64-7': 'Now note that [EQUATION]', '0706.3147-1-64-8': 'It therefore follows that there exist functions [MATH] and co-ordinates [MATH] such that [EQUATION]', '0706.3147-1-64-9': 'Suppose that the remaining co-ordinates on the base are [MATH].', '0706.3147-1-64-10': 'As [MATH] and [MATH] are orthogonal to [MATH], there exist functions [MATH] such that [EQUATION] for [MATH].', '0706.3147-1-65-0': 'Similarly, as [EQUATION] it also follows that there exist functions [MATH], [MATH] and [MATH] for [MATH] such that [EQUATION] for [MATH].', '0706.3147-1-65-1': 'As [MATH] form an orthonormal basis for the base space, we can without loss of generality take [MATH] to be co-ordinates on the base space.', '0706.3147-1-66-0': 'In principle, the functions [MATH] can depend on all the co-ordinates.', '0706.3147-1-66-1': 'However, note that [EQUATION]', '0706.3147-1-66-2': 'Imposing the constraint [MATH] thus gives the conditions [EQUATION]', '0706.3147-1-66-3': 'One therefore can set [EQUATION] with [MATH] satisfying [EQUATION]', '0706.3147-1-66-4': 'We also set [EQUATION] where [MATH] satisfies [EQUATION]', '0706.3147-1-66-5': 'Here [MATH] and [MATH] are functions of [MATH].', '0706.3147-1-66-6': 'Next we define [EQUATION]', '0706.3147-1-66-7': 'Note that [MATH] are an orthonormal basis of the Kahler base with the property that [EQUATION] and [EQUATION] with [EQUATION]', '0706.3147-1-66-8': 'These constraints therefore imply that [EQUATION]', '0706.3147-1-66-9': 'Hence the Kahler base is a product of two 2-manifolds [MATH], [MATH], with metric [EQUATION]', '0706.3147-1-66-10': 'Taking the orthonormal basis [MATH], the metrics on [MATH] and [MATH] are [EQUATION]', '0706.3147-1-66-11': 'It is then straightforward to compute the curvature in this basis.', '0706.3147-1-66-12': 'We find that the only non-vanishing components are fixed by [EQUATION]', '0706.3147-1-66-13': 'Therefore we conclude that [MATH] is [MATH], and [MATH] is [MATH], [MATH] or [MATH] depending on whether [MATH], [MATH] or [MATH] respectively.', '0706.3147-1-67-0': '## Solutions with [MATH] and [MATH]', '0706.3147-1-68-0': 'In the next case, we shall assume that [MATH] and [MATH] do not both vanish, and define the vector fields [MATH], [MATH] on the Kahler base via [EQUATION]', '0706.3147-1-68-1': 'Then note that [MATH] and [MATH], and ([REF]), ([REF]) and ([REF]) can be rewritten in terms of [MATH] and [MATH] as [EQUATION]', '0706.3147-1-68-2': 'Therefore we find that [MATH] is a holomorphic Killing vector on the base and satisfies [EQUATION]', '0706.3147-1-68-3': 'Moreover, [MATH] preserves the complex structure [EQUATION]', '0706.3147-1-68-4': 'In contrast, [MATH] defines a closed 1-form on the base, which is conformally Killing with [EQUATION]', '0706.3147-1-68-5': 'Here [MATH] denotes the metric of the Kahler base.', '0706.3147-1-68-6': 'From ([REF] ) it is clear that [MATH] and [MATH] commute, so that locally one can choose co-ordinates [MATH], [MATH] so that [EQUATION] and let the remaining two co-ordinates of the base space be [MATH], [MATH].', '0706.3147-1-69-0': 'Note that [MATH] and [MATH] satisfy [EQUATION]', '0706.3147-1-69-1': 'Then, one can write the metric on the Kahler base locally as [EQUATION] where [MATH], [MATH] and [MATH].', '0706.3147-1-69-2': 'By making a co-ordinate transformation of the form [EQUATION] one can without loss of generality set [MATH] in ([REF]) and drop primes throughout.', '0706.3147-1-70-0': 'Then it is straightforward to show that the condition [MATH] implies that [MATH], so that [EQUATION] where [MATH], and [EQUATION]', '0706.3147-1-70-1': 'The necessary and sufficient condition in order for [MATH] to be a covariantly constant complex structure is [EQUATION]', '0706.3147-1-70-2': 'In fact, we can take [MATH] without loss of generality, (this can be obtained, if necessary, by making the re-definition [MATH] and [MATH]).', '0706.3147-1-70-3': 'So [EQUATION]', '0706.3147-1-70-4': 'In addition, ([REF]) and ([REF]) can be rewritten as [EQUATION]', '0706.3147-1-70-5': 'The real portions of ([REF]) imply that [EQUATION] and hence [EQUATION]', '0706.3147-1-70-6': 'Therefore [EQUATION] for some function [MATH].', '0706.3147-1-70-7': 'It should be noted that although the K ahler metric [MATH] has a conformal dependence on [MATH], the portion of the metric [MATH] which appears in the five dimensional metric does not depend on either [MATH] or [MATH].', '0706.3147-1-71-0': 'In order to examine the behaviour of the scalars, note that ([REF]) implies that [EQUATION]', '0706.3147-1-71-1': 'To proceed further, we introduce the following holomorphic basis for the K ahler base space [EQUATION] with [MATH], [MATH] obtained by complex conjugation.', '0706.3147-1-71-2': 'It is straightforward to show that in this basis [EQUATION] as expected, and [EQUATION]', '0706.3147-1-71-3': 'Using this basis, one can compute explicitly the following components of the spin connection: [EQUATION]', '0706.3147-1-72-0': 'Moreover, it is straightforward to show that the imaginary portion of ([REF]) implies that [EQUATION] and [EQUATION]', '0706.3147-1-72-1': 'The imaginary parts of ([REF]) and ([REF]) can then be rewritten as [EQUATION]', '0706.3147-1-72-2': 'Note that as the [MATH] do not all vanish in the gauged theory, it follows that the imaginary parts of [MATH] and [MATH] do not depend on [MATH] , and depend on [MATH] via the factor [MATH].', '0706.3147-1-72-3': 'It is therefore convenient to define [EQUATION] and rewrite ([REF]), ([REF]) and ([REF]) as [EQUATION] and [EQUATION]', '0706.3147-1-72-4': 'We next consider the constraints ([REF]), ([REF]) and ([REF]).', '0706.3147-1-72-5': 'These are equivalent to [EQUATION] and [EQUATION]', '0706.3147-1-72-6': 'Note that ([REF]) implies the integrability condition associated with ([REF]), and ([REF]) implies that [MATH] and [MATH] satisfy the Cauchy-Riemann equations.', '0706.3147-1-72-7': 'Hence, [MATH] is a holomorphic function of [MATH].', '0706.3147-1-73-0': 'The components of [MATH] are also fixed by ([REF]), ([REF]) and ([REF]) to be [EQUATION] with the remaining components determined by complex conjugation; this exhausts the content of ([REF]), ([REF]) and ([REF]).', '0706.3147-1-73-1': 'Using these components, it is straightforward to compute [MATH] in the co-ordinate basis; we find [EQUATION]', '0706.3147-1-73-2': 'Using ([REF]) and ([REF]), the gauge field strengths for these solutions can be written as [EQUATION] which satisfy [MATH] automatically.', '0706.3147-1-74-0': 'Next consider the integrability condition associated with ([REF]): this can be written as [EQUATION]', '0706.3147-1-74-1': 'This can be evaluated to give the constraint [EQUATION] where [MATH] is the Laplacian on [MATH].', '0706.3147-1-74-2': 'In fact, this constraint enables ([REF]) to be solved for [MATH] (up to a total derivative); we find [EQUATION] and the constraint ([REF]) implies that the scalars [MATH] are given by [EQUATION] for constant [MATH].', '0706.3147-1-75-0': 'Lastly, consider the equations ([REF]) and ([REF] ).', '0706.3147-1-75-1': 'These are equivalent to [EQUATION] and [EQUATION] where [MATH] denotes the restriction of the exterior derivative to hypersurfaces of constant [MATH].', '0706.3147-1-75-2': 'The integrability conditions of these two equations are [EQUATION] and [EQUATION] which hold automatically.', '0706.3147-1-76-0': '## Solutions with [MATH]', '0706.3147-1-77-0': 'We now turn to the class of solutions with [MATH].', '0706.3147-1-77-1': 'It is clear that ([REF]), ([REF]) and ([REF]) imply that [MATH] and [MATH] are covariantly constant.', '0706.3147-1-77-2': 'In particular, this implies that [MATH] is constant and without loss of generality we can set [MATH].', '0706.3147-1-77-3': 'One can choose co-ordinates [MATH], [MATH] so that locally [EQUATION] and two additional co-ordinates [MATH], [MATH] can be chosen on the base so that the Kahler metric takes the form [EQUATION] where [MATH].', '0706.3147-1-78-0': 'Recall that [MATH] is anti-self-dual, so taking positive orientation on the base with respect to [MATH], we have [EQUATION]', '0706.3147-1-78-1': 'This complex structure is automatically covariantly constant.', '0706.3147-1-78-2': 'Then the real portions of ([REF]) and ([REF]) imply that [EQUATION] so that [MATH] is only a function of [MATH] and [MATH].', '0706.3147-1-78-3': 'Also, as in the previous case, the real part of ([REF]) and ([REF]) implies that [EQUATION] so [EQUATION]', '0706.3147-1-78-4': 'It is convenient to define [EQUATION]', '0706.3147-1-78-5': 'Then the remaining portions of ([REF]), ([REF]), ( [REF]) and ([REF]) can be rewritten as [EQUATION]', '0706.3147-1-78-6': 'As the [MATH] do not all vanish, these constraints imply that [EQUATION]', '0706.3147-1-78-7': 'We take the following holomorphic basis for the Kahler base space: [EQUATION] with [MATH], [MATH] fixed by complex conjugation.', '0706.3147-1-79-0': 'In this basis, we obtain the following spin connection components: [EQUATION] and the components of [MATH] and [MATH] are: [EQUATION]', '0706.3147-1-79-1': 'To proceed, we turn to the constraint ([REF]).', '0706.3147-1-79-2': 'This implies that [EQUATION] and [EQUATION]', '0706.3147-1-79-3': 'The constraints ([REF]) and ([REF]) then imply that [EQUATION] together with [EQUATION] and the constraints [EQUATION] and [EQUATION]', '0706.3147-1-79-4': 'Just as in the previous section, these constraints imply that [MATH] and [MATH] satisfy the Cauchy-Riemann equations; [MATH] is a holomorphic function of [MATH] .', '0706.3147-1-80-0': 'In these co-ordinates [MATH] takes the form [EQUATION]', '0706.3147-1-80-1': 'The gauge field strengths for these solutions can be written as [EQUATION] which automatically satisfy the Bianchi identity [MATH].', '0706.3147-1-81-0': 'Finally the integrability condition associated with ([REF]) can be written as [EQUATION]', '0706.3147-1-81-1': 'It is then straightforward to show that these constraints imply that the integrability condition [MATH] associated with the expression in ( [REF]) holds automatically.', '0706.3147-1-81-2': 'Lastly, the constraints ([REF]) and ([REF]) fix [MATH] and [MATH] in terms of constant linear combinations of [MATH] and [MATH]; these conditions do not impose any further constraints on the geometry.', '0706.3147-1-82-0': '# Summary and Discussion', '0706.3147-1-83-0': 'In this paper we have employed the spinorial geometry method for the task of classifying [MATH] supersymmetric solutions with at least one time-like Killing spinor of the theory of [MATH] supergravity.', '0706.3147-1-83-1': 'Our results provide a general framework for the explicit construction of many new black holes and the investigation of their physical properties and relevance to AdS/CFT correspondence and holography.', '0706.3147-1-84-0': 'In general, supersymmetric solutions in five dimensional theories must preserve either [MATH], [MATH], [MATH] or [MATH] of the supersymmetries.', '0706.3147-1-84-1': 'This is because the Killing spinor equations are linear over [MATH].', '0706.3147-1-84-2': 'However in the ungauged theories, it was found that supersymmetric solutions can only preserve [MATH] or [MATH] of supersymmetries [CITATION].', '0706.3147-1-84-3': 'Moreover, to find time-like supersymmetric solutions in the ungauged theory, one must solve the gauge equations and the Bianchi identities in addition to the Killing spinor equations.', '0706.3147-1-84-4': 'In the null case one must additionally solve one of the components of the Einstein equations of motion.', '0706.3147-1-84-5': 'A similar situation arises for [MATH] supersymmetric solutions in the gauged theory.', '0706.3147-1-84-6': 'However, for the solutions with [MATH] supersymmetry considered in our present work, we have demonstrated that if one of the Killing spinors is time-like, then supersymmetry and Bianchi identities alone imply that all components of Einstein and gauge equations together with the scalar equations are automatically satisfied.', '0706.3147-1-85-0': 'Maximally supersymmetric solutions (preserving all 8 of the supersymmetries) of the five dimensional gauged supergravity theory have vanishing gauge field strengths and constant scalars and are locally isometric to [MATH].', '0706.3147-1-85-1': 'Moreover, it has been demonstrated in [CITATION] that all solutions of [MATH], [MATH] supergravity preserving [MATH] of supersymmetry must be locally isometric to [MATH], with vanishing gauge field strengths and constant scalars.', '0706.3147-1-85-2': 'An analogous situation also arises in the case of [MATH] supergravity, where it has been shown that all solutions with [MATH] supersymmetry must be locally isometric to a maximally supersymmetric solution [CITATION].', '0706.3147-1-85-3': 'However, in the case of [MATH] supergravity, it has been shown that one cannot obtain [MATH] supersymmetric solutions by taking quotients of the maximally supersymmetric solutions [CITATION].', '0706.3147-1-85-4': 'In contrast, there is a [MATH]-supersymmetric supersymmetric solution of [MATH], [MATH] gauged supergravity which is obtained by taking a certain quotient of [MATH] [CITATION].', '0706.3147-1-86-0': 'One future direction is the completion of the classification of supersymmetric solutions in [MATH], [MATH] supergravity by classifying [MATH] supersymmetric solutions with two null Killing spinors (i.e., two Killing spinors with associated null Killing vectors).', '0706.3147-1-86-1': 'In addition, it would be interesting to investigate whether there are any regular asymptotically [MATH] black ring solutions (see [CITATION] for a recent discussion).', '0706.3147-1-86-2': 'Supersymmetric black rings are known to exist in the ungauged theory [CITATION].', '0706.3147-1-86-3': 'For asymptotically flat supersymmetric black rings of [MATH], [MATH] ungauged supergravity, supersymmetry is enhanced from [MATH] supersymmetry to maximal supersymmetry at the horizon.', '0706.3147-1-86-4': 'If there do exist [MATH] black rings in the gauged theory, it may be reasonable to expect that the supersymmetry will be enhanced from [MATH] to [MATH] at the horizon.', '0706.3147-1-86-5': 'We hope that the classification of [MATH] supersymmetric solutions can provide a method of determining whether there exists a [MATH] supersymmetric solution corresponding to the near-horizon geometry of a black ring.', '0706.3147-1-86-6': 'It should be noted that in [CITATION], black rings with two [MATH] symmetries have already been excluded.', '0706.3147-1-86-7': 'However, as we have seen, [MATH] supersymmetric solutions in general only have one additional [MATH] symmetry; so more general ring solutions may be possible.'}

{'0706.3147-2-0-0': '# Introduction', '0706.3147-2-1-0': 'There has been a considerable research effort in recent years devoted to the analysis and study of solutions of supergravity theories in various dimensions and in particular those obtained as low energy limits of superstring and M-theory.', '0706.3147-2-1-1': 'This effort is motivated by the important role that black holes and domain walls have played in some of the recent developments that took place in superstring theory.', '0706.3147-2-1-2': 'These include the conjectured equivalence between string theory on anti-de Sitter (AdS) spaces and certain superconformal gauge theories living on the boundary [CITATION] known as the AdS/CFT correspondence, the understanding of the microscopic analysis of black hole entropy [CITATION] and the understanding of various duality symmetries relating string theories to each other and to M-theory.', '0706.3147-2-1-3': 'An interesting possibility that arises from the conjectured AdS/CFT correspondence is the ability to obtain information of the nonperturbative structure of field theories by studying dual classical gravitational configurations.', '0706.3147-2-1-4': 'A notable example in this direction is the Hawking-Page phase transition [CITATION] which was interpreted in [CITATION] as a thermal phase transition from a confining to a deconfining phase in the dual [MATH], [MATH] super Yang-Mills theory.', '0706.3147-2-1-5': 'Various interesting results using Anti-de Sitter black holes and their CFT duals have been obtained in recent years (see for example [CITATION]).', '0706.3147-2-2-0': 'In this paper we will focus on the study of supersymmetric solutions in five-dimensional [MATH] gauged supergravity coupled to abelian vector multiplets [CITATION].', '0706.3147-2-2-1': 'These solutions are relevant for the holographic descriptions of four dimensional field theories with less than maximal supersymmetry.', '0706.3147-2-2-2': 'Explicit supersymmetric black holes for these theories were constructed in [CITATION].', '0706.3147-2-2-3': 'However, these solutions have naked singularities or naked closed time-like curves.', '0706.3147-2-2-4': 'Domain walls and magnetic strings were also constructed in [CITATION].', '0706.3147-2-2-5': 'Obviously one would like to study the general structure of supersymmetric solutions in five dimensions rather than some specific solutions based on a certain ansatz.', '0706.3147-2-2-6': 'The main purpose of this paper is to construct a systematic classification of half-supersymmetric solutions.', '0706.3147-2-3-0': 'The first systematic classification of supersymmetric solutions, following the results of [CITATION], was performed in [CITATION] for minimal [MATH] supergravity in [MATH].', '0706.3147-2-3-1': 'In [CITATION] it was shown that supersymmetric solutions fall into two classes which depend on whether the Killing vector obtained from the Killing spinor is time-like or null.', '0706.3147-2-3-2': 'For the time-like case, one obtains the Israel-Wilson-Perjes class of solutions and the null solutions are pp-waves.', '0706.3147-2-3-3': 'Further generalizations were presented in [CITATION].', '0706.3147-2-3-4': 'More recently and motivated by the results of Tod, purely bosonic supersymmetric solutions of minimal [MATH] were classified in [CITATION].', '0706.3147-2-3-5': 'The basic idea in this analysis is to assume the existence of a Killing spinor, (i.e., to assume that the solution preserves at least one supersymmetry) and construct differential forms as bilinears in the Killing spinor.', '0706.3147-2-3-6': 'Then Fierz identities and the vanishing of the supersymmetry transformation of the fermionic fields in a bosonic background provide a set of algebraic and differential equations for the spinor bilinear differential forms which can be used to deduce the form of the metric and gauge fields.', '0706.3147-2-3-7': 'Such a general framework provides a powerful method for obtaining many new solutions, in contrast to the earlier methods that start with an ansatz for the metric and assume certain symmetries for the solution from the outset.', '0706.3147-2-3-8': 'The strategy of [CITATION] was used later to perform similar classifications of supersymmetric solutions in various supergravity theories.', '0706.3147-2-3-9': 'In particular, in [CITATION] the classification of [MATH] supersymmetric solutions of the minimal gauged [MATH] supergravity was performed.', '0706.3147-2-4-0': 'Explicit supersymmetric asymptotically anti-de Sitter black hole solutions with no closed time loops or naked singularities were constructed for the minimal supergravity theory in [CITATION].', '0706.3147-2-4-1': 'The results of [CITATION] for the time-like solutions were generalized in [CITATION] to the non-minimal case where the scalar fields live on symmetric spaces and explicit solutions for the [MATH] theory (with three [MATH] -charges) were also constructed.', '0706.3147-2-4-2': 'The constraint of symmetric spaces was relaxed in [CITATION], where solutions with a null Killing vector in both gauged and ungauged theories were also obtained.', '0706.3147-2-5-0': 'In this paper we focus on the classification of half supersymmetric solutions in gauged [MATH] supergravity with vector multiplets.', '0706.3147-2-5-1': 'Half supersymmetric solutions have two Killing spinors from which one can construct two Killing vectors as bilinears in the Killing spinors.', '0706.3147-2-5-2': 'These vectors could be either time-like or null.', '0706.3147-2-5-3': 'Therefore one has to consider three cases depending on the nature of the Killing spinors and vectors considered.', '0706.3147-2-5-4': 'In our present work we will focus on the cases where the solutions contain at least one Killing spinor with an associated time-like Killing vector.', '0706.3147-2-5-5': 'In order to investigate supersymmetric solutions with more than one Killing spinor, it is very useful to express the Killing spinors in terms of differential forms [CITATION], [CITATION], [CITATION].', '0706.3147-2-5-6': 'Such a method, known now by the spinorial geometry method, has been very efficient in classifying solutions of supergravity theories in ten and eleven dimensions [CITATION], [CITATION], [CITATION] [CITATION].', '0706.3147-2-5-7': 'The spinorial geometry method has also been recently used to classify half-supersymmetric solutions in [MATH], [MATH] supergravity [CITATION].', '0706.3147-2-6-0': 'We organize our work as follows.', '0706.3147-2-6-1': 'In section two, we present the basic structure of the theory of [MATH], [MATH] gauged supergravity coupled to abelian vector multiplets and the equations of motion.', '0706.3147-2-6-2': 'In section three we express spinors in five dimensions as differential forms on [MATH].', '0706.3147-2-6-3': 'We start with the generic form of the spinor and then use the gauge symmetries ([MATH] and [MATH] preserving the symplectic Majorana condition to write down two canonical forms for a single symplectic Majorana spinor corresponding to time-like and null Killing vectors.', '0706.3147-2-6-4': 'In section four, we derive the conditions for quarter supersymmetric solutions with time-like Killing vector.', '0706.3147-2-6-5': 'In section five, the [MATH] Killing constraints, i. e., the conditions for a time-like quarter supersymmetric solution, are then substituted into the generic Killing spinor equations and the resulting equations are rewritten in the form of constraints on the Kahler base.', '0706.3147-2-6-6': 'Section six contains a detailed classification of half-supersymmetric solutions.', '0706.3147-2-6-7': 'Our paper ends with two appendices.', '0706.3147-2-6-8': 'Appendix A deals with the determination of the linear system obtained from the Killing spinor equations.', '0706.3147-2-6-9': 'Appendix B discusses the integrability conditions of the Killing spinor equations.', '0706.3147-2-6-10': 'There it is demonstrated that for a given background preserving at least half of the supersymmetry, where at least one of the Killing spinors generates a time-like Killing vector, all of the Einstein, gauge and scalar field equations of motion hold automatically provided that the Bianchi identity is satisfied.', '0706.3147-2-7-0': '# [MATH] supergravity', '0706.3147-2-8-0': 'In this section, we review briefly some aspects of the [MATH], [MATH] gauged supergravity coupled to abelian vector multiplets is [CITATION].', '0706.3147-2-8-1': 'The bosonic action of the theory is [EQUATION] where [MATH] take values [MATH] and [MATH] are the two-forms representing gauge field strengths (one of the gauge fields corresponds to the graviphoton).', '0706.3147-2-8-2': 'The metric has mostly negative signature.', '0706.3147-2-8-3': 'The constants [MATH] are symmetric in [MATH] and are not assumed to satisfy the non-linear adjoint identity which arises when the scalars lie in a symmetric space [CITATION]; though we will assume that [MATH] is invertible, with inverse [MATH].', '0706.3147-2-8-4': 'The [MATH] are scalar fields subject to the constraint [EQUATION]', '0706.3147-2-8-5': 'The fields [MATH] can thus be regarded as being functions of [MATH] unconstrained scalars [MATH].', '0706.3147-2-8-6': 'It is convenient to define [EQUATION] so that the condition ([REF]) becomes [EQUATION]', '0706.3147-2-8-7': 'In addition, the coupling [MATH] depends on the scalars via [EQUATION] so in particular', '0706.3147-2-9-0': '[EQUATION] where [MATH] denotes differentiation with respect to [MATH].', '0706.3147-2-9-1': 'The scalar potential can be written as [EQUATION] where [MATH] are constants.', '0706.3147-2-10-0': 'Bosonic backgrounds are said to be supersymmetric if there exists a spinor [MATH] for which the supersymmetry variations of the gravitino and dilatino vanish in the given background.', '0706.3147-2-10-1': 'For the gravitino this requires [EQUATION] and for the dilatino it requires [EQUATION]', '0706.3147-2-10-2': 'The Einstein equation derived from ([REF]) is given by [EQUATION]', '0706.3147-2-10-3': 'The Maxwell equations (varying [MATH]) are [EQUATION]', '0706.3147-2-10-4': 'The scalar equations (varying [MATH]) are [EQUATION]', '0706.3147-2-10-5': 'If a quantity [MATH] satisfies [MATH] then there must be a function [MATH] such that [MATH].', '0706.3147-2-10-6': 'This implies that the dilatino equation ([REF]) can be simplified to [EQUATION] and the scalar equation can be written as [EQUATION]', '0706.3147-2-11-0': '# Spinors in Five Dimensions', '0706.3147-2-12-0': 'Following [CITATION] , we write spinors in five dimensions as forms on [MATH].', '0706.3147-2-12-1': 'We represent a generic spinor [MATH] in the form [EQUATION] where [MATH], [MATH] are 1-forms on [MATH], [MATH] and [MATH] and [MATH] are complex functions.', '0706.3147-2-13-0': 'The action of [MATH]-matrices on these forms is given by [EQUATION]', '0706.3147-2-13-1': 'We define [MATH] by [MATH].', '0706.3147-2-13-2': 'This satisfies [EQUATION]', '0706.3147-2-13-3': 'The charge conjugation operator [MATH] is defined by [EQUATION] where [MATH] is antisymmetric with [MATH].', '0706.3147-2-14-0': 'The Killing spinors [MATH] of the theory satisfy a symplectic Majorana constraint which is [EQUATION] so if one writes [EQUATION] then [MATH] is fixed via [EQUATION]', '0706.3147-2-14-1': 'We note the useful identity [EQUATION]', '0706.3147-2-14-2': 'It will be particularly useful in our work to complexify the gamma operators.', '0706.3147-2-14-3': 'Therefore we write [EQUATION]', '0706.3147-2-15-0': '## Gauge transformations and [MATH] spinors', '0706.3147-2-16-0': 'There are two types of gauge transformation that preserve the symplectic Majorana condition ([REF]).', '0706.3147-2-16-1': 'First, we have the [MATH] gauge transformations described by [EQUATION] and there are also [MATH] gauge transformations of the form [EQUATION] for real functions [MATH].', '0706.3147-2-17-0': 'Note in particular that [MATH], [MATH] and [MATH] generate a [MATH] which leaves [MATH] and [MATH] invariant and acts on [MATH] , [MATH]; whereas [MATH], [MATH] and [MATH] generate another [MATH] which leaves the [MATH] invariant but acts on [MATH] and [MATH].', '0706.3147-2-17-1': 'In addition, [MATH] generates a [MATH] which acts (simultaneously) on [MATH] and [MATH] , whereas [MATH] generates another [MATH] which acts (simultaneously) on [MATH] and [MATH].', '0706.3147-2-18-0': 'Therefore, for a single symplectic Majorana spinor, one can always use [MATH] gauge transformations to write [EQUATION] or [EQUATION] or [EQUATION] for some real function [MATH].', '0706.3147-2-18-1': 'However, under the transformation [EQUATION] the spinor in ([REF]) transforms as [EQUATION] and [EQUATION] and [MATH] is unchanged.', '0706.3147-2-18-2': 'This transformation corresponds to reflections in the [MATH] directions.', '0706.3147-2-18-3': 'Moreover, the spinor in ([REF]) is equivalent to that in ([REF]) under a [MATH] gauge transformation.', '0706.3147-2-18-4': 'The spinors corresponding to ([REF]) and ([REF]) are therefore equivalent under these transformations.', '0706.3147-2-18-5': 'Hence, for a single spinor, one need only consider the cases ([REF]) and ([REF]).', '0706.3147-2-19-0': '## Differential Forms from Spinors', '0706.3147-2-20-0': 'In order to define differential forms, we first define a Hermitian inner product on [MATH] by [EQUATION]', '0706.3147-2-20-1': 'Then [MATH] gauge-invariant [MATH]-forms are obtained from spinors [MATH], [MATH] via [EQUATION]', '0706.3147-2-20-2': 'In particular, for the generic Majorana spinor given in ([REF]) and ([REF]) one finds [EQUATION]', '0706.3147-2-20-3': 'The scalars are then given by [EQUATION]', '0706.3147-2-20-4': 'Hence, by comparing with [CITATION], it is clear that the spinor given in ([REF]) corresponds to the time-like class of solutions, whereas that in ([REF]) is in the null class of solutions.', '0706.3147-2-20-5': 'With a slight abuse of notation, we shall refer to the corresponding Killing spinors as being either time-like or null.', '0706.3147-2-21-0': '## Canonical [MATH] spinors', '0706.3147-2-22-0': 'We will now assume that there are two linearly independent symplectic Majorana Killing spinors [MATH], [MATH], where [MATH] is time-like.', '0706.3147-2-22-1': 'In appendix B it is demonstrated that the existence of such spinors is sufficient to ensure that the scalar, gauge and Einstein equations of motion hold automatically from the integrability conditions, provided one assumes that the Bianchi identities are satisfied.', '0706.3147-2-22-2': 'So the only equations which must be solved are the Killing spinor equations together with the Bianchi identity.', '0706.3147-2-23-0': 'From the previous reasoning, we can take [MATH] to have the canonical form.', '0706.3147-2-23-1': '[EQUATION] for [MATH].', '0706.3147-2-23-2': 'Next consider [MATH] given by [EQUATION] for complex [MATH].', '0706.3147-2-23-3': 'It is possible to simplify [MATH] a little using gauge transformations which leave [MATH] invariant.', '0706.3147-2-23-4': 'In particular, by using an appropriate [MATH] transformation, one could for example set [MATH] with [MATH].', '0706.3147-2-23-5': 'However, we will not make this gauge choice.', '0706.3147-2-24-0': '## The 1/4 Supersymmetric time-like Solution', '0706.3147-2-25-0': 'In this section we obtain the time-like solutions preserving a quarter of the supersymmetry using the spinorial geometry method.', '0706.3147-2-25-1': 'These solutions were derived in [CITATION].', '0706.3147-2-25-2': 'In order to obtain 1/4 supersymmetric solutions with time-like Killing spinor, it suffices to consider the equations ([REF])-([REF]) and set [MATH] and [MATH].', '0706.3147-2-25-3': 'Then from the dilatino equation, we find [EQUATION] whereas from the gravitino equation we find [EQUATION]', '0706.3147-2-25-4': 'To analyze this linear system, we will first consider the gravitino equations.', '0706.3147-2-25-5': 'Note that ([REF]) implies that [EQUATION] and [EQUATION]', '0706.3147-2-25-6': 'Next, consider ([REF]), ([REF]) and ([REF]).', '0706.3147-2-25-7': 'These imply [EQUATION]', '0706.3147-2-25-8': 'From ([REF]) and ([REF]) we find [EQUATION] and', '0706.3147-2-26-0': '[EQUATION]', '0706.3147-2-26-1': 'From ([REF]) we find [EQUATION] and from ([REF]) and ([REF]) we obtain [EQUATION] and [EQUATION]', '0706.3147-2-26-2': 'Hence, to summarize, we obtain the following purely geometric constraints [EQUATION] together with [EQUATION]', '0706.3147-2-26-3': 'It is straightforward to show that the constraints ([REF]) are the necessary and sufficient conditions for the 1-form [EQUATION] to define a Killing vector [MATH].', '0706.3147-2-26-4': 'This form is, as expected, the 1-form spinor bilinear which is obtained from [MATH].', '0706.3147-2-26-5': 'Note that this Killing vector satisfies [EQUATION]', '0706.3147-2-26-6': 'In fact, one can choose a gauge in which the Lie derivative of the vielbein with respect to [MATH] vanishes.', '0706.3147-2-26-7': 'To see this, note that [EQUATION]', '0706.3147-2-26-8': 'From ([REF]) we note that [MATH].', '0706.3147-2-26-9': 'Without loss of generality we can make a [MATH] gauge transformation in order to set [EQUATION]', '0706.3147-2-26-10': 'This gauge transformation alters the form of the Killing spinors via the transformation given in ([REF]).', '0706.3147-2-26-11': 'However, the Killing spinors can be restored to their original form by making a [MATH] gauge transformation generated by [MATH].', '0706.3147-2-26-12': 'Working in this gauge, ([REF]) implies that [MATH] and hence [EQUATION] where the constraints in ([REF]), ([REF]) and ([REF]) imply that [MATH].', '0706.3147-2-26-13': 'By making a further [MATH] gauge transformation generated by [MATH] which leaves [MATH] invariant and maps [MATH] for [MATH], we can without loss of generality take [MATH].', '0706.3147-2-26-14': 'In this basis the vielbein is time-independent.', '0706.3147-2-27-0': 'Equation ([REF]) also has a simple geometric interpretation.', '0706.3147-2-27-1': 'First note that the only [MATH] gauge-invariant 2-form which can be obtained from [MATH] is the real part of the 2-form [EQUATION]', '0706.3147-2-27-2': 'The real part of this form (denoted by [MATH]) is then given by [EQUATION]', '0706.3147-2-27-3': 'It is convenient to make a conformal rescaling of the complexified basis and define [EQUATION]', '0706.3147-2-27-4': 'We shall refer to the 4-manifold with metric [EQUATION] as the base space [MATH].', '0706.3147-2-27-5': 'Then it is clear that [MATH] defines an almost complex structure on this 4-manifold.', '0706.3147-2-27-6': 'In fact, ([REF]) implies that [MATH] is covariantly constant with respect to the Levi-civita connection of the base manifold, and hence [MATH] is a Kahler manifold (as expected) with Ka hler form [MATH].', '0706.3147-2-28-0': 'Also, from ([REF]) we have [EQUATION]', '0706.3147-2-28-1': 'We remark that ([REF]) implies that [EQUATION] where [MATH] is (locally) the potential for the Ricci form of the K ahler base [MATH] .', '0706.3147-2-29-0': 'The remaining constraints on the [MATH]-flux are then [EQUATION]', '0706.3147-2-29-1': 'Finally, we substitute these constraints into the dilatino equations.', '0706.3147-2-29-2': 'From ( [REF])-([REF]), we find [EQUATION]', '0706.3147-2-30-0': '# Killing spinor in [MATH] background', '0706.3147-2-31-0': 'In this section, we substitute the constraints obtained in the previous section back into the generic Killing spinor equation ([REF])-([REF]) and simplify as much as possible.', '0706.3147-2-31-1': 'We find from the dilatino equation: [EQUATION]', '0706.3147-2-31-2': 'And from the gravitino equations we find [EQUATION] and [EQUATION]', '0706.3147-2-31-3': 'Note that these equations admit a solution of the form [MATH] , [MATH], [MATH] with [MATH], [MATH] real constants, and no additional constraints on the fluxes or geometry.', '0706.3147-2-31-4': 'Hence we observe that the generic time-like solution preserves [MATH] supersymmetry.', '0706.3147-2-31-5': 'More generally, if [MATH], [MATH] are symplectic Majorana Killing spinors, then so are [EQUATION] which is just a special case of ([REF]) with [MATH].', '0706.3147-2-31-6': 'In particular, the equations computed above are invariant under the transformations [EQUATION] therefore it is clear that the Killing spinors arise in pairs.', '0706.3147-2-32-0': '## Solutions with [MATH]', '0706.3147-2-33-0': 'Suppose we consider the case when [MATH].', '0706.3147-2-33-1': 'Then, assuming that [MATH], we find from ([REF]) and ([REF]) that [EQUATION]', '0706.3147-2-33-2': 'If, however, [MATH] then ([REF]) and ([REF]) again imply ([REF]), as we assume that not all of the [MATH] vanish.', '0706.3147-2-34-0': 'Hence, from ([REF]), ([REF]), ([REF]) and ([REF]) it follows directly that [MATH], [MATH], [MATH] with [MATH], [MATH] real constants.', '0706.3147-2-34-1': 'The solution is therefore only 1/4-supersymmetric.', '0706.3147-2-34-2': 'Thus, to find new solutions with enhanced supersymmetry, one must take [MATH]; henceforth we shall assume that [MATH].', '0706.3147-2-35-0': '## Constraints on the base space', '0706.3147-2-36-0': 'It will be particularly useful to rewrite the equations ([REF])-( [REF]) in terms of constraints on the Kahler base.', '0706.3147-2-36-1': 'Throughout this section, unless stated otherwise, tensor indices are evaluated with respect to the 4-dimensional complex basis [MATH], [MATH]; so we shall drop the [MATH] from all expressions.', '0706.3147-2-36-2': 'It is convenient to define a real vector field [MATH] on the Kahler base as follows [EQUATION]', '0706.3147-2-36-3': 'In order to rewrite the constraints, we define a time co-ordinate [MATH] so that the Killing vector field associated with the Killing spinor [MATH] is [EQUATION] and set [EQUATION] where [MATH] is a 1-form defined on the Kahler base.', '0706.3147-2-37-0': 'Then ([REF]) is equivalent to [EQUATION] where here [MATH] denotes the Levi-civita connection of the Kahler base metric given in ([REF]).', '0706.3147-2-37-1': 'Also, ([REF]) can be rewritten as [EQUATION]', '0706.3147-2-37-2': 'It is also useful to define [EQUATION]', '0706.3147-2-37-3': 'It is then straightforward to show that [EQUATION]', '0706.3147-2-37-4': 'The commutator is given by [EQUATION]', '0706.3147-2-37-5': 'Next, ([REF]) and ([REF]) are equivalent to [EQUATION]', '0706.3147-2-37-6': 'These equations simply imply that [EQUATION]', '0706.3147-2-37-7': 'In addition, ([REF]) and ([REF]) can be rewritten as [EQUATION] and [EQUATION]', '0706.3147-2-37-8': 'In order to simplify the remainder of the equations, observe that for indices [MATH], [EQUATION] and [EQUATION] where on the LHS of ([REF]) and ([REF]), spatial indices are taken with respect to the original five-dimensional basis, whereas on the RHS, they are taken with respect to the conformally rescaled K ahler basis; and [MATH] denotes the spin connection of the K ahler base space.', '0706.3147-2-37-9': 'From henceforth, the hat will be dropped, and we will work solely on the Kahler base space.', '0706.3147-2-38-0': 'Then ([REF]) is equivalent to [EQUATION] and using this, ([REF]) can be rewritten as [EQUATION]', '0706.3147-2-38-1': 'It is also useful to rewrite ([REF]) as [EQUATION]', '0706.3147-2-38-2': 'In addition, ([REF]), ([REF]), ([REF]) and ([REF]) can be rewritten as [EQUATION]', '0706.3147-2-39-0': '# Half Supersymmetric Solutions', '0706.3147-2-40-0': 'Suppose that the solution preserves exactly four of the supersymmetries.', '0706.3147-2-40-1': 'Then the four linearly independent Killing spinors are [MATH], [MATH] and [EQUATION]', '0706.3147-2-40-2': 'As all of the scalars, gauge field strengths and components of the spin connection are [MATH]-independent, it follows that [MATH] is also a Killing spinor.', '0706.3147-2-40-3': 'As the solution is exactly half-supersymmetric, it follows that there must be real constants [MATH] such that [EQUATION] or equivalently [EQUATION]', '0706.3147-2-40-4': 'On substituting these constraints into ([REF]) we find that [EQUATION]', '0706.3147-2-40-5': 'Contracting this expression with [MATH] we find that [EQUATION]', '0706.3147-2-40-6': 'The real part of this expression implies that [MATH].', '0706.3147-2-41-0': 'Suppose now that [MATH].', '0706.3147-2-41-1': 'From ([REF]) we find that [EQUATION] and hence [EQUATION] where [MATH].', '0706.3147-2-41-2': 'Note that by making a redefinition of the type [EQUATION] we can without loss of generality set [MATH] and drop the primes on [MATH] and [MATH].', '0706.3147-2-42-0': 'It will be convenient to split the solutions into three classes.', '0706.3147-2-42-1': 'For the first class [MATH] and [MATH], for the second [MATH] but [MATH] and for the third [MATH].', '0706.3147-2-43-0': '## Solutions with [MATH], [MATH]', '0706.3147-2-44-0': 'For this class of solutions we have [EQUATION] where [MATH].', '0706.3147-2-44-1': 'Here we have the conditions [MATH] and [MATH].', '0706.3147-2-45-0': 'To proceed with the analysis for these solutions, we define the 1-forms [MATH], [MATH] and [MATH] on the Kahler base via [EQUATION]', '0706.3147-2-45-1': 'The components of these 1-forms can be easily shown to be [MATH]-independent [EQUATION]', '0706.3147-2-45-2': 'For convenience we set [MATH] and [MATH].', '0706.3147-2-46-0': 'In order to evaluate various integrability constraints, it is useful to compute the components of the covariant derivatives: [EQUATION] and [EQUATION] and [EQUATION]', '0706.3147-2-46-1': 'It immediately follows that [MATH] and [MATH] defines a Killing vector on the Kahler base space.', '0706.3147-2-46-2': 'In particular, setting [MATH], [MATH] is exact and satisfies [EQUATION]', '0706.3147-2-46-3': 'The first integrability condition we shall examine is obtained by considering the constraints ([REF]), ([REF]), ([REF]) and ([REF]).', '0706.3147-2-46-4': 'These are equivalent to [EQUATION]', '0706.3147-2-46-5': 'Taking the exterior derivative of this equation, we obtain the constraint [EQUATION]', '0706.3147-2-46-6': 'Hence, using ([REF]) and ([REF]) we obtain the constraints [EQUATION] and [EQUATION]', '0706.3147-2-46-7': 'Observe also that ([REF]) is equivalent to [EQUATION]', '0706.3147-2-46-8': 'Using ([REF]) and ([REF]), we compute [EQUATION]', '0706.3147-2-46-9': 'To proceed, we impose the integrability condition, [MATH].', '0706.3147-2-46-10': 'We note the following useful identities: [EQUATION] and [EQUATION] from whence we obtain [EQUATION]', '0706.3147-2-46-11': 'Using this expression, the constraint [MATH] implies that [EQUATION] for real constant [MATH].', '0706.3147-2-47-0': 'Next we consider the integrability condition [MATH].', '0706.3147-2-47-1': 'It is straightforward to show that [MATH] satisfies [EQUATION]', '0706.3147-2-47-2': 'Hence the integrability condition [MATH] implies that [EQUATION]', '0706.3147-2-47-3': 'We observe that [EQUATION]', '0706.3147-2-47-4': 'It is then straightforward but tedious to show that ([REF]) implies [EQUATION]', '0706.3147-2-47-5': 'Using this condition the above relations simplify and we obtain [EQUATION]', '0706.3147-2-48-0': 'The components of [MATH] are therefore given by [EQUATION]', '0706.3147-2-48-1': 'Using the expressions for [MATH] which we have obtained, we next examine ([REF]) and ([REF]).', '0706.3147-2-48-2': 'These may be rewritten as [EQUATION] and [EQUATION]', '0706.3147-2-48-3': 'We note the useful identities: [EQUATION] where we have set [MATH].', '0706.3147-2-48-4': 'Then from the integrability condition [MATH] we find the constraint [EQUATION]', '0706.3147-2-48-5': 'Note that [EQUATION] then upon differentiating ([REF]) with respect to [MATH] gives [EQUATION]', '0706.3147-2-48-6': 'It turns out that the constraints ([REF]) and ([REF]) are also sufficient to ensure that [EQUATION]', '0706.3147-2-48-7': 'Next, note that the constraints ([REF]) and ([REF]) can be used to write the gauge field strengths [MATH] as [EQUATION] and note that as [MATH] it follows that [EQUATION]', '0706.3147-2-48-8': 'It is then straightforward to show that the Bianchi identity [MATH] follows automatically from the constraints we have obtained.', '0706.3147-2-48-9': 'To proceed further, it is useful to consider the cases for which [MATH] and [MATH] separately.', '0706.3147-2-48-10': 'Observe that [MATH] implies that [MATH] and [MATH] are either both real or both imaginary.', '0706.3147-2-49-0': '### Solutions with [MATH]', '0706.3147-2-50-0': 'In order to introduce a local co-ordinate system for solutions with [MATH], recall that [MATH] is a Killing vector on the Kahler base space.', '0706.3147-2-50-1': 'Furthermore, as [MATH], the closure of [MATH] implies that [MATH] preserves the Kahler form; [EQUATION]', '0706.3147-2-50-2': 'It is also straightforward to show that [EQUATION]', '0706.3147-2-50-3': 'Hence it follows that [MATH] defines a symmetry of the full five dimensional solution.', '0706.3147-2-50-4': 'As [MATH], ([REF]) and ([REF]) can be inverted to obtain [EQUATION]', '0706.3147-2-50-5': 'It is convenient to define the real 1-forms [MATH] and [MATH] by [EQUATION]', '0706.3147-2-50-6': 'It is then straightforward, but tedious, to show that [EQUATION] and also that [EQUATION]', '0706.3147-2-50-7': 'In addition, we find that [EQUATION]', '0706.3147-2-50-8': 'Hence we define the following orthonormal basis on the Kahler base space [EQUATION] where [EQUATION]', '0706.3147-2-50-9': 'As [MATH], [MATH] are commuting vector fields, we can choose co-ordinates [MATH], [MATH] such that [EQUATION]', '0706.3147-2-50-10': 'Then, defining [EQUATION] we have [EQUATION] and from ([REF]) we see that there must be a function [MATH] such that [EQUATION]', '0706.3147-2-50-11': 'As [MATH], [MATH], [MATH], [MATH] are orthogonal, it follows that [MATH] form a local co-ordinate system on the base space.', '0706.3147-2-50-12': 'One can then write [EQUATION]', '0706.3147-2-50-13': 'As [MATH] is a Killing vector, the functions [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] do not depend on [MATH] (or [MATH]).', '0706.3147-2-50-14': 'Furthermore, [EQUATION]', '0706.3147-2-50-15': 'Therefore [MATH], [MATH] and [MATH] are functions of [MATH] and [MATH] only.', '0706.3147-2-50-16': 'However, there is a non-trivial [MATH] -dependence in [MATH]; [MATH] is not a Killing vector.', '0706.3147-2-51-0': 'It is also useful to observe that the identity [MATH] implies [EQUATION] and hence we shall set [EQUATION]', '0706.3147-2-51-1': 'Here [MATH] is a real function which satisfies [EQUATION] which implies [MATH].', '0706.3147-2-51-2': 'With these conventions, it is straightforward to compute [EQUATION] and [EQUATION]', '0706.3147-2-51-3': 'Also note that ([REF]) can be rewritten as [EQUATION]', '0706.3147-2-51-4': 'If [MATH] then this constraint can be integrated up to give [EQUATION] for constant [MATH].', '0706.3147-2-52-0': 'To proceed, consider the equation ([REF]), which can be rewritten as [EQUATION]', '0706.3147-2-52-1': 'Taking the self-dual projection of ([REF]) yields the constraints [EQUATION] together with [EQUATION]', '0706.3147-2-52-2': 'Using these constraints, the anti-self-dual projection of ([REF] ) fixes [MATH] to be given by [EQUATION]', '0706.3147-2-52-3': 'Imposing the covariant constancy condition [MATH] imposes two additional constraints: [EQUATION] and [EQUATION]', '0706.3147-2-52-4': 'Finally, we compare the spin connection components [MATH] computed in this basis with the expression given in ([REF]), noting that [MATH].', '0706.3147-2-53-0': 'This implies that [EQUATION]', '0706.3147-2-53-1': 'Then from ([REF]) and ([REF]) we obtain [EQUATION]', '0706.3147-2-53-2': 'Then ([REF]) and ([REF]) can be integrated up to give [EQUATION] and ([REF]) and ([REF]) then imply [EQUATION]', '0706.3147-2-53-3': 'Hence, in these co-ordinates, the orthonormal basis of the Kahler base space is [EQUATION] and if [MATH], then [MATH] can be written as [EQUATION]', '0706.3147-2-53-4': 'By considering ([REF]), [MATH] is fixed (up to a total derivative) by [EQUATION] and by considering ([REF]) we find the gauge field strengths are given by [EQUATION]', '0706.3147-2-54-0': '### Solutions with [MATH], [MATH]', '0706.3147-2-55-0': 'If [MATH] and [MATH] are imaginary but non-vanishing, then from ([REF]) we obtain [MATH], and the constraints on [MATH] and [MATH] as given in ([REF]) and ([REF]) imply that [EQUATION]', '0706.3147-2-55-1': 'Moreover, ([REF]) can be integrated up to give [EQUATION] for constants [MATH].', '0706.3147-2-55-2': 'The Kahler form can be expressed by [EQUATION]', '0706.3147-2-55-3': 'It can also be demonstrated that [MATH] is given in these cases by [EQUATION] for some real function [MATH] with [EQUATION] and hence [EQUATION] for some real function [MATH] where [MATH] satisfy [EQUATION]', '0706.3147-2-55-4': 'Note that [MATH] with [MATH].', '0706.3147-2-55-5': 'Without loss of generality, we can work in a gauge for which [MATH] and [MATH].', '0706.3147-2-56-0': 'It is then straightforward to prove the following identities: [EQUATION]', '0706.3147-2-56-1': 'There are then three cases to consider.', '0706.3147-2-56-2': '[EQUATION]', '0706.3147-2-56-3': 'It is then straightforward to show that [EQUATION] where', '0706.3147-2-57-0': '[EQUATION] so that [EQUATION]', '0706.3147-2-57-1': 'Hence the 3-manifold with metric [MATH] is either [MATH], the Nil-manifold or [MATH] according as to whether [MATH], [MATH] or [MATH] respectively.', '0706.3147-2-58-0': '### Solutions with [MATH], [MATH]', '0706.3147-2-59-0': 'If [MATH] and [MATH] are real but non-vanishing, then [MATH], and from ([REF]) and ([REF]) we obtain [EQUATION]', '0706.3147-2-59-1': 'Moreover, from ([REF]) we obtain [EQUATION]', '0706.3147-2-59-2': 'This implies that [MATH], and hence without loss of generality we can set [MATH] and the scalars are therefore constants.', '0706.3147-2-59-3': 'Furthermore, we also find that [EQUATION] and [EQUATION] with [EQUATION] and hence we can work in a gauge for which [EQUATION]', '0706.3147-2-59-4': 'It is then straightforward to prove the following identities: [EQUATION]', '0706.3147-2-59-5': 'There are then three cases to consider.', '0706.3147-2-60-0': 'It is then straightforward to show that [EQUATION] where [EQUATION] so that [EQUATION]', '0706.3147-2-60-1': 'Hence the 3-manifold with metric [MATH] is either [MATH], the Nil-manifold or [MATH] according as to whether [MATH], [MATH] or [MATH] respectively.', '0706.3147-2-61-0': '### Solutions with [MATH]', '0706.3147-2-62-0': 'If [MATH], then [MATH] and the scalars [MATH] are constant and [MATH] is constant.', '0706.3147-2-62-1': 'Without loss of generality, we set [MATH].', '0706.3147-2-62-2': 'The following constraints also hold: [EQUATION] so that [MATH] for these solutions.', '0706.3147-2-62-3': 'Furthermore, we also find [EQUATION] and the gauge field strengths are given by [EQUATION]', '0706.3147-2-62-4': 'Note also that [MATH] and [MATH] satisfy [EQUATION] where here [MATH] denotes the covariant derivative restricted to the base space, and [MATH], [MATH] are base space indices.', '0706.3147-2-63-0': 'It is convenient to define [EQUATION]', '0706.3147-2-63-1': 'Then [EQUATION] and [EQUATION] where [MATH] is defined by [EQUATION]', '0706.3147-2-63-2': 'Note that ([REF]) implies that [EQUATION] and hence [EQUATION] where [MATH] is fixed by comparing the integrability condition associated with ([REF]) with the expression for the gauge field strengths ([REF]).', '0706.3147-2-63-3': 'We find [EQUATION]', '0706.3147-2-63-4': 'Next we define [EQUATION] so that [EQUATION]', '0706.3147-2-63-5': 'We also define [EQUATION]', '0706.3147-2-63-6': 'Then [MATH] form an orthonormal time-independent basis for the Kahler base space, such that [EQUATION] and [EQUATION]', '0706.3147-2-63-7': 'Now note that [EQUATION]', '0706.3147-2-63-8': 'It therefore follows that there exist functions [MATH] and co-ordinates [MATH] such that [EQUATION]', '0706.3147-2-63-9': 'Suppose that the remaining co-ordinates on the base are [MATH].', '0706.3147-2-63-10': 'As [MATH] and [MATH] are orthogonal to [MATH], there exist functions [MATH] such that [EQUATION] for [MATH].', '0706.3147-2-64-0': 'Similarly, as [EQUATION] it also follows that there exist functions [MATH], [MATH] and [MATH] for [MATH] such that [EQUATION] for [MATH].', '0706.3147-2-64-1': 'As [MATH] form an orthonormal basis for the base space, we can without loss of generality take [MATH] to be co-ordinates on the base space.', '0706.3147-2-65-0': 'In principle, the functions [MATH] can depend on all the co-ordinates.', '0706.3147-2-65-1': 'However, note that [EQUATION]', '0706.3147-2-65-2': 'Imposing the constraint [MATH] thus gives the conditions [EQUATION]', '0706.3147-2-65-3': 'One therefore can set [EQUATION] with [MATH] satisfying [EQUATION]', '0706.3147-2-65-4': 'We also set [EQUATION] where [MATH] satisfies [EQUATION]', '0706.3147-2-65-5': 'Here [MATH] and [MATH] are functions of [MATH].', '0706.3147-2-65-6': 'Next we define [EQUATION]', '0706.3147-2-65-7': 'Note that [MATH] are an orthonormal basis of the Kahler base with the property that [EQUATION] and [EQUATION] with [EQUATION]', '0706.3147-2-65-8': 'These constraints therefore imply that [EQUATION]', '0706.3147-2-65-9': 'Hence the Kahler base is a product of two 2-manifolds [MATH], [MATH], with metric [EQUATION]', '0706.3147-2-65-10': 'Taking the orthonormal basis [MATH], the metrics on [MATH] and [MATH] are [EQUATION]', '0706.3147-2-65-11': 'It is then straightforward to compute the curvature in this basis.', '0706.3147-2-65-12': 'We find that the only non-vanishing components are fixed by [EQUATION]', '0706.3147-2-65-13': 'Therefore we conclude that [MATH] is [MATH], and [MATH] is [MATH], [MATH] or [MATH] depending on whether [MATH], [MATH] or [MATH] respectively.', '0706.3147-2-66-0': '## Solutions with [MATH] and [MATH]', '0706.3147-2-67-0': 'In the next case, we shall assume that [MATH] and [MATH] do not both vanish, and define the vector fields [MATH], [MATH] on the Kahler base via [EQUATION]', '0706.3147-2-67-1': 'Then note that [MATH] and [MATH], and ([REF]), ([REF]) and ([REF]) can be rewritten in terms of [MATH] and [MATH] as [EQUATION]', '0706.3147-2-67-2': 'Therefore we find that [MATH] is a holomorphic Killing vector on the base and satisfies [EQUATION]', '0706.3147-2-67-3': 'Moreover, [MATH] preserves the complex structure [EQUATION]', '0706.3147-2-67-4': 'In contrast, [MATH] defines a closed 1-form on the base, which is conformally Killing with [EQUATION]', '0706.3147-2-67-5': 'Here [MATH] denotes the metric of the Kahler base.', '0706.3147-2-67-6': 'From ([REF] ) it is clear that [MATH] and [MATH] commute, so that locally one can choose co-ordinates [MATH], [MATH] so that [EQUATION] and let the remaining two co-ordinates of the base space be [MATH], [MATH].', '0706.3147-2-68-0': 'Note that [MATH] and [MATH] satisfy [EQUATION]', '0706.3147-2-68-1': 'Then, one can write the metric on the Kahler base locally as [EQUATION] where [MATH], [MATH] and [MATH].', '0706.3147-2-68-2': 'By making a co-ordinate transformation of the form [EQUATION] one can without loss of generality set [MATH] in ([REF]) and drop primes throughout.', '0706.3147-2-69-0': 'Then it is straightforward to show that the condition [MATH] implies that [MATH], so that [EQUATION] where [MATH], and [EQUATION]', '0706.3147-2-69-1': 'The necessary and sufficient condition in order for [MATH] to be a covariantly constant complex structure is [EQUATION]', '0706.3147-2-69-2': 'In fact, we can take [MATH] without loss of generality, (this can be obtained, if necessary, by making the re-definition [MATH] and [MATH]).', '0706.3147-2-69-3': 'So [EQUATION]', '0706.3147-2-69-4': 'In addition, ([REF]) and ([REF]) can be rewritten as [EQUATION]', '0706.3147-2-69-5': 'The real portions of ([REF]) imply that [EQUATION] and hence [EQUATION]', '0706.3147-2-69-6': 'Therefore [EQUATION] for some function [MATH].', '0706.3147-2-69-7': 'It should be noted that although the K ahler metric [MATH] has a conformal dependence on [MATH], the portion of the metric [MATH] which appears in the five dimensional metric does not depend on either [MATH] or [MATH].', '0706.3147-2-70-0': 'In order to examine the behaviour of the scalars, note that ([REF]) implies that [EQUATION]', '0706.3147-2-70-1': 'To proceed further, we introduce the following holomorphic basis for the K ahler base space [EQUATION] with [MATH], [MATH] obtained by complex conjugation.', '0706.3147-2-70-2': 'It is straightforward to show that in this basis [EQUATION] as expected, and [EQUATION]', '0706.3147-2-70-3': 'Using this basis, one can compute explicitly the following components of the spin connection: [EQUATION]', '0706.3147-2-71-0': 'Moreover, it is straightforward to show that the imaginary portion of ([REF]) implies that [EQUATION] and [EQUATION]', '0706.3147-2-71-1': 'The imaginary parts of ([REF]) and ([REF]) can then be rewritten as [EQUATION]', '0706.3147-2-71-2': 'Note that as the [MATH] do not all vanish in the gauged theory, it follows that the imaginary parts of [MATH] and [MATH] do not depend on [MATH] , and depend on [MATH] via the factor [MATH].', '0706.3147-2-71-3': 'It is therefore convenient to define [EQUATION] and rewrite ([REF]), ([REF]) and ([REF]) as [EQUATION] and [EQUATION]', '0706.3147-2-71-4': 'We next consider the constraints ([REF]), ([REF]) and ([REF]).', '0706.3147-2-71-5': 'These are equivalent to [EQUATION] and [EQUATION]', '0706.3147-2-71-6': 'Note that ([REF]) implies the integrability condition associated with ([REF]), and ([REF]) implies that [MATH] and [MATH] satisfy the Cauchy-Riemann equations.', '0706.3147-2-71-7': 'Hence, [MATH] is a holomorphic function of [MATH].', '0706.3147-2-72-0': 'The components of [MATH] are also fixed by ([REF]), ([REF]) and ([REF]) to be [EQUATION] with the remaining components determined by complex conjugation; this exhausts the content of ([REF]), ([REF]) and ([REF]).', '0706.3147-2-72-1': 'Using these components, it is straightforward to compute [MATH] in the co-ordinate basis; we find [EQUATION]', '0706.3147-2-72-2': 'Using ([REF]) and ([REF]), the gauge field strengths for these solutions can be written as [EQUATION] which satisfy [MATH] automatically.', '0706.3147-2-73-0': 'Next consider the integrability condition associated with ([REF]): this can be written as [EQUATION]', '0706.3147-2-73-1': 'This can be evaluated to give the constraint [EQUATION] where [MATH] is the Laplacian on [MATH].', '0706.3147-2-73-2': 'In fact, this constraint enables ([REF]) to be solved for [MATH] (up to a total derivative); we find [EQUATION] and the constraint ([REF]) implies that the scalars [MATH] are given by [EQUATION] for constant [MATH].', '0706.3147-2-74-0': 'Lastly, consider the equations ([REF]) and ([REF] ).', '0706.3147-2-74-1': 'These are equivalent to [EQUATION] and [EQUATION] where [MATH] denotes the restriction of the exterior derivative to hypersurfaces of constant [MATH].', '0706.3147-2-74-2': 'The integrability conditions of these two equations are [EQUATION] and [EQUATION] which hold automatically.', '0706.3147-2-75-0': '## Solutions with [MATH]', '0706.3147-2-76-0': 'We now turn to the class of solutions with [MATH].', '0706.3147-2-76-1': 'It is clear that ([REF]), ([REF]) and ([REF]) imply that [MATH] and [MATH] are covariantly constant.', '0706.3147-2-76-2': 'In particular, this implies that [MATH] is constant and without loss of generality we can set [MATH].', '0706.3147-2-76-3': 'One can choose co-ordinates [MATH], [MATH] so that locally [EQUATION] and two additional co-ordinates [MATH], [MATH] can be chosen on the base so that the Kahler metric takes the form [EQUATION] where [MATH].', '0706.3147-2-77-0': 'Recall that [MATH] is anti-self-dual, so taking positive orientation on the base with respect to [MATH], we have [EQUATION]', '0706.3147-2-77-1': 'This complex structure is automatically covariantly constant.', '0706.3147-2-77-2': 'Then the real portions of ([REF]) and ([REF]) imply that [EQUATION] so that [MATH] is only a function of [MATH] and [MATH].', '0706.3147-2-77-3': 'Also, as in the previous case, the real part of ([REF]) and ([REF]) implies that [EQUATION] so [EQUATION]', '0706.3147-2-77-4': 'It is convenient to define [EQUATION]', '0706.3147-2-77-5': 'Then the remaining portions of ([REF]), ([REF]), ( [REF]) and ([REF]) can be rewritten as [EQUATION]', '0706.3147-2-77-6': 'As the [MATH] do not all vanish, these constraints imply that [EQUATION]', '0706.3147-2-77-7': 'We take the following holomorphic basis for the Kahler base space: [EQUATION] with [MATH], [MATH] fixed by complex conjugation.', '0706.3147-2-78-0': 'In this basis, we obtain the following spin connection components: [EQUATION] and the components of [MATH] and [MATH] are: [EQUATION]', '0706.3147-2-78-1': 'To proceed, we turn to the constraint ([REF]).', '0706.3147-2-78-2': 'This implies that [EQUATION] and [EQUATION]', '0706.3147-2-78-3': 'The constraints ([REF]) and ([REF]) then imply that [EQUATION] together with [EQUATION] and the constraints [EQUATION] and [EQUATION]', '0706.3147-2-78-4': 'Just as in the previous section, these constraints imply that [MATH] and [MATH] satisfy the Cauchy-Riemann equations; [MATH] is a holomorphic function of [MATH] .', '0706.3147-2-79-0': 'In these co-ordinates [MATH] takes the form [EQUATION]', '0706.3147-2-79-1': 'The gauge field strengths for these solutions can be written as [EQUATION] which automatically satisfy the Bianchi identity [MATH].', '0706.3147-2-80-0': 'Finally the integrability condition associated with ([REF]) can be written as [EQUATION]', '0706.3147-2-80-1': 'It is then straightforward to show that these constraints imply that the integrability condition [MATH] associated with the expression in ( [REF]) holds automatically.', '0706.3147-2-80-2': 'Lastly, the constraints ([REF]) and ([REF]) fix [MATH] and [MATH] in terms of constant linear combinations of [MATH] and [MATH]; these conditions do not impose any further constraints on the geometry.', '0706.3147-2-81-0': '# Summary and Discussion', '0706.3147-2-82-0': 'In this paper we have employed the spinorial geometry method for the task of classifying [MATH] supersymmetric solutions with at least one time-like Killing spinor of the theory of [MATH] supergravity.', '0706.3147-2-82-1': 'Our results provide a general framework for the explicit construction of many new black holes and the investigation of their physical properties and relevance to AdS/CFT correspondence and holography.', '0706.3147-2-83-0': 'In general, supersymmetric solutions in five dimensional theories must preserve either [MATH], [MATH], [MATH] or [MATH] of the supersymmetries.', '0706.3147-2-83-1': 'This is because the Killing spinor equations are linear over [MATH].', '0706.3147-2-83-2': 'However in the ungauged theories, it was found that supersymmetric solutions can only preserve [MATH] or [MATH] of supersymmetries [CITATION].', '0706.3147-2-83-3': 'Moreover, to find time-like supersymmetric solutions in the ungauged theory, one must solve the gauge equations and the Bianchi identities in addition to the Killing spinor equations.', '0706.3147-2-83-4': 'In the null case one must additionally solve one of the components of the Einstein equations of motion.', '0706.3147-2-83-5': 'A similar situation arises for [MATH] supersymmetric solutions in the gauged theory.', '0706.3147-2-83-6': 'However, for the solutions with [MATH] supersymmetry considered in our present work, we have demonstrated that if one of the Killing spinors is time-like, then supersymmetry and Bianchi identities alone imply that all components of Einstein and gauge equations together with the scalar equations are automatically satisfied.', '0706.3147-2-84-0': 'Maximally supersymmetric solutions (preserving all 8 of the supersymmetries) of the five dimensional gauged supergravity theory have vanishing gauge field strengths and constant scalars and are locally isometric to [MATH].', '0706.3147-2-84-1': 'Moreover, it has been demonstrated in [CITATION] that all solutions of [MATH], [MATH] supergravity preserving [MATH] of supersymmetry must be locally isometric to [MATH], with vanishing gauge field strengths and constant scalars.', '0706.3147-2-84-2': 'An analogous situation also arises in the case of [MATH] supergravity, where it has been shown that all solutions with [MATH] supersymmetry must be locally isometric to a maximally supersymmetric solution [CITATION].', '0706.3147-2-84-3': 'However, in the case of [MATH] supergravity, it has been shown that one cannot obtain [MATH] supersymmetric solutions by taking quotients of the maximally supersymmetric solutions [CITATION].', '0706.3147-2-84-4': 'In contrast, there is a [MATH]-supersymmetric supersymmetric solution of [MATH], [MATH] gauged supergravity which is obtained by taking a certain quotient of [MATH] [CITATION].', '0706.3147-2-85-0': 'One future direction is the completion of the classification of supersymmetric solutions in [MATH], [MATH] supergravity by classifying [MATH] supersymmetric solutions with two null Killing spinors (i.e., two Killing spinors with associated null Killing vectors).', '0706.3147-2-85-1': 'In addition, it would be interesting to investigate whether there are any regular asymptotically [MATH] black ring solutions (see [CITATION] for a recent discussion).', '0706.3147-2-85-2': 'Supersymmetric black rings are known to exist in the ungauged theory [CITATION].', '0706.3147-2-85-3': 'For asymptotically flat supersymmetric black rings of [MATH], [MATH] ungauged supergravity, supersymmetry is enhanced from [MATH] supersymmetry to maximal supersymmetry at the horizon.', '0706.3147-2-85-4': 'If there do exist [MATH] black rings in the gauged theory, it may be reasonable to expect that the supersymmetry will be enhanced from [MATH] to [MATH] at the horizon.', '0706.3147-2-85-5': 'We hope that the classification of [MATH] supersymmetric solutions can provide a method of determining whether there exists a [MATH] supersymmetric solution corresponding to the near-horizon geometry of a black ring.', '0706.3147-2-85-6': 'It should be noted that in [CITATION], black rings with two [MATH] symmetries have already been excluded.', '0706.3147-2-85-7': 'However, as we have seen, [MATH] supersymmetric solutions in general only have one additional [MATH] symmetry; so more general ring solutions may be possible.', '0706.3147-2-86-0': 'W. Sabra would like to thank Queen Mary University of London and Khuri Lab at Rockefeller University for hospitality during which some of this work was completed.', '0706.3147-2-86-1': 'The work of W. Sabra was supported in part by the National Science Foundation under grant number PHY-0601213.', '0706.3147-2-87-0': '# Systematic treatment of the Killing spinor equation', '0706.3147-2-88-0': 'In this appendix we will evaluate the linear system obtained from the Killing spinor acting on [MATH] given in ([REF]), keeping the parameters arbitrary.', '0706.3147-2-88-1': 'It would naively appear that we have to evaluate two sets of Killing spinor equations, according to the two choices of symplectic index [MATH].', '0706.3147-2-88-2': 'However, making use of the symplectic Majorana condition, together with the fact that the gauge field strengths and scalars are real, it is straightforward to show that it suffices just to consider the case when [MATH], the [MATH] equations are then implied automatically.', '0706.3147-2-88-3': 'In the following, it will be convenient to define [MATH] and [MATH].', '0706.3147-2-89-0': 'From the dilatino equation we obtain [EQUATION]', '0706.3147-2-89-1': 'Then from the gravitino part of the Killing spinor equations we obtain the following constraints:', '0706.3147-2-90-0': 'From along the [MATH]-direction of the supercovariant derivative- [EQUATION]', '0706.3147-2-90-1': 'From along the [MATH]-direction of the supercovariant derivative- [EQUATION]', '0706.3147-2-90-2': 'From along the [MATH]-direction of the supercovariant derivative- [EQUATION]', '0706.3147-2-90-3': 'Throughout these equations, spatial indices of [MATH], [MATH], [MATH], [MATH] and [MATH] have been raised with [MATH].', '0706.3147-2-91-0': '# Integrability Conditions and Equations of Motion', '0706.3147-2-92-0': 'In this appendix we examine the integrability conditions of the Killing spinor equations.', '0706.3147-2-92-1': 'It will be shown that, if a background preserves at least half of the supersymmetry, and admits a Killing spinor for which the associated Killing vector is time-like, and the Bianchi identity holds, then all components of the Einstein, gauge and scalar equations hold automatically.', '0706.3147-2-93-0': 'First we consider the integrability condition associated with the gravitino equation ([REF]).', '0706.3147-2-93-1': 'After some gamma matrix manipulation we find [EQUATION]', '0706.3147-2-93-2': 'Next consider the dilatino equation ([REF]).', '0706.3147-2-93-3': 'This gives the integrability condition [EQUATION]', '0706.3147-2-93-4': 'It will be convenient to define [EQUATION] so that [MATH], [MATH] and [MATH] correspond to the Einstein, gauge field and scalar equations of motion respectively.', '0706.3147-2-94-0': 'To proceed we act on the gravitino integrability condition ([REF]) from the left with [MATH] and contract over the index [MATH].', '0706.3147-2-94-1': 'We assume that the Bianchi identity [MATH] holds.', '0706.3147-2-94-2': 'After some considerable gamma matrix manipulation (and making use of ([REF]) to simplify the expressions further), we find the constraint [EQUATION]', '0706.3147-2-94-3': 'Also, on contracting the dilatino integrability condition ([REF]) with [MATH], and again assuming the Bianchi identity [MATH] holds, we find [EQUATION]', '0706.3147-2-94-4': 'To proceed, we evaluate the constraints ([REF]) and ([REF]) on a background which preserves at least half of the supersymmetry, and which admits a Killing spinor for which the associated Killing vector is time-like.', '0706.3147-2-94-5': 'In particular, we first consider a generic Killing spinor [EQUATION]', '0706.3147-2-94-6': 'Substituting this expression into ([REF]) for [MATH] gives the constraints [EQUATION]', '0706.3147-2-94-7': 'Evaluating these constraints on the canonical form of the [MATH] time-like Killing spinor by setting [MATH] we obtain the constraints [EQUATION]', '0706.3147-2-94-8': 'Now substitute these expressions back into ([REF]) and eliminate the [MATH] terms to find [EQUATION]', '0706.3147-2-94-9': 'Assuming that the background is at least half-supersymmetric, we take [MATH], and hence find [EQUATION] and [EQUATION]', '0706.3147-2-94-10': 'Next consider ([REF]) for [MATH] evaluated on the generic spinor [MATH].', '0706.3147-2-94-11': 'Using the constraints [MATH] and [MATH] which we have already obtained, this expression simplifies to [EQUATION] from which we find the constraints [EQUATION] and taking ([REF]) with [MATH] we find [EQUATION]', '0706.3147-2-94-12': 'Evaluating these constraints on the canonical [MATH] time-like spinor by taking [MATH] we obtain the constraints [EQUATION]', '0706.3147-2-94-13': 'Hence we have shown that for solutions with at least half supersymmetry, the constraint ([REF]) implies that [EQUATION]', '0706.3147-2-94-14': 'Next consider the constraint ([REF]) obtained from the dilatino integrability conditions.', '0706.3147-2-94-15': 'On using [MATH] this constraint simplifies to [EQUATION]', '0706.3147-2-94-16': 'Evaluating this expression on the generic Killing spinor [MATH], one obtains [EQUATION]', '0706.3147-2-94-17': 'Evaluating these constraints on the canonical [MATH] time-like spinor by taking [MATH], the following conditions are obtained [EQUATION]', '0706.3147-2-94-18': 'Now substitute these constraints back into ([REF]) to find [EQUATION]', '0706.3147-2-94-19': 'Assuming that the background is at least half-supersymmetric, we take [MATH], and hence [EQUATION]', '0706.3147-2-94-20': 'Hence from the constraint ([REF]) we have found the constraints [EQUATION]'}

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'0706.3147-2-71-1'], ['0706.3147-1-72-2', '0706.3147-2-71-2'], ['0706.3147-1-72-3', '0706.3147-2-71-3'], ['0706.3147-1-72-4', '0706.3147-2-71-4'], ['0706.3147-1-72-5', '0706.3147-2-71-5'], ['0706.3147-1-72-6', '0706.3147-2-71-6'], ['0706.3147-1-72-7', '0706.3147-2-71-7'], ['0706.3147-1-79-0', '0706.3147-2-78-0'], ['0706.3147-1-79-1', '0706.3147-2-78-1'], ['0706.3147-1-79-2', '0706.3147-2-78-2'], ['0706.3147-1-79-3', '0706.3147-2-78-3'], ['0706.3147-1-79-4', '0706.3147-2-78-4'], ['0706.3147-1-86-0', '0706.3147-2-85-0'], ['0706.3147-1-86-1', '0706.3147-2-85-1'], ['0706.3147-1-86-2', '0706.3147-2-85-2'], ['0706.3147-1-86-3', '0706.3147-2-85-3'], ['0706.3147-1-86-4', '0706.3147-2-85-4'], ['0706.3147-1-86-5', '0706.3147-2-85-5'], ['0706.3147-1-86-6', '0706.3147-2-85-6'], ['0706.3147-1-86-7', '0706.3147-2-85-7'], ['0706.3147-1-19-0', '0706.3147-2-18-0'], ['0706.3147-1-19-1', '0706.3147-2-18-1'], ['0706.3147-1-19-2', '0706.3147-2-18-2'], ['0706.3147-1-19-3', '0706.3147-2-18-3'], 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'0706.3147-2-59-5'], ['0706.3147-1-34-0', '0706.3147-2-33-0'], ['0706.3147-1-34-1', '0706.3147-2-33-1'], ['0706.3147-1-34-2', '0706.3147-2-33-2'], ['0706.3147-1-69-0', '0706.3147-2-68-0'], ['0706.3147-1-69-1', '0706.3147-2-68-1'], ['0706.3147-1-69-2', '0706.3147-2-68-2'], ['0706.3147-1-39-0', '0706.3147-2-38-0'], ['0706.3147-1-39-1', '0706.3147-2-38-1'], ['0706.3147-1-42-0', '0706.3147-2-41-0'], ['0706.3147-1-42-1', '0706.3147-2-41-1'], ['0706.3147-1-42-2', '0706.3147-2-41-2'], ['0706.3147-1-7-0', '0706.3147-2-6-0'], ['0706.3147-1-7-1', '0706.3147-2-6-1'], ['0706.3147-1-7-2', '0706.3147-2-6-2'], ['0706.3147-1-7-3', '0706.3147-2-6-3'], ['0706.3147-1-7-4', '0706.3147-2-6-4'], ['0706.3147-1-7-5', '0706.3147-2-6-5'], ['0706.3147-1-7-6', '0706.3147-2-6-6'], ['0706.3147-1-7-7', '0706.3147-2-6-7'], ['0706.3147-1-7-8', '0706.3147-2-6-8'], ['0706.3147-1-7-9', '0706.3147-2-6-9'], ['0706.3147-1-7-10', '0706.3147-2-6-10']]

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[]

[]

[]

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0706.3147

hep-ph-0012175

{'hep-ph-0012175-1-0-0': 'In some supersymmetric extensions of the Standard Model fairly light superpartner of t-quark is predicted, which may form bound states (stoponiums) under certain conditions.', 'hep-ph-0012175-1-0-1': 'We estimate potentials of TESLA linear collider in search for stoponium, considering the basic electron-positron option and the [MATH] option (Photon Linear Collider - PLC).', 'hep-ph-0012175-1-1-0': 'It is found that PLC could be the best machine for discovery and study of these new narrow strong resonances.', 'hep-ph-0012175-1-1-1': 'It can produce thousands stoponiums per 100 fb[MATH] integrated [MATH] luminosity in the high energy peak.', 'hep-ph-0012175-1-1-2': 'In the case of scenarios when stoponium decays mainly into two gluons the signal/background ratio is about 1/4.', 'hep-ph-0012175-1-1-3': 'In addition the decay channel [MATH] into two lightest Higgs bosons could be seen with high significance.', 'hep-ph-0012175-1-1-4': 'Thus, several weeks run is sufficient for the discovery of stoponium, if its mass is approximately known (for example from observation of direct stops production at LHC).', 'hep-ph-0012175-1-1-5': 'Then, in MSSM scenarios with dominant [MATH] decay PLC shows excellent possibilities to discover bound state of stops, practically immediately after beginning of operating.', 'hep-ph-0012175-1-1-6': 'Thus, PLC has good possibilities to study strong interaction of top quark superpartners in nonperturbative regime.', 'hep-ph-0012175-1-2-0': 'The e[MATH]e[MATH] option also has good discovery prospects but only in the case of scenarios with dominant decay into two lightest Higgs bosons, with hundreds of events per 100 [MATH].', 'hep-ph-0012175-1-2-1': 'Interesting possibility appears in the case when the resonance is seated on 0.1% width luminosity peak - one could resolve the stoponium exited states.', 'hep-ph-0012175-1-3-0': '# Introduction', 'hep-ph-0012175-1-4-0': 'The broken supersymmetry is favorite among the different extensions of the Standard Model.', 'hep-ph-0012175-1-4-1': 'It can happen that superpartners of top-quarks (stops, [MATH]) are long-living enough to compose (colorless) bound states, stoponiums, denoted as [MATH] in what follows.', 'hep-ph-0012175-1-4-2': 'In this scenario experimental study of the corresponding resonances could provide precise value of stop mass and stoponium partial widths, consequently yielding precise values of various parameters of SUSY Lagrangian.', 'hep-ph-0012175-1-4-3': 'Then, if the difference between stop and LSP masses is very small, the search for stop evidence in collisions at high energy could be problematic.', 'hep-ph-0012175-1-4-4': 'Observation of stoponium bound states will be the signature of such models confirming the existence of stop.', 'hep-ph-0012175-1-5-0': 'There are theoretical motivations for stop to be fairly light.', 'hep-ph-0012175-1-5-1': 'First one appeals to the renormalization group behavior of soft mass terms.', 'hep-ph-0012175-1-5-2': 'Indeed, gauge couplings raise while Yukawa couplings reduce these terms when energy scale evolves down, with Yukawa contributions being very large for stop.', 'hep-ph-0012175-1-5-3': 'The next motivation concerns left-right mixing in squark sector, which is proportional to Yukawa coupling and decreases the mass of the lightest stop.', 'hep-ph-0012175-1-5-4': 'Therefore, light stop may appear in different SUSY models ( see, e.g., Refs. [CITATION] for examples in the frameworks of supergravity and gauge mediation).', 'hep-ph-0012175-1-5-5': 'Experimental bound on stoponium mass comes from searches for stop at LEP2 and TEVATRON.', 'hep-ph-0012175-1-5-6': 'The concrete number depends on the MSSM spectrum [CITATION]: lower bound is about 90 GeV for sneutrino masses larger than 45-50 GeV or for neutralino masses larger than 50 GeV (ALEPH), while CDF excludes stop mass up to 130 GeV for smaller sneutrino masses.', 'hep-ph-0012175-1-5-7': 'The limitation is weaker if stop and neutralino masses are degenerate, it is about 60 GeV (ALEPH).', 'hep-ph-0012175-1-6-0': 'Stoponium was studied in Refs. [CITATION] in detail, in particular its effective couplings and partial widths were calculated, and prospects to be discovered at LHC were estimated.', 'hep-ph-0012175-1-6-1': 'In Ref. [CITATION] it was briefly mentioned also the possibility to observe stoponiums in photon collisions, however, without analyzing this phenomenology.', 'hep-ph-0012175-1-6-2': 'The first look at the PLC prospects on the stoponium search was done in [CITATION].', 'hep-ph-0012175-1-7-0': 'Now, when main parameters of TESLA project are under technical discussion one should understand clearly signatures of stop bound states in [MATH] and [MATH] collisions, and compare potentials of these two options of future linear collider.', 'hep-ph-0012175-1-7-1': 'In present analysis we will use year integrated luminosity for the [MATH] option equal to 300 fb[MATH] for [MATH] GeV, and one should rescale this figure for lower collision energies approximately as [MATH] [CITATION].', 'hep-ph-0012175-1-7-2': 'Beamstrahlung and ISR will affect considerably the stoponium production rate due to a condition to tune the collision energy at the resonance point.', 'hep-ph-0012175-1-7-3': 'According to the current TESLA reference parameter set the average energy loss due to beamstrahlung is about 3.', 'hep-ph-0012175-1-7-4': 'The average energy loss due to ISR is about 5 [CITATION].', 'hep-ph-0012175-1-7-5': 'Correspondingly, the luminosity distribution has a characteristic width about 5.', 'hep-ph-0012175-1-8-0': 'However, stoponium is very narrow resonance, even more narrow than the initial beam energy spread.', 'hep-ph-0012175-1-8-1': 'Its production rate is proportional to the differential luminosity, [MATH], at the [MATH] peak, where [MATH] is a collision energy for hard subprocess.', 'hep-ph-0012175-1-8-2': 'The initial (beam) energy spread for energies 200-500 GeV is about 0.07-0.1 and it is determined by the bunch compression system and undulator for the positron production.', 'hep-ph-0012175-1-8-3': 'The fraction of the luminosity in this [MATH]0.1 peak is determined by the ISR and beamstrahlung.', 'hep-ph-0012175-1-8-4': 'The ISR leaves in this peak about 50 of the luminosity [CITATION].', 'hep-ph-0012175-1-8-5': 'The average number of beamstrahlung photons with the energy more than [MATH] is about [MATH] per electron for TESLA conditions.', 'hep-ph-0012175-1-8-6': 'Thus, the probability of the e[MATH]e[MATH] collision without such beamstrahlung photons can be estimated as [MATH].', 'hep-ph-0012175-1-8-7': 'The probability of events without any photons (ISR or beamstrahlung ones) with the energies greater 0.1 is [MATH].', 'hep-ph-0012175-1-8-8': 'So, about 17 of the luminosity is concentrated in [MATH]0.1 range.', 'hep-ph-0012175-1-8-9': 'The differential luminosity in this peak is higher than in 5 interval by a factor of [MATH] times (here factor 0.8 is due to ISR in the 5 region).', 'hep-ph-0012175-1-8-10': 'This peak could be very important factor for increasing significance of the resonance (when its mass is known), for measurement of its mass and even for resolving close excited states of the stoponium.', 'hep-ph-0012175-1-8-11': 'Note, however, that for a search for the stoponium this peak will not help, because in presence of large background the scanning time in some wide energy interval, required for the resonance observation, has the following dependence on the bin width [MATH] and the differential luminosity, [MATH] (here bin [MATH] corresponds to the effective width of the luminosity distribution in the peak).', 'hep-ph-0012175-1-8-12': 'So, the ratio of scanning times for the 5% and the 0.1% width peaks is [MATH], almost the same.', 'hep-ph-0012175-1-9-0': 'Photon colliders based on Compton backscattering of laser photons on high energy electrons has been proposed a long time ago [CITATION].', 'hep-ph-0012175-1-9-1': 'This option has been included in the TESLA Conceptual Design Report [CITATION] and work on the Technical Design Report is under way.', 'hep-ph-0012175-1-9-2': 'Since the CDR parameters of TESLA were changed and luminosities have grown both in e[MATH]e[MATH] and [MATH] collisions.', 'hep-ph-0012175-1-10-0': 'At the present workshop it was reported [CITATION] that PLC luminosity can be further increased by a factor of 2.5 due to possible decrease of the horizontal beam emittance at the TESLA damping ring (however this has not improved e[MATH]e[MATH] luminosity because it is restricted by collision effects).', 'hep-ph-0012175-1-10-1': 'At present the [MATH] luminosity within the 20% interval just below [MATH] could be about 40% of luminosity in the e[MATH]e[MATH] collisions (where [MATH]^+[MATH]^-[MATH] at [MATH] GeV).', 'hep-ph-0012175-1-10-2': 'In the analysis we will consider 60 fb[MATH] for PLC year luminosity at [MATH] GeV.', 'hep-ph-0012175-1-11-0': 'The most bright evidence of the narrow resonance is its direct s-channel production in [MATH] annihilation or [MATH] fusion.', 'hep-ph-0012175-1-11-1': 'One can note, that high powers of the coupling constants, [MATH], emerge in the squared matrix elements in the [MATH] annihilation into stoponium.', 'hep-ph-0012175-1-11-2': 'Indeed, [MATH] arises from two electroweak vertices, and [MATH] comes from squared derivative of the stoponium wave function (scalar stoponium can be created there only in P-wave by propagation of neutral vector particle: photon or [MATH] boson).', 'hep-ph-0012175-1-11-3': 'At the same time two powers of [MATH] are eliminated in the case of [MATH] fusion mechanism because the stoponium production can be proceeded in S-wave there.', 'hep-ph-0012175-1-11-4': 'This circumstance makes for the relative enhancement of the stoponium production rate at PLC in comparison with [MATH] option.', 'hep-ph-0012175-1-11-5': 'Two powers of [MATH] are eliminated also in the case of associated production of stoponiums, for example in the Higgs-like reactions [MATH] and [MATH].', 'hep-ph-0012175-1-11-6': 'Hereafter we denote stoponium as [MATH].', 'hep-ph-0012175-1-12-0': 'To complete this brief review of possible production mechanisms one can note that in hadron collisions the stoponium resonance production is available in S-wave through the gluon fusion.', 'hep-ph-0012175-1-12-1': 'Here the effective [MATH] vertex includes [MATH] and one can anticipate large stoponium cross sections.', 'hep-ph-0012175-1-12-2': 'However, main decay channel [MATH] is too dirty due to huge QCD [MATH] background.', 'hep-ph-0012175-1-12-3': 'Then, the most promising decay channel at LHC is [MATH] [CITATION], but in order to discover stoponiums one year of LHC operating at high luminosity is needed, or even more depending on SUSY scenario.', 'hep-ph-0012175-1-13-0': 'We consider stoponium mass range [MATH] GeV, which could be surely probed by first TESLA run.', 'hep-ph-0012175-1-13-1': 'It is worth to note that the same interval is not an exceptional case for SUSY models with stoponiums as a quasistationary state, as we discuss briefly in the next section.', 'hep-ph-0012175-1-14-0': 'Cross sections for various tree-level background processes were evaluated with the help of CompHEP package [CITATION].', 'hep-ph-0012175-1-15-0': '# Stop bound states', 'hep-ph-0012175-1-16-0': 'It is clear that gluons try to bind two stops as well as ordinary quarks.', 'hep-ph-0012175-1-16-1': 'The corresponding bound state can be described as a quasistationary system with energy levels [MATH]) and masses [MATH] similarly to quarkonium.', 'hep-ph-0012175-1-16-2': 'For stoponium mass [MATH] GeV the binding energies [MATH] are of order 1 GeV [CITATION].', 'hep-ph-0012175-1-16-3': 'This treatment is valid if the formation process (time scale [MATH]) is faster than destroying one.', 'hep-ph-0012175-1-17-0': 'Among destroying mechanisms the obvious ones are the stop decays: [MATH]LSP, [MATH] and [MATH].', 'hep-ph-0012175-1-17-1': 'At first, let us consider the third decay.', 'hep-ph-0012175-1-17-2': 'It proceeds only through loop diagrams, if Universal boundary condition on soft terms is imposed (that is motivated by the absence of FCNC).', 'hep-ph-0012175-1-17-3': 'So, partial width is highly reduced by a factor of [MATH] in comparison with the first two tree-level decay processes [CITATION].', 'hep-ph-0012175-1-17-4': 'The rates of latter decays depend on the parameters of the model.', 'hep-ph-0012175-1-17-5': 'As an example, in the framework of gravity mediation, where LSP is neutralino, these decays proceed at the tree level and the corresponding partial widths are of order O[MATH].', 'hep-ph-0012175-1-17-6': 'In the framework of models with gauge mediated supersymmetry breaking [CITATION], where LSP is gravitino, the first process is strongly suppressed by supersymmetry breaking scale, but remaining one has the same partial width as in gravity mediation.', 'hep-ph-0012175-1-17-7': 'Hence, the possibility of existence of stoponium is a subject of special study in each concrete model.', 'hep-ph-0012175-1-17-8': 'For instance, in models with the lightest chargino being mostly wino and the lightest stop being mostly right stop (i.e., [MATH]), decay into chargino is damped and stoponium could exist if [MATH], i.e., when the first decay channel is kinematically forbidden.', 'hep-ph-0012175-1-17-9': 'One can state that SUSY scenario, where tree-level decays, [MATH]LSP and [MATH], are somehow suppressed and, therefore, stop decay can not destroy the stoponium formation, is not an exceptional case.', 'hep-ph-0012175-1-18-0': 'Next destroying mechanism is related to the stop annihilation.', 'hep-ph-0012175-1-18-1': 'Here two gluon channel is always open with partial width about 1 MeV.', 'hep-ph-0012175-1-18-2': 'Generally the gluon channel is dominant.', 'hep-ph-0012175-1-18-3': 'However, for the certain choice of model parameters, partial width into two lightest Higgs bosons, [MATH], can be larger, increasing the stoponium total width by a factor of [MATH].', 'hep-ph-0012175-1-18-4': 'In Ref. [CITATION] these figures were analyzed and found that quasistationary description is valid for [MATH] GeV in models with forbidden stop tree-level decays and neutralino being mostly bino.', 'hep-ph-0012175-1-18-5': 'The worst case is a model with chargino and neutralino states are both higgsino-like.', 'hep-ph-0012175-1-18-6': 'Here the stoponium total width increases rapidly with [MATH] and quasistationary treatment fails for [MATH] GeV.', 'hep-ph-0012175-1-19-0': '# Stoponium in [MATH] collisions', 'hep-ph-0012175-1-20-0': 'Let us begin with study of stoponium events in photon-photon scattering.', 'hep-ph-0012175-1-20-1': 'The main effect associated with stoponium would be a direct resonance production, where stoponium is produced in spin 0 state.', 'hep-ph-0012175-1-20-2': 'The corresponding cross section is described by the Breit-Wigner formula (similarly to the light Higgs production discussed in Ref. [CITATION]) [EQUATION] where [MATH] is squared colision energy for hard subprocess, [MATH] is the stoponium partial width for the decay into state [MATH], [MATH] is stoponium total width, [MATH] are helicities of initial photons.', 'hep-ph-0012175-1-21-0': 'At photon colliders the width of the luminosity distribution is much wider than that of the stoponium.', 'hep-ph-0012175-1-21-1': 'After integration over the luminosity distribution we obtain the effective cross section [EQUATION]', 'hep-ph-0012175-1-21-2': 'The differential cross section, [MATH][MATH], at [MATH] can be estimated as [MATH][MATH] according to 15% width of the high energy luminosity peak (note, we define the [MATH] luminosity as the luminosity in the interval [MATH]%).', 'hep-ph-0012175-1-21-3': 'So the effective cross section is [EQUATION]', 'hep-ph-0012175-1-21-4': 'Photon beams are planned to be highly polarized.', 'hep-ph-0012175-1-21-5': 'Hence, as stoponium is a scalar the production cross section will be enhanced by factor two if initial photons have total helicity equal to zero.', 'hep-ph-0012175-1-21-6': 'Hereafter we assume initial total helicity 0 ([MATH]) in numerical estimates.', 'hep-ph-0012175-1-21-7': 'Finally, we parameterize the stoponium cross sections as follows [EQUATION]', 'hep-ph-0012175-1-21-8': 'One should take into account that squared stoponium wave function at the origin, attending in [MATH], scales as a square root of its mass [CITATION].', 'hep-ph-0012175-1-22-0': 'As it has been stressed above one can discuss two main variants of the SUSY models, one with dominant [MATH] decay mode and another with stoponium total width being saturated by [MATH] mode.', 'hep-ph-0012175-1-22-1': 'Let us make qualitative signal/background analysis for different decay channels within these two variants.', 'hep-ph-0012175-1-22-2': 'The signal significance can be evaluated by ratio [MATH] because one deals with resonance and background rate in the signal bin can be fixed as average cross section in neighboring bins (here [MATH] are numbers of signal and background events).', 'hep-ph-0012175-1-22-3': 'We used the results for stoponium width and branching ratios calculated in Ref. [CITATION] with some corrections [CITATION].', 'hep-ph-0012175-1-23-0': '1.', 'hep-ph-0012175-1-23-1': 'In the first scenario stoponium total width [MATH] MeV and photon-photon branching is [MATH].', 'hep-ph-0012175-1-23-2': 'By making use of Eq. ([REF]) one obtains the signal rate at the level of 110 fb for [MATH] GeV.', 'hep-ph-0012175-1-23-3': 'So, more than six thousand stoponiums will be produced per year if L[MATH] fb[MATH].', 'hep-ph-0012175-1-23-4': 'Background is two jet production, where subprocess [MATH] gives main contribution with very large unpolarized cross section, [MATH] pb, if optimal cut on the jet angle of 45[MATH] is applied.', 'hep-ph-0012175-1-23-5': 'However, production of fermions is suppressed in collisions of photons with the total helicity 0 [CITATION]: [EQUATION] where [MATH] is velocity of the quarks.', 'hep-ph-0012175-1-23-6': 'So, [MATH] pairs are produced only in collisions of photons with the total helicity 2.', 'hep-ph-0012175-1-23-7': 'This fact is used for suppression of similar background in the analysis of the Higgs production in [MATH] collisions [CITATION].', 'hep-ph-0012175-1-23-8': 'The corresponding luminosity spectra for total helicity 0 ([MATH]) and 2 ([MATH]) can be found elsewhere [CITATION].', 'hep-ph-0012175-1-23-9': 'In the region [MATH]10% near the maximum energy (detector resolution or intrinsic resolution of the PLC for two collinear jets) one can obtain the ratio of the luminosities [MATH], that assumes at least one order suppression of the [MATH] background.', 'hep-ph-0012175-1-23-10': 'Note that the cross section 6 pb for background corresponds to the case of unpolarized beams.', 'hep-ph-0012175-1-23-11': 'It is zero for collisions of photons with the total helicity 0 and is equal to 12 pb for the total helicity 2, therefore the remaining cross section is 1.2 pb.', 'hep-ph-0012175-1-23-12': 'Note that due to gluon emission [MATH] pairs can be produced even in collisions of photons with the total helicity 0.', 'hep-ph-0012175-1-23-13': 'Detailed studies have shown that with proper cuts this process is not important [CITATION] and contribution from resolved photons is also not significant [CITATION].', 'hep-ph-0012175-1-24-0': 'Furthermore, the cross section is proportional to the fourth power of the electric charge of quarks, so the main contribution is given by [MATH] and [MATH] quarks.', 'hep-ph-0012175-1-24-1': 'The later can be easily suppressed by the vertex detector.', 'hep-ph-0012175-1-24-2': 'This gives additional factor of 2 in the background suppression.', 'hep-ph-0012175-1-24-3': 'Additional improvement can give the detector energy resolution which is at least factor of 2 smaller than the width of the [MATH] luminosity peak.', 'hep-ph-0012175-1-24-4': 'All three methods give a suppression factor of 40.', 'hep-ph-0012175-1-24-5': 'The remaining background is about 0.3 pb, while the isotropic signal is smaller by factor 1.4 only if the cut on the jet angle of 45[MATH] is applied.', 'hep-ph-0012175-1-24-6': 'Hence in the scenario with dominant [MATH] channel the signal/background ratio is [MATH].', 'hep-ph-0012175-1-24-7': 'The signal significance is about 35 for 60 fb[MATH] and [MATH] GeV.', 'hep-ph-0012175-1-25-0': 'These figures can be related to the first year of PLC operation if the stoponium mass is known approximately, for example from the observation of the direct stops production at LHC.', 'hep-ph-0012175-1-25-1': 'In opposite case, for a search of a stoponium one should make scanning with the energy bin [MATH] 10%.', 'hep-ph-0012175-1-25-2': 'It is clear that stoponium can be found in this scenario during several month work in the whole energy region under discussion, 200-400 GeV.', 'hep-ph-0012175-1-26-0': 'For two photon channel the background process, [MATH], proceeds through one-loop diagrams, so the corresponding cross section is small, about 10 fb [CITATION].', 'hep-ph-0012175-1-26-1': 'One should note that the photon-photon invariant mass bin can be taken equal to [MATH] for CMS-like crystal electromagnetic calorimeter [CITATION].', 'hep-ph-0012175-1-26-2': 'Thus, for [MATH] GeV window the background rate can be estimated at the level of 1 fb.', 'hep-ph-0012175-1-26-3': 'The signal rate is [MATH] fb for [MATH] GeV, providing the signal significance about [MATH] for statistics 60 fb[MATH].', 'hep-ph-0012175-1-27-0': 'Some other decay channels in the framework of the first scenario should be discussed.', 'hep-ph-0012175-1-27-1': 'First note, that [MATH] final state has no chance for the detection of stoponiums due to huge SM background, [MATH] pb at [MATH] GeV.', 'hep-ph-0012175-1-27-2': 'More promising are decay channels with background processes emerging due to the higher order corrections from perturbation theory.', 'hep-ph-0012175-1-27-3': 'For instance, SM background to [MATH] and [MATH] final states comes from 1) one-loop [MATH] processes [MATH] fb [CITATION]) and [MATH] fb [CITATION]), and 2) from tree-level [MATH] processes (e.g. [MATH] for [MATH]), with total cross section smaller than 1 fb within cuts on final [MATH] and jets reasonably motivated by 2-body ([MATH]) kinematics of the signal events.', 'hep-ph-0012175-1-28-0': 'As to signal [MATH] rate one can get from Ref. [CITATION] the branching [MATH], so [MATH] fb already for [MATH] GeV.', 'hep-ph-0012175-1-28-1': 'It means very low level of the signal significance, lower than 0.5 for statistics 60 fb[MATH].', 'hep-ph-0012175-1-29-0': 'Natural level of [MATH] branching is about [MATH] for stoponium masses far from the threshold, 250-400 GeV, although in some points it could fall down due to opening of new channels or degeneration of stoponium and Higgs masses.', 'hep-ph-0012175-1-29-1': 'This provides signal rate [MATH] fb for [MATH] GeV and significance at the level of 2.7 if one uses formula ([REF]).', 'hep-ph-0012175-1-29-2': 'However, the threshold effect is significant still for this value of the stoponium mass, and these figures should be improved to 1.8 fb for signal rate and to 2 for the significance.', 'hep-ph-0012175-1-30-0': 'The [MATH] decay channel, where [MATH] is the lightest Higgs boson, is open if [MATH].', 'hep-ph-0012175-1-30-1': 'As current limit on [MATH] mass is about 80-100 GeV this channel could exist for [MATH] GeV.', 'hep-ph-0012175-1-30-2': 'If consider mass region not very close to the threshold (say [MATH] GeV for [MATH] GeV) the [MATH] branching is about [MATH] or even higher.', 'hep-ph-0012175-1-30-3': 'In this case the signal rate is about 1 fb or larger, if one uses formula ([REF]), and if take into account the threshold factor one gets signal cross section at the level of 0.2 fb or larger.', 'hep-ph-0012175-1-30-4': 'The background from direct double [MATH] production through one-loop diagrams can be estimated by the cross section of this process in SM, [MATH] fb [CITATION].', 'hep-ph-0012175-1-30-5': 'There are no reasons for very large additional contributions to this process in supersymmetric models.', 'hep-ph-0012175-1-30-6': 'Then, we found that direct electroweak production of four b quarks ([MATH]) together with contribution from [MATH] final state (assuming 10 of [MATH] misidentification) has the rate smaller than 0.1 fb.', 'hep-ph-0012175-1-30-7': 'These cross sections for background processes correspond to 15 width of the [MATH] luminosity spectrum.', 'hep-ph-0012175-1-30-8': 'The detector resolution is at least factor of two better (full width), therefore the total cross section of background processes can be estimated as (0.2+0.1)/2 = 0.15 fb.', 'hep-ph-0012175-1-30-9': 'We see that [MATH] decay can also be studied in the considered scenario.', 'hep-ph-0012175-1-30-10': 'For [MATH] GeV, [MATH] GeV and 60 fb[MATH] integrated luminosity about 20 stoponium events will be produced in the decay channel [MATH] with [MATH] ratio about 2.3 and statistical significance about 7.', 'hep-ph-0012175-1-31-0': 'In Fig. [REF] the signal significances are represented for main stoponium decay channels in the case of the first scenario.', 'hep-ph-0012175-1-31-1': 'As a resume for this scenario we conclude that stoponium can be found at PLC during several months scan of the 200-400 GeV region in [MATH] and [MATH] decay modes.', 'hep-ph-0012175-1-31-2': 'If its mass is known approximately, it will be found during first weeks.', 'hep-ph-0012175-1-31-3': 'More than a year is necessary in order to observe stoponium in [MATH] and [MATH] channels.', 'hep-ph-0012175-1-32-0': 'Note that LHC (at the high luminosity operating stage) has good prospects to observe light stoponium in [MATH] mode in this scenario [CITATION].', 'hep-ph-0012175-1-32-1': 'Thus, these two colliders could be complementary in study of different effective stoponium couplings, [MATH] and [MATH] at LHC and the photon collider, respectively and [MATH] at PLC if this decay channel is open kinematically.', 'hep-ph-0012175-1-33-0': '2.', 'hep-ph-0012175-1-33-1': 'In the second scenario stoponium total width could be about 10 MeV or even larger.', 'hep-ph-0012175-1-33-2': 'The photon-photon branching in this case is smaller, [MATH].', 'hep-ph-0012175-1-33-3': 'So, about thousand of stoponiums will be produced per year and almost all of them will decay to pairs of lightest Higgs bosons.', 'hep-ph-0012175-1-33-4': 'This result suggests, that stoponium will be discovered practically immediately after PLC start, since the background ([MATH], [MATH] ...) is very small.', 'hep-ph-0012175-1-33-5': 'Again the scanning over the energy interval is necessary if the stoponium mass is not known.', 'hep-ph-0012175-1-33-6': 'In Fig. [REF] the year yields of stoponiums are represented for different masses of lightest Higgs bosons.', 'hep-ph-0012175-1-34-0': 'Due to rather high statistics for the signal and absence (practically) of the background one gets PLC as a stoponium factory.', 'hep-ph-0012175-1-34-1': 'The detailed study of the stoponium characteristics will be available in this case, in particular measurement of its mass and total width and effective couplings [MATH] and [MATH].', 'hep-ph-0012175-1-34-2': 'One can stress the importance of study the [MATH] coupling, which relates directly to the stop-Higgs interaction and, thus, to the mechanism of the superpartner mass generation.', 'hep-ph-0012175-1-35-0': 'Note that in this scenario several years of operating at high luminosity is needed in order to observe stoponium at LHC [CITATION].', 'hep-ph-0012175-1-36-0': '# Stoponium in [MATH] collisions', 'hep-ph-0012175-1-37-0': '1.', 'hep-ph-0012175-1-37-1': 'First we discuss direct resonant production of stoponiums in [MATH] collisions where initial electron and positron would be in P-state (thus, stoponium is produced in spin 1 state).', 'hep-ph-0012175-1-37-2': 'As in case of [MATH] collisions one gets for narrow resonance after the integration of Breit-Wigner distribution the following formula for production cross section [EQUATION] where [MATH] are helicities of initial electrons and positrons.', 'hep-ph-0012175-1-37-3': 'The differential luminosity at [MATH] can be estimated as [MATH], where [MATH] is a fraction of the [MATH] luminosity corresponding to some interval [MATH] near the [MATH] peak.', 'hep-ph-0012175-1-37-4': 'Let us consider two methods of the stoponium detection:', 'hep-ph-0012175-1-38-0': '1) [MATH]% [MATH]^+[MATH]^-[MATH], [MATH]%;', 'hep-ph-0012175-1-39-0': '2) [MATH]% [MATH]^+[MATH]^-[MATH], [MATH]%.', 'hep-ph-0012175-1-40-0': 'Then, the stoponium partial width into [MATH] pair, [MATH], is equal to [CITATION] [EQUATION] with [MATH] being wave function of P-state.', 'hep-ph-0012175-1-40-1': 'In correspondence with Ref. [CITATION] one can approximate [MATH] GeV.', 'hep-ph-0012175-1-41-0': 'We assume, also, 80% circular polarization for the electron beam and 60% for the positron beam, and neglect fairly weak enhancement by [MATH]-boson pole for light stoponiums.', 'hep-ph-0012175-1-42-0': 'Finally, for the first method of the stoponium detection one gets for the production rate in [MATH] collisions the following estimate [EQUATION] while it is about 5 times larger in case of second method.', 'hep-ph-0012175-1-43-0': 'Let us now consider background for the direct resonance production of stoponiums in [MATH] collisions.', 'hep-ph-0012175-1-44-0': 'In the case of first MSSM scenario with [MATH] decay channel being dominant the main background is [MATH] process.', 'hep-ph-0012175-1-44-1': 'It has a cross section about 10 pb for [MATH] GeV.', 'hep-ph-0012175-1-44-2': 'So, this channel is too dirty with rather low significance, less than 1 for 100 fb[MATH] statistics.', 'hep-ph-0012175-1-44-3': 'In other channels the stoponium production has too small rate: few events per year in [MATH] and [MATH] channels.', 'hep-ph-0012175-1-45-0': 'In the case of second MSSM scenario with [MATH] dominant decay mode almost all stoponiums will decay into pair of lightest Higgs bosons, and the number of events for [MATH] GeV and 100 [MATH] is 50 (250) (the second number is for the case of seating on the 0.1% luminosity peak).', 'hep-ph-0012175-1-45-1': 'As direct pair production of Higgs bosons, [MATH], is negligible since the corresponding [MATH] coupling includes electron mass, this channel is should be practically free of background.', 'hep-ph-0012175-1-45-2': 'Indeed, direct production of four b-quarks in electron-positron collisions has very small cross section if exclude [MATH] peaks (less than 0.05fb if apply the cut [MATH] GeV).', 'hep-ph-0012175-1-45-3': 'So, stoponium can be easily discovered at [MATH] machine if the collision energy is tuned at [MATH].', 'hep-ph-0012175-1-46-0': 'Note, that in this scenario the yield of stoponiums at PLC with [MATH][MATH]^+[MATH]^-[MATH] is higher than that in e[MATH]e[MATH] collisions by a factor of 25(5) (second number for the case of seating on the 0.1% peak of the [MATH] luminosity).', 'hep-ph-0012175-1-46-1': 'Thus, the scanning time in e[MATH]e[MATH] mode will be about factor of 50 longer than at PLC due to smaller cross section and smaller energy bin (see also discussion on the scanning time in the Introduction).', 'hep-ph-0012175-1-47-0': '2.', 'hep-ph-0012175-1-47-1': 'The second effect related to stoponium at [MATH] collider would be a production in Higgs-like channels.', 'hep-ph-0012175-1-47-2': 'In present analysis we neglect Higgs-stoponium mixing as well as Higgs influence on stoponium-involved processes.', 'hep-ph-0012175-1-47-3': 'First, let us evaluate effective coupling constants between stoponium and weak bosons, [EQUATION] where we used stoponium wave function evaluated at the origin in Ref. [CITATION]; [MATH] and [MATH] are corresponding amplitudes of [MATH] processes evaluated in Ref. [CITATION].', 'hep-ph-0012175-1-47-4': 'Certainly, these effective couplings are obtained with all particles being on-shell, that is rather rough approximation.', 'hep-ph-0012175-1-47-5': 'However, one can hope that it is acceptable for the estimate.', 'hep-ph-0012175-1-48-0': 'The calculation with the effective couplings above gives the cross section about 0.03 fb for [MATH] (in case of no mixing in stop sector and [MATH] TeV).', 'hep-ph-0012175-1-48-1': 'So, only a few signal events will be produced for 100 fb[MATH] statistics.', 'hep-ph-0012175-1-48-2': 'The [MATH] background is too heavy for this level of the signal, so only the second scenario could have some prospects.', 'hep-ph-0012175-1-48-3': 'The main background will come from associated double Higgs bosons production, [MATH], the corresponding cross section is of order [MATH] fb for [MATH] [CITATION].', 'hep-ph-0012175-1-48-4': 'It gives the signal significance less than 1.', 'hep-ph-0012175-1-49-0': 'In the case of [MATH]-fusion the signal cross section is very small, less than [MATH] fb, that closes this channel.', 'hep-ph-0012175-1-50-0': '# Conclusions', 'hep-ph-0012175-1-51-0': 'In two considered scenarios of supersymmetric extension of the Standard Model (1st one with dominant [MATH] decay mode, and [MATH] being dominant in 2nd scenario) Photon Linear Collider will be the best machine to discover and study bound state of stops, if it exists.', 'hep-ph-0012175-1-52-0': 'In case of 1st scenario stoponium will be observed at PLC in the [MATH] and [MATH] (later if permitted kinematically) in the beginning of the operation - several months for scanning of whole energy region or several weeks if the stoponium mass is approximately known.', 'hep-ph-0012175-1-53-0': 'In case of 2nd scenario, about thousand of stoponiums will be produced free of background.', 'hep-ph-0012175-1-53-1': 'It means that stoponium can be discovered at PLC practically immediately.', 'hep-ph-0012175-1-54-0': 'Study of the effective stoponium couplings with photons, gluons and lightest Higgs bosons will be available at PLC, latter two depending on the MSSM scenario.', 'hep-ph-0012175-1-54-1': 'Measurement of the stoponium mass and total width (extracted from the measured signal rate) will be possible also.', 'hep-ph-0012175-1-55-0': 'Stoponium discovery mass range will be limited only by attainable values of the [MATH] collision energy, which is discussed up to [MATH] GeV.', 'hep-ph-0012175-1-55-1': 'The tuning of PLC collision energy at the resonance point is necessary within the 15% window.', 'hep-ph-0012175-1-55-2': 'These estimates were done for the case of PLC year luminosity being 60 fb[MATH].', 'hep-ph-0012175-1-56-0': 'In [MATH] collisions stoponium can be observed and studied only in the scenario with [MATH] decay channel being dominant, but with the rate lower than in the PLC case by a factor of 25 (5) (the second number for seating on the 0.1 luminosity peak).', 'hep-ph-0012175-1-56-1': 'In this comparison it was assumed that [MATH] luminosity in the high energy peak is equal 40 of e[MATH]e[MATH] luminosity.', 'hep-ph-0012175-1-56-2': 'Search time here is about 50 times longer than in [MATH] collisions.', 'hep-ph-0012175-1-56-3': 'However, there is one important advantage of e[MATH]e[MATH] collisions: by use of very monochromatic part of the luminosity spectrum (0.1%) one can make precise measurement of the stoponium mass and resolve its excited states.', 'hep-ph-0012175-1-56-4': 'Although this is possible only in the scenario when [MATH] decay dominates.', 'hep-ph-0012175-1-57-0': 'A few further comments can be made.', 'hep-ph-0012175-1-57-1': 'The first one is related to the circumstance that at photon colliders ground state of stoponium could not be distinguished from excited states due to the detector resolutions.', 'hep-ph-0012175-1-57-2': 'Therefore, the resonance peak will include contributions from ground state and all excitations, leading to enhancement factor of about 2 in all cross sections [CITATION].', 'hep-ph-0012175-1-57-3': 'At the same time there is a big uncertainty because of poor understanding of the stoponium wave function, that results in 30-50% error when the stoponium rates are estimated [CITATION].', 'hep-ph-0012175-1-58-0': 'Then, stoponium has the same quantum numbers as neutral Higgs bosons.', 'hep-ph-0012175-1-58-1': 'Thus, interesting phenomena could appear due to the interference of stoponium with Higgs sector.', 'hep-ph-0012175-1-58-2': 'This point was discussed briefly in Ref. [CITATION].', 'hep-ph-0012175-1-59-0': 'The authors are indebted to I.F. Ginzburg, H.J. Schreiber and other participants of the present Workshop for useful discussions.', 'hep-ph-0012175-1-59-1': 'The work of D.G. was supported in part under RFBR grant 99-01-18410, CRDF grant (award RP1-2103), Swiss Science Foundation grant 7SUPJ062239, Russian Academy of Science JRP grant 37 and by ISSEP fellowship.', 'hep-ph-0012175-1-59-2': 'The work of V.I. has been partially supported by the CERN-INTAS 99-377 and RFBR-DFG 99-02-04011 grants.'}

{'hep-ph-0012175-2-0-0': 'In some supersymmetric extensions of the Standard Model fairly light superpartner of t-quark is predicted, which may form bound states (stoponiums) under certain conditions.', 'hep-ph-0012175-2-0-1': 'We estimate potentials of TESLA linear collider in search for stoponium, considering the basic electron-positron option and the [MATH] option (Photon Linear Collider - PLC).', 'hep-ph-0012175-2-1-0': 'It is found that PLC could be the best machine for discovery and study of these new narrow strong resonances.', 'hep-ph-0012175-2-1-1': 'It can produce thousands stoponiums per 100 fb[MATH] integrated [MATH] luminosity in the high energy peak.', 'hep-ph-0012175-2-1-2': 'In the case of scenarios when stoponium decays mainly into two gluons the signal/background ratio is about 1/4.', 'hep-ph-0012175-2-1-3': 'In addition the decay channel [MATH] into two lightest Higgs bosons could be seen with high significance.', 'hep-ph-0012175-2-1-4': 'Thus, several weeks run is sufficient for the discovery of stoponium, if its mass is approximately known (for example from observation of direct stops production at LHC).', 'hep-ph-0012175-2-1-5': 'Then, in MSSM scenarios with dominant [MATH] decay PLC shows excellent possibilities to discover bound state of stops, practically immediately after beginning of operating.', 'hep-ph-0012175-2-1-6': 'Thus, PLC has good possibilities to study strong interaction of top quark superpartners in nonperturbative regime.', 'hep-ph-0012175-2-2-0': 'The e[MATH]e[MATH] option also has good discovery prospects but only in the case of scenarios with dominant decay into two lightest Higgs bosons, with hundreds of events per 100 [MATH].', 'hep-ph-0012175-2-2-1': 'Interesting possibility appears in the case when the resonance is seated on 0.1% width luminosity peak - one could resolve the stoponium exited states.', 'hep-ph-0012175-2-3-0': '# Introduction', 'hep-ph-0012175-2-4-0': 'The broken supersymmetry is favorite among the different extensions of the Standard Model.', 'hep-ph-0012175-2-4-1': 'It can happen that superpartners of top-quarks (stops, [MATH]) are long-living enough to compose (colorless) bound states, stoponiums, denoted as [MATH] in what follows.', 'hep-ph-0012175-2-4-2': 'In this scenario experimental study of the corresponding resonances could provide precise value of stop mass and stoponium partial widths, consequently yielding precise values of various parameters of SUSY Lagrangian.', 'hep-ph-0012175-2-4-3': 'Then, if the difference between stop and LSP masses is very small, the search for stop evidence in collisions at high energy could be problematic.', 'hep-ph-0012175-2-4-4': 'Observation of stoponium bound states will be the signature of such models confirming the existence of stop.', 'hep-ph-0012175-2-5-0': 'There are theoretical motivations for stop to be fairly light.', 'hep-ph-0012175-2-5-1': 'First one appeals to the renormalization group behavior of soft mass terms.', 'hep-ph-0012175-2-5-2': 'Indeed, gauge couplings raise while Yukawa couplings reduce these terms when energy scale evolves down, with Yukawa contributions being very large for stop.', 'hep-ph-0012175-2-5-3': 'The next motivation concerns left-right mixing in squark sector, which is proportional to Yukawa coupling and decreases the mass of the lightest stop.', 'hep-ph-0012175-2-5-4': 'Therefore, light stop may appear in different SUSY models ( see, e.g., Refs. [CITATION] for examples in the frameworks of supergravity and gauge mediation).', 'hep-ph-0012175-2-5-5': 'Experimental bound on stoponium mass comes from searches for stop at LEP2 and TEVATRON.', 'hep-ph-0012175-2-5-6': 'The concrete number depends on the MSSM spectrum [CITATION]: lower bound is about 90 GeV for sneutrino masses larger than 45-50 GeV or for neutralino masses larger than 50 GeV (ALEPH), while CDF excludes stop mass up to 130 GeV for smaller sneutrino masses.', 'hep-ph-0012175-2-5-7': 'The limitation is weaker if stop and neutralino masses are degenerate, it is about 60 GeV (ALEPH).', 'hep-ph-0012175-2-6-0': 'Stoponium was studied in Refs. [CITATION] in detail, in particular its effective couplings and partial widths were calculated, and prospects to be discovered at LHC were estimated.', 'hep-ph-0012175-2-6-1': 'In Ref. [CITATION] it was briefly mentioned also the possibility to observe stoponiums in photon collisions, however, without analyzing this phenomenology.', 'hep-ph-0012175-2-6-2': 'The first look at the PLC prospects on the stoponium search was done in [CITATION].', 'hep-ph-0012175-2-7-0': 'Now, when main parameters of TESLA project are under technical discussion one should understand clearly signatures of stop bound states in [MATH] and [MATH] collisions, and compare potentials of these two options of future linear collider.', 'hep-ph-0012175-2-7-1': 'In present analysis we will use year integrated luminosity for the [MATH] option equal to 300 fb[MATH] for [MATH] GeV, and one should rescale this figure for lower collision energies approximately as [MATH] [CITATION].', 'hep-ph-0012175-2-7-2': 'Beamstrahlung and ISR will affect considerably the stoponium production rate due to a condition to tune the collision energy at the resonance point.', 'hep-ph-0012175-2-7-3': 'According to the current TESLA reference parameter set the average energy loss due to beamstrahlung is about 3.', 'hep-ph-0012175-2-7-4': 'The average energy loss due to ISR is about 5 [CITATION].', 'hep-ph-0012175-2-7-5': 'Correspondingly, the luminosity distribution has a characteristic width about 5.', 'hep-ph-0012175-2-8-0': 'However, stoponium is very narrow resonance, even more narrow than the initial beam energy spread.', 'hep-ph-0012175-2-8-1': 'Its production rate is proportional to the differential luminosity, [MATH], at the [MATH] peak, where [MATH] is a collision energy for hard subprocess.', 'hep-ph-0012175-2-8-2': 'The initial (beam) energy spread for energies 200-500 GeV is about 0.07-0.1 and it is determined by the bunch compression system and undulator for the positron production.', 'hep-ph-0012175-2-8-3': 'The fraction of the luminosity in this [MATH]0.1 peak is determined by the ISR and beamstrahlung.', 'hep-ph-0012175-2-8-4': 'The ISR leaves in this peak about 50 of the luminosity [CITATION].', 'hep-ph-0012175-2-8-5': 'The average number of beamstrahlung photons with the energy more than [MATH] is about [MATH] per electron for TESLA conditions.', 'hep-ph-0012175-2-8-6': 'Thus, the probability of the e[MATH]e[MATH] collision without such beamstrahlung photons can be estimated as [MATH].', 'hep-ph-0012175-2-8-7': 'The probability of events without any photons (ISR or beamstrahlung ones) with the energies greater 0.1 is [MATH].', 'hep-ph-0012175-2-8-8': 'So, about 17 of the luminosity is concentrated in [MATH]0.1 range.', 'hep-ph-0012175-2-8-9': 'The differential luminosity in this peak is higher than in 5 interval by a factor of [MATH] times (here factor 0.8 is due to ISR in the 5 region).', 'hep-ph-0012175-2-8-10': 'This peak could be very important factor for increasing significance of the resonance (when its mass is known), for measurement of its mass and even for resolving close excited states of the stoponium.', 'hep-ph-0012175-2-8-11': 'Note, however, that for a search for the stoponium this peak will not help, because in presence of large background the scanning time in some wide energy interval, required for the resonance observation, has the following dependence on the bin width [MATH] and the differential luminosity, [MATH] (here bin [MATH] corresponds to the effective width of the luminosity distribution in the peak).', 'hep-ph-0012175-2-8-12': 'So, the ratio of scanning times for the 5% and the 0.1% width peaks is [MATH], almost the same.', 'hep-ph-0012175-2-9-0': 'Photon colliders based on Compton backscattering of laser photons on high energy electrons has been proposed a long time ago [CITATION].', 'hep-ph-0012175-2-9-1': 'This option has been included in the TESLA Conceptual Design Report [CITATION] and work on the Technical Design Report is under way.', 'hep-ph-0012175-2-9-2': 'Since the CDR parameters of TESLA were changed and luminosities have grown both in e[MATH]e[MATH] and [MATH] collisions.', 'hep-ph-0012175-2-10-0': 'At the present workshop it was reported [CITATION] that PLC luminosity can be further increased by a factor of 2.5 due to possible decrease of the horizontal beam emittance at the TESLA damping ring (however this has not improved e[MATH]e[MATH] luminosity because it is restricted by collision effects).', 'hep-ph-0012175-2-10-1': 'At present the [MATH] luminosity within the 20% interval just below [MATH] could be about 40% of luminosity in the e[MATH]e[MATH] collisions (where [MATH]^+[MATH]^-[MATH] at [MATH] GeV).', 'hep-ph-0012175-2-10-2': 'In the analysis we will consider 60 fb[MATH] for PLC year luminosity at [MATH] GeV.', 'hep-ph-0012175-2-11-0': 'The most bright evidence of the narrow resonance is its direct s-channel production in [MATH] annihilation or [MATH] fusion.', 'hep-ph-0012175-2-11-1': 'One can note, that high powers of the coupling constants, [MATH], emerge in the squared matrix elements in the [MATH] annihilation into stoponium.', 'hep-ph-0012175-2-11-2': 'Indeed, [MATH] arises from two electroweak vertices, and [MATH] comes from squared derivative of the stoponium wave function (scalar stoponium can be created there only in P-wave by propagation of neutral vector particle: photon or [MATH] boson).', 'hep-ph-0012175-2-11-3': 'At the same time two powers of [MATH] are eliminated in the case of [MATH] fusion mechanism because the stoponium production can be proceeded in S-wave there.', 'hep-ph-0012175-2-11-4': 'This circumstance makes for the relative enhancement of the stoponium production rate at PLC in comparison with [MATH] option.', 'hep-ph-0012175-2-11-5': 'Two powers of [MATH] are eliminated also in the case of associated production of stoponiums, for example in the Higgs-like reactions [MATH] and [MATH].', 'hep-ph-0012175-2-11-6': 'Hereafter we denote stoponium as [MATH].', 'hep-ph-0012175-2-12-0': 'To complete this brief review of possible production mechanisms one can note that in hadron collisions the stoponium resonance production is available in S-wave through the gluon fusion.', 'hep-ph-0012175-2-12-1': 'Here the effective [MATH] vertex includes [MATH] and one can anticipate large stoponium cross sections.', 'hep-ph-0012175-2-12-2': 'However, main decay channel [MATH] is too dirty due to huge QCD [MATH] background.', 'hep-ph-0012175-2-12-3': 'Then, the most promising decay channel at LHC is [MATH] [CITATION], but in order to discover stoponiums one year of LHC operating at high luminosity is needed, or even more depending on SUSY scenario.', 'hep-ph-0012175-2-13-0': 'We consider stoponium mass range [MATH] GeV, which could be surely probed by first TESLA run.', 'hep-ph-0012175-2-13-1': 'It is worth to note that the same interval is not an exceptional case for SUSY models with stoponiums as a quasistationary state, as we discuss briefly in the next section.', 'hep-ph-0012175-2-14-0': 'Cross sections for various tree-level background processes were evaluated with the help of CompHEP package [CITATION].', 'hep-ph-0012175-2-15-0': '# Stop bound states', 'hep-ph-0012175-2-16-0': 'It is clear that gluons try to bind two stops as well as ordinary quarks.', 'hep-ph-0012175-2-16-1': 'The corresponding bound state can be described as a quasistationary system with energy levels [MATH]) and masses [MATH] similarly to quarkonium.', 'hep-ph-0012175-2-16-2': 'For stoponium mass [MATH] GeV the binding energies [MATH] are of order 1 GeV [CITATION].', 'hep-ph-0012175-2-16-3': 'This treatment is valid if the formation process (time scale [MATH]) is faster than destroying one.', 'hep-ph-0012175-2-17-0': 'Among destroying mechanisms the obvious ones are the stop decays: [MATH]LSP, [MATH] and [MATH].', 'hep-ph-0012175-2-17-1': 'At first, let us consider the third decay.', 'hep-ph-0012175-2-17-2': 'It proceeds only through loop diagrams, if Universal boundary condition on soft terms is imposed (that is motivated by the absence of FCNC).', 'hep-ph-0012175-2-17-3': 'So, partial width is highly reduced by a factor of [MATH] in comparison with the first two tree-level decay processes [CITATION].', 'hep-ph-0012175-2-17-4': 'The rates of latter decays depend on the parameters of the model.', 'hep-ph-0012175-2-17-5': 'As an example, in the framework of gravity mediation, where LSP is neutralino, these decays proceed at the tree level and the corresponding partial widths are of order O[MATH].', 'hep-ph-0012175-2-17-6': 'In the framework of models with gauge mediated supersymmetry breaking [CITATION], where LSP is gravitino, the first process is strongly suppressed by supersymmetry breaking scale, but remaining one has the same partial width as in gravity mediation.', 'hep-ph-0012175-2-17-7': 'Hence, the possibility of existence of stoponium is a subject of special study in each concrete model.', 'hep-ph-0012175-2-17-8': 'For instance, in models with the lightest chargino being mostly wino and the lightest stop being mostly right stop (i.e., [MATH]), decay into chargino is damped and stoponium could exist if [MATH], i.e., when the first decay channel is kinematically forbidden.', 'hep-ph-0012175-2-17-9': 'One can state that SUSY scenario, where tree-level decays, [MATH]LSP and [MATH], are somehow suppressed and, therefore, stop decay can not destroy the stoponium formation, is not an exceptional case.', 'hep-ph-0012175-2-18-0': 'Next destroying mechanism is related to the stop annihilation.', 'hep-ph-0012175-2-18-1': 'Here two gluon channel is always open with partial width about 1 MeV.', 'hep-ph-0012175-2-18-2': 'Generally the gluon channel is dominant.', 'hep-ph-0012175-2-18-3': 'However, for the certain choice of model parameters, partial width into two lightest Higgs bosons, [MATH], can be larger, increasing the stoponium total width by a factor of [MATH].', 'hep-ph-0012175-2-18-4': 'In Ref. [CITATION] these figures were analyzed and found that quasistationary description is valid for [MATH] GeV in models with forbidden stop tree-level decays and neutralino being mostly bino.', 'hep-ph-0012175-2-18-5': 'The worst case is a model with chargino and neutralino states are both higgsino-like.', 'hep-ph-0012175-2-18-6': 'Here the stoponium total width increases rapidly with [MATH] and quasistationary treatment fails for [MATH] GeV.', 'hep-ph-0012175-2-19-0': '# Stoponium in [MATH] collisions', 'hep-ph-0012175-2-20-0': 'Let us begin with study of stoponium events in photon-photon scattering.', 'hep-ph-0012175-2-20-1': 'The main effect associated with stoponium would be a direct resonance production, where stoponium is produced in spin 0 state.', 'hep-ph-0012175-2-20-2': 'The corresponding cross section is described by the Breit-Wigner formula (similarly to the light Higgs production discussed in Ref. [CITATION]) [EQUATION] where [MATH] is squared colision energy for hard subprocess, [MATH] is the stoponium partial width for the decay into state [MATH], [MATH] is stoponium total width, [MATH] are helicities of initial photons.', 'hep-ph-0012175-2-21-0': 'At photon colliders the width of the luminosity distribution is much wider than that of the stoponium.', 'hep-ph-0012175-2-21-1': 'After integration over the luminosity distribution we obtain the effective cross section [EQUATION]', 'hep-ph-0012175-2-21-2': 'The differential luminosity, [MATH][MATH], at [MATH] can be estimated as [MATH][MATH] according to 15% width of the high energy luminosity peak (note, we define the [MATH] luminosity as the luminosity in the interval [MATH]%).', 'hep-ph-0012175-2-21-3': 'So the effective cross section is [EQUATION]', 'hep-ph-0012175-2-21-4': 'Photon beams are planned to be highly polarized.', 'hep-ph-0012175-2-21-5': 'Hence, as stoponium is a scalar the production cross section will be enhanced by factor two if initial photons have total helicity equal to zero.', 'hep-ph-0012175-2-21-6': 'Hereafter we assume initial total helicity 0 ([MATH]) in numerical estimates.', 'hep-ph-0012175-2-21-7': 'Finally, we parameterize the stoponium cross sections as follows [EQUATION]', 'hep-ph-0012175-2-21-8': 'One should take into account that squared stoponium wave function at the origin, attending in [MATH], scales as a square root of its mass [CITATION].', 'hep-ph-0012175-2-22-0': 'As it has been stressed above one can discuss two main variants of the SUSY models, one with dominant [MATH] decay mode and another with stoponium total width being saturated by [MATH] mode.', 'hep-ph-0012175-2-22-1': 'Let us make qualitative signal/background analysis for different decay channels within these two variants.', 'hep-ph-0012175-2-22-2': 'The signal significance can be evaluated by ratio [MATH] because one deals with resonance and background rate in the signal bin can be fixed as average cross section in neighboring bins (here [MATH] are numbers of signal and background events).', 'hep-ph-0012175-2-22-3': 'We used the results for stoponium width and branching ratios calculated in Ref. [CITATION] with some corrections [CITATION].', 'hep-ph-0012175-2-23-0': '1.', 'hep-ph-0012175-2-23-1': 'In the first scenario stoponium total width [MATH] MeV and photon-photon branching is [MATH].', 'hep-ph-0012175-2-23-2': 'By making use of Eq. ([REF]) one obtains the signal rate at the level of 110 fb for [MATH] GeV.', 'hep-ph-0012175-2-23-3': 'So, more than six thousand stoponiums will be produced per year if L[MATH] fb[MATH].', 'hep-ph-0012175-2-23-4': 'Background is two jet production, where subprocess [MATH] gives main contribution with very large unpolarized cross section, [MATH] pb, if optimal cut on the jet angle of 45[MATH] is applied.', 'hep-ph-0012175-2-23-5': 'However, production of fermions is suppressed in collisions of photons with the total helicity 0 [CITATION]: [EQUATION] where [MATH] is velocity of the quarks.', 'hep-ph-0012175-2-23-6': 'So, [MATH] pairs are produced only in collisions of photons with the total helicity 2.', 'hep-ph-0012175-2-23-7': 'This fact is used for suppression of similar background in the analysis of the Higgs production in [MATH] collisions [CITATION].', 'hep-ph-0012175-2-23-8': 'The corresponding luminosity spectra for total helicity 0 ([MATH]) and 2 ([MATH]) can be found elsewhere [CITATION].', 'hep-ph-0012175-2-23-9': 'In the region [MATH]10% near the maximum energy (detector resolution or intrinsic resolution of the PLC for two collinear jets) one can obtain the ratio of the luminosities [MATH], that assumes at least one order suppression of the [MATH] background.', 'hep-ph-0012175-2-23-10': 'Note that the cross section 6 pb for background corresponds to the case of unpolarized beams.', 'hep-ph-0012175-2-23-11': 'It is zero for collisions of photons with the total helicity 0 and is equal to 12 pb for the total helicity 2, therefore the remaining cross section is 1.2 pb.', 'hep-ph-0012175-2-23-12': 'Note that due to gluon emission [MATH] pairs can be produced even in collisions of photons with the total helicity 0.', 'hep-ph-0012175-2-23-13': 'Detailed studies have shown that with proper cuts this process is not important [CITATION] and contribution from resolved photons is also not significant [CITATION].', 'hep-ph-0012175-2-24-0': 'Furthermore, the cross section is proportional to the fourth power of the electric charge of quarks, so the main contribution is given by [MATH] and [MATH] quarks.', 'hep-ph-0012175-2-24-1': 'The later can be easily suppressed by the vertex detector.', 'hep-ph-0012175-2-24-2': 'This gives additional factor of 2 in the background suppression.', 'hep-ph-0012175-2-24-3': 'Additional improvement can give the detector energy resolution which is at least factor of 2 smaller than the width of the [MATH] luminosity peak.', 'hep-ph-0012175-2-24-4': 'All three methods give a suppression factor of 40.', 'hep-ph-0012175-2-24-5': 'The remaining background is about 0.3 pb, while the isotropic signal is smaller by factor 1.4 only if the cut on the jet angle of 45[MATH] is applied.', 'hep-ph-0012175-2-24-6': 'Hence in the scenario with dominant [MATH] channel the signal/background ratio is [MATH].', 'hep-ph-0012175-2-24-7': 'The signal significance is about 35 for 60 fb[MATH] and [MATH] GeV.', 'hep-ph-0012175-2-25-0': 'These figures can be related to the first year of PLC operation if the stoponium mass is known approximately, for example from the observation of the direct stops production at LHC.', 'hep-ph-0012175-2-25-1': 'In opposite case, for a search of a stoponium one should make scanning with the energy bin [MATH] 10%.', 'hep-ph-0012175-2-25-2': 'It is clear that stoponium can be found in this scenario during several month work in the whole energy region under discussion, 200-400 GeV.', 'hep-ph-0012175-2-26-0': 'For two photon channel the background process, [MATH], proceeds through one-loop diagrams, so the corresponding cross section is small, about 10 fb [CITATION].', 'hep-ph-0012175-2-26-1': 'One should note that the photon-photon invariant mass bin can be taken equal to [MATH] for CMS-like crystal electromagnetic calorimeter [CITATION].', 'hep-ph-0012175-2-26-2': 'Thus, for [MATH] GeV window the background rate can be estimated at the level of 1 fb.', 'hep-ph-0012175-2-26-3': 'The signal rate is [MATH] fb for [MATH] GeV, providing the signal significance about [MATH] for statistics 60 fb[MATH].', 'hep-ph-0012175-2-27-0': 'Some other decay channels in the framework of the first scenario should be discussed.', 'hep-ph-0012175-2-27-1': 'First note, that [MATH] final state has no chance for the detection of stoponiums due to huge SM background, [MATH] pb at [MATH] GeV.', 'hep-ph-0012175-2-27-2': 'More promising are decay channels with background processes emerging due to the higher order corrections from perturbation theory.', 'hep-ph-0012175-2-27-3': 'For instance, SM background to [MATH] and [MATH] final states comes from 1) one-loop [MATH] processes [MATH] fb [CITATION]) and [MATH] fb [CITATION]), and 2) from tree-level [MATH] processes (e.g. [MATH] for [MATH]), with total cross section smaller than 1 fb within cuts on final [MATH] and jets reasonably motivated by 2-body ([MATH]) kinematics of the signal events.', 'hep-ph-0012175-2-28-0': 'As to signal [MATH] rate one can get from Ref. [CITATION] the branching [MATH], so [MATH] fb already for [MATH] GeV.', 'hep-ph-0012175-2-28-1': 'It means very low level of the signal significance, lower than 0.5 for statistics 60 fb[MATH].', 'hep-ph-0012175-2-29-0': 'Natural level of [MATH] branching is about [MATH] for stoponium masses far from the threshold, 250-400 GeV, although in some points it could fall down due to opening of new channels or degeneration of stoponium and Higgs masses.', 'hep-ph-0012175-2-29-1': 'This provides signal rate [MATH] fb for [MATH] GeV and significance at the level of 2.7 if one uses formula ([REF]).', 'hep-ph-0012175-2-29-2': 'However, the threshold effect is significant still for this value of the stoponium mass, and these figures should be improved to 1.8 fb for signal rate and to 2 for the significance.', 'hep-ph-0012175-2-30-0': 'The [MATH] decay channel, where [MATH] is the lightest Higgs boson, is open if [MATH].', 'hep-ph-0012175-2-30-1': 'As current limit on [MATH] mass is about 80-100 GeV this channel could exist for [MATH] GeV.', 'hep-ph-0012175-2-30-2': 'If consider mass region not very close to the threshold (say [MATH] GeV for [MATH] GeV) the [MATH] branching is about [MATH] or even higher.', 'hep-ph-0012175-2-30-3': 'In this case the signal rate is about 1 fb or larger, if one uses formula ([REF]), and if take into account the threshold factor one gets signal cross section at the level of 0.2 fb or larger.', 'hep-ph-0012175-2-30-4': 'The background from direct double [MATH] production through one-loop diagrams can be estimated by the cross section of this process in SM, [MATH] fb [CITATION].', 'hep-ph-0012175-2-30-5': 'There are no reasons for very large additional contributions to this process in supersymmetric models.', 'hep-ph-0012175-2-30-6': 'Then, we found that direct electroweak production of four b quarks ([MATH]) together with contribution from [MATH] final state (assuming 10 of [MATH] misidentification) has the rate smaller than 0.1 fb.', 'hep-ph-0012175-2-30-7': 'These cross sections for background processes correspond to 15 width of the [MATH] luminosity spectrum.', 'hep-ph-0012175-2-30-8': 'The detector resolution is at least factor of two better (full width), therefore the total cross section of background processes can be estimated as (0.2+0.1)/2 = 0.15 fb.', 'hep-ph-0012175-2-30-9': 'We see that [MATH] decay can also be studied in the considered scenario.', 'hep-ph-0012175-2-30-10': 'For [MATH] GeV, [MATH] GeV and 60 fb[MATH] integrated luminosity about 20 stoponium events will be produced in the decay channel [MATH] with [MATH] ratio about 2.3 and statistical significance about 7.', 'hep-ph-0012175-2-31-0': 'In Fig. [REF] the signal significances are represented for main stoponium decay channels in the case of the first scenario.', 'hep-ph-0012175-2-31-1': 'As a resume for this scenario we conclude that stoponium can be found at PLC during several months scan of the 200-400 GeV region in [MATH] and [MATH] decay modes.', 'hep-ph-0012175-2-31-2': 'If its mass is known approximately, it will be found during first weeks.', 'hep-ph-0012175-2-31-3': 'More than a year is necessary in order to observe stoponium in [MATH] and [MATH] channels.', 'hep-ph-0012175-2-32-0': 'Note that LHC (at the high luminosity operating stage) has good prospects to observe light stoponium in [MATH] mode in this scenario [CITATION].', 'hep-ph-0012175-2-32-1': 'Thus, these two colliders could be complementary in study of different effective stoponium couplings, [MATH] and [MATH] at LHC and the photon collider, respectively and [MATH] at PLC if this decay channel is open kinematically.', 'hep-ph-0012175-2-33-0': '2.', 'hep-ph-0012175-2-33-1': 'In the second scenario stoponium total width could be about 10 MeV or even larger.', 'hep-ph-0012175-2-33-2': 'The photon-photon branching in this case is smaller, [MATH].', 'hep-ph-0012175-2-33-3': 'So, about thousand of stoponiums will be produced per year and almost all of them will decay to pairs of lightest Higgs bosons.', 'hep-ph-0012175-2-33-4': 'This result suggests, that stoponium will be discovered practically immediately after PLC start, since the background ([MATH], [MATH] ...) is very small.', 'hep-ph-0012175-2-33-5': 'Again the scanning over the energy interval is necessary if the stoponium mass is not known.', 'hep-ph-0012175-2-33-6': 'In Fig. [REF] the year yields of stoponiums are represented for different masses of lightest Higgs bosons.', 'hep-ph-0012175-2-34-0': 'Due to rather high statistics for the signal and absence (practically) of the background one gets PLC as a stoponium factory.', 'hep-ph-0012175-2-34-1': 'The detailed study of the stoponium characteristics will be available in this case, in particular measurement of its mass and total width and effective couplings [MATH] and [MATH].', 'hep-ph-0012175-2-34-2': 'One can stress the importance of study the [MATH] coupling, which relates directly to the stop-Higgs interaction and, thus, to the mechanism of the superpartner mass generation.', 'hep-ph-0012175-2-35-0': 'Note that in this scenario several years of operating at high luminosity is needed in order to observe stoponium at LHC [CITATION].', 'hep-ph-0012175-2-36-0': '# Stoponium in [MATH] collisions', 'hep-ph-0012175-2-37-0': '1.', 'hep-ph-0012175-2-37-1': 'First we discuss direct resonant production of stoponiums in [MATH] collisions where initial electron and positron would be in P-state (thus, stoponium is produced in spin 1 state).', 'hep-ph-0012175-2-37-2': 'As in case of [MATH] collisions one gets for narrow resonance after the integration of Breit-Wigner distribution the following formula for production cross section [EQUATION] where [MATH] are helicities of initial electrons and positrons.', 'hep-ph-0012175-2-37-3': 'The differential luminosity at [MATH] can be estimated as [MATH], where [MATH] is a fraction of the [MATH] luminosity corresponding to some interval [MATH] near the [MATH] peak.', 'hep-ph-0012175-2-37-4': 'Let us consider two methods of the stoponium detection:', 'hep-ph-0012175-2-38-0': '1) [MATH]% [MATH]^+[MATH]^-[MATH], [MATH]%;', 'hep-ph-0012175-2-39-0': '2) [MATH]% [MATH]^+[MATH]^-[MATH], [MATH]%.', 'hep-ph-0012175-2-40-0': 'Then, the stoponium partial width into [MATH] pair, [MATH], is equal to [CITATION] [EQUATION] with [MATH] being wave function of P-state.', 'hep-ph-0012175-2-40-1': 'In correspondence with Ref. [CITATION] one can approximate [MATH] GeV.', 'hep-ph-0012175-2-41-0': 'We assume, also, 80% circular polarization for the electron beam and 60% for the positron beam, and neglect fairly weak enhancement by [MATH]-boson pole for light stoponiums.', 'hep-ph-0012175-2-42-0': 'Finally, for the first method of the stoponium detection one gets for the production rate in [MATH] collisions the following estimate [EQUATION] while it is about 5 times larger in case of second method.', 'hep-ph-0012175-2-43-0': 'Let us now consider background for the direct resonance production of stoponiums in [MATH] collisions.', 'hep-ph-0012175-2-44-0': 'In the case of first MSSM scenario with [MATH] decay channel being dominant the main background is [MATH] process.', 'hep-ph-0012175-2-44-1': 'It has a cross section about 10 pb for [MATH] GeV.', 'hep-ph-0012175-2-44-2': 'So, this channel is too dirty with rather low significance, less than 1 for 100 fb[MATH] statistics.', 'hep-ph-0012175-2-44-3': 'In other channels the stoponium production has too small rate: few events per year in [MATH] and [MATH] channels.', 'hep-ph-0012175-2-45-0': 'In the case of second MSSM scenario with [MATH] dominant decay mode almost all stoponiums will decay into pair of lightest Higgs bosons, and the number of events for [MATH] GeV and 100 [MATH] is 50 (250) (the second number is for the case of seating on the 0.1% luminosity peak).', 'hep-ph-0012175-2-45-1': 'As direct pair production of Higgs bosons, [MATH], is negligible since the corresponding [MATH] coupling includes electron mass, this channel is should be practically free of background.', 'hep-ph-0012175-2-45-2': 'Indeed, direct production of four b-quarks in electron-positron collisions has very small cross section if exclude [MATH] peaks (less than 0.05fb if apply the cut [MATH] GeV).', 'hep-ph-0012175-2-45-3': 'So, stoponium can be easily discovered at [MATH] machine if the collision energy is tuned at [MATH].', 'hep-ph-0012175-2-46-0': 'Note, that in this scenario the yield of stoponiums at PLC with [MATH][MATH]^+[MATH]^-[MATH] is higher than that in e[MATH]e[MATH] collisions by a factor of 25(5) (second number for the case of seating on the 0.1% peak of the [MATH] luminosity).', 'hep-ph-0012175-2-46-1': 'Thus, the scanning time in e[MATH]e[MATH] mode will be about factor of 50 longer than at PLC due to smaller cross section and smaller energy bin (see also discussion on the scanning time in the Introduction).', 'hep-ph-0012175-2-47-0': '2.', 'hep-ph-0012175-2-47-1': 'The second effect related to stoponium at [MATH] collider would be a production in Higgs-like channels.', 'hep-ph-0012175-2-47-2': 'In present analysis we neglect Higgs-stoponium mixing as well as Higgs influence on stoponium-involved processes.', 'hep-ph-0012175-2-47-3': 'First, let us evaluate effective coupling constants between stoponium and weak bosons, [EQUATION] where we used stoponium wave function evaluated at the origin in Ref. [CITATION]; [MATH] and [MATH] are corresponding amplitudes of [MATH] processes evaluated in Ref. [CITATION].', 'hep-ph-0012175-2-47-4': 'Certainly, these effective couplings are obtained with all particles being on-shell, that is rather rough approximation.', 'hep-ph-0012175-2-47-5': 'However, one can hope that it is acceptable for the estimate.', 'hep-ph-0012175-2-48-0': 'The calculation with the effective couplings above gives the cross section about 0.03 fb for [MATH] (in case of no mixing in stop sector and [MATH] TeV).', 'hep-ph-0012175-2-48-1': 'So, only a few signal events will be produced for 100 fb[MATH] statistics.', 'hep-ph-0012175-2-48-2': 'The [MATH] background is too heavy for this level of the signal, so only the second scenario could have some prospects.', 'hep-ph-0012175-2-48-3': 'The main background will come from associated double Higgs bosons production, [MATH], the corresponding cross section is of order [MATH] fb for [MATH] [CITATION].', 'hep-ph-0012175-2-48-4': 'It gives the signal significance less than 1.', 'hep-ph-0012175-2-49-0': 'In the case of [MATH]-fusion the signal cross section is very small, less than [MATH] fb, that closes this channel.', 'hep-ph-0012175-2-50-0': '# Conclusions', 'hep-ph-0012175-2-51-0': 'In two considered scenarios of supersymmetric extension of the Standard Model (1st one with dominant [MATH] decay mode, and [MATH] being dominant in 2nd scenario) Photon Linear Collider will be the best machine to discover and study bound state of stops, if it exists.', 'hep-ph-0012175-2-52-0': 'In case of 1st scenario stoponium will be observed at PLC in the [MATH] and [MATH] (later if permitted kinematically) in the beginning of the operation - several months for scanning of whole energy region or several weeks if the stoponium mass is approximately known.', 'hep-ph-0012175-2-53-0': 'In case of 2nd scenario, about thousand of stoponiums will be produced free of background.', 'hep-ph-0012175-2-53-1': 'It means that stoponium can be discovered at PLC practically immediately.', 'hep-ph-0012175-2-54-0': 'Study of the effective stoponium couplings with photons, gluons and lightest Higgs bosons will be available at PLC, latter two depending on the MSSM scenario.', 'hep-ph-0012175-2-54-1': 'Measurement of the stoponium mass and total width (extracted from the measured signal rate) will be possible also.', 'hep-ph-0012175-2-55-0': 'Stoponium discovery mass range will be limited only by attainable values of the [MATH] collision energy, which is discussed up to [MATH] GeV.', 'hep-ph-0012175-2-55-1': 'The tuning of PLC collision energy at the resonance point is necessary within the 15% window.', 'hep-ph-0012175-2-55-2': 'These estimates were done for the case of PLC year luminosity being 60 fb[MATH].', 'hep-ph-0012175-2-56-0': 'In [MATH] collisions stoponium can be observed and studied only in the scenario with [MATH] decay channel being dominant, but with the rate lower than in the PLC case by a factor of 25 (5) (the second number for seating on the 0.1 luminosity peak).', 'hep-ph-0012175-2-56-1': 'In this comparison it was assumed that [MATH] luminosity in the high energy peak is equal 40 of e[MATH]e[MATH] luminosity.', 'hep-ph-0012175-2-56-2': 'Search time here is about 50 times longer than in [MATH] collisions.', 'hep-ph-0012175-2-56-3': 'However, there is one important advantage of e[MATH]e[MATH] collisions: by use of very monochromatic part of the luminosity spectrum (0.1%) one can make precise measurement of the stoponium mass and resolve its excited states.', 'hep-ph-0012175-2-56-4': 'Although this is possible only in the scenario when [MATH] decay dominates.', 'hep-ph-0012175-2-57-0': 'A few further comments can be made.', 'hep-ph-0012175-2-57-1': 'The first one is related to the circumstance that at photon colliders ground state of stoponium could not be distinguished from excited states due to the detector resolutions.', 'hep-ph-0012175-2-57-2': 'Therefore, the resonance peak will include contributions from ground state and all excitations, leading to enhancement factor of about 2 in all cross sections [CITATION].', 'hep-ph-0012175-2-57-3': 'At the same time there is a big uncertainty because of poor understanding of the stoponium wave function, that results in 30-50% error when the stoponium rates are estimated [CITATION].', 'hep-ph-0012175-2-58-0': 'Then, stoponium has the same quantum numbers as neutral Higgs bosons.', 'hep-ph-0012175-2-58-1': 'Thus, interesting phenomena could appear due to the interference of stoponium with Higgs sector.', 'hep-ph-0012175-2-58-2': 'This point was discussed briefly in Ref. [CITATION].', 'hep-ph-0012175-2-59-0': 'The authors are indebted to I.F. Ginzburg, H.J. Schreiber and other participants of the present Workshop for useful discussions.', 'hep-ph-0012175-2-59-1': 'The work of D.G. was supported in part under RFBR grant 99-01-18410, CRDF grant (award RP1-2103), Swiss Science Foundation grant 7SUPJ062239, Russian Academy of Science JRP grant 37 and by ISSEP fellowship.', 'hep-ph-0012175-2-59-2': 'The work of V.I. has been partially supported by the CERN-INTAS 99-377 and RFBR-DFG 99-02-04011 grants.'}

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'hep-ph-0012175-4-8-8'], ['hep-ph-0012175-3-8-9', 'hep-ph-0012175-4-8-9'], ['hep-ph-0012175-3-8-10', 'hep-ph-0012175-4-8-10'], ['hep-ph-0012175-3-8-11', 'hep-ph-0012175-4-8-11'], ['hep-ph-0012175-3-8-12', 'hep-ph-0012175-4-8-12'], ['hep-ph-0012175-3-25-0', 'hep-ph-0012175-4-25-0'], ['hep-ph-0012175-3-25-1', 'hep-ph-0012175-4-25-1'], ['hep-ph-0012175-3-25-2', 'hep-ph-0012175-4-25-2'], ['hep-ph-0012175-3-29-0', 'hep-ph-0012175-4-29-0'], ['hep-ph-0012175-3-29-1', 'hep-ph-0012175-4-29-1'], ['hep-ph-0012175-3-29-2', 'hep-ph-0012175-4-29-2'], ['hep-ph-0012175-3-48-0', 'hep-ph-0012175-4-48-0'], ['hep-ph-0012175-3-48-1', 'hep-ph-0012175-4-48-1'], ['hep-ph-0012175-3-48-2', 'hep-ph-0012175-4-48-2'], ['hep-ph-0012175-3-48-3', 'hep-ph-0012175-4-48-3'], ['hep-ph-0012175-3-48-4', 'hep-ph-0012175-4-48-4'], ['hep-ph-0012175-3-43-0', 'hep-ph-0012175-4-43-0'], ['hep-ph-0012175-3-43-1', 'hep-ph-0012175-4-43-1'], ['hep-ph-0012175-3-43-2', 'hep-ph-0012175-4-43-2'], ['hep-ph-0012175-3-43-3', 'hep-ph-0012175-4-43-3'], ['hep-ph-0012175-3-43-4', 'hep-ph-0012175-4-43-4']]

[['hep-ph-0012175-1-21-2', 'hep-ph-0012175-2-21-2'], ['hep-ph-0012175-3-46-0', 'hep-ph-0012175-4-46-0'], ['hep-ph-0012175-3-46-1', 'hep-ph-0012175-4-46-1'], ['hep-ph-0012175-3-56-0', 'hep-ph-0012175-4-56-0'], ['hep-ph-0012175-3-56-2', 'hep-ph-0012175-4-56-2'], ['hep-ph-0012175-3-2-0', 'hep-ph-0012175-4-2-0'], ['hep-ph-0012175-3-5-6', 'hep-ph-0012175-4-5-5'], ['hep-ph-0012175-3-45-0', 'hep-ph-0012175-4-45-0'], ['hep-ph-0012175-3-45-3', 'hep-ph-0012175-4-45-3']]

[]

[['hep-ph-0012175-3-44-2', 'hep-ph-0012175-4-44-2']]

[]

['hep-ph-0012175-1-23-0', 'hep-ph-0012175-1-33-0', 'hep-ph-0012175-1-37-0', 'hep-ph-0012175-1-37-4', 'hep-ph-0012175-1-38-0', 'hep-ph-0012175-1-39-0', 'hep-ph-0012175-1-47-0', 'hep-ph-0012175-2-23-0', 'hep-ph-0012175-2-33-0', 'hep-ph-0012175-2-37-0', 'hep-ph-0012175-2-37-4', 'hep-ph-0012175-2-38-0', 'hep-ph-0012175-2-39-0', 'hep-ph-0012175-2-47-0', 'hep-ph-0012175-3-23-0', 'hep-ph-0012175-3-33-0', 'hep-ph-0012175-3-37-0', 'hep-ph-0012175-3-37-4', 'hep-ph-0012175-3-38-0', 'hep-ph-0012175-3-39-0', 'hep-ph-0012175-3-47-0', 'hep-ph-0012175-4-23-0', 'hep-ph-0012175-4-33-0', 'hep-ph-0012175-4-37-0', 'hep-ph-0012175-4-37-4', 'hep-ph-0012175-4-38-0', 'hep-ph-0012175-4-39-0', 'hep-ph-0012175-4-47-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/', '4': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0012175

{'hep-ph-0012175-3-0-0': 'In some supersymmetric extensions of the Standard Model fairly light superpartner of t-quark is predicted, which may form bound states (stoponiums) under certain conditions.', 'hep-ph-0012175-3-0-1': 'We estimate potentials of TESLA linear collider in search for stoponium, considering the basic electron-positron option and the [MATH] option (Photon Linear Collider - PLC).', 'hep-ph-0012175-3-1-0': 'It is found that PLC could be the best machine for discovery and study of these new narrow strong resonances.', 'hep-ph-0012175-3-1-1': 'It can produce thousands stoponiums per 100 fb[MATH] integrated [MATH] luminosity in the high energy peak.', 'hep-ph-0012175-3-1-2': 'In the case of scenarios when stoponium decays mainly into two gluons the signal/background ratio is about 1/4.', 'hep-ph-0012175-3-1-3': 'In addition the decay channel [MATH] into two lightest Higgs bosons could be seen with high significance.', 'hep-ph-0012175-3-1-4': 'Thus, several weeks run is sufficient for the discovery of stoponium, if its mass is approximately known (for example from observation of direct stops production at LHC).', 'hep-ph-0012175-3-1-5': 'Then, in MSSM scenarios with dominant [MATH] decay PLC shows excellent possibilities to discover bound state of stops, practically immediately after beginning of operating.', 'hep-ph-0012175-3-1-6': 'Thus, PLC has good possibilities to study strong interaction of top quark superpartners in nonperturbative regime.', 'hep-ph-0012175-3-2-0': 'The e[MATH]e[MATH] option also has good discovery prospects but only in the case of scenarios with dominant decay into two lightest Higgs bosons, with hundreds of events per 100 [MATH].', 'hep-ph-0012175-3-2-1': 'Interesting possibility appears in the case when the resonance is seated on 0.1% width luminosity peak - one could resolve the stoponium exited states.', 'hep-ph-0012175-3-3-0': '# Introduction', 'hep-ph-0012175-3-4-0': 'The broken supersymmetry is favorite among the different extensions of the Standard Model.', 'hep-ph-0012175-3-4-1': 'It can happen that superpartners of top-quarks (stops, [MATH]) are long-living enough to compose (colorless) bound states, stoponiums, denoted as [MATH] in what follows.', 'hep-ph-0012175-3-4-2': 'In this scenario experimental study of the corresponding resonances could provide precise value of stop mass and stoponium partial widths, consequently yielding precise values of various parameters of SUSY Lagrangian.', 'hep-ph-0012175-3-4-3': 'Then, if the difference between stop and LSP masses is very small, the search for stop evidence in collisions at high energy could be problematic.', 'hep-ph-0012175-3-4-4': 'Observation of stoponium bound states will be the signature of such models confirming the existence of stop.', 'hep-ph-0012175-3-5-0': 'There are theoretical motivations for stop to be fairly light.', 'hep-ph-0012175-3-5-1': 'First one appeals to the renormalization group behavior of soft mass terms.', 'hep-ph-0012175-3-5-2': 'Indeed, gauge couplings raise while Yukawa couplings reduce these terms when energy scale evolves down, with Yukawa contributions being very large for stop.', 'hep-ph-0012175-3-5-3': 'The next motivation concerns left-right mixing in squark sector, which is proportional to Yukawa coupling and decreases the mass of the lightest stop.', 'hep-ph-0012175-3-5-4': 'Therefore, light stop may appear in different SUSY models ( see, e.g., Refs. [CITATION] for examples in the frameworks of supergravity and gauge mediation).', 'hep-ph-0012175-3-5-5': 'Experimental bound on stoponium mass comes from searches for stop at LEP2 and TEVATRON.', 'hep-ph-0012175-3-5-6': 'The concrete number depends on the MSSM spectrum [CITATION]: lower bound is about 90 GeV for sneutrino masses larger than 45-50 GeV or for neutralino masses larger than 50 GeV (ALEPH), while CDF excludes stop mass up to 130 GeV for smaller sneutrino masses.', 'hep-ph-0012175-3-5-7': 'The limitation is weaker if stop and neutralino masses are degenerate, it is about 60 GeV (ALEPH).', 'hep-ph-0012175-3-6-0': 'Stoponium was studied in Refs. [CITATION] in detail, in particular its effective couplings and partial widths were calculated, and prospects to be discovered at LHC were estimated.', 'hep-ph-0012175-3-6-1': 'In Ref. [CITATION] it was briefly mentioned also the possibility to observe stoponiums in photon collisions, however, without analyzing this phenomenology.', 'hep-ph-0012175-3-6-2': 'The first look at the PLC prospects on the stoponium search was done in [CITATION].', 'hep-ph-0012175-3-7-0': 'Now, when main parameters of TESLA project are under technical discussion one should understand clearly signatures of stop bound states in [MATH] and [MATH] collisions, and compare potentials of these two options of future linear collider.', 'hep-ph-0012175-3-7-1': 'In present analysis we will use year integrated luminosity for the [MATH] option equal to 300 fb[MATH] for [MATH] GeV, and one should rescale this figure for lower collision energies approximately as [MATH] [CITATION].', 'hep-ph-0012175-3-7-2': 'Beamstrahlung and ISR will affect considerably the stoponium production rate due to a condition to tune the collision energy at the resonance point.', 'hep-ph-0012175-3-7-3': 'According to the current TESLA reference parameter set the average energy loss due to beamstrahlung is about 3.', 'hep-ph-0012175-3-7-4': 'The average energy loss due to ISR is about 5 [CITATION].', 'hep-ph-0012175-3-7-5': 'Correspondingly, the luminosity distribution has a characteristic width about 5.', 'hep-ph-0012175-3-8-0': 'However, stoponium is very narrow resonance, even more narrow than the initial beam energy spread.', 'hep-ph-0012175-3-8-1': 'Its production rate is proportional to the differential luminosity, [MATH], at the [MATH] peak, where [MATH] is a collision energy for hard subprocess.', 'hep-ph-0012175-3-8-2': 'The initial (beam) energy spread for energies 200-500 GeV is about 0.07-0.1 and it is determined by the bunch compression system and undulator for the positron production.', 'hep-ph-0012175-3-8-3': 'The fraction of the luminosity in this [MATH]0.1 peak is determined by the ISR and beamstrahlung.', 'hep-ph-0012175-3-8-4': 'The ISR leaves in this peak about 50 of the luminosity [CITATION].', 'hep-ph-0012175-3-8-5': 'The average number of beamstrahlung photons with the energy more than [MATH] is about [MATH] per electron for TESLA conditions.', 'hep-ph-0012175-3-8-6': 'Thus, the probability of the e[MATH]e[MATH] collision without such beamstrahlung photons can be estimated as [MATH].', 'hep-ph-0012175-3-8-7': 'The probability of events without any photons (ISR or beamstrahlung ones) with the energies greater 0.1 is [MATH].', 'hep-ph-0012175-3-8-8': 'So, about 17 of the luminosity is concentrated in [MATH]0.1 range.', 'hep-ph-0012175-3-8-9': 'The differential luminosity in this peak is higher than in 5 interval by a factor of [MATH] times (here factor 0.8 is due to ISR in the 5 region).', 'hep-ph-0012175-3-8-10': 'This peak could be very important factor for increasing significance of the resonance (when its mass is known), for measurement of its mass and even for resolving close excited states of the stoponium.', 'hep-ph-0012175-3-8-11': 'Note, however, that for a search for the stoponium this peak will not help, because in presence of large background the scanning time in some wide energy interval, required for the resonance observation, has the following dependence on the bin width [MATH] and the differential luminosity, [MATH] (here bin [MATH] corresponds to the effective width of the luminosity distribution in the peak).', 'hep-ph-0012175-3-8-12': 'So, the ratio of scanning times for the 5% and the 0.1% width peaks is [MATH], almost the same.', 'hep-ph-0012175-3-9-0': 'Photon colliders based on Compton backscattering of laser photons on high energy electrons has been proposed a long time ago [CITATION].', 'hep-ph-0012175-3-9-1': 'This option has been included in the TESLA Conceptual Design Report [CITATION] and work on the Technical Design Report is under way.', 'hep-ph-0012175-3-9-2': 'Since the CDR parameters of TESLA were changed and luminosities have grown both in e[MATH]e[MATH] and [MATH] collisions.', 'hep-ph-0012175-3-10-0': 'At the present workshop it was reported [CITATION] that PLC luminosity can be further increased by a factor of 2.5 due to possible decrease of the horizontal beam emittance at the TESLA damping ring (however this has not improved e[MATH]e[MATH] luminosity because it is restricted by collision effects).', 'hep-ph-0012175-3-10-1': 'At present the [MATH] luminosity within the 20% interval just below [MATH] could be about 40% of luminosity in the e[MATH]e[MATH] collisions (where [MATH]^+[MATH]^-[MATH] at [MATH] GeV).', 'hep-ph-0012175-3-10-2': 'In the analysis we will consider 60 fb[MATH] for PLC year luminosity at [MATH] GeV.', 'hep-ph-0012175-3-11-0': 'The most bright evidence of the narrow resonance is its direct s-channel production in [MATH] annihilation or [MATH] fusion.', 'hep-ph-0012175-3-11-1': 'One can note, that high powers of the coupling constants, [MATH], emerge in the squared matrix elements in the [MATH] annihilation into stoponium.', 'hep-ph-0012175-3-11-2': 'Indeed, [MATH] arises from two electroweak vertices, and [MATH] comes from squared derivative of the stoponium wave function (scalar stoponium can be created there only in P-wave by propagation of neutral vector particle: photon or [MATH] boson).', 'hep-ph-0012175-3-11-3': 'At the same time two powers of [MATH] are eliminated in the case of [MATH] fusion mechanism because the stoponium production can be proceeded in S-wave there.', 'hep-ph-0012175-3-11-4': 'This circumstance makes for the relative enhancement of the stoponium production rate at PLC in comparison with [MATH] option.', 'hep-ph-0012175-3-11-5': 'Two powers of [MATH] are eliminated also in the case of associated production of stoponiums, for example in the Higgs-like reactions [MATH] and [MATH].', 'hep-ph-0012175-3-11-6': 'Hereafter we denote stoponium as [MATH].', 'hep-ph-0012175-3-12-0': 'To complete this brief review of possible production mechanisms one can note that in hadron collisions the stoponium resonance production is available in S-wave through the gluon fusion.', 'hep-ph-0012175-3-12-1': 'Here the effective [MATH] vertex includes [MATH] and one can anticipate large stoponium cross sections.', 'hep-ph-0012175-3-12-2': 'However, main decay channel [MATH] is too dirty due to huge QCD [MATH] background.', 'hep-ph-0012175-3-12-3': 'Then, the most promising decay channel at LHC is [MATH] [CITATION], but in order to discover stoponiums one year of LHC operating at high luminosity is needed, or even more depending on SUSY scenario.', 'hep-ph-0012175-3-13-0': 'We consider stoponium mass range [MATH] GeV, which could be surely probed by first TESLA run.', 'hep-ph-0012175-3-13-1': 'It is worth to note that the same interval is not an exceptional case for SUSY models with stoponiums as a quasistationary state, as we discuss briefly in the next section.', 'hep-ph-0012175-3-14-0': 'Cross sections for various tree-level background processes were evaluated with the help of CompHEP package [CITATION].', 'hep-ph-0012175-3-15-0': '# Stop bound states', 'hep-ph-0012175-3-16-0': 'It is clear that gluons try to bind two stops as well as ordinary quarks.', 'hep-ph-0012175-3-16-1': 'The corresponding bound state can be described as a quasistationary system with energy levels [MATH]) and masses [MATH] similarly to quarkonium.', 'hep-ph-0012175-3-16-2': 'For stoponium mass [MATH] GeV the binding energies [MATH] are of order 1 GeV [CITATION].', 'hep-ph-0012175-3-16-3': 'This treatment is valid if the formation process (time scale [MATH]) is faster than destroying one.', 'hep-ph-0012175-3-17-0': 'Among destroying mechanisms the obvious ones are the stop decays: [MATH]LSP, [MATH] and [MATH].', 'hep-ph-0012175-3-17-1': 'At first, let us consider the third decay.', 'hep-ph-0012175-3-17-2': 'It proceeds only through loop diagrams, if Universal boundary condition on soft terms is imposed (that is motivated by the absence of FCNC).', 'hep-ph-0012175-3-17-3': 'So, partial width is highly reduced by a factor of [MATH] in comparison with the first two tree-level decay processes [CITATION].', 'hep-ph-0012175-3-17-4': 'The rates of latter decays depend on the parameters of the model.', 'hep-ph-0012175-3-17-5': 'As an example, in the framework of gravity mediation, where LSP is neutralino, these decays proceed at the tree level and the corresponding partial widths are of order O[MATH].', 'hep-ph-0012175-3-17-6': 'In the framework of models with gauge mediated supersymmetry breaking [CITATION], where LSP is gravitino, the first process is strongly suppressed by supersymmetry breaking scale, but remaining one has the same partial width as in gravity mediation.', 'hep-ph-0012175-3-17-7': 'Hence, the possibility of existence of stoponium is a subject of special study in each concrete model.', 'hep-ph-0012175-3-17-8': 'For instance, in models with the lightest chargino being mostly wino and the lightest stop being mostly right stop (i.e., [MATH]), decay into chargino is damped and stoponium could exist if [MATH], i.e., when the first decay channel is kinematically forbidden.', 'hep-ph-0012175-3-17-9': 'One can state that SUSY scenario, where tree-level decays, [MATH]LSP and [MATH], are somehow suppressed and, therefore, stop decay can not destroy the stoponium formation, is not an exceptional case.', 'hep-ph-0012175-3-18-0': 'Next destroying mechanism is related to the stop annihilation.', 'hep-ph-0012175-3-18-1': 'Here two gluon channel is always open with partial width about 1 MeV.', 'hep-ph-0012175-3-18-2': 'Generally the gluon channel is dominant.', 'hep-ph-0012175-3-18-3': 'However, for the certain choice of model parameters, partial width into two lightest Higgs bosons, [MATH], can be larger, increasing the stoponium total width by a factor of [MATH].', 'hep-ph-0012175-3-18-4': 'In Ref. [CITATION] these figures were analyzed and found that quasistationary description is valid for [MATH] GeV in models with forbidden stop tree-level decays and neutralino being mostly bino.', 'hep-ph-0012175-3-18-5': 'The worst case is a model with chargino and neutralino states are both higgsino-like.', 'hep-ph-0012175-3-18-6': 'Here the stoponium total width increases rapidly with [MATH] and quasistationary treatment fails for [MATH] GeV.', 'hep-ph-0012175-3-19-0': '# Stoponium in [MATH] collisions', 'hep-ph-0012175-3-20-0': 'Let us begin with study of stoponium events in photon-photon scattering.', 'hep-ph-0012175-3-20-1': 'The main effect associated with stoponium would be a direct resonance production, where stoponium is produced in spin 0 state.', 'hep-ph-0012175-3-20-2': 'The corresponding cross section is described by the Breit-Wigner formula (similarly to the light Higgs production discussed in Ref. [CITATION]) [EQUATION] where [MATH] is squared colision energy for hard subprocess, [MATH] is the stoponium partial width for the decay into state [MATH], [MATH] is stoponium total width, [MATH] are helicities of initial photons.', 'hep-ph-0012175-3-21-0': 'At photon colliders the width of the luminosity distribution is much wider than that of the stoponium.', 'hep-ph-0012175-3-21-1': 'After integration over the luminosity distribution we obtain the effective cross section [EQUATION]', 'hep-ph-0012175-3-21-2': 'The differential luminosity, [MATH][MATH], at [MATH] can be estimated as [MATH][MATH] according to 15% width of the high energy luminosity peak (note, we define the [MATH] luminosity as the luminosity in the interval [MATH]%).', 'hep-ph-0012175-3-21-3': 'So the effective cross section is [EQUATION]', 'hep-ph-0012175-3-21-4': 'Photon beams are planned to be highly polarized.', 'hep-ph-0012175-3-21-5': 'Hence, as stoponium is a scalar the production cross section will be enhanced by factor two if initial photons have total helicity equal to zero.', 'hep-ph-0012175-3-21-6': 'Hereafter we assume initial total helicity 0 ([MATH]) in numerical estimates.', 'hep-ph-0012175-3-21-7': 'Finally, we parameterize the stoponium cross sections as follows [EQUATION]', 'hep-ph-0012175-3-21-8': 'One should take into account that squared stoponium wave function at the origin, attending in [MATH], scales as a square root of its mass [CITATION].', 'hep-ph-0012175-3-22-0': 'As it has been stressed above one can discuss two main variants of the SUSY models, one with dominant [MATH] decay mode and another with stoponium total width being saturated by [MATH] mode.', 'hep-ph-0012175-3-22-1': 'Let us make qualitative signal/background analysis for different decay channels within these two variants.', 'hep-ph-0012175-3-22-2': 'The signal significance can be evaluated by ratio [MATH] because one deals with resonance and background rate in the signal bin can be fixed as average cross section in neighboring bins (here [MATH] are numbers of signal and background events).', 'hep-ph-0012175-3-22-3': 'We used the results for stoponium width and branching ratios calculated in Ref. [CITATION] with some corrections [CITATION].', 'hep-ph-0012175-3-23-0': '1.', 'hep-ph-0012175-3-23-1': 'In the first scenario stoponium total width [MATH] MeV and photon-photon branching is [MATH].', 'hep-ph-0012175-3-23-2': 'By making use of Eq. ([REF]) one obtains the signal rate at the level of 110 fb for [MATH] GeV.', 'hep-ph-0012175-3-23-3': 'So, more than six thousand stoponiums will be produced per year if L[MATH] fb[MATH].', 'hep-ph-0012175-3-23-4': 'Background is two jet production, where subprocess [MATH] gives main contribution with very large unpolarized cross section, [MATH] pb, if optimal cut on the jet angle of 45[MATH] is applied.', 'hep-ph-0012175-3-23-5': 'However, production of fermions is suppressed in collisions of photons with the total helicity 0 [CITATION]: [EQUATION] where [MATH] is velocity of the quarks.', 'hep-ph-0012175-3-23-6': 'So, [MATH] pairs are produced only in collisions of photons with the total helicity 2.', 'hep-ph-0012175-3-23-7': 'This fact is used for suppression of similar background in the analysis of the Higgs production in [MATH] collisions [CITATION].', 'hep-ph-0012175-3-23-8': 'The corresponding luminosity spectra for total helicity 0 ([MATH]) and 2 ([MATH]) can be found elsewhere [CITATION].', 'hep-ph-0012175-3-23-9': 'In the region [MATH]10% near the maximum energy (detector resolution or intrinsic resolution of the PLC for two collinear jets) one can obtain the ratio of the luminosities [MATH], that assumes at least one order suppression of the [MATH] background.', 'hep-ph-0012175-3-23-10': 'Note that the cross section 6 pb for background corresponds to the case of unpolarized beams.', 'hep-ph-0012175-3-23-11': 'It is zero for collisions of photons with the total helicity 0 and is equal to 12 pb for the total helicity 2, therefore the remaining cross section is 1.2 pb.', 'hep-ph-0012175-3-23-12': 'Note that due to gluon emission [MATH] pairs can be produced even in collisions of photons with the total helicity 0.', 'hep-ph-0012175-3-23-13': 'Detailed studies have shown that with proper cuts this process is not important [CITATION] and contribution from resolved photons is also not significant [CITATION].', 'hep-ph-0012175-3-24-0': 'Furthermore, the cross section is proportional to the fourth power of the electric charge of quarks, so the main contribution is given by [MATH] and [MATH] quarks.', 'hep-ph-0012175-3-24-1': 'The later can be easily suppressed by the vertex detector.', 'hep-ph-0012175-3-24-2': 'This gives additional factor of 2 in the background suppression.', 'hep-ph-0012175-3-24-3': 'Additional improvement can give the detector energy resolution which is at least factor of 2 smaller than the width of the [MATH] luminosity peak.', 'hep-ph-0012175-3-24-4': 'All three methods give a suppression factor of 40.', 'hep-ph-0012175-3-24-5': 'The remaining background is about 0.3 pb, while the isotropic signal is smaller by factor 1.4 only if the cut on the jet angle of 45[MATH] is applied.', 'hep-ph-0012175-3-24-6': 'Hence in the scenario with dominant [MATH] channel the signal/background ratio is [MATH].', 'hep-ph-0012175-3-24-7': 'The signal significance is about 35 for 60 fb[MATH] and [MATH] GeV.', 'hep-ph-0012175-3-25-0': 'These figures can be related to the first year of PLC operation if the stoponium mass is known approximately, for example from the observation of the direct stops production at LHC.', 'hep-ph-0012175-3-25-1': 'In opposite case, for a search of a stoponium one should make scanning with the energy bin [MATH] 10%.', 'hep-ph-0012175-3-25-2': 'It is clear that stoponium can be found in this scenario during several month work in the whole energy region under discussion, 200-400 GeV.', 'hep-ph-0012175-3-26-0': 'For two photon channel the background process, [MATH], proceeds through one-loop diagrams, so the corresponding cross section is small, about 10 fb [CITATION].', 'hep-ph-0012175-3-26-1': 'One should note that the photon-photon invariant mass bin can be taken equal to [MATH] for CMS-like crystal electromagnetic calorimeter [CITATION].', 'hep-ph-0012175-3-26-2': 'Thus, for [MATH] GeV window the background rate can be estimated at the level of 1 fb.', 'hep-ph-0012175-3-26-3': 'The signal rate is [MATH] fb for [MATH] GeV, providing the signal significance about [MATH] for statistics 60 fb[MATH].', 'hep-ph-0012175-3-27-0': 'Some other decay channels in the framework of the first scenario should be discussed.', 'hep-ph-0012175-3-27-1': 'First note, that [MATH] final state has no chance for the detection of stoponiums due to huge SM background, [MATH] pb at [MATH] GeV.', 'hep-ph-0012175-3-27-2': 'More promising are decay channels with background processes emerging due to the higher order corrections from perturbation theory.', 'hep-ph-0012175-3-27-3': 'For instance, SM background to [MATH] and [MATH] final states comes from 1) one-loop [MATH] processes [MATH] fb [CITATION]) and [MATH] fb [CITATION]), and 2) from tree-level [MATH] processes (e.g. [MATH] for [MATH]), with total cross section smaller than 1 fb within cuts on final [MATH] and jets reasonably motivated by 2-body ([MATH]) kinematics of the signal events.', 'hep-ph-0012175-3-28-0': 'As to signal [MATH] rate one can get from Ref. [CITATION] the branching [MATH], so [MATH] fb already for [MATH] GeV.', 'hep-ph-0012175-3-28-1': 'It means very low level of the signal significance, lower than 0.5 for statistics 60 fb[MATH].', 'hep-ph-0012175-3-29-0': 'Natural level of [MATH] branching is about [MATH] for stoponium masses far from the threshold, 250-400 GeV, although in some points it could fall down due to opening of new channels or degeneration of stoponium and Higgs masses.', 'hep-ph-0012175-3-29-1': 'This provides signal rate [MATH] fb for [MATH] GeV and significance at the level of 2.7 if one uses formula ([REF]).', 'hep-ph-0012175-3-29-2': 'However, the threshold effect is significant still for this value of the stoponium mass, and these figures should be improved to 1.8 fb for signal rate and to 2 for the significance.', 'hep-ph-0012175-3-30-0': 'The [MATH] decay channel, where [MATH] is the lightest Higgs boson, is open if [MATH].', 'hep-ph-0012175-3-30-1': 'As current limit on [MATH] mass is about 80-100 GeV this channel could exist for [MATH] GeV.', 'hep-ph-0012175-3-30-2': 'If consider mass region not very close to the threshold (say [MATH] GeV for [MATH] GeV) the [MATH] branching is about [MATH] or even higher.', 'hep-ph-0012175-3-30-3': 'In this case the signal rate is about 1 fb or larger, if one uses formula ([REF]), and if take into account the threshold factor one gets signal cross section at the level of 0.2 fb or larger.', 'hep-ph-0012175-3-30-4': 'The background from direct double [MATH] production through one-loop diagrams can be estimated by the cross section of this process in SM, [MATH] fb [CITATION].', 'hep-ph-0012175-3-30-5': 'There are no reasons for very large additional contributions to this process in supersymmetric models.', 'hep-ph-0012175-3-30-6': 'Then, we found that direct electroweak production of four b quarks ([MATH]) together with contribution from [MATH] final state (assuming 10 of [MATH] misidentification) has the rate smaller than 0.1 fb.', 'hep-ph-0012175-3-30-7': 'These cross sections for background processes correspond to 15 width of the [MATH] luminosity spectrum.', 'hep-ph-0012175-3-30-8': 'The detector resolution is at least factor of two better (full width), therefore the total cross section of background processes can be estimated as (0.2+0.1)/2 = 0.15 fb.', 'hep-ph-0012175-3-30-9': 'We see that [MATH] decay can also be studied in the considered scenario.', 'hep-ph-0012175-3-30-10': 'For [MATH] GeV, [MATH] GeV and 60 fb[MATH] integrated luminosity about 20 stoponium events will be produced in the decay channel [MATH] with [MATH] ratio about 2.3 and statistical significance about 7.', 'hep-ph-0012175-3-31-0': 'In Fig. [REF] the signal significances are represented for main stoponium decay channels in the case of the first scenario.', 'hep-ph-0012175-3-31-1': 'As a resume for this scenario we conclude that stoponium can be found at PLC during several months scan of the 200-400 GeV region in [MATH] and [MATH] decay modes.', 'hep-ph-0012175-3-31-2': 'If its mass is known approximately, it will be found during first weeks.', 'hep-ph-0012175-3-31-3': 'More than a year is necessary in order to observe stoponium in [MATH] and [MATH] channels.', 'hep-ph-0012175-3-32-0': 'Note that LHC (at the high luminosity operating stage) has good prospects to observe light stoponium in [MATH] mode in this scenario [CITATION].', 'hep-ph-0012175-3-32-1': 'Thus, these two colliders could be complementary in study of different effective stoponium couplings, [MATH] and [MATH] at LHC and the photon collider, respectively and [MATH] at PLC if this decay channel is open kinematically.', 'hep-ph-0012175-3-33-0': '2.', 'hep-ph-0012175-3-33-1': 'In the second scenario stoponium total width could be about 10 MeV or even larger.', 'hep-ph-0012175-3-33-2': 'The photon-photon branching in this case is smaller, [MATH].', 'hep-ph-0012175-3-33-3': 'So, about thousand of stoponiums will be produced per year and almost all of them will decay to pairs of lightest Higgs bosons.', 'hep-ph-0012175-3-33-4': 'This result suggests, that stoponium will be discovered practically immediately after PLC start, since the background ([MATH], [MATH] ...) is very small.', 'hep-ph-0012175-3-33-5': 'Again the scanning over the energy interval is necessary if the stoponium mass is not known.', 'hep-ph-0012175-3-33-6': 'In Fig. [REF] the year yields of stoponiums are represented for different masses of lightest Higgs bosons.', 'hep-ph-0012175-3-34-0': 'Due to rather high statistics for the signal and absence (practically) of the background one gets PLC as a stoponium factory.', 'hep-ph-0012175-3-34-1': 'The detailed study of the stoponium characteristics will be available in this case, in particular measurement of its mass and total width and effective couplings [MATH] and [MATH].', 'hep-ph-0012175-3-34-2': 'One can stress the importance of study the [MATH] coupling, which relates directly to the stop-Higgs interaction and, thus, to the mechanism of the superpartner mass generation.', 'hep-ph-0012175-3-35-0': 'Note that in this scenario several years of operating at high luminosity is needed in order to observe stoponium at LHC [CITATION].', 'hep-ph-0012175-3-36-0': '# Stoponium in [MATH] collisions', 'hep-ph-0012175-3-37-0': '1.', 'hep-ph-0012175-3-37-1': 'First we discuss direct resonant production of stoponiums in [MATH] collisions where initial electron and positron would be in P-state (thus, stoponium is produced in spin 1 state).', 'hep-ph-0012175-3-37-2': 'As in case of [MATH] collisions one gets for narrow resonance after the integration of Breit-Wigner distribution the following formula for production cross section [EQUATION] where [MATH] are helicities of initial electrons and positrons.', 'hep-ph-0012175-3-37-3': 'The differential luminosity at [MATH] can be estimated as [MATH], where [MATH] is a fraction of the [MATH] luminosity corresponding to some interval [MATH] near the [MATH] peak.', 'hep-ph-0012175-3-37-4': 'Let us consider two methods of the stoponium detection:', 'hep-ph-0012175-3-38-0': '1) [MATH]% [MATH]^+[MATH]^-[MATH], [MATH]%;', 'hep-ph-0012175-3-39-0': '2) [MATH]% [MATH]^+[MATH]^-[MATH], [MATH]%.', 'hep-ph-0012175-3-40-0': 'Then, the stoponium partial width into [MATH] pair, [MATH], is equal to [CITATION] [EQUATION] with [MATH] being wave function of P-state.', 'hep-ph-0012175-3-40-1': 'In correspondence with Ref. [CITATION] one can approximate [MATH] GeV.', 'hep-ph-0012175-3-41-0': 'We assume, also, 80% circular polarization for the electron beam and 60% for the positron beam, and neglect fairly weak enhancement by [MATH]-boson pole for light stoponiums.', 'hep-ph-0012175-3-42-0': 'Finally, for the first method of the stoponium detection one gets for the production rate in [MATH] collisions the following estimate [EQUATION] while it is about 5 times larger in case of second method.', 'hep-ph-0012175-3-43-0': 'Let us now consider background for the direct resonance production of stoponiums in [MATH] collisions.', 'hep-ph-0012175-3-43-1': 'One should note that stoponium, being produced in excited state with spin 1, will be transfered to the basic state with spin 0 by the emission of a photon.', 'hep-ph-0012175-3-43-2': 'Therefore, one can consider decay modes of scalar stoponium in case of [MATH] production as well.', 'hep-ph-0012175-3-43-3': 'Then, additional photon emmited by excited stoponium could be used for further suppression of the background.', 'hep-ph-0012175-3-43-4': 'However, the detection of this photon could be not easy because its energy is rather small, of order 1 GeV.', 'hep-ph-0012175-3-44-0': 'In the case of first MSSM scenario with [MATH] decay channel being dominant the main background is [MATH] process.', 'hep-ph-0012175-3-44-1': 'It has a cross section about 10 pb for [MATH] GeV.', 'hep-ph-0012175-3-44-2': 'So, this channel is too dirty with rather low significance, less than 1 for 100 fb[MATH] statistics.', 'hep-ph-0012175-3-44-3': 'In other channels the stoponium production has too small rate: few events per year in [MATH] and [MATH] channels.', 'hep-ph-0012175-3-45-0': 'In the case of second MSSM scenario with [MATH] dominant decay mode almost all stoponiums will decay into pair of lightest Higgs bosons, and the number of events for [MATH] GeV and 100 [MATH] is 50 (250) (the second number is for the case of seating on the 0.1% luminosity peak).', 'hep-ph-0012175-3-45-1': 'As direct pair production of Higgs bosons, [MATH], is negligible since the corresponding [MATH] coupling includes electron mass, this channel is should be practically free of background.', 'hep-ph-0012175-3-45-2': 'Indeed, direct production of four b-quarks in electron-positron collisions has very small cross section if exclude [MATH] peaks (less than 0.05fb if apply the cut [MATH] GeV).', 'hep-ph-0012175-3-45-3': 'So, stoponium can be easily discovered at [MATH] machine if the collision energy is tuned at [MATH].', 'hep-ph-0012175-3-46-0': 'Note, that in this scenario the yield of stoponiums at PLC with [MATH][MATH]^+[MATH]^-[MATH] is higher than that in e[MATH]e[MATH] collisions by a factor of 25(5) (second number for the case of seating on the 0.1% peak of the [MATH] luminosity).', 'hep-ph-0012175-3-46-1': 'Thus, the scanning time in e[MATH]e[MATH] mode will be about factor of 50 longer than at PLC due to smaller cross section and smaller energy bin (see also discussion on the scanning time in the Introduction).', 'hep-ph-0012175-3-47-0': '2.', 'hep-ph-0012175-3-47-1': 'The second effect related to stoponium at [MATH] collider would be a production in Higgs-like channels.', 'hep-ph-0012175-3-47-2': 'In present analysis we neglect Higgs-stoponium mixing as well as Higgs influence on stoponium-involved processes.', 'hep-ph-0012175-3-47-3': 'First, let us evaluate effective coupling constants between stoponium and weak bosons, [EQUATION] where we used stoponium wave function evaluated at the origin in Ref. [CITATION]; [MATH] and [MATH] are corresponding amplitudes of [MATH] processes evaluated in Ref. [CITATION].', 'hep-ph-0012175-3-47-4': 'Certainly, these effective couplings are obtained with all particles being on-shell, that is rather rough approximation.', 'hep-ph-0012175-3-47-5': 'However, one can hope that it is acceptable for the estimate.', 'hep-ph-0012175-3-48-0': 'The calculation with the effective couplings above gives the cross section about 0.03 fb for [MATH] (in case of no mixing in stop sector and [MATH] TeV).', 'hep-ph-0012175-3-48-1': 'So, only a few signal events will be produced for 100 fb[MATH] statistics.', 'hep-ph-0012175-3-48-2': 'The [MATH] background is too heavy for this level of the signal, so only the second scenario could have some prospects.', 'hep-ph-0012175-3-48-3': 'The main background will come from associated double Higgs bosons production, [MATH], the corresponding cross section is of order [MATH] fb for [MATH] [CITATION].', 'hep-ph-0012175-3-48-4': 'It gives the signal significance less than 1.', 'hep-ph-0012175-3-49-0': 'In the case of [MATH]-fusion the signal cross section is very small, less than [MATH] fb, that closes this channel.', 'hep-ph-0012175-3-50-0': '# Conclusions', 'hep-ph-0012175-3-51-0': 'In two considered scenarios of supersymmetric extension of the Standard Model (1st one with dominant [MATH] decay mode, and [MATH] being dominant in 2nd scenario) Photon Linear Collider will be the best machine to discover and study bound state of stops, if it exists.', 'hep-ph-0012175-3-52-0': 'In case of 1st scenario stoponium will be observed at PLC in the [MATH] and [MATH] (later if permitted kinematically) in the beginning of the operation - several months for scanning of whole energy region or several weeks if the stoponium mass is approximately known.', 'hep-ph-0012175-3-53-0': 'In case of 2nd scenario, about thousand of stoponiums will be produced free of background.', 'hep-ph-0012175-3-53-1': 'It means that stoponium can be discovered at PLC practically immediately.', 'hep-ph-0012175-3-54-0': 'Study of the effective stoponium couplings with photons, gluons and lightest Higgs bosons will be available at PLC, latter two depending on the MSSM scenario.', 'hep-ph-0012175-3-54-1': 'Measurement of the stoponium mass and total width (extracted from the measured signal rate) will be possible also.', 'hep-ph-0012175-3-55-0': 'Stoponium discovery mass range will be limited only by attainable values of the [MATH] collision energy, which is discussed up to [MATH] GeV.', 'hep-ph-0012175-3-55-1': 'The tuning of PLC collision energy at the resonance point is necessary within the 15% window.', 'hep-ph-0012175-3-55-2': 'These estimates were done for the case of PLC year luminosity being 60 fb[MATH].', 'hep-ph-0012175-3-56-0': 'In [MATH] collisions stoponium can be observed and studied only in the scenario with [MATH] decay channel being dominant, but with the rate lower than in the PLC case by a factor of 25 (5) (the second number for seating on the 0.1 luminosity peak).', 'hep-ph-0012175-3-56-1': 'In this comparison it was assumed that [MATH] luminosity in the high energy peak is equal 40 of e[MATH]e[MATH] luminosity.', 'hep-ph-0012175-3-56-2': 'Search time here is about 50 times longer than in [MATH] collisions.', 'hep-ph-0012175-3-56-3': 'However, there is one important advantage of e[MATH]e[MATH] collisions: by use of very monochromatic part of the luminosity spectrum (0.1%) one can make precise measurement of the stoponium mass and resolve its excited states.', 'hep-ph-0012175-3-56-4': 'Although this is possible only in the scenario when [MATH] decay dominates.', 'hep-ph-0012175-3-57-0': 'A few further comments can be made.', 'hep-ph-0012175-3-57-1': 'The first one is related to the circumstance that at photon colliders ground state of stoponium could not be distinguished from excited states due to the detector resolutions.', 'hep-ph-0012175-3-57-2': 'Therefore, the resonance peak will include contributions from ground state and all excitations, leading to enhancement factor of about 2 in all cross sections [CITATION].', 'hep-ph-0012175-3-57-3': 'At the same time there is a big uncertainty because of poor understanding of the stoponium wave function, that results in 30-50% error when the stoponium rates are estimated [CITATION].', 'hep-ph-0012175-3-58-0': 'Then, stoponium has the same quantum numbers as neutral Higgs bosons.', 'hep-ph-0012175-3-58-1': 'Thus, interesting phenomena could appear due to the interference of stoponium with Higgs sector.', 'hep-ph-0012175-3-58-2': 'This point was discussed briefly in Ref. [CITATION].', 'hep-ph-0012175-3-59-0': 'The authors are indebted to I.F. Ginzburg, H.J. Schreiber and other participants of the present Workshop for useful discussions.', 'hep-ph-0012175-3-59-1': 'The work of D.G. was supported in part under RFBR grant 99-01-18410, CRDF grant (award RP1-2103), Swiss Science Foundation grant 7SUPJ062239, Russian Academy of Science JRP grant 37 and by ISSEP fellowship.', 'hep-ph-0012175-3-59-2': 'The work of V.I. has been partially supported by the CERN-INTAS 99-377 and RFBR-DFG 99-02-04011 grants.'}

{'hep-ph-0012175-4-0-0': 'In some supersymmetric extensions of the Standard Model fairly light superpartner of t-quark is predicted, which may form bound states (stoponiums) under certain conditions.', 'hep-ph-0012175-4-0-1': 'We estimate potentials of TESLA linear collider in search for stoponium, considering the basic electron-positron option and the [MATH] option (Photon Linear Collider - PLC).', 'hep-ph-0012175-4-1-0': 'It is found that PLC could be the best machine for discovery and study of these new narrow strong resonances.', 'hep-ph-0012175-4-1-1': 'It can produce thousands stoponiums per 100 fb[MATH] integrated [MATH] luminosity in the high energy peak.', 'hep-ph-0012175-4-1-2': 'In the case of scenarios when stoponium decays mainly into two gluons the signal/background ratio is about 1/4.', 'hep-ph-0012175-4-1-3': 'In addition the decay channel [MATH] into two lightest Higgs bosons could be seen with high significance.', 'hep-ph-0012175-4-1-4': 'Thus, several weeks run is sufficient for the discovery of stoponium, if its mass is approximately known (for example from observation of direct stops production at LHC).', 'hep-ph-0012175-4-1-5': 'Then, in MSSM scenarios with dominant [MATH] decay PLC shows excellent possibilities to discover bound state of stops, practically immediately after beginning of operating.', 'hep-ph-0012175-4-1-6': 'Thus, PLC has good possibilities to study strong interaction of top quark superpartners in nonperturbative regime.', 'hep-ph-0012175-4-2-0': 'The e[MATH]e[MATH] option also has some prospects to observe stoponium but only in the case of scenarios with dominant decay into two lightest Higgs bosons, with tens of events per 100 [MATH].', 'hep-ph-0012175-4-2-1': 'Interesting possibility appears in the case when the resonance is seated on 0.1% width luminosity peak - one could resolve the stoponium exited states.', 'hep-ph-0012175-4-3-0': '# Introduction', 'hep-ph-0012175-4-4-0': 'The broken supersymmetry is favorite among the different extensions of the Standard Model.', 'hep-ph-0012175-4-4-1': 'It can happen that superpartners of top-quarks (stops, [MATH]) are long-living enough to compose (colorless) bound states, stoponiums, denoted as [MATH] in what follows.', 'hep-ph-0012175-4-4-2': 'In this scenario experimental study of the corresponding resonances could provide precise value of stop mass and stoponium partial widths, consequently yielding precise values of various parameters of SUSY Lagrangian.', 'hep-ph-0012175-4-4-3': 'Then, if the difference between stop and LSP masses is very small, the search for stop evidence in collisions at high energy could be problematic.', 'hep-ph-0012175-4-4-4': 'Observation of stoponium bound states will be the signature of such models confirming the existence of stop.', 'hep-ph-0012175-4-5-0': 'There are theoretical motivations for stop to be fairly light.', 'hep-ph-0012175-4-5-1': 'First one appeals to the renormalization group behavior of soft mass terms.', 'hep-ph-0012175-4-5-2': 'Indeed, gauge couplings raise while Yukawa couplings reduce these terms when energy scale evolves down, with Yukawa contributions being very large for stop.', 'hep-ph-0012175-4-5-3': 'The next motivation concerns left-right mixing in squark sector, which is proportional to Yukawa coupling and decreases the mass of the lightest stop.', 'hep-ph-0012175-4-5-4': 'Therefore, light stop may appear in different SUSY models ( see, e.g., Refs. [CITATION] for examples in the frameworks of supergravity and gauge mediation).', 'hep-ph-0012175-4-5-5': 'Experimental bound on stoponium mass comes from searches for stop at LEP2 and TEVATRON, it depends on the MSSM spectrum [CITATION]: lower bound is about 90 GeV for sneutrino masses larger than 45-50 GeV or for neutralino masses larger than 50 GeV (ALEPH), while CDF excludes stop mass up to 130 GeV for smaller sneutrino masses.', 'hep-ph-0012175-4-5-6': 'The limitation is weaker if stop and neutralino masses are degenerate, it is about 60 GeV (ALEPH).', 'hep-ph-0012175-4-6-0': 'Stoponium was studied in Refs. [CITATION] in detail, in particular its effective couplings and partial widths were calculated, and prospects to be discovered at LHC were estimated.', 'hep-ph-0012175-4-6-1': 'In Ref. [CITATION] it was briefly mentioned also the possibility to observe stoponiums in photon collisions, however, without analyzing this phenomenology.', 'hep-ph-0012175-4-6-2': 'The first look at the PLC prospects on the stoponium search was done in [CITATION].', 'hep-ph-0012175-4-7-0': 'Now, when main parameters of TESLA project are under technical discussion one should understand clearly signatures of stop bound states in [MATH] and [MATH] collisions, and compare potentials of these two options of future linear collider.', 'hep-ph-0012175-4-7-1': 'In present analysis we will use year integrated luminosity for the [MATH] option equal to 300 fb[MATH] for [MATH] GeV, and one should rescale this figure for lower collision energies approximately as [MATH] [CITATION].', 'hep-ph-0012175-4-7-2': 'Beamstrahlung and ISR will affect considerably the stoponium production rate due to a condition to tune the collision energy at the resonance point.', 'hep-ph-0012175-4-7-3': 'According to the current TESLA reference parameter set the average energy loss due to beamstrahlung is about 3.', 'hep-ph-0012175-4-7-4': 'The average energy loss due to ISR is about 5 [CITATION].', 'hep-ph-0012175-4-7-5': 'Correspondingly, the luminosity distribution has a characteristic width about 5.', 'hep-ph-0012175-4-8-0': 'However, stoponium is very narrow resonance, even more narrow than the initial beam energy spread.', 'hep-ph-0012175-4-8-1': 'Its production rate is proportional to the differential luminosity, [MATH], at the [MATH] peak, where [MATH] is a collision energy for hard subprocess.', 'hep-ph-0012175-4-8-2': 'The initial (beam) energy spread for energies 200-500 GeV is about 0.07-0.1 and it is determined by the bunch compression system and undulator for the positron production.', 'hep-ph-0012175-4-8-3': 'The fraction of the luminosity in this [MATH]0.1 peak is determined by the ISR and beamstrahlung.', 'hep-ph-0012175-4-8-4': 'The ISR leaves in this peak about 50 of the luminosity [CITATION].', 'hep-ph-0012175-4-8-5': 'The average number of beamstrahlung photons with the energy more than [MATH] is about [MATH] per electron for TESLA conditions.', 'hep-ph-0012175-4-8-6': 'Thus, the probability of the e[MATH]e[MATH] collision without such beamstrahlung photons can be estimated as [MATH].', 'hep-ph-0012175-4-8-7': 'The probability of events without any photons (ISR or beamstrahlung ones) with the energies greater 0.1 is [MATH].', 'hep-ph-0012175-4-8-8': 'So, about 17 of the luminosity is concentrated in [MATH]0.1 range.', 'hep-ph-0012175-4-8-9': 'The differential luminosity in this peak is higher than in 5 interval by a factor of [MATH] times (here factor 0.8 is due to ISR in the 5 region).', 'hep-ph-0012175-4-8-10': 'This peak could be very important factor for increasing significance of the resonance (when its mass is known), for measurement of its mass and even for resolving close excited states of the stoponium.', 'hep-ph-0012175-4-8-11': 'Note, however, that for a search for the stoponium this peak will not help, because in presence of large background the scanning time in some wide energy interval, required for the resonance observation, has the following dependence on the bin width [MATH] and the differential luminosity, [MATH] (here bin [MATH] corresponds to the effective width of the luminosity distribution in the peak).', 'hep-ph-0012175-4-8-12': 'So, the ratio of scanning times for the 5% and the 0.1% width peaks is [MATH], almost the same.', 'hep-ph-0012175-4-9-0': 'Photon colliders based on Compton backscattering of laser photons on high energy electrons has been proposed a long time ago [CITATION].', 'hep-ph-0012175-4-9-1': 'This option has been included in the TESLA Conceptual Design Report [CITATION] and work on the Technical Design Report is under way.', 'hep-ph-0012175-4-9-2': 'Since the CDR parameters of TESLA were changed and luminosities have grown both in e[MATH]e[MATH] and [MATH] collisions.', 'hep-ph-0012175-4-10-0': 'At the present workshop it was reported [CITATION] that PLC luminosity can be further increased by a factor of 2.5 due to possible decrease of the horizontal beam emittance at the TESLA damping ring (however this has not improved e[MATH]e[MATH] luminosity because it is restricted by collision effects).', 'hep-ph-0012175-4-10-1': 'At present the [MATH] luminosity within the 20% interval just below [MATH] could be about 40% of luminosity in the e[MATH]e[MATH] collisions (where [MATH]^+[MATH]^-[MATH] at [MATH] GeV).', 'hep-ph-0012175-4-10-2': 'In the analysis we will consider 60 fb[MATH] for PLC year luminosity at [MATH] GeV.', 'hep-ph-0012175-4-11-0': 'The most bright evidence of the narrow resonance is its direct s-channel production in [MATH] annihilation or [MATH] fusion.', 'hep-ph-0012175-4-11-1': 'One can note, that high powers of the coupling constants, [MATH], emerge in the squared matrix elements in the [MATH] annihilation into stoponium.', 'hep-ph-0012175-4-11-2': 'Indeed, [MATH] arises from two electroweak vertices, and [MATH] comes from squared derivative of the stoponium wave function (scalar stoponium can be created there only in P-wave by propagation of neutral vector particle: photon or [MATH] boson).', 'hep-ph-0012175-4-11-3': 'At the same time two powers of [MATH] are eliminated in the case of [MATH] fusion mechanism because the stoponium production can be proceeded in S-wave there.', 'hep-ph-0012175-4-11-4': 'This circumstance makes for the relative enhancement of the stoponium production rate at PLC in comparison with [MATH] option.', 'hep-ph-0012175-4-11-5': 'Two powers of [MATH] are eliminated also in the case of associated production of stoponiums, for example in the Higgs-like reactions [MATH] and [MATH].', 'hep-ph-0012175-4-11-6': 'Hereafter we denote stoponium as [MATH].', 'hep-ph-0012175-4-12-0': 'To complete this brief review of possible production mechanisms one can note that in hadron collisions the stoponium resonance production is available in S-wave through the gluon fusion.', 'hep-ph-0012175-4-12-1': 'Here the effective [MATH] vertex includes [MATH] and one can anticipate large stoponium cross sections.', 'hep-ph-0012175-4-12-2': 'However, main decay channel [MATH] is too dirty due to huge QCD [MATH] background.', 'hep-ph-0012175-4-12-3': 'Then, the most promising decay channel at LHC is [MATH] [CITATION], but in order to discover stoponiums one year of LHC operating at high luminosity is needed, or even more depending on SUSY scenario.', 'hep-ph-0012175-4-13-0': 'We consider stoponium mass range [MATH] GeV, which could be surely probed by first TESLA run.', 'hep-ph-0012175-4-13-1': 'It is worth to note that the same interval is not an exceptional case for SUSY models with stoponiums as a quasistationary state, as we discuss briefly in the next section.', 'hep-ph-0012175-4-14-0': 'Cross sections for various tree-level background processes were evaluated with the help of CompHEP package [CITATION].', 'hep-ph-0012175-4-15-0': '# Stop bound states', 'hep-ph-0012175-4-16-0': 'It is clear that gluons try to bind two stops as well as ordinary quarks.', 'hep-ph-0012175-4-16-1': 'The corresponding bound state can be described as a quasistationary system with energy levels [MATH]) and masses [MATH] similarly to quarkonium.', 'hep-ph-0012175-4-16-2': 'For stoponium mass [MATH] GeV the binding energies [MATH] are of order 1 GeV [CITATION].', 'hep-ph-0012175-4-16-3': 'This treatment is valid if the formation process (time scale [MATH]) is faster than destroying one.', 'hep-ph-0012175-4-17-0': 'Among destroying mechanisms the obvious ones are the stop decays: [MATH]LSP, [MATH] and [MATH].', 'hep-ph-0012175-4-17-1': 'At first, let us consider the third decay.', 'hep-ph-0012175-4-17-2': 'It proceeds only through loop diagrams, if Universal boundary condition on soft terms is imposed (that is motivated by the absence of FCNC).', 'hep-ph-0012175-4-17-3': 'So, partial width is highly reduced by a factor of [MATH] in comparison with the first two tree-level decay processes [CITATION].', 'hep-ph-0012175-4-17-4': 'The rates of latter decays depend on the parameters of the model.', 'hep-ph-0012175-4-17-5': 'As an example, in the framework of gravity mediation, where LSP is neutralino, these decays proceed at the tree level and the corresponding partial widths are of order O[MATH].', 'hep-ph-0012175-4-17-6': 'In the framework of models with gauge mediated supersymmetry breaking [CITATION], where LSP is gravitino, the first process is strongly suppressed by supersymmetry breaking scale, but remaining one has the same partial width as in gravity mediation.', 'hep-ph-0012175-4-17-7': 'Hence, the possibility of existence of stoponium is a subject of special study in each concrete model.', 'hep-ph-0012175-4-17-8': 'For instance, in models with the lightest chargino being mostly wino and the lightest stop being mostly right stop (i.e., [MATH]), decay into chargino is damped and stoponium could exist if [MATH], i.e., when the first decay channel is kinematically forbidden.', 'hep-ph-0012175-4-17-9': 'One can state that SUSY scenario, where tree-level decays, [MATH]LSP and [MATH], are somehow suppressed and, therefore, stop decay can not destroy the stoponium formation, is not an exceptional case.', 'hep-ph-0012175-4-18-0': 'Next destroying mechanism is related to the stop annihilation.', 'hep-ph-0012175-4-18-1': 'Here two gluon channel is always open with partial width about 1 MeV.', 'hep-ph-0012175-4-18-2': 'Generally the gluon channel is dominant.', 'hep-ph-0012175-4-18-3': 'However, for the certain choice of model parameters, partial width into two lightest Higgs bosons, [MATH], can be larger, increasing the stoponium total width by a factor of [MATH].', 'hep-ph-0012175-4-18-4': 'In Ref. [CITATION] these figures were analyzed and found that quasistationary description is valid for [MATH] GeV in models with forbidden stop tree-level decays and neutralino being mostly bino.', 'hep-ph-0012175-4-18-5': 'The worst case is a model with chargino and neutralino states are both higgsino-like.', 'hep-ph-0012175-4-18-6': 'Here the stoponium total width increases rapidly with [MATH] and quasistationary treatment fails for [MATH] GeV.', 'hep-ph-0012175-4-19-0': '# Stoponium in [MATH] collisions', 'hep-ph-0012175-4-20-0': 'Let us begin with study of stoponium events in photon-photon scattering.', 'hep-ph-0012175-4-20-1': 'The main effect associated with stoponium would be a direct resonance production, where stoponium is produced in spin 0 state.', 'hep-ph-0012175-4-20-2': 'The corresponding cross section is described by the Breit-Wigner formula (similarly to the light Higgs production discussed in Ref. [CITATION]) [EQUATION] where [MATH] is squared colision energy for hard subprocess, [MATH] is the stoponium partial width for the decay into state [MATH], [MATH] is stoponium total width, [MATH] are helicities of initial photons.', 'hep-ph-0012175-4-21-0': 'At photon colliders the width of the luminosity distribution is much wider than that of the stoponium.', 'hep-ph-0012175-4-21-1': 'After integration over the luminosity distribution we obtain the effective cross section [EQUATION]', 'hep-ph-0012175-4-21-2': 'The differential luminosity, [MATH][MATH], at [MATH] can be estimated as [MATH][MATH] according to 15% width of the high energy luminosity peak (note, we define the [MATH] luminosity as the luminosity in the interval [MATH]%).', 'hep-ph-0012175-4-21-3': 'So the effective cross section is [EQUATION]', 'hep-ph-0012175-4-21-4': 'Photon beams are planned to be highly polarized.', 'hep-ph-0012175-4-21-5': 'Hence, as stoponium is a scalar the production cross section will be enhanced by factor two if initial photons have total helicity equal to zero.', 'hep-ph-0012175-4-21-6': 'Hereafter we assume initial total helicity 0 ([MATH]) in numerical estimates.', 'hep-ph-0012175-4-21-7': 'Finally, we parameterize the stoponium cross sections as follows [EQUATION]', 'hep-ph-0012175-4-21-8': 'One should take into account that squared stoponium wave function at the origin, attending in [MATH], scales as a square root of its mass [CITATION].', 'hep-ph-0012175-4-22-0': 'As it has been stressed above one can discuss two main variants of the SUSY models, one with dominant [MATH] decay mode and another with stoponium total width being saturated by [MATH] mode.', 'hep-ph-0012175-4-22-1': 'Let us make qualitative signal/background analysis for different decay channels within these two variants.', 'hep-ph-0012175-4-22-2': 'The signal significance can be evaluated by ratio [MATH] because one deals with resonance and background rate in the signal bin can be fixed as average cross section in neighboring bins (here [MATH] are numbers of signal and background events).', 'hep-ph-0012175-4-22-3': 'We used the results for stoponium width and branching ratios calculated in Ref. [CITATION] with some corrections [CITATION].', 'hep-ph-0012175-4-23-0': '1.', 'hep-ph-0012175-4-23-1': 'In the first scenario stoponium total width [MATH] MeV and photon-photon branching is [MATH].', 'hep-ph-0012175-4-23-2': 'By making use of Eq. ([REF]) one obtains the signal rate at the level of 110 fb for [MATH] GeV.', 'hep-ph-0012175-4-23-3': 'So, more than six thousand stoponiums will be produced per year if L[MATH] fb[MATH].', 'hep-ph-0012175-4-23-4': 'Background is two jet production, where subprocess [MATH] gives main contribution with very large unpolarized cross section, [MATH] pb, if optimal cut on the jet angle of 45[MATH] is applied.', 'hep-ph-0012175-4-23-5': 'However, production of fermions is suppressed in collisions of photons with the total helicity 0 [CITATION]: [EQUATION] where [MATH] is velocity of the quarks.', 'hep-ph-0012175-4-23-6': 'So, [MATH] pairs are produced only in collisions of photons with the total helicity 2.', 'hep-ph-0012175-4-23-7': 'This fact is used for suppression of similar background in the analysis of the Higgs production in [MATH] collisions [CITATION].', 'hep-ph-0012175-4-23-8': 'The corresponding luminosity spectra for total helicity 0 ([MATH]) and 2 ([MATH]) can be found elsewhere [CITATION].', 'hep-ph-0012175-4-23-9': 'In the region [MATH]10% near the maximum energy (detector resolution or intrinsic resolution of the PLC for two collinear jets) one can obtain the ratio of the luminosities [MATH], that assumes at least one order suppression of the [MATH] background.', 'hep-ph-0012175-4-23-10': 'Note that the cross section 6 pb for background corresponds to the case of unpolarized beams.', 'hep-ph-0012175-4-23-11': 'It is zero for collisions of photons with the total helicity 0 and is equal to 12 pb for the total helicity 2, therefore the remaining cross section is 1.2 pb.', 'hep-ph-0012175-4-23-12': 'Note that due to gluon emission [MATH] pairs can be produced even in collisions of photons with the total helicity 0.', 'hep-ph-0012175-4-23-13': 'Detailed studies have shown that with proper cuts this process is not important [CITATION] and contribution from resolved photons is also not significant [CITATION].', 'hep-ph-0012175-4-24-0': 'Furthermore, the cross section is proportional to the fourth power of the electric charge of quarks, so the main contribution is given by [MATH] and [MATH] quarks.', 'hep-ph-0012175-4-24-1': 'The later can be easily suppressed by the vertex detector.', 'hep-ph-0012175-4-24-2': 'This gives additional factor of 2 in the background suppression.', 'hep-ph-0012175-4-24-3': 'Additional improvement can give the detector energy resolution which is at least factor of 2 smaller than the width of the [MATH] luminosity peak.', 'hep-ph-0012175-4-24-4': 'All three methods give a suppression factor of 40.', 'hep-ph-0012175-4-24-5': 'The remaining background is about 0.3 pb, while the isotropic signal is smaller by factor 1.4 only if the cut on the jet angle of 45[MATH] is applied.', 'hep-ph-0012175-4-24-6': 'Hence in the scenario with dominant [MATH] channel the signal/background ratio is [MATH].', 'hep-ph-0012175-4-24-7': 'The signal significance is about 35 for 60 fb[MATH] and [MATH] GeV.', 'hep-ph-0012175-4-25-0': 'These figures can be related to the first year of PLC operation if the stoponium mass is known approximately, for example from the observation of the direct stops production at LHC.', 'hep-ph-0012175-4-25-1': 'In opposite case, for a search of a stoponium one should make scanning with the energy bin [MATH] 10%.', 'hep-ph-0012175-4-25-2': 'It is clear that stoponium can be found in this scenario during several month work in the whole energy region under discussion, 200-400 GeV.', 'hep-ph-0012175-4-26-0': 'For two photon channel the background process, [MATH], proceeds through one-loop diagrams, so the corresponding cross section is small, about 10 fb [CITATION].', 'hep-ph-0012175-4-26-1': 'One should note that the photon-photon invariant mass bin can be taken equal to [MATH] for CMS-like crystal electromagnetic calorimeter [CITATION].', 'hep-ph-0012175-4-26-2': 'Thus, for [MATH] GeV window the background rate can be estimated at the level of 1 fb.', 'hep-ph-0012175-4-26-3': 'The signal rate is [MATH] fb for [MATH] GeV, providing the signal significance about [MATH] for statistics 60 fb[MATH].', 'hep-ph-0012175-4-27-0': 'Some other decay channels in the framework of the first scenario should be discussed.', 'hep-ph-0012175-4-27-1': 'First note, that [MATH] final state has no chance for the detection of stoponiums due to huge SM background, [MATH] pb at [MATH] GeV.', 'hep-ph-0012175-4-27-2': 'More promising are decay channels with background processes emerging due to the higher order corrections from perturbation theory.', 'hep-ph-0012175-4-27-3': 'For instance, SM background to [MATH] and [MATH] final states comes from 1) one-loop [MATH] processes [MATH] fb [CITATION]) and [MATH] fb [CITATION]), and 2) from tree-level [MATH] processes (e.g. [MATH] for [MATH]), with total cross section smaller than 1 fb within cuts on final [MATH] and jets reasonably motivated by 2-body ([MATH]) kinematics of the signal events.', 'hep-ph-0012175-4-28-0': 'As to signal [MATH] rate one can get from Ref. [CITATION] the branching [MATH], so [MATH] fb already for [MATH] GeV.', 'hep-ph-0012175-4-28-1': 'It means very low level of the signal significance, lower than 0.5 for statistics 60 fb[MATH].', 'hep-ph-0012175-4-29-0': 'Natural level of [MATH] branching is about [MATH] for stoponium masses far from the threshold, 250-400 GeV, although in some points it could fall down due to opening of new channels or degeneration of stoponium and Higgs masses.', 'hep-ph-0012175-4-29-1': 'This provides signal rate [MATH] fb for [MATH] GeV and significance at the level of 2.7 if one uses formula ([REF]).', 'hep-ph-0012175-4-29-2': 'However, the threshold effect is significant still for this value of the stoponium mass, and these figures should be improved to 1.8 fb for signal rate and to 2 for the significance.', 'hep-ph-0012175-4-30-0': 'The [MATH] decay channel, where [MATH] is the lightest Higgs boson, is open if [MATH].', 'hep-ph-0012175-4-30-1': 'As current limit on [MATH] mass is about 80-100 GeV this channel could exist for [MATH] GeV.', 'hep-ph-0012175-4-30-2': 'If consider mass region not very close to the threshold (say [MATH] GeV for [MATH] GeV) the [MATH] branching is about [MATH] or even higher.', 'hep-ph-0012175-4-30-3': 'In this case the signal rate is about 1 fb or larger, if one uses formula ([REF]), and if take into account the threshold factor one gets signal cross section at the level of 0.2 fb or larger.', 'hep-ph-0012175-4-30-4': 'The background from direct double [MATH] production through one-loop diagrams can be estimated by the cross section of this process in SM, [MATH] fb [CITATION].', 'hep-ph-0012175-4-30-5': 'There are no reasons for very large additional contributions to this process in supersymmetric models.', 'hep-ph-0012175-4-30-6': 'Then, we found that direct electroweak production of four b quarks ([MATH]) together with contribution from [MATH] final state (assuming 10 of [MATH] misidentification) has the rate smaller than 0.1 fb.', 'hep-ph-0012175-4-30-7': 'These cross sections for background processes correspond to 15 width of the [MATH] luminosity spectrum.', 'hep-ph-0012175-4-30-8': 'The detector resolution is at least factor of two better (full width), therefore the total cross section of background processes can be estimated as (0.2+0.1)/2 = 0.15 fb.', 'hep-ph-0012175-4-30-9': 'We see that [MATH] decay can also be studied in the considered scenario.', 'hep-ph-0012175-4-30-10': 'For [MATH] GeV, [MATH] GeV and 60 fb[MATH] integrated luminosity about 20 stoponium events will be produced in the decay channel [MATH] with [MATH] ratio about 2.3 and statistical significance about 7.', 'hep-ph-0012175-4-31-0': 'In Fig. [REF] the signal significances are represented for main stoponium decay channels in the case of the first scenario.', 'hep-ph-0012175-4-31-1': 'As a resume for this scenario we conclude that stoponium can be found at PLC during several months scan of the 200-400 GeV region in [MATH] and [MATH] decay modes.', 'hep-ph-0012175-4-31-2': 'If its mass is known approximately, it will be found during first weeks.', 'hep-ph-0012175-4-31-3': 'More than a year is necessary in order to observe stoponium in [MATH] and [MATH] channels.', 'hep-ph-0012175-4-32-0': 'Note that LHC (at the high luminosity operating stage) has good prospects to observe light stoponium in [MATH] mode in this scenario [CITATION].', 'hep-ph-0012175-4-32-1': 'Thus, these two colliders could be complementary in study of different effective stoponium couplings, [MATH] and [MATH] at LHC and the photon collider, respectively and [MATH] at PLC if this decay channel is open kinematically.', 'hep-ph-0012175-4-33-0': '2.', 'hep-ph-0012175-4-33-1': 'In the second scenario stoponium total width could be about 10 MeV or even larger.', 'hep-ph-0012175-4-33-2': 'The photon-photon branching in this case is smaller, [MATH].', 'hep-ph-0012175-4-33-3': 'So, about thousand of stoponiums will be produced per year and almost all of them will decay to pairs of lightest Higgs bosons.', 'hep-ph-0012175-4-33-4': 'This result suggests, that stoponium will be discovered practically immediately after PLC start, since the background ([MATH], [MATH] ...) is very small.', 'hep-ph-0012175-4-33-5': 'Again the scanning over the energy interval is necessary if the stoponium mass is not known.', 'hep-ph-0012175-4-33-6': 'In Fig. [REF] the year yields of stoponiums are represented for different masses of lightest Higgs bosons.', 'hep-ph-0012175-4-34-0': 'Due to rather high statistics for the signal and absence (practically) of the background one gets PLC as a stoponium factory.', 'hep-ph-0012175-4-34-1': 'The detailed study of the stoponium characteristics will be available in this case, in particular measurement of its mass and total width and effective couplings [MATH] and [MATH].', 'hep-ph-0012175-4-34-2': 'One can stress the importance of study the [MATH] coupling, which relates directly to the stop-Higgs interaction and, thus, to the mechanism of the superpartner mass generation.', 'hep-ph-0012175-4-35-0': 'Note that in this scenario several years of operating at high luminosity is needed in order to observe stoponium at LHC [CITATION].', 'hep-ph-0012175-4-36-0': '# Stoponium in [MATH] collisions', 'hep-ph-0012175-4-37-0': '1.', 'hep-ph-0012175-4-37-1': 'First we discuss direct resonant production of stoponiums in [MATH] collisions where initial electron and positron would be in P-state (thus, stoponium is produced in spin 1 state).', 'hep-ph-0012175-4-37-2': 'As in case of [MATH] collisions one gets for narrow resonance after the integration of Breit-Wigner distribution the following formula for production cross section [EQUATION] where [MATH] are helicities of initial electrons and positrons.', 'hep-ph-0012175-4-37-3': 'The differential luminosity at [MATH] can be estimated as [MATH], where [MATH] is a fraction of the [MATH] luminosity corresponding to some interval [MATH] near the [MATH] peak.', 'hep-ph-0012175-4-37-4': 'Let us consider two methods of the stoponium detection:', 'hep-ph-0012175-4-38-0': '1) [MATH]% [MATH]^+[MATH]^-[MATH], [MATH]%;', 'hep-ph-0012175-4-39-0': '2) [MATH]% [MATH]^+[MATH]^-[MATH], [MATH]%.', 'hep-ph-0012175-4-40-0': 'Then, the stoponium partial width into [MATH] pair, [MATH], is equal to [CITATION] [EQUATION] with [MATH] being wave function of P-state.', 'hep-ph-0012175-4-40-1': 'In correspondence with Ref. [CITATION] one can approximate [MATH] GeV.', 'hep-ph-0012175-4-41-0': 'We assume, also, 80% circular polarization for the electron beam and 60% for the positron beam, and neglect fairly weak enhancement by [MATH]-boson pole for light stoponiums.', 'hep-ph-0012175-4-42-0': 'Finally, for the first method of the stoponium detection one gets for the production rate in [MATH] collisions the following estimate [EQUATION] while it is about 5 times larger in case of second method.', 'hep-ph-0012175-4-43-0': 'Let us now consider background for the direct resonance production of stoponiums in [MATH] collisions.', 'hep-ph-0012175-4-43-1': 'One should note that stoponium, being produced in excited state with spin 1, will be transfered to the basic state with spin 0 by the emission of a photon.', 'hep-ph-0012175-4-43-2': 'Therefore, one can consider decay modes of scalar stoponium in case of [MATH] production as well.', 'hep-ph-0012175-4-43-3': 'Then, additional photon emmited by excited stoponium could be used for further suppression of the background.', 'hep-ph-0012175-4-43-4': 'However, the detection of this photon could be not easy because its energy is rather small, of order 1 GeV.', 'hep-ph-0012175-4-44-0': 'In the case of first MSSM scenario with [MATH] decay channel being dominant the main background is [MATH] process.', 'hep-ph-0012175-4-44-1': 'It has a cross section about 10 pb for [MATH] GeV.', 'hep-ph-0012175-4-44-2': 'So, this channel is too dirty with negligible significance.', 'hep-ph-0012175-4-45-0': 'In the case of second MSSM scenario with [MATH] dominant decay mode almost all stoponiums will decay into pair of lightest Higgs bosons, and the number of events for [MATH] GeV and 100 [MATH] is 5 (25) (the second number is for the case of seating on the 0.1% luminosity peak).', 'hep-ph-0012175-4-45-1': 'As direct pair production of Higgs bosons, [MATH], is negligible since the corresponding [MATH] coupling includes electron mass, this channel is should be practically free of background.', 'hep-ph-0012175-4-45-2': 'Indeed, direct production of four b-quarks in electron-positron collisions has very small cross section if exclude [MATH] peaks (less than 0.05fb if apply the cut [MATH] GeV).', 'hep-ph-0012175-4-45-3': 'So, stoponium can be observed at [MATH] machine if the collision energy is tuned at [MATH].', 'hep-ph-0012175-4-46-0': 'Note, that in this scenario the yield of stoponiums at PLC with [MATH][MATH]^+[MATH]^-[MATH] is higher than that in e[MATH]e[MATH] collisions by a factor of 250(50) (second number for the case of seating on the 0.1% peak of the [MATH] luminosity).', 'hep-ph-0012175-4-46-1': 'Thus, the scanning time in e[MATH]e[MATH] mode will be about factor of 500 longer than at PLC due to smaller cross section and smaller energy bin (see also discussion on the scanning time in the Introduction).', 'hep-ph-0012175-4-47-0': '2.', 'hep-ph-0012175-4-47-1': 'The second effect related to stoponium at [MATH] collider would be a production in Higgs-like channels.', 'hep-ph-0012175-4-47-2': 'In present analysis we neglect Higgs-stoponium mixing as well as Higgs influence on stoponium-involved processes.', 'hep-ph-0012175-4-47-3': 'First, let us evaluate effective coupling constants between stoponium and weak bosons, [EQUATION] where we used stoponium wave function evaluated at the origin in Ref. [CITATION]; [MATH] and [MATH] are corresponding amplitudes of [MATH] processes evaluated in Ref. [CITATION].', 'hep-ph-0012175-4-47-4': 'Certainly, these effective couplings are obtained with all particles being on-shell, that is rather rough approximation.', 'hep-ph-0012175-4-47-5': 'However, one can hope that it is acceptable for the estimate.', 'hep-ph-0012175-4-48-0': 'The calculation with the effective couplings above gives the cross section about 0.03 fb for [MATH] (in case of no mixing in stop sector and [MATH] TeV).', 'hep-ph-0012175-4-48-1': 'So, only a few signal events will be produced for 100 fb[MATH] statistics.', 'hep-ph-0012175-4-48-2': 'The [MATH] background is too heavy for this level of the signal, so only the second scenario could have some prospects.', 'hep-ph-0012175-4-48-3': 'The main background will come from associated double Higgs bosons production, [MATH], the corresponding cross section is of order [MATH] fb for [MATH] [CITATION].', 'hep-ph-0012175-4-48-4': 'It gives the signal significance less than 1.', 'hep-ph-0012175-4-49-0': 'In the case of [MATH]-fusion the signal cross section is very small, less than [MATH] fb, that closes this channel.', 'hep-ph-0012175-4-50-0': '# Conclusions', 'hep-ph-0012175-4-51-0': 'In two considered scenarios of supersymmetric extension of the Standard Model (1st one with dominant [MATH] decay mode, and [MATH] being dominant in 2nd scenario) Photon Linear Collider will be the best machine to discover and study bound state of stops, if it exists.', 'hep-ph-0012175-4-52-0': 'In case of 1st scenario stoponium will be observed at PLC in the [MATH] and [MATH] (later if permitted kinematically) in the beginning of the operation - several months for scanning of whole energy region or several weeks if the stoponium mass is approximately known.', 'hep-ph-0012175-4-53-0': 'In case of 2nd scenario, about thousand of stoponiums will be produced free of background.', 'hep-ph-0012175-4-53-1': 'It means that stoponium can be discovered at PLC practically immediately.', 'hep-ph-0012175-4-54-0': 'Study of the effective stoponium couplings with photons, gluons and lightest Higgs bosons will be available at PLC, latter two depending on the MSSM scenario.', 'hep-ph-0012175-4-54-1': 'Measurement of the stoponium mass and total width (extracted from the measured signal rate) will be possible also.', 'hep-ph-0012175-4-55-0': 'Stoponium discovery mass range will be limited only by attainable values of the [MATH] collision energy, which is discussed up to [MATH] GeV.', 'hep-ph-0012175-4-55-1': 'The tuning of PLC collision energy at the resonance point is necessary within the 15% window.', 'hep-ph-0012175-4-55-2': 'These estimates were done for the case of PLC year luminosity being 60 fb[MATH].', 'hep-ph-0012175-4-56-0': 'In [MATH] collisions stoponium could be observed only in the scenario with [MATH] decay channel being dominant, but with the rate lower than in the PLC case by a factor of 250 (50) (the second number for seating on the 0.1 luminosity peak).', 'hep-ph-0012175-4-56-1': 'In this comparison it was assumed that [MATH] luminosity in the high energy peak is equal 40 of e[MATH]e[MATH] luminosity.', 'hep-ph-0012175-4-56-2': 'Search time here is about 500 times longer than in [MATH] collisions.', 'hep-ph-0012175-4-56-3': 'However, there is one important advantage of e[MATH]e[MATH] collisions: by use of very monochromatic part of the luminosity spectrum (0.1%) one can make precise measurement of the stoponium mass and resolve its excited states.', 'hep-ph-0012175-4-56-4': 'Although this is possible only in the scenario when [MATH] decay dominates.', 'hep-ph-0012175-4-57-0': 'A few further comments can be made.', 'hep-ph-0012175-4-57-1': 'The first one is related to the circumstance that at photon colliders ground state of stoponium could not be distinguished from excited states due to the detector resolutions.', 'hep-ph-0012175-4-57-2': 'Therefore, the resonance peak will include contributions from ground state and all excitations, leading to enhancement factor of about 2 in all cross sections [CITATION].', 'hep-ph-0012175-4-57-3': 'At the same time there is a big uncertainty because of poor understanding of the stoponium wave function, that results in 30-50% error when the stoponium rates are estimated [CITATION].', 'hep-ph-0012175-4-58-0': 'Then, stoponium has the same quantum numbers as neutral Higgs bosons.', 'hep-ph-0012175-4-58-1': 'Thus, interesting phenomena could appear due to the interference of stoponium with Higgs sector.', 'hep-ph-0012175-4-58-2': 'This point was discussed briefly in Ref. [CITATION].', 'hep-ph-0012175-4-59-0': 'The authors are indebted to I.F. Ginzburg, H.J. Schreiber and other participants of the present Workshop for useful discussions.', 'hep-ph-0012175-4-59-1': 'The work of D.G. was supported in part under RFBR grant 99-01-18410, CRDF grant (award RP1-2103), Swiss Science Foundation grant 7SUPJ062239, Russian Academy of Science JRP grant 37 and by ISSEP fellowship.', 'hep-ph-0012175-4-59-2': 'The work of V.I. has been partially supported by the CERN-INTAS 99-377 and RFBR-DFG 99-02-04011 grants.'}

1908.04120

{'1908.04120-1-0-0': 'As an extension of the Ivanov-Zupnik approach to self-dual nonlinear electrodynamics in four dimensions [CITATION], we reformulate U(1) duality-invariant nonlinear models for a gauge [MATH]-form in [MATH] dimensions as field theories with manifestly U(1) invariant self-interactions.', '1908.04120-1-0-1': 'This reformulation is suitable to generate arbitrary duality-invariant nonlinear systems including those with higher derivatives.', '1908.04120-1-1-0': '# Introduction', '1908.04120-1-2-0': 'As an extension of the seminal work by Gaillard and Zumino [CITATION], the general formalism of duality-invariant models for nonlinear electrodynamics in four dimensions was developed in the mid-1990s [CITATION].', '1908.04120-1-2-1': 'The Gaillard-Zumino-Gibbons-Rasheed (GZGR) approach was generalised to off-shell [MATH] and [MATH] globally [CITATION] and locally [CITATION] supersymmetric theories.', '1908.04120-1-2-2': 'In particular, the first consistent perturbative scheme to construct the [MATH] supersymmetric Born-Infeld action was given in [CITATION] (this approach was further pursued in [CITATION]).', '1908.04120-1-2-3': 'The GZGR formalism was also extended to higher dimensions [CITATION].', '1908.04120-1-3-0': 'Nonlinear electrodynamics with U(1) duality symmetry is described by a Lorentz invariant Lagrangian [MATH] which is a solution to the self-duality equation [EQUATION] where [EQUATION]', '1908.04120-1-3-1': 'In the case of theories with higher derivatives, this scheme is generalised in accordance with the two rules given in [CITATION].', '1908.04120-1-3-2': 'Firstly, the definition of [MATH] is replaced with [EQUATION]', '1908.04120-1-3-3': 'Secondly, the self-duality equation [REF] is replaced with [EQUATION]', '1908.04120-1-3-4': 'Duality-invariant theories with higher derivative theories naturally occur in [MATH] supersymmetry [CITATION].', '1908.04120-1-3-5': 'Further aspects of duality-invariant theories with higher derivatives were studied in, e.g., [CITATION].', '1908.04120-1-4-0': 'Self-duality equation [REF] is nonlinear, and therefore its general solutions are difficult to find.', '1908.04120-1-4-1': 'In the early 2000s, Ivanov and Zupnik [CITATION] proposed a reformulation of duality-invariant electrodynamics involving an auxiliary antisymmetric tensor [MATH], which is equivalent to a symmetric spinor [MATH] and its conjugate [MATH].', '1908.04120-1-4-2': 'The new Lagrangian [MATH] is at most quadratic in the electromagnetic field strength [MATH], while the self-interaction is described by a nonlinear function of the auxiliary variables, [MATH], [EQUATION]', '1908.04120-1-4-3': 'The original theory [MATH] is obtained from [MATH] by integrating out the auxiliary variables.', '1908.04120-1-4-4': 'In terms of [MATH], the condition of U(1) duality invariance was shown [CITATION] to be equivalent to the requirement that the self-interaction [EQUATION] is invariant under linear U(1) transformations [MATH], with [MATH], and thus [EQUATION] where [MATH] is a real function of one real variable.', '1908.04120-1-5-0': 'The above discussion shows that the Ivanov-Zupnik (IZ) approach is a universal formalism to generate U(1) duality-invariant models for nonlinear electrodynamics.', '1908.04120-1-5-1': 'Some time ago, there was a revival of interest in duality-invariant dynamical systems [CITATION] inspired by the desire to achieve a better understanding of the UV properties of extended supergravity theories.', '1908.04120-1-5-2': 'The authors of [CITATION] have put forward the so-called "twisted self-duality constraint," which was further advocated in [CITATION], as a systematic procedure to generate manifestly duality-invariant theories.', '1908.04120-1-5-3': 'However, these approaches have been demonstrated [CITATION] to be variants of the IZ scheme [CITATION] developed a decade earlier.', '1908.04120-1-6-0': 'The IZ approach has been generalised to off-shell [MATH] and [MATH] globally and locally supersymmetric theories [CITATION].', '1908.04120-1-6-1': 'In this note we provide a generalisation of the approach to higher dimensions, [MATH].', '1908.04120-1-6-2': 'In even dimensions, [MATH], the maximal duality group for a system of [MATH] gauge [MATH]-forms depends on the dimension of spacetime.', '1908.04120-1-6-3': 'The duality group is U[MATH] if [MATH] is even, and [MATH] if [MATH] is odd [CITATION] (see, e.g, section 8 of [CITATION] for a review).', '1908.04120-1-6-4': 'This is why we choose [MATH].', '1908.04120-1-6-5': 'The fact that the maximal duality group depends on the dimension of space-time was discussed in the mid-1980s [CITATION] and also in the late 1990s [CITATION].', '1908.04120-1-7-0': '# New formulation', '1908.04120-1-8-0': 'In Minkowski space of even dimension [MATH], with [MATH] a positive integer, we consider a self-interacting theory of a gauge [MATH]-form [MATH] with the property that the Lagrangian, [MATH], is a function of the field strengths [MATH].', '1908.04120-1-8-1': 'We assume that the theory possesses U(1) duality invariance.', '1908.04120-1-8-2': 'This means that the Lagrangian is a solution to the self-duality equation [CITATION] [EQUATION] where we have introduced [EQUATION]', '1908.04120-1-8-3': 'As usual, the notation [MATH] is used for the Hodge dual of [MATH], [EQUATION]', '1908.04120-1-8-4': 'We now introduce a reformulation of the above theory.', '1908.04120-1-8-5': 'Along with the field strength [MATH], our new Lagrangian [MATH] is defined to depend on an auxiliary rank-[MATH] antisymmetric tensor [MATH] which is unconstrained.', '1908.04120-1-8-6': 'We choose [MATH] to have the form [EQUATION] where we have denoted [EQUATION]', '1908.04120-1-8-7': 'The last term in [REF], [MATH], is at least quartic in [MATH].', '1908.04120-1-8-8': 'It is assumed that the equation of motion for [MATH], [EQUATION] allows one to integrate out the auxiliary field [MATH] to result with [MATH].', '1908.04120-1-9-0': 'It may be shown that the self-duality equation [REF] is equivalent to the following condition on the self-interaction in [REF] [EQUATION]', '1908.04120-1-9-1': 'Introducing (anti) self-dual components of [MATH], [EQUATION] the above condition turns into [EQUATION]', '1908.04120-1-9-2': 'This means that [MATH] is invariant under U(1) phase transformations, [EQUATION]', '1908.04120-1-9-3': 'In four dimensions, the most general solution to this condition is given by eq. [REF].', '1908.04120-1-9-4': 'Similar solutions exist in higher dimensions, [MATH], with [MATH] a real function of one variable.', '1908.04120-1-9-5': 'However more general self-interactions become possible beyond four dimensions.', '1908.04120-1-10-0': 'It is worth pointing out that an infinitesimal U(1) duality transformation [EQUATION]', '1908.04120-1-10-1': 'Equation [REF] tells us that [MATH] duality invariant.', '1908.04120-1-11-0': 'There are several interesting generalisations of the construction described.', '1908.04120-1-11-1': 'They include (i) coupling to gravity; (ii) coupling to a dilaton with enhanced SL[MATH] duality; (iii) duality-invariant systems with higher derivatives; and (iv) U[MATH] duality-invariant systems of [MATH] gauge [MATH]-forms in [MATH] dimensions.', '1908.04120-1-12-0': 'Recently, U(1) duality-invariant theories of a gauge [MATH]-form in [MATH] dimensions have been described [CITATION] within the Pasti-Sorokin-Tonin approach [CITATION].', '1908.04120-1-12-1': 'It was argued in [CITATION] that the approach of [CITATION] is the most efficient method to determine all possible manifestly U(1) duality invariant self-interactions provided Lorentz invariance is kept manifest.', '1908.04120-1-12-2': 'Our analysis has provided an alternative formalism.'}

{'1908.04120-2-0-0': 'As an extension of the Ivanov-Zupnik approach to self-dual nonlinear electrodynamics in four dimensions [CITATION], we reformulate U(1) duality-invariant nonlinear models for a gauge [MATH]-form in [MATH] dimensions as field theories with manifestly U(1) invariant self-interactions.', '1908.04120-2-0-1': 'This reformulation is suitable to generate arbitrary duality-invariant nonlinear systems including those with higher derivatives.', '1908.04120-2-1-0': '# Introduction', '1908.04120-2-2-0': 'As an extension of the seminal work by Gaillard and Zumino [CITATION], the general formalism of duality-invariant models for nonlinear electrodynamics in four dimensions was developed in the mid-1990s [CITATION].', '1908.04120-2-2-1': 'The Gaillard-Zumino-Gibbons-Rasheed (GZGR) approach was generalised to off-shell [MATH] and [MATH] globally [CITATION] and locally [CITATION] supersymmetric theories.', '1908.04120-2-2-2': 'In particular, the first consistent perturbative scheme to construct the [MATH] supersymmetric Born-Infeld action was given in [CITATION] (this approach was further pursued in [CITATION]).', '1908.04120-2-2-3': 'The GZGR formalism was also extended to higher dimensions [CITATION].', '1908.04120-2-3-0': 'Nonlinear electrodynamics with U(1) duality symmetry is described by a Lorentz invariant Lagrangian [MATH] which is a solution to the self-duality equation [EQUATION] where [EQUATION]', '1908.04120-2-3-1': 'In the case of theories with higher derivatives, this scheme is generalised in accordance with the two rules given in [CITATION].', '1908.04120-2-3-2': 'Firstly, the definition of [MATH] is replaced with [EQUATION]', '1908.04120-2-3-3': 'Secondly, the self-duality equation [REF] is replaced with [EQUATION]', '1908.04120-2-3-4': 'Duality-invariant theories with higher derivative theories naturally occur in [MATH] supersymmetry [CITATION].', '1908.04120-2-3-5': 'Further aspects of duality-invariant theories with higher derivatives were studied in, e.g., [CITATION].', '1908.04120-2-4-0': 'Self-duality equation [REF] is nonlinear, and therefore its general solutions are difficult to find.', '1908.04120-2-4-1': 'In the early 2000s, Ivanov and Zupnik [CITATION] proposed a reformulation of duality-invariant electrodynamics involving an auxiliary antisymmetric tensor [MATH], which is equivalent to a symmetric spinor [MATH] and its conjugate [MATH].', '1908.04120-2-4-2': 'The new Lagrangian [MATH] is at most quadratic in the electromagnetic field strength [MATH], while the self-interaction is described by a nonlinear function of the auxiliary variables, [MATH], [EQUATION]', '1908.04120-2-4-3': 'The original theory [MATH] is obtained from [MATH] by integrating out the auxiliary variables.', '1908.04120-2-4-4': 'In terms of [MATH], the condition of U(1) duality invariance was shown [CITATION] to be equivalent to the requirement that the self-interaction [EQUATION] is invariant under linear U(1) transformations [MATH], with [MATH], and thus [EQUATION] where [MATH] is a real function of one real variable.', '1908.04120-2-4-5': 'The Ivanov-Zupnik (IZ) approach [CITATION] has been used by Novotny [CITATION] to establish the relation between helicity conservation for the tree-level scattering amplitudes and the electric-magnetic duality.', '1908.04120-2-5-0': 'The above discussion shows that the IZ approach is a universal formalism to generate U(1) duality-invariant models for nonlinear electrodynamics.', '1908.04120-2-5-1': 'Some time ago, there was a revival of interest in duality-invariant dynamical systems [CITATION] inspired by the desire to achieve a better understanding of the UV properties of extended supergravity theories.', '1908.04120-2-5-2': 'The authors of [CITATION] have put forward the so-called "twisted self-duality constraint," which was further advocated in [CITATION], as a systematic procedure to generate manifestly duality-invariant theories.', '1908.04120-2-5-3': 'However, these approaches have been demonstrated [CITATION] to be variants of the IZ scheme [CITATION] developed a decade earlier.', '1908.04120-2-6-0': 'The IZ approach has been generalised to off-shell [MATH] and [MATH] globally and locally supersymmetric theories [CITATION].', '1908.04120-2-6-1': 'In this note we provide a generalisation of the approach to higher dimensions, [MATH].', '1908.04120-2-6-2': 'In even dimensions, [MATH], the maximal duality group for a system of [MATH] gauge [MATH]-forms depends on the dimension of spacetime.', '1908.04120-2-6-3': 'The duality group is U[MATH] if [MATH] is even, and [MATH] if [MATH] is odd [CITATION] (see, e.g, section 8 of [CITATION] for a review).', '1908.04120-2-6-4': 'This is why we choose [MATH].', '1908.04120-2-6-5': 'The fact that the maximal duality group depends on the dimension of space-time was discussed in the mid-1980s [CITATION] and also in the late 1990s [CITATION].', '1908.04120-2-7-0': '# New formulation', '1908.04120-2-8-0': 'In Minkowski space of even dimension [MATH], with [MATH] a positive integer, we consider a self-interacting theory of a gauge [MATH]-form [MATH] with the property that the Lagrangian, [MATH], is a function of the field strengths [MATH].', '1908.04120-2-8-1': 'We assume that the theory possesses U(1) duality invariance.', '1908.04120-2-8-2': 'This means that the Lagrangian is a solution to the self-duality equation [CITATION] [EQUATION] where we have introduced [EQUATION]', '1908.04120-2-8-3': 'As usual, the notation [MATH] is used for the Hodge dual of [MATH], [EQUATION]', '1908.04120-2-8-4': 'We now introduce a reformulation of the above theory.', '1908.04120-2-8-5': 'Along with the field strength [MATH], our new Lagrangian [MATH] is defined to depend on an auxiliary rank-[MATH] antisymmetric tensor [MATH] which is unconstrained.', '1908.04120-2-8-6': 'We choose [MATH] to have the form [EQUATION] where we have denoted [EQUATION]', '1908.04120-2-8-7': 'The last term in [REF], [MATH], is at least quartic in [MATH].', '1908.04120-2-8-8': 'It is assumed that the equation of motion for [MATH], [EQUATION] allows one to integrate out the auxiliary field [MATH] to result with [MATH].', '1908.04120-2-9-0': 'It may be shown that the self-duality equation [REF] is equivalent to the following condition on the self-interaction in [REF] [EQUATION]', '1908.04120-2-9-1': 'Introducing (anti) self-dual components of [MATH], [EQUATION] the above condition turns into [EQUATION]', '1908.04120-2-9-2': 'This means that [MATH] is invariant under U(1) phase transformations, [EQUATION]', '1908.04120-2-9-3': 'In four dimensions, the most general solution to this condition is given by eq. [REF].', '1908.04120-2-9-4': 'Similar solutions exist in higher dimensions, [MATH], with [MATH] a real function of one variable.', '1908.04120-2-9-5': 'However more general self-interactions become possible beyond four dimensions.', '1908.04120-2-10-0': 'It is worth pointing out that an infinitesimal U(1) duality transformation [EQUATION]', '1908.04120-2-10-1': 'Equation [REF] tells us that [MATH] duality invariant.', '1908.04120-2-11-0': 'There are several interesting generalisations of the construction described.', '1908.04120-2-11-1': 'They include (i) coupling to gravity; (ii) coupling to a dilaton with enhanced SL[MATH] duality; (iii) duality-invariant systems with higher derivatives; and (iv) U[MATH] duality-invariant systems of [MATH] gauge [MATH]-forms in [MATH] dimensions.', '1908.04120-2-12-0': 'Recently, U(1) duality-invariant theories of a gauge [MATH]-form in [MATH] dimensions have been described [CITATION] within the Pasti-Sorokin-Tonin approach [CITATION].', '1908.04120-2-12-1': 'It was argued in [CITATION] that the approach of [CITATION] is the most efficient method to determine all possible manifestly U(1) duality invariant self-interactions provided Lorentz invariance is kept manifest.', '1908.04120-2-12-2': 'Our analysis has provided an alternative formalism.'}

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[['1908.04120-1-5-0', '1908.04120-2-5-0']]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1908.04120